INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2009; 19:1016–1035 Published online 15 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1361
Model predictive control for networked control systems Jing Wu1 , Liqian Zhang2 and Tongwen Chen1, ∗, † 1 Department
of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 2 Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
SUMMARY This paper investigates the problem of model predictive control for a class of networked control systems. Both sensor-to-controller and controller-to-actuator delays are considered and described by Markovian chains. The resulting closed-loop systems are written as jump linear systems with two modes. The control scheme is characterized as a constrained delay-dependent optimization problem of the worst-case quadratic cost over an infinite horizon at each sampling instant. A linear matrix inequality approach for the controller synthesis is developed. It is shown that the proposed state feedback model predictive controller guarantees the stochastic stability of the closed-loop system. Copyright q 2008 John Wiley & Sons, Ltd. Received 13 November 2006; Revised 15 October 2007; Accepted 9 June 2008 KEY WORDS:
model predictive control (MPC); networked control systems (NCSs); linear matrix inequalities (LMIs); jump linear systems; stochastic stability
1. INTRODUCTION With the development of large-scale or complex industrial systems, communication networks play a more and more important role, by which a tremendous amount of information is sensed, processed, and transmitted. Such control systems, where control loops are closed through real-time communication networks, are referred to as networked control systems (NCSs). Examples include high-speed paper production, power generation plants, petrochemical processing facilities, and so on. The presence of the network brings new functionalities that were not available in the past, such as low cost, reduced system wiring, simple system diagnosis and maintenance, and increased system agility. However, the data exchanged between NCS components are exposed to stochastic or deterministic ∗ Correspondence
to: Tongwen Chen, Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4. † E-mail:
[email protected] Contract/grant sponsor: Natural Sciences and Engineering Research Council of Canada (NSERC)
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delays [1, 2], losses [3–5], and asynchronization [6, 7], which may degrade performance and even cause instability of the feedback control loops. To solve these problems, various methods and many results have been developed. Network-induced delay, as one of the main issues, has been the focus of attention [8–12]. In [11], the stability analysis and control design of NCSs were studied when the network-induced delay at each sampling instant is random and less than one sampling time. The results in [11] have been extended to the case with longer delays in [8]. The stability of NCSs was also formulated, respectively, by a hybrid system approach with deterministic delays in [2], by a switched system approach with constant controller gain in [10], and by a jump linear system approach with random delays in [9, 12]. Moreover, some optimization and compensation methods were presented, see [13, 14] and the references therein. Model predictive control (MPC), also known as moving horizon control or receding horizon control, has received much attention in the past decades due to its extensive applications in the control of industrial processes such as distillation and oil fractionation, pulp and paper processing, and so on [15–17]. Its essence is as follows: at every sampling instant, solutions to an optimization problem over a fixed number of future time instants, known as the time horizon, are obtained; only the first optimal control move is implemented as the current control law; at the next sampling time, the measurement is used to update the state estimate and the same procedure is repeated. This feature renders the MPC approach very appropriate to incorporate the input/output constraints into the on-line optimization as well as to compensate time delays, which increases the possibility of its application in the synthesis and analysis