Path-By-Path Output Regulation of Switched Systems With a Receding Horizon of Modal Knowledge

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1 24 American Control Conference (ACC) June 4-6, 24. Portlan, Oregon, USA Path-By-Path Output Regulation of Switche Systems With a Receing Horizon of Moal Knowlege Ray Essick Ji-Woong Lee Geir Dulleru Abstract We aress a iscrete-time LQG control problem over a fixe performance winow an apply a receing-horizon type control strategy, resulting in an exact solution to the problem in terms of semiefinite programming. The systems consiere take parameters from a finite set, an switch between them accoring to an automaton. The controller has a finite preview of future parameters, beyon which only the set of parameters is known. We provie necessary an sufficient convex conitions for the existence of a controller which guarantees both exponential stability an finite-horizon performance levels for the system; the performance levels may iffer accoring to the particular parameter sequence within the performance winow. A simple, physics-base example is provie to illustrate the main results. I. INTRODUCTION This paper consiers the stabilization an output regulation of switche linear systems; i.e. systems whose state space parameters take values in a finite set. Such systems are governe by equations of the form x(t + ) = A θ(t) x(t) + B,θ(t) w(t) + B 2,θ(t) u(t) z(t) = C,θ(t) x(t) + D,θ(t) w(t) + D 2,θ(t) u(t) y(t) = C 2,θ(t) x(t) + D 2,θ(t) w(t) () where the switching sequence θ takes values in {,..., N} an the corresponing system parameters come from a finite set. The amissible sequences θ are generate by the output of a finite automaton (such as in the left half of Fig. 3); this is a stanar moel for iscrete transition systems. For convenience, we will enote by θ (s:t) the switching sequence (θ(s),..., θ(t)) when t s. We take a receing-horizon type approach by giving the controller finite memory of L past parameters as well as finite preview of H future parameters. The use R. Essick an G. Dulleru are with the Deparment of Mechanical Science an Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 68 USA. These authors were partially supporte by grants NSA SoS W9NSF-3-86 an AFOSR MURI FA J.W. Lee is with the Department of Mechanical an Nuclear Engineering, Pennsylvania State University, University Park, PA 682 USA. This author was partially supporte by NSF grant ECCS θ(t L) θ(t) θ(t + H) θ(t + T ) Fig.. The performance winow (re) an the winow of information available to the controller (blue). of a fixe-length winow of future moes suggests a comparison with moel preictive control (MPC); such problems have been stuie in numerous engineering contexts (see ], 4], 5], 6], 3], 2], 4], 6] an references). While these techniques can be practically use to overcome the computational ifficulties of infinite-horizon performance, they require a sufficient horizon length as well as constraints on the terminal state or cost, which are typically not known a priori. Further, these methos typically require extensive online computation. Our control objective is output regulation over a finite-length winow, subject to a Gaussian isturbance. Specifically, we require the system output to satisfy (2) for t t+t E z(s) 2] < γ θ(t:t+t ). (2) s=t Fig. emonstrates the relationship between this performance winow an controller information winow. The performance boun γ θ(t:t+t ) may explicitly epen on the sequence of parameters taken over this