Vector Fields, Control, and Input-Output Systems Bronis law - - PowerPoint PPT Presentation

Vector Fields, Control, and Input-Output Systems

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw XXVIII International Fall Workshop of Geometry and Physics Madrid, September 2-6, 2019

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Overview

In system theory dynamic variables consist of: input, state, output. Input variable is also called control, output is called observation. A system description may consists of: state variable and dynamical equations - a dynamical system; input and state variables, and dynamical equations

• a control system;

input, state and output variables, and dynamical and output equations

• a controlled and observed system.

In all these cases the system description contains the state. A different description, without state variable, consists of Input and output variables, and a causal operator F : input → output - an input-output system (black box).

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Example: discrete time, automata theory

In automata theory: u(t) ∈ U -input, x(t) ∈ X - state x(t + 1) = f (x(t), u(t)), x(0) = x0. y(t) = h(x(t)) = h(x(t), u(t)) where y(t) ∈ Y - output. U, X and Y are called input, state, and output spaces (finite sets). An input-output system is given by a causal operator u(·)

F

− → y(·) where u(t) ∈ U, y(t) ∈ Y .

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Contents of lectures

Lecture 1

• Families of vector fields and polydynamical systems:

how a family of vector fields acts on a manifold (state space) Lecture 2

• Control systems:

controllability, optimal control, feedback equivalence Lecture 3

• Controlled and observed systems:
• bservability, input-state-output systems, input-output maps

Lecture 4

• Realizations of input-output maps:

finding the state space description of a ”black box”

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Overview II

Control systems can be defined as multi-dynamical systems. This means that, roughly, at each moment of time the system can ”choose” one of possible dynamical rules according to which it will evolve. The choice could be random (Nature) or up to a steering rule, or up to a human or other decision. In our lectures the dynamical rules will be given by ODEs described by vector fields on a state space (a manifold). Our aim: Present geometric description and analysis of nonlinear control systems. Formulate basic problems and state general mathematical results of geometric control theory. All geometric objects will be of class C ∞, sometimes of class C ω.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Plan

Lecture 1: Families of vector fields and poly-dynamical systems

• Dynamical systems, flows
• Non-commuting flows
• Lie bracket
• Poly-dynamical systems:
• Chow-Rashevskii theorem
• Orbits, orbit theorem
• Equivalence of families of v. fields

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Dynamical system: vector field and its flow

X will denote the state space of a system. We assume: X - an open subset of Rn or a connected C ∞ manifold, dim X = n. Let f be a smooth vector field on X. The differential equation ˙ x = f (x), x(0) = p defines, for fixed initial point p, a unique trajectory denoted x(t) = f t(p). Let all trajectories be defined for all t ∈ R, i.e., f is complete. Then for each t ∈ R we have a map f t : X → X, p → f t(p). The flow f t of f is the family of maps f t : X → X. It has the group property f t1 ◦ f t2 = f t1+t2, f −t = (f t)−1, f 0 = id.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Poly-dynamical system: a family of vector fields and its poly-flow

Let F = {fu}u∈U be a family of vector fields on a state space X (manifold). Each of f t

u defines the flow of fu, i.e., the family of maps

f t

u : X → X,

t ∈ R.

• Definition. A poly-dynamical system is the family of flows f t

u : X → X defined

by the family F = {fu}u∈U.

• Convention. We assume that all vector fields are complete, i.e. there is no

”escape to infinity in finite time” of trajectories. Thus, f t

u are defined for all t.

Each flow f t

u defines a parametrized group of diffeomorphisms of X, a subgroup

• f Diff ∞(X). We will analyze the action of the group generated by all these

subgroups. Such parametrized families of diffeomorphisms are compositions f tk

uk ◦ · · · f t2 u2 ◦ f t1 u1 : X −

→ X, (⋆) where k ≥ 1, u1, . . . , uk ∈ U, t1, . . . , tk ∈ R. We will call them the poly-flow of the family F.

• Remark. All results will be valid without the completeness assumption. Then
• ne should use pseudogroups of local diffeomorphisms instead of groups of

global diffeomorphisms.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

In many problems it is enough to fix an initial point x0 ∈ X and consider ”broken trajectories” starting from x0 and defined by x(t) = f tk

uk ◦ · · · ◦ f t2 u2 ◦ f t1 u1 (x0),

t = t1 + · · · + tk.

• Question. How can we analyze properties of poly-dynamical systems?

What can we say on the set of points reachable from x0 by broken trajectories? A basic tool will be the Lie bracket of vector fields (the commutator). Using it many questions can be answered without solving differential equations. Note that in the flow and poly-flow the times tj can be positive and negative. Thus a poly-dynamical system defines a group of diffeomorphisms of X. If we take only tj ≥ o then the diffeomorphisms (⋆) form a semigroup. This makes several problems more difficult compared to the case of the group.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Lie bracket of vector fields

If f = (f1, . . . , fn)T, g = (g1, . . . , gn)T are vector fields in local coordinates on Rn, then their Lie bracket is the vector field [f , g](x) = ∂g ∂x (x)f (x) − ∂f ∂x (x)g(x). If f =

• i

fi ∂ ∂xi , g =

• i

gi ∂ ∂xi are treated as differential operators, then the Lie bracket is the commutator, [f , g] = f g − g f =

• i

 

j

∂gi ∂xj fj −

• j

∂fi ∂xj gj   ∂ ∂xi . Basic property: vector fields commute iff their flows commute: [f , g] ≡ 0 ⇐ ⇒ f t ◦ gs = gs ◦ f t.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Commutator of flows

Definition If G is a group and a, b ∈ G, the commutator [a, b] is [a, b] := a−1b−1ab. Claim If vector fields f , g do not commute, applying repeatedly the commutator [g s, f t] = g −s ◦ f −t ◦ g s ◦ f t

• f their flows with small s = t gives movement along the Lie bracket [g, f ].

To see this, for given p ∈ X consider the curve α(t) = f −t ◦ g −t ◦ f t ◦ g t(p). Then α′(0) = 0 and α′′(0) = 2[g, f ](p). Define a map ψt : X → X, ψt = f −t ◦ g −t ◦ f t ◦ g t. Composing this commutator k2 times with t replaced by t/k gives curves βk(t) = ψ t

k ◦ · · · ◦ ψ t k (p),

k2-times which converge to a reparametrized trajectory of [g, f ], namely, with exp denoting the flow, βk(t) − → exp(t2[g, f ])(p) as k − → ∞.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Example: Car un-parking

Let x = (x1, x2, θ) denote position of a car. Its kinematic movements with left-most (L) and rightmost (R) position of the steering wheel describe vector fields on R2 × S1 written in natural coordinates f L = (r cos θ, r sin θ, b)T, f R = (r cos θ, r sin θ, −b)T, where r > 0, b ∈ R are constants. Their Lie bracket is: [f L, f R] = r(−2 sin θ , 2b , 0)T. Approximate movement along the commutator: t − → (x1(0), x2(0) + t2rb, 0) from the initial condition (x1(0), x2(0), 0), if θ ≈ 0. Precisely, (x(t), θ(t)) = (x1(0), x2(0) + t2rb, 0) + O(t3). Illustration of the group commutator moves and the approximate trajectory:

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Example: rotations in R3

Example: two rotations in R3. Consider the vector fields on R3 f1 = x2 ∂3 − x3 ∂2 f2 = x3 ∂1 − x1 ∂3 representing rotations around the axes Ox1 and Ox2, respectively, where ∂i =

∂ ∂xi . Then their Lie bracket is the rotation around Ox3:

[f1, f2] = f3 = x1 ∂2 − x2 ∂1. Conclusion: Repeated commutator of small rotations around Ox1 and Ox2 produces approximate rotation around Ox3.

