Delay and Its Time-derivative Dependent Robust Stability of Uncertain Neutral Systems with Saturating Actuators

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1 International Journal of Automation and Computing 74, November 200, DOI: 0.007/s Delay and Its ime-derivative Dependent Robust Stability of Uncertain Neutral Systems with Saturating Actuators Fatima El Haoussi El Houssaine issir 2 Presidence of University Sidi Mohammed Ben Abdellah, Fès 30000, Morocco 2 Laboratory of Electronics, Signals Systems and Data Processing, Department of Physics, Faculty of Sciences, University Sidi Mohammed Ben Abdellah, Fès 30000, Morocco Abstract: his note concerns the problem of the robust stability of uncertain neutral systems with time-varying delay and saturating actuators. he system considered is continuous in time with norm bounded parametric uncertainties. By incorporating the free weighing matrix approach developed recently, some new delay-dependent stability conditions in terms of linear matrix inequalities LMIs with some tuning parameters are obtained. An estimate of the domain of attraction of the closed-loop system under a priori designed controller is proposed. he approach is based on a polytopic description of the actuator saturation nonlinearities and the Lyapunov- Krasovskii method. Numerical examples are used to demonstrate the effectiveness of the proposed design method. Keywords: Actuator saturation, neutral systems, uncertainty, delay-dependent condition, linear matrix inequality LMI, time varying delay systems. Introduction he problem of delayed systems has been investigated over the years because the phenomena of time-delay are very often encountered in different technical systems, such as electric, pneumatic, and hydraulic networks, chemical processes, long transmission lines, etc. ime delay is often a source of instability and oscillation in practical systems. he robust stability of uncertain systems with time delays has received considerable attention. Existing criteria for asymptotic stability of time-delay systems can be classified into two types: delay-independent stability [, 2] and delaydependent stability [3 0]. he former does not include any information on the size of delay while the latter employs such information. It is well known that delay-independent criteria tend to be conservative, especially when the size of a delay is small. A major problem in the control of linear dynamical systems with time delay is that the actuator saturations are unavoidable. he actuator saturation not only deteriorates the control system performance, but also leads to undesirable stability effects. he stability analysis and stabilization of time delay systems with saturating actuator have been widely investigated by many researchers. References [ 3] have treated this problem and obtained saturating control laws that govern the system stability when the initial state belongs to an estimated domain. By using the Lyapunov method, a number of delaydependent robust stabilization techniques for a class of uncertain state-delayed systems have been investigated via predicted-based transformation [8]. Fridman and Shaked [4] combined Park s and Moon s inequalities with a descriptor model transform and found rather efficient criteria for systems with polytopic-type uncertainties. Recently, a free weighting matrix approach to overcoming the conservativeness of methods involving a fixed model transformation [6, 4] is expected to be able to further improve the performance when applied to delay-dependent stability with some tun- ing parameters. his has motivated the work in this paper. In the more general case of neutral-type systems where the delay appears in the state derivative and in the state, several sufficient conditions have been obtained for delayindependent [5] and the delay-dependent [6 23] cases. Note that unlike retarded-type systems, neutral systems may be destabilized by small changes in delays [24]. his work aims to develop some delay-dependent methods for neutral systems with a time-varying delay, actuators constraints, and norm-bounded parametric uncertainties