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Robust H∞ state-feedback control for linear systems Hao Chen1,2 , Zhenzhen Zhang1 and Huazhang Wang1

Research Cite this article: Chen H, Zhang Z, Wang H. 2017 Robust H∞ state-feedback control for linear systems. Proc. R. Soc. A 473: 20160934. http://dx.doi.org/10.1098/rspa.2016.0934 Received: 22 December 2016 Accepted: 14 March 2017

Subject Areas: applied mathematics, electrical engineering Keywords: convex combination, disturbance load, geometric sequence division, robust H∞ control Author for correspondence: Hao Chen e-mail: [email protected]

1 College of Electrical and Information Engineering, Southwest

University for Nationalities, Chengdu 610041, People’s Republic of China 2 Research Centre for Applied Science, Computing and Engineering, Glyndwr University, Wrexham LL11 2AW, UK HC, 0000-0002-4983-6488 This paper investigates the problem of robust H∞ control for linear systems. First, the statefeedback closed-loop control algorithm is designed. Second, by employing the geometric progression theory, a modified augmented Lyapunov–Krasovskii functional (LKF) with the geometric integral interval is established. Then, parameter uncertainties and the derivative of the delay are flexibly described by introducing the convex combination skill. This technique can eliminate the unnecessary enlargement of the LKF derivative estimation, which gives less conservatism. In addition, the designed controller can ensure that the linear systems are globally asymptotically stable with a guaranteed H∞ performance in the presence of a disturbance input and parameter uncertainties. A liquid monopropellant rocket motor with a pressure feeding system is evaluated in a simulation example. It shows that this proposed state-feedback control approach achieves the expected results for linear systems in the sense of the prescribed H∞ performance.

1. Introduction Time delays are often encountered in many dynamic systems, such as communication networks, chemical systems and aeronautical dynamics. Naturally, delays are sometimes time-varying, which commonly brings a negative influence on systems, causing either poor performance or even instability [1–4]. Various methods have been developed in recent years for stability analysis and synthesis of delayed systems. These investigations into the stability conditions for time-delayed systems are commonly classified into either delay-independent or

2017 The Author(s) Published by the Royal Society. All rights reserved.

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(i) A linear system with time-varying delays, parameter uncertainties and disturbances is investigated related to the robust H∞ stabilization problems. New delay-dependent criteria with differentiable and non-differentiable time-varying delays are derived for stability and robustness analyses. (ii) A geometric sequence-based delay-partitioning method is employed to separate the interval [h0 , hN ] into several unequal segments. Comparing with some existing equivalently partitioning methods, the proposed LKF is modified with geometric integral intervals and the matrix dimensions of the augmented LKF are simplified. For the control of linear systems with disturbance load and uncertainties, the expected H∞ performance can be obtained. (iii) A convex combination skill is introduced not only to overcome parameter uncertainties, but also to deal with the derivative of the time-varying delay. Based on a convex combination approach, the derivative of the LKF is estimated without using extra inequalities and constraint conditions. So unnecessary enlargement is avoidable.

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delay-dependent criteria. As much of the work into delay has demonstrated, the delay-dependent criteria are more useful for reducing the conservatism [5,6]. Delay partitioning skills have been developed recently that can dramatically improve the stability conditions [7,8]. Less conservative stability conditions can be derived for a thinner separated subinterval [7]. In [9,10], a new delay partition method was developed by using the geometric sequence theory to deal with the interval time-varying delay and mixed delays, respectively. This geometric sequence division (GSD) approach can efficiently analyse the stability of the delayed Takagi–Sugeno fuzzy systems with less conservatism. In [8], by further dividing primarily separated segments into secondary-level segments, good stability results are presented. However, this design requires too many adjustable parameters that need to be determined. Thus, it costs extra computation work. Nowadays, there are two potential strategies to improve the stability criteria: Lyapunov– Krasovskii functional (LKF) construction and the estimation of its time derivative [11,12]. Various investigations into the construction of the LKF have been proposed to derive the stability conditions for delayed systems. In [13], a discretized LKF was constructed by dividing the delay interval into multiple segments. Then, by using the discretized and augmented LKF together, a delay-partitioning LKF is proposed in [14–17], which is related to all the subintervals in the delay interval and augmented terms based on the systems. This kind of LKF improves the results. In fact, by applying the same LKF, a better result may be obtained if well-developed inequalities and techniques are used for the estimation of the LKF derivative. Therefore, it is another important means of reducing conservatism with respect to the precise estimation. Three main approaches are employed: the free-weighting matrix [18,19], the Jensen inequality [20,21] and the convex optimization method, as well as their combinations [22–25]. However, in recent developments some terms have been neglected due to the derivative of the LKF being estimated, which may lead to conservative conditions. So, the question of how to estimate the upper bound of the LKF derivative with less conservatism is a hot topic and full of challenges. Over the past few decades, complex mathematical modelling with higher order has frequently been used in many engineering applications, which may cause nonlinearity in dynamic systems [26,27]. Parameter uncertainties and disturbance loading inevitably occur in physical systems. A number of studies were developed to control uncertain systems [28,29]. In recent developments, H∞ related control methods have been widely investigated to overcome the input disturbance [30,31]. Based on the H∞ performance theory, some studies are proposed in the case of external disturbance to design a suitable controller [32–36]. In this paper, the performance of robust H∞ control for a linear system is investigated. Delay-dependent stability results are derived such that the delayed linear system is globally asymptotically stable, and can be stabilized with the external disturbance corresponding to the guaranteed H∞ performance. The main contributions of this paper include the following.