of NCSs [18–22]. In [18], an MPC strategy for multivariable plants was presented. The sensor-to-controller delays were described by stochastic and deterministic quantities, respectively, but controller-to-actuator delays were assumed to be known and fixed; a communication constraint was imposed to restrict that all transmitted data were specified into a region in which the measurements lied and that at any time, only one plant and one actuator were permitted to be addressed. The choice between these alternatives was a function of prior knowledge about the nature of the noise and computational requirements. In [19], two protocols, TCP and UDP, were considered for NCSs with packet losses. The MPC method was used to compensate the packets dropped at the sensor-to-controller side, while zero control was applied when the control packet was lost. In this case, control signal equals zero when a packet is dropped at the controller-to-actuator side. In [20–22], modified MPC methods were introduced to compensate the delayed or missing control signals. In [21], both current and future control increment signals were used to update control signals of the plant; an adaptive predictive controller with variable horizon was designed, but no stability was considered. In [20, 22], future control move was chosen from the received control sequences to compensate the delayed control signals. The stability of the system was discussed with the consideration of fixed delay in the sensor-tocontroller side and fixed or random delay in the controller-to-actuator side [20] and a constant control gain was designed with the assumption that the delay increases at most 1 at each step [22]. The goal of this study is to design a predictive control strategy for an NCS such that at each sampling instant the infinite horizon quadratic objective is minimized while guaranteeing the stochastic stability of the closed-loop system. The actuator will implement most recently received signal directly to the plant and only the first control move will be used. The networked communication delays are assumed to be random and bounded, without loss of generality, which are described by Markovian chains. Delay-dependent conditions for the existence of such controllers are given and a linear matrix inequality (LMI) approach is developed. Moreover, a set of corresponding time-varying control laws is presented, which is different from the existing References [20–22]. A numerical example [23] is given to show the feasibility and efficiency of the proposed method. Copyright q
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u (k)
x(k)
τk
dk
v ( k)
Figure 1. The setup of the networked control system.
The paper is organized as follows. Section 2 introduces the basic setup in Figure 1, namely, closed-loop NCS modeling and the MPC problem formulation. Section 3 considers the feasibility of the optimization problem presented in Section 2. Section 4 discusses the stability of the resulting closed-loop NCS. Section 5 provides a numerical example to illustrate the design procedure. Finally, Section 6 gives some concluding remarks. Notation Throughout this paper, for symmetric matrices X and Y , the notation X Y (respectively, X >Y ) means that the matrix X −Y is positive semi-definite (respectively, positive definite); I is an identity matrix with appropriate dimensions; M T represents the transpose of the matrix M; E{·} denotes the expectation operator with respect to some probability measure P. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. PROBLEM FORMULATION Consider the networked control setup in Figure 1, where the plant is a linear time-invariant discretetime system, k 0 is the random time delay from the sensor to the controller, dk 0 is the random time delay from the controller to the actuator, and the controller Fk is to be designed by the MPC method at time instant k. Suppose that the buffer is long enough to hold all the data arrived and that the delay introduced by the buffer can be omitted compared to the network-induced delays. By this assumption, all the past control signals will be available and the most recently received control signal will be used by the actuator with the rule of the buffer, namely, first-in-last-out. v(k) is the output of the controller and satisfies v(k) = Fk x(k −k )
