A Simplified Design for Strict Lyapunov Functions under Matrosov Conditions

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1 1 A Simpified Design for Strict Lyapunov Functions under Matrosov Conditions Frederic Mazenc Michae Maisoff Oivier Bernard Abstract We construct strict Lyapunov functions for broad casses of noninear systems satisfying Matrosov type conditions. Our new constructions are simper than the designs avaiabe in the iterature. We iustrate the practica interest of our designs using a gobay asymptoticay stabe bioogica mode. I. INTRODUCTION Lyapunov functions pay an essentia roe in modern noninear systems anaysis and controer design. Oftentimes, non-strict Lyapunov functions are readiy avaiabe. However, strict (i.e., strong) Lyapunov functions are preferabe since they can be used to quantify the effects of disturbances; see the precise definitions beow. Strict Lyapunov functions have been used in severa bioogica contexts e.g. to quantify the effects of actuator noise and other uncertainty on the steady state concentrations of competing species in chemostats [13], but their expicit construction can be chaenging. For some arge casses of systems, there are mechanisms for transforming non-strict Lyapunov functions into the required strict Lyapunov functions e.g. [4], [11], [1], [14], [15]. For systems satisfying conditions of Matrosov s type [8], [10], strict Lyapunov functions were constructed in [15], under very genera conditions. However, the generaity of the assumptions in [15] makes its constructions compicated and therefore difficut to appy. Moreover, the Lyapunov functions provided by [15] are not ocay bounded from beow by positive definite quadratic Mazenc is with Projet MERE INRIA-INRA, UMR Anayse des Systèmes et Biométrie, INRA, p. Viaa, Montpeier, France, mazenc@supagro.inra.fr, Te: 33 (0) , Fax: 33 (0) He acknowedges enightening discussions with D. Nesic about the importance of oca properties of Lyapunov functions. Maisoff is with the Dept. of Mathematics, Louisiana State University, Baton Rouge, LA USA, maisoff@su.edu, Te: (5) , Fax: (5) Maisoff was supported by NSF/DMS Grants and Bernard is with Projet COMORE INRIA Sophia-Antipois, 004 route des Lucioes, BP 93, 0690 Sophia-Antipois, France, oivier.bernard@inria.fr, Te: , Fax:

2 functions, even for asymptoticay stabe inear systems, which admit a quadratic strict Lyapunov function. The shape of Lyapunov functions, their oca properties and their simpicity matter when they are used to investigate robustness and construct feedbacks and gains. In the present work, we revisit the probem of constructing strict Lypaunov functions under Matrosov s conditions. Our resuts have the foowing desirabe features. First, they ead to simpified constructions of strict Lyapunov functions; see Remark 3 beow. For a arge famiy of systems, the Lyapunov functions we construct have the added advantage of being ocay bounded from beow by positive definite quadratic functions, with time derivatives aong the trajectories that are ocay bounded from above by negative definite quadratic functions. Second, our work does not require a non-strict positive definite radiay unbounded Lyapunov function. Rather, we ony require a non-strict positive definite function whose derivative aong the trajectories is non-positive. One of our motivations is that one can frequenty find non-strict Lyapunov-ike functions which are not proper but which make it possibe to estabish goba asymptotic stabiity of an equiibrium point. For instance, the ceebrated Lyapunov function from [5] for a muti-species chemostat (aso reported in [16]) is not proper. In such cases, the stabiity proof is often based on the fact that the modes are derived from mass baance properties [1] eading to the boundedness of the trajectories. Our work yieds robustness in the sense of input-to-state stabiity (ISS). The ISS notion is a fundamenta paradigm of noninear contro that makes it possibe to quantify the effects of uncertainty [17], [18]. Whie our assumptions are more restrictive than those used in [8], [15], they are genera in the sense that, to the best of our knowedge, they are satisfied by a exampes whose stabiity can be estabished by the generaized Matrosov s theorem; specificay, see e.g. the exampes in [15] whose auxiiary functions satisfy our Assumptions 1- beow. II. DEFINITIONS AND NOTATION We omit the arguments of our functions when they are cear. We use the standard casses of comparison functions K and KL; see [18] for their we known definitions. We aways assume that D R n is an open set for which 0 D. A function V : D R R is positive definite on D provided V (0, t) 0 and inf t V (x, t) > 0 for a x D \ {0}. A function V is negative definite provided V is positive definite. Let (resp., ) denote the standard Eucidean norm (resp., essentia supremum). We aways assume that our functions are sufficienty smooth.

