Optimal switching control design for polynomial systems: an LMI approach
OOptimal switching control design forpolynomial systems: an LMI approach
Didier Henrion , , , Jamal Daafouz , Mathieu Claeys Draft of October 31, 2018
Abstract
We propose a new LMI approach to the design of optimal switching sequences forpolynomial dynamical systems with state constraints. We formulate the switchingdesign problem as an optimal control problem which is then relaxed to a linearprogramming (LP) problem in the space of occupation measures. This infinite-dimensional LP can be solved numerically and approximately with a hierarchy ofconvex finite-dimensional LMIs. In contrast with most of the existing work on LMImethods, we have a guarantee of global optimality, in the sense that we obtainan asympotically converging (i.e. with vanishing conservatism) hierarchy of lowerbounds on the achievable performance. We also explain how to construct an almostoptimal switching sequence.
A switched system is a particular class of a hybrid system that consists of a set of dynam-ical subsystems, one of which is active at any instant of time, and a policy for activatingand deactivating the subsystems. One may encounter such dynamical systems in a widevariety of application domains such as automotive industry, power systems, aircraft andtraffic control, and more generally the area of embedded systems. Switched systems havebeen the concern of many researchers and many results are available for stability anal-ysis and control design. They put in evidence the important fact that it is possible toorchestrate the subsystems through an adequate switching strategy in order to imposeglobal stability. Interested readers may refer to the survey papers [10, 20, 29, 22] and theinteresting and useful books [21, 30] and the references therein.In this context, switching plays a major role for stability and performance properties.Indeed, switched systems are generally controlled by switched controllers and the control CNRS; LAAS; 7 avenue du colonel Roche, F-31077 Toulouse; France. [email protected] Universit´e de Toulouse; UPS, INSA, INP, ISAE; UT1, UTM, LAAS; F-31077 Toulouse; France Faculty of Electrical Engineering, Czech Technical University in Prague, Technick´a 2, CZ-16626Prague, Czech Republic Universit´e de Lorraine, CRAN, CNRS, IUF, 2 avenue de la fort de Haye, 54516 Vandœuvre cedex,France. [email protected] a r X i v : . [ m a t h . O C ] M a r ignal is intrinsically discontinuous. As far as optimality is concerned, several results arealso available in two main different contexts: • the first category of methods exploits necessary optimality conditions, in the formof Pontryagin’s maximum principle (the so-called indirect approaches), or througha large nonlinear discretization of the problem (the so-called direct approaches), see[2, 3, 6, 15, 23, 24, 25, 26, 27, 28, 31] for details. Therefore only local optimalitycan be guaranteed for general nonlinear problems, even when discretization can beproperly controlled; • the second category collects extensions of the performance indexes H and H ∞ orig-inally developped for linear time invariant systems without switching, and use theflexibility of Lyapunov’s approach, see for instance [13, 9] and references therein.Even for linear switched systems, the proposed results are based on nonconvex op-timization problems (e.g. bilinear matrix inequality conditions) difficult to solvedirectly. Sufficient linear matrix inequality (LMI) design conditions may be ob-tained, but at the price of introducing a conservatism (pessimism) which is hard,if not impossible, to evaluate. Since the computation of this optimal strategy is adifficult task, a suboptimal solution is of interest only when it is proved to be con-sistent, meaning that it imposes to the switched system a performance not worsethan the one produced by each isolated subsystem [14].Despite the interest of these existing approaches, the optimal control problem is notcompletely solved for switched