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features of the proposed adaptive sliding mode controller are as follows: (i) The structured (or parametric) uncertainties and unstructured uncertainties (un-modeled dynamics, unknown external disturbances) are synthesized into a single type uncertainty term called lumped uncertainty. Therefore, a linearly parameterized dynamic model of the system is not required, and the simple structure and computationally efficient properties of this approach make it suitable for the real-time control applications. (ii) The adaptive sliding mode control scheme design relies on the online estimated uncertainty vector rather than relying on the worst-case scenario (i.e., bounds of uncertainties). Therefore, a-priory knowledge of the bounds of uncertainties is not required, and at each time instant, the control input compensates for the uncertainty that exists. (iii) The developed continuous control law using fundamentals of the sliding mode control theory eliminates the chattering phenomena without trade-off between performance and robustness, which is prevalent in boundary-layer approach.
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16476:{\displaystyle {\begin{aligned}{\dot {\hat {\mathbf {x} }}}&=A{\hat {\mathbf {x} }}+B\mathbf {u} +LM\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +{\begin{bmatrix}-1\\L_{2}\end{bmatrix}}M\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+{\begin{bmatrix}B&{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\end{bmatrix}}{\begin{bmatrix}\mathbf {u} \\\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\end{bmatrix}}\\&=A_{\text{obs}}{\hat {\mathbf {x} }}+B_{\text{obs}}\mathbf {u} _{\text{obs}}\end{aligned}}}
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306:, and the sliding mode along the surface commences after the finite time when system trajectories have reached the surface. In the theoretical description of sliding modes, the system stays confined to the sliding surface and need only be viewed as sliding along the surface. However, real implementations of sliding mode control approximate this theoretical behavior with a high-frequency and generally non-deterministic switching control signal that causes the system to "chatter" in a tight neighborhood of the sliding surface. Chattering can be reduced through the use of
9032:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=\overbrace {{\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}{\dot {\mathbf {x} }}} ^{{\dot {\sigma }}(\mathbf {x} )}={\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u\right)} ^{\dot {\mathbf {x} }}=\overbrace {} ^{\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}\underbrace {\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u\right)} ^{\dot {\mathbf {x} }}} _{{\text{( i.e., an }}n\times 1{\text{ vector )}}}}
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97:
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6962:{\displaystyle {\dot {\mathbf {x} }}=\overbrace {f(\mathbf {x} ,t)-B(\mathbf {x} ,t)\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}f(\mathbf {x} ,t)} ^{f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u}=f(\mathbf {x} ,t)\left(\mathbf {I} -B(\mathbf {x} ,t)\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}\right)}
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2059:) exists and is reachable along system trajectories. The principle of sliding mode control is to forcibly constrain the system, by suitable control strategy, to stay on the sliding surface on which the system will exhibit desirable features. When the system is constrained by the sliding control to stay on the sliding surface, the system dynamics are governed by reduced-order system obtained from Equation (
2707:
15462:{\displaystyle {\mathord {\overbrace {\begin{bmatrix}{\dot {e}}_{2}\\{\dot {e}}_{3}\\\vdots \\{\dot {e}}_{n}\end{bmatrix}} ^{{\dot {\mathbf {e} }}_{2}}}}=A_{2}{\mathord {\overbrace {\begin{bmatrix}e_{2}\\e_{3}\\\vdots \\e_{n}\end{bmatrix}} ^{\mathbf {e} _{2}}}}+L_{2}v(e_{1})=A_{2}\mathbf {e} _{2}+L_{2}v_{\text{eq}}=A_{2}\mathbf {e} _{2}+L_{2}A_{12}\mathbf {e} _{2}=(A_{2}+L_{2}A_{12})\mathbf {e} _{2}.}
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11658:. In these sliding mode observers, the order of the observer dynamics are reduced by one when the system enters the sliding mode. In this particular example, the estimator error for a single estimated state is brought to zero in finite time, and after that time the other estimator errors decay exponentially to zero. However, as first described by Drakunov, a
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2892:{\displaystyle \underbrace {\overbrace {\sigma ^{\intercal }} ^{\tfrac {\partial V}{\partial \sigma }}\overbrace {\dot {\sigma }} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} _{\tfrac {\operatorname {d} V}{\operatorname {d} t}}<0\qquad {\text{(i.e., }}{\tfrac {\operatorname {d} V}{\operatorname {d} t}}<0{\text{)}}}
9240:
3521:{\displaystyle {\dot {\sigma }}={\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\dot {\mathbf {x} }} ^{\tfrac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}={\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\mathbf {u} \right)} ^{\dot {\mathbf {x} }}}
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13371:{\displaystyle {\begin{aligned}{\dot {\mathbf {e} }}&={\dot {\hat {\mathbf {x} }}}-{\dot {\mathbf {x} }}\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +Lv({\hat {x}}_{1}-x_{1})-A\mathbf {x} -B\mathbf {u} \\&=A({\hat {\mathbf {x} }}-\mathbf {x} )+Lv({\hat {x}}_{1}-x_{1})\\&=A\mathbf {e} +Lv(e_{1})\end{aligned}}}
4359:{\displaystyle {\mathord {\underbrace {D^{+}{\Bigl (}{\mathord {\underbrace {2{\mathord {\overbrace {\sqrt {V}} ^{{}\propto \|\sigma \|_{2}}}}} _{W}}}{\Bigr )}} _{D^{+}W\,\triangleq \,{\mathord {{\text{Upper right-hand }}{\dot {W}}}}}}}={\frac {1}{\sqrt {V}}}{\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu }
5050:{\displaystyle \underbrace {\overbrace {\sigma ^{\intercal }} ^{\tfrac {\partial V}{\partial \sigma }}\overbrace {\dot {\sigma }} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} _{\tfrac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\mathord {\overbrace {\|\sigma \|_{2}} ^{\sqrt {V}}}})^{\alpha }}
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or a similar technique of applying a continuous signal to an output that can only take discrete states. Sliding mode control has many applications in robotics. In particular, this control algorithm has been used for tracking control of unmanned surface vessels in simulated rough seas with high degree
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One application of sliding mode controller is the control of electric drives operated by switching power converters. Because of the discontinuous operating mode of those converters, a discontinuous sliding mode controller is a natural implementation choice over continuous controllers that may need to
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that have more moderate control action. In particular, because actuators have delays and other imperfections, the hard sliding-mode-control action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics. Continuous control design methods are not as susceptible to these
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origin. One of the compensatory methods is the adaptive sliding mode control method proposed in which uses estimated uncertainty to construct continuous control law. In this method chattering is eliminated while preserving accuracy (for more details see references and ). The three distinguished
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1444:, it has the ability to drive trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is better than asymptotic). However, once the trajectories reach the sliding surface, the system takes on the character of the sliding mode (e.g., the origin
366:. Because the control can be as simple as a switching between two states (e.g., "on"/"off" or "forward"/"reverse"), it need not be precise and will not be sensitive to parameter variations that enter into the control channel. Additionally, because the control law is not a
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having finite frequency and amplitude. Chattering is a harmful phenomenon because it leads to low control accuracy, high wear of moving mechanical parts, and high heat losses in power circuits. For more details, see Utkin, Vadim; Lee, Jason Hoon (July 2006),
17116: – A topology that combines four switches forming the four legs of an "H". Can be used to drive a motor (or other electrical device) forward or backward when only a single supply is available. Often used in actuator in sliding-mode controlled systems.
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method. The multiple control structures are designed so that trajectories always move toward an adjacent region with a different control structure, and so the ultimate trajectory will not exist entirely within one control structure. Instead, it will
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Although various theories exist for sliding mode control system design, there is a lack of a highly effective design methodology due to practical difficulties encountered in analytical and numerical methods. A reusable computing paradigm such as a
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is uniformly bounded away from zero, the sliding mode will be maintained as if there was no disturbance. The invariance property of sliding mode control to certain disturbances and model uncertainties is its most attractive feature; it is strongly
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8347:{\displaystyle {\dot {V}}(\sigma (\mathbf {x} ))=\overbrace {\sigma (\mathbf {x} )^{\text{T}}} ^{\tfrac {\partial V}{\partial \mathbf {x} }}\overbrace {{\dot {\sigma }}(\mathbf {x} )} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}}
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11646:. These non-linear high-gain observers have the ability to bring coordinates of the estimator error dynamics to zero in finite time. Additionally, switched-mode observers have attractive measurement noise resilience that is similar to a
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It follows from these theorems that the sliding motion is invariant (i.e., insensitive) to sufficiently small disturbances entering the system through the control channel. That is, as long as the control is large enough to ensure that
1998:
14908:{\displaystyle {\mathord {\underbrace {v_{\text{eq}}} _{\text{scalar}}}}={\mathord {\underbrace {A_{12}} _{1\times (n-1) \atop {\text{ vector}}}}}{\mathord {\underbrace {\mathbf {e} _{2}} _{(n-1)\times 1 \atop {\text{ vector}}}}}.}
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1735:(i.e., it switches from one state to another as trajectories move across this surface), the surface is reached in finite time. Once a trajectory reaches the surface, it will slide along it and may, for example, move toward the
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or boundary layers around the sliding surface, or other compensatory methods. Although the system is nonlinear in general, the idealized (i.e., non-chattering) behavior of the system in Figure 1 when confined to the
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can, however, be utilized to transform a 'unsolvable problem' of optimal design into a practically solvable 'non-deterministic polynomial problem'. This results in computer-automated designs for sliding model control.
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11221:{\displaystyle u(x,{\dot {x}})={\begin{cases}|{\dot {x}}|+k+1&{\text{if }}\underbrace {x+{\dot {x}}} <0,\\-\left(|{\dot {x}}|+k+1\right)&{\text{if }}\overbrace {x+{\dot {x}}} ^{\sigma }>0\end{cases}}}
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to slide along the restricted sliding mode subspace. Trajectories from this reduced-order sliding mode have desirable properties (e.g., the system naturally slides along it until it comes to rest at a desired
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must be more strongly bounded away from zero. That is, if it vanishes too quickly, the attraction to the sliding mode will only be asymptotic. To ensure that the sliding mode is entered in finite time,
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14739:{\displaystyle 0={\dot {\sigma }}=a_{11}{\mathord {\overbrace {e_{1}} ^{{}=0}}}+A_{12}\mathbf {e} _{2}-{\mathord {\overbrace {v_{\text{eq}}} ^{v(\sigma )}}}=A_{12}\mathbf {e} _{2}-v_{\text{eq}}.}
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Continuous Nested
Algorithms: The Fifth Generation of Sliding Mode Controllers. In: Recent Advances in Sliding Modes: From Control to Intelligent Mechatronics, X. Yu, O. Efe (Eds.), pp. 5–35
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equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control
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9235:{\displaystyle u(\mathbf {x} )={\begin{cases}u^{+}(\mathbf {x} )&{\text{if }}\sigma (\mathbf {x} )>0\\u^{-}(\mathbf {x} )&{\text{if }}\sigma (\mathbf {x} )<0\end{cases}}}
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6499:{\displaystyle \mathbf {u} =-\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}f(\mathbf {x} ,t)}
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17128: – Another (feedback) method of encoding a continuous range of values in a signal that rapidly switches between two states (i.e., a kind of specialized sliding-mode control)
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8142:-dimensional sliding mode simplex). This surface may have favorable properties (e.g., when the plant dynamics are forced to slide along this surface, they move toward the origin
1422:; however, the sliding mode is enforced by creasing the vector field with high-gain feedback. Like a marble rolling along a crack, trajectories are confined to the sliding mode.
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1410:. Sliding mode control forces the system trajectories into this subspace and then holds them there so that they slide along it. This reduced-order subspace is referred to as a
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6361:{\displaystyle {\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\mathbf {u} \right)} ^{\dot {\mathbf {x} }}=\mathbf {0} }
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11792:{\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=A\mathbf {x} +B\mathbf {u} \\y={\begin{bmatrix}1&0&0&\cdots &\end{bmatrix}}\mathbf {x} =x_{1}\end{cases}}}
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produces outputs that only assume two states, but the effective output swings through a continuous range of motion. These complications can be avoided by using a different
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11399:{\displaystyle u(x,{\dot {x}})=-(|{\dot {x}}|+k+1)\underbrace {\operatorname {sgn} (\overbrace {{\dot {x}}+x} ^{\sigma })} _{{\text{(i.e., tests }}\sigma >0{\text{)}}}}
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So once the system reaches the sliding mode, the system's 2-dimensional dynamics behave like this 1-dimensional system, which has a globally exponentially stable
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must be greater than a scaled version of the maximum possible estimator errors for the system (i.e., the initial errors, which are assumed to be bounded so that
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condition after some initial transient during the period while the system finds the sliding mode. The same motion is approximately maintained when the equality
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2545:{\displaystyle V(\sigma (\mathbf {x} ))={\frac {1}{2}}\sigma ^{\intercal }(\mathbf {x} )\sigma (\mathbf {x} )={\frac {1}{2}}\|\sigma (\mathbf {x} )\|_{2}^{2}}
47:
control signal (or more rigorously, a set-valued control signal) that forces the system to "slide" along a cross-section of the system's normal behavior. The
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of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a
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design method that produces a continuous controller. In some cases, sliding-mode control designs can be approximated by other continuous control designs.
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1402:, then the output will track the desired sinusoid. In sliding-mode control, the designer knows that the system behaves desirably (e.g., it has a stable
972:
10289:{\displaystyle {\dot {\sigma }}={\dot {x}}_{1}+{\dot {x}}_{2}={\dot {x}}+{\ddot {x}}={\dot {x}}\,+\,\overbrace {a(t,x,{\dot {x}})+u} ^{\ddot {x}}}
5363:{\displaystyle \operatorname {sgn} (\sigma )\neq \operatorname {sgn} ({\dot {\sigma }})\qquad {\text{and}}\qquad |{\dot {\sigma }}|\geq \mu >0}
104:
trajectory of a system being stabilized by a sliding mode controller. After the initial reaching phase, the system states "slides" along the line
16910:). However, a similar procedure can be used to design a sliding mode observer for a vector of weighted combinations of states (i.e., when output
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8523:{\displaystyle {\begin{cases}{\dot {\sigma }}<0&{\text{if }}\sigma >0\\{\dot {\sigma }}>0&{\text{if }}\sigma <0\end{cases}}}
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forces the estimate of the measured state to have zero error in finite time. Additionally, white zero-mean symmetric measurement noise (e.g.,
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First-Order
Continuous Adaptive Sliding Mode Control for Robot Manipulators with Finite-Time Convergence of Trajectories to Real sliding Mode
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that ensures that there is always a control that can move a trajectory to move closer to the sliding mode. Then, once the sliding mode where
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890:{\displaystyle \mathbf {u} (t)\triangleq {\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\\vdots \\u_{m-1}(t)\\u_{m}(t)\end{bmatrix}}\in \mathbb {R} ^{m}}
707:{\displaystyle \mathbf {x} (t)\triangleq {\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\\vdots \\x_{n-1}(t)\\x_{n}(t)\end{bmatrix}}\in \mathbb {R} ^{n}}
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14030:{\displaystyle {\dot {\sigma }}={\dot {e}}_{1}=a_{11}e_{1}+A_{12}\mathbf {e} _{2}-v(e_{1})=a_{11}e_{1}+A_{12}\mathbf {e} _{2}-v(\sigma )}
14247:
1271:. That is, under the control law, whenever the system is started away from the origin, it will return to it. For example, the component
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or a manifold (i.e., the sliding surface) such that the system trajectory exhibits desirable behavior when confined to this manifold.
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of this distance. So the sliding-mode control acts like a stiff pressure always pushing in the direction of the sliding mode where
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For simplicity, this example assumes that the sliding mode observer has access to a measurement of a single state (i.e., output
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16716:{\displaystyle u_{\text{obs}}\triangleq {\begin{bmatrix}\mathbf {u} \\\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\end{bmatrix}}.}
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of the closed-loop system. Trajectories along this subspace can be likened to trajectories along eigenvectors (i.e., modes) of
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with respect to this differential equation, then the system will slide along the sliding mode surface toward the equilibrium.
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must always be bounded firmly away from zero. To ensure this condition, sliding mode controllers are discontinuous across the
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17264:. 15th International Workshop on Variable Structure Systems and Sliding Mode Control. Graz University of Technology, Austria.
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may represent the difference some output is away from a known signal (e.g., a desirable sinusoidal signal); if the control
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5759:{\displaystyle \{\mathbf {x} \in \mathbb {R} ^{n}:\sigma ^{\intercal }(\mathbf {x} ){\dot {\sigma }}(\mathbf {x} )<0\}}
68:
along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a
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can be built that brings the estimation error for all estimated states to zero in a finite (and arbitrarily small) time.
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3952:{\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\sqrt {V}})^{\alpha }\leq -\mu {\sqrt {V}}.}
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17312:"Autonomous Navigation and Obstacle Avoidance of Unmanned Vessels in Simulated Rough Sea States - Villanova University"
17106: – Sliding mode control is often implemented as a bang–bang control. In some cases, such control is necessary for
16612:{\displaystyle B_{\text{obs}}\triangleq {\begin{bmatrix}B&{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\end{bmatrix}},}
7865:
6158:
5863:. Moreover, if the reachability conditions from Theorem 1 are satisfied, the sliding mode will enter the region where
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Figure 1 shows an example trajectory of a system under sliding mode control. The sliding surface is described by
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as the system both flows through a continuous state space but also moves through different discrete control modes.
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17231:
Zeinali M.; Notash L. (2010). "Adaptive sliding mode control with uncertainty estimator for robot manipulators".
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13384:
6207:
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1215:
17323:
Mahini; et al. (2013). "An experimental setup for autonomous operation of surface vessels in rough seas".
12522:{\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},\dots ,{\hat {x}}_{n})\in \mathbb {R} ^{n}}
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The vital part of SMC design is to choose a control law so that the sliding mode (i.e., this surface given by
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8073:{\displaystyle s_{1}{\dot {x}}_{1}+s_{2}{\dot {x}}_{2}+\cdots +s_{n-1}{\dot {x}}_{n-1}+s_{n}{\dot {x}}_{n}=0}
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surface is chosen because it has desirable reduced-order dynamics when constrained to it. In this case, the
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4842:, this result implies that the rate of approach to the sliding mode must be firmly bounded away from zero.
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12645:{\displaystyle {\dot {\hat {\mathbf {x} }}}=A{\hat {\mathbf {x} }}+B\mathbf {u} +Lv({\hat {x}}_{1}-x_{1})}
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7726:{\displaystyle \sigma (\mathbf {x} )\triangleq s_{1}x_{1}+s_{2}x_{2}+\cdots +s_{n-1}x_{n-1}+s_{n}x_{n}}
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60:
17275:
Utkin, Vadim I. (1993). "Sliding Mode
Control Design Principles and Applications to Electric Drives".
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So the system can be feedback stabilized (to return to the sliding mode) by means of the control law
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16809:, the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal
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4701:
4063:{\displaystyle {\frac {1}{\sqrt {V}}}{\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ,}
3751:{\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\sqrt {V}})^{\alpha }}
1853:
17289:
17140: – Another modulation scheme that produces continuous motion through discontinuous switching.
16981:
16729:
13831:
12353:
11985:
11659:
11545:{\displaystyle {\dot {x}}=-x\qquad {\text{(i.e., }}\sigma (x,{\dot {x}})=x+{\dot {x}}=0{\text{)}}}
10529:
10344:
9428:
9328:
6512:
6118:
5805:
5462:
4403:
2071:
1831:
1805:
1559:
1533:
1323:
1301:
1176:
17548:
17538:
9826:{\displaystyle {\begin{cases}{\dot {x}}_{1}=x_{2}\\{\dot {x}}_{2}=a(t,x_{1},x_{2})+u\end{cases}}}
4761:
4677:
14503:
7757:
5866:
5772:
5769:
That is, when initial conditions come entirely from this space, the
Lyapunov function candidate
4543:
3965:
17462:
17284:
17176:
17137:
17125:
17057:{\displaystyle \sigma (\mathbf {x} )\triangleq {\hat {\mathbf {y} }}-\mathbf {y} =\mathbf {0} }
11233:
9619:
7539:
7352:
5994:
5895:
5552:
5377:
5185:
3764:
2383:
2379:
909:
722:
391:
48:
14140:{\displaystyle \mathbf {e} _{2}\triangleq (e_{2},e_{3},\ldots ,e_{n})\in \mathbb {R} ^{(n-1)}}
12087:{\displaystyle A\triangleq {\begin{bmatrix}a_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}}}
9386:
is some control (e.g., possibly extreme, like "off" or "reverse") that ensures
Equation (
4430:
1414:, and when closed-loop feedback forces trajectories to slide along it, it is referred to as a
17591:. Lecture Notes in Control and Information Sciences. Vol. 193. London: Springer-Verlag.
17103:
16880:
14352:
13769:
12730:
12375:
9625:
9492:
back and forth along this sliding surface (i.e., the true trajectory may not smoothly follow
9286:
is some control (e.g., possibly extreme, like "on" or "forward") that ensures
Equation (
5442:
5097:
4825:
3055:
1372:
518:{\displaystyle {\dot {\mathbf {x} }}(t)=f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\,\mathbf {u} (t)}
379:
15720:
15007:
12103:
15542:
14980:
14948:
14434:
14147:
is the collection of the estimator errors for all of the unmeasured states. To ensure that
13547:
12130:
8113:
4372:
3830:
2947:
2266:
1993:{\displaystyle \left\{\mathbf {x} \in \mathbb {R} ^{n}:\sigma _{k}(\mathbf {x} )=0\right\}}
1885:
1345:
1274:
1210:
1087:
342:
273:
17757:
Fridman, L.; Moreno, J.; Bandyopadhyay, B.; Kamal Asif
Chalanga, S.; Chalanga, S. (2015).
1436:
Finding feedback gains so that the system trajectory intersects and stays on the manifold.
8:
17244:
17119:
15839:
11651:
10834:
10676:
9894:
8087:
7555:
6532:
6071:
is constant, and so sliding mode trajectories are described by the differential equation
6040:
is achieved, the system will stay on that sliding mode. Along sliding mode trajectories,
4735:
2953:
2262:
1775:
1732:
1441:
1034:
367:
314:
283:
133:
107:
73:
56:
17518:
Drakunov, S.V. (1983). "An adaptive quasioptimal filter with discontinuous parameters".
13727:{\displaystyle \sigma ({\hat {x}}_{1},{\hat {x}})\triangleq e_{1}={\hat {x}}_{1}-x_{1}.}
17829:
17786:
Advances in
Variable Structure Systems and Sliding Mode Control—Theory and Applications
17570:
17340:
6142:
5660:{\displaystyle \{\mathbf {x} \in \mathbb {R} ^{n}:\sigma (\mathbf {x} )=\mathbf {0} \}}
1196:
350:
12210:
is a row vector corresponding to the influence of the first state on the other states,
17854:
17833:
17819:
17803:
17789:
17770:
17743:
17724:
17705:
17686:
17667:
17648:
17625:
17600:
17574:
17560:
17495:
17450:
17429:
17404:
17379:
17211:
17088:
15846:, and hence the noise will have little effect on the equivalent sliding mode control
13825:
12346:
is a column vector representing the influence of the other states on the first state.
11971:{\displaystyle \mathbf {u} \triangleq (u_{1},u_{2},\dots ,u_{r})\in \mathbb {R} ^{r}}
11883:{\displaystyle \mathbf {x} \triangleq (x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}}
11650:. For simplicity, the example here uses a traditional sliding mode modification of a
11630:
8173:
7225:
As discussed in an example below, a sliding mode control law can keep the constraint
5799:
2407:
2387:
399:
32:
17845:. Studies in Systems, Decision and Control. Vol. 271. London: Springer-Verlag.
17451:"Genetic algorithm automated approach to the design of sliding mode control systems"
17344:
5983:{\displaystyle {\frac {\partial \sigma }{\partial {\mathbf {x} }}}B(\mathbf {x} ,t)}
17892:
17884:
17846:
17811:
17788:. Studies in Systems, Decision and Control. Vol. 24. London: Springer-Verlag.
17762:
17761:. Studies in Systems, Decision and Control. Vol. 24. London: Springer-Verlag.