3 3 Consider a genera noninear system ẋ = F(x, t, δ(t)) (1) evoving on a forward invariant open set G that is diffeomorphic to R n, with disturbances δ in the set L (C) of a measurabe essentiay bounded functions vaued in a given subset C of Eucidean space. Assume 0 G, 0 C, F C 1 with F(0, t, 0) 0, where C 1 means continuousy differentiabe. A C 1 function V : D R R is a Lyapunov-ike function for (1) with δ 0 provided V is positive definite and V (x, t) := V V (x, t) + (x, t)f(x, t, 0) 0 for a x D and t 0. If in t x addition V (x, t) is negative definite, then V is a strict Lyapunov-ike function for (1) with δ 0. A function W : D R is radiay unbounded (or proper) provided im x D, x + W (x) = +. A (strict) Lyapunov-ike function is a (strict) Lyapunov function provided it is aso proper. Let t φ(t; t o, x o, δ) denote the soution for (1) with arbitrary initia condition x(t 0 ) = x o and any δ L (C), which we aways assume to be uniquey defined on [t o, + ). Let M(G) denote the set of a continuous functions M : G [0, ) for which (A) M(0) = 0 and (B) M(x) + as x boundary(g) or x + whie remaining in G. We say that (1) is ISS on G with disturbances in C (or just ISS when G and C are cear) [17], [18] provided there exist β KL, M M(G) and γ K such that φ(t; t o, x o, δ) β(m(x o ), t t o ) + γ( δ ) for a t t o 0, x o G, and δ L (C). When G = R n and M(x) = x, this becomes the usua ISS definition. The ISS property reduces to the standard (uniformy) gobay asymptoticay stabe condition when δ 0 but is far more genera because it quantifies the effects of disturbances. A. Statement of Assumptions and Resut III. MAIN RESULT For simpicity, we first state our main resut for time invariant systems ẋ = f(x) evoving on D; see Section IV for generaizations to time-varying systems ẋ = f(x, t). We assume: Assumption 1: There exist an integer j ; functions V i : D R, N i : D [0, + ), and φ i : [0, + ) (0, + ); and constants a i (0, 1] such that (a) V i (0) = N i (0) = 0 for a i, (b) V 1 (x)f(x) N 1 (x) and V i (x)f(x) N i (x) + φ i (V 1 (x)) i 1 i =,..., j for a x D, and (c) the function V 1 is positive definite on D. =1 N a i (x)v 1 a i 1 (x) for

4 4 Assumption : (i) There exists a function ρ : [0, + ) (0, + ) such that j =1 N (x) ρ(v 1 (x))v 1 (x) for a x D. (ii) There exist functions p,..., p j : [0, + ) [0, + ) such that V i (x) p i (V 1 (x))v 1 (x) for a x D hods for i =, 3,..., j. Theorem 1: Assume that there exist j and functions satisfying Assumptions 1-. Then one can buid expicit functions k, Ω K C 1 such that S(x) = j =1 Ω (k (V 1 (x)) + V (x)) satisfies S(x) V 1 (x) and S(x)f(x) 1 4 ρ(v 1(x))V 1 (x) () for a x D. Remark 1: The proof of Theorem 1 uses the trianguar structure of the inequaities in Assumption 1(b) in an essentia way. The differences between Assumptions 1- and the assumptions from [15] are these. First, whie Assumption 1 above ensures that V 1 is positive definite but not necessariy proper, [15] requires a radiay unbounded non-strict Lyapunov function. Second, our Assumption 1 is a restrictive version of [15, Assumption ] because we specify the oca properties of the functions which correspond to the χ i s of [15, Assumption ]. Finay, our Assumption imposes reations between the functions N i and V 1, which are not required in [15]. In Section IV, we extend our resut to time-varying systems. Note that we do not require V,..., V j to be nonnegative. Remark : If D = R n and V 1 is radiay unbounded, then () impies that S is a strict Lyapunov function for ẋ = f(x). If V 1 is not radiay unbounded, then one cannot concude from Lyapunov s theorem that the origin is gobay asymptoticay stabe. However, in many cases, goba asymptotic stabiity can be proved through a Lyapunov-ike function and extra arguments. We iustrate this in Section V. If V 1 is bounded from beow by a positive definite quadratic form in a neighborhood of 0, then we get positive definite quadratic ower bounds on S (by ()) and 1 ρ(v 4 1(x))V 1 (x) near 0. B. Proof of Theorem 1 Step 1: Construction of the k i s and Ω i s. Fix j and functions satisfying Assumptions 1-. Fix k,..., k j C 1 K such that k i (s) s+p i (s)s and k i(s) 1 for a s 0 for i =, 3,..., j. Lemma 1: The functions U 1 (x) = V 1 (x) and U i (x) = k i (V 1 (x)) + V i (x) satisfy k i (V 1 (x)) U i (x) V 1 (x) for a i =,..., j and a x D. In fact, Assumption (ii) and our choices of the k i s give U i (x) V 1 (x) + p i (V 1 (x))v 1 (x) p i (V 1 (x))v 1 (x) = V 1 (x) and U i (x) k i (V 1 (x)) + p i (V 1 (x))v 1 (x) k i (V 1 (x)) for i, which