systems and new strategies are more than welcome, ascomputationally viable design techniques are missing. In this paper, we consider theproblem of designing optimal switching rules in the case of polynomial switched dynamicalsystems. Classically, we formulate the optimal control switching problem as an optimalcontrol problem with controls being functions of time valued in { , } , and we relax itinto a control problem with controls being functions of time valued in [0 , Consider the optimal control problem p ∗ = inf (cid:82) T l σ ( t ) ( t, x ( t )) dt s . t . ˙ x ( t ) = f σ ( t ) ( t, x ( t )) , σ ( t ) ∈ { , , . . . , m } x (0) ∈ X , x ( T ) ∈ X T x ( t ) ∈ X, t ∈ [0 , T ] (1)with given polynomial velocity field f σ ( t ) ∈ R [ t, x ] n and given polynomial Lagrangian l σ ( t ) ∈ R [ t, x ] indexed by an integer-valued signal σ : [0 , T ] → { , , . . . , m } . System state x ( t ) belongs to a given compact semialgebraic set X ⊂ R n for all t ∈ [0 , T ] and the initialstate x (0) resp. final state x ( T ) are constrained to a given compact semialgebraic set X ⊂ X resp. X T ⊂ X . In problem (1) the infimum is w.r.t. sequence σ and terminaltime T .In this paper, for the sake of simplicity, we assume that the terminal time T is finite, thatis, we do not consider the asymptotic behavior. Typically, if a solution to problem (1) isexpected for a very large or infinite terminal time, we must reformulate the problem byrelaxing the state constraints.Optimal control problem (1) can then be equivalently written as p ∗ = inf u (cid:82) T (cid:80) mk =1 l k ( t, x ( t )) u k ( t ) dt s . t . dx ( t ) = (cid:80) mk =1 f k ( t, x ( t )) u k ( t ) dtx (0) ∈ X , x ( T ) ∈ X T x ( t ) ∈ X, u ( t ) ∈ U, t ∈ [0 , T ] (2)where the infimum is with respect to a time-varying vector u ( t ) which belongs for all t ∈ [0 , T ] to the (nonconvex) discrete set U := { (1 , , . . . , , (0 , , . . . , , . . . , (0 , , . . . , } ⊂ R m . In general the infimum in problem (2) is not attained (see our numerical examples lateron) and the problem is relaxed to p ∗ R = inf (cid:82) T (cid:80) mk =1 l k ( t, x ( t )) u k ( t ) dt s . t . dx ( t ) = (cid:80) mk =1 f k ( t, x ( t )) u k ( t ) dtx (0) ∈ X , x ( T ) ∈ X T x ( t ) ∈ X, u ( t ) ∈ conv U, t ∈ [0 , T ] (3)where the minimization is now with respect to a time-varying vector u ( t ) which belongsfor all t ∈ [0 , T ] to the (convex) simplexconv U = { u ∈ R m : m (cid:88) k =1 u k = 1 , u k ≥ , k = 1 , . . . , m } .
3n [2], problem (3) is called the embedding of problem (1), and it is proved that the set oftrajectories of problem (1) is dense (w.r.t. the uniform norm in the space of continuousfunctions of time) in the set of trajectories of embedded problem (3). Note however thatthese authors consider the more general problem of switching design in the presence ofadditional bounded controls in each individual dynamics. To cope with chattering effectsdue to the simultaneous presence of controls and (initial and terminal) state constraints,they have to introduce a further relaxation of the embedded control problem. In thispaper, we do not have controls in the dynamics, and the only design parameter is theswitching sequence.An equivalent way of writing the dynamics in problem (3) is via a differential inclusion˙ x ( t ) ∈ conv { f ( t, x ( t )) , . . . , f m ( t, x ( t )) } . (4)By this it is meant that at time t , the state velocity ˙ x ( t ) can be any convex combina-tion of the vector fields f k ( t, x ( t )), k = 1 , . . . , m , see e.g. [4, Section 3.1] for a tutorialintroduction. For this reason, problem (3) is also sometimes called the convexification ofproblem (1).Since problem (3) is a relaxation of problem (1), it holds p ∗ R ≤ p ∗ . For most of thephysically relevant problems, and especially when the state constraints in problem (1) arenot overly stringent, it actually holds that p ∗ R = p ∗ . For a discussion about the cases forwhich p ∗ R < p ∗ , please refer to [16, Appendix C] and references therein. Given an initial condition