Nonlinear L 2 -gain analysis via a cascade

Transcription

1 9th IEEE Conference on Decision and Control December -7, Hilton Atlanta Hotel, Atlanta, GA, USA Nonlinear L -gain analysis via a cascade Peter M Dower, Huan Zhang and Christopher M Kellett Abstract A nonlinear L -gain property that generalizes the conventional (linear) notion of L -gain for finite dimensional dynamical systems is studied In particular, an optimizationbased approximation strategy that facilitates computation of gain bounds via a cascade is presented The cascade in question consists of the dynamical system under test, driven by a nonlinear energy saturation operator that limits the energy delivered to that system The main advantage of this approach is demonstrated to be improved regularity properties of the value function This is demonstrated via a specific energy saturation operator and a simple example Index Terms L -gain analysis, nonlinear systems, cascade, optimization, dynamic programming I INTRODUCTION Nonlinear L -gain [], [9] is a finite gain property that generalizes the conventional (linear) notion of L -gain (see, for example, [8]) to include nonlinear gain functions of the form commonly employed in the input-to-state stability framework [7] The motivation for this generalization is to admit the application of L -gain analysis tools of a broader class of nonlinear systems, and to facilitate the broader application of small-gain based control design tools (cf []) via the synthesis of tighter gain bounds A significant problem with the concept of nonlinear L - gain introduced lies with the computation of the nonlinear gain functions involved Bisection methods based on verification results for Hamilton-Jacobi-Bellman equations cannot be extended from the linear gain case due to a lack of ordering on general classes of nonlinear gain functions Instead, previous work involving the authors [] involves the development of an optimization-based characterization for this nonlinear L -gain property It was shown that the value function involved can in principle be used to approximate bounds for the nonlinear L -gain of a nonlinear system The main technical difficulty explored in [] concerns the development of a problem reformulation that avoids an inequality constraint present in the aforementioned optimization problem In this work, an alternative formulation is provided that is defined with respect to a particular cascade This new formulation proves to be useful in that it admits a nonlinear scaling of the value function involved, thereby PM Dower and H Zhang are with the Department of Electrical & Electronic Engineering, University of Melbourne, Melbourne, Victoria, Australia pdower,hzhang@unimelbeduau Dower and Zhang are supported by AFOSR grant FA CM Kellett is with the School of Electrical Engineering & Computer Science, University of Newcastle, Newcastle, New South Wales, Australia chriskellett@newcastleeduau Kellett is supported by ARC grant DP773, The authors acknowledge M Cantoni at the University of Melbourne for a number of helpful discussions in relation to this research ameliorating some of the more serious regularity issues observed in [] In terms of organization, Section II recalls the nonlinear L -gain property of [] along with a summary of some of the salient features of the associated optimization-based characterization of nonlinear L -gain introduced there An alternative formulation using a generic cascade is provided