• Observation. The three vector fields f1, f2, f3 are closed under the Lie bracket:

[f1, f2] = f3, [f2, f3] = f1, [f3, f1] = f2. (⋆) Therefore, the Lie algebra generated by f1 and f2 is Lie {f1, f2} = span {f1, f2, f3}. The vector fields in the Lie algebra span integrable 2-dimensional distribution

• n R3 \ {0}:

D(x) = span {f1(x), f2(x), f3(x)}, x = 0.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

A family of vector fields, its Lie algebra and Chow-Rashevskii theorem

Let F = {fu}u∈U be a family of vector fields on X. Denote the Lie algebra of vector fields generated by F by L = Lie {fu}u∈U which is the smallest family of vector fields on X which contains the family F and is closed under taking linear combinations and Lie bracket. Let L(p) ⊂ TpX be the subspace of tangent vectors L(p) := {g(p) | g ∈ Lie {fu }u∈U}. Theorem (Chow and Rashevskii) If X is connected, n = dim X, and dim L(p) = n, ∀ p ∈ X, then any point of X is reachable from any other point piecewise by trajectories

• f vector fields in F = {fu}u∈U (with forward and backward movements).

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Orbit

Let F = {fu}u∈U be a family of vector fields on X, possibly not complete.

• Definition. The orbit of a point p ∈ X of the family F is the set of points

reachable from p piecewise by trajectories (forward and backward) of the vector fields fu. Precisely, Orb (p) = { f tk

uk ◦ · · · ◦ f t1 u1 (p) | u1, . . . , uk ∈ U,

t1, . . . , tk ∈ R, k ≥ 1 }, where f t

u denotes the flow of the vector field fu and we take t1, . . . , tk so that

the above point is well defined. Then The relation “q belongs to the orbit of p” is an equivalence relation on X. The space X is a disjoint union of orbits (equivalence classes). The orbits are immersive submanifolds of X (next slide), possibly of different dimensions. Thus, the family F defines a singular foliation on X, with the leaves being the orbits of F.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Orbit Theorem

Orbit Theorem (P. Stefan, H. Sussmann) S = Orb (p) is an immersed submanifold of X. If fu are analytic, then its tangent space is TpS = L(p), where L(p) := {g(p) | g ∈ Lie {fu }u∈U}. Lie {fu}u∈U is the Lie algebra of vector fields generated by the family {fu}u∈U. The orbits define the integral foliation (with singularities) of X of the involutive distribution p → L(p). In the smooth case: fu ∈ C ∞, u ∈ U, the tangent space is TpS = Γ(p), where Γ is the smallest distribution on X which contains the vector fields in F (i.e. fu(p) ∈ Γ(p) for all u ∈ U) and is invariant under any flow f t

u , u ∈ U.

In the smooth case L(p) ⊂ Γ(p). In the analytic case L(p) = Γ(p).

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Example of a singular foliation of orbits

Consider again two vector fields on R3 representing rotations around the axes Ox1 and Ox2, f1 = x2 ∂3 − x3 ∂2, f2 = x3 ∂1 − x1 ∂3. Their Lie bracket is the rotation around Ox3: [f1, f2] = x1 ∂2 − x2 ∂1 = f3. Since the three vector fields f1, f2, f3 are closed under the Lie bracket, [f1, f2] = f3, [f2, f3] = f1, [f3, f1] = f2, the Lie algebra generated by f1 and f2 is Lie {f1, f2} = span {f1, f2, f3}. The vector fields in the Lie algebra span 2-dimensional distribution on R3 \ {0}: L(x) = span {f1(x), f2(x), f3(x)}, x = 0, and L(0) = {0}. The Orbit Theorem implies that the orbits of points in R3 \ {0} are 2-dimensional submanifolds. They are spheres centered at {0}. This follows from the fact that f1 and f2 are rotations. The singular foliation of orbits consists of such spheres of radius r > 0 and of a 0-dimensional orbit (the origin). Illustrating figure - next slide

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Foliation of orbits - spheres

p Orb(p)

L(q)

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Positive orbit

For a family of vector fields F = {fu}u∈U we define positive orbit of a point p as the set of points reachable from p with broken forward trajectories of F: Orb +(p) = { f tk

uk ◦ · · · ◦ f t1 u1 (p) | u1, . . . , uk ∈ U,

t1 ≥ 0, . . . , tk ≥ 0, k ≥ 1 }. Recall that the orbit consists of points reachable from p with broken forward-backward trajectories: Orb (p) = { f tk

uk ◦ · · · ◦ f t1 u1 (p) | u1, . . . , uk ∈ U,

t1, . . . , tk ∈ R, k ≥ 1, }. Clearly, we have Orb +(p) ⊂ Orb (p). Let L(p) := {g(p) | g ∈ Lie {fu }u∈U}. Nonempty Interior Theorem 1. If dim L(p) = n for a given p ∈ X, then the positive orbit from p has a nonempty interior in X, int Orb +(p) = ∅.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Interior of Orb+(p) in Orb(p)

Nonempty Interior Theorem 2. If fu are analytic for any u ∈ U, then intS Orb +(p) = ∅, where S = Orb (p) and ”intS” means ”interior in S”. This is not true for fu ∈ C ∞.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Krener’s proof of nonempty interior Thm.

By continuity we have the implication (A) : dim L(p) = n = ⇒ (A′) : dim L(x) for x ∈ nbhd. of p. Step 1 (A) = ⇒ ∃ u1 ∈ U: fu1(p) = 0. Then the trajectory t → f t

u1(p) is a 1-dim. submanifold S1 of X.

Step 2 (A’) = ⇒ ∃ u2 ∈ U s.t. fu2 is not tangent to S1 at a point x1 ∈ S1 arbitrarily close to p. Define the map (t1, t2) − → x = f t2

u2 ◦ f t1 u1 (p).

Its image with small t1, t2 contains a submanifold S2 of dimension 2. Step 3 (A’) = ⇒ ∃ u3 ∈ U and x2 ∈ S2 s.t. fu3 is not tangent to S2 at x2. Thus the image of the map (t1, t2, t3) − → x = f t3

u3 ◦ f t2 u2 ◦ f t1 u1 (x0)

with arbitrarily small t1, t2, t3 contains a submanifold S3 of X of dimension 3. After n steps we obtain a submanifold Sn ⊂ X of dimension n which is contained in the positive orbit. Thus the positive orbit has a nonempty interior.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

When Orb +(p) = Orb (p)? Sometimes negative time trajectories of vector fields in a family F = {fu}u∈U can be replaced or approximated by positive time trajectories (broken trajectories) of other vector fields in the family F. Then Orb +(p) = Orb (p). The following conditions are sufficient for Orb +(p) = Orb (p) ∀ p ∈ X: For any p ∈ X and any u ∈ U ∃ v ∈ U such that −fu(p) = fv(p). More generally, for any p ∈ X and u ∈ U −fu(p) ∈ convex cone generated by {fv(p) : v ∈ U}. If the family F satisfies one of such conditions then all properties of the

• rbit are inherited by the positive orbit. In particular

Corollary of Chow-Rashevskii If F satisfies one of above conditions and X is connected then dim L(p) = dim X for all p ∈ X implies that Orb +(p) = X.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Equivalence of families of vector fields

Let F = {fu}u∈U and F ′ = {f ′

u }u∈U be families of real-analytic vector fields

• n manifolds X and X ′ which are bracket generating at fixed points

p ∈ X and p′ ∈ X, respectively.

• Thm. (Krener 1973) Given points p ∈ X and p′ ∈ X ′, the families F and F ′

are related by a local real-analytic diffeomorphism Φ : (X, p) → (X ′, p′) iff their iterated Lie brackets taken at p and p′ are related by a linear invertible transformation L : TpX → Tp′X ′. It is enough to take the right-iterated Lie brackets relation L [· · · [[fu1, fu2], fu3] · · · , fuk ](p) = [· · · [[fu1, fu2], fu3] · · · , fuk ](p′) for all u1, . . . , uk ∈ U, k ≥ 1. If this holds then the local diffeomorphism Φ is unique and L = dΦ(p).

• Thm. (Sussmann 1974) If the manifolds X and X ′ are connected and simply

connected and the vector fields in the families F and F ′ are complete then the diffeomorphism Φ : X → X ′ is global. Conclusion. If the iterated Lie brackets generate the whole tangent space then they contain all information on the family F up to a (local or global) diffeomorphism.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Lecture 2: Control systems

Plan:

• Basic classes of control systems
• Lie algebraic criteria for controllability properties
• Feedback equivalence (gauge equivalence)
• Optimal control problems
• Pontryagin Maximum Principle
• Sub-Riemannian problems, abnormal extremals

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Control systems: different classes

General nonlinear systems: Σ : ˙ x = f (x, u), x(t) ∈ X, u(t) ∈ U where: X ⊂ Rn - open, connected, or X - differentiable manifold, dim X = n; U - usually a subset of Rm (more generally, of a metric space). Assumed regularity: f (x, u) smooth in both arguments, or at least smooth in x and continuous in u. Special classes of systems: Control-affine: ˙ x = f (x) +

m

• i=1

uigi(x)). Control-linear: ˙ x =

• uigi(x)).

