Misplaced Pages

#2997

104-440: At any given time, a dynamical system has a state representing a point in an appropriate state space . This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic , that is, for a given time interval only one future state follows from

208-455: A flow , the vector field v( x ) is an affine function of the position in the phase space, that is, with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b  ≠ 0 with A  = 0 is just a straight line in the direction of  b : State (controls) In control engineering and system identification ,

312-475: A locally compact and Hausdorff topological space X , it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X . Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system ( R , X* , Φ*). In compact dynamical systems the limit set of any orbit is non-empty , compact and simply connected . A dynamical system may be defined formally as

416-441: A measure space ( X , Σ, μ ) and suppose ƒ is a μ -integrable function, i.e. ƒ ∈ L ( μ ). Then we define the following averages : Time average: This is defined as the average (if it exists) over iterations of T starting from some initial point x : Space average: If μ ( X ) is finite and nonzero, we can consider the space or phase average of ƒ: In general the time average and space average may be different. But if

520-401: A monoid action of T on X . The function Φ( t , x ) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t , called the evolution parameter . X is called phase space or state space , while the variable x represents an initial state of the system. We often write if we take one of

624-408: A single-input single-output (SISO) system , the transfer function is defined as the ratio of output and input G ( s ) = Y ( s ) / U ( s ) {\displaystyle G(s)=Y(s)/U(s)} . For a multiple-input multiple-output (MIMO) system , however, this ratio is not defined. Therefore, assuming zero initial conditions, the transfer function matrix

728-423: A state-space representation is a mathematical model of a physical system specified as a set of input, output, and variables related by first-order differential equations or difference equations . Such variables, called state variables , 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

832-404: A symplectic structure . When T is taken to be the reals, the dynamical system is called global or a flow ; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow . A discrete dynamical system , discrete-time dynamical system is a tuple ( T , M , Φ), where M is a manifold locally diffeomorphic to a Banach space , and Φ is a function. When T

936-454: A time series 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 Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations. The internal state variables are

1040-445: A unitary operator on a Hilbert space H ; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying ‖ Ux ‖ = ‖ x ‖ for all x in H , or equivalently, satisfying U * U = I, but not necessarily UU * = I). Let P be the orthogonal projection onto { ψ  ∈  H  |  Uψ  = ψ} = ker( I  −  U ). Then, for any x in H , we have: where

1144-1145: A bit: G ( s ) = s 2 + 3 s + 3 s 2 + 2 s + 1 = s + 2 s 2 + 2 s + 1 + 1 {\displaystyle \mathbf {G} (s)={\frac {s^{2}+3s+3}{s^{2}+2s+1}}={\frac {s+2}{s^{2}+2s+1}}+1} which yields the following controllable realization x ˙ ( t ) = [ − 2 − 1 1 0 ] x ( t ) + [ 1 0 ] u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)={\begin{bmatrix}-2&-1\\1&0\\\end{bmatrix}}\mathbf {x} (t)+{\begin{bmatrix}1\\0\end{bmatrix}}\mathbf {u} (t)} y ( t ) = [ 1 2 ] x ( t ) + [ 1 ] u ( t ) {\displaystyle \mathbf {y} (t)={\begin{bmatrix}1&2\end{bmatrix}}\mathbf {x} (t)+{\begin{bmatrix}1\end{bmatrix}}\mathbf {u} (t)} Notice how

SECTION 10

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1248-404: A constant. G ( s ) = G S P ( s ) + G ( ∞ ) . {\displaystyle \mathbf {G} (s)=\mathbf {G} _{\mathrm {SP} }(s)+\mathbf {G} (\infty ).} The strictly proper transfer function can then be transformed into a canonical state-space realization using techniques shown above. The state-space realization of