winow. This particular performance problem an path-epenent control scheme were examine in 7], 8] with controllers epening only on memory of past moes; our results agree exactly with these results when the number of future moes available to the controller is zero. II. ANALYSIS OF TIME-VARYING SYSTEMS We require some results about linear time-varying (LTV) systems; the results of this section are conveniently represente using block-iagonal operators on sequences in l 2 (R n ) (in the style of 2]). Let A be a block-iagonal operator such that the t-th block A t = θ /$3. 24 AACC 265

2 A(t) for every t. Then A takes the form A() A() A = A(2).... Similarly efine block-iagonal operators B, C, D. Also let Z enote the unilateral shift operator such that for x = (x, x, x 2,... ) we have Zx = (, x, x,... ). We write Q when the inner prouct x, Qx c x for some c. The exponent Q k enotes repeate multiplication, an Q enotes the ajoint operator. Consier the system escribe by x(t + ) = A(t)x(t) + B(t)w(t) z(t) = C(t)x(t) + D(t)w(t) (3) where A(t) R n n, B(t) R n m, C(t) R l n, an D(t) R l m for all t. This system has the operator representation x = ZAx + ZBw; z = Cx + Dw. (4) Definition : The system of (4) is uniformly exponentially stable if there exist constants c an λ (, ) such that whenever w =, (ZA) k cλ k. Lemma 2: The system of (4) is uniformly exponentially stable if an only if there exists a block iagonal operator Y such that (ZA)Y (ZA) T Y. (5) Proof: See 2]. Lemma 3: If the conitions of Lemma 2 hol with α, β, γ > such that αi Y βi, (ZA)Y (ZA) T Y γi then the constants of Definition are given by β c = α, λ = γ β. (6) Proof: See ], or 3]. We consier performance as the average output variance over a forwar winow of length T. Let the isturbance w be an i.i.. sequence of R m -value Gaussian ranom variables; i.e. for all s, t : { I for t = s Ew(t)] = ; Ew(t)w(s)] = for t s. (7) Definition 4: Let T an let Γ = {γ t : t } be an inexe collection of positive numbers uniformly boune below. The system of (4) satisfies T-step performance levels Γ if for w satisfying (7) an x() = the output satisfies (8) for all t t+t E z(s) 2] 2 < γ 2 t. (8) s=t If γ > is such that γ γ t for every γ t Γ, then the system satisfies T-step uniform performance level γ. For a uniformly exponentially stable system, there exists a unique solution Y to the equation (ZA)Y (ZA) T Y = (ZB)(ZB) T. (9) The block structure of Y is given by t (Y ) t = Φ(t +, s + )B s Bs T Φ(t +, s + ) T s= where Φ(t, s) is the state transition matrix efine by { I t = s Φ(t, s) = A(t )... A(s) t > s. Remark 5: It is immeiate from the efinition of Y that any block-iagonal matrix X satisfying (ZA)X(ZA) T X (ZB)(ZB) T will also satisfy X Y. Using (4) an the properties of (7), one can show that E z(t) 2] = tr ( CY C T + DD T ) t. () We introuce a winowe trace for operators: tr (t,t ) (X) = t+t tr(x s ). () s=t Using the winowe trace, (4) can be written as tr (t,t ) ( CY C T + DD T ) < γ 2 t t. Remark 6: The winowe trace preserves orer; i.e. if X Y, then tr(t, t, X) tr(t, t, Y ) for all t. Lemma 7: Consier the system of (4): (a) The system is uniformly exponentially stable an satisfies T-step performance levels Γ if there exists a Y such that the following hol: (ZA)Y (ZA) Y (ZB)(ZB) T (2) tr (t,t ) (CY C T + DD T ) < γ 2 t t. (3) (b) If the system is uniformly exponentially stable an satisfies T-step performance levels Γ, then there exists a Y satisfying (2) an (3) whose blocks epen on a finite number of past parameters. 265