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We consider a dynamic system that can be described as follows: ˙ = A(t)x(t) + Ad (t)x(t − h(t)) + B(t)u(t) + Bω (t)ω(t), x(t) z(t) = C(t)x(t) + Cd (t)x(t − h(t)) + D(t)u(t) + Dω (t)ω(t) and

x(t) = φ(t),

t ∈ [−hN , 0],

⎫ t ≥ 0,⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(2.1)

where x(t) ∈ Rn is the state vector; A(t), Ad (t) ∈ Rm×n are the real matrices with appropriate dimensions; h(t) is the time-varying delay; ω(t) ∈ Rk is the disturbance input which belongs to L2 [0, +∞); φ(t) ∈ C([−hN , 0], Rn ) is the initial function. The time-varying delay h(t) is considered as the following two cases. Case 2.1. h(t) is a differentiable function satisfying 0 ≤ h0 ≤ h(t) ≤ hN ,

˙ ≤ μ2 , ∀t ≥ 0; μ1 ≤ h(t)

(2.2)

Case 2.2. h(t) is a continuous function satisfying 0 ≤ h0 ≤ h(t) ≤ hN ,

∀t ≥ 0,

(2.3)

where h0 , hN , μ1 , μ2 are constants. The state-feedback controller is formed as u(t) = Kx(t).

(2.4)

Therefore, the compact form of the closed loop is given by ˙ = (A(t) + B(t)K)x(t) + Ad (t)x(t − h(t)) + Bω (t)ω(t), x(t) and

z(t) = (C(t) + D(t)K)x(t) + Cd (t)x(t − h(t)) + Dω (t)ω(t).

t≥0

(2.5)

Considering the uncertainties that exist in the system, the parameters are represented as Uv (t), (v = 1, . . . , 8), and U1 (t) = A(t), U2 (t) = Ad (t), U3 (t) = B(t), U4 (t) = Bω (t), U5 (t) = C(t), U6 (t) = Cd (t), U7 (t) = D(t), U8 (t) = Dω (t), which are not exactly known and might be taken from an interval Uv (t) ∈ [Uv1 , Uv2 ]. Then the parameters with uncertainties satisfy Uv (t) = Γ1 (t)Uv1 + Γ2 (t)Uv2 =

2

Γo (t)Uvo ,

(2.6)

o=1

with any constant Γ1 (t) ≥ 0, Γ2 (t) ≥ 0 satisfying Γ1 (t) + Γ2 (t) = 1. Definition 2.3. Given a scalar ρ > 0, if there exists a control law for the dynamic system (2.5) (i) such that when ω(t) = 0 system (2.5) is asymptotically stable and (ii) under zero initial conditions, ∞ the output z(t) satisfies z(t)2 ≤ ρω(t)2 , that is, 0 [zT (t)z(t) − ρ 2 ωT (t)ω(t)] dt ≤ 0 for any nonzero ω(t) ∈ L2 [0, +∞). If the above two conditions are satisfied, the dynamic system (2.5) is said to be asymptotically stable with a guaranteed H∞ performance ρ. Some lemmas are employed for the design implementation as follows. Lemma 2.4 ([37], free-matrix-based integral inequality). Let r : [γ − , γ + ] → Rn be a differentiable function, Z ∈ Rn×n and T1 , T3 ∈ R3n×3n be symmetric matrices, and T2 ∈ R3n×3n , S1 , S2 ∈ R3n×n

...................................................

2. Problem statements and preliminaries

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Notations. Rn is the n-dimensional Euclidean space. P > (≥)0 means that the matrix P is positive (semi-positive) definite. In (0n ) is the identity (zero) matrix with n-dimensions; AT denotes the transpose and He(A) = A + AT . The symbol ∗ denotes the elements below the main diagonal of a symmetric block matrix. • is the Euclidean norm in Rn .

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satisfy the condition

⎡

−

γ+ γ−

⎤ S1 ⎥ S2 ⎦ ≥ 0, Z

4 (2.7)

˙ ds ≤ T Ξ , r˙T (s)Z r(s)

(2.8)

γ + where = [rT (γ + ) rT (γ − ) 1/(γ + − γ − ) γ − rT (s) ds]T , Ξ = (γ + − γ − )(T1 + 13 T3 ) + He(S1 ς1 + S2 ς2 ), ς1 = eˆ1 − eˆ2 , ς2 = 2eˆ3 − eˆ1 − eˆ2 , eˆ1 = [I 0 0], eˆ2 = [0 I 0], eˆ3 = [0 0 I]. Lemma 2.5 ([8], extended reciprocal convex combination (RCC)). For any vectors f1 , . . . , fN with appropriate dimensions, scalars βi (t) ∈ [0, 1], N ¯ i > 0, and there exists matrix i=1 βi (t) = 1, and matrices h Sij (i = 1, . . . , N − 1, j = i + 1, . . . , N) that satisfies h¯ i Sij ≥ 0, ∗ h¯ j then the following inequality holds:

−

N i=1

⎡ ⎤T ⎡ f1 h¯ 1 ⎢.⎥ ⎢ 1 T ⎥ ⎢ ⎢ f h¯ i fi ≤ − ⎣ .. ⎦ ⎣ ∗ βi (t) i ∗ fN

··· .. . ∗

⎤⎡ ⎤ f1 S1,N ⎢ ⎥ .. ⎥ ⎥ ⎢ .. ⎥ . . ⎦⎣ . ⎦ h¯ N fN

Lemma 2.6 ([38], auxiliary function-based integral inequality). For a symmetric positive matrix h¯ ∈ Rn×n and differentiable function r : [γ − , γ + ] → Rn , then the next double integral inequality holds γ+ γ+ ˙ ds dθ ≤ T (t)Π (t), r˙T (s)h¯ r(s) − (2.9) γ−

θ

where (t) = [rT (γ + ), 1/(γ + − γ − ) −6h¯ −6h¯ 24h¯ ∗ −18h¯ 48h¯ . ∗

∗

γ + γ−

rT (s) ds, 1/(γ + − γ − )2

γ + γ + γ−

θ

rT (s) ds dθ]T and Π =

−144h¯

Lemma 2.7 ([39], Finsler’s lemma). Let η ∈ Rn , Θ = Θ T ∈ Rn×n , and D ∈ Rm×n with rank(D) < n. The following statements are equivalent: (i) ηT Θη < 0, ∀ Dη = 0, η = 0; T (ii) D⊥ ΘD⊥ < 0; (iii) ∃D ∈ Rn×m : Θ + He(DD) < 0; where D⊥ ∈ Rn×(n−rank(D)) is the right orthogonal complement of D.

3. Main results The GSD-based delay-partitioning method is applied in figure 1. For any integral N ≥ 1, the delay interval [h0 , hN ] is separated into N unequal geometric subintervals as ⎫ ⎪ ϑi = i ⎪ ⎪ ⎬ i (3.1) ⎪ and hi = h0 + β , i = 1, . . . , N,⎪ ⎪ ⎭ β=1

where N is the subintervals in [h0 , hN ], is a real positive number, and ϑi is the length of the ith subinterval which is equal to i . It is obtained as [h0 , hN ] = N k=1 Sk . There exists an integer

...................................................

then it holds that

T2 T3 ∗

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T1 ⎢ ⎣∗ ∗

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ϑ1 h 1

ϑ2

N h2

5

ϑN

hN−1

hN

Figure 1. Delay-partitioning approach using GSD.

k ∈ {1, . . . , N} ∀t ≥ 0, such that h(t) ∈ Sk , ⎤T

⎡ ⎢ ej = ⎣0n , . . . , 0n ,

In

⎥ 0n , . . . , 0n ⎦ ∈ R(4N+6)n×n ,

j−1

j = 1, . . . , 4N + 6.

(3.2)

4N−j+6

The augmented vector is defined as ˙T

T

T

ψ(t) = x (t), x (t), x (t − h0 ),

ψ1T (t),

T

x (t − h(t)),

t

T

x (s) ds, t−h(t)

ψ2T (t),

ψ3T (t),

ψ4T (t),

T

ω (t) , (3.3)

where ψ1 (t) = [xT (t − h1 ), . . . , xT (t − hN )]T , t T t ψ2 (t) = xT (s) ds, . . . , xT (s) ds ,

and

t−h1

t−hN

T 1 1 t−hN−1 T T ψ3 (t) = x (s) ds, . . . , x (s)ds ϑ1 t−h1 ϑN t−hN T −h0 t−h0 −hN−1 t−hN−1 1 1 T T ψ4 (t) = x (s) ds dθ , . . . , x (s) ds dθ . (ϑ1 )2 −h1 t+θ (ϑN )2 −hN t+θ t−h0

Next, the new delay-dependent stability criteria with H∞ performance ρ are given for system (2.5). Theorem 3.1. Given a positive integer N, and ϑi = i . Consider a time-varying delay satisfying case 2.1. System (2.5) is globally asymptotically stable with H∞ performance ρ if there exist symmetric ˜ i ∈ Rn×n (i = 1, . . . , N), X ˜ i ∈ Rn×n (i = 1, . . . , k), T˜ i ∈ Rn×n (i = k, . . . , N), positive definite matrices Qˆ i , Z (2N+1)n×(2N+1)n 3n×3n , symmetric matrices T1 , T3 ∈ R , matrices T2 ∈ R3n×3n , S1 , S2 ∈ R3n×n , J ∈ U∈R n×n (4N+6)n×n (4N+6)n×1 and U ∈ R , H∈R , such that the following linear matrix inequalities (LMIs) R hold: ⎤ ⎡ T1 T2 S1 ⎥ ⎢ Ti = ⎣ ∗ T3 S2 ⎦ ≥ 0 (3.4) ˆ ∗ ∗ Qi and 2 o=1

Γo (t)

2 s1,2 =1

s1,2 (t)