x ∈ X and an admissible control u ( t ), denote by x ( t | x , u ), t ∈ [0 , T ], the corresponding admissible trajectory, an absolutely continuous function oftime with values in X . Define the occupation measure µ ( A × B | x , u ) := (cid:90) T I A × B ( t, x ( t | x , u )) dt for all subsets A × B in the Borel σ -algebra of subsets of [0 , T ] × X , where I A ( x ) is theindicator function of set A , equal to one if x ∈ A , and zero otherwise. We write x ( t | x , u )resp. µ ( dt, dx | x , u ) to emphasize the dependence of x resp. µ on initial condition x and control u , but for conciseness we also use the notation x ( t ) resp. µ ( dt, dx ). Theoccupation measure can be disintegrated into µ ( A × B ) = (cid:90) A ξ ( B | t ) ω ( dt )where ξ ( dx | t ) is the distribution of x ∈ R n , conditional on t , and ω ( dt ) is the marginalw.r.t. time t , which models the control action as a measure on [0 , T ]. The conditional ξ is a stochastic kernel, in the sense that for all t ∈ [0 , T ], ξ ( . | t ) is a probability measure on X , and for every B in the Borel σ -algebra of subsets of X , ξ ( B | . ) is a Borel measurablefunction on [0 , T ]. An equivalent definition is ξ ( B | t ) = I B ( x ( t )) = δ x ( t ) ( B ) where δ isthe Dirac measure. The occupation measure encodes the system trajectory, and the value4 T µ ( dt, B ) = µ ([0 , T ] × B ) is equal to the total time spent by the trajectory in set B ⊂ X .Note also that time integration of any smooth test function v : [0 , T ] × X → R along atrajectory becomes a time and space integration against µ , i.e. (cid:90) T v ( t, x ( t )) dt = (cid:90) T (cid:90) X v ( t, x ) µ ( dt, dx ) = (cid:90) vµ. In optimal control problem (3), we associate an occupation measure µ k ( dt, dx ) = ξ k ( dx | t ) ω k ( t )for each system mode k = 1 , . . . , m , so that globally m (cid:88) k =1 µ k = µ is the occupation measure of a system trajectory subject to switching. The marginal ω k is the control, modeled as a measure which is absolutely continuous w.r.t. the Lebesguemeasure, i.e. such that (cid:90) X µ k ( dt, dx ) = ω k ( dt ) = u k ( t ) dt for some measurable control function u k ( t ), k = 1 , . . . , m . The system dynamics inproblem (3) can then be expressed as dx ( t ) = m (cid:88) k =1 f k ( x ( t )) ω k ( dt ) (5)where the controls are now measures ω k . To enforce that u ( t ) ∈ conv U for almost alltimes t ∈ [0 , T ], we add the constraint m (cid:88) k =1 ω k ( dt ) = I [0 ,T ] ( t ) dt (6)where the right hand side is the Lebesgue measure, or uniform measure on [0 , T ].Given a smooth test function v : [0 , T ] × X → R and an admissible trajectory x ( t ) withoccupation measure µ ( dt, dx ), it holds (cid:82) T dv ( t, x ( t )) = v ( T, x ( T )) − v (0 , x (0))= (cid:82) T (cid:0) ∂v∂t ( t, x ( t )) + (cid:80) k grad v ( t, x ( t )) f k ( t, x ( t )) u k ( t ) (cid:1) dt = (cid:82) T (cid:82) X (cid:0) ∂v∂t ( t, x ) µ ( dt, dx )+ (cid:80) k grad v ( t, x ) f k ( t, x ) µ k ( dt, dx ))= (cid:80) k (cid:82) ∂v∂t µ k + grad vf k µ k . Now, consider that the initial state is not a single vector x but a random vector whosedistribution is ruled by a probability measure µ , so that at time t the state x ( t ) is alsomodeled by a probability measure µ t ( . ) := ξ ( . | t ), not necessarily equal to δ x ( t ) . Theinterpretation is that µ t ( B ) is the probability that the state x ( t ) belongs to a set B ⊂ . Optimal control problem (3) can then be formulated as a linear programming (LP)problem: p ∗ M = inf (cid:80) k (cid:82) l k µ k s . t . (cid:82) vµ T − (cid:82) vµ = (cid:80) k (cid:82) ∂v∂t µ k + grad vf k µ k (cid:80) k (cid:82) wµ k = (cid:82) w ∀ v ∈ C ([0 , T ] × X ) , ∀ w ∈ C ([0 , T ]) (7)where the infimum is w.r.t. measures µ ∈ M + ( X ), µ T ∈ M + ( X T ), µ k ∈ M + ([0 , T ] × X ), k = 1 , . . . , m with M + ( A ) denoting the cone of finite nonnegative measures supported on A , identified as the dual of the cone of nonnegative continuous functions supported on A .It follows readily