17592:
17552:
17472:
17369:
17365:
17332:
17294:
17240:
11557:
10820:{\displaystyle |{\dot {x}}|+|a(t,x,{\dot {x}})|\geq |{\dot {x}}+a(t,x,{\dot {x}})|}
6138:
1731:
trajectories will approach the sliding surface, and because the control law is not
1403:
1200:
412:
359:
96:
40:
16947:). In each case, the sliding mode will be the manifold where the estimated output
12795:
is an observer gain vector that serves a similar purpose as in the typical linear
17107:
17098:
12350:
The goal is to design a high-gain state observer that estimates the state vector
5998:
5439:
toward the switching surface should have a non-zero lower bound. So, even though
2378:
however, this smooth behavior may not be truly realizable. Similarly, high-speed
1766:
1094:
1038:
1026:{\displaystyle B:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n\times m}}
383:
375:
17766:
5892:
is more strongly bounded away from zero in finite time. Hence, the sliding mode
17640:
17617:
17093:
15714:
15673:
13443:
is the estimator error for the first state estimate. The nonlinear control law
12796:
11643:
9535:
7464:
7219:
5091:
4783:
2603:
2325:
1769:
with a contour of constant height along which trajectories are forced to move.
374:
time (i.e., better than asymptotic behavior). Under certain common conditions,
363:
354:
36:
20:
17897:
17888:
17875:
Drakunov S.V., Utkin V.I.. (1992). "Sliding mode control in dynamic systems".
17544:[1992] Proceedings of the 31st IEEE Conference on Decision and Control
17476:
17336:
17311:
7463:). That is, when the system is constrained this way, it behaves like a simple
17909:
17756:
17375:
17083:
15874:
15681:
12284:
is a matrix representing the influence of the other states on themselves, and
11647:
7467:
1659:
The sliding-mode-control law switches from one state to another based on the
725:
85:
44:
17815:
17556:
12999:{\displaystyle \mathbf {e} =(e_{1},e_{2},\dots ,e_{n})\in \mathbb {R} ^{n}}
1430:
3596:
2398:
The following theorems form the foundation of variable structure control.
17158:
5374:
That is, the system should always be moving toward the switching surface
962:{\displaystyle f:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}}
101:
17259:
8572:
because it is the product of a negative and a positive number. Note that
398:
Sliding mode control must be applied with more care than other forms of
17850:
17596:
15677:
11655:
5579:
from one non-zero value to another as trajectories cross the manifold.
5257:{\displaystyle \operatorname {sgn} (\sigma ){\dot {\sigma }}\leq -\mu }
4074:
2285:
sense. Roughly speaking, the resulting closed-loop system moving along
1419:
338:
218:
17683:
Modern
Sliding Mode Control Theory - New perspectives and applications
17298:
9451:
The resulting trajectory should move toward the sliding surface where
2701:), a sufficient condition for the existence of a sliding mode is that
15713:
system can be stabilized in exactly the same way as a typical linear
3827:
This condition ensures that for the neighborhood of the sliding mode
1083:
10978:{\displaystyle |{\dot {x}}|+k+1>|{\dot {x}}|+|a(t,x,{\dot {x}})|}
6972:
The resulting system matches the sliding mode differential equation
5824:
trajectories are sure to move toward the sliding mode surface where
5171:{\displaystyle \sigma {\dot {\sigma }}\leq -\mu |\sigma |^{\alpha }}
17113:
9488:. Because real systems have delay, sliding mode trajectories often
7470:, and so it has a globally exponentially stable equilibrium at the
6581:
Likewise, the system trajectories on the sliding mode behave as if
4758:(i.e., the system enters the sliding mode) in finite time. Because
905:
307:
52:
14349:
is sufficiently large, it can be assumed that the system achieves
14327:{\displaystyle M>\max\{|a_{11}e_{1}+A_{12}\mathbf {e} _{2}|\}.}
9529:, but it will always return to the sliding mode after leaving it).
8084:
which is a reduced-order system (i.e., the new system is of order
7378:
has a finite upper bound. In this case, the sliding mode is where
4850:
In the context of sliding mode control, this condition means that
11410:
Assuming that the system trajectories are forced to move so that
10027:
By the previous example, we must choose the feedback control law
7820:
84:, like a system under SMC, may be viewed as a special case of a
17122: – Uses switching-mode control to drive continuous outputs
13046:{\displaystyle \mathbf {e} ={\hat {\mathbf {x} }}-\mathbf {x} }
9925:
that is known). For this system, choose the switching function
1772:
The sliding (hyper)surface/manifold is typically of dimension
17645:
Variable
Structure Systems: From Principles to Implementation
12691:
is a nonlinear function of the error between estimated state
10016:{\displaystyle \sigma (x_{1},x_{2})=x_{1}+x_{2}=x+{\dot {x}}}
2214:, the control action is capable of maintaining the system at
17718:
7860:. When trajectories are forced to slide along this surface,
7272:
in order to asymptotically stabilize any system of the form
2371:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=\mathbf {0} ;}
1523:{\displaystyle \sigma :\mathbb {R} ^{n}\to \mathbb {R} ^{m}}
403:
problems and can be made to mimic sliding-mode controllers.
349:
Intuitively, sliding mode control uses practically infinite
17719:
Shtessel, Y.; Edwards, C.; Fridman, L.; Levant, A. (2014).
17539:"Sliding-mode observers based on equivalent control method"
17157:'Chatter' or 'chattering' is the undesirable phenomenon of
11785:
11214:
9819:
9228:
8516:
7016:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=\mathbf {0} }
17874:
17802:
17680:
17661:
17401:
Differential Equations with Discontinuous Right-hand Sides
16863:{\displaystyle \operatorname {sgn} ({\hat {x}}_{1}-x_{1})}
16802:{\displaystyle \operatorname {sgn} ({\hat {x}}_{1}-x_{1})}
12277:{\displaystyle A_{22}\in \mathbb {R} ^{(n-1)\times (n-1)}}
12127:
is a scalar representing the influence of the first state
1828:
is the number of input signals (i.e., control signals) in
17740:
Sliding Mode Control. The Delta-Sigma Modulation Approach
17704:. London: The Institution of Engineering and Technology.
17681:
Bartolini, G.; Fridman, L.; Pisano, A.; Usai, E. (2008).
15831:{\displaystyle v=M\operatorname {sgn} ({\hat {x}}_{1}-x)}
14231:{\displaystyle v(\sigma )=M\operatorname {sgn} (\sigma )}
12854:{\displaystyle L={\begin{bmatrix}-1\\L_{2}\end{bmatrix}}}
7181:{\displaystyle \sigma ^{\intercal }{\dot {\sigma }}<0}
4112:{\displaystyle \operatorname {d} W/{\operatorname {d} t}}
3675:{\displaystyle \operatorname {d} V/{\operatorname {d} t}}
3014:{\displaystyle \sigma ^{\intercal }{\dot {\sigma }}<0}
17638:
4845:
17784:
Li, S.; Yu, X.; Fridman, L.; Man, Z.; Wang, X. (2017).
17664:
Advances in Variable Structure and Sliding Mode Control
14529:
may be replaced with the equivalent continuous control
3597:
Reachability: Attaining sliding manifold in finite time
2401:
1406:) provided that it is constrained to a subspace of its
17840:
16646:
16564:
16551:
16346:
16299:
16286:
16173:
16049:
15842:) only affects the switching frequency of the control
15185:
15050:
13620:). Hence, the sliding mode control switching function
12820:
12339:{\displaystyle A_{12}\in \mathbb {R} ^{1\times (n-1)}}
12025:
11734:
8316:
8258:
4959:
4924:
4883:
3382:
2851:
2810:
2775:
2734:
2269:, and so existence and uniqueness of solutions to the
904:-dimensional input vector that will be used for state
760:
577:
17841:
Steinberger, M.; Horn, M.; Fridman, L., eds. (2020).
17233:
International Journal of Mechanism and Machine Theory
17134: – A generalized form of delta-sigma modulation.
17006:
16984:
16953:
16916:
16883:
16815:
16754:
16732:
16627:
16532:
16495:
15889:
15852:
15781:
15754:
15723:
15690:
15622:
15572:
15545:
15507:
15478:
15040:
15010:
14983:
14951:
14924:
14758:
14565:
14535:
14506:
14464:
14437:
14388:
14355:
14250:
14194:
14153:
14046:
13859:
13834:
13792:
13772:
13743:
13629:
13577:
13550:
13514:
13456:
13387:
13062:
13012:
12924:
12870:
12808:
12766:
12733:
12697:
12661:
12542:
12411:
12378:
12356:
12292:
12218:
12162:
12133:
12106:
12013:
11988:
11896:
11808:
11674:
11566:
11455:
11416:
11244:
10997:
10869:
10837:
10687:
10593:
10558:
10532:
10491:
10408:
10373:
10347:
10306:
10117:
10077:
10033:
9933:
9897:
9843:
9705:
9661:
9628:
9549:
9498:
9457:
9431:
9398:
9354:
9331:
9298:
9254:
9099:
9066:
8588:
8540:
8432:
8399:
8364:
8190:
8148:
8116:
8090:
7920:
7868:
7829:
7793:
7760:
7598:
7558:
7476:
7428:
7387:
7355:
7281:
7234:
7194:
7149:
7106:
7071:
7032:
6981:
6590:
6541:
6515:
6380:
6246:
6210:
6161:
6121:
6080:
6046:
6007:
5935:
5898:
5869:
5830:
5808:
5775:
5676:
5605:
5555:
5526:
5487:
5465:
5445:
5406:
5380:
5276:
5217:
5188:
5123:
5100:
5066:
4859:
4828:
4791:
4764:
4738:
4704:
4680:
4650:
4588:
4546:
4508:
4457:
4433:
4406:
4375:
4161:
4125:
4083:
4009:
3968:
3874:
3833:
3793:
3767:
3692:
3646:
3607:
3570:
3537:
3323:
3285:
3245:
3214:
3181:
3141:
3110:
3078:
3058:
3027:
2982:
2956:
2912:
2710:
2656:
2650:
is the distance away from the sliding manifold where
2612:
2578:
2425:
2333:
2291:
2256:
2220:
2181:
2139:
2096:
2074:
2026:
1929:
1888:
1856:
1834:
1808:
1778:
1741:
1706:
1669:
1624:
1584:
1562:
1536:
1530:
that represents a kind of "distance" that the states
1486:
1474:
may only have asymptotic stability on this surface).
1450:
1375:
1348:
1326:
1304:
1277:
1218:
1179:
1144:
1102:
1051:
975:
917:
737:
554:
430:
317:
286:
226:
162:
136:
110:
17808:
Advanced and Optimization Based Sliding Mode Control
17699:
17490:
Utkin, Vadim; Guldner, Jürgen; Shi, Jingxin (1999).
10664:{\displaystyle u<-|{\dot {x}}+a(t,x,{\dot {x}})|}
6535:, the rapid switching across the sliding mode where
362:). The main strength of sliding mode control is its
17843:
Variable-Structure Systems and Sliding-Mode Control
17662:Edwards, C.; Fossas Colet, E.; Fridman, L. (2006).
17230:
15031:acts like an output vector for the error subsystem
10476:{\displaystyle u>|{\dot {x}}+a(t,x,{\dot {x}})|}
5114:is scalar valued, the sufficient condition becomes
17164:Chattering Problem in Sliding Mode Control Systems
17056:
16992:
16970:
16935:
16902:
16862:
16801:
16740:
16715:
16611:
16517:
16475:
15865:
15830:
15767:
15736:
15705:
15664:
15608:
15558:
15531:
15493:
15461:
15023:
14996:
14969:
14937:
14907:
14738:
14548:
14521:
14492:
14450:
14423:
14374:
14326:
14230:
14177:
14139:
14029:
13842:
13816:
13778:
13758:
13726:
13612:
13563:
13536:
13497:
13435:
13370:
13045:
12998:
12910:
12853:
12787:
12752:
12719:
12683:
12644:
12521:
12397:
12364:
12338:
12276:
12202:
12146:
12119:
12086:
11996:
11970:
11882:
11791:
11642:Sliding mode control can be used in the design of
11611:
11544:
11439:
11398:
11220:
10977:
10853:
10819:
10663:
10579:
10544:
10518:
10475:
10394:
10359:
10333:
10288:
10101:
10063:
10015:
9913:
9883:
9825:
9689:
9647:
9606:
9521:
9480:
9439:
9413:
9378:
9339:
9313:
9278:
9234:
9083:
9031:
8564:
8522:
8416:
8385:
8346:
8164:
8134:
8102:
8072:
7900:
7852:
7811:
7779:
7725:
7570:
7521:
7455:
7411:
7370:
7338:
7261:
7209:
7180:
7132:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
7131:
7092:
7058:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
7057:
7015:
6961:
6567:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
6566:
6523:
6498:
6360:
6229:
6193:
6129:
6103:
6063:
6033:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
6032:
5982:
5910:
5884:
5856:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
5855:
5816:
5790:
5758:
5659:
5582:
5567:
5541:
5513:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
5512:
5473:
5451:
5431:
5392:
5362:
5256:
5200:
5170:
5106:
5082:
5049:
4834:
4814:
4774:
4750:
4720:
4690:
4666:
4632:
4574:
4532:
4494:
4439:
4419:
4388:
4358:
4144:
4111:
4062:
3992:
3951:
3857:
3822:
3811:
3779:
3750:
3674:
3633:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
3632:
3585:
3556:
3520:
3302:
3271:
3231:
3198:
3167:
3127:
3093:
3064:
3044:
3013:
2968:
2935:
2891:
2695:) and the sliding surface given by Equation (
2682:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2681:
2642:
2594:
2544:
2370:
2317:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2316:
2246:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2245:
2207:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2206:
2165:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2164:
2122:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2121:
2082:
2052:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} }
2051:
1992:
1906:
1874:
1842:
1816:
1790:
1757:
1723:
1692:
1647:
1607:
1570:
1544:
1522:
1466:
1394:
1361:
1334:
1312:
1290:
1263:
1187:
1161:
1130:
1068:
1025:
961:
889:
706:
517:
329:
298:
264:
209:
148:
122:
17492:Sliding Mode Control in Electromechanical Systems
17423:
15501:for the unmeasured states converges to zero, the
6578:as if it were driven by this continuous control.
4251:
4180:
2281:. Thus the solutions are to be understood in the
17907:
17700:Fridman, L.; Barbot, J.-P.; Plestan, F. (2016).
17647:. London: The Insitite of Electrical Engineers.
14257:
13447:can be designed to enforce the sliding manifold
9844:
7901:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=0}
6194:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=0}
17615:
17494:. Philadelphia, PA: Taylor & Francis, Inc.
17489:
17175:Other pulse-type modulation techniques include
9607:{\displaystyle {\ddot {x}}=a(t,x,{\dot {x}})+u}
7339:{\displaystyle {\ddot {x}}=a(t,x,{\dot {x}})+u}
4633:{\displaystyle 2{\sqrt {V(t)}}\leq V_{0}-\mu t}
80:. In the context of modern control theory, any
17783:
15680:must be negative). Hence, provided that it is
12684:{\displaystyle v:\mathbb {R} \to \mathbb {R} }
12203:{\displaystyle A_{21}\in \mathbb {R} ^{(n-1)}}
5593:) and sliding surface given by Equation (
4502:which is represented by differential equation
3272:{\displaystyle {\dot {\sigma }}(\mathbf {x} )}
3168:{\displaystyle {\dot {\sigma }}(\mathbf {x} )}
2133:Ensure that the system is capable of reaching
17737:
17622:Sliding Mode Control. Theory and Applications
17513:
17511:
17198:
17196:
17166:, vol. 10.1109/VSS.2006.1644542., pp. 346–350
14977:states to the trajectory of the output state
12911:{\displaystyle L_{2}\in \mathbb {R} ^{(n-1)}}
11623:
6104:{\displaystyle {\dot {\sigma }}=\mathbf {0} }
2643:{\displaystyle \|\sigma (\mathbf {x} )\|_{2}}
1131:{\displaystyle \mathbf {u} (\mathbf {x} (t))}
14318:
14260:
14178:{\displaystyle \sigma {\dot {\sigma }}<0}
13817:{\displaystyle \sigma {\dot {\sigma }}<0}
12372:using only information from the measurement
11982:is a scalar equal to the first state of the
11660:sliding mode observer for non-linear systems
10102:{\displaystyle \sigma {\dot {\sigma }}<0}
9872:
9847:
8565:{\displaystyle \sigma {\dot {\sigma }}<0}
6152:on the sliding mode can be found by solving
5921:
5753:
5677:
5654:
5606:
5077:
5067:
5012:
5005:
4803:
4792:
4222:
4215:
2631:
2613:
2589:
2579:
2528:
2510:
1765:origin. So the switching function is like a
1618:A state that is on this sliding surface has
1578:that is outside of this sliding surface has
1093:A common task is to design a state-feedback
382:; hence, sliding mode control describes the
17277:IEEE Transactions on Industrial Electronics
16518:{\displaystyle A_{\text{obs}}\triangleq A,}
14945:represents the contribution from the other
14458:is constant (i.e., 0) along this manifold,
10587:, the control law should be picked so that
10402:, the control law should be picked so that
4815:{\displaystyle \|{\mathord {\cdot }}\|_{2}}
1608:{\displaystyle \sigma (\mathbf {x} )\neq 0}
76:consisting of the boundaries is called the
17806:; Incremona, G.P.; Cucuzzella, M. (2019).
17530:
17508:
17392:
17360:
17358:
17356:
17354:
17251:
17224:
17208:Deterministic control of uncertain systems
17193:
17000:with zero error (i.e., the manifold where
16936:{\displaystyle \mathbf {y} =C\mathbf {x} }
16726:That is, by augmenting the control vector
15880:The final version of the observer is thus
14500:as well. Hence, the discontinuous control
13436:{\displaystyle e_{1}={\hat {x}}_{1}-x_{1}}
9618:which can be expressed in a 2-dimensional
8110:because the system is constrained to this
7532:
6230:{\displaystyle \mathbf {u} (\mathbf {x} )}
5267:which is equivalent to the condition that
5208:, the scalar sufficient condition becomes
3557:{\displaystyle \mathbf {u} (\mathbf {x} )}
2689:). For the system given by Equation (
1440:Because sliding mode control laws are not
1264:{\displaystyle \mathbf {x} =^{\intercal }}
17896:
17483:
17466:
17316:
17288:
14115:
12986:
12886:
12775:
12677:
12669:
12509:
12308:
12234:
12178:
11958:
11870:
10223:
10219:
9884:{\displaystyle \sup\{|a(\cdot )|\}\leq k}
8165:{\displaystyle \mathbf {x} =\mathbf {0} }
5690:
5619:
4280:
4276:
2393:
2261:Note that because the control law is not
1945:
1758:{\displaystyle \mathbf {x} =\mathbf {0} }
1510:
1495:
1467:{\displaystyle \mathbf {x} =\mathbf {0} }
1425:The sliding-mode control scheme involves
1007:
998:
984:
949:
940:
926:
877:
694:
500:
17536:
17517:
17442:
17398:
17374:(3rd ed.). Upper Saddle River, NJ:
11637:
6509:That is, even though the actual control
4145:{\displaystyle W\triangleq 2{\sqrt {V}}}
210:{\displaystyle s=x_{1}+{\dot {x}}_{1}=0}
95:
17589:Variable Structure and Lyapunov Control
17586:
17351:
17257:
17202:
15873:. Hence, the sliding mode observer has
15744:is viewed as the output matrix (i.e., "
13786:must always have opposite signs (i.e.,
13006:be the state estimator error. That is,
11440:{\displaystyle \sigma (\mathbf {x} )=0}
9522:{\displaystyle \sigma (\mathbf {x} )=0}
9481:{\displaystyle \sigma (\mathbf {x} )=0}
8575:
7853:{\displaystyle \sigma (\mathbf {x} )=0}
7585:
5587:For the system given by Equation (
5094:. For the case when switching function
5083:{\displaystyle \|{\mathord {\cdot }}\|}
2936:{\displaystyle \sigma (\mathbf {x} )=0}
2595:{\displaystyle \|{\mathord {\cdot }}\|}
2412:
1916:
1693:{\displaystyle \sigma (\mathbf {x} )=0}
1648:{\displaystyle \sigma (\mathbf {x} )=0}
1045:can be used to guarantee that solution
417:
217:surface corresponds to the first-order
17908:
17424:Perruquetti, W.; Barbot, J.P. (2002).
17364:
17322:
17268:
13498:{\displaystyle 0={\hat {x}}_{1}-x_{1}}
12918:is a column vector. Additionally, let
1914:-dimensional sliding surface given by
17702:Recent Trends in Sliding Mode Control
17274:
16971:{\displaystyle {\hat {\mathbf {y} }}}
12788:{\displaystyle L\in \mathbb {R} ^{n}}
10580:{\displaystyle {\dot {\sigma }}<0}
10395:{\displaystyle {\dot {\sigma }}>0}
6064:{\displaystyle \sigma (\mathbf {x} )}
4846:Consequences for sliding mode control
3303:{\displaystyle \sigma (\mathbf {x} )}
3199:{\displaystyle \sigma (\mathbf {x} )}
370:, the sliding mode can be reached in
265:{\displaystyle {\dot {x}}_{1}=-x_{1}}
17721:Sliding Mode Control and Observation
17245:10.1016/j.mechmachtheory.2009.08.003
15676:(i.e., the real part of each of its
14424:{\displaystyle {\hat {x}}_{1}=x_{1}}
14345:can be picked large enough; al). If
13613:{\displaystyle {\hat {x}}_{1}=x_{1}}
12533:states. The observer takes the form
11612:{\displaystyle (x,{\dot {x}})=(0,0)}
8579:
7589:
7522:{\displaystyle (x,{\dot {x}})=(0,0)}
5997:. That is, the system has a kind of
2416:
2402:Theorem 1: Existence of sliding mode
1920:
1138:(i.e., a mapping from current state
421:
386:for a broad set of dynamic systems.
17426:Sliding Mode Control in Engineering
15665:{\displaystyle (A_{2}+L_{2}A_{12})}
9379:{\displaystyle u^{-}(\mathbf {x} )}
9279:{\displaystyle u^{+}(\mathbf {x} )}
13:
17868:
17448:
17210:. London: Peter Peregrinus Press.
15472:So, to ensure the estimator error
14868:
14811:
11978:is a vector of inputs, and output
8906:
8885:
8729:
8708:
8643:
8622:
8330:
8319:
8269:
8261:
7093:{\displaystyle {\dot {\sigma }}=0}
6943:
6935:
6887:
6879:
6728:
6720:
6672:
6664:
6465:
6457:
6409:
6401:
6258:
6250:
5947:
5939:
5432:{\displaystyle |{\dot {\sigma }}|}
4973:
4962:
4938:
4927:
4894:
4886:
4335:
4324:
4099:
4084:
4036:
4025:
3889:
3878:
3812:{\displaystyle 0<\alpha \leq 1}
3707:
3696:
3662:
3647:
3426:
3418:
3398:
3385:
3350:
3342:
2865:
2854:
2824:
2813:
2789:
2778:
2745:
2737:
2257:Existence of closed-loop solutions
1477:The sliding-mode designer picks a
16:Method in nonlinear control theory
14:
17927:
15609:{\displaystyle (n-1)\times (n-1)}
10519:{\displaystyle x+{\dot {x}}>0}
10334:{\displaystyle x+{\dot {x}}<0}
5918:will be attained in finite time.
5667:surface is reachable is given by
4728:in finite time, which means that
4667:{\displaystyle {\sqrt {V}}\geq 0}
4451:. So, by comparison to the curve
1552:are away from a sliding surface.
406:
17877:International Journal of Control
17587:Zinober, Alan S.I., ed. (1994).