1352-1597: A continuous time-invariant linear state-space model can be derived in the following way: First, taking the Laplace transform of x ˙ ( t ) = A x ( t ) + B u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)} yields s X ( s ) − x ( 0 ) = A X ( s ) + B U ( s ) . {\displaystyle s\mathbf {X} (s)-\mathbf {x} (0)=\mathbf {A} \mathbf {X} (s)+\mathbf {B} \mathbf {U} (s).} Next, we simplify for X ( s ) {\displaystyle \mathbf {X} (s)} , giving ( s I − A ) X ( s ) = x ( 0 ) + B U ( s ) {\displaystyle (s\mathbf {I} -\mathbf {A} )\mathbf {X} (s)=\mathbf {x} (0)+\mathbf {B} \mathbf {U} (s)} and thus X ( s ) = ( s I − A ) − 1 x ( 0 ) + ( s I − A ) − 1 B U ( s ) . {\displaystyle \mathbf {X} (s)=(s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {x} (0)+(s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} \mathbf {U} (s).} Substituting for X ( s ) {\displaystyle \mathbf {X} (s)} in

1456-463: A dynamical system. In 1913, George David Birkhoff proved Poincaré's " Last Geometric Theorem ", a special case of the three-body problem , a result that made him world-famous. In 1927, he published his Dynamical Systems . Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem . Combining insights from physics on the ergodic hypothesis with measure theory , this theorem solved, at least in principle,

1560-469: A dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the Krylov–Bogolyubov theorem ) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction

1664-459: A fundamental problem of statistical mechanics . The ergodic theorem has also had repercussions for dynamics. Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on

1768-615: A given measure of the state space is summed for all future points of a trajectory, assuring the invariance. Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems , chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on

1872-403: A local subregion of the oatmeal, but will distribute the syrup evenly throughout. At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density. The formal definition is as follows: Let T  : X → X be a measure-preserving transformation on a measure space ( X , Σ , μ ) , with μ ( X ) = 1 . Then T

1976-491: A major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ \  G , where G is a Lie group and Γ is a lattice in  G . In the last 20 years, there have been many works trying to find a measure-classification theorem similar to Ratner 's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis . An important partial result (solving those conjectures with an extra assumption of positive entropy)

2080-542: A measure-preserving transformation of a measure space , the triplet ( T , ( X , Σ, μ ), Φ). Here, T is a monoid (usually the non-negative integers), X is a set , and ( X , Σ, μ ) is a probability space , meaning that Σ is a sigma-algebra on X and μ is a finite measure on ( X , Σ). A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ − 1 σ ∈ Σ {\displaystyle \Phi ^{-1}\sigma \in \Sigma } . A map Φ

2184-554: A probability space with a measure preserving transformation T , and let 1 ≤ p ≤ ∞. The conditional expectation with respect to the sub-σ-algebra Σ T of the T -invariant sets is a linear projector E T of norm 1 of the Banach space L ( X , Σ, μ ) onto its closed subspace L ( X , Σ T , μ ). The latter may also be characterized as the space of all T -invariant L -functions on X . The ergodic means, as linear operators on L ( X , Σ, μ ) also have unit operator norm; and, as

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2288-418: A set of measure zero, where χ A is the indicator function of A . The occurrence times of a measurable set A is defined as the set k 1 , k 2 , k 3 , ..., of times k such that T ( x ) is in A , sorted in increasing order. The differences between consecutive occurrence times R i = k i − k i −1 are called the recurrence times of A . Another consequence of

2392-580: A simple consequence of the Birkhoff–Khinchin theorem, converge to the projector E T in the strong operator topology of L if 1 ≤ p ≤ ∞, and in the weak operator topology if p = ∞. More is true if 1 < p ≤ ∞ then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of ƒ ∈ L are dominated in L ; however, if ƒ ∈ L , the ergodic means may fail to be equidominated in L . Finally, if ƒ

2496-435: A single complex number of unit length (which we think of as U ), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of U will converge to 0. Also, 0 is the only fixed point of U , and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described). Let ( X , Σ, μ ) be as above

2600-445: A small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: Many people regard French mathematician Henri Poincaré as

2704-569: A strictly proper system D equals zero. Another fairly common situation is when all states are outputs, i.e. y = x , which yields C = I , the identity matrix . This would then result in the simpler equations x ˙ ( t ) = ( A + B K ) x ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=\left(A+BK\right)\mathbf {x} (t)} y ( t ) = x ( t ) {\displaystyle \mathbf {y} (t)=\mathbf {x} (t)} This reduces