Linear: ˙ x = Ax + Bu, x(t) ∈ Rn, u(t) ∈ U ⊂ Rm. Linear systems are often used as local models in engineering applications.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Lie algebras of control system

Given Σ : ˙ x = f (x, u), x(t) ∈ X, u(t) ∈ U denote fu = f (·, u) and introduce two families of vector fields on X: F = {fu}u∈U, G = {fu − fv | u, v ∈ U}. Claim: analyzing the above families we will be able to determine several properties of Σ without solving the differential equation Σ.

• Definition. The Lie algebra L = Lie F of Σ is the smallest linear space of

vector fields on X which contains F and is closed under Lie bracket.

• Definition. The Lie ideal L0 of system Σ is the smallest linear space of vector

fields on X which contains the family G and is closed under the Lie bracket of elements from L and L0, i.e., (f1 ∈ L, f2 ∈ L0) = ⇒ [f1, f2] ∈ L (equivalently, from F and L0, i.e., (f1 ∈ F, f2 ∈ L0) = ⇒ [f1, f2] ∈ L0 ). Clearly, L0 is closed under the Lie bracket and is an ideal of the Lie algebra L.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Computing Lie algebra and Lie ideal of control-affine system

For control-affine system Σaff : ˙ x = f (x) +

• uigi(x))

the Lie algebra is L = Lie {f , g1, . . . , gm} and the Lie ideal of Σaff is L0 = Lie ideal in L generated by g1, . . . , gm. Equivalently, L0 = the smallest vector space of C ∞ v.f. containing g1, . . . , gm and closed under brackets with f and gi.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Reachable sets and accessibility criteria

Assume the system Σ : ˙ x = f (x, u) be given. Denote by Rt(p) the set of points in X reachable from p in time t, and by R≤t(p) =

• 0≤s≤t

Rs(p)

• the set of points in X reachable from p in time ≤ t.

The class of admissible controls u : [0, T] → U may be:

• piecewise constant controls – no regularity of f with respect to u required,
• piecewise continuous controls – continuity of f (x, u) with respect to (x, u),

together with all iterated partial derivatives with respect to x. Theorem (Sussmann and Jurdjevic 1972) If fu = f (·, u) are C ∞ then, for any t > 0, dim L(p) = n = ⇒ int R≤t(p) = ∅, dim L0(p) = n = ⇒ int Rt(p) = ∅, If fu = f (·, u) are real-analytic then the converse implications also hold. Instead of L = Lie F one may take the Lie module generated by F.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Dimension of reachable sets

Without assuming dim L(p) = n or dim L0(p) = n we can estimate the dimension of the reachable sets. Namely Theorem If fu = f (·, u) are real analytic then, for any t > 0, dim R≤t(p) = dim L(p) and int SR≤t(p) = ∅, where the interior is taken in the orbit S = Orb (p) of the family F = {fu = f (·, u)}u∈U of the system. Similarly, dim Rt(p) = dim L0(p) and int St Rt(p) = ∅, where the interior is taken in the orbit St = Orb t(p) of points reachable in summarized time t (counted with ± signs) with trajectories of the family F. Such orbit Orb t(p) is an immersed submanifold of dimension dim L0(p). Conclusion: the dimensions dim L(p) and dim L0(p) give crucial information on the reachable sets from p.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Lie algebra and Lie ideal of linear systems

Assume U ⊂ Rm, span U = Rm and consider a linear system ˙ x = Ax + Bu = Ax +

• uibi,

x(t) ∈ Rn, u(t) ∈ U where bi - columns of B. The family G is spanned by constant vector fields G = span {b1, ˙ ,bm} and F = {Ax + G}. The only nonzero iterated Lie brackets are [Ax, bi] = −Abi, [Ax, [Ax, bi]] = [Ax, −Abi] = A2bi, . . . , adj

Axbi = [Ax, [Ax, · · · , [Ax, bi] · · · ]] = (−1)jAjbi.

Thus the Lie ideal L0 consists of constant vector fields L0 = span {Ajbi | 0 ≤ j ≤ n − 1, 1 ≤ i ≤ m}, and the Lie algebra is L = span {Ax, L0}.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Reachable sets for linear systems

For linear systems ˙ x = Ax + Bu, x(t) ∈ Rn, u(t) ∈ U ⊂ Rm the reachable points from x(0) = p are given by the formula x(t) = etAp + etA t e−sABu(s)ds. If U = Rm then admissible controls u : [0, t] → U form a linear space and the reachable set Rt(p) is an affine subspace of X = Rn, Rt(p) = etAp + L0, t > 0. Using the expansion esA = si

i! Ai and applying the Cayley-Hamilton theorem

that An = ajAj with j = 0, . . . , n − 1 it is then easy to conclude that Theorem The linear system with U = Rm is completely controllable, i.e., any point is reachable from any other point iff dim L0 = n or, equivalently, iff rank [B, AB, . . . , An−1B] = n. Moreover, for any p and t > 0 the same condition is equivalent to Rt(p) = Rm. This elementary criterion was proved by R. Kalman in the beginning of 60-ties.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Complete controllability and time reversible systems

Recall: a system ˙ x = f (x, u), x ∈ X, u ∈ U, is completely controllable or controllable if any point x1 ∈ X can be connected with any other point x2 ∈ X by an admissible (forward) trajectory. Generally, the set of reachable points R(p) = ∪t≥0Rt(p) is difficult to determine. When the reversibility property (R) −fu(p) ∈ convex cone generated by {fv(p) : v ∈ U}. holds for all p ∈ X, u ∈ U, then the positive orbit and the obit from p of the family F = {f (·, u)}u∈U coincide and criteria for Orb (p) = X can be used: Corollary of Chow-Rashevskii If X is connected and the reversibility condition (R) holds then condition dim L(p) = n implies that the system is controllable. Recall that a vector field f is called Poisson stable at p ∈ X if for any neighbourhood V of p and T > 0 there exist t1, t2 > T such that f t1(p) ∈ V and f −t2(p) ∈ V . This can be used as another reversibility condition. Theorem (B. Bonnard 1981) Assume that f is complete and consider a system ˙ x = f (x) +

• uigi,

u ∈ U ⊂ Rm, 0 ∈ int U. If X is connected and the set of Poisson stable points for f is dense in X then dim Lie {f , g1, . . . , gm}(p) = n for all p ∈ X implies complete controllability.

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Equivalence problems

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

State and feedback equivalence

Better understanding of control systems give natural notions of equivalence. Consider two systems Σ : ˙ x = f (x, u),

• Σ :

˙ ˜ x = ˜ f (˜ x, ˜ u). Definition 1. Σ and Σ are state equivalent if Σ can be transformed into Σ via ˜ x = Φ(x), ˜ u = u where Φ : X → X is a diffeomorphism of the state space. Definition 2. Σ and Σ are feedback equivalent if Σ can be transformed into Σ via an invertible transformation ˜ x = Φ(x), ˜ u = Ψ(x, u). For control-affine systems Ψ(x, u) = α(x) + β(x)u. The affine transformation ˜ u = Ψ(x, u) = α(x) + β(x)u can be called control gauge transformation, similarly as the nonlinear one ˜ u = Ψ(x, u).

Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Criteria for state equivalence of control systems

Consider two real-analytic in x systems with the same control space U, Σ : ˙ x = f (x, u),

• Σ :

˙ ˜ x = ˜ f (˜ x, ˜ u) They are (locally, globally) state equivalent iff the families of vector fields F = {fu}u∈U and ˜ F = {˜ fu}u∈U are related by a (local, global) diffeomorphism. Corollary of Thm. on equivalence of F and ˜ F: Let dim L(p) = n = dim ˜ L(˜ p). A criterion of local (global) equivalence around points p ∈ X and ˜ p ∈ ˜ X is the relation of iterated Lie brackets at p and ˜ p by a linear L : TpX → T˜

p ˜

X. The same problem considered for control-affine systems Σ : ˙ x = f (x) +

• i=1,...,m

uigi(x),

• Σ :

˙ ˜ x = ˜ f (˜ x) +

• i=1,...,m

ui ˜ gi(˜ x) reduces to equivalence of finite families F = {g0, . . . , gm} and ˜ F = {˜ g0, . . . , ˜ gm}, where g0 = f and ˜ g0 = ˜ f . Again, the criterion of iterated Lie brackets at points p and ˜ p (Lecture 1) is a criterion for state equivalence around these points.