in Section III Section IV presents a specific cascade, along with further analysis and an example A summary of conclusions is provided in Section V Throughout the paper, R and R denote the reals and non-negative reals respectively R m denotes m-dimensional Euclidean space L (I;R m ) denotes the space of square integrable functions mapping the real interval I R to R m This space is equipped with an L -norm, denoted L(I;R m ) Specifically, w L (I;R m ) w L(I;R m ) = w(s) ds < I Here, denotes the Euclidean norm on R m Specific variants of this space include L,loc (R m ) = L ([,T];R m ), T [, ) L, (R m ) = L ([, );R m ) Note that L,loc (R m ) L, (R m ) Elements of L ([,T];R m ), T R fixed, may be extended to L, (R m ) L,loc (R m ) via an operator χ T, where χ T : L ([,T];R m ) L, (R m ), χ T [w] = w(s) s [,T], s > T Where appropriate, explicit reference to the codomain R m will be dropped In other notation, a function γ : R R is of class K if it is continuous, non-decreasing, and satisfies γ() = Function γ is of class K if it is of class K, strictly increasing, and unbounded A Class of systems II A NONLINEAR L -GAIN PROPERTY Attention is restricted to finite dimensional nonlinear systems of the form ẋ(t) = f(x(t)) + g(x(t))w(t), x() = x R n Σ : z(t) = h(x(t)) () where t R denotes time, x R n denotes the initial condition, and w : R R m, z : R R p, and x : () //$ IEEE

2 R R n denote the input, output, and state trajectories (all functions of time) respectively For convenience, also define the function l : R n R by l(x) = h(x) In order to simplify the analysis, functions f, g, and l (and consequently h) are assumed to satisfy a number of basic properties, consistent with [] (Chapter 3) In particular, it is assumed that there exist constants C,C,C 3,C < such that f C (R n ;R n ) f(x) f(y) C x y x,y R n (3) f(x) C ( + x ) x R n ( g C R n ;R n m) g(x) C 3 x R n () h C (R n ;R p ), l C (R n ;R) () l(x) l(y) C ( + x + y ) x y Note that these assumptions imply that f() = R n and h() = R p They also imply closeness of solutions in the following sense Lemma : With T R, let x(t) R n, t [,T], denote the solution of () with initial state x() = x R n, driven by an input w L [,T], where w L[,T] ω R Then, there exist constants B, B, B 3 R dependent on T and ω such that x(t) x B t + B t + B3 x t () for all t [,T] B Nonlinear L -gain property System () has nonlinear L -gain [] with transient and gain bound (β,γ) K K if ( ) z L β ( x [,T] ) + γ w L [,T] (7) for all initial conditions x R n, all inputs w L [,T], and all time horizons T A tight lower bound for all nonlinear gains γ for which property (7) holds is denoted by γ Formally, γ (s) = inf γ(s) β K st property (7) holds with transient / gain bound (β,γ) Nonlinear systems () with the nonlinear L -gain property (7) naturally define an operator on L That is, (7) = Σ : L ([,T];R m ) L ([,T];R p ) T R and all initial states x R n Given that the L -norm measures a generalized notion of energy, this is a particularly compelling property for physical systems Nonlinear systems () with the nonlinear L -gain property (7) enjoy an obvious stability property Proposition : Suppose system () is -state detectable [3] and satisfies the nonlinear L -gain property (7) Then, for all x R n, w L, (R m z L, (R ) = p ), (9) lim t x(t) = (8) Bounds for the nonlinear L -gain of system () can be characterized via a constrained optimization problem [] Consistent with the notation used in [], the