2808-400: A time-step of a discrete dynamical system. The ergodic theorem then asserts that the average behavior of a function ƒ over sufficiently large time-scales is approximated by the orthogonal component of ƒ which is time-invariant. In another form of the mean ergodic theorem, let U t be a strongly continuous one-parameter group of unitary operators on H . Then the operator converges in

2912-462: Is controllable if and only if rank ⁡ [ B A B A 2 B ⋯ A n − 1 B ] = n , {\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,} where rank

3016-438: Is ergodic if for every E in Σ with μ( T ( E ) Δ E ) = 0 (that is, E is invariant ), either μ ( E ) = 0 or μ ( E ) = 1 . The operator Δ here is the symmetric difference of sets, equivalent to the exclusive-or operation with respect to set membership. The condition that the symmetric difference be measure zero is called being essentially invariant . Let T : X → X be a measure-preserving transformation on

3120-835: Is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system): Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form: G ( s ) = 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 . {\displaystyle \mathbf {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}}}.} The coefficients can now be inserted directly into

3224-478: Is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain T {\displaystyle {\mathcal {T}}} into the space of diffeomorphisms of the manifold to itself. In other terms, f ( t ) is a diffeomorphism, for every time t in the domain T {\displaystyle {\mathcal {T}}} . A real dynamical system , real-time dynamical system , continuous time dynamical system , or flow

Dynamical system - Misplaced Pages Continue

3328-464: Is a matrix with the dimension q × p {\displaystyle q\times p} 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 Rosenbrock system matrix provides a bridge between the state-space representation and its transfer function . Any given transfer function which

3432-406: Is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval. More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of ƒ exists for almost every x and that the (almost everywhere defined) limit function f ^ {\displaystyle {\hat {f}}}

3536-446: Is a tuple ( T , M , Φ) with T an open interval in the real numbers R , M a manifold locally diffeomorphic to a Banach space , and Φ a continuous function . If Φ is continuously differentiable we say the system is a differentiable dynamical system . If the manifold M is locally diffeomorphic to R , the dynamical system is finite-dimensional ; if not, the dynamical system is infinite-dimensional . This does not assume

3640-496: Is assumed to be in the Zygmund class, that is |ƒ| log (|ƒ|) is integrable, then the ergodic means are even dominated in L . Let ( X , Σ, μ ) be a measure space such that μ ( X ) is finite and nonzero. The time spent in a measurable set A is called the sojourn time . An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative measure of A is equal to the mean sojourn time : for all x except for

3744-1370: Is called controllable canonical form 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). The transfer function coefficients can also be used to construct another type of canonical form x ˙ ( t ) = [ 0 0 0 − d 4 1 0 0 − d 3 0 1 0 − d 2 0 0 1 − d 1 ] x ( t ) + [ n 4 n 3 n 2 n 1 ] u ( t ) {\displaystyle {\dot {\mathbf {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}}\mathbf {x} (t)+{\begin{bmatrix}n_{4}\\n_{3}\\n_{2}\\n_{1}\end{bmatrix}}\mathbf {u} (t)} y ( t ) = [ 0 0 0 1 ] x ( t ) . {\displaystyle \mathbf {y} (t)={\begin{bmatrix}0&0&0&1\end{bmatrix}}\mathbf {x} (t).} This state-space realization

3848-402: Is called observable canonical form 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). Transfer functions which are only proper (and not strictly proper ) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and

3952-684: Is characterized by the algebraization of general system theory , which makes it possible to use Kronecker vector-matrix structures . The capacity of these structures can be efficiently applied to research systems with or without modulation. The state-space representation (also known as the " time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With p {\displaystyle p} inputs and q {\displaystyle q} outputs, we would otherwise have to write down q × p {\displaystyle q\times p} Laplace transforms to encode all

4056-592: Is derived from Y ( s ) = G ( s ) U ( s ) {\displaystyle \mathbf {Y} (s)=\mathbf {G} (s)\mathbf {U} (s)} using the method of equating the coefficients which yields G ( s ) = C ( s I − A ) − 1 B + D . {\displaystyle \mathbf {G} (s)=\mathbf {C} (s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} +\mathbf {D} .} Consequently, G ( s ) {\displaystyle \mathbf {G} (s)}

