On the Equivalence Between Dissipativity and Optimality of Discontinuous Nonlinear Regulators for Filippov Dynamical Systems

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY On the Equivalence Between Dissipativity and Optimality of Discontinuous Nonlinear Regulators for Filippov Dynamical Systems Teymur Sadikhov Student Member IEEE and Wassim M Haddad Fellow IEEE Abstract In this paper we derive guaranteed gain sector and disk margins for nonlinear optimal and inverse optimal discontinuous feedback regulators that minimize a nonlinear-nonquadratic performance functional for Filippov dynamical systems Specifically we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving Filippov set-valued maps In addition we extend classical dissipativitytheorytoaddressthe problem of dissipative discontinuous dynamical systems These results are then used to derive extended Kalman-Yakubovich-Popov conditions for characterizing necessary and sufficient conditions for dissipativity of discontinuous systems using Clarke gradients and locally Lipschitz continuous storage functions Furthermore using the newly developed dissipativity notions we develop a return difference inequality to provide connections between dissipativity and optimality of nonlinear discontinuous controllers for Filippov dynamical systems Specifically using the extended Kalman-Yakubovich-Popov conditions we show that our discontinuous feedback control law satisfies a return difference inequality if and only if the controller is dissipative with respect to a quadratic supply rate Index Terms Differential inclusions discontinuous systems disk margins dissipativity theory Filippov solutions gain margins inverse optimality Kalman Yakubovich-Popov conditions nonlinear control optimal control sector margins I INTRODUCTION F OR continuous-time nonlinear dynamical systems with continuously differentiable flows the problem of guaranteed stability margins for optimal and inverse optimal regulators is well known [1] [3] Specifically nonlinear inverse optimal controllers that minimize a (in the terminology of [3]) meaningful nonlinear-nonquadratic performance criterion involving a nonlinear-nonquadratic nonnegative-definite function of the state and a quadratic positive definite function of the control are known to possess sector margin guarantees to component decoupled memoryless input nonlinearities lying in the conic sector These results also hold for disk margin guarantees where asymptotic stability of the closed-loop system is guaranteed in the face of a dissipative dynamic input operator Manuscript received April ; revised November May and August ; accepted September Date of publication September ; date of current version January This research was supported in part by the Air Force Office of Scientific Research under Grant FA Recommended by Associate Editor K Morris The authors are with the School of Aerospace Engineering Georgia Institute of Technology Atlanta GA USA ( tsadikhov@gatechedu; wmhaddad@aerospacegatechedu) Digital Object Identifier /TAC In addition using a certain return difference condition closely related to loop gain concepts in linear systems theory an equivalence between dissipativity with respect to a quadratic supply rate and optimality of a nonlinear feedback regulator also holds [1] In a two-part paper [4] [5] the authors extend the results of [1] [3] to develop a general framework for hybrid feedback systems by addressing stability dissipativity optimality and inverse optimality of impulsive dynamical systems In particular [5] considers a hybrid feedback optimal control problem over an infinite horizon involving a hybrid nonlinear-nonquadratic performance functional In [6] sufficient conditions for hybrid gain sector and disk margins guarantees for nonlinear hybrid dynamical systems were developed In [7] the authors provide asufficient condition for discontinuous -gain stabilizability of a nonlinear affine system with respect to Filippov solutions Their sufficient condition requires the existence of a viscosity supersolution of a Hamiton Jacobi Bellman equation In the recent paper [8] we developed input-output and state dissipativity notions for dynamical systems with discontinuous vector fields Specifically we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving set-valued maps specifying a set of directions for the system velocity and admitting solutions with absolutely continuous curves In particular we introduce a generalized definition of dissipativity for discontinuous dynamical systems in terms of set-valued supply rate maps and set-valued storage maps consisting of locally Lebesgue integrable supply rates and Lipschitz continuous storage functions respectively In addition we introduce the notion of a set-valued available storage map and a set-valued required supply rate map andshowthat if these maps have closed convex images they specialize to single-valued maps corresponding to the smallest available storage and the largest required supply of the differential inclusion respectively Furthermore we show that all system storage functions are bounded from above by the largest required supply and bounded from below by the smallest available storage and hence as in the case for systems with continuously differentiable flows a dissipative differential inclusion can deliver to its surroundings only a fraction of its generalized stored energy and can store only a fraction of the generalized work done to it Numerous engineering applications give rise to discontinuous dynamical systems Specifically in impact mechanics the motion of a dynamical system is subject to velocity jumps IEEE Personal use is permitted but republication/redistribution requires IEEE permission See for more information

