Receding horizon control for multiplicative noise stochastic systems with input delay

Automatica 81 (2017) 390–396

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Receding horizon control for multiplicative noise stochastic systems with input delay✩ Rong Gao a,b , Juanjuan Xu a , Huanshui Zhang a,1 a

School of Control Science and Engineering, Shandong University, Jinan 250061, China

b

School of Mathematics and Statistics Science, Ludong University, Yantai, 264025, China

article

info

Article history: Received 17 October 2015 Received in revised form 10 January 2017 Accepted 5 March 2017

Keywords: Receding horizon control Stochastic system Multiplicative noise Mean square stabilization Input delay

abstract This paper is concerned with finite horizon stabilization control for a class of discrete time stochastic systems subject to multiplicative noise and input delay. By constructing a new cost function, a complete solution to the problem of finite horizon stabilization is given for the first time based on previous work Zhang et al. (2015). It is shown that the system can be stabilized in the mean square sense with the receding horizon control (RHC) if and only if two new inequalities on the terminal weighting matrices are satisfied. Moreover, the two inequalities can be solved by using iterative algorithm. The explicit stabilizing controller is derived by solving a finite horizon optimal control problem. Simulations demonstrate the effectiveness of the proposed method. © 2017 Published by Elsevier Ltd.

1. Introduction In the past few decades, receding horizon control (RHC, also known as model predictive control) has attracted interest from the control community because of its applicability in chemical, automotive, and aerospace processes. A considerable amount of research effort has been devoted to RHC, e.g., see Garcia, Prett, and Morari (1989), Mayne (2014), Mayne, Rawlings, Rao, and Scokaert (2000), Richalet, Rault, Testud, and Papon (1978) and the references therein. The basic concept of RHC is to solve an optimization problem on the finite horizon at the current time and implement only the first solution as the current control. This procedure is then repeated at the next time step. The stabilization problem as one of fundamental problems has been studied extensively based on RHC. Kwon et al. (Kwon & Pearson, 1977) originally studied the stabilizing property of the RHC law for linear systems. This idea was then generalized

✩ This work is supported by the Taishan Scholar Construction Engineering by Shandong Government and the National Natural Science Foundation of China under Grants 61120106011, 61573221. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (R. Gao), [email protected] (J. Xu), [email protected] (H. Zhang). 1 Fax: +86 531 88399038.

http://dx.doi.org/10.1016/j.automatica.2017.04.002 0005-1098/© 2017 Published by Elsevier Ltd.

to stochastic systems and time delay systems (Bernardini & Bemporad, 2012; Cannon, Kouvaritakis, & Wu, 2009a,b; Chatterjee, Hokayem, & Lygeros, 2011; Chatterjee & Lygeros, 2015; Hessem & Bosgra, 2003; Hokayem, Cinquemani, Chatterjee, Ramponi, & Lygeros, 2012; Kwon, Lee, & Han, 2004; Lee & Han, 2015; Park, Yoo, Han, & Kwon, 2008; Perez & Goodwin, 2001; Primbs & Sung, 2009; Wei & Visintini, 2014). Chatterjee et al. (2011) and Hokayem et al. (2012) investigated the RHC for additive noise systems. In Chatterjee et al. (2011), the optimization problem was solved by using a vector space method that ensured the variance of state was bounded. In Hokayem et al. (2012), incomplete state information was considered and a Kalman filter was used to estimate the optimal state. RHC bounded the state of the overall systems in the mean square sense. Refs. Cannon et al. (2009a) and Primbs and Sung (2009) studied RHC for systems with multiplicative noise. In Cannon et al. (2009a), the concept of probability invariance was introduced to ensure the stability of a closed-loop system, whereas Primbs and Sung (2009) used semi-definite programming to solve the optimization problem and the stability of the closed-loop system was ensured under a specific terminal weight and terminal constraint. In Cannon et al. (2009b), a system with both additive and multiplicative noise was considered based on RHC. Other related RHC stochastic problems can be found in Bernardini and Bemporad (2012), Chatterjee and Lygeros (2015), Hessem and Bosgra (2003), Perez and Goodwin (2001) and Wei and Visintini (2014). Systems with time delay have also been subjected to RHC (Kwon et al., 2004; Lee & Han, 2015; Park et al., 2008). For