via linear memoryless state feedback control law. he control law serves to guarantee the local stability of the closed loop system when the initial states are taken in a predetermined region of attraction. In the following, a linear matrix inequality LMI optimization approach will be proposed to design the state feedback gain for maximizing this estimate of the domain of attraction. A less conservative estimate of the region of attraction will be derived based on the Lyapunov-Krasovskii functional approach. he conditions are given in terms of LMIs. Note that the LMIs approach has the advantage that it can be solved numerically very efficiently using the interior point algorithm developed in [25, 26] and using Matlab LMI toolbox [27]. hree numerical examples are given to illustrate that the results are less conservative than previous work. 2 Problem formulation and definitions Consider the following uncertain neutral system with time-varying delay: ẋt Cẋt τt = A 0 + A 0txt + A + A t xt ht + B + Btsatut, t > 0 Manuscript received May 3, 2009; revised August 9, 2009

2 456 International Journal of Automation and Computing 74, November 200 where xt R n is the state vector; ut R m is the control input; C, A 0, A, and B are known real constant matrices. he delays τ and h are assumed to be some unknown functions of time and are continuously differentiable, with their respective rates of change bounded as follows: and 0 ht h m, 0 τt < 2 0 ḣt d <, 0 τt d2 < 3 where h m, d, and d 2 are given positive constants. Note that condition 3 should be satisfied so as to ensure that system admits a physical meaningful solution compatible with the causality principle see [28] for more discussions on varying delays and derivative bounds. he initial condition of system is given by xθ = φθ, θ [ h, 0] where h = max{τt, ht}, t 0, and φ is a differentiable vector valued initial function. Define the operator : C [ h, 0] R n as x t = xt Cxt τ. Assumption. All the eigenvalues of matrix C are inside the unit circle. In this paper, we assume that the uncertainties can be described as follows: [ A 0t A t Bt] = DF t[e 0 E E 2] 4 where D, E 0, E, and E 2 are known constant real matrices of appropriate dimensions, and F t denotes timevarying parameter uncertainties and is assumed to be of block diagonal form F t = diag{f t,, F rt}, where F it R p i q i ; i =,, r are unknown real time-varying matrices satisfying Fi tf it I, t 0. he saturation function is defined by and satut = [satu t, satu 2t,, satu mt] 5 u i, if u i > u i satu it = u i, if u i u i u i u i, if u i < u i. Lemma [29]. Let D, E, and F t be real matrices of appropriate dimensions with F = diag{f,, F r}, Fi F i I, i =,, r. hen, for any real matrix Λ = diag{µ I,, µ ri} > 0, the following inequality is true: DF te + E F t D DΛD + E Λ E. In this paper, we consider the stabilization of system using a linear state feedback ut = Kxt. 6 For an initial condition x 0 = φ C [ h; 0], denote the state trajectory of system by xt, φ. Suppose that the solution xt = 0 is asymptotically stable for all delays satisfying 2. hen, the domain of attraction of the origin is Ψ = {φ C [ h; 0] : lim xt, φ = 0}. t Moreover, we are interested in obtaining an estimate Ξ δ Ψ of the domain of attraction, where Ξ δ = {φ C [ h; 0] : max φ δ} [ h;0] and where δ > 0 is a scalar to be maximized in the sequel. Define the polyhedron DK, ū = {x R n ; k ix ū i, i =,, m} where k i denotes the i-th row of K. We exploit the idea of [] in the development of the results of this paper. Denote the set of all diagonal matrices in R m m with diagonal elements that are or 0 by N. hen, there are 2 m elements D i in N, where N denotes the set of diagonal matrices with diagonal elements are 0 or, and for every i =,, 2 m, D i = I D i is also an element in N. Lemma 2 []. Given K and H in R m n, we have satkxt Co{D ikx + D i Hx, i =,, 2 m } for all x R n that satisfy h ix ū i, i =,, m. herefore, for x S c, any compact set of R n, let H be in R m n such that h ix ū i. hen, the motion of the system 4 can be described by the following system: ẋt Cẋt τt = 2 m j= λ j  jxt + A xt ht 7 where Âj = BDjK + D j H + A0, 2 m j= λj =, λj 0 A 0 = A 0 + DF te 0, A = A + DF te, and B = B + DF te 2. Choose a Lyapunov functional candidate to be V t = x tp xt + 0 h m t+θ t ht ẋ srẋsdsdθ + x sqxsds+ t τt ẋ sw ẋsds 8 where P = P > 0, Q = Q > 0, R = R > 0, W = W > 0. A convenient choice of the set S c can be defined from a symmetric positive definite matrix P as D e = {xt R n ; x tp xt β } 9 where β is a positive scalar. 