Co-design of aperiodic sampled-data min-jumping rules for linear impulsive, switched impulsive and sampled-data systems
CCo-design of aperiodic sampled-data min-jumping rulesfor linear impulsive, switched impulsive andsampled-data systems
Corentin Briat ∗ ‘ Abstract
Co-design conditions for the design of a jumping-rule and a sampled-data controllaw for impulsive and impulsive switched systems subject to aperiodic sampled-datameasurements are provided. Semi-infinite discrete-time Lyapunov-Metzler conditionsare first obtained. As these conditions are difficult to check and generalize to morecomplex systems, an equivalent formulation is provided in terms of clock-dependent(infinite-dimensional) matrix inequalities. These conditions are then, in turn, approx-imated by a finite-dimensional optimization problem using a sum of squares basedrelaxation. It is proven that the sum of squares relaxation is non conservative providedthat the degree of the polynomials is sufficiently large. It is emphasized that acceptableresults are obtained for low polynomial degrees in the considered examples.
Keywords.
Hybrid systems; sampled-data control; switching/jumping rules; clock-dependent conditions
The co-design problem in control theory amounts to simultaneously determining two compo-nents of the considered control system. Important examples are, for instance, the co-design ofa sampling-rule together with a corresponding controller [1, 2], the co-design of a continuouscontrol law and a switching rule for a switched system [3, 4], the co-design of communicationprotocols and controllers in networked control systems [5, 6] or the co-design of multiplecontinuous/sampled-data controllers along with a switching rule in order to stabilize a givensystem [7, 8].In the context of linear switched systems, a popular switching rule is the so-called ”min-switching rule” [4, 9, 10]. In this setup, a quadratic Lyapunov function is consider for each ∗ a r X i v : . [ m a t h . O C ] N ov ode and the mode is chosen in real-time in a way that makes the value of the Lyapunovfunction minimum. It can be shown that this corresponds to finding a certain partitioningof the state-space. An alternative approach [11] is based on the use of a common quadraticLyapunov function for all the subsystems and the switching rule is chosen such that thederivative of the Lyapunov function is minimum at any time. Although slightly different,these approaches share the underlying assumption that the state/output of the system iscontinuously measured, which may be impractical from an implementation point of viewor may lead to undesired chattering. A potential workaround to these issues would bethe consideration of sampled measurements instead of continuous-time ones. Establishingevent-triggered or self-triggered sampling strategies are possible solutions but have beenalmost exclusively considered in the sampled-data control of continuous-time systems. Theco-design problem adds a layer of difficulty since one has to, on the top of the switching-/sampling-rule, simultaneously design a controller. In the context of switched systems, thatwas recently solved in [4] using timer-/clock-dependent Lyapunov functions which leads toinfinite-dimensional stability and stabilization conditions that are affine in the matrices ofthe system and, hence, amenable to be solved using polynomial methods and semidefiniteprogramming techniques [12–14]. Analogous conditions for impulsive and switched systemswere also proposed in [15–17] using looped-functionals and in [18–20] using clock-dependentLyapunov functions.Most of the results on the design of switching rules pertain on the assumption of contin-uous measurements of the state/output. The case of sampled periodic/aperiodic measure-ments has been relatively few studied in the literature in spite of its importance. For instance,a sampled-data switching rule is designed for stabilizing switched affine systems is consideredin [21]. The approach is based on minimizing the derivative of the Lyapunov function at eachsampling instants, which is another an alternative approach to the min-switching rule. Per-haps more closely, a periodic sampled-data switching rule for switched systems is developedin [22] using a min-switching approach. However, the approach does not consider aperiodicsampling, the presence of state jumps and the design of a sampled-data control law. Sincetime-varying delays, jitter and packet loss induced by communication channels in commu-nication networks destroy the periodicity of the arrival times of the sampled measurements,making the sampling scheme apparently aperiodic (see e.g. [23, 24]), it seemed important toincorporate this phenomenon in the current setup. Several methods have been proposed todeal with aperiodic sampling. One is based on the reformulation of a sampled-data systemas an input-delay system that can be analyzed using Lyapunov-Krasovskii functionals; seee.g. [25]. A second approach is a robust one where the sampling operator is considered asan uncertainty and the interconnection is analyzed using robust analysis techniques [26–28].Direct discrete-time approaches have also been considered in [29, 30]. Finally, the hybrid for-mulation [31] of a sampled-data system has been considered in various works and analyzedusing Lyapunov functionals [32], Looped-functionals [15, 33, 34] and clock-/timer- dependentLyapunov functions [18, 20, 31, 35].The objective of this paper is to address an analogous problem for linear impulsive sys-tems, a general class of hybrid systems that encompasses switched and sampled-data systems2s particular instances [18, 36]. We consider here a set-up where both a min-jumping rule (an analogue to the min-switching rule considered in the context of switched systems) and acontinuous-time.sampled-data state-feedback controller in the context of unpredictable sam-pling instants. This has to be contrasted with the setup where the future sampling instant isknown (periodic sampling or event-/self- triggered sampling strategies) and the setup wheremeasurements are continuous. Such a scenario occurs, for instance, in networked controlsystems where actuation decisions are made upon reception of measurements from sensors,which may happen aperiodically due to the presence of sampling, jitter, delays and packetloss [37, 38]. The advantage of such an approach is twofold: it rules out the chattering phe-nomenon of the continuous-time methods and is realistic from an implementation viewpointsince no continuous measurement is assumed. It is assumed here that the measurements fromsensors arrive at discrete time instants which are assumed to satisfy a mild range dwell-timecondition [15, 18]. Hence, both periodic and aperiodic measurements are considered andcan be easily defined in a way to incorporate jitter, small delays and self/event-triggeredsampling mechanisms [24]. The proposed approach is then applied to impulsive systems andswitched-impulsive systems which can be used to model networked control systems subjectto delays and communication outages, systems controlled by multiple controllers or systemswith limited actuation resources [37]. To the author’s knowledge, this is the first time thatconditions for the co-design of asynchronous switching/jumping rules and a sampled-datastate-feedback controller are obtained for impulsive and switched impulsive systems.Sufficient stabilization conditions for the co-design of min-jumping rule and, possibly, asampled-data controller are first stated as discrete-time Lyapunov-Metzler conditions [10,39].These conditions are extended to the co-design of min-jumping/switching rule and sampled-data controllers for impulsive switched systems. The conditions are stated as robust linearmatrix inequalities with a scalar uncertainty at the exponential, which makes them difficultto check; [15, 16] for a similar discussion. These conditions are also difficult to considerwhen additional uncertainties are involved in the system expression, when the system isnot time-invariant or even nonlinear. Looped-functionals [15–17, 33] and clock-dependentLyapunov functions [18–20,40–43] were introduced as a workaround to equivalently representa discrete-time stability condition into a form that is affine in the matrices of the system.The price to pay is the infinite dimensionality of the affine conditions meaning that theycannot be directly checked. However, finite-dimensional approximations can be obtainedby relying on sum of squares (SOS) [12, 18, 19, 41], Handelman’s Theorem [35, 44–47] ordiscretization methods [40]. In this regard, the infinite-dimensional conditions have theadvantage of being readily applicable for design purposes, robustness/performance analysis;see e.g. [15–20, 33, 40–43]. In this paper, SOS relaxations are considered as they often leadto more efficient finite-dimensional conditions than the others [41]. For all the obtainedco-design conditions, converse results are obtained meaning that if the original Lyapunov-Metzler conditions are feasible, then there exists a solution to the SOS program. In thisregard, considering polynomial can be seen as non restrictive in the present cases. Severalexamples illustrate the efficiency of the proposed conditions. Notations.