4160-406: 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,

4264-442: Is integrable: Furthermore, f ^ {\displaystyle {\hat {f}}} is T -invariant, that is to say holds almost everywhere, and if μ ( X ) is finite, then the normalization is the same: In particular, if T is ergodic, then f ^ {\displaystyle {\hat {f}}} must be a constant (almost everywhere), and so one has that almost everywhere. Joining

Dynamical system - Misplaced Pages Continue

4368-466: Is invariant in all the partial sums as N {\displaystyle N} grows, while for the latter part, from the telescoping series one would have: This theorem specializes to the case in which the Hilbert space H consists of L functions on a measure space and U is an operator of the form where T is a measure-preserving endomorphism of X , thought of in applications as representing

4472-410: Is often concerned with ergodic transformations . The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not allow the syrup to remain in

4576-513: Is played by the various notions of entropy for dynamical systems. The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry , methods of ergodic theory have been used to study

4680-418: Is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory. For the special class of ergodic systems , this time average

4784-415: Is realized. The study of dynamical systems is the focus of dynamical systems theory , which has applications to a wide variety of fields such as mathematics, physics, biology , chemistry , engineering , economics , history , and medicine . Dynamical systems are a fundamental part of chaos theory , logistic map dynamics, bifurcation theory , the self-assembly and self-organization processes, and

4888-399: Is said to preserve the measure if and only if, for every σ in Σ, one has μ ( Φ − 1 σ ) = μ ( σ ) {\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )} . Combining the above, a map Φ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it

4992-512: Is taken to be the integers, it is a cascade or a map . If T is restricted to the non-negative integers we call the system a semi-cascade . A cellular automaton is a tuple ( T , M , Φ), with T a lattice such as the integers or a higher-dimensional integer grid , M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents

5096-466: Is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of ( A + B K ( I − D K ) − 1 C ) {\displaystyle \left(A+BK\left(I-DK\right)^{-1}C\right)} . This assumes that the closed-loop system is controllable or that the unstable eigenvalues of A can be made stable through appropriate choice of K . For

5200-435: Is the conditional expectation given the σ-algebra C {\displaystyle {\mathcal {C}}} of invariant sets of T . Corollary ( Pointwise Ergodic Theorem ): In particular, if T is also ergodic, then C {\displaystyle {\mathcal {C}}} is the trivial σ-algebra, and thus with probability 1: Von Neumann's mean ergodic theorem , holds in Hilbert spaces. Let U be

5304-445: Is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. M {\displaystyle {\mathcal {M}}} is a manifold , i.e. locally a Banach space or Euclidean space, or in the discrete case a graph . f is an evolution rule t  →  f (with t ∈ T {\displaystyle t\in {\mathcal {T}}} ) such that f

SECTION 50

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5408-516: Is the number of linearly independent rows in a matrix, and where n is the number of state variables. 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 final state, observability provides that knowing an output trajectory provides enough information to predict

5512-425: Is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger properties, such as mixing and equidistribution , have also been extensively studied. The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes

5616-424: Is then ( T , M , Φ). Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy where G : ( T × M ) M → C {\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} } is a functional from the set of evolution functions to

5720-434: Is usually used instead of t {\displaystyle t} . Hybrid systems 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: Stability and natural response characteristics of a continuous-time LTI system (i.e., linear with matrices that are constant with respect to time) can be studied from

5824-700: Is written in the following form: x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)} y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) {\displaystyle \mathbf {y} (t)=\mathbf {C} (t)\mathbf {x} (t)+\mathbf {D} (t)\mathbf {u} (t)} where: In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in

5928-535: Is Σ-measurable, and is measure-preserving. The triplet ( T , ( X , Σ, μ ), Φ), for such a Φ, is then defined to be a dynamical system . The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates Φ n = Φ ∘ Φ ∘ ⋯ ∘ Φ {\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi } for every integer n are studied. For continuous dynamical systems,

6032-463: The Poincaré recurrence theorem , which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of

6136-602: The attractor , but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution. For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of

