Reachability and Controllability

Transcription

1 Capitolo. INTRODUCTION 4. Reachability and Controllability Reachability. The reachability problem is to find the set of all the final states x(t ) reachable starting from a given initial state x(t ) : A state x(t ) of a dynamic system is reachable from the state x(t ) in the time interval [t,t if it exists an input function u( ) U such that x(t ) = ψ(t,t,x(t ),u( )). LetX + (t,t,x(t ))denotethe set of all the final states x(t ) reachable at time t starting from the initial state x(t ). Controllability. The controllability problem is to find the set of all the initial states x(t ) controllable to a given final state x(t ): A state x(t ) of a dynamic system is controllable to state x(t ) in the time interval [t,t if it exists an input function u( ) U such that x(t ) = ψ(t,t,x(t ),u( )). Let X (t,t,x(t )) the set of all the initial states x(t ) controllable to the final state x(t ) at time t. X (t,t,x(t )) 3 x(t ) X + (t,t,x(t )) x(t ) t t For time-invariant systems one can use t = and t = t: X + (t,t,x(t )) X + (t,x()), X (t,t,x(t )) X (t,x(t)) For discrete-time systems it is t k: X + (t,x()) X + (k,x()), X (t,x(t)) X (k,x(k))

2 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.2 Discrete time-invariant linear systems let us consider the following discrete time-invariant linear system: Reachability x(k +) = Ax(k)+Bu(k) The set X + (k) of all the states reachable from the origin in k steps is equal to the set of all the states x(k) obtained starting from the initial condition x() = and considering only the forced evolution of the system: x(k) = k j= A (k j ) Bu(j) = [B AB... A k B u(k ) u(k 2). u() and varying the input u(),u(),...,u(k ) in all the possible ways. Definition. Reachability matrix in k steps: R + (k) = [B AB... A k B The set X + (k) of all the states reachable from the origin in k steps is a vectorial space which is equal to the image of matrix R + (k): X + (k) = Im[R + (k) The subspaces X + (k) reachable in,2,...,k steps satisfy the following chain of inclusions (n is the dimension of the state space): X + () X + (2)... X + (n) = X + (n+) =... The maximum reachable subspace X + (n) is obtained, at the most, in n steps.

3 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.3 Definition. Reachability matrix of the system: R + = R + (k) k=n = R + (n) = [B AB... A n B The subspace X + of all the state reachable from the origin in a time interval however long is equal to the image of matrix R + : X + = Im[B AB... A n B = ImR + Definition. A system is reachable if the subspace X + of all the reachable states from the origin is equal to the whole state space X: X + = X Necessary and sufficient condition for a system to be reachable is: rank(r + ) = n For discrete, time-invariantlinear systems the set X + (k,x ) has the structure of a linear variety : Graphical representation: X + (k,x ) = A k x +ImR + (k) x 2 X + (k,x ) R + (k) A k x x

4 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.4 Controllability A state x = x() is controllable to zero in k steps if it exists an input sequence u(),u(),...,u(k ) which brings the initial state x to a final state equal to the origin x(k) = in the time interval [,k: that is: A k x = k j= = x(k) = A k x + k j= A (k j ) Bu(j) A (k j ) Bu(j) A k x X + (k) = ImR + if the state A k x() is reachable from the origin in k steps. Property. A system is controllable if and only if the following relation holds: ImA n X + (n) = ImR + where R + is the reachability matrix of the system. For discrete linear systems the reachability and controllability properties are NOT equivalent: ) The reachability implies the controllability. reachability controllability In fact the reachability implies X + = X from which it follows that: ImA n X + = X, that is the system is surely controllable. 2) The controllability does not imply the reachability: controllability reachability In fact if, for example A = and rank(b) < n, then: rank([b AB... A n B) = rank([b... < n that is the system is not reachable even if it is controllable. IfAisafullrankmatrix, thenthereachabilityandthecontrollabilityimply one another.

5 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.5 Invariant time-continuous linear systems Let us consider the following invariant time-continuous linear system: Reachability: ẋ = Ax(t)+Bu(t) A state x(t) is reachable at time t starting from zero if it exists an input function u( ) such that: x(t) = t e A(t τ) Bu(τ)dτ Let X + (t) be the set of all the states reachable from the origin x() = in the time interval [, t and let X + denote be the set of all the states reachable from the origin x() = in the time interval [,. Let R t denote the linear function R t : U X defined as follows: R t : u( ) x(t) = t e A(t τ) Bu(τ)dτ The set U is infinite dimensional. The states x(t) reachable at time t are all the states which belongs to the image of the linear function R t : X + (t) = {x : x ImR t } ThesetX + (t), beingtheimageofalinearfunction, isavectorialsubspace of the state space X. Property. For each t >, the reachable subspace X + (t) is the image of the reachability matrix R + : X + (t) = X + = ImR + For continuous-time systems, the reachable subspace does NOT depend on the length of the time interval [,t. The smaller is the time interval [,t the larger is the control action u(t).

6 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.6 Controllability. For invariant time-continuous linear systems, the controllable subspace X does not depend on the amplitude of the time interval [,t and it is equal to the reachable subspace X +. Example. Let us consider the following electrical network: L C V C J(t) I L V R R The dynamic equations of the systems are: L di L = V C +R(J I L ) RI L dt C dv C = J I L dt V = V C +R(J I L ) wherei L isthecurrentwhichflowsintheinductance, V C isthevoltageacrossthecapacitor, J is the input current and V is the output voltage. In matrix form, the system dynamics can be represented as follows: ẋ = [ 2R L L C x+ [ R L C V = [ R x+[rj The reachability matrix of the system is R + = [ R L C LC 2R2 L 2 R LC J, detr + = LC x = [ R 2 [ IL V C L C The system is reachable only if R + is a full rank matrix. The system is NOT completely reachable if: R 2 = L RC = L C R that is if the inductance time constant L R is equal to the capacitor time constant RC. In this case the two system eigenvalues are coincident: λ,2 = LC.

7 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.7 Example. Compute the reachability matrix R + of the following system: ẋ(t) = x(t)+ u(t) Reachability matrix R + and computation of the subspace X + : R + = [ b Ab A 2 b =, X + = ImR + = Im The system is reachable. Example. Compute the reachability matrix R + of the following system: ẋ(t) = x(t)+ u(t) Reachability matrix R + and computation of the subspace X + : R + = [ b Ab A 2 b =, X + = Im [ R + = Im The system is NOT completely reachable. Example. Compute the reachability matrix R + of the following system: ẋ(t) = x(t)+ u(t) Reachability matrix R + and computation of the subspace X + : R + = [ B AB A 2 B =, X + = Im [ R + = Im The system is NOT completely reachable...

8 Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.8 Equivalent systems Property. Discrete or continuous-time linear systems which are algebraically equivalent have the same reachability properties. Let T be a full rank transformation matrix which links two algebraically equivalent time-invariant linear systems S = (A, B, C, D) and S = (A, B, C, D): x = Tx { A = T AT, B = T B C = CT, D = D The reachability subspaces in k steps of the two systems S and S, that is X + (k) and X + (k), are linked by the following relation: X + (k) = Im[B...A k B = Im(T [B...A k B) = T X + (k) ThesubspaceX + isinvariantwithrespecttoastatespacetransformation: x = Tx X + = TX + Let R + and R + be the reachability matrices of the two systems. The following relation holds: R + = T R + R + = TR + If the two systems have only one input, R + and R + are squared full rank matrices which satisfy the following relation: T = R + (R + )