The set of symmetric matrices of dimension n is denoted by S n and for A, B ∈ n , A (cid:22) B means that A − B is negative semidefinite. The cone of symmetric positive(semi)definite matrices of dimension n is denoted by ( S n (cid:23) ) S n (cid:31) . The n -dimensional vectorof ones is denoted by n . We consider in this paper the following general class of switched impulsive systems withmultiple jump maps and sampled-data state-feedback:˙ x ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 u ( t ) = K σ ( t + k ) ,σ ( t k ) x ( t k ) + K σ ( t + k ) ,σ ( t k ) u ( t k ) , t ∈ ( t k , t k +1 ] x ( t + k ) = J σ ( t + k ) ,σ ( t k ) x ( t k ) , k ∈ Z ≥ (1)where x, x ∈ R n and u ∈ R m are the state of the system, the initial condition and the controlinput, respectively. The notation x ( t + k ) is defined as x ( t + k ) := lim s ↓ t k x ( s ) and we have that x ( t k ) = lim s ↑ t k x ( s ), i.e. the trajectories are left-continuous. The signal σ ( t ) ∈ { , . . . , N } isassumed to be piecewise constant and to only change value on { t k } ∞ k =0 where the sequence { t k } ∞ k =0 satisfies the range dwell-time condition T k := t k +1 − t k ∈ [ T min , T max ], 0 < T min ≤ T max < ∞ , for all k ∈ Z ≥ . It is assumed throughout this paper that this sequence is notknown a priori and is decided, for example, at the sensor level through some event-basedalgorithm that is not known from the controller and scheduling side. Interestingly, this mayoccur in the case of periodic sampling which is subject to a sensor-to-controller time-varyingdelay or data loss; see e.g. [23, 24]. Note, however, that at any time t , the sequence { t k } k ∈ κ ( t ) with κ ( t ) := { k ∈ Z ≥ : t k ≤ t } is known.In this paper, we will address the following general co-design problem for the system (1): Problem 1
Assuming aperiodic sampled-data measurements x ( t k ) satisfying a range dwell-time condition, solve the co-design problem for the system (1) , that is the goal is to design(a) a sampled-data state-dependent sampled-data switching law σ , and(b) a mode-dependent sampled-data control law u such that the closed-loop system is asymptotically stable under the same range dwell-timeconstraint. Co-design problems are, in general, not easy as they can be often non-convex and, therefore,difficult to solve, unless in some particular cases. In the present paper, we aim at proposingsolutions which can be checked using convex programming techniques.To clarify the ideas, we briefly give now several interesting examples covered by theaforementioned general setup. The first one is the sampled-data controller co-design for LTIsystems . This problem is captured by the system˙ x ( t ) = Ax ( t ) + Bu ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 u ( t ) = K σ ( t + k ) ,σ ( t k ) x ( t k ) + K σ ( t + k ) ,σ ( t k ) u ( t k ) , t ∈ ( t k , t k +1 ] (2)4nd the aim is to design N sampled-data controllers (i.e. the gains K i,j , K i,j , i, j =1 , . . . , N ) together with a selecting rule σ deciding on what controllers to use upon sampled-measurements arrivals.The second example concerns the control of jump systems˙ x ( t ) = A ( t ) x ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 x ( t + k ) = J σ ( t k ) x ( t k ) , k ∈ Z ≥ (3)where the aim is to select which jump matrix to use upon sampled-measurements arrivals.Note that there is no co-design problem here unless some jump matrices can be partiallyor fully designed. For instance, if J i =: J i + J i X i for some i , where the X i ’s are matricesto be designed and the others are fixed and known. In such a case, the problem becomesa co-design problem which can be solved using convex optimization techniques. Note thatthis may not be the case when the matrices J i have a different structure.Finally, the third example pertains on the co-design of sampled-data controllers alongwith a switching rule based on sampled-data measurements for switched systems. Thisproblem is well captured by the following model˙ x ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 u ( t ) = K σ ( t + k ) ,σ ( t k ) x ( t k ) + K σ ( t + k ) ,σ ( t k ) u ( t k ) , t ∈ ( t k , t k +1 ] (4)where the goal is to design N sampled-data controllers (i.e. the gains K i,j , K i,j , i, j =1 , . . . , N ) together with a switching rule σ for the switched system as well for selecting theright sampled-data controller. We start with the results on impulsive systems. We first give some preliminaries, followedby the main theoretical results. Due to the structure of the conditions, a section pertainingon solving those conditions is provided.
