BIBO stabilization of feedback control systems with time dependent delays

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1 to appear in Applied Mathematics and Computation BIBO stabilization of feedback control systems with time dependent delays Essam Awwad a,b, István Győri a and Ferenc Hartung a a Department of Mathematics, University of Pannonia, Hungary b Department of Mathematics, Benha University, Benha, Egypt Abstract his paper investigates the bounded input bounded output BIBO stability in a class of control system of nonlinear differential equations with time-delay. he proofs are based on our studies on the boundedness of the solutions of a general class of nonlinear Volterra integral equations. Keywords: boundedness, Volterra integral equations, bounded input bounded output BIBO stability, differential equations with delays. 1 Introduction Differential and integral equations with time delays appear frequently as mathematical models in natural sciences, economics and engineering, and they are used to describe propagation and transport phenomena see, e.g., [1, 6, 7, 8, 9, 11, 15, 17, 22]. In control engineering the time delay appears in the control naturally, since time is needed to sense information and to react to it. he first control applications using time delays go back to the 3s [4, 5], and since that it is an extensively studied field see, e.g., [6, 7, 23, 25, 29]. Stability of a differential equation is a central question in engineering applications, especially in control theory. Many different concepts of stability has been investigated in the literature, e.g., stability of an equilibrium of the system, orbital stability of the system output trajectory, or structural stability of the system. Because of its simplicity and weakness, the notion of bounded-input bounded-output BIBO stability was extensively studied and used in control theory for different classes of dynamical systems see, e.g., his research was partially ported by the Hungarian National Foundation for Scientific Research Grant No. K

2 [2, 1, 16, 18, 19, 2, 26, 27, 28]. We say that a control system is BIBO stable, if any bounded input produces a bounded output. It is known see, e.g., [2] that a linear control system ẋ = Ax + Bu is BIBO stable, if the trivial solution of ẋ = Ax is asymptotically stable, i.e., all eigenvalues of A have negative real parts. Using a novel approach, in this paper we study the BIBO stability of a feedback control system with time delays. First we rewrite the control system as a Volterra integral equation, then, motivated by the results obtained for discrete Volterra systems [12], we formulate sufficient conditions for the boundedness of nonlinear Volterra integral equations, and finally we apply our general results to obtain BIBO stability of the control system. Also we introduce a new notion of BIBO stability, which we call local BIBO stability. he rest of the article is organized as follows. In Section 2 we give the system description, some notations and definitions which are used throughout the paper. In Section 3 we state our main results. In Section 4 we give sufficient conditions for the boundedness of the solution of Volterra integral equations. In Section 5 some special estimates are studied. In Section 6 we give the proofs of our main results. 2 System description and preliminaries hroughout this paper R, R +, R n and R n n denote the set of real numbers, nonnegative real numbers, n-dimensional real column vectors and n n-dimensional real matrices, respectively. he maximum norm on R n is denoted by, i.e., x := max 1 i n x i, where x = x 1,...,x n. he matrix norm on R n n generated by the maximum vector norm will be denoted by, as well. L n will denote the set of bounded functions r : R + R n n with norm r := t rt. CX,Y denotes the set of continuous functions mapping from X to Y. In this paper we consider the nonlinear control system ẋt = gt,xt σt + ut, t, 2.1 where xt, ut are the state vector, control input, respectively, σt is a continuous function and g : R + R n R n satisfies gs, x bsφ x, s R +, x R n, 2.2 where bs > is a continuous function, φ : R + R + is a monotone nondecreasing mapping. hroughout the paper, we assume that the output of the system is the whole state xt, so we use the state in the feedback control. We assume that the uncontrolled system, i.e., 2.1 with u has unbounded solutions. Our goal is to find a control law of the form ut = Dxt τ + rt 2.3 2