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Linear systems

We say that a system Σ is state linearizable if it is state equivalent to a controllable linear system ˙ x = Ax + Bu. Theorem (Respondek, Sussmann 1981) System Σ : ˙ x = f (x) +

• uigi(x)

is locally state linearizable to a controllable system iff the vector fields adi

f gj,

i = 0, . . . , n − 1, j = 1, . . . , m commute and span the whole tangent space. Here ad0

f gj = gj,

adf gj = [f , gj], . . . , adi

f gj = [f , · · · , [f , gj]] with f appearing i times. Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Involutive distributions, feedback linearization

Recall that a distribution D(x) = span {f1(x), . . . , fr(x)} generated by linearly

• indep. vector fields f1, . . . , fr is involutive if ∃ functions ψk

ij such that

[fi, fj] =

• k

ψk

ijfk,

∀ i, j.

• Definition. We say that a system Σ is feedback linearizable if it is feedback

equivalent to a controllable linear system. Theorem (W. Respondek + B.J. 1980) System Σ : ˙ x = f (x) +

• uigi(x)

is locally feedback linearizable iff the distributions Dk(x) = span {adi

f gj, 0 ≤ i ≤ k, j = 1, . . . , m}

are involutive and of constant dimension, for all k = 0, 1, . . . , n − 2, and Dn−1(x) is the whole tangent space. Similar conditions were obtained by Hunt, Su, Mayer (1983). The distributions D0 ⊂ D1 ⊂ · · · are important invariants of system Σ!

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Optimal control problems

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Optimal control problems

Suppose we are given a control system Σ : ˙ x = f (x, u), x(t) ∈ X, u(t) ∈ U where U ⊂ Rm is a closed subset. Problem 1. Given x0, x1 ∈ X, find control u : [0, T] → U such that the pair (x(·), u(·)) solving Σ minimizes J(u) = T L(x(t), u(t))dt and satisfies end-point conditions x(0) = x0, x(T) = x1. Problem 2 (more general). Given initial and final submanifolds M0, M1 ⊂ X, find a pair (x, u) : [0, T] → X × U fulfilling Σ which minimizes J(x, u) = ϕ(X(T)) + T L(x(t), u(t))dt and satisfies x(0) ∈ M0, x(T) ∈ M1.

• Remark. If the system equation is ˙

x = u, then both problems are classical variational problems of minimizing ϕ(X(T)) + T

0 L(x(t), ˙

x(t))dt with given boundary conditions.

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Necessary conditions of optimality - definitions

Necessary conditions for optimality in the classical problem of minimization of T

0 L(x(t), ˙

x(t))dt are the Euler-Lagrange equations. For an optimal control problem, where the set U is closed (often compact)

• ne can not use usual variations as in the case of open U.

Often optimal control takes values in the boundary ∂U. New needle variations were introduced giving new type of necessary conditions.

• Definition. The Hamiltonian H : T ∗X × U → R of the problem is

H(p, x, u) = p f (x, u) + ψ0L(x, u) where ψ0 ∈ R is a constant and p ∈ T ∗

x M is called adjoint variable.

(Here p f denotes duality product of covector p with vector f .)

• Definition. Hamiltonian equations of the problem can be written

˙ x = ∂H(p, x, u) ∂p = f (x, u) ˙ p = −∂H(p, x, u) ∂x = −p ∂f (x, u) ∂x − ψ0 ∂L(x, u) ∂x where ψ0 ∈ R is a constant and, implicitly, we use local coordinates.

• Definition. Given (p(t), x(t), u(t)) ∈ T ∗X × U, the maximum condition is

max

v∈U H(p(t), x(t), v) = H(p(t), x(t), u(t)),

t ∈ [0, T].

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Necessary conditions of optimality - Pontryagin Maximum Principle

Consider the problem Minimize T L(x(t), u(t))dt subject to ˙ x = f (x, u), u(t) ∈ U. Pontryagin Maximum Principle. If (u(t), x(t)), t ∈ [0, T], solves the minimization problem with given boundary conditions x(0) = x0, x(T) = x1 then there exists ψ0 ≤ 0 and the adjoint function p(t) ∈ T ∗

x(t)X such that:

• 1. (ψ0, p(t)) = 0 on [0, T].
• 2. The adjoint variable p(t) satisfies the adjoint equation

˙ p = −p ∂f (x, u) ∂x − ψ0 ∂L(x, u) ∂x .

• 3. The maximum condition is fulfilled

max

v∈U H(p(t), x(t), v) = H(p(t), x(t), u(t)),

t ∈ [0, T].

• 4. The Hamiltonian is constant along the extremal:

H(p(t), x(t), u(t)) = const, t ∈ [0, T].

• 5. If the boundary conditions are x(0) ∈ M0 and x(T) ∈ M1, then p satisfies

the transversality conditions p(0) ∈ (Tx(0)M0)⊥ and p(T) ∈ (Tx(T)M1)⊥.

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Pontryagin Maximum Principle - remarks

The state equation ˙ x = f (x, u) and the adjoint equation ˙ p = −p ∂f (x, u) ∂x − ψ0 ∂L(x, u) ∂x , taken together, are Hamiltonian equations for the Hamiltonian H(p, x, u) = p f (x, u) + ψ0L(x, u). The control value u(t) can often be determined from the maximum condition max

v∈U H(p(t), x(t), v) = H(p(t), x(t), u(t)),

t ∈ [0, T]. It is then a function u(x, p), precisely, u(t) = u(x(t), p(t)). When u = u(x, p) is plugged to Hamiltonian equations, one obtains a system

• f 2n differential equations for 2n unknown functions x(t) and p(t).

These equations together with boundary conditions x(0) ∈ M0, x(T) ∈ M1, p(0) ∈ (Tx(0)M0)⊥, p(T) ∈ (Tx(T)M1)⊥ have to be solved in order to determine the extremals. One may verify that there are here 2n scalar boundary conditions.

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Pontryagin Maximum Principle - remarks In its full form the Maksimum Principle was published in the beginning of 60-ties. Its was initialized by L.S. Pontryagin in the end of 50-ties. Its full version was stated and proved by his young co-workers: R.V. Gamkrelidze for linear systems; V.G. Boltianskii in the general case. Boltianskii introduced the needle variations uτ,v,ε(t) =

• v

if t ∈ (τ − ε, τ] u(t)

• therwise

depending on τ ∈ (0, T], v ∈ U and ǫ ≥ 0. These variations are admissible even if u(t) ∈ ∂U. This enabled him to prove PMP.

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Example: sub-Riemannian energy

Suppose D is a smooth distribution of rank m < n on X given by the field of tangent planes x → D(x) ⊂ TxX. Let g be a Riemannian metric on D. Then for any curve x : [0, T] → M everywhere tangent to D we can define its length and energy L(x(·)) = T

• g( ˙

x(t), ˙ x(t))dt, E(x(·)) = T 1 2g( ˙ x(t), ˙ x(t))dt. By definition, a geodesic of the sub-Riemannian structure defined by (D, g) is a smooth curve that is everywhere tangent to D and locally minimizes the length L or, equivalently, the energy E. We can apply the Maximum Principle in order to try to find a geodesic joining two points x0 and x1 in X. Let the distribution be spanned by orthonormal with respect to g vector fields g1, . . . , gm so that D(x) = span {g1(x), . . . , gm(x)}. Any C 1 curve tangent to D can be written as a trajectory of the control system ˙ x =

• ujgj(x),

u(t) = (u1, . . . , um) ∈ Rm. Any local piece of geodesic can be defined as an optimal trajectory of the control system joining x0 to x1. According to PMP, it should satisfy the Hamiltonian equations with the Hamiltonian, corresponding to the energy H =

• ujp gj + 1

2ψ0g(x, x) =

• ujp gj + 1

2ψ0

• u2

j . Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Example: sub-Riemannian energy

Given the Hamiltonian of the problem H =

• ujp gj + 1

2ψ0

• u2

j

we can write the necessary conditions of PMP as existence of ψ0 ≤ 0 and p(t) ∈ T ∗

x(t)X such that (ψ0, p(t)) never vanishes and the Hamiltonian

equations hold ˙ x =

• ujgj(x)

= ∂H/∂p, ˙ p = −

• ujp ∂gj

∂x (x) = −∂H/∂x. Multiplying (ψ0, p) by a nonzero positive constant does not change the conditions, thus we may assume that ψ0 = −1 (normal extremal) or ψ0 = 0 (abnormal extremal). The maximum condition implies that the derivatives ∂H/∂uj = pgj + ψ0uj vanish along the extremal. Thus, for the normal extremal uj = pgj. This control can be plugged to the Hamiltonian which gives H = 1/2

• (pgj(x))2.