2 424 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY 2014 and force discontinuities leading to nonsmooth dynamical systems [9] [10] In mechanical systems subject to unilateral constraints on system positions [11] discontinuities occur naturally through system-environment interactions Alternatively control of networks and control over networks with dynamic topologies also give rise to discontinuous systems [12] Specifically link failures or creations in network systems result in switchings of the communication topology leading to dynamical systems with discontinuous right-hand sides In addition open-loop and feedback controllers also give rise to discontinuous dynamical systems In particular bang-bang controllers discontinuously switch between maximum and minimum control input values to generate minimum-time system trajectories [13] whereas sliding mode controllers [14] [15] use discontinuous feedback control for system stabilization In switched systems [16] [17] switching algorithms are used to select an appropriate plant (or controller) from a given finite parameterized family of plants (or controllers) giving rise to discontinuous systems As for dynamical systems with continuously differentiable flows [18] dissipativity theory can play a fundamental role in addressing robustness disturbance rejection stability of feedback interconnections and optimality for discontinuous dynamical systems Even though passivity notions for the specific problem of the control of mechanical systems with discontinuous friction-type nonlinearities are considered in [19] [21] using input-to-state stability notions and set-valued nonlinearity extensions of the circle and Popov criterion the general problem of dissipativity theory in the sense of Willems [22] [23] for discontinuous dynamical systems and its connections to nonlinear discontinuous feedback regulator theory and inverse optimal control has not been addressed in the literature It is important to note however that the problem of stabilization for discontinuous systems with nonsmooth control Lyapunov functions has been extensively addressed in the literature; see [24] [29] and the references therein However with the exception of [7] [30] that address the specific problem of -gain stabilizability these results do not explore the underlying connections between steadystate viscosity supersolutions of the Hamilton-Jacobi-Bellman equation and nonsmooth closed-loop Lyapunov functions for guaranteeing both stability and optimality for discontinuous dynamical systems In addition gain sector and disk margin guarantees are not provided in the aforementioned references by exploiting connections between dissipativity theory discontinuous nonlinear regulator theory and an inverse optimal control problem In this paper we use the results of [8] to develop extended Kalman-Yakubovich-Popov conditions in terms of the discontinuous system dynamics characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions In addition we develop sufficient conditions for gain sector and disk margins guarantees for Filippov nonlinear dynamical systems controlled by optimal and inverse optimal discontinuous regulators Furthermore we develop a counterpart to the classical return difference inequality for continuous-time systems with continuously differentiable flows [1] [31] for Filippov dynamical systems and provide connections between dissipativity and optimality for discontinuous nonlinear controllers In particular we show an equivalence between dissipativity and optimality of discontinuous controllers holds for Filippov dynamical systems Specifically we show that an optimal nonlinear controller satisfying a return difference condition is equivalent to the fact that the Filippov dynamical system with input and output is dissipative with respect to a supply rate of the form Finally we note that a preliminary and considerably shorter and incomplete conference version of this paper appeared in [32] However the value added material of this paper over [32] is that no proofs of any of the optimality theorems are given in [32] nor any examples or motivation and applications are discussed in [32] In addition we note that a preliminary proof of Theorem 31 is given in [33] Here we give a complete proof of this theorem II NOTATION AND MATHEMATICAL PRELIMINARIES Thenotationusedinthispaper is fairly standard Specifically denotes the set of real numbers denotes the set of real column vectors denotes the set of nonnegative integers and denotes transpose We write and to denote the boundary and the closure of the subset respectively Furthermore we write for the Euclidean vector norm on fortheopen ball centered at with radius for the distance from a point to the set thatis and as to denote that approaches the set thatisfor every there exists such that for all Finally the notions of openness convergence continuity and compactness that we use throughout the paper refer to the topology generated on by the norm In this paper we consider nonlinear dynamical systems of the form (1) (2) where for every is Lebesgue measurable and locally essentially bounded [34] with respect to thatis is bounded on a bounded neighborhood of every point and continuous with respect to admits an equilibrium point at for some ;thatis and The control in (1) is restricted to the class of admissible controls consisting of measurable and locally essentially bounded functions such that A measurable function satisfying is called a control law If where is a control law and satisfies (1) then we call a feedback control law Note that the feedback control law is an admissible control since has values in Given a control law and a feedback control law theclosed-loop system is given by (3)

3 SADIKHOV AND HADDAD: ON THE EQUIVALENCE BETWEEN DISSIPATIVITY AND OPTIMALITY OF DISCONTINUOUS NONLINEAR REGULATORS 425 Note that an arc (ie an absolutely continuous function from to )satisfies (1) for an admissible control if and only if [34 p 152] where thatis Here is a set-valued map that assigns sets to points and denotes the collection of all subsets of The set captures all of the directions in that can be generated at with inputs The inputs can be selected as either or We assume that is an upper semicontinuous nonempty convex and compact set for all That is for every and every there exists such that for all satisfying This assumption is mainly used to guarantee the existence of Filippov solutions to (3) [34] An absolutely continuous function is said to be a Filippov solution [34] of (3) on the interval with initial condition if satisfies where the Filippov set-valued map by (4) (5) is defined denotes the Lebesgue measure in denotes convex closure and denotes the intersection over all sets of Lebesgue measure zero 1 Note that since is locally essentially bounded is upper semicontinuous and has nonempty compact and convex values Thus Filippov solutions are limits of solutions to with averaged over progressively smaller neighborhoods around the solution point and hence allow solutions to be defined at points where itself is not defined Hence the tangent vector to a Filippov solution when it exists lies in the convex closure of the limiting