Feedback stabilization of Bernoulli jump nonlinear systems: a passivity-based approach

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1 1 Feedback stabilization of Bernoulli jump nonlinear systems: a passivity-based approach Yingbo Zhao, Student Member, IEEE, Vijay Gupta, Member, IEEE Abstract We study feedback stabilization of a Bernoulli jump nonlinear system. First, the relationship between stochastic passivity and stochastic stability is established. A state feedback controller that stabilizes a wide class of Bernoulli jump nonlinear systems is then designed using passivity-based tools. The advantage of our approach is that the nonlinear dynamics can be non-affine in the control input. I. INTRODUCTION We study feedback stabilization of Bernoulli jump nonlinear systems (BJNSs) using passivity-based tools. A Bernoulli jump system (BJS) is a special class of hybrid systems. Such a system consists of m modes that describe the possible dynamics according to which the system state evolves. The particular mode that is active at any time is described by a discrete variable that takes value in the set M ={1, 2,..., m} stochastically. Specifically, the discrete variable chooses the active mode at any time according to a given probability distribution, and independently from one time step to the next. BJS is a simplified version of the more general class of Markovian jump systems (MJS), in which the system mode is chosen according to a Markovian random process. MJSs have been used to characterize random abrupt changes in system dynamics caused by, for instance, changes of subsystem interconnections, random component failures or repairs, and change of operating points [16]. BJSs have gained importance in the modeling of networked control systems where the communication channel is assumed to be memoryless. If the random packet losses due to the communication channel can be modelled as an i.i.d. Bernoulli sequence with a known probability distribution, the system can be examined as a BJS [18], [27]. Moreover, feedback control through a communication network that introduces random delays has also been modelled as a Bernoulli jump system in the literature [15]. A vast literature is available for MJSs for problems such as stability and stabilization [10], [29], H 2 and H control [6], [11], and filtering [22], [25], [28]. However, most of these results deal either with Markovian jump linear systems (MJLSs), in which the dynamics of every mode is linear, or MJSs in which every mode has dynamics that is linear with additive unknown nonlinearities. These nonlinearities are assumed to be Lipschitz and norm-bounded and usually considered as fictitious uncertainties, so that the robust control The authors are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN s: yzhao5@nd.edu; vgupta2@nd.edu. Research supported by the NSF under grants CCF and ECCS framework applies [8], [14]. Despite significant work for particular MJNSs [2], [9], [13], to our best knowledge, the feedback stabilization problem for general MJNSs is still open. Even for the special case of BJNSs, the design of feedback stabilizing controllers is, in general, an open problem. Some results based on LMI approaches and T-S fuzzy models are available in [9], [24]. However, a key assumption in these approaches is that the nonlinear jump system can be approximated by T-S fuzzy models. In this paper, we use a passivity based approach to solve the feedback stabilization problem of a particular, but broad, class of BJNSs systems in which the dynamics of every individual mode is arbitrary except for the constraint that passivity indices [4] exist for the dynamics (although the dynamics need not be passive). This