307:, is usually equal to the order of the system's defining differential equation, but not necessarily. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction. It is important to understand that converting a state-space realization to a transfer function form may lose some internal information about the system, and may provide a description of a system which is stable, when the state-space realization is unstable at certain points. In electric circuits, the number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as
6013:
36:
324:
9142:
12530:
8530:
12510:
9572:
8890:
5255:
4840:
8267:
10764:
9365:
9137:{\displaystyle {\begin{bmatrix}B&AB\end{bmatrix}}={\begin{bmatrix}{\begin{bmatrix}0\\{\frac {1}{m}}\end{bmatrix}}&{\begin{bmatrix}0&1\\-{\frac {k}{m}}&-{\frac {b}{m}}\end{bmatrix}}{\begin{bmatrix}0\\{\frac {1}{m}}\end{bmatrix}}\end{bmatrix}}={\begin{bmatrix}0&{\frac {1}{m}}\\{\frac {1}{m}}&-{\frac {b}{m^{2}}}\end{bmatrix}}}
3098:
4974:
4008:
4591:
8525:{\displaystyle {\begin{bmatrix}{\dot {\mathbf {x} }}_{1}(t)\\{\dot {\mathbf {x} }}_{2}(t)\end{bmatrix}}={\begin{bmatrix}0&1\\-{\frac {k}{m}}&-{\frac {b}{m}}\end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{1}(t)\\\mathbf {x} _{2}(t)\end{bmatrix}}+{\begin{bmatrix}0\\{\frac {1}{m}}\end{bmatrix}}\mathbf {u} (t)}
10494:
3125:
Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals (i.e., as controllability provides that an input is available that brings any initial state to any desired
5849:
7631:
3235:
3717:
9567:{\displaystyle {\begin{bmatrix}C\\CA\end{bmatrix}}={\begin{bmatrix}{\begin{bmatrix}1&0\end{bmatrix}}\\{\begin{bmatrix}1&0\end{bmatrix}}{\begin{bmatrix}0&1\\-{\frac {k}{m}}&-{\frac {b}{m}}\end{bmatrix}}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}
5966:
4580:
5250:{\displaystyle {\dot {\textbf {x}}}(t)={\begin{bmatrix}0&0&0&-d_{4}\\1&0&0&-d_{3}\\0&1&0&-d_{2}\\0&0&1&-d_{1}\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}n_{4}\\n_{3}\\n_{2}\\n_{1}\end{bmatrix}}{\textbf {u}}(t)}
2989:
4956:
3843:
134:, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables. Output variables’ values depend on the state variable values and may also depend on the input variable values.
2770:
7767:
2976:
The state controllability condition implies that it is possible – by admissible inputs – to steer the states from any initial value to any final value within some finite time window. A continuous time-invariant linear state-space model is
3445:
2444:
2260:
3574:
4282:
1701:
494:
4835:{\displaystyle {\dot {\mathbf {x} }}(t)={\begin{bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\-d_{4}&-d_{3}&-d_{2}&-d_{1}\end{bmatrix}}\mathbf {x} (t)+{\begin{bmatrix}0\\0\\0\\1\end{bmatrix}}\mathbf {u} (t)}
5347:
10296:
10902:
9951:
6551:
3341:
1518:
2061:
5697:
2154:
1794:
588:
7311:
10759:{\displaystyle {\dot {\mathbf {x} }}(t)={\begin{bmatrix}{\dot {x}}_{1}(t)\\{\dot {x}}_{2}(t)\end{bmatrix}}=\mathbf {f} (t,x(t))={\begin{bmatrix}x_{2}(t)\\-{\frac {g}{\ell }}\sin {x_{1}}(t)-{\frac {k}{m\ell }}{x_{2}}(t)\end{bmatrix}}.}
7879:
5459:
1881:
3832:
2519:
2335:
1956:
1593:
5708:
7406:
7448:
3139:
6325:
7130:
6164:
6850:
6942:
1286:
5530:
6635:
4345:
which contains transfer functions for each input output combination. Due to the simplicity of this matrix notation, the state-space representation is commonly used for multiple-input, multiple-output systems. The
3585:
270:
into trend and cycle, compose individual indicators into a composite index, identify turning points of the business cycle, and estimate GDP using latent and unobserved time series. Many applications rely on the
6400:
7202:
7047:
6236:
9671:
4192:
5857:
9750:
8047:
2878:
6724:
287:
are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system,
8840:
829:
750:
671:
7931:
6073:
3093:{\displaystyle \operatorname {rank} {\begin{bmatrix}\mathbf {B} &\mathbf {A} \mathbf {B} &\mathbf {A} ^{2}\mathbf {B} &\cdots &\mathbf {A} ^{n-1}\mathbf {B} \end{bmatrix}}=n,}
6899:
5464:
The strictly proper transfer function can then be transformed into a canonical state-space realization using techniques shown above. The state-space realization of the constant is trivially
4381:
10482:
4050:
11043:
10422:
10168:
8762:
8651:
2814:
315:. The state variables defined must be linearly independent, i.e., no state variable can be written as a linear combination of the other state variables, or the system cannot be solved.
4003:{\displaystyle \mathbf {Y} (s)=\mathbf {C} ((s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {x} (0)+(s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} \mathbf {U} (s))+\mathbf {D} \mathbf {U} (s).}
4848:
9794:
6002:
2571:
7637:
1157:
1126:
1038:
950:
862:
783:
704:
625:
10353:
1370:
1342:
3352:
2941:
2351:
8874:
7440:
6434:
4317:
3751:
3479:
2170:
9290:
8158:
4119:
3487:
1416:
allow for time domains that have both continuous and discrete parts. Depending on the assumptions made, the state-space model representation can assume the following forms:
1212:
1093:
1005:
917:
9252:
8120:
4203:
1604:
397:
10810:
2559:
12404:
9986:
5263:
10174:
4343:
250:
10818:
8693:
2958:) even though it is not internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if those singularities in the transfer function are
9842:
6442:
3262:
1439:
1967:
11325:
10074:
9823:
9214:
8189:
8082:
6976:
6931:
5557:
2066:
1706:
500:
7213:
5356:
because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
182:. The capacity of these structures can be efficiently applied to research systems with or without modulation. The state-space representation (also known as the "
7782:
10094:
10052:
10030:
10008:
9350:
9330:
9310:
9185:
9165:
8255:
8233:
8211:
4965:
because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
1410:
1390:
1314:
389:
369:
349:
305:
224:
204:
5378:
1805:
3759:
11075:
Stock, J.H.; Watson, M.W. (2016), "Dynamic Factor Models, Factor-Augmented Vector
Autoregressions, and Structural Vector Autoregressions in Macroeconomics",
5844:{\displaystyle {\dot {\textbf {x}}}(t)={\begin{bmatrix}-2&-1\\1&0\\\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}1\\0\end{bmatrix}}{\textbf {u}}(t)}
2449:
2265:
1886:
1523:
7317:
7626:{\displaystyle {\dot {\mathbf {x} }}(t)=\left(A-BK\left(I+DK\right)^{-1}C\right)\mathbf {x} (t)+B\left(I-K\left(I+DK\right)^{-1}D\right)\mathbf {r} (t)}
11652:
3230:{\displaystyle \operatorname {rank} {\begin{bmatrix}\mathbf {C} \\\mathbf {C} \mathbf {A} \\\vdots \\\mathbf {C} \mathbf {A} ^{n-1}\end{bmatrix}}=n.}
11239:
Bańbura, Marta; Modugno, Michele (2012-11-12). "Maximum
Likelihood Estimation of Factor Models on Datasets with Arbitrary Pattern of Missing Data".
12246:
262:
The state-space model can be applied in subjects such as economics, statistics, computer science and electrical engineering, and neuroscience. In
6247:
5372:) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant.
4375:
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
7055:
6089:
4372:
can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system):
6776:
3126:
final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the system).
3712:{\displaystyle \mathbf {X} (s)=(s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {x} (0)+(s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} \mathbf {U} (s).}
1219:
5467:
6557:
1292:
In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in the common
6331:
7136:
6981:
6170:
17:
9591:
5961:{\displaystyle {\textbf {y}}(t)={\begin{bmatrix}1&2\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}1\end{bmatrix}}{\textbf {u}}(t)}
6083:. The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results.
4135:
259:
approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions.
11762:
11645:
11506:
9677:
7951:
2903:. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's
2822:
6655:
11594:
11564:
11495:
11438:
11393:
11309:
11116:
11092:
11026:
10999:
12435:
11229:," NBER Chapters, in: NBER Macroeconomics Annual 1989, Volume 4, pages 351-409, National Bureau of Economic Research, Inc.
8769:
12536:
12087:
11824:
788:
709:
630:
4575:{\displaystyle {\textbf {G}}(s)={\frac {n_{1}s^{3}+n_{2}s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}}.}
7885:
6027:
12564:
12348:
11975:
11782:
11638:
11545:
11519:
11476:
11457:
6856:
4053:
186:
approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With
79:
57:
10429:
50:
12303:
275:
or a state observer to produce estimates of the current unknown state variables using their previous observations.
9796:
is a linear combination of states and inputs then the equations can be written in matrix notation like above. The
4016:
1128:
is the "feedthrough (or feedforward) matrix" (in cases where the system model does not have a direct feedthrough,
4951:{\displaystyle \mathbf {y} (t)={\begin{bmatrix}n_{4}&n_{3}&n_{2}&n_{1}\end{bmatrix}}\mathbf {x} (t).}
10360:
10106:
8700:
8536:
2895:
where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is
2786:
12569:
12490:
12430:
12028:
2765:{\displaystyle {\textbf {G}}(s)=k{\frac {(s-z_{1})(s-z_{2})(s-z_{3})}{(s-p_{1})(s-p_{2})(s-p_{3})(s-p_{4})}}.}
12574:
12023:
11712:
9758:
7762:{\displaystyle \mathbf {y} (t)=\left(I+DK\right)^{-1}C\mathbf {x} (t)+\left(I+DK\right)^{-1}D\mathbf {r} (t)}
97:
5974:
12465:
11862:
11819:
11772:
11767:
11624:
11175:
Smith, Anne C.; Brown, Emery N. (2003). "Estimating a State-Space Model from Point
Process Observations".
1131:
1100:
1012:
924:
836:
757:
678:
599:
12579:
12516:
11812:
11738:
10307:
3440:{\displaystyle s\mathbf {X} (s)-\mathbf {x} (0)=\mathbf {A} \mathbf {X} (s)+\mathbf {B} \mathbf {U} (s).}
2439:{\displaystyle z\mathbf {X} (z)-z\mathbf {x} (0)=\mathbf {A} \mathbf {X} (z)+\mathbf {B} \mathbf {U} (z)}
2255:{\displaystyle s\mathbf {X} (s)-\mathbf {x} (0)=\mathbf {A} \mathbf {X} (s)+\mathbf {B} \mathbf {U} (s)}
12140:
12075:
11676:
11620:
7942:
3569:{\displaystyle (s\mathbf {I} -\mathbf {A} )\mathbf {X} (s)=\mathbf {x} (0)+\mathbf {B} \mathbf {U} (s)}
1347:
1319:
4277:{\displaystyle \mathbf {G} (s)=\mathbf {C} (s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} +\mathbf {D} }
2913:
1696:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)}
489:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)}
12541:
12399:
12038:
11869:
11692:
8848:
7414:
6408:
4291:
3725:
3453:
2776:
9257:
8125:
6941:
4585:
The coefficients can now be inserted directly into the state-space model by the following approach:
4063:
1162:
1043:
955:
867:
12440:
11697:
9219:
8087:
5365:
5342:{\displaystyle {\textbf {y}}(t)={\begin{bmatrix}0&0&0&1\end{bmatrix}}{\textbf {x}}(t).}
4347:
4126:
2892:
2561:. The stability of a time-invariant state-space model can be determined by looking at the system's
44:
10780:
10291:{\displaystyle {\dot {x}}_{2}(t)=-{\frac {g}{\ell }}\sin {x_{1}}(t)-{\frac {k}{m\ell }}{x_{2}}(t)}
2542:
12485:
12470:
12123:
12118:
12018:
11886:
11667:
11616:
10897:{\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}n\pi \\0\end{bmatrix}}}
11044:"Modeling of dynamic systems with modulation by means of Kronecker vector-matrix representation"
9962:
9946:{\displaystyle m\ell ^{2}{\ddot {\theta }}(t)=-m\ell g\sin \theta (t)-k\ell {\dot {\theta }}(t)}
6546:{\displaystyle {\dot {\mathbf {x} }}(t)=\left(A+BK\left(I-DK\right)^{-1}C\right)\mathbf {x} (t)}
4968:
The transfer function coefficients can also be used to construct another type of canonical form
3336:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}
1513:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}
12445:
12205:
11924:
11919:
10770:
4322:
3249:" of a continuous time-invariant linear state-space model can be derived in the following way:
2056:{\displaystyle \mathbf {x} (k+1)=\mathbf {A} (k)\mathbf {x} (k)+\mathbf {B} (k)\mathbf {u} (k)}
229:
119:
61:
10989:
8662:
12475:
12460:
12425:
12113:
12013:
11881:
11016:
9825:
argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).
5692:{\displaystyle {\textbf {G}}(s)={\frac {s^{2}+3s+3}{s^{2}+2s+1}}={\frac {s+2}{s^{2}+2s+1}}+1}
4363:
3104:
2959:
2896:
2149:{\displaystyle \mathbf {y} (k)=\mathbf {C} (k)\mathbf {x} (k)+\mathbf {D} (k)\mathbf {u} (k)}
1789:{\displaystyle \mathbf {y} (t)=\mathbf {C} (t)\mathbf {x} (t)+\mathbf {D} (t)\mathbf {u} (t)}
583:{\displaystyle \mathbf {y} (t)=\mathbf {C} (t)\mathbf {x} (t)+\mathbf {D} (t)\mathbf {u} (t)}
123:
107:
12343:
7306:{\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)-BK\mathbf {y} (t)+B\mathbf {r} (t)}
2535:(i.e., linear with matrices that are constant with respect to time) can be studied from the
12495:
12450:
11896:
11841:
11687:
11682:
10059:
7941:
A classical linear system is that of one-dimensional movement of an object (e.g., a cart).
7874:{\displaystyle {\dot {\mathbf {x} }}(t)=\left(A-BKC\right)\mathbf {x} (t)+B\mathbf {r} (t)}
171:
152:
9799:
9190:
8165:
8058:
6952:
6907:
6012:
8:
12070:
12048:
11797:
11792:
11750:
11702:
11060:
10921:
5454:{\displaystyle {\textbf {G}}(s)={\textbf {G}}_{\mathrm {SP} }(s)+{\textbf {G}}(\infty ).}
1876:{\displaystyle \mathbf {x} (k+1)=\mathbf {A} \mathbf {x} (k)+\mathbf {B} \mathbf {u} (k)}
127:
103:
93:
11488:
Mathematical
Systems Theory I, Modelling, State Space Analysis, Stability and Robustness
3827:{\displaystyle \mathbf {Y} (s)=\mathbf {C} \mathbf {X} (s)+\mathbf {D} \mathbf {U} (s),}
12455:
12033:
11353:
11274:
11208:
10079:
10037:
10015:
9993:
9755:
The first is the state equation and the latter is the output equation. If the function
9335:
9315:
9295:
9170:
9150:
8240:
8218:
8196:
4125:, however, this ratio is not defined. Therefore, assuming zero initial conditions, the
2904:
2900:
2514:{\displaystyle \mathbf {Y} (z)=\mathbf {C} \mathbf {X} (z)+\mathbf {D} \mathbf {U} (z)}
2330:{\displaystyle \mathbf {Y} (s)=\mathbf {C} \mathbf {X} (s)+\mathbf {D} \mathbf {U} (s)}
1951:{\displaystyle \mathbf {y} (k)=\mathbf {C} \mathbf {x} (k)+\mathbf {D} \mathbf {u} (k)}
1588:{\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)}
1395:
1375:
1299:
374:
354:
334:
290:
209:
189:
167:
115:
9577:
which also has full rank. Therefore, this system is both controllable and observable.