values of the system vector field in progressively smaller neighborhoods around the solution point Dynamical systems of the form given by (4) and (5) are called differential inclusions in the literature [36] and for every state they specify a set of possible evolutions of rather than a single one Since the Filippov set-valued map given by (6) is upper semicontinuous with nonempty convex and compact values and is also locally bounded [34 p 85] it follows that Filippov solutions to (3) exist [34 Thm 1 p 77] Recall that the Filippov solution to (3) is a right maximal solution if it cannot be extended (either uniquely or nonuniquely) forward in time We assume that all right maximal Filippov solutions to (3) exist on and hence we assume that (3) is forward complete Recall that (3) is forward complete if and only if the Filippov solutions to (3) are uniformly globally sliding 1 Alternatively we can consider Krasovskii solutions of (3) wherein the possible misbehavior of the derivative of the state on null measure sets is not ignored; that is is replaced with and where is assumed to be locally bounded (6) time stable [37 Lem 1 p 182] An equilibrium point of (3) is a point such that It is easy to see that is an equilibrium point of (3) if and only if the constant function is a Filippov solution of (3) We denote the set of equilibrium points of (3) by Since the set-valued map is upper semicontinuous it follows that is closed To develop discontinuous optimal controllers for discontinuous dynamical systems given by (1) we need to introduce the notion of generalized derivatives and gradients Here we focus on Clarke generalized derivatives and gradients [28] Definition 21: ([28] [29]) Let be a locally Lipschitz continuous function The Clarke upper generalized derivative of at in the direction of is defined by The Clarke generalized gradient of at is the set where co denotes the convex hull denotes the nabla operator is the set of measure zero of points where does not exist is any subset of of measure zero and the increasing unbounded sequence converges to Note that (7) always exists Furthermore note that it follows from Definition 21 that the generalized gradient of at consists of all convex combinations of all the possible limits of the gradient at neighboring points where is differentiable In addition note that since is Lipschitz continuous it follows from Rademacher s theorem [38 Thm 6 p 281] that the gradient of exists almost everywhere and hence is bounded Thus since for every is convex closed and bounded it follows that is compact In order to state the main results of this paper we need some additional notation and definitions Given a locally Lipschitz continuous function theset-valued Lie derivative of with respect to at [29] [39] is defined as If is convex with compact values then is a closed and bounded possibly empty interval in If is continuously differentiable at then In the case where is nonempty we use the notion (resp ) to denote the largest (resp smallest) element of Furthermore we adopt the convention Finally recall that a function is regular at [28 Def 234] if for all the right directional derivative exists and is called regular on if it is regular at every (7) (8) (9)

4 426 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY 2014 III DISSIPATIVITY AND STABILITY OF DISCONTINUOUS FEEDBACK SYSTEMS In this section we extend the notion of classical dissipativity [22] [23] of dynamical systems with continuously differentiable flows to discontinuous systems The following definition is needed for the main results of this section Definition 31: The discontinuous dynamical system given by (1) and (2) is weakly (resp strongly) dissipative with respect to the (locally Lebesgue integrable) supply rate if there exists a locally Lipschitz continuous and regular storage function such that the dissipation inequality (10) is satisfied for at least one (resp every) Filippov solution of with Since is assumed to be locally Lipschitz continuous and regular an equivalent statement for the dissipativeness of involving supply rates is or equivalently where (11) is is averaged over progressively smaller neighborhoods around with Analogously for fixed and a measurable and locally essentially bounded define the set-valued Lie derivative by that is we fix and apply the Filippov construction over Notethatif is a Filippov solution to (1) with then In addition note that is a closed and bounded possibly empty interval in Lemma 31: Let be a Filippov solution of (1) corresponding to the input and let be locally Lipschitz continuous and regular Then exists for almost all and for almost all Proof: The proof is similar to the proof of Lemma 1 of [29] and hence is omitted Next consider the special case of dissipative systems with quadratic supply rates [18] [23] Specifically set let and be given and assume where denotes the set of symmetric matrices Furthermore we assume that there exists a function such that and so that as shown by Theorem 32 of [8] all storage functions for are positive definite Next define for every denotes the upper right directional Dini derivative of along the Filippov state trajectories of (1) through with at Next we show that dissipativeness of discontinuous nonlinear affine dynamical systems of the form (12) (13) where is an open set with and can be characterized in terms of the system functions and Here we assume that and are Lebesgue measurable and locally essentially bounded The following lemma is necessary for the next theorem For the statement of this lemma we require some additional notation Specifically given a locally Lipschitz continuous function define the set-valued Lie derivative of with respect to at and by where denotes the Filippov set-valued map of over for each admissible input That where and denotes the intersection over all sets of Lebesgue measure zero Finally we assume that the set is singlevalued 2 for almost all modulo The following definition is necessary for the statement of the next result Definition 32: ([8]) The nonlinear dynamical system given by (1) and (2) is weakly (resp strongly) completely reachable if for every there exists a finite time and an admissible input defined on such that at least one (resp every) Filippov solution of can be driven from to is weakly (resp strongly) zero-state observable if and implies for at least one (resp every) Filippov solution of 2 The assumption that is single-valued is necessary for obtaining Kalman Yakubovich Popov conditions for (12) and (13) with Lebesgue measurable and locally essentially bounded system functions and and with locally Lipschitz continuous storage functions Specifically as will be seen in the proof of Theorem 31 the requirement that there exists (resp ) such that for all (resp ) used in the proof of Theorem 31 holds if and only if is a singleton To see this let with and assume ad absurdum the required exists Then either or Assume and let Then Hence which leads to a contradiction A similar construction shows the result for

