# Networked guaranteed cost control for a class of industrial processes with state delay.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 Published online 13 September 2007 in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/apj.092 Research Article Networked, guaranteed cost control for a class of industrial processes with state delay Chen Peng1,2 * 1 2 School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu, 210042, P. R. China Faculty of Information Technology, Queensland University of Technology, Brisbane QLD 4001, Australia Received 4 February 2007; Revised 15 May 2007; Accepted 8 August 2007 ABSTRACT: This paper addresses a class of industrial processes with state delay and controlled over communication networks. With consideration of network-induced delay and data packet dropout, a model is proposed for networked control systems (NCSs). With the introduction of free weighting matrices, a method for suboptimal guaranteed cost controller design is proposed in the form of matrix inequalities. A nonconvex minimization problem is then formulated for finding suboptimal allowable equivalent delay bound (SAEDB) and the feedback gain of a memoryless controller. The proposed method is demonstrated through networked injection velocity control in thermoplastic injection molding. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: guaranteed cost; state delay; NCSs; LMIs; suboptimal allowable equivalent delay bound INTRODUCTION Many industrial processes have input and/or output delay. Basically, there are two methods to describe those industrial processes: the first- or second-order plus delay model, and the state-space model with state delay. To deal with those systems with delay, many control schemes have been presented, such as the double controller scheme (Tian and Gao, 1999), the model predictive control approach (Tellez et al ., 2005), and the fault-tolerant control approach (Su et al ., 2006). However, most of them are local control systems (LCSs), implying that the physical architecture of communication networks for process control is point-to-point with asynchronous communications. The advantages of this architecture are obvious, such as small transmission delay (which usually is neglected in system analysis and design), high reliability, and deterministic communication behavior. However, with the growing complexity of industrial processes and increasing demands on optimization of conflicting objectives, safe operation, and minimum environment requirements, more and more components and devices are being introduced into communication networks of control systems (Tian et al ., 2006a), which are termed networked control systems (NCSs). NCSs involve communication patterns in *Correspondence to: Chen Peng, School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu, 210042, P. R. China. E-mail: pchme@163.com 2007 Curtin University of Technology and John Wiley & Sons, Ltd. which both informational and physical control loops are closed through a real-time network. Recently, much attention has been paid to the study of stability analysis and control design of NCSs (Zhang et al ., 2001; Yue et al ., 2005; Peng and Tian, 2006a) owing to their low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. Compared to LCS (Han, 2004; Jiang and Han, 2006), one of the important issues in NCSs is the effect of network-induced delay on the system performance. Performance of the NCSs is directly dependent upon the network-induced delay. Time-varying characteristics of the network-induced delay not only degrade the control performance but also introduce distortion of the controller signal (Hong, 1995). There has been much effort to assign certain performance criteria when designing a controller, such as quadratic cost minimization, pole placement, and H2 /H∞ norm minimization. Among those methods, the guaranteed cost control aims to stabilize the systems while maintaining an adequate level of performance represented by the quadratic cost. The existing guaranteed cost control design methods