approach, in particular, allows us to consider dynamics that are not affine in the control input, which seems to be a major assumption in works such as [2], [13], [21]. Dissipativity, in general, and passivity, in particular, have now become an important analysis and design tool, especially for nonlinear systems. For a thorough overview of passivity, we refer the readers to [23]. Two properties are of particular interest for passive systems. First, Lyapunov function candidates can be obtained naturally from the storage functions induced by passivity. As a result, stability and stabilization problems can often be solved immediately once passivity is guaranteed. Second, passivity is preserved under negative feedback and parallel interconnections, thereby making it an effective approach for the analysis and design of largescale systems. Passivity indices provide a way to extend the passivity based framework to systems that may be non-passive. Intuitively, passivity indices provide a way to quantify the excess or shortage of passivity for a system. A lot of literature has considered the problem of obtaining these indices and both analytic and experimental methods are available [3], [26]. Of particular interest to the present paper is the work of Lin et al. [13], who studied the stabilization problem for interconnected continuous time MJNSs, with dynamics for every mode affine in the control input, by introducing a stochastic passivity concept. However, much less work has been done for discrete-time stochastic systems with regard to passivity [5], [17]. The results for continuous-time systems cannot be immediately applied for discrete-time systems since it is well known that passivity may not be preserved under sampling. We borrow the definition of stochastic passivity from Wang et al. [21], which can be viewed as a generalization of the classical passivity for discrete time MJSs. The main contribution of [21] is the generalization of the KYP lemma for discrete-time MJNSs, which gives a necessary and sufficient

2 2 condition for a MJNS to be passive. In this paper, we extend the theory of stochastic passivity by using it to solve the feedback stabilization problem of BJNSs in which passivity indices exist for each mode. We also establish the relation between the stochastic passivity concept proposed in [21] and various notions of stochastic stability. The contributions of this paper are now summarized. First, we develop the stochastic passivity concept from [21] for BJNSs, and provide the relation between stochastic passivity and various notions of stochastic stability. Second, we propose a passivity-based feedback stabilization method for the class of BJNSs for which the dynamics in every mode has passivity indices. Unlike existing methods, our approach does not require the dynamics to be affine in the control input. Finally, we address the problem of feedback stabilization of BJNSs across a communication network that introduces delays. We obtain a novel scattering transformation method for the case when both the plant and the controller are BJNSs. As opposed to the traditional scattering transformation approach, our approach also applies to the case when the plant and the controller are non-passive. Further, it does not require the mode information from both the plant and the controller in real time, which is a desirable property in the presence of communication delays. The rest of the paper is organized as follows. The problem formulation is presented in Section II, where the basic definitions of stochastic passivity and stochastic stability are also given. Section III focuses on the feedback stabilization problem of BJNSs using passivity indices. Section IV concludes the paper. The proofs are provided in the appendices. Notations: Throughout this paper, we denote random variables using boldface letters, e.g. x, and its realization as x. E[x] denotes the expectation of random variable x. For any 0 k j we use the notation x j k = ( x(k), x(k 1),..., x(j) ) to denote a finite segment of a sequence x = {x(0), x(1),... } and we omit the subscript k when it is equal to 0. The space of square-summable vector functions over Z is denoted by l 2 [0, ), and for a sequence x = {x(0), x(1),...