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12509:
12313:
11965:
11836:
11829:
11590:
11560:
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11515:
11491:
11472:
11453:
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11389:
11357:
11345:
11305:
11301:
11266:
11200:
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11157:
11122:
11112:
11088:
11022:
10995:
6080:
4351:
4057:
3253:
3246:
2562:
252:
179:
11278:
11212:
151:
in which the axes are the state variables. The system state can be represented as a
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12063:
11857:
11807:
11802:
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11733:
11612:
11337:
11297:
11256:
11248:
11184:
11149:
11080:
11055:
10774:
7945:
for an object moving horizontally on a plane and attached to a wall with a spring:
7401:{\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)-DK\mathbf {y} (t)+D\mathbf {r} (t)}
256:
163:
11084:
11048:
Scientific and
Technical Journal of Information Technologies, Mechanics and Optics
6750:
equals zero. Another fairly common situation is when all states are outputs, i.e.
12379:
12323:
12145:
11787:
11707:
10941:
8881:
6767:
6727:
5369:
4369:
2971:
148:
11140:
Roesser, R. (1975). "A discrete state-space model for linear image processing".
10955:
for information about phase state (like state space) in physics and mathematics.
12353:
12318:
12308:
12133:
11891:
11717:
11188:
10985:
10946:
10931:
10926:
2955:
2160:
284:
174:
form. The state-space method is characterized by the algebraization of general
11014:
9585:
The more general form of a state-space model can be written as two functions.
12558:
12298:
12278:
12195:
11874:
11349:
11270:
11196:
11161:
11153:
11126:
10964:
10936:
9356:
5971:
Notice how the output also depends directly on the input. This is due to the
3120:
2944:
1413:
272:
175:
166:
is linear, time-invariant, and finite-dimensional, then the differential and
323:
12384:
12215:
11630:
11226:
11204:
10961:
for information about state space with discrete states in computer science.
6320:{\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+BK\mathbf {y} (t)}
2888:
263:
7125:{\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)}
6159:{\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)}
12480:
12251:
12160:
12155:
11777:
11755:
10958:
10952:
6845:{\displaystyle {\dot {\mathbf {x} }}(t)=\left(A+BK\right)\mathbf {x} (t)}
6641:
2780:
267:
183:
143:
11261:
1281:{\displaystyle {\dot {\mathbf {x} }}(t):={\frac {d}{dt}}\mathbf {x} (t)}
12374:
12333:
12328:
12241:
12150:
12058:
11970:
11950:
10010:
is the mass of the pendulum (pendulum rod's mass is assumed to be zero)
5525:{\displaystyle {\textbf {y}}(t)={\textbf {G}}(\infty ){\textbf {u}}(t)}
2884:
2536:
2532:
1293:
11341:
6630:{\displaystyle \mathbf {y} (t)=\left(I-DK\right)^{-1}C\mathbf {x} (t)}
12369:
12338:
12236:
12080:
12043:
11980:
11934:
11929:
11914:
11508:
Mathematical
Control Theory: Deterministic Finite Dimensional Systems
11374:. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107049994
11252:
9988:
is the angle of the pendulum with respect to the direction of gravity
308:
10984:
10076:
is the radius of the pendulum (to the center of gravity of the mass
2531:
Stability and natural response characteristics of a continuous-time
331:
The most general state-space representation of a linear system with
12271:
12103:
10812:
and so the equilibrium points of a pendulum are those that satisfy
9834:
6020:
A common method for feedback is to multiply the output by a matrix
2341:
312:
6395:{\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+DK\mathbf {y} (t)}
3107:
is the number of linearly independent rows in a matrix, and where
12394:
12231:
12185:
12108:
12008:
12003:
11955:
7197:{\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)}
7042:{\displaystyle \mathbf {u} (t)=-K\mathbf {y} (t)+\mathbf {r} (t)}
6231:{\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)}
4350:
provides a bridge between the state-space representation and its
11372:
Forecasting, Structural Time Series Models and the Kalman Filter
11015:
Katalin M. Hangos; József Bokor & Gábor
Szederkényi (2004).
9666:{\displaystyle \mathbf {\dot {x}} (t)=\mathbf {f} (t,x(t),u(t))}
9352:
that could move the cart into any other position in the system.
327:
Block diagram representation of the linear state-space equations
12409:
12389:
12261:
12053:
11292:"State-Space Models with Markov Switching and Gibbs-Sampling",
10488:
Instead, the state equation can be written in the general form
5532:. Together we then get a state-space realization with matrices
4187:{\displaystyle \mathbf {Y} (s)=\mathbf {G} (s)\mathbf {U} (s)}
118:
of a physical system specified as a set of input, output, and
12210:
12190:
12180:
12175:
12170:
12165:
12128:
11960:
11538:
Control System Design: An
Introduction to State-Space Methods
266:, for example, state-space models can be used to decompose a
11326:"A New Approach to Linear Filtering and Prediction Problems"
9187:. This means, that if initial state of the system is known (
7772:
One fairly common simplification to this system is removing
12200:
6936:
4197:
using the method of equating the coefficients which yields
4122:
11227:"New Indexes of Coincident and Leading Economic Indicators
10991:
Intelligent
Control Systems: An Introduction with Examples
11576:
On the applications of state-space models in econometrics
11041:
9745:{\displaystyle \mathbf {y} (t)=\mathbf {h} (t,x(t),u(t))}
8042:{\displaystyle m{\ddot {y}}(t)=u(t)-b{\dot {y}}(t)-ky(t)}
2981:
2943:
can similarly be used to determine whether the system is
2873:{\displaystyle \lambda (s)=|s\mathbf {I} -\mathbf {A} |.}
2775:
The denominator of the transfer function is equal to the
2565:
in factored form. It will then look something like this:
1296:
case, matrices will be time invariant. The time variable
255:
to encode all the information about a system. Unlike the
6719:{\displaystyle \left(A+BK\left(I-DK\right)^{-1}C\right)}
3129:
A continuous time-invariant linear state-space model is
10870:
10827:
10640:
10529:
9533:
9466:
9442:
9414:
9406:
9374:
9072:
9028:
8972:
8937:
8929:
8899:
8586:
8562:
8480:
8415:
8359:
8276:
6904:
This reduces the necessary eigendecomposition to just
6079:
are unrestricted the values can easily be negated for
5928:
5885:
5804:
5743:
5291:
5168:
5009:
4874:
4783:
4626:
3154:
3004:
11486:
Hinrichsen, Diederich; Pritchard, Anthony J. (2005).
11413:(1st ed.). Quantum Publishers, Inc. p. 172.
10821:
10783:
10497:
10432:
10363:
10310:
10177:
10109:
10082:
10062:
10040:
10018:
9996:
9965:
9845:
9802:
9761:
9680:
9594:
9368:
9338:
9318:
9298:
9260:
9222:
9193:
9173:
9153:
8893:
8851:
8772:
8703:
8665:
8539:
8270:
8243:
8221:
8199:
8168:
8128:
8090:
8061:
7954:
7888:
7785:
7640:
7451:
7417:
7320:
7216:
7139:
7058:
6984:
6955:
6910:
6859:
6779:
6658:
6560:
6445:
6411:
6334:
6250:
6173:
6092:
6030:
5977:
5860:
5711:
5560:
5470:
5381:
5266:
4977:
4851:
4594:
4384:
4325:
4294:
4206:
4138:
4066:
4019:
3846:
3762:
3728:
3588:
3490:
3456:
3355:
3265:
3142:
2992:
2916:
2825:
2789:
2574:
2545:
2452:
2354:
2268:
2173:
2069:
1970:
1889:
1808:
1709:
1607:
1526:
1442:
1398:
1378:
1350:
1322:
1302:
1222:
1165:
1134:
1103:
1046:
1015:
958:
927:
870:
839:
791:
760:
712:
681:
633:
602:
503:
400:
377:
357:
337:
293:
232:
212:
192:
27:
Mathematical model of a system in control engineering
8835:{\displaystyle {\dot {x}}_{2}(t)={\ddot {x}}_{1}(t)}
5702:
which yields the following controllable realization
2526:
11485:
7442:and substituting in the state equation results in
5544:determined by the strictly proper part, and matrix
824:{\displaystyle \mathbf {u} (t)\in \mathbb {R} ^{p}}
745:{\displaystyle \mathbf {y} (t)\in \mathbb {R} ^{q}}
666:{\displaystyle \mathbf {x} (t)\in \mathbb {R} ^{n}}
10896:
10804:
10758:
10476:
10416:
10347:
10290:
10162:
10088:
10068:
10046:
10024:
10002:
9980:
9945:
9817:
9788:
9744:
9665:
9566:
9344:
9324:
9304:
9284:
9246:
9208:
9179:
9159:
9136:
8868:
8834:
8756:
8687:
8645:
8524:
8249:
8227:
8205:
8183:
8152:
8114:
8076:
8041:
7925:
7873:
7761:
7625:
7434:
7400:
7305:
7196:
7124:
7041:
6970:
6925:
6893:
6844:
6770:. This would then result in the simpler equations
6718:
6629:
6545:
6436:and substituting in the state equation results in
6428:
6394:
6319:
6230:
6158:
6067:
5996:
5960:
5843:
5691:
5524:
5453:
5341:
5249:
4950:
4834:
4574:
4337:
4311:
4276:
4186:
4113:
4044:
4002:
3826:
3745:
3711:
3568:
3473:
3439:
3335:
3229:
3092:
2935:
2872:
2808:
2764:
2553:
2513:
2438:
2329:
2254:
2148:
2055:
1950:
1875:
1788:
1695:
1587:
1512:
1404:
1384:
1364:
1336:
1308:
1280:
1206:
1151:
1120:
1087:
1032:
999:
944:
911:
856:
823:
777:
744:
698:
665:
619:
582:
488:
391:state variables is written in the following form:
383:
363:
343:
299:
244:
218:
198:
11018:Analysis and Control of Nonlinear Process Systems
6734:can be made stable through appropriate choice of
785:is called the "input (or control) vector",
12556:
9833:A classic nonlinear system is a simple unforced
7926:{\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)}
6068:{\displaystyle \mathbf {u} (t)=K\mathbf {y} (t)}
11428:
6894:{\displaystyle \mathbf {y} (t)=\mathbf {x} (t)}
226:outputs, we would otherwise have to write down
11238:
10484:is the rotational acceleration of the pendulum
10477:{\displaystyle {\dot {x}}_{2}={\ddot {x}}_{1}}
6726:. This assumes that the closed-loop system is
11646:
11584:
11106:
10054:is coefficient of friction at the pivot point
6024:and setting this as the input to the system:
5551:Here is an example to clear things up a bit:
11660:
11555:Zadeh, Lotfi A.; Desoer, Charles A. (1979).
11388:(6th ed.). John Wiley & Sons, Inc.
6652:appropriately through eigendecomposition of
5359:
4123:multiple-input multiple-output (MIMO) system
4060:is defined as the ratio of output and input
4045:{\displaystyle \mathbf {x} (0)=\mathbf {0} }
11587:Time series analysis by state space methods
11554:
11225:James H. Stock & Mark W. Watson, 1989.
11109:Time series analysis by state space methods
11079:, vol. 2, Elsevier, pp. 415–525,
11074:
11653:
11639:
11174:
10424:is the rotational velocity of the pendulum
10417:{\displaystyle x_{2}(t)={\dot {x}}_{1}(t)}
10163:{\displaystyle {\dot {x}}_{1}(t)=x_{2}(t)}
8757:{\displaystyle x_{2}(t)={\dot {x}}_{1}(t)}
8646:{\displaystyle \mathbf {y} (t)=\left\left}
2809:{\displaystyle s\mathbf {I} -\mathbf {A} }
11535:
11452:(3rd ed.). Oxford University Press.
11260:
11107:Durbin, James; Koopman, Siem Jan (2012).
11059:
4357:
1372:). In the latter case, the time variable
1358:
1330:
864:is the "state (or system) matrix",
811:
732:
653:
80:Learn how and when to remove this message
11429:Antsaklis, P. J.; Michel, A. N. (2007).
11294:State-Space Models with Regime Switching
7936:
6940:
6937:Feedback with setpoint (reference) input
6011:
4054:single-input single-output (SISO) system
322:
43:This article includes a list of general
11589:. Oxford, UK: Oxford University Press.
11139:
9789:{\displaystyle f(\cdot ,\cdot ,\cdot )}
6016:Typical state-space model with feedback
5352:This state-space realization is called
4961:This state-space realization is called
14:
12557:
11504:
11466:
11408:
11323:
11142:IEEE Transactions on Automatic Control
5997:{\displaystyle {\textbf {G}}(\infty )}
11634:
10988:; R. Lakner & M. Gerzson (2001).
9332:are constants, then there is a force
8695:represents the position of the object
8261:The state equation would then become
4368:Any given transfer function which is
2887:) are the system transfer function's
706:is called the "output vector",
12491:Generative adversarial network (GAN)
11447:
11383:
11061:10.17586/2226-1494-2015-15-5-839-848
11042:Vasilyev A.S.; Ushakov A.V. (2015).
9580:
6730:or that the unstable eigenvalues of
3240:
2910:The zeros found in the numerator of
1152:{\displaystyle \mathbf {D} (\cdot )}
1121:{\displaystyle \mathbf {D} (\cdot )}
1033:{\displaystyle \mathbf {C} (\cdot )}
945:{\displaystyle \mathbf {B} (\cdot )}
857:{\displaystyle \mathbf {A} (\cdot )}
778:{\displaystyle \mathbf {u} (\cdot )}
699:{\displaystyle \mathbf {y} (\cdot )}
627:is called the "state vector",
620:{\displaystyle \mathbf {x} (\cdot )}
29:
10348:{\displaystyle x_{1}(t)=\theta (t)}
9828:
8213:is the viscous friction coefficient
6949:In addition to feedback, an input,
6004:constant in the transfer function.
5980:
5944:
5906:
5863:
5827:
5782:
5716:
5563:
5508:
5492:
5473:
5434:
5404:
5384:
5322:
5269:
5233:
5146:
4982:
4387:
2919:
2577:
24:
11420:
6640:The advantage of this is that the
5988:
5500:
5442:
5414:
5411:
5364:Transfer functions which are only
3111:is the number of state variables.
2965:
2883:The roots of this polynomial (the
278:
180:Kronecker vector-matrix structures
49:it lacks sufficient corresponding
25:
12591:
11606:
10032:is the gravitational acceleration
8842:is the acceleration of the object
7776:, which reduces the equations to
4013:Assuming zero initial conditions
2527:Example: continuous-time LTI case
1365:{\displaystyle t\in \mathbb {Z} }
1337:{\displaystyle t\in \mathbb {R} }
318:
178:, which makes it possible to use
12529:
12528:
12508:
11585:Durbin, J.; Koopman, S. (2001).
10604:
10502:
9699:
9682:
9620:
9601:
9598:
8853:
8620:
8616:
8595:
8591:
8541:
8509:
8445:
8420:
8318:
8284:
7910:
7890:
7858:
7838:
7790:
7746:
7694:
7642:
7610:
7536:
7456:
7419:
7411:solving the output equation for
7385:
7365:
7342:
7322:
7290:
7270:
7247:
7221:
7181:
7161:
7141:
7109:
7089:
7063:
7026:
7009:
6986:
6878:
6861:
6829:
6784:
6614:
6562:
6530:
6450:
6413:
6405:solving the output equation for
6379:
6356:
6336:
6304:
6281:
6255:
6215:
6195:
6175:
6143:
6123:
6097:
6052:
6032:
4932:
4853:
4819:
4762:
4599:
4296:
4270:
4262:
4244:
4236:
4225:
4208:
4171:
4157:
4140:
4038:
4021:
3984:
3979:
3959:
3954:
3936:
3928:
3905:
3887:
3879:
3865:
3848:
3808:
3803:
3786:
3781:
3764:
3730:
3693:
3688:
3670:
3662:
3639:
3621:
3613:
3590:
3553:
3548:
3531:
3514:
3506:
3498:
3458:
3421:
3416:
3399:
3394:
3377:
3360:
3320:
3315:
3298:
3293:
3270:
3194:
3188:
3172:
3167:
3158:
3114:
3069:
3052:
3039:
3028:
3020:
3015:
3008:
2936:{\displaystyle {\textbf {G}}(s)}
2858:
2850:
2802:
2794:
2547:
2498:
2493:
2476:
2471:
2454:
2423:
2418:
2401:
2396:
2379:
2359:
2314:
2309:
2292:
2287:
2270:
2239:
2234:
2217:
2212:
2195:
2178:
2133:
2119:
2102:
2088:
2071:
2040:
2026:
2009:
1995:
1972:
1935:
1930:
1913:
1908:
1891:
1860:
1855:
1838:
1833:
1810:
1800:Explicit discrete time-invariant
1773:
1759:
1742:
1728:
1711:
1680:
1666:
1649:
1635:
1612:
1572:
1567:
1550:
1545:
1528:
1497:
1492:
1475:
1470:
1447:
1418:
1265:
1227:
1176:
1136:
1105:
1057:
1017:
969:
929:
881:
841:
793:
762:
714:
683:
635:
604:
567:
553:
536:
522:
505:
473:
459:
442:
428:
405:
34:
11471:(3rd ed.). Prentice Hall.
