BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS. Mauro Boccadoro Magnus Egerstedt Paolo Valigi Yorai Wardi

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1 BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS Mauro Boccaoro Magnus Egerstet Paolo Valigi Yorai Wari Dipartimento i Ingegneria Elettronica e ell Informazione Università i Perugia 06125, Perugia Italy {magnus,ywari}@ece.gatech.eu Electrical an Computer Engineering Georgia Institute of Technology Atlanta, GA Abstract: In this paper we report of a technique to esign optimal feeback control laws for hybri systems with autonomous (continuous) moes. Existing techniques esign the optimal switching surfaces base on a singular sample evolution of the system; hence proviing a solution epenent on the initial conitions. On the other han, the optimal switching times can be foun, proviing an an open loop control to the system, but those also are epenent on the initial conitions. The technique presente relies on a variational approach, giving the erivative of the switching times with respect to the initial conitions, thus proviing a tool to esign programs/algorithms generating switching surfaces which are optimal for any possible execution of the system. Keywors: Hybri Systems, Switching Surfaces, Optimal Control, Variational Methos 1. INTRODUCTION Consier a switche system with autonomous continuous ynamics, ẋ(t) f q(t) (x(t)), (1) q + (t) s(x(t), q(t)). (2) where (1) escribes the continuous ynamics of the state variable x X R n an (2) escribes the iscrete event ynamics of the system. Given an initial conition x 0 : x(t 0 ), the switching law (2) etermines the switching instants t i, i 1, 2,..., an thus the intervals where a certain moal function is active, as well as the initial conition for the o..e. which efines the evolution uner the next moe. The iscrete variable q is piecewise constant in time an belongs to a finite or countable set Q, hence, it can be expresse in terms of the inex i as q(i). In terms of such inex the ynamics of a switche system is: ẋ(t) f i (x(t)), t (t i 1, t i ] (3) i + s(x(t), i, t). (4) with the unerstaning that f i : f q(i), for a given map q(i), i.e., in this case (4) only expresses the occurrence of the i th switch, the specification of the next active moe being given by the map q(i). Since the continuous moes are autonomous, the evolution of the system is etermine by the

2 active moes, accoring to (4). When the function s oes not epen by the (continuous) state variable x, the switching instants are etermine as exogenous inputs, an the system is controlle in open loop (timing control); when s is epenent only on the state variables, the switching law is given in a feeback form, an it may be efine by switching surfaces in the state space. To formulate the problem we are intereste with, consier a simple execution of (3,4) with only one switch, starting at x(t 0 ) x 0 with moe 1, switching to moe 2 at time, an exogenous switch, an terminating either at a fixe final time t 2 or in corresponence of a terminal manifol efine by a function g(x), so that t 2 satisfies g(x(t 2 )) 0. For ease of reference, enote such two sets of possible executions by χ t an χ g, respectively. To fix notation, let the explicit representation of the evolution etermine by moe i be given by x(t) ϕ i (t, s, x(s)), hence, { ϕ1 (t, t x(t) 0, x 0 ) t [t 0, ] (5) ϕ 2 (t,, x( )) t (, t 2 ] Also, let x i : x(t i ), an R : f 1 (x 1 ) f 2 (x 1 ). In this paper the following conventions will be use: 1) vectors are column vectors; 2) the erivative of a scalar, e.g. L, w.r.t. a vector x is a row vector: L x : L x [ L x 1,..., L x n ]. (6) (hence L T x is a column vector). The Hessian matrix is enote by L xx. If f is a (column) vector, function of the vector x i.e., then f [f (1) (x),..., f (n) (x)] T f x : f x f (1) f (1)... x 1 x n f (n) (n) f... x 1 x n Accoring