5 SADIKHOV AND HADDAD: ON THE EQUIVALENCE BETWEEN DISSIPATIVITY AND OPTIMALITY OF DISCONTINUOUS NONLINEAR REGULATORS 427 The following theorem gives necessary and sufficient Kalman Yakubovich Popov conditions for dynamical systems with Lebesgue measurable and locally essentially bounded system functions Theorem 31: Let andlet be weakly zero-state observable and weakly completely reachable If there exist functions and such that is locally Lipschitz continuous regular and positive definite andforalmostall (14) (15) (18) where satisfies (12) Next using the sum rule for computing the generalized gradient of locally Lipschitz continuous functions [40] it follows that for almost all Now it follows from Lemma 31 that for almost all Hence (16) (17) Next note that (19) then is weakly dissipative with respect to the supply rate Converselyif is weakly dissipative with respect to the supply rate thenthere exist functions and such that is locally Lipschitz continuous regular and positive definite andforalmostall (14) (16) hold Proof: First suppose that there exist functions and such that is locally Lipschitz continuous regular and positive definite and (14) (17) are satisfied Then for every admissible input it follows from (14) (17) that where the integral in (20) is the Lebesgue integral Using (19) and (20) it follows from (18) that (20) The assertion now follows from Definition 31 Conversely suppose that is weakly dissipative with respect to the supply rate Now it follows from Theorem 31 of [8] that the available storage of (see [8] for details) is finite for all and (21) for almost all and Dividing (21) by and letting it follows that: (22) where is a solution satisfying (12) and Now with it follows from (22) that Next let be such that (23)

6 428 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY 2014 Now it follows from (22) that Since for almost all it follows that: and hence it follows from (23) and (24) that: (24) (25) Since the left-hand side of (25) is quadratic in thereexist functions and such that Now equating coefficients of equal powers yields (14) (16) with and with the positive definiteness of following from Theorem 32 of [8] Remark 31: Note that if is invertible for all then inequality (17) can be equivalently written as (26) which is free of This follows from the fact that (17) holds if and only if: (27) holds A similar expression to (26) involving generalized inverses also holds in the case where is singular for some The following stability theorems are needed for the next set of results as well as the results of the next sections In addressing the stability properties of a Filippov solution of a discontinuous dynamical system the usual stability definitions are valid [18] [41] Here we state the stability theorems for only the local case; the global stability theorems are similar except for the additional assumption of properness on the Lyapunov function and nonrestricting the domain of analysis For the remainder of the paper the adjective weak is used in reference to a stability property when the stability property is satisfied by at least one Filippov solution starting from every initial condition in whereas strong is used when the stability property is satisfied by all Filippov solutions starting from every initial condition in Theorem 32: ([29] [42]) Consider the discontinuous nonlinear dynamical system given by (3) Let be an equilibrium point of and let be an open and connected set with If is a positive definite locally Lipschitz continuous and regular function such that (resp ) for almost all such that then is strongly Lyapunov (resp strongly asymptotically) stable The following definitions are needed for the statement of the next result We say a set is weakly positively invariant (resp strongly positively invariant) with respect to (3) if for every contains a right maximal solution (resp all right maximal solutions) of (3) [29] Theorem 33: ([29] [42]) Consider the discontinuous nonlinear dynamical system given by (3) Let be an equilibrium point of let be an open strongly positively invariant set with respect to (3) such that andlet be locally Lipschitz continuous and regular on Assume that for every and every Filippov solution satisfying there exists a compact subset of containing for all Furthermoreassumethat for almost all such that Finally define and let be the largest weakly positively invariant subset of If then as If alternatively contains no invariant set other than then the Filippov solution of is strongly asymptotically stable for all Finally we consider feedback interconnections of dissipative discontinuous dynamical systems Specifically consider the discontinuous nonlinear dynamical system given by (12) and (13) with the discontinuous nonlinear feedback system given by (28) (29) where and We assume that and are Lebesgue measurable and locally essentially bounded (28) and (29) has at least one equilibrium point and the required properties for the existence of solutions of the feedback interconnection of and are satisfied Note that with a negative feedback interconnection and We assume that the negative feedback interconnection of and is well posed that is for all and The next theorem provides sufficient conditions for global asymptotic stability of discontinuous dissipative feedback systems with quadratic supply rates For the remainder of the paper we assume that the forward path and the feedback path are strongly dissipative systems The obtained stability results also hold for the case where and are weakly dissipative In this case however the set-valued Lie derivative operator should be replaced with the upper right Dini directional derivative in the proofs of the stability theorems The following lemma is necessary for the next theorem Lemma 32: ([29]) Let be a Filippov solution of the discontinuous dynamical system (3) and let be locally Lipschitz continuous and regular Then exists for almost all and for almost all