11450:Linear System Theory and Design
11402:
11377:
11364:
11317:
11285:
11241:Journal of Applied Econometrics
8869:{\displaystyle \mathbf {y} (t)}
7435:{\displaystyle \mathbf {y} (t)}
6429:{\displaystyle \mathbf {y} (t)}
4319:is a matrix with the dimension
4312:{\displaystyle \mathbf {G} (s)}
3746:{\displaystyle \mathbf {X} (s)}
3474:{\displaystyle \mathbf {X} (s)}
12441:Recurrent neural network (RNN)
12431:Differentiable neural computer
11302:10.7551/mitpress/6444.003.0013
11232:
11219:
11168:
11133:
11100:
11068:
11035:
11008:
10978:
10967:for a statistical application.
10742:
10736:
10703:
10697:
10659:
10653:
10629:
10626:
10620:
10608:
10589:
10583:
10557:
10551:
10518:
10512:
10411:
10405:
10380:
10374:
10342:
10336:
10327:
10321:
10285:
10279:
10246:
10240:
10203:
10197:
10157:
10151:
10135:
10129:
9975:
9969:
9940:
9934:
9910:
9904:
9877:
9871:
9812:
9806:
9783:
9765:
9739:
9736:
9730:
9721:
9715:
9703:
9692:
9686:
9660:
9657:
9651:
9642:
9636:
9624:
9613:
9607:
9285:{\displaystyle {\ddot {y}}(t)}
9279:
9273:
9241:
9235:
9203:
9197:
8863:
8857:
8829:
8823:
8798:
8792:
8751:
8745:
8720:
8714:
8682:
8676:
8632:
8626:
8607:
8601:
8551:
8545:
8519:
8513:
8461:
8455:
8436:
8430:
8340:
8334:
8306:
8300:
8178:
8172:
8153:{\displaystyle {\ddot {y}}(t)}
8147:
8141:
8109:
8103:
8071:
8065:
8036:
8030:
8018:
8012:
7991:
7985:
7976:
7970:
7920:
7914:
7900:
7894:
7868:
7862:
7848:
7842:
7806:
7800:
7756:
7750:
7704:
7698:
7652:
7646:
7620:
7614:
7546:
7540:
7472:
7466:
7429:
7423:
7395:
7389:
7375:
7369:
7352:
7346:
7332:
7326:
7300:
7294:
7280:
7274:
7257:
7251:
7237:
7231:
7191:
7185:
7171:
7165:
7151:
7145:
7119:
7113:
7099:
7093:
7079:
7073:
7036:
7030:
7019:
7013:
6996:
6990:
6965:
6959:
6945:Output feedback with set point
6888:
6882:
6871:
6865:
6839:
6833:
6800:
6794:
6624:
6618:
6572:
6566:
6540:
6534:
6466:
6460:
6423:
6417:
6389:
6383:
6366:
6360:
6346:
6340:
6314:
6308:
6291:
6285:
6271:
6265:
6225:
6219:
6205:
6199:
6185:
6179:
6153:
6147:
6133:
6127:
6113:
6107:
6062:
6056:
6042:
6036:
5991:
5985:
5955:
5949:
5917:
5911:
5874:
5868:
5838:
5832:
5793:
5787:
5732:
5726:
5574:
5568:
5519:
5513:
5503:
5497:
5484:
5478:
5445:
5439:
5426:
5420:
5395:
5389:
5333:
5327:
5280:
5274:
5244:
5238:
5157:
5151:
4998:
4992:
4942:
4936:
4863:
4857:
4829:
4823:
4772:
4766:
4615:
4609:
4398:
4392:
4306:
4300:
4249:
4229:
4218:
4212:
4181:
4175:
4167:
4161:
4150:
4144:
4114:{\displaystyle G(s)=Y(s)/U(s)}
4108:
4102:
4091:
4085:
4076:
4070:
4031:
4025:
3994:
3988:
3972:
3969:
3963:
3941:
3921:
3915:
3909:
3892:
3872:
3869:
3858:
3852:
3818:
3812:
3796:
3790:
3774:
3768:
3740:
3734:
3703:
3697:
3675:
3655:
3649:
3643:
3626:
3606:
3600:
3594:
3563:
3557:
3541:
3535:
3524:
3518:
3510:
3491:
3468:
3462:
3431:
3425:
3409:
3403:
3387:
3381:
3370:
3364:
3330:
3324:
3308:
3302:
3286:
3280:
2930:
2924:
2863:
2842:
2835:
2829:
2753:
2734:
2731:
2712:
2709:
2690:
2687:
2668:
2663:
2644:
2641:
2622:
2619:
2600:
2588:
2582:
2508:
2502:
2486:
2480:
2464:
2458:
2433:
2427:
2411:
2405:
2389:
2383:
2369:
2363:
2324:
2318:
2302:
2296:
2280:
2274:
2249:
2243:
2227:
2221:
2205:
2199:
2188:
2182:
2143:
2137:
2129:
2123:
2112:
2106:
2098:
2092:
2081:
2075:
2050:
2044:
2036:
2030:
2019:
2013:
2005:
1999:
1988:
1976:
1962:Explicit discrete time-variant
1945:
1939:
1923:
1917:
1901:
1895:
1870:
1864:
1848:
1842:
1826:
1814:
1783:
1777:
1769:
1763:
1752:
1746:
1738:
1732:
1721:
1715:
1690:
1684:
1676:
1670:
1659:
1653:
1645:
1639:
1628:
1622:
1582:
1576:
1560:
1554:
1538:
1532:
1507:
1501:
1485:
1479:
1463:
1457:
1275:
1269:
1243:
1237:
1207:{\displaystyle \dim=q\times p}
1189:
1186:
1180:
1172:
1146:
1140:
1115:
1109:
1088:{\displaystyle \dim=q\times n}
1070:
1067:
1061:
1053:
1040:is the "output matrix",
1027:
1021:
1000:{\displaystyle \dim=n\times p}
982:
979:
973:
965:
939:
933:
912:{\displaystyle \dim=n\times n}
894:
891:
885:
877:
851:
845:
803:
797:
772:
766:
724:
718:
693:
687:
645:
639:
614:
608:
577:
571:
563:
557:
546:
540:
532:
526:
515:
509:
483:
477:
469:
463:
452:
446:
438:
432:
421:
415:
13:
1:
12486:Variational autoencoder (VAE)
12446:Long short-term memory (LSTM)
11713:Computational learning theory
11085:10.1016/bs.hesmac.2016.04.002
10972:
10100:The state equations are then
9247:{\displaystyle {\dot {y}}(t)}
8876:is the position of the object
8764:is the velocity of the object
8115:{\displaystyle {\dot {y}}(t)}
6746:For a strictly proper system
6648:can be controlled by setting
952:is the "input matrix",
98:configuration space (physics)
12466:Convolutional neural network
11625:nonlinear state-space models
11330:Journal of Basic Engineering
11324:Kalman, R. E. (1960-03-01).
10805:{\displaystyle {\dot {x}}=0}
10355:is the angle of the pendulum
9147:which has full rank for all
5548:determined by the constant.
2554:{\displaystyle \mathbf {A} }
7:
12461:Multilayer perceptron (MLP)
11536:Friedland, Bernard (2005).
11505:Sontag, Eduardo D. (1999).
11409:Brogan, William L. (1974).
11386:Control Systems Engineering
11111:. Oxford University Press.
10914:
6007:
4963:controllable canonical form
1392:is usually used instead of
1159:is the zero matrix),
10:
12596:
12537:Artificial neural networks
12451:Gated recurrent unit (GRU)
11677:Differentiable programming
11514:(2nd ed.). Springer.
11467:Khalil, Hassan K. (2001).
11370:Harvey, Andrew C. (1990).
11189:10.1162/089976603765202622
11077:Handbook of Macroeconomics
9981:{\displaystyle \theta (t)}
6741:
4361:
3118:
2969:
2339:
2158:
1960:
1798:
1597:
1432:
1420:
112:state-space representation
91:
18:State space representation
12504:
12418:
12362:
12291:
12224:
12096:
11996:
11989:
11943:
11907:
11870:Artificial neural network
11850:
11726:
11693:Automatic differentiation
11666:
11621:affine state-space models
11617:linear state-space models
10994:. Springer. p. 254.
8257:is the mass of the object
6978:, can be added such that
5360:Proper transfer functions
5354:observable canonical form
4338:{\displaystyle q\times p}
2777:characteristic polynomial
2165:continuous time-invariant
1434:Continuous time-invariant
245:{\displaystyle q\times p}
130:. Such variables, called
12565:Classical control theory
11698:Neuromorphic engineering
11661:Differentiable computing
11448:Chen, Chi-Tsong (1999).
11384:Nise, Norman S. (2010).
11154:10.1109/tac.1975.1100844
11021:. Springer. p. 25.
8688:{\displaystyle x_{1}(t)}
4348:Rosenbrock system matrix
4127:transfer function matrix
2950:The system may still be
1316:can be continuous (e.g.
12471:Residual neural network
11887:Artificial Intelligence
11431:A Linear Systems Primer
11296:, The MIT Press, 2017,
7943:Newton's laws of motion
3753:in the output equation
2346:discrete time-invariant
1599:Continuous time-variant
122:related by first-order
64:more precise citations.
10898:
10806:
10760:
10478:
10418:
10349:
10292:
10164:
10090:
10070:
10048:
10026:
10004:
9982:
9947:
9819:
9790:
9746:
9667:
9568:
9346:
9326:
9306:
9286:
9248:
9210:
9181:
9161:
9138:
8870:
8836:
8758:
8689:
8647:
8526:
8251:
8235:is the spring constant
8229:
8207:
8185:
8154:
8116:
8078:
8043:
7927:
7875:
7763:
7627:
7436:
7402:
7307:
7198:
7126:
7043:
6972:
6946:
6927:
6895:
6846:
6720:
6631:
6547:
6430:
6396:
6321:
6232:
6160:
6075:. Since the values of
6069:
6017:
5998:
5962:
5845:
5693:
5526:
5455:
5343:
5251:
4952:
4836:
4576:
4358:Canonical realizations
4339:
4313:
4278:
4188:
4115:
4046:
4004:
3828:
3747:
3713:
3570:
3475:
3450:Next, we simplify for
3441:
3337:
3231:
3094:
2937:
2874:
2810:
2766:
2555:
2515:
2440:
2331:
2256:
2150:
2057:
1952:
1877:
1790:
1697:
1589:
1514:
1406:
1386:
1366:
1338:
1310:
1282:
1208:
1153:
1122:
1089:
1034:
1001:
946:
913:
858:
825:
779:
746:
700:
667:
621:
584:
490:
385:
365:
345:
328:
301:
246:
220:
200:
124:differential equations
12570:Mathematical modeling
12426:Neural Turing machine
12014:Human image synthesis
11411:Modern Control Theory
10949:of state-space models
10899:
10807:
10777:of a system are when
10761:
10479:
10419:
10350:
10293:
10165:
10091:
10071:
10069:{\displaystyle \ell }
10049:
10027:
10005:
9983:
9948:
9820:
9791:
9747:
9668:
9569:
9347:
9327:
9307:
9287:
9249:
9211:
9182:
9162:
9139:
8871:
8837:
8759:
8690:
8648:
8527:
8252:
8230:
8208:
8186:
8155:
8117:
8079:
8044:
7937:Moving object example
7928:
7876:
7764:
7628:
7437:
7403:
7308:
7199:
7127:
7044:
6973:
6944:
6928:
6896:
6847:
6721:
6632:
6548:
6431:
6397:
6322:
6233:
6161:
6070:
6015:
5999:
5963:
5846:
5694:
5527:
5456:
5344:
5252:
4953:
4837:
4577:
4364:Realization (systems)
4340:
4314:
4279:
4189:
4116:
4047:
4005:
3829:
3748:
3714:
3571:
3476:
3442:
3338:
3232:
3095:
2938:
2897:asymptotically stable
2875:
2811:
2767:
2556:
2516:
2441:
2332:
2257:
2151:
2058:
1953:
1878:
1791:
1698:
1590:
1515:
1407:
1387:
1367:
1339:
1311:
1283:
1209:
1154:
1123:
1090:
1035:
1002:
947:
914:
859:
826:
780:
747:
701:
668:
622:
585:
491:
386:
366:
346:
326:
302:
247:
221:
201:
108:system identification
12575:Time domain analysis
12517:Computer programming
12496:Graph neural network
12071:Text-to-video models
12049:Text-to-image models
11897:Large language model
11882:Scientific computing
11688:Statistical manifold
11683:Information geometry
11557:Linear System Theory
10819:
10781:
10495:
10430:
10361:
10308:
10175:
10107:
10080:
10060:
10038:
10016:
9994:
9963:
9843:
9818:{\displaystyle u(t)}
9800:
9759:
9678:
9592:
9366:
9336:
9316:
9296:
9258:
9220:
9209:{\displaystyle y(t)}
9191:
9171:
9151:
8891:
8849:
8770:
8701:
8663:
8537:
8268:
8241:
8219:
8197:
8184:{\displaystyle u(t)}
8166:
8126:
8088:
8077:{\displaystyle y(t)}
8059:
7952:
7886:
7783:
7638:
7449:
7415:
7318:
7214:
7137:
7056:
6982:
6971:{\displaystyle r(t)}
6953:
6926:{\displaystyle A+BK}
6908:
6857:
6777:
6656:
6558:
6443:
6409:
6332:
6248:
6171:
6090:
6028:
5975:
5858:
5709:
5558:
5468:
5379:
5264:
4975:
4849:
4592:
4382:
4323:
4292:
4204:
4136:
4064:
4017:
3844:
3760:
3726:
3586:
3488:
3454:
3353:
3263:
3140:
2990:
2914:
2823:
2787:
2779:found by taking the
2572:
2543:
2450:
2352:
2266:
2171:
2067:
1968:
1887:
1806:
1707:
1605:
1524:
1440:
1396:
1376:
1348:
1344:) or discrete (e.g.
1320:
1300:
1220:
1163:
1132:
1101:
1044:
1013:
956:
925:
868:
837:
789:
758:
710:
679:
631:
600:
501:
398:
375:
355:
335:
291:
230:
210:
190:
128:difference equations
11863:In-context learning
11703:Pattern recognition
10922:Control engineering
8191:is an applied force
2952:input–output stable
168:algebraic equations
104:control engineering
94:quantum state space
12580:Time series models
12456:Echo state network
12344:JĂĽrgen Schmidhuber
12039:Facial recognition
12034:Speech recognition
11944:Software libraries
11559:. Krieger Pub Co.