corresponding infinite horizon value function W : R n R R of interest is given by W(x,s) = sup sup T w L [,T] s z L [,T] () holds, x() = x () A candidate for the nonlinear L -gain of system (), given a specific transient bound β K, is then given by γ,β(s) = sup x R n max (W(x,s) β ( x ), γ (s)) () where γ (s) = W(,s) () Lemma 3: Suppose that system () satisfies the nonlinear L -gain property (7) with transient / gain bound (β,γ) K K Then, the gain candidates γ,β and γ of (), () are zero at zero, non-decreasing, and satisfy the inequality γ (s) γ,β(s) γ(s), s R (3) III NONLINEAR L -GAIN ANALYSIS VIA A GENERIC A Concept CASCADE Bounds on the nonlinear L -gain of system () can be analysed via the cascade of Figure There, Σ : L,loc (R m ) L,loc (R p ), S ξ : L,loc (R m ) L, (R m ), in which Σ denotes the input / output operator defined by system (), and S ξ denotes an auxiliary operator parameterized by ξ R The role of this auxiliary operator is to precisely limit the energy passed from the auxiliary input v L,loc (R m ) through to the system input w L,loc (R m ) of system Σ, whilst ensuring that the dynamics of Σ can be sufficiently excited via the cascade to generate maximal output energy in characterizing the nonlinear L -gain More precisely, S ξ is to have the following properties: (I) S ξ [] = ξ R; (II) S ξ [v] = S ξ [v] = S ξ [ v] v L,loc (R m ); (III) S ξ [v] ξ L[, ) v L,loc (R m ), ξ R; (IV) For each w L, (R m ), w L[, ) < ξ, ξ R, v L,loc (R m ) such that w = S ξ [v] (V) a state-space representation for S ξ, with state ξ : R R satisfying ξ() = ξ () Given such an operator S, the maximum output energy obtainable from the cascade of Figure may be characterized in terms of the value Ŵ : Rn R R of a specific optimization problem Explicitly, Ŵ(x,ξ ) = sup sup T v L [,T] z L [,T] (), () hold, x() = x, w = S ξ χ T [v] () 3

3 z Σ w S ξ Nonlinear system Energy saturation under test w L [, ) ξ Fig System Σ cascaded with the energy saturation operator S ξ (Here, χ is the extension operator of ()) It may be noted that unlike (), the variational problem of () does not involve an explicit integral constraint on the input Instead, the role of operator S is to implement this constraint implicitly via some internal dynamics Furthermore, under a continuity assumption on Ŵ, the value functions of () and () may be shown to be equivalent, provided that operator S satisfies the properties () This facilitates nonlinear L - gain analysis of system () to be formulated in terms of the inequality constraint-free value function of () Theorem 3: Suppose there exists an operator S satisfying properties (I) (IV) of () Then, the following properties concerning the value functions W and Ŵ of () and () hold: ) Ŵ(x, ) is an even and non-decreasing function for each fixed x R n, in the sense that for all ξ ξ ; ) W and Ŵ satisfy the inequality ξ Ŵ(x,ξ ) Ŵ(x,ξ ) () Ŵ(x,ξ ) W(x, ξ ) (7) for all x R n, ξ R; 3) Continuity of Ŵ(x, ) for each x R n fixed implies that the value functions W and Ŵ of () and () are equivalent in the sense that W ( x, ξ ) = Ŵ(x,ξ ) (8) for all x R n and all ξ R Proof: Fix x R n By definition (), property (II) of () implies that for any ξ R, Ŵ(x, ξ ) = sup sup T v L [,T] (), () hold, x() = x, z = Σ S ξ χ T [ v] = sup sup T v L [,T] (), () hold, x() = x, z = Σ S ξ χ T [ v] = Ŵ(x,ξ ), where = z L [,T] That is, Ŵ(x, ) is an even function Suppose that ξ = Fix any ξ R, and let v φ ( ) = ae denote the zero input in L, (R m ) Properties (I) and (III) imply that S ξ [v φ ] = v φ = S χ T [v] for any v v L [,T], T As v φ is sub-optimal in the definition () of Ŵ(x,ξ ), Ŵ(x,ξ ) sup z L [,T] z = Σ S ξ [v φ ] T = sup sup z L [,T] z = Σ S χ T [v] T v L [,T] = Ŵ(x,), so that () holds in this special case In the definition () of W(x,), w L[,T] implies that w = v φ ae on [,T] Hence, W(x,) = sup T z L z [,T] = Σ[v φ ] That is, W(x,) = Ŵ(x,) Ŵ(x,ξ ) for all (x,ξ ) R n+ This proves (), (7), and (8) in the special case where ξ = Now suppose that ξ R \ Then, the set I ξ = δ R ξ + δ > ξ > δ defines a non-empty, bounded interval whose closure contains Fix any δ I ξ For any ǫ >, let T ǫ and v ǫ L [,T ǫ ] be such that Ŵ(x,ξ ) < z ǫ L [,T ǫ ] + ǫ, where z ǫ = Σ[w ǫ ], and property (III) of () has been applied to generate w ǫ = Sξ χ T ǫ[v ǫ ] L, (R m ), w ǫ L [,T ǫ ] ξ Consequently, T ǫ and w ǫ are suboptimal in the definition () of W(x, ξ ) That is, Ŵ(x,ξ ) ǫ < z ǫ L [,T ǫ ] W ( x, ξ ) (9) Next, let T ǫ and w ǫ L [,T] ǫ be such that w ǫ L [,T ǫ] ξ and W(x, ξ ) ǫ < z ǫ L [,T ǫ], where z ǫ = Σ[ w ], ǫ and w ǫ = χ T ǫ [w] ǫ L, (R m ) Note that w ǫ L[, ) = w ǫ L[,T ǫ] ξ < ξ + δ, where the last inequality follows by definition of δ I ξ Property (IV) of () then implies that there exists a v ǫ L, (R m ) such that w ǫ S ξ +δ [ v ] ǫ As v ǫ (restricted to [,T]) ǫ and T, ǫ are sub-optimal in the definition () of Ŵ (x,ξ + δ), W(x, ξ ) ǫ z ǫ L [,T ǫ ] Ŵ (x,ξ + δ) () Combining inequalities (9) and () implies that Ŵ(x,ξ ) ǫ < W(x, ξ ) Ŵ (x,ξ + δ) + ǫ As ǫ > is arbitrary, Ŵ(x,ξ ) W(x, ξ ) Ŵ (x,ξ + δ) () As ξ < ξ +δ and δ I ξ is arbitrary, () clearly holds locally Repeated applications on overlapping intervals I, R, generalizes this to a global result This proves inequalities () and (7) Furthermore, continuity of Ŵ(x, ), inequality (), and the fact that I ξ implies that (8) holds This completes the proof Remark 3: Theorem 3 is the key result required for the development of the approximation strategy for nonlinear L -gain bounds such as () and () presented in this paper This strategy differs from that developed in [] in that the value function computed there was W(x,s) Ŵ (x, s)

4 Regularity issues associated with the s dependence were clearly evident in the explicit examples presented in that work By computing Ŵ instead, it is demonstrated later (see Section IV-C) that these issues can largely be avoided It is also noted that W(x,s) R is only defined for (x,s) R n R, limiting the application of other approximation approaches such as max-plus methods [], [] This is not the case for Ŵ(x,ξ), which is defined for all (x,ξ) Rn+ B Dynamic programming principle A standard application of dynamic programming yields a dynamic programming principle for the optimization problem defined by Ŵ of () Theorem 33: The value function Ŵ of () satisfies the dynamic programming principle (DPP) z L Ŵ (x,ξ ) = sup + (), () hold [,τ] v L [,τ] Ŵ (x(τ),ξ(τ)) x() = x ξ() = ξ () for all τ R, x R n, and ξ R Proof: The proof follows the same argument as that of Lemma 3 of [] The details are omitted C Stability of the worst case dynamics The DPP () can be employed to gain an understanding of how the output energy of system () is maximized In particular, it is possible to determine how the worst case dynamics associated with the value function Ŵ of () evolve in order to maximize the output energy Here, it is shown that near-optimal trajectories of system () always converge to zero To show this, a number of