Automatic Control 1 - Reachability Analysis

Lecture: Reachability AnalysisAutomatic Control 1Reachability AnalysisProf. Alberto BemporadUniversity of TrentoAcademic year 2010-2011Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 1 / 23

Lecture: Reachability AnalysisReachability problemReachabilityConsider the linear discrete-time systemx(k + 1) = Ax(k) + Bu(k)with x ∈ n , u ∈ m and initial condition x(0) = x 0 ∈ n∑k−1The solution is x(k) = A k x 0 + A j Bu(k − 1 − j)Definitionj=0The system x(k + 1) = Ax(k) + Bu(k) is (completely) reachable if ∀x 1 , x 2 ∈ n thereexist k ∈ and u(0), u(1), . . ., u(k − 1) ∈ m such that∑k−1x 2 = A k x 1 + A j Bu(k − 1 − j)j=0In simple words: a system is completely reachable if from any state x 1 we canreach any state x 2 at some time k, by applying a suitable input sequenceProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 2 / 23

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Lecture: Reachability AnalysisLinear algebra recallsAlgebraically equivalent systemsConsider the linear system x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) + Du(k)x(0) = x 0Let T be invertible and define the change of coordinates x = Tz, z = T −1 x z(k + 1) = T −1 x(k + 1) = T −1 (Ax(k) + Bu(k)) = T −1 ATz(k) + T −1 Bu(k)y(k) = CTz(k) + Du(k)z 0 = T −1 x 0and hence z(k + 1) = Ãz(k) + ˜Bu(k)y(k) = ˜Cz(k) + ˜Du(k)z(0) = T −1 x 0The dynamical systems (A, B, C, D) and (Ā, ¯B, ¯C, ¯D) are called algebraicallyequivalentProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 4 / 23

Lecture: Reachability AnalysisReachability AnalysisReachabilityLet’s focus on the problem of determining a sequence of n inputs transferringthe state vector from x 1 to x 2 after n steps⎡x 2 − A n x} {{ } 1 = B AB . . . A n−1 B } {{ }⎢⎣XRu(n − 1)u(n − 2).u(0)⎤⎥⎦} {{ }UThis is equivalent to solve with respect to U the linear system of equationsRU = XThe matrix R ∈ n×nm is called the reachability matrix of the systemA solution U exists if and only if X ∈ Im(R)(Rouché-Capelli theorem: a solution exists ⇔ rank([R X]) = rank(R))Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 8 / 23

Lecture: Reachability AnalysisReachability AnalysisReachabilityTheoremThe system (A, B) is completely reachable ⇔ rank(R) = nProof:(⇒) Assume (A, B) reachable, choose x 1 = 0 and x 2 = x. Then ∃k ≥ 0 such that∑k−1x = A j Bu(k − 1 − j)j=0If k ≤ n, then clearly x ∈ Im(R). If k > n, by Cayley-Hamilton theorem we haveagain x ∈ Im(R). Since x is arbitrary, Im(R) = n , so rank(R) = n.(⇐) If rank(R) = n, then Im(R) = n . Let X = x 2 − A n x 1 andU = u(n − 1) ′ . . . u(1) ′ u(0) ′ ′ . The system X = RU can be solved with respect toU, ∀X, so any state x 1 can be transferred to x 2 in k = n steps. Therefore, thesystem (A, B) is completely reachable.Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 9 / 23

Lecture: Reachability AnalysisReachability AnalysisComments on the reachability propertyThe reachability property of a system only depends on A and BWe therefore say that a pair (A, B) is reachable ifrank B AB . . . A n−1 B = nMATLAB» R=ctrb(A,B)» rank(R)Im(R) is the set of states that are reachable from the origin, that is the set ofstates x ∈ n for which there exists k ∈ and u(0), u(1), . . ., u(k − 1) ∈ msuch that∑k−1x = A j Bu(k − 1 − j)j=0If Im(R) = n , a system is completely reachable ⇔ all the states arereachable from the origin in n steps (proof: set x = x 2 − A n x 1 )Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 10 / 23