17455:International Journal of Control
17050:
17042:
17028:
17014:
16986:
16958:
16929:
16918:
16734:
16650:
16459:
16434:
16350:
16268:
16161:
16144:
16037:
16020:
15946:
15929:
15900:
15706:{\displaystyle \mathbf {e} _{2}}
15693:
15494:{\displaystyle \mathbf {e} _{2}}
15481:
15446:
15392:
15357:
15309:
15248:
15143:
14852:
14710:
14645:
14493:{\displaystyle {\dot {e}}_{1}=0}
14303:
14049:
14002:
13932:
13836:
13759:{\displaystyle {\dot {\sigma }}}
13737:To attain the sliding manifold,
13335:
13267:
13253:
13229:
13218:
13160:
13143:
13116:
13094:
13071:
13039:
13025:
13014:
12926:
12591:
12574:
12549:
12416:
12358:
11990:
11898:
11810:
11765:
11715:
11704:
11687:
11424:
9690:{\displaystyle x_{2}={\dot {x}}}
9506:
9465:
9433:
9414:{\displaystyle {\dot {\sigma }}}
9369:
9333:
9314:{\displaystyle {\dot {\sigma }}}
9269:
9212:
9191:
9160:
9139:
9107:
9074:
8991:
8961:
8938:
8911:
8896:
8810:
8780:
8757:
8734:
8719:
8693:
8660:
8648:
8633:
8605:
8407:
8302:
8273:
8237:
8213:
8172:). Taking the derivative of the
8158:
8150:
7885:
7837:
7606:
7210:{\displaystyle {\dot {\sigma }}}
7125:
7114:
7051:
7040:
7009:
6998:
6947:
6905:
6891:
6857:
6843:
6824:
6796:
6773:
6746:
6732:
6690:
6676:
6642:
6619:
6595:
6560:
6549:
6517:
6483:
6469:
6427:
6413:
6382:
6354:
6339:
6321:
6307:
6284:
6262:
6220:
6212:
6178:
6123:
6097:
6054:
6026:
6015:
5967:
5952:
5849:
5838:
5810:
5740:
5717:
5681:
5650:
5639:
5610:
5542:{\displaystyle {\dot {\sigma }}}
5506:
5495:
5467:
5459:may become vanishingly small as
4533:{\displaystyle {\dot {z}}=-\mu }
4495:{\displaystyle z(t)=z_{0}-\mu t}
3626:
3615:
3601:To ensure that the sliding mode
3586:{\displaystyle {\dot {\sigma }}}
3547:
3539:
3531:and so the feedback control law
3507:
3489:
3475:
3452:
3430:
3392:
3367:
3354:
3293:
3262:
3222:
3189:
3158:
3118:
3094:{\displaystyle {\dot {\sigma }}}
3035:
2946:Roughly speaking (i.e., for the
2920:
2675:
2664:
2623:
2520:
2490:
2476:
2439:
2361:
2350:
2310:
2299:
2239:
2228:
2200:
2189:
2158:
2147:
2115:
2104:
2076:
2045:
2034:
1972:
1936:
1836:
1810:
1751:
1743:
1708:
1677:
1632:
1592:
1564:
1538:
1460:
1452:
1328:
1306:
1220:
1181:
1146:
1112:
1104:
1053:
739:
556:
502:
487:
464:
435:
17738:Sira- Ramirez, Hebertt (2015).
17417:
11477:
10064:{\displaystyle u(x,{\dot {x}})}
9084:{\displaystyle u(\mathbf {x} )}
8417:{\displaystyle u(\mathbf {x} )}
8386:{\displaystyle {\dot {V}}<0}
6204:for the equivalent control law
5583:Theorem 2: Region of attraction
5325:
5319:
3823:Explanation by comparison lemma
3232:{\displaystyle u(\mathbf {x} )}
3128:{\displaystyle u(\mathbf {x} )}
3045:{\displaystyle u(\mathbf {x} )}
2844:
1724:{\displaystyle \mathbf {x} (t)}
1162:{\displaystyle \mathbf {x} (t)}
1069:{\displaystyle \mathbf {x} (t)}
353:to force the trajectories of a
91:
17624:. London: Taylor and Francis.
17305:
17169:
17151:
17032:
17018:
17010:
16962:
16857:
16832:
16822:
16796:
16771:
16761:
16699:
16674:
16664:
16438:
16399:
16374:
16364:
16272:
16248:
16223:
16213:
16148:
16124:
16099:
16089:
16024:
16000:
15975:
15965:
15933:
15904:
15825:
15807:
15797:
15659:
15623:
15603:
15591:
15585:
15573:
15520:
15508:
15441:
15405:
15291:
15278:
14964:
14952:
14882:
14870:
14831:
14819:
14688:
14682:
14516:
14510:
14396:
14314:
14264:
14225:
14219:
14204:
14198:
14132:
14120:
14107:
14062:
14024:
14018:
13961:
13948:
13696:
13670:
13664:
13643:
13633:
13585:
13571:after some finite time (i.e.,
13537:{\displaystyle {\hat {x}}_{1}}
13522:
13470:
13408:
13361:
13348:
13318:
13293:
13283:
13271:
13257:
13246:
13208:
13183:
13173:
13147:
13098:
13053:. The error dynamics are then
13029:
12978:
12933:
12903:
12891:
12720:{\displaystyle {\hat {x}}_{1}}
12705:
12673:
12639:
12614:
12604:
12578:
12553:
12501:
12489:
12461:
12439:
12429:
12420:
12331:
12319:
12269:
12257:
12251:
12239:
12195:
12183:
11950:
11905:
11862:
11817:
11665:Here, consider the LTI system
11606:
11594:
11588:
11567:
11507:
11486:
11428:
11420:
11365:
11328:
11315:
11299:
11282:
11278:
11269:
11248:
11143:
11126:
11054:
11037:
11022:
11001:
10971:
10967:
10940:
10933:
10925:
10908:
10888:
10871:
10847:
10839:
10813:
10809:
10782:
10760:
10752:
10748:
10721:
10714:
10706:
10689:
10657:
10653:
10626:
10604:
10469:
10465:
10438:
10416:
10258:
10231:
10058:
10037:
9963:
9937:
9907:
9899:
9868:
9864:
9858:
9851:
9807:
9775:
9595:
9568:
9510:
9502:
9469:
9461:
9373:
9365:
9273:
9265:
9216:
9208:
9195:
9187:
9164:
9156:
9143:
9135:
9111:
9103:
9078:
9070:
8971:
8957:
8948:
8934:
8900:
8892:
8873:
8828:
8790:
8776:
8767:
8753:
8723:
8715:
8697:
8689:
8637:
8629:
8609:
8601:
8411:
8403:
8306:
8298:
8242:
8233:
8220:
8217:
8209:
8203:
8129:
8117:
7889:
7881:
7841:
7833:
7610:
7602:
7516:
7504:
7498:
7477:
7456:{\displaystyle {\dot {x}}+x=0}
7365:
7359:
7327:
7300:
7262:{\displaystyle {\dot {x}}+x=0}
7118:
7110:
7044:
7036:
7002:
6994:
6915:
6901:
6867:
6853:
6834:
6820:
6806:
6792:
6783:
6769:
6756:
6742:
6700:
6686:
6652:
6638:
6629:
6615:
6553:
6545:
6493:
6479:
6437:
6423:
6371:and so the equivalent control
6317:
6303:
6294:
6280:
6224:
6216:
6182:
6174:
6058:
6050:
6019:
6011:
5977:
5963:
5842:
5834:
5785:
5779:
5744:
5736:
5721:
5713:
5643:
5635:
5599:), the subspace for which the
5499:
5491:
5425:
5408:
5344:
5327:
5316:
5301:
5289:
5283:
5230:
5224:
5158:
5149:
5038:
4996:
4603:
4597:
4556:
4550:
4467:
4461:
3987:
3975:
3921:
3910:
3852:
3840:
3739:
3728:
3619:
3611:
3551:
3543:
3485:
3471:
3462:
3448:
3297:
3289:
3266:
3258:
3226:
3218:
3193:
3185:
3162:
3154:
3122:
3114:
3101:have opposite signs. That is,
3039:
3031:
2924:
2916:
2668:
2660:
2627:
2619:
2524:
2516:
2494:
2486:
2480:
2472:
2446:
2443:
2435:
2429:
2354:
2346:
2324:is approximated by the smooth
2303:
2295:
2265:, it is certainly not locally
2232:
2224:
2193:
2185:
2151:
2143:
2108:
2100:
2038:
2030:
1976:
1968:
1901:
1889:
1718:
1712:
1681:
1673:
1636:
1628:
1596:
1588:
1505:
1252:
1227:
1156:
1150:
1125:
1122:
1116:
1108:
1063:
1057:
1002:
944:
861:
855:
838:
832:
802:
796:
779:
773:
749:
743:
678:
672:
655:
649:
619:
613:
596:
590:
566:
560:
512:
506:
497:
483:
474:
460:
451:
445:
1:
17520:Automation and Remote Control
17449:Li, Yun; et al. (1996).
17186:
15866:{\displaystyle v_{\text{eq}}}
15768:{\displaystyle v_{\text{eq}}}
15532:{\displaystyle (n-1)\times 1}
14938:{\displaystyle v_{\text{eq}}}
14549:{\displaystyle v_{\text{eq}}}
7819:. The sliding surface is the
7812:{\displaystyle 1\leq i\leq n}
7578:). The switching function is
7412:{\displaystyle {\dot {x}}=-x}
4721:{\displaystyle {\sqrt {V}}=0}
1875:{\displaystyle 1\leq k\leq m}
16993:{\displaystyle \mathbf {y} }
16978:follows the measured output
16748:with the switching function
16741:{\displaystyle \mathbf {u} }
13843:{\displaystyle \mathbf {x} }
12365:{\displaystyle \mathbf {x} }
11997:{\displaystyle \mathbf {x} }
10831:and by the assumption about
10545:{\displaystyle \sigma >0}
10360:{\displaystyle \sigma <0}
9440:{\displaystyle \mathbf {x} }
9340:{\displaystyle \mathbf {x} }
7582:to be the linear combination
7542:described by Equation (
6524:{\displaystyle \mathbf {u} }
6130:{\displaystyle \mathbf {x} }
5817:{\displaystyle \mathbf {x} }
5474:{\displaystyle \mathbf {x} }
4420:{\displaystyle 2{\sqrt {V}}}
3640:is attained in finite time,
2083:{\displaystyle \mathbf {x} }
1843:{\displaystyle \mathbf {u} }
1817:{\displaystyle \mathbf {x} }
1571:{\displaystyle \mathbf {x} }
1545:{\displaystyle \mathbf {x} }
1335:{\displaystyle \mathbf {u} }
1313:{\displaystyle \mathbf {x} }
1188:{\displaystyle \mathbf {u} }
7:
17767:10.1007/978-3-319-18290-2_2
17685:. Berlin: Springer Verlag.
17666:. Berlin: Springer Verlag.
17428:. Marcel Dekker Hardcover.
17067:
16874:to the 2-input LTI system.
15566:must be chosen so that the
14337:That is, positive constant
9388:
9288:
9045:
8393:, the feedback control law
8178:
7739:
7544:
7026:, the sliding mode surface
5595:
5589:
4775:{\displaystyle {\sqrt {V}}}
4691:{\displaystyle {\sqrt {V}}}
4582:, it must be the case that
4398:upper right-hand derivative
3021:, the feedback control law
2697:
2691:
2558:
2068:To force the system states
2061:
2006:
1802:is the number of states in
1205:
1078:
531:
10:
17932:
17643:; Spurgeon, Sarah (2004).
17074:Variable structure control
14522:{\displaystyle v(\sigma )}
11624:Automated design solutions
11232:which can be expressed in
8578:
7780:{\displaystyle s_{i}>0}
7588:
7139:only approximately holds.
5885:{\displaystyle {\dot {V}}}
5791:{\displaystyle V(\sigma )}
4822:of the switching function
4575:{\displaystyle z(0)=z_{0}}
3993:{\displaystyle V\in (0,1]}
2415:
2172:from any initial condition
1919:
420:
413:nonlinear dynamical system
61:variable structure control
17889:10.1080/00207179208934270
17477:10.1080/00207179608921865
17337:10.1017/s0263574712000720
17079:Variable structure system
15004:. In particular, the row
9921:has a finite upper bound
7371:{\displaystyle a(\cdot )}
5922:Theorem 3: Sliding motion
5911:{\displaystyle \sigma =0}
5568:{\displaystyle \sigma =0}
5393:{\displaystyle \sigma =0}
5201:{\displaystyle \alpha =1}
3780:{\displaystyle \mu >0}
1850:. For each control index
82:variable structure system
17258:Zeinali, Meysar (2018).
17144:
17132:Pulse-density modulation
14918:This equivalent control
12529:be the estimates of the
12405:. Hence, let the vector
4440:{\displaystyle \propto }
2906:of the surface given by
17816:10.1137/1.9781611975840
17557:10.1109/CDC.1992.371368
17537:Drakunov, S.V. (1992).
17399:Filippov, A.F. (1988).
16903:{\displaystyle y=x_{1}}
16870:is viewed as a control
14375:{\displaystyle e_{1}=0}
13779:{\displaystyle \sigma }
12753:{\displaystyle y=x_{1}}
12398:{\displaystyle y=x_{1}}
9648:{\displaystyle x_{1}=x}
8424:must be chosen so that
7533:Control design examples
5452:{\displaystyle \sigma }
5107:{\displaystyle \sigma }
4835:{\displaystyle \sigma }
4782:is proportional to the
4540:with initial condition
3564:has a direct impact on
3065:{\displaystyle \sigma }
2279:Picard–Lindelöf theorem
1395:{\displaystyle x_{1}=0}
1043:Picard–Lindelöf theorem
390:be applied by means of
86:hybrid dynamical system
35:method that alters the
17177:delta-sigma modulation
17138:Pulse-width modulation
17126:Delta-sigma modulation
17058:
16994:
16972:
16943:uses a generic matrix
16937:
16904:
16864:
16803:
16742:
16717:
16613:
16519:
16477:
15877:–like features.
15867:
15832:
15769:
15738:
15737:{\displaystyle A_{12}}
15707:
15666:
15610:
15560:
15533:
15495:
15463:
15025:
15024:{\displaystyle A_{12}}
14998:
14971:
14939:
14909:
14740:
14550:
14523:
14494:
14452:
14425:
14376:
14328:
14232:
14179:
14141:
14031:
13844:
13818:
13780:
13760:
13728:
13614:
13565:
13544:tracks the real state
13538:
13499:
13437:
13372:
13047:
13000:
12912:
12855:
12789:
12754:
12721:
12685:
12646:
12523:
12399:
12366:
12340:
12278:
12204:
12148:
12121:
12120:{\displaystyle a_{11}}
12088:
11998:
11972:
11884:
11793:
11613:
11546:
11441:
11400:
11222:
10979:
10855:
10821:
10665:
10581:
10546:
10520:
10477:
10396:
10361:
10335:
10290:
10103:
10065:
10017:
9915:
9885:
9827:
9691:
9649:
9608:
9523:
9482:
9441:
9415:
9380:
9341:
9315:
9280:
9236:
9085:
9033:
8566:
8524:
8418:
8387:
8348:
8166:
8136:
8104:
8074:
7902:
7854:
7813:
7781:
7727:
7572:
7523:
7457:
7413:
7372:
7340:
7263:
7211:
7182:
7133:
7094:
7059:
7017:
6963:
6568:
6525:
6500:
6362:
6231:
6195:
6150:equivalent control law
6131:
6105:
6065:
6034:
5984:
5912:
5886:
5857:
5818:
5792:
5760:
5661:
5569:
5543:
5514:
5475:
5453:
5433:
5394:
5364:
5258:
5202:
5172:
5108:
5084:
5051:
4836:
4816:
4776:
4752:
4722:
4692:
4668:
4634:
4576:
4534:
4496:
4441:
4421:
4390:
4360:
4285:Upper right-hand
4146:
4113:
4064:
3994:
3953:
3859:
3813:
3781:
3752:
3676:
3634:
3587:
3558:
3522:
3304:
3273:
3233:
3200:
3169:
3129:
3095:
3066:
3046:
3015:
2970:
2937:
2893:
2683:
2644:
2596:
2546:
2394:Theoretical foundation
2384:delta-sigma modulation
2380:pulse-width modulation
2372:
2318:
2247:
2208:
2166:
2123:
2084:
2053:
1994:
1908:
1876:
1844:
1818:
1792:
1759:
1725:
1694:
1649:
1609:
1572:
1546:
1524:
1468:
1412:sliding (hyper)surface
1396:
1363:
1336:
1314:
1292:
1265:
1189:
1163:
1132:
1070:
1027:
963:
891:
708:
519:
392:pulse-width modulation
331:
300:
277:
266:
211:
150:
124:
78:sliding (hyper)surface
17742:. Basel: Birkhauser.
17723:. Basel: Birkhauser.
17059:
16995:
16973:
16938:
16905:
16865:
16804:
16743:
16718:
16614:
16520:
16478:
15868:
15833:
15770:
15739:
15708:
15667:
15611:
15561:
15559:{\displaystyle L_{2}}
15534:
15496:
15464:
15026:
14999:
14997:{\displaystyle x_{1}}
14972:
14970:{\displaystyle (n-1)}
14940:
14910:
14741:
14551:
14524:
14495:
14453:
14451:{\displaystyle e_{1}}
14426:
14377:
14329:
14233:
14180:
14142:
14032:
13845:
13819:
13781:
13761:
13729:
13615:
13566:
13564:{\displaystyle x_{1}}
13539:
13500:
13438:
13373:
13048:
13001:
12913:
12856:
12790:
12755:
12722:
12686:
12647:
12524:
12400:
12367:
12341:
12279:
12205:
12149:
12147:{\displaystyle x_{1}}
12122:
12089:
11999:
11973:
11885:
11794:
11638:Sliding mode observer
11614:
11547:
11442:
11401:
11223:
10980:
10856:
10822:
10666:
10582:
10547:
10521:
10478:
10397:
10362:
10336:
10291:
10104:
10066:
10018:
9916:
9886:
9828:
9692:
9650:
9609:
9524:
9483:
9442:
9416:
9381:
9342:
9316:
9281:
9237:
9086:
9034:
8567:
8525:
8419:
8388:
8349:
8167:
8137:
8135:{\displaystyle (n-1)}
8105:
8075:
7903:
7855:
7814:
7782:
7728:
7573:
7524:
7458:
7414:
7373:
7341:
7264:
7212:
7183:
7134:
7095:
7060:
7018:
6964:
6574:forces the system to
6569:
6526:
6501:
6363:
6232:
6196:
6132:
6106:
6066:
6035:
5985:
5913:
5887:
5858:
5819:
5793:
5761:
5662:
5570:
5544:
5515:
5476:
5454:
5434:
5395:
5365:
5259:
5203:
5173:
5109:
5085:
5052:
4837:
4817:
4777:
4753:
4723:
4693:
4669:
4635:
4577:
4535:
4497:
4442:
4422:
4391:
4389:{\displaystyle D^{+}}
4361:
4147:
4114:
4065:
3995:
3954:
3860:
3858:{\displaystyle V\in }
3814:
3782:
3753:
3677:
3635:
3588:
3559:
3523:
3305:
3274:
3234:
3201:
3170:
3130:
3096:
3067:
3047:
3016:
2971:
2938:
2894:
2684:
2645:
2597:
2547:
2373:
2319:
2248:
2209:
2167:
2124:
2085:
2054:
1995:
1909:
1907:{\displaystyle (n-1)}
1877:
1845:
1819:
1793:
1760:
1726:
1695:
1650:
1610:
1573:
1547:
1525:
1469:
1397:
1364:
1362:{\displaystyle x_{1}}
1337:
1315:
1293:
1291:{\displaystyle x_{1}}
1266:
1190:
1164:
1133:
1071:
1028:
964:
892:
709:
520:
332:
301:
267:
212:
151:
125:
99:
55:control law is not a
17004:
16982:
16951:
16914:
16881:
16813:
16752:
16730:
16625:
16530:
16493:
15887:
15850:
15779:
15752:
15721:
15688:
15620:
15570:
15543:
15505:
15476:
15038:
15008:
14981:
14949:
14922:
14756:
14563:
14533:
14504:
14462:
14435:
14386:
14353:
14248:
14192:
14151:
14044:
13857:
13832:
13790:
13770:
13741:
13627:
13575:
13548:
13512:
13454:
13385:
13060:
13010:
12922:
12868:
12806:
12764:
12731:
12695:
12659:
12540:
12409:
12376:
12354:
12290:
12216:
12160:
12131:
12104:
12011:
11986:
11894:
11806:
11672:
11564:
11453:
11414:
11242:
10995:
10867:
10835:
10685:
10591:
10556:
10530:
10489:
10406:
10371:
10345:
10304:
10115:
10075:
10031:
9931:
9895:
9841:
9703:
9659:
9626:
9547:
9496:
9455:
9429:
9396:
9352:
9329:
9296:
9252:
9097:
9064:
8586:
8538:
8430:
8397:
8362:
8188:
8146:
8114:
8088:
7918:
7866:
7827:
7791:
7758:
7596:
7556:
7548:) with single input
7474:
7426:
7385:
7353:
7279:
7232:
7192:
7147:
7104:
7069:
7030:
6979:
6588:
6539:
6513:
6378:
6244:
6208:
6159:
6119:
6078:
6044:
6005:
5933:
5896:
5867:
5828:
5806:
5773:
5674:
5603:
5553:
5524:
5485:
5463:
5443:
5404:
5378:
5274:
5215:
5186:
5121:
5098:
5064:
4857:
4826:
4789:
4762:
4736:
4702:
4678:
4648:
4644:. Moreover, because
4586:
4544:
4506:
4455:
4431:
4404:
4373:
4159:
4123:
4081:
4007:
3966:
3872:
3831:
3791:
3765:
3690:
3644:
3605:
3568:
3535:
3321:
3283:
3243:
3212:
3179:
3139:
3108:
3076:
3056:
3025:
2980:
2954:
2910:
2708:
2654:
2610:
2576:
2423:
2331:
2289:
2267:Lipschitz continuous
2218:
2179:
2137:
2094:
2072:
2024:
1927:
1886:
1854:
1832:
1806:
1776:
1739:
1704:
1667:
1622:
1582:
1560:
1534:
1484:
1448:
1373:
1346:
1324:
1302:
1298:of the state vector
1275:
1216:
1177:
1142:
1100:
1049:
973:
915:
735:
552:
428:
378:requires the use of
343:exponentially stable
315:
284:
274:exponentially stable
224:
160:
134:
108:
72:and the geometrical
25:sliding mode control
17120:Switching amplifier
12797:Luenberger observer
11802:where state vector
11652:Luenberger observer
10854:{\displaystyle |a|}
10677:triangle inequality
9914:{\displaystyle |a|}
8534:Hence, the product
8103:{\displaystyle n-1}
7571:{\displaystyle m=1}
4751:{\displaystyle V=0}
2969:{\displaystyle m=1}
2541:
1791:{\displaystyle n-m}
1408:configuration space
1369:quickly returns to
368:continuous function
330:{\displaystyle s=0}
299:{\displaystyle s=0}
149:{\displaystyle s=0}
123:{\displaystyle s=0}
57:continuous function
17898:10338.dmlcz/135339
17851:10.1007/BFb0033675
17597:10.1007/BFb0033675
17054:
16990:
16968:
16933:
16900:
16860:
16799:
16738:
16713:
16704:
16609:
16600:
16592:
16515:
16473:
16471:
16404:
16335:
16327:
16201:
16074:
15863:
15828:
15765:
15734:
15703:
15662:
15606:
15556:
15529:
15491:
15459:
15235:
15127:
15021:
14994:
14967:
14935:
14905:
14899:
14865:
14842:
14808:
14784:
14777:
14736:
14546:
14519:
14490:
14448:
14421:
14372:
14324:
14228:
14175:
14137:
14027:
13840:
13814:
13776:
13756:
13724:
13610:
13561:
13534:
13495:
13433:
13368:
13366:
13043:
12996:
12908:
12851:
12845:
12785:
12750:
12717:
12681:
12642:
12519:
12395:
12362:
12336:
12274:
12200:
12144:
12117:
12084:
12078:
12004:state vector. Let
11994:
11968:
11880:
11789:
11784:
11758:
11609:
11542:
11437:
11396:
11395:
11377:(i.e., tests
11372:
11218:
11213:
11102:
10975:
10851:
10817:
10661:
10577:
10542:
10516:
10473:
10392:
10357:
10331:
10286:
10099:
10061:
10013:
9911:
9881:
9823:
9818:
9687:
9645:
9604:
9519:
9478:
9437:
9411:
9376:
9337:
9311:
9276:
9232:
9227:
9091:is chosen so that
9081:
9029:
9028:
9005:
8562:
8520:
8515:
8414:
8383:
8344:
8341:
8279:
8176:in Equation (
8162:
8132:
8100:
8070:
7898:
7850:
7809:
7777:
7723:
7568:
7519:
7453:
7409:
7368:
7336:
7259:
7207:
7178:
7129:
7090:
7055:
7013:
6959:
6564:
6521:
6496:
6358:
6227:
6191:
6127:
6101:
6061:
6030:
5980:
5908:
5882:
5853:
5814:
5788:
5756:
5657:
5565:
5539:
5510:
5471:
5449:
5429:
5390:
5360:
5254:
5198:
5168:
5104:
5080:
5047:
4986:
4984:
4956:
4949:
4902:
4832:
4812:
4772:
4748:
4718:
4688:
4664:
4630:
4572:
4530:
4492:
4437:
4417:
4386:
4356:
4303:
4260:
4246:
4239:
4142:
4109:
4060:
3990:
3949:
3855:
3809:
3777:
3748:
3672:
3630:
3583:
3554:
3518:
3409:
3300:
3269:
3229:
3196:
3165:
3125:
3091:
3062:
3052:is picked so that
3042:
3011:
2966:
2950:control case when
2933:
2889:
2876:
2837:
2835:
2807:
2800:
2753:
2679:
2640:
2592:
2542:
2527:
2368:
2314:
2277:guaranteed by the
2271:closed-loop system
2243:
2204:
2162:
2119:
2080:
2049:
1990:
1904:
1872:
1840:
1814:
1788:
1755:
1721:
1690:
1645:
1605:
1568:
1542:
1520:
1479:switching function
1464:
1392:
1359:
1332:
1310:
1288:
1261:
1203:in Equation (
1185:
1159:
1128:
1076:to Equation (
1066:
1033:are assumed to be
1023:
959:
887:
866:
704:
683:
515:
384:optimal controller
327:
296:
278:
262:
207:
146:
120:
17916:Nonlinear control
17860:978-3-030-36620-9
17795:978-3-319-62895-0
17776:978-3-319-18289-6
17749:978-3-319-17256-9
17711:978-1-78561-076-9
17692:978-3-540-79016-7
17673:978-3-540-32800-1
17639:Sabanovic, Asif;
17631:978-0-7484-0601-2
17606:978-3-540-19869-7
17566:978-0-7803-0872-5
17501:978-0-7484-0116-1
17435:978-0-8247-0671-5
17410:978-90-277-2699-5
17385:978-0-13-067389-3
17371:Nonlinear Systems
17299:10.1109/41.184818
17217:978-0-86341-170-0
17104:Bang–bang control
17089:Nonlinear control
17035:
16965:
16835:
16774:
16677:
16635:
16540:
16503:
16466:
16454:
16441:
16427:
16377:
16275:
16226:
16151:
16102:
16027:
15978:
15936:
15912:
15907:
15860:
15810:
15762:
15748:"). That is, the
15339:
15259:
15243:
15160:
15150:
15135:
15116:
15086:
15063:
14932:
14897:
14895:
14849:
14847:
14840:
14838:
14794:
14792:
14782:
14771:
14763:
14761:
14730:
14692:
14676:
14670:
14627:
14615:
14581:
14543:
14475:
14399:
14166:
13885:
13869:
13805:
13753:
13699:
13667:
13646:
13588:
13525:
13508:so that estimate
13473:
13411:
13296:
13260:
13186:
13150:
13123:
13106:
13101:
13078:
13032:
12708:
12617:
12581:
12561:
12556:
12492:
12464:
12442:
12423:
11694:
11631:genetic algorithm
11585:
11540:
11528:
11504:
11481:
11465:
11392:
11378:
11364:
11357:
11344:
11320:
11318:
11295:
11266:
11204:
11197:
11190:
11169:
11139:
11095:
11078:
11075:
11050:
11019:
10964:
10921:
10884:
10806:
10773:
10745:
10702:
10650:
10617:
10568:
10507:
10462:
10429:
10383:
10322:
10285:
10282:
10271:
10255:
10216:
10201:
10186:
10165:
10143:
10127:
10090:
10055:
10010:
9837:Also assume that
9760:
9724:
9684:
9592:
9559:
9408:
9308:
9203:
9151:
9057:
9056:
9053:
9052:
9025:
9011:
9001:
8998:
8985:
8922:
8920:
8919:
8917:
8880:
8820:
8817:
8804:
8740:
8701:
8686:
8674:
8667:
8654:
8598:
8553:
8502:
8488:
8464:
8450:
8374:
8343:
8340:
8313:
8295:
8281:
8278:
8255:
8248:
8200:
8174:Lyapunov function
8055:
8017:
7973:
7941:
7878:
7754:where the weight
7751:
7750:
7747:
7746:
7495:
7438:
7397:
7324:
7291:
7244:
7204:
7169:
7081:
6991:
6952:
6896:
6813:
6763:
6737:
6681:
6602:
6474:
6418:
6349:
6346:
6333:
6267:
6171:
6090:
5958:
5879:
5800:Lyapunov function
5733:
5536:
5421:
5340:
5323:
5313:
5242:
5136:
5034:
5032:
5025:
4983:
4951:
4948:
4921:
4916:
4904:
4901:
4880:
4862:
4860:
4770:
4710:
4686:
4656:
4606:
4518:
4415:
4345:
4319:
4318:
4297:
4286:
4232:
4207:
4203:
4189:
4187:
4166:
4164:
4140:
4046:
4020:
4019:
3944:
3918:
3899:
3736:
3717:
3580:
3517:
3514:
3501:
3435:
3411:
3408:
3379:
3374:
3359:
3333:
3255:
3151:
3088:
3002:
2887:
2875:
2848:
2834:
2802:
2799:
2772:
2767:
2755:
2752:
2731:
2713:
2711:
2570:
2569:
2566:
2565:
2508:
2460:
2408:Lyapunov function
2388:nonlinear control
2343:
2018:
2017:
2014:
2013:
1037:and sufficiently
543:
542:
539:
538:
442:
400:nonlinear control
380:bang–bang control
237:
192:
130:. The particular
33:nonlinear control
17923:
17902:
17900:
17883:(4): 1029–1037.
