Hybrid decentralized maximum entropy control for large-scale dynamical systems

Transcription

1 Nonlinear Analysis: Hybrid Systems 1 (2007) Hybrid decentralized maximum entropy control for large-scale dynamical systems Wassim M. Haddad a,, Qing Hui a,1, VijaySekhar Chellaboina b,2, Sergey G. Nersesov c,3 a School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA , United States b Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN , United States c Department of Mechanical Engineering, Villanova University, Villanova, PA , United States Received 6 April 2006; accepted 6 April 2006 Abstract In the analysis of complex, large-scale dynamical systems it is often essential to decompose the overall dynamical system into a collection of interacting subsystems. Because of implementation constraints, cost, and reliability considerations, a decentralized controller architecture is often required for controlling large-scale interconnected dynamical systems. In this paper, a novel class of fixed-order, energy-based hybrid decentralized controllers is proposed as a means for achieving enhanced energy dissipation in large-scale lossless and dissipative dynamical systems. These dynamic decentralized controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that each subsystem subcontroller pair of the hybrid closed-loop system is consistent with basic thermodynamic principles. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative combustion control example is given to demonstrate the efficacy of the proposed approach. c 2006 Elsevier Ltd. All rights reserved. Keywords: Hybrid control; Hybrid systems; Decentralized dynamic compensation; Impulsive dynamical systems; Large-scale systems; Dissipative systems; Maximum entropy control 1. Introduction Modern complex dynamical systems 4 are highly interconnected and mutually interdependent, both physically and through a multitude of information and communication network constraints. The sheer size (i.e., dimensionality) This research was supported in part by the Air Force Office of Scientific Research under Grant F and the National Science Foundation under Grant ECS Corresponding author. Tel.: ; fax: addresses: wm.haddad@aerospace.gatech.edu (W.M. Haddad), qing hui@ae.gatech.edu (Q. Hui), chellaboina@utk.edu (V. Chellaboina), sergey.nersesov@villanova.edu (S.G. Nersesov). 1 Tel.: ; fax: Tel.: ; fax: Tel.: ; fax: Here we have in mind large flexible space structures, aerospace systems, electric power systems, network systems, economic systems, and ecological systems, to cite but a few examples X/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi: /j.nahs

2 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) and complexity of these large-scale dynamical systems often necessitates a decentralized architecture for analyzing and controlling these systems. Specifically, in the control-system design of complex large-scale dynamical systems it is often desirable to treat the overall system as a collection of interconnected subsystems. The behavior of the composite (i.e., large-scale) system can then be predicted from the behaviors of the individual subsystems and their interconnections. The need for decentralized control design of large-scale systems is a direct consequence of the physical size and complexity of the dynamical model. In particular, computational complexity may be too large for model analysis while severe constraints on communication links between system sensors, actuators, and processors may render centralized control architectures impractical. Moreover, even when communication constraints do not exist, decentralized processing may be more economical. The complexity of modern controlled large-scale dynamical systems is further exacerbated by the use of hierarchical embedded control subsystems within the feedback control system, that is, abstract decision-making units performing logical checks that identity system mode operation and specify the continuous-variable subcontroller to be activated. Such systems typically possess a multiechelon hierarchical hybrid decentralized control architecture characterized by continuous-time dynamics at the lower levels of the hierarchy and discrete-time dynamics at the higher levels of the hierarchy. The lower-level units directly interact with the dynamical system to be controlled while the higher-level units receive information from the lower-level units as inputs and provide (possibly discrete) output commands which serve to coordinate and reconcile the (sometimes competing) actions of the lower-level units. The hierarchical controller organization reduces processor cost and controller complexity by breaking up the processing task into relatively small pieces and decomposing the fast and slow control functions. Typically, the higher-level units perform logical checks that determine system mode operation, while the lower-level units execute continuous-variable commands for a given system mode of operation. Since implementation constraints, cost, and reliability considerations often require decentralized controller architectures for controlling large-scale systems, decentralized control has received considerable attention in the literature [1 17]. A straightforward decentralized control design technique is that of sequential optimization [2,12, 17], wherein a sequential centralized subcontroller design procedure is applied to an augmented closed-loop plant composed of the actual plant and the remaining subcontrollers. Clearly, a key difficulty with decentralized control predicated on sequential optimization is that of dimensionality. An alternative approach to sequential optimization for decentralized control is based on subsystem decomposition with centralized design procedures applied to the individual subsystems of the large-scale system [1,3 8,10,13 16]. Decomposition techniques exploit subsystem interconnection data and in many cases, such as in the presence of very high system dimensionality, is absolutely essential for designing decentralized controllers. In this paper, we develop a novel energy-based hybrid decentralized control framework for lossless and dissipative large-scale dynamical systems [18] based on subsystem decomposition. The notion of energy here refers to abstract energy notions for which a physical system energy interpretation is not necessary. These dynamical systems cover a very broad spectrum of applications including mechanical