} l 2 [0, ), we denote the norms x(k) xt (k)x(k) and x ( x T (k)x(k) ) 1 2. For a real matrix A, σ max (A) and σ min (A) denote its largest and smallest singular values, respectively. Moreover, if A = A T, then we use λ max (A) and λ min (A) to denote its largest and smallest eigenvalues, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for the stated algebraic operations. In symmetric block matrices or complex matrix expressions, we use an asterisk * to represent a term that is readily induced by symmetry. For a finite set M, we use M to denote its cardinality. Three random variables x, y and z form a Markov chain, denoted by x y z, if the random variables x and z are conditionally mutually independent when conditioned on y. II. PROBLEM FORMULATION Consider the discrete-time Bernoulli jump nonlinear system x(k 1) = f θ(k) (x(k), u(k)), y(k) = h θ(k) (x(k), u(k)), k 0, (1) where x(k) R n is the system state, u(k) R q is the control input and y(k) R q is the output. The random variable θ(k) : Z M {1, 2,..., m} is referred to as the system mode. We assume that the random process θ = {θ(k)} is a time homogeneous stationary process that consists of independent and identically distributed (i.i.d.) random variables taking values in the finite set M. The initial state x(0) is assumed to be mutually independent with θ(k) for all k Z and the input sequence u = {u(0), u(1),... } is deterministic. System (1) evolves as x(k 1) = f i (x(k), u(k)) and y(k) = h i (x(k), u(k)) when θ(k) = i. It is assumed that functions f i and h i are smooth maps and the origin (x, u ) = (0, 0) is an isolated equilibrium for all i = 1, 2,..., m. The function h i ( ), i M is assumed to be locally Lipschitz near the equilibrium, satisfying h i ( x(k) u(k) ) h i (x(k), u(k)) h i ( x(k) u(k) ), (2) for any (x(k), u(k)) in a neighborhood of (x, u ), where h i h i 0 are scalar constants. In this paper, all the results and discussions are restricted to an open set X U which is a neighbourhood near the origin (x, u ) = (0, 0) where the Lipschitz condition is satisfied for f i and h i for any i M. This is a natural restriction since, in general, only local properties are available for nonlinear systems. The following definitions are standard. Definition 1: Consider the discrete-time Bernoulli jump nonlinear system (1) and an open set X in the state space containing the origin x = 0. Let u(k) = 0 for all k Z. The equilibrium of system (1) is said to be locally stochastically stable [19], if for every initial state (x(0), θ(0)) X M, E( x 2 x(0), θ(0)) <. Definition 2: System (1) is said to be input-output l 2 stable if u l 2 [0, ), γ R and β R such that the inequality E y 2 γ 2 u 2 β holds. The infimum of γ is called the l 2 gain, denoted by γ = inf γ. The concept of passivity as defined originally pertained to single mode deterministic systems. Recently there have been attempts to generalize this definition to stochastic systems [1], [12] and MJSs, in particular [21]. We borrow the following definition from [21]. Definition 3: The discrete-time Bernoulli jump nonlinear system (1) is said to be locally stochastically passive if there exists a set of positive definite functions V (x(k), i) : X M R corresponding to every mode i M, called the storage functions, satisfying V (0, i) = 0 and α i ( x(k) ) V (x(k), i) ᾱ i ( x(k) ) where α i ( ) and ᾱ i ( ) are class K functions, such that for all k Z, all θ(k) M and all (x(k), u(k)) in a neighborhood of the equilibrium point (0, 0), E[V (x(k 1), θ(k 1)) x(k), θ(k)] V (x(k), θ(k)) y T (k)u(k). (3)