to this convention, for the scalars c, t an the vectors x, y, z, the usual chain rule applies to c(x(t)) an c(x(y)), i.e. t c xẋ, c y c x x y ( v stays for v t ); also: x T y z 1.1 Problem formulation y T x z + x T y z, (7) When the optimal control problem to minimize a cost function J t 0 L(x(t))t (8) is formulate, for some continuously ifferentiable function L, an such that L xx is symmetric, then it is known that when t 1, a (locally) optimal switching time, it satisfies the following conition, see e.g. (Egerstet et al., 2003): c(t 1 ) : pt (t 1 )R(x 1 ) 0 (9) where p T (t), for t [t 1, t 2 ] is given by: p T (t) t L x (x(s))φ 2 (s, t)s + p T (t 2 )Φ 2 (t 2, t) (10) with Φ i the transition matrix of the linearize time-varying system ż(t) fi(x(t)) x z(t), an p T (t 2 ) 0 for fixe final time an p T (t 2 ) L(x 2 )g x (x 2 )/L 2, for an evolution ening at a terminal manifol, where L 2 : g x (x 2 )f(x 2 ), the Lie erivative of g along f 2 evaluate at x 2. Assuming to start from a perturbe initial conition x 0 x 0 + δx 0 ; it is possible to use the information of optimality of t 1, as a switching time, to etermine t 1; in other wors: what is the epenence of the optimal switching time on the initial conitions? This problem is motivate by the etermination of optimal switching surfaces, which ten to solve optimal control problems for autonomous system via the synthesis of feeback laws, which may be pursue for specifications of stability or optimal control. Relevant application of such technique may arise in many areas such as behavior base robotics (Arkin, 1998), or manufacturing systems (Khmelnitsky an Caramanis, 1998) to cite a few. Computational methos exist an are base on the optimization of parametrize switching surfaces (Boccaoro et al., 2005). However, the choice of the optimal values for such parameters epen on the particular trajectory chosen to run an optimization program, an thus, funamentally, on the initial conitions (remin that the we are consiering a system with no continuous inputs). An interesting reference for this type of approach is (Giua et al., 2001), which aresse a timing optimization problem, an iscovere the special structure of the solution for linear quaratic problems. Inee, in that case it is possible to ientify homogeneous regions in the continuous state space, whose bounaries, when reache, etermine the optimal switches, thus proviing a feeback solution to a problem which is formulate in terms of an open loop strategy. Here we explicitly investigate the relation existing between optimal switching times an initial conitions, stuying how the conition of optimality (9) that switching times must satisfy, vary in epenence of the initial conitions.

3 This paper reports the work in progress towar this goal, which is still being pursue. 2. OPTIMAL SWITCHING TIMES V/S INITIAL CONDITIONS It is well known that, uner mil assumptions, executions of switche systems are continuous w.r.t. the initial conitions (Broucke an Arapostathis, 2002). If we assume that also the epenence of c on t 1 as well as t 1 on x 0 is such, we may characterize function t 1 by eriving (9) w.r.t. x 0 an setting this erivative to zero. In fact, if starting from x 0 x 0 + δx 0, it results t 1 t 1 + δt 1 ; then, by continuity, 0 c( t 1 ) c(t 1 ) + δx 0 + o(δx 0 ). Hence, set 0, to satisfy optimality conition for t 1. As we will see this yiels a formula for the variational epenence of t 1 on x 0. To go further, the superscript will be roppe (hence assuming that, x 1 etc. are relative to optimal executions) in orer to reuce the notational buren. By (7) we have that R T p() + p T ( ) R (11) To calculate p(t1), account for the following result, which is reaily verifie: x a t(x) f(s, x)s a t f x (s, x)s f(t, t)t x (12) Then, consiering first the simpler case of fixe final time, by (10, 7, 12) p( ) Φ T 2 (s, )L xx (x(s)) x(s) + Φ T 