systems, fluid systems, electromechanical systems, electrical systems, combustion systems, structural vibration systems, biological systems, physiological systems, power systems, telecommunications systems, and economic systems, to cite but a few examples. The concept of an energybased hybrid decentralized controller can be viewed as a feedback control technique that exploits the coupling between a physical large-scale dynamical system and an energy-based decentralized controller to efficiently remove energy from the physical large-scale system. Specifically, if a dissipative or lossless large-scale system is at a high energy level, and a lossless feedback decentralized controller at a low energy level is attached to it, then subsystem energy will generally tend to flow from each subsystem into the corresponding subcontroller, decreasing the subsystem energy and increasing the subcontroller energy [19]. Of course, emulated energy, and not physical energy, is accumulated by each subcontroller. Conversely, if each attached subcontroller is at a high energy level and the corresponding subsystem is at a low energy level, then energy can flow from each subcontroller to each corresponding subsystem, since each subcontroller can generate real, physical energy to effect the required energy flow. Hence, if and when the subcontroller states coincide with a high emulated energy level, then we can reset these states to remove the emulated energy so that the emulated energy is not returned to the plant. In this case, the overall closed-loop system consisting of the plant and the controller possesses discontinuous flows since it combines logical switchings with continuous dynamics, leading to impulsive differential equations [20 26]. The contents of the paper are as follows. In Section 2, we establish definitions, notation, and review some basic results on impulsive differential equations which provide the mathematical foundation for designing fixed-order,

3 246 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) energy-based hybrid decentralized controllers. In Section 3, we present a general state-dependent, energy-based hybrid control framework for lossless large-scale dynamical systems. We then turn our attention to thermodynamic stabilization in Section 4. In particular, we construct hybrid decentralized controllers that guarantee that each closedloop subsystem subcontroller pair is consistent with basic thermodynamic principles. In addition, for both statedependent hybrid decentralized control architectures we show that each decentralized controller corresponds to a maximum entropy controller. In Section 5, we apply the proposed approach to the control of thermoacoustic instabilities in combustion processes. The overall framework demonstrates that hybrid controllers provide an extremely effective way for dissipating energy in combustion systems. Finally, we draw conclusions in Section Hybrid decentralized control and large-scale impulsive dynamical systems In this section, we introduce notation, several definitions, and some key results needed for analyzing large-scale impulsive dynamical systems. Let R denote the set of real numbers, let R + denote the set of nonnegative real numbers, let R n denote the set of n 1 real column vectors, let Z + denote the set of nonnegative integers, let Z + denote the set of positive integers, let R q + denote the positive orthant of Rq, that is, if v R q +, then every entry of v is positive, let ( ) T denote transpose, and let I n denote the n n identity matrix. Furthermore, let S, S, and S denote the boundary, the interior, and the closure of the subset S R n, respectively. We write for the Euclidean vector norm, B ε (α), α R n, ε > 0, for the open ball centered at α with radius ε, and V (x) for the Fréchet derivative of V at x. Finally, we write x(t) M as t to denote that x(t) approaches the set M, that is, for each ε > 0 there exists T > 0 such that dist(x(t), M) < ε for all t > T, where dist(p, M) inf x M p x. In this paper, we consider continuous-time nonlinear dynamical systems G of the form ẋ(t) = F(x(t), u(t)), x(0) = x 0, t 0, (1) y(t) = H(x(t)), where t 0, x(t) D R n, u(t) R m, y(t) R l, F : D R m R n, H : D R l, and D is an open set with 0 D. Here, we assume that G represents a large-scale dynamical system composed of q interconnected controlled subsystems G i so that, for all i = 1,..., q, F i (x, u) = f i (x i ) + I i (x) + G i (x i )u i, H i (x) = h i (x i ), where x i D i R n i, u i R m i, y i h i (x i ) R l i, (u i, y i ) is the input output pair for the ith subsystem, f i : R n i R n i and I i : D R n i are smooth (i.e., infinitely differentiable) and satisfy f i (0) = 0 and I i (0) = 0, G i : R n i R n i m i is smooth, h i : R n i R l i and satisfies h i (0) = 0, q i=1 n i = n, q i=1 m i = m, and q i=1 l i = l. Here, f i : D i R n i R n i defines the vector field of each isolated subsystem of (1) and I i : D R n i defines the structure of the interconnection dynamics of the ith subsystem with all other subsystems. Furthermore, for the large-scale dynamical system G we assume that the required properties for the existence and uniqueness of solutions are satisfied, that is, for every i {1,..., q}, u i ( ) satisfies sufficient regularity conditions such that the system (1) has a unique solution forward in time. We define the composite input and composite output for the large-scale system G as u [u T 1,..., ut q ]T and y [y1 T,..., yt q ]T, respectively. Next, we consider state-dependent hybrid (resetting) decentralized dynamic controllers G ci, i = 1,..., q, of the form ẋ ci (t) = f ci (x ci (t), y i (t)), x ci (0) = x ci0, (x ci (t), y i (t)) Z ci, t 0, (5) x ci (t) = f di (x ci (t), y i (t)), (x ci (t), y i (t)) Z ci, (6) u i (t) = h ci (x ci (t), y i (t)), where x ci D ci R n ci, D ci is an open set with 0 D ci, y ci h ci (x ci, y i ) R m i, f ci : D ci R l i R n ci is smooth and satisfies f ci (0, 0) = 0, f di : D ci R l i R n ci is continuous, h ci : D ci R l i R m i is smooth and satisfies h ci (0, 0) = 0, x ci (t) x ci (t + ) x ci (t), Z ci D ci R l i is the resetting set, and q i=1 n ci = n c. Note that the hybrid decentralized controller (5) (7) represents an impulsive dynamical system G c composed of q impulsive subsystems G ci involving multiple hybrid processors operating independently, with each processor receiving a subset (2) (3) (4) (7)