2

s (t)(Ωks1 s2 + He(UΥo )) < 0,

s=1

where Υo = (A(t) + B(t)K)eT2 + Ad (t)eTN+4 + Bω (t)eT4N+6 − eT1 , Ωks1 s2 = Ω1s + Ω2ks1 s2 + Ω3,k + Ω4 + Ωω + eT1 YeT1 ,

k = 1, . . . , N,

(3.5)

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h0

1

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⎧⎡ ⎤T ⎡ ⎤⎫ T − eT T ⎪ ⎪ e e ⎪ ⎪ 2 4 N+6 ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ ⎪ ⎪ . . ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ . ⎪ . ⎪ ⎪⎢ . ⎥ ⎢ ⎥⎪ . ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ ⎪ ⎢T ⎪ ⎪ ⎢ ⎥ ⎥⎪ T T ⎪ e2 − eN+3 ⎢e2N+5 ⎥ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎬ ⎨⎢ ⎢ ⎥ ⎥⎪ ⎢T ⎢ ⎥ ⎥ 1 T T) (e − e Ω1s = He ⎢e2N+6 ⎥ U ⎢ ⎥ , ϑ 3 4 1 ⎢ ⎪ ⎢ ⎥ ⎥⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥ ⎥⎪ .. ⎪ ⎪ ⎪⎢ .. ⎥ ⎢ ⎥⎪ ⎪ ⎪ . . ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ ⎪⎢ ⎪ ⎪ ⎢ ⎥ ⎥⎪ ⎪ ⎢eT ⎪ ⎪ ⎢ 1 (eT − eT ) ⎥⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎣ ⎦ ⎦ ϑ N+2 N+3 3N+5 ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ ⎩ eT T T e2 − (1 − μs )eN+4 ⎭ N+5

...................................................

k−1

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Ω2ks1 s2 =

6

˜ i eT − ei+3 X ˜ i eT ) + ek+2 X ˜ k eT − (1 − μs1 )eN+4 X ˜ k eT (ei+2 X N+4 i+2 i+3 k+2

i=1

+ (1 − μs2 )eN+4 T˜ k eTN+4 − ek+3 T˜ k eTk+3 + ⎡ N

eTi+2

⎤T

⎡

eTi+2

⎤

⎡

N

(ei+2 T˜ i eTi+2 − ei+3 T˜ i eTi+3 ),

i=k+1

eTk+2

⎤T

⎡

eTk+2

⎤

⎢ T ⎥ ⎢ T ⎥ ⎢T ⎥ ⎢T ⎥ ⎢ e ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ i+3 ⎦ Ξ1 ⎣ ei+3 ⎦ − ⎣eN+4 ⎦ Ξ2 ⎣eN+4 ⎦ , i=1,i =k eTk+3 eTk+3 eT2N+5+i eT2N+5+i ⎡ ⎤T ⎡ ⎤ eTi+2 eTi+2 N ⎢T ⎥ ⎢T ⎥ ⎢e ⎥ ⎢ ⎥ Ω4 = ⎣ 2N+5+i ⎦ Ξ3 ⎣e2N+5+i ⎦ i=1 eT3N+5+i eT3N+5+i ⎤T ⎡ ⎤⎡ ⎤ ⎡ eT2 eT ΞωT Cd (t) ΞωT Dω (t) ΞωT Ξω ⎥ ⎢ ⎥ ⎢ T2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ eT ⎥ , Ωω = ⎢ CTd (t)Cd (t) CTd (t)Dω (t) ⎦ ⎣ N+4 ⎦ ⎣ eN+4 ⎦ ⎣ ∗ ∗ ∗ Dω (t)T Dω (t) − In ∗ ρ 2 eT4N+6 eT4N+6

Ω3,k =

and

with

Y=

N i=1

⎡ ⎛⎛ ⎞2 ⎛ ⎞2 ⎞ ⎤ N i i−1 1 ⎢ ⎜⎝ ⎟˜ ⎥ ˜1 + 2i Qˆ i + ⎣(h21 − h20 )Z β ⎠ − ⎝h0 + β ⎠ ⎠ Z ⎝ h0 + i⎦ , 2 i=2

! " 1 Ξ1 = (i )2 T1 + T3 + i He(S1 ς1 + S2 ς2 ), 3 ⎡

˜i −6Z ⎢ Ξ3 = ⎣ ∗ ∗

−6Z˜ i −18Z˜ i ∗

⎤ 24Z˜ i ⎥ 48Z˜ i ⎦ ˜i −144Z

β=1

β=1

⎡

Qˆ k ⎢ Ξ2 = ⎣ ∗ ∗

−Qˆ k + J ˆ 2Qk − JT − J ∗

⎤ −J ⎥ −Qˆ k + J⎦ , ˆ Qk

and Ξω = C(t) + D(t)K.

Moreover, the controller gain matrix in (2.4) can be designed as K = (B+ B)−1 B+ (U + U)−1 U + H, where U + is the Moore–Penrose generalized inverse of matrix B, U.

