Reachable Set Analysis for Dynamic Neural Networks with Polytopic Uncertainties

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1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 5, May 15, 2012 Reachable Set Analysis for Dynamic Neural Networks with Polytopic Uncertainties ZUO Zhi-Qiang ( ãö), CHEN Yin-Ping (í ), an WANG Yi-Jing ( è ) Tianjin Key Laboratory of Process Measurement an Control, School of Electrical Engineering an Automation, Tianjin University, Tianjin , China (Receive November 7, 2011; revise manuscript receive December 22, 2011) Abstract n this paper, the reachable set estimation problem is stuie for a class of ynamic neural networks subject to polytopic uncertainties. The problem aresse here is to fin a set as small as possible to boun the states starting from the origin by inputs with peak values. The maximal Lyapunov functional is propose to erive a sufficient conition for the existence of a non-ellipsoial boun to estimate the states of neural networks. t is theoretically shown that this metho is superior to the traitional one base on the common Lyapunov function. Finally, two examples illustrate the avantages of our propose result. PACS numbers: i Key wors: neural networks, reachable set, polytopic uncertainties, maximal Lyapunov functional 1 ntrouction Dynamic neural networks analysis has attracte the increasing attention of researchers in the past ecaes ue to the successful applications in solving pattern recognition, associative memory, optimization problem, ientification an control, static image processing. Different kins of neural networks moels have been propose an stuie. To mention a few, we refer to [1] [3] references therein. Up to now, most works on ynamic neural networks have focuse on the stability analysis problem an a lot of results have been reporte for ifferent kins of neural networks, see, for example Refs. [4] [10]. Since neural networks can be viewe as a special class of nonlinear ynamic systems, many concepts originally appeare in control theory have been introuce to analyze the neural networks, such as, controllability, [11 12] state estimation, [13 15] input-tostate stability (SS), [16 17] passivity, [18 19] an so on. On the other han, uncertainties are inevitably encountere in practice. Generally speaking, there are mainly two kins of uncertainties, namely, the normboune uncertainty an the polytopic ones. t is well known the existence of uncertainty may lea to instability of neural networks. Therefore, much attention is evote to robust stability analysis of neural networks, for instance Refs. [20] [22]. As we known, the common Lyapunov function can lea to easily tractable matrix inequalities or linear matrix inequalities conitions. However, it probably yiels conservative evaluation of stability an performance. For example, systems with polytopic uncertainties may be stable without having a common Lyapunov function. As an important approach for state estimation an parameter estimation in control theory, a lot of works have been evote to the reachable set bouning, for example Refs. [23] [25]. All the above results on reachable set estimation are expresse in the form of ellipsoi. f there are polytopic uncertainties for systems, the common Lyapunov function has to be use for all the vertices of the polytope. To the best of our knowlege, there is no ocument available concerning reachable set estimation for uncertain neural networks, which motivates our stuy. n this paper, we evelop the reachable set bouning for ynamic neural networks subject to both polytopic uncertainties an boune peak inputs. We propose a maximal Lyapunov functional, which is obtaine by taking pointwise maximum over a family of Lyapunov functionals, to erive a tighter non-ellipsoial boun of the reachable sets. This metho shows great avantages over the traitional one base on common Lyapunov function. Finally, two numerical examples are provie to illustrate the usefulness of our theoretical results. Notations The notations use throughout the paper are stanar. R n enotes the n-imensional Eucliean space. The superscript T enotes matrix transposition, an an 0 enote the ientity an zero matrix with appropriate imension. The notation P > 0 (P 0) means that P is symmetric an positive efinite (positive semi-efinite). Co{ } enotes a convex hull. iag{l 1, l 2,...,l n } stans for a iagonal matrix with iagonal elements being the scalars l 1, l 2,...,l n. The symmetric terms in a symmetric matrix are enote by. Matri- Supporte by the National Natural Science Founation of China uner Grant Nos , , , , Program for New Century Excellent Talents in University uner Grant No. NCET , an the nepenent nnovation Founation of Tianjin University Corresponing author, zqzuo@tju.eu.cn c 2011 Chinese Physical Society an OP Publishing Lt

