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Robust Delay-Dependent Stabilizing Control of Time-Delay Systems with State and Input Delays: Augmented L.K. Functional Approach
Robust Delay-Dependent Stabilizing Control of Time-Delay Systems with State and Input Delays: Augmented L.K. Functional Approach M.N.A. Parlak¸ cı ∗ I.B. K¨ u¸ cu ¨ kdemiral ∗∗ ∗
Department of Computer Science, Istanbul Bilgi University, Dolapdere, 34440, Istanbul, Turkey (e-mail: [email protected]). ∗∗ Department of Electrical Engineering, Yildiz Technical University, 34349, Be¸sikta¸s, Istanbul, Turkey (e-mail: [email protected]) Abstract: In this paper, we investigate the design problem of a stabilizing control for a class of linear uncertain time-delay systems with time-varying state and input delays. The control law is selected to be a state-feedback controller. Adopting to employ an augmented type of Lyapunov-Krasovskii functional, for the nominal case, we first derive some sufficient delaydependent stabilization criteria which can be solved using a convex optimization technique with interior-point algorithms. The stabilization synthesis is then extended to the case when the time-delay system is subject to the norm-bounded uncertainties which affect state and input matrices. Several numerical examples are presented to demonstrate the application of the proposed synthesis of a stabilizing controller. The numerical results on the maximum allowable delay bound and the uncertainty bound seem to be quite less conservative in comparison to the existing methods from the literature. Keywords: Input delay, state delay, norm-bounded uncertainty, stability, stabilization. 1. INTRODUCTION Time-delay phenomenon concerning feedback control systems is often a source of instability and poor performance in many engineering systems. From this point of view, a considerable amount of attention has been paid to the problem of stabilization and control of these systems. The time delay usually affects the system state, control input and state derivatives which lead to different classes of timedelay systems. The primary reasons for a delay to occur are in general due to the long transmission lines used for remote control systems, finite processing rate of computers, and/or inexact modeling of physical or dynamical systems. One can refer to References Hale (1977), Richard (2003), Gu et al. (2003) and Mahmoud (2000) for an earlier and recent investigations and surveys on time-delay systems. In stabilization point of view, most of the attention has been focused on the study of delay-dependent stabilization problem of systems having state and input delays, e.g. see Yue and Han (2005), Zhang et al. (2005), Zhang et al. (2009), Wang et al. (2007), as it is well-known that delaydependent methods generally are less conservative than delay independent ones. In real physical systems, input delays are frequently encountered because both measurement delays which can be a result of transmission of measurement information and computational delays are represented by input delay (Yue and Han (2005)). Moreover, the actuators that are employed for controlling the physical systems dynamically involve time-delays. For example, systems that use hy-
draulic or pneumatic actuators are usually subject to the effects of time-delay more extensively. From this point of view, control systems involving state delays can as well be considered to be exposed to actuator delays which justify to be investigated as a class of input delayed timedelay systems. In fact, retarded type of time-delay control systems studied in the literature can be regarded as a class of time-delay systems for which input delays are typically ignored. In order to examplify the occurrence of input delay explicitly in engineering systems, we can take into account a chemical plant in which several reactors are linked by some pipelines. In the chemical process, different kinds of liquids flow from one reactor to another. The behaviour of the entire chemical plant is governed by the chemical processes in each of these reactors. In particular, the output of one reactor affects the process in another reactor by acting as an input to it. In other words, there is a need to specify the dynamical behavior of the first reactor so that the input to the second reactor is known under the circumstance that a certain amount of time delay should be taken into account. This time delay in the input results as an access time required for the liquid flow to cross the pipeline from the first reactor to the second. In addition, the reactors may involve inherent time delays as a result of mixing and reaction of two different liquids. Consequently, an accurate modeling of such a chemical plant requires to consider simultaneously both input delays and most likely state delays which might be identical or different in length, (Habets (1994)). Hence, this gives the basic motivation for us to consider the stabilization problem for a class of time-delay systems with state and input delays.
For the sake of simplicity, we have initially considered the case of the same time-varying delay function appearing on the state and input leaving the work of considering such systems with noncoincident state and input delays as a future work. In recent years, the stabilization of input delayed systems are taken into account. A robust stabilizing controller design problem for uncertain time-delay systems having only input delay is considered in Yue and Han (2005). In particular, Yue and Han (2005) propose a novel design method that is based on reduction, first order transformation, cone-complementary linearization and memory type feedback. However, the memory controller scheme is strongly in need of the use of the exact delay information which is not always possible for any physical systems. Delay-dependent stabilization of linear time-delay systems having both state and input delays is also considered in Zhang et al. (2005) and Zhang et al. (2009). It is claimed by these authors that less conservative results can be obtained by use of different type of Lyapunov-Krasovskii functionals and some integral inequalities. For example, Zhang et al. (2005) introduces a new integral inequality for quadratic terms which significantly reduces conservatism. However, requirements for tuning four parameters makes the controller design process very crucial and impractical. On the other hand, the method of Zhang et al. (2009) utilizes a new type of augmented Lyapunov-Krasovskii functional together with the combination of free-weighting-matrix (FWM) technique and integral inequality to overcome the stabilization problem with delay-dependent methods. Nevertheless, this method is only valid for the stabilization of time-delay systems with constant delays. The robust stability and stabilization of a class of linear uncertain time-varying state delayed systems are studied in Parlak¸cı (2006) based on augmented type Lyapunov-Krasovskii functional. However, the author(s) do not consider any delay in the control input. Finally a novel LyapunovKrasovskii functional is also introduced in Wang et al. (2007) which leads to less conservative robust stabilization results for input-delay system having time-varying, ratebounded delays. In this paper, the problem of robust stabilization with memoryless state-feedback control is investigated for both nominal and uncertain state and input-delayed systems. The main objective of the paper is to seek for an improved linear matrix inequality (LMI) type conditions to ensure larger bounds for time-varying delays and uncertainties. The contribution of proposed methodology is based on several ideas. The first one is selecting a novel quasi-fullsize augmented Lyapunov Krasovskii functional to obtain less conservative results. The second aim is to employ the method of introducing relaxation with slack matrices and finally the third one is the use of Jensen’s inequality which allow us not to eliminate the useful negative terms that come from completion to squares in taking the timederivative of Lyapunov-Krasovskii functional. The rest of the paper is organized as follows: Section 2 presents the problem formulation. Main results are introduced in Section 3. Section 4 extends the idea to the robust stabilization of time-delay systems having norm bounded uncertainties. In Section 5, through a vast number of numerical examples, it is illustrated in large that
the proposed approach considerably improves the existing methods and gives better upper delay and norm bounds than those obtained earlier. Finally Section 6 concludes the paper. Notation: The notation to be used in the paper is fairly standard. R stands for the set of real numbers, Rm×n is the set of m × n dimensional real matrices. diag denotes the diagonal matrices, Trace stands for the trace operator, R+ symbolizes the set of positive real numbers. The identity and null matrices are denoted by I and 0, respectively. X ≻ 0(, ≺ 0) denotes that X is a positive definite (positive semi-definite, negative definite) matrix. X ≻ Y means that X −Y is positive definite. Finally, the notation ’∗’ denotes off diagonal block completion of a symmetric matrix. 2. PROBLEM FORMULATION Consider a class of time-delay system with time-varying state and input delays given as x(t) ˙ = Ax(t) + Ah x(t − h(t)) + Bu u(t) + Bh u(t − h(t)) ¯ 0] x(t) = φ(t), t ∈ [−h, (1) where x(t) ∈ Rn is the state vector, u(t) ∈ Rmu is the control input. Then A, Ah , Bu , and Bh are known real constant state space matrices of appropriate dimensions. On the other hand, the delay h(t) is assumed to be a continuous, time-varying function which satisfies ˙ 0 ≤ h(t) < ¯h, |h(t)| ≤ µ, ∀t ≥ 0. (2) ¯ and µ are known positive constants. Besides, where, h φ(t) is a continuous and differentiable vector valued initial condition function for states. Then our goal is to find a suitable state-feedback control law in the form of u(t) = Kx(t), such that the closed-loop system exhibits an asymptotically stable behavior for any h(t) satisfying (2). Substituting the state-feedback control law into (1) yields the closed-loop system ¯ 0](3) ¯ x(t) ˙ = Ax(t) + A¯h x(t − h(t)), x(t) = φ(t), t ∈ [−h, where A¯ , A + Bu K and A¯h , Ah + Bh K. 3. MAIN RESULTS ¯ µ, the Theorem 1. Given nonnegative scalar constants h, closed-loop system (3) is asymptotically stable for all h(t) satisfyingh (2), ifi there exist h i symmetric positive definite ¯ ¯ ¯ ¯ W ¯ and matrices matrices X P¯12 , Z11 Z¯12 , Z¯1 , Z¯2 , T , Q, ⋆ P22
⋆
Z22
¯i , (i = 1, . . . , 5), S¯i , (i = 1, . . . , 5), with appropriate L, N dimensions such that ¯¯ ¯¯ ¯ ¯ T ¯ ¯T
Ψ11 Ψ12 Ψ13 ∗ Ψ22 Ψ23 ∗ ∗ Ψ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ψ14 Ψ24 Ψ34 ¯11 −Z ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ψ15 hZ11 hZ12 Ψ25 0 0 Ψ35 0 0 0 0 0 ¯11 −Z 0 0 ¯11 −Z ¯12 ∗ −Z ¯22 ∗ ∗ −Z ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
hS1 Ψ110 ¯S ¯T Ψ210 h 2 ¯S ¯T h 0 3 ¯S ¯T h 0 4 ¯S ¯T h 0 5 5 0 0 0 0 0 0 ¯Z ¯1 −h 0 0 ¯ ¯2 0 ∗ − hZ ∗ ∗ −T ∗ ∗ ∗ hN 1 ¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯N ¯T h
Z¯22 − Z¯1 ≻ 0
0 0 0 T P¯12 T P¯12 ¯Z ¯12 h ¯Z ¯22 h 0 0 0 Ψ1111
≺0 (4) (5)
Z¯22 − Z¯2 ≻ 0 (6) T where Ψ11 = AX + XAT + Bu L + LT BuT + P¯12 + P¯12 + Z¯1 − ¯ W ¯ +N ¯ T +N ¯1 , Ψ12 = Ah X+Bh L+Z¯22 −Z¯1 −N ¯T+ Z¯22 +Q+ 1 1 T ¯2 + S¯1T , Ψ13 = −P¯12 + N ¯3 − S¯1T , Ψ14 = P¯22 − Z¯12 ¯4 , N +N ¯5 , Ψ110 = XAT +LT B T , Ψ22 = −(1−µ)Q− ¯ Ψ15 = P¯22 + N u ¯ T −N ¯2 + S¯T + S¯2 + Z¯2 − Z¯1 , Ψ23 = Z¯22 − Z¯2 − N ¯3 − S¯T + S¯3 , N 2 2 2 T T ¯4 +S¯4 , Ψ25 = −Z¯12 −N ¯5 +S¯5 , Ψ210 = XAT + Ψ24 = Z¯12 −N h ¯ − S¯T − S¯3 , Ψ34 = −P¯22 − S¯4 , LT BhT , Ψ33 = Z¯2 − Z¯22 − W 3 T Ψ35 = −P¯22 + Z¯12 − S¯5 , Ψ1111 = −XT −1X. Moreover, the control law u(t) = LX −1 x(t) is a stabilizing controller for the resulting closed-loop system (3). Proof: Let us choose a Lyapunov Krasovskii functional candidate as V (t, xt ) = V1 + V2 + V3 + V4 where Rt V1 = η T (t)P η(t), η T (t) = [xT (t) t−h¯ xT (s)ds], P = P11 P12 R R ¯ 0 ¯ t ζ T (s)Zζ(s)dsdβ, where ζ T (s) = ∗ P22 , V2 = h −h t+β Rt T 12 [xT (s) x˙ T (s)] and Z = Z∗11 Z Z22 , V3 = t−h(t) x (s)Qx(s)ds, Rt V4 = t−h¯ xT (s)W x(s)ds where xt is the function x(θ) ¯ t]. Taking the time derivative of V with θ ∈ [t − h, along the state trajectory of system (1), (2), (3) yields P4 V˙ (t, xt ) = i=1 V˙ i . On the other hand, let us define an extended state vector as T ¯ χ (t) , xT (t) xT (t − h(t)) xT (t − h) Z
t
xT (s)ds
t−h(t)
Z
t−h(t)
¯ t−h
xT (s)ds .