that p ∗ ≥ p ∗ R ≥ p ∗ M , and under some additional assumptions it shouldbe possible to prove that p ∗ R = p ∗ M and that the marginal densities u k extracted fromsolutions µ k of problem (7) are optimal for problem (2) and hence problem (1). We leavethe rigorous statement and its proof for an extended version of this paper.Note that the use of relaxations and LP formulations of optimal control problems (onordinary differential equations and partial differential equations) is classical, and can betraced back to the work by L. C. Young, Filippov, and then Warga and Gamkrelidze,amongst many others. For more details and a historical survey, see e.g. [12, Part III]. To summarize, we have formulated our relaxed optimal switching control problem (3)as the convex LP (7) in the space of measures. This can be seen as an extension ofthe approach of [18] which was originally designed for classical optimal control problems.Alternatively, this can also be understood as an application of the approach of [7] where thecontrol measures are restricted to be absolutely continuous w.r.t. time. Indeed, absolutecontinuity of the control measures is enforced by relation (6). The infinite-dimensional LPon measures (7) can be solved approximately by a hierarchy of finite-dimensional linearmatrix inequality (LMI) problems, see [18, 7, 16] for details (not reproduced here). Themain idea behind the hierarchy is to manipulate each measure via its moments truncatedto degree 2 d , where d is a relaxation order, and to use necessary LMI conditions for avector to contain moments of a measure. The hierarchy then consists of LMI problems ofincreasing sizes, and it provides a sequence of lower bounds p ∗ d ≤ p ∗ which is monotonicallyincreasing, i.e. p ∗ d ≤ p ∗ d +1 and asymptotically converging, i.e. lim d →∞ p ∗ d = p ∗ .The number of variables N d at the LMI relaxation of order d grows linearly in m (thenumber of modes), and polynomially in d , but the exponent is a linear function of n (thenumber of states). In practice, given the current state-of-the-art in general-purpose LMIsolvers and personal computers, we can expect an LMI problem to be solved in a matterof a few minutes provided the problem is reasonably well-conditioned and N d ≤ Optimal switching sequence
Let y k,α := (cid:90) T (cid:90) X t α µ k ( dt, dx ) = (cid:90) T t α ω k ( dt ) , α = 0 , , . . . denote the moments of measure ω k , k = 1 , . . . , m + 1. Solving the LMI relaxation of order d yields real numbers { y dk,α } α =0 , ,..., d which are approximations to y k,α .In particular, the zero order moment of each measure µ k is an approximation of its mass,and hence of the time t k = (cid:82) µ k ∈ [0 , T ] spent by an optimal switching sequence on mode k . At each LMI relaxation d , it holds (cid:80) m +1 k =1 y dk, = 1 for all d , and lim d →∞ y dk, = t k , sothat in practice, it is expected that good approximations of t k are obtained at relativelysmall relaxation orders.The (approximate) higher order moments { y dk,α } α =1 ,..., d allow to recover (approximately)the densities u k ( t ) of each measure ω k ( dt ), for k = 1 , . . . , m +1. The problem of recoveringa density from its moments is a well-studied inverse problem of numerical analysis. Sincewe expect in many cases the density to be piecewise constant, with possible discontinuitieshere corresponding to commutations between system modes, we propose the followingstrategy.Let us assume that we have the moments y α := (cid:90) T t α ω ( dt )of a (nonnegative) measure with piecewise constant density ω ( dt ) = u ( t ) dt := N (cid:88) k =1 u k I [ t k − ,t k ] ( t ) dt such that the boundary values are zero, i.e. u = 0 and u N +1 = 0. The Radon-Nikodymderivative of this measure reads u (cid:48) ( dt ) = N (cid:88) k =1 ( u k +1 − u k ) δ t k ( dt )where δ t k denotes the Dirac measure at t = t k . Let y (cid:48) α := (cid:90) T t α u (cid:48) ( dt ) = N (cid:88) k =1 ( u k +1 − u k ) t αk denote the moments of the (signed) derivative measure. By integration