3 3 G P r 1 - u 1 G 1 G F and h i and h i have been defined in (2). Moreover, an upper bound of the l 2 gain γ can be obtained by solving the following optimization problem. min γ(a), s.t. E[γ y(θ(k))h 2 γ a y < 0, (6) i:a i>0,i M y y 2 G 2 u 2 r 2 where a = [a 1,, a m ], (b) feedback Fig. 1. Feedback stabilization of Bernoulli jump nonlinear systems, where G 1 is the plant and G 2 is the controller. Note that E[V (x(k 1), θ(k 1)) x(k), θ(k)] is a real number since (x(k), θ(k)) is a realization of the random variable (x(k), θ(k)) that belongs to the support set X M. Further, the output y(k) is a deterministic vector once (x(k), θ(k)) is observed. Finally, by assumption, the input u(k) is deterministic. When the BJNS has only one mode, (3) recovers the traditional definition of passivity [7]. Due to the stochasticity inherent in the system dynamics, instead of requiring the increase of the storage function to be upper bounded by the supply rate y T (k)u(k), we add the expectation operator to the storage function with respect to the system mode. The implication is that the average increase of the storage function is now upper bounded by the supply rate y T (k)u(k). We can also generalize the concept of passivity indices from deterministic systems to BJNSs. Definition 4: System (1) is said to have mode dependent passivity indices (ν i, ρ i ) if there exists a set of storage functions V (x(k), i) : X M R such that for all k Z, θ(k) M and all (x(k), u(k)) in a neighborhood of the equilibrium point, E[V (x(k 1), θ(k 1)) x(k), θ(k)] V (x(k), θ(k)) (1 ρ θ(k) ν θ(k) )y T (k)u(k) ρ θ(k) y T (k)y(k) ν θ(k) u T (k)u(k). (4) In the sequel, with a slight abuse of notation, we use passivity indices (ν i, ρ i ) to refer to the entire set of mode dependent passivity indices {(ν i, ρ i ) i M}. Note that passivity indices may exist even for a non-passive system. III. FEEDBACK STABILIZATION OF BERNOULLI JUMP NONLINEAR SYSTEMS WITH KNOWN PASSIVITY INDICES In this section, we propose a method to design a feedback controller that ensures input-output l 2 stability of the closedloop system if the passivity indices of the BJNS are known. We first present the following result. Theorem 1: Suppose that the discrete-time Bernoulli jump nonlinear system (1) has passivity indices (ν i, ρ i ). Then the system is input-output l 2 stable if where h 2 ρ(i) = E[ρ θ(k) h 2 ρ > 0, (5) { h2 i, if ρ i 0, h 2 i, if ρ i > 0, i M, γ(a) = ( E[γ u E h 2 θ(k) ) 1 2 E[γ y (θ(k))h 2, γ y { h2 h 2 γ y (i) = i, if γ y (i) 0, h 2 i, if γ y(i) < 0, γ u (i) = ν i 0.5a 1 i 1 ρ i ν i, γ y (i) = ρ i 0.5a i 1 ρ i ν i, i M. Proof. See Appendix V-A. Remark 1: The condition for input-output l 2 stability of a one mode deterministic system with passivity indices (ρ, ν) requires ρ > 0 while ν can be arbitrary [4, Equation (2.85)]. In contrast, a Bernoulli jump nonlinear system that is inputoutput l 2 stable need not contain modes that satisfy ρ i > 0 for all i M. Thus, the dynamics of some of the modes may be input-output unstable. Theorem 1 provides a sufficient condition to guarantee the input-output stability of a Bernoulli system with both stable and unstable modes. Remark 2: Consider the special case where the output of system (1) is independent of the system mode θ, i.e., x(k 1) = f θ(k) (x(k), u(k)), y(k) = h(x(k), u(k)). Then, the condition