Normal extremals can be found as solutions of Hamiltonian equations for H.

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Abnormal extremals

For ψ0 = 0 (abnormal extremal ) the maximum condition implies that, along the extremal, pgj = 0, j = 1, . . . , m. This system of equations does not immediately allow to find the optimal

• control. However, differentiating them along the extremal gives
• i

uip[gi, gj] = 0 which is a linear homogeneous system for u. Its solutions depend on singularities of the matrix (p[gj, gj]). We have to stop our discussion here. Abnormal extremals do not appear in Riemannian geometry. They appear in sub-Riemannian geometry but it was long believed that they can not be minimizing, that is, they do not describe sub-Riemannian geodesics. Several wrong proofs of this fact were published until R. Montgomery (1994) found an abnormal minimizing extremal with a rank 2 distribution. It was next proved by W.S.Liu and H. Sussmann (1995) that for generic problems with 2-dimensional distribution abnormal extremals do minimize! Finally, in 2006 Y. Chitour, F. Jean and E. Tr´ elat showed that for rank ≥ 3 generic sub-Riemannian structures minimizing abnormal extremals do not exist!

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Plan - Lecture 3

Lecture 3: Controlled and observed systems Input-state-output systems (I-S-O systems) Part I

• Observed dynamical systems
• Controlled and observed systems
• Observability

Part II

• Input-output maps of I-S-O systems
• Taylor series of I-O map, Volterra series
• Black box input-output systems
• Realization problem

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Observed dynamical systems

Let f be a smooth vector field on a manifold M and let f t : X → X be its flow. Let h : X → Y = Rp be a smooth function. An observed system consists of ˙ x = f (x) y = h(x) where: x(t) ∈ X is its state; y(t) ∈ Y = Rp is observation. Such systems model problems in applications where only certain part of the state, or only some functions of it, can be observed or measured. The observation problem is the problem of determining the state function x(t) while observing only y(t). This is nontrivial when dim Y < dim X. It is enough to think of the case where the observation is scalar as the general case is analogous.

• Example. Suppose that two masses, joined by an ideal spring, oscillate.

The vertical position y(t) = x1(t) of one of them is observed. Is it possible to determine the position x2(t) of the other one?

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Observed dynamical systems - Example

Example. The dynamic equations of two masses, joined by an ideal spring, are ˙ x1 = v1, ˙ v1 = k m1 (x2 − x1 + c) ˙ x2 = v2, ˙ v2 = k m2 (x1 − x2 − c) with m1, m2 - masses of the balls, k > 0 - the Hook constant of the spring, c - the neutral length of the spring. The whole state is x = (x1, v1, x2, v2) but only one component y = h(x) = x1 is observed. However, knowing x1(t) on an arbitrarily short time interval it is possible , using the system equations, to determine x1(t), v1(t), x2(t) and v2(t) for all t. Important: to determine them we have to know the equations of the system!

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Observability of dynamical systems

Consider an observed system ˙ x = f (x), y = h(x). Given an initial point p ∈ X and the trajectory xp(t) = f t(p) starting from p, the observation function equals to yp(t) = h(xp(t)) = h ◦ f t(p).

• Definition. Two states p1, p2 ∈ X are called indistinguishable on [0, T) if

yp1(t) = yp2(t), for t ∈ [0, T). States p1, p2 ∈ X are called small time indistinguishable if ∃ ε > 0 such that yp1(t) = yp2(t), for t ∈ [0, ε). Otherwise the two states are called distinguishable on [0, T) (resp. small time distinguishable). The system is called observable on [0, T) (resp. small time observable) if any two states p1 = p2 in X are distinguishable on [0, T) (resp. small time distinguishable). Above 0 < T ≤ ∞. Observability on [0, T) requires that the trajectories of f are defined on [0, T).

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Observability of dynamical systems

Assume Y = Rp. Then the function h : X → Rp can be differentiated along f . Writing in local coordinates x1, . . . , xn f (x) =

• i

f i(x) ∂ ∂xi and f (h) =

• i

f i(x) ∂h ∂xi we can define the k-th iterated derivative f k(h) = f · · · f (h). The following facts are direct consequences of definitions of observability.

• If two points p1 and p2 are small time indistinguishable then

f k(h)(p1) = f k(h)(p2) for k ≥ 0. In the analytic case the converse also holds.

• The observed dynamical system is observable on [0, T) iff the family of

time-shifted functions {h ◦ f t}t∈[0,T) on X distinguishes points of X. Conclusions: • Even a scalar observation y(t), known on an interval of time, may be enough for distinguishing different multidimensional states.

• If we measure the observation function y(t) = h(x(t)) with some delays τi:

y(t + τi) = h(f τi (x(t))), then taking enough such measurements we may distinguish all states.

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Observability of linear systems

Consider a linear observed system Ω : ˙ x = Ax, y = Cx, x ∈ Rn, y ∈ Rp. Define the observability matrix      C CA . . . CAn      Theorem (Kalman) The following conditions are equivalent: System Ω is observable on [0, ∞). System Ω is small time observable. The rank of the n × np observability matrix is n (maximal). If y is scalar then C is a row vector and the observability matrix is n × n.

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Observability of control systems - definitions

Consider a control system with observation (controlled and observed system) ∆ : ˙ x = f (x, u), y = h(x). Given an initial point p ∈ X, the trajectory x(t, p, u(·)) starting from p depends also on the control function t → u(t) ∈ U. The observation function equals to yp,u(t) = h(x(t, p, u(·))).

• Definition. Two states p1, p2 ∈ X are indistinguishable on [0, T) (resp.

small time indistinguishable) if the observation functions yp1,u(t) and yp2,u(t) corresponding to starting points p1 and p2 coincide on [0, T) (resp. on an interval [0, ε) with some ε > 0) for any admissible control function u on [0, T) (resp. on [0, ε)). Otherwise they are called distinguishable on [0, T) (resp. small time distinguishable). The system is called observable on [0, T) (resp. small time observable) if any two states p1, p2 in X are distinguishable on [0, T) (resp. on [0, ε)).

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Observability of linear control systems

Consider a linear controlled and observed system ˙ x = Ax + Bu, y = Cx, x ∈ Rn, u ∈ Rm, y ∈ Rp.

• Theorem. The following conditions are equivalent:

Linear observed dynamical system ˙ x = Ax, y = Cx is observable on [0, T). Linear controlled and observed system ˙ x = Ax + Bu, y = Cx is observable on [0, T). Above 0 < T ≤ ∞ is arbitrary. The equivalence also holds for small time

• bservability.
• Conclusion. For linear systems the rank condition for observability of a system

without control applies also to the system with control. This will not be true for nonlinear systems!

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Criteria for observability of control systems

Consider again a controlled and observed system of class C r, r ∈ {∞, ω}, ∆ : ˙ x = f (x, u), y = h(x). The following two objects are useful in studying observability of ∆. Consider again the family vector fields defined by the control system F = {fu = f (·, u)}u∈U. Assume Y = Rp and consider the family of iterated Lie derivatives of the function h along the vector fields in F. We obtain the family of functions IO = { fuk · · · fu1h : k ≥ 0, u1, . . . , uk ∈ U }. Theorem 1. If the family IO distinguishes points on X then system ∆ is small time observable. Consider another family of functions OO = { h ◦ f t1

u1 ◦ · · · ◦ f tk uk

: k ≥ 0, u0, . . . , uk ∈ U, t1, . . . , tk ≥ 0 }. The next result is, essentially, a reformulation of the definition of observability. Theorem 2. The system ∆ is observable on [0, ∞) if and only if the family of functions OO distinguishes points on X.