can be classified into two types: delay-independent stabilization (Moheimani and Moheimani, 1997; Yu and Gao, 2001) in which a state feedback controller is obtained irrespective of the size of the delay, and delay-dependent stabilization (Lee et al ., 2001; Yue et al ., 2006) in which the controller is designed with consideration of the size of the delay. Most of the existing reports on guaranteed cost control of NCSs in the open literature have dealt with signal Asia-Pacific Journal of Chemical Engineering COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES quantization in Yue et al . (2006), networked control for T–S fuzzy systems in Zhang et al . (2007), discretetime model with structural uncertainties in Huang and Nguang (2006) and the existence of a state feedback controller in terms of the solvability of bilinear matrix inequalities (BMIs) in Li et al . (2004). To the best of our knowledge, no results have been found on the statedelay-guaranteed cost control of NCSs, an important issue to practical industrial processes with delay. This has greatly motivated this research. This paper is structured as follows: Modeling of NCSs and the definition of suboptimal allowable equivalent delay bound (SAEDB) are presented in Section 2. The SAEDB and guaranteed cost controller design for NCSs with state delay are considered and an optimal problem is formulated in Section 3. Numerical examples are given to demonstrate the effectiveness of our method in Section 4. Section 5 concludes the paper. Notation: n denotes the n-dimensional Euclidean space, n×m is the set of n × m real matrices, I is the identity matrix of appropriate dimensions, || · || stands for the Euclidean vector norm or the induced matrix 2norm as appropriate. The notation X > 0 (respectively, X ≥ 0), for X ∈ n×n means that the matrix X is a real symmetric positive definite (respectively, positive semidefinite). λmax (P )(λmin (P )) denotes the maximum (minimum) of eigenvalue of real symmetric matrix P . For an arbitrary matrix B and two symmetric matrices A B A and C , denotes a symmetric matrix, where ∗ C ∗ denotes the entries implied by symmetry. PROBLEM STATEMENTS u(t + ) = u(ik h + τik ), t ∈ [ik h + τik , ik +1 h + τik +1 ) (4) where h is the sampling period, ik (k = 1, 2, 3, . . .) are some integers and {i1 , i2 , i3 , . . .} ⊂ {0, 1, 2, 3, . . .}. τik is the network-induced delay, which denotes the time from the instant ik h when sensor nodes sample from the plant to the instant when actuator nodes transmit data to the plant. It follows that ∪∞ k =1 [ik h + τik , ik +1 h + τik +1 ] = + τisc , where τica [t0 , ∞), t0 ≥ 0. Assume that τik = τica k k k sc is the sensor to controller delay, τik is controller to actuator delay and computational and overhead delays . are included in τisc k Further assume that there is no error code in the transmission and the state feedback controller is chosen. From above analysis, Eqn (4) can be replaced by u(t + ) = Kx (t − τik ), t ∈ {ik h + τik , k = 1, 2, . . .} (5) where K is the state feedback gain. According to Eqn (5), the real input u(t) realized through the zero-order hold is a piecewise constant function and the real control system can be modeled as ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d) Consider the following system with state delay and parameter uncertainties ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d) + (B + B )u(t) (1) x (t) = ϕ(t), t ∈ [−d, 0] (2) where x (t) ∈ n and u(t) ∈ m are the state vector and the control input vector, respectively. A, A1 , and B are constant matrices with appropriate dimensions. A, A1 and B are uncertain matrices which can be time varying. d is a scalar representing the delay in the industry system. Assume that the parameter uncertainties A, A1 , and B of system model are norm-bounded and satisfy [A A1 B ] = DF (t)[E1 E2 E3 ] Also assume that the sensors are clock-driven and controllers and actuators are event-driven; the data is transmitted with a single-packet and the full state variables are available for measurements; the real input u(t) realizes through zero-order hold. The time interval between the instant i(k ) h + τ(i(k ) ) , at which a packet arrives at the actuator, and the next arrival instant i(k +1) h + τ(i(k +1) ) is the effect duration of the Zero-hold operation, which can be described as (3) where F (t) ∈ i ×j is an uncertain matrix which satisfies F T (t)F (t) ≤ I . D, E1 , E2 and E3 are constant matrices of appropriate dimensions. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. + (B + B )u(t) (6) u(t + ) = Kx (t − τik ), t ∈ {ik h + τik , k = 1, 2, . . .} (7) The closed-loop system (Eqn (6)) with network communication can be expressed as ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d) + (B + B )Kx (ik h), t ∈ [ik h + τik , ik +1 h + τik +1 ) (8) Remark 1 In Eqn (8), {i1 , i2 , i3 , . . .} is a subset of {0, 1, 2, 3, . . .}. Moreover, it is not required that ik +1 > ik . If |ik +1 − ik | = 1, it means no data packet dropout in the transmission. If ik +1 > ik + 1, it means some data packet dropout. If ik +1 < ik , it means that unordered data arrival sequence occurs. Therefore, Eqn (8) can be viewed as a general form of NCSs, in which the effect of the networked-induced delay, data packet dropout, and unordered data arrival sequence are simultaneously considered. Detailed explanation can be found in Peng Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj 651 652 C. PENG Asia-Pacific Journal of Chemical Engineering and Tian (2006a), Yue et al . (2004), Peng and Yue (2006b). are satisfied, where 1 + M 1T , 11 = S + AX T + XAT + M 2T , 12 = A1 X T + λ2 XAT + M 1 + M 3T , 13 = BY + λ3 XAT − M Given positive definite symmetric matrices Q1 and R1 , we will consider the cost function ∞ [x T (t)Q1 x (t) + u T (t)R1 u(t)]dt (9) J = 4T , 14 = P + λ4 XAT − X T + M 22 = − S + λ2 A1 X T + λ2 XAT1 , 2 , 23 = λ2 BY + λ3 XAT1 − M 0 Associated with the cost function (9), the guaranteed cost controller is defined as follows: Definition 1 Consider the uncertain time-delay system of NCSs (8). If there exists a control law u(t) and a positive scalar γ , such that for all admissible uncertainties the closed-loop system is stable and the value of the cost function (9) satisfies J ≤ γ , then γ is said to be a guaranteed cost and u(t) is said to be a guaranteed cost controller for the system (8). To facilitate development, we first introduce the following definition. Definition 2 SAEDB of NCSs, denoted by η, which satisfies (ik +1 − ik )h + τik +1 ≤ η, k = 1, 2, 3, . . . Remark 2 In Definition 2, |(ik +1 − ik )| > 1 implies that there are some packet dropouts. If ik +1 < ik , there are non-ordered data sequences. SAEDB is directly related with the number of data packet dropouts (|(ik +1 − ik )| − 1), network-induced delay τik +1 , and the sampling period h. Therefore, the SAEDB η can be used as a parameter for co-design of NCS controller and network QoS. MAIN RESULTS 24 = −λ2 X T + λ4 XAT1 , 3 − M 3T , 33 = λ3 BY + λ3 Y T B T − M 4T , 34 = −λ3 X T + λ4 Y T B T − M 44 = η R − λ4 X T − λ4 X , then the system (8) is asymptotically stable with the feedback gain K = YX −T , the cost function J in (9) satisfies the following bound: P X −T ϕ(0) + J ≤ϕ T (0)X −1 + 1 12 13 14 ηM X 2 0 22 23 24 ηM ∗ 3 33 34 ηM ∗ 0 ∗ ∗ 4 44 ηM ∗ ∗ 0 ∗ ∗ ∗ ∗ η R 0 ∗ ∗ ∗ ∗ ∗ −Q1−1 ∗ ∗ ∗ ∗ ∗ ∗ ((ik +1 − ik )h + τik +1 ) ≤ η, k = 1, 2, . . . 11 0 0 YT 0 < 0 (10) 0 0 −R1−1 (11) 2007 Curtin University of Technology and John Wiley & Sons, Ltd. −η 0 0 S X −T ϕ(t)dt ϕ T (t)X −1 −d RX −T ϕ̇(v )dvds ϕ̇ T (v )X −1 (12) s Proof: We set y(t) = ẋ (t) and construct a Lyapunov– Krasovskii functional candidate as (13) V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ) t where V1 (xt ) = x T (t)Px (t), V2 (xt ) = t−d x T (v )Sx t t T (v )dv , V3 (xt ) = t−η s y (v )Ry(v )dvds, and P > 0, S > 0 and R > 0. Taking the time derivative of V (xt ) along the state trajectory (8) yields that, for t ∈ [ik h + τik , ik +1 h + τik +1 ), V̇1 (xt ) = 2x T (t)Py(t) In this section, we will develop some practically computable criteria to