by parts it holds y (cid:48) α = − αy α − , α = 0 , , , . . . which shows that the moments of u (cid:48) can be obtained readily from the moments of u . Since u (cid:48) is a sum of N Dirac measures, the moment matrix of u (cid:48) is a (signed) sum of N rank-onemoment matrices, and the atoms t k as well as the weights u k +1 − u k , k = 1 , . . . , N can7e obtained readily from an eigenvalue decomposition of the moment matrix as explainede.g. in [19, Section 4.3].More generally, the reader interested in numerical methods for reconstructing a measurefrom the knowledge of its moments is referred to [17] and references therein, as well asto the recent works [11, 1, 5] which deal with the problem of reconstructing a piecewise-smooth function from its Fourier coefficients. To make the connection between momentsand Fourier coefficients, let us just mention that the moments y α = (cid:82) t α u ( t ) dt of a smoothdensity u ( t ) are (up to scaling) the Taylor coefficients of the Fourier transform ˆ u ( s ) := (cid:82) e − πist u ( t ) dt = (cid:80) α ( − πi ) α α ! y α s α . If the y α are given, then ˆ u ( s ) is given by its Taylor series,and the density u ( t ) is recovered with the inverse Fourier transform u ( t ) = (cid:82) e πist ˆ u ( s ) ds .Numerically, an approximate density can be obtained by applying the inverse fast Fouriertransform to the (suitably scaled) sequence { y dα } α =0 , ,..., d of moments. Consider the scalar ( n = 1) optimal control problem (1): p ∗ = inf (cid:82) x ( t ) dt s . t . ˙ x ( t ) = a σ ( t ) x ( t ) x (0) = , x (1) ∈ [ − , x ( t ) ∈ [ − , , ∀ t ∈ [0 , σ : [0 , (cid:55)→ { , } and a := − , a := 1 . In Table 1 we report the lower bounds p ∗ d on the optimal value p ∗ obtained by solving LMIrelaxations of increasing orders d , rounded to 5 significant digits. We also indicate thenumber of variables (i.e. total number of moments) of each LMI problem, as well as thezeroth order moment of each occupation measure (recall that these are approximations ofthe time spent on each mode). We observe that the values of the lower bounds and themasses stabilize quickly.In this simple case, it is easy to obtain analytically the optimal switching sequence:it consists of driving the state from x (0) = to x ( ) = 0 with the first mode, i.e. u ( t ) = 1 , u ( t ) = 0 for t ∈ [0 , [, and then chattering between the first and second modewith equal proportion so as to keep x ( t ) = 0, i.e. u ( t ) = , u ( t ) = for t ∈ ] , p ∗ = (cid:90) / (cid:18) − t (cid:19) dt = 124 ≈ . · − . Because of chattering, the infimum in problem (1) is not attained by an admissible switch-ing sequence. It is however attained in the convexified problem (3).8 p ∗ d N d y d , y d , − . · −
18 0.74056 0.259442 4 . · −
45 0.75170 0.248303 4 . · −
84 0.74632 0.253684 4 . · −
135 0.74918 0.250825 4 . · −
198 0.74974 0.250266 4 . · −
273 0.74990 0.250107 4 . · −
360 0.74996 0.25004Table 1: Lower bounds p ∗ d on the optimal value p ∗ obtained by solving LMI relaxationsof increasing orders d ; N d is the number of variables in the LMI problem; y dk, is theapproximate time spent on each mode k = 1 , y ,α = (cid:90) t α dt + 12 (cid:90) t α dt = 2 + 2 − α α ,y ,α = 12 (cid:90) t α dt = 2 − − α α and they can be compared with the following moment vectors obtained numerically atthe 7th LMI relaxation: y = [0 . . . . . · · · ] ,y = [0 . . . . . · · · ] ,y = [0 . . . . . · · · ] ,y = [0 . . . . . · · · ] . We observe that the approximate moments y k closely match the optimal moments y k , sothat the approximate control law u k extracted from y k will be almost optimal. We revisit the double integrator example with state constraint studied in [18], formulatedas the following optimal switching problem: p ∗ = inf T s . t . ˙ x ( t ) = f σ ( t ) ( x ( t )) x (0) = [1 , , x ( T ) = [0 , x ( t ) ≥ − , ∀ t ∈ [0 , T ]where the infimum is w.r.t. to a switching sequence σ : [0 , T ] (cid:55)→ { , } with free terminaltime T ≥ f := (cid:20) x − (cid:21) , f := (cid:20) x (cid:21) .