in (5) implies that the system is inputoutput l 2 stable if E[ρ θ(k) ] > 0. Based on Theorem 1, we are now ready to present a passivity-index based controller design method for BJNSs which ensures the input-output l 2 stability of the closed-loop system. Theorem 2: Consider the feedback interconnection of two discrete-time BJNSs with passivity indices (ν 1,i, ρ 1,i ) (i M 1 ={1, 2,..., m 1 }) and (ν 2,j, ρ 2,j ) (j M 2 ={1, 2,..., m 2 }), respectively, as shown in Fig. 1. Assume that the random process θ =(θ 1, θ 2 ) is Bernoulli. The interconnection is input-output l 2 stable if E[λ max (Q θ(k) )h 2 Q < 0, (7) where for θ(k) = (i, j) M 1 M 2, h 2 Q(i, j) = { h2 (i,j), if λ max (Q (i,j) ) 0, h 2 (i,j), if λ max (Q (i,j) ) < 0. Furthermore, an upper bound on the l 2 gain can be obtained by solving the following problem min γ(a), s.t. E[γ y(θ(k))h 2 γ a (1,1),a (1,2),...,a y < 0, (m1,m 2 )>0 (8)

4 4 where G 2 υ j I γ(a) = ( E[γ r E h 2 θ(k) ) 1 2 E[γ y (θ(k))h 2, γ y { h2 (i,j), if γ y (i, j) 0, h 2 γ y (i, j) = h 2 (i,j), if γ y (i, j) < 0, γ r (i, j) = λ max (R (i,j) ) 0.5a 1 (i,j) σ max(s (i,j) ), γ y (i, j) = λ max (Q (i,j) ) 0.5a (i,j) σ max (S (i,j) ), (9) and [ (ρ Q (i,j) = 1,i ν 2,j )I q 1 2 (ρ 2,jν 2,j ρ 1,i ν 1,i )I q (ρ 2,j ν 1,i )I q [ ] (1 ρ1,i ν S (i,j) = 1,i )I q 2ν 1,i I q, 2v 2,j I q (1 ρ 2,j ν 2,j )I q [ ] ν1,i I R (i,j) = q 0. 0 ν 2,j I q Proof. See Appendix V-B. Theorem 2 can be used to design a state feedback stabilizing controller. In particular, system G 1 can be viewed as the plant to be controlled, and system G 2 as the controller to be designed. Suppose that the plant G 1 is unstable. Using Theorem 2, we can obtain passivity indices of the controller G 2 such that inequality (7) is satisfied, which guarantees that the closed-loop system is input-output l 2 stable with a guaranteed upper bound on the l 2 gain. We provide one possible choice of passivity indices for the controller G 2, which is amenable to a readily realizable structure as shown later. For simplicity, assume that θ 2 (k) = θ 1 (k) for all k Z, i.e., the controller switches its mode synchronously with the mode of the plant. In this case, Q (i,j) = Q (i,i) Q i for any (i, j). Further, the constraint Q i < 0 identifies the set of passivity indices ( ν i, ρ i ) allowed for the controller G 2 given the plant G 1 s passivity indices (ν 1,i, ρ 1,i ). Denote this set as Λ(ν 1,i, ρ 1,i ) {( ν i, ρ i ) R 2 Q i < 0}. Note that Λ(ν 1,i, ρ 1,i ) may be empty for certain combinations of (ν 1,i, ρ 1,i ). Our controller design assumes that there exists at least one index i 0 M such that Λ(ν 1,i0, ρ i0 ) is nonempty. We let ν 2,j = ρ 1,i and ρ 2,j = ν 1,i for all i M, i i 0, and select any arbitrary element from this set Λ( ν i0, ρ i0 ) as our choice of the controller s passivity indices ( ν i0, ρ i0 ). This particular design of passivity indices for G 2 ensures that Q i = 0 for all i M, i i 0 and Q i0 < 0. As a result, inequality (7) must hold for all modes. Remark 3: Note that the above method assumed θ 2 (k) = θ 1 (k) for all k Z. However, Theorem 2 is more general. If we assume that the intersection set Λ i M Λ(ν 1,i, ρ 1,i ) is not empty, then each element of Λ, denoted by ( ν, ρ), defines a mode-independent controller with passivity indices ( ν, ρ) that feedback stabilizes the BJNS G 1. In this case, the controller does not need to know the mode of the plant in real time. Many structures are possible that achieve the passivity indices (ν 2,j, ρ 2,j ) of the controller G 2 obtained via Theorem 2 and the discussion above. For instance, we propose the controller design method with the configuration depicted in Fig. 2. In Fig. 2, the system H can be any passive, linear ], ɶu ɶy 1 (1 