Lemmas are first presented, based on the assumption that Ŵ of () is proper in the following sense: Ŵ is continuous, Ŵ(,) =, with Ŵ(x,ξ) > (x,ξ) Rn+, Ŵ(,ξ) Ŵ(x,ξ) (x,ξ) Rn+, α K st α( x ) Ŵ(x,) x Rn (3) Lemma 3: Suppose that Ŵ is proper in the sense of (3), and that (x,ξ ) R n+ is fixed Then, given any δ R >, there exists an ǫ R > such that for any ǫ- optimal time horizon T ǫ R and input v ǫ L [,T ǫ ], ǫ (,ǫ ], max ( x ǫ (T ǫ ), ξ ǫ (T ǫ ) ) < δ, () in which x ǫ denotes the trajectory of system () driven by input S ξ χ T ǫ[v ǫ ] L, [, ) α(δ),ŵ(,δ) Proof: Fix any δ R > Select ǫ = min >, where positivity follows from α K and properness of Ŵ For any ǫ (,ǫ ], let T ǫ R and v ǫ L [,T ǫ ] be such that z ǫ L [,T ǫ ] > Ŵ(x,ξ ) ǫ Ŵ(x,ξ ) ǫ As T ǫ and v ǫ are suboptimal in the DPP (), Ŵ(x,ξ ) z ǫ L [,T ǫ ] + Ŵ(x ǫ (T ǫ ),ξ ǫ (T ǫ )) Combining these two inequalities yields that Ŵ(xǫ (T ǫ ),ξ ǫ (T ǫ )) ǫ Using the definition of ǫ, Theorem 3 (in particular, equation ()), and properness of Ŵἀ(δ) ǫ Ŵ(xǫ (T ǫ ),ξ ǫ (T ǫ )) Ŵ(xǫ (T ǫ ),) α( x ǫ (T ǫ ) ), so that x ǫ (T ǫ ) δ Similarly, Ŵ(,δ) ǫ Ŵ(x ǫ (T ǫ ),ξ ǫ (T ǫ )) Ŵ(,ξǫ (T ǫ )), so that ξ ǫ (T ǫ ) δ As ǫ (,ǫ ] is arbitrary, the result follows Lemma 3: Given (x,ξ ) R n+, let T ǫ denote the set T ǫ = T R > Ŵ(x,ξ ) ǫ < sup v L [,T] z L [,T] of all ǫ-optimal time horizons in the definition of Ŵ(x,ξ ) Let T : R > R denote the function T (ǫ) = inf T ǫ, () Then, ) Any ǫ-optimal time horizon can be made arbitrarily longer, so that T T ǫ = [T, ) T ǫ () ) T ( ) is non-increasing; 3) Ŵ proper, x, implies that there exists ǫ, T R > such that T (ǫ) T, ǫ (,ǫ ]; (7) Proof: Fix (x,ξ ) R n+ Assertion ): Suppose T T ǫ Then for any δ R, Ŵ(x,ξ ) ǫ < sup z L [,T] v L [,T] sup z L [,T+δ] v L [,T+δ] That is, T +δ T ǫ, where δ R is arbitrary Hence, () holds Assertion ): Fix any ˆǫ R >, ǫ (, ˆǫ] Then, any time horizon that is ǫ-optimal is by definition also ˆǫ-optimal That is, T ǫ Tˆǫ, so that inf T ǫ inf Tˆǫ, as required Assertion 3): Fix x and suppose that the assertion is false Then, for every ǫ, T R >, there exists an ǫ (,ǫ ] such that T (ǫ) < T In particular, select sequences ˆǫ n, ˆT n R >, n Z >, such that ˆǫ n and ˆT n as n Then, there exists ǫ n (, ˆǫ n ] such that T (ǫ n ) < ˆT n, n Z > That is, there exists an ǫ n -optimal time horizon T ǫn [, ˆT n ] and input v ǫn L [,T ǫn ] such that z ǫn L [,T ǫn] + ǫ n > Ŵ(x,ξ ) z ǫn L [,T ǫn] + Ŵ( xǫn (T ǫn ), ξ ǫn (T ǫn )), where ( x ǫn ( ), ξ ǫn ( ) ) R n+ denotes the trajectory of the cascade Σ x S ξ, driven by the input v ǫn = χ T ǫn[v ǫn ], and z ǫn denotes the corresponding output Consequently, Ŵ ( x ǫn (T ǫn ),) Ŵ ( x ǫn (T ǫn ), ξ ǫn (T ǫn ) ) < ǫ n ˆǫ n, (8)

5 where the left-hand inequality comes from Theorem 3 Define w ǫn = Sξ [ v ǫn ], and note that by property (III) of (), w ǫn ξ L[, ) Consequently, Lemma implies that given any δ R >, there exists a N Z > such that x ǫn (T ǫn ) x δ n Z >N That is, lim n x ǫn (T ǫn ) = x Combining this limit with inequality (8), the definition of sequence ǫ n, and continuity of Ŵ implies that W(x,) =, which by properness (3) yields that x =, which is a contradiction Theorem 3: Assume that Ŵ is proper in the sense of (3) Given (x,ξ ) R n+, suppose that an optimal input v L,loc (R m ) exists for Ŵ of () Then, the corresponding trajectory denoted by (x ( ),ξ ( )) : R R n+ is stable in the sense that lim t x (t) = = lim ξ (t) (9) t Proof: Fix (x,ξ ) R n+ By Lemma 3, T ( ) of () is a non-increasing function, which