obtain the SAEDB and relevant controller for system (8) based on a Lyapunov–Krasovskii functional method. First, let us consider the simple case without the parameter uncertainties A, A1 and B . Theorem 1 For given scalars η and λi (i = 2, 3, 4), if i (i = there exist matrices P, S and R > 0, X ,Y and M 1, 2, 3, 4) with appropriate dimensions such that 0 (14) V̇2 (xt ) = x (t)Sx (t) − x (t − d)Sx (t − d) (15) t T V̇3 (xt ) = ηy (t)Ry(t) − y T (s)Ry(s)ds (16) T T t−η the Newton–Leibnitz formula x (t) − x (ik h) − Using t y(s)ds = 0 and Eqn (8), for arbitrary matrices Ni ik h and Mi (i = 1, 2, 3, 4) of appropriate dimensions, we have t T T y(s)ds = 0 (17) ξ (t)M x (t) − x (ik h) − ik h and ξ T (t)N T [Ax (t) + A1 x (t − d) + BKx (ik h) − y(t)] = 0 (18) Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES where ξ T (t) = [x T (t)x T (t − d)x T (ik h)y T (t)], M = [M1T M2T M3T M4T ], N = [N1T N2T N3T N4T ]. From (11), it can be seen that, when t ∈ [ik h + τik , ik +1 h + τik +1 ), t t y T (s)Ry(s)ds ≤ − y T (v )Ry(v )dv (19) − t−η ik h and − 2ξ (t)M T + t T y(s)ds ≤ ηξ T (t)M T R −1 M ξ(t) Defining N2 = λ2 N1 , N3 = λ3 N1 , N4 = λ4 N1 and Y = P = XPX T , R = XRX T , S = KX T , defining X = N −1 , T T XSX and Mi = XMi X (i = 1, 2, 3, 4) then pre-postmultiplying both sides of (24) with diag(X , X , X , X ) and its transpose,respectively, (23) can be derived by the (24). + Q 11 1 ∗ ∗ ∗ ∗ 12 22 ∗ ∗ ∗ 13 23 33 + K T R1 K ∗ ∗ 14 24 34 44 ∗ ηM1 ηM2 ηM3 <0 ηM4 −ηR (24) ik h t y T (v )Ry(v )dv (20) ik h Then combining Eqns (17)–(20) and considering the performance index (9) for t ∈ [ik h + τik , ik +1 h + τik +1 ), we obtain 11 12 13 14 ∗ 22 23 24 V̇ (xt ) ≤ ξ T (t) ∗ ∗ 33 34 ∗ ∗ ∗ 44 × ξ(t) + ηξ T (t)M T R −1 M ξ(t) 11 + Q1 12 13 ∗ 22 23 T ≤ ξ (t) ∗ ∗ 33 + K T R1 K ∗ ∗ ∗ × ξ(t) + ηξ T (t)M T R −1 M ξ(t) (21) 14 24 34 44 (22) From (24) and by Schur complements, we have for t ∈ [ik h + τik , ik +1 h + τik +1 ) V̇ (xt ) ≤ −x T (t)Q1 x (t) − x T (ik h)K T R1 Kx (ik h) < 0 Since ∞ k =1 [ik h + τik , ik +1 h + τik +1 ) = [t0 , ∞) and x (t) is continuous in t, t0 ≥ 0 and V (xt ) is continuous in t ∈ [t0 , ∞) (Yue et al ., 2005). Therefore by using Lyapunov–Krasovskii theorem, the closed-loop system (8) is asymptotically stable. Furthermore, by integrating both sides of (25) from 0 to T and using the initial condition, we obtain T [x T (t)Q1 x (t) + u T (t)R1 u(t)]dt − 0 ≥ x T (T )Px (T ) − x T (0)Px (0) 0 T x T (v )Sx (v )dv − x T (v )Sx (v )dv + T −d where 11 = S + N1 A + AT N1T + M1 + M1T 12 = N1 A1 + AT N2T + M2T 13 = N1 BK + A T N3T − M1 + M3T 44 = 12 22 ∗ ∗ ∗ y T (v )Ry(v )dvds −η − M4T 13 23 33 + Y T R1 Y ∗ ∗ (26) s T T →∞ T −d By Schur complements, we can show that the solvability of (10) implies that of (23). 11 + XQ1 X T ∗ ∗ ∗ ∗ 0 lim 33 = N3 BK + K B N3T − M3 − M3T 34 = s 0 lim x T (T )Px (T ) = 0, AT1 N4T T T −N3 + K B N4T ηR − N4 − N4T T −η T →∞ 23 = N2 BK + AT1 N3T − M2 T T As the closed-loop system (8) is asymptotically stable, 22 = −S + N2 A1 + AT1 N2T T −d y T (v )Ry(v )dvds − 14 = P + AT N4T − N1 + M4T 24 = −N2 + + T 14 24 34 44 ∗ 1 ηM 2 ηM 3 ηM <0 4 ηM −η R (23) 2007 Curtin University of Technology and John Wiley & Sons, Ltd. T x T (v )Sx (v )dv = 0, lim T →∞ T −η T y T (v )Ry(v )dvds = 0 (27) s Hence we obtain ∞ [x T (t)Q1 x (t) 0 + u T (t)R1 u(t)]dt ≤ ϕ T (0)P ϕ(0) 0 ϕ T (v )S ϕ(v )dv + + −d 0 0 ϕ̇ T (v )R ϕ̇(v )dvds −η (28) s Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj 653 654 C. PENG Asia-Pacific Journal of Chemical Engineering From the definition P = XPX T , R = XRX T , S = XSX T , −1 −T −1 −T S we have P = X P X , R = X RX , S = X −1 X −T . Substituting them into (28), we obtain (12). It follows that this sufficient condition for the existence of guaranteed cost controllers is satisfied from the Definition 2. This completes the proof. Remark 3 From the proof of Theorem 1, we define N2 = λ2 N1 , N3 = λ3 N1 and N4 = λ4 N1 to derive condition (10), showing that (24) and (10) are not strictly equivalent. Therefore, it can be expected that the controller designed by condition (24) can guarantee the asymptotic stability of system (8) with a larger bound η than one derived on the basis of (10). where P X −T ϕ(0), J1 = ϕ T (0)X −1 0 S X −T ϕ(v )dv , ϕ T (v )X −1 J2 = J3 = + 0 −d ϕ̇ T (v )R ϕ̇(v )dvds −η Algorithm 1 Obtain η: 1. Given αi and βi as upper and lower bounds of λi , respectively, and ζi (i = 1, 2, 3) as step increment. Set η0 = 0 and K = 0. 