6240-473: The edge of chaos concept. The concept of a dynamical system has its origins in Newtonian mechanics . There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation , difference equation or other time scale .) To determine

6344-800: The eigenvalues of the matrix A {\displaystyle \mathbf {A} } . The stability of a time-invariant state-space model can be determined by looking at the system's transfer function in factored form. It will then look something like this: G ( s ) = k ( s − z 1 ) ( s − z 2 ) ( s − z 3 ) ( s − p 1 ) ( s − p 2 ) ( s − p 3 ) ( s − p 4 ) . {\displaystyle \mathbf {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})}}.} The denominator of

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6448-455: The geodesic flow on Riemannian manifolds , starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory . Ergodic theory has fruitful connections with harmonic analysis , Lie theory ( representation theory , lattices in algebraic groups ), and number theory (the theory of diophantine approximations , L-functions ). Ergodic theory

6552-423: The "space" lattice, while the one in T represents the "time" lattice. Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems . Such systems are useful for modeling, for example, image processing . Given a global dynamical system ( R , X , Φ) on

6656-448: The behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise , etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory , is based on general notions of measure theory. Its initial development

6760-460: The behavior of all orbits classified. In a linear system the phase space is the N -dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle : if u ( t ) and w ( t ) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u ( t ) +  w ( t ). For

6864-408: The common LTI case, matrices will be time invariant. The time variable t {\displaystyle t} can be continuous (e.g. t ∈ R {\displaystyle t\in \mathbb {R} } ) or discrete (e.g. t ∈ Z {\displaystyle t\in \mathbb {Z} } ). In the latter case, the time variable k {\displaystyle k}

6968-404: The constant is trivially y ( t ) = G ( ∞ ) u ( t ) {\displaystyle \mathbf {y} (t)=\mathbf {G} (\infty )\mathbf {u} (t)} . Together we then get a state-space realization with matrices A , B and C determined by the strictly proper part, and matrix D determined by the constant. Here is an example to clear things up

7072-898: The construction and maintenance of machines and structures that are common in daily life, such as ships , cranes , bridges , buildings , skyscrapers , jet engines , rocket engines , aircraft and spacecraft . In the most general sense, a dynamical system is a tuple ( T , X , Φ) where T is a monoid , written additively, X is a non-empty set and Φ is a function with and for any x in X : for t 1 , t 2 + t 1 ∈ I ( x ) {\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)} and   t 2 ∈ I ( Φ ( t 1 , x ) ) {\displaystyle \ t_{2}\in I(\Phi (t_{1},x))} , where we have defined

7176-485: The current state. However, some systems are stochastic , in that random events also affect the evolution of the state variables. In physics , a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation

7280-481: The development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in the semisimple Lie group SO(n,1) . Ergodicity of the geodesic flow on Riemannian symmetric spaces was demonstrated by F. I. Mautner in 1957. In 1967 D. V. Anosov and Ya. G. Sinai proved ergodicity of the geodesic flow on compact manifolds of variable negative sectional curvature . A simple criterion for

7384-452: The ergodic theorem is that the average recurrence time of A is inversely proportional to the measure of A , assuming that the initial point x is in A , so that k 0 = 0. (See almost surely .) That is, the smaller A is, the longer it takes to return to it. The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact manifolds of constant negative curvature of any dimension

7488-446: The ergodicity of a homogeneous flow on a homogeneous space of a semisimple Lie group was given by Calvin C. Moore in 1966. Many of the theorems and results from this area of study are typical of rigidity theory . In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity of the flow was established by Hillel Furstenberg in 1972. Ratner's theorems provide

7592-408: The field of the complex numbers. This equation is useful when modeling mechanical systems with complicated constraints. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces —in which case the differential equations are partial differential equations . Linear dynamical systems can be solved in terms of simple functions and

7696-399: The first to the last claim and assuming that μ ( X ) is finite and nonzero, one has that for almost all x , i.e., for all x except for a set of measure zero. For an ergodic transformation, the time average equals the space average almost surely. As an example, assume that the measure space ( X , Σ, μ ) models the particles of a gas as above, and let ƒ( x ) denote the velocity of