3 Proof: For (a), suppose a solution to (2) an (3) exists. Then by Lemma 2 the system is uniformly exponentially stable. Construct Y as in (9); then Remarks 5 an 6 show that θ(t L) θ(t) θ(t + H) θ tr (t,t ) (CY C T + DD T ) < γ 2 t so by () the system satisfies performance level Γ. For (b), let ɛ > an consier the sequence Y (ɛ,m+) = (ZA)Y (ɛ,m) (ZA) T + (ZB)(ZB) T + ɛi with Y (ɛ,) = ɛi. The block structure of Y (ɛ,m) shows that each block epens exactly on the past M system parameters. Also, for any t the blocks Y (ɛ,m) Y (ɛ,m+) t t = for all M t. Let the block-wise (weak) limit of this sequence be Y (ɛ). This limit satisfies Y (ɛ) = (ZA)Y (ɛ) (ZA) T + (ZB)(ZB) T + ɛi an ɛi Y (ɛ,m) Y (ɛ,m+) Y (ɛ) for M. From the efinition of Y (ɛ) is is easy to see that Y (ɛ) t = Y + ɛỹ where Ỹ is the solution to the Lyapunov equation (ZA)Ỹ (ZA) Ỹ = I. Since both Y an Ỹ are boune operators, there exists a β > such that Y (ɛ) βi. Further, we have tr (t,t ) (CY (ɛ) C T + DD T ) = tr (t,t ) (CY C T + DD T ) + ɛ tr (t,t ) (CỸ CT ). The system satisfies T-step performance levels Γ, so tr (t,t ) (CY C T + DD T, t) < γ 2 t ; choose α sufficiently small such that tr (t,t ) (CY (α) C T + DD T, t) < γ 2 t also. Since the system is uniformly exponentially stable, there exist c, λ as in Definition. Choose M large enough that c 2 λ 2M < α/(β α). Then we have (ZA)Y (α,m) (ZA) T Y (α,m) = (ZA) Y (α,m) Y (α,m )] (ZA) T αi = (ZA) M Y (ɛ,) Y (ɛ,)] ((ZA) T ) M αi (c 2 λ 2M )(β α)i αi an because Y (ɛ,m) Y (ɛ) we have as neee. tr (t,t ) (CY (ɛ,m) C T + DD T ) < γ 2 t III. OUTPUT REGULATION FOR SWITCHED SYSTEMS Consier the finite collection {(A i, B i, C i, D i )} for i {,..., N}, an let Q {, } N N be the θl H(t) θh L (t + ) Fig. 2. The finite-length switching paths at time t (blue) an t + (re), an the resulting inuce switching moes at these times. ajacency matrix of a strongly connecte irecte graph. A sequence (finite or infinte) is calle amissible if it represents a vali walk in this irecte graph. This switche system is escribe by x(t + ) = A θ(t) x(t) + B θ(t) w(t) z(t) = C θ(t) x(t) + D θ(t) w(t). θ H L (4) Remark 8: For any fixe switching sequence θ, the switche system reuces to the LTV system x = ZA θ x + ZB θ w; z = C θ x + D θ w. (5) Finite path epenence leas to the notion of inuce switche systems. Inuce system moes correspon to amissible finite-length switching paths (see Fig. 2); amissible transitions are exactly those which preserve switching knowlege. The inuce switching graph for a small system is shown in Fig. 3. In the worst case the number of inuce moes grows combinatorially in path length; nevertheless fining the inuce switching graph is straightforwar. In our main result, inuce systems arise via path-epenent controller gains. For systems without feeback, the inuce moes have parameters etermine by the current switching moe. We introuce the function φ : {,..., N} L+H+ {,..., N}, efine below (as appropriate): φ(θ (t L:t+H) ) = θ(t), φ((i L,..., i,..., i H )) = i. (6) In essence, φ selects the current moe from a sequence of past an/or future moes. This correspons to the moe marke in angle brackets in Fig. 3. Definition 9: A switche linear system is uniformly exponentially stable if for every amissible sequence θ the system is stable in the sense of Definition. Definition : A switche linear system satisfies T- step performance levels Γ = {γ j : j {,..., N} T + } if for each amissible sequence θ the system satisfies the conition of Definition 4, where γ t = γ θt:t+t is etermine exactly by the switching sequence. 2652