B+ ,

Proof. For any t ≥ 0, there should exist an integer k ∈ {1, . . . , N}, such that h(t) ∈ Sk . The LKF is as follows: V(xt , k)|h(t)∈Sk = V1 (xt ) + V2 (xt , k) + V3 (xt , k) + V4 (xt ),

(3.6)

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where

7

k−1 t i=1

+

V3 (xt , k) =

t−i

˜ i x(s − hi−1 ) ds + xT (s − hi−1 )X

t−h(t)

N

t−hk

i

i=1

xT (s)T˜ k x(s) ds +

i=k+1

−(h0 +i−1

β=1

V4 (xt ) =

N −(h0 + i=1

β

) t

−(h0 + iβ=1 β ) i−1

and

N t

β=1

t+θ

θ

t−h(t)

˜ k x(s) ds xT (s)X

xT (s − hi−1 )T˜ i x(s − hi−1 ) ds,

˙ ds dθ x˙T (s)Qˆ i x(s)

β ) 0 t

−(h0 + iβ=1 β )

t−i

t−hk−1

t+β

˜ i x(s) ˙ ds dβ dθ. x˙T (s)Z

˙ ≤ μ2 , for any 1 (t) ≥ 0, 2 (t) ≥ 0 satisfying 1 (t) + 2 (t) = 1, let h(t) ˙ = In the case of μ1 ≤ h(t) 2 1 (t)μ1 + 2 (t)μ2 = s=1 s (t)μs . Remark 3.2. In the proposed LKF, V1 (xt ) contains slack variables by using the augmented state vectors, which gives more flexibility with respect to the adjustable partition number. Here V2 (xt ), V3 (xt ) and V4 (xt ) are novelly constructed using the GSD expression with unfixed integral intervals. In the second and third terms of the V2 (xt ), the time-varying delay h(t) is used in the upper and lower bound of the integral forms. By using the convex combination method, when dealing with the derivative of the LKF, these integral forms can be estimated without unnecessary enlargement comparing with the conventional approaches. Thus, the new proposed LKF can provide less conservative results. The derivative of the Lyapunov functional V(xt , k)|h(t)∈Sk along the trajectory of the system described in (2.5) is given as ˙ t , k)|h(t)∈S = V˙ 1 (xt ) + V˙ 2 (xt , k) + V˙ 3 (xt , k) + V˙ 4 (xt ), V(x k

(3.7)

where ⎡

⎤T

⎤ ⎡ ψ˙ 2 (t) ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ψ3 (t) ⎥ U⎢ ⎥ V˙ 1 (xt ) = 2 ⎢ ψ˙ 3 (t) ⎢ t ⎥ ⎦ ⎣ ⎣ ⎦ T ˙ x (s)ds x(t) − (1 − h(t))x(t − h(t)) ψ2 (t)

t−h(t)

⎡ ⎢ ⎢ =2⎢ ⎢ t ⎣

ψ2 (t)

2 s=1

⎡

ψ˙ 2 (t) ψ˙ 3 (t)

⎤

⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ U⎢ # $ ⎥ ⎥ 2 ⎢ ⎥ ⎦ T ⎣ x (s) ds s (t)μs x(t − h(t))⎦ x(t) − 1 −

ψ3 (t)

t−h(t)

= ψ T (t)

⎤T

s=1

s (t)Ω1s ψ(t).

(3.8)

...................................................

V2 (xt , k) =

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⎤T ⎡ ⎤ ψ2 (t) ψ2 (t) ⎥ ⎥ ⎢ ⎢ ψ3 (t) V1 (xt ) = ⎣ ⎦ U ⎣ ψ3 (t) ⎦, t t T T x (s) ds x (s) ds t−h(t) t−h(t) ⎡

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V˙ 2 (xt , k) is derived as

8

i=1

⎛

˜ k x(t − hk−1 ) − ⎝1 − + x (t − hk−1 )X T

⎛ + ⎝1 −

2

2

⎞ ˜ k x(t − h(t)) s1 μs1 ⎠ xT (t − h(t))X

s1 =1

⎞

s2 μs2 ⎠ xT (t − h(t))T˜ k x(t − h(t)) − xT (t − hk )T˜ k x(t − hk )

s2 =1 N

+

(xT (t − hi−1 )T˜ i x(t − hi−1 ) − xT (t − hi )T˜ i x(t − hi ))

i=k+1

= ψ T (t)

2 2

s1 s2 Ω2ks1 s2 ψ(t).

(3.9)

s1 =1 s2 =1

The derivative of V3 (xt , k) is deduced as #N $ t−(h0 +i−1 β ) N β=1 T 2i i ˆ ˙ ˙ − ˙ ds. x˙T (s)Qˆ i x(s) Qi x(t) V 3 (xt , k) = x˙ (t) i=1

i=1

t−(h0 + iβ=1 β )

(3.10)

Using lemmas 2.4 and 2.5, the third term of (3.10) is derived as −

N

i

t−(h0 +i−1

β=1

t−(h0 + iβ=1 β )

i=1

N

=−

i

N i=1,i =k

˙ ds x˙T (s)Qˆ i x(s)

t−(h0 +i−1

i=1,i =k

≤

β )

β=1

β )

t−(h0 + iβ=1 β )

˙ ds − ϑk x˙T (s)Qˆ i x(s)

t−hk−1 t−hk

˙ ds x˙T (s)Qˆ k x(s)

T ζ1iT (t)Ξ1 ζ1i (t) − ζ2k (t)Ξ2 ζ2k (t),

(3.11)

where T

T

1 ϑi

T

t−hi−1

T

ζ1i (t) = x (t − hi−1 )

x (t − hi )