3 which guarantees that the closed-loop system { ẋt = gt,xt σt Dxt τ + rt, t >, xt = ψt, t t 2.4 is BIBO stable. We pose that the feedback gain in 2.3 is a positive diagonal matrix, i.e., D = diagd 1,...,d n, d i >, i = 1,...,n. { } r L n is the reference input, t := max τ, inf{t σt}, ψ is a continuous vectorvalued initial function and ψ t := max t t ψt. t In applications the time delay τ in the control 2.3 appears naturally, since time is needed to sense information and react to it. Using diagonal feedback 2.3 is the simplest possible control law, and in heorem 3.1 and 3.3 we give sufficient conditions on how to select the feedback gain and the time delay τ to guarantee the boundedness of the solutions. Based on the definition of BIBO stability in [24], we introduce the next definition. Definition 2.1. he control system 2.4 with reference input r : R + R n is BIBO stable if there exist positive constants γ 1 and γ 2 = γ 2 ψ t satisfying for every reference input r L n. xt γ 1 r + γ 2, t In heorem 3.3 below we need a weaker version of BIBO stability. Next we introduce this new notion, which we call local BIBO stability. Definition 2.2. he control system 2.4 with reference input r : R + R n is locally BIBO stable if there exist positive constants δ 1, δ 2 and γ satisfying provided that ψ t < δ 1 and r < δ 2. xt γ, t Our approach is the following. We associate the linear system with the constant delay τ and the initial condition żt = Dzt τ, t 2.5 zt = ψt, τ t 2.6 to 2.4. hen the state equation 2.4 can be considered as the nonlinear perturbation of 2.5, so the variation of constants formula [14] yields xt = zt + Wt s [gs,xs σs + rs]ds, t

4 Here zt is the solution of and Wt is the fundamental solution of 2.5 i.e. the solution of initial value problem Ẇt = DWt τ, t, {, τ t <, Wt = I, t =, where I is the identity matrix and is the zero matrix. Since D is a diagonal matrix, it is easy to see that Wt is diagonal matrix too, and Wt = diagw 1 t,...,w n t, where for i = 1,...,n ẇ i t = d i w i t τ, t, {, τ t <, w i t = 1, t =. 2.8 We can rewrite the equation 2.7 as a Volterra integral equation where and xt = ft,s,x ds + ht, t, 2.9 ht := zt + Wt srsds, 2.1 ft,s,x := Wt sgs,xs σs In Section 4 we will study the boundedness of the solutions of nonlinear Volterra integral equations, and apply our results for the boundedness of Main results Our main goal in this section is to formulate sufficient conditions which grantee BIBO stability of the control system 2.4. Our first result is given for the case when g in 2.1 has a sub-linear estimate, i.e., when φt = t p, with < p < 1 in 2.2. heorem 3.1. Let g : R + R n R n be a continuous function which satisfies inequality 2.2 with φt = t p, < p < 1, t. he feedback control system 2.4 is BIBO stable if hold. b := bt <, < d i < π, t, i = 1,...,n 3.1 t 2τ 4

5 he following theorem gives a sufficient condition for the BIBO stability in the case of a linear estimate. heorem 3.2. Let g : R + R n R n be a continuous function which satisfies inequality 2.2 with φt = t, t. he feedback control system 2.4 is BIBO stable if holds. b < d i 1, t, i = 1,...,n 3.2 eτ In the er-linear case, when φt = t p, p > 1 in 2.2, we show that the control system 2.4 is locally BIBO stable. heorem 3.3. Let g : R + R n R n be a continuous function which satisfies inequality 2.2 with φt = t p, p > 1, t. hen the solution x of the feedback control system 2.4 is locally BIBO stable if 3.2 holds. he proofs of the main results are given in Section 6. 4 Volterra integral equation In this section we obtain sufficient conditions for the boundedness of the solutions of nonlinear Volterra integral equations. We consider the nonlinear Volterra integral equation xt = ft,s,x ds + ht, t, 4.1 with initial condition xt = ψt, t [ t, ], 4.2 where t is fixed, and the following conditions are satisfied. A1 he function f : R + R + C [ t,, R n R n is a Volterra-type functional, i.e., f is continuous and ft,s,x = ft,s,y for all s t and x, y C [ t,, R n if x µ = y µ, t µ s. A2 For any s t and x C [ t,, R n, f i t,s,x k i t,sφ max xθ, t θ s i = 1,...,n holds, where ft = f 1 t,...,f n t. Here k i : {t,s : s t} R +, i = 1,...,n is continuous, and φ : R + R + is a monotone nondecreasing mapping such that φṽ >, ṽ >, and φ =. 5