Let us consider in this section the following impulsive system with multiple jump maps andsampled-data state-feedback:˙ x ( t ) = Ax ( t ) + Bu ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 u ( t ) = K σ ( t + k ) x ( t k ) + K σ ( t + k ) u ( t k ) , t ∈ ( t k , t k +1 ] x ( t + k ) = J σ ( t + k ) x ( t k ) , k ∈ Z ≥ (5)where x ( · ) , x ∈ R n , u ( · ) ∈ R m are the state of the system, the initial condition and thecontrol input. The signal σ ( t ) ∈ { , . . . , N } as well as the sequence { t k } ∞ k =0 satisfy theconditions stated in Section 2. It is assumed throughout this paper that this sequence is not5nown a priori and is decided, for example, at the sensor level through some event-basedalgorithm that is not known from the controller and scheduling side. Interestingly, this mayoccur in the case of periodic sampling which is subject to a sensor-to-controller time-varyingdelay or data loss; see e.g. [23, 24].The objective of this section is to obtain co-design conditions for the simultaneous designof the state-feedback control gains K i ∈ R m × n , K i ∈ R m × m , i = 1 , . . . , N and the jumpscheduling law σ ( t ) that is actuated only at the times in { t k } ∞ k =0 using only current state-information. In order to solve this problem, we first rewrite the impulsive sampled-datasystem (5) into an impulsive system with augmented state-space˙ χ ( t ) = (cid:20) A B (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ¯ A χ ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 χ ( t + k ) = (cid:34) J σ ( t + k ) K σ ( t + k ) K σ ( t + k ) (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) ¯ J σ ( t + k ) χ ( t k ) , k ∈ Z ≥ (6)where χ ( t ) := col( x ( t ) , u ( t )), ¯ J i =: ¯ J i + ¯ J i K i , K i := [ K i K i ], i = 1 , . . . , N , and for whichwe propose the following min-jumping rule σ ( t + k ) = arg min i ∈{ ,...,N } (cid:8) χ ( t k ) T P i χ ( t k ) (cid:9) (7)where the matrices P i ∈ S n + m (cid:31) have to be designed. This rule is clearly inspired by thecontinuous min-switching law considered in [9] in the context of switched systems.The following result states a sufficient condition for the stability of the system (5) con-trolled with the min-jumping rule (7): Proposition 2
Let < T min ≤ T max < ∞ and assume that there exist matrices P i ∈ S n + m (cid:31) , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N verifying TN Π = TN and a scalar ε > suchthat the condition ¯ J Ti e ¯ A T θ (cid:32) N (cid:88) j =1 π ji P j (cid:33) e ¯ Aθ ¯ J i − P i + ε I (cid:22) holds for all i = 1 , . . . , N and all θ ∈ [ T min , T max ] .Then, the system (5) controlled with the rule (7) is asymptotically stable for any sequence { t k } ∞ k =0 satisfying the range dwell-time condition t k +1 − t k ∈ [ T min , T max ] .Proof : The proof of this result is based on the consideration of the linear discrete-timeswitched system χ ( t k +1 ) = e ¯ AT k ¯ J σ ( t + k ) χ ( t k ) (9) A nonnegative matrix is a matrix containing nonnegative entries. ♦ The condition (8) is not an LMI because of the products between P j and π ji but becomesso whenever the π ji ’s are chosen a priori. An approach based on stationary distributionsof Markov processes is proposed in [9, 10]. Another one relies on the simplification of thematrix Π (i.e. Lemma 1 in [10]) or on an equivalent simpler condition [22]. However, theseapproaches do not directly apply to the current problem because of the uncertain parameter θ and the presence of the controller gains that need to be designed. It is unclear at this timewhether there exists a similar way to solve this problem and a brute force approach may benecessary to appropriately select this matrix. A possible solution would be the considerationof a θ -dependent matrix Π. This is left for future research.It is also worth mentioning that the approach based on Lyapunov-Metzler inequalitieshas been favored here as it is computationally more appealing than an approach based on the S -procedure [39] which is known to be less conservative but involves more decision variables(i.e. 2 N ( N −
1) instead of N ( N −
1) for the current one) and more inequality constraints.In this regard, the practical advantage of the approach based on the S -procedure is unclear.In spite of that, the conditions in the above theorem can be easily be extended to this moregeneral setting without any difficulty.Finally, when the state-feedback gains K i are to be computed, then the condition (8) isnot appropriate since it cannot be easily turned into a form that is convenient for designpurposes. At last, the presence of the uncertain parameter θ at the exponential adds somecomputational complexity to the overall approach. However, this latter problem has nowbeen extensively studied; see e.g. [15,18,33,40,48] and subsequent works of the same authors. Inspired by the results in [18, 19], the following co-design result is obtained:
Theorem 3
Let < T min ≤ T max < ∞ . Then, the following statements are equivalent:(a) There exist matrices P i ∈ S n + m (cid:31) , K i ∈ R m × ( n + m ) , i = 1 , . . . , N , and a nonnegative matrix Π ∈ R N × N verifying TN Π = TN such that the condition (8) holds for all i = 1 , . . . , N and all θ ∈ [ T min , T max ] .(b) There exist differentiable matrix-valued functions S i : [0 , T max ] (cid:55)→ S n + m , matrices P i ∈ S n + m (cid:31) , K i ∈ R m × ( n + m ) , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N verifying TN Π = TN and a scalar ε > such that the conditions − ˙ S i ( τ ) + ¯ A T S i ( τ ) + S i ( τ ) ¯ A (cid:22) − P i + ¯ J Ti S i ( θ ) ¯ J i + ε I (cid:22) and N (cid:88) j =1 π ji P j − S i (0) (cid:22) old for all i = 1 , . . . , N , all τ ∈ [0 , T max ] and all θ ∈ [ T min , T max ] .(c) There exist some differentiable matrix-valued functions ˜ S i : [0 , T max ] (cid:55)→ S n + m , matrices ˜ P i ∈ S n + m (cid:31) , U i ∈ R m × ( n + m ) , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N verifying TN Π = TN and a scalar ε > such that the conditions ˙˜ S i ( τ ) + ˜ S i ( τ ) ¯ A T + ¯ A ˜ S i ( τ ) (cid:22) (cid:20) − ˜ P i (cid:63) ¯ J i ˜ P i + ¯ J i U i − ˜ S i ( θ ) (cid:21) ≺ and − N diag j =1 { ˜ P j } + V i ˜ S i (0) V Ti (cid:22) where V i = N col j =1 { π / ji I n + m } hold for all i = 1 , . . . , N , all τ ∈ [0 , T max ] and all θ ∈ [ T min , T max ] .Moreover, when the conditions of statement (c) hold, then the conditions in Proposition 2 holdwith K i = U i ˜ P − i and P i = ˜ P − i , i = 1 , . . . , N . As a result, the system (5) with the controllergains K i = U i ˜ P − i , i = 1 , . . . , N , and the rule (7) with P i = ˜ P − i is asymptotically stable forany sequence { t k } ∞ k =0 satisfying the range dwell-time condition t k +1 − t k ∈ [ T min , T max ] .Proof : Proof that (a) implies (b).