9e know from [18] that the optimal sequence consists of starting with mode 1, i.e. u ( t ) =1, u ( t ) = 0 for t ∈ [0 , u ( t ) = u ( t ) = for t ∈ [2 , ] and then eventually driving the state to the originwith mode 2, i.e. u ( t ) = 0, u ( t ) = 1 for t ∈ [ , ]. Here too the infimum p ∗ = is notattained for problem (1), whereas it is attained with the above controls for problem (3).In Table 1 we report the lower bounds p ∗ d on the optimal value p ∗ obtained by solving LMIrelaxations of increasing orders d , rounded to 5 significant digits. We also indicate thenumber of variables (i.e. total number of moments) of each LMI problem, as well as thezeroth order moment of each occupation measure (recall that these are approximations ofthe time spent on each mode). We observe that the values of the lower bounds and themasses stabilize quickly to the optimal values p ∗ = , y , = , y , = . d p ∗ d N d y d , y d , . . . . . . . p ∗ d on the optimal value p ∗ obtained by solving LMI relaxationsof increasing orders d ; N d is the number of variables in the LMI problem; y dk, is theapproximate time spent on each mode k = 1 , ω k ( dt ) withpiecewise constant densities, is obtained numerically as explained in Section 5, by consid-ering the moments of the (weak) derivative of control measures. Consider the optimal control problem (1): p ∗ = inf (cid:82) ∞ (cid:107) x ( t ) (cid:107) dt s . t . ˙ x ( t ) = A σ ( t ) x ( t ) x (0) = [0 , − σ : [0 , ∞ ) (cid:55)→ { , } and A := (cid:20) − − (cid:21) , A := (cid:20) − − − (cid:21) . Since our framework cannot directly accomodate infinite-horizon problems, we introducea terminal condition (cid:107) x ( T ) (cid:107) ≤ − so that terminal time T is finite. It means that theswitching sequence should drive the state in a small ball around the origin.In Table 3 we report the lower bounds p ∗ d on the optimal value p ∗ obtained by solving LMIrelaxations of increasing orders d , rounded to 5 significant digits. We also indicate the10 p ∗ d N d y d , y d , . . . . . . . p ∗ d on the optimal value p ∗ obtained by solving LMI relaxationsof increasing orders d ; N d is the number of variables in the LMI problem; y dk, is theapproximate time spent on each mode k = 1 , SourceTargetChatteringMode 1
Figure 1: Suboptimal trajectory starting at source point x = ( − ,
0) with mode 1, thenchattering between modes 1 and 2 to reach the target, a neighborhood of the origin.In Figure 1 we plot an almost optimal trajectory inferred from the moments of the occu-11ation measure, for an LMI relaxation of order d = 8. The trajectory consists in startingfrom x = ( − ,
0) with mode 1 during 0.065 time units, and then chattering between mode1 and mode 2 with respective proportions 49.3/50.7 until x reaches the neighborhood ofthe origin (cid:107) x ( T ) (cid:107) ≤ − for T = 3 .
84. This trajectory is slightly suboptimal, as it yieldsa cost of 0 . . In this paper we address the problem of designing an optimal switching sequence for ahybrid system with polynomial Lagrangian (objective function) and polynomial vectorfields (dynamics). With the help of occupation measures, we relax the problem from(control) functions with values in { , } to (control) measures which are absolutely con-tinuous w.r.t. time and summing up to one. This allows for a convex linear programming(LP) formulation of the optimal control problem that can be solved numerically which aclassical hierarchy of finite-dimensional convex linear matrix inequality (LMI) relaxations.We can think of two simple extensions of our approach: • Open-loop versus closed-loop. In problem (1) the control signal is the switchingsequence σ ( t ) which is a function of time: this is an open-loop control, similarly towhat was proposed in [7] for impulsive control design. In addition, constrain theswitching sequence to be an explicit or implicit function of the state, i.e. σ ( x ( t )), aclosed-loop control signal. In this case, each occupation measure will be explicitlydepending on time, state and control, and it will disintegrate as µ ( dt, dx, du ) = ξ ( dt | t, u ) ω ( du | t ) dt , and we should follow the framework described originally in[18]. • Switching and impulsive