2 No. 5 Communications in Theoretical Physics 905 ces, if not explicitly state, are assume to have compatible imensions. For a matrix P > 0, enote E(P, 1) : = {x R n : x T Px 1}. For a general Lyapunov function V, enote its 1-level set as L V : = {x R n : V (x) 1}. 2 Problem Formulation an Preliminaries Consier the following ynamic neural networks ẋ(t) = Cx(t) + Af(x(t)) + J(t), (1) where x(t) = [x 1 (t) x 2 (t) x n (t)] T R n is the state vector of the neural networks. C = iag{c 1, c 2,...,c n } is a constant iagonal matrix with c i > 0; f(x(t)) = [f 1 (x 1 (t)), f 2 (x 2 (t)),..., f n (x n (t))] T R n is the activation of neurons. A = (a ij ) n n is the connection weight matrix; J(t) represents external inputs with the form J T (t)j(t). (2) Furthermore, the activation function f i ( ) (i = 1, 2,...,n) is assume to satisfy the following conition 0 f i(s 1 ) f i (s 2 ) s 1 s 2 l i, (3) for any s 1, s 2 R, s 1 s 2, where l i > 0 (i = 1, 2,...,n) are positive scalars. Matrices C, A are uncertain but belong to a given polytope Φ = [C A], (4) Φ = θ i Φ i, θ i 0, θ i = 1, (5) where the N vertices of the polytope are escribe by Φ i = [C i A i ], (i = 1, 2,..., N). By (3), it is easy to see f j (x j (t))[f j (x j (t)) l j x j (t)] 0 (j = 1, 2,..., n). (6) From the above inequality (6), for any non-negative iagonal matrix H = iag{h 1, h 2,..., h n }, the following conition hols n 2 h j f j (x j (t))[l j x j (t) f j (x j (t))] 0, j=1 which can be rewritten as 2[x T (t)lhf(x(t)) f T (x(t))hf(x(t))] 0. (7) n this paper, we will construct a function which is the maximum of a family of Lyapunov functions. Each Lyapunov function correspons to a vertex of the polytope (i.e., Φ i ). Given a family of matrices P j > 0, (j = 1, 2,..., N), the pointwise maximum quaratic function is efine as V max = max{x T P j x}. The 1-level set of V max is enote as L Vmax : = {x R n : V max 1}. Since V max is strict convex an not ifferentiable everywhere, we introuce the efinitions of subgraient an subifferential to characterize the behavior at nonifferentiable points. Let g : R n R be a real-value convex function in R n, a vector v is calle a subgraient at a point x 0 if g(x) g(x 0 ) v T (x x 0 ), x R n. The set of all subgraients at x 0 is calle the subifferential at x 0 an is enote as f(x 0 ). The function f(x) is ifferentiable at x 0 if an only if the subifferential is mae up of only one point, which is the erivative at x 0. Consier x 0 R n. Suppose that there exists an integer s (1 < s N) such that V max (x 0 ) = x T 0 P jx 0 for j = 1,...,s an V max (x 0 ) > x T 0 P j x 0 for j > s. Base on the results in [26], we have the following properties: (i) for a vector ζ R n, the irectional erivative of V max at x 0 along ζ is ζ V max (x 0 ) = lim t 0 + V max (x 0 + tζ) V max (x 0 ) t = max ξ V max(x 0) {ξt ζ} ; (ii) V max (x 0 ) = Co{2P j x 0, j = 1,...,s}. The following lemma is an extension of the reachable set estimation for ynamic systems to that for neural networks (1). Lemma [27] Let V be a Lyapunov function for neural networks (1). f V (x) + αv (x) α J T (t)j(t) 0, t then we have V (x) 1. 3 Main Results Theorem 1 f there exist matrices P j > 0 with compatible imensions, a non-negative iagonal matrix H, an scalars α > 0, β ij 0 for all i, j = 1, 2,..., N such that Γ = Γ 11 P j A i + LH P j 2H 0 α Γ 11 = P j C i Ci T P j + αp j + β ij (P j P i ), < 0, (8) then for all the amissible polytopic uncertainties, the intersection of a family of ellipsois, i.e., E(P j, 1) (9) j=1, 2,..., N is a set which bouns the reachable set of neural networks (1), where E(P j, 1) = {x R n : x T P j x 1}. Proof Given a family of matrices P j > 0, the pointwise maximum Lyapunov functional is chosen as where V max (x) = max{v j (x)}, j = 1, 2,...,N, (10) For x 0, we efine V j (x) = x T (t)p j x(t). N max (x) : = {j {1, 2,..., N} : V j (x) = V max (x)}.