4 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) of the available system measurements and updating a subset of the system actuators. Furthermore, for generality, we allow the hybrid decentralized dynamic controller to be of fixed dimension n c which may be less than the plant order n. In addition, we define the composite input and composite output for the impulsive decentralized dynamic compensator G c as u c y = [u T c1,..., ut cq ]T and y c u = [y T c1,..., yt cq ]T, respectively. The equations of motion for each closed-loop dynamical subsystem G i, i = 1,..., q, have the form xi (t) = f ci ( x i (t)) + Ĩ i (x), x i (0) = x i0, x i (t) Z i, t 0, (8) x i (t) = f di ( x i (t)), x i (t) Z i, where [ ] [ ] xi x i x, f fi (x ci ( x i ) i ) + G i (x i )h ci (x ci, h i (x i )), ci f ci (x ci, h i (x i )) (10) [ ] [ ] Ii (x) Ĩ i (x), f 0 0 di ( x i ), f di (x ci, h i (x i )) (11) and Z i { x i D i : (x ci, h i (x i )) Z ci }, with ñ i n i + n ci and D i D i D ci, i = 1,..., q. Hence, the equations of motion for the closed-loop dynamical system G have the form x(t) = f c ( x(t)), x(0) = x 0, x(t) Z, t 0, (12) x(t) = f d ( x(t)), x(t) Z, where x(t) = [ x 1 T(t),..., xt q (t)]t, f c ( x) [ f c1 T ( x 1) + Ĩ1 T(x),..., f cq T ( x q) + Ĩq T(x)]T, Z q i=1 { x D : x i Z i }, D q D i=1 i, and f d1 ( x 1 )χz ( x 1 1 ) f d ( x)., f dq ( x q )χz q ( x q ) (9) (13) { 1, χ xi Z Z ( x i i ) = i 0, x i Z i = 1,..., q. (14) i, We refer to the differential equation (12) as the continuous-time dynamics, and we refer to the difference equation (13) as the resetting law. Note that although the closed-loop state vector consists of plant states and controller states, it is clear from (11) that only those states associated with the controller are reset. A function x : I x0 D is a solution to the impulsive dynamical system (12) and (13) on the interval I x0 R with initial condition x(0) = x 0 if x( ) is left-continuous and x(t) satisfies (12) and (13) for all t I x0. For further discussion on solutions to impulsive differential equations, see [20 29]. For convenience, we use the notation s(t, x 0 ) to denote the solution x(t) of (12) and (13) at time t 0 with initial condition x(0) = x 0. For a particular closed-loop trajectory x(t), we let t k τ k ( x 0 ) denote the kth instant of time at which x(t) intersects Z, and we call the times t k the resetting times. Thus, the trajectory of the closed-loop system G from the initial condition x(0) = x 0 is given by ψ(t, x 0 ) for 0 < t t 1, where ψ(t, x 0 ) denotes the solution to the continuous-time dynamics of the closed-loop system G. If and when the trajectory reaches a state x(t 1 ) satisfying x(t 1 ) Z, then the state is instantaneously transferred to x(t + 1 ) x(t 1) + f d ( x(t 1 )) according to the resetting law (13). The trajectory x(t), t 1 < t t 2, is then given by ψ(t t 1, x(t + 1 )), and so on. Our convention here is that the solution x(t) of G is left-continuous, that is, it is continuous everywhere except at the resetting times t k, and x k x(t k ) = lim ε 0 + x(t k ε), x + k x(t k) + f d ( x(t k )) = lim ε 0 + x(t k + ε), (16) for k = 1, 2,.... To ensure the well-posedness of the resetting times, we make the following additional assumptions: Assumption 1. If x Z \ Z, then there exists ε > 0 such that, for all 0 < δ < ε, s(δ, x) Z. (15)

5 248 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) Assumption 2. If x Z, then x + f d ( x) Z. Assumption 1 ensures that if a trajectory reaches the closure of Z at a point that does not belong to Z, then the trajectory must be directed away from Z, that is, a trajectory cannot enter Z through a point that belongs to the closure of Z but not to Z. Furthermore, Assumption 2 ensures that when a trajectory intersects the resetting set Z, it instantaneously exits Z. Finally, we note that if x 0 Z, then the system initially resets to x + 0 = x 0 + f d ( x 0 ) Z, which serves as the initial condition for the continuous-time dynamics (12). It follows from Assumptions 1 and 2 that, for a particular initial condition, the resetting times t k = τ k ( x 0 ) are distinct and well defined [24]. Since the resetting set Z is a subset of the state space and is independent of time, impulsive dynamical systems of the form (12) and (13) are time-invariant systems. These systems are called statedependent impulsive dynamical systems [24]. Since the resetting times are well defined and distinct, and since the solution to (12) exists and is unique, it follows that the solution of the impulsive dynamical system (12) and (13) also exists and is unique over a forward time interval. However, it is important to note that the analysis of impulsive dynamical systems can be quite involved. In particular, such systems can exhibit Zenoness and beating, as well as confluence, wherein solutions exhibit infinitely many resettings in a finite-time, encounter the same resetting surface a finite or infinite number of times in zero time, and coincide after a certain point in time [24,26]. In this paper we allow for the possibility of confluence and Zeno solutions; however, Assumption 2 precludes the possibility of beating. Furthermore, since not every bounded solution of an impulsive dynamical system over a forward time interval can be extended to infinity due to Zeno solutions, we assume that existence and uniqueness of solutions are satisfied in forward time. For details, see [20 23]. For the statement of the next result the following key assumption is needed. Assumption 3. Consider the closed-loop impulsive dynamical system G. Then for every x 0 Z and every ε > 0 and t t k, there exists δ(ε, x 0, t) > 0 such that if x 0 y < δ(ε, x 0, t), y D, then s(t, x 0 ) s(t, y) < ε. Assumption 3 is a weakened version of the quasi-continuous dependence assumption given in [24,26], and is a generalization of the standard continuous dependence property for dynamical systems with continuous flows to dynamical systems with left-continuous flows. Specifically, by letting t [0, ), Assumption 3 specializes to the classical continuous dependence of solutions of a given dynamical system with respect to the system s initial conditions x 0 D. Since solutions of impulsive dynamical systems are not continuous in time and solutions are not continuous functions of the system initial conditions, Assumption 3 involving pointwise continuous dependence is needed to apply the hybrid invariance principle developed in [24,26] to hybrid closed-loop systems. Sufficient conditions