Assume that the conditions in statement (a) holdand define S ∗ i ( τ ) := e ¯ A T τ (cid:34) N (cid:88) j =1 π ji P j (cid:35) e ¯ Aτ . (16)We show now that S i = S ∗ i verifies the conditions in statement (b). Differentiating S ∗ i ( τ )with respect to τ yields ˙ S ∗ i ( τ ) = ¯ A T S ∗ i ( τ ) + S ∗ i ( τ ) ¯ A (17)and, hence, the condition (10) is verified with S i = S ∗ i . Similarly, evaluating S ∗ i ( τ ) at τ = 0yields S ∗ i (0) = N (cid:88) j =1 π ji P j (18)and, therefore, the condition (12) is satisfied with S i = S ∗ i . Finally, substituting S i = S ∗ i inthe LHS of (11) yields ¯ J Ti e ¯ A T θ (cid:32) N (cid:88) j =1 π ji P j (cid:33) e ¯ Aθ ¯ J i − P i + ε I (19)which is negative semidefinite since it was assumed that the statement (a) holds or, equiva-lently, that the condition (8) holds. The proof is completed.8 roof that (b) implies (a). Assume that the conditions of statement (b) hold. Then,integrating (10) over [0 , θ ] yields [49] e ¯ A T θ S i (0) e ¯ Aθ − S i ( θ ) (cid:22) e ¯ A T θ (cid:34) N (cid:88) j =1 π ji P j (cid:35) e ¯ Aθ − S i ( θ ) (cid:22) . By pre- and post-multiplying the above inequality by ¯ J Ti and ¯ J i , respectively, and using (11),we finally obtain (8). The proof is completed. Proof that (b) is equivalent to (c).
This follows from the changes of variables U i = K i ˜ P i ,˜ P i = P − i , ˜ S i ( τ ) = S i ( τ ) − and Schur complements. ♦ The conditions stated in statement (c) are more convenient to consider than those statedin Proposition 2 since the uncertainty θ is not at the exponential anymore and the conditionsare convex in the decision matrices P i ’s and K i ’s whenever the matrix Π is chosen a priori.More specifically, these conditions are convex infinite-dimensional LMI conditions that canbe solved in an efficient way using discretization methods [40–42] or sum of squares methods[12, 18, 19, 41], which can be both shown to be asymptotically exact. On the other hand,when Π has to be designed, the problem becomes nonlinear and may be difficult to solve;see [9, 10] for some discussion on how to potentially solve such a problem.When only the scheduling law needs to be designed, the following immediate corollaryshould be considered: Corollary 4
Assume that B = 0 in (5) and let < T min ≤ T max < ∞ . Then, the followingstatements are equivalent:(a) There exist matrices P i ∈ S n (cid:31) , i = 1 , . . . , N , and a nonnegative matrix Π ∈ R N × N verifying TN Π = TN such that the condition J Ti e A T θ (cid:32) N (cid:88) j =1 π ji P j (cid:33) e Aθ J i − P i ≺ holds for all i = 1 , . . . , N and all θ ∈ [ T min , T max ] .(b) There exist some differentiable matrix-valued functions S i : [0 , T max ] (cid:55)→ S n , matrices P i ∈ S n (cid:31) , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N verifying TN Π = TN and ascalar ε > such that the conditions − ˙ S i ( τ ) + A T S i ( τ ) + S i ( τ ) A (cid:22) − P i + J Ti S i ( θ ) J i + ε I (cid:22) nd N (cid:88) j =1 π ji P j − S i (0) (cid:22) hold for all i = 1 , . . . , N , all τ ∈ [0 , T max ] and all θ ∈ [ T min , T max ] .Moreover, when the conditions of statement (b) are verified, then the system (28) with B = 0 controlled with the rule σ ( t + k ) = arg min i ∈{ ,...,N } (cid:8) x ( t k ) T P i x ( t k ) (cid:9) (24) is asymptotically stable for any sequence { t k } ∞ k =0 satisfying the range dwell-time condition t k +1 − t k ∈ [ T min , T max ] . There are multiple ways for solving the conditions in Theorem 3. A first approach is toassume that the matrix-valued functions S i are piecewise linear, which would then resultin a finite set of finite-dimensional LMI conditions; see e.g. [4, 40]. This approach is easyto use but may not be efficient because of its high computational complexity and its poorconvergence properties as the number of pieces increase [41]. An alternative one is based onpolynomial optimization techniques either relying on Handelman’s Theorem [35, 44–47] orPutinar’s Positivstellensatz [12–14]. Both of these methods will result in a finite-dimensionalsemidefinite program which can be solved using standard solvers such as SeDuMi [50]. Wepropose to use here an approach based on sums of squares. The conversion to a semidefi-nite program can be performed using the package SOSTOOLS [13] to which we input theSOS program corresponding to the considered conditions. We illustrate below how an SOSprogram associated with some given conditions can be obtained. We define the followingintervals [0 , T max ] = { τ ∈ R : g ( τ ) := τ ( T max − τ ) ≥ } . (25)and [ T min , T max ] = { τ ∈ R : h ( τ ) := ( τ − T min )( T max − τ ) ≥ } . (26)In what follows, we say that a symmetric polynomial matrix Θ( · ) is a sum of squares matrix(SOS matrix) or is SOS, for simplicity, if there exists a polynomial matrix Ξ( · ) such thatΘ( · ) = Ξ( · ) T Ξ( · ). Proposition 5 ( [12])
A univariate polynomial matrix is positive semidefinite if and onlyif it is SOS.