11177:Neural Computation
10894:
10888:
10856:
10802:
10756:
10747:
10594:
10474:
10414:
10345:
10288:
10160:
10086:
10066:
10044:
10022:
10000:
9978:
9943:
9815:
9786:
9742:
9663:
9564:
9558:
9519:
9511:
9455:
9427:
9392:
9342:
9322:
9302:
9282:
9244:
9206:
9177:
9157:
9134:
9128:
9058:
9050:
9017:
8959:
8915:
8866:
8832:
8754:
8685:
8643:
8637:
8575:
8522:
8502:
8466:
8404:
8345:
8247:
8225:
8203:
8181:
8150:
8112:
8074:
8039:
7923:
7871:
7759:
7623:
7432:
7398:
7303:
7194:
7122:
7039:
6968:
6947:
6923:
6891:
6842:
6716:
6627:
6543:
6426:
6392:
6317:
6228:
6156:
6065:
6018:
5994:
5958:
5936:
5898:
5841:
5819:
5774:
5689:
5522:
5451:
5339:
5314:
5247:
5225:
5138:
4948:
4925:
4832:
4812:
4755:
4572:
4335:
4309:
4274:
4184:
4111:
4042:
4000:
3824:
3743:
3709:
3566:
3471:
3437:
3333:
3252:First, taking the
3227:
3212:
3090:
3075:
2933:
2905:Lyapunov stability
2870:
2806:
2762:
2551:
2511:
2436:
2327:
2252:
2146:
2053:
1948:
1873:
1786:
1693:
1585:
1510:
1402:
1382:
1362:
1334:
1306:
1278:
1204:
1149:
1118:
1085:
1030:
997:
942:
909:
854:
821:
775:
742:
696:
663:
617:
580:
486:
381:
361:
341:
329:
297:
253:Laplace transforms
242:
216:
196:
170:may be written in
116:mathematical model
12552:
12551:
12314:Stephen Grossberg
12287:
12286:
11596:978-0-19-852354-3
11566:978-0-88275-809-1
11497:978-3-540-44125-0
11469:Nonlinear Systems
11440:978-0-8176-4460-4
11395:978-0-470-54756-4
11342:10.1115/1.3662552
11311:978-0-262-27711-2
11118:978-0-19-964117-8
11094:978-0-444-59487-7
11028:978-1-85233-600-4
11001:978-1-4020-0134-5
10986:Katalin M. Hangos
10793:
10775:stationary points
10722:
10677:
10574:
10542:
10509:
10465:
10443:
10396:
10265:
10220:
10188:
10120:
10089:{\displaystyle m}
10047:{\displaystyle k}
10025:{\displaystyle g}
10003:{\displaystyle m}
9931:
9868:
9604:
9581:Nonlinear systems
9507:
9492:
9345:{\displaystyle u}
9325:{\displaystyle m}
9305:{\displaystyle b}
9270:
9232:
9180:{\displaystyle m}
9160:{\displaystyle b}
9124:
9102:
9088:
9046:
9013:
8998:
8955:
8814:
8783:
8736:
8498:
8400:
8385:
8325:
8291:
8250:{\displaystyle m}
8228:{\displaystyle k}
8206:{\displaystyle b}
8138:
8100:
8009:
7967:
7797:
7463:
7228:
7070:
6791:
6457:
6262:
6104:
6081:negative feedback
5982:
5946:
5908:
5865:
5829:
5784:
5723:
5718:
5681:
5636:
5565:
5510:
5494:
5475:
5436:
5406:
5386:
5324:
5271:
5235:
5148:
4989:
4984:
4606:
4567:
4389:
4352:transfer function
4058:transfer function
3722:Substituting for
3277:
3254:Laplace transform
3247:transfer function
3241:Transfer function
2921:
2901:marginally stable
2757:
2579:
2563:transfer function
2524:
2523:
1619:
1454:
1428:State-space model
1405:{\displaystyle t}
1385:{\displaystyle k}
1309:{\displaystyle t}
1262:
1234:
412:
384:{\displaystyle n}
364:{\displaystyle q}
344:{\displaystyle p}
300:{\displaystyle n}
219:{\displaystyle q}
199:{\displaystyle p}
90:
89:
82:
16:(Redirected from
12587:
12542:Machine learning
12532:
12531:
12512:
12267:Action selection
12257:Self-driving car
12064:Stable Diffusion
12029:Speech synthesis
11994:
11993:
11858:Machine learning
11734:Gradient descent
11655:
11648:
11641:
11632:
11631:
11613:Wolfram language
11600:
11570:
11551:
11532:
11530:
11528:
11513:
11501:
11482:
11463:
11444:
11415:
11414:
11406:
11400:
11399:
11381:
11375:
11368:
11362:
11361:
11321:
11315:
11314:
11289:
11283:
11282:
11264:
11253:10.1002/jae.2306
11236:
11230:
11223:
11217:
11216:
11172:
11166:
11165:
11137:
11131:
11130:
11104:
11098:
11097:
11072:
11066:
11065:
11063:
11039:
11033:
11032:
11012:
11006:
11005:
10982:
10903:
10901:
10900:
10895:
10893:
10892:
10861:
10860:
10853:
10852:
10839:
10838:
10811:
10809:
10808:
10803:
10795:
10794:
10786:
10765:
10763:
10762:
10757:
10752:
10751:
10735:
10734:
10733:
10723:
10721:
10710:
10696:
10695:
10694:
10678:
10670:
10652:
10651:
10607:
10599:
10598:
10582:
10581:
10576:
10575:
10567:
10550:
10549:
10544:
10543:
10535:
10511:
10510:
10505:
10500:
10483:
10481:
10480:
10475:
10473:
10472:
10467:
10466:
10458:
10451:
10450:
10445:
10444:
10436:
10423:
10421:
10420:
10415:
10404:
10403:
10398:
10397:
10389:
10373:
10372:
10354:
10352:
10351:
10346:
10320:
10319:
10297:
10295:
10294:
10289:
10278:
10277:
10276:
10266:
10264:
10253:
10239:
10238:
10237:
10221:
10213:
10196:
10195:
10190:
10189:
10181:
10169:
10167:
10166:
10161:
10150:
10149:
10128:
10127:
10122:
10121:
10113:
10095:
10093:
10092:
10087:
10075:
10073:
10072:
10067:
10053:
10051:
10050:
10045:
10031:
10029:
10028:
10023:
10009:
10007:
10006:
10001:
9987:
9985:
9984:
9979:
9952:
9950:
9949:
9944:
9933:
9932:
9924:
9870:
9869:
9861:
9858:
9857:
9829:Pendulum example
9824:
9822:
9821:
9816:
9795:
9793:
9792:
9787:
9751:
9749:
9748:
9743:
9702:
9685:
9672:
9670:
9669:
9664:
9623:
9606:
9605:
9597:
9573:
9571:
9570:
9565:
9563:
9562:
9524:
9523:
9516:
9515:
9508:
9500:
9493:
9485:
9460:
9459:
9432:
9431:
9397:
9396:
9351:
9349:
9348:
9343:
9331:
9329:
9328:
9323:
9311:
9309:
9308:
9303:
9291:
9289:
9288:
9283:
9272:
9271:
9263:
9253:
9251:
9250:
9245:
9234:
9233:
9225:
9215:
9213:
9212:
9207:
9186:
9184:
9183:
9178:
9166:
9164:
9163:
9158:
9143:
9141:
9140:
9135:
9133:
9132:
9125:
9123:
9122:
9110:
9103:
9095:
9089:
9081:
9063:
9062:
9055:
9054:
9047:
9039:
9022:
9021:
9014:
9006:
8999:
8991:
8964:
8963:
8956:
8948:
8920:
8919:
8875:
8873:
8872:
8867:
8856:
8841:
8839:
8838:
8833:
8822:
8821:
8816:
8815:
8807:
8791:
8790:
8785:
8784:
8776:
8763:
8761:
8760:
8755:
8744:
8743:
8738:
8737:
8729:
8713:
8712:
8694:
8692:
8691:
8686:
8675:
8674:
8652:
8650:
8649:
8644:
8642:
8638:
8625:
8624:
8623:
8600:
8599:
8598:
8580:
8576:
8544:
8531:
8529:
8528:
8523:
8512:
8507:
8506:
8499:
8491:
8471:
8470:
8454:
8453:
8448:
8429:
8428:
8423:
8409:
8408:
8401:
8393:
8386:
8378:
8350:
8349:
8333:
8332:
8327:
8326:
8321:
8316:
8299:
8298:
8293:
8292:
8287:
8282:
8256:
8254:
8253:
8248:
8234:
8232:
8231:
8226:
8212:
8210:
8209:
8204:
8190:
8188:
8187:
8182:
8159:
8157:
8156:
8151:
8140:
8139:
8131:
8121:
8119:
8118:
8113:
8102:
8101:
8093:
8083:
8081:
8080:
8075:
8048:
8046:
8045:
8040:
8011:
8010:
8002:
7969:
7968:
7960:
7932:
7930:
7929:
7924:
7913:
7893:
7880:
7878:
7877:
7872:
7861:
7841:
7836:
7832:
7799:
7798:
7793:
7788:
7768:
7766:
7765:
7760:
7749:
7741:
7740:
7732:
7728:
7697:
7689:
7688:
7680:
7676:
7645:
7632:
7630:
7629:
7624:
7613:
7608:
7604:
7600:
7599:
7591:
7587:
7539:
7534:
7530:
7526:
7525:
7517:
7513:
7465:
7464:
7459:
7454:
7441:
7439:
7438:
7433:
7422:
7407:
7405:
7404:
7399:
7388:
7368:
7345:
7325:
7312:
7310:
7309:
7304:
7293:
7273:
7250:
7230:
7229:
7224:
7219:
7203:
7201:
7200:
7195:
7184:
7164:
7144:
7131:
7129:
7128:
7123:
7112:
7092:
7072:
7071:
7066:
7061:
7048:
7046:
7045:
7040:
7029:
7012:
6989:
6977:
6975:
6974:
6969:
6932:
6930:
6929:
6924:
6900:
6898:
6897:
6892:
6881:
6864:
6851:
6849:
6848:
6843:
6832:
6827:
6823:
6793:
6792:
6787:
6782:
6725:
6723:
6722:
6717:
6715:
6711:
6707:
6706:
6698:
6694:
6636:
6634:
6633:
6628:
6617:
6609:
6608:
6600:
6596:
6565:
6552:
6550:
6549:
6544:
6533:
6528:
6524:
6520:
6519:
6511:
6507:
6459:
6458:
6453:
6448:
6435:
6433:
6432:
6427:
6416:
6401:
6399:
6398:
6393:
6382:
6359:
6339:
6326:
6324:
6323:
6318:
6307:
6284:
6264:
6263:
6258:
6253:
6237:
6235:
6234:
6229:
6218:
6198:
6178:
6165:
6163:
6162:
6157:
6146:
6126:
6106:
6105:
6100:
6095:
6074:
6072:
6071:
6066:
6055:
6035:
6003:
6001:
6000:
5995:
5984:
5983:
5967:
5965:
5964:
5959:
5948:
5947:
5941:
5940:
5910:
5909:
5903:
5902:
5867:
5866:
5850:
5848:
5847:
5842:
5831:
5830:
5824:
5823:
5786:
5785:
5779:
5778:
5725:
5724:
5719:
5714:
5698:
5696:
5695:
5690:
5682:
5680:
5664:
5663:
5653:
5642:
5637:
5635:
5619:
5618:
5608:
5592:
5591:
5581:
5567:
5566:
5531:
5529:
5528:
5523:
5512:
5511:
5496:
5495:
5477:
5476:
5460:
5458:
5457:
5452:
5438:
5437:
5419:
5418:
5417:
5408:
5407:
5388:
5387:
5348:
5346:
5345:
5340:
5326:
5325:
5319:
5318:
5273:
5272:
5256:
5254:
5253:
5248:
5237:
5236:
5230:
5229:
5222:
5221:
5208:
5207:
5194:
5193:
5180:
5179:
5150:
5149:
5143:
5142:
5135:
5134:
5103:
5102:
5071:
5070:
5039:
5038:
4991:
4990:
4985:
4980:
4957:
4955:
4954:
4949:
4935:
4930:
4929:
4922:
4921:
4910:
4909:
4898:
4897:
4886:
4885:
4856:
4841:
4839:
4838:
4833:
4822:
4817:
4816:
4765:
4760:
4759:
4752:
4751:
4737:
4736:
4722:
4721:
4707:
4706:
4608:
4607:
4602:
4597:
4581:
4579:
4578:
4573:
4568:
4566:
4565:
4564:
4549:
4548:
4536:
4535:
4526:
4525:
4513:
4512:
4503:
4502:
4490:
4489:
4479:
4478:
4477:
4462:
4461:
4449:
4448:
4439:
4438:
4426:
4425:
4416:
4415:
4405:
4391:
4390:
4344:
4342:
4341:
4336:
4318:
4316:
4315:
4310:
4299:
4283:
4281:
4280:
4275:
4273:
4265:
4260:
4259:
4247:
4239:
4228:
4211:
4193:
4191:
4190:
4185:
4174:
4160:
4143:
4129:is derived from
4120:
4118:
4117:
4112:
4098:
4051:
4049:
4048:
4043:
4041:
4024:
4009:
4007:
4006:
4001:
3987:
3982:
3962:
3957:
3952:
3951:
3939:
3931:
3908:
3903:
3902:
3890:
3882:
3868:
3851:
3833:
3831:
3830:
3825:
3811:
3806:
3789:
3784:
3767:
3752:
3750:
3749:
3744:
3733:
3718:
3716:
3715:
3710:
3696:
3691:
3686:
3685:
3673:
3665:
3642:
3637:
3636:
3624:
3616:
3593:
3575:
3573:
3572:
3567:
3556:
3551:
3534:
3517:
3509:
3501:
3480:
3478:
3477:
3472:
3461:
3446:
3444:
3443:
3438:
3424:
3419:
3402:
3397:
3380:
3363:
3342:
3340:
3339:
3334:
3323:
3318:
3301:
3296:
3279:
3278:
3273:
3268:
3236:
3234:
3233:
3228:
3217:
3216:
3209:
3208:
3197:
3191:
3175:
3170:
3161:
3099:
3097:
3096:
3091:
3080:
3079:
3072:
3067:
3066:
3055:
3042:
3037:
3036:
3031:
3023:
3018:
3011:
2942:
2940:
2939:
2934:
2923:
2922:
2879:
2877:
2876:
2871:
2866:
2861:
2853:
2845:
2815:
2813:
2812:
2807:
2805:
2797:
2771:
2769:
2768:
2763:
2758:
2756:
2752:
2751:
2730:
2729:
2708:
2707:
2686:
2685:
2666:
2662:
2661:
2640:
2639:
2618:
2617:
2598:
2581:
2580:
2560:
2558:
2557:
2552:
2550:
2520:
2518:
2517:
2512:
2501:
2496:
2479:
2474:
2457:
2445:
2443:
2442:
2437:
2426:
2421:
2404:
2399:
2382:
2362:
2336:
2334:
2333:
2328:
2317:
2312:
2295:
2290:
2273:
2261:
2259:
2258:
2253:
2242:
2237:
2220:
2215:
2198:
2181:
2155:
2153:
2152:
2147:
2136:
2122:
2105:
2091:
2074:
2062:
2060:
2059:
2054:
2043:
2029:
2012:
1998:
1975:
1957:
1955:
1954:
1949:
1938:
1933:
1916:
1911:
1894:
1882:
1880:
1879:
1874:
1863:
1858:
1841:
1836:
1813:
1795:
1793:
1792:
1787:
1776:
1762:
1745:
1731:
1714:
1702:
1700:
1699:
1694:
1683:
1669:
1652:
1638:
1621:
1620:
1615:
1610:
1594:
1592:
1591:
1586:
1575:
1570:
1553:
1548:
1531:
1519:
1517:
1516:
1511:
1500:
1495:
1478:
1473:
1456:
1455:
1450:
1445:
1419:
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1371:
1369:
1368:
1363:
1361:
1343:
1341:
1340:
1335:
1333:
1315:
1313:
1312:
1307:
1287:
1285:
1284:
1279:
1268:
1263:
1261:
1250:
1236:
1235:
1230:
1225:
1213:
1211:
1210:
1205:
1179:
1158:
1156:
1155:
1150:
1139:
1127:
1125:
1124:
1119:
1108:
1094:
1092:
1091:
1086:
1060:
1039:
1037:
1036:
1031:
1020:
1006:
1004:
1003:
998:
972:
951:
949:
948:
943:
932:
918:
916:
915:
910:
884:
863:
861:
860:
855:
844:
830:
828:
827:
822:
820:
819:
814:
796:
784:
782:
781:
776:
765:
751:
749:
748:
743:
741:
740:
735:
717:
705:
703:
702:
697:
686:
672:
670:
669:
664:
662:
661:
656:
638:
626:
624:
623:
618:
607:
589:
587:
586:
581:
570:
556:
539:
525:
508:
495:
493:
492:
487:
476:
462:
445:
431:
414:
413:
408:
403:
390:
388:
387:
382:
370:
368:
367:
362:
350:
348:
347:
342:
306:
304:
303:
298:
257:frequency domain
251:
249:
248:
243:
225:
223:
222:
217:
205:
203:
202:
197:
164:dynamical system
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
12595:
12594:
12590:
12589:
12588:
12586:
12585:
12584:
12555:
12554:
12553:
12548:
12500:
12414:
12380:Google DeepMind
12358:
12324:Geoffrey Hinton
12283:
12220:
12146:Project Debater
12092:
11990:Implementations
11985:
11939:
11903:
11846:
11788:Backpropagation
11722:
11708:Tensor calculus
11662:
11659:
11609:
11604:
11603:
11597:
11573:
11567:
11548:
11526:
11524:
11522:
11511:
11498:
11479:
11460:
11441:
11423:
11421:Further reading
11418:
11407:
11403:
11396:
11382:
11378:
11369:
11365:
11322:
11318:
11312:
11291:
11290:
11286:
11237:
11233:
11224:
11220:
11173:
11169:
11138:
11134:
11119:
11105:
11101:
11095:
11073:
11069:
11040:
11036:
11029:
11013:
11009:
11002:
10983:
10979:
10975:
10970:
10942:Controllability
10917:
10887:
10886:
10880:
10879:
10866:
10865:
10855:
10854:
10848:
10844:
10841:
10840:
10834:
10830:
10823:
10822:
10820:
10817:
10816:
10785:
10784:
10782:
10779:
10778:
10746:
10745:
10729:
10725:
10724:
10714:
10709:
10690:
10686:
10685:
10669:
10663:
10662:
10647:
10643:
10636:
10635:
10603:
10593:
10592:
10577:
10566:
10565:
10564:
10561:
10560:
10545:
10534:
10533:
10532:
10525:
10524:
10501:
10499:
10498:
10496:
10493:
10492:
10468:
10457:
10456:
10455:
10446:
10435:
10434:
10433:
10431:
10428:
10427:
10399:
10388:
10387:
10386:
10368:
10364:
10362:
10359:
10358:
10315:
10311:
10309:
10306:
10305:
10272:
10268:
10267:
10257:
10252:
10233:
10229:
10228:
10212:
10191:
10180:
10179:
10178:
10176:
10173:
10172:
10145:
10141:
10123:
10112:
10111:
10110:
10108:
10105:
10104:
10081:
10078:
10077:
10061:
10058:
10057:
10039:
10036:
10035:
10017:
10014:
10013:
9995:
9992:
9991:
9964:
9961:
9960:
9923:
9922:
9860:
9859:
9853:
9849:
9844:
9841:
9840:
9831:
9801:
9798:
9797:
9760:
9757:
9756:
9698:
9681:
9679:
9676:
9675:
9619:
9596:
9595:
9593:
9590:
9589:
9583:
9557:
9556:
9551:
9545:
9544:
9539:
9529:
9528:
9518:
9517:
9510:
9509:
9499:
9494:
9484:
9478:
9477:
9472:
9462:
9461:
9454:
9453:
9448:
9438:
9437:
9434:
9433:
9426:
9425:
9420:
9410:
9409:
9402:
9401:
9391:
9390:
9381:
9380:
9370:
9369:
9367:
9364:
9363:
9337:
9334:
9333:
9317:
9314:
9313:
9297:
9294:
9293:
9262:
9261:
9259:
9256:
9255:
9224:
9223:
9221:
9218:
9217:
9192:
9189:
9188:
9172:
9169:
9168:
9152:
9149:
9148:
9127:
9126:
9118:
9114:
9109:
9104:
9094:
9091:
9090:
9080:
9078:
9068:
9067:
9057:
9056:
9049:
9048:
9038:
9035:
9034:
9024:
9023:
9016:
9015:
9005:
9000:
8990:
8984:
8983:
8978:
8968:
8967:
8965:
8958:
8957:
8947:
8944:
8943:
8933:
8932:
8925:
8924:
8914:
8913:
8905:
8895:
8894:
8892:
8889:
8888:
8882:controllability
8852:
8850:
8847:
8846:
8817:
8806:
8805:
8804:
8786:
8775:
8774:
8773:
8771:
8768:
8767:
8739:
8728:
8727:
8726:
8708:
8704:
8702:
8699:
8698:
8670:
8666:
8664:
8661:
8660:
8636:
8635:
8619:
8615:
8614:
8611:
8610:
8594:
8590:
8589:
8585:
8581:
8574:
8573:
8568:
8561:
8557:
8540:
8538:
8535:
8534:
8508:
8501:
8500:
8490:
8487:
8486:
8476:
8475:
8465:
8464:
8449:
8444:
8443:
8440:
8439:
8424:
8419:
8418:
8411:
8410:
8403:
8402:
8392:
8387:
8377:
8371:
8370:
8365:
8355:
8354:
8344:
8343:
8328:
8317:
8315:
8314:
8313:
8310:
8309:
8294:
8283:
8281:
8280:
8279:
8272:
8271:
8269:
8266:
8265:
8242:
8239:
8238:
8220:
8217:
8216:
8198:
8195:
8194:
8167:
8164:
8163:
8160:is acceleration
8130:
8129:
8127:
8124:
8123:
8092:
8091:
8089:
8086:
8085:
8060:
8057:
8056:
8001:
8000:
7959:
7958:
7953:
7950:
7949:
7939:
7909:
7889:
7887:
7884:
7883:
7857:
7837:
7816:
7812:
7789:
7787:
7786:
7784:
7781:
7780:
7745:
7733:
7715:
7711:
7710:
7693:
7681:
7663:
7659:
7658:
7641:
7639:
7636:
7635:
7609:
7592:
7574:
7570:
7569:
7559:
7555:
7535:
7518:
7500:
7496:
7495:
7482:
7478:
7455:
7453:
7452:
7450:
7447:
7446:
7418:
7416:
7413:
7412:
7384:
7364:
7341:
7321:
7319:
7316:
7315:
7289:
7269:
7246:
7220:
7218:
7217:
7215:
7212:
7211:
7180:
7160:
7140:
7138:
7135:
7134:
7108:
7088:
7062:
7060:
7059:
7057:
7054:
7053:
7025:
7008:
6985:
6983:
6980:
6979:
6954:
6951:
6950:
6939:
6909:
6906:
6905:
6877:
6860:
6858:
6855:
6854:
6828:
6810:
6806:
6783:
6781:
6780:
6778:
6775:
6774:
6768:Identity matrix
6758:, which yields
6744:
6699:
6681:
6677:
6676:
6663:
6659:
6657:
6654:
6653:
6613:
6601:
6583:
6579:
6578:
6561:
6559:
6556:
6555:
6529:
6512:
6494:
6490:
6489:
6476:
6472:
6449:
6447:
6446:
6444:
6441:
6440:
6412:
6410:
6407:
6406:
6378:
6355:
6335:
6333:
6330:
6329:
6303:
6280:
6254:
6252:
6251:
6249:
6246:
6245:
6214:
6194:
6174:
6172:
6169:
6168:
6142:
6122:
6096:
6094:
6093:
6091:
6088:
6087:
6051:
6031:
6029:
6026:
6025:
6010:
5979:
5978:
5976:
5973:
5972:
5943:
5942:
5935:
5934:
5924:
5923:
5905:
5904:
5897:
5896:
5891:
5881:
5880:
5862:
5861:
5859:
5856:
5855:
5826:
5825:
5818:
5817:
5811:
5810:
5800:
5799:
5781:
5780:
5773:
5772:
5767:
5761:
5760:
5752:
5739:
5738:
5715:
5713:
5712:
5710:
5707:
5706:
5659:
5655:
5654:
5643:
5641:
5614:
5610:
5609:
5587:
5583:
5582:
5580:
5562:
5561:
5559:
5556:
5555:
5507:
5506:
5491:
5490:
5472:
5471:
5469:
5466:
5465:
5433:
5432:
5410:
5409:
5403:
5402:
5401:
5383:
5382:
5380:
5377:
5376:
5370:strictly proper
5362:
5321:
5320:
5313:
5312:
5307:
5302:
5297:
5287:
5286:
5268:
5267:
5265:
5262:
5261:
5232:
5231:
5224:
5223:
5217:
5213:
5210:
5209:
5203:
5199:
5196:
5195:
5189:
5185:
5182:
5181:
5175:
5171:
5164:
5163:
5145:
5144:
5137:
5136:
5130:
5126:
5121:
5116:
5111:
5105:
5104:
5098:
5094:
5089:
5084:
5079:
5073:
5072:
5066:
5062:
5057:
5052:
5047:
5041:
5040:
5034:
5030:
5025:
5020:
5015:
5005:
5004:
4981:
4979:
4978:
4976:
4973:
4972:
4931:
4924:
4923:
4917:
4913:
4911:
4905:
4901:
4899:
4893:
4889:
4887:
4881:
4877:
4870:
4869:
4852:
4850:
4847:
4846:
4818:
4811:
4810:
4804:
4803:
4797:
4796:
4790:
4789:
4779:
4778:
4761:
4754:
4753:
4747:
4743:
4738:
4732:
4728:
4723:
4717:
4713:
4708:
4702:
4698:
4692:
4691:
4686:
4681:
4676:
4670:
4669:
4664:
4659:
4654:
4648:
4647:
4642:
4637:
4632:
4622:
4621:
4598:
4596:
4595:
4593:
4590:
4589:
4560:
4556:
4544:
4540:
4531:
4527:
4521:
4517:
4508:
4504:
4498:
4494:
4485:
4481:
4480:
4473:
4469:
4457:
4453:
4444:
4440:
4434:
4430:
4421:
4417:
4411:
4407:
4406:
4404:
4386:
4385:
4383:
4380:
4379:
4370:strictly proper
4366:
4360:
4324:
4321:
4320:
4295:
4293:
4290:
4289:
4269:
4261:
4252:
4248:
4243:
4235:
4224:
4207:
4205:
4202:
4201:
4170:
4156:
4139:
4137:
4134:
4133:
4094:
4065:
4062:
4061:
4037:
4020:
4018:
4015:
4014:
3983:
3978:
3958:
3953:
3944:
3940:
3935:
3927:
3904:
3895:
3891:
3886:
3878:
3864:
3847:
3845:
3842:
3841:
3807:
3802:
3785:
3780:
3763:
3761:
3758:
3757:
3729:
3727:
3724:
3723:
3692:
3687:
3678:
3674:
3669:
3661:
3638:
3629:
3625:
3620:
3612:
3589:
3587:
3584:
3583:
3552:
3547:
3530:
3513:
3505:
3497:
3489:
3486:
3485:
3457:
3455:
3452:
3451:
3420:
3415:
3398:
3393:
3376:
3359:
3354:
3351:
3350:
3319:
3314:
3297:
3292:
3269:
3267:
3266:
3264:
3261:
3260:
3243:
3211:
3210:
3198:
3193:
3192:
3187:
3184:
3183:
3177:
3176:
3171:
3166:
3163:
3162:
3157:
3150:
3149:
3141:
3138:
3137:
3133:if and only if
3123:
3117:
3074:
3073:
3068:
3056:
3051:
3050:
3048:
3043:
3038:
3032:
3027:
3026:
3024:
3019:
3014:
3012:
3007:
3000:
2999:
2991:
2988:
2987:
2974:
2972:Controllability
2968:
2966:Controllability
2918:
2917:
2915:
2912:
2911:
2862:
2857:
2849:
2841:
2824:
2821:
2820:
2801:
2793:
2788:
2785:
2784:
2747:
2743:
2725:
2721:
2703:
2699:
2681:
2677:
2667:
2657:
2653:
2635:
2631:
2613:
2609:
2599:
2597:
2576:
2575:
2573:
2570:
2569:
2546:
2544:
2541:
2540:
2529:
2497:
2492:
2475:
2470:
2453:
2451:
2448:
2447:
2446:
2422:
2417:
2400:
2395:
2378:
2358:
2353:
2350:
2349:
2345:
2313:
2308:
2291:
2286:
2269:
2267:
2264:
2263:
2262:
2238:
2233:
2216:
2211:
2194:
2177:
2172:
2169:
2168:
2164:
2132:
2118:
2101:
2087:
2070:
2068:
2065:
2064:
2063:
2039:
2025:
2008:
1994:
1971:
1969:
1966:
1965:
1934:
1929:
1912:
1907:
1890:
1888:
1885:
1884:
1883:
1859:
1854:
1837:
1832:
1809:
1807:
1804:
1803:
1772:
1758:
1741:
1727:
1710:
1708:
1705:
1704:
1703:
1679:
1665:
1648:
1634:
1611:
1609:
1608:
1606:
1603:
1602:
1571:
1566:
1549:
1544:
1527:
1525:
1522:
1521:
1520:
1496:
1491:
1474:
1469:
1446:
1444:
1443:
1441:
1438:
1437:
1397:
1394:
1393:
1377:
1374:
1373:
1357:
1349:
1346:
1345:
1329:
1321:
1318:
1317:
1301:
1298:
1297:
1264:
1254:
1249:
1226:
1224:
1223:
1221:
1218:
1217:
1175:
1164:
1161:
1160:
1135:
1133:
1130:
1129:
1104:
1102:
1099:
1098:
1056:
1045:
1042:
1041:
1016:
1014:
1011:
1010:
968:
957:
954:
953:
928:
926:
923:
922:
880:
869:
866:
865:
840:
838:
835:
834:
815:
810:
809:
792:
790:
787:
786:
761:
759:
756:
755:
736:
731:
730:
713:
711:
708:
707:
682:
680:
677:
676:
657:
652:
651:
634:
632:
629:
628:
603:
601:
598:
597:
566:
552:
535:
521:
504:
502:
499:
498:
472:
458:
441:
427:
404:
402:
401:
399:
396:
395:
376:
373:
372:
356:
353:
352:
336:
333:
332:
321:
292:
289:
288:
285:state variables
281:
279:State variables
231:
228:
227:
211:
208:
207:
191:
188:
187:
149:geometric space
132:state variables
100:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
12593:
12583:
12582:
12577:
12572:
12567:
12550:
12549:
12547:
12546:
12545:
12544:
12539:
12526:
12525:
12524:
12519:
12505:
12502:
12501:
12499:
12498:
12493:
12488:
12483:
12478:
12473:
12468:
12463:
12458:
12453:
12448:
12443:
12438:
12433:
12428:
12422:
12420:
12416:
12415:
12413:
12412:
12407:
12402:
12397:
12392:
12387:
12382:
12377:
12372:
12366:
12364:
12360:
12359:
12357:
12356:
12354:Ilya Sutskever
12351:
12346:
12341:
12336:
12331:
12326:
12321:
12319:Demis Hassabis
12316:
12311:
12309:Ian Goodfellow
12306:
12301:
12295:
12293:
12289:
12288:
12285:
12284:
12282:
12281:
12276:
12275:
12274:
12264:
12259:
12254:
12249:
12244:
12239:
12234:
12228:
12226:
12222:
12221:
12219:
12218:
12213:
12208:
12203:
12198:
12193:
12188:
12183:
12178:
12173:
12168:
12163:
12158:
12153:
12148:
12143:
12138:
12137:
12136:
12126:
12121:
12116:
12111:
12106:
12100:
12098:
12094:
12093:
12091:
12090:
12085:
12084:
12083:
12078:
12068:
12067:
12066:
12061:
12056:
12046:
12041:
12036:
12031:
12026:
12021:
12016:
12011:
12006:
12000:
11998:
11991:
11987:
11986:
11984:
11983:
11978:
11973:
11968:
11963:
11958:
11953:
11947:
11945:
11941:
11940:
11938:
11937:
11932:
11927:
11922:
11917:
11911:
11909:
11905:
11904:
11902:
11901:
11900:
11899:
11892:Language model
11889:
11884:
11879:
11878:
11877:
11867:
11866:
11865:
11854:
11852:
11848:
11847:
11845:
11844:
11842:Autoregression
11839:
11834:
11833:
11832:
11822:
11820:Regularization
11817:
11816:
11815:
11810:
11805:
11795:
11790:
11785:
11783:Loss functions
11780:
11775:
11770:
11765:
11760:
11759:
11758:
11748:
11743:
11742:
11741:
11730:
11728:
11724:
11723:
11721:
11720:
11718:Inductive bias
11715:
11710:
11705:
11700:
11695:
11690:
11685:
11680:
11672:
11670:
11664:
11663:
11658:
11657:
11650:
11643:
11635:
11629:
11628:
11615:functions for
11608:
11607:External links
11605:
11602:
11601:
11595:
11581:
11580:
11579:
11577:
11572:
11571:
11565:
11552:
11546:
11533:
11520:
11502:
11496:
11483:
11477:
11464:
11458:
11445:
11439:
11433:. Birkhauser.
11425:
11424:
11422:
11419:
11417:
11416:
11401:
11394:
11376:
11363:
11316:
11310:
11284:
11247:(1): 133–160.
11231:
11218:
11183:(5): 965–991.
11167:
11132:
11117:
11099:
11093:
11067:
11054:(5): 839–848.