that guarantee that the impulsive dynamical system G satisfies a stronger version of Assumption 3 are given in [26] (see also [30]). The following result provides a generalization of the results given in [26] for establishing sufficient conditions for guaranteeing that the impulsive dynamical system G satisfies Assumption 3. Proposition 2.1. Consider the large-scale impulsive dynamical system G given by the feedback interconnection of G and G c. Assume that Assumptions 1 and 2 hold, τ 1 ( ) is continuous at every x Z such that 0 < τ 1 ( x) <, and if x Z, then x + f d ( x) Z \ Z. Furthermore, for every x Z \ Z such that 0 < τ 1 ( x) <, assume that the following statements hold: (i) If a sequence { x (i) } i=1 D is such that lim i x (i) = x and lim i τ 1 ( x (i) ) exists, then either f d ( x) = 0 and lim i τ 1 ( x (i) ) = 0, or lim i τ 1 ( x (i) ) = τ 1 ( x). (ii) If a sequence { x (i) } i=1 Z \ Z is such that lim i x (i) = x and lim i τ 1 ( x (i) ) exists, then lim i τ 1 ( x (i) ) = τ 1 ( x). Then G satisfies Assumption 3. Proof. The proof is similar to the proof of Proposition 2.1 given in [31] and, hence, is omitted. The following result provides sufficient conditions for establishing continuity of τ 1 ( ) at x 0 Z and sequential continuity of τ 1 ( ) at x 0 Z \ Z, that is, lim i τ 1 ( x (i) ) = τ 1 ( x 0 ) for { x (i) } i=1 Z and lim i x (i) = x 0. For this result, the following definition is needed. First, however, recall that the Lie derivative of a smooth function X : D R along the vector field of the continuous-time dynamics f c ( x) is given by L f c X ( x) dt d X ( ψ(t, x)) t=0 =

6 X ( x) x W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) f c ( x), and the zeroth and higher-order Lie derivatives are, respectively, defined by L 0 X ( x) X ( x) and f c f c (L k 1 f X ( x)), where k 1. c L k f c X ( x) L Definition 2.1. Let M q i=1 { x D : X i ( x) = 0}, where X i : D R, i = 1,..., q, are infinitely differentiable functions. A point x M such that f c ( x) 0 is transversal to (12) if there exists k i {1, 2,...}, i = 1,..., q, such that L r X f i ( x) = 0, r = 0,..., 2k i 2, L 2k i 1 c f X i ( x) 0, i = 1,..., q. (17) c Proposition 2.2. Consider the large-scale impulsive dynamical system G given by the feedback interconnection of G and G c. Let X i : D R, i = 1,..., q, be infinitely differentiable functions such that Z = q i=1 { x D : X i ( x) = 0}, and assume that every x Z is transversal to (12). Then at every x 0 Z such that 0 < τ 1 ( x 0 ) <, τ 1 ( ) is continuous. Furthermore, if x 0 Z \ Z is such that τ 1 ( x 0 ) (0, ) and (i) { x (i) } i=1 Z \ Z or (ii) lim i τ 1 ( x (i) ) > 0, where { x (i) } i=1 Z is such that lim i x (i) = x 0 and lim i τ 1 ( x (i) ) exists, then lim i τ 1 ( x (i) ) = τ 1 ( x 0 ). Proof. The proof is similar to the proof of Proposition 2.2 given in [31] and, hence, is omitted. Remark 2.1. Let x 0 Z be such that lim i τ 1 ( x (i) ) τ 1 ( x 0 ) for some sequence { x (i) } i=1 Z with lim i x (i) = x 0. Then it follows from Proposition 2.2 that lim i τ 1 ( x (i) ) = 0. Remark 2.2. Proposition 2.2 is a nontrivial generalization of Proposition 4.2 of [26] and Lemma 3 of [30]. Specifically, Proposition 2.2 establishes the continuity of τ 1 ( ) in the case where the resetting set Z is not a closed set. In addition, the transversality condition given in Definition 2.1 is also a generalization of the conditions given in [26] and [30] by considering higher-order derivatives of the function X i ( ) rather than simply considering the firstorder derivative as in [26,30]. For the case k i = 1, i = 1,..., q, this condition guarantees that the solution of the closed-loop system (8) and (9) is not tangent to the closure of the resetting set Z at the intersection with Z. The next result characterizes impulsive dynamical system limit sets in terms of continuously differentiable functions. In particular, we show that the system trajectories of a state-dependent impulsive dynamical system converge to an invariant set contained in a union of level surfaces characterized by the continuous-time system dynamics and the resetting system dynamics. For the next result assume that f c ( ), f d ( ), Ĩ( ), and Z are such that the dynamical system G given by (12) and (13) satisfies Assumptions 1 3. Note that for addressing the stability of the zero solution of an impulsive dynamical system the usual stability definitions are valid. For details, see [20 26]. Theorem 2.1 ([31]). Consider the impulsive dynamical system (12) and (13) and assume Assumptions 1 3 hold. Assume D ci D is a compact positively invariant set with respect to (12) and (13), assume that if x 0 Z then x 0 + f d ( x 0 ) Z \ Z, and assume that there exists a continuously differentiable function V : D ci R such that V ( x) f c ( x) 0, V ( x + f d ( x)) V ( x), x D ci, x Z, x D ci, x Z. (19) Let R { x D ci : x Z, V ( x) f c ( x) = 0} { x D ci : x Z, V ( x + f d ( x)) V ( x) = 0} and let M denote the largest invariant set contained in R. If x 0 D ci, then x(t) M as t. Furthermore, if 0 D ci, V (0) = 0, V ( x) > 0, x 0, and the set R contains no invariant set other than the set {0}, then the zero solution x(t) 0 to (12) and (13) is asymptotically stable and D ci is a subset of the domain of attraction of (12) and (13). Remark 2.3. Setting D = R n and requiring V ( x) as x in Theorem 2.1, it follows that the zero solution x(t) 0 to (12) and (13) is globally asymptotically stable. A similar remark holds for Theorem 2.2 below. (18)