17864:
17837:
17799:
17780:
17753:
17734:
17730:978-0-81764-8923
17715:
17696:
17677:
17658:
17635:
17610:
17579:
17578:
17534:
17528:
17527:
17515:
17506:
17505:
17487:
17481:
17480:
17470:
17446:
17440:
17439:
17421:
17415:
17414:
17396:
17390:
17389:
17362:
17349:
17348:
17320:
17314:
17309:
17303:
17302:
17292:
17272:
17266:
17265:
17255:
17249:
17248:
17228:
17222:
17221:
17200:
17180:
17173:
17167:
17155:
17063:
17061:
17060:
17055:
17053:
17045:
17037:
17036:
17031:
17026:
17017:
16999:
16997:
16996:
16991:
16989:
16977:
16975:
16974:
16969:
16967:
16966:
16961:
16956:
16946:
16942:
16940:
16939:
16934:
16932:
16921:
16909:
16907:
16906:
16901:
16899:
16898:
16869:
16867:
16866:
16861:
16856:
16855:
16843:
16842:
16837:
16836:
16828:
16808:
16806:
16805:
16800:
16795:
16794:
16782:
16781:
16776:
16775:
16767:
16747:
16745:
16744:
16739:
16737:
16722:
16720:
16719:
16714:
16709:
16708:
16698:
16697:
16685:
16684:
16679:
16678:
16670:
16653:
16637:
16636:
16633:
16618:
16616:
16615:
16610:
16605:
16604:
16597:
16596:
16586:
16585:
16542:
16541:
16538:
16524:
16522:
16521:
16516:
16505:
16504:
16501:
16482:
16480:
16479:
16474:
16472:
16468:
16467:
16464:
16462:
16456:
16455:
16452:
16443:
16442:
16437:
16432:
16429:
16428:
16425:
16413:
16409:
16408:
16398:
16397:
16385:
16384:
16379:
16378:
16370:
16353:
16340:
16339:
16332:
16331:
16321:
16320:
16277:
16276:
16271:
16266:
16254:
16247:
16246:
16234:
16233:
16228:
16227:
16219:
16206:
16205:
16195:
16194:
16164:
16153:
16152:
16147:
16142:
16130:
16123:
16122:
16110:
16109:
16104:
16103:
16095:
16079:
16078:
16071:
16070:
16040:
16029:
16028:
16023:
16018:
16006:
15999:
15998:
15986:
15985:
15980:
15979:
15971:
15949:
15938:
15937:
15932:
15927:
15914:
15913:
15908:
15903:
15898:
15896:
15872:
15870:
15869:
15864:
15862:
15861:
15858:
15845:
15837:
15835:
15834:
15829:
15818:
15817:
15812:
15811:
15803:
15774:
15772:
15771:
15766:
15764:
15763:
15760:
15747:
15743:
15741:
15740:
15735:
15733:
15732:
15712:
15710:
15709:
15704:
15702:
15701:
15696:
15671:
15669:
15668:
15663:
15658:
15657:
15648:
15647:
15635:
15634:
15615:
15613:
15612:
15607:
15565:
15563:
15562:
15557:
15555:
15554:
15538:
15536:
15535:
15530:
15500:
15498:
15497:
15492:
15490:
15489:
15484:
15468:
15466:
15465:
15460:
15455:
15454:
15449:
15440:
15439:
15430:
15429:
15417:
15416:
15401:
15400:
15395:
15389:
15388:
15379:
15378:
15366:
15365:
15360:
15354:
15353:
15341:
15340:
15337:
15331:
15330:
15318:
15317:
15312:
15306:
15305:
15290:
15289:
15274:
15273:
15261:
15260:
15258:
15257:
15256:
15251:
15244:
15239:
15232:
15231:
15211:
15210:
15197:
15196:
15180:
15178:
15175:
15174:
15162:
15161:
15159:
15158:
15157:
15152:
15151:
15146:
15141:
15136:
15131:
15124:
15123:
15118:
15117:
15109:
15094:
15093:
15088:
15087:
15079:
15071:
15070:
15065:
15064:
15056:
15045:
15043:
15030:
15028:
15027:
15022:
15020:
15019:
15003:
15001:
15000:
14995:
14993:
14992:
14976:
14974:
14973:
14968:
14944:
14942:
14941:
14936:
14934:
14933:
14930:
14914:
14912:
14911:
14906:
14901:
14900:
14898:
14896:
14893:
14891:
14866:
14861:
14860:
14855:
14844:
14843:
14841:
14839:
14836:
14834:
14809:
14804:
14803:
14786:
14785:
14783:
14780:
14778:
14773:
14772:
14769:
14745:
14743:
14742:
14737:
14732:
14731:
14728:
14719:
14718:
14713:
14707:
14706:
14694:
14693:
14691:
14677:
14672:
14671:
14668:
14662:
14660:
14654:
14653:
14648:
14642:
14641:
14629:
14628:
14626:
14619:
14616:
14611:
14610:
14601:
14599:
14596:
14595:
14583:
14582:
14574:
14555:
14553:
14552:
14547:
14545:
14544:
14541:
14528:
14526:
14525:
14520:
14499:
14497:
14496:
14491:
14483:
14482:
14477:
14476:
14468:
14457:
14455:
14454:
14449:
14447:
14446:
14430:
14428:
14427:
14422:
14420:
14419:
14407:
14406:
14401:
14400:
14392:
14381:
14379:
14378:
14373:
14365:
14364:
14348:
14344:
14340:
14333:
14331:
14330:
14325:
14317:
14312:
14311:
14306:
14300:
14299:
14287:
14286:
14277:
14276:
14267:
14237:
14235:
14234:
14229:
14184:
14182:
14181:
14176:
14168:
14167:
14159:
14146:
14144:
14143:
14138:
14136:
14135:
14118:
14106:
14105:
14087:
14086:
14074:
14073:
14058:
14057:
14052:
14036:
14034:
14033:
14028:
14011:
14010:
14005:
13999:
13998:
13986:
13985:
13976:
13975:
13960:
13959:
13941:
13940:
13935:
13929:
13928:
13916:
13915:
13906:
13905:
13893:
13892:
13887:
13886:
13878:
13871:
13870:
13862:
13849:
13847:
13846:
13841:
13839:
13823:
13821:
13820:
13815:
13807:
13806:
13798:
13785:
13783:
13782:
13777:
13765:
13763:
13762:
13757:
13755:
13754:
13746:
13733:
13731:
13730:
13725:
13720:
13719:
13707:
13706:
13701:
13700:
13692:
13685:
13684:
13669:
13668:
13660:
13654:
13653:
13648:
13647:
13639:
13619:
13617:
13616:
13611:
13609:
13608:
13596:
13595:
13590:
13589:
13581:
13570:
13568:
13567:
13562:
13560:
13559:
13543:
13541:
13540:
13535:
13533:
13532:
13527:
13526:
13518:
13504:
13502:
13501:
13496:
13494:
13493:
13481:
13480:
13475:
13474:
13466:
13446:
13442:
13440:
13439:
13434:
13432:
13431:
13419:
13418:
13413:
13412:
13404:
13397:
13396:
13377:
13375:
13374:
13369:
13367:
13360:
13359:
13338:
13324:
13317:
13316:
13304:
13303:
13298:
13297:
13289:
13270:
13262:
13261:
13256:
13251:
13236:
13232:
13221:
13207:
13206:
13194:
13193:
13188:
13187:
13179:
13163:
13152:
13151:
13146:
13141:
13129:
13125:
13124:
13119:
13114:
13108:
13107:
13102:
13097:
13092:
13090:
13080:
13079:
13074:
13069:
13052:
13050:
13049:
13044:
13042:
13034:
13033:
13028:
13023:
13017:
13005:
13003:
13002:
12997:
12995:
12994:
12989:
12977:
12976:
12958:
12957:
12945:
12944:
12929:
12917:
12915:
12914:
12909:
12907:
12906:
12889:
12880:
12879:
12860:
12858:
12857:
12852:
12850:
12849:
12842:
12841:
12799:. Likewise, let
12794:
12792:
12791:
12786:
12784:
12783:
12778:
12759:
12757:
12756:
12751:
12749:
12748:
12726:
12724:
12723:
12718:
12716:
12715:
12710:
12709:
12701:
12690:
12688:
12687:
12682:
12680:
12672:
12651:
12649:
12648:
12643:
12638:
12637:
12625:
12624:
12619:
12618:
12610:
12594:
12583:
12582:
12577:
12572:
12563:
12562:
12557:
12552:
12547:
12545:
12532:
12528:
12526:
12525:
12520:
12518:
12517:
12512:
12500:
12499:
12494:
12493:
12485:
12472:
12471:
12466:
12465:
12457:
12450:
12449:
12444:
12443:
12435:
12425:
12424:
12419:
12414:
12404:
12402:
12401:
12396:
12394:
12393:
12371:
12369:
12368:
12363:
12361:
12345:
12343:
12342:
12337:
12335:
12334:
12311:
12302:
12301:
12283:
12281:
12280:
12275:
12273:
12272:
12237:
12228:
12227:
12209:
12207:
12206:
12201:
12199:
12198:
12181:
12172:
12171:
12153:
12151:
12150:
12145:
12143:
12142:
12126:
12124:
12123:
12118:
12116:
12115:
12093:
12091:
12090:
12085:
12083:
12082:
12075:
12074:
12063:
12062:
12049:
12048:
12037:
12036:
12003:
12001:
12000:
11995:
11993:
11981:
11977:
11975:
11974:
11969:
11967:
11966:
11961:
11949:
11948:
11930:
11929:
11917:
11916:
11901:
11889:
11887:
11886:
11881:
11879:
11878:
11873:
11861:
11860:
11842:
11841:
11829:
11828:
11813:
11798:
11796:
11795:
11790:
11788:
11787:
11781:
11780:
11768:
11763:
11762:
11756:
11718:
11707:
11696:
11695:
11690:
11685:
11618:
11616:
11615:
11610:
11587:
11586:
11578:
11551:
11549:
11548:
11543:
11541:
11538:
11530:
11529:
11521:
11506:
11505:
11497:
11482:
11479:
11467:
11466:
11458:
11446:
11444:
11443:
11438:
11427:
11405:
11403:
11402:
11397:
11394:
11393:
11390:
11379:
11376:
11373:
11368:
11363:
11358:
11353:
11346:
11345:
11337:
11333:
11331:
11302:
11297:
11296:
11288:
11285:
11268:
11267:
11259:
11227:
11225:
11224:
11219:
11217:
11216:
11203:
11198:
11193:
11192:
11191:
11183:
11173:
11171:
11170:
11167:
11163:
11159:
11146:
11141:
11140:
11132:
11129:
11103:
11098:
11097:
11096:
11088:
11076:
11073:
11057:
11052:
11051:
11043:
11040:
11021:
11020:
11012:
10984:
10982:
10981:
10976:
10974:
10966:
10965:
10957:
10936:
10928:
10923:
10922:
10914:
10911:
10891:
10886:
10885:
10877:
10874:
10860:
10858:
10857:
10852:
10850:
10842:
10826:
10824:
10823:
10818:
10816:
10808:
10807:
10799:
10775:
10774:
10766:
10763:
10755:
10747:
10746:
10738:
10717:
10709:
10704:
10703:
10695:
10692:
10675:However, by the
10670:
10668:
10667:
10662:
10660:
10652:
10651:
10643:
10619:
10618:
10610:
10607:
10586:
10584:
10583:
10578:
10570:
10569:
10561:
10551:
10549:
10548:
10543:
10525:
10523:
10522:
10517:
10509:
10508:
10500:
10482:
10480:
10479:
10474:
10472:
10464:
10463:
10455:
10431:
10430:
10422:
10419:
10401:
10399:
10398:
10393:
10385:
10384:
10376:
10366:
10364:
10363:
10358:
10340:
10338:
10337:
10332:
10324:
10323:
10315:
10295:
10293:
10292:
10287:
10284:
10283:
10275:
10272:
10267:
10257:
10256:
10248:
10226:
10224:
10218:
10217:
10209:
10203:
10202:
10194:
10188:
10187:
10179:
10173:
10172:
10167:
10166:
10158:
10151:
10150:
10145:
10144:
10136:
10129:
10128:
10120:
10108:
10106:
10105:
10100:
10092:
10091:
10083:
10070:
10068:
10067:
10062:
10057:
10056:
10048:
10022:
10020:
10019:
10014:
10012:
10011:
10003:
9991:
9990:
9978:
9977:
9962:
9961:
9949:
9948:
9924:
9920:
9918:
9917:
9912:
9910:
9902:
9890:
9888:
9887:
9882:
9871:
9854:
9832:
9830:
9829:
9824:
9822:
9821:
9806:
9805:
9793:
9792:
9768:
9767:
9762:
9761:
9753:
9745:
9744:
9732:
9731:
9726:
9725:
9717:
9696:
9694:
9693:
9688:
9686:
9685:
9677:
9671:
9670:
9654:
9652:
9651:
9646:
9638:
9637:
9613:
9611:
9610:
9605:
9594:
9593:
9585:
9561:
9560:
9552:
9528:
9526:
9525:
9520:
9509:
9487:
9485:
9484:
9479:
9468:
9446:
9444:
9443:
9438:
9436:
9420:
9418:
9417:
9412:
9410:
9409:
9401:
9385:
9383:
9382:
9377:
9372:
9364:
9363:
9346:
9344:
9343:
9338:
9336:
9320:
9318:
9317:
9312:
9310:
9309:
9301:
9285:
9283:
9282:
9277:
9272:
9264:
9263:
9241:
9239:
9238:
9233:
9231:
9230:
9215:
9204:
9201:
9194:
9186:
9185:
9163:
9152:
9149:
9142:
9134:
9133:
9110:
9090:
9088:
9087:
9082:
9077:
9060:The control law
9047:
9038:
9036:
9035:
9030:
9027:
9026:
9023:
9012:
9010:( i.e., an
9009:
9006:
9000:
8999:
8994:
8989:
8986:
8981:
8977:
8964:
8941:
8925:
8923:
8918:
8916:
8915:
8914:
8904:
8903:
8899:
8883:
8881:
8876:
8872:
8871:
8853:
8852:
8840:
8839:
8826:
8824:
8819:
8818:
8813:
8808:
8805:
8800:
8796:
8783:
8760:
8744:
8742:
8741:
8739:
8738:
8737:
8727:
8726:
8722:
8706:
8700:
8696:
8688:
8687:
8679:
8675:
8670:
8669:
8668:
8663:
8658:
8655:
8653:
8652:
8651:
8641:
8640:
8636:
8620:
8617:
8615:
8608:
8600:
8599:
8591:
8580:
8576:
8571:
8569:
8568:
8563:
8555:
8554:
8546:
8529:
8527:
8526:
8521:
8519:
8518:
8503:
8500:
8490:
8489:
8481:
8465:
8462:
8452:
8451:
8443:
8423:
8421:
8420:
8415:
8410:
8392:
8390:
8389:
8384:
8376:
8375:
8367:
8353:
8351:
8350:
8345:
8342:
8339:
8328:
8317:
8314:
8309:
8305:
8297:
8296:
8288:
8284:
8282:
8280:
8277:
8276:
8267:
8259:
8256:
8251:
8250:
8249:
8246:
8240:
8228:
8226:
8216:
8202:
8201:
8193:
8171:
8169:
8168:
8163:
8161:
8153:
8141:
8139:
8138:
8133:
8109:
8107:
8106:
8101:
8079:
8077:
8076:
8071:
8063:
8062:
8057:
8056:
8048:
8044:
8043:
8031:
8030:
8019:
8018:
8010:
8006:
8005:
7981:
7980:
7975:
7974:
7966:
7962:
7961:
7949:
7948:
7943:
7942:
7934:
7930:
7929:
7907:
7905:
7904:
7899:
7888:
7880:
7879:
7871:
7859:
7857:
7856:
7851:
7840:
7818:
7816:
7815:
7810:
7786:
7784:
7783:
7778:
7770:
7769:
7741:
7732:
7730:
7729:
7724:
7722:
7721:
7712:
7711:
7699:
7698:
7683:
7682:
7658:
7657:
7648:
7647:
7635:
7634:
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7609:
7590:
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5470:
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5455:
5450:
5438:
5436:
5435:
5430:
5428:
5423:
5422:
5414:
5411:
5400:, and its speed
5399:
5397:
5396:
5391:
5369:
5367:
5366:
5361:
5347:
5342:
5341:
5333:
5330:
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5199:
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5105:
5089:
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3301:
3296:
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3257:
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3248:
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3225:
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3144:
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3100:
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3063:
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3038:
3020:
3018:
3017:
3012:
3004:
3003:
2995:
2992:
2991:
2975:
2973:
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2967:
2942:
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2939:
2934:
2923:
2898:
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2863:
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2833:
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2811:
2808:
2803:
2801:
2798:
2787:
2776:
2773:
2768:
2760:
2758:
2756:
2754:
2751:
2743:
2735:
2732:
2727:
2726:
2717:
2715:
2688:
2686:
2685:
2680:
2678:
2667:
2649:
2647:
2646:
2641:
2639:
2638:
2626:
2601:
2599:
2598:
2593:
2588:
2587:
2560:
2551:
2549:
2548:
2543:
2540:
2535:
2523:
2509:
2501:
2493:
2479:
2471:
2470:
2461:
2453:
2442:
2417:
2413:
2377:
2375:
2374:
2369:
2364:
2353:
2345:
2344:
2336:
2323:
2321:
2320:
2315:
2313:
2302:
2252:
2250:
2249:
2244:
2242:
2231:
2213:
2211:
2210:
2205:
2203:
2192:
2171:
2169:
2168:
2163:
2161:
2150:
2128:
2126:
2125:
2120:
2118:
2107:
2089:
2087:
2086:
2081:
2079:
2058:
2056:
2055:
2050:
2048:
2037:
2008:
1999:
1997:
1996:
1991:
1989:
1985:
1975:
1967:
1966:
1954:
1953:
1948:
1939:
1921:
1917:
1913:
1911:
1910:
1905:
1881:
1879:
1878:
1873:
1849:
1847:
1846:
1841:
1839:
1827:
1823:
1821:
1820:
1815:
1813:
1801:
1797:
1795:
1794:
1789:
1764:
1762:
1761:
1756:
1754:
1746:
1730:
1728:
1727:
1722:
1711:
1699:
1697:
1696:
1691:
1680:
1654:
1652:
1651:
1646:
1635:
1614:
1612:
1611:
1606:
1595:
1577:
1575:
1574:
1569:
1567:
1551:
1549:
1548:
1543:
1541:
1529:
1527:
1526:
1521:
1519:
1518:
1513:
1504:
1503:
1498:
1473:
1471:
1470:
1465:
1463:
1455:
1401:
1399:
1398:
1393:
1385:
1384:
1368:
1366:
1365:
1360:
1358:
1357:
1342:can ensure that
1341:
1339:
1338:
1333:
1331:
1319:
1317:
1316:
1311:
1309:
1297:
1295:
1294:
1289:
1287:
1286:
1270:
1268:
1267:
1262:
1260:
1259:
1223:
1201:dynamical system
1194:
1192:
1191:
1186:
1184:
1172:
1168:
1166:
1165:
1160:
1149:
1137:
1135:
1134:
1129:
1115:
1107:
1075:
1073:
1072:
1067:
1056:
1032:
1030:
1029:
1024:
1022:
1021:
1010:
1001:
993:
992:
987:
968:
966:
965:
960:
958:
957:
952:
943:
935:
934:
929:
903:
896:
894:
893:
888:
886:
885:
880:
871:
870:
854:
853:
831:
830:
795:
794:
772:
771:
742:
720:
713:
711:
710:
705:
703:
702:
697:
688:
687:
671:
670:
648:
647:
612:
611:
589:
588:
559:
533:
524:
522:
521:
516:
505:
490:
467:
444:
443:
438:
433:
422:
418:
336:
334:
333:
328:
305:
303:
302:
297:
271:
269:
268:
263:
261:
260:
245:
244:
239:
238:
230:
216:
214:
213:
208:
200:
199:
194:
193:
185:
178:
177:
155:
153:
152:
147:
129:
127:
126:
121:
41:nonlinear system
17931:
17930:
17926:
17925:
17924:
17922:
17921:
17920:
17906:
17905:
17871:
17869:Further reading
17861:
17826:
17796:
17777:
17750:
17731:
17712:
17693:
17674:
17655:
17641:Fridman, Leonid
17632:
17607:
17583:
17582:
17567:
17535:
17531:
17526:(9): 1167–1175.