11034:
11027:
11007:
11000:
10976:
10974:
10971:
10969:
10968:
10962:
10956:
10950:
10947:Discretization
10944:
10939:
10934:
10932:State observer
10929:
10927:Control theory
10924:
10918:
10916:
10913:
10905:
10904:
10891:
10885:
10882:
10881:
10878:
10875:
10872:
10871:
10869:
10864:
10859:
10851:
10847:
10843:
10842:
10837:
10833:
10829:
10828:
10826:
10801:
10798:
10792:
10789:
10767:
10766:
10755:
10750:
10744:
10741:
10738:
10732:
10728:
10720:
10717:
10713:
10708:
10705:
10702:
10699:
10693:
10689:
10684:
10681:
10676:
10673:
10668:
10665:
10664:
10661:
10658:
10655:
10650:
10646:
10642:
10641:
10639:
10634:
10631:
10628:
10625:
10622:
10619:
10616:
10613:
10610:
10606:
10602:
10597:
10591:
10588:
10585:
10580:
10573:
10570:
10563:
10562:
10559:
10556:
10553:
10548:
10541:
10538:
10531:
10530:
10528:
10523:
10520:
10517:
10514:
10508:
10504:
10486:
10485:
10471:
10464:
10461:
10454:
10449:
10442:
10439:
10425:
10413:
10410:
10407:
10402:
10395:
10392:
10385:
10382:
10379:
10376:
10371:
10367:
10356:
10344:
10341:
10338:
10335:
10332:
10329:
10326:
10323:
10318:
10314:
10299:
10298:
10287:
10284:
10281:
10275:
10271:
10263:
10260:
10256:
10251:
10248:
10245:
10242:
10236:
10232:
10227:
10224:
10219:
10216:
10211:
10208:
10205:
10202:
10199:
10194:
10187:
10184:
10170:
10159:
10156:
10153:
10148:
10144:
10140:
10137:
10134:
10131:
10126:
10119:
10116:
10098:
10097:
10085:
10065:
10055:
10043:
10033:
10021:
10011:
9999:
9989:
9977:
9974:
9971:
9968:
9954:
9953:
9942:
9939:
9936:
9930:
9927:
9921:
9918:
9915:
9912:
9909:
9906:
9903:
9900:
9897:
9894:
9891:
9888:
9885:
9882:
9879:
9876:
9873:
9867:
9864:
9856:
9852:
9848:
9830:
9827:
9814:
9811:
9808:
9805:
9785:
9782:
9779:
9776:
9773:
9770:
9767:
9764:
9753:
9752:
9741:
9738:
9735:
9732:
9729:
9726:
9723:
9720:
9717:
9714:
9711:
9708:
9705:
9701:
9697:
9694:
9691:
9688:
9684:
9673:
9662:
9659:
9656:
9653:
9650:
9647:
9644:
9641:
9638:
9635:
9632:
9629:
9626:
9622:
9618:
9615:
9612:
9609:
9603:
9600:
9582:
9579:
9575:
9574:
9561:
9555:
9552:
9550:
9547:
9546:
9543:
9540:
9538:
9535:
9534:
9532:
9527:
9522:
9514:
9506:
9503:
9498:
9495:
9491:
9488:
9483:
9480:
9479:
9476:
9473:
9471:
9468:
9467:
9465:
9458:
9452:
9449:
9447:
9444:
9443:
9441:
9436:
9435:
9430:
9424:
9421:
9419:
9416:
9415:
9413:
9408:
9407:
9405:
9400:
9395:
9389:
9386:
9383:
9382:
9379:
9376:
9375:
9373:
9341:
9321:
9301:
9292:), and if the
9281:
9278:
9275:
9269:
9266:
9243:
9240:
9237:
9231:
9228:
9205:
9202:
9199:
9196:
9176:
9156:
9145:
9144:
9131:
9121:
9117:
9113:
9108:
9105:
9101:
9098:
9093:
9092:
9087:
9084:
9079:
9077:
9074:
9073:
9071:
9066:
9061:
9053:
9045:
9042:
9037:
9036:
9033:
9030:
9029:
9027:
9020:
9012:
9009:
9004:
9001:
8997:
8994:
8989:
8986:
8985:
8982:
8979:
8977:
8974:
8973:
8971:
8966:
8962:
8954:
8951:
8946:
8945:
8942:
8939:
8938:
8936:
8931:
8930:
8928:
8923:
8918:
8912:
8909:
8906:
8904:
8901:
8900:
8898:
8878:
8877:
8865:
8862:
8859:
8855:
8843:
8831:
8828:
8825:
8820:
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8803:
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8794:
8789:
8782:
8779:
8765:
8753:
8750:
8747:
8742:
8735:
8732:
8725:
8722:
8719:
8716:
8711:
8707:
8696:
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8669:
8654:
8653:
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8634:
8631:
8628:
8622:
8618:
8613:
8612:
8609:
8606:
8603:
8597:
8593:
8588:
8587:
8584:
8579:
8572:
8569:
8567:
8564:
8563:
8560:
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8532:
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8518:
8515:
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8505:
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8494:
8489:
8488:
8485:
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8435:
8432:
8427:
8422:
8417:
8416:
8414:
8407:
8399:
8396:
8391:
8388:
8384:
8381:
8376:
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8366:
8364:
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8348:
8342:
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8336:
8331:
8324:
8320:
8312:
8311:
8308:
8305:
8302:
8297:
8290:
8286:
8278:
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8275:
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8026:
8023:
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7999:
7996:
7993:
7990:
7987:
7984:
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7669:
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7331:
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7288:
7285:
7282:
7279:
7276:
7272:
7268:
7265:
7262:
7259:
7256:
7253:
7249:
7245:
7242:
7239:
7236:
7233:
7227:
7223:
7205:
7204:
7193:
7190:
7187:
7183:
7179:
7176:
7173:
7170:
7167:
7163:
7159:
7156:
7153:
7150:
7147:
7143:
7132:
7121:
7118:
7115:
7111:
7107:
7104:
7101:
7098:
7095:
7091:
7087:
7084:
7081:
7078:
7075:
7069:
7065:
7038:
7035:
7032:
7028:
7024:
7021:
7018:
7015:
7011:
7007:
7004:
7001:
6998:
6995:
6992:
6988:
6967:
6964:
6961:
6958:
6938:
6935:
6922:
6919:
6916:
6913:
6902:
6901:
6890:
6887:
6884:
6880:
6876:
6873:
6870:
6867:
6863:
6852:
6841:
6838:
6835:
6831:
6826:
6822:
6819:
6816:
6813:
6809:
6805:
6802:
6799:
6796:
6790:
6786:
6743:
6740:
6714:
6710:
6705:
6702:
6697:
6693:
6690:
6687:
6684:
6680:
6675:
6672:
6669:
6666:
6662:
6638:
6637:
6626:
6623:
6620:
6616:
6612:
6607:
6604:
6599:
6595:
6592:
6589:
6586:
6582:
6577:
6574:
6571:
6568:
6564:
6553:
6542:
6539:
6536:
6532:
6527:
6523:
6518:
6515:
6510:
6506:
6503:
6500:
6497:
6493:
6488:
6485:
6482:
6479:
6475:
6471:
6468:
6465:
6462:
6456:
6452:
6425:
6422:
6419:
6415:
6403:
6402:
6391:
6388:
6385:
6381:
6377:
6374:
6371:
6368:
6365:
6362:
6358:
6354:
6351:
6348:
6345:
6342:
6338:
6327:
6316:
6313:
6310:
6306:
6302:
6299:
6296:
6293:
6290:
6287:
6283:
6279:
6276:
6273:
6270:
6267:
6261:
6257:
6239:
6238:
6227:
6224:
6221:
6217:
6213:
6210:
6207:
6204:
6201:
6197:
6193:
6190:
6187:
6184:
6181:
6177:
6166:
6155:
6152:
6149:
6145:
6141:
6138:
6135:
6132:
6129:
6125:
6121:
6118:
6115:
6112:
6109:
6103:
6099:
6064:
6061:
6058:
6054:
6050:
6047:
6044:
6041:
6038:
6034:
6009:
6006:
5993:
5990:
5987:
5969:
5968:
5957:
5954:
5951:
5939:
5933:
5930:
5929:
5927:
5922:
5919:
5916:
5913:
5901:
5895:
5892:
5890:
5887:
5886:
5884:
5879:
5876:
5873:
5870:
5852:
5851:
5840:
5837:
5834:
5822:
5816:
5813:
5812:
5809:
5806:
5805:
5803:
5798:
5795:
5792:
5789:
5777:
5771:
5768:
5766:
5763:
5762:
5759:
5756:
5753:
5751:
5748:
5745:
5744:
5742:
5737:
5734:
5731:
5728:
5722:
5700:
5699:
5688:
5685:
5679:
5676:
5673:
5670:
5667:
5662:
5658:
5652:
5649:
5646:
5640:
5634:
5631:
5628:
5625:
5622:
5617:
5613:
5607:
5604:
5601:
5598:
5595:
5590:
5586:
5579:
5576:
5573:
5570:
5521:
5518:
5515:
5505:
5502:
5499:
5489:
5486:
5483:
5480:
5462:
5461:
5450:
5447:
5444:
5441:
5431:
5428:
5425:
5422:
5416:
5413:
5400:
5397:
5394:
5391:
5361:
5358:
5350:
5349:
5338:
5335:
5332:
5329:
5317:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5292:
5290:
5285:
5282:
5279:
5276:
5258:
5257:
5246:
5243:
5240:
5228:
5220:
5216:
5212:
5211:
5206:
5202:
5198:
5197:
5192:
5188:
5184:
5183:
5178:
5174:
5170:
5169:
5167:
5162:
5159:
5156:
5153:
5141:
5133:
5129:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5106:
5101:
5097:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5074:
5069:
5065:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5042:
5037:
5033:
5029:
5026:
5024:
5021:
5019:
5016:
5014:
5011:
5010:
5008:
5003:
5000:
4997:
4994:
4988:
4959:
4958:
4947:
4944:
4941:
4938:
4934:
4928:
4920:
4916:
4912:
4908:
4904:
4900:
4896:
4892:
4888:
4884:
4880:
4876:
4875:
4873:
4868:
4865:
4862:
4859:
4855:
4843:
4842:
4831:
4828:
4825:
4821:
4815:
4809:
4806:
4805:
4802:
4799:
4798:
4795:
4792:
4791:
4788:
4785:
4784:
4782:
4777:
4774:
4771:
4768:
4764:
4758:
4750:
4746:
4742:
4739:
4735:
4731:
4727:
4724:
4720:
4716:
4712:
4709:
4705:
4701:
4697:
4694:
4693:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4671:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4649:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4625:
4620:
4617:
4614:
4611:
4605:
4601:
4583:
4582:
4571:
4563:
4559:
4555:
4552:
4547:
4543:
4539:
4534:
4530:
4524:
4520:
4516:
4511:
4507:
4501:
4497:
4493:
4488:
4484:
4476:
4472:
4468:
4465:
4460:
4456:
4452:
4447:
4443:
4437:
4433:
4429:
4424:
4420:
4414:
4410:
4403:
4400:
4397:
4394:
4362:Main article:
4359:
4356:
4334:
4331:
4328:
4308:
4305:
4302:
4298:
4288:Consequently,
4286:
4285:
4272:
4268:
4264:
4258:
4255:
4251:
4246:
4242:
4238:
4234:
4231:
4227:
4223:
4220:
4217:
4214:
4210:
4195:
4194:
4183:
4180:
4177:
4173:
4169:
4166:
4163:
4159:
4155:
4152:
4149:
4146:
4142:
4110:
4107:
4104:
4101:
4097:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4040:
4036:
4033:
4030:
4027:
4023:
4011:
4010:
3999:
3996:
3993:
3990:
3986:
3981:
3977:
3974:
3971:
3968:
3965:
3961:
3956:
3950:
3947:
3943:
3938:
3934:
3930:
3926:
3923:
3920:
3917:
3914:
3911:
3907:
3901:
3898:
3894:
3889:
3885:
3881:
3877:
3874:
3871:
3867:
3863:
3860:
3857:
3854:
3850:
3835:
3834:
3823:
3820:
3817:
3814:
3810:
3805:
3801:
3798:
3795:
3792:
3788:
3783:
3779:
3776:
3773:
3770:
3766:
3742:
3739:
3736:
3732:
3720:
3719:
3708:
3705:
3702:
3699:
3695:
3690:
3684:
3681:
3677:
3672:
3668:
3664:
3660:
3657:
3654:
3651:
3648:
3645:
3641:
3635:
3632:
3628:
3623:
3619:
3615:
3611:
3608:
3605:
3602:
3599:
3596:
3592:
3577:
3576:
3565:
3562:
3559:
3555:
3550:
3546:
3543:
3540:
3537:
3533:
3529:
3526:
3523:
3520:
3516:
3512:
3508:
3504:
3500:
3496:
3493:
3470:
3467:
3464:
3460:
3448:
3447:
3436:
3433:
3430:
3427:
3423:
3418:
3414:
3411:
3408:
3405:
3401:
3396:
3392:
3389:
3386:
3383:
3379:
3375:
3372:
3369:
3366:
3362:
3358:
3344:
3343:
3332:
3329:
3326:
3322:
3317:
3313:
3310:
3307:
3304:
3300:
3295:
3291:
3288:
3285:
3282:
3276:
3272:
3242:
3239:
3238:
3237:
3226:
3223:
3220:
3215:
3207:
3204:
3201:
3196:
3190:
3186:
3185:
3182:
3179:
3178:
3174:
3169:
3165:
3164:
3160:
3156:
3155:
3153:
3148:
3145:
3119:Main article:
3116:
3113:
3101:
3100:
3089:
3086:
3083:
3078:
3071:
3065:
3062:
3059:
3054:
3049:
3047:
3044:
3041:
3035:
3030:
3025:
3022:
3017:
3013:
3010:
3006:
3005:
3003:
2998:
2995:
2982:if and only if
2970:Main article:
2967:
2964:
2932:
2929:
2926:
2881:
2880:
2869:
2865:
2860:
2856:
2852:
2848:
2844:
2840:
2837:
2834:
2831:
2828:
2804:
2800:
2796:
2792:
2773:
2772:
2761:
2755:
2750:
2746:
2742:
2739:
2736:
2733:
2728:
2724:
2720:
2717:
2714:
2711:
2706:
2702:
2698:
2695:
2692:
2689:
2684:
2680:
2676:
2673:
2670:
2665:
2660:
2656:
2652:
2649:
2646:
2643:
2638:
2634:
2630:
2627:
2624:
2621:
2616:
2612:
2608:
2605:
2602:
2596:
2593:
2590:
2587:
2584:
2549:
2539:of the matrix
2528:
2525:
2522:
2521:
2510:
2507:
2504:
2500:
2495:
2491:
2488:
2485:
2482:
2478:
2473:
2469:
2466:
2463:
2460:
2456:
2435:
2432:
2429:
2425:
2420:
2416:
2413:
2410:
2407:
2403:
2398:
2394:
2391:
2388:
2385:
2381:
2377:
2374:
2371:
2368:
2365:
2361:
2357:
2347:
2338:
2337:
2326:
2323:
2320:
2316:
2311:
2307:
2304:
2301:
2298:
2294:
2289:
2285:
2282:
2279:
2276:
2272:
2251:
2248:
2245:
2241:
2236:
2232:
2229:
2226:
2223:
2219:
2214:
2210:
2207:
2204:
2201:
2197:
2193:
2190:
2187:
2184:
2180:
2176:
2166:
2161:Laplace domain
2157:
2156:
2145:
2142:
2139:
2135:
2131:
2128:
2125:
2121:
2117:
2114:
2111:
2108:
2104:
2100:
2097:
2094:
2090:
2086:
2083:
2080:
2077:
2073:
2052:
2049:
2046:
2042:
2038:
2035:
2032:
2028:
2024:
2021:
2018:
2015:
2011:
2007:
2004:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1978:
1974:
1963:
1959:
1958:
1947:
1944:
1941:
1937:
1932:
1928:
1925:
1922:
1919:
1915:
1910:
1906:
1903:
1900:
1897:
1893:
1872:
1869:
1866:
1862:
1857:
1853:
1850:
1847:
1844:
1840:
1835:
1831:
1828:
1825:
1822:
1819:
1816:
1812:
1801:
1797:
1796:
1785:
1782:
1779:
1775:
1771:
1768:
1765:
1761:
1757:
1754:
1751:
1748:
1744:
1740:
1737:
1734:
1730:
1726:
1723:
1720:
1717:
1713:
1692:
1689:
1686:
1682:
1678:
1675:
1672:
1668:
1664:
1661:
1658:
1655:
1651:
1647:
1644:
1641:
1637:
1633:
1630:
1627:
1624:
1618:
1614:
1600:
1596:
1595:
1584:
1581:
1578:
1574:
1569:
1565:
1562:
1559:
1556:
1552:
1547:
1543:
1540:
1537:
1534:
1530:
1509:
1506:
1503:
1499:
1494:
1490:
1487:
1484:
1481:
1477:
1472:
1468:
1465:
1462:
1459:
1453:
1449:
1435:
1431:
1430:
1425:
1414:Hybrid systems
1401:
1381:
1360:
1356:
1353:
1332:
1328:
1325:
1305:
1290:
1289:
1277:
1274:
1271:
1267:
1260:
1257:
1253:
1248:
1245:
1242:
1239:
1233:
1229:
1215:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1178:
1174:
1171:
1168:
1148:
1145:
1142:
1138:
1117:
1114:
1111:
1107:
1096:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1059:
1055:
1052:
1049:
1029:
1026:
1023:
1019:
1008:
996:
993:
990:
987:
984:
981:
978:
975:
971:
967:
964:
961:
941:
938:
935:
931:
920:
908:
905:
902:
899:
896:
893:
890:
887:
883:
879:
876:
873:
853:
850:
847:
843:
832:
818:
813:
808:
805:
802:
799:
795:
774:
771:
768:
764:
753:
739:
734:
729:
726:
723:
720:
716:
695:
692:
689:
685:
674:
660:
655:
650:
647:
644:
641:
637:
616:
613:
610:
606:
591:
590:
579:
576:
573:
569:
565:
562:
559:
555:
551:
548:
545:
542:
538:
534:
531:
528:
524:
520:
517:
514:
511:
507:
496:
485:
482:
479:
475:
471:
468:
465:
461:
457:
454:
451:
448:
444:
440:
437:
434:
430:
426:
423:
420:
417:
411:
407:
380:
360:
340:
320:
319:Linear systems
317:
296:
280:
277:
241:
238:
235:
215:
195:
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
12592:
12581:
12578:
12576:
12573:
12571:
12568:
12566:
12563:
12562:
12560:
12543:
12540:
12538:
12535:
12534:
12527:
12523:
12520:
12518:
12515:
12514:
12511:
12507:
12506:
12503:
12497:
12494:
12492:
12489:
12487:
12484:
12482:
12479:
12477:
12474:
12472:
12469:
12467:
12464:
12462:
12459:
12457:
12454:
12452:
12449:
12447:
12444:
12442:
12439:
12437:
12434:
12432:
12429:
12427:
12424:
12423:
12421:
12419:Architectures
12417:
12411:
12408:
12406:
12403:
12401:
12398:
12396:
12393:
12391:
12388:
12386:
12383:
12381:
12378:
12376:
12373:
12371:
12368:
12367:
12365:
12363:Organizations
12361:
12355:
12352:
12350:
12347:
12345:
12342:
12340:
12337:
12335:
12332:
12330:
12327:
12325:
12322:
12320:
12317:
12315:
12312:
12310:
12307:
12305:
12302:
12300:
12299:Yoshua Bengio
12297:
12296:
12294:
12290:
12280:
12279:Robot control
12277:
12273:
12270:
12269:
12268:
12265:
12263:
12260:
12258:
12255:
12253:
12250:
12248:
12245:
12243:
12240:
12238:
12235:
12233:
12230:
12229:
12227:
12223:
12217:
12214:
12212:
12209:
12207:
12204:
12202:
12199:
12197:
12196:Chinchilla AI
12194:
12192:
12189:
12187:
12184:
12182:
12179:
12177:
12174:
12172:
12169:
12167:
12164:
12162:
12159:
12157:
12154:
12152:
12149:
12147:
12144:
12142:
12139:
12135:
12132:
12131:
12130:
12127:
12125:
12122:
12120:
12117:
12115:
12112:
12110:
12107:
12105:
12102:
12101:
12099:
12095:
12089:
12086:
12082:
12079:
12077:
12074:
12073:
12072:
12069:
12065:
12062:
12060:
12057:
12055:
12052:
12051:
12050:
12047:
12045:
12042:
12040:
12037:
12035:
12032:
12030:
12027:
12025:
12022:
12020:
12017:
12015:
12012:
12010:
12007:
12005:
12002:
12001:
11999:
11995:
11992:
11988:
11982:
11979:
11977:
11974:
11972:
11969:
11967:
11964:
11962:
11959:
11957:
11954:
11952:
11949:
11948:
11946:
11942:
11936:
11933:
11931:
11928:
11926:
11923:
11921:
11918:
11916:
11913:
11912:
11910:
11906:
11898:
11895:
11894:
11893:
11890:
11888:
11885:
11883:
11880:
11876:
11875:Deep learning
11873:
11872:
11871:
11868:
11864:
11861:
11860:
11859:
11856:
11855:
11853:
11849:
11843:
11840:
11838:
11835:
11831:
11828:
11827:
11826:
11823:
11821:
11818:
11814:
11811:
11809:
11806:
11804:
11801:
11800:
11799:
11796:
11794:
11791:
11789:
11786:
11784:
11781:
11779:
11776:
11774:
11771:
11769:
11766:
11764:
11763:Hallucination
11761:
11757:
11754:
11753:
11752:
11749:
11747:
11744:
11740:
11737:
11736:
11735:
11732:
11731:
11729:
11725:
11719:
11716:
11714:
11711:
11709:
11706:
11704:
11701:
11699:
11696:
11694:
11691:
11689:
11686:
11684:
11681:
11679:
11678:
11674:
11673:
11671:
11669:
11665:
11656:
11651:
11649:
11644:
11642:
11637:
11636:
11633:
11626:
11622:
11618:
11614:
11611:
11610:
11598:
11592:
11588:
11583:
11582:
11578:
11575:
11574:
11568:
11562:
11558:
11553:
11549:
11547:0-486-44278-0
11543:
11539:
11534:
11523:
11521:0-387-98489-5
11517:
11510:
11509:
11503:
11499:
11493:
11489:
11484:
11480:
11478:0-13-067389-7
11474:
11470:
11465:
11461:
11459:0-19-511777-8
11455:
11451:
11446:
11442:
11436:
11432:
11427:
11426:
11412:
11405:
11397:
11391:
11387:
11380:
11373:
11367:
11359:
11355:
11351:
11347:
11343:
11339:
11335:
11331:
11327:
11320:
11313:
11307:
11303:
11299:
11295:
11288:
11280:
11276:
11272:
11268:
11263:
11258:
11254:
11250:
11246:
11242:
11235:
11228:
11222:
11214:
11210:
11206:
11202:
11198:
11194:
11190:
11186:
11182:
11178:
11171:
11163:
11159:
11155:
11151:
11147:
11143:
11136:
11128:
11124:
11120:
11114:
11110:
11103:
11096:
11090:
11086:
11082:
11078:
11071:
11062:
11057:
11053:
11049:
11045:
11038:
11030:
11024:
11020:
11019:
11011:
11003:
10997:
10993:
10992:
10987:
10981:
10977:
10966:
10965:Kalman filter
10963:
10960:
10957:
10954:
10951:
10948:
10945:
10943:
10940:
10938:
10937:Observability
10935:
10933:
10930:
10928:
10925:
10923:
10920:
10919:
10912:
10910:
10907:for integers
10889:
10883:
10876:
10873:
10867:
10862:
10857:
10849:
10845:
10835:
10831:
10824:
10815:
10814:
10813:
10799:
10796:
10790:
10787:
10776:
10772:
10753:
10748:
10739:
10730:
10726:
10718:
10715:
10711:
10706:
10700:
10691:
10687:
10682:
10679:
10674:
10671:
10666:
10656:
10648:
10644:
10637:
10632:
10623:
10617:
10614:
10611:
10600:
10595:
10586:
10578:
10571:
10568:
10554:
10546:
10539:
10536:
10526:
10521:
10515:
10506:
10491:
10490:
10489:
10469:
10462:
10459:
10452:
10447:
10440:
10437:
10426:
10408:
10400:
10393:
10390:
10383:
10377:
10369:
10365:
10357:
10339:
10333:
10330:
10324:
10316:
10312:
10304:
10303:
10302:
10282:
10273:
10269:
10261:
10258:
10254:
10249:
10243:
10234:
10230:
10225:
10222:
10217:
10214:
10209:
10206:
10200:
10192:
10185:
10182:
10171:
10154:
10146:
10142:
10138:
10132:
10124:
10117:
10114:
10103:
10102:
10101:
10083:
10063:
10056:
10041:
10034:
10019:
10012:
9997:
9990:
9972:
9966:
9959:
9958:
9957:
9937:
9928:
9925:
9919:
9916:
9913:
9907:
9901:
9898:
9895:
9892:
9889:
9886:
9883:
9880:
9874:
9865:
9862:
9854:
9850:
9846:
9839:
9838:
9837:
9836:
9826:
9809:
9803:
9780:
9777:
9774:
9771:
9768:
9762:
9733:
9727:
9724:
9718:
9712:
9709:
9706:
9695:
9689:
9674:
9654:
9648:
9645:
9639:
9633:
9630:
9627:
9616:
9610:
9588:
9587:
9586:
9578:
9559:
9553:
9548:
9541:
9536:
9530:
9525:
9520:
9512:
9504:
9501:
9496:
9489:
9486:
9481:
9474:
9469:
9463:
9456:
9450:
9445:
9439:
9428:
9422:
9417:
9411:
9403:
9398:
9393:
9387:
9384:
9377:
9371:
9362:
9361:
9360:
9359:test is then
9358:
9357:observability
9353:
9339:
9319:
9299:
9276:
9267:
9264:
9238:
9229:
9226:
9200:
9194:
9174:
9154:
9129:
9119:
9115:
9111:
9106:
9099:
9096:
9085:
9082:
9075:
9069:
9064:
9059:
9051:
9043:
9040:
9031:
9025:
9018:
9010:
9007:
9002:
8995:
8992:
8987:
8980:
8975:
8969:
8960:
8952:
8949:
8940:
8934:
8926:
8921:
8916:
8910:
8907:
8902:
8896:
8887:
8886:
8885:
8884:test is then
8883:
8860:
8844:
8826:
8818:
8811:
8808:
8801:
8795:
8787:
8780:
8777:
8766:
8748:
8740:
8733:
8730:
8723:
8717:
8709:
8705:
8697:
8679:
8671:
8667:
8659:
8658:
8657:
8639:
8629:
8604:
8582:
8577:
8570:
8565:
8558:
8554:
8548:
8533:
8516:
8503:
8495:
8492:
8483:
8477:
8472:
8467:
8458:
8450:
8433:
8425:
8412:
8405:
8397:
8394:
8389:
8382:
8379:
8374:
8367:
8362:
8356:
8351:
8346:
8337:
8329:
8322:
8303:
8295:
8288:
8273:
8264:
8263:
8262:
8244:
8237:
8222:
8215:
8200:
8193:
8175:
8169:
8162:
8144:
8135:
8132:
8122:is velocity;
8106:
8097:
8094:
8084:is position;
8068:
8062:
8055:
8054:
8053:
8033:
8027:
8024:
8021:
8015:
8006:
8003:
7997:
7994:
7988:
7982:
7979:
7973:
7964:
7961:
7955:
7948:
7947:
7946:
7944:
7917:
7906:
7903:
7897:
7882:
7865:
7854:
7851:
7845:
7833:
7829:
7826:
7823:
7820:
7817:
7813:
7809:
7803:
7794:
7779:
7778:
7777:
7775:
7753:
7742:
7737:
7734:
7729:
7725:
7722:
7719:
7716:
7712:
7707:
7701:
7690:
7685:
7682:
7677:
7673:
7670:
7667:
7664:
7660:
7655:
7649:
7634:
7617:
7605:
7601:
7596:
7593:
7588:
7584:
7581:
7578:
7575:
7571:
7566:
7563:
7560:
7556:
7552:
7549:
7543:
7531:
7527:
7522:
7519:
7514:
7510:
7507:
7504:
7501:
7497:
7492:
7489:
7486:
7483:
7479:
7475:
7469:
7460:
7445:
7444:
7443:
7426:
7392:
7381:
7378:
7372:
7361:
7358:
7355:
7349:
7338:
7335:
7329:
7314:
7297:
7286:
7283:
7277:
7266:
7263:
7260:
7254:
7243:
7240:
7234:
7225:
7210:
7209:
7208:
7188:
7177:
7174:
7168:
7157:
7154:
7148:
7133:
7116:
7105:
7102:
7096:
7085:
7082:
7076:
7067:
7052:
7051:
7050:
7033:
7022:
7016:
7005:
7002:
6999:
6993:
6962:
6956:
6943:
6934:
6920:
6917:
6914:
6911:
6885:
6874:
6868:
6853:
6836:
6824:
6820:
6817:
6814:
6811:
6807:
6803:
6797:
6788:
6773:
6772:
6771:
6769:
6765:
6761:
6757:
6753:
6749:
6739:
6737:
6733:
6729:
6712:
6708:
6703:
6700:
6695:
6691:
6688:
6685:
6682:
6678:
6673:
6670:
6667:
6664:
6660:
6651:
6647:
6643:
6621:
6610:
6605:
6602:
6597:
6593:
6590:
6587:
6584:
6580:
6575:
6569:
6554:
6537:
6525:
6521:
6516:
6513:
6508:
6504:
6501:
6498:
6495:
6491:
6486:
6483:
6480:
6477:
6473:
6469:
6463:
6454:
6439:
6438:
6437:
6420:
6386:
6375:
6372:
6369:
6363:
6352:
6349:
6343:
6328:
6311:
6300:
6297:
6294:
6288:
6277:
6274:
6268:
6259:
6244:
6243:
6242:
6222:
6211:
6208:
6202:
6191:
6188:
6182:
6167:
6150:
6139:
6136:
6130:
6119:
6116:
6110:
6101:
6086:
6085:
6084:
6082:
6078:
6059:
6048:
6045:
6039:
6023:
6014:
6005:
5952:
5937:
5931:
5925:
5920:
5914:
5899:
5893:
5888:
5882:
5877:
5871:
5854:
5853:
5835:
5820:
5814:
5807:
5801:
5796:
5790:
5775:
5769:
5764:
5757:
5754:
5749:
5746:
5740:
5735:
5729:
5720:
5705:
5704:
5703:
5686:
5683:
5677:
5674:
5671:
5668:
5665:
5660:
5656:
5650:
5647:
5644:
5638:
5632:
5629:
5626:
5623:
5620:
5615:
5611:
5605:
5602:
5599:
5596:
5593:
5588:
5584:
5577:
5571:
5554:
5553:
5552:
5549:
5547:
5543:
5539:
5535:
5516:
5487:
5481:
5448:
5429:
5423:
5398:
5392:
5375:
5374:
5373:
5371:
5367:
5357:
5355:
5336:
5330:
5315:
5309:
5304:
5299:
5294:
5288:
5283:
5277:
5260:
5259:
5241:
5226:
5218:
5214:
5204:
5200:
5190:
5186:
5176:
5172:
5165:
5160:
5154:
5139:
5131:
5127:
5123:
5118:
5113:
5108:
5099:
5095:
5091:
5086:
5081:
5076:
5067:
5063:
5059:
5054:
5049:
5044:
5035:
5031:
5027:
5022:
5017:
5012:
5006:
5001:
4995:
4986:
4971:
4970:
4969:
4966:
4964:
4945:
4939:
4926:
4918:
4914:
4906:
4902:
4894:
4890:
4882:
4878:
4871:
4866:
4860:
4845:
4844:
4826:
4813:
4807:
4800:
4793:
4786:
4780:
4775:
4769:
4756:
4748:
4744:
4740:
4733:
4729:
4725:
4718:
4714:
4710:
4703:
4699:
4695:
4688:
4683:
4678:
4673:
4666:
4661:
4656:
4651:
4644:
4639:
4634:
4629:
4623:
4618:
4612:
4603:
4588:
4587:
4586:
4569:
4561:
4557:
4553:
4550:
4545:
4541:
4537:
4532:
4528:
4522:
4518:
4514:
4509:
4505:
4499:
4495:
4491:
4486:
4482:
4474:
4470:
4466:
4463:
4458:
4454:
4450:
4445:
4441:
4435:
4431:
4427:
4422:
4418:
4412:
4408:
4401:
4395:
4378:
4377:
4376:
4373:
4371:
4365:
4355:
4353:
4349:
4332:
4329:
4326:
4303:
4266:
4256:
4253:
4240:
4232:
4221:
4215:
4200:
4199:
4198:
4178:
4164:
4153:
4147:
4132:
4131:
4130:
4128:
4124:
4105:
4099:
4095:
4088:
4082:
4079:
4073:
4067:
4059:
4055:
4034:
4028:
3997:
3991:
3975:
3966:
3948:
3945:
3932:
3924:
3918:
3912:
3899:
3896:
3883:
3875:
3861:
3855:
3840:
3839:
3838:
3821:
3815:
3799:
3793:
3777:
3771:
3756:
3755:
3754:
3737:
3706:
3700:
3682:
3679:
3666:
3658:
3652:
3646:
3633:
3630:
3617:
3609:
3603:
3597:
3582:
3581:
3580:
3560:
3544:
3538:
3527:
3521:
3502:
3494:
3484:
3483:
3482:
3465:
3434:
3428:
3412:
3406:
3390:
3384:
3373:
3367:
3356:
3349:
3348:
3347:
3327:
3311:
3305:
3289:
3283:
3274:
3259:
3258:
3257:
3255:
3250:
3248:
3224:
3221:
3218:
3213:
3205:
3202:
3199:
3180:
3151:
3146:
3143:
3136:
3135:
3134:
3132:
3127:
3122:
3121:Observability
3115:Observability
3112:
3110:
3106:
3087:
3084:
3081:
3076:
3063:
3060:
3057:
3045:
3033:
3001:
2996:
2993:
2986:
2985:
2984:
2983:
2980:
2973:
2963:
2961:
2957:
2953:
2948:
2946:
2945:minimum phase
2927:
2908:
2906:
2902:
2898:
2894:
2893:singularities
2890:
2886:
2867:
2854:
2846:
2838:
2832:
2826:
2819:
2818:
2817:
2798:
2790:
2782:
2778:
2759:
2748:
2744:
2740:
2737:
2726:
2722:
2718:
2715:
2704:
2700:
2696:
2693:
2682:
2678:
2674:
2671:
2658:
2654:
2650:
2647:
2636:
2632:
2628:
2625:
2614:
2610:
2606:
2603:
2594:
2591:
2585:
2568:
2567:
2566:
2564:
2538:
2534:
2505:
2489:
2483:
2467:
2461:
2430:
2414:
2408:
2392:
2386:
2375:
2372:
2366:
2355:
2348:
2343:
2340:
2321:
2305:
2299:
2283:
2277:
2246:
2230:
2224:
2208:
2202:
2191:
2185:
2174:
2167:
2162:
2159:
2140:
2126:
2115:
2109:
2095:
2084:
2078:
2047:
2033:
2022:
2016:
2002:
1991:
1985:
1982:
1979:
1964:
1961:
1942:
1926:
1920:
1904:
1898:
1867:
1851:
1845:
1829:
1823:
1820:
1817:
1802:
1799:
1780:
1766:
1755:
1749:
1735:
1724:
1718:
1687:
1673:
1662:
1656:
1642:
1631:
1625:
1616:
1601:
1598:
1579:
1563:
1557:
1541:
1535:
1504:
1488:
1482:
1466:
1460:
1451:
1436:
1433:
1429:
1426:
1424:
1421:
1417:
1415:
1399:
1379:
1354:
1351:
1326:
1323:
1303:
1295:
1272:
1258:
1255:
1251:
1246:
1240:
1231:
1216:
1201:
1198:
1195:
1192:
1183:
1169:
1166:
1143:
1112:
1097:
1082:
1079:
1076:
1073:
1064:
1050:
1047:
1024:
1009:
994:
991:
988:
985:
976:
962:
959:
936:
921:
906:
903:
900:
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273:Kalman Filter
269:
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12385:Hugging Face
12349:David Silver
11997:Audio–visual
11851:Applications
11830:Augmentation
11675:
11586:
11556:
11537:
11525:. Retrieved
11507:
11490:. Springer.