7 250 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) Theorem 2.2. Consider the impulsive dynamical system G (12) and (13) and assume Assumptions 1 3 hold. Assume D ci D is a compact positively invariant set with respect to (12) and (13) such that 0 D ci, assume that if x 0 Z then x 0 + f d ( x 0 ) Z \ Z, and assume that for all x 0 D ci, x 0 0, there exists τ 0 such that x(τ) Z, where x(t), t 0, denotes the solution to (12) and (13) with the initial condition x 0. Furthermore, assume that there exists a continuously differentiable vector function V = [v 1,..., v q ] T : D R q + and a positive vector p Rq + such that V (0) = 0, the scalar function v : D R + defined by v( x) p T V ( x), x D, is such that v( x) > 0, x D, x 0, and v ( x) f c ( x) 0, v( x + f d ( x)) < v( x), x D ci, x Z, x D ci, x Z. (21) Then the zero solution x(t) 0 to (12) and (13) is asymptotically stable and D ci is a subset of the domain of attraction of (12) and (13). Proof. It follows from (21) that R = { x D ci : x Z, v ( x) f c ( x) = 0}. Since for all x 0 D ci, x 0 0, there exists τ 0 such that x(τ) Z, it follows that the largest invariant set contained in R is {0}. Now, the result is a direct consequence of Theorem Hybrid decentralized control for large-scale dynamical systems In this section, we present a hybrid decentralized controller design framework for large-scale dynamical systems. Specifically, we consider nonlinear large-scale dynamical systems G of the form given by (1) and (2) where u( ) satisfies sufficient regularity conditions such that (1) has a unique solution forward in time. Furthermore, we consider hybrid decentralized dynamic controllers G ci, i = 1,..., q, of the form ẋ ci (t) = f ci (x ci (t), y i (t)), x ci (0) = x c0i, (x ci (t), y i (t)) Z ci, (22) x ci (t) = η i (y i (t)) x ci (t), (x ci (t), y i (t)) Z ci, (23) y ci (t) = h ci (x ci (t), y i (t)), where x ci (t) D ci R n ci, D ci is an open set with 0 D ci, y i (t) R l i, y ci (t) R m i, f ci : D ci R l i R n ci is smooth on D ci and satisfies f ci (0, 0) = 0, η i : R l i D ci is continuous and satisfies η i (0) = 0, h ci : D ci R l i R m i is smooth and satisfies h ci (0, 0) = 0, q i=1 l i = l, and q i=1 m i = m. Recall that for the dynamical system G given by (1) and (2), a vector function S(u, y) [s 1 (u 1, y 1 ),..., s q (u q, y q )] T, where S : U Y R q is such that S(0, 0) = 0, is called a vector supply rate [32,33] if it is componentwise locally integrable for all input output pairs satisfying (1) and (2), that is, for every i {1,..., q} and for all input output pairs (u i, y i ) U i Y i satisfying (1) and (2), s i (, ) satisfies t 2 t 1 s i (u i (σ ), y i (σ )) dσ <, t 2 t 1 0. Here, U = U 1 U q and Y = Y 1 U q are input and output spaces, respectively, that are assumed to be closed under the shift operator. Furthermore, we assume that G is vector lossless with respect to the vector supply rate S(u, y), and hence, there exist a continuous, nonnegative definite vector storage function V s = [v s1,..., v sq ] T : D R q + and a Kamke function w : Rq + Rq such that V s (0) = 0, w(0) = 0, the zero solution z(t) 0 to the comparison system ż(t) = w(z(t)), z(0) = z 0, t 0, (25) is Lyapunov stable, and the vector dissipation equality V s (x(t)) = V s (x(t 0 )) + t t 0 w(v s (x(σ )))dσ + t (20) (24) t 0 S(u(σ ), y(σ ))dσ, (26) is satisfied for all t t 0 0, where x(t), t t 0, is the solution to G with u U. In this case, it follows from Theorem 3.2 of [32] that there exists a nonnegative vector p R q +, p 0, such that G is lossless with respect to the supply rate p T S(u, y) and with the storage function v s (x) = p T V s (x), x D. In addition, we assume that the nonlinear large-scale dynamical system G is completely reachable [18] and zero-state observable [18], and there exist functions

8 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) κ i : Y i U i such that κ i (0) = 0 and s i (κ i (y i ), y i ) < 0, y i 0, for all i = 1,..., q, so that all storage functions v s (x) = p T V s (x), x D, are positive definite, that is, p T V s (x) > 0, x D, x 0 [32]. Finally, we assume that V s ( ) is component decoupled, that is, V s (x) = [v s1 (x 1 ),..., v sq (x q )] T, x D, and continuously differentiable. Note that if each disconnected subsystem G i (i.e., I i (x) 0, i {1,..., q}) of G is lossless with respect to the supply rate s i (u i, y i ), then V s ( ) is component decoupled. Consider the negative feedback interconnection of G and G c given by y i = u ci and u i = y ci, i = 1,..., q. In this case, the closed-loop system G can be written in terms of the subsystems G i, i = 1,..., q, given by xi (t) = f ci ( x i (t)) + Ĩ i (x), x i (0) = x i0, x i (t) Z i, t 0, (27) x i (t) = f di ( x i (t)), x i (t) Z i, where t 0, x i (t) [xi T(t), xt ci (t)]t, Z i { x i D i : (x ci, h i (x i )) Z ci }, [ ] [ f fi (x ci ( x i ) i ) G i (x i )h ci (x ci, h i (x i )) Ii (x), Ĩ f ci (x ci, h i (x i )) i (x) 0 [ f di ( x i ) ] 0. η i (h i (x i )) x ci Hence, the equations of the motion for the closed-loop system G have the form (28) ], (29) x(t) = f c ( x(t)), x(t 0 ) = x 0, x(t) Z, t t 0, (31) x(t) = f d ( x(t)), x(t) Z, where x(t) = [ x 1 T(t),..., xt q (t)]t, f c ( x) [ f c1 T ( x 1) + Ĩ1 T(x),..., f cq T ( x q) + Ĩq T(x)]T, Z q i=1 { x D : x i Z i }, D q D i=1 i, and f d1 ( x 1 )χz ( x 1 1 ) { f d ( x) 1,., χ xi Z Z ( x i i ) = i 0, x i i = 1,..., q. (33) Z i, f dq ( x q )χz q ( x q ) Assume that there exist infinitely differentiable functions v ci : D ci R l i R +, i = 1,..., q, such that v ci (x ci, y i ) 0, x ci D ci, y i R l i, and v ci (x ci, y i ) = 0 if and only if x ci = η i (y i ) and v ci (x ci (t), y i (t)) = s ci (u ci (t), y ci (t)), (30) (32) (x ci (t), y i (t)) Z i, t 0, (34) where s ci : R l i R m i R is such that s ci (0, 0) = 0, i = 1,..., q. We associate with the plant a positive-definite, continuously differentiable function v p (x) p T V s (x), which we will refer to as the plant energy composed of the subsystem energies v si (x i ), i = 1,..., q. Furthermore, we associate with the controller a nonnegative-definite, infinitely differentiable function v c (x c, y) p T V c (x c, y), where V c (x c, y) [v c1 (x c1, y 1 ),..., v cq (x cq, y q )] T, called the controller emulated energy composed of the subcontroller emulated energies v ci (x ci, y i ), i = 1,..., q. Finally, we associate with the closed-loop system the function v( x) v p (x) + v c (x c, H(x)), called the total energy composed of the total subsystem energies v si (x i ) + v ci (x ci, y i ), i = 1,..., q. Next, we construct the resetting set for each subsystem G i, i = 1,..., q, of the closed-loop system G in the following form Z i = {(x i, x ci ) D D ci : L f c v ci (x ci, h i (x i )) = 0 and v ci (x ci, h i (x i )) > 0} = {(x i, x ci ) D D ci : s ci (h i (x i ), h ci (x ci, h i (x i ))) = 0 and v ci (x ci, h i (x i )) > 0}, (36) where i = 1,..., q. The resetting sets Z i, i = 1,..., q, are thus defined to be the sets of all points in the closedloop state space that correspond to decreasing subcontroller emulated energy. By resetting the subcontroller states, the subsystem energy can never increase after the first resetting event. Furthermore, if the closed-loop subsystem total (35)