7800-602: The flow through x must be defined for all time for every element of S . More commonly there are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor. In the geometrical definition, a dynamical system is the tuple ⟨ T , M , f ⟩ {\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle } . T {\displaystyle {\mathcal {T}}}

7904-440: The following: where There is no need for higher order derivatives in the equation, nor for the parameter t in v ( t , x ), because these can be eliminated by considering systems of higher dimensions. Depending on the properties of this vector field, the mechanical system is called The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above The dynamical system

8008-407: The founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included

8112-416: The information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. The state-space model can be applied in subjects such as economics, statistics, computer science and electrical engineering, and neuroscience. In econometrics , for example, state-space models can be used to decompose

8216-501: The initial state of the system). A continuous time-invariant linear state-space model is observable if and only if rank ⁡ [ C C A ⋮ C A n − 1 ] = n . {\displaystyle \operatorname {rank} {\begin{bmatrix}\mathbf {C} \\\mathbf {C} \mathbf {A} \\\vdots \\\mathbf {C} \mathbf {A} ^{n-1}\end{bmatrix}}=n.} The " transfer function " of

8320-657: The limit is with respect to the norm on H . In other words, the sequence of averages converges to P in the strong operator topology . Indeed, it is not difficult to see that in this case any x ∈ H {\displaystyle x\in H} admits an orthogonal decomposition into parts from ker ⁡ ( I − U ) {\displaystyle \ker(I-U)} and ran ⁡ ( I − U ) ¯ {\displaystyle {\overline {\operatorname {ran} (I-U)}}} respectively. The former part

8424-513: The map Φ is understood to be a finite time evolution map and the construction is more complicated. The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have

8528-1364: The necessary eigendecomposition to just A + B K {\displaystyle A+BK} . In addition to feedback, an input, r ( t ) {\displaystyle r(t)} , can be added such that u ( t ) = − K y ( t ) + r ( t ) {\displaystyle \mathbf {u} (t)=-K\mathbf {y} (t)+\mathbf {r} (t)} . x ˙ ( t ) = A x ( t ) + B u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)} y ( t ) = C x ( t ) + D u ( t ) {\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)} becomes x ˙ ( t ) = A x ( t ) − B K y ( t ) + B r ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)-BK\mathbf {y} (t)+B\mathbf {r} (t)} y ( t ) = C x ( t ) − D K y ( t ) + D r ( t ) {\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)-DK\mathbf {y} (t)+D\mathbf {r} (t)} solving

8632-588: The number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as capacitors and inductors . 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. The most general state-space representation of a linear system with p {\displaystyle p} inputs, q {\displaystyle q} outputs and n {\displaystyle n} state variables

8736-464: The observed statistics of hyperbolic systems. The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems . But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as

8840-448: The output also depends directly on the input. This is due to the G ( ∞ ) {\displaystyle \mathbf {G} (\infty )} constant in the transfer function. A common method for feedback is to multiply the output by a matrix K and setting this as the input to the system: u ( t ) = K y ( t ) {\displaystyle \mathbf {u} (t)=K\mathbf {y} (t)} . Since

8944-894: The output equation Y ( s ) = C X ( s ) + D U ( s ) , {\displaystyle \mathbf {Y} (s)=\mathbf {C} \mathbf {X} (s)+\mathbf {D} \mathbf {U} (s),} giving Y ( s ) = C ( ( s I − A ) − 1 x ( 0 ) + ( s I − A ) − 1 B U ( s ) ) + D U ( s ) . {\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).} Assuming zero initial conditions x ( 0 ) = 0 {\displaystyle \mathbf {x} (0)=\mathbf {0} } and

9048-470: The output equation for y ( t ) {\displaystyle \mathbf {y} (t)} and substituting in the state equation results in Ergodic theorem Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems ; it is the study of ergodicity . In this context, "statistical properties" refers to properties which are expressed through

9152-714: The output equation for y ( t ) {\displaystyle \mathbf {y} (t)} and substituting in the state equation results in x ˙ ( t ) = ( A + B K ( I − D K ) − 1 C ) x ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=\left(A+BK\left(I-DK\right)^{-1}C\right)\mathbf {x} (t)} y ( t ) = ( I − D K ) − 1 C x ( t ) {\displaystyle \mathbf {y} (t)=\left(I-DK\right)^{-1}C\mathbf {x} (t)} The advantage of this