17516:
17509:
17502:
17488:
17484:
17447:
17443:
17436:
17422:
17418:
17411:
17397:
17393:
17386:
17363:
17352:
17321:
17317:
17310:
17306:
17273:
17269:
17256:
17252:
17229:
17225:
17218:
17204:Zinober, A.S.I.
17201:
17194:
17189:
17184:
17183:
17174:
17170:
17156:
17152:
17147:
17099:Optimal control
17070:
17049:
17041:
17027:
17025:
17024:
17013:
17005:
17002:
17001:
16985:
16983:
16980:
16979:
16957:
16955:
16954:
16952:
16949:
16948:
16944:
16928:
16917:
16915:
16912:
16911:
16894:
16890:
16882:
16879:
16878:
16851:
16847:
16838:
16827:
16826:
16825:
16814:
16811:
16810:
16790:
16786:
16777:
16766:
16765:
16764:
16753:
16750:
16749:
16733:
16731:
16728:
16727:
16703:
16702:
16693:
16689:
16680:
16669:
16668:
16667:
16655:
16654:
16649:
16642:
16641:
16632:
16628:
16626:
16623:
16622:
16599:
16598:
16591:
16590:
16581:
16577:
16574:
16573:
16560:
16559:
16557:
16547:
16546:
16537:
16533:
16531:
16528:
16527:
16500:
16496:
16494:
16491:
16490:
16470:
16469:
16463:
16458:
16457:
16451:
16447:
16433:
16431:
16430:
16424:
16420:
16411:
16410:
16403:
16402:
16393:
16389:
16380:
16369:
16368:
16367:
16355:
16354:
16349:
16342:
16341:
16334:
16333:
16326:
16325:
16316:
16312:
16309:
16308:
16295:
16294:
16292:
16282:
16281:
16267:
16265:
16264:
16252:
16251:
16242:
16238:
16229:
16218:
16217:
16216:
16200:
16199:
16190:
16186:
16183:
16182:
16169:
16168:
16160:
16143:
16141:
16140:
16128:
16127:
16118:
16114:
16105:
16094:
16093:
16092:
16073:
16072:
16066:
16062:
16059:
16058:
16045:
16044:
16036:
16019:
16017:
16016:
16004:
16003:
15994:
15990:
15981:
15970:
15969:
15968:
15945:
15928:
15926:
15925:
15915:
15899:
15897:
15895:
15894:
15890:
15888:
15885:
15884:
15857:
15853:
15851:
15848:
15847:
15843:
15813:
15802:
15801:
15800:
15780:
15777:
15776:
15759:
15755:
15753:
15750:
15749:
15745:
15728:
15724:
15722:
15719:
15718:
15697:
15692:
15691:
15689:
15686:
15685:
15653:
15649:
15643:
15639:
15630:
15626:
15621:
15618:
15617:
15571:
15568:
15567:
15550:
15546:
15544:
15541:
15540:
15506:
15503:
15502:
15485:
15480:
15479:
15477:
15474:
15473:
15450:
15445:
15444:
15435:
15431:
15425:
15421:
15412:
15408:
15396:
15391:
15390:
15384:
15380:
15374:
15370:
15361:
15356:
15355:
15349:
15345:
15336:
15332:
15326:
15322:
15313:
15308:
15307:
15301:
15297:
15285:
15281:
15269:
15265:
15252:
15247:
15246:
15245:
15234:
15233:
15227:
15223:
15220:
15219:
15213:
15212:
15206:
15202:
15199:
15198:
15192:
15188:
15181:
15179:
15177:
15176:
15170:
15166:
15153:
15142:
15140:
15139:
15138:
15137:
15126:
15125:
15119:
15108:
15107:
15106:
15103:
15102:
15096:
15095:
15089:
15078:
15077:
15076:
15073:
15072:
15066:
15055:
15054:
15053:
15046:
15044:
15042:
15041:
15039:
15036:
15035:
15015:
15011:
15009:
15006:
15005:
14988:
14984:
14982:
14979:
14978:
14950:
14947:
14946:
14929:
14925:
14923:
14920:
14919:
14892:
14869:
14867:
14856:
14851:
14850:
14848:
14846:
14845:
14835:
14812:
14810:
14799:
14795:
14793:
14791:
14790:
14779:
14768:
14764:
14762:
14760:
14759:
14757:
14754:
14753:
14727:
14723:
14714:
14709:
14708:
14702:
14698:
14678:
14667:
14663:
14661:
14659:
14658:
14649:
14644:
14643:
14637:
14633:
14618:
14617:
14606:
14602:
14600:
14598:
14597:
14591:
14587:
14573:
14572:
14564:
14561:
14560:
14540:
14536:
14534:
14531:
14530:
14505:
14502:
14501:
14478:
14467:
14466:
14465:
14463:
14460:
14459:
14442:
14438:
14436:
14433:
14432:
14415:
14411:
14402:
14391:
14390:
14389:
14387:
14384:
14383:
14360:
14356:
14354:
14351:
14350:
14346:
14342:
14338:
14313:
14307:
14302:
14301:
14295:
14291:
14282:
14278:
14272:
14268:
14263:
14249:
14246:
14245:
14193:
14190:
14189:
14158:
14157:
14152:
14149:
14148:
14119:
14114:
14113:
14101:
14097:
14082:
14078:
14069:
14065:
14053:
14048:
14047:
14045:
14042:
14041:
14006:
14001:
14000:
13994:
13990:
13981:
13977:
13971:
13967:
13955:
13951:
13936:
13931:
13930:
13924:
13920:
13911:
13907:
13901:
13897:
13888:
13877:
13876:
13875:
13861:
13860:
13858:
13855:
13854:
13835:
13833:
13830:
13829:
13797:
13796:
13791:
13788:
13787:
13771:
13768:
13767:
13745:
13744:
13742:
13739:
13738:
13715:
13711:
13702:
13691:
13690:
13689:
13680:
13676:
13659:
13658:
13649:
13638:
13637:
13636:
13628:
13625:
13624:
13604:
13600:
13591:
13580:
13579:
13578:
13576:
13573:
13572:
13555:
13551:
13549:
13546:
13545:
13528:
13517:
13516:
13515:
13513:
13510:
13509:
13489:
13485:
13476:
13465:
13464:
13463:
13455:
13452:
13451:
13444:
13427:
13423:
13414:
13403:
13402:
13401:
13392:
13388:
13386:
13383:
13382:
13365:
13364:
13355:
13351:
13334:
13322:
13321:
13312:
13308:
13299:
13288:
13287:
13286:
13266:
13252:
13250:
13249:
13234:
13233:
13228:
13217:
13202:
13198:
13189:
13178:
13177:
13176:
13159:
13142:
13140:
13139:
13127:
13126:
13115:
13113:
13112:
13093:
13091:
13089:
13088:
13081:
13070:
13068:
13067:
13063:
13061:
13058:
13057:
13038:
13024:
13022:
13021:
13013:
13011:
13008:
13007:
12990:
12985:
12984:
12972:
12968:
12953:
12949:
12940:
12936:
12925:
12923:
12920:
12919:
12890:
12885:
12884:
12875:
12871:
12869:
12866:
12865:
12844:
12843:
12837:
12833:
12830:
12829:
12816:
12815:
12807:
12804:
12803:
12779:
12774:
12773:
12765:
12762:
12761:
12744:
12740:
12732:
12729:
12728:
12727:and the output
12711:
12700:
12699:
12698:
12696:
12693:
12692:
12676:
12668:
12660:
12657:
12656:
12633:
12629:
12620:
12609:
12608:
12607:
12590:
12573:
12571:
12570:
12548:
12546:
12544:
12543:
12541:
12538:
12537:
12530:
12513:
12508:
12507:
12495:
12484:
12483:
12482:
12467:
12456:
12455:
12454:
12445:
12434:
12433:
12432:
12415:
12413:
12412:
12410:
12407:
12406:
12389:
12385:
12377:
12374:
12373:
12357:
12355:
12352:
12351:
12312:
12307:
12306:
12297:
12293:
12291:
12288:
12287:
12238:
12233:
12232:
12223:
12219:
12217:
12214:
12213:
12182:
12177:
12176:
12167:
12163:
12161:
12158:
12157:
12138:
12134:
12132:
12129:
12128:
12111:
12107:
12105:
12102:
12101:
12077:
12076:
12070:
12066:
12064:
12058:
12054:
12051:
12050:
12044:
12040:
12038:
12032:
12028:
12021:
12020:
12012:
12009:
12008:
11989:
11987:
11984:
11983:
11979:
11962:
11957:
11956:
11944:
11940:
11925:
11921:
11912:
11908:
11897:
11895:
11892:
11891:
11874:
11869:
11868:
11856:
11852:
11837:
11833:
11824:
11820:
11809:
11807:
11804:
11803:
11783:
11782:
11776:
11772:
11764:
11757:
11755:
11750:
11745:
11740:
11730:
11729:
11720:
11719:
11714:
11703:
11686:
11684:
11683:
11676:
11675:
11673:
11670:
11669:
11644:state observers
11640:
11626:
11577:
11576:
11565:
11562:
11561:
11537:
11520:
11519:
11496:
11495:
11478:
11457:
11456:
11454:
11451:
11450:
11423:
11415:
11412:
11411:
11389:
11375:
11374:
11359:
11336:
11335:
11334:
11332:
11321:
11319:
11298:
11287:
11286:
11281:
11258:
11257:
11243:
11240:
11239:
11212:
11211:
11199:
11182:
11181:
11174:
11172:
11166:
11164:
11142:
11131:
11130:
11125:
11124:
11120:
11114:
11113:
11087:
11086:
11079:
11077:
11072:
11070:
11053:
11042:
11041:
11036:
11029:
11028:
11011:
11010:
10996:
10993:
10992:
10970:
10956:
10955:
10932:
10924:
10913:
10912:
10907:
10887:
10876:
10875:
10870:
10868:
10865:
10864:
10846:
10838:
10836:
10833:
10832:
10812:
10798:
10797:
10765:
10764:
10759:
10751:
10737:
10736:
10713:
10705:
10694:
10693:
10688:
10686:
10683:
10682:
10656:
10642:
10641:
10609:
10608:
10603:
10592:
10589:
10588:
10560:
10559:
10557:
10554:
10553:
10531:
10528:
10527:
10499:
10498:
10490:
10487:
10486:
10468:
10454:
10453:
10421:
10420:
10415:
10407:
10404:
10403:
10375:
10374:
10372:
10369:
10368:
10346:
10343:
10342:
10314:
10313:
10305:
10302:
10301:
10274:
10273:
10247:
10246:
10227:
10225:
10208:
10207:
10193:
10192:
10178:
10177:
10168:
10157:
10156:
10155:
10146:
10135:
10134:
10133:
10119:
10118:
10116:
10113:
10112:
10082:
10081:
10076:
10073:
10072:
10047:
10046:
10032:
10029:
10028:
10002:
10001:
9986:
9982:
9973:
9969:
9957:
9953:
9944:
9940:
9932:
9929:
9928:
9922:
9906:
9898:
9896:
9893:
9892:
9867:
9850:
9842:
9839:
9838:
9817:
9816:
9801:
9797:
9788:
9784:
9763:
9752:
9751:
9750:
9747:
9746:
9740:
9736:
9727:
9716:
9715:
9714:
9707:
9706:
9704:
9701:
9700:
9676:
9675:
9666:
9662:
9660:
9657:
9656:
9633:
9629:
9627:
9624:
9623:
9584:
9583:
9551:
9550:
9548:
9545:
9544:
9505:
9497:
9494:
9493:
9464:
9456:
9453:
9452:
9432:
9430:
9427:
9426:
9400:
9399:
9397:
9394:
9393:
9368:
9359:
9355:
9353:
9350:
9349:
9332:
9330:
9327:
9326:
9300:
9299:
9297:
9294:
9293:
9268:
9259:
9255:
9253:
9250:
9249:
9226:
9225:
9211:
9200:
9198:
9190:
9181:
9177:
9174:
9173:
9159:
9148:
9146:
9138:
9129:
9125:
9118:
9117:
9106:
9098:
9095:
9094:
9073:
9065:
9062:
9061:
9022:
9008:
9007:
8990:
8988:
8987:
8960:
8937:
8930:
8926:
8924:
8921:
8910:
8909:
8905:
8895:
8888:
8884:
8882:
8867:
8863:
8848:
8844:
8835:
8831:
8827:
8825:
8809:
8807:
8806:
8779:
8756:
8749:
8745:
8743:
8733:
8732:
8728:
8718:
8711:
8707:
8705:
8692:
8678:
8677:
8676:
8659:
8657:
8656:
8647:
8646:
8642:
8632:
8625:
8621:
8619:
8618:
8616:
8604:
8590:
8589:
8587:
8584:
8583:
8545:
8544:
8539:
8536:
8535:
8514:
8513:
8499:
8497:
8480:
8479:
8476:
8475:
8461:
8459:
8442:
8441:
8434:
8433:
8431:
8428:
8427:
8406:
8398:
8395:
8394:
8366:
8365:
8363:
8360:
8359:
8329:
8318:
8315:
8301:
8287:
8286:
8285:
8283:
8272:
8268:
8260:
8257:
8245:
8241:
8236:
8229:
8227:
8212:
8192:
8191:
8189:
8186:
8185:
8157:
8149:
8147:
8144:
8143:
8115:
8112:
8111:
8089:
8086:
8085:
8058:
8047:
8046:
8045:
8039:
8035:
8020:
8009:
8008:
8007:
7995:
7991:
7976:
7965:
7964:
7963:
7957:
7953:
7944:
7933:
7932:
7931:
7925:
7921:
7919:
7916:
7915:
7884:
7870:
7869:
7867:
7864:
7863:
7836:
7828:
7825:
7824:
7792:
7789:
7788:
7765:
7761:
7759:
7756:
7755:
7717:
7713:
7707:
7703:
7688:
7684:
7672:
7668:
7653:
7649:
7643:
7639:
7630:
7626:
7620:
7616:
7605:
7597:
7594:
7593:
7557:
7554:
7553:
7549:
7535:
7487:
7486:
7475:
7472:
7471:
7430:
7429:
7427:
7424:
7423:
7389:
7388:
7386:
7383:
7382:
7354:
7351:
7350:
7316:
7315:
7283:
7282:
7280:
7277:
7276:
7236:
7235:
7233:
7230:
7229:
7196:
7195:
7193:
7190:
7189:
7161:
7160:
7154:
7150:
7148:
7145:
7144:
7124:
7113:
7105:
7102:
7101:
7073:
7072:
7070:
7067:
7066:
7050:
7039:
7031:
7028:
7027:
7008:
6997:
6983:
6982:
6980:
6977:
6976:
6946:
6942:
6934:
6932:
6923:
6904:
6890:
6886:
6878:
6876:
6875:
6871:
6870:
6856:
6842:
6841:
6837:
6823:
6795:
6772:
6765:
6745:
6731:
6727:
6719:
6717:
6708:
6689:
6675:
6671:
6663:
6661:
6660:
6656:
6655:
6641:
6618:
6611:
6609:
6594:
6592:
6591:
6589:
6586:
6585:
6559:
6548:
6540:
6537:
6536:
6516:
6514:
6511:
6510:
6482:
6468:
6464:
6456:
6454:
6445:
6426:
6412:
6408:
6400:
6398:
6397:
6393:
6392:
6381:
6379:
6376:
6375:
6353:
6338:
6336:
6335:
6320:
6306:
6283:
6276:
6272:
6270:
6261:
6257:
6249:
6247:
6245:
6242:
6241:
6219:
6211:
6209:
6206:
6205:
6177:
6163:
6162:
6160:
6157:
6156:
6122:
6120:
6117:
6116:
6096:
6082:
6081:
6079:
6076:
6075:
6053:
6045:
6042:
6041:
6025:
6014:
6006:
6003:
6002:
5999:controllability
5966:
5951:
5950:
5946:
5938:
5936:
5934:
5931:
5930:
5924:
5897:
5894:
5893:
5871:
5870:
5868:
5865:
5864:
5848:
5837:
5829:
5826:
5825:
5809:
5807:
5804:
5803:
5774:
5771:
5770:
5739:
5725:
5724:
5716:
5707:
5703:
5694:
5689:
5688:
5680:
5675:
5672:
5671:
5649:
5638:
5623:
5618:
5617:
5609:
5604:
5601:
5600:
5585:
5575:manifold; they
5554:
5551:
5550:
5528:
5527:
5525:
5522:
5521:
5505:
5494:
5486:
5483:
5482:
5481:approaches the
5466:
5464:
5461:
5460:
5444:
5441:
5440:
5424:
5413:
5412:
5407:
5405:
5402:
5401:
5379:
5376:
5375:
5343:
5332:
5331:
5326:
5320:
5305:
5304:
5275:
5272:
5271:
5234:
5233:
5216:
5213:
5212:
5187:
5184:
5183:
5162:
5157:
5156:
5148:
5128:
5127:
5122:
5119:
5118:
5099:
5096:
5095:
5071:
5070:
5065:
5062:
5061:
5041:
5037:
5027:
5015:
5011:
5004:
5002:
5000:
4999:
4972:
4961:
4958:
4937:
4926:
4923:
4908:
4906:
4893:
4885:
4882:
4871:
4867:
4865:
4863:
4861:
4858:
4855:
4854:
4848:
4827:
4824:
4823:
4806:
4802:
4796:
4795:
4790:
4787:
4786:
4765:
4763:
4760:
4759:
4737:
4734:
4733:
4729:
4705:
4703:
4700:
4699:
4681:
4679:
4676:
4675:
4651:
4649:
4646:
4645:
4641:
4615:
4611:
4592:
4587:
4584:
4583:
4566:
4562:
4545:
4542:
4541:
4510:
4509:
4507:
4504:
4503:
4477:
4473:
4456:
4453:
4452:
4449:proportionality
4432:
4429:
4428:
4427:and the symbol
4410:
4405:
4402:
4401:
4380:
4376:
4374:
4371:
4370:
4334:
4323:
4321:
4309:
4289:
4288:
4283:
4282:
4281:
4267:
4263:
4262:
4250:
4249:
4241:
4225:
4221:
4210:
4209:
4197:
4195:
4194:
4190:
4188:
4186:
4185:
4179:
4178:
4172:
4168:
4167:
4165:
4163:
4162:
4160:
4157:
4156:
4135:
4124:
4121:
4120:
4098:
4093:
4082:
4079:
4078:
4035:
4024:
4022:
4010:
4008:
4005:
4004:
3967:
3964:
3963:
3939:
3924:
3920:
3913:
3888:
3877:
3875:
3873:
3870:
3869:
3832:
3829:
3828:
3825:
3819:are constants.
3792:
3789:
3788:
3766:
3763:
3762:
3742:
3738:
3731:
3706:
3695:
3693:
3691:
3688:
3687:
3661:
3656:
3645:
3642:
3641:
3625:
3614:
3606:
3603:
3602:
3599:
3572:
3571:
3569:
3566:
3565:
3546:
3538:
3536:
3533:
3532:
3506:
3504:
3503:
3488:
3474:
3451:
3444:
3440:
3438:
3429:
3425:
3417:
3415:
3397:
3391:
3384:
3381:
3366:
3364:
3362:
3353:
3349:
3341:
3339:
3325:
3324:
3322:
3319:
3318:
3292:
3284:
3281:
3280:
3261:
3247:
3246:
3244:
3241:
3240:
3221:
3213:
3210:
3209:
3188:
3180:
3177:
3176:
3157:
3143:
3142:
3140:
3137:
3136:
3117:
3109:
3106:
3105:
3080:
3079:
3077:
3074:
3073:
3057:
3054:
3053:
3034:
3026:
3023:
3022:
2994:
2993:
2987:
2983:
2981:
2978:
2977:
2955:
2952:
2951:
2919:
2911:
2908:
2907:
2884:
2864:
2853:
2850:
2845:
2823:
2812:
2809:
2788:
2777:
2774:
2759:
2757:
2744:
2736:
2733:
2722:
2718:
2716:
2714:
2712:
2709:
2706:
2705:
2674:
2663:
2655:
2652:
2651:
2634:
2630:
2622:
2611:
2608:
2607:
2583:
2582:
2577:
2574:
2573:
2536:
2531:
2519:
2500:
2489:
2475:
2466:
2462:
2452:
2438:
2424:
2421:
2420:
2404:
2396:
2360:
2349:
2335:
2334:
2332:
2329:
2328:
2309:
2298:
2290:
2287:
2286:
2259:
2238:
2227:
2219:
2216:
2215:
2199:
2188:
2180:
2177:
2176:
2175:Having reached
2157:
2146:
2138:
2135:
2134:
2114:
2103:
2095:
2092:
2091:
2075:
2073:
2070:
2069:
2044:
2033:
2025:
2022:
2021:
1971:
1962:
1958:
1949:
1944:
1943:
1935:
1934:
1930:
1928:
1925:
1924:
1887:
1884:
1883:
1855:
1852:
1851:
1835:
1833:
1830:
1829:
1825:
1809:
1807:
1804:
1803:
1799:
1777:
1774:
1773:
1767:topographic map
1750:
1742:
1740:
1737:
1736:
1707:
1705:
1702:
1701:
1676:
1668:
1665:
1664:
1631:
1623:
1620:
1619:
1591:
1583:
1580:
1579:
1563:
1561:
1558:
1557:
1537:
1535:
1532:
1531:
1514:
1509:
1508:
1499:
1494:
1493:
1485:
1482:
1481:
1459:
1451:
1449:
1446:
1445:
1429:Selection of a
1380:
1376:
1374:
1371:
1370:
1353:
1349:
1347:
1344:
1343:
1327:
1325:
1322:
1321:
1305:
1303:
1300:
1299:
1282:
1278:
1276:
1273:
1272:
1255:
1251:
1219:
1217:
1214:
1213:
1180:
1178:
1175:
1174:
1170:
1145:
1143:
1140:
1139:
1111:
1103:
1101:
1098:
1097:
1052:
1050:
1047:
1046:
1011:
1006:
1005:
997:
988:
983:
982:
974:
971:
970:
953:
948:
947:
939:
930:
925:
924:
916:
913:
912:
901:
881:
876:
875:
865:
864:
849:
845:
842:
841:
820:
816:
813:
812:
806:
805:
790:
786:
783:
782:
767:
763:
756:
755:
738:
736:
733:
732:
718:
698:
693:
692:
682:
681:
666:
662:
659:
658:
637:
633:
630:
629:
623:
622:
607:
603:
600:
599:
584:
580:
573:
572:
555:
553:
550:
549:
501:
486:
463:
434:
432:
431:
429:
426:
425:
409:
316:
313:
312:
285:
282:
281:
272:, which has an
256:
252:
240:
229:
228:
227:
225:
222:
221:
195:
184:
183:
182:
173:
169:
161:
158:
157:
135:
132:
131:
109:
106:
105:
100:Figure 1:
94:
21:control systems
17:
12:
11:
5:
17929:
17919:
17918:
17904:
17903:
17870:
17867:
17866:
17865:
17859:
17838:
17825:978-1611975833
17824:
17800:
17794:
17781:
17775:
17754:
17748:
17735:
17729:
17716:
17710:
17697:
17691:
17678:
17672:
17659:
17653:
17636:
17630:
17612:
17611:
17605:
17581:
17580:
17565:
17529:
17507:
17500:
17482:
17468:10.1.1.43.1654
17461:(3): 721–739.
