From Two-Dimensional Linear Quadratic Optimal Control to Iterative

Ind. Eng. Chem. Res. 2006, 45, 4603-4616

4603

PROCESS DESIGN AND CONTROL From Two-Dimensional Linear Quadratic Optimal Control to Iterative Learning Control. Paper 1. Two-Dimensional Linear Quadratic Optimal Controls and System Analysis Jia Shi,†,‡ Furong Gao,*,† and Tie-Jun Wu‡ Department of Chemical Engineering, Hong Kong UniVersity of Science & Technology, Clear Water Bay, Kowloon, Hong Kong, and Institute of Intelligent Systems and Decision Making, Zhejiang UniVersity, Hangzhou 310027, PRC

This paper develops two-dimensional linear quadratic (2DLQ) optimal control schemes that can be applied for iterative learning control (ILC) design. For a class of repetitive 2D processes, based on control objectives defined over a single cycle or multiple cycles, two different 2DLQ optimal control schemes, single-cycle 2DLQ (SC-2DLQ) and multicycle 2DLQ (MC-2DLQ), are derived. Furthermore, to balance control performance and computational load of MC-2DLQ optimal control, a cyclewise receding horizon 2DLQ (CWRH-2DLQ) optimal control strategy is introduced based on a quadratic performance index with a cyclewise receding horizon. The cyclewise stability and robustness analysis are conducted based on the cyclewise dynamics of the closed-loop system. The simulation shows that the proposed algorithms are effective. On the basis of the results of this paper, new ILC schemes for batch processes will be presented as a sequel to this paper. 1. Introduction Many batch processes have dynamic behavior along two directions (time and batch). These types of processes can be considered as 2D systems, which, mathematically, may be described by the following two types of 2D state-space models:1 (i) Roesser type (R-type)

{

(

)

δ1xh(t1,t2) ) f(xh(t1,t2),xV(t1,t2),u(t1,t2),t1,t2) δ2xV(t1,t2) y(t1,t2) ) g(xh(t1,t2),xV(t1,t2),u(t1,t2),t1,t2)

(1)

(ii) Fornasini-Marchesini type (FM-type)

{

δ1δ2x(t1,t2) ) f(δ1x(t1,t2),δ2x(t1,t2),x(t1,t2),u(t1,t2),t1,t2) y(t1,t2) ) g(x(t1,t2),u(t1,t2),t1,t2)

(2)

where δ1 and δ2, respectively, represent the derivative operators for a continuous-time system or the unit forward-shift operators for a discrete-time system of the two axes. It is clear that their inputs, outputs, and internal states are all defined as functions of two independent time axes, and the evolutions of the system states along the two axes, continuous or discrete, are determined by past information in the 2D sense. In comparison to a 1D model, a 2D model (eq 1or 2) can better describe a complex process such as a repetitive process or a batch process. Over the past decades, 2D systems have received considerable attention in both theoretical and practical research. On the theoretical front, the early works2-5 focused mainly on the * To whom correspondence should be addressed. Telephone: +8522358 7139. Fax: +852-2358 0054. E-mail: [email protected]. † Hong Kong University of Science & Technology. ‡ Zhejiang University.

stability issues of linear 2D systems which are more complicated than those of the 1D case. Recently, a number of results on the robust control of 2D systems has been reported. On the basis of the linear matrix inequality, Du and Zie6 conducted a robust stability analysis and synthesized a robust controller for an FMtype 2D model with norm-bounded parameter uncertainties. Guan et al.7 and Xie et al.8 extended robust guaranteed cost control and robust H∞ control of a 1D system to FM-type 2D discrete systems. For an R-type 2D model, Du et al.9 proposed robust H∞ control, and Lam et al.10 synthesized a robust output feedback control. All of the above works, however, were conducted based on system stabilization, leading to conservative linear time-invariant 2D state feedback controls. Different from the above stability based methods, LQ optimal control and MPC can be introduced to 2D system to design controllers that optimize a given control performance index. For an FM-type discrete 2D linear system, Bisiacco et al.11 solved the 1D infinite horizon optimal control problem and investigated the stability of the closed-loop system in the frequency domain. Sˇ ebek and Kraus12 investigated linear 2D stochastic LQ control via a 2D polynomial technique. For the R-type linear 2D system, most LQ optimal control works transformed a 2D system into a 1D system; for examples, see Rostan and Lee,13 Jagannathan and Syrmos,14 and Tsai et al..15 On the application front, 2D modeling and system theories have been successfully applied in 2D digital signal processing,16 image data processing,17 and mining processes.18 Galkowski et al.19-22 have conducted a comprehensive analysis and control synthesis, in both the discrete and continuous context, for a class of repetitive processes described by an R-type 2D model. Another driving force for 2D system theory development comes from iterative learning control (ILC) design, which is undergoing rapid development with many successful applications for solving practical control engineering problems. An ILC system may be considered as a special 2D system with dynamic

10.1021/ie051297i CCC: $33.50 © 2006 American Chemical Society Published on Web 05/26/2006

4604

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006

evolution along both time and iteration directions. The key difference from a general 2D system is that an ILC system has dynamic evolution repeated along time over a finite duration. This does not prohibit researchers from treating an ILC problem as a 2D system. Geng et al.,23 in 1990, first pointed out the relationship between an ILC system and an R-type linear 2D system and conducted the design and analysis of an ILC system from a 2D system viewpoint. A direct ILC synthesis based on linear 2D system theory was proposed by Kurek and Zaremba.24 On the basis of the robust control theory devolved for an R-type 2D system, the authors25 extended this idea by proposing an integrated design method for an ILC system. This design allows investigation of robust convergences along both time and cycle directions in an ILC system sense. The authors would like to echo again that the benefits of treating ILC problems from a 2D system viewpoint: (i) In a 2D sense, the properties of an ILC system along the time and iteration directions can be separately investigated both in controller design and convergence analysis. (ii) Design of ILC in a 2D system frame can result in a cyclewise iterative learning law combined with timewise feedback control law to ensure the control performance not only along cycles but also over time. (iii) A 2D system is a maturing area26,27 with many designs and methodologies that can be borrowed for ILC design. All the above-mentioned ILC system designs are based on the stabilization requirement; they result in typically conservative ILC laws of linear time-invariant form. For better control performance, 2D control methods based on performance optimization should be explored for ILC system design. Few results have been reported in this area with the exception of ref 28 where a 2D optimal control method is used. The resulting control, however, is only feed-forward ILC, and the convergence analysis is lacking. The objective of this two part series of papers is to explore 2D optimal control schemes for ILC system design and analysis. The 1st of the series focuses on 2DLQ optimal control problems and system analysis. To allow the design methods proposed in this paper to be migrated smoothly from 2D systems to ILC systems, the 2D plants considered are assumed to be of a class of linear repetitive processes described by a FM-type linear 2D model. Two