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Part II Input-output approach

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Input-output approach

Consider an initialized controlled and observed system ˙ x = f (x, u), x(0) = x0 y = h(x), where u(t) ∈ U, x(t) ∈ X and y(t) ∈ Y . With given initial point x0 it defines a map u(·)

F

− → y(·) from the space of controls (inputs) u : [0, T] → U to observations (outputs) y : [0, T] → Y . The map is called the input-output map of the system. (Here we assume that for any u(·) the trajectory x(t) is defined on [0, T].) This map is important for applications and for theoretical reasons.

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Explicit formulas for I-O maps: Linear systems

Consider a linear system Λ : ˙ x = Ax + Bu, x(0) = 0 y = Cx, where x(t) ∈ X = Rn, u(t) ∈ Rm and y(t) ∈ Rp. Here A, B and C are constant matrices of appropriate size. The solution x(t) and then y(t) can be expressed explicitly in terms of u using the Cauchy formula. We obtain the input-output description of the system given by the integral operator y(t) = (Ku)(t) = t k(t − s)u(s)ds. Here k is the p × m matrix valued function: k(t) = CetAB. If x(0) = x0 = 0 is fixed then (Ku)(t) = Ce(tA)x0 + t

0 k(t − s)u(s)ds.

The state variable x(t) is eliminated in this description. For fixed x(0) the operator K describes completely the input-output behaviour u(·) − → y(·).

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I-O map of control affine system Consider an initialized control-affine controlled and observed system ˙ x = f (x) +

m

• i=1

uigi(x), x(0) = x0, y = h(x). For regular controls it defines the input output map u(·) − → y(·). Controls can be taken piecewise continuous or measurable, bounded. In general, the input-output map can not be computed explicitly. However: We can compute its Taylor series at u ≡ 0! This will require finding the flow f t of f and some differentiation and integration.

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Input-output maps: Volterra series

Consider an analytic control-affine system with scalar control u(t) ∈ R, ˙ x = f (x) + ug(x), x(0) = x0, y = h(x). If ˙ x = f (x), x(0) = x0, has a solution on [0, T] then its input-output map on [0, T] can be expressed via a Volterra series (converging if u(t) ≤ M - small) y(t) = (Fu)(t) =

∞

• k=0

t

• sk
• · · ·

s2

• Wk(t, s1, . . . , sk)u(s1) · · · u(sk)ds1 · · · dsk,

Here the iterated integral is taken on the simplex ∆ = {0 ≤ s1 ≤ · · · ≤ sk ≤ t}. The series is the Taylor series of the map u[0,t] → y(t) for fixed t, at u ≡ 0 !! The first terms of the series are y(t) = W0(t)+

t

• Wk(t, s1)u(s1)ds1+

t

• s2
• W2(t, s1, s2)u(s1)u(s2)ds1ds2 + . . . .

Here W0(t) = h(f t(x0)). Further Volterra kernels W1(t, s1), W2(t, s1, s2) e.t.c. will be given by explicit formulas.

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Input-output maps: Volterra series

For linear systems W0(t) = CetAx0, W1(t, s1) = Cet−s1B and Wk = 0 for k > 1. For control affine system the Volterra kernels Wk(t, s1, . . . , sk) are defined by the system data f , g and h. Namely, we modify h and g using the flow f t of f introducing ˜ ht(z) = h ◦ f t(z) = h(f t(z)) and ˜ gs(z) = ((df s)−1g) ◦ f s(z) which are h and g in the ”moving coordinate system” z = f −t(x). The Volterra kernels are equal to the iterated derivatives of the function ˜ ht along the vector fields ˜ gs, evaluated at x0: Wk(t, s1, . . . , sk) = (˜ gs1 · · · ˜ gsk ˜ ht)(x0). The use of Volterra series in the theory of input-output systems and in control theory was promoted by R. Brockett, P. Crouch, A. Krener, C. Lesiak, ....

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Volterra expansion for multi-dimensional control

In the case of analytic multi-dimensional control system ˙ x = f (x) +

m

• i=1

uigi(x), x(0) = x0, y = h(x) the Volterra expansion of the input-output map is similar, (Fu)(t) =

∞

• k=0
• i1...ik

t

• sk
• · · ·

s2

• W i1...ik

k

(t, s1, . . . , sk)ui1(s1) · · · uik (sk)ds1 · · · dsk.

• Thm. The series converges absolutely for t ∈ [0, T] and measurable |u(t)| ≤ M

with M small enough. Here the Volterra kernels are expressed in terms of the system by the formulas W i1...ik

k

(t, s1, . . . , sk) = (˜ gi1,s1 · · · ˜ gik ,sk ˜ ht)(x0), where 1 ≤ i1, . . . , ik ≤ m and ˜ ht(x) = h ◦ f t(x) and ˜ gi,s(x) = (df s(x))−1gi(f s(x)). Here again we use the notation gh for the Lie derivative of the function h along a vector field g (also denoted Lgh).

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Control-linear systems

A simpler case is that of analytic control-linear systems ˙ x =

• uigi(x),

x(0) = x0, y = h(x). For such system the drift f is zero, thus ˜ ht = h and the vector fields ˜ gi,s = (df s(x))−1gi(f s(x)) = gi are independent of s. Then the Volterra kernels are constant and can be taken outside the integral, (Fu)(t) =

∞

• k=0
• i1...ik

W i1...ik

k

t sk · · · s2 ui1(s1) · · · uik (sk)ds1 · · · dsk, W i1...ik

k

= (gi1 · · · gik h)(x0). This form of the expansion is especially convenient for algebraic manipulations. In the case of control-affine system ˙ x = f (x) +

m

• i=1

uigi(x), x(0) = x0, y = h(x) we can use a simple trick of adding an additional ”control” component u0 ≡ 1. Denoting f = g0 we can replace the control-affine system by the control-linear

• system. The resulting expansion with u0 ≡ 1 is then called Chen-Fliess series.

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Convergence

It is easy to check that if the absolute values of the inputs ui(·), i = 1, . . . , m, are bounded by a constant M then the multiple integrals I i1···ik

t

(u) = t sk · · · s2 ui1(s1) · · · uik (sk)ds1 · · · dsk in the series satisfy the estimates |I i1···ik

t

| ≤ Mktk(k!)−1. If the Volterra kernels satisfy the estimates |W i1···ik

k

| ≤ CRkk!, for some C > 0 and R > 0 (which hold for analytic systems) then the Volterra series converges for all t < (MR)−1.

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Example: unicycle

Consider the system, called unicycle, described by the equations ˙ x1 = u1 cos θ, ˙ x2 = u1 sin θ, ˙ θ = u2 and let the observation (output) be scalar y = x1. The system is analytic and the drift term is zero, therefore it is represented by two vector fields g1, g2, an

• bservation function h, where

g1 = cos θ ∂ ∂x1 + sin θ ∂ ∂x2 , g2 = ∂ ∂θ , and h = x1. Earlier claims are applicable here with ˜ gi,t = gi and the Volterra kernels are constant (because the drift term is zero). Computing the Lie derivatives we get (g2)2rg1h = (−1)r cos φ, (g2)2r+1g1h = (−1)r+1 sin φ, r ≥ 0, with all others being zero. Choosing the initial point as x0 = (0, 0, 0) we obtain the Volterra kernels W 2···21

2r+1 = (−1)r

and all other vanishing. Thus the Volterra expansion is, with k = 2r + 1,

∞

• r=0

(−1)r t sk · · · s2 u2(s1) · · · u2(sk−1)u1(sk)ds1 · · · dsk−1dsk and it converges for all t.

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The example is a particular case of a broad class of nonlinear strictly causal

• perators which are expressed in the form of Volterra series.