3 906 Communications in Theoretical Physics Vol. 57 t follows that V j (x) < V max if j / N max (x). Without loss of generality, we assume that N max (x) = {1, 2,..., n 1 } for some integer n 1 N. Here n 1 is the number of ellipsois E(P j, 1) s intersecte at x. Therefore, V j (x) = V max (x) for j n 1 an V j (x) < V max (x) for j > n 1. Now we know x T (P j P i )x 0, j n 1, i {1, 2,..., N}. (11) To better illustrate the thoughts of our proof, we efine a set Υ j = {x R n : V j (x) V i (x), i j}, an consier two ifferent cases for x. (i) f x Υ j \ i =j Υ i (i.e., x is ifferentiable), then where V max = x T (t)p j x(t), t V max + αv max α J T (t)j(t) ζ T (t) ζ T (t) = [ x T (t) f T (x(t)) J T (t) ]. f (8) is satisfie, then using convexity property, we have: t V max + αv max α J T (t)j(t) β ij x T (P i P j )x 0. The last inequality hols because x T (P i P j )x 0 if x Υ j \ i =j Υ i. This implies that V max 1 by Lemma 1. (ii) f x n1 j=1 Υ j\ N j=n Υ 1+1 j, it is clear that x belongs to the intersection of Υ j s (j = 1, 2,...,n 1). This means that x is nonifferentiable. As a result, V max = x T P j x, j N max (x) an V max = Co{2P j x : j N max (x)}. Therefore, provie that (8) is met, we obtain ẋv max (x) + αv max (x) α J T (t)j(t) max β ij x T (P i P j )x. j N max(x) Note that (11) is satisfie in such a case. Therefore we can raw the conclusion that V max 1. t is easy to see that L Vmax is the intersection of the ellipsois E(P j, 1) s. This point can be verifie as follows. On the one han, if x belongs to the intersection of E(P j, 1) s, then x belongs to any ellipsoi E(P j, 1), j = 1, 2,..., N. Therefore, x belongs to L Vmax. On the other han, we suppose that x L Vmax. Since V max = max{x T P j x}, it is obvious that x T P j x 1 for all j = 1, 2,..., N. This implies that x belongs to the intersection of E(P j, 1) s. Remark 1 We have constructe a function which is the maximum of a family of Lyapunov functions in Theorem 1. t V max = 2 x T (t)p j ẋ(t) Then we have = 2 x(t) T P j [ Cx(t) + Af(x(t)) + J(t)] 2x T (t)p j [ Cx(t) + Af(x(t)) + J(t)] + 2 [ x T (t)lhf(x(t)) f T (x(t))hf(x(t))]. (12) P j C C T P j + αp j P j A + LH P j 2H 0 α ζ(t), (13) t is well known that the common Lyapunov function will lea to easily tractable matrix inequality conitions an simplify computational issues even though it may bring conservatism to some extent. Remark 2 The neural networks with polytopic uncertainties of the form (4) an (5) is consiere in this paper. Both the maximum Lyapunov functional propose in (11) an the common Lyapunov function can be use to boun all the states starting from the origin. As for the robust boune uncertainties (i.e. norm-boune uncertainties), the neural networks can be presente by ẋ(t) = [C + C(t)]x(t) + [A + A(t)]f(x(t)) + J(t) with the parameter uncertainties C(t), A(t) which are normboune an satisfy [ C(t) A(t)] = DF(t)[E 1 E 2 ], where D, E 1, an E 2 are known constant real matrices an F(t) is an unknown matrix function with Lebesgue measurable elements an such that F T (t)f(t). For this case, we can only apply the common Lyapunov function to erive the reachable set of the networks. By setting P j = P, β ij = 0, (i, j = 1, 2,...,N), we can easily obtain the ellipsoial boun of a reachable set for neural network (1) by using the common Lyapunov function. Theorem 1 reuces to a simple form of set reachable criterion. Corollary 1 f there exists a matrix P > 0 with compatible imension, a non-negative iagonal matrix H, an a scalar α > 0 such that PC i Ci TP + αp PA i + LH P 2H 0 α < 0, (14) then for all the amissible polytopic uncertainties, the ellipsoi E(P, 1) = {x R n : x T Px 1}