17441:
17434:
17416:
17409:
17391:
17384:
17350:
17331:(5): 703–715.
17315:
17304:
17267:
17250:
17223:
17216:
17206:, ed. (1990).
17191:
17190:
17188:
17185:
17182:
17181:
17168:
17149:
17148:
17146:
17143:
17142:
17141:
17135:
17129:
17123:
17117:
17111:
17101:
17096:
17094:Robust control
17091:
17086:
17081:
17076:
17069:
17066:
17052:
17048:
17044:
17040:
17034:
17030:
17023:
17020:
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17009:
16988:
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16931:
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16897:
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16886:
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16854:
16850:
16846:
16841:
16834:
16831:
16824:
16821:
16818:
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16793:
16789:
16785:
16780:
16773:
16770:
16763:
16760:
16757:
16736:
16724:
16723:
16712:
16707:
16701:
16696:
16692:
16688:
16683:
16676:
16673:
16666:
16663:
16660:
16657:
16656:
16652:
16648:
16647:
16645:
16640:
16631:
16620:
16608:
16603:
16595:
16589:
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16580:
16576:
16575:
16572:
16569:
16566:
16565:
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16558:
16556:
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16552:
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16545:
16536:
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16514:
16511:
16508:
16499:
16484:
16483:
16461:
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16446:
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16436:
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16401:
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16392:
16388:
16383:
16376:
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16366:
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16356:
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16347:
16345:
16338:
16330:
16324:
16319:
16315:
16311:
16310:
16307:
16304:
16301:
16300:
16298:
16293:
16291:
16288:
16287:
16285:
16280:
16274:
16270:
16263:
16260:
16257:
16255:
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16250:
16245:
16241:
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16225:
16222:
16215:
16212:
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16108:
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15989:
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15935:
15931:
15924:
15921:
15918:
15916:
15911:
15906:
15902:
15893:
15892:
15856:
15840:Gaussian noise
15827:
15824:
15821:
15816:
15809:
15806:
15799:
15796:
15793:
15790:
15787:
15784:
15758:
15731:
15727:
15715:state observer
15700:
15695:
15661:
15656:
15652:
15646:
15642:
15638:
15633:
15629:
15625:
15605:
15602:
15599:
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15584:
15581:
15578:
15575:
15553:
15549:
15528:
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15522:
15519:
15516:
15513:
15510:
15488:
15483:
15470:
15469:
15458:
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15443:
15438:
15434:
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15424:
15420:
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15411:
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15399:
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15387:
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15373:
15369:
15364:
15359:
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15348:
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15329:
15325:
15321:
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15311:
15304:
15300:
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15280:
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15255:
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15230:
15226:
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15218:
15215:
15214:
15209:
15205:
15201:
15200:
15195:
15191:
15187:
15186:
15184:
15173:
15169:
15165:
15156:
15149:
15145:
15134:
15130:
15122:
15115:
15112:
15105:
15104:
15101:
15098:
15097:
15092:
15085:
15082:
15075:
15074:
15069:
15062:
15059:
15052:
15051:
15049:
15018:
15014:
14991:
14987:
14966:
14963:
14960:
14957:
14954:
14928:
14916:
14915:
14904:
14890:
14887:
14884:
14881:
14878:
14875:
14872:
14864:
14859:
14854:
14833:
14830:
14827:
14824:
14821:
14818:
14815:
14807:
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14798:
14789:
14776:
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14747:
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14735:
14726:
14722:
14717:
14712:
14705:
14701:
14697:
14690:
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14609:
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14571:
14568:
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14512:
14509:
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14471:
14445:
14441:
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14410:
14405:
14398:
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14371:
14368:
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14335:
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14323:
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14310:
14305:
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14285:
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14271:
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14262:
14259:
14256:
14253:
14239:
14238:
14227:
14224:
14221:
14218:
14215:
14212:
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14206:
14203:
14200:
14197:
14174:
14171:
14165:
14162:
14156:
14134:
14131:
14128:
14125:
14122:
14117:
14112:
14109:
14104:
14100:
14096:
14093:
14090:
14085:
14081:
14077:
14072:
14068:
14064:
14061:
14056:
14051:
14038:
14037:
14026:
14023:
14020:
14017:
14014:
14009:
14004:
13997:
13993:
13989:
13984:
13980:
13974:
13970:
13966:
13963:
13958:
13954:
13950:
13947:
13944:
13939:
13934:
13927:
13923:
13919:
13914:
13910:
13904:
13900:
13896:
13891:
13884:
13881:
13874:
13868:
13865:
13838:
13813:
13810:
13804:
13801:
13795:
13775:
13752:
13749:
13735:
13734:
13723:
13718:
13714:
13710:
13705:
13698:
13695:
13688:
13683:
13679:
13675:
13672:
13666:
13663:
13657:
13652:
13645:
13642:
13635:
13632:
13607:
13603:
13599:
13594:
13587:
13584:
13558:
13554:
13531:
13524:
13521:
13506:
13505:
13492:
13488:
13484:
13479:
13472:
13469:
13462:
13459:
13430:
13426:
13422:
13417:
13410:
13407:
13400:
13395:
13391:
13379:
13378:
13363:
13358:
13354:
13350:
13347:
13344:
13341:
13337:
13333:
13330:
13327:
13325:
13323:
13320:
13315:
13311:
13307:
13302:
13295:
13292:
13285:
13282:
13279:
13276:
13273:
13269:
13265:
13259:
13255:
13248:
13245:
13242:
13239:
13237:
13235:
13231:
13227:
13224:
13220:
13216:
13213:
13210:
13205:
13201:
13197:
13192:
13185:
13182:
13175:
13172:
13169:
13166:
13162:
13158:
13155:
13149:
13145:
13138:
13135:
13132:
13130:
13128:
13122:
13118:
13111:
13105:
13100:
13096:
13087:
13084:
13082:
13077:
13073:
13066:
13065:
13041:
13037:
13031:
13027:
13020:
13016:
12993:
12988:
12983:
12980:
12975:
12971:
12967:
12964:
12961:
12956:
12952:
12948:
12943:
12939:
12935:
12932:
12928:
12905:
12902:
12899:
12896:
12893:
12888:
12883:
12878:
12874:
12862:
12861:
12848:
12840:
12836:
12832:
12831:
12828:
12825:
12822:
12821:
12819:
12814:
12811:
12782:
12777:
12772:
12769:
12747:
12743:
12739:
12736:
12714:
12707:
12704:
12679:
12675:
12671:
12667:
12664:
12653:
12652:
12641:
12636:
12632:
12628:
12623:
12616:
12613:
12606:
12603:
12600:
12597:
12593:
12589:
12586:
12580:
12576:
12569:
12566:
12560:
12555:
12551:
12516:
12511:
12506:
12503:
12498:
12491:
12488:
12481:
12478:
12475:
12470:
12463:
12460:
12453:
12448:
12441:
12438:
12431:
12428:
12422:
12418:
12392:
12388:
12384:
12381:
12360:
12348:
12347:
12333:
12330:
12327:
12324:
12321:
12318:
12315:
12310:
12305:
12300:
12296:
12285:
12271:
12268:
12265:
12262:
12259:
12256:
12253:
12250:
12247:
12244:
12241:
12236:
12231:
12226:
12222:
12211:
12197:
12194:
12191:
12188:
12185:
12180:
12175:
12170:
12166:
12155:
12141:
12137:
12114:
12110:
12095:
12094:
12081:
12073:
12069:
12065:
12061:
12057:
12053:
12052:
12047:
12043:
12039:
12035:
12031:
12027:
12026:
12024:
12019:
12016:
11992:
11965:
11960:
11955:
11952:
11947:
11943:
11939:
11936:
11933:
11928:
11924:
11920:
11915:
11911:
11907:
11904:
11900:
11877:
11872:
11867:
11864:
11859:
11855:
11851:
11848:
11845:
11840:
11836:
11832:
11827:
11823:
11819:
11816:
11812:
11800:
11799:
11786:
11779:
11775:
11771:
11767:
11761:
11754:
11751:
11749:
11746:
11744:
11741:
11739:
11736:
11735:
11733:
11728:
11725:
11722:
11721:
11717:
11713:
11710:
11706:
11702:
11699:
11693:
11689:
11682:
11681:
11679:
11639:
11636:
11625:
11622:
11621:
11620:
11608:
11605:
11602:
11599:
11596:
11593:
11590:
11584:
11581:
11575:
11572:
11569:
11554:
11553:
11552:
11536:
11533:
11527:
11524:
11518:
11515:
11512:
11509:
11503:
11500:
11494:
11491:
11488:
11485:
11476:
11473:
11470:
11464:
11461:
11436:
11433:
11430:
11426:
11422:
11419:
11408:
11407:
11406:
11388:
11385:
11382:
11371:
11367:
11362:
11356:
11352:
11349:
11343:
11340:
11330:
11327:
11324:
11317:
11314:
11311:
11308:
11305:
11301:
11294:
11291:
11284:
11280:
11277:
11274:
11271:
11265:
11262:
11256:
11253:
11250:
11247:
11230:
11229:
11228:
11215:
11210:
11207:
11202:
11196:
11189:
11186:
11180:
11177:
11165:
11162:
11158:
11155:
11152:
11149:
11145:
11138:
11135:
11128:
11123:
11119:
11116:
11115:
11112:
11109:
11106:
11101:
11094:
11091:
11085:
11082:
11071:
11069:
11066:
11063:
11060:
11056:
11049:
11046:
11039:
11035:
11034:
11032:
11027:
11024:
11018:
11015:
11009:
11006:
11003:
11000:
10987:
10986:
10985:
10973:
10969:
10963:
10960:
10954:
10951:
10948:
10945:
10942:
10939:
10935:
10931:
10927:
10920:
10917:
10910:
10906:
10903:
10900:
10897:
10894:
10890:
10883:
10880:
10873:
10849:
10845:
10841:
10829:
10828:
10827:
10815:
10811:
10805:
10802:
10796:
10793:
10790:
10787:
10784:
10781:
10778:
10772:
10769:
10762:
10758:
10754:
10750:
10744:
10741:
10735:
10732:
10729:
10726:
10723:
10720:
10716:
10712:
10708:
10701:
10698:
10691:
10673:
10672:
10671:
10659:
10655:
10649:
10646:
10640:
10637:
10634:
10631:
10628:
10625:
10622:
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10606:
10602:
10599:
10596:
10576:
10573:
10567:
10564:
10541:
10538:
10535:
10515:
10512:
10506:
10503:
10497:
10494:
10483:
10471:
10467:
10461:
10458:
10452:
10449:
10446:
10443:
10440:
10437:
10434:
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10391:
10388:
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10356:
10353:
10350:
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10321:
10318:
10312:
10309:
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10260:
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10023:
10009:
10006:
10000:
9997:
9994:
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9965:
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9956:
9952:
9947:
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9939:
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9909:
9905:
9901:
9880:
9877:
9874:
9870:
9866:
9863:
9860:
9857:
9853:
9849:
9846:
9835:
9834:
9833:
9820:
9815:
9812:
9809:
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9796:
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9783:
9780:
9777:
9774:
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9766:
9759:
9756:
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9603:
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9588:
9582:
9579:
9576:
9573:
9570:
9567:
9564:
9558:
9555:
9539:
9538:
9536:dynamic system
9531:
9530:
9518:
9515:
9512:
9508:
9504:
9501:
9477:
9474:
9471:
9467:
9463:
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9147:
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9137:
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9128:
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9105:
9102:
9080:
9076:
9072:
9069:
9055:
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9050:
9041:
9039:
9024: vector )
9021:
9018:
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7112:
7109:
7089:
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7053:
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6779:
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6762:
6758:
6755:
6752:
6748:
6744:
6741:
6734:
6730:
6725:
6722:
6714:
6711:
6706:
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6699:
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6692:
6688:
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6678:
6674:
6669:
6666:
6659:
6654:
6651:
6648:
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6640:
6637:
6634:
6631:
6628:
6625:
6621:
6617:
6614:
6607:
6601:
6597:
6562:
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6506:
6495:
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6467:
6462:
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6411:
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6396:
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6388:
6384:
6369:
6368:
6356:
6352:
6345:
6341:
6332:
6328:
6323:
6319:
6316:
6313:
6309:
6305:
6302:
6299:
6296:
6293:
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6282:
6279:
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6264:
6260:
6255:
6252:
6226:
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6218:
6214:
6202:
6201:
6190:
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6184:
6180:
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6170:
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6125:
6113:
6112:
6099:
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6089:
6086:
6060:
6056:
6052:
6049:
6028:
6024:
6021:
6017:
6013:
6010:
5991:
5990:
5979:
5976:
5973:
5969:
5965:
5962:
5954:
5949:
5944:
5941:
5923:
5920:
5907:
5904:
5901:
5878:
5875:
5851:
5847:
5844:
5840:
5836:
5833:
5812:
5787:
5784:
5781:
5778:
5767:
5766:
5755:
5752:
5749:
5746:
5742:
5738:
5732:
5729:
5723:
5719:
5715:
5710:
5706:
5702:
5697:
5692:
5687:
5683:
5679:
5656:
5652:
5648:
5645:
5641:
5637:
5634:
5631:
5626:
5621:
5616:
5612:
5608:
5584:
5581:
5564:
5561:
5558:
5535:
5532:
5508:
5504:
5501:
5497:
5493:
5490:
5469:
5448:
5427:
5420:
5417:
5410:
5389:
5386:
5383:
5372:
5371:
5359:
5356:
5353:
5350:
5346:
5339:
5336:
5329:
5318:
5312:
5309:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5265:
5264:
5253:
5250:
5247:
5241:
5238:
5232:
5229:
5226:
5223:
5220:
5197:
5194:
5191:
5180:
5179:
5165:
5160:
5155:
5151:
5147:
5144:
5141:
5135:
5132:
5126:
5103:
5092:Euclidean norm
5079:
5074:
5069:
5058:
5057:
5044:
5040:
5031:
5024:
5018:
5014:
5010:
5007:
4998:
4995:
4992:
4989:
4981:
4978:
4975:
4970:
4967:
4964:
4955:
4946:
4943:
4940:
4935:
4932:
4929:
4920:
4915:
4912:
4899:
4896:
4891:
4888:
4879:
4874:
4870:
4847:
4844:
4831:
4809:
4805:
4799:
4794:
4784:Euclidean norm
4769:
4747:
4744:
4741:
4717:
4714:
4709:
4685:
4663:
4660:
4655:
4629:
4626:
4623:
4618:
4614:
4610:
4605:
4602:
4599:
4596:
4591:
4569:
4565:
4561:
4558:
4555:
4552:
4549:
4529:
4526:
4523:
4517:
4514:
4491:
4488:
4485:
4480:
4476:
4472:
4469:
4466:
4463:
4460:
4436:
4414:
4409:
4383:
4379:
4367:
4366:
4355:
4352:
4349:
4343:
4340:
4337:
4332:
4329:
4326:
4317:
4313:
4308:
4296:
4293:
4279:
4275:
4270:
4266:
4259:
4253:
4244:
4238:
4228:
4224:
4220:
4217:
4214:
4206:
4202:
4193:
4182:
4175:
4171:
4139:
4134:
4131:
4128:
4107:
4104:
4101:
4096:
4092:
4089:
4086:
4073:which, by the
4071:
4070:
4059:
4056:
4053:
4050:
4044:
4041:
4038:
4033:
4030:
4027:
4018:
4014:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3960:
3959:
3948:
3943:
3938:
3935:
3932:
3927:
3923:
3917:
3912:
3909:
3906:
3903:
3897:
3894:
3891:
3886:
3883:
3880:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3824:
3821:
3808:
3805:
3802:
3799:
3796:
3776:
3773:
3770:
3759:
3758:
3745:
3741:
3735:
3730:
3727:
3724:
3721:
3715:
3712:
3709:
3704:
3701:
3698:
3670:
3667:
3664:
3659:
3655:
3652:
3649:
3628:
3624:
3621:
3617:
3613:
3610:
3598:
3595:
3579:
3576:
3553:
3549:
3545:
3541:
3529:
3528:
3513:
3509:
3500:
3496:
3491:
3487:
3484:
3481:
3477:
3473:
3470:
3467:
3464:
3461:
3458:
3454:
3450:
3447:
3443:
3432:
3428:
3423:
3420:
3414:
3406:
3403:
3400:
3394:
3390:
3387:
3378:
3373:
3369:
3356:
3352:
3347:
3344:
3338:
3332:
3329:
3312:
3311:
3299:
3295:
3291:
3288:
3279:positive when
3268:
3264:
3260:
3254:
3251:
3228:
3224:
3220:
3217:
3207:
3195:
3191:
3187:
3184:
3175:negative when
3164:
3160:
3156:
3150:
3147:
3124:
3120:
3116:
3113:
3087:
3084:
3061:
3041:
3037:
3033:
3030:
3010:
3007:
3001:
2998:
2990:
2986:
2976:), to achieve
2965:
2962:
2959:
2932:
2929:
2926:
2922:
2918:
2915:
2900:
2899:
2883:
2880:
2873:
2870:
2867:
2862:
2859:
2856:
2843:
2840:
2832:
2829:
2826:
2821:
2818:
2815:
2806:
2797:
2794:
2791:
2786:
2783:
2780:
2771:
2766:
2763:
2750:
2747:
2742:
2739:
2730:
2725:
2721:
2677:
2673:
2670:
2666:
2662:
2659:
2637:
2633:
2629:
2625:
2621:
2618:
2615:
2604:Euclidean norm
2591:
2586:
2581:
2568:
2567:
2564:
2563:
2554:
2552:
2539:
2534:
2530:
2526:
2522:
2518:
2515:
2512:
2507:
2504:
2499:
2496:
2492:
2488:
2485:
2482:
2478:
2474:
2469:
2465:
2459:
2456:
2451:
2448:
2445:
2441:
2437:
2434:
2431:
2428:
2403:
2400:
2395:
2392:
2367:
2363:
2359:
2356:
2352:
2348:
2342:
2339:
2312:
2308:
2305:
2301:
2297:
2294:
2258:
2255:
2254:
2253:
2241:
2237:
2234:
2230:
2226:
2223:
2202:
2198:
2195:
2191:
2187:
2184:
2173:
2160:
2156:
2153:
2149:
2145:
2142:
2117:
2113:
2110:
2106:
2102:
2099:
2078:
2047:
2043:
2040:
2036:
2032:
2029:
2016:
2015:
2012:
2011:
2002:
2000:
1988:
1984:
1981:
1978:
1974:
1970:
1965:
1961:
1957:
1952:
1947:
1942:
1938:
1933:
1903:
1900:
1897:
1894:
1891:
1882:, there is an
1871:
1868:
1865:
1862:
1859:
1838:
1812:
1787:
1784:
1781:
1753:
1749:
1745:
1720:
1717:
1714:
1710:
1689:
1686:
1683:
1679:
1675:
1672:
1657:
1656:
1644:
1641:
1638:
1634:
1630:
1627:
1616:
1604:
1601:
1598:
1594:
1590:
1587:
1566:
1540:
1517:
1512:
1507:
1502:
1497:
1492:
1489:
1462:
1458:
1454:
1438:
1437:
1434:
1391:
1388:
1383:
1379:
1356:
1352:
1330:
1308:
1285:
1281:
1258:
1254:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1222:
1183:
1158:
1155:
1152:
1148:
1127:
1124:
1121:
1118:
1114:
1110:
1106:
1065:
1062:
1059:
1055:
1020:
1017:
1014:
1009:
1004:
1000:
996:
991:
986:
981:
978:
956:
951:
946:
942:
938:
933:
928:
923:
920:
898:
897:
884:
879:
874:
869:
863:
860:
857:
852:
848:
844:
843:
840:
837:
834:
829:
826:
823:
819:
815:
814:
811:
808:
807:
804:
801:
798:
793:
789:
785:
784:
781:
778:
775:
770:
766:
762:
761:
759:
754:
751:
748:
745:
741:
715:
714:
701:
696:
691:
686:
680:
677:
674:
669:
665:
661:
660:
657:
654:
651:
646:
643:
640:
636:
632:
631:
628:
625:
624:
621:
618:
615:
610:
606:
602:
601:
598:
595:
592:
587:
583:
579:
578:
576:
571:
568:
565:
562:
558:
541:
540:
537:
536:
527:
525:
514:
511:
508:
504:
499:
496:
493:
489:
485:
482:
479:
476:
473:
470:
466:
462:
459:
456:
453:
450:
447:
441:
437:
408:
407:Control scheme
405:
355:dynamic system
337:surface is an
326:
323:
320:
295:
292:
289:
259:
255:
251:
248:
243:
236:
233:
206:
203:
198:
191:
188:
181:
176:
172:
168:
165:
145:
142:
139:
119:
116:
113:
93:
90:
43:by applying a
15:
9:
6:
4:
3:
2:
17928:
17917:
17914:
17913:
17911:
17899:
17894:
17890:
17886:
17882:
17878:
17873:
17872:
17862:
17856:
17852:
17848:
17844:
17839:
17835:
17831:
17827:
17821:
17817:
17813:
17809:
17805:
17801:
17797:
17791:
17787:
17782:
17778:
17772:
17768:
17764:
17760:
17755:
17751:
17745:
17741:
17736:
17732:
17726:
17722:
17717:
17713:
17707:
17703:
17698:
17694:
17688:
17684:
17679:
17675:
17669:
17665:
17660:
17656:
17654:0-86341-350-1
17650:
17646:
17642:
17637:
17633:
17627:
17623:
17619:
17616:Edwards, C.;
17614:
17613:
17608:
17602:
17598:
17594:
17590:
17585:
17584:
17576:
17572:
17568:
17562:
17558:
17554:
17550:
17546:
17545:
17540:
17533:
17525:
17521:
17514:
17512:
17503:
17497:
17493:
17486:
17478:
17474:
17469:
17464:
17460:
17456:
17452:
17445:
17437:
17431:
17427:
17420:
17412:
17406:
17402:
17395:
17387:
17381:
17377:
17376:Prentice Hall
17373:
17372:
17367:
17361:
17359:
17357:
17355:
17346:
17342:
17338:
17334:
17330:
17326:
17319:
17313:
17308:
17300:
17296:
17291:
17290:10.1.1.477.77
17286:
17282:
17278:
17271:
17263:
17262:
17254:
17246:
17242:
17238:
17234:
17227:
17219:
17213:
17209:
17205:
17199:
17197:
17192:
17178:
17172:
17165:
17160:
17154:
17150:
17139:
17136:
17133:
17130:
17127:
17124:
17121:
17118:
17115:
17112:
17109:
17105:
17102:
17100:
17097:
17095:
17092:
17090:
17087:
17085:
17084:Hybrid system
17082:
17080:
17077:
17075:
17072:
17071:
17065:
17046:
17038:
17021:
17007:
16925:
16922:
16895:
16891:
16887:
16884:
16875:
16873:
16852:
16848:
16844:
16839:
16829:
16819:
16816:
16791:
16787:
16783:
16778:
16768:
16758:
16755:
16710:
16705:
16694:
16690:
16686:
16681:
16671:
16661:
16658:
16643:
16638:
16629:
16621:
16606:
16601:
16593:
16587:
16582:
16578:
16570:
16567:
16561:
16554:
16548:
16543:
16534:
16526:
16512:
16509:
16506:
16497:
16489:
16488:
16487:
16448:
16444:
16421:
16417:
16415:
16405:
16394:
16390:
16386:
16381:
16371:
16361:
16358:
16343:
16336:
16328:
16322:
16317:
16313:
16305:
16302:
16296:
16289:
16283:
16278:
16261:
16258:
16256:
16243:
16239:
16235:
16230:
16220:
16210:
16207:
16202:
16196:
16191:
16187:
16179:
16176:
16170:
16165:
16157:
16154:
16137:
16134:
16132:
16119:
16115:
16111:
16106:
16096:
16086:
16083:
16080:
16075:
16067:
16063:
16055:
16052:
16046:
16041:
16033:
16030:
16013:
16010:
16008:
15995:
15991:
15987:
15982:
15972:
15962:
15959:
15956:
15953:
15950:
15942:
15939:
15922:
15919:
15917:
15909:
15883:
15882:
15881:
15878:
15876:
15875:Kalman filter
15854:
15841:
15822:
15819:
15814:
15804:
15794:
15791:
15788:
15785:
15782:
15756:
15729:
15725:
15716:
15698:
15683:
15679:
15675:
15654:
15650:
15644:
15640:
15636:
15631:
15627:
15600:
15597:
15594:
15588:
15582:
15579:
15576:
15551:
15547:
15526:
15523:
15517:
15514:
15511:
15486:
15456:
15451:
15436:
15432:
15426:
15422:
15418:
15413:
15409:
15402:
15397:
15385:
15381:
15375:
15371:
15367:
15362:
15350:
15346:
15342:
15333:
15327:
15323:
15319:
15314:
15302:
15298:
15294:
15286:
15282:
15275:
15270:
15266:
15262:
15253:
15240:
15236:
15228:
15224:
15216:
15207:
15203:
15193:
15189:
15182:
15171:
15167:
15163:
15154:
15147:
15132:
15128:
15120:
15113:
15110:
15099:
15090:
15083:
15080:
15067:
15060:
15057:
15047:
15034:
15033:
15032:
15016:
15012:
14989:
14985:
14961:
14958:
14955:
14926:
14902:
14888:
14885:
14879:
14876:
14873:
14862:
14857:
14828:
14825:
14822:
14816:
14813:
14805:
14800:
14796:
14787:
14774:
14765:
14752:
14751:
14750:
14733:
14724:
14720:
14715:
14703:
14699:
14695:
14685:
14679:
14673:
14664:
14655:
14650:
14638:
14634:
14630:
14623:
14620:
14612:
14607:
14603:
14592:
14588:
14584:
14578:
14575:
14569:
14566:
14559:
14558:
14557:
14537:
14513:
14507:
14487:
14484:
14479:
14472:
14469:
14443:
14439:
14416:
14412:
14408:
14403:
14393:
14369:
14366:
14361:
14357:
14321:
14308:
14296:
14292:
14288:
14283:
14279:
14273:
14269:
14254:
14251:
14244:
14243:
14242:
14222:
14216:
14213:
14210:
14207:
14201:
14195:
14188:
14187:
14186:
14172:
14169:
14163:
14160:
14154:
14129:
14126:
14123:
14110:
14102:
14098:
14094:
14091:
14088:
14083:
14079:
14075:
14070:
14066:
14059:
14054:
14021:
14015:
14012:
14007:
13995:
13991:
13987:
13982:
13978:
13972:
13968:
13964:
13956:
13952:
13945:
13942:
13937:
13925:
13921:
13917:
13912:
13908:
13902:
13898:
13894:
13889:
13882:
13879:
13872:
13866:
13863:
13853:
13852:
13851:
13827:
13811:
13808:
13802:
13799:
13793:
13773:
13750:
13747:
13721:
13716:
13712:
13708:
13703:
13693:
13686:
13681:
13677:
13673:
13661:
13655:
13650:
13640:
13630:
13623:
13622:
13621:
13605:
13601:
13597:
13592:
13582:
13556:
13552:
13529:
13519:
13490:
13486:
13482:
13477:
13467:
13460:
13457:
13450:
13449:
13448:
13428:
13424:
13420:
13415:
13405:
13398:
13393:
13389:
13356:
13352:
13345:
13342:
13339:
13331:
13328:
13326:
13313:
13309:
13305:
13300:
13290:
13280:
13277:
13274:
13263:
13243:
13240:
13238:
13225:
13222:
13214:
13211:
13203:
13199:
13195:
13190:
13180:
13170:
13167:
13164:
13156:
13153:
13136:
13133:
13131:
13120:
13109:
13103:
13085:
13083:
13075:
13056:
13055:
13054:
13035:
13018:
12991:
12981:
12973:
12969:
12965:
12962:
12959:
12954:
12950:
12946:
12941:
12937:
12930:
12900:
12897:
12894:
12881:
12876:
12872:
12846:
12838:
12834:
12826:
12823:
12817:
12812:
12809:
12802:
12801:
12800:
12798:
12780:
12770:
12767:
12745:
12741:
12737:
12734:
12712:
12702:
12665:
12662:
12634:
12630:
12626:
12621:
12611:
12601:
12598:
12595:
12587:
12584:
12567:
12564:
12558:
12536:
12535:
12534:
12514:
12504:
12496:
12486:
12479:
12476:
12473:
12468:
12458:
12451:
12446:
12436:
12426:
12390:
12386:
12382:
12379:
12328:
12325:
12322:
12316:
12313:
12303:
12298:
12294:
12286:
12266:
12263:
12260:
12254:
12248:
12245:
12242:
12229:
12224:
12220:
12212:
12192:
12189:
12186:
12173:
12168:
12164:
12156:
12139:
12135:
12112:
12108:
12100:
12099:
12098:
12079:
12071:
12067:
12059:
12055:
12045:
12041:
12033:
12029:
12022:
12017:
12014:
12007:
12006:
12005:
11963:
11953:
11945:
11941:
11937:
11934:
11931:
11926:
11922:
11918:
11913:
11909:
11902:
11875:
11865:
11857:
11853:
11849:
11846:
11843:
11838:
11834:
11830:
11825:
11821:
11814:
11777:
11773:
11769:
11759:
11752:
11747:
11742:
11737:
11731:
11726:
11723:
11711:
11708:
11700:
11697:
11691:
11677:
11668:
11667:
11666:
11663:
11661:
11657:
11653:
11649:
11648:Kalman filter
11645:
11635:
11632:
11603:
11600:
11597:
11591:
11582:
11579:
11573:
11570:
11559:
11555:
11534:
11531:
11525:
11522:
11516:
11513:
11510:
11501:
11498:
11492:
11489:
11483:
11474:
11471:
11468:
11462:
11459:
11449:
11448:
11434:
11431:
11417:
11409:
11386:
11383:
11380:
11369:
11360:
11354:
11350:
11347:
11341:
11338:
11325:
11322:
11312:
11309:
11306:
11303:
11292:
11289:
11275:
11272:
11263:
11260:
11254:
11251:
11245:
11238:
11237:
11235:
11231:
11208:
11205:
11200:
11194:
11187:
11184:
11178:
11175:
11160:
11156:
11153:
11150:
11147:
11136:
11133:
11121:
11117:
11110:
11107:
11104:
11099:
11092:
11089:
11083:
11080:
11067:
11064:
11061:
11058:
11047:
11044:
11030:
11025:
11016:
11013:
11007:
11004:
10998:
10991:
10990:
10988:
10961:
10958:
10952:
10949:
10946:
10943:
10937:
10929:
10918:
10915:
10904:
10901:
10898:
10895:
10892:
10881:
10878:
10863:
10862:
10843:
10830:
10803:
10800:
10794:
10791:
10788:
10785:
10779:
10776:
10770:
10767:
10756:
10742:
10739:
10733:
10730:
10727:
10724:
10718:
10710:
10699:
10696:
10681:
10680:
10678:
10674:
10647:
10644:
10638:
10635:
10632:
10629:
10623:
10620:
10614:
10611:
10600:
10597:
10594:
10574:
10571:
10565:
10562:
10539:
10536:
10533:
10513:
10510:
10504:
10501:
10495:
10492:
10484:
10459:
10456:
10450:
10447:
10444:
10441:
10435:
10432:
10426:
10423:
10412:
10409:
10389:
10386:
10380:
10377:
10354:
10351:
10348:
10328:
10325:
10319:
10316:
10310:
10307:
10299:
10298:
10279:
10276:
10268:
10264:
10261:
10252:
10249:
10243:
10240:
10237:
10234:
10228:
10220:
10213:
10210:
10204:
10198:
10195:
10189:
10183:
10180:
10174:
10169:
10162:
10159:
10152:
10147:
10140:
10137:
10130:
10124:
10121:
10111:
10110:
10096:
10093:
10087:
10084:
10078:
10052:
10049:
10043:
10040:
10034:
10026:
10007:
10004:
9998:
9995:
9992:
9987:
9983:
9979:
9974:
9970:
9966:
9958:
9954:
9950:
9945:
9941:
9934:
9927:
9926:
9903:
9878:
9875:
9861:
9855:
9836:
9813:
9810:
9802:
9798:
9794:
9789:
9785:
9781:
9778:
9772:
9769:
9764:
9757:
9754:
9741:
9737:
9733:
9728:
9721:
9718:
9708:
9699:
9698:
9681:
9678:
9672:
9667:
9663:
9642:
9639:
9634:
9630:
9621:
9617:
9601:
9598:
9589:
9586:
9580:
9577:
9574:
9571:
9565:
9562:
9556:
9553:
9543:
9542:
9541:
9540:
9537:
9534:Consider the
9533:
9532:
9516:
9513:
9499:
9491:
9475:
9472:
9458:
9450:
9424:
9405:
9402:
9391:
9390:
9360:
9356:
9348:
9324:
9305:
9302:
9291:
9290:
9260:
9256:
9248:
9247:
9245:
9222:
9219:
9205:
9182:
9178:
9170:
9167:
9153:
9130:
9126:
9119:
9114:
9100:
9093:
9092:
9067:
9059:
9058:
9049:
9042:
9040:
9019:
9016:
9013:
9002:
8995:
8982:
8978:
8974:
8968:
8965:
8954:
8951:
8945:
8942:
8931:
8927:
8889:
8877:
8868:
8864:
8860:
8857:
8854:
8849:
8845:
8841:
8836:
8832:
8821:
8814:
8801:
8797:
8793:
8787:
8784:
8773:
8770:
8764:
8761:
8750:
8746:
8712:
8702:
8683:
8680:
8671:
8664:
8626:
8612:
8595:
8592:
8582:
8581:
8577:
8559:
8556:
8550:
8547:
8541:
8533:
8510:
8507:
8504:
8494:
8491:
8485:
8482:
8472:
8469:
8466:
8456:
8453:
8447:
8444:
8435:
8426:
8425:
8400:
8380:
8377:
8371:
8368:
8357:
8336:
8333:
8325:
8322:
8310:
8292:
8289:
8264:
8252:
8230:
8223:
8206:
8197:
8194:
8184:
8183:
8181:
8180:
8175:
8154:
8126:
8123:
8120:
8097:
8094:
8091:
8083:
8067:
8064:
8059:
8052:
8049:
8040:
8036:
8032:
8027:
8024:
8021:
8014:
8011:
8002:
7999:
7996:
7992:
7988:
7985:
7982:
7977:
7970:
7967:
7958:
7954:
7950:
7945:
7938:
7935:
7926:
7922:
7914:
7913:
7911:
7895:
7892:
7875:
7872:
7862:
7861:
7847:
7844:
7830:
7822:
7806:
7803:
7800:
7797:
7794:
7774:
7771:
7766:
7762:
7753:
7752:
7743:
7736:
7734:
7718:
7714:
7708:
7704:
7700:
7695:
7692:
7689:
7685:
7679:
7676:
7673:
7669:
7665:
7662:
7659:
7654:
7650:
7644:
7640:
7636:
7631:
7627:
7621:
7617:
7613:
7599:
7592:
7591:
7587:
7581:
7565:
7562:
7559:
7547:
7546:
7541:
7537:
7536:
7530:
7513:
7510:
7507:
7501:
7492:
7489:
7483:
7480:
7469:
7468:linear system
7466:
7450:
7447:
7444:
7441:
7435:
7432:
7422:(i.e., where
7406:
7403:
7400:
7394:
7391:
7381:
7380:
7379:
7362:
7356:
7333:
7330:
7321:
7318:
7312:
7309:
7306:
7303:
7297:
7294:
7288:
7285:
7275:
7274:
7273:
7256:
7253:
7250:
7247:
7241:
7238:
7228:
7227:
7226:
7223:
7221:
7201:
7198:
7175:
7172:
7166:
7163:
7155:
7151:
7140:
7121:
7107:
7087:
7084:
7078:
7075:
7047:
7033:
7005:
6988:
6985:
6975:
6974:
6973:
6955:
6938:
6927:
6924:
6919:
6912:
6909:
6898:
6882:
6872:
6864:
6861:
6850:
6847:
6838:
6831:
6828:
6817:
6814:
6809:
6803:
6800:
6789:
6786:
6780:
6777:
6766:
6760:
6753:
6750:
6739:
6723:
6712:
6709:
6704:
6697:
6694:
6683:
6667:
6657:
6649:
6646:
6635:
6632:
6626:
6623:
6612:
6605:
6599:
6584:
6583:
6582:
6579:
6577:
6556:
6542:
6534:
6490:
6487:
6476:
6460:
6449:
6446:
6441:
6434:
6431:
6420:
6404:
6394:
6389:
6386:
6374:
6373:
6372:
6350:
6343:
6330:
6326:
6314:
6311:
6300:
6297:
6291:
6288:
6277:
6273:
6253:
6240:
6239:
6238:
6188:
6185:
6168:
6165:
6155:
6154:
6153:
6151:
6146:
6144:
6140:
6093:
6087:
6084:
6074:
6073:
6072:
6047:
6022:
6008:
6000:
5996:
5974:
5971:
5960:
5942:
5929:
5928:
5927:
5919:
5905:
5902:
5899:
5876:
5873:
5845:
5831:
5801:
5782:
5776:
5750:
5747:
5730:
5727:
5708:
5704:
5700:
5695:
5685:
5670:
5669:
5668:
5646:
5632:
5629:
5624:
5614:
5598:
5597:
5592:
5591:
5580:
5578:
5562:
5559:
5556:
5533:
5530:
5502:
5488:
5446:
5418:
5415:
5387:
5384:
5381:
5357:
5354:
5351:
5348:
5337:
5334:
5310:
5307:
5298:
5295:
5292:
5286:
5280:
5277:
5270:
5269:
5268:
5251:
5248:
5245:
5239:
5236:
5227:
5221:
5218:
5211:
5210:
5209:
5195:
5192:
5189:
5163:
5153:
5145:
5142:
5139:
5133:
5130:
5124:
5117:
5116:
5115:
5101:
5093:
5072:
5042:
5029:
5022:
5016:
5008:
4993:
4990:
4987:
4979:
4976:
4968:
4965:
4953:
4944:
4941:
4933:
4930:
4918:
4913:
4910:
4897:
4889:
4877:
4872:
4868:
4853:
4852:
4851:
4843:
4829:
4807:
4797:
4785:
4767:
4745:
4742:
4739:
4715:
4712:
4707:
4683:
4661:
4658:
4653:
4627:
4624:
4621:
4616:
4612:
4608:
4600:
4594:
4589:
4567:
4563:
4559:
4553:
4547:
4527:
4524:
4521:
4515:
4512:
4489:
4486:
4483:
4478:
4474:
4470:
4464:
4458:
4450:
4434:
4412:
4407:
4399:
4381:
4377:
4353:
4350:
4347:
4341:
4338:
4330:
4327:
4315:
4311:
4306:
4294:
4291:
4277:
4273:
4268:
4264:
4257:
4242:
4236:
4226:
4218:
4212:
4204:
4200:
4191:
4173:
4169:
4155:
4154:
4153:
4137:
4132:
4129:
4126:
4105:
4102:
4094:
4090:
4087:
4076:
4057:
4054:
4051:
4048:
4042:
4039:
4031:
4028:
4016:
4012:
4003:
4002:
4001:
3984:
3981:
3978:
3972:
3969:
3946:
3941:
3936:
3933:
3930:
3925:
3915:
3907:
3904:
3901:
3895:
3892:
3884:
3881:
3868:
3867:
3866:
3849:
3846:
3843:
3837:
3834:
3820:
3806:
3803:
3800:
3797:
3794:
3774:
3771:
3768:
3743:
3733:
3725:
3722:
3719:
3713:
3710:
3702:
3699:
3686:
3685:
3684:
3668:
3665:
3657:
3653:
3650:
3622:
3608:
3594:
3577:
3574:
3511:
3498:
3494:
3482:
3479:
3468:
3465:
3459:
3456:
3445:
3441:
3421:
3412:
3404:
3401:
3388:
3376:
3371:
3345:
3336:
3330:
3327:
3317:
3316:
3315:
3286:
3252:
3249:
3215:
3208:
3182:
3148:
3145:
3111:
3104:
3103:
3102:
3085:
3082:
3059:
3028:
3008:
3005:
2999:
2996:
2988:
2984:
2963:
2960:
2957:
2949:
2944:
2930:
2927:
2913:
2905:
2881:
2878:
2871:
2868:
2860:
2857:
2841:
2838:
2830:
2827:
2819:
2816:
2804:
2795:
2792:
2784:
2781:
2769:
2764:
2761:
2748:
2740:
2728:
2723:
2719:
2704:
2703:
2702:
2700:
2699:
2694:
2693:
2671:
2657:
2635:
2616:
2605:
2584:
2562:
2555:
2553:
2537:
2532:
2513:
2505:
2502:
2497:
2483:
2467:
2463:
2457:
2454:
2449:
2432:
2426:
2419:
2418:
2414:
2411:
2409:
2399:
2391:
2389:
2385:
2381:
2365:
2357:
2340:
2337:
2327:
2306:
2292:
2284:
2280:
2276:
2272:
2268:
2264:
2235:
2221:
2196:
2182:
2174:
2154:
2140:
2132:
2131:
2130:
2111:
2097:
2066:
2064:
2063:
2041:
2027:
2010:
2003:
2001:
1986:
1982:
1979:
1963:
1959:
1955:
1950:
1940:
1931:
1923:
1922:
1918:
1915:
1898:
1895:
1892:
1869:
1866:
1863:
1860:
1857:
1785:
1782:
1779:
1770:
1768:
1747:
1734:
1715:
1687:
1684:
1670:
1662:
1642:
1639:
1625:
1617:
1602:
1599:
1585:
1555:
1554:
1553:
1515:
1500:
1490:
1487:
1480:
1475:
1456:
1443:
1435:
1432:
1428:
1427:
1426:
1423:
1421:
1417:
1413:
1409:
1405:
1389:
1386:
1381:
1377:
1354:
1350:
1283:
1279:
1256:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1224:
1212:
1209:) around the
1208:
1207:
1202:
1198:
1173:to the input
1153:
1119:
1096:
1091:
1089:
1085:
1081:
1080:
1060:
1044:
1040:
1036:
1018:
1015:
1012:
994:
989:
979:
976:
954:
936:
931:
921:
918:
911:
907:
882:
872:
867:
858:
850:
846:
835:
827:
824:
821:
817:
809:
799:
791:
787:
776:
768:
764:
757:
752:
746:
731:
730:
729:
727:
724:
721:-dimensional
699:
689:
684:
675:
667:
663:
652:
644:
641:
638:
634:
626:
616:
608:
604:
593:
585:
581:
574:
569:
563:
548:
547:
546:
535:
528:
526:
509:
494:
491:
480:
477:
471:
468:
457:
454:
448:
439:
424:
423:
419:
416:
415:described by
414:
404:
401:
396:
393:
387:
385:
381:
377:
373:
369:
365:
361:
356:
352:
347:
344:
340:
324:
321:
318:
309:
293:
290:
287:
275:
257:
253:
249:
246:
241:
234:
231:
220:
204:
201:
196:
189:
186:
179:
174:
170:
166:
163:
143:
140:
137:
117:
114:
111:
103:
98:
89:
87:
83:
79:
75:
71:
67:
62:
58:
54:
50:
46:
45:discontinuous
42:
38:
34:
30:
26:
22:
17880:
17876:
17842:
17807:
17785:
17758:
17739:
17720:
17701:
17682:
17663:
17644:
17621:
17618:Spurgeon, S.
17588:
17543:
17532:
17523:
17519:
17491:
17485:
17458:
17454:
17444:
17425:
17419:
17400:
17394:
17370:
17366:Khalil, H.K.
17328:
17324:
17318:
17307:
17283:(1): 23–36.
17280:
17276:
17270:
17260:
17253:
17239:(1): 80–90.
17236:
17232:
17226:
17207:
17171:
17163:
17159:oscillations
17153:
16876:
16871:
16725:
16485:
15879:
15471:
14917:
14894: vector
14837: vector
14748:
14336:
14240:
14039:
13850:). However,
13736:
13507:
13380:
12863:
12654:
12349:
12096:
11801:
11664:
11641:
11627:
11480:(i.e.,
10526:(i.e., when
10341:(i.e., when
9489:
9422:
9387:
9322:
9287:
9043:
8177:
7737:
7579:
7543:
7421:
7348:
7271:
7224:
7141:
7025:
6971:
6580:
6575:
6508:
6370:
6203:
6149:
6147:
6114:
5992:
5925:
5768:
5594:
5588:
5586:
5576:
5373:
5266:
5181:
5059:
4849:
4368:
4072:
3961:
3826:
3760:
3600:
3530:
3313:
3310:is negative.
3206:is positive.
2945:
2904:neighborhood
2901:
2847:(i.e.,
2696:
2690:
2571:
2556:
2405:
2397:
2274:
2260:
2129:, one must:
2067:
2060:
2019:
2004:
1771:
1700:. Desirable
1660:
1658:
1478:
1476:
1439:
1431:hypersurface
1424:
1416:sliding mode
1415:
1411:
1204:
1092:
1077:
1041:so that the
899:
716:
544:
529:
410:
397:
395:of success.
388:
371:
348:
279:
92:Introduction
77:
70:sliding mode
69:
65:
28:
24:
18:
17804:Ferrara, A.
17547:. pp.
15678:eigenvalues
14431:). Because
13826:essentially
11558:equilibrium
11234:closed form
10552:), to make
10367:), to make
9620:state space
8182:), we have
7538:Consider a
6237:. That is,
6139:equilibrium
5995:nonsingular
4732:must reach
4698:must reach
2406:Consider a
2090:to satisfy
1420:LTI systems
1404:equilibrium
1095:control law
411:Consider a
360:equilibrium
102:Phase plane
17403:. Kluwer.