7 SADIKHOV AND HADDAD: ON THE EQUIVALENCE BETWEEN DISSIPATIVITY AND OPTIMALITY OF DISCONTINUOUS NONLINEAR REGULATORS 429 Theorem 34: Let and Consider the closed-loop system consisting of the discontinuous nonlinear dynamical systems given by (12) and (13) and given by (28) and (29); and assume and are strongly zero-state observable Furthermore assume is strongly dissipative with respect to the supply rate and has a locally Lipschitz continuous regular and radially unbounded storage function and is strongly dissipative with respect to the supply rate and has a locally Lipschitz continuous regular and radially unbounded storage function Finally assume there exists such that (30) Then the negative feedback interconnection of and is globally strongly asymptotically stable Proof: Note that the closed-loop dynamics of the negative feedback interconnection of and are given by (31) Now the proof follows from Lemma 32 and Theorems 32 and 33 using the Lyapunov function candidate and the fact that for almost all The following corollary to Theorem 34 is necessary for the results of Section VI Corollary 31: Consider the closed-loop system consisting of the discontinuous nonlinear dynamical systems given by (12) and (13) and given by (28) and (29) Let be such that and let be positive definite and assume and are strongly zero-state observable If is strongly dissipative with respect to the supply rate and has a locally Lipschitz continuous regular and radially unbounded storage function and is strongly dissipative with respect to the supply rate and has a locally Lipschitz continuous regular and radially unbounded storage function then the negative feedback interconnection of and is globally strongly asymptotically stable Proof: The proof is a direct consequence of Theorem 34 with and Specifically let be such that In this case givenby(30)satisfies so that all the conditions of Theorem 34 are satisfied IV STABILITY MARGINS FOR DISCONTINUOUS FEEDBACK REGULATORS To develop relative stability margins for discontinuous nonlinear regulators consider the discontinuous nonlinear dynamical system given by (32) (33) where and are Lebesgue measurable and locally essentially bounded and is a discontinuous feedback controller such that is weakly (resp strongly) asymptotically stable with Furthermore assume that the system is weakly (resp strongly) zero-state observable Next we define the relative stability margins for given by (32) and (33) Specifically let and consider the negative feedback interconnection of and where is either a linear operator a nonlinear static operator or a dynamic nonlinear operator with input and output Furthermoreweassumethatin the nominal case so that the nominal closed-loop system is weakly (resp strongly) asymptotically stable Definition 41: Let be such that Then the discontinuous nonlinear dynamical system given by (32) and (33) is said to have a weak (resp strong) gain margin if the negative feedback interconnection of and is globally weakly (resp strongly) asymptotically stable for all where Definition 42: Let be such that Then the discontinuous nonlinear dynamical system given by (32) and (33) is said to have a weak (resp strong) sector margin if the negative feedback interconnection of and is globally weakly (resp strongly) asymptotically stable for all nonlinearities such that and forall Definition 43: Let be such that Then the discontinuous nonlinear dynamical system given by (32) and (33) is said to have a weak (resp strong) disk margin if the negative feedback interconnection of and is globally weakly (resp strongly) asymptotically stable for all dynamic operators such that is weakly (resp strongly) zero-state observable and weakly (resp strongly) dissipative with respect to the supply rate where and such that Definition 44: Let be such that Then the discontinuous nonlinear dynamical system given by (32) and (33) is said to have a weak (resp strong) structured disk margin if the negative feedback interconnection of and is globally weakly (resp strongly) asymptotically stable for all dynamic operators such that is weakly (resp strongly) zero-state observable and is weakly (resp strongly) dissipative with respect to the supply rate

8 430 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY 2014 where and such that Remark 41: Note that if has a weak (resp strong) disk margin then has weak (resp strong) gain and sector margins where (38) (39) (40) V NONLINEAR-NONQUADRATIC OPTIMAL REGULATORS FOR DISCONTINUOUS DYNAMICAL SYSTEMS In this section we consider a control problem involving a notion of optimality with respect to a nonlinear-nonquadratic cost functional To address the optimal control problem let be an open set and let where and Next consider the controlled nonlinear discontinuous dynamical system (1) where is restricted to the class of admissible controls consisting of measurable and locally essentially bounded functions such that for almost all and the constraint set is given Given a control law and a feedback control the closed-loop dynamical is given by (3) Next we present a main theorem for characterizing feedback controllers that guarantee stability of the controlled discontinuous dynamical system and minimize a nonlinear-nonquadratic performance functional For the statement of this result let be Lipschitz continuous and define the set of regulation controllers by (41) Then with the feedback control the zero Filippov solution of the closed-loop system (3) is locally strongly asymptotically stable and there exists a neighborhood of the origin such that In addition if minimizes Finally if and then the feedback control in the sense that (42) (43) (44) Note that restricting our minimization problem to that is inputs corresponding to null convergent solutions can be interpreted as incorporating a system detectability condition through the cost Theorem 51: Consider the controlled discontinuous nonlinear dynamical system (1) with performance functional 3 then the zero Filippov solution of the closed-loop system (3) is globally strongly asymptotically stable Proof: Local and global strong asymptotic stability follow from (35) (38) by applying Theorem 32 to the closed-loop system (3) Next with where is measurable and locally essentially bounded let beanyfilippov solution of (1) Then it follows that for almost every Moreover it follows from Lemma 31 that for almost every Nowsince and are arbitrary it follows that: (34) where (34) is defined with respect to absolutely continuous state arcs and measurable control functions Assume that there exists a locally Lipschitz continuous and regular function and a control law such that (45) Next let let andlet for almost all be the Filippov solution of (1) Then it follows from (45) that (35) (36) (37) 3 Since solutions to (1) are not necessarily unique givenby(34) depends on the particular state trajectory along which we integrate Alternatively if we assume that is essentially one-sided Lipschitz on for some then there exists a unique Filippov solution to (1) with initial condition and [34] Furthermore note that (46) (47)