Forgetting about a system with input, state and output, we may consider nonlinear maps u(·) − → y(·) describing a system with inputs and outputs, only. They can be described by Volterra series. Assume that the input and output are scalar, i.e., U = Y = R. A general Volterra series operator is given by the expansion (Fu)(t) =

N

• k=0

t sk · · · s2 Wk(s1, . . . , sk, t)u(s1) · · · u(sk)ds1 · · · dsk−1dsk, where N is finite or N = ∞. In the case of N = ∞ we assume convergence of the expansion. For example, this is guaranteed when T < ∞, input u is bounded, and the Volterra kernels Wk are continuous and satisfy the growth estimate |Wk(s1, . . . , sk, t)| ≤ CRkk! for some positive constants C and R, with R depending on T and bounds on u.

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Recapitulation

Controlled and observed systems, also called input-state-output systems, (ISO) : ˙ x = f (x, u), y = h(x) form the most general class of finite dimensional deterministic systems. From theoretical point of view controllability and observability are dual notions as it will be seen in the next lecture. The duality is a consequence of duality between points x ∈ X and functions h : X → R given by (x, h) → h(x). In applications often the most important variables are the input u and output y while x is an auxiliary variable needed for system equations. In the next lecture we will study general input-output maps (no expansion) u(·)

F

− → y(·) where the state variable x is absent. We already saw that an initialized input-state-output (ISO) system above defines an input-output operator. Of central theoretical importance is the converse realization problem, to be discussed in the next lecture:

• Given an input-output map F, does there exist an I-S-O system which has F

as its input-output map?

• If it does, how can we construct it? How to construct the state x?

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Realization problem

The realization problem can be stated, in engineering terms, as follows. Given a black box, with input functions u(t) ∈ U and outputs y(t) ∈ Y which behaves according to a nonlinear map F : u(·) → y(·).

• Find its state description:

a state space X (a differentiable manifold) and equations ˙ x = f (x, u), x(0) = x0 y = h(x, u) which have the same input-output map F as the black box.

• Find a minimal state space X which has this property.

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Plan - Lecture 4

Lecture 4: Input-output maps and their state realizations

• Realization problem
• Realizations of linear input-output maps
• Nonlinear causal maps, input semigroup and group
• Realizability conditions
• Construction of a realizations

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Realization problem

The realization problem can be stated, in engineering terms, as follows. Given an input-output system (black box) with input functions u(t) ∈ U and

• utputs y(t) ∈ Y , which behaves according to a nonlinear map

F : u(·) − → y(·).

• Find, if exists, its state description:

a state space X (a differentiable manifold) and equations ˙ x = f (x, u), x(0) = x0 y = h(x) which have the same input-output map F as the black box.

• Find a minimal state space X which has this property.

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Causal F If the operator F : u(·) → y(·) is going to have a realization ˙ x = f (x, u), x(0) = x0 y = h(x) then it has to be causal , i.e. its output at time t should depend on the input up to time t, only: u|[0,t) = v|[0,t) = ⇒ (Fu)(t) = (Fv)(t).

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Linear realization problem

A general linear input-output system can be given by an integral operator y(t) = (Fu)(t) = t k(t, s)u(s)ds. (⋆) For autonomous (i.e. time invariant) systems: k(t, s) = ˜ k(t − s). A linear input-state-output system Λ : ˙ x = Ax + Bu, x(0) = 0, y = Cx has the input-output map given by the Volterra integral (⋆) with the kernel k(t, s) = ˜ k(t − s) = Ce(t−s)AB. (⋆⋆) .

The linear realization problem: Given a general F as in (⋆), find, if exist, matrices A, B and C such that (⋆⋆) holds.

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Solution of linear realization problem - existence Define Hankel matrix of i-o operator Ku = k0u(t) + t

0 k(t, s)u(s)ds :

H =      k1 k2 k3 . . . k2 k3 k4 . . . k3 k4 k5 . . . . . . . . . . . . ...      , where ki = ∂i−1 ∂ti−1 k(t, 0)|t=0 = ∂i−1 ∂ti−1 ˜ k(0), i ≥ 1. The following result follows from a theory developed by Rudolf Kalman. Theorem (existence). The operator K has a realization Λ if and only if (a) k(t, s) is shift invariant, i.e., k(t, s) = ˜ k(t − s), (b) k(t, s) is analytic for t ≥ s, (c) rank H is finite. If these conditions are satisfied, then the minimal dimension of the realization is equal to n = rank H.

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Solution of linear realization problem - uniqueness Theorem (uniqueness). Two realizations of minimal dimension of the same operator K are related by a linear isomorphism of the state spaces. A realization is of minimal dimension iff it is controllable and

• bservable.

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Construction of linear realization

If the i-o operator K has a realization Λ then ˜ k(t) = CetAB, thus ki = CAi−1B and the Hankel matrix of K is H =      CB CAB CA2B . . . CAB CA2B CA3B . . . CA2B CA3B CA4B . . . . . . . . . . . . ...      , Such H suggests a construction of realization when H is defined by a general K. For simplicity we assume that the input and output are 1-dimensional. Then:

• The state space X is the span (in the space of infinite column vectors) of all

columns of H and has dimension n = rank H, thus X ≃ Rn.

• The linear operator A is defined by the assignment

Hi − → AHi := Hi+1, where Hi is the i-th column of H.

• The operator B : R → X is defined by 1 ∈ R → H1 - the first column in H.
• The operator C : X → R is the projection from infinite column vectors to

their first components, (h1, h2, . . . )T − → h1.

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General realization problem again Given an input-output system (black box) with inputs u(t) ∈ U and

• utputs y(t) ∈ Y , represented by a nonlinear map

F : u(·) − → y(·). Find, if exists, its state description, i.e.: a state space X (a differentiable manifold), and equations ˙ x = f (x, u), x(0) = x0, y = h(x), which give the same input-output map F. Find a minimal state space X which has this property.

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Recall that an operator F : u(·) → y(·) is called causal if the value

• f y(t) is uniquely defined by the values of u on the interval [0, t),

u|[0,t) = v|[0,t) = ⇒ F(u)(t) = F(v)(t) for any t in the domain of u and v. We assume that the output functions y(·) are continuous and, if they are defined on right-open interval [0, T), they have a continuous limit at T. In particular, if the input u(·) is defined on an interval [0, Tu), only, we assume that the value F(u)(Tu) is defined.

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Causal F and its response function

• Notation. For causal F and input function u : [0, Tu) → U we denote by

F, u the value of y = F(u) at the end of the action of u, F, u = (Fu)(Tu). If F is causal then the following function, called response function, u − → F, u =: R(u) is well defined on finite inputs u : [0, Tu) → U. Note that two finite inputs u : [0, Tu) → U and v : [0, Tv) → V can be concatenated, giving a new finite input denoted u v, (u v)(t) =

• u(t)

for t ∈ [0, Tu), v(t − Tu) for t ∈ [Tu, Tu + Tv).

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Semigroup of inputs

For simplicity we shall assume that admissible inputs are piecewise constant, with values in a set U which has at least two elements. Denote by a = (t1, u1) · · · (tk, uk) a piecewise constant function a : [0, Ta) → U defined as follows (Fig. 1) a(t) = ui, for t ∈ [Ti−1, Ti), where T0 = 0 and Ti = t1 + · · · + ti, i = 1, . . . , k, and Ta = Tk. When b = (s1, v1) · · · (sm, vm) then a b = (t1, u1) · · · (tk, uk)(s1, v1) · · · (sm, vm). The family SU of such inputs with the above product is called

input semigroup. The empty input is a unit in SU.

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Existence of realizations

Assume that: U ⊂ Rm is compact; the time horizon is 0 < T ≤ ∞; U is the set of piecewise constant functions u : [0, T) → U; Y is the set of continuous functions y : [0, T) → Y = Rp having continuous limits at T, if T < ∞. Consider a causal operator F : U → Y.

• Definitions. F is analytic if the maps Ek → Y

(t1, . . . , tk) − → F, (t1, u1) · · · (tk, uk) are analytic, for any k ≥ 1 and any u1, . . . , uk ∈ U. Here R+ = [0, ∞) and Ek = {(t1, . . . , tk) ∈ Rk

+ : t1 + · · · + tk < T}.

F is jointly analytic if the maps Ek × Uk → Y (t1, . . . , tk, u1, . . . , uk) − → F, (t1, u1) · · · (tk, uk) are analytic, for any k ≥ 1. The following result was proven in 1980 and 1986.