(1)
u(k) is the control input of the plant, which equals to u(k) = v(k −dk ) = Fk−dk x(k −dk −k−dk )
(2)
Assume that p is the prediction horizon and q is the control horizon. q control moves u(k + m|k), m = 0, 1, . . . , q −1, are computed by minimizing a nominal cost J p (k) over a prediction Copyright q
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horizon p as follows: min
u(k+m|k),m=0,1,...,q−1
J p (k)
(3)
subjecting to some constraints on the control input u(k +m|k), m = 0, 1, . . . , q −1 and on the state x(k +m|k), m = 0, 1, . . . , p, where x(k +m|k) is the state predicted at time k +m based on the measurements of time k, x(k|k) is the state measured at time k, u(k +m|k) is the corresponding predicted control input at time k +m, and u(k|k) is the control move to be implemented at time k. In this paper, we consider the case as p = q = ∞, which is refereed to as the infinite horizon. Let Fk = {x0 , 0 , d0 , . . . , xk , k , dk } be the -algebra generated by {(xl , l , dl ), 0lk}. The quadratic objective is J∞ (k) =
∞
E{(x(k +m|k)T Qx(k +m|k)+u(k +m|k)T Ru(k +m|k))|Fk }
(4)
m=0
where Q>0, R0 are symmetric weighting matrices. It is assumed that both k and dk are bounded, that is, k ,
ddk d
Without loss of generality, we assume that = 0 and d = 0. Since current time delays are usually correlated with the previous time delays, it is reasonable to model two random delays k and dk as two independent homogeneous Markov chains that take values in M = {0, 1, . . . , } and N = {0, 1, . . . , d}, and their transition probability matrices are = [i j ] and = [r s ], respectively [1, 11, 12]. That is, k and dk jump from mode i to j and from mode r to s, respectively, with probabilities i j and r s , which are defined by i j = Pr(k+1 = j|k = i),
r s = Pr(dk+1 = s|dk =r )
(5)
where i j , r s 0 and ¯
i j = 1,
j=0
d¯
r s = 1
(6)
s=0
for all i, j ∈ M and r, s ∈ N. Suppose that the model of the plant is a linear time-invariant discrete-time model as follows: x(k +1) = Ax(k)+ Bu(k)
(7)
and that the exact measurement of the system state is available at each sampling time k, i.e. x(k|k) = x(k)
(8)
The controller design scheme can be stated as follows: At each sampling time k, 1. measure the state x(k); 2. compute the state-feedback gain Fk in u(k +m|k) = Fk−dk+m|k x(k +m −k+m−dk+m|k −dk+m|k |k) Copyright q
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such that the performance objective in (4) is minimized; k+m−dk+m|k |k is the sensor-tocontroller delay, which is an m-step ahead prediction based on the measurement of time k, namely, k−dk , where dk+m|k is the predicted controller-to-actuator delay at time k +m; 3. implement the first control move u(k|k), that is, u(k) = u(k|k) = Fk−dk x(k −k−dk −dk )
(10)
By the control input given in (10), the resulting closed-loop system can be written as x(k +1) = Ax(k)+ B Fk−dk x(k −k−dk −dk )
(11)
With the modelings of k−dk and dk as two Markov chains, it can be seen that the system in (11) is a Markovian jump linear delay system with two modes, and the time delays are mode-dependent. Furthermore, it can be seen from the second step and the plant model in (7) that the predicted state x(k +m|k) satisfies the following difference equation: x(k +m +1|k) = Ax(k +m|k)+ Bu(k +m|k) = Ax(k +m|k)+ B Fk−dk+m|k x(k +m −k+m−dk+m|k |k −dk+m|k |k)
(12)
The key to solving the MPC problem is to find a way to solve the optimization problem in step 2 at each sampling time k. In the following, we give the sufficient conditions for the -suboptimal problem J∞ (k)0. At each sampling time, define X (k +m|k) as T ¯ X (k +m|k) = [x T (k +m|k) x T (k +m −1|k) · · · x T (k +m − ¯ − d|k)]
Consider the quadratic function which is given by V (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) = V1 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )+ V2 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) + V3 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )+ V4 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) where V1 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) = x T (k +m|k)P(k+m−dk+m|k |k , dk+m|k )x(k +m|k) V2 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) =