4 a b c b c f b a e a a a e f Fig. 3. A switching constraint graph with four moes, an the inuce switching graph generate by finite-path epenence on one past state an one future state. For each inuce moe, the current moe of the unerlying system is ientifie in brackets. Theorem : Fix H an L. The system of (5) is uniformly exponentially stable an satisfies performance levels Γ if an only if there exists an integer M an matrices Y j for j {,..., N} L+M+H such that A φ(i( L:H) )Y i( L M:H ) A T φ(i ( L:H) ) Y (i( L M+:H) B φ(i( L:H) )B T φ(i ( L:H) ) (7) for every amissible i = (i L M,..., i H ); an T ( tr C T+ φ(î(t L:t+H) ) Y î (t L M:t+H ) C T φ(î (t L:t+H) ) t= ) + D φ(î(t L:t+H) ) DT < γ 2 φ(î (t L:t+H) ) î (,T )(8) for every amissible î = (î L M,..., î H+T ). Proof: For sufficiency, suppose matrices Y j satisfy (7) an (8). The set j {,..., N} L+M+H+ is finite, so there exist α, β, η > such that αi Y j βi an the left sie of (8) is boune by ηi. Let θ be an amissible switching sequence. Select i M,..., i such that (i M,..., i, θ()) is an amissible path (this is possible because Q is strongly connecte). Define θ( M) = i M ;... ; θ( ) = i. Construct block-iagonal operators A, B, C, D with blocks efine by A t = A φ(θ(t L:t+H) ); B t = B φ(θ(t L:t+H) ) C t = C φ(θ(t L:t+H) ); D t = D φ(θ(t L:t+H) ) an also Y efine by Y t = Y θt M:t+H. Then we apply (7) an (8) to show that Y satisfies the conitions of Lemma 7. This hols for any amissible switching sequence, so the system satisfies T-step performance c levels Γ; because α, β, η o not epen on the switching sequence, Lemma 3 gives provies c, λ inepenent of the particular switching sequence. For necessity, suppose the system is both uniformly exponentially stable an satisfies T-step performance. Lemma 7 guarantees the existence of an M an a solution Y to (2) an (3) whose blocks are epenent only on M past parameters A(t ),..., A(t M); B(t ),..., B(t M). Applying φ shows A(t) = A θ(t) = A φ(θ(t L:t+H) ) (likewise for B(t)). These parameters are etermine exactly by θ (t L M:t+H ) ; relabel Y t = Y θ(t LM :t+h ) for each t. This hols for any switching sequence, so select a θ which is recurrent, i.e. one in which every amissible sequence of length M + appears infinitely often, an the block structure of (2) an (3) generates every inequality specifie in (7) an (8) as neee. Now consier the system of () connecte in feeback with a controller of the form ˆx(t + ) = Â(t)ˆx(t) + ˆB(t)y(t) u(t) = Ĉ(t)ˆx(t) + ˆD(t)y(t) (9) where ˆx R n k. The controller will be etermine by the switching sequence θ (t L:t+H) for each time (i.e. Â(t) = Âθ(t L:t+H). For amissible j {,..., N} L+H+, let ] Âj ˆBj K j = ˆDj an efine ] ] ] Ai B,i B2,i à i =, B,i =, B2,i = I C,i = C,i ] ] I, C2,i = C 2,i D 2,i = ] ] D 2,i, D2,i = D 2,i Ĉ j from which we construct the close-loop matrices A C (j) = Ãj + B 2,j K j C2,j B C (j) = B,j + B 2,j K j D2,j C C (j) = C,j + D 2,j K j C2,j D C (j) = D,j + D 2,j K j D2,j an the corresponing close-loop equations x C (t+) = A C (θ (t L:t+H) )x C (t)+b C (θ (t L:t+H) )w(t) z(t) = C C (θ (t L:t+H) )x C (t)+d C (θ (t L:t+H) )w(t). (2) Theorem 2: Fix H an L. The close- 2653