3 Main results In this section, we will give some sufficient conditions for 4 to be robustly stable. Lemma 3. Under Assumption, the system described by 4 is robustly stable if there exist P = P > 0, Q = Q > 0, R = R > 0, W = W > 0, and appropriately dimensioned matrices Y, Y 2,, and 2 such that the

3 F. E. Haoussi and E. H. issir / Delay and Its ime-derivative Dependent Robust Stability of 457 following LMIs hold: Γ j = Γ j Γ 2j Γ 3 h my C Γ 2j Γ 22 Γ 32 h my 2 2C Γ 3 Γ 32 Γ < 0 h my h my2 0 h mr 0 C C d 2W where Γ j = Â j + Â j + Y + Y + Q Γ 2j = P + 2 Â j + Y 2 Γ 22 = h mr + W 2 2 Γ 3 = A Y Γ 32 = A 2 Y2 Γ 33 = d Q. j =,, 2 m 0 Proof. Calculating the derivative of V t along the solution of system and using 2 yields V t 2x tp ẋt + x tqxt d x t htq xt ht + h mẋ trẋt t ht ẋ srẋsds+ ẋ tw ẋt d 2ẋ τtw ẋ τt. Using the free weighting matrix approach introduced in [6], for appropriately matrices Y, Y 2,, and 2, we have 2[x t + ẋ t 2] [ ẋt + Cẋ τt + Âjxt+ A xt ht] = 0, j =,, 2 m 2[x ty + ẋ ty 2] [xt xt ht t ht ẋsds] = 0. For any positive semi-definite matrix Z Z2 Z3 Z = Z 2 Z 22 Z32 0 Z 3 Z 32 Z 33 and ηt = xt ẋt xt ht we also have h mη tzηt η tzηtds 0. t ht hen, adding those terms to the right-hand side of, letting Ωt, s = ξt, s ẋt τt, Z = Y Y 2 0 R Y Y 2 0 and using the Schur complement we obtain V t 2 m j= λjω t, sγ jωt, s. hen, there exists π such that V t π xt 2, which ensures the asymptotic stability of system 4 according to [30]. heorem. For given ε, ε 2 R, if there exist Q = Q > 0, R = R > 0, W = W > 0, X = X > 0, X 2, X 3 R n n, U, G R m n, Λ = diag{µ I,, µ ri} > 0, and positive scalars β, δ, satisfying the following linear matrix inequalities : where β gi ū 2 0, i =,, m 3 i X Σ = X 2 + X2 + ε X A + A X Σ 2j = X3 X 2 + A 0 + ε 2A X + BD ju + D j G Σ 22 = X 3 X 3 + DΛD and g i denotes the i-th row of G, then control law 6 with K = UX stabilizes system 4 for every initial condition in Ξ δ with Σ Σ 2j Σ 22 ε QA ε 2QA d Q h mε RA h mε 2RA 0 h mr 0 W C 0 0 d 2W h mx 2 h mx h mr X 2 X W X Q E 0X + E 2D ju + D j G 0 EQ Λ < 0 j =,, 2 m 2 he symbol stands for symmetric block in matrix inequalities.

4 458 International Journal of Automation and Computing 74, November 200 Using 6, we obtain the following inequality: h m δ 2 max{λ maxx + 2 d λmaxq ; 2h 2 mλ maxq + d 2 λmaxw + h mλ maxr } β. 4 Proof. From the requirement that P = P > 0, and the fact that in 0, 2 2 must be negative definite, it follows that P is non-singular with P = X = P 0 2 = X 0 X 2 X 3 hen, multiply both sides of 0 by diag{x, I, I, I} and diag{x, I, I, I}, and introduce some changes of variables such that X = P, Q = Q, R = R, W = W, U = KX, N X Y + X2 Y 2 G = HX, = N 2 X3 X. Y 2 By the Schur complement [25], LMI 0 implies 5 with Π 2j = X3 X 2 + N 2 + A 0X + BD ju + D j G. he main difficulty in the application of condition 5 is the presence of some nonlinearities such as X, N, and N2. Unfortunately, the condition cannot be directly solved and there is the need to tune the variables. o overcome this, we choose N = ε A X, N 2 = ε 2A X 6 where ε and ε 2 are some decision variables, and use the simple procedure presented in Remark 3 in this section.. M j + DF te + E F td < 0, j =,, 2 m 7 where M j is shown at the bottom of this page, D and Ē are as follows: D = 0 D E = E 0X + Z 0 E Q where Z = E 2D ju + D jg. According to Lemma, 7 holds if there exists Λ = diag{µ I,, µ ri} > 0 such that M j + DΛD + E Λ E < 0. 8 hus, by the Schur complement, 8 is equivalent to 2 of heorem. Moreover, the satisfaction of LMI 3 guarantees that h ix ū i, x D e, i =,, m. his can be proven in the same manner as in [ 3]. Furthermore, following [22], the Lyapunov functional defined in 8 can be shown to satisfy with π = λ minx and π Dφ 2 V φ π 2 max φ 2 [ h,0] h m π 2 = max{λ maxx + 2 d λmaxq ; 2h 2 mλ maxq + d 2 λmaxw + h mλ maxr }. From V < 0, it follows that V t < V 0 and therefore x tx xt V t < V 0 max φθ 2 π 2 β. θ [ h,0] X 2 + X2 + N + N Π 2j X 3 X3 X N A X N 2 d Q h mx N h mx N 2 0 h mr 0 W C 0 0 d 2W h mx 2 h mx h mr X 2 X W X Q < 0 M j = j =,, 2 m 5 Σ Σ 2j Σ 22 ε QA ε 2QA d Q h mε RA h mε 2RA 0 h mr 0 W C 0 0 d 2W h mx 2 h mx h mr X 2 X W X Q