6 A3 h : R + R n is continuous and ht = h 1 t,...,h n t, t. A4 ψ C [ t, ], R n. he next definitions will be useful in the proofs of our results on Volterra integral equations. Definition 4.1. Let the functions φ and k i be defined by assumption A2. We say that the nonnegative constant µ has property P with if there is v µ, such that φµ k i t,sds + φv k i t,sds + h i t < v, t, i = 1,...,n 4.3 holds. Definition 4.2. We say that initial function ψ C [ t, ], R n belongs to the set S if there exists such that µ := max xt t t has property P, where x : [ t, R n is the solution of Remark 4.3. If there exist a and two positive constants µ and v such that 4.3 holds, then I : = max 1 i n t J : = max 1 i n t H : = max 1 i n t Conditions 4.4 and 4.5 are equivalent to J := max 1 i n t and if = condition 4.6 becomes H := max k i t,sds <, 4.4 k i t,sds <, 4.5 h i t < i n t k i t,sds <, 4.7 h i t <. 4.8 he following result gives a sufficient condition for the boundedness of the solution of 4.1. heorem 4.4. Let A1-A4 be satisfied and the initial function ψ belongs to the set S. hen the solution x of is bounded. 6

7 Proof. Since the initial function ψ belongs to the set S, by Definition 4.2 there exists such that µ := max t t xt has property P, where x is the solution of From 4.1 we get xt = ft,s,x ds + Hence A2 yields for i = 1,...,n x i t φµ f i t,s,x ds + k i t,sφ max t µ k i t,sds + ft,s,x ds + ht, t. f i t,s,x ds + h i t x µ ds + k i t,sφ max x µ ds + h i t t µ s k i t,sφ max x µ ds + h i t, t. 4.9 t µ s Let v µ be such that 4.3 holds with µ = µ. hen, in particular, φµ k i,sds + h i < v, i = 1,...,n, so 4.9 with t = implies x i < v for all i = 1,...,n, hence x < v. Now we show that xt is bounded for all t >. For sake of contradiction, assume that xt is an unbounded function. Since it is continuous and x < v, there exists t 1 > such that xt 1 > v. Let t := inf{t > : xt > v}. hen the continuity of x and µ v implies max t µ t x µ = x t = v. herefore there exists i such that x i t = v. hen from 4.9 with t = t, the monotonicity of φ and using that µ has property P, we obtain v = x i t φµ φµ k i t,sds + t k i t,sds + φv k i t,sφ max t t µ s x µ ds + h i t k i t,sds + h i t < v, which is a contradiction. So the solution x of 4.1 is bounded by v. 5 Some special estimates In this section, we give some applications of our heorem 4.4. hroughout this section we assume that the nonlinear function f in 4.1 can be estimated by the function φt = t p, t > with p > in A2. here are three cases: 1. Sub-linear estimate < p < 1; 2. Linear estimate p = 1; 3. Super-linear estimate p > 1. 7

8 5.1 Sub-linear estimate Our aim in this subsection is to establish a sufficient, as well as a necessary and sufficient condition for the boundedness of all solutions of 4.1 and for the scalar case of 4.1, respectively. he next result provides a sufficient condition for the boundedness of the solutions of 4.1 in the sub-linear case. heorem 5.1. Let A1-A4 be satisfied and φt = t p, t, with a fixed p, 1. If 4.7 and 4.8 hold, then all solutions of 4.1 are bounded. Proof. It follows from 4.7 and 4.8 that J < and H <. Let ψ C [ t, ], R n, and x be the corresponding solution of For µ := max t t ψt, there exists v µ large enough such that herefore By the definitions of J and H, we get v p 1 J + 1 v H < v p J + H < v. v P k i t,sds + h i t v p J + H < v t, i = 1,...,n. Hence by Definition 4.1, we obtain µ has property P with =. Since A1-A4 are satisfied and µ has property P, then, by heorem 4.4, the solution x of 4.1 is bounded. We consider the scalar Volterra integral equation with initial condition where xt = kt,sx p s τsds + ht, t, 5.2 xt = ψt, t [ t, ], 5.3 B k : {t,s : s t} R + and h : R + R + are continuous, < p < 1, τ C R +, R +, t := inf t {t τt} > is finite and ψ C [ t, ],,. he condition B implies that the solutions of are positive. he following result provides a necessary and sufficient condition for the boundedness of the positive solutions of 5.2. he necessary part of the next theorem was motivated by a similar result of Lipovan [21] proved for convolution-type integral equation. 8