11487:
11468:
11449:
11430:
11410:
11404:
11385:
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11366:
11336:(1): 35–45.
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6731:
6728:controllable
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2979:controllable
2978:
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2530:
1427:
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371:outputs and
330:
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264:econometrics
261:
161:
157:state vector
156:
142:
138:
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131:
111:
101:
76:
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48:
12533:Categories
12481:Autoencoder
12436:Transformer
12304:Alex Graves
12252:OpenAI Five
12156:IBM Watsonx
11778:Convolution
11756:Overfitting
11148:(1): 1–10.
10959:State space
10953:Phase space
10771:equilibrium
8845:the output
6642:eigenvalues
2956:BIBO stable
2891:(i.e., the
2885:eigenvalues
2781:determinant
2537:eigenvalues
1423:System type
268:time series
206:inputs and
184:time-domain
144:phase space
139:state space
62:introducing
12559:Categories
12522:Technology
12375:EleutherAI
12334:Fei-Fei Li
12329:Yann LeCun
12242:Q-learning
12225:Decisional
12151:IBM Watson
12059:Midjourney
11951:TensorFlow
11798:Activation
11751:Regression
11746:Clustering
10973:References
3131:observable
2533:LTI system
309:capacitors
92:See also:
45:references
12405:MIT CSAIL
12370:Anthropic
12339:Andrew Ng
12237:AlphaZero
12081:VideoPoet
12044:AlphaFold
11981:MindSpore
11935:SpiNNaker
11930:Memristor
11837:Diffusion
11813:Rectifier
11793:Batchnorm
11773:Attention
11768:Adversary
11540:. Dover.
11358:259115248
11350:0021-9223
11271:0883-7252
11197:0899-7667
11162:0018-9286
11127:794591362
10877:π
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10719:ℓ
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5368:(and not
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4741:−
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4711:−
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4330:×
4254:−
4241:−
3946:−
3933:−
3897:−
3884:−
3680:−
3667:−
3631:−
3618:−
3579:and thus
3503:−
3481:, giving
3374:−
3275:˙
3203:−
3181:⋮
3147:
3061:−
3046:⋯
2997:
2960:removable
2855:−
2827:λ
2799:−
2741:−
2719:−
2697:−
2675:−
2651:−
2629:−
2607:−
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1355:∈
1327:∈
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1199:×
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807:∈
770:⋅
728:∈
691:⋅
649:∈
612:⋅
410:˙
313:inductors
237:×
120:variables
12513:Portals
12272:Auto-GPT
12104:Word2vec
11908:Hardware
11825:Datasets
11727:Concepts
11527:June 28,
11279:14231301
11213:10020032
11205:12803953
10915:See also
9835:pendulum
7207:becomes
6241:becomes
6008:Feedback
4121:. For a
2342:Z-domain
351:inputs,
70:May 2009
12395:Meta AI
12232:AlphaGo
12216:PanGu-ÎŁ
12186:ChatGPT
12161:Granite
12109:Seq2seq
12088:Whisper
12009:WaveNet
12004:AlexNet
11976:Flux.jl
11956:PyTorch
11808:Sigmoid
11803:Softmax
11668:General
6742:Example
3837:giving
3346:yields
593:where:
162:If the
147:is the
58:improve
12410:Huawei
12390:OpenAI
12292:People
12262:MuZero
12124:Gemini
12119:Claude
12054:DALL-E
11966:Theano
11623:, and
11593:
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9956:where
8656:where
8052:where
6766:, the
5366:proper
4056:, the
4052:and a
3103:where
172:matrix
155:, the
153:vector
47:, but
12476:Mamba
12247:SARSA
12211:LLaMA
12206:BLOOM
12191:GPT-J
12181:GPT-4
12176:GPT-3
12171:GPT-2
12166:GPT-1
12129:LaMDA
11961:Keras
11512:(PDF)
11354:S2CID
11275:S2CID
11209:S2CID
3245:The "
2954:(see
2889:poles
114:is a
12400:Mila
12201:PaLM
12134:Bard
12114:BERT
12097:Text
12076:Sora
11591:ISBN
11561:ISBN
11542:ISBN
11529:2012
11516:ISBN
11492:ISBN
11473:ISBN
11454:ISBN
11435:ISBN
11390:ISBN
11346:ISSN
11306:ISBN
11267:ISSN
11201:PMID
11193:ISSN
11158:ISSN
11123:OCLC
11113:ISBN
11089:ISBN
11023:ISBN
10996:ISBN
10769:The
9355:The
9312:and
9167:and
8880:The
5540:and
3256:of
3144:rank
3105:rank
2994:rank
311:and
137:The
110:, a
106:and
96:and
12141:NMT
12024:OCR
12019:HWR
11971:JAX
11925:VPU
11920:TPU
11915:IPU
11739:SGD
11338:doi
11298:doi
11257:hdl
11249:doi
11185:doi
11150:doi
11081:doi
11056:doi
10680:sin
10223:sin
9896:sin
6644:of
2962:).
2899:or
2783:of
1294:LTI
1167:dim
1048:dim
960:dim
872:dim
141:or
126:or
102:In
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6451:x
6424:)
6421:t
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6390:)
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6384:(
6380:y
6376:K
6373:D
6370:+
6367:)
6364:t
6361:(
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6350:=
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6344:t
6341:(
6337:y
6315:)
6312:t
6309:(
6305:y
6301:K
6298:B
6295:+
6292:)
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6286:(
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6275:=
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6223:t
6220:(
6216:u
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6189:=
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6098:x
6077:K
6063:)
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6057:(
6053:y
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6046:=
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5992:)
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5981:G
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5875:)
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5839:)
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5828:u
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5546:D
5542:C
5538:B
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5520:)
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5509:u
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5449:.
5446:)
5440:(
5435:G
5430:+
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5424:s
5421:(
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5412:S
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5399:=
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5337:.
5334:)
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5323:x
5316:]
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5234:u
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4216:s
4213:(
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4145:(
4141:Y
4109:)
4106:s
4103:(
4100:U
4096:/
4092:)
4089:s
4086:(
4083:Y
4080:=
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4074:s
4071:(
4068:G
4039:0
4035:=
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4022:x
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3980:D
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3973:)
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3910:(
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3900:1
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3888:A
3880:I
3876:s
3873:(
3870:(
3866:C
3862:=
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3856:s
3853:(
3849:Y
3822:,
3819:)
3816:s
3813:(
3809:U
3804:D
3800:+
3797:)
3794:s
3791:(
3787:X
3782:C
3778:=
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3772:s
3769:(
3765:Y
3741:)
3738:s
3735:(
3731:X
3707:.
3704:)
3701:s
3698:(
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3689:B
3683:1
3676:)
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3659:s
3656:(
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3650:)
3647:0
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3634:1
3627:)
3622:A
3614:I
3610:s
3607:(
3604:=
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3595:(
3591:X
3564:)
3561:s
3558:(
3554:U
3549:B
3545:+
3542:)
3539:0
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3528:=
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3519:(
3515:X
3511:)
3507:A
3499:I
3495:s
3492:(
3469:)
3466:s
3463:(
3459:X
3435:.
3432:)
3429:s
3426:(
3422:U
3417:B
3413:+
3410:)
3407:s
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3400:X
3395:A
3391:=
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3385:0
3382:(
3378:x
3371:)
3368:s
3365:(
3361:X
3357:s
3331:)
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3281:(
3271:x
3225:.
3222:n
3219:=
3214:]
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3200:n
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3189:C
3173:A
3168:C
3159:C
3152:[
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3088:,
3085:n
3082:=
3077:]
3070:B
3064:1
3058:n
3053:A
3040:B
3034:2
3029:A
3021:B
3016:A
3009:B
3002:[
2931:)
2928:s
2925:(
2920:G
2868:.
2864:|
2859:A
2851:I
2847:s
2843:|
2839:=
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2833:s
2830:(
2803:A
2795:I
2791:s
2760:.
2754:)
2749:4
2745:p
2738:s
2735:(
2732:)
2727:3
2723:p
2716:s
2713:(
2710:)
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2688:)
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2669:(
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2648:s
2645:(
2642:)
2637:2
2633:z
2626:s
2623:(
2620:)
2615:1
2611:z
2604:s
2601:(
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2592:=
2589:)
2586:s
2583:(
2578:G
2548:A
2509:)
2506:z
2503:(
2499:U
2494:D
2490:+
2487:)
2484:z
2481:(
2477:X
2472:C
2468:=
2465:)
2462:z
2459:(
2455:Y
2434:)
2431:z
2428:(
2424:U
2419:B
2415:+
2412:)
2409:z
2406:(
2402:X
2397:A
2393:=
2390:)
2387:0
2384:(
2380:x
2376:z
2370:)
2367:z
2364:(
2360:X
2356:z
2325:)
2322:s
2319:(
2315:U
2310:D
2306:+
2303:)
2300:s
2297:(
2293:X
2288:C
2284:=
2281:)
2278:s
2275:(
2271:Y
2250:)
2247:s
2244:(
2240:U
2235:B
2231:+
2228:)
2225:s
2222:(
2218:X
2213:A
2209:=
2206:)
2203:0
2200:(
2196:x
2189:)
2186:s
2183:(
2179:X
2175:s
2144:)
2141:k
2138:(
2134:u
2130:)
2127:k
2124:(
2120:D
2116:+
2113:)
2110:k
2107:(
2103:x
2099:)
2096:k
2093:(
2089:C
2085:=
2082:)
2079:k
2076:(
2072:y
2051:)
2048:k
2045:(
2041:u
2037:)
2034:k
2031:(
2027:B
2023:+
2020:)
2017:k
2014:(
2010:x
2006:)
2003:k
2000:(
1996:A
1992:=
1989:)
1986:1
1983:+
1980:k
1977:(
1973:x
1946:)
1943:k
1940:(
1936:u
1931:D
1927:+
1924:)
1921:k
1918:(
1914:x
1909:C
1905:=
1902:)
1899:k
1896:(
1892:y
1871:)
1868:k
1865:(
1861:u
1856:B
1852:+
1849:)
1846:k
1843:(
1839:x
1834:A
1830:=
1827:)
1824:1
1821:+
1818:k
1815:(
1811:x
1784:)
1781:t
1778:(
1774:u
1770:)
1767:t
1764:(
1760:D
1756:+
1753:)
1750:t
1747:(
1743:x
1739:)
1736:t
1733:(
1729:C
1725:=
1722:)
1719:t
1716:(
1712:y
1691:)
1688:t
1685:(
1681:u
1677:)
1674:t
1671:(
1667:B
1663:+
1660:)
1657:t
1654:(
1650:x
1646:)
1643:t
1640:(
1636:A
1632:=
1629:)
1626:t
1623:(
1613:x
1583:)
1580:t
1577:(
1573:u
1568:D
1564:+
1561:)
1558:t
1555:(
1551:x
1546:C
1542:=
1539:)
1536:t
1533:(
1529:y
1508:)
1505:t
1502:(
1498:u
1493:B
1489:+
1486:)
1483:t
1480:(
1476:x
1471:A
1467:=
1464:)
1461:t
1458:(
1448:x
1400:t
1380:k
1359:Z
1352:t
1331:R
1324:t
1304:t
1288:.
1276:)
1273:t
1270:(
1266:x
1259:t
1256:d
1252:d
1244:)
1241:t
1238:(
1228:x
1214:,
1202:p
1196:q
1193:=
1190:]
1187:)
1181:(
1177:D
1173:[
1147:)
1141:(
1137:D
1116:)
1110:(
1106:D
1095:,
1083:n
1077:q
1074:=
1071:]
1068:)
1062:(
1058:C
1054:[
1028:)
1022:(
1018:C
1007:,
995:p
989:n
986:=
983:]
980:)
974:(
970:B
966:[
940:)
934:(
930:B
919:,
907:n
901:n
898:=
895:]
892:)
886:(
882:A
878:[
852:)
846:(
842:A
831:;
817:p
812:R
804:)
801:t
798:(
794:u
773:)
767:(
763:u
752:;
738:q
733:R
725:)
722:t
719:(
715:y
694:)
688:(
684:y
673:;
659:n
654:R
646:)
643:t
640:(
636:x
615:)
609:(
605:x
578:)
575:t
572:(
568:u
564:)
561:t
558:(
554:D
550:+
547:)
544:t
541:(
537:x
533:)
530:t
527:(
523:C
519:=
516:)
513:t
510:(
506:y
484:)
481:t
478:(
474:u
470:)
467:t
464:(
460:B
456:+
453:)
450:t
447:(
443:x
439:)
436:t
433:(
429:A
425:=
422:)
419:t
416:(
406:x
379:n
359:q
339:p
295:n
240:p
234:q
214:q
194:p
83:)
77:(
72:)
68:(
54:.
20:)
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