2. According to Theorem 1, under the constraint of ζi , αi , and βi , group different λi to obtain corresponding η and K based on LMI, subject to (10) and (11). If η > η0 , set η = η0 and K = K0 , else return to step 2. 3. Based on Corollary 1, substitute K = K0 into (24) and find the maximum allowable bound of η. 4. Output η and stop. However, it is worth mentioning that the corresponding upper bound (12) for the cost function (9) is not a convex function in P, S , and R. In the following, we propose a method to find a suboptimal value for this upper bound. In order to achieve the least guaranteed cost value γ among all possible choices of P, S, i (i = 1, 2, 3, 4), we have to solve the R, X , Y , and M following minimization problem: Min J1 + J2 + J3 subject to (10) and (11) 3 = (30) 2007 Curtin University of Technology and John Wiley & Sons, Ltd. RX −T ϕ̇(v )dvds. ϕ̇ T (v )X −1 s −d 0 0 ϕ̇ T (v )ϕ̇(v )dvds −η (31) s According to tr(AB ) = tr(BA), the following relationships hold: J1 = ϕ T (0)X −1 P X −T ϕ(0) = tr(1 X −1 P X −T ) 0 ϕ T (v )X −1 S X −T ϕ(v )dv = tr(2 X −1 S X −T ) J2 = J3 = To obtain the η, we present following search algorithm: −η 0 (29) s 1 = ϕ T (0)ϕ(0), 0 ϕ T (v )ϕ(v )dv , 2 = −d 0 0 Before moving on, let us define matrix values 1 , 2 , and 3 such that From the proof of Theorem 1, we have the following corollary. Corollary 1 For given scalars η and K , if there exist matrices P , S and R > 0, X , Ni and Mi (i = 1, 2, 3, 4) with appropriate dimensions, such that (11) and (24) are satisfied, then system (8) with the feedback gain K is asymptotically stable, and the cost function J in (9) satisfies the following bound: 0 T ϕ T (t)S ϕ(t)dt J ≤ ϕ (0)P ϕ(0) + −d 0 −η 0 ϕ̇ T (v )X −1 RX −T s RX −T ) ϕ̇(v )dvds = tr(3 X −1 (32) Therefore, the minimization of the upper bound of the cost in (30) is formulated as follows: P X −T ) + tr(2 X −1 S X −T ) Min tr(1 X −1 RX −T ) + tr(3 X −1 subject to (10) and (11) (33) Let us derive the upper bounds on the cost function (9). Assume that there exist new variables 1 = 1T , 2 = 2T , 3 = 3T which satisfy X −1 P X −T < 1 , X −1 S X −T < 2 , X −1 RX −T < 3 (34) By Schur complement, (34) is equivalent to (35). 1 X −T 3 X −T 2 X −1 > 0, X −T P −1 X −1 >0 −1 R X −1 > 0, S −1 (35) Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES By introducing new variables L, T , W and H , and P −1 , W = S −1 and H = R −1 , defining L = X −1 , T = we can replace condition (35) by (36). 2 L 3 L 1 L > 0, > 0, > 0 (36) LT T LT W LT H For finding solution of (39), we present the following iterative algorithm, which are similar to those in Ghaoui et al . (1997) and Lee et al . (2001). Here, the (34) is used as a termination criterion in the algorithm 2. Assuming (36) is satisfied, we can conclude that the following inequalities hold: 1. Choose different λi (i = 2, 3, 4) to obtain the maximum η based on Algorithm 1, such that there exists a feasible solution to LMI conditions in (10) and (11). 2. Choose a sufficiently large initial γ0 , search a feai (i = 1, 2, 3, 4), 1 , 2 , sible set ( P, S, R, X , Y , M 3 , L, T , W , H ) satisfying LMI in (39). Set k = 1, γ = γ0 . 