ζ2k (t) = [xT (t − hk−1 )

xT (t − h(t)) xT (t − hk )]T ;

x (s) ds

,

t−hi

Ξ1 and Ξ2 are defined in theorem 3.1. Then, it follows from (3.10) and (3.11) that #N $ T 2i ˆ ˙ ˙ + ψ T (t)Ω3,k ψ(t). ˙ Qi x(t) V 3 (xt , k) ≤ x (t)

(3.12)

i=1

The derivative of V4 (xt ) is deduced as ⎡ ⎛⎛ ⎞2 ⎛ ⎞2 ⎞ ⎤ N i i−1 1 ⎢ ⎜⎝ ⎟˜ ⎥ ˜1 + ˙ V˙ 4 (xt ) = x˙T (t) ⎣(h21 − h20 )Z β ⎠ − ⎝h0 + β ⎠ ⎠ Z ⎝ h0 + i ⎦ x(t) 2 β=1

i=2

−

−h0 t −h1

t+θ

˜ 1 x(s) ˙ ds dθ − x˙T (s)Z

β=1

N −(h0 +i−1 β ) t β=1 i=2

−(h0 + iβ=1 β )

t+θ

˜ i x(s) ˙ ds dθ. x˙T (s)Z

(3.13)

...................................................

˜ i x(t − hi−1 ) − xT (t − hi )X ˜ i x(t − hi )) (xT (t − hi−1 )X

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V˙ 2 (xt , k) =

k−1

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Using lemma 2.6, the last two terms are deduced as

≤

N

t+θ

˜ 1 x(s) ˙ ds dθ − x˙T (s)Z

i=2

−(h0 + iβ=1 β )

t+θ

9 ˜ i x(s) ˙ ds dθ x˙T (s)Z

ζ3iT (t)Ξ3 ζ3i (t),

(3.14)

i=1

where ζ3i (t) = [xT (t − hi−1 ) (1/ϑi ) in theorem 3.1. Then it is derived as ⎡

t−hi−1 t−hi

xT (s) ds (1/(ϑi )2 )

−hi−1 t−hi−1 −hi

t+θ

xT (s) ds dθ]T ; Ξ3 is defined

⎛⎛ ⎞2 ⎛ ⎞2 ⎞ ⎤ N i i−1 1 ⎢ ⎜⎝ ⎟ ⎥ ˙ β ⎠ − ⎝h0 + β ⎠ ⎠ Z˜ i ⎦ x(t) V˙ 4 (xt ) ≤ x˙T (t) ⎣(h21 − h20 )Z˜ 1 + ⎝ h0 + 2 β=1

i=2

β=1

+ ψ T (t)Ω4 ψ(t).

(3.15)

Under the zero-initial condition, it is apparent that V(x(t))|t=0 = 0. Let ∞ [zT (t)z(t) − ρ 2 ωT (t)ω(t)] dt. J=

(3.16)

0

Then for any non-zero ω(t) ∈ L2 [0, +∞), ∞ ˙ t , k)] dt, [ψ T (t)Ωω ψ(t) + V(x J≤

(3.17)

0

where ψ T (t)Ωω ψ(t) = zT (t)z(t) − ρ 2 ωT (t)ω(t). Therefore, the following inequality holds: ˙ t , k)|h(t)∈S + ψ T (t)Ωω ψ(t) ≤ ψ T (t) V(x k

2 s1,2 =1

s1,2 (t)

2

s (t)Ωks1 s2 ψ(t).

(3.18)

s=1

Using the augmented vector (3.3) with the simplification expression (3.2), system (2.5) is represented as 0 = Υo ψ(t),

(3.19)

where Υo is defined in theorem 3.1. Hence, the asymptotic stability condition for system (2.5) is expressed as ψ T (t)

2

s1,2 (t)

s1,2 =1

2

s (t)Ωks1 s2 ψ(t) < 0 subject to : 0 = Υo ψ(t).

(3.20)

s=1

Consequently, by means of lemma 2.6, there exists a matrix U with appropriate dimensions such that (3.20) is equivalent to ψ T (t)

2 o=1

Γo (t)

2 s1,2 =1

s1,2 (t)

2

s (t)(Ωks1 s2 + He(UΥo ))ψ(t) < 0.

(3.21)

s=1

As a result, the derivatives of the newly proposed Lyapunov functionals are deduced as ˙ t , k)|h(t)∈S < 0. This means that V(x ˙ t , k)|h(t)∈S < ξ x(t)2 for sufficiently small ξ > 0. Hence V(x k k system (2.5) is ensured of being globally asymptotically stable with the guaranteed H∞ performance given in definition 2.3. This completes the proof. Theorem 3.3. Given a positive integer N, and ϑi = i . Consider a time-varying delay satisfying case 2.2. System (2.5) is globally asymptotically stable with H∞ performance ρ if there exist symmetric ˜ i, X ˜ i , ∈ Rn×n (i = 1, . . . , N), Uˆ ∈ R(2N)n×(2N)n , symmetric matrices T1 , T3 ∈ positive definite matrices Qˆ i , Z

...................................................