(7)
¯ 2 ζ T (t)Zζ(t) V˙ 2 ≤ h Z t Z t Z Z12 − ζ T (s)ds 11 ζ T (s)ds ∗ Z − Z 22 1 t−h(t) t−h(t) Z t−h(t) Z t−h(t) Z Z12 − ζ T (s)ds 11 ζ T (s)ds ∗ Z22 − Z2 t−h¯ ¯ t−h Note that with the following definition I 0 00 0 Γ3 , ¯ ¯ A Ah 0 0 0
(13)
we can rewrite we define Γ4 , ζ(t) = Γ3 χ(t). Similarly, 0 0 0I 0 0 0 0 0 I and Γ5 , . Then we can write I −I 0 0 0 0 I −I 0 0 Z t Z t−h(t) x(s) x(s) ds = Γ4 χ(t), ds = Γ5 χ(t). x(s) ˙ x(s) ˙ ¯ t−h(t) t−h
Thus, in view of the definitions of Γ4 and Γ5 we get V˙ 2 ≤ χT (t)Ω2 χ(t) (14) where Z12 T Z11 ¯ 2 ΓT ZΓ3 −ΓT Z11 Z12 Ω2 = h 3 4 ∗ Z22 −Z1 Γ4 −Γ5 ∗ Z22 −Z2 Γ5 . The time-derivative of V3 is computed as V˙ 3 ≤ xT (t)Qx(t) − (1 − µ)xT (t − h(t))Qx(t − h(t)) Describing Γ6 , [I 0 0 0 0], Γ7 , [0 I 0 0 0] and Ω3 = ΓT6 QΓ6 − (1 − µ)Γ7 QΓ7 leads to write V˙ 3 ≤ χT (t)Ω3 χ(t). (15) ˙ Finally, we compute V4 as follows:
Then, defining Γ1 , [ I0 00 00 0I 0I ] allows to hwrite η(t) =i ¯A ¯h 0 0 0 Γ1 χ(t). In a similar manner, we define Γ2 , A I 0 −I 0 0 to obtain η(t) ˙ = Γ2 χ(t). Then we obtain V˙ 1 = 2η T (t)P η(t) ˙ = χT (t)Ω1 χ(t) (8)
¯ ¯ V˙ 4 = xT (t)W x(t) − x(t − h)W x(t − h) (16) ¯ = Γ8 χ(t), where Γ8 , [0 0 I 0 0], We can define x(t − h) then (16) can be rewritten as V˙ 4 = χT (t)Ω4 χ(t). (17)
where
where Ω4 , ΓT6 W Γ6 − ΓT8 W Γ8 . It follows from NewtonLeibnitz relation that we can always construct the following null equations: " # Z
Ω1 = ΓT1 P Γ2 + ΓT2 P Γ1 . On the other hand Z t ¯ 2 ζ T (t)Zζ(t) − h ¯ ζ T (s)Zζ(s)ds V˙ 2 = h ¯ 2 ζ T (t)Zζ(t) − h ¯ =h
¯ t−h Z t
(9)
t
2χT (t)N T Γ9 χ(t) −
¯ −h
t−h(t)
¯ t−h
T
ζ (s)Zζ(s)ds
T
2χ (t)S (10)
Note that from Jensen’s inequality we have Z t Z11 Z12 T ¯ −h ζ (s) ζ(s)ds ∗ Z22 t−h(t) Z t Z t Z Z ≤− ζ T (s)ds 11 12 ζ T (s)ds (11) ∗ Z 22 t−h(t) t−h(t)
and similarly
t−h(t)
Z11 Z12 ζ(s)ds ∗ Z22 ¯ t−h Z t−h(t) Z t−h(t) Z11 Z12 T ≤− ζ (s)ds ζ T (s)ds. (12) ∗ Z22 t−h¯ ¯ t−h Then using the relations given in (11), (12) and in view of (5) and (6), we obtain ¯ −h
Z
ζ T (s)
(18)
#
(19)
and
ζ T (s)Zζ(s)ds
t−h(t)
Z
x(s)ds ˙ =0
t−h(t)
T
"
Γ10 χ(t) −
Z
t−h(t) ¯ t−h
x(s)ds ˙ = 0,
where N , [N1 N2 N3 N4 N5 ], S , [S1 S2 S3 S4 S5 ], Γ9 = [I −I 0 0 0 0], and Γ10 = [0 I −I 0 0]. Then completing (18) and (19) to squares gives us the following: 0 ≤ χT (t)Ω5 χ(t), T
(20)
0 ≤ χ (t)Ω6 χ(t) (21) T T −1 T ¯ where Ω5 , N Γ9 + Γ9 N + hN Z1 N and Ω6 , S Γ10 + ¯ T Z −1 S. As a result, substituting V˙ i , i = 1, . . . , 4 ΓT10 S + hS 2 computed in (8), (14), (15) and (17) into V˙ (t, xt ) and adding relaxation terms of (18) and (19) allows to obtain V˙ (t, xt ) ≤ χT (t)Ωχ(t) (22) P6 where Ω = i=1 Ωi . It appears that if Ω ≺ 0 is established, then V˙ (t, xt ) ≤ χT (t)Ωχ(t) < 0 is achieved and the closedloop system is guaranteed to be globally asymptotically stable for every h(t) satisfying (2). T
Using Schur complement formula, Ω ≺ 0 is congruent to ¯ T ¯ T Φ11 Φ12 Φ13 ∗ Φ22 Φ23 ∗ ∗ Φ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Φ14 Φ24 Φ34 Φ44 ∗ ∗ ∗ ∗ ∗
Φ15 Φ16 Φ17 hN1 hS1 ¯ T ¯ T hS Φ25 Φ26 Φ27 hN 2 2 ¯ T ¯ T hS Φ35 0 0 hN 3 3 ¯ T ¯ T hS 0 0 0 hN 4 4 ¯ T ≺ 0 ¯ T hS Φ55 0 0 hN 5 5 ∗ −Z11 −Z12 0 0 ∗ ∗ −Z22 0 0 ¯ ∗ ∗ ∗ −hZ1 0 ¯ 2 ∗ ∗ ∗ ∗ −hZ T P11 A¯ + A¯T P11 + P12 + P12 + Z1 −
(23)
where Φ11 = Z22 + Q + W + N1T + N1 , Φ12 = P11 A¯h + Z22 − Z1 − N1T + N2 + S1T , Φ13 = −P12 + N3 − S1T , Φ14 = A¯T P12 + P22 − T Z12 + N4 , Φ15 = A¯T P12 + P22 + N5 , Φ22 = −(1 − µ)Q − T ¯ A¯T Z T , N2 − N2 + S2T + S2 + Z2 − Z1 , Φ16 = ¯ hZ11 + h 12 T ¯ ¯ ¯ Φ17 = hZ12 + hA Z22 , Φ23 = Z22 − Z2 − N3 − S2T + S3 ,, T T Φ24 = A¯Th P12 + Z12 − N4 + S4 , Φ25 = A¯Th P12 − Z12 − T T T ¯ ¯ N5 + S5 , Φ33 = Z2 − Z22 − W − S3 − S3 , Φ26 = hAh Z12 , ¯ A¯T Z22 , Φ34 = −P22 − S4 , Φ35 = −P22 + Z T − S5 , Φ27 = h 12 h Φ44 = −Z11 , Φ55 = −Z11 . Pre- and post multiplying (23) by diag{X, X, X, X, X, X, X, X, X} −1 where X , P11 and employing the variable changes (·) = X(·)X lead to Φ¯ Φ¯ Φ¯ Φ¯ Φ¯ Φ¯ ¯ ¯N ¯T h ¯S ¯T T Φ h 11
12
13
14
15
17
1 ¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯ T hN 5
1 ¯S ¯T h 2 ¯S ¯T h 3 ¯S ¯T h 4 ¯S ¯T h 5
¯ 24 Φ ¯ 25 Φ ¯ 26 Φ ¯ 27 Φ ¯ 34 Φ ¯ 35 Φ 0 0 ¯ 44 0 Φ 0 0 ≺ 0 (24) ¯ ∗ Φ55 0 0 ¯11 −Z ¯12 ∗ ∗ −Z 0 0 ¯22 ∗ ∗ ∗ −Z 0 0 ¯Z ¯1 ∗ ∗ ∗ ∗ −h 0 ¯Z ¯2 ∗ ∗ ∗ ∗ ∗ −h ¯ 11 = AX ¯ + X A¯T + P¯12 + P¯ T + Z¯1 − Z¯22 + Q ¯+ where Φ 12 ¯ +N ¯T + N ¯1 , Φ ¯ 12 = A¯h X + Z¯22 − Z¯1 − N ¯T + N ¯2 + S¯T , W 1 1 1 ¯ 13 = −P¯12 + N ¯3 − S¯1T , Φ ¯ 14 = X A¯T X −1 P¯12 + P¯22 − Φ T ¯5 , Φ ¯ 22 = −(1 − ¯4 , Φ ¯ 15 = X A¯T X −1 P¯12 + P¯22 + N Z¯12 +N ¯ Z¯11 + ¯−N ¯T − N ¯2 + S¯T + S¯2 + Z¯2 − Z¯1 , Φ ¯ 16 = h µ)Q 2 2 ¯ Z¯12 + hX ¯ A¯T X −1 Z¯22 , Φ ¯ A¯T X −1 Z¯ T , Φ17 = h ¯ = Z¯22 − hX 23 12 T T T −1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Z2 − N3 − S2 + S3 , Φ24 = X Ah X P12 + Z12 − N4 + S¯4 , T ¯5 + S¯5 , Φ ¯ 33 = Z¯2 − Z¯22 − ¯ 25 = X A¯T X −1 P¯12 − Z¯12 −N Φ h T T −1 ¯ T ¯ ¯ ¯ ¯ ¯ ¯ W − S3 − S3 , Φ26 = hX Ah X Z12 , Φ27 = ¯ hX A¯Th X −1 Z¯22 , T ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 44 = −Z¯11 , Φ34 = −P22 − S4 , Φ35 = −P22 + Z12 − S5 , Φ
¯ Φ=
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯ 22 Φ ¯ 23 Φ ¯ 33 ∗ Φ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
16
¯ 55 = −Z¯11 . Φ ¯ can be decomposed as Note that Φ ¯ =Φ ¯0 + Φ ¯1 + Φ ¯T ≺ 0 Φ 1 where ¯ 11 Φ ¯ 12 Φ ¯ 13 Φ ∗ Φ¯ 22 Φ¯ 23 ∗ ∗ Φ¯ 33
¯0 = Φ
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
˜ ¯ 14 Φ ˜ ¯ Φ24 ¯ 34 Φ ¯ 44 Φ ∗ ∗ ∗ ∗ ∗
˜ ˜ ˜ ¯ 15 Φ ¯ 16 Φ ¯ 17 Φ ˜ ¯ Φ25 0 0 ¯ 35 Φ 0 0 0 0 0 ¯ 55 Φ 0 0 ¯11 −Z ¯12 ∗ −Z ¯22 ∗ ∗ −Z ∗ ∗ ∗ ∗ ∗ ∗
(25)
¯N ¯T h 1 ¯ ¯ hN2T ¯N ¯T h 3 ¯ ¯ hN4T ¯ T hN 5
¯S ¯T h 1 ¯ ¯ hS2T ¯S ¯T h 3 ¯ ¯ hS4T ¯S ¯T h 5
0 0 0 0 ¯Z ¯1 −h 0 ¯ ¯2 ∗ −hZ
(26)
˜¯ ¯ ¯T ¯ ˜ ¯ ¯ ¯ ˜¯ with Φ 14 = P22 − Z12 + N4 , Φ15 = P22 + N5 , Φ16 = ˜¯ = ¯hZ¯ , Φ ˜ ˜ ¯ Z¯11 , Φ ¯ 24 = Z¯ T − N ¯4 + S¯4 , Φ ¯ 25 = −Z¯ T − h 17 12 12 12 ¯ ¯ N5 + S5 and ¯ ¯ ¯ ¯ ¯1 = Φ
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
¯ 16 Φ ¯ 17 ¯ 14 Φ ¯ 15 Φ 0Φ ¯ ¯ ¯ ¯ 0Φ 24 Φ25 Φ26 Φ27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
(27)
¯ = X A¯T X −1 P¯ , Φ ¯ = ¯ 15 = X A¯T X −1 P¯12 , Φ with Φ 14 12 16 ¯ T −1 ¯ T ¯ T −1 ¯ T ¯ ¯ ¯ ¯ ¯ hX A X Z12 , Φ17 = hX A X Z22 , Φ24 = X Ah X −1 P¯12 , ¯ ¯ ¯ ¯ ¯T −1 Z¯ T , Φ ¯T −1 P¯12 , Φ Φ 25 = X Ah X 26 = hX Ah X 27 = 12 T −1 ¯ A¯ X Z¯22 . Besides Φ ¯ 1 can be rewritten as Φ ¯1 = hX h ΠT1 X −1 Π2 where ¯ A¯h X 0 0 0 0 0 0 0 , Π1 = AX T ¯ ¯T Π2 = 0 0 0 P¯12 P¯12 ¯hZ¯12 hZ12 0 0 . Also note that for any symmetric positive definite matrix T , the following bounding inequality always holds: ΠT1 X −1 Π2 + ΠT1 X −1 Π2