5 5 proves Lemma 1. Set k 1 (s) s. Returning to the proof of the theorem, define the functions U i according to Lemma 1. Let Ω 1,..., Ω j K C 1 be functions for which Ω i (s) 1 for a s 0 and a i, and such that Ω i(u i ) Φ(V 1 ) j =1+i Ω (U ) 1 a, where { [ ] } Φ(V 1 ) = max i=,..,j φ i (V 1 ) 1 (1 ai )/a a 4(j 1)(i 1) i i ρ(v 1 ) (3) for i = 1,,..., j. Specificay, take Ω i (p) = p 0 µ i(r)dr where the nondecreasing functions µ i : [0, ) [1, ) are from Lemma A.1 in Appendix A beow. In particuar, we take Ω j (p) p. Step : Stabiity Anaysis. Since Ω 1 (s) 1 everywhere, Ω 1(U 1 (x)) U 1 (x) = V 1 (x) everywhere. Hence, S(x) = Ω 1 (V 1 (x)) + j i= Ω i(u i (x)) satisfies the first requirement in (). To check the decay estimate in (), first note that Assumption 1(b) and our choices of the k i s give S(x)f(x) = Ω 1(U 1 ) V 1 + j i= Ω i(u i ) [k i(v 1 ) V 1 + V ] i j i=1 Ω i(u i )N i + j i= Ω i(u i ) ( φ i (V 1 ) i 1 =1 N a i aong the trajectories of ẋ = f(x). Define the positive functions Γ,..., Γ j 1)(i 1)Ω i (U i(x))φ i (V 1 (x))]/ρ(v 1 (x)). For any i for which 0 < a i j i=1 Ω i(u i ) V i ) (4) V 1 a i 1 by Γ i (x) = [4(j < 1, we can appy Young s Inequaity v 1 v v p 1 + vq with p = 1/a i, q = 1/(1 a i ), v 1 = Γ i (x) 1 a i N (x) a i, and v = {V 1 (x)/γ i (x)} 1 a i to get N (x) a i V 1 (x) 1 a i Γ i (x) (1 a i)/a i N (x)+v 1 (x)/γ i (x) for a x D. The preceding inequaity aso hods when a i = 1, so we can substitute it into (4) to get S(x)f(x) j i=1 Ω i(u i )N i + ( 1 a i ) j i= Ω a i(u i )φ i (V 1 )Γ i i 1 i =1 N + ( j i= Ω i(u i ) φ i(v 1 )(i 1) Γ i ) V 1 j i=1 Ω i(u i )N i + 1 ρ(v 4 1)V 1 + ( [ j 4(j 1)(i 1)Ω i= Ω i(u i )φ i (V 1 ) i (U i )φ i (V 1 ) ρ(v 1 ) j i=1 Ω i (U i)n i ρ(v 1)V 1 + Φ(V 1 ) j i= by our choices of the Γ i s and the formua for Φ in (3). ] 1 a i a i i 1 =1 N ) ( Ω i (U i) 1 a i i 1 =1 N Since Ω i 1 for a i, Assumption (i) gives j i=1 Ω i(u i )N i ρ(v 1 )V 1. Hence, (5) gives S(x)f(x) 1 ρ(v 4 1)V 1 1 j i=1 Ω i(u i )N i + Φ(V 1 ) ( j i= Ω i(u i ) ) 1/a i i 1 =1 N. It foows that S(x)f(x) 1 4 ρ(v 1)V 1 + j 1 i=1 [ 1 Ω i(u i ) + Φ(V 1 ) j =1+i Ω (U ) 1/a ]Ni, by an easy switching, ) (5)