Lecture: Reachability AnalysisReachability AnalysisMinimum-energy controlLet (A, B) reachable and consider steering the state from x(0) = x 1 intox(k) = x 2 , k > n⎡ ⎤u(k − 1)x 2 − A k x} {{ } 1 = B AB . . . A k−1 B u(k − 2)⎢} {{ }⎥⎣ . ⎦XR ku(0)} {{ }U(R k ∈ n×km is the reachability matrix for k steps)Since rank(R k ) = rank(R) = n, ∀k > n (Cayley-Hamilton), we getrank R k = rank[R k X] = nHence the system X = R k U admits solutions UProblemDetermine the input sequence {u(j)} k−1j=0that brings the state fromx(0) = x 1 to x(k) = x 2 with minimum energy 1 ∑k−1‖u(j)‖ 2 = 1 22 U′ Uj=0Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 11 / 23

Lecture: Reachability AnalysisReachability AnalysisMinimum-energy controlThe problem is equivalent to finding the solution U of the system of equationsX = R k Uwith minimum norm ‖U‖We must solve the optimization problemU ∗ = arg min 1 2 ‖U‖2 subject to X = R k ULet’s apply the method of Lagrange multipliers: (U, λ) = 1 2 ‖U‖2 + λ ′ (X − R k U)Lagrangean function∂ ∂ U = U − R′ k λ = 0∂ ∂ λ = X − R kU = 0⇒ U ∗ = R ′ k (R kR ′ k} {{ )−1 ·X}R # k = pseudoinverse matrixMATLABU=pinv(Rk)*XNote that R k R ′ k is invertible because rank(R k) = rank(R) = n, ∀k ≥ nProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 12 / 23

Lecture: Reachability AnalysisCanonical reachability decompositionCanonical reachability decompositionGoal: Make a change of coordinates to separate reachable and unreachablestatesLet rank(R) = n c < n and consider the change of coordinatesT = w nc +1 . . . w n v 1 . . . v ncwhere v 1 , . . . , v nc is a basis of Im(R), and wnc +1, . . . , w n is a completion toobtain a basis of nAs Im(R) is A-invariant (Ax ∈ Im(R) ∀x ∈ Im(R) follows fromCayley-Hamilton theorem), Av i has no components along the basis vectorsw nc +1, . . . , w n . Since T −1 Av i are the new coordinates of Av i , the first n − n ccomponents of T −1 Av i are zeroThe columns of B also have zero components along w nc +1, . . . , w n , becauseIm(B) ⊆ Im(R)In the new coordinates, the system has matrices à = T −1 AT, ˜B = T −1 B e˜C = CT in the canonical reachability form (a.k.a. controllability staircase form) Auc 0 0à =˜B = ˜C = MATLABCA c B uc C c[At,Bt,Ct,Tinv]=c ctrbf(A,B,C)A 21Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 13 / 23

Lecture: Reachability AnalysisCanonical reachability decompositionReachability and transfer functionWhy are the eigenvalues of A uc not appearing in the transfer function G(z) ?Remember: G(z) explains the forced response, i.e., the response for x(0) = 0Expressed in canonical decomposition, the system evolution is⎧⎨ x uc (k + 1) = A uc x uc (k)x⎩ c (k + 1) = A c x c (k) + B c u(k) + A 21 x uc (k)y(k) = C uc x uc (k) + C c x c (k) + Du(k)For x uc (0) = 0, x c (0) = 0, we get x uc (k) ≡ 0 and⎧⎨ x c (k + 1) = A c x c (k) + B c u(k)y(k) = C⎩c x c (k) + Du(k)x c (0) = 0so the forced response does not depend at all on A uc !The input u(k) only affects the output y(k) through the reachable subsystem(A c , B c , C c , D), not through the unreachable part A ucProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 15 / 23

Lecture: Reachability AnalysisCanonical reachability decompositionCanonical reachability decompositionPropositionA c ∈ n c×n c and Bc ∈ nc×m are a completely reachable pairProof:Consider the reachability matrix˜R = ˜B ØB . . . à n−1˜B 0 0 . . . 0=B c A c B c . . . A n−1cB cand˜R = T −1 B T −1 ATT −1 B . . . T −1 A n−1 TT −1 B = T −1 RSince T is nonsingular, rank(˜R) = rank(R) = n c , sorank B c A c B c . . . A n c−1c B c = ncthat is (A c , B c ) is completely reachable□Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 16 / 23