T
ΠT1 T −1 Π1 + ΠT2 XT −1 X Hence, substituting (28) into (25) gives
−1
Π2 .
(28)
¯ Φ ¯ 0 + ΠT1 T −1 Π1 + ΠT2 XT −1 X −1 Π2 0 (29) Φ Then applying Schur complement formula to (29), replacing A¯ with A + Bu K and A¯h with Ah + Bh K, and defining L , KX immediately lead to the matrix inequality given in (4) In the light of the previous theorem, in order to achieve maximum ¯h such that the closed-loop system stays stable, one can solve the following optimization problem for the symmetric, positive definite decision matrices X, Z¯11 , Z¯22 , ¯ W ¯ , matrices L, P¯12 , Z¯12 , N ¯i , (i = 1, . . . , 5), Z¯1 , Z¯2 , T , Q, S¯i , (i = 1, . . . , 5): ¯ max h subject to (4) − (6)
(30)
If the problem described in (30) has a feasible solution set, then state-feedback control law, u = LX −1 x(t), is said to be the suboptimal stabilizing controller for the given stabilization problem. Note that the matrix inequality condition (4) is not in the form of an LMI so that it can be solved using convex optimization algorithms. In other words, we can not find a global maxima for (30) using convex optimization algorithms. However, if one affords more computational efforts such as cone-complementary algorithm, one may still obtain a suboptimal controller for the problem defined in Section 2 using an iterative algorithm presented in the following sequel. First, we define a new variable R = RT ≻ 0 such that R XT −1 X and replace the condition (4) with ¯ X ¯ ¯ I ¯ I R R X T¯ I ¯ T¯ 0, I R 0, I X 0, I T 0 (31) X ¯¯ ¯¯ ¯ ¯ T ¯ ¯T
Ψ11 Ψ12 Ψ13 ∗ Ψ22 Ψ23 ∗ ∗ Ψ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ψ14 Ψ15 hZ11 hZ12 Ψ24 Ψ25 0 0 Ψ34 Ψ35 0 0 ¯11 −Z 0 0 0 ¯11 ∗ −Z 0 0 ¯11 −Z ¯12 ∗ ∗ −Z ¯22 ∗ ∗ ∗ −Z ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
hS1 Ψ110 ¯S ¯T Ψ210 h 2 ¯S ¯T h 0 3 ¯S ¯T h 0 4 ¯S ¯T h 0 5 5 0 0 0 0 0 0 ¯Z ¯1 −h 0 0 ¯Z ¯2 0 ∗ −h ∗ ∗ −T ∗ ∗ ∗ hN 1 ¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯N ¯T h
0 0 0 T P¯12 T P¯12 ¯Z ¯12 h ¯Z ¯T h
≺ 0, 22 0 0
0 −R
(32) which can be justified following the inequality of R−1 − X −1 T X −1 0 and using Schur complement formula
¯ , R−1 , X ¯ , X −1 and T¯ , T −1 , and defining R allows to obtain the inequalities in (31). Then, in order to achieve a maximum allowable delay bound for the closedloop system (3), one may use the following optimization problem: ¯ + XX ¯ + T¯T ) min Trace (RR subject to the LMIs (5), (6), (32), (31), Finally to achieve the sub-optimal controller, one can use the following algorithm to solve the cone-complementary algorithm (El Ghaoui et al. (1997)): Algorithm: ¯ and µ such that (1) Choose a sufficiently small initial h there exists a feasible solution set ¯ 0 , X0 , R ¯ 0 , R0 , T¯0 , T0 , } {X to the LMI conditions in (5), (6), (32), (31) and set k = 0. (2) Solve the following LMI optimization problem for the ¯ X, R, ¯ R, T¯, T, } variables {X, ¯k R + X ¯ k X + T¯k T + RR ¯ k + XX ¯ k + T¯Tk ) min Trace (R subject to the LMIs (5), (6), (32), (31) ¯ k+1 := X, ¯ Xk+1 := X, R ¯ k+1 := R, ¯ and set: X Rk+1 := R, T¯k+1 := T¯, Tk+1 := T . (3) If R XT −1 X is feasible for the above solution set ¯o = h ¯ and µo = µ then return to Step (1) then set h ¯ and µ to some extent. Otherwise, after increasing h set k = k + 1 and go to Step (2) and repeat the optimization for a pre-specified number of iterations, say kmax , until finding a feasible solution satisfying R XT −1X. If such a solution does not exist, then exit. ¯ o = h, ¯ to If one can find a feasible solution set, µo = µ and h ¯ o is said to be the maximum allowable this algorithm than h delay bound for the closed-loop system (3). Moreover, suboptimal stabilizing control law for the system can be constructed as u(t) = LX −1 x(t). 4. ROBUST STABILIZATION
¯ µ, the Theorem 2. Given nonnegative scalar constants h, uncertain system (33) can be stabilized by a memoryless state feedback control law u(t) = LX −1 x(t) for all h(t) satisfyingh(2), ifi there exist h i symmetric positive definite ¯11 Z ¯12 X P¯12 Z ¯ ¯ ¯ ¯ matrices ⋆ P¯22 , ¯22 , Z1 , Z2 , T , Q, W matrices ⋆ Z ¯i , (i = 1, . . . , 5), S¯i , (i = 1, . . . , 5) all in appropriate L, N dimensions, and positive scalar ǫ, such that Ψ ¯ ¯Z ¯ ¯Z ¯ ¯N ¯T h ¯S ¯T Ψ ¯ ¯ h h h 0 Ψ Ψ Ψ Ψ Ψ 11
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
12
13
14
15
11
12
Ψ22 Ψ23 Ψ24 Ψ25 0 0 ∗ Ψ33 Ψ34 Ψ35 0 0 ¯11 ∗ ∗ −Z 0 0 0 ¯11 ∗ ∗ ∗ −Z 0 0 ¯11 −Z ¯12 ∗ ∗ ∗ ∗ −Z ¯22 ∗ ∗ ∗ ∗ ∗ −Z ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
1
¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯N ¯T h
1
110
¯S ¯T Ψ210 h 2 ¯S ¯T h 0 3 ¯S ¯T h 0 4 ¯S ¯T h 0 5 5 0 0 0 0 0 0 ¯Z ¯1 −h 0 0 ¯ ¯ ∗ −hZ2 0 ¯ ∗ ∗ Ψc ∗ ∗ ∗ ∗ ∗ ∗
a
0 0 T P¯12 T P¯12 ¯Z ¯12 h ¯Z ¯22 h 0 0 0 ¯ Ψe ∗
¯b Ψ 0 0 0 0 0 0 0 ¯d Ψ 0 −ǫI
≺0
(34) T ¯ 11 = AX + XAT + Bu L + LT BuT + P¯12 + P¯12 where Ψ + Z¯1 − ¯ +W ¯ +N ¯1T + N ¯1 + ǫGGT , Ψ ¯ 110 = ǫGGT + XAT + Z¯22 + Q ¯ a = XE T + LT E T , Ψb = XE T + LT E T , LT BuT , Ψ A Bu Ah Bh ¯ c = ǫGGT − T , Ψ ¯ d = XE T + LT E T , Ψ ¯ e = −XT −1X. Ψ A Bu Proof: Replacing A with A + ∆A, Bu with Bu + ∆Bu , Ah with Ah + ∆Ah and finally Bh with Bh + ∆B h in (4), and defining J T , GT 0 0 0 0 0 0 0 0 GT 0 and H , [ EA X + EBu L EAh X + EBh L 0 0 0 0 0 0 0 0 0 ] we obtain the condition Ψ + JF (t)H + H T F T (t)J T ≺ 0 instead of (4). Note that Ψ + JF (t)H + H T F T (t)J T ≺ 0 holds for all F (t) satisfying F T (t)F (t) ≺ I if there exist ǫ > 0 such that Ψ + ǫJJ T + ǫ−1 H T H ≺ 0. Hence, using Schur complement formula immediately leads to the matrix inequality condition (34) which concludes the proof.
As described in the previous section, replacing the term −XT −1X ≺ 0 with R in (34) such that R XT −1 X, one can still use the cone-complementary linearization algo¯ and γ such that rithm to identify the maximum allowable h the closed-loop system exhibits globally asymptotically stable behaviour.
Now consider a class of uncertain time-delay system with time-varying state and input delays given as
5. NUMERICAL EXAMPLES
x(t) ˙ = [A + ∆A] + x(t) + [Ah + ∆Ah ]x(t − h(t))
Now we present three numerical examples to illustrate the design procedure:
+[Bu + ∆Bu ]u(t) + [Bh + ∆Bh ]u(t − h(t)) ¯ 0] x(t) = φ(t), t ∈ [−h, (33)
Example 1.
where x(t) ∈ Rn is the state vector, u(t) ∈ Rmu is the control input. Then A, Ah , Bu , Bh are known real constant matrices. On the other hand ∆A, ∆Ah , ∆Bu and ∆Bh are matrix valued functions representing time varying uncertainties. These parameter uncertainties are assumed to be norm bounded and of the form, [∆A ∆Ah ∆Bu ∆Bh ] = GF (t) [EA EAh EBu EBh ] where G, EA , EAh and EBh are known constant real matrices in appropriate dimensions which represent the structure of uncertainties, and F (t) is an unknown matrix function with Lebesgue measurable elements and satisfies F T (t)F (t) I for all t ≥ 0. Note that all matrices are with appropriate dimensions.
Consider system (1) with
−1 −1 A= −0.5 0 −2 −0.2 Ah = 0.5 0
1 0.2 0.5 0
0.5 1 0.3 0
−0.5 −1 0 0
0.5 0 , 0.5 1.2
1 1 Bu = 1 0
0 −0.5 0.5 0 , −2 −0.5 0 −1
1 1 Bh = . 1 1
Applying Theorem 1 to this system by using conecomplementary algorithm presented in Section 3 and setting µ = 0, we find that the system can be stabilized
¯ = 0.73. This result is obfor any h satisfying h ≤ h tained in 224 iterations by cone-complementary algorithm. When h = 0.73, the controller gain is obtained as K = [0.4547 0.0294 −0.2736 −0.2399] . This result is less conservative than the results obtained by Zhang et al. (2005) and Zhang et al. (2009). For this system, using the method of Zhang et al. (2005), we calculate the upper bound, ¯h as 0.27 using the parameters λ1 = −1.1436, λ2 = −0.9341, µ1 = 1.0116, µ2 = 0.9324. Zhang et al. (2009) calculate ¯ as 0.6 with the controller the maximum delay bound, h, gain K = [0.038 0.003 − 0.033 − 0.551]. Example 2. The second example is concerned with stabilizing the input delay system (1) without any state delays. The system matrices of (33) given appropriately as follows: 1 0 0 0 . , Bh = , Bu = A= 0 0 1 −5 The matrices in description of system uncertainties are given by EA = I, EAh = EBu = EBh = 0, and G = γI, where γ > 0 is some scalar. In order to make a fair ¯ = 0.4 in conecomparison, if we stop updating h at h complementary algorithm presented in previous section, we calculate the uncertainty bound γmax = 2.01. However, applying Theorem 1 of Chen and Zheng (2006) to this ¯ = 0.4. example, it is found that γmax = 1.412 for h Using the proposed method of this note, we obtain the stabilizing controller gain as K = [−2.5626 −0.0948] . Similarly applying Theorem 12 in Yue and Han (2005) to this example, it is found that γmax = 0.5998. For this system, setting γ to the upper bound 1.412 as found ¯ = in Chen and Zheng (2006), we find delay-bound as h 0.5050 at 2 steps which is much bigger than 0.4. For this value of γ, we obtain the stabilizing controller gain as K = [−1.8037 −0.0426] . During all these simulations, maximum number of iterations is chosen to be as 30 and step sizes for h and γ are set to 0.005. Example 3. Consider system (33) with uncertainties and input time-varying delay where 0 1 0 0 0 A= , Ah = , Bh = , |q| ≤ γ −1.25 −3 q 0 1 We consider the case where 0 ≤ h(t) ≤ 0.2. Table 1 presents the maximum allowable γ values, γmax , for different values of µ both obtained by our method and recent methods presented in literature. The results show that our approach is significantly less conservative than the existing methods. Discarding V3 in the Lyapunov-Krasovskii functional by simply setting Q = 0, it is obtained that the controller gain K = [−47.5 − 7.96] guarantees robust stabilization of this system for unknown µ. Table 1. Comparison of proposed method with some previous results in terms of uncertainty bound γmax . µ 0 0.1 0.9 10 Unknown
Yue and Han (2005) 9.8615 9.8538 9.8506 Not reported Not reported
Wang et al. (2007) 26.4502 20.9157 13.4176 13.4176 Not reported
Theorem 2 34.45 34.45 34.45 34.45 34.45
As a result, the above three examples clearly indicate that the proposed stabilization method of this note is less conservative than the results given in literature. 6. CONCLUSION This paper proposes an improved stabilization technique for time-delay systems having both time-varying state and input delays. The stabilization criteria are derived by constructing a novel augmented Lyapunov-Krakovskii functional and employing a relaxation approach by use of slack matrices and Jensen’s inequality. Moreover, the method is extended to the robust stabilization problem of time-delay systems subject to norm bounded uncertainties. Finally, the effectiveness of the proposed method is demonstrated through a number of numerical examples, which illustrate that the presented method is quite less conservative in comparison to those of some earlier reported ones in the recent literature. REFERENCES Chen, W. and Zheng, W. (2006). On improved robust stabilisation of uncertain systems with unknown input delay. Automatica, 42, 1067–1072. El Ghaoui, L., Qustry, F., and Ait Rami, M. (1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42(8), 1171–1176. Gu, K., Kharitonov, V., and Chen, J. (2003). Stability of time delay systems. Birkhauser:Basel, Boston. Habets, L.C.G.J.M. (1994). Algebraic and computational aspects of time-delays. . Eindhoven University, Eindhoven. Hale, J. (1977). Theory of Functional Differential Equations. Springer:Berlin. Mahmoud, M.S. (2000). Robust control and Filtering for Time-Delay Systems. Marcel Dekker Inc. Parlak¸cı, M.N.A. (2006). Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems. Int. J. of Robust Nonlinear Control, 16, 599–636. Richard, J.P. (2003). Time-delay systems: an overview of some recent advances and open problems. Automatica, 39, 1667–1694. Wang, Z., Goldsmith, P., and Tan, D. (2007). Improvement on robust control of uncertain systems with timevarying input delays. IET Control Theory Appl., 1(1), 189–194. Yue, D. and Han, Q.L. (2005). Delayed feedback control of uncertain systems with time-varying input delay. Automatica, 41, 233–240. Zhang, X.M., Wu, M., She, J., and He, Y. (2005). Delay dependent stabilisation of linear systems with timevarying state and input delays. Automatica, 41, 1405– 1412. Zhang, X., Li, M., Wu, M., and She, J. (2009). Further results on stability and stabilisation of linear systems with state and input delays. Int. Journal of Systems Science, 40(1), 1–10.