9 heorem 5.2. Assume B, and lim inf t kt,sds + ht >, 5.4 kt,sds + ht >, t. 5.5 hen the solution of is bounded for all ψ C [ t, ],,, if and only if 4.7 and 4.8 are satisfied. Proof. Suppose 4.7 and 4.8 are satisfied. Clearly, by heorem 5.1 the solution of is bounded for all ψ C [ t, ],,. Conversely, let the solution x of be bounded on R +. First we prove that the solution x >. Assume for the sake of contradiction that xt for some t. hen there exists λ > such that xt >, t [,λ and xλ =. From 5.5 it follows that there exists an ǫ > such that From 5.2 with t = λ we have = xλ = λ λ ǫ λ ǫ kλ,sds + hλ >. kλ,sx p s τsds + hλ min t µ λ ǫ λ ǫ >, kλ,sx p s τsds + hλ x p µ λ ǫ kλ,sds + hλ which is a contradiction. Clearly, the positivity of x and 5.2 yield kλ,sds + hλ min xt ht, for all t, hence condition 4.8 is satisfied. Next we prove 4.7. For any t > we get xt kt,sx p s τsds 9 min x p µ, 1 t µ λ ǫ min x p µ kt, sds. t µ

10 herefore the boundedness of x implies t kt, sds <. 5.6 Define now m := lim inf xt, t which is finite. We show that m >. Assume for the sake of contradiction that m =. In this case we can find a strictly increasing sequence t n n 1, such that xt n = From 5.2 with t = t n we obtain Since xt n >, we have inf xt >, and xt n as n. t t t n xt n = n n = x p t n x 1 p t n kt n,sx p s τsds + ht n kt n,s n n inf x p µds + ht n t µ t n kt n,sds + ht n. For n large enough such that < x p t n 1, we get x 1 p t n n lim n n kt n,sds + ht n x p t n. kt n,sds + ht n. aking limit of the last inequality, we obtain kt n,sds + ht n =, which contradicts to 5.4. So m >. herefore there exists such that xt 1 2 m >, t. Hence xt = kt,sx p s τsds + ht kt,sx p s τsds 1 2 p mp kt,sds, t. 1

11 By the boundedness of the solution x we get t his and 5.6 imply condition Linear estimate kt,sds <. Our aim in this subsection is to obtain a sufficient condition for the boundedness of the solutions of a linear Volterra integral equation. he following result gives a sufficient condition for the boundedness. heorem 5.3. Assume A1-A4 are satisfied and φt = t, t. hen all solutions of 4.1 are bounded, if for some one of the following conditions hold: i 4.4 and 4.6 hold, and ii for all t, i = 1,...,n, J := max 1 i n t k i t,sds < 1, i = 1,...,n. 5.7 and J := max 1 i n t 1 t 1 t k i t,sds = 1, k i t,sds < 1 1 k i t,sds k i t,sds <, 5.8 k i t,sds 1 h i t <. 5.9 Proof. Let ψ C [ t, ], R n, and x be the solution of We show that x is bounded if i or ii holds. Let µ := max xt. t t First we prove that µ has property P under condition i. By 4.4 and 4.6 we have I < and H <. herefore there exists v µ such that 1 J 1 µ I + H < v, so µ I + H < 1 J v. 11

12 Hence for all t, i = 1,...,n, we obtain therefore µ k i t,sds + h i t µ I + H < 1 J v µ k i t,sds + v 1 k i t,sds v, k i t,sds + h i t < v, t, i = 1,...,n. hen µ has property P, and hence, by heorem 4.4, the solution x of 4.1 is bounded. Next we prove that µ has property P if ii is satisfied. For all t 5.8 and 5.9 imply 1 t 1 hen there exists v µ large enough such that which yields 1 µ k i t,sds + v k i t,sds 1 >, i = 1,...,n, 1 } k i t,sds {µ k i t,sds + h i t <. 1 } k i t,sds {µ k i t,sds + h i t < v, t, k i t,sds + h i t < v, t, i = 1,...,n. hen µ has property P, and hence, by heorem 4.4, the solution x of 4.1 is bounded. he next example illustrates the applicability of heorem 5.3 in a case when condition ii holds for a large enough. Example 5.4. We consider the scalar Volterra integral equation xt = 2 1.5e 2 t ln 2 e 2t s xsds + ce 2t, t. Here ht = ce 2t, c R and kt,s = 2 1.5e 2 t ln 2 e 2t s, t =. Clearly t ht = c <, and kt,sds = 2 1.5e 2 t ln 2 e 2t s ds = 1 1.5e 2 t ln 2 1 e 2t, 12