is either bounded above, or not, as ǫ + First suppose that it is That is, there exists T R > such that T (ǫ) T for all ǫ R >, and lim ǫ + T (ǫ) = T < Hence, () of Lemma 3 implies that T + δ T ǫ for any ǫ, δ R > That is, Ŵ(x,ξ ) ǫ < z ǫ L [,T +δ], where z ǫ = Σx S ξ χ T +δ[v ǫ ], with v ǫ L [,T +δ] denoting the corresponding ǫ-optimal input As T +δ and v ǫ are also sub-optimal in the DPP (), Ŵ(x,ξ ) z ǫ L [,T +δ] + Ŵ(x ǫ (T + δ),ξ ǫ (T + δ)) Hence, combining the above two inequalities, Ŵ(x ǫ (T + δ),ξ ǫ (T + δ)) < ǫ As ǫ, δ R > are arbitrary, properness (3) of Ŵ, and the assumed existence of the optimal trajectory, implies that (x (T ),ξ (T )) = R n+ Assumptions (3) and () together with property (III) of () then imply that (x (t),ξ (t)) = R n+, t T, which proves (9) in the case of bounded T Now suppose that T is not bounded, so that T (ǫ) as ǫ + Then, Lemma 3 and () imply that for any δ R >, δ lim T max ( x ǫ (T), ξ ǫ (T) ) As ǫ, δ R > are arbitrary, the result (9) again follows D A sufficient condition for continuity of Ŵ(x, ) In the general statement of Theorem 3, continuity of the function Ŵ(x, ) is required for each fixed x R n, where Ŵ is the value function of () This property may be demonstrated in the special case where specific incremental gain properties hold for both the system Σ of () and the cascaded operator S of Figure These properties are shown to be sufficient to demonstrate continuity of the gain bound γ of () Lemma 37: Suppose that system Σ of () and the cascaded operator S of Figure satisfy respectively the incremental nonlinear gain properties ) Σ x [w ] Σ x [w ] L γ [,T] ( w w L [,T] ) S ξ [v ] S ξ [v ] L β [,T] ( v v L [,T] for all w,, v, L [,T], T, where β, γ K Suppose additionally that system Σ satisfies the nonlinear L -gain property (7) with transient / gain bound pair in K K Then, the value function Ŵ of () satisfies the property that Ŵ(x, ) : R R defines a continuous function for any fixed x R n The proof relies on closeness of solutions [] (Lemma 3) and Theorem 3 The details are omitted for brevity Lemma 37 may be applied in demonstrating that the gain candidate γ is continuous Corollary 38: Suppose that the assumptions of Lemma 37 hold Then, gain bound γ of () satisfies γ K IV NONLINEAR L -GAIN ANALYSIS VIA A SPECIFIC CASCADE Further insight into the analysis problem of the preceding section can be obtained by considering a specific energy saturation operator S to be cascaded with system () A An explicit energy saturation operator S A candidate energy saturation operator is denoted by S ξ : L,loc (R m ) L, (R m ), ξ R, where ξ(t) = ξ(t) v(t), ξ() = ξ S ξ : w(t) = (3) ξ(t)v(t) in which ξ(t) R, v(t), w(t) R m denote respectively the state, input, and output of the nonlinear dynamics of operator S ξ, with initial state ξ R Theorem : The operator S of (3) satisfies properties (I) (V) of () B Dynamic programming equation The dynamic programming principle () may be employed to derive a dynamic programming equation (DPE) This DPE takes the form of a Hamilton-Jacobi-Bellman style PDE In particular, = H (x,ξ, x W(x,ξ), ξ W(x,ξ)) (3) for all (x,ξ) R n+, where H(x,ξ,p,q) = h(x) p,f(x) sup p, ξ g(x)v q ξ v (3) v R m For the specific cascade constructed using (3), it may be shown that the value function Ŵ of () is a viscosity solution of the DPE (3), provided suitable assumptions on system () and the regularity of Ŵ hold Uniqueness of this viscosity solution has yet to be demonstrated However, for the purpose of illustrative computation in the remainder of this paper, uniqueness of this viscosity solution is assumed This facilitates the approximation of Ŵ via the application of standard numerical methods to (3) Here, an approximating Markov chain method [] is employed