9 SADIKHOV AND HADDAD: ON THE EQUIVALENCE BETWEEN DISSIPATIVITY AND OPTIMALITY OF DISCONTINUOUS NONLINEAR REGULATORS 431 where the integral in (47) is the Lebesgue integral Now using (40) (46) (47) and the fact that it follows that: locally essentially bounded Furthermore we consider performance integrands of the form (49) where and with denoting the set of positive definite matrices so that (34) becomes which yields (43) Note that (39) is the steady-state Hamilton Jacobi Bellman equation for the discontinuous dynamical system (1) with the cost Since we are not imposing that solutions to (39) be smooth the Hamilton Jacobi Bellman equation (39) should be interpreted in the viscosity sense (ie a viscosity supersolution) [43] [44] or equivalently as in the proximal analysis formalism of [35] Specifically since where denotes the subdifferential of at [28] [35] it follows from (40) that is a viscosity supersolution of (39) However in general is not a viscosity subsolution of (39) which shows that the equivalence between optimal regulation solvability of the Hamilton Jacobi Bellman equation and feedback stabilizability breaks down for nonsmooth value functions It is important to note that Theorem 51 provides constructive sufficient conditions for optimality of a feedback controller Furthermore this controller is stabilizing and its optimality is independent of the system initial condition Finally necessary conditions for optimality of nonsmooth regulation and existence of viscosity solutions of the resulting Hamilton Jacobi Bellman equation are discussed in [45] [46] Next we specialize Theorem 51 to discontinuous affine dynamicalsystemsspecificallyweconstructdiscontinuousnonlinear feedback controllers using an optimal control framework that minimizes a nonlinear-nonquadratic performance criterion Thisisaccomplishedbychoosing the controllersuchthatthetotal generalizedderivativeofthelyapunovfunctionisnegativealong theclosed-loopsystemtrajectorieswhileprovidingsufficientconditions for the existence of asymptotically stabilizing viscosity supersolutions to the Hamilton Jacobi Bellman equation Thus these results provide a family of globally stabilizing controllers parameterizedbythecostfunctionalthatisminimized The controllers obtained in this section are predicated on an inverse optimal control problem [3] [18] In particular to avoid the complexity in solving the steady-state Hamilton Jacobi Bellman equation we do not attempt to minimize a given cost functional but rather we parameterize a family of stabilizing controllers that minimize some derived cost functional that provides flexibility in specifying the control law Consider the discontinuous nonlinear affine dynamical system given by (48) where and We assume that and are Lebesgue measurable and (50) Theorem 52: Consider the discontinuous nonlinear controlled affine dynamical system (48) with performance functional (50) Assume that there exists a locally Lipschitz continuous and regular function such that and Then the zero Filippov solution discontinuous dynamical system (51) (52) (53) (54) (55) of the closed-loop (56) is globally asymptotically stable with the feedback control law and the performance functional (50) with is minimized in the sense that Finally (57) (58) (59) (60) Proof: The result is a direct consequence of Theorem 51 with and Specifically with (49) the Hamiltonian has the form

10 432 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY 2014 Now the feedback control law (57) is obtained by setting With (57) it follows that (51) (53) (54) and (55) imply (35) (36) (38) and (44) respectively Next since is locally Lipschitz continuous and regular and is a local minimum of it follows that and hence since by assumption it follows that which implies (37) Next with given by (58) and given by (57) (39) holds Finally since and is positive definite for almost all condition (40) holds The result now follows as a direct consequence of Theorem 51 Example 51: To illustrate the utility of Theorem 52 we consider a controlled nonsmooth harmonic oscillator with nonsmooth friction given by ([29]) (61) (62) where and To construct an inverse optimal globally stabilizing control law for (61) and (62) let and note that Hence which implies that and for almost all Next it follows that: where with Let Now satisfies (54) so that the inverse optimal control law (57) is given by (63) In this case the performance functional (50) with (64) is minimized in the sense of (59) Furthermore using the feedback control law (63) it follows that: Note that Nowlet and note that if and only if Hence since is the largest strongly positively invariant set contained in it follows from Theorem 33 that as for all Filippov solutions of (61) and (62) Now since is radially unbounded the feedback control law (63) is globally strongly stabilizing VI GAIN SECTOR AND DISK MARGINS OF NONLINEAR-NONQUADRATIC OPTIMAL REGULATORS FOR DISCONTINUOUS DYNAMICAL SYSTEMS In this section we derive guaranteed gain sector and disk margins for nonlinear optimal and inverse optimal regulators that minimize a nonlinear-nonquadratic performance criterion for discontinuous dynamical systems Specifically sufficient conditions that guarantee gain sector and disk margins are given in terms of the state control and cross-weighting nonlinear-nonquadratic weighting functions In particular we consider the discontinuous nonlinear dynamical system given by (65) (66) where and with a nonquadratic performance criterion (67) where and are given such that and Once again we assume that and are Lebesgue measurable and locally essentially bounded In this case the optimal nonlinear feedback controller that minimizes the nonlinear-nonquadratic performance criterion (67) is given by the following result Theorem 61: Consider the discontinuous nonlinear dynamical system given by (65) and (66) with performance functional (67) Assume that there exists a locally Lipschitz continuous and regular function such that (68) (69)