V3 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) = Copyright q
2008 John Wiley & Sons, Ltd.
0
k+m−1
=−k+m−dk+m|k |k −dk+m|k +1
l=k+m+−1
k+m−1 l=k+m−k+m−dk+m|k |k −dk+m|k
y T (l|k)W y(l|k)
x T (l|k)Sx(l|k)
Int. J. Robust Nonlinear Control 2009; 19:1016–1035 DOI: 10.1002/rnc
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V4 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ) −1
k+m−1
=−−d+1
l=k+m+
= (1−)
[y T (l|k)W y(l|k)(l −k −m −+1)+ x T (l|k)Sx(l|k)]
and y(k +m|k) = x(k +m +1|k)− x(k +m|k). P(k+m−dk+m|k |k , dk+m|k ), W , and S are positive definite matrices with appropriate dimensions. At the sampling time k, suppose that the following inequality holds for all x(k +m|k) and u(k +m|k), m0 satisfying (12): E{V (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } −E{x(k +m|k)T Qx(k +m|k)+u(k +m|k)T Ru(k +m|k)|Fk }
(14)
For the control performance J∞ (k) to be finite, we must have E{V (X (∞|k), ∞|k , d∞|k )} = 0. Thus, from (14), we obtain −V (X (k|k), k−dk|k , dk|k )− J∞ (k)
(15)
Before proceeding, we introduce the following definition and lemma. Definition 1 (Chen et al. [24]) ¯ 0] and initial The system in (11) is stochastically stable if for all finite x(k) = defined on [−¯ − d, ˜ mode 0 , d0 , there exists a finite number (, 0 , d0 )>0 such that N 2 ˜ lim E 0 , d0 ) x(k) , 0 , d0 0, P1 (i,r ), P2 (i,r ), Y (i,r ) W >0, S>0, Z (i,r ), and a scalar >0 such that the following optimization problem is feasible: min
(17)
subject to x(k|k)T P(i,r )x(k|k)+ + (1−)
−1
0
k−1
y(l|k)T W y(l|k)+
=−i−r +1 l=k+−1 k−1
=−−d+1 l=k+
k−1
x T (l|k)Sx(l|k)
l=k−i−r
[y T (l|k)W y(l|k)(l −k −+1)+ x T (l|k)Sx(l|k)]
(18)
and
Z (i,r ) Y (i,r )
0
(19)
−Y (i,r )⎥ ⎥0, Fk−r is known, and thus, (18)–(20) are LMIs. However, when r = 0, since Fk is unknown, inequality (20) is not linear with unknown variables. In the following theorem, we give an equivalent LMI condition for (20) for any i ∈ M and r ∈ N. Theorem 3 The matrix inequalities in (18)–(20) are equivalent to the following LMIs: ⎡ ⎤ T T − x T (k|k) xpast ypast ⎢ ⎥ ⎢ ∗ −X (i,r ) 0 0 ⎥ ⎢ ⎥ ⎢ ⎥0 ⎢ ∗ ⎥ ∗ T 0 1 ⎣ ⎦ ∗ ∗ ∗ T2 ⎡ ⎤ Z˜ 1 (i,r ) Z˜ 2 (i,r ) 0 ⎢ ⎥ ⎢ ∗ ⎥ 0 Z˜ 3 (i,r ) (i,r )W ⎣ ⎦ ∗ ∗ W ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
11
12
0
[X (i,r )+ X 1T (i,r )]T
X 1T (i,r )
0
X (i,r )
∗
22
BU − (i,r ) S
X 2T (i,r )T
X 2T (i,r )
0
0
∗
∗
− S
0
0
U T BT
0
∗
∗
∗
0
0
0
0
0
∗
∗
∗
∗
1 W −
(i,r )
∗
∗
∗
∗
∗
−R
0
∗
∗
∗
∗
∗
∗
−Q
∗
∗
∗
∗
∗
∗
∗
(21)
(22)
⎤ X (i,r ) ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ k. Proof Let us assume that the optimization problem in Theorem 2 is feasible at the sampling time k. The only LMI in the problem that depends explicitly on the measured state x(k|k) = x(k) of the system is the constraint in (18). Inequalities (19) and (20) can be easily proved by setting the decision variables at time k +1 equal to be the optimal values computed at time k since the parameters are independent of the state of x(k|k). Thus, to prove this theorem, we only need to prove that inequality (18) is feasible for all the future measured states X (k +m|k +m) = X (k +m), m1. We will show that the theorem holds at the sampling time k +1 first. By Q>0 and R0 in (14), we have E{V (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k ) −V (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk }0, P1k+1 (i,r ), P2k+1 (i,r ), Y k+1 (i,r ), W k+1 >0, S k+1 >0, and Z k+1 (i,r ), we have E{V (X (k +1|k +1), k+1−dk+1 , dk+1 )|Fk+1 } E{V (X (k +1|k +1), k+1−dk+1|k |k , dk+1|k )|Fk }