We also need the following result:
Proposition 6 ( [45, 51])
Let us consider a univariate polynomial symmetric matrix M ( · ) .Then, the following statements are equivalent: S i , Γ i , ∆ i : R (cid:55)→ S n + m , i = 1 , . . . , N ,constant symmetric matrices P i ∈ S n + m , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N suchthat T Π = T and a scalar ε > • Γ i ( · ) , ∆ i ( · ), i = 1 , . . . , N , are SOS matrices, • P i − ε I , i = 1 , . . . , N , are SOS matrices, • ˙ S i ( τ ) − Sym[ S i ( τ ) ¯ A i ] − Γ i ( τ ) g ( τ ) is an SOS matrix for all i = 1 , . . . , N , • P i − ¯ J Ti S i ( θ ) ¯ J i − ε I − ∆ i ( θ ) h ( θ ) is an SOS matrix for all i = 1 , . . . , N , • − N (cid:88) j =1 π ji P j + S i (0) is an SOS matrix for all i = 1 , . . . , N . (a) The matrix M ( θ ) is positive semidefinite for all θ ∈ [ T min , T max ] .(b) There exists an SOS matrix N ( · ) such that the matrix M ( θ ) − N ( θ ) h ( θ ) is SOS. The following result provides the sum of squares formulation of Theorem Corollary 4,(b):
Proposition 7
Let ε and < T min ≤ T max < ∞ be given. The conditions of Corollary 4,(b), are feasible with polynomial matrices S i if and only if the sum of squares program inBox 1 is feasible. Moreover, we have that the conditions in Corollary 4, (a) hold.Proof : Clearly the first statement in Box 1 is equivalent to saying that the matrix-valuedfunctions Γ i ( · ) , ∆ i ( · ) are positive semidefinite for all i = 1 , . . . , N . The second statementis equivalent to saying that the matrices P i are positive definite for all i = 1 , . . . , N . SinceΓ i ( τ ) (cid:23) τ ∈ R and g ( τ ) ≥ τ ∈ [0 , ¯ T ], then, from Proposition 6, we cansee that the second statement is equivalent to saying that − ˙ S i ( τ ) + Sym[ S i ( τ ) ¯ A i ] (cid:23) τ ∈ [0 , ¯ T ], which coincides with the condition (10) in Corollary 4, (b). Similarly, the fourthstatement is equivalent to saying that − P i + ¯ J Ti S i ( θ ) ¯ J i + ε I (cid:22) θ ∈ [ T min , T max ],which coincides with the condition (11) in Corollary 4, (b). Finally, the last statement isequivalent to the condition (12) in Corollary 4, (b). This proves the first part of the result.The second part follows from the fact that (b) implies (a) in Corollary 4. ♦ We have proved above that the conditions of Corollary 4, (b) can be verified by solvinga semidefinite program if we restrict ourselves to polynomial matrices. In this regard, wehave proved that we have a sufficient SOS condition for assessing whether the statement (a)of Corollary 4 holds. The following result proves that we still have the equivalence between(a) and (b) when considering polynomial matrices.11 roposition 8 (Asymptotic exactness)
Let < T min ≤ T max < ∞ be given. Assumethat the conditions of Corollary 4, (a) hold. Then, there exist ε > and d ∈ Z ≥ such thatthe SOS program in Box 1 is feasible using polynomial matrices of degree d .Proof : The general closed form for the S i ’s which solve the conditions in Corollary 4, (b),is given by S ∗ i ( τ ) = e ¯ A T τ (cid:34) N (cid:88) j =1 π ji P j (cid:35) e ¯ Aτ + (cid:90) τ ¯ J Ti e ¯ A T ( τ − s ) W i ( s ) e ¯ A ( τ − s ) ¯ J i ds, τ ∈ [0 , T max ] . (27)where W i ( s ) (cid:23) i = 1 , . . . , N . Note that only the case W i ≡ τ as closely as desired by a polynomial matrix. This proves the result. ♦ Theorem 9
Let < T min ≤ T max < ∞ be given. Then, the following statements areequivalent:(a) The conditions of Corollary 4, (a) hold.(b) There exist some scalars ε > and d ∈ Z ≥ such that the SOS program in Box 1 isfeasible using polynomial matrices of degree d .Proof : The proof follows from Corollary 4 and Proposition 8. ♦ We have the analogous result for Theorem 3, (c):
Theorem 10
Let < T min ≤ T max < ∞ be given. Then, the following statements areequivalent:(a) The conditions of Theorem 3, (a) hold.(b) There exist some scalars ε > and d ∈ Z ≥ such that the SOS program in Box 2 isfeasible using polynomial matrices of degree d .Proof : The proof is identical to that of Theorem 9. ♦ S i , Γ i , ∆ i : R (cid:55)→ S n + m , i = 1 , . . . , N ,constant symmetric matrices ˜ P i ∈ S n + m , i = 1 , . . . , N , constant matrices U i ∈ R m × ( n + m ) , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N such that T Π = T and a scalar ε > • Γ i ( · ) , ∆ i ( · ), i = 1 , . . . , N , are SOS matrices, • ˜ P i − ε I , i = 1 , . . . , N , are SOS matrices, • − ˙˜ S i ( τ ) − Sym[ ˜ S i ( τ ) ¯ A Ti ] − Γ i ( τ ) g ( τ ) is an SOS matrix for all i = 1 , . . . , N , • (cid:20) ˜ P i − ε I ( ¯ J i ˜ P i + ¯ J i U i ) T ¯ J i ˜ P i + ¯ J i U i ˜ S i ( θ ) − ∆ i ( θ ) h ( θ ) (cid:21) is an SOS matrix for all i = 1 , . . . , N , • N diag j =1 [ ˜ P j ] − V i ˜ S i (0) V Ti is an SOS matrix for all i = 1 , . . . , N . Let us now consider the following class of switched impulsive systems with multiple jumpmaps and sampled-data state-feedback:˙ x ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 u ( t ) = K σ ( t + k ) ,σ ( t k ) x ( t k ) + K σ ( t + k ) ,σ ( t k ) u ( t k ) , t ∈ ( t k , t k +1 ] x ( t + k ) = J σ ( t + k ) ,σ ( t k ) x ( t k ) , k ∈ Z ≥ (28)where x, x ∈ R n and u ∈ R m are the state of the system, the initial condition and thecontrol input, respectively. As before, the switching signal σ : R ≥ (cid:55)→ { , . . . , N } is assumedto be piecewise constant and to only change values on { t k } ∞ k =0 .The above system can be equivalently represented by the impulsive-switched system˙ χ ( t ) = ¯ A σ ( t ) χ ( t ) , t ∈ R ≥ \{ t k } ∞ k =0 χ ( t + k ) = ¯ J σ ( t + k ) ,σ ( t k ) χ ( t k ) , k ∈ Z ≥ (29)with χ = col( x, u ), K j,i := [ K j,i K j,i ],¯ A i = (cid:20) A i B i (cid:21) , ¯ J j,i = (cid:20) J j,i K j,i K j,i (cid:21) =: ¯ J j,i + ¯ J j,i K j,i (30)for i, j = 1 , . . . , N and for which we propose the jump rule: σ ( t + k ) = arg min j ∈{ ,...,N } (cid:8) χ ( t k ) T ¯ J Tj,σ ( t k ) P j ¯ J j,σ ( t k ) χ ( t k ) (cid:9) . (31)13 .2 Main results The following result states a sufficient condition for the stability of the system (28) controlledwith the min-jumping rule (31):
Proposition 11
Let < T min ≤ T max < ∞ and assume that there exist matrices P i ∈ S n + m (cid:31) and a nonnegative matrix Π ∈ R N × N verifying TN Π = TN such that the condition e ¯ A Ti θ (cid:32) N (cid:88) j =1 π ji ¯ J Tj,i P j ¯ J j,i (cid:33) e ¯ A i θ − P i ≺ holds for all i = 1 , . . . , N and all θ ∈ [ T min , T max ] .Then, the system (28) controlled with the min-jumping rule (31) is asymptotically stablefor any sequence { t k } ∞ k =0 satisfying the range dwell-time condition t k +1 − t k ∈ [ T min , T max ] .Proof : The proof consists of an extension of a proof in [10] and is given below for com-pleteness. Let σ ( t k ) = σ ( t + k − ) = i and assume that the conditions of Proposition 11 hold.Define then the function V ( χ ( t + k )) := min j ∈ ,...,N (cid:8) χ ( t + k ) T P j χ ( t + k ) (cid:9) = min j ∈ ,...,N (cid:8) χ ( t k ) T ¯ J Tj,i P j ¯ J j,i χ ( t k ) (cid:9) (33)which is consistent with the rule (11). Note that the function V ( χ ( t + k )) is radially unboundedsince all the matrices P j ’s are positive definite. Then, we have that V ( χ ( t + k )) = min λ ≥ TN λ =1 (cid:40) χ ( t k ) T (cid:32) N (cid:88) j =1 λ j ¯ J Tj,i P j ¯ J j,i (cid:33) χ ( t k ) (cid:41) ≤ χ ( t + k − ) T e A Ti T k (cid:32) N (cid:88) j =1 π ji ¯ J Tj,i P j ¯ J j,i (cid:33) e A i T k χ ( t + k − ) < χ ( t + k − ) T P i χ ( t + k − ) = V ( χ ( t + k − )) (34)provided that T k ∈ [ T min , T max ]. So, V is a discrete-time Lyapunov function for the system χ ( t + k ) = ¯ J σ ( t + k ) ,σ ( t k ) e ¯ A σ ( tk ) T k χ ( t + k − ) (35)controlled with the rule (31). Since the stability of the above discrete-time system is equiv-alent to that of (28) under the min-jumping rule (31) then we can conclude that the system(28) controlled with the min-jumping rule (31) is asymptotically stable for any sequence { t k } ∞ k =0 satisfying the range dwell-time condition t k +1 − t k ∈ [ T min , T max ]. This completesthe proof. ♦ As for Proposition 2, the conditions in Proposition 11 are not directly tractable and needto be reformulated in this regard. This is stated in the following result:14 heorem 12
Let < T min ≤ T max < ∞ . Then, the following statements are equivalent:(a) There exist matrices P i ∈ S n + m (cid:31) , K ij ∈ R m × ( n + m ) , i, j = 1 , . . . , N , and a nonnegativematrix Π ∈ R N × N verifying TN Π = TN such that the conditions in Proposition 11 hold.(b) There exist some differentiable matrix-valued functions S i : [0 , T max ] (cid:55)→ S n + m , matrices P i ∈ S n + m (cid:31) , K i ∈ R m × ( n + m ) , i = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N verifying TN Π = TN and a scalar ε > such that the conditions − ˙ S i ( τ ) + ¯ A Ti S i ( τ ) + S i ( τ ) ¯ A i (cid:22) − P i + S i ( θ ) + ε I (cid:22) and N (cid:88) j =1 π ji ¯ J Tj,i P j ¯ J j,i − S i (0) (cid:22) hold for all i = 1 , . . . , N , all τ ∈ [0 , T max ] and all θ ∈ [ T min , T max ] .