5 F. E. Haoussi and E. H. issir / Delay and Its ime-derivative Dependent Robust Stability of 459 Inequality 4 guarantees that for all initial functions φ Ξ δ, the trajectories of xt remain within D e and V < 0 along the trajectories of the closed loop system 7 which implies that lim t xt = 0. When C = 0, A 0 = 0, A = 0, B = 0, and the matrices Z 3, Z 32, Z 33 are zeros, the matrix Z can be reduced to Z =. Let Z Z2 Z 2 Z Z Z Z = 2 = 22 Z22 X ZX. hen, following the same steps as in the proof of heorem, we obtain the following corollary. Corollary. Consider the nominal system of 4. For given ε, ε 2 R, if there exist Q = Q > 0, R = R > 0, W = W > 0, X = X > 0, X 2, X 3 R n n, U, G R m n, and positive scalars β, δ, satisfying 3 and the following LMIs: Z 2 Ω Ω 2j Ω 22 ε QA ε 2QA d Q h mx 2 h mx 3 0 h mr X Q j =,, 2 m Z Z 2 Z22 ε RA ε 2RA R 0 where Ω = X 2 + X2 + ε X A + A X + h m Z Ω 2j = X3 X 2 + A 0 + ε 2A X + BD ju + D jg + h m Z2 Ω 22 = X3 X 3 + h m Z22 < 0 then the system is asymptotically stabilized by K = UX for any initial condition in Ξ δ with h m δ 2 max{λ maxx + 2 d λmaxq ; 2h 2 mλ maxq + h mλ maxr } β. Remark. By Corollary, if we take ε = 0, our result reduces to heorem in [4, 2]. Clearly, ε = 0 is not the best choice. his implies that our result is less conservative than that of [4, 2]. his is an advantage of our result since generally a comparison between results in terms of LMIs is made by numerical examples while in this paper it is established theoretically that our result is less conservative than those of [4, 2]. Remark 2. he global stability cannot be ensured in general. Furthermore, in general, when it is possible to compute a global stabilizing control law [3], it is difficult to simultaneously guarantee good performance and robustness for the closed-loop system. heorem provides a condition allowing us to compute both a control law and a domain of attraction in which the closed loop neutral system is robustly stable. It is interesting to come up with a solution such that the domain of initial conditions is the largest possible. However, from the nonlinearity of 4, this is very difficult or even impossible. Assume the following conditions: X σ I, Q σ 2I, R σ 3I, W σ 4I. By Schur complement, the following LMIs are obtained: σ I I σ 2I I 0, 0, 9 I X I Q σ 3I I σ 4I I 0, I R I W It follows that condition 4 is satisfied if δ 2 h m max{σ + 2 d σ2; 2h2 mσ 2 + h mσ 3 + d σ4} 2 β holds. Combining the facts derived above, we can construct a feasibility problem for given h m as follows: Find Q, R, W, X, X 2, X 3, U, G, Λ, β, ε, ε 2, δ, σ i, i =,, 4 subject to X > 0, Q > 0, R > 0, W > 0, Λ > 0, β > 0, δ > 0, σ i > 0, i =,, 4, and 2 4, 9, Given h m, if the above problem has a solution, we say that there exists a controller ut = UX xt that guarantees stability of the saturated neutral system 4. Remark 3. In the derivation of heorem, two tuning parameters ε and ε 2 are introduced. An interesting question is how to find the desired values of these parameters. In this case, the desired values for ε and ε 2 obtained correspond to the largest bound of time delay for which it is possible to find a feasible solution. A way to overcome the computational difficulties of solving this problem consists in considering the following algorithm: Algorithm. Step. Let β = and fix h m0, h mstep, ε 0, ε 20, δ 0 small enough to have a feasible solution Q, R, W, X, X 2, X 3, U, G, Λ, σ i, i =,, 4 for 2. Step 2. Let h m = h m0 + h mstep, ε = ε 0, ε 2 = ε 20, and solve 2. Step 3. If 2 is feasible, let h m0 = h m, ε 0 = ε, ε 20 = ε 2, and go to Step 2. Otherwise, h m = h m0 h mstep. Step 4. If ε and ε 2 are sufficiently large go to Step 5, otherwise change ε and ε 2, and go to Step 3. Step 5. Stop, the desired values of ε and ε 2 are ε 0 and ε 20, respectively. 4 Numerical examples In this section, we provide three numerical examples to demonstrate that the proposed method gives less conservative results than the existing ones. Example. Consider the nominal neutral system provided in [3] in the form of with B = 0, A 0t = 0, A t = 0, Bt = 0, d = 0, d 2 = 0, and A 0 = , A = ,