13 hence Let 1 := ln 2, then lim t kt,sds = 1. his means that 1 k 1,sds = t 1 1 e 2ln 2 = > 1. kt,sds = M > 1, therefore conditions of heorem 5.3 do not hold with =. If we take := ln 5, then we obtain and Moreover 1 and 1 kt,sds = 1 25e 2t 1 2e 2t < 1, t t kt,sds = 1. 1 kt, sds kt,sds = 24e 2t 1 2e 2t 23e 2t 1 2e 2t = 24 23, t, 1 kt, sds ht = 1 2e 2t c e 2t < c 23e 2t 23, t. hen condition ii in heorem 5.3 is satisfied with this, therefore x is bounded. Here our result is applicable but the several results in the literature are not applicable see, e.g., Proposition in [3]. 5.3 Super-linear estimate Our aim in this subsection is to obtain a sufficient condition for the boundedness in the er-linear case. heorem 5.5. Assume that conditions A1-A4 are satisfied with φt = t p, t >, where p > 1, 4.7, 4.8 hold, J > and let ψ t := max t t ψt. hen the solution x of is bounded if one of the following conditions is satisfied i ψ t 1 1 p 1 pj and H < p 1 p 1 1 p 1 pj ; 13

14 ii H < ψ t J ψ t p. Proof. Assume 4.7 and 4.8 are satisfied. i Since ψ t then v := 1 1 p 1 pj 1 1 p 1 pj and H < p 1 p 1 pj satisfies inequality 1 p 1 = 1 pj 1 p p 1 1 p 1 J, pj H < v J v p. 5.1 ii Since H < ψ t J ψ t p, v = ψ t satisfies inequality 5.1. Hence in both cases the condition 4.3 is satisfied with =, therefore µ := ψ t has property P. hen the conditions of heorem 4.4 hold, so the solution of 4.1 is bounded. 6 he proof of the main results In this section we give the proofs our main results heorems 3.1, 3.2 and 3.3 using our results obtained for Volterra integral equations. First we give the proof of heorem 3.1 for the sub-linear case. Proof of heorem 3.1. First note that the feedback control system 2.4 is equivalent to the Volterra integral equation 4.1, where h and f are defined by 2.1 and 2.11, respectively. Clearly, A1 is satisfied with t = t. By [14] and under our conditions, for i = 1,...,n, the solution z i of the initial value problem z i t = d i z i t τ, t 6.1 with initial condition satisfies the inequality z i t = ψ i t, τ t 6.2 z i t M ψ i τ e α it, t τ, 6.3 where M and α i are positive constants. Hence every solution of 6.1 tends to zero as t, and this implies that every solution of 2.5 tends to the zero vector as t, and z := zt M ψ τ <. 6.4 t 14

15 From 3.1 it follows see, e.g., Proposition 2.1 in [13] that the fundamental solution w i of satisfies C i := where C i is positive constant. From 2.7, for all i = 1,...,n, we have x i t = z i t + w i t dt <, i = 1,...,n, 6.5 w i t s[g i s,xs σs + r i s]ds, t, i = 1,...,n, where xt = x 1 t,...,x n t, gt,x = g 1 t,x,...,g n t,x, rt = r 1 t,...,r n t and zt = z 1 t,...,z n t. herefore 2.1 and 2.11 imply f i t,s,x = w i t sg i s,xs σs and h i t = z i t + w i t sr i sds. Hence by 2.2 f i t,s,x w i t s g i s,xs σs w i t s bsφ max xµ, t µ s so A2 holds with k i t,s := w i t s bs, s t. By 3.1, 6.4, 6.5 and the definition of the infinity norm, we obtain H := max 1 i n t max 1 i n t max 1 i n t h i t = z + C r z i t + max 1 i n t z i t + r max 1 i n w i t s rs ds w i t dt 6.6 M ψ τ + C r M ψ t + C r, 6.7 { } where C := max i=1,...,n C i and t := max τ, inf{t σt} t, therefore 4.8 holds. By condition 3.1 we get Hence k i t,sds = J := max 1 i n t w i t s bsds, t, i = 1,...,n. k i t,sds b max 1 i n w i t dt C b <,