control. We may also combine switching control and im-pulsive control if we extend the system dynamics (5) to dx ( t ) = m (cid:88) k =1 f k ( x ( t )) ω k ( dt ) + p (cid:88) j =1 g j ( t ) τ j ( dt )where g j are given continuous vector functions of time and τ j are signed measuresto be found, jointly with the switching measures ω k . Whereas switching controlmeasures ω k are restricted by (6) to be absolutely continuous w.r.t. the Lebesguemeasure of time, impulsive control measure τ j can concentrate in time. For example,for a dynamical system dx ( t ) = g ( t ) τ ( dt ), a Dirac measure τ ( dt ) = δ s enforces attime t = s a state jump x + ( s ) = x − ( s ) + g ( s ). In this case, to avoid trivial solutions,the objective function should penalize the total variation of the impulsive controlmeasures, see [7]. 12 Removing the states in the occupation measures. In the case that all dynamics f k ( t, x ), k = 1 , . . . , m are affine in x , we can numerically integrate the state trajec-tory and approximate the arcs by polynomials of time. It follows that the occupationmeasures µ k ( dt, dx ), once integrated, do not depend on x anymore. They depend ontime t only. We can then use finite-dimensional LMI conditions which are necessaryand sufficient for a vector to contain the moments of a univariate measure, thereis no need to construct a hierarchy of LMI relaxations. There is however still ahierarchy of LMI problems to be solved, now indexed by the degree of the polyno-mial approximation of the arcs of the state trajectory. To cope with high degreeunivariate polynomials, alternative bases than monomials are recommended (e.g.Chebyshev polynomials), see [8] for more details. Acknowledgments
This work benefited from discussions with Milan Korda, Jean-Bernard Lasserre and LucaZaccarian.
References [1] D. Batenkov. Complete algebraic reconstruction of piecewise-smooth functions fromFourier data. arXiv:1211.0680 , Nov. 2012.[2] S. C. Bengea, R. A. DeCarlo. Optimal control of switching systems. Automatica,41:11-27, 2005.[3] M. S. Branicky, V. S. Borkar, S. K. Mitter. A unified framework for hybrid control:model and optimal control theory. IEEE Trans. Autom. Control, 43:31–45, 1998.[4] A. Bressan, B. Piccoli. Introduction to the mathematical theory of control. Amer.Inst. Math. Sci., Springfield, MO, 2007.[5] E. J. Cand`es, C. Fernandez-Granda. Towards a mathematical theory of super-resolution. arXiv:1203.5871 , Mar. 2012.[6] C. Cassandras, D. L. Pepyne, Y. Wardi. Optimal control of a class of hybrid systems.IEEE Trans. Autom. Control, 46:398–415, 2001.[7] M. Claeys, D. Arzelier, D. Henrion, J. B. Lasserre. Measures and LMI for impul-sive optimal control with applications to space rendezvous problems. Proc. Amer.Control Conf., Montr´eal, Canada, 2012.[8] M. Claeys, D. Arzelier, D. Henrion, J. B. Lasserre. Moment LMI approach to LTVimpulsive control. Work in progress, Feb. 2013.[9] G. S. Deaecto, J. C. Geromel, J. Daafouz. Dynamic output feedback H ∞ control ofswitched linear systems. Automatica, 47:1713–1720, 2011.1310] R. A. DeCarlo, M. S. Branicky, S. Pettersson, B. Lennartson. Perspectives andresults on the stability and stabilizability of hybrid systems. Proc. of the IEEE,88:1069-1082, 2000.[11] Y. de Castro, F. Gamboa. Exact reconstruction using Beurling minimal extrapola-tion. J. Math. Anal. Appl. 395(1):336-354, 2012.[12] H. O. Fattorini. Infinite dimensional optimization and control theory. CambridgeUniv. Press, Cambridge, UK, 1999.[13] J. C. Geromel, P. Colaneri, P. Bolzern. Dynamic output feedback control of switchedlinear systems. IEEE Trans. Autom. Control, 53:720-733, 2008.[14] J.C. Geromel, G. Deaecto, J. Daafouz. Suboptimal switching control consistencyanalysis for switched linear systems. To appear in IEEE Trans. Autom. Control,2013.[15] S. Hedlund, A. Rantzer. Optimal control of hybrid systems. Proc. IEEE Conf. De-cision and Control, Pheonix, Arizona, 1999.[16] D. Henrion, M. Korda. Convex computation of the region of attraction of polynomialcontrol systems. arXiv:1208.1751 , Aug. 2012.[17] D. Henrion, J. B. Lasserre, M. Mevissen. Mean squared error minimization forinverse moment problems. arXiv:1208.6398arXiv:1208.6398