3. Solve the following LMI problem for the variables (L, X , T , P, W , S, H, R) 0 −η P X −T ϕ(0) < tr(1 1 ) ϕ T (0)X −1 0 −d 0 ϕ T (v )X −1 S X −T ϕ(v )dv < tr(2 2 ) ϕ̇ T (v )X −1 RX −T ϕ̇(v )dvds < tr(3 3 ) (37) s Algorithm 2 Pk T + Tk P + Wk S Min tr(Xk L + Lk X + Rk H + Hk R) (41) + Sk W + For some constant γ > 0, assume tr(1 1 ) + tr(2 2 ) + tr(3 3 ) < γ (38) Therefore, we can construct a feasibility problem as follows: Given η and λi (i = 2, 3, 4), i Find P, S, R, X , Y , M (i = 1, 2, 3, 4), 1 , 2 , 3 , L, T , W , H subject to : P > 0, S > 0, R > 0, T > 0, W > 0, H > 0 and (10), (11), (36), (38) (39) If the above problem has a solution, we can say that there exists a feedback gain K = YX −T which guarantees the cost function (9) with the cost less than γ . But (36) includes nonlinear conditions. It cannot be solved directly by using LMIs. However, using the idea of cone complementary linearization algorithm in Lee et al . (2001), Ghaoui et al . (1997), the above feasibility problem can be solved iteratively. So we present the following nonlinear minimization problem instead of the original nonconvex minimization in (30) Min tr(XL + PT + SW + RH ) subject to : P > 0, S > 0, R > 0, T > 0, W > 0, H > 0 and (10), (11), (36), (38), X I P I S I > 0, > 0, > 0, I L I T I W R I >0 I H (40) Although it is only a suboptimal solution of (33), it is easier to solve than the original nonconvex minimization problem (33). We can easily find a suboptimal minimum of the guaranteed cost based on the cone complementary linearization method (Ghaoui et al ., 1997). 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Obtain the suboptimal upper bound γ : subject to LMI in (40) Set Xk +1 = X , Lk +1 = L, Tk +1 = T , Wk +1 = W , P, Sk +1 = S, Rk +1 = R. Hk +1 = H , P̃+1 = 4. If the (34) is satisfied, then return to Step 2 after decreasing γ0 to some extent. If the (34) is not satisfied within a specified number of iterations, then exit. Otherwise, set k = k + 1 and go to Step 3. 5. Output suboptimal guaranteed cost γ and corresponding guaranteed cost feedback gain K = YX −T . The above algorithm gives the controller feedback gain K = YX −T under the constraint of guaranteed cost γ , and corresponding suboptimal guaranteed cost γ . In Section 5, we will illustrate the effectiveness of the above algorithm. In the following section, we extend the obtained stabilization conditions for uncertain systems (8). Considering the effect of the parameter uncertainties A, A1 , and B , the following theorem provides the condition for existence of the guaranteed cost controller for the uncertain time-delay systems (8). Theorem 2 For given scalars η and λi (i = 2, 3, 4), 1 + λ2 + λ3 + λ4 > 0, if there exist scalars εi > 0 (i = i (i = 1, 2, 3), matrices P, S and R > 0, X , Y and M 1, 2, 3, 4) with appropriate dimension, if 11 11 <0 (42) ∗ 22 where 11 = 11 + ∗ ∗ ∗ ∗ 12 13 14 22 + λ2 23 24 33 + λ3 34 ∗ 44 + λ4 ∗ ∗ ∗ ∗ ∗ 1 ηM 2 ηM 3 ηM 4 ηM η R Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj 655 656 C. PENG 12 = Asia-Pacific Journal of Chemical Engineering λXE1T 0 λXE2T 0 0 0 0 0 0 0 0 0 X 0 0 0 0 λY T E3T 0 0 0 0 YT 0 0 The example system , From Tian and Gao (1999), when choosing the secondorder plus delay model parameters (K , , ξ, d) = (0.858 m · s −1 × 103 , 128.5 rad/s, 0.953, 15 ms), we can convert the second-order plus delay model of the thermoplastic injection molding system described in Tian and Gao (1999) into the following to resemble the state-space model: λ = 1 + λ2 + λ3 + λ4 22 = diag{−λε1 I , −λε2 I , −λε3 I , −Q1−1 , −R1−1 }, = (ε1 + ε2 + ε3 )DD T . Then the system (8) with the feedback gain K = YX −T is asymptotically stable and the cost function J in (9) satisfies the following bound: J ≤ ϕ (0)X T + 0 −η 0 −1 PX −T ϕ(0) + 0 T ϕ (t)X −1 −T SX ϕ(t)dt −d ϕ̇ T (v )X −1 RX −T ϕ̇(v )dvds (43) ∗ ∗ ∗ ∗ (45) y(t) = Cx (t) (46) 0.1 −0.2449 −0.0165 , A1 = 0 1.000 0 1 0 B= ,C = 0 0.0142 A= 0 , 0.1 (47) s Proof: Replace A, A1 , and B with A + DF (t)E1 , A1 + DF (t)E2 and B + DF (t)E3 in (21). Then, following a similar procedure as in the proof of theorem 1, we have where ẋ (t) = Ax (t) + A1 x (t − d) + Bu(t) 12 ∗ ∗ ∗ 13 23 ∗ ∗ 14 24 34 44 + λ4 ∗ 1 ηM 2 ηM 3 ηM <0 4 ηM η R Then, when considering the uncertainty in system identification, we have the following model based on (45)–(46) ẋ (t) = (A + A)x (t) + (A1 + A1 )x (t − d) + (B + B )u(t) (44) y(t) = Cx (t) 22 + λ2 + λε−1 XE2T E2 X T , = 2 33 + λ3 + = λε3−1 Y T E3T E3 Y D= E2 = + Y T R1 Y . Using Schur complement, we can obtain (42) from (44) for λ > 0. This completes the proof. Remark 4 To obtain the suboptimal guaranteed cost γ of the system (8) with parameter uncertainties, we can construct the same nonlinear minimization problem as (30), and the corresponding solution is similar to Algorithm 2; it is omitted here. NUMERICAL EXAMPLES In this section, we consider networked injection velocity control for Thermoplastic Injection Molding taken from (Tian and Gao, 1999). Our objective is to control the injection velocity over Ethernet-based networks to guarantee the given performance index γ defined in (9). 