• Theorem. F has an analytic (resp. jointly analytic) realization on [0, T) iff

(i) F is analytic (resp. jointly analytic), (ii) rank F < ∞. If these conditions hold then F has also a minimal realization.

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Recall that a realization ˙ x = f (x, u), x(0) = x0, y = h(x) is analytic (resp. jointly analytic) if: X is analytic and; f (x, u) and h(x, u) are analytic in x (resp. jointly analytic in (x, u)). Recall also that a function defined on a subset of W ⊂ RN is called analytic if it has an analytic extension to an open subset containing W .

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Uniqueness of minimal realizations

Consider a causal map F : U → Y and its realization which is an initialized input-state-output system ˙ x = f (x, u), x(0) = x0 y = h(x), where u(t) ∈ U, x(t) ∈ X and y(t) ∈ Y .

• Definition. The realization is minimal if it is weakly controllable and observable.

Weakly controllable (transitive) means that any point in X can be reached from any other point using forward and backward trajectories of the system. It was proved by H. Sussmann in 1977 that: If a causal map F has a realization then it also has a minimal realization. Two minimal realizations of the same F are diffeomorphic (equivalent up to a diffeomorphism of their state spaces). The proof of the first statement consists of two steps:

• 1. Replace the original state space by the orbit of the point x0 (a submanifold).
• 2. If the system reduced to the orbit is not observable, use the relation of

indistinguishability (for forward-backward controls) to define new, minimal state space by taking the quotient (it is again a manifold).

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Rank of causal operator F

Assume, for simplicity, that Y = R. Consider the output of F after applying a ”control input” a = (t1, u1) · · · (tk, uk) followed by an observation input b ∈ SU: F, (t1, u1) · · · (tk, uk)b . In the definition of the rank we use several observation inputs b1, . . . , bk ∈ SU and take the rank of the Jacobian matrix of the map Rk

+ → Rk given by

(t1, . . . , tk) − → (F, ab1, . . . , F, abk). rank F is defined as the supremum of ranks of k × k matrices (aij) of the form rank F = sup rank ∂ ∂ti F, (t1, u1) · · · (tk, uk)bj

• where the supremum is taken over: all k ≥ 1; all t1, . . . , tk ≥ 0; all bi ∈ SU.

Because of analyticity it is enough to take the supremum

• ver arbitrarily small times in

a = (t1, u1) · · · (tk, uk) and in

• bservation inputs bi ∈ SU.

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Idea behind definition of rank F

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Group of ”inputs”

For constructing a realization we extend the semigroup SU to a group GU. Namely, the semigroup of piecewise constant inputs SU = {(t1, u1) · · · (tk, uk) : t1, . . . , tk ≥ 0, u1, . . . , uk ∈ U, k ≥ 0} can be extended to a group GU. Consider a semigroup of formal sequences ˜ GU = {(t1, u1) · · · (tk, uk) : t1, . . . , tk ∈ R, u1, . . . , uk ∈ U, k ≥ 0} with concatenation product, where we admit also negative ti. Let ”≃” denote the equivalence relation in ˜ G generated by the relations (t1, u)(t2, u) ≃ (t1 + t2, u), (0, u) ≃ e where e denotes the empty sequence. The input group is defined as the quotient GU = ˜ GU/ ≃ . We will write (t1, u1) · · · (tk, uk) ∈ GU, omitting equivalence classes in our

• notation. We have

SU ⊂ GU.

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Construction of a realization from causal operator F

We will construct a realization under an additional assumption (1980): The maps (t1, . . . , tk) − → F, (t1, u1) · · · (tk, uk) have analytic extensions from Rk

+ to Rk.

This assumption is necessary and sufficient when the realization is required to be complete (i.e. state trajectories are defined for all t ∈ R). The construction works under a weaker assumption that F is of class C r and the function R : SU → R has a C r extension to the group GU such that the rank remains the same. Here r ∈ {1, 2, . . . , ∞, ω}. (A construction which does not require this assumption but consists of finding local realizations first and then gluing them together was given in 1986.) Preparing the construction. We begin with extending the response function R : SU → R to the group GU: Step 0. The causal F defines response function R : SU → R, R(a) = F, a. As the maps (t1, . . . , tk) − → F, (t1, u1) · · · (tk, uk) have analytic extensions from Rk

+ to Rk, the response function R has an extension to the group,

denoted again R : GU → R.

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Construction of a realization from causal operator F

The construction is based on the single datum: the function R : GU → R. Step 1. Define the equivalence relation on GU: a ≃ b ⇐ ⇒ R(ac) = R(bc) ∀ c ∈ GU. The state space (a set) is the quotient space X = GU/ ≃ . Denote x = [a] - equivalence class of a ∈ GU and define x0 = [e], where e is the neutral element in GU. Step 2. Define flows (families of maps X → X parametrized by t ∈ R): f t

u (x) := [a · (t, u)],

where · denotes concatenation. Step 3. Define the output map h(x) = R(a). So defined representation (X, f t

u , h, x0) of the response function R is well

defined on the level of sets with no regularity whatsoever. We need to introduce a topology and differential structure in X. Once we know how to differentiate, we will define the vector fields f (x, u) = d dt f t

u (x)|t=0. Bronis law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Topology and differential structure on X

Topology in X: the strongest topology such that the maps ψ¯

u : Rk → X

ψ¯

u(t1, . . . , tk) := [(t1, u1) · · · (tk, uk)]

are continuous for any k ≥ 1 and any ¯ u = (u1, . . . , uk) ∈ Uk. Differential structure on X defined by specifying differentiable functions on X: A function ϕ : X → R is of class C r, r ∈ {1, 2, . . . , ∞, ω} if the composed map φ ◦ ψ¯

u : Rk → R

is of class C r for any k ≥ 1 and ¯ u = (u1, . . . , uk) ∈ Uk. Using the definition of the equivalence relation [ · ] on GU, defined by the function R : Gu → R, and assuming that the maps (t1, . . . , tk) → R((t1, u1) · · · (tk, uk)) are of class C r and rank F < ∞ it can be proved that X with such topology and differentiable structure becomes a differentiable manifold of class C r and dim X = rank F. Furthermore, one can prove that f (x, u) and h(x) are of class C r in x, which ends the proof of existence of realization. If F is jointly analytic then one gets joint analyticity of f (x, u) with respect to (x, u). Note that the proof is constructive, while being abstract. If U is not compact then the manifold X may be not paracompact (not countable at infinity).

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Local realizations and gluing together

Our construction presented above was used in 1980 for proving existence

• f complete (with complete vector fields fu) global realizations of abstract

input-output maps, analytic as well as of class C r, 1 ≤ r ≤ ∞. Solving general local versions of the realization problem required different tools and was done later.

• M. Fliess proposed a local existence theorem using formal power series

(Inv. Math. 1983) but the proof was not correct. (The formal power series approach is the same as the one with Volterra series with constant kernels.) General local realization theorems were published in 1986 (papers by BJ and by J.P. Gauthier and G. Bornard) and in 1987 (paper by F. Celle and J.P. Gauthier). These results and a ”gluing together” technique of Sussmann were used for proving global existence theorems without assuming completeness (1986, 1987). A review of other results, including theorems on bilinear realizations, can be found in the paper ”Observability (Deterministic Systems) and Realization Theory” by J.P. Gauthier 2011.

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Other approaches

The causal map describing the behaviour of a black-box system may be given by a Volterra series (Fu)(t) =

∞

• k=0
• i1...ik

t

• sk
• · · ·

s2

• W i1...ik

k

(t, s1, . . . , sk)ui1(s1) · · · uik (sk)ds1 · · · dsk. In this case we may ask if there exists its control-affine realization. For finite Volterra series with regular kernels such realizations always exist as shown by R. Brockett. A detailed analysis of such minimal global realizations was given by P. Crouch (1981) where the state space was a nilpotent Lie group. The situation is simpler when the Volterra series has constant kernels, (Fu)(t) =

∞

• k=0
• i1...ik

W i1...ik

k

t sk · · · s2 ui1(s1) · · · uik (sk)ds1 · · · dsk. In this case it is enough to assume that the kernels satisfy the estimates |W i1...ik

k

| ≤ CRkk! and certain rank defined by them is finite. Under these assumption one may prove that there exists a local realization with the state space being an open subset of Rn, with n equal to the rank of the series.

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