9 252 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) energy is conserved between resetting events, then a decrease in subsystem energy is accompanied by a corresponding increase in subsystem emulated energy. Hence, this approach allows the subsystem energy to flow to the subcontroller, where it increases the subcontroller emulated energy but does not allow the subcontroller emulated energy to flow back to the subsystem after the first resetting event. This energy dissipating hybrid decentralized controller effectively enforces a one-way energy transfer between each subsystem and corresponding subcontroller after the first resetting event. For practical implementation, knowledge of x ci and y i is sufficient to determine whether or not the closed-loop state vector is in the set Z i, i = 1,..., q. The next theorem gives sufficient conditions for asymptotic stability of the closed-loop system G using statedependent hybrid decentralized controllers. Theorem 3.1. Consider the closed-loop impulsive dynamical system G given by (31) and (32). Assume that D ci D is a compact positively invariant set with respect to G such that 0 D ci, assume that G is vector lossless with respect to the vector supply rate S(u, y) [s 1 (u 1, y 1 ),..., s q (u q, y q )] T and with a positive, continuously differentiable vector storage function V s (x) = [v s1 (x 1 ),..., v sq (x q )] T, x D. In addition, assume there exist smooth functions v ci : D ci R l i R + such that v ci (x ci, y i ) 0, x ci D ci, y i R l i, v ci (x ci, y i ) = 0 if and only if x ci = η i (y i ), and (34) holds. Finally, assume that every x 0 Z is transversal to (27) and s i (u i, y i ) + s ci (u ci, y ci ) = 0, x i Z i, i = 1,..., q, (37) where y i = u ci = h i (x i ), u i = y ci = h ci (x ci, h i (x i )), and Z i, i = 1,..., q, is given by (36). Then the zero solution x(t) 0 to the closed-loop system G is asymptotically stable. In addition, the total energy function v( x) of G given by (35) is strictly decreasing across resetting events. Finally, if D = R n, D c = R n c, and v( ) is radially unbounded, then the zero solution x(t) 0 to G is globally asymptotically stable. Proof. First, note that since v ci (x ci, y i ) 0, x ci D ci, y i R l i, i = 1,..., q, it follows that Z i = {(x i, x ci ) D D ci : L f c v ci (x ci, h i (x i )) = 0 and v ci (x ci, h i (x i )) 0} = {(x i, x ci ) D D ci : X i ( x i ) = 0}, (38) where X i ( x i ) = L f c v ci (x ci, h i (x i )), i = 1,..., q. Next, we show that if the transversality condition (17) holds, then Assumptions 1 3 hold and, for every x 0 D ci there exists τ 0 such that x(τ) Z. Note that if x 0 Z \ Z, that is, v ci (x ci (0), h i (x i (0))) = 0 and L f c v ci (x ci (0), h i (x i (0))) = 0, i {1,..., q}, it follows from the transversality condition that there exists δ i > 0 such that for all t (0, δ i ], L f c v ci (x ci (t), h i (x i (t))) 0. Hence, since v ci (x ci (t), h i (x i (t))) = v ci (x ci (0), h i (x i (0))) + t L f c v ci (x ci (τ), h i (x i (τ))) for some τ (0, t] and v ci (x ci, y i ) 0, x ci D ci, y i R l i, i {1,..., q}, it follows that v ci (x ci (t), h i (x i (t))) > 0, t (0, δ], which implies that Assumption 1 is satisfied. Furthermore, if x Z then, since v ci (x ci, y i ) = 0 if and only if x ci = η(y i ), it follows from (34) that x i + f di ( x i ) Z i \ Z { i, i {1,..., q}. Hence, Assumption } 2 holds. Next, consider the set M γ q i=1 x D ci : v ci (x ci, h i (x i )) = γ i, where γ i 0, i = 1,..., q, and γ [γ 1,..., γ q ] T. It follows from the transversality condition that for every γ i 0, M γ does not contain any nontrivial trajectory of G, i = 1,..., q. To see this, suppose, ad absurdum, there exists a nontrivial trajectory x(t) M γ, t 0, for some γ i 0 and for some i {1,..., q}. In this case, it follows that dk v dt k ci (x ci (t), h i (x i (t))) = L k v f ci (x ci (t), h i (x i (t))) 0, k = 1, 2,..., i {1,..., q}, which contradicts the transversality condition. c Next, we show that for every x 0 Z, x 0 0, there exists τ > 0 such that x(τ) Z. To see this, suppose, ad absurdum, x i (t) Z i for all i = 1,..., q, t 0, which implies that or d dt v ci(x ci (t), h i (x i (t))) 0, t 0, i = 1,..., q, (39) v ci (x ci (t), h i (x i (t))) = 0, t 0, i = 1,..., q. (40) If (39) holds, then it follows that v ci (x ci (t), h i (x i (t))) is a (decreasing or increasing) monotonic function of time. Hence, v ci (x ci (t), h i (x i (t))) γ i as t, where γ i 0 is a constant for i = 1,..., q, which implies that

10 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) the positive limit set of the closed-loop system is contained in M γ for some γ i 0, i = 1,..., q, and hence, is a contradiction. Similarly, if (40) holds then M 0 contains a nontrivial trajectory of G also leading to a contradiction. Hence, for every x 0 Z, there exists τ > 0 such that x(τ) Z. Thus, it follows that for every x 0 Z, 0 < τ 1 ( x 0 ) <. Now, it follows from Proposition 2.2 that τ 1 ( ) is continuous at x 0 Z. Furthermore, for all x 0 Z \ Z and for every sequence { x (i) } i=1 Z \ Z converging to x 0 Z \ Z, it follows from the transversality condition and Proposition 2.2 that lim i τ 1 ( x (i) ) = τ 1 ( x 0 ). Next, let x 0 Z \ Z and let { x (i) } i=1 D ci be such that lim i x (i) = x 0 and lim i τ 1 ( x (i) ) exists. In this case, it follows from Proposition 2.2 that either lim i τ 1 ( x (i) ) = 0 or lim i τ 1 ( x (i) ) = τ 1 ( x 0 ). Furthermore, since x 0 Z \ Z corresponds to the case where v ci (x ci0, h i (x i0 )) = 0, i {1,..., q}, it follows that x ci0 = η i (h i (x i0 )), and hence, f di ( x i0 ) = 0, i {1,..., q}. Now, it follows from Proposition 2.1 that Assumption 3 holds. To show that the zero solution x(t) 0 to G is asymptotically stable, consider the Lyapunov function candidate corresponding to the total energy function v( x) given by (35). Since G is vector lossless with respect to the vector supply rate S(u, y), and hence, lossless with respect to the supply rate p T S(u, y), where p R q +, and (34) and (37) hold, it follows that q v( x(t)) = p i [s i (u i (t), y i (t)) + s ci (u ci (t), y ci (t))] = 0, x(t) Z, (41) i=1 where p i, i = 1,..., q, denotes the ith element of p R q +. Furthermore, it follows from (30) and (38) that v( x(t k )) = v c (x c (t k + ), H(x(t+ k ))) v c(x c (t k ), H(x(t k ))) = q i=1 p i v ci (x ci (t k ), h i (x i (t k )))χ Z i ( x i (t k )) < 0, x(t k ) Z, k Z +. Thus, it follows from Theorem 2.2 that the zero solution x(t) 0 to G is asymptotically stable. Finally, if D = R n, D c = R n c, and v( ) is radially unbounded, then global asymptotic stability is immediate. Remark 3.1. If v ci = v ci (x