2 (s, ) L T x (x s) s L T x (x 1)Φ 2 (, ) (13) To get x(s) notice that x( ) ϕ 1 (, t 0, x 0 ), hence x(s) ϕ 2 (s,, ϕ 1 (, t 0, x 0 )) for s [, t 2 ], thus, x(s) x(s) + x(s) x 1 + x(s) x 1 x 1 x 1 x 0 (14) Now, x(s)/ f 2 (x(s)) 1, x(s)/ x 1 Φ 2 (s, ), x 1 / x 0 Φ 1 (, t 0 ), x 1 / f 1 (x 1 ), Φ 2 (, ) I, Φ 2 (s, ) Φ 2 (s, ) f 2(x 1 ) x (15) 1 For time invariant ynamics, [ϕ(s, t + h, x) ϕ(s, t, x)]/h [ϕ(s h, t, x) ϕ(s, t, x)]/h f(x(s)) + o(h). (to be transpose). It results: where I 1 I 3 p( ) I 1 I 2 + I 3 I 4 K (16) I 2 Φ T 2 (s, )L xx (x(s))φ 2 (s, )f 1 (x 1 ) I 4 Φ T 2 (s, )L xx (x(s))f 2 (x(s)) Φ T 2 (s, )L xx (x(s))φ 2 (s, )Φ 1 (, t 0 )s f T 2x (x 1)Φ T 2 (s, )L T x (x(s)) s K L T x (x 1 ) (17) To hanle these, integrate by parts I 2 (letting ), taking into account that L xx (x(s))f 2 (x(s))s L T x (x(s)) we have t2 Φ T 2 (s, )L xx (x(s))f 2 (x(s))s I 4 + Φ T 2 (s, )L T x (x(s)) t2 I 4 + Φ T 2 (t 2, )L T x (x 2) K (18) This leas to the cancellation of I 4 an K in (16). To complete, let s compute R(x 1 )/. Again, notice that x 1 x( ) x[ (x 0 ), x 0 ], hence, R(x 1 ) R [ x (x x1 1) + x ] 1 x [ 0 R x (x 1) f 1 (x 1 ) t ] 1 + Φ 1 (, t 0 ) (19) Multiplying this by p T ( ), (16) by R T from the left an summing up we finally obtain: ( ) [ R T (Qf 1 Φ T 2 (t 2, )L T x (x 2 )) + p T ( )R x f 1 ] + [ R T Q p T ( )R x ] Φ1 (, t 0 ) (20) where f 1 : f 1 (x 1 ), an Q : Φ T 2 (s, )L xx (x(s))φ 2 (s, )s (21) which is a kin of quaratic form co-costate. Notice that the term multiplying t1 above, is a scalar. So, if we know that t 1 is a local optimum for an evolution starting from x 0, then, assuming to start from x 0 x 0 +δx 0, we simply must switch at t 1 + δt 1 + o(δx 0 ). Accoring to (20), δt 1 [R T Q p T ( )R x ]Φ 1 (, t 0 ) δx 0 R T (Qf 1 Φ T 2 (t 2, )L T x (x 2)) + p T ( )R x f 1 (22)

4 3. TOWARD THE CONSTRUCTION OF THE OPTIMAL SWITCHING SURFACES To put in use Eq. (22) assume that one optimal switching time has been erive for a certain sample evolution of the system, e.g. one starting in ˆx 0. Then the optimal switching surfaces are efine by the optimal switching states yiele by the variation on the optimal switching times when initial conitions ifferent than ˆx 0 are consiere. However, it must be pai attention to the fact that the formula erive above works for a fixe final time: inee for the case of evolution ening at a terminal manifol the following result hols, Theorem 1. Consier a nominal an a perturbe execution of the set χ g, x( ) an y( ), respectively, the first starting at x 0 an the latter starting from a point y 0 which lies on the nominal trajectory; i.e., assume that it exists an interval δt 0 such that y 0 ϕ 1 (t 0 + δt 0, t 0, x 0 ). Then, the optimal switching time for all δt 0 < t 1 t 0 t 1(y 0 ) t 1(x 0 ) δt 0 (23) Proof Denote by a b a trajectory from point a to b, an let x(t 1 ) x 1 an x(t 2 ) x 2 where t 2 is the terminal time if the switching time from moe 1 to moe 2 is t 1. If (23) i not hol then assume t 1 (y 0) t 1 (x 0) δt 0 + ɛ (24) for some ɛ (assume with no loss of generality ɛ > 0). Let the nominal trajectory that switches at t 1 (x 0)+ɛ terminate at x ɛ 2. Denote A x(t 0) x(t 0 + δt 0 ), B x(t 0 + δt 0 ) x 1, C x 1 x 2, D x(t 1 ) x(t 1 + ɛ) an E x(t 1 + ɛ) xɛ 2. Accoring to (23) the optimal nominal trajectory is A B C paying for this the cost J(A) +J(B) + J(C). On the other han by (24) the perturbe trajectory B C incurs in a greater cost than B D E, which implies that J(C) > J(D)+J(E). This in turn means that if the nominal trajectory switches at t 1(x 0 ) + ɛ then it pays less than if the switch take place at t 1 Notice that Theorem 1 easily extens to negative δt 0, i.e., if y 0 is chosen such that the evolution starting from y 0 will reach x 0 we must a the time neee to reach x 0 from y 0 to the optimal (nominal) switching time. In case of fixe terminal time the optimal switching state