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) = x 0 . 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 F u ( · ) − → 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: observability, 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 R n 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 − t = ( f t ) − 1 , f 0 = id . f t 1 ◦ f t 2 = f t 1 + t 2 , 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 = { f u } u ∈ U be a family of vector fields on a state space X (manifold). Each of f t u defines the flow of f u , 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 = { f u } 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 of Diff ∞ ( X ). We will analyze the action of the group generated by all these subgroups. Such parametrized families of diffeomorphisms are compositions f t k u k ◦ · · · f t 2 u 2 ◦ f t 1 u 1 : X − → X , ( ⋆ ) where k ≥ 1 , u 1 , . . . , u k ∈ U , t 1 , . . . , t k ∈ R . We will call them the poly-flow of the family F . Remark. All results will be valid without the completeness assumption. Then one 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 x 0 ∈ X and consider ”broken trajectories” starting from x 0 and defined by x ( t ) = f t k u k ◦ · · · ◦ f t 2 u 2 ◦ f t 1 u 1 ( x 0 ) , t = t 1 + · · · + t k . Question. How can we analyze properties of poly-dynamical systems? What can we say on the set of points reachable from x 0 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 t j can be positive and negative. Thus a poly-dynamical system defines a group of diffeomorphisms of X . If we take only t j ≥ 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 = ( f 1 , . . . , f n ) T , g = ( g 1 , . . . , g n ) T are vector fields in local coordinates on R n , then their Lie bracket is the vector field [ f , g ]( x ) = ∂ g ∂ x ( x ) f ( x ) − ∂ f ∂ x ( x ) g ( x ) . If ∂ ∂ � � f = f i ∂ x i , g = g i ∂ x i i i are treated as differential operators, then the Lie bracket is the commutator,   ∂ g i ∂ f i  ∂ � � � [ f , g ] = f g − g f = ∂ x j f j − ∂ x j g j ∂ x i . i j j Basic property: vector fields commute iff their flows commute: ⇒ f t ◦ g s = g s ◦ f t . [ f , g ] ≡ 0 ⇐ 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 − 1 b − 1 ab . 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 of 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 ψ t : X → X , ψ t = f − t ◦ g − t ◦ f t ◦ g t . Composing this commutator k 2 times with t replaced by t / k gives curves k 2 -times β k ( t ) = ψ t k ◦ · · · ◦ ψ t k ( p ) , which converge to a reparametrized trajectory of [ g , f ], namely, with exp denoting the flow, → exp( t 2 [ g , f ])( p ) as k − β k ( t ) − → ∞ . Bronis� law Jakubczyk Institute of Mathematics Polish Academy of Sciences, Warsaw Vector Fields, Control, and Input-Output Systems

Example: Car un-parking Let x = ( x 1 , x 2 , 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 R 2 × S 1 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 , 2 b , 0) T . Approximate movement along the commutator: → ( x 1 (0) , x 2 (0) + t 2 rb , 0) t �− from the initial condition ( x 1 (0) , x 2 (0) , 0), if θ ≈ 0. Precisely, ( x ( t ) , θ ( t )) = ( x 1 (0) , x 2 (0) + t 2 rb , 0) + O ( t 3 ). 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