9256-511: The particle at position x . Then the pointwise ergodic theorems says that the average velocity of all particles at some given time is equal to the average velocity of one particle over time. A generalization of Birkhoff's theorem is Kingman's subadditive ergodic theorem . Birkhoff–Khinchin theorem . Let ƒ be measurable, E (|ƒ|) < ∞, and T be a measure-preserving map. Then with probability 1 : where E ( f | C ) {\displaystyle E(f|{\mathcal {C}})}

9360-563: The periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in

9464-544: The set I ( x ) := { t ∈ T : ( t , x ) ∈ U } {\displaystyle I(x):=\{t\in T:(t,x)\in U\}} for any x in X . In particular, in the case that U = T × X {\displaystyle U=T\times X} we have for every x in X that I ( x ) = T {\displaystyle I(x)=T} and thus that Φ defines

9568-434: 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, n {\displaystyle n} , 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

9672-511: The state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system . If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit . Before the advent of computers , finding an orbit required sophisticated mathematical techniques and could be accomplished only for

9776-438: The state variable values and may also depend on the input variable values. The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method

9880-1124: The state-space model by the following approach: x ˙ ( t ) = [ 0 1 0 0 0 0 1 0 0 0 0 1 − d 4 − d 3 − d 2 − d 1 ] x ( t ) + [ 0 0 0 1 ] u ( t ) {\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)} y ( t ) = [ n 4 n 3 n 2 n 1 ] x ( t ) . {\displaystyle \mathbf {y} (t)={\begin{bmatrix}n_{4}&n_{3}&n_{2}&n_{1}\end{bmatrix}}\mathbf {x} (t).} This state-space realization

9984-412: The strong operator topology as T → ∞. In fact, this result also extends to the case of strongly continuous one-parameter semigroup of contractive operators on a reflexive space. Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick

10088-525: The system is minimum phase . The system may still be input–output stable (see BIBO stable ) 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 removable ). 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

10192-532: The system transfer function's poles (i.e., the singularities where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is asymptotically stable or marginally stable . An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability . The zeros found in the numerator of G ( s ) {\displaystyle \mathbf {G} (s)} can similarly be used to determine whether

10296-441: The transfer function is equal to the characteristic polynomial found by taking the determinant of s I − A {\displaystyle s\mathbf {I} -\mathbf {A} } , λ ( s ) = | s I − A | . {\displaystyle \lambda (s)=\left|s\mathbf {I} -\mathbf {A} \right|.} The roots of this polynomial (the eigenvalues ) are

10400-422: The transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere . This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff . (Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The equidistribution theorem

10504-1049: The values of K are unrestricted the values can easily be negated for negative feedback . The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results. x ˙ ( t ) = A x ( t ) + B u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)} y ( t ) = C x ( t ) + D u ( t ) {\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)} becomes x ˙ ( t ) = A x ( t ) + B K y ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+BK\mathbf {y} (t)} y ( t ) = C x ( t ) + D K y ( t ) {\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+DK\mathbf {y} (t)} solving

10608-496: The variables as constant. The function is called the flow through x and its graph is called the trajectory through x . The set is called the orbit through x . The orbit through x is the image of the flow through x . A subset S of the state space X is called Φ- invariant if for all x in S and all t in T Thus, in particular, if S is Φ- invariant , I ( x ) = T {\displaystyle I(x)=T} for all x in S . That is,

10712-548: Was motivated by problems of statistical physics . A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem , which claims that almost all points in any subset of the phase space eventually revisit the set. Systems for which the Poincaré recurrence theorem holds are conservative systems ; thus all ergodic systems are conservative. More precise information

10816-460: Was proved by Eberhard Hopf in 1939, although special cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924). The relation between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2, R ) was described in 1952 by S. V. Fomin and I. M. Gelfand . The article on Anosov flows provides an example of ergodic flows on SL(2, R ) and on Riemann surfaces of negative curvature. Much of

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