−h1

N −(h0 +i−1 β ) t β=1

rspa.royalsocietypublishing.org Proc. R. Soc. A 473: 20160934

−

−h0 t

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2

Γo (t)(Ωˆ k + He(UΥo )) < 0,

(3.23)

o=1

where Ωks1 s2 is modified to Ωˆ k by changing Ω2ks1 s2 and Ω1s to Ωˆ 1 and Ωˆ 2 as follows: Ωˆ k = Ωˆ 1 + Ωˆ 2 + Ω3,k + Ω4 + Ωω + eT1 YeT1 , ⎧⎡ ⎤T ⎡ ⎤⎫ eT2 − eT4 ⎪ ⎪ eTN+6 ⎪ ⎪ ⎪ ⎢ . ⎥ ⎪ ⎪ ⎢ ⎥⎪ .. ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ . ⎪ . ⎪ ⎪⎢ . ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ T T ⎨⎢eT ⎢ e2 − eN+3 ⎥⎬ ⎥ 2N+5 ⎥ U⎢ ⎥ , Ωˆ 1 = He ⎢ ⎢eT ⎢ ⎥ ⎥ ⎪ ⎢ 2N+6 ⎥ ⎪ ⎪ ⎢ eT3 − eT4 ⎥⎪ ⎪ ⎪ ⎪⎢ . ⎥ ⎢ ⎥⎪ ⎪ ⎪ . ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ . ⎪ ⎪⎣ . ⎦ .. ⎪ ⎣ ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ eT T T eN+2 − eN+3 3N+5 Ωˆ 2 =

N

˜ i eT − ei+3 X ˜ i eT ). (ei+2 X i+2 i+3

i=1

Υo and the design of controller gain K are given in theorem 3.1. Proof. For system (2.5), modify the Lyapunov functionals (3.6) by amending V1 (xt ) and V2 (xt , k) as

V(xt , k)|h(t)∈Sk = Vˆ 1 (xt ) + Vˆ 2 (xt ) + V3 (xt , k) + V4 (xt ),

where

and

(3.24)

T ψ2 (t) ψ2 (t) ˆ U V 1 (xt ) = ψ3 (t) ψ3 (t) Vˆ 2 (xt , k) =

N t i=1

t−i

˜ i x(s − hi−1 ) ds. xT (s − hi−1 )X

Then, following a similar process to the proof of theorem 3.1, the asymptotic stability condition with H∞ performance ρ for system (2.5) is equivalent to ψ T (t)

2

Γo (t)(Ωˆ k + He(UΥo ))ψ(t) < 0.

(3.25)

o=1

This completes the proof.

Remark 3.4. Although the expected results are obtained in this proposed work, the application of the delay partitioning approach leads to a high complexity. Meanwhile, the partition number N can be increased to reduce the conservatism. But this causes an extra computational load. How to choose an appropriate number N and how to reduce the complexity are full of challenges. The trade-off will be the subject of future works.

4. Numerical example A liquid monopropellant rocket motor with a pressure feeding system is considered in this section. The dynamic model of the rocket motor was originally given in [40]. Then, this practical

...................................................

and

10

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R3n×3n , matrices T2 ∈ R3n×3n , S1 , S2 ∈ R3n×n , J ∈ Rn×n and U ∈ R(4N+5)n×n , such that the following LMIs hold: ⎤ ⎡ T1 T2 S1 ⎥ ⎢ Ti = ⎣ ∗ T3 S2 ⎦ ≥ 0 (3.22) ∗ ∗ Qˆ i

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Table 1. Upper bounds of hN for h0 = 0. [41] 1.0723

theorem 3.1 (n = 3) 1.0812

theorem 3.3 (n = 3) 0.9069

..........................................................................................................................................................................................................

and complex system is further investigated in [36,41]. The feeding system and combustion chamber equations in the absence of steady flow are linearized as ⎫ x˙1 (t) = (σ − 1)x1 (t) − σ x1 (t − hN ) + x3 (t − hN ),⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ˙x2 (t) = [−x4 (t) + u(t) + ω2 (t)], ⎪ ⎪ ψJ1 ⎬ (4.1) 1 ⎪ ⎪ x˙3 (t) = [−x3 (t) + x4 (t) − J2 x1 (t)] ⎪ ⎪ (1 − ψ)J1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎭ and x˙4 (t) = [x2 (t) − x3 (t) + ω4 (t)], J3 where x1 (t), x2 (t), x3 (t) and x4 (t) are the non-dimensional instantaneous pressure in the combustion chamber, the instantaneous mass flow upstream of the capacitance, the instantaneous mass rate of injected propellant and the instantaneous pressure at the place in the feeding line, respectively. hN is the state-steady operation delay. u(t), ω2 (t) and ω4 (t) are the control input and the disturbance inputs, respectively. Considering the numerical values σ = 1, ψ = 0.5, J1 = 2, J2 = 1, J3 = 1, the parameter matrices of system (4.1) are given as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ σ −1 0 0 0 σ 0 1 0 0 0 ⎢ 0 ⎢ 0 0 0 0⎥ ⎢1⎥ ⎢1⎥ 0 0 −1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A1 = ⎢ ⎥ , A2 = ⎢ ⎥ , B1 = ⎢ ⎥ , B2 = ⎢ ⎥ , 0 −1 1⎦ ⎣ −1 ⎣ 0 0 0 0⎦ ⎣0⎦ ⎣0⎦ 0 1 −1 0 0 0 0 0 0 1 C1 = [1 0

0 0],

C2 = 0,

D1 = 1,

D2 = 0.