6 6 of the order of summation. Since the N i s are nonnegative, () now readiy foows from (3). IV. EXTENSION TO TIME-VARYING SYSTEMS One can prove an anaog of Theorem 1 for ẋ = f(x, t), as foows. We assume that there exists R K such that f(x, t) R( x ) everywhere, and that time-varying anaogs of Assumptions 1- hod. These anaogs of Assumptions 1- are obtained by repacing their arguments x by (x, t), and V i (x)f(x) by V i (x, t) = V i t (x, t) + V i x (x, t)f(x, t), assuming N i(0, t) V i (0, t) 0. More generay, assume that these time-varying versions of Assumptions 1- hod except that the ower bound on Σ i N i is repaced by a reation of the form S j (x, t) := j =1 N (x, t) p(t)ρ(v 1 (x, t))v 1 (x, t). (6) Here ρ is again positive, and p(t) is assumed to be non-negative and admit constants B, T, p m > 0 such that T +t t p(s)ds > p m and p(t) B for a t. This aows p(t) = 0 for some t s (e.g., p(t) = cos (t) and T = π), so S j (x, t) is not necessariy positive definite. Nevertheess, we can prove an anaog of Theorem 1 in this situation, as foows. [ t ] t Set V j+1 (x, t) = p() d ds V t T s 1 (x, t). Since V 1 (x, t) 0 and p and V 1 are nonnegative, V j+1 = V 1 (x, t) t p()d + T p(t)v t T 1(x, t) + V t t 1 p()dds t T s (7) p m V 1 (x, t) + T p(t)v 1 (x, t) p m V 1 (x, t) + T S j(x,t). ρ(v 1 (x,t)) aong the trajectories of ẋ = f(x, t), where the first inequaity is by our choice of p m. Therefore, V i N i (x, t) + φ i (V 1 (x, t)) i 1 =1 N (x, t) a i V 1 (x, t) 1 a i for i j, and V j+1 N j+1 (x, t) + φ j+1 (V 1 (x, t)) j =1 N (x, t) (8) with N j+1 (x, t) = p m V 1 (x, t) and φ j+1 (V 1 (x, t)) = T. Aso, j+1 ρ(v 1 (x,t)) =1 N (x, t) p m V 1 (x, t), V 1 N 1 (x, t), and V j+1 (x, t) T BV1 (x, t). Therefore, the properties required to appy the time-varying version of Theorem 1 are satisfied by V 1, V,..., V j+1. V. APPLICATION TO A REAL BIOTECHNOLOGICAL SYSTEM Severa adaptive contro probems for bioreactors have been soved [1], [6], [9]. The proofs in these works construct non-strict Lyapunov functions. Here we use Theorem 1 to construct a strict

7 7 Lyapunov-ike function for the system and corresponding adaptive controer in [9]. We then use our strict Lyapunov-ike function to quantify the robustness of the controer to uncertainty. Consider an experimenta anaerobic digester used to treat waste water [3], [9], [19]. This process degrades a pouting organic substrate s with the anaerobic bacteria x and produces a methane fow rate y 1. The methane and substrate can generay be measured, so the system is ṡ = u(s in s) kr(s, x), ẋ = r(s, x) αux, y = (λr(s, x), s) (9) where the biomass growth rate r is any nonnegative C 1 function that admits positive functions and such that s (s, x) r(s, x) xs (s, x); (10) u is the nonnegative input (i.e. diution rate); α is a known positive rea number representing the fraction of the biomass in the iquid phase; and λ, k, and s in are positive constants representing methane production and substrate consumption yieds and the infuent substrate concentration, respectivey. Hence, y 1 = λr(s, x). We wish to reguate the variabe s to a prescribed positive rea number s (0, s in ). We assume that there are known constants γ M > γ m > 0 such that γ := k/[λ(s in s )] (γ m, γ M ) and k/[λs in ] < γ m. (11) We introduce the notation v = s in s and x = v kα. The work [9] eads to a non-strict Lyapunov-ike function and an adaptive controer for an error dynamics associated with (9). We next review these earier resuts. In Section V-A, we use them to buid a strict Lyapunov-ike function for the error dynamics, and in Section V-B we use our Lyapunov construction in a robustness anaysis. We introduce the dynamics γ = y 1 (γ γ m )(γ M γ)ν evoving on (γ m, γ M ), where ν is a function to be seected that is independent of x. With u = γy 1, the system (9) with its dynamic extension becomes ṡ = y 1 [γ(s in s) k/λ], ẋ = y 1 α[1/(αλ) γx], γ = y 1 (γ γ m )(γ M γ)ν (1) by the definition of y 1, with the same output y as before. The dynamics (1) evoves on the invariant domain E = (0, + ) (0, + ) (γ m, γ M ). The foowing is easiy checked:

8 8 Lemma : For each initia vaue (s(t 0 ), x(t 0 ), γ(t 0 )) E, there is a compact set K o (0, ) so that the corresponding soution of (1) is such that (s(t), x(t)) K o for a t t 0. It foows from Lemma and (10) that we can re-parameterize (1) in terms of τ = t t o y 1 ()d. Doing so and setting x = x x, s = s s, and γ = γ γ yieds the error dynamics s = γ s + γv, x = α [ γ x γx ], γ = (γ γ m )(γ M γ)ν (13) for t ( s, x, γ)(τ 1 (t)). The state space of (13) is D = ( s, + ) ( x, + ) (γ m γ, γ M γ ). The system (13) has an uncouped trianguar structure; i.e., its ( s, γ)-subsystem does not depend on x, and the x-subsystem is gobay input-to-state stabe with respect to γ with the ISS Lyapunov function x [18]. Therefore (13) is gobay asymptoticay stabe to 0 if and ony if the system s = γ s + γv, γ = (γ γ m )(γ M γ)ν, (14) with state space F = ( s, + ) (γ m γ, γ M γ ) is gobay asymptoticay stabe to 0. Therefore, we may imit our anaysis to (14) in the seque. For a given a tuning parameter K > 0, [9] uses the Lyapunov-ike function V 1 ( s, γ) = 1 γ m s + v γ Kγ m 0 d (15) (+γ γ m)(γ M γ ) for (14), which is positive definite on D := ( s, + ) (γ m γ, γ M γ ). Aso, V1 = 1 γ m [ γ s + s γv ] + v Kγ m γν aong the trajectories of (14). Choosing ν( s) = K s gives V 1 = γ γ m s N 1 ( s), where N 1 ( s) = s (because γ(t) (γ m, γ M ) for a t). Using the LaSae Invariance Principe [7], [9] shows that (14) is gobay asymptoticay stabe to 0 when ν( s) = K s. A. Construction of a Strict Lyapunov-Like Function for System (14) Set V ( s, γ) = s γ. Aong the trajectories of (14), in cosed oop with ν( s) = K s, simpe cacuations yied V = γ s γ γ v + (γ γ m )(γ M γ)k s. From the reation γ s γ v γ / + γ s /(v ) and the fact that the maximum vaue of (γ γ m )(γ M γ) over γ [γ m, γ M ] is [ ] (γ M γ m ) γ /4, we get V N ( γ) + M v + K(γ M γ m) N 4 1 ( s), where N ( γ) = v γ. Moreover, since V 1 is bounded from above by a positive definite quadratic function near 0, we can find a positive function ρ so that ρ(v 1 )V 1 min{1, v }( s + γ ) on D. (In fact, we can choose ρ so that

9 9 outside a neighborhood of zero, ρ(v) = (i) are satisfied. Aso, c 1+v for a suitabe constant c.) Thus, Assumptions 1 and V 1 ( s, γ) 1 γ m s v + Kγ m(γ M γ m) γ (16) hods on D. This gives V ( s, γ) s γ γm K(γM γ m) v V 1 ( s, γ). (Our choice of V was motivated by our desire to have the preceding estimate.) Hence, Assumption (ii) is satisfied as we, so Theorem 1 appies with p (V 1 ) [γ m K(γM γ m )]/ v. We now expicity buid the strict Lyapunov-ike function from Theorem 1. Since j = and a = 1, U ( s, γ) = Υ 1 V 1 ( s, γ) + V ( s, γ) where Υ 1 = 1+[γ m K(γM γ m )]/ v. Since Ω 1 (s) = [γ M /v +K(γ M γ m ) /]s and Ω (s) s, S( s, γ) = U ( s, γ) + [ γ M = V ( s, γ) + v + K(γ M γ m) ] V 1 ( s, γ) [ Υ 1 + γ M v + K(γ M γ m ) ] V 1 ( s, γ) is a strict Lyapunov-ike function for (14) in cosed oop with ν( s) = K s. In fact, (17) Ṡ W ( s, γ), where W ( s, γ) = N ( γ) + Υ 1 N 1 ( s) = v γ + Υ 1 s (18) aong the cosed oop trajectories. Remark 3: Since V 1 is not gobay proper on R, we cannot construct the required expicit strong Lyapunov function for (14) using the resuts of [15]. Notice that (17) is a simpe inear combination of V 1 and V. By contrast, the strong Lyapunov functions provided by [15, Theorem 3] for the j = time invariant case have the form S(x) = Q 1 (V 1 (x))v 1 (x) + Q (V 1 (x))v (x) where Q 1 is nonnegative, and where the positive definite function Q needs to gobay satisfy Q (V 1 ) φ 1 (ω(x)/{ρ( x )}) where V (x)f(x) N (x) + φ(n 1 (x))ρ( x ) for some φ K and some positive nondecreasing function ρ and the positive definite function ω needs to satisfy N 1 (x) + N (x) ω(x) everywhere. In particuar, we cannot take Q to be constant to get a inear combination of the V i s, so the construction of [15] is more compicated than the one we provide here. Simiar remarks appy to the other constructions in [15]. B. Robustness Resut It is important to assess the robustness of a contro design to bounded uncertainties before impementing the controer. Indeed, bioogica systems are known to have highy uncertain dynamics.