(c) There exist some differentiable matrix-valued functions ˜ S i : [0 , T max ] (cid:55)→ S n + m , matrices ˜ P i ∈ S n + m (cid:31) , U i,j ∈ R m × ( n + m ) , i, j = 1 , . . . , N , a nonnegative matrix Π ∈ R N × N verifying TN Π = TN and a scalar ε > such that the conditions ˙˜ S i ( τ ) + ˜ S i ( τ ) ¯ A Ti + ¯ A i ˜ S i ( τ ) (cid:22) P i − ˜ S i ( θ ) + ε I (cid:22) and − ˜ S i (0) V Ti V i − N diag j =1 { ˜ P j } (cid:22) hold for all i = 1 , . . . , N and all θ ∈ [ T min , T max ] where V i = N col j =1 { π / ji [ ¯ J j,i ˜ S i (0) + ¯ J j,i U j,i ] } .Moreover, when the conditions of statement (c) hold, then the conditions of Proposition 11hold with K j,i = U j,i ˜ S i (0) − , i, j = 1 , . . . , N , i (cid:54) = j , and P i = ˜ P − i , i = 1 , . . . , N . As aresult, the system (28) with controller gains K j,i = U j,i ˜ S i (0) − and min-jumping rule (31) is asymptotically stable for any sequence { t k } ∞ k =0 satisfying the range dwell-time condition t k +1 − t k ∈ [ T min , T max ] .Proof : This result can be proven in the same way as Theorem 3. ♦ S i , Γ i , ∆ i : R (cid:55)→ S n + m , i = 1 , . . . , N ,constant symmetric matrices ˜ P i , i = 1 , . . . , N , constant matrices U i ∈ R m × ( n + m ) , i =1 , . . . , N , a nonnegative matrix Π ∈ R N × N such that T Π = T and a scalar ε > • Γ i ( · ) , ∆ i ( · ), i = 1 , . . . , N , are SOS matrices, • P i − ε I , i = 1 , . . . , N , are SOS matrices, • − ˙˜ S i ( τ ) − Sym[ ˜ S i ( τ ) ¯ A Ti ] − Γ i ( τ ) g ( τ ) is an SOS matrix for all i = 1 , . . . , N , • − ˜ P i + ˜ S i ( θ ) − ε I − ∆ i ( θ ) h ( θ ) is an SOS matrix for all i = 1 , . . . , N , • ˜ S i (0) V Ti V i N diag j =1 [ ˜ P j ] is an SOS matrix for all i = 1 , . . . , N . Theorem 13
Let < T min ≤ T max < ∞ be given. Then, the following statements areequivalent:(a) The conditions of Theorem 12, (a) hold.(b) There exist some scalars ε > and d ∈ Z ≥ such that the SOS program in Box 3 isfeasible using polynomial matrices of degree d .Proof : The proof is identical to that of Theorem 9 and Theorem 10. ♦ We provide several simple illustrative examples. Although simple, some of these resultsare interesting from an educational viewpoint as the impact of the different parts of thesystem onto its stability is immediately clear from their formulation. The infinite-dimensionalconditions stated in the main results of the paper are checked using the package SOSTOOLS[13] and the semidefinite programming solver SeDuMi [50]. Examples of sum of squaresprograms can be found in [18, 19, 41]. 16 .1 Example 1. Sampled-data control or state jumps
Let us consider the system (5) with the matrices A = (cid:20) (cid:21) , B = (cid:20) (cid:21) (42)and we define two jump matrices¯ J = (cid:20) I K K (cid:21) and ¯ J = . . . (43)The rationale for considering this system lies in the fact that the pair ( A, B ) is non-stabilizable as the first state is not affected by the sampled-data control input while thesecond one is. To overcome this situation, we assume the existence of a jump map thatcan have a stabilizing effect on the first state with the caveat that it slightly destabilizesthe second one. Hence, stabilizing the system should be possible provided that one can findsuitable controller gains K and K as well as a suitable jumping rule of the form (7). It isinteresting to see that a game needs to be played between the two jump matrices as applyingthe first one stabilizes the second state but leaves the dynamics of the first one unchanged(i.e. exponentially increasing) whereas applying the second jump matrix will result in thedecrease in magnitude of the first state but an increase of the first one.Applying then the conditions of Theorem 3(c) with T min = 10ms, T max = 50ms, π = π = 0 . P = . . . . . . . . . , P = . . . . . . . . . and the following controller gain (cid:2) K K (cid:3) = (cid:2) − . − . − . (cid:3) . (44)Simulation results are depicted in Fig. 1 where we can see that the co-designed sampled-data jump rule and state-feedback control law are effectively able to stabilize the open-loopunstable system. We consider here the system (5) with the matrices A = (cid:20) (cid:21) , J = (cid:20) . (cid:21) and J = (cid:20) . (cid:21) (45)17 x ( t ) x ( t ) u ( t ) < ( t ) Figure 1: Simulation results for the system (5), (42), (43).where we can see that, as before, a game needs to be played between the two jump matrices.Applying the conditions of Theorem 3(c) (adapted to the case where no state-feedbackcontroller has to be designed) with π = π = 0 . T min = T max = 0 .