11 SADIKHOV AND HADDAD: ON THE EQUIVALENCE BETWEEN DISSIPATIVITY AND OPTIMALITY OF DISCONTINUOUS NONLINEAR REGULATORS 433 (70) (71) and (72) (73) Then the zero Filippov solution discontinuous dynamical system of the closed-loop (74) is globally strongly asymptotically stable with the feedback control law (75) (81) Next using the sum rule for the generalized gradient of locally Lipschitz continuous functions [40] it follows that for almost all Now it follows from Lemma 31 that for almost all Hence and the performance functional (67) is minimized in the sense that Finally (76) (77) It follows from (81) and (82) that for all all : (82) and almost Proof: The proof is a direct consequence of Theorem 51 The following key lemma is needed Lemma 61: Consider the discontinuous nonlinear dynamical system given by (65) and (66) where is a strongly stabilizing feedback control law given by (75) Suppose satisfies (78) (79) with such that Then for almost all and the solution to(65)satisfies (80) Proof: Note that it follows from (75) (78) and (79) that for almost all and Now integrating over and using (47) yields (80) Note that with condition (80) is the counterpart for discontinuous dynamical systems of the return difference condition for continuous-time and discrete-time systems [1] [31] [47] Next using the extended nonlinear Kalman- Yakubovich-Popov conditions for discontinuous dynamical systems given by Theorem 31 we show that for a given nonlinear dynamical system given by (65) and (66) there exists an equivalence between optimality and dissipativity For the following result we assume that for the given discontinuous nonlinear system (65) if there exists a feedback control law that minimizes the performance functional (67) with and then there exists a locally Lipschitz continuous regular and positive-definite function such that (78) and (79) are satisfied Necessary and sufficient conditions such that the aforementioned statement holds modulo (79) holding are given in Theorem 376 of [28] Theorem 62: Consider the discontinuous nonlinear dynamical system given by (65) and (66) The feedback control law is optimal with respect to a performance functional (67) with and if and only if the nonlinear system is strongly dissipative with respect to the supply rate and has a

12 434 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL 59 NO 2 FEBRUARY 2014 locally Lipschitz continuous regular positive-definite and radially unbounded storage function Proof: If the control law is optimal with respect to a performance functional (67) with and then by assumption there exists a locally Lipschitz continuous regular and positive-definite function such that (78) and (79) are satisfied Hence it follows fromlemma61thatthesolution to(65)satisfies which implies that is strongly dissipative with respect to the supply rate Conversely if is strongly dissipative with respect to the supply rate and has a locally Lipschitz continuous regular and positive-definite storage function then with and it follows from Theorem 31 that there exists a function such that and for almost all : Now the result follows from Theorem 61 with Example 61: Consider the discontinuous nonlinear dynamical system given in Example 51 Note that with and it follows from the analysis given in Example 51 that the optimal control law minimizes the cost functional Now it follows from Theorem 62 that the discontinuous nonlinear dynamical system is strongly dissipative with respect to the supply rate where To show this consider the storage function Next with and the extended Kalman-Yakubovich-Popov conditions given in Theorem 31 become for almost all Next we present disk margins for the nonlinear-nonquadratic optimal regulator given by Theorem 61 First we consider the case in which is a constant diagonal matrix Theorem 63: Consider the discontinuous nonlinear dynamical system given by (65) and (66) where is a strongly stabilizing feedback control law given by (75) and where satisfies (78) and (79) with such that If where then the discontinuous nonlinear system has a strong structured disk margin If in addition then the discontinuous nonlinear system has a strong disk margin Proof: Note that for all and almost all it follows from Lemma 61 that the solution to (65) satisfies Hence with the storage function is strongly dissipative with respect to the supply rate Nowtheresult is a direct consequence of Corollary 31 and Definitions 44 and 43 with and Next we consider the case in which is not a diagonal constant matrix For the following result define: (87) where is such that and Theorem 64: Consider the discontinuous nonlinear dynamical system given by (65) and (66) where is a strongly stabilizing feedback control law given by (75) and suppose satisfies (78) and (79) with such that Then the discontinuous nonlinear system has a strong disk margin where Proof: Note that for almost all and it follows from Lemma 61 that the solution to (65) satisfies (83) (84) (85) (86) Next it was shown in Example 51 that and Now with and conditions (83) (85) are satisfied Furthermore (86) is equivalent to (79) which is satisfied since is optimal Hence it follows from Theorem 31 that is strongly dissipative with respect to the supply rate which implies that Hence with the storage function is strongly dissipative with respect to the supply rate Nowtheresultisa direct consequence of Corollary 31 and Definition 43 with and