17187:References
17108:optimality
15682:observable
12154:on itself,
11656:LTI system
8358:To ensure
6533:continuous
4075:chain rule
3314:Note that
2410:candidate
2263:continuous
1733:continuous
1442:continuous
1035:continuous
376:optimality
364:robustness
339:LTI system
219:LTI system
17834:198313071
17575:120072463
17549:2368–2370
17463:CiteSeerX
17285:CiteSeerX
17039:−
17033:^
17022:≜
17008:σ
16963:^
16845:−
16833:^
16820:
16784:−
16772:^
16759:
16687:−
16675:^
16662:
16639:≜
16568:−
16544:≜
16507:≜
16439:^
16387:−
16375:^
16362:
16303:−
16273:^
16236:−
16224:^
16211:
16177:−
16149:^
16112:−
16100:^
16087:
16053:−
16025:^
15988:−
15976:^
15963:
15934:^
15910:˙
15905:^
15820:−
15808:^
15795:
15598:−
15589:×
15580:−
15524:×
15515:−
15241:⏞
15217:⋮
15148:˙
15133:⏞
15114:˙
15100:⋮
15084:˙
15061:˙
14959:−
14886:×
14877:−
14863:⏟
14826:−
14817:×
14806:⏟
14775:⏟
14721:−
14686:σ
14674:⏞
14656:−
14613:⏞
14579:˙
14576:σ
14514:σ
14473:˙
14397:^
14223:σ
14217:
14202:σ
14164:˙
14161:σ
14155:σ
14127:−
14111:∈
14092:…
14060:≜
14022:σ
14013:−
13943:−
13883:˙
13867:˙
13864:σ
13803:˙
13800:σ
13794:σ
13774:σ
13751:˙
13748:σ
13709:−
13697:^
13674:≜
13665:^
13644:^
13631:σ
13586:^
13523:^
13483:−
13471:^
13421:−
13409:^
13306:−
13294:^
13264:−
13258:^
13223:−
13212:−
13196:−
13184:^
13148:^
13121:˙
13110:−
13104:˙
13099:^
13076:˙
13036:−
13030:^
12982:∈
12963:…
12898:−
12882:∈
12824:−
12771:∈
12706:^
12674:→
12627:−
12615:^
12579:^
12559:˙
12554:^
12505:∈
12490:^
12477:…
12462:^
12440:^
12421:^
12326:−
12317:×
12304:∈
12264:−
12255:×
12246:−
12230:∈
12190:−
12174:∈
12018:≜
11954:∈
11935:…
11903:≜
11866:∈
11847:…
11815:≜
11753:⋯
11692:˙
11583:˙
11526:˙
11502:˙
11484:σ
11472:−
11463:˙
11418:σ
11381:σ
11370:⏟
11361:σ
11355:⏞
11342:˙
11326:
11293:˙
11276:−
11264:˙
11201:σ
11195:⏞
11188:˙
11137:˙
11118:−
11100:⏟
11093:˙
11048:˙
11017:˙
10962:˙
10919:˙
10882:˙
10804:˙
10771:˙
10757:≥
10743:˙
10700:˙
10648:˙
10615:˙
10601:−
10566:˙
10563:σ
10534:σ
10505:˙
10460:˙
10427:˙
10381:˙
10378:σ
10349:σ
10320:˙
10280:¨
10269:⏞
10253:˙
10214:˙
10199:¨
10184:˙
10163:˙
10141:˙
10125:˙
10122:σ
10088:˙
10085:σ
10079:σ
10053:˙
10008:˙
9935:σ
9876:≤
9862:⋅
9758:˙
9722:˙
9682:˙
9590:˙
9557:¨
9500:σ
9459:σ
9406:˙
9403:σ
9392:) (i.e.,
9361:−
9306:˙
9303:σ
9292:) (i.e.,
9206:σ
9183:−
9154:σ
9017:×
9003:⏟
8996:˙
8983:⏞
8907:∂
8890:σ
8886:∂
8878:⏞
8858:…
8815:˙
8802:⏞
8730:∂
8713:σ
8709:∂
8684:˙
8681:σ
8672:⏞
8665:˙
8644:∂
8627:σ
8623:∂
8596:˙
8593:σ
8551:˙
8548:σ
8542:σ
8505:σ
8486:˙
8483:σ
8467:σ
8448:˙
8445:σ
8372:˙
8334:
8326:σ
8323:
8311:⏞
8293:˙
8290:σ
8270:∂
8262:∂
8253:⏞
8231:σ
8207:σ
8198:˙
8124:−
8095:−
8053:˙
8025:−
8015:˙
8000:−
7986:⋯
7971:˙
7939:˙
7876:˙
7873:σ
7831:σ
7804:≤
7798:≤
7693:−
7677:−
7663:⋯
7614:≜
7600:σ
7493:˙
7436:˙
7404:−
7395:˙
7363:⋅
7322:˙
7289:¨
7242:˙
7202:˙
7199:σ
7167:˙
7164:σ
7156:⊺
7152:σ
7108:σ
7079:˙
7076:σ
7034:σ
6989:˙
6986:σ
6944:∂
6939:σ
6936:∂
6925:−
6888:∂
6883:σ
6880:∂
6848:−
6761:⏞
6729:∂
6724:σ
6721:∂
6710:−
6673:∂
6668:σ
6665:∂
6633:−
6600:˙
6543:σ
6466:∂
6461:σ
6458:∂
6447:−
6410:∂
6405:σ
6402:∂
6390:−
6344:˙
6331:⏞
6259:∂
6254:σ
6251:∂
6169:˙
6166:σ
6088:˙
6085:σ
6048:σ
6009:σ
5948:∂
5943:σ
5940:∂
5900:σ
5877:˙
5832:σ
5783:σ
5731:˙
5728:σ
5709:⊺
5705:σ
5686:∈
5633:σ
5615:∈
5557:σ
5534:˙
5531:σ
5520:surface,
5489:σ
5447:σ
5419:˙
5416:σ
5382:σ
5352:μ
5349:≥
5338:˙
5335:σ
5311:˙
5308:σ
5299:
5293:≠
5287:σ
5281:
5252:μ
5249:−
5246:≤
5240:˙
5237:σ
5228:σ
5222:
5190:α
5164:α
5154:σ
5146:μ
5143:−
5140:≤
5134:˙
5131:σ
5125:σ
5102:σ
5078:‖
5073:⋅
5068:‖
5043:α
5023:⏞
5013:‖
5009:σ
5006:‖
4994:μ
4991:−
4988:≤
4977:
4966:
4954:⏟
4942:
4934:σ
4931:
4919:⏞
4914:˙
4911:σ
4898:σ
4895:∂
4887:∂
4878:⏞
4873:⊺
4869:σ
4830:σ
4804:‖
4798:⋅
4793:‖
4659:≥
4625:μ
4622:−
4609:≤
4528:μ
4525:−
4516:˙
4487:μ
4484:−
4435:∝
4354:μ
4351:−
4348:≤
4339:
4328:
4295:˙
4278:≜
4258:⏟
4237:⏟
4223:‖
4219:σ
4216:‖
4213:∝
4205:⏞
4152:), means
4130:≜
4103:
4088:
4055:μ
4052:−
4049:≤
4040:
4029:
3973:∈
3937:μ
3934:−
3931:≤
3926:α
3908:μ
3905:−
3902:≤
3893:
3882:
3838:∈
3804:≤
3801:α
3769:μ
3744:α
3726:μ
3723:−
3720:≤
3711:
3700:
3666:
3651:
3609:σ
3578:˙
3575:σ
3512:˙
3499:⏞
3427:∂
3422:σ
3419:∂
3402:
3389:
3377:⏞
3372:˙
3351:∂
3346:σ
3343:∂
3331:˙
3328:σ
3287:σ
3253:˙
3250:σ
3183:σ
3149:˙
3146:σ
3086:˙
3083:σ
3060:σ
3000:˙
2997:σ
2989:⊺
2985:σ
2914:σ
2869:
2858:
2828:
2817:
2805:⏟
2793:
2785:σ
2782:
2770:⏞
2765:˙
2762:σ
2749:σ
2746:∂
2738:∂
2729:⏞
2724:⊺
2720:σ
2658:σ
2632:‖
2617:σ
2614:‖
2590:‖
2585:⋅
2580:‖
2529:‖
2514:σ
2511:‖
2484:σ
2468:⊺
2464:σ
2433:σ
2341:˙
2338:σ
2293:σ
2222:σ
2183:σ
2141:σ
2098:σ
2028:σ
1960:σ
1941:∈
1896:−
1867:≤
1861:≤
1783:−
1671:σ
1626:σ
1600:≠
1586:σ
1506:→
1488:σ
1257:⊺
1243:…
1197:stabilize
1016:×
1003:→
995:×
945:→
937:×
910:functions
873:∈
825:−
810:⋮
753:≜
690:∈
642:−
627:⋮
570:≜
440:˙
308:deadbands
250:−
235:˙
190:˙
17910:Category
17810:. SIAM.
17620:(1998).
17368:(2002).
17345:31903795
17325:Robotica
17114:H-bridge
17068:See also
11168:if
11074:if
10109:. Here,
10071:so that
9423:positive
9323:negative
9202:if
9150:if
8501:if
8463:if
7787:for all
7529:origin.
4640:for all
4447:denotes
3962:So, for
2326:dynamics
2283:Filippov
1556:A state
1169:at time
906:feedback
341:with an
53:feedback
37:dynamics
15684:, this
15674:Hurwitz
15616:matrix
15539:vector
14382:(i.e.,
11654:for an
11447:, then
9891:(i.e.,
9490:chatter
7912:and so
7821:simplex
7552:(i.e.,
6531:is not
5182:Taking
5090:is the
4396:is the
4077:(i.e.,
2606:(i.e.,
2602:is the
1086:and is
276:origin.
31:) is a
17857:
17832:
17822:
17792:
17773:
17746:
17727:
17708:
17689:
17670:
17651:
17628:
17603:
17573:
17563:
17498:
17465:
17432:
17407:
17382:
17343:
17287:
17214:
16486:where
14781:scalar
14556:where
14241:where
14185:, let
14040:where
13381:where
12864:where
12760:, and
12655:where
12097:where
9622:(with
9246:where
7823:where
7580:picked
7465:stable
7220:robust
6143:stable
6115:If an
5577:switch
5060:where
4369:where
3761:where
3239:makes
3135:makes
2948:scalar
2572:where
1798:where
1211:origin
1088:unique
1084:exists
1039:smooth
908:. The
900:is an
726:vector
717:is an
545:where
372:finite
17830:S2CID
17571:S2CID
17341:S2CID
17145:Notes
16872:input
15717:when
10485:When
10300:When
9697:) as
9421:) is
9321:) is
7540:plant
7349:when
5798:is a
4119:with
2902:in a
1195:) to
723:state
74:locus
66:slide
49:state
39:of a
17855:ISBN
17820:ISBN
17790:ISBN
17771:ISBN
17744:ISBN
17725:ISBN
17706:ISBN
17687:ISBN
17668:ISBN
17649:ISBN
17626:ISBN
17601:ISBN
17561:ISBN
17496:ISBN
17430:ISBN
17405:ISBN
17380:ISBN
17212:ISBN
14255:>
14170:<
13828:all
13824:for
13809:<
13766:and
11384:>
11206:>
11105:<
10905:>
10598:<
10572:<
10537:>
10511:>
10413:>
10387:>
10352:<
10326:<
10094:<
9655:and
9220:<
9168:>
8557:<
8508:<
8492:>
8470:>
8454:<
8378:<
7772:>
7188:and
7173:<
6148:The
5926:Let
5802:and
5748:<
5355:>
3798:<
3787:and
3772:>
3072:and
3006:<
2879:<
2839:<
1824:and
1661:sign
1199:the
969:and
728:and
351:gain
17893:hdl
17885:doi
17847:doi
17812:doi
17763:doi
17593:doi
17553:doi
17473:doi
17333:doi
17295:doi
17241:doi
17064:).
16817:sgn
16756:sgn
16659:sgn
16634:obs
16619:and
16539:obs
16502:obs
16465:obs
16453:obs
16426:obs
16359:sgn
16208:sgn
16084:sgn
15960:sgn
15792:sgn
15672:is
14749:So
14258:max
14214:sgn
11560:at
11323:sgn
11236:as
9845:sup
9425:at
9325:at
6576:act
6141:is
5993:be
5322:and
5296:sgn
5278:sgn
5219:sgn
4400:of
2382:or
2275:not
2273:is
2065:).
29:SMC
19:In
17912::
17891:.
17881:55
17879:.
17853:.
17828:.
17818:.
17769:.
17599:.
17569:.
17559:.
17551:.
17541:.
17524:44
17522:.
17510:^
17471:.
17459:64
17457:.
17453:.
17378:.
17353:^
17339:.
17329:31
17327:.
17293:.
17281:40
17279:.
17237:45
17235:.
17195:^
15859:eq
15761:eq
15730:12
15655:12
15437:12
15386:12
15338:eq
15017:12
14931:eq
14801:12
14770:eq
14729:eq
14704:12
14669:eq
14639:12
14593:11
14542:eq
14297:12
14274:11
13996:12
13973:11
13926:12
13903:11
12299:12
12225:22
12169:21
12113:11
12072:22
12060:21
12046:12
12034:11
11890:,
10861:,
10679:,
7222:.
4674:,
4000:,
3865:,
3593:.
2943:.
1090:.
1082:)
23:,
17901:.
17895::
17887::
17863:.
17849::
17836:.
17814::
17798:.
17779:.
17765::
17752:.
17733:.
17714:.
17695:.
17676:.
17657:.
17634:.
17609:.
17595::
17577:.
17555::
17504:.
17479:.
17475::
17438:.
17413:.
17388:.
17347:.
17335::
17301:.
17297::
17247:.
17243::
17220:.
17179:.
17110:.
17051:0
17047:=
17043:y
17029:y
17019:)
17015:x
17011:(
16987:y
16959:y
16945:C
16930:x
16926:C
16923:=
16919:y
16896:1
16892:x
16888:=
16885:y
16858:)
16853:1
16849:x
16840:1
16830:x
16823:(
16797:)
16792:1
16788:x
16779:1
16769:x
16762:(
16735:u
16711:.
16706:]
16700:)
16695:1
16691:x
16682:1
16672:x
16665:(
16651:u
16644:[
16630:u
16607:,
16602:]
16594:]
16588:M
16583:2
16579:L
16571:M
16562:[
16555:B
16549:[
16535:B
16513:,
16510:A
16498:A
16460:u
16449:B
16445:+
16435:x
16422:A
16418:=
16406:]
16400:)
16395:1
16391:x
16382:1
16372:x
16365:(
16351:u
16344:[
16337:]
16329:]
16323:M
16318:2
16314:L
16306:M
16297:[
16290:B
16284:[
16279:+
16269:x
16262:A
16259:=
16249:)
16244:1
16240:x
16231:1
16221:x
16214:(
16203:]
16197:M
16192:2
16188:L
16180:M
16171:[
16166:+
16162:u
16158:B
16155:+
16145:x
16138:A
16135:=
16125:)
16120:1
16116:x
16107:1
16097:x
16090:(
16081:M
16076:]
16068:2
16064:L
16056:1
16047:[
16042:+
16038:u
16034:B
16031:+
16021:x
16014:A
16011:=
16001:)
15996:1
15992:x
15983:1
15973:x
15966:(
15957:M
15954:L
15951:+
15947:u
15943:B
15940:+
15930:x
15923:A
15920:=
15901:x
15855:v
15844:v
15826:)
15823:x
15815:1
15805:x
15798:(
15789:M
15786:=
15783:v
15757:v
15746:C
15726:A
15699:2
15694:e
15660:)
15651:A
15645:2
15641:L
15637:+
15632:2
15628:A
15624:(
15604:)
15601:1
15595:n
15592:(
15586:)
15583:1
15577:n
15574:(
15552:2
15548:L
15527:1
15521:)
15518:1
15512:n
15509:(
15487:2
15482:e
15457:.
15452:2
15447:e
15442:)
15433:A
15427:2
15423:L
15419:+
15414:2
15410:A
15406:(
15403:=
15398:2
15393:e
15382:A
15376:2
15372:L
15368:+
15363:2
15358:e
15351:2
15347:A
15343:=
15334:v
15328:2
15324:L
15320:+
15315:2
15310:e
15303:2
15299:A
15295:=
15292:)
15287:1
15283:e
15279:(
15276:v
15271:2
15267:L
15263:+
15254:2
15249:e
15237:]
15229:n
15225:e
15208:3
15204:e
15194:2
15190:e
15183:[
15172:2
15168:A
15164:=
15155:2
15144:e
15129:]
15121:n
15111:e
15091:3
15081:e
15068:2
15058:e
15048:[
15013:A
14990:1
14986:x
14965:)
14962:1
14956:n
14953:(
14927:v
14903:.
14889:1
14883:)
14880:1
14874:n
14871:(
14858:2
14853:e
14832:)
14829:1
14823:n
14820:(
14814:1
14797:A
14788:=
14766:v
14734:.
14725:v
14716:2
14711:e
14700:A
14696:=
14689:)
14683:(
14680:v
14665:v
14651:2
14646:e
14635:A
14631:+
14624:0
14621:=
14608:1
14604:e
14589:a
14585:=
14570:=
14567:0
14538:v
14517:)
14511:(
14508:v
14488:0
14485:=
14480:1
14470:e
14444:1
14440:e
14417:1
14413:x
14409:=
14404:1
14394:x
14370:0
14367:=
14362:1
14358:e
14347:M
14343:M
14339:M
14322:.
14319:}
14315:|
14309:2
14304:e
14293:A
14289:+
14284:1
14280:e
14270:a
14265:|
14261:{
14252:M
14226:)
14220:(
14211:M
14208:=
14205:)
14199:(
14196:v
14173:0
14133:)
14130:1
14124:n
14121:(
14116:R
14108:)
14103:n
14099:e
14095:,
14089:,
14084:3
14080:e
14076:,
14071:2
14067:e
14063:(
14055:2
14050:e
14025:)
14019:(
14016:v
14008:2
14003:e
13992:A
13988:+
13983:1
13979:e
13969:a
13965:=
13962:)
13957:1
13953:e
13949:(
13946:v
13938:2
13933:e
13922:A
13918:+
13913:1
13909:e
13899:a
13895:=
13890:1
13880:e
13873:=
13837:x
13812:0
13722:.
13717:1
13713:x
13704:1
13694:x
13687:=
13682:1
13678:e
13671:)
13662:x
13656:,
13651:1
13641:x
13634:(
13606:1
13602:x
13598:=
13593:1
13583:x
13557:1
13553:x
13530:1
13520:x
13491:1
13487:x
13478:1
13468:x
13461:=
13458:0
13445:v
13429:1
13425:x
13416:1
13406:x
13399:=
13394:1
13390:e
13362:)
13357:1
13353:e
13349:(
13346:v
13343:L
13340:+
13336:e
13332:A
13329:=
13319:)
13314:1
13310:x
13301:1
13291:x
13284:(
13281:v
13278:L
13275:+
13272:)
13268:x
13254:x
13247:(
13244:A
13241:=
13230:u
13226:B
13219:x
13215:A
13209:)
13204:1
13200:x
13191:1
13181:x
13174:(
13171:v
13168:L
13165:+
13161:u
13157:B
13154:+
13144:x
13137:A
13134:=
13117:x
13095:x
13086:=
13072:e
13040:x
13026:x
13019:=
13015:e
12992:n
12987:R
12979:)
12974:n
12970:e
12966:,
12960:,
12955:2
12951:e
12947:,
12942:1
12938:e
12934:(
12931:=
12927:e
12904:)
12901:1
12895:n
12892:(
12887:R
12877:2
12873:L
12847:]
12839:2
12835:L
12827:1
12818:[
12813:=
12810:L
12781:n
12776:R
12768:L
12746:1
12742:x
12738:=
12735:y
12713:1
12703:x
12678:R
12670:R
12666::
12663:v
12640:)
12635:1
12631:x
12622:1
12612:x
12605:(
12602:v
12599:L
12596:+
12592:u
12588:B
12585:+
12575:x
12568:A
12565:=
12550:x
12531:n
12515:n
12510:R
12502:)
12497:n
12487:x
12480:,
12474:,
12469:2
12459:x
12452:,
12447:1
12437:x
12430:(
12427:=
12417:x
12391:1
12387:x
12383:=
12380:y
12359:x
12332:)
12329:1
12323:n
12320:(
12314:1
12309:R
12295:A
12270:)
12267:1
12261:n
12258:(
12252:)
12249:1
12243:n
12240:(
12235:R
12221:A
12196:)
12193:1
12187:n
12184:(
12179:R
12165:A
12140:1
12136:x
12109:a
12080:]
12068:A
12056:A
12042:A
12030:a
12023:[
12015:A
11991:x
11980:y
11964:r
11959:R
11951:)
11946:r
11942:u
11938:,
11932:,
11927:2
11923:u
11919:,
11914:1
11910:u
11906:(
11899:u
11876:n
11871:R
11863:)
11858:n
11854:x
11850:,
11844:,
11839:2
11835:x
11831:,
11826:1
11822:x
11818:(
11811:x
11778:1
11774:x
11770:=
11766:x
11760:]
11748:0
11743:0
11738:1
11732:[
11727:=
11724:y
11716:u
11712:B
11709:+
11705:x
11701:A
11698:=
11688:x
11678:{
11619:.
11607:)
11604:0
11601:,
11598:0
11595:(
11592:=
11589:)
11580:x
11574:,
11571:x
11568:(
11539:)
11535:0
11532:=
11523:x
11517:+
11514:x
11511:=
11508:)
11499:x
11493:,
11490:x
11487:(
11475:x
11469:=
11460:x
11435:0
11432:=
11429:)
11425:x
11421:(
11391:)
11387:0
11366:)
11351:x
11348:+
11339:x
11329:(
11316:)
11313:1
11310:+
11307:k
11304:+
11300:|
11290:x
11283:|
11279:(
11273:=
11270:)
11261:x
11255:,
11252:x
11249:(
11246:u
11209:0
11185:x
11179:+
11176:x
11161:)
11157:1
11154:+
11151:k
11148:+
11144:|
11134:x
11127:|
11122:(
11111:,
11108:0
11090:x
11084:+
11081:x
11068:1
11065:+
11062:k
11059:+
11055:|
11045:x
11038:|
11031:{
11026:=
11023:)
11014:x
11008:,
11005:x
11002:(
10999:u
10972:|
10968:)
10959:x
10953:,
10950:x
10947:,
10944:t
10941:(
10938:a
10934:|
10930:+
10926:|
10916:x
10909:|
10902:1
10899:+
10896:k
10893:+
10889:|
10879:x
10872:|
10848:|
10844:a
10840:|
10814:|
10810:)
10801:x
10795:,
10792:x
10789:,
10786:t
10783:(
10780:a
10777:+
10768:x
10761:|
10753:|
10749:)
10740:x
10734:,
10731:x
10728:,
10725:t
10722:(
10719:a
10715:|
10711:+
10707:|
10697:x
10690:|
10658:|
10654:)
10645:x
10639:,
10636:x
10633:,
10630:t
10627:(
10624:a
10621:+
10612:x
10605:|
10595:u
10575:0
10540:0
10514:0
10502:x
10496:+
10493:x
10470:|
10466:)
10457:x
10451:,
10448:x
10445:,
10442:t
10439:(
10436:a
10433:+
10424:x
10417:|
10410:u
10390:0
10355:0
10329:0
10317:x
10311:+
10308:x
10277:x
10265:u
10262:+
10259:)
10250:x
10244:,
10241:x
10238:,
10235:t
10232:(
10229:a
10221:+
10211:x
10205:=
10196:x
10190:+
10181:x
10175:=
10170:2
10160:x
10153:+
10148:1
10138:x
10131:=
10097:0
10059:)
10050:x
10044:,
10041:x
10038:(
10035:u
10005:x
9999:+
9996:x
9993:=
9988:2
9984:x
9980:+
9975:1
9971:x
9967:=
9964:)
9959:2
9955:x
9951:,
9946:1
9942:x
9938:(
9923:k
9908:|
9904:a
9900:|
9879:k
9873:}
9869:|
9865:)
9859:(
9856:a
9852:|
9848:{
9814:u
9811:+
9808:)
9803:2
9799:x
9795:,
9790:1
9786:x
9782:,
9779:t
9776:(
9773:a
9770:=
9765:2
9755:x
9742:2
9738:x
9734:=
9729:1
9719:x
9709:{
9679:x
9673:=
9668:2
9664:x
9643:x
9640:=
9635:1
9631:x
9602:u
9599:+
9596:)
9587:x
9581:,
9578:x
9575:,
9572:t
9569:(
9566:a
9563:=
9554:x
9517:0
9514:=
9511:)
9507:x
9503:(
9476:0
9473:=
9470:)
9466:x
9462:(
9434:x
9389:5
9374:)
9370:x
9366:(
9357:u
9334:x
9289:5
9274:)
9270:x
9266:(
9261:+
9257:u
9223:0
9217:)
9213:x
9209:(
9196:)
9192:x
9188:(
9179:u
9171:0
9165:)
9161:x
9157:(
9144:)
9140:x
9136:(
9131:+
9127:u
9120:{
9115:=
9112:)
9108:x
9104:(
9101:u
9079:)
9075:x
9071:(
9068:u
9048:)
9046:5
9044:(
9020:1
9014:n
8992:x
8979:)
8975:u
8972:)
8969:t
8966:,
8962:x
8958:(
8955:B
8952:+
8949:)
8946:t
8943:,
8939:x
8935:(
8932:f
8928:(
8912:x
8901:)
8897:x
8893:(
8874:]
8869:n
8865:s
8861:,
8855:,
8850:2
8846:s
8842:,
8837:1
8833:s
8829:[
8822:=
8811:x
8798:)
8794:u
8791:)
8788:t
8785:,
8781:x
8777:(
8774:B
8771:+
8768:)
8765:t
8762:,
8758:x
8754:(
8751:f
8747:(
8735:x
8724:)
8720:x
8716:(
8703:=
8698:)
8694:x
8690:(
8661:x
8649:x
8638:)
8634:x
8630:(
8613:=
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27:(
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