(25)
This is because P k+1 (i,r )>0, P1k+1 (i,r ), P2k+1 (i,r ), Y k+1 (i,r ), W k+1 >0, S k+1 >0, and Z k+1 (i,r ) are optimal values at time k +1, while P k (i,r )>0, P1k (i,r ), P2k (i,r ), Y k (i,r ), W k >0, S k >0, and Z k (i,r ) are only feasible at time k +1. By (14), we have E{V (X (k +1|k), k+1−dk+1|k |k , dk+1|k )− V (X (k|k), k−dk , dk )|Fk } T R Fk−dk x(k −k−dk −dk )|Fk } −E{x T (k|k)Qx(k|k)+ x T (k −k−dk −dk )Fk−d k
−E{x T (k|k)Qx(k|k)|Fk } Copyright q
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with m = 0. Since the measured state x(k +1|k +1) = x(k +1) equals Ax(k)+ B Fk−dk x(k −k−dk − dk ), x(k +1|k) = x(k +1|k +1). Combining the inequalities (25) and (26), we have E{V (X (k +1|k +1), k+1−dk+1 , dk+1 )− V (X (k|k), k−dk , dk )|Fk } −E{x T (k|k)Qx(k|k)|Fk }−min {Q}E{x(k|k)2 |Fk } From the above inequality, if letting = min {Q}>0 (since Q>0), we can see that for any N 1, N E{V (X (N ))|, 0 , d0 }−E{V (X (0))|, 0 , d0 }− E x(k|k)2 |, 0 , d0 k=0
or
E
N
1 x(k|k) |, 0 , d0 {E{V (X (0))|, 0 , d0 }−E{V (X (N ))|, 0 , d0 }} k=0 2
1 E{V (X (0))|, 0 , d0 } which implies that lim E
N →∞
1 ˜ x(k)2 |, 0 , d0 E{V (X (0))|, 0 , d0 } = (, 0 , d0 ) k=0 N
From Definition 1, the stochastic stability is obtained.
(27)
Remark 1 It is noted that the LMIs given in (18)–(20) can be solved numerically, efficiently by using standard LMI techniques, and no tuning of parameters are involved. The solution derived is general since it is solved under consideration of all the value of i and r at each time k. Therefore, the MPC scheme proposed in Theorem 2 makes the on-line implementation possible. Our MPC formulation of NCSs can also be extended to the case with input/output constraints. Assume that the output of the linear time-invariant system in (7) is given by z(k) = C x(k) Then we have the following input and output variance constraints: lim E{(u(k +m|k)−E{u(k +m|k)})T (u(k +m|k)−E{u(k +m|k)})|Fk } u max
m→∞
lim E{(z(k +m|k)−E{z(k +m|k)})T (z(k +m|k)−E{z(k +m|k)})|Fk } z max
m→∞
where z(k +m|k) = C x(k +m|k) is the predicted output, u max >0, and z max >0. They can be rewritten in LMIs as follows: 2 −z max I +C SC T 0 2 −u max I U 0 − S UT
Copyright q
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(28) (29)
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For simplicity, the proof is omitted, which is similar to that in [23]. Thus, we have the following corollary. Corollary 6 ¯ be the measured Consider the stochastic system in (11) and let x(k|k), x(k −1|k), . . . , x(k − ¯ − d|k) ¯ state x at time instant k, k −1, . . . , k − ¯ − d, respectively. Then there exists a state feedback controller in (10) such that both (13) and (14) hold if there exist matrices P(i,r )>0, P1 (i,r ), P2 (i,r ), Y (i,r ) W >0, S>0, Z (i,r ), and a scalar >0 such that the following optimization problem holds: min subject to (18)–(20) and (28)–(29) for all i ∈ M and r ∈ N. Remark 2 The input and output variance constraints have been often considered for control designs of stochastic systems, see [26] and the reference therein. The variance constraints on the inputs refer to the control energy or power constraints, and the mission requirements can be naturally stated in terms of the variance constraints on the outputs.