10 10 This is especiay the case for waste water treatment processes made up of a compex mixture of bacteria. In [9], good performance of the controer was observed but coud not be expained by a theoretica approach. Here we prove that an appropriate adaptive controer gives ISS of the reevant error dynamics to disturbances; see Section II for the definitions and motivations for ISS. We focus on the system (9) for cases where s in is repaced by H in (t) = s in + δ 1 (t) and, for an arbitrary positive constant K > 0, the adaptive contro is given by u = (γ + δ (t))y 1, γ = Ky 1 (γ γ m )(γ M γ)( s + δ 3 (t)) (19) where the disturbances δ 1 (t) and δ 3 (t) are bounded in absoute vaue by a constant δ 1 and the disturbance δ (t) is bounded by a constant δ ; we specify the δ i s beow. We maintain the assumptions and notation from the preceding subsections. We aso assume k/λ < (γ m δ )(s in δ 1 ), δ 1 < s in, and δ < γ m. (0) In particuar, we keep the definitions of x and γ from (11) and the first sentence after (11) unchanged; we do not repace s in by H in (t) in the expressions for x and γ. Our anaysis wi use the function S from (17) extensivey. To specify our bounds δ i, we use the constants Ξ = [Υ 1 + γ M /v + K(γ M γ m ) ] /γ m and Υ = min{γ γ m, γ M γ } (1) where Υ 1 is from Section V-A. See Section V-C for an exampe with specific bounds δ i. Repacing s in with H in in (9), and using u from (19) and the expression for y 1, we get [ [ ṡ = y 1 (γ + δ (t))(s in + δ 1 (t) s) λ] k, ẋ = 1 y1 α(γ + δ λ (t))x ], γ = Ky 1 (γ γ m )(γ M γ)( s + δ 3 (t)). () For simpicity, we restrict to the attractive and invariant domain where 0 < s < s in which, in practice, is the domain of interest (but a resut on the entire set where s > 0 can be proved). Using (0) and arguing as we did to obtain Lemma, the time re-scaing τ = t t o y 1 ()d yieds s = γ s + γv + δ (t)(s in s) + (γ + δ (t))δ 1 (t), x = α(γ + δ (t)) x α(δ (t) + γ)x, γ = K(γ γ m )(γ M γ) s K(γ γ m )(γ M γ)δ 3 (t) (3)