Department of Computer Science, Istanbul Bilgi University, Dolapdere, 34440, Istanbul, Turkey (e-mail: [email protected]). ∗∗ Department of Electrical Engineering, Yildiz Technical University, 34349, Be¸sikta¸s, Istanbul, Turkey (e-mail: [email protected]) Abstract: In this paper, we investigate the design problem of a stabilizing control for a class of linear uncertain time-delay systems with time-varying state and input delays. The control law is selected to be a state-feedback controller. Adopting to employ an augmented type of Lyapunov-Krasovskii functional, for the nominal case, we first derive some sufficient delaydependent stabilization criteria which can be solved using a convex optimization technique with interior-point algorithms. The stabilization synthesis is then extended to the case when the time-delay system is subject to the norm-bounded uncertainties which affect state and input matrices. Several numerical examples are presented to demonstrate the application of the proposed synthesis of a stabilizing controller. The numerical results on the maximum allowable delay bound and the uncertainty bound seem to be quite less conservative in comparison to the existing methods from the literature. Keywords: Input delay, state delay, norm-bounded uncertainty, stability, stabilization. 1. INTRODUCTION Time-delay phenomenon concerning feedback control systems is often a source of instability and poor performance in many engineering systems. From this point of view, a considerable amount of attention has been paid to the problem of stabilization and control of these systems. The time delay usually affects the system state, control input and state derivatives which lead to different classes of timedelay systems. The primary reasons for a delay to occur are in general due to the long transmission lines used for remote control systems, finite processing rate of computers, and/or inexact modeling of physical or dynamical systems. One can refer to References Hale (1977), Richard (2003), Gu et al. (2003) and Mahmoud (2000) for an earlier and recent investigations and surveys on time-delay systems. In stabilization point of view, most of the attention has been focused on the study of delay-dependent stabilization problem of systems having state and input delays, e.g. see Yue and Han (2005), Zhang et al. (2005), Zhang et al. (2009), Wang et al. (2007), as it is well-known that delaydependent methods generally are less conservative than delay independent ones. In real physical systems, input delays are frequently encountered because both measurement delays which can be a result of transmission of measurement information and computational delays are represented by input delay (Yue and Han (2005)). Moreover, the actuators that are employed for controlling the physical systems dynamically involve time-delays. For example, systems that use hy-
draulic or pneumatic actuators are usually subject to the effects of time-delay more extensively. From this point of view, control systems involving state delays can as well be considered to be exposed to actuator delays which justify to be investigated as a class of input delayed timedelay systems. In fact, retarded type of time-delay control systems studied in the literature can be regarded as a class of time-delay systems for which input delays are typically ignored. In order to examplify the occurrence of input delay explicitly in engineering systems, we can take into account a chemical plant in which several reactors are linked by some pipelines. In the chemical process, different kinds of liquids flow from one reactor to another. The behaviour of the entire chemical plant is governed by the chemical processes in each of these reactors. In particular, the output of one reactor affects the process in another reactor by acting as an input to it. In other words, there is a need to specify the dynamical behavior of the first reactor so that the input to the second reactor is known under the circumstance that a certain amount of time delay should be taken into account. This time delay in the input results as an access time required for the liquid flow to cross the pipeline from the first reactor to the second. In addition, the reactors may involve inherent time delays as a result of mixing and reaction of two different liquids. Consequently, an accurate modeling of such a chemical plant requires to consider simultaneously both input delays and most likely state delays which might be identical or different in length, (Habets (1994)). Hence, this gives the basic motivation for us to consider the stabilization problem for a class of time-delay systems with state and input delays.
For the sake of simplicity, we have initially considered the case of the same time-varying delay function appearing on the state and input leaving the work of considering such systems with noncoincident state and input delays as a future work. In recent years, the stabilization of input delayed systems are taken into account. A robust stabilizing controller design problem for uncertain time-delay systems having only input delay is considered in Yue and Han (2005). In particular, Yue and Han (2005) propose a novel design method that is based on reduction, first order transformation, cone-complementary linearization and memory type feedback. However, the memory controller scheme is strongly in need of the use of the exact delay information which is not always possible for any physical systems. Delay-dependent stabilization of linear time-delay systems having both state and input delays is also considered in Zhang et al. (2005) and Zhang et al. (2009). It is claimed by these authors that less conservative results can be obtained by use of different type of Lyapunov-Krasovskii functionals and some integral inequalities. For example, Zhang et al. (2005) introduces a new integral inequality for quadratic terms which significantly reduces conservatism. However, requirements for tuning four parameters makes the controller design process very crucial and impractical. On the other hand, the method of Zhang et al. (2009) utilizes a new type of augmented Lyapunov-Krasovskii functional together with the combination of free-weighting-matrix (FWM) technique and integral inequality to overcome the stabilization problem with delay-dependent methods. Nevertheless, this method is only valid for the stabilization of time-delay systems with constant delays. The robust stability and stabilization of a class of linear uncertain time-varying state delayed systems are studied in Parlak¸cı (2006) based on augmented type Lyapunov-Krasovskii functional. However, the author(s) do not consider any delay in the control input. Finally a novel LyapunovKrasovskii functional is also introduced in Wang et al. (2007) which leads to less conservative robust stabilization results for input-delay system having time-varying, ratebounded delays. In this paper, the problem of robust stabilization with memoryless state-feedback control is investigated for both nominal and uncertain state and input-delayed systems. The main objective of the paper is to seek for an improved linear matrix inequality (LMI) type conditions to ensure larger bounds for time-varying delays and uncertainties. The contribution of proposed methodology is based on several ideas. The first one is selecting a novel quasi-fullsize augmented Lyapunov Krasovskii functional to obtain less conservative results. The second aim is to employ the method of introducing relaxation with slack matrices and finally the third one is the use of Jensen’s inequality which allow us not to eliminate the useful negative terms that come from completion to squares in taking the timederivative of Lyapunov-Krasovskii functional. The rest of the paper is organized as follows: Section 2 presents the problem formulation. Main results are introduced in Section 3. Section 4 extends the idea to the robust stabilization of time-delay systems having norm bounded uncertainties. In Section 5, through a vast number of numerical examples, it is illustrated in large that
the proposed approach considerably improves the existing methods and gives better upper delay and norm bounds than those obtained earlier. Finally Section 6 concludes the paper. Notation: The notation to be used in the paper is fairly standard. R stands for the set of real numbers, Rm×n is the set of m × n dimensional real matrices. diag denotes the diagonal matrices, Trace stands for the trace operator, R+ symbolizes the set of positive real numbers. The identity and null matrices are denoted by I and 0, respectively. X ≻ 0(, ≺ 0) denotes that X is a positive definite (positive semi-definite, negative definite) matrix. X ≻ Y means that X −Y is positive definite. Finally, the notation ’∗’ denotes off diagonal block completion of a symmetric matrix. 2. PROBLEM FORMULATION Consider a class of time-delay system with time-varying state and input delays given as x(t) ˙ = Ax(t) + Ah x(t − h(t)) + Bu u(t) + Bh u(t − h(t)) ¯ 0] x(t) = φ(t), t ∈ [−h, (1) where x(t) ∈ Rn is the state vector, u(t) ∈ Rmu is the control input. Then A, Ah , Bu , and Bh are known real constant state space matrices of appropriate dimensions. On the other hand, the delay h(t) is assumed to be a continuous, time-varying function which satisfies ˙ 0 ≤ h(t) < ¯h, |h(t)| ≤ µ, ∀t ≥ 0. (2) ¯ and µ are known positive constants. Besides, where, h φ(t) is a continuous and differentiable vector valued initial condition function for states. Then our goal is to find a suitable state-feedback control law in the form of u(t) = Kx(t), such that the closed-loop system exhibits an asymptotically stable behavior for any h(t) satisfying (2). Substituting the state-feedback control law into (1) yields the closed-loop system ¯ 0](3) ¯ x(t) ˙ = Ax(t) + A¯h x(t − h(t)), x(t) = φ(t), t ∈ [−h, where A¯ , A + Bu K and A¯h , Ah + Bh K. 3. MAIN RESULTS ¯ µ, the Theorem 1. Given nonnegative scalar constants h, closed-loop system (3) is asymptotically stable for all h(t) satisfyingh (2), ifi there exist h i symmetric positive definite ¯ ¯ ¯ ¯ W ¯ and matrices matrices X P¯12 , Z11 Z¯12 , Z¯1 , Z¯2 , T , Q, ⋆ P22
⋆
Z22
¯i , (i = 1, . . . , 5), S¯i , (i = 1, . . . , 5), with appropriate L, N dimensions such that ¯¯ ¯¯ ¯ ¯ T ¯ ¯T
Ψ11 Ψ12 Ψ13 ∗ Ψ22 Ψ23 ∗ ∗ Ψ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ψ14 Ψ24 Ψ34 ¯11 −Z ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ψ15 hZ11 hZ12 Ψ25 0 0 Ψ35 0 0 0 0 0 ¯11 −Z 0 0 ¯11 −Z ¯12 ∗ −Z ¯22 ∗ ∗ −Z ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
hS1 Ψ110 ¯S ¯T Ψ210 h 2 ¯S ¯T h 0 3 ¯S ¯T h 0 4 ¯S ¯T h 0 5 5 0 0 0 0 0 0 ¯Z ¯1 −h 0 0 ¯ ¯2 0 ∗ − hZ ∗ ∗ −T ∗ ∗ ∗ hN 1 ¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯N ¯T h
Z¯22 − Z¯1 ≻ 0
0 0 0 T P¯12 T P¯12 ¯Z ¯12 h ¯Z ¯22 h 0 0 0 Ψ1111
≺0 (4) (5)
Z¯22 − Z¯2 ≻ 0 (6) T where Ψ11 = AX + XAT + Bu L + LT BuT + P¯12 + P¯12 + Z¯1 − ¯ W ¯ +N ¯ T +N ¯1 , Ψ12 = Ah X+Bh L+Z¯22 −Z¯1 −N ¯T+ Z¯22 +Q+ 1 1 T ¯2 + S¯1T , Ψ13 = −P¯12 + N ¯3 − S¯1T , Ψ14 = P¯22 − Z¯12 ¯4 , N +N ¯5 , Ψ110 = XAT +LT B T , Ψ22 = −(1−µ)Q− ¯ Ψ15 = P¯22 + N u ¯ T −N ¯2 + S¯T + S¯2 + Z¯2 − Z¯1 , Ψ23 = Z¯22 − Z¯2 − N ¯3 − S¯T + S¯3 , N 2 2 2 T T ¯4 +S¯4 , Ψ25 = −Z¯12 −N ¯5 +S¯5 , Ψ210 = XAT + Ψ24 = Z¯12 −N h ¯ − S¯T − S¯3 , Ψ34 = −P¯22 − S¯4 , LT BhT , Ψ33 = Z¯2 − Z¯22 − W 3 T Ψ35 = −P¯22 + Z¯12 − S¯5 , Ψ1111 = −XT −1X. Moreover, the control law u(t) = LX −1 x(t) is a stabilizing controller for the resulting closed-loop system (3). Proof: Let us choose a Lyapunov Krasovskii functional candidate as V (t, xt ) = V1 + V2 + V3 + V4 where Rt V1 = η T (t)P η(t), η T (t) = [xT (t) t−h¯ xT (s)ds], P = P11 P12 R R ¯ 0 ¯ t ζ T (s)Zζ(s)dsdβ, where ζ T (s) = ∗ P22 , V2 = h −h t+β Rt T 12 [xT (s) x˙ T (s)] and Z = Z∗11 Z Z22 , V3 = t−h(t) x (s)Qx(s)ds, Rt V4 = t−h¯ xT (s)W x(s)ds where xt is the function x(θ) ¯ t]. Taking the time derivative of V with θ ∈ [t − h, along the state trajectory of system (1), (2), (3) yields P4 V˙ (t, xt ) = i=1 V˙ i . On the other hand, let us define an extended state vector as T ¯ χ (t) , xT (t) xT (t − h(t)) xT (t − h) Z
t
xT (s)ds
t−h(t)
Z
t−h(t)
¯ t−h
xT (s)ds .