5 loop system (2) is uniformly exponentially stable an satisfies T-step performance levels Γ if an only if there exists an M an a collection of matrices Y j for each amissible j {,..., N} L+M+H such that A C (i ( L:H) )Y i( L M:H ) A C (i ( L:H) ) T Y (i( L M+:H) B C (i ( L:H) )B C (i ( L:H) ) T (2) for every amissible i = (i L M,..., i H ); an T ( tr C C (î (t L:t+H) )Yî(t L M:t+H ) C C (î (t L:t+H) ) T T+ t= + D C (î (t L:t+H) )D C (î (t L:t+H) ) T ) < γ 2 î (:T ) (22) for every amissible î = (î L M,..., î H+T ). Proof: Apply Theorem to (2). We now erive equivalent convex conitions by applying a change-of-variable formula similar to that of 8], 5]. A Schur complement argument gives as equivalent conitions to (2) an (22) the inequalities Y i ( L M:H ) A T C (i ( L:H)) A C (i ( L:H) ) Y i( L M+:H) B C (i ( L:H) ) BC T (i ( L:H)) I (23) Y C T î ( L M:H ) C (î ( L:H) ) C C (î ( L:H) ) Zî( L M:H ) D C (î ( L:H) ) DC T (î ( L:H) ) I (24) T tr Zî(t L M:t+H ) < γ 2. (25) î (:T ) t= Partition the matrices Y j an Y j as ] ] Rj T Y j = j, Y Sj U j = j T T j U T j (26) where R j, S j R n n, U j, T j R n n k. Define ] ] I Si( L M+:H) I Ri( L M:H ) M i = Ui T ; Mi = ( L M+:H) Ti T ( L M:H ) for every i = (i L M,..., i H ), an also Si( L M+:H) W i = A ] j R i( L M:H ) Ui( L M+:H) S + B i( L M+:H) 2,i I T T ] i ( L M:H ). C 2,i R i( L M:H ) I ] K i( L:H) (27) Now apply a congruence transformation to (23) with M i M i I, an to (24) with M i I I. Algebraic manipulation leas to the following result. Theorem 3: There exists a controller with memory L an horizon H achieving uniform stabilization an T-step uniform performance level γ for the close-loop system (2) if an only if there exist an integer M, matrices R j, S j for j {,..., N} L+M+H, an matrices Z i, W i for i {,..., N} L+M+H+ such that H i + F T i W i G i + G T i W T i F i (28) Ĥ i + ˆF T i W i Ĝ i + ĜT i W T i for all amissible i = (i L M,..., i H ); an ˆF i (29) T tr Zî(t L M:t+H ) < γi 2 (:T ) (3) t= for all amissible î = (î L M,..., î H+T ), where i = i ( L M:H ), i + = i ( L M+:H), an ] ] I F i = B2,i T ; ˆFi = D2,i T ] ] I I G i = ; C 2,i D Ĝi = 2,i C 2,i D 2,i S i I A T i A T i S i+ R i R i A T i H i = R i+ I B,i S i+ S i+ B,i I S i I C,i T Ĥ i = R i R i C,i T Z i D,j. I IV. APPLICATION TO A PHYSICAL SYSTEM Consier the system of Fig. 4, which epicts a oublepenulum system with a barbell connecte to the upper linkage using the operating point where all linkages are vertical. Each link has length m an mass Kg. A small. Kg mass may jump between the ens of the barbell at each time step, proucing two operating moes. The nonlinear ynamics of each operating moe are linearize with all linkages vertical (transitions between the two operating moes correspon to a iscontinuous rop or rise in the potential energy of the system via the motion of the small mass). A controlle torque τ u is applie about the bottom hinge, an a isturbance torque about the top hinge. The continuous ynamics are iscretize using an interval of t =.5s. 2654

6 In the configuration shown in Fig. 4, the barbell can be stabilize by riving the topmost hinge (i.e., the upper en of the mile link) left if the jumping mass is at the blue en, or right if the jumping mass is at the re en; the correct control action epens on the position of the jumping mass. However, the intermeiate linkage elays the effect of an input choice on the position of this hinge. If the mass switches position after a choice has been mae, this action chosen will push the system away from equilibrium. Knowlege of the next switching moe allows the controller to stabilize the system. We apply the result of Theorem 2 when L = H = with a performance horizon T = 3 to this system. The γ j are taken as free variables with the objective of minimizing their sum. For a controller