11 11 evoving on ˆD = ( s, s in s ) ( x, + ) (γ m γ, γ M γ ). The foowing ISS resut impies that the trajectories of () satisfy (x(t), s(t)) (x, s ) gobay uniformy, with an additiona term that is sma when the disturbances δ i are sma; see Section II for the precise ISS definition. For the proof of this resut, see Appendix B beow. Theorem : Assume that the system (3) satisfies (11) and (0). Define the constants { δ 1 = Υ v min 4 ( 4 } v Υ1 Ξv + 5γ ), and 4 M K(γ M γ m ) + 5Ξγ δ γ M δ1 = M 4(s in + δ 1 ). (4) Assume that δ 1 and δ 3 are bounded in absoute vaue by δ 1 and that the disturbance δ is bounded in absoute vaue by δ. Then the cosed oop error dynamics (3) is ISS on ˆD. 1 C. Numerica Exampe We simuated an anaerobic digestion process used to process waste water and produce biogas. We caibrated the mode using rea experimenta data [] to get reaistic parameter vaues. We took the infuent concentration s in to be piecewise constant with respect to time, foowing the rea profie experimented in [3]; see the figure beow. Finay, we simuated white noises for the δ i, introducing a 1% noise in the controer u and noises of standard deviation 0.5 g/ and 0.1 g/ to perturb s in and s; see (19). Our set point s was 1.5 g/. Foowing [9], we appied our controer successivey on the intervas on which s in is constant. s (g/) s in (g/) y 1 (/h) Time (days) Time (days) Time (days) Fig. 1. Controed substrate s (eft), infuent substrate concentration s in (midde) and methane fow rate y 1 (right). Our controed simuation iustrates the robustness of the controer. The controer was successfuy impemented on a rea process in [9], where it was asserted that the feedback ent itsef to 1 The constant can be repaced by any constant in (0, 1) that is cose enough to 1; we use the constant to get (A.7).

12 1 reaistic settings in which there are noisy signas. However, [9] did not provide any theoretica argument to prove this assertion. Our theory and simuation confirm and vaidate the robustness of the controer under reaistic vaues of noise. VI. CONCLUSIONS We provided new strict Lyapunov function constructions for noninear systems that satisfy Matrosov s conditions. The advantages of our constructions ie in their simpicity and their appicabiity to the various exampes whose stabiity can be estabished by the generaized Matrosov theorem. We demonstrated the efficacy of our methods through a cass of biotechnoogica modes with disturbances, which are of compeing engineering interest. Lemma A.1: Let k 1,..., k j APPENDIX A: TWO LEMMAS K C 1 ; Φ be a continuous nonnegative function defined on [0, ); and a 1, a,..., a j (0, 1] be constants. Then one can construct C 1 functions µ 1,..., µ j : [0, ) [1, ) such that for a s 0, µ j (s) = 1 and µ i (s) Φ(s) j µ i(s) 0 for i = 1 to j 1. =1+i µ1/a (k (s)) and Proof: Let Φ be a positive increasing function such that Φ Φ everywhere, and set k = (k 1 + k +... k j ). To prove the emma, it suffices to find µ i s such that µ i (s) Φ(s) j 1 =1+i µ a (k(s)) and µ i(s) 0 for i = 1 to j 1 and a s 0, where µ j (s) = 1. Let us construct these functions by induction. Assume that m functions µ j, µ j 1,..., µ j m+1 are avaiabe. Then µ j m (s) := 1 + s + Φ(s) j =1+j m µ 1 a (k(s)) Φ(s) j =1+j m 1 a µ (k(s)) (A.1) and µ j m is nondecreasing, because Φ, k, and µ for = j to j m + 1 are nondecreasing. Lemma A.: Let X(t) be a soution of a system Ẋ = F (X) defined over [0, + ). Let M be a positive definite function and c a, c b, and c c be positive constants such that for each t 0, the time derivative of M aong the soution X(t) satisfies either Then, is satisfied for a t 0. M(X(t)) e min{ca, c b }t ( e M(X(0)) 1 ) + M c a or M c b M(X(t)) + c c. e cc c b c c min { } c a, c (A.) b