5. NUMERICAL EXAMPLE In order to demonstrate the proposed MPC algorithm, a simple example will be investigated in this section. Consider a classical angular positioning system in Figure 2 [23], which consists of a rotating antenna at the origin of the plane and is driven by an electric motor. The control target is using the motor to rotate the antenna so that it always points to the direction of a moving object in the plane. Assume that the angular position of the antenna (rad), the angular position of the moving object r (rad), and the angular velocity of the antenna ˙ (rad s−1 ) are measurable. The motion of the antenna can be described by the following discrete-time counterparts by discretization, using a sampling time of 0.1 s and the Euler’s first-order approximation for the derivative (k +1) 1 0.1 0 x(k +1) = = x(k)+ u(k) (30) ˙ +1) 0 1−0.1 (k) 0.1 (k
Target object Goal : θ ≅ θ r Antenna Motor
θr
θ
u
Figure 2. The angular positioning system. Copyright q
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-3
x 10
γ
2 1 0
(a)
0
10
20
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0
10
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0
10
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τ
k
2 1 0
(b)
d
k
1 0.5 0
state trajectory
(c) 0.01 0 -0.01
(d)
u
0.1 0 -0.1
(e)
step
Figure 3. The angular positioning system.
where = 0.787 rad−1 V −1 s−2 and 0.1 s−1 (k)10 s−1 . The parameter (k) is proportional to the coefficient of viscous in the rotating parts of the antenna. Assume that (k) = 0.1 and friction 0.01 the initial state x(k) = 0 , the infinite MPC optimization to be solved at each time k is min
u(k+m|k)
J∞ (k) =
∞
E{x T (k +m|k)Qx(k +m|k)+u T (k +m|k)Ru(k +m|k)|Fk }
m=0
subject to |u(k +m|k)|0.1 V with Q = I2×2 and R = 0.1. The system dynamics is shown in Figure 3, where Figure 3(a) is the upper bound of the above quadratic function. Figures 3(b) and (c) give the changes of network-induced delays. k = i, i ∈ {0, 1, 2} means data are delayed by i Ts on the sensor-to-controller side at time k during the transmission, where Ts is the sampling time. Similarly, dk =r, r ∈ {0, 1} means that data are delayed by r Ts on the controller-to-actuator Copyright q
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Int. J. Robust Nonlinear Control 2009; 19:1016–1035 DOI: 10.1002/rnc
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side. By these two random serials, we assume their transition probability matrices as ⎡ ⎤ 0.5 0.2 0.3 0.5 0.5 ⎢ ⎥ ⎥ =⎢ ⎣ 0.4 0.5 0.1⎦ , = 0.8 0.2 0.3 0.2 0.5 Figure 3(d) is the state trajectories, which show that the closed-loop system is stable by the controller we designed. Moreover, it is easy to see that our designed control sequence, Figure 3(e), lies in the area [−0.1, 0.1].
6. CONCLUSIONS In this paper, we have studied the problem of MPC for NCSs. Based on the minimization of an upper bound of the worst-case infinite horizon quadratic cost function at each sampling instant, a state feedback model predictive controller has been proposed by using the LMI approach. Only first control move is implemented to the plant. The stochastic delay-dependent stability conditions with and without input/output constraints have been presented, respectively, for the closed-loop system resulting from the proposed controller. The numerical example shows the effectiveness of our method.