R. Gao et al. / Automatica 81 (2017) 390–396

instance, Kwon et al. (2004) studied RHC for a system with state delay and used a linear matrix inequality (LMI) condition on the terminal weighting matrices to guarantee the stability of a closedloop system. In Park et al. (2008), a system with input delay was investigated based on RHC, and a stabilization condition was derived based on LMI. Ref. Lee and Han (2015) also considered RHC stabilization for a system with state delay. By proposing a more generalized cost function, a delay dependent stability condition was obtained. However, it is notable that time delay and multiplicative noise have been considered separately in all of the aforementioned studies and the references therein. When a system has both time delay and multiplicative noise, the control problem is particularly difficult. One of the obstacles is that the separation principle does not hold for stochastic systems with multiplicative noise. In this article, we discuss the RHC stabilization of discrete time linear systems with both multiplicative noise and input delay. Our aim is to determine the RHC stabilization condition, and derive the RHC stabilization controller when this condition is met. The main contributions of this paper are three-fold: First, we construct a novel cost function that includes two terminal weighting matrices. An explicit stabilizing controller is obtained by solving this finite horizon optimal control problem. Second, a necessary and sufficient condition for the stabilization of delayed stochastic systems is developed. Under some mild assumptions, it is shown that the system can be stabilized in the mean square sense if and only if two inequalities regarding terminal weighting matrices are satisfied. Third, by introducing a slack variable, an iterative algorithm has been proposed to solve the two inequalities. The remainder of this paper is organized as follows. Section 2 presents the formulation of the problem for stochastic systems with multiplicative noise and input delay. In Section 3 the corresponding RHC law and the necessary and sufficient condition for the asymptotic mean square stability of the closed-loop system are derived. The iterative algorithm to solve the two inequalities is also discussed in Section 3. A numerical example to validate the performance of the proposed RHC is provided in Section 4. Finally, our conclusions are given in Section 5. The following notations are used throughout the paper. Rn denotes the n dimensional Euclidean space. The subscript ′ represents the matrix transpose; a symmetric matrix M > 0(≥ 0) means that it is strictly positive definite (positive semidefinite). {Ω , F , P , {Fk }k≥0 } denotes a complete probability space on which some scalar white noise ωk is defined such that {Fk }k≥0 is the natural filtration generated by ωk , i.e., Fk = σ {ω0 , . . . , ωk }. Let xˆ k|m = Em−1 (xk ), where Em−1 (xk ) is the conditional expectation of xk with respect to Fm−1 . E (·) denotes the mathematical expectation over the noise {ωk , k ≥ 0}. 2. Problem statement Consider the following linear discrete time stochastic system with input delay: xk+1

= (A + ωk A¯ )xk + (B + ωk B¯ )uk−d ,

(1)

with the initial condition x0 , u−d , u−d+1 , . . . , u−1 . For the conve¯ Bk = B + ωk B. ¯ Then, nience of later discussion, let Ak = A + ωk A, system (1) becomes xk+1 = Ak xk + Bk uk−d ,

(2)

where xk ∈ Rn is the state; uk ∈ Rm is the input with delay d > 0; ¯ A, and B are matrices of appropriate dimensions; and ωk is a A¯ , B, scalar random white noise with zero mean and variance σ . The problem to be solved in this paper is formulated as follows: Find the Fk−d−1 -measurable controller uk−d = H xˆ k|k−d , k ≥ d, such that the closed-loop system xk+1 = Ak xk + Bk H xˆ k|k−d is asymptotically mean square stable, i.e., limk→∞ E (x′k xk ) = 0.

391

Remark 1. Note that the results presented in this paper are applicable to more general systems of multiple multiplicative noises with no substantial difference: (1) (2) xk+1 = (A + ωk A¯ )xk + (B + ωk B¯ )uk−d , (1)

where ωk

̸= ωk(2) .

3. Receding horizon control for discrete time stochastic systems with input delay In this section, we present results for the asymptotic mean square stability for discrete time stochastic systems with input delay (1). The RHC solution is given first. 3.1. Receding horizon control To solve the problem formulated in Section 2, we first introduce the following function: (d)

J (xk , Uk , k, k + N , Uk )

=

N 

x′k+i Qxk+i +

N −d 

i=0

u′k+i Ruk+i + (xk+N +1 )′

i=0

× P (1) xk+N +1 + (xk+N +1 )′

d+1 

P (i) xˆ k+N +1|k+N +i−d−1 ,

i =2

(d)

where xk and Uk = (uk−1 , . . . , uk−d ) are known values at time k; Uk = (uk , . . . , uk+N −d ) is the control to be determined; Q ≥ 0, R > 0, and N is a finite positive integer. For the convenience of discussions in the below, we denote the cost function as: (d)

Jk−1 (xk , Uk , k, k + N , Uk )

  = Ek−1 J (xk , U(kd) , k, k + N , Uk ) ,

(3)

(d)

where Ek−1 [J (xk , Uk , k, k + N , Uk )] is the conditional mathematical expectation given Fk−1 = σ {ω0 , . . . , ωk−1 }. It is assumed that the weighting matrices P (i) , i = 1, 2, . . . , d + 1 satisfy P (1) > 0, P (2) ≤ 0, and P (i) = (A′ )i−2 P (2) Ai−2 , d+1 

i = 3, . . . , d + 1,

(4)

P (i) > 0.