13 13 e M(X) Proof: The time derivative of the function Ω(X) = e M(X) 1 aong the soution X(t) is Ω = M. Therefore, either Ω e M(X) c a Ω(X)c a or Ω e M(X) c b M(X) + e M(X) c c. In the atter case, M(X) cc c b e cc Ω 1 em(x) c b M(X) whie M(X) cc c b Ω e M(X) c b M(X) + c b c c. Hence, either Ω e M(X) c a Ω(X)c a or Ω c b e M(X) M(X) + e cc c b c c. (A.3) Notice that e A 1 = A 0 em dm Ae A for a A 0. We deduce that if (A.3) hods, then Ω c b ( e M(X) 1 ) + e cc/c b cc = c b Ω(X) + e cc/c b cc. Therefore, in the two cases, we have Ω min { c a, c b } Ω(X) + e cc c b c c. By integrating over any interva [0, t] and using the definition of Ω, we deduce that for a t 0, M(X(t)) n (1 + e min{ca, c ( b }t e M(X(0)) 1 ) ) cc + e c b cc. (A.4) min{c a, c b } Noting that n(1 + A) A is vaid for a A 0, we deduce that (A.) is satisfied. APPENDIX B: PROOF OF THEOREM Since the x sub-dynamics in (3) is ISS when ( γ, δ ) is viewed as its disturbance (because d dt x b x + b( δ + γ ) aong its trajectories for appropriate constants b, b > 0 combined with standard ISS arguments [18]), it suffices to check that the ( s, γ) sub-dynamics is ISS with respect to δ [18]. (In other words, the seria connection of ISS systems is ISS.) Hence, we focus on the ( s, γ) sub-dynamics in the rest of the proof. It foows from eementary cacuations and (18) that aong the trajectories of (3), Ṡ W ( s, γ) + T 1 ( s, γ) + T ( s, γ) (A.5) with T 1 ( s, γ) = S s ( s, γ) [δ s in s + (γ + δ )δ 1 ] and T ( s, γ) = S γ ( s, γ) K(γ γ m)(γ M γ)δ 1. Reca the constants Ξ and Υ defined in (1). One can use the formuas for S s S and γ to get T 1 ( s, γ) γ Ξ s [δ s in s + γδ 1 + δ 1 δ ] γ Ξ s [δ (s in + δ 1 ) + γ M δ 1 ], T ( s, γ) s Ξ v γ K(γ γ m )(γ M γ)δ 1 K ( γ+γ γ m)(γ M γ γ) K 4 (γ M γ m ) δ 1 s + Ξv δ 1 γ. (A.6)

14 14 Therefore, T 1 ( s, γ) + T ( s, γ) E 1 γ + E s with E 1 = Ξv δ 1 + (s in + δ 1 )δ + γ M δ 1 and E = K (γ 4 M γ m ) δ 1 + Ξ[δ (s in + δ 1 ) + γ M δ 1 ]. From (4), we deduce that E 1 99 Υ 100 v and 4 E ( K (γ ) 4 M γ m ) + Ξ 5γ M δ1 99 Υ v Υ1. Therefore, (A.5) and (A.6) give Ṡ v γ Υ 1 s Υ v 99 γ Υ v Υ1 s, (A.7) by our choice (18) of W. We introduce the constants Υ 3 = 397v Υ, Υ 4 = 1 + Ξ v K ( and Υ 6 = 1 Ξv v + 5γ M 4 (10 4 ), Υ (γ γ m)(γ M γ ) 5 = min{ v,υ 1} max{ 1+Ξ ) [ + 1 K (γ Υ 1 4 M γ m ) + Ξ 5γ M 4,Υ 4}, ] (A.8). We consider two cases. Case 1: If γ ( γ m γ, (γ m γ ) ] or γ [ (γ M γ ), γ M γ ), then γ Υ. From pq p + q /4 with p = Υ 1 s and q =.99Υ v and (A.7), Ṡ v 4 ( 199 ) Υ 00 Υ 1 s + 99 Υ 100 v Υ1 s v 4 Case : If γ [ (γ m γ ), (γ M γ ) ], then [ ( 199 ) ( ) ] 99 Υ Υ 3. (A.9) γ 0 d (10 4 ) γ (+γ γ m)(γ M γ ) (γ γ m)(γ M γ ). By our formua (17) for S, we get S( s, γ) 1+Ξ s + Υ 4 γ max { 1+Ξ, Υ 4} [ s + γ ]. We aso have W ( s, γ) min { v, Υ 1 } ( s + γ ). Therefore, 1 W ( s, γ) Υ 5S( s, γ). It foows from (A.5) that { } Ṡ Υ 5 S( s, γ) v 4 γ 1Υ 1 s E + v 1 } { } { γ v + Υ1 E } { s Υ1 (A.10) Υ 5 S( s, γ) + E 1 v + E Υ 1 Υ 5 S( s, γ) + Υ 6 δ 1. (We used the formuas for E 1 and E and (4) and appied pq p q to the terms in braces.) Therefore, in the first case we get (A.9), whie (A.10) hods in the second case. Hence, aong the cosed-oop trajectories, we have either Ṡ Υ 3 or Ṡ Υ 5S( s, γ) + Υ 6 δ 1. Using (16) and (17) and the fact that V (1 Υ 1 )V 1 everywhere, one easiy checks that S admits a constant Q o > 0 such that Q o ( s + γ ) S( s, γ) on D. We readiy concude by appying Lemma A., combined with the reation p + q p+ q (by taking square roots of both sides of (A.) with M = S). The constant (10 4 ) enters the anaysis by finding the minimum vaues of the denominator in the integrand.

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