(7)
¯ 2 ζ T (t)Zζ(t) V˙ 2 ≤ h Z t Z t Z Z12 − ζ T (s)ds 11 ζ T (s)ds ∗ Z − Z 22 1 t−h(t) t−h(t) Z t−h(t) Z t−h(t) Z Z12 − ζ T (s)ds 11 ζ T (s)ds ∗ Z22 − Z2 t−h¯ ¯ t−h Note that with the following definition I 0 00 0 Γ3 , ¯ ¯ A Ah 0 0 0
(13)
we can rewrite we define Γ4 , ζ(t) = Γ3 χ(t). Similarly, 0 0 0I 0 0 0 0 0 I and Γ5 , . Then we can write I −I 0 0 0 0 I −I 0 0 Z t Z t−h(t) x(s) x(s) ds = Γ4 χ(t), ds = Γ5 χ(t). x(s) ˙ x(s) ˙ ¯ t−h(t) t−h
Thus, in view of the definitions of Γ4 and Γ5 we get V˙ 2 ≤ χT (t)Ω2 χ(t) (14) where Z12 T Z11 ¯ 2 ΓT ZΓ3 −ΓT Z11 Z12 Ω2 = h 3 4 ∗ Z22 −Z1 Γ4 −Γ5 ∗ Z22 −Z2 Γ5 . The time-derivative of V3 is computed as V˙ 3 ≤ xT (t)Qx(t) − (1 − µ)xT (t − h(t))Qx(t − h(t)) Describing Γ6 , [I 0 0 0 0], Γ7 , [0 I 0 0 0] and Ω3 = ΓT6 QΓ6 − (1 − µ)Γ7 QΓ7 leads to write V˙ 3 ≤ χT (t)Ω3 χ(t). (15) ˙ Finally, we compute V4 as follows:
Then, defining Γ1 , [ I0 00 00 0I 0I ] allows to hwrite η(t) =i ¯A ¯h 0 0 0 Γ1 χ(t). In a similar manner, we define Γ2 , A I 0 −I 0 0 to obtain η(t) ˙ = Γ2 χ(t). Then we obtain V˙ 1 = 2η T (t)P η(t) ˙ = χT (t)Ω1 χ(t) (8)
¯ ¯ V˙ 4 = xT (t)W x(t) − x(t − h)W x(t − h) (16) ¯ = Γ8 χ(t), where Γ8 , [0 0 I 0 0], We can define x(t − h) then (16) can be rewritten as V˙ 4 = χT (t)Ω4 χ(t). (17)
where
where Ω4 , ΓT6 W Γ6 − ΓT8 W Γ8 . It follows from NewtonLeibnitz relation that we can always construct the following null equations: " # Z
Ω1 = ΓT1 P Γ2 + ΓT2 P Γ1 . On the other hand Z t ¯ 2 ζ T (t)Zζ(t) − h ¯ ζ T (s)Zζ(s)ds V˙ 2 = h ¯ 2 ζ T (t)Zζ(t) − h ¯ =h
¯ t−h Z t
(9)
t
2χT (t)N T Γ9 χ(t) −
¯ −h
t−h(t)
¯ t−h
T
ζ (s)Zζ(s)ds
T
2χ (t)S (10)
Note that from Jensen’s inequality we have Z t Z11 Z12 T ¯ −h ζ (s) ζ(s)ds ∗ Z22 t−h(t) Z t Z t Z Z ≤− ζ T (s)ds 11 12 ζ T (s)ds (11) ∗ Z 22 t−h(t) t−h(t)
and similarly
t−h(t)
Z11 Z12 ζ(s)ds ∗ Z22 ¯ t−h Z t−h(t) Z t−h(t) Z11 Z12 T ≤− ζ (s)ds ζ T (s)ds. (12) ∗ Z22 t−h¯ ¯ t−h Then using the relations given in (11), (12) and in view of (5) and (6), we obtain ¯ −h
Z
ζ T (s)
(18)
#
(19)
and
ζ T (s)Zζ(s)ds
t−h(t)
Z
x(s)ds ˙ =0
t−h(t)
T
"
Γ10 χ(t) −
Z
t−h(t) ¯ t−h
x(s)ds ˙ = 0,
where N , [N1 N2 N3 N4 N5 ], S , [S1 S2 S3 S4 S5 ], Γ9 = [I −I 0 0 0 0], and Γ10 = [0 I −I 0 0]. Then completing (18) and (19) to squares gives us the following: 0 ≤ χT (t)Ω5 χ(t), T
(20)
0 ≤ χ (t)Ω6 χ(t) (21) T T −1 T ¯ where Ω5 , N Γ9 + Γ9 N + hN Z1 N and Ω6 , S Γ10 + ¯ T Z −1 S. As a result, substituting V˙ i , i = 1, . . . , 4 ΓT10 S + hS 2 computed in (8), (14), (15) and (17) into V˙ (t, xt ) and adding relaxation terms of (18) and (19) allows to obtain V˙ (t, xt ) ≤ χT (t)Ωχ(t) (22) P6 where Ω = i=1 Ωi . It appears that if Ω ≺ 0 is established, then V˙ (t, xt ) ≤ χT (t)Ωχ(t) < 0 is achieved and the closedloop system is guaranteed to be globally asymptotically stable for every h(t) satisfying (2). T
Using Schur complement formula, Ω ≺ 0 is congruent to ¯ T ¯ T Φ11 Φ12 Φ13 ∗ Φ22 Φ23 ∗ ∗ Φ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Φ14 Φ24 Φ34 Φ44 ∗ ∗ ∗ ∗ ∗
Φ15 Φ16 Φ17 hN1 hS1 ¯ T ¯ T hS Φ25 Φ26 Φ27 hN 2 2 ¯ T ¯ T hS Φ35 0 0 hN 3 3 ¯ T ¯ T hS 0 0 0 hN 4 4 ¯ T ≺ 0 ¯ T hS Φ55 0 0 hN 5 5 ∗ −Z11 −Z12 0 0 ∗ ∗ −Z22 0 0 ¯ ∗ ∗ ∗ −hZ1 0 ¯ 2 ∗ ∗ ∗ ∗ −hZ T P11 A¯ + A¯T P11 + P12 + P12 + Z1 −
(23)
where Φ11 = Z22 + Q + W + N1T + N1 , Φ12 = P11 A¯h + Z22 − Z1 − N1T + N2 + S1T , Φ13 = −P12 + N3 − S1T , Φ14 = A¯T P12 + P22 − T Z12 + N4 , Φ15 = A¯T P12 + P22 + N5 , Φ22 = −(1 − µ)Q − T ¯ A¯T Z T , N2 − N2 + S2T + S2 + Z2 − Z1 , Φ16 = ¯ hZ11 + h 12 T ¯ ¯ ¯ Φ17 = hZ12 + hA Z22 , Φ23 = Z22 − Z2 − N3 − S2T + S3 ,, T T Φ24 = A¯Th P12 + Z12 − N4 + S4 , Φ25 = A¯Th P12 − Z12 − T T T ¯ ¯ N5 + S5 , Φ33 = Z2 − Z22 − W − S3 − S3 , Φ26 = hAh Z12 , ¯ A¯T Z22 , Φ34 = −P22 − S4 , Φ35 = −P22 + Z T − S5 , Φ27 = h 12 h Φ44 = −Z11 , Φ55 = −Z11 . Pre- and post multiplying (23) by diag{X, X, X, X, X, X, X, X, X} −1 where X , P11 and employing the variable changes (·) = X(·)X lead to Φ¯ Φ¯ Φ¯ Φ¯ Φ¯ Φ¯ ¯ ¯N ¯T h ¯S ¯T T Φ h 11
12
13
14
15
17
1 ¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯ T hN 5
1 ¯S ¯T h 2 ¯S ¯T h 3 ¯S ¯T h 4 ¯S ¯T h 5
¯ 24 Φ ¯ 25 Φ ¯ 26 Φ ¯ 27 Φ ¯ 34 Φ ¯ 35 Φ 0 0 ¯ 44 0 Φ 0 0 ≺ 0 (24) ¯ ∗ Φ55 0 0 ¯11 −Z ¯12 ∗ ∗ −Z 0 0 ¯22 ∗ ∗ ∗ −Z 0 0 ¯Z ¯1 ∗ ∗ ∗ ∗ −h 0 ¯Z ¯2 ∗ ∗ ∗ ∗ ∗ −h ¯ 11 = AX ¯ + X A¯T + P¯12 + P¯ T + Z¯1 − Z¯22 + Q ¯+ where Φ 12 ¯ +N ¯T + N ¯1 , Φ ¯ 12 = A¯h X + Z¯22 − Z¯1 − N ¯T + N ¯2 + S¯T , W 1 1 1 ¯ 13 = −P¯12 + N ¯3 − S¯1T , Φ ¯ 14 = X A¯T X −1 P¯12 + P¯22 − Φ T ¯5 , Φ ¯ 22 = −(1 − ¯4 , Φ ¯ 15 = X A¯T X −1 P¯12 + P¯22 + N Z¯12 +N ¯ Z¯11 + ¯−N ¯T − N ¯2 + S¯T + S¯2 + Z¯2 − Z¯1 , Φ ¯ 16 = h µ)Q 2 2 ¯ Z¯12 + hX ¯ A¯T X −1 Z¯22 , Φ ¯ A¯T X −1 Z¯ T , Φ17 = h ¯ = Z¯22 − hX 23 12 T T T −1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Z2 − N3 − S2 + S3 , Φ24 = X Ah X P12 + Z12 − N4 + S¯4 , T ¯5 + S¯5 , Φ ¯ 33 = Z¯2 − Z¯22 − ¯ 25 = X A¯T X −1 P¯12 − Z¯12 −N Φ h T T −1 ¯ T ¯ ¯ ¯ ¯ ¯ ¯ W − S3 − S3 , Φ26 = hX Ah X Z12 , Φ27 = ¯ hX A¯Th X −1 Z¯22 , T ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 44 = −Z¯11 , Φ34 = −P22 − S4 , Φ35 = −P22 + Z12 − S5 , Φ
¯ Φ=
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯ 22 Φ ¯ 23 Φ ¯ 33 ∗ Φ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
16
¯ 55 = −Z¯11 . Φ ¯ can be decomposed as Note that Φ ¯ =Φ ¯0 + Φ ¯1 + Φ ¯T ≺ 0 Φ 1 where ¯ 11 Φ ¯ 12 Φ ¯ 13 Φ ∗ Φ¯ 22 Φ¯ 23 ∗ ∗ Φ¯ 33
¯0 = Φ
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
˜ ¯ 14 Φ ˜ ¯ Φ24 ¯ 34 Φ ¯ 44 Φ ∗ ∗ ∗ ∗ ∗
˜ ˜ ˜ ¯ 15 Φ ¯ 16 Φ ¯ 17 Φ ˜ ¯ Φ25 0 0 ¯ 35 Φ 0 0 0 0 0 ¯ 55 Φ 0 0 ¯11 −Z ¯12 ∗ −Z ¯22 ∗ ∗ −Z ∗ ∗ ∗ ∗ ∗ ∗
(25)
¯N ¯T h 1 ¯ ¯ hN2T ¯N ¯T h 3 ¯ ¯ hN4T ¯ T hN 5
¯S ¯T h 1 ¯ ¯ hS2T ¯S ¯T h 3 ¯ ¯ hS4T ¯S ¯T h 5
0 0 0 0 ¯Z ¯1 −h 0 ¯ ¯2 ∗ −hZ
(26)
˜¯ ¯ ¯T ¯ ˜ ¯ ¯ ¯ ˜¯ with Φ 14 = P22 − Z12 + N4 , Φ15 = P22 + N5 , Φ16 = ˜¯ = ¯hZ¯ , Φ ˜ ˜ ¯ Z¯11 , Φ ¯ 24 = Z¯ T − N ¯4 + S¯4 , Φ ¯ 25 = −Z¯ T − h 17 12 12 12 ¯ ¯ N5 + S5 and ¯ ¯ ¯ ¯ ¯1 = Φ
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
¯ 16 Φ ¯ 17 ¯ 14 Φ ¯ 15 Φ 0Φ ¯ ¯ ¯ ¯ 0Φ 24 Φ25 Φ26 Φ27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
(27)
¯ = X A¯T X −1 P¯ , Φ ¯ = ¯ 15 = X A¯T X −1 P¯12 , Φ with Φ 14 12 16 ¯ T −1 ¯ T ¯ T −1 ¯ T ¯ ¯ ¯ ¯ ¯ hX A X Z12 , Φ17 = hX A X Z22 , Φ24 = X Ah X −1 P¯12 , ¯ ¯ ¯ ¯ ¯T −1 Z¯ T , Φ ¯T −1 P¯12 , Φ Φ 25 = X Ah X 26 = hX Ah X 27 = 12 T −1 ¯ A¯ X Z¯22 . Besides Φ ¯ 1 can be rewritten as Φ ¯1 = hX h ΠT1 X −1 Π2 where ¯ A¯h X 0 0 0 0 0 0 0 , Π1 = AX T ¯ ¯T Π2 = 0 0 0 P¯12 P¯12 ¯hZ¯12 hZ12 0 0 . Also note that for any symmetric positive definite matrix T , the following bounding inequality always holds: ΠT1 X −1 Π2 + ΠT1 X −1 Π2