APPENDIX Proof of Theorem 2 Assume that k+m−dk+m|k |k = i, dk+m|k =r . First, we will show that (19) and (20) implies (14). Following [24, 25], we introduce the following equations: y(k +m|k) = x(k +m +1|k)− x(k +m|k) 0 = −y(k +m|k)+(A − I )x(k +m|k)+ B Fk−r x(k +m −i −r |k) x(k +m −i −r |k) = x(k +m|k)−
k+m−1
(A1)
y(l|k)
l=k+m−i−r
Notice from (12) and (A1) that E{V1 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V1 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } = E{E{[x(k+m|k)+y(k+m|k)]T P(k+m+1−dk+m+1|k |k , dk+m+1|k )[x(k +m|k)+y(k+m|k)]|Fk } − x T (k +m|k)P(k+m−dk+m|k |k , dk+m|k )x(k +m|k)|Fk } = E{[x(k +m|k)+ y(k +m|k)]T P(i,r )[x(k +m|k)+ y(k +m|k)] − x T (k +m|k)P(i,r )x(k +m|k)|Fk } Copyright q
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Int. J. Robust Nonlinear Control 2009; 19:1016–1035 DOI: 10.1002/rnc
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= E x T (k +m|k)[P(i,r )− P(i,r )]x(k +m|k)+ y T (k +m|k)P(i,r )y(k +m|k) y(k +m|k) + 2T (k +m|k)G T (i,r ) Fk 0
where P(i,r ) =
d
i j r s P( j, s),
(k +m|k) =
j=0 s=0
x(k +m|k) y(k +m|k)
,
G(i,r ) =
P(i,r )
0
P1 (i,r )
P2 (i,r )
P1 (i,r ) and P2 (i,r ) are constant matrices with appropriate dimensions. Thus, by the relation in (19), (A1) and Lemma 1, we have y(k +m|k) T T E 2 (k +m|k)G (i,r ) Fk 0 ⎧ ⎧ T ⎨ ⎨ 0 I 0 I T T + G(i,r )+[Y (i,r ) 0] E (k +m|k) G (i,r ) ⎩ ⎩ A − I −I A − I −I ⎫ ⎬
+[Y (i,r ) 0]T (k +m|k)+(i +r )T (k +m|k)Z (i,r )(k +m|k) ⎭ +
k+m−1
y T (l|k)W y(l|k)
l=k+m−i−r
+2T (k +m|k) −Y (i,r )+ G T (i,r )
0 B Fk−r
x(k +m −i −r |k)|Fk
⎫ ⎬ ⎭
Therefore, E{V1 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V1 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } 0 I P(i,r )− P(i,r ) 0 T + G T (i,r ) E (k +m|k) A − I −I 0 P(i,r ) T 0 I Y T (i,r ) + (k +m|k) G(i,r )+[Y (i,r ) 0]+ A − I −I 0 +(i +r )T (k +m|k)Z (i,r )(k +m|k)+ +2T (k +m|k) −Y (i,r )+ G T (i,r ) Copyright q
2008 John Wiley & Sons, Ltd.
k+m−1
y T (l|k)W y(l|k)
l=k+m−i−r
0 B Fk−r
x(k +m −i −r |k) Fk
(A2)
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For V2 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ), we have E{V2 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V2 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } = E{E{V2 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )|Fk } −V2 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } =E
d
i j r s
0
k+m
=− j−s+1 l=k+m+
j=0 s=0
y T (l|k)W y(l|k)
y (l|k)W y(l|k) Fk − =−i−r +1 l=k+m+−1 0
=E
k+m−1
d
T
d¯
+ ii
y T (l|k)W y(l|k)
l=k+m−i−r
j=0 s=0
k+m−1
i j r s ( j +s)y T (k +m|k)W y(k +m|k)−
s=0,s=r
r s +rr
¯ j=0, j=i
i j +
d¯
¯
s=0,s=r j=0, j=i
i j r s
y (l|k)W y(l|k)− y (l|k)W y(l|k) Fk × =− j−s+1 l=k+m+ =−i−r +1 l=k+m+