ci, y i ) is only a function of x ci and v ci (x ci ) is a positive-definite function, i {1,..., q}, then we can choose η i (y i ) 0. In this case, v ci (x ci ) = 0 if and only if x ci = 0. Remark 3.2. In the proof of Theorem 3.1, we assume that x 0 Z for x 0 0. This proviso is necessary since it may be possible to reset the states of the closed-loop system to the origin, in which case x(s) = 0 for a finite value of s. In this case, for t > s, we have v( x(t)) = v( x(s)) = v(0) = 0. This situation does not present a problem, however, since reaching the origin in finite time is a stronger condition than reaching the origin as t. Remark 3.3. Theorem 3.1 can be generalized to the case where G is vector dissipative with respect to the vector supply rate S(u, y) with the component decoupled vector storage function V s (x) = [v s1 (x 1 ),..., v sq (x q )] T, x D. Specifically, in this case (41) becomes v( x(t)) = q i=1 p id i (x i (t)) 0, x(t) Z, where d i : D i R, i = 1,..., q, is a continuous, nonnegative-definite dissipation rate function. Now, Theorem 3.1 holds with the additional assumption that the only invariant set contained in R q i=1 { x D ci : d i (x i ) = 0} is M = {0}. 4. Quasi-thermodynamic stabilization and maximum entropy control In this section, we use the recently developed notion of system thermodynamics [34] to develop thermodynamically consistent hybrid decentralized controllers for large-scale systems. Specifically, since our energy-based hybrid controller architecture involves the exchange of energy with conservation laws describing transfer, accumulation, and dissipation of energy between the subcontrollers and the plant subsystems, we construct a modified hybrid controller that guarantees that each subsystem subcontroller pair (G i, G ci ) is consistent with basic thermodynamic principles after the first resetting event. To develop thermodynamically consistent hybrid decentralized controllers consider the closed-loop subsystem subcontroller pair (G i, G ci ) given by (27) and (28) with Z i given by Z i { x i D i : φ i ( x i )(v pi ( x i ) v ci ( x i )) = 0 and v ci (x ci, h i (x i )) > 0}, i = 1,..., q, (43) (42)

11 254 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) where φ i ( x i ) s ci (h i (x i ), h ci (x ci, h i (x i ))), v pi ( x i ) v si (x i ), and v ci ( x i ) v ci (x ci, h i (x i )). We refer to φ i ( ) as the net energy flow function. We assume that the energy flow function φ i ( x i ) is infinitely differentiable and the transversality condition (17) holds with X i ( x i ) = φ i ( x i )(v pi ( x i ) v ci ( x i )) for all i = 1,..., q. To ensure a thermodynamically consistent energy flow between the subsystem G i and subcontroller G ci after the first resetting event, each subcontroller resetting logic must be designed in such a way so as to satisfy three key thermodynamic axioms. Namely, between resettings the energy flow function φ i ( ) must satisfy the following two axioms [34,35]: Axiom 1. For the connectivity matrix C R 2 2 [34, p. 56] associated with the subsystem G l defined by { 0, if φl ( x C (i, j) l (t)) 0 i j, i, j = 1, 2, l = 1,..., q, t t + 1, otherwise, 1, (44) and C (i,i) = C (k,i), i k, i, k = 1, 2, rank C = 1, and for C (i, j) = 1, i j, φ l ( x l (t)) = 0 if and only if v pl ( x l ) = v cl ( x l ), x l (t) Z l, l = 1,..., q, t t + 1. Axiom 2. φ l ( x i (t))(v pl ( x i ) v cl ( x i )) 0, x i (t) Z i, i = 1,..., q, t t + 1. Furthermore, across resettings the energy difference between the subsystem and the subcontroller must satisfy the following axiom [36,37]: Axiom 3. [v pi ( x i + f di ( x i )) v ci ( x i + f di ( x i ))][v pi ( x i ) v ci ( x i )] 0, i = 1,..., q, x i Z i. The fact that φ i ( x i (t)) = 0 if and only if v pi ( x i (t)) = v ci ( x i (t)), x i (t) Z i, t t + 1, implies that the ith subsystem and the ith subcontroller are connected; alternatively, φ i ( x i (t)) 0, t t + 1, implies that the ith subsystem and the ith subcontroller are disconnected. Axiom 1 implies that if the energies in the ith subsystem and the ith subcontroller are equal, then energy exchange between the ith subsystem G i and the ith subcontroller G ci is not possible unless a resetting event occurs. This statement is consistent with the zeroth law of thermodynamics, which postulates that temperature equality is a necessary and sufficient condition for thermal equilibrium of an isolated system. Axiom 2 implies that energy flows from a more energetic subsystem to a less energetic subsystem and is consistent with the second law of thermodynamics, which states that heat (energy) must flow in the direction of lower temperatures. Finally, Axiom 3 implies that the energy difference between the ith subsystem G i and the ith subcontroller G ci across resetting instants is monotonic, that is, [v pi ( x i (t k + )) v ci( x i (t k + ))][v pi( x i (t k )) v ci ( x i (t k ))] 0 for all v pi ( x i ) v ci ( x i ), x i Z i, i = 1,..., q, k Z +. With the resetting law given by (43), it follows that each ith subsystem G i of the closed-loop dynamical system G satisfies Axioms 1 3 for all t t 1. To see this, note that since φ i ( x i ) 0, the connectivity matrix C is given by [ ] 1 1 C =, (46) 1 1 and hence rank C = 1. The second condition in Axiom 1 need not be satisfied since the case where φ i ( x i ) = 0 or v pi ( x i ) = v ci ( x i ), corresponds to a resetting instant. Furthermore, it follows from the definition of the resetting set (43) that Axiom 2 is satisfied for each closed-loop subsystem pairs (G i, G ci ) for all t t + 1. Finally, since v ci ( x i + f di ( x i )) = 0 and v pi ( x i + f di ( x i )) = v pi ( x i ), x i Z i, it follows from the definition of the resetting set that [v pi ( x i + f di ( x i )) v ci ( x i + f di ( x i ))][v pi ( x i ) v ci ( x i )] = v pi ( x i )[v pi ( x i ) v ci ( x i )] 0, x i Z i, i = 1,..., q, and hence, Axiom 3 is satisfied across resettings. Hence, each ith closed-loop subsystem G i of the closed-loop system G is thermodynamically consistent after the first resetting event in the sense of [34 37]. Note that this statement is only true for each closed-loop subsystem G i. For the hybrid closed-loop system G, Axioms 1 3 may not hold since the interconnection function I(x) defining G may not necessarily correspond to a thermodynamically consistent model. (45) (47)