ρυ j j ) I H (1 ρυ) - j j I u ρ j I Fig. 2. Configuration of the controller G 2, where ρ j and ν j are its passivity indices ( ρ j ν j 1 for the sake of well-posedness). The system H can be any passive system with ũ R q as the input and ỹ R q as the output. time invariant, q-input q-output system 1. We can show that the system G 2 has passivity indices (ν 2,j, ρ 2,j ) as follows. According to [20], the storage function for a passive linear time invariant system can be written as a quadratic function of its state x h. Let W (x h (k)) = x T h (k)p x h(k) : R q R, with P > 0 denote the storage function of H. Notice that since the feedforward ν 2,j I and feedback ρ 2,j I are static, their state spaces are null, which means that W (x h (k)) remains the storage function of the overall system G 2. Since H is passive, according to definition (3), for all k Z, any x h (k) R q and any θ 2 (k) = j M 2 it holds that E[W (x h (k 1), θ 2 (k 1)) x h (k), θ 2 (k)] W (x h (k), θ 2 (k)) (a) = W (x h (k 1)) W (x h (k)) (b) ỹ T (k)ũ(k) = (y(k) ν 2,j u(k)) T (u(k) ρ 2,j y(k)) = (1 ρ 2,j ν 2,j )y T (k)u(k) ρ 2,j y(k)y T (k) ν 2,j u T (k)u(k), (10) where (a) holds because, x h (k1) is determined given x h (k) and θ 2 (k), and W (x h (k 1)) does not depend on θ 2 (k 1), (b) holds due to the passivity of H. Finally, since W (x h (k)) is also a storage function for G 2, the inequality (10) implies that G 2 has passivity indices (ν 2,j, ρ 2,j ), as desired. We now provide an example to illustrate the application of Theorem 2. Example 1: Consider a BJNS with two modes and Pr{θ(k) = 1} = Pr{θ(k) = 2} = 0.5. When θ(k) = 1, the system evolves as x 1 (k 1) = ( sin(x 2 (k)))x 1 (k), x 2 (k 1) = 1.25x 2 (k) 0.75u(k), y(k) = 1.25x 2 (k) 0.5u(k), and when θ(k) = 2 it evolves as x 1 (k 1) = 0, x 2 (k 1) = 1.5x 2 (k) 1.25u(k), y(k) = 1.25x 2 (k) u(k). First, by choosing a mode independent storage function V (x(k), θ(k)) = x 2 (k) it can be verified that the dynamics in mode 1 has global passivity indices (0, 0.5), and the dynamics in mode 2 has global passivity indices (0, 1). Thus, the plant is unstable in open-loop. For simplicity, we assume 1 Here for simplicity, we let H i = H, which means that the dynamics of H does not depend on θ 2 (k). But the design continues to hold when H has a dynamics that changes with the mode index θ 2 (k). y

5 5 that the controller switches simultaneously with the plant. The passivity indices for the controller are chosen as (1, 0.5) for mode 1 and (1.5, 0.5) for mode 2, according to the method discussed earlier. As a result, the matrices Q (i,j) in (7) in Theorem 2 become Q (1,1) = [ ] [ , Q (2,2) = which are both negative definite. Thus, the inequality (7) is satisfied for any distributions of the jumping probability. Notice that the matrices Q (1,2) and Q (2,1) need not be considered because of the assumption that the controller switches simultaneously with the plant. We then employ the controller design method depicted in Fig. 2 with ( ν 1, ρ 1 ) = (1, 0.5) and ( ν 2, ρ 2 ) = (1.5, 0.5), the system H given by x H (k 1) = 0.5x H (k) 0.2ũ H (k), ], ỹ H (k) = x H (k) 0.8ũ H (k), (11) which is passive and the corresponding closed-loop system is input-output l 2 stable. IV. CONCLUSION In this paper, we studied the problem of feedback stabilization for discrete time Bernoulli jump nonlinear systems using a passivity-based method. We borrowed the concept of stochastic passivity for BJNSs from [21]. First, the relation between stochastic passivity and stochastic stability was established. Then, using passivity-based techniques we proposed a controller design such that the closed-loop system is inputoutput l 2 stable. This work can be extended in many directions. More general dissipativity concepts such as QSR dissipativity, which includes passivity (indices) as a special case can be considered. Second, Markovian jump nonlinear systems may be studied where the jumping modes of the system form a Markovian process instead of a Bernoulli process. REFERENCES [1] M. Aliyu. Dissipative analysis and stability of nonlinear stochastic statedelayed systems. Nonlinear Dynamics and Systems Theory, 4(3): , [2] M. Aliyu and E. Boukas. 