By using the Matlab-based LMI toolbox [42], different methods are considered to compare with existing results. The maximum values of hN are given in table 1 with h0 = 0 and ρ = 0.5302. In table 1, according to a differentiable function h(t) satisfying (2.2), theorem 3.1 shows a less conservative result than the developments in [36,41]. Referring to remark 4.1, and considering the case of a continuous function h(t) satisfying (2.3) given in case 2.2, these two works [36,41] did not provide any further information. However, theorem 3.3 is derived to show a good result. It means that the designed control method can overcome the time-varying delay h(t) with an unknown derivative bound. Remark 4.1. Both the lower and upper bounds of delay h(t) are considered in cases 2.1 and 2.2. Actually, case 2.1 is a special circumstance of case 2.2, which means less conservative results are obtained by using case 2.1 for a differentiable function of h(t). However, case 2.2 is employed to overcome some scenarios in which h(t) is not differentiable [43]. For a time-varying delay, that is, h(t) = 0.5 + 0.5 sin(t), considering ρ = 0.5302, h0 = 0 and hN = 1.0812, the state response of systems (2.1) is shown in figure 2. Now considering the disturbance ω(t) = cos(t)e−0.1t with hN = 1.0812, the guaranteed H∞ performance is solved as γ = 0.5302 using the LMIs, then the gain matrix K is designed as K = [−0.0082 − 0.4038 − 1.0742 1.0000].

(4.2)

The guaranteed H∞ performance is displayed in figure 3. We consider ±20% parameter uncertainties with regard to (2.6), that is, Uv (t) ∈ [Uv (t) × 80%, Uv (t) × 120%]. Then, to ensure global asymptotic stability based on the obtained maximum value

...................................................

[36] 1.0000

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method ρ = 0.5302

11

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2.0

12

x3(t), r = 0.5302, hN = 1.0812 x4(t), r = 0.5302, hN = 1.0812

x (t)

0.5 0 −0.5 −1.0 −1.5 −2.0

0

100

200

300

400

500

600

700

t

Figure 2. State response of systems (2.1). (Online version in colour.)

0.7 0.6 r ||w||2

amplitude

0.5

||x||2

0.4 0.3 0.2 0.1

0

10

20

t

30

40

50

Figure 3. H∞ performance of systems (2.1).

in theorems 3.1 and 3.3, the guaranteed H∞ performance index is solved as ρ = 2.1905 and ρ = 0.7056, respectively. Remark 4.2. Using the GSD method can dramatically improve the efficiency of obtaining hN with less computational complexity. For example, the sum formulation of the geometric progression is a1 (1 − N )/(1 − ), where a1 is the first item and = 1. Let a1 = 1, = 2 and the partitioning number N = 4, that is, a1 (1 − N )/(1 − ) = 15. This means that the partition number N = 15 in the case of the equivalent division method. However, if we use this GSD approach, the

...................................................

x2(t), r = 0.5302, hN = 1.0812 1.0

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x1(t), r = 0.5302, hN = 1.0812

1.5

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In this paper, robust H∞ control for linear systems with disturbance and uncertainties is proposed. In terms of the geometric progression method, the modified LKF is developed with less slack variables. The decision variables are dramatically reduced due to the GSD approach. Then the designed state-feedback controller can successfully nullify the effect of the disturbance input and parameter uncertainties. In addition, by employing the convex combination skill parameter uncertainties, the derivatives of delay are flexibly described. This technique can reduce the unnecessary enlargement of the LKF derivative estimation. Finally, a liquid monopropellant rocket motor with a pressure feeding system is applied to illustrate that the implemented control algorithm can ensure that the pressure feeding system is globally asymptotically stable with a guaranteed H∞ performance with disturbance and uncertainties. Owing to the complex dynamics of the unstable system, the control of the perturbed nonlinear unstable systems is full of challenges. This recently became a hot topic. Future works will thus focus on the robust H∞ control of unstable systems with stochastic uncertainties. Authors’ contributions. H.C. and Z.Z. conceived and designed the study. H.W. and H.C. performed the simulation. All authors read and approved the manuscript.

Competing interests. The authors declare that there are no competing interests regarding the publication of this paper.

Funding. This work was partially supported by the National Nature Science Foundation of China (61673016), the Sichuan Youth Science and Technology Innovation Research Team (2017TD0028), the Scientific Research Fund of the Sichuan Provincial Education Department (15ZB0483, 17ZB0459), the Innovative Research Team of the Education Department of Sichuan Province (15TD0050) and the SWUN Innovation Teams (14CXTD03). Acknowledgements. The authors greatly appreciate the editor’s and the anonymous referees’ constructive and valuable comments that improved the quality of this paper.

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...................................................

5. Conclusion

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partition number N = 4, which is 30% less than the equivalent partitioning method. Hence, the decision variables are decreased and the entire computational load can be reduced. In addition, if the common ratio = 1, the number of subintervals will be N × a1 . So the length of each segment becomes equal. Then previously developed studies using the equivalent partition method in [7,18] can be considered as a special case of this development.

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