12 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) If D ci D is a compact positively invariant set with respect to the closed-loop dynamical system G given by (31) and (32) such that 0 D ci, and the transversality condition (17) holds with X i ( x i ) = φ i ( x i )(v pi ( x i ) v ci ( x i )) for all i = 1,..., q, then using similar arguments as in the proof of Theorem 3.1 it can be shown that the zero solution x(t) 0 to the closed-loop system G, with resetting set Z i given by (43), is asymptotically stable. Furthermore, in this case, the hybrid decentralized controller (22) and (23), with resetting set (43), is a quasi-thermodynamically stabilizing compensator. Finally, we show that the hybrid decentralized controllers developed in this section and Section 3 are maximum entropy controllers. To do this, the following hybrid definition of entropy is needed. Definition 4.1. For each decentralized subcontroller G ci given by (22) (24), a function S ci : R + R, i = 1,..., q, satisfying S ci (v ci ( x i (T ))) S ci (v ci ( x i (t 1 ))) 1 c i k Z [t1,t ) v ci ( x i (t k )), T t 1, i = 1,..., q, (48) where k Z [t1,t ) {k : t 1 t k < T }, c i > 0, is called an entropy function of G ci, i = 1,..., q. The next result gives necessary and sufficient conditions for establishing the existence of an entropy function of G ci, i = 1,..., q, over an interval t (t k, t k+1 ] involving the consecutive resetting times t k and t k+1, k Z +. Theorem 4.1. For each decentralized subcontroller G ci given by (22) (24), a function S ci : R + R, i = 1,..., q, is an entropy function of G ci if and only if S ci (v ci ( x i (ˆt ))) S ci (v ci ( x i (t))), t k < t ˆt t k+1, i = 1,..., q, (49) S ci (v ci ( x i (t k ) + f di ( x i (t k )))) S ci (v ci ( x i (t k ))) 1 c i v ci ( x i (t k )), k Z +, i = 1,..., q. (50) Proof. Let k Z + and suppose S ci (v ci ) is an entropy function of G ci. Then, (48) holds. Now, since for t k < t ˆt t k+1, Z [t,ˆt ) = Ø, (49) is immediate. Next, note that S ci (v ci ( x i (t + k ))) S ci(v ci ( x i (t k ))) 1 c i v ci ( x i (t k )), i = 1,..., q, (51) which, since Z [tk,t k + ) = k, implies (50). Conversely, suppose (49) and (50) hold, and let ˆt t t 1 and Z [t,ˆt ) = {i, i + 1,..., j}. (Note that if Z [t,ˆt ) = Ø the converse result is a direct consequence of (49).) If Z [t,ˆt ) Ø, it follows from (49) and (50) that S cl (v cl ( x l (ˆt ))) S cl (v cl ( x l (t))) = S cl (v cl ( x l (ˆt ))) S cl (v cl ( x l (t + j ))) + + j i m=0 j i 1 m=0 S cl (v cl ( x l (t j m ) + f dl ( x l (t j m )))) S cl (v cl ( x l (t j m ))) S cl (v cl ( x l (t j m ))) S cl (v cl ( x l (t + j m 1 ))) + S cl (v cl ( x l (t i ))) S cl (v cl ( x l (t))) 1 c l j i v cl ( x l (t j m )) m=0 = 1 c l k Z [t,ˆt ) v cl ( x l (t k )), l = 1,..., q, (52) which implies that S ci (v ci ) is an entropy function of G ci, i = 1,..., q.

13 256 W.M. Haddad et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) The next theorem establishes the existence of an entropy function for G ci, i = 1,..., q. Theorem 4.2. Consider the hybrid decentralized subcontrollers G ci given by (22) (24), with Z i given by (36) or (43). Then the function S ci : R + R, i = 1,..., q, given by S ci (v ci ) = log e (c i + v ci ) log e c i, v ci R +, i = 1,..., q, (53) where c i > 0, is an entropy function of G ci, i = 1,..., q. In addition, for i = 1,..., q, Ṡ ci (v ci ( x i (t))) > 0, x i (t) Z i, t k < t t k+1, (54) 1 c i v ci ( x i (t k )) < S ci (v ci ( x i (t k ))) < v ci( x i (t k )) c i + v ci ( x i (t k )), Proof. Since v ci ( x i (t)) > 0, x i (t) Z i, i = 1,..., q, t (t k, t k+1 ], k Z +, it follows that Ṡ ci (v ci ( x i (t))) = v ci( x i (t)) c i + v ci ( x i (t)) > 0, x i(t k ) Z i, k Z +. (55) x i(t) Z i, i = 1,..., q. (56) Furthermore, since v ci ( x i (t k ) + f di ( x i (t k ))) = 0, x i (t k ) Z i, i = 1,..., q, k Z +, it follows that, for i = 1,..., q, [ S ci (v ci ( x i (t k ))) = log e 1 v ] ci( x i (t k )) > 1 v ci ( x i (t k )), x i (t k ) Z i, k Z +, (57) c i + v ci ( x i (t k )) c i and [ S ci (v ci ( x i (t k ))) = log e 1 v ] ci( x i (t k )) < v ci( x i (t k )) c i + v ci ( x i (t k )) c i + v ci ( x i (t k )), x i(t k ) Z i, k Z +, (58) x where in (57) and (58) we used the fact that 1+x < log e (1 + x) < x, x > 1, x 0. The result is now an immediate consequence of Theorem 4.1. Using (56), the resetting set Z i, i = 1,..., q, given by (36) can be rewritten as { } Z d i x i D i : dt S ci(v ci ( x i )) = 0 and v ci ( x i ) > 0, i = 1,..., q, (59) where dt d S ci(v ci ( x i (t))) lim τ t t τ 1 [S ci(v ci ( x i (t))) S ci (v ci ( x i (τ)))] whenever the limit on the right-hand side exists, and S ci = log e (c i + v ci ) log e c i denotes the continuously differentiable ith subcontroller entropy. Hence, each decentralized controller G ci corresponds to a maximum entropy controller. Alternatively, for i = 1,..., q, the resetting set Z i given by (43) can be rewritten as { x i (t k ) : k Z + }, where t k is the maximum final time such that S ci (v ci ( x i (t))) S ci (v ci ( x i (t 1 ))) (or S ci (v ci ( x i (t))) S ci (v ci ( x i (t 1 )))) holds under the constraint v pi ( x i (t)) v ci ( x i (t)) (or v pi ( x i (t)) v ci ( x i (t))) for 0 t < t 1, and S ci (v ci ( x i (t))) S ci (v ci ( x i (t k+1 ))) holds under the constraint v pi ( x i (t)) v ci ( x i (t)) for all t k t < t k+1 and k Z +. Hence, each decentralized controller G ci corresponds to a constrained maximum entropy controller. 5. Hybrid decentralized control for combustion systems High performance aeroengine afterburners and ramjets often experience combustion instabilities at some operating condition. Combustion in these high energy density engines is highly susceptible to flow disturbances, resulting in fluctuations to the instantaneous rate of heat release in the combustor. This unsteady combustion provides an acoustic source resulting in self-excited oscillations. In particular, unsteady combustion generates acoustic pressure and velocity oscillations which in turn perturb the combustion even further [38,39]. These pressure oscillations, known as thermoacoustic instabilities, often lead to high vibration levels causing mechanical failures, high levels of acoustic noise, high burn rates, and even component melting. Hence, the need for active control to mitigate combustion induced pressure instabilities is crucial. In this section we apply the results developed in this paper to the control of thermoacoustic instabilities in combustion processes. We stress that the combustion model we use can be stabilized by conventional nonlinear