16 i.e., 4.7 holds with =. hen the conditions of heorem 5.1 for < p < 1 with = are satisfied, hence the solution x of 2.4 is bounded. Now we show that the inequality 5.1 is satisfied with v := max 3M ψ t + C r, 3C b 1 1 p. We have v > 2C b 1 1 p, and so 6.8 yields v p 1 J < J 2C b 1 2. Similarly, v > 2M ψ t + C r and 6.7 imply 1 v H < H 2M ψ t + C r 1 2. herefore 5.1 is satisfied, hence by heorem 5.1 we get xt < v = max 3M ψ t + C r, 3C b 1 1 p, t. So xt < 3C r + max 3M ψ t, 3C b 1 1 p = γ 1 r + γ 2, t, where γ 1 := 3C and γ 2 := max 3M ψ t, 3C b 1 1 p. Hence the feedback control system 2.4 is BIBO stable. he next Lemma will be useful in the proof of heorem 3.2. Lemma 6.1. Assume the conditions of heorem 3.2 are satisfied. he inequalities are satisfied with h i t + v k i t,sds < v, t, i = 1,...,n 6.9 v := max r z +, 1, d 1 b 1 b d d where d := mind 1,...,d n, W = diagw 1,...,w n is the fundamental solution of 2.5, z is the solution of the initial value problem , z := t zt and k i t,s := w i t sbs, s t. 16

17 Proof. From 3.2 it follows see, e.g., heorem 3.1 in [13] that the fundamental solution w i of is positive and Hence w i tdt = 1 d i and w i sds < 1 d i, i = 1,...,n, t. 6.1 J := max 1 i n t = max 1 i n t max 1 i n b k i t,sds w i t sbsds By 3.2 and 6.11, we have for t, i = 1,...,n herefore 1 Hence v > r z + d 1 b 1 b d d w i tdt = b d r d iw i t sds d 1 + t w it sbsds w it s rs ds 1 w i t sbsds v > v > zt 1 w it sbsds w it sbsds + zt 1 w it sbsds, t. w i t s rs ds + zt, t, i = 1,...,n. w i t s rs ds + zt + v h i t + v w i t sbsds k i t,sds, t, i = 1,...,n, where k i t,s = w i t sbs and ht = h 1 t,...,h n t is defind by

18 Proof of heorem 3.2. According to 6.6 and 6.1 and the positivity of w i, we have H := max 1 i n t h i t z + r d M ψ t + r d <, 6.12 { } where d := mind 1,...,d n and t := max τ, inf{t σt}. By 6.11, we obtain t J < 1, so condition i in heorem 5.3 holds with =, hence the solution x of 2.4 is bounded. Lemma 6.1 yields that relation 6.9 is satisfied with v := max herefore the boundedness of the solution and 6.12 gives r xt < v := max + d z, 1, t. d b d b r d b + z 1 b d, 1. So 6.4 yields r xt < max, 1 d b d b + d z 1 d b r + max = γ 1 r + γ 2, t, d M ψ t d b, 1 where γ 1 := 1 d M ψ t and γ 2 := max, 1. d b d b Hence the feedback control system 2.4 is BIBO stable. Proof of heorem 3.3. Suppose ψ t δ 1 and r δ 2, where δ 1 and δ 2 will be specified later. According to 6.4 z M ψ t Mδ 1. So 6.12 yields H := max 1 i n t h i t z + r d Mδ 1 + δ 2 d. Relation 6.11 implies J < 1. If the positive constants δ 1 and δ 2 are selected so that Mδ 1 + δ 2 p 1 d p 1 1 p 1, p 18

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