6 C Example c (x, ξ) W x Fig 3 ξ c for system (33) Infinite horizon value function W V C ONCLUSIONS x(t) ξ(t) * w (t) Ew(t) Ez(t) Time Fig Optimal trajectory and input for system (33) 7 γ (s) An optimization approach to the characterization of a nonlinear L -gain property was presented Key to this approach was the introduction of a cascaded input operator that precisely limits the input energy delivered to the system under test The resulting optimization problem was studied, yielding a dynamic programming principle and stability results in the presence of worst case inputs A dynamic programming equation was proposed A numerical method was employed to demonstrate the application of this approach to a simple example Trajectory, optimal input, and energies Consider a scalar nonlinear system () with f (x) = x + φ(x), g(x) =, h(x) = x, (33) where φ(x) = x x + x, and x, w R In the absence of inputs, the nonlinearity φ decreases the rate of convergence of the state towards the origin when x > It is expected that the tight lower bound γ of (8) should satisfy γ (s) s, s > Figure illustrates the computed infinite horizon value function W (x, s), x [, ], s [, 8], using the approach of [] For small s >, the s behaviour alluded to in Remark 3 is evident for x = Recall that W (, ξ) = for ξ R< Figure 3 illustrates c (x, ξ), x [, ], ξ [, ] Observe the function W c that W (, ξ) is defined for ξ R< due to (8) Figure illustrates the optimal trajectory of system () generated as result of the application of the optimal input v defined by () The signal w shown is defined as the per Figure, ie, w = Sξ [v ] Note that the internal dynamics of the cascade are stable, as expected by Theorem 3 The obtained gain bound γ of () is illustrated in Figure 3 8 W (x, s) 7 Fig 3 x Fig s Infinite horizon value function W for system (33) R EFERENCES [] PM Dower and CM Kellett A dynamic programming approach to the approximation of nonlinear L -gain In IEEE Conference on Decision and Control (Cancun, Mexico), pages IEEE, 8 [] PM Dower and WM McEneaney A max-plus affine power method for approximation of a class of mixed l-infinity / l- value functions In Proc nd IEEE Conference on Decision & Control (Maui, Hawaii), pages IEEE, 3 [3] A Isidori Nonlinear control systems II Springer-Verlag, 999 s 8 for system (33) Gain bound γ [] ZP Jiang, AR Teel, and L Praly Small-gain theorem for ISS systems and applications Math Control Signals Systems, 7:9, 99 [] HJ Kushner and PG Dupuis Numerical methods for stochastic control problems in continuous time Applications of Mathematics: Stochastic Modelling and Applied Probability Springer-Verlag, New York, 99 [] WM McEneaney Max-plus methods for nonlinear control and estimation Systems & Control: Foundations & Application Birkhauser, [7] ED Sontag New characterizations of input to state stability IEEE Trans Automatic Control, :83 9, 99 [8] AJ van der Schaft L -gain and passivity techniques in nonlinear control, volume 8 of Lecture notes in control and information sciences Springer-Verlag, 99 [9] H Zhang and PM Dower A max-plus method for the approximation of transient bounds for systems with nonlinear L -gain In MTNS (Hungary, Budapest), 7