0
k+m−1
0
T
k+m−1
T
Notice that E
0
k+m−1
=− j−s+1 l=k+m+
E
−1
y (l|k)W y(l|k)− T
k+m−1
=−−d+1 l=k+m+
0
k+m−1
=−i−r +1 l=k+m+
y (l|k)W y(l|k)|Fk T
y (l|k)W y(l|k)|Fk T
Thus, it can be seen that E{V2 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V2 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } E
d
k+m−1
i j r s ( j +s)y T (k +m|k)W y(k +m|k)−
j=0 s=0
y T (l|k)W y(l|k)
l=k+m−i−r
k+m−1 −1 T y (l|k)W y(l|k) Fk + (1−) =−−d+1 l=k+m+
(A3)
since 1−ii rr 1−. Copyright q
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Similarly, for V3 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ), we have E{V3 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V3 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } k+m d k+m−1 T T i j r s x (l|k)Sx(l|k)− x (l|k)Sx(l|k) Fk =E l=k+m−i−r j=0 s=0 l=k+m+1− j−s E x T (k +m|k)Sx(k +m|k)− x T (k +m −i −r |k)Sx(k +m −i −r |k) x (l|k)Sx(l|k) Fk + (1−) l=k+m+1−−d k+m
T
(A4)
For V4 (X (k +m|k), k+m−dk+m|k |k , dk+m|k ), we have E{V4 (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V4 (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } = E (1−)
−1
k+m
[y T (l|k)W y(l|k)(l −k −m −)+ x T (l|k)Sx(l|k)]
=−−d+1 l=k+m+1+
[y (l|k)W y(l|k)(l −k −m −+1)+ x (l|k)Sx(l|k)] Fk − (1−) =−−d+1 l=k+m+ −1
= E (1−)
k+m−1
T
T
(+d)(+d −1) T y (k +m|k)W y(k +m|k) 2
+(1−)(+d −1)x T (k +m|k)Sx(k +m|k)−(1−) × y T (l|k)W y(l|k)−(1−) x T (l|k)Sx(l|k) Fk l=k+m+ =−−d+1 l=k+m−−d+1 −1
k+m−1
k+m−1
(A5)
Combining (A2)–(A5) together, we obtain E{V (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk }
E (k +m|k)(k +m|k)+2 (k +m|k) G (i,r ) T
T
T
0 B Fk−r
−Y (i,r ) x(k +m −i −r |k)
−x (k +m −i −r |k)Sx(k +m −i −r |k)|Fk T
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Int. J. Robust Nonlinear Control 2009; 19:1016–1035 DOI: 10.1002/rnc
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=E
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎡
0
⎢ G (i,r ) B Fk−r (k +m|k)T ⎢ ⎣ T
∗
−S
where
⎫ ⎪ ⎪ ⎬ −Y (i,r )⎥ ⎥ (k +m|k) Fk ⎦ ⎪ ⎪ ⎭ ⎤
(k +m|k) =
(k +m|k)
(A6)
x(k +m −i −r |k)
As a result, it can be seen that E{V (X (k +m +1|k), k+m+1−dk+m+1|k |k , dk+m+1|k )− V (X (k +m|k), k+m−dk+m|k |k , dk+m|k )|Fk } +E{x T (k +m|k)Qx(k +m|k)+u T (k +m|k)Ru(k +m|k)|Fk } E{T (k +m|k)(i,r )(k +m|k)|Fk } On the other hand, from (15), it follows that (13) holds if V (X (k|k), k|k , dk|k ) that is, x(k|k)T P(k−dk , dk )x(k|k)+
+ (1−)
−1
k−1
0
k−1
y(l|k)T W y(l|k)+
=−k−dk −dk +1 l=k+−1
k−1 l=k−k−dk −dk
x T (l|k)Sx(l|k)
[y T (l|k)W y(l|k)(l −k −+1)+ x T (l|k)Sx(l|k)]
=−−d+1 l=k+
Since (18) holds for i ∈ M and r ∈ N, the proof is obtained.
ACKNOWLEDGEMENTS
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