with L = H =, this system is feasible with the best γ j given by γ =.98, γ 2 =.82, γ 2 =.95, γ 22 = 2.23 γ 2 = 2.36, γ 22 = 2.29, γ 22 = 2.69, γ 222 = 2.9 These results also emonstrate an avantage of path-bypath performance measure. The best uniform gain that can be place on this system is 2.9; esigning for this level of performance woul result in a 5% larger boun on the performance over path 2 than was achieve with path-by-path control. V. CONCLUDING REMARKS We have examine a receing-horizon-type output regulation problem an evelope exact, convex conitions for the existence an synthesis of controllers which achieve this performance. The performance level can be uniform or allowe to vary path-by-path, an can be optimize as part of solving the existence conitions. The resulting controllers are compute offline, so the implementation of the controller requires only selecting the appropriate gain for the observe switching sequence at each time step. The example system presente emonstrates that a path-epenent controller can outperform a moal controller, an that foreknowlege of the system moes can outperform past-epenent controllers. REFERENCES ] A. Bempora, M. Morari, V. Dua, an E. N. Pistikopoulos, The explicit linear quaratic regulator for constraine systems, Automatica, vol. 38, no., pp. 3 2, 22. 2] G. Dulleru an S. Lall, A new approach for analysis an synthesis of time-varying systems, Automatic Control, IEEE Transactions on, vol. 44, no. 8, pp , ] R. Essick, J.-W. Lee, an G. Dulleru, An exact convex solution to receing horizon control, in American Control Conference (ACC), 22. 4] R. Jungers, The Joint Spectral Raius: Theory an Applications. Berlin, Germany: Springer, 29. 5] S. S. Keerthi an E. G. Gilbert, Optimal infinite-horizon feeback laws for a general class of constraine iscrete-time systems: Stability an moving-horizon approximations, Journal of Optimization Theory an Applications, vol. 57, pp , ] W. H. Kwon an S. Han, Receing Horizon Control. New York: Springer-Verlag, 25. 7] J.-W. Lee, G. Dulleru, an P. Khargonekar, An output regulation problem for switche linear systems in iscrete time, in Decision an Control, 28. CDC 28. 6th IEEE Conference on, 27, pp ], Path-by-path optimal control of switche an markovian jump linear systems, in Decision an Control, 28. CDC th IEEE Conference on, 28, pp ] J.-W. Lee an G. E. Dulleru, Optimal isturbance attenuation for iscrete-time switche an markovian jump linear systems, SIAM Journal on Control an Optimization, vol. 45, no. 4, pp , 26. ], Uniform stabilization of iscrete-time switche an Markovian jump linear systems, Automatica, vol. 42, no. 2, pp , 26. ] J.-W. Lee an P. Khargonekar, Optimal output regulation for iscrete-time switche an markovian jump linear systems, SIAM Journal on Control an Optimization, vol. 47, pp. 4 72, 28. 2] D. Liberzon an A. Morse, Basic probleems in stability an esign of switche systems, Control Systems Magazine, vol. 9, pp. 59 7, ] D. Liberzon, Switching in systems an control. Boston: Birkhäuser, 23. 4] M. Mariton, Jump Linear Systems in Automatic Control. New York, NY: Marcel Dekker, 99. 5] C. Scherer, P. Gahinet, an M. Chilali, Multiobjective outputfeeback control via lmi optimization, Automatic Control, IEEE Transactions on, vol. 42, no. 7, pp , ] Y. Wang an S. Boy, Fast moel preictive control using online optimization, in IFAC Worl Congress, 28, pp τ u Fig. 4. A barbell ouble-penulum system with a movable mass locate at either the upper (blue) or lower (re) position. 2655