2007 Curtin University of Technology and John Wiley & Sons, Ltd. (49) where uncertain matrices A, A1 , and B with form (3) and the following parameters: where 11 + + λε−1 XE1T E1 X T + XQ1 X T , = 1 (48) 0.1 0 0.01 0 , E1 = 0 0 0 0.01 T 0.1 0 , E3 = [ 0 0 0 ]T . 0.1 0 0.01 0 0 0 0 0 T , (50) Assume that the initial conditions are x1 (t) = 0.5e t+1 and x2 (t) = −0.5e t+1 for t ∈ [−d, 0], the system delay is included in state delay x (t − d), the network-induced delay will be considered in the controller design. Controller design with stability First, we consider the stabilization problem to derive the SAEDB η. When the controller is implemented through a network, the problem can be written in the form of (8). Suppose that the full state variables are available for measurement. Applying Algorithm 1 with λ2 = −0.2, λ3 = 1.9, λ4 = 7.8, we can find that the SAEDB that guarantees the stability of system (48) is 3.8 ms and the corresponding feedback gain K = [−0.0754 − 0.0261]. It means that, if h = 1 ms and the data packet dropout can be neglected in the transmission, the maximum allowable network-induced delay τik ≤ 2.8 ms. The designed controller can stabilize the Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering COST CONTROL FOR A CLASS OF INDUSTRIAL PROCESSES system (6) as long as the upper bound of the networkinduced delay is less than 2.8 ms. 50 control point set point r Controller design with guaranteed cost Now we consider the above system with guaranteed cost under networked control. According to algorithm 2, Table 1 shows the obtained suboptimal guaranteed cost and corresponding controller feedback gain with η = 2.8, Q1 = 0.1I2×2 and R1 = 0.1. It shows that under the constraint of guaranteed cost performance index γ , the designed controller can stabilize system (6). The state responses of system (48) with controller feedback gain K = [−0.2565 −0.0728] are shown in Fig. 1. Figure 1 also shows that the system is asymptotically stable with given controller. Given injection velocity set-point as solid line in Fig. 2, the tracing curve based on controller feedback gain K = [−0.2565 −0.0728] is shown as dotted line in Fig. 2. It can been seen that the method developed in this paper can trace the varying injection velocity. Furthermore, compared to the keeping time of injection velocity(about 0.8 s), the adjust time of our method (about 0.2 s) is less. Table 1. Suboptimal guaranteed cost and corresponding controller feedback gain. γ Number of iterations State feedback gain, K 50 45 40 34 36 46 [−0.2571 −0.0713] [−0.2575 −0.0720] [−0.2565 −0.0728] 2 x1 x2 1 Ram Velocity r* 103 (m/s) 40 30 20 10 0 10 0 0.5 1 1.5 2 Simulation Time t (s) 2.5 3 Figure 2. Trace of injection velocity set-points with K = [−0.2565 −0.0728]. CONCLUSION When industrial processes with delay are controlled over communication networks, the time-varying characteristics of the nonideal network conditions cannot be neglected, which not only degrade control performance but also introduce distortion of the controller signal. In this paper, we have presented a solution to the delaydependent guaranteed cost control problem via a memoryless state feedback for a class of uncertain state-delay industrial systems interconnected over data network. First, an NCS model is presented to include nonideal network conditions, such as data packet dropout. Then, the guaranteed cost control for nominal and uncertain systems with state delay has been introduced, and a cone complementary linearization algorithm is presented to solve the nonconvex problem of NCS controller design and to obtain a suboptimal upper bound of the cost. Numerical examples have been given to demonstrate the effectiveness of the proposed method. 