i =1

Note that once P (2) is given, P (i) , i = 3, . . . , d + 1 are determined by (4). Thus, there are only two independent terminal weighting matrices, P (1) and P (2) . We shall show that the two matrices P (1) and P (2) play a key role in designing the RHC to guarantee mean square stability. Remark 2. The cost function (3) is nonnegative. Because



Ek−1 (xk+N +1 )′ P (1) xk+N +1 + (xk+N +1 )′

d+1 

P (i)

i=2

× xˆ k+N +1|k+N +i−d−1



d+1    ≥ Ek−1 (xk+N +1 )′ P (i) xk+N +1 ≥ 0, i =1

Q ≥ 0 and R > 0, we have that the cost function (3) is nonnegative. Remark 3. Considering the input delay in the stochastic system (1), the terminal terms of the cost function herein are given by

(xk+N +1 )′ P (1) xk+N +1 + (xk+N +1 )′

d+1  i=2

P (i) xˆ k+N +1|k+N +i−d−1

392

R. Gao et al. / Automatica 81 (2017) 390–396

as in (3) to derive the RHC, which is in the form of the optimal cost value of optimal control for system (1) with cost function J =E

 N 

′

xi Qxi +

N −d 

i =d

In this subsection, we investigate the property of cost function (3), which will play an important role in the stabilization of the system (1).

 ′

′

ui Rui + xN +1 P

(1)

x N +1 .

i =0

This is different from the delay free case where there is only one terminal term involved in the cost, i.e., P (i) = 0, for i = 2, 3, . . . , d + 1. The following lemma gives the solution to the finite horizon linear quadratic optimal control problem of (3) subject to (1). Lemma 1. The optimal control of system (1) with the cost function (3) is unique if and only if the recursion (6)–(10) is well defined, i.e., Υs , s = k + N , k + N − 1, . . . , k + d are all invertible. If this condition holds, then the optimal controller us is given by 1 ˆ s+d|s us = −Υs− +d Ms+d x

(5)

for s = k, k + 1, . . . , k + N − d, where xˆ s+d|s = Es−1 (xs+d ) = Ad xs +

d  i=1

(i)

Υs , Ms , and Ps , i = 1, 2, . . . , d + 1 satisfy the following backwards recursion for s = k + N , k + N − 1, . . . , k + d: (1)

(1)

Υs = R + B′ Ps+1 B + σ B¯ ′ Ps+1 B¯ + B′



(j)

Ps+1 B,

d+1 

¯ )′ P (1) (A¯ + BH ¯ ) (A + BH )′ Θ (A + BH ) + σ (A¯ + BH − Θ + Q + H ′ RH ≤ 0, A′ P (1) A + σ A¯ ′ P (1) A¯ − P (1) + (Ad )′ P (2) Ad + Q ≤ 0, d−1 where Θ = P (1) + i=0 (A′ )i P (2) Ai . Then

(j)

Ps+1 A,

(7)

(d)

− Jk∗−1 (xk , U(kd) , k, k + N , Uk ) ≤ 0 holds, where Jk∗−1 (xk , Uk , k, k + N , Uk ) is the optimal cost of (3); (d) Uk (k ≥ d) is the RHC as (11) at time k − i, i = 1, . . . , d; Uk is the optimal control as (5) at time k + i, i = 0, . . . , k + N − d; xk+1 and xk are related by Eq. (1) where ωk is a scalar random white noise, uk−d is the RHC at time k − d. Proof. In view of the cost function (3), we have Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

− Jk∗−1 (xk , U(kd) , k, k + N , Uk ) N N −d    = Ek−1 Ek (x¯¯ k+1+i )′ Q x¯¯ k+1+i + (u¯¯ k+i+1 )′ R i=0

(d+1) (1) (1) Ps(1) = A′ Ps+1 A + σ A¯ ′ Ps+1 A¯ + Q + A′ Ps+1 A,

(8)

Ps(2) = −Ms′ Υs−1 Ms ,

(9)

Ps =

(i−1) A′ Ps+1 A,

i = 3, . . . , d + 1,

(10)

with the terminal value (i) Pk+N +1 = P (i) ,

(13)

Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

j =2

(i)

(12)

(d)

(6)

j =2

(1) (1) Ms = B′ Ps+1 A + σ B¯ ′ Ps+1 A¯ + B′

Lemma 2. Assume there exist P (1) and P (2) in (3) satisfying the following matrix inequality for some H:

(d)

Ai−1 Bus−i ,

d+1

3.2. Asymptotic mean square stability

i =2

i=0

× x¯ k+i +

i = 1, . . . , d + 1.

i =0

× u¯¯ k+i+1 + (x¯¯ k+N +2 )′ P (1) x¯¯ k+N +2 + (x¯¯ k+N +2 )′ d+1 N    × P (i) xˆ¯¯ k+N +2|k+N +i−d − Ek−1 x¯ ′k+i Q N −d 

u¯ ′k+i Ru¯ k+i + (¯xk+N +1 )′ P (1) x¯ k+N +1

i =0

Moreover, the optimal costate λs−1 and state xs satisfy the following non-homogeneous relationship