13 SADIKHOV AND HADDAD: ON THE EQUIVALENCE BETWEEN DISSIPATIVITY AND OPTIMALITY OF DISCONTINUOUS NONLINEAR REGULATORS 435 Next using Theorem 33 we provide an alternative result that guarantees sector and gain margins for the case in which is diagonal Theorem 65: Consider the discontinuous nonlinear dynamical system given by (65) and (66) where is a strongly stabilizing feedback control law given by (75) and suppose satisfies (78) and (79) with such that Furthermore let where If is strongly zero-state observable then the discontinuous nonlinear system has a strong sector (and hence gain) margin Proof: Let where is a static nonlinearity such that and forall where and ; or equivalently forall In this case the closed-loop discontinuous system (65) and (66) with is given by (88) Next consider the locally Lipschitz continuous and regular Lyapunov function candidate Now it follows from (78) (79) and (82) that: is globally strongly asymptotically stable for all such that which implies that the discontinuous nonlinear system given by (65) and (66) has strong sector (and hence gain) margin Note that in the case where is diagonal Theorem 65 guarantees larger strong gain and sector margins to the strong gain and sector margin guarantees provided by Theorem 64 However Theorem 65 does not provide strong disk margin guarantees VII CONCLUSION In this paper we extended the notion of dissipativity theory for continuous dynamical systems with continuously differentiable flows to discontinuous dynamical systems whose solutions are characterized by Filippov set-valued maps Furthermore extended Kalman-Yakubovich-Popov conditions in terms of the discontinuous system dynamics for characterizing dissipativity via generalized Clarke gradients of locally Lipschitz continuous storage functions were developed In addition sufficient conditions for gain sector and disk margin guarantees for discontinuous nonlinear systems controlled by nonlinear optimal and inverse optimal regulators that minimize a nonlinearnonquadratic performance criterion were derived Using these results connections between dissipativity and optimality of discontinuous nonlinear systems were established These results provide a generalization of the meaningful inverse optimal nonlinear regulator stability margins as well as the classical linearquadratic optimal regulator gain and phase margins to discontinuous nonlinear regulators ACKNOWLEDGMENT The authors would like to thank Dr S Bhat for several fruitful discussions and suggestions which implies that the closed-loop discontinuous system (88) is strongly Lyapunov stable Next let and note that if and only if Nowsince is strongly zero-state observable it follows that is the largest weakly positively invariant set contained in Hence it follows from Theorem 33 that as Thus the closed-loop discontinuous system (88) REFERENCES [1] P J Moylan and B D O Anderson Nonlinear regulator theory and an inverse optimal control problem IEEE Trans Autom Control vol 18 pp [2] P J Moylan Implications of passivity in a class of nonlinear systems IEEE Trans Autom Control vol 19 pp [3] R Freeman and P Kokotović Inverse optimality in robust stabilization SIAM J Control Optim vol 34 pp [4] W M Haddad V Chellaboina and N A Kablar Nonlinear impulsive dynamical systems Part I: Stability and dissipativity Int J Control vol 74 pp [5] W M Haddad V Chellaboina and N A Kablar Nonlinear impulsive dynamical systems Part II: Stability of feedback interconnections and optimality Int J Control vol 74 pp [6] W M Haddad V Chellaboina and S G Nersesov On the equivalence between dissipativity and optimality of nonlinear hybrid controllers Int J Hybrid Sys vol 1 pp [7] A Bacciotti and F Ceragioli -gain stabilizability with respect to Filippov solutions in Proc IEEE Conf Dec ControlLasVegasNV December 2002 pp [8] W M Haddad and T Sadikhov Dissipative differential inclusions set-valued energy storage and supply rate maps and discontinuous dynamical systems in Proc Amer Control Conf Montreal QC Canada June 2012 pp [9] B Brogliato Nonsmooth Mechanics 2nd ed London UK: Springer-Verlag 1999 [10] F Pfeiffer and C Glocker Multibody Dynamics With Unilateral Contacts New York: Wiley 1996 [11] G A S Pereira M F M Campos and V Kumar Decentralized algorithms for multi-robot manipulation via caging Int J Robotics Res vol 23 pp

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Chellaboina and W M Haddad Stability margins of discrete-time nonlinear-nonquadratic optimal regulators Int J Syst Sci vol33 pp Teymur Sadikhov (S 10) was born in Baku Azerbaijan He received the BS degree in aeronautical engineering from Istanbul Technical University Istanbul Turkey in 2008 the MS degree in engineering sciences (mechanical engineering option) from the University of California San Diego in 2010 and is currently pursuing the PhD degree in aerospace engineering at the Georgia Institute of Technology Atlanta From May to August of 2013 he was a Research Intern with the Autonomous and Intelligent Robotics Laboratory United Technologies Research Center where he worked on trajectory planning for autonomous helicopters His research interests include stability and control of discontinuous dynamical systems adaptive estimation and control control of multi-agent systems model predictive control motion planning and verification and validation of adaptive controllers Mr Sadikhov received the BP ITU Foundation Nippon Foundation and the Azerbaijan the Turkey government scholarship award and the Powell Fellowship Wassim M Haddad (S 87 M 87 SM 01 F 09) received the BS MS and PhD degrees in mechanical engineering from the Florida Institute of Technology (Florida Tech) Melbourne in and 1987 respectively In 1988 he joined the faculty of the Mechanical and Aerospace Engineering Department Florida Tech Since 1994 he has been with the School of Aerospace Engineering Georgia Institute of Technology (Georgia Tech) Atlanta where he holds the rank of Professor the Andrew and David Lewis Chair in Dynamical Systems and Control and serves as Chair of the Flight Mechanics and Control Discipline He also holds joint Professor appointments with the Schools of Biomedical Engineering and Electrical and Computer Engineering Georgia Tech His interdisciplinary research contributions in systems and control are documented in over 540 archival journal and conference publications and seven books in the areas of science mathematics medicine and engineering His research is on nonlinear robust and adaptive control nonlinear systems large-scale systems hierarchical control impulsive and hybrid systems system thermodynamics network systems system biology and mathematical neuroscience His secondary interests include the history of science and mathematics as well as natural philosophy Dr Haddad received an Outstanding Alumni Achievement Award for his contributions to nonlinear dynamical systems and control recognition for outstanding contributions to joint university and industry programs and the 2014 AIAA Pendray Aerospace Literature Award for paramount and fundamental contributions to the literature of dynamic systems and control for large-scale aerospace systems He is an NSF Presidential Faculty Fellow in recognition for his demonstrated excellence and continued promise in scientific and engineering research; and a member of the Academy of Nonlinear Sciences for contributions to nonlinear stability theory dynamical systems and control