4 No. 5 Communications in Theoretical Physics 907 is a set which bouns the reachable set of neural networks (1). Remark 3 As can be seen from Corollary 1, the conition obtaine by common Lyapunov function is a special case for that by maximal Lyapunov one. Therefore, we can expect that a tighter bouning set can be obtaine by using the maximal functional approach. However, the better result is achieve at the cost of a bigger computational complexity. Therefore, there is a traeoff between better performance an less computational buren for solving such a problem. n orer to fin the smallest boun for the reachable set of neural network (1), one may propose a simple optimization problem. That is, maximize δ subject to δ P j which can be transforme to the following optimization problem: minimize δ ( δ = 1/δ), [ ] δ (a) 0, s.t. P j (15) (b) (8) or (14). 4 llustrative Examples n this section, two numerical examples are provie to show the effectiveness of the metho evelope in this paper. Example 1 Consier the neural networks (1) with parameters [ ] C =, ρ [ ] A =, L = iag{3, 4}, ρ where ρ is a scalar parameter satisfying ρ 0.3 an = 2. Note that the uncertainty polytope in this case possesses N = 2 vertices with [ ] [ ] C 1 =, C 2 =, [ A 1 = ], A 2 = [ ] By solving the optimization problems (15), the resulting δ s for ifferent values of α are liste in Table 1. t is verifie that the maximal Lyapunov functional metho yiels much tighter bouns than the common Lyapunov approach. We cannot fin any feasible solution when α is larger than by using the common Lyapunov metho. The reachable set correspons to the system is epicte in Fig. 1. As we can see from the figure, the shae area compute by Theorem 1 is much tighter than the ellipsoi obtaine by the common Lyapunov metho. Table 1 The resulting δ for ifferent α of Example 1. α Corollary Theorem function. However, we can get the result of δ = by Theorem 1. Therefore, for this example, the maximal Lyapunov functional leas to a wier application range comparing with the common Lyapunov one. Fig. 1 The bouning reachable sets compute by ifferent methos for α = 0.3. Example 2 Consier the polytopic uncertain neural networks (1) with parameters: [ ] A =, ρ [ ] C =, L = iag{3, 4}, ρ where ρ 0.3, = 2. t is verifie that there is no feasible solution using the metho of common Lyapunov 5 Conclusion The reachable set bouning for neural networks in the presence of polytopic uncertainties is consiere in this paper. By utilizing the maximal Lyapunov functional approach, we have erive a sufficient conition which results in a non-ellipsoial set to boun all the states of the neural networks starting from the origin by inputs with peak values. Since there exist some nonifferentiable points, the sub-graient an sub-ifferential concepts have been introuce to characterize the behavior at non-ifferentiable points. Our results have emonstrate much flexibility an less conservatism compare with the common Lyapunov function. The simulation illustrates the valiness an merit of the propose metho. The future work in this eneavor will focus on the following issues. First, reachable set estimation for uncertain neural networks with time elay. Time elay is not

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