T
ΠT1 T −1 Π1 + ΠT2 XT −1 X Hence, substituting (28) into (25) gives
−1
Π2 .
(28)
¯ Φ ¯ 0 + ΠT1 T −1 Π1 + ΠT2 XT −1 X −1 Π2 0 (29) Φ Then applying Schur complement formula to (29), replacing A¯ with A + Bu K and A¯h with Ah + Bh K, and defining L , KX immediately lead to the matrix inequality given in (4) In the light of the previous theorem, in order to achieve maximum ¯h such that the closed-loop system stays stable, one can solve the following optimization problem for the symmetric, positive definite decision matrices X, Z¯11 , Z¯22 , ¯ W ¯ , matrices L, P¯12 , Z¯12 , N ¯i , (i = 1, . . . , 5), Z¯1 , Z¯2 , T , Q, S¯i , (i = 1, . . . , 5): ¯ max h subject to (4) − (6)
(30)
If the problem described in (30) has a feasible solution set, then state-feedback control law, u = LX −1 x(t), is said to be the suboptimal stabilizing controller for the given stabilization problem. Note that the matrix inequality condition (4) is not in the form of an LMI so that it can be solved using convex optimization algorithms. In other words, we can not find a global maxima for (30) using convex optimization algorithms. However, if one affords more computational efforts such as cone-complementary algorithm, one may still obtain a suboptimal controller for the problem defined in Section 2 using an iterative algorithm presented in the following sequel. First, we define a new variable R = RT ≻ 0 such that R XT −1 X and replace the condition (4) with ¯ X ¯ ¯ I ¯ I R R X T¯ I ¯ T¯ 0, I R 0, I X 0, I T 0 (31) X ¯¯ ¯¯ ¯ ¯ T ¯ ¯T
Ψ11 Ψ12 Ψ13 ∗ Ψ22 Ψ23 ∗ ∗ Ψ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ψ14 Ψ15 hZ11 hZ12 Ψ24 Ψ25 0 0 Ψ34 Ψ35 0 0 ¯11 −Z 0 0 0 ¯11 ∗ −Z 0 0 ¯11 −Z ¯12 ∗ ∗ −Z ¯22 ∗ ∗ ∗ −Z ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
hS1 Ψ110 ¯S ¯T Ψ210 h 2 ¯S ¯T h 0 3 ¯S ¯T h 0 4 ¯S ¯T h 0 5 5 0 0 0 0 0 0 ¯Z ¯1 −h 0 0 ¯Z ¯2 0 ∗ −h ∗ ∗ −T ∗ ∗ ∗ hN 1 ¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯N ¯T h
0 0 0 T P¯12 T P¯12 ¯Z ¯12 h ¯Z ¯T h
≺ 0, 22 0 0
0 −R
(32) which can be justified following the inequality of R−1 − X −1 T X −1 0 and using Schur complement formula
¯ , R−1 , X ¯ , X −1 and T¯ , T −1 , and defining R allows to obtain the inequalities in (31). Then, in order to achieve a maximum allowable delay bound for the closedloop system (3), one may use the following optimization problem: ¯ + XX ¯ + T¯T ) min Trace (RR subject to the LMIs (5), (6), (32), (31), Finally to achieve the sub-optimal controller, one can use the following algorithm to solve the cone-complementary algorithm (El Ghaoui et al. (1997)): Algorithm: ¯ and µ such that (1) Choose a sufficiently small initial h there exists a feasible solution set ¯ 0 , X0 , R ¯ 0 , R0 , T¯0 , T0 , } {X to the LMI conditions in (5), (6), (32), (31) and set k = 0. (2) Solve the following LMI optimization problem for the ¯ X, R, ¯ R, T¯, T, } variables {X, ¯k R + X ¯ k X + T¯k T + RR ¯ k + XX ¯ k + T¯Tk ) min Trace (R subject to the LMIs (5), (6), (32), (31) ¯ k+1 := X, ¯ Xk+1 := X, R ¯ k+1 := R, ¯ and set: X Rk+1 := R, T¯k+1 := T¯, Tk+1 := T . (3) If R XT −1 X is feasible for the above solution set ¯o = h ¯ and µo = µ then return to Step (1) then set h ¯ and µ to some extent. Otherwise, after increasing h set k = k + 1 and go to Step (2) and repeat the optimization for a pre-specified number of iterations, say kmax , until finding a feasible solution satisfying R XT −1X. If such a solution does not exist, then exit. ¯ o = h, ¯ to If one can find a feasible solution set, µo = µ and h ¯ o is said to be the maximum allowable this algorithm than h delay bound for the closed-loop system (3). Moreover, suboptimal stabilizing control law for the system can be constructed as u(t) = LX −1 x(t). 4. ROBUST STABILIZATION
¯ µ, the Theorem 2. Given nonnegative scalar constants h, uncertain system (33) can be stabilized by a memoryless state feedback control law u(t) = LX −1 x(t) for all h(t) satisfyingh(2), ifi there exist h i symmetric positive definite ¯11 Z ¯12 X P¯12 Z ¯ ¯ ¯ ¯ matrices ⋆ P¯22 , ¯22 , Z1 , Z2 , T , Q, W matrices ⋆ Z ¯i , (i = 1, . . . , 5), S¯i , (i = 1, . . . , 5) all in appropriate L, N dimensions, and positive scalar ǫ, such that Ψ ¯ ¯Z ¯ ¯Z ¯ ¯N ¯T h ¯S ¯T Ψ ¯ ¯ h h h 0 Ψ Ψ Ψ Ψ Ψ 11
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
12
13
14
15
11
12
Ψ22 Ψ23 Ψ24 Ψ25 0 0 ∗ Ψ33 Ψ34 Ψ35 0 0 ¯11 ∗ ∗ −Z 0 0 0 ¯11 ∗ ∗ ∗ −Z 0 0 ¯11 −Z ¯12 ∗ ∗ ∗ ∗ −Z ¯22 ∗ ∗ ∗ ∗ ∗ −Z ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
1
¯N ¯T h 2 ¯N ¯T h 3 ¯N ¯T h 4 ¯N ¯T h
1
110
¯S ¯T Ψ210 h 2 ¯S ¯T h 0 3 ¯S ¯T h 0 4 ¯S ¯T h 0 5 5 0 0 0 0 0 0 ¯Z ¯1 −h 0 0 ¯ ¯ ∗ −hZ2 0 ¯ ∗ ∗ Ψc ∗ ∗ ∗ ∗ ∗ ∗
a
0 0 T P¯12 T P¯12 ¯Z ¯12 h ¯Z ¯22 h 0 0 0 ¯ Ψe ∗
¯b Ψ 0 0 0 0 0 0 0 ¯d Ψ 0 −ǫI
≺0
(34) T ¯ 11 = AX + XAT + Bu L + LT BuT + P¯12 + P¯12 where Ψ + Z¯1 − ¯ +W ¯ +N ¯1T + N ¯1 + ǫGGT , Ψ ¯ 110 = ǫGGT + XAT + Z¯22 + Q ¯ a = XE T + LT E T , Ψb = XE T + LT E T , LT BuT , Ψ A Bu Ah Bh ¯ c = ǫGGT − T , Ψ ¯ d = XE T + LT E T , Ψ ¯ e = −XT −1X. Ψ A Bu Proof: Replacing A with A + ∆A, Bu with Bu + ∆Bu , Ah with Ah + ∆Ah and finally Bh with Bh + ∆B h in (4), and defining J T , GT 0 0 0 0 0 0 0 0 GT 0 and H , [ EA X + EBu L EAh X + EBh L 0 0 0 0 0 0 0 0 0 ] we obtain the condition Ψ + JF (t)H + H T F T (t)J T ≺ 0 instead of (4). Note that Ψ + JF (t)H + H T F T (t)J T ≺ 0 holds for all F (t) satisfying F T (t)F (t) ≺ I if there exist ǫ > 0 such that Ψ + ǫJJ T + ǫ−1 H T H ≺ 0. Hence, using Schur complement formula immediately leads to the matrix inequality condition (34) which concludes the proof.
As described in the previous section, replacing the term −XT −1X ≺ 0 with R in (34) such that R XT −1 X, one can still use the cone-complementary linearization algo¯ and γ such that rithm to identify the maximum allowable h the closed-loop system exhibits globally asymptotically stable behaviour.
Now consider a class of uncertain time-delay system with time-varying state and input delays given as
5. NUMERICAL EXAMPLES
x(t) ˙ = [A + ∆A] + x(t) + [Ah + ∆Ah ]x(t − h(t))
Now we present three numerical examples to illustrate the design procedure:
+[Bu + ∆Bu ]u(t) + [Bh + ∆Bh ]u(t − h(t)) ¯ 0] x(t) = φ(t), t ∈ [−h, (33)
Example 1.
where x(t) ∈ Rn is the state vector, u(t) ∈ Rmu is the control input. Then A, Ah , Bu , Bh are known real constant matrices. On the other hand ∆A, ∆Ah , ∆Bu and ∆Bh are matrix valued functions representing time varying uncertainties. These parameter uncertainties are assumed to be norm bounded and of the form, [∆A ∆Ah ∆Bu ∆Bh ] = GF (t) [EA EAh EBu EBh ] where G, EA , EAh and EBh are known constant real matrices in appropriate dimensions which represent the structure of uncertainties, and F (t) is an unknown matrix function with Lebesgue measurable elements and satisfies F T (t)F (t) I for all t ≥ 0. Note that all matrices are with appropriate dimensions.
Consider system (1) with
−1 −1 A= −0.5 0 −2 −0.2 Ah = 0.5 0
1 0.2 0.5 0
0.5 1 0.3 0
−0.5 −1 0 0
0.5 0 , 0.5 1.2
1 1 Bu = 1 0
0 −0.5 0.5 0 , −2 −0.5 0 −1
1 1 Bh = . 1 1
Applying Theorem 1 to this system by using conecomplementary algorithm presented in Section 3 and setting µ = 0, we find that the system can be stabilized
¯ = 0.73. This result is obfor any h satisfying h ≤ h tained in 224 iterations by cone-complementary algorithm. When h = 0.73, the controller gain is obtained as K = [0.4547 0.0294 −0.2736 −0.2399] . This result is less conservative than the results obtained by Zhang et al. (2005) and Zhang et al. (2009). For this system, using the method of Zhang et al. (2005), we calculate the upper bound, ¯h as 0.27 using the parameters λ1 = −1.1436, λ2 = −0.9341, µ1 = 1.0116, µ2 = 0.9324. Zhang et al. (2009) calculate ¯ as 0.6 with the controller the maximum delay bound, h, gain K = [0.038 0.003 − 0.033 − 0.551]. Example 2. The second example is concerned with stabilizing the input delay system (1) without any state delays. The system matrices of (33) given appropriately as follows: 1 0 0 0 . , Bh = , Bu = A= 0 0 1 −5 The matrices in description of system uncertainties are given by EA = I, EAh = EBu = EBh = 0, and G = γI, where γ > 0 is some scalar. In order to make a fair ¯ = 0.4 in conecomparison, if we stop updating h at h complementary algorithm presented in previous section, we calculate the uncertainty bound γmax = 2.01. However, applying Theorem 1 of Chen and Zheng (2006) to this ¯ = 0.4. example, it is found that γmax = 1.412 for h Using the proposed method of this note, we obtain the stabilizing controller gain as K = [−2.5626 −0.0948] . Similarly applying Theorem 12 in Yue and Han (2005) to this example, it is found that γmax = 0.5998. For this system, setting γ to the upper bound 1.412 as found ¯ = in Chen and Zheng (2006), we find delay-bound as h 0.5050 at 2 steps which is much bigger than 0.4. For this value of γ, we obtain the stabilizing controller gain as K = [−1.8037 −0.0426] . During all these simulations, maximum number of iterations is chosen to be as 30 and step sizes for h and γ are set to 0.005. Example 3. Consider system (33) with uncertainties and input time-varying delay where 0 1 0 0 0 A= , Ah = , Bh = , |q| ≤ γ −1.25 −3 q 0 1 We consider the case where 0 ≤ h(t) ≤ 0.2. Table 1 presents the maximum allowable γ values, γmax , for different values of µ both obtained by our method and recent methods presented in literature. The results show that our approach is significantly less conservative than the existing methods. Discarding V3 in the Lyapunov-Krasovskii functional by simply setting Q = 0, it is obtained that the controller gain K = [−47.5 − 7.96] guarantees robust stabilization of this system for unknown µ. Table 1. Comparison of proposed method with some previous results in terms of uncertainty bound γmax . µ 0 0.1 0.9 10 Unknown
Yue and Han (2005) 9.8615 9.8538 9.8506 Not reported Not reported
Wang et al. (2007) 26.4502 20.9157 13.4176 13.4176 Not reported
Theorem 2 34.45 34.45 34.45 34.45 34.45
As a result, the above three examples clearly indicate that the proposed stabilization method of this note is less conservative than the results given in literature. 6. CONCLUSION This paper proposes an improved stabilization technique for time-delay systems having both time-varying state and input delays. The stabilization criteria are derived by constructing a novel augmented Lyapunov-Krakovskii functional and employing a relaxation approach by use of slack matrices and Jensen’s inequality. Moreover, the method is extended to the robust stabilization problem of time-delay systems subject to norm bounded uncertainties. Finally, the effectiveness of the proposed method is demonstrated through a number of numerical examples, which illustrate that the presented method is quite less conservative in comparison to those of some earlier reported ones in the recent literature. REFERENCES Chen, W. and Zheng, W. (2006). On improved robust stabilisation of uncertain systems with unknown input delay. Automatica, 42, 1067–1072. El Ghaoui, L., Qustry, F., and Ait Rami, M. (1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42(8), 1171–1176. Gu, K., Kharitonov, V., and Chen, J. (2003). Stability of time delay systems. Birkhauser:Basel, Boston. Habets, L.C.G.J.M. (1994). Algebraic and computational aspects of time-delays. . Eindhoven University, Eindhoven. Hale, J. (1977). Theory of Functional Differential Equations. Springer:Berlin. Mahmoud, M.S. (2000). Robust control and Filtering for Time-Delay Systems. Marcel Dekker Inc. Parlak¸cı, M.N.A. (2006). Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems. Int. J. of Robust Nonlinear Control, 16, 599–636. Richard, J.P. (2003). Time-delay systems: an overview of some recent advances and open problems. Automatica, 39, 1667–1694. Wang, Z., Goldsmith, P., and Tan, D. (2007). Improvement on robust control of uncertain systems with timevarying input delays. IET Control Theory Appl., 1(1), 189–194. Yue, D. and Han, Q.L. (2005). Delayed feedback control of uncertain systems with time-varying input delay. Automatica, 41, 233–240. Zhang, X.M., Wu, M., She, J., and He, Y. (2005). Delay dependent stabilisation of linear systems with timevarying state and input delays. Automatica, 41, 1405– 1412. Zhang, X., Li, M., Wu, M., and She, J. (2009). Further results on stability and stabilisation of linear systems with state and input delays. Int. Journal of Systems Science, 40(1), 1–10.