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6 6 ρ i ν i 1 ɛ and obtain E[γ u = E[ ν θ(k) (4ɛ) 1 1 ρ θ(k) ν θ(k) 2 ], = E[ρ θ(k) h 2 γ y ɛ E h 2 γ y (θ(k)), where any 0 < ɛ < E[ρ θ(k) h 2 γ y (E h 2 γ y (θ(k))) 1 ensures that < 0. Next, by using Definition 4 and taking expectation on both sides of (4) with respect to x(k), θ(k), we obtain that for any k Z, and any u(k) R q near the origin, E[V (x(k 1), θ(k 1)) V (x(k), θ(k))] E[(1 ρ θ(k) ν θ(k) )y T (k)u(k) ρ θ(k) y T (k)y(k) ν θ(k) u T (k)u(k)]. Sum over k ranging from 0 to N 1, E[V (x(n), θ(n)) V (x(0), θ(0))] N 1 E[(1 ρ θ(k) ν θ(k) )y T (k)u(k) ρ θ(k) y T (k)y(k) ν θ(k) u T (k)u(k)]. Let N, and use the assumption that V (, ) is a positive definite function, we obtain E[V (x(0), θ(0))] (θ(k))( x(k) u(k) ) 2 γ u (θ(k)) u(k) 2 ] (a) = E[γ u u 2 E( x(k) u(k) ) 2 (b) (E h 2 θ(k) ) 1 E y 2 E[γ u u 2, where (a) holds because γ y (θ(k))h 2 γ y (θ(k)) and ( x(k) u(k) ) 2 are mutually independent and the definition that u 2 = u(k) 2, (b) follows from the assumption that < 0. It is also worth noticing that the assumption of the mode random variables θ(k), k Z being i.i.d. makes E[γ u and time-invariant functions of a i. Moreover, notice that for every i M, γ y (i) < 0 implies that γ u (i) > 0. As a result, < 0 E[γ y < 0 E[γ u > 0. Therefore, E y 2 γ 2 (a) u 2 ( ) 1 V (x(0), θ(0)) E h 2 θ(k), which means that the system is input-output stable with the l 2 gain smaller than or equal to γ(a). B. Proof of Theorem 2 The proof is similar to that of Theorem 1. First, we show that if E[λ max (Q θ(k) )h 2 Q < 0, then there exist a i s such that < 0. It is obvious that E[λ max (Q θ(k) )h 2 γ y E[λ max (Q θ(k) )h 2 Q < 0 by the definition of h 2 Q (θ(k)) and h2 γ y (θ(k)). Moreover, we let a i = 2(σ max (S i )) 1 ɛ and obtain E[γ r = E[λ max (R θ(k) ) 4 1 ɛ 1 σ 2 max(s θ(k) )], = E[λ max (Q θ(k) )h 2 γ y ɛ E h 2 γ y (θ(k)), and 0 < ɛ < E[λ max (Q θ(k) )h 2 γ y (E h 2 γ y (θ(k))) 1 ensures that < 0. According to Definition 4, there exist positive definite functions V i (x i (k), j), such that E[V i (x i (k 1), θ i (k 1)) x i (k), θ i (k)] V i (x i (k), θ i (k)) (1 ρ θi(k)ν θi(k))y T i (k)u i (k) ρ θi(k)y T i (k)y i (k) ν θi(k)u T i (k)u i (k). Then, by defining x [x T 1 x T 2 ] T, y [y T 1 y T 2 ] T, r [r T 1 r T 2 ] T, V (x(k), θ(k)) = V 1 (x 1 (k), θ 1 (k)) V 2 (x 2 (k), θ 2 (k)), and using the signal relations due to interconnection, which is u 1 = r 1 y 2, u 2 = r 2 y 1, we have for any x(k) near the origin and any θ(k) M 1 M 2, E[V (x(k 1), θ(k 1)) x(k), θ(k)] V (x(k), θ(k)) y T (k)q θ(k) y(k) r T (k)s θ(k) y(k) r T (k)r θ(k) r(k) γ y (θ(k))h 2 γ y (θ(k))( x(k) r(k) ) 2 γ r (θ(k)) r(k) 2. Take expectation on both sides with respect to x(k), θ(k), and sum over k ranging from 0 to N 1, letting N, V (x(0), θ(0)) (θ(k))( x(k) r(k) ) 2 (a) = γ r (θ(k)) r(k) 2 ] E[( x(k) r(k) ) 2 ] E[γ r r(k) 2 (b) (E h 2 θ(k) ) 1 E y 2 E[γ r r(k) 2, where (a) holds because γ y (θ(k))h 2 γ y (θ(k)) and ( x(k) r(k) ) 2 are mutually independent, (b) follows from the assumption that < 0. Therefore, E y 2 γ 2 (a) r 2 ( ) 1 V (x(0), θ(0)) E h 2 θ(k), which means that the closed-loop system with input r and output y is input-output stable with the l 2 gain smaller than or equal to γ(a).