+ (¯xk+N +1 )′

d+1 



P (i) xˆ¯ k+N +1|k+N +i−d−1 ,

(14)

i=2

λs−1 = Ps(1) xs +

d+1 

Ps(i) xˆ s|s+i−d−2 ,

where u¯¯ k+i+1 , u¯ k+i , i = 0, . . . , N − d, are the optimal control (d) sequences that minimize the cost function Jk (xk+1 , Uk+1 , k +

i =2

s = k + N + 1, . . . , k + d.

(d)

Proof. By applying Pontryagin’s maximum principle (Zhang, Wang, & Li, 2012) to system (2) with cost function (3), we can obtain the following costate equations:

λk+N = P (1) xk+N +1 +

d+1 



P (i) xˆ k+N +1|k+N +i−d−1 ,

u˜ k+i+1 =

i =2

λj−1 = Ej−1 (A′j λj ) + Qxj , 0 = Ej−d−1 (B′j λj ) + Ruj−d ,

j = k, . . . , k + N , j = k + d, . . . , k + N ,

where λj is the costate. Then the results can be obtained by using similar procedure to that in Theorem 1 of Zhang, Lin, Xu, and Fu (2015). We omitted the detailed proof to save space. The RHC at time k is given by (5) with s = k as uk = −Υk−+1d Mk+d xˆ k+d|k .

1, k + N + 1, Uk+1 ) and Jk−1 (xk , Uk , k, k + N , Uk ), respectively. x¯¯ k+i+1 , x¯ k+i are the respective optimal state trajectories generated when the system is controlled by u¯¯ k+i+1 , u¯ k+i , i = 0, . . . , N − d. Let us replace the control u¯¯ k+i+1 in (14) by u˜ k+i+1 , which is given by

(11)

u¯ k+i+1 ,

i = 0, . . . , N − d − 1,

H xˆ˜ k+N +1|k+N +1−d ,

i = N − d.

Note that, since u˜ k+N +1−d may not be optimal on k + N + 1 − d, the resulting state trajectory on k + N + 2 is neither x¯¯ k+N +2 nor x¯ k+N +2 , and is thus denoted by x˜ k+N +2 . Then, it follows from (14) that (d)

Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

− Jk∗−1 (xk , U(kd) , k, k + N , Uk )  ≤ Ek−1 (˜xk+N +1 )′ Q x˜ k+N +1 + (˜uk+N −d+1 )′ R × u˜ k+N −d+1 + (˜xk+N +2 )′ P (1) x˜ k+N +2 + (˜xk+N +2 )′

R. Gao et al. / Automatica 81 (2017) 390–396

×

d+1 

× xˆ¯ k+N +1|k+N +1−d + Ek−1 [(x˘¯ k+N +1 )′ (A′ P (1) A

P (i) xˆ˜ k+N +2|k+N −i−d − x¯ ′k Q x¯ k − u¯ ′k Ru¯ k

+ σ A¯ ′ P (1) A¯ − P (1) + (Ad )′ P (2) Ad + Q )x˘¯ k+N +1 ].

i=2

′ (1)

−(¯xk+N +1 ) P

x¯ k+N +1 − (¯xk+N +1 )

′

d+1 

P

(i)

(d)

Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

 × xˆ¯ k+N +1|k+N +i−d−1  = Ek−1 (¯xk+N +1 )′ Q x¯ k+N +1 + (xˆ¯ k+N +1|k+N +1−d )′

− Jk∗−1 (xk , U(kd) , k, k + N , Uk ) ≤ 0. This completes the proof.

′ (1)

× H RH xˆ¯ k+N +1|k+N +1−d + (˜xk+N +2 ) P x˜ k+N +2 d+1  + (˜xk+N +2 )′ P (i) xˆ˜ k+N +2|k+N −i−d − x¯ ′k Q x¯ k ′

(16)

Using (12)–(13), we obtain

i=2

The following lemma is the basis of the next main theorem. Lemma 3. Given Q > 0 and R > 0, suppose that system (1) can be stabilized in the mean square sense by RHC. Then, the following coupled Lyapunov equations (17)–(22) admit a solution with Z > 0.

i =2

− u¯ ′k Ru¯ k − (¯xk+N +1 )′ P (1) x¯ k+N +1 − (¯xk+N +1 )′ d+1   × P (i) xˆ¯ k+N +1|k+N +i−d−1 ,

¯ )′ X (A¯ + BH ¯ ) (A + BH )′ Z (A + BH ) + σ (A¯ + BH + Q + H ′ RH = Z ,

i=2

(17)

d−1

X =Z+

where we have used the fact x˜ k+N +1 = x¯ k+N +1 . Note that

H = −Υ −1 M ,

Then, we have

L=MΥ ′

(d)

Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

Υ = R + B ZB + σ B X B¯ ,

(21)

M = B ZA + σ B X A¯ .