Lecture: Reachability AnalysisControllabilityControllabilityIf the system is completely reachable, we have seen that we can bring thestate vector from any value x(0) = x 1 to any other value x(n) = x 2Let’s focus on the subproblem of determining a finite sequence of inputs thatbrings the state to the final value x(n) = 0DefinitionA system x(k + 1) = Ax(k) + Bu(k) is controllable to the origin in k steps if∀x 0 ∈ n there exists a sequence u(0), u(1), . . ., u(k − 1) ∈ m such that0 = A k x 0 + ∑ k−1j=0 Aj Bu(k − 1 − j)Controllability is a weaker condition than reachabilityProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 17 / 23

Lecture: Reachability AnalysisControllabilityControllabilityThe linear system of equations⎡−A k x 0 = B AB . . . A k−1 B } {{ }⎢⎣R ku(k − 1)u(k − 2).u(0)admits a solution if and only if A k x 0 ∈ Im(R k ), ∀x 0 ∈ nTheoremThe system is controllable to the origin in k steps if and only ifIm(A k ) ⊆ Im(R k )⎤⎥⎦If a system is controllable in n steps, it is also controllable in k steps for eachk > n (just set u(n) = u(n + 1) = . . . = u(k − 1) = 0)For the same reason, if a system is controllable in k steps with k < n, it is alsocontrollable in n steps (just set u(k) = u(k + 1) = . . . = u(n − 1) = 0)Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 18 / 23

Lecture: Reachability AnalysisControllabilityControllabilityDefinitionA system is said completely controllable if it is controllable in n stepsTheoremA system x(k + 1) = Ax(k) + Bu(k) is completely controllable if and only ifall the eigenvalues of its unreachable part A uc are nullProof:Given an initial state x 0 ∈ n , we must solve with respect U the linear systemof equations⎡ ⎤−A n x 0 = B AB . . . A n−1 B U = RU, U = ⎣u(n−1).u(0)⎦Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 19 / 23

Lecture: Reachability AnalysisControllabilityControllabilityLet (Ã, ˜B) be a canonical reachability decomposition of (A, B). Then−A n x 0 = −Tà n z 0 = T˜B TØB . . . Tà n−1˜B U = T˜RUwhere we set z 0 = T −1 x 0 =zucand T = [wz nr +1 . . . w n v 1 . . . v nr]csince T is invertible, the system T˜RU = −Tà n z 0 is equivalent to ˜RU = −à n z 0 ,that is 0 0 . . . 0AnB c A c B c . . . A n−1 U = −uc0 zuccB c ⋆ A n cz cas the pair (A c , B c ) is completely reachable, the system above has a solution ifand only is A n uc z uc = 0 for any initial condition z uc , that is A n uc = 0if A uc is a nilpotent matrix (= all its n − n c eigenvalues are zero), byCayley-Hamilton A n−n cuc = 0 and thereforeA n uc = An cuc An−n cuc= A n cuc · 0 = 0Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 20 / 23□

Lecture: Reachability AnalysisControllabilityStabilizabilityDefinitionA linear system x(k + 1) = Ax(k) + Bu(k) is called stabilizable if can bedriven asymptotically to the originStabilizability is a weaker condition than controllabilityTheoremA linear system x(k + 1) = Ax(k) + Bu(k) is stabilizable if and only if allthe eigenvalues of its unreachable part have moduli < 1Proof: zuc (0)Take any z 0 =z c (0)∈ n and an input sequence {u(k)} ∞ k=0that makes thereachable component z c (k) → 0 for k → ∞If A k uc → 0 for k → ∞, then z uc(k) = A k uc z uc(0) also converges to zero fork → ∞Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 21 / 23□

Lecture: Reachability AnalysisContinuous-time systemsReachability analysis of continuous-time systemsSimilar definitions of reachability, controllability, and stabilizability can begiven for continuous-time systemsẋ(t) = Ax(t) + Bu(t)No distinction between controllability and reachability in continuous-time(because no finite-time convergence of modal response exists)Reachability matrix and canonical reachability decomposition are identical todiscrete-timerank R = n is also a necessary and sufficient condition for reachabilityA uc asymptotically stable (all eigenvalues with negative real part) is also anecessary and sufficient condition for stabilizabilityProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 22 / 23

Lecture: Reachability AnalysisContinuous-time systemsEnglish-Italian Vocabularyreachabilitycontrollabilitystabilizabilitycontrollability staircase formraggiungibilitàcontrollabilitàstabilizzabilitàdecomposizione canonica di raggiungibilitàTranslation is obvious otherwise.Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 23 / 23