02 yields the matrices P = (cid:20) . . . . (cid:21) and P = (cid:20) . . . . (cid:21) . (46)The simulation results in Fig. 2 illustrate the ability of the min-switching rule to efficientlystabilize the system. Let us consider the system (28) with the matrices A = A = (cid:20) (cid:21) , B = (cid:20) (cid:21) and B = (cid:20) (cid:21) . (47)This system represents a system for which only one actuator can be updated at a time whilethe other maintains its previous control input. The first one can act on both states (i.e.the pair ( A, B ) is controllable) whereas the second one can only act on the second state.Applying the conditions of Theorem 12(c) with π = π = 0 . T min = 10ms, T max = 50ms,18 x ( t ) x ( t ) < ( t ) Figure 2: Simulation results for the system (5), (45).with polynomials of order 2 in the associated sum of squares conditions yields the matrices P = 10 . − . − . − . − .
06 3 .
51 0 . − . − .
14 0 .
03 7 .
45 0 . − . − .
64 0 .
59 203 . P = 10 . − . − . − . − .
05 4 .
03 0 . − . − .
53 0 .
05 191 .
30 0 . − . − .
53 0 .
61 23 . together with K , = (cid:2) − . − . − . − . (cid:3) ,K , = (cid:2) − . − . − . − . (cid:3) ,K , = (cid:2) − . − . . − . (cid:3) ,K , = (cid:2) − . − . . . (cid:3) . The simulation results depicted in Fig. 3 demonstrate the stabilization effect of the proposedco-design approach. 19 x ( t ) x ( t ) u ( t ) u ( t ) < ( t ) Figure 3: Simulation results for the system (28), (47).
Let us consider here the continuous-time system˙ x = (cid:20) (cid:21) x + (cid:20) (cid:21) u (48)for which we aim to design a sampled-data state-feedback control law together with a decisionrule on whether to update the value of the control input or keep the previous one when asampled measurement arrives. This can be modeled using the augmented system˙ z = z (49)where z = col( x, u ) which is subject to the jump matrices J = I and J = (cid:20) I K K (cid:21) (50)where the controller matrices K , K need to be designed. Choosing now Π = (cid:20) . . . . (cid:21) and T min = 0 .
01, polynomials of degree 4, we can show that we can stabilize such a systemwith aperiodic measurements such that T k ∈ [0 . , .
13] with the controller matrices K = (cid:2) − . − . (cid:3) and K = − . . (51)20 x ( t ) x ( t ) u ( t ) Control law updates
Figure 4: Simulation results for the system (28), (48).The SOS program has 1803 primal and 480 dual variables. It takes less than 3 seconds tobe solved. We also get the matrices P = . − . − . − . . − . − . − . . , P = . − . − . − . . − . − . − . . (52) Let us start with a switched-system of the form (28) with the matrices A = , B = , A = , B = (53)and A = , B = . (54)The spectrum of all the above matrices is located in the right-half plane. Moreover, thecontrollability matrix of each subsystem has rank one, which means that only one state canbe controlled at a time, namely, state 3,1 and 2 for the subsystem 1,2 and 3, respectively.21he goal is then to find a sampled-data control law that allows for the stabilization of thecontrollable state and a switching law allowing to stabilize the overall system. We hence,define the augmented system ˙ z = (cid:20) A σ B σ (cid:21) z (55)where z = col( x, u ). We also select the following parameters d = 2, T min = 0 . T max = 0 . . . . . . . . . . The SDP problem has 2200 primal and 738 dual variables and is solved in less than 2seconds. The following Lyapunov matrices are obtained: P = . . − . − . . . − . − . − . − . . − . − . − . − . . , (56) P = . . − . − . . . − . − . − . − . . . − . − . . . , (57) P = . . − . − . . . − . − . − . − . . . − . − . . . (58)as well as the following state-feedback gains K , = (cid:2) − . − . − . . (cid:3) ,K , = (cid:2) − . − . − . . (cid:3) ,K , = (cid:2) − . − . − . . (cid:3) ,K , = (cid:2) − . − . − . − . (cid:3) ,K , = (cid:2) − . − . − . − . (cid:3) ,K , = (cid:2) − . − . − . . (cid:3) ,K , = (cid:2) − . − . − . . (cid:3) ,K , = (cid:2) − . − . − . . (cid:3) ,K , = (cid:2) − . − . − . − . (cid:3) . (59)The simulation results depicted in Fig. 5 demonstrate the stabilization effect of the pro-posed co-design approach. 22 x ( t ) x ( t ) x ( t ) u ( t ) < ( t ) Figure 5: Simulation results for the system (28), (53).
A co-design approach for the simultaneous design of a sampled-data control law and a min-jumping rule for jump systems subject to sampled measurements has been provided. Theapproach naturally extends to address the co-design problem for switched systems sub-ject to sampled measurements. The stability and stabilization conditions are expressed asinfinite-dimensional Lyapunov-Metzler conditions that can reduce to infinite-dimensionalsemidefinite programs when the Lyapunov-Metzler variable is set to a specific value. Asthe conditions are infinite-dimensional, they cannot be directly checked. Relaxed resultsbased on sum of squares (SOS) are provided and result in finite-dimensional semidefiniteprograms when the degree of the polynomials are fixed. Converse results are then givento demonstrate that SOS relaxations are, in fact, non conservative if we allow the degreeof the polynomials to be arbitrarily large. Numerical results tend to suggest that satisfyingresults are already obtained for low polynomial degrees. Examples are given for illustrations.Potential extensions include the consideration of system uncertainties and the inclusion ofperformance measures such that the L -performance measure. These extensions are, in fact,rather straightforward due to the affine dependence of the conditions in terms of the matricesof the system. 23 eferences [1] A. Anta and P. Tabuada, “To sample or not to sample: Self-triggered control for non-linear systems,” IEEE Transactions on Automatic Control , vol. 55(9), pp. 2030–2042,2010.[2] A. Seuret and C. Prieur, “Event-triggered sampling algorithms based on a lyapunovfunction,” in , Orlando, USA, 2011, pp.6128–6133.[3] D. Liberzon,
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