(22)

(23)

d−1  X =Z+ (A′ )i LAi ,

(24)

i=0

with

× xˆ¯ k+N +1|k+N +1−d + (xˆ¯ k+N +1|k+N +1−d )′   d+1  ′ ′ (1) ′ ¯ ′ (1) ¯ ′ ′ (i) × H B P A + σH B P A + H B P A

L = M ′ Υ −1 M ,

¯ ) + Q + H ′ RH − × (A¯ + BH

 i =1

P (i)



M = B ZA + σ B X A¯ .

(27)

Z = A′ ZA + σ A¯ ′ X A¯ + Q − L

(15)

We can see that x¯ k+N +1 = x˘¯ k+N +1 + xˆ¯ k+N +1|k+N +1−d , where x˘¯ k+N +1 is the estimation error of x¯ k+N +1 . Furthermore, (15) gives

d+1

(26)

¯′

According to the algebraic Riccati-ZXL equations (23)–(27), we have

i =2

i=1

Υ = R + B ZB + σ B X B¯ , ′

× xˆ¯ k+N +1|k+N +1−d + (xˆ¯ k+N +1|k+N +1−d )′   d+1  ′ ′ (1) ′ ¯ ′ (1) ¯ ′ ′ (i) × H B P BH + σ H B P BH + H B P BH

¯ )′ P (1) P (i) (A + BH ) + σ (A¯ + BH

(25)

¯′

′

i=2

× (A + BH )′

¯′

Z = A′ ZA + σ A¯ ′ X A¯ + Q − L,

i=2

d+1 

(20)

¯′

Proof. Because system (1) can be stabilized in the mean square sense, we know from Zhang et al. (2015) that the following RiccatiZXL equations (23)–(27) admit a unique solution with Z > 0.

+ (Ad )′ P (2) Ad )¯xk+N +1 + (xˆ¯ k+N +1|k+N +1−d )′   d+1  ′ (1) ′ (1) ¯ ′ (i) ¯ × A P BH + σ A P BH + A P BH

− Jk∗−1 (xk , U(kd) , k, k + N , Uk )  ≤ −¯x′k Q x¯ k − u¯ ′k Ru¯ k + Ek−1 (xˆ¯ k+N +1|k+N +1−d )′

(19)

M,

′

≤ Ek−1 [−¯x′k Q x¯ k − u¯ ′k Ru¯ k + (¯xk+N +1 )′ Q x¯ k+N +1 + (xˆ¯ k+N +1|k+N +1−d )′ H ′ RH xˆ¯ k+N +1|k+N +1−d ]  + Ek−1 (¯xk+N +1 )′ (A′ P (1) A + σ A¯ ′ P (1) A¯ − P (1)

(d)

−1 ′

− Jk∗−1 (xk , U(kd) , k, k + N , Uk )

Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

(18)

with

= Ak+N +1 x¯ k+N +1 + Bk+N +1 H xˆ¯ k+N +1|k+N +1−d .

× xˆ¯ k+N +1|k+N +1−d − (xˆ¯ k+N +1|k+N +1−d )′ P (2)  × xˆ¯ k+N +1|k+N +1−d .

 (A′ )i LAi , i=0

x˜ k+N +2 = Ak+N +1 x˜ k+N +1 + Bk+N +1 u˜ k+N +1−d



393



= A′ ZA + σ A¯ ′ X A¯ + Q − M ′ Υ −1 M = A′ ZA + σ A¯ ′ X A¯ − M ′ Υ −1 M − M ′ Υ −1 M + M ′ Υ −1 Υ Υ −1 M + Q .

(28)

If we let H = −Υ −1 M, then (28) becomes

¯ )′ X (A¯ + BH ¯ ) Z = (A + BH )′ Z (A + BH ) + σ (A¯ + BH + Q + H ′ RH . Thus, the coupled Lyapunov equations (17)–(22) admit a solution with Z > 0. This completes the proof. It is obvious that the Riccati-ZXL equations (23)–(27) can be rewritten as: A′ XA + σ A¯ ′ X A¯ + Q − X − (Ad )′ LAd = 0,

(29)

where L = M ′ Υ −1 M ,

(30)

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R. Gao et al. / Automatica 81 (2017) 390–396





Υ = R + B′ X −

d−1  (A′ )i LAi B + σ B¯ ′ X B¯ ,

(31)

i =0

 M = B′

xk+1 = (A + ωk A¯ )xk + (B + ωk B¯ )uk .

 d−1  ′ i i X− (A ) LA A + σ B¯ ′ X A¯ .