6 460 International Journal of Automation and Computing 74, November 200 C = he results are compared in able. It can be seen that the delay-dependent stability condition of Lemma 3 is less conservative in the sense that the computed maximum delay bound is larger. able Comparison of maximal delay bounds of h m Methods h m Lien et al. [3] 0.3 Chen et al. [7] Fridman [9] 0.74 Lien and Chen [20] Park and Kwon [2].378 Yang et al. [23].533 Chen [6].5497 Lemma 3.79 Example 2. We consider a state-feedback example, taken from [4, 7] C = 0, A 0t = 0, A t = 0, Bt = 0 where A 0 =, A =, B = Now, we address the problem of finding a state-feedback controller for guaranteeing stability of the above system. Applying Corollary of this paper, we take ε = 0.36, ε 2 = 0.43, d = 0, and d 2 = 0. able 2 gives a comparison of several results of the maximum allowable bound of delay and corresponding control gain. able 2 Stability bound of h m and control gain K. B =, ū = 5. When d = 0, the delay is time-invariant. In [32], for A =, the upper bound on the time delay was found to be h m = In [], stabilization by a saturated memoryless state feedback law was accomplished for h m 0.35 with a maximum radius of the stability ball of Fridman et al. [2] gave a bound of h m =.854 according to heorem therein, while by using Matlab LMI oolbox we obtain h m =.7543, with a stability radius δ = By heorem, for d = 0, d 2 = 0, ε = 0., ε 2 = 0.88, β =, and h m.89, the closed-loop system is found to be asymptotically stable with the stabilizing gain K = and the stability ball radius δ = 0.9. For ε = 0.46, ε 2 = 0.82, and β =, the upper bound on the time delay is found to be h m 2. For h m = 2, the stabilizing gain is K = , with a stability ball radius of δ = able 3 gives a comparison of stability ball radii for various constant delay values. From able 3, we find that δ increases when the system time delay h m decreases, and our approach gives larger estimation of the domain of attraction. By numerical simulation, we show in Fig. the trajectories of the saturated closed-loop system and the domain of attraction for the state trajectories for h m = he outer ellipsoid is D e and the inner ellipsoid has a circle radius of δ = Methods h m Control gain Li and Souza [7] h m K = Fridman and Shaked [4] h m.5 K = Cho et al. [3] h m.6 K = heorem 3 in this paper h m.824 K = Example 3. Consider the example given in []. he system is described by with C = 0, A 0t = 0, A t = 0, Bt = 0 and A 0 = , A = , Fig. rajectories and estimate of the domain of attraction for h m = 0.35 able 3 Comparison of stability ball radii δ Methods δ = 0.35 δ = δ =.7543 δ =.8 δ =.89 Proposed method ε = 0., ε 2 = Fridman et al. [2] Infeasibility ε = 0, ε 2 = 0.95 Cao et al. [] Infeasibility

7 F. E. Haoussi and E. H. issir / Delay and Its ime-derivative Dependent Robust Stability of 46 5 Conclusions In this paper, we have presented several new delaydependent robust stabilization conditions for neutral systems with saturating actuators. he method is based on a transformation of the actuator saturation nonlinearities into a convex combination of polytope, Lyapunov-Krasovskii functional, and the free weighting matrices technique. A state feedback control law governing the stability of the system against perturbations is constructed by solving LMIs depending on two tuning parameters. o overcome the difficulty in finding the best values of the tuning parameters that give the largest bound of the delay ht, an iterative algorithm is proposed. An estimation of the domain of attraction is also proposed. It is also shown that our result is more general and less conservative than those of [4, 2]. 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