0 Acknowledgment State −1 This work is partially supported by Australian Research Council (ARC) under the Discovery Projects Grant Scheme (grant ID: DP0559111), Natural Science Foundation of China (grant ID: 60474079), and Natural Science Foundation of Jiangsu Province of China (grant ID: BK2006573). −2 −3 −4 −5 −6 0 0.1 0.2 0.3 0.4 Simulation Time t (s) 0.5 0.6 Figure 1. Simulation of system (9) with γ = 40, K = [−0.2565 − 0.0728]. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. REFERENCES Ghaoui LE, Oustry F, AitRami M. A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans. Automat. Control 1997; 42(8): 1171–1176. Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj 657 658 C. PENG Han QL. A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 2004; 40: 1791–1796. Hong SH. Scheduling algorithm of data sampling times in the integrated communication and control systems. IEEE Trans. Control Syst. Technol. 1995; 3: 225–231. Huang D, Nguang SK. State feedback guaranteed cost control of uncertain networked control systems. In Systems and Control in Aerospace and Astronautics, 2006. ISSCAA 2006. 1st International Symposium on, 2006; 19–21, Harbin, China. Jiang XF, Han QL. Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica 2006; 42: 1059–1065. Lee YS, Moon YS, Kwon WH. Delay-dependent guaranteed cost control for uncertain state-delayed stystems. In Proceedings of the American Control Conference, June 25–27; 2001, Arlington, U.S. Li SB, Wang Z, Sun YX. Guaranteed cost control and its application to networked control systems. In Industrial Electronics, 2004 IEEE International Symposium on, 2004; 591–596, Ajcaccio, France. Moheimani SOR, Moheimani IR. Optimal quadratic guaranteed cost control of a class of uncertain time delay systems. IEE Proc. Control Theory Appl. 1997; 144: 1838–1845. Peng C, Tian Y-C. Robust H∞ control of networked control systems with parameter uncertainty and state-delay. Eur. J. Control 2006a; 12(5): 471–480. Peng C, Yue D. State feedback controller design of networked control systems with parameter uncertainty and state delay. Asian Journal of Control 2006; 8(4): 385–392. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering Su SW, Bao J, Lee PL. A hybrid active-passive fault-tolerant control approach. Asia Pac. J. Chem. Eng. 2006; 1(2): 54–62. Tellez R, Svrcek WY, Yong BR. Dynamic simulation and advanced control of a heat intergated plant. Dev. Chem. Eng. Mineal Procees. 2005; 13: 279–288. Tian YC, Gao FR. Injection velocity control of thermoplastic injection molding via a double controller scheme. Ind. Eng. Chem. Res. 1999; 38: 3396–3406. Tian Y-C, Levy D, Tadé MO, Gu T, Fidge C. Queuing packets in communication networks networked control systems. In Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA’06), Dalian, 2006a; 210–214. Yu L, Gao FR. Output feedback guaranteed cost control for uncertain discrete-time systems using linear matrix inequalities. J. Optim. Theory Appl. 2001; 113: 621–634. Yue D, Han Q-L, Peng C. State feedback controller design of networked control systems. IEEE Trans. Circuits Syst. II: Express Briefs. 2004; 51: 640–644. Yue D, Han Q-L, Lam J. Network-based robust H∞ control of systems with uncertainty. Automatica 2005; 41: 999–1007. Yue D, Peng C, Tang GY. Guaranteed cost control of linear systems over networks with state and input quantisations. IEE Proc. Control Theory Appl. 2006; 153(6): 658–664. Zhang W, Branicky MS, Phillips SM. Stability of networked control systems. IEEE Control Syst. Mag. 2001; 21: 84–99. Zhang HG, Yang DD, Chai TY. Guaranteed cost networked control for TCS fuzzy systems with time delays. IEEE Trans. Syst. Man Cybern. Part C: Appl. Rev. 2007; 37(2): 160–172. Asia-Pac. J. Chem. Eng. 2007; 2: 650–658 DOI: 10.1002/apj

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