(32)

i=0

By using Lemmas 2 and 3, the main result of the paper can be presented in the following theorem. Theorem 1. Given Q > 0 and R > 0, system (1) with the RHC (11) is asymptotically mean square stable if and only if there exist P (1) > 0, P (2) ≤ 0 such that

Θ = P (1) +

d−1  (A′ )i P (2) Ai > 0,

and H satisfying (12)–(13). Proof. Sufficiency: According to Lemma 2, there exist P (1) , P (2) , and H satisfying (12)–(13). Thus, we obtain

¯ )′ P (1) (A + BH )′ P (1) (A + BH ) + σ (A¯ + BH ¯ ) − P (1) + Q + H ′ RH ≤ 0, × (A¯ + BH ′ (1)

AP

− Jk∗−1 (xk , U(kd) , k, k + N , Uk ) ≤ 0.

+P

+ Q ≤ 0.

(36) (37)

Further, note that for any P > 0, any A, A¯ , Q > 0, and σ > 0, there always exists some P 2 ≤ 0 that satisfies (37). Then, (36) and (37) reduce to

Hence, Theorem 1 contains the result for delay free stochastic systems as a special case.

AP

(d)

E {Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

A−P

(1)

d ′ (2) d

+ (A ) P

A + Q ≤ 0.

(38) (39)

It can be shown that condition (39) is naturally satisfied. In fact, d−1 note the relationship P (1) = Θ − i=0 (A′ )i P (2) Ai , with which (39) can be equivalently rewritten as

− Jk∗−1 (xk , U(kd) , k, k + N , Uk )} = E [Jk∗ (xk+1 , U(kd+)1 , k + 1, k + N + 1, Uk+1 )] − E [Jk∗−1 (xk , U(kd) , k, k + N , Uk )],

A′ Θ A − Θ + Q + P (2) ≤ 0.

(40)

Suppose there exist Θ , H satisfying (38). We can choose P (2) ≤ 0 such that (40) holds. Since P (2) is not independent of Θ , denote d−1 ′ i (2) i P (1) = Θ − A . Thus, we have proved that if (38) i=0 (A ) P holds, there exists a decomposition of Θ such that (39) holds. Then, (38)–(39) reduce to

then

+ N , Uk )]

is nonincreasing. Considering (d)

E [Jk∗−1 (xk , Uk , k, k + N , Uk )] > 0,

(A + BH )′ Θ (A + BH ) − Θ + Q + H ′ RH ≤ 0,

we have that

which is the special case for deterministic systems (Kim, 2002; Lee, Kwon, & Choi, 1998).

(d)

lim E [Jk∗−1 (xk , Uk , k, k + N , Uk )]

k→∞

exists, and

3.3. Solving of the two inequalities (12) and (13)

(d)

lim {E [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

k→∞

− E [Jk∗−1 (xk , U(kd) , k, k + N , Uk )]} = 0.

(33)

By virtue of (12), (13), and (16), we obtain (d)

E [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )] ′

In this subsection we will investigate how to solve the two inequalities (12) and (13). An iterative algorithm is proposed below. By introducing a slack variable Q˜ ≥ 0, it follows from (12) and (13) that we have

¯ )′ P (1) (A¯ + BH ¯ ) (A + BH )′ Θ (A + BH ) + σ (A¯ + BH ′ ¯ − Θ + Q + H RH = 0,

+ N , Uk )] ≤ −E (¯xk Q x¯ k + u¯ k Ru¯ k ). ′

A′ P (1) A + σ A¯ ′ P (1) A¯ − P (1) + (Ad )′ P (2) Ad + Q¯ = 0,

Combined with (33), this yields (34)

Note that R > 0. Thus, we have that E (¯x′k Q x¯ k ) ≤ E (¯x′k Q x¯ k + u¯ ′k Ru¯ k ).

A−P

(2)

(1)

′ (1)

lim E (¯x′k Q x¯ k + u¯ ′k Ru¯ k ) = 0.

A + σA P

(1)

(A + BH )′ Θ (A + BH ) − Θ + Q + H ′ RH ≤ 0,

Because

k→∞

¯ ′ (1) ¯

Remark 5. For a deterministic system, i.e., A¯ = 0, B¯ = 0 in (1), (12)–(13) reduce to the following equations:

(d)

Ek−1 [Jk∗ (xk+1 , Uk+1 , k + 1, k + N + 1, Uk+1 )]

− E [Jk∗−1 (xk , U(kd) , k, k

The corresponding stabilizability conditions in Theorem 1 become

¯ )′ P (1) (A + BH )′ P (1) (A + BH ) + σ (A¯ + BH (1) ′ ¯ ¯ × (A + BH ) − P + Q + H RH ≤ 0.

i =0

(d) E [Jk∗−1 (xk , Uk , k, k

Remark 4. If we let d = 0, system (1) reduces to a stochastic system without time delay:

(35)

Combining (34) and (35), we have limk→∞ E (¯x′k Q x¯ k ) = 0. Note that Q > 0, then limk→∞ E (¯x′k x¯ k ) = 0. Necessity: Suppose system (1) can be stabilized with the RHC. According to Lemma 3, the coupled Lyapunov equation admits a solution with Z > 0. Then, there exist P (1) = X , P (2) = −L, H = −Υ −1 M such that inequality (12) holds. According to (29)–(32), the system can be stabilized. Hence, there exist P (1) = X , P (2) = −L such that (13) holds. This completes the proof.

where Θ = P (1) + i=0 (A′ )i P (2) Ai , Q¯ = Q + Q˜ > 0. Notice Q¯ > 0 and R > 0, the iterative algorithm is defined as follows:

d−1

Zk = A′ Zk+1 A + σ A¯ ′ Xk+1 A¯ + Q¯ − Lk ,

(41)

d−1

Xk = Zk +

 (A′ )i Lk+i Ai ,

(42)

i =0

where Lk = Mk′ Υk−1 Mk ,

Υk = B Zk+1 B + σ B¯ ′ Xk+1 B¯ + R, Mk = B′ Zk+1 A + σ B¯ ′ Xk+1 A¯ , ′

(43) (44) (45)

R. Gao et al. / Automatica 81 (2017) 390–396

395

with the terminal value ZN +1 = PN +1 , XN +1 = PN +1 , where PN +1 is a given positive definite matrix. By using the stabilizability of (1) and the positive definiteness of Q¯ , the above iterative algorithm will converge to Z and X (Zhang et al., 2015), i.e., limk→∞ Zk = Z , limk→∞ Xk = X , and the above Eqs. (41)–(45) become: Z = A′ ZA + σ A¯ ′ X A¯ + Q¯ − L, X =Z+

d−1 

(A′ )i LAi ,

i=0

where L = M ′ Υ −1 M ,

Υ = B′ ZB + σ B¯ ′ X B¯ + R, M = B′ ZA + σ B¯ ′ X A¯ . Thus, the solutions to the inequalities (12) and (13) can be given by P (1) = X , P (2) = −L, H = −Υ −1 M. 4. Numerical example In this section, a numerical example is presented to illustrate the proposed method. Example. Consider the discrete time stochastic system (1) with

    −1 −1.1 −0.1 0.1 , A¯ = , 0.5 1.1 −0.5 −0.1   −0.6 B= , 0.4     0.6 1.2 B¯ = , d = 2, σ = 1, x0 = , −0.5 1.8 A=

u−1 = −0.1, u−2 = 0.2, and the cost function (3) with 0.1 Q = 0





0 , 0.1

R = 1,

Take the slack variable Q˜ =



10 0

X N +1 =

0 10



N = 50. 0.9 0



0 0.9

, the terminal value ZN +1 =



. By using the iterative algorithm proposed in

Section 3.3, we can obtain



5.4633 , 8.7678

H = −0.6466

 −0.7734 .

P (1) =

9.3416 5.4633





P (2) =

 −2.1560 −2.5788

 −2.5788 , −3.0845

By applying Theorem 1, it can easily be verified that the conditions (12)–(13) are satisfied. The controller is given by (11) as uk = − 0.6345



0.7585 xˆ k+d|k .



The state trajectory E (x′k xk ) of the closed-loop system with this controller is shown in Fig. 1, from which it is clear that the proposed RHC stabilizes the discrete time stochastic system with input delay. 5. Conclusion This paper has proposed an RHC approach for the stabilization of linear discrete time stochastic systems with input delay. An explicit stabilization controller has been obtained by solving a coupled Riccati equation. By designing a proper cost function, a necessary and sufficient condition on the two terminal weighting

Fig. 1. State trajectory of E (x′k xk ) under the proposed RHC.