may vary because the perturbe trajectory escribe in Theorem 1 above, switching at t 1 δt 0, reaches the point x(t 2 ) (of the nominal trajectory) at time instant t 2 δt 0, thence visits aitional states from t 2 δt 0 to t 2 (in other wors x( ) (t2 δt 0,t 2] is a set of states not visite by x( )). Such remnants of the perturbe trajectory a further costs, so that two ifferent trajectories, even if the starting point of one of them lies in the trajectory of the other, cannot really be properly compare, in terms of optimal switching states. This can be actually seen: take an i.c. y 0 ϕ 1 (t 0 + δt 0, t 0, x 0 ) very close to x 0, so that δx 0 f 1 (x 0 )δt 0 + o(δt 0 ). Multiplying (22) by such δx 0, we have that its numerator (plus higher orer terms) is: [R T Q + p T ( )R x ]Φ 1 (, t 0 )δx 0 [R T Qf 1 p T ( )R x f 1 ]δt 0 (25) where Φ 1 (, t 0 )f 1 (x 0 ) f 1 (x1) is ue to the fact that vector fiels obey their variational ynamics 2. Hence in this case δt 1 [R T Qf 1 p T ( )R x f 1 ] δt 0 R T (Qf 1 Φ T 2 (t 2, )L T x (x 2)) + p T ( )R x f 1 (26) In this case conition (23) is equivalent to δt 1 δt 0, so that to be verifie, enominator an numerator shoul have ha the same terms, oppose in sign. Here, the only term making the ifference, preventing (23) to hol (as expecte) is R T Φ T 2 (t 2, )L T x (x 2 ). Accoring to Theorem 1, an to the above verification, the proceure to buil optimal switching surfaces shoul be better pursue consiering evolution ening at terminal manifols, since variations in the switching times efine sounly optimal switching states as well. Theorem 1 also gives an hint about the set of initial conitions that shoul be consiere to set such proceure. Inee, it seems reasonable account only for that set of initial conitions which are transversal to the flow efine by the vector fiel of the initial ynamics (here f 1 ) which contains ˆx 0. Such set of initial conition is a surface itself an can be escribe by s(x) 0 where s is a R-value function such that s(ˆx 0 ) 0 an such that s x (x) is collinear with f 1 (x), so that s woul be a kin of potential of the vector fiel. This choice is justifie by Theorem 1, since the components of the variation δx 0 on some x 0 which are tangent to the flow yiel no ifference on the optimal switching state, hence giving no contribution to the construction of an optimal switching surface which is optimal for the executions etermine by any possible initial conition. 4. CONCLUSION AND FUTURE WORKS This paper presents the first steps to etermine optimal switching surfaces for hybri systems 2 Inee, the variational system ż(t) f(x(t)) z(t) has x the solution z(t) f(x(t)), which can be seen from the chain rule f f xf

5 with autonomous moes. Future work will be evote to the erivation of an analogue formula of (22) for executions ening at a terminal manifol. This in orer to pursue the program outline above, about the investigation of the impact of transverse variations in the initial conition on the switching states. REFERENCES Arkin, R.C. (1998). Behavior Base Robotics. The MIT Press. Cambrige, MA. Boccaoro, M., Y. Wari, M. Egerstet an E. Verriest (2005). Optimal control of switching surfaces in hybri ynamical systems. JD- EDS 15(4), Broucke, M. an A. Arapostathis (2002). Continuous selections of trajectories of hybri systems. Systems an Control Letters 47, Egerstet, M., Y. Wari an F. Delmotte (2003). Optimal control of switching times in switche ynamical systems. In: 42n IEEE Conference on Decision an Control (CDC 03). Maui, Hawaii, USA. Giua, A., C. Seatzu an C. Van Der Mee (2001). Optimal control of switche autonomous linear systems. In: 40 th IEEE Conf. on Decision an Control (CDC 2001). Orlano, FL, USA. pp Khmelnitsky, E. an M. Caramanis (1998). Onemachine n-part-type optimal setup scheuling: analytical characterization of switching surfaces. IEEE Trans. on Automatic Control 43(11),