matrices was proposed, so as to guarantee the asymptotic mean square stability of the closed-loop system. The two inequalities which guarantee the mean square stability have been solved by using iterative algorithm. Some desirable extensions would be to stochastic systems with multiple input delays. References Bernardini, D., & Bemporad, A. (2012). Stabilizing model predictive control of stochastic constrained linear systems. IEEE Transactions on Automatic Control, 57(6), 1468–1480. Cannon, M., Kouvaritakis, B., & Wu, X. (2009a). Model predictive control for systems with stochastic multiplicative uncertainty and probabilistic constraints. Automatica, 45(1), 167–172. Cannon, M., Kouvaritakis, B., & Wu, X. (2009b). Probabilistic constrained MPC for systems with multiplicative and additive stochastic uncertainty. IEEE Transactions on Automatic Control, 54(7), 1626–1632. Chatterjee, D., Hokayem, P., & Lygeros, J. (2011). Stochastic receding horizon control with bounded control inputs: a vector-space approach. IEEE Transactions on Automatic Control, 56(11), 2704–2711. Chatterjee, D., & Lygeros, J. (2015). On stability and performance of stochastic predictive control techniques. IEEE Transactions on Automatic Control, 60(2), 509–514. Garcia, C. E., Prett, D. M., & Morari, M. (1989). Model predictive control: theory and practice-a survey. Automatica, 25(3), 335–348. Hessem, D.H.V., & Bosgra, O.H. (2003). A full solution to the constrained stochastic closed-loop MPC problem via state and innovations feedback and its receding horizon implementation. In Proceedings of the 42nd IEEE conference decision and control, Mad, Hawaii, USA, (pp. 929–934). Hokayem, P., Cinquemani, E., Chatterjee, D., Ramponi, F., & Lygeros, J. (2012). Stochastic receding horizon control with output feedback and bounded controls. Automatica, 48(1), 77–88. Kim, K. B. (2002). Implementation of stabilizing receding horizon controls for linear time-varying systems. Automatica, 38(10), 1705–1711. Kwon, W. H., Lee, Y. S., & Han, S. H. (2004). General receding horizon control for linear time-delay systems. Automatica, 40(9), 1603–1611. Kwon, W. H., & Pearson, A. E. (1977). A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Transactions on Automatic Control, 22(5), 838–842. Lee, Y. S., & Han, S. (2015). An improved receding horizon control for time-delay systems. Journal of Optimization Theory & Applications, 165(2), 627–638. Lee, J. W., Kwon, W. H., & Choi, J. H. (1998). On stability of constrained receding horizon control with finite terminal weighting matrix. Automatica, 34(12), 1607–1612. Mayne, D. Q. (2014). Model predictive control: recent developments and future promise. Automatica, 50(12), 2967–2986. Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. M. (2000). Constrained model predictive control: stability and optimality. Automatica, 36(6), 789–814. Park, J. H., Yoo, H. W., Han, S., & Kwon, W. H. (2008). Receding horizon control for input delayed systems. IEEE Transactions on Automatic Control, 53(7), 1746–1752. Perez, T., & Goodwin, G.C. (2001). Stochastic output feedback model predictive control. In Proceedings of the American control conference, Arlington, USA, (pp. 2412–2417).

396

R. Gao et al. / Automatica 81 (2017) 390–396

Primbs, J. A., & Sung, C. H. (2009). Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise. IEEE Transactions on Automatic Control, 54(2), 221–230. Richalet, J., Rault, A., Testud, J. L., & Papon, J. (1978). Model predictive heuristic control: applications to industrial processes. Automatica, 14(5), 413–428. Wei, F. J., & Visintini, A. L. (2014). On the stability of receding horizon control for continous-time stochastic systems. Systems & Control Letters, 63, 43–49. Zhang, H. S., Lin, L., Xu, J. J., & Fu, M. Y. (2015). Linear quadratic regulation and stabilization of discrete-time systems with delay and multiplicative noise. IEEE Transactions on Automatic Control, 60(10), 2599–2613. Zhang, H.S., Wang, H.X., & Li, L. (2012). Adapted and cansual maximum principle and analytical solution to optimal control for stochastic multiplicative noise systems with multiple input-delays. In Proceedings of the 51st IEEE conference decision and control, Maui, Hawaii, USA, (pp. 2122–2177).

Rong Gao received the B.E. degree in mathematics from Yantai University in 1999, the M.E. degree in mathematics in 2005 from Ludong University. Since 2013 she has been pursuing his Ph.D. degree at the School of Control Science and Engineering, Shandong University, Jinan, China. His research interests include receding horizon control, optimal control, stochastic systems, time-delay systems.

Juanjuan Xu received the B.E. degree in mathematics from Qufu Normal University in 2006, the M.E. degree in mathematics in 2009 from Shandong University, and the Ph.D. degree in control science and engineering in 2013 from Shandong University. She then did postdoctoral research at the School of Control Science and Engineering, Shandong University. Her research interests include distributed consensus, optimal control, game theory, stochastic systems, and time-delay systems.

Huanshui Zhang graduated in mathematics from the Qufu Normal University in 1986 and received his M.Sc. and Ph.D. degrees in control theory from Heilongjiang University, China, and Northeastern University, China, in 1991 and 1997, respectively. He worked as a postdoctoral fellow at Nanyang Technological University from 1998 to 2001 and Research Fellow at Hong Kong Polytechnic University from 2001 to 2003. He is currently a Changjiang Professorship at Shandong University, China. He held Professor in Harbin Institute of Technology from 2003 to 2006. He also held visiting appointments as Research Scientist and Fellow with Nanyang Technological University, Curtin University of Technology and Hong Kong City University from 2003 to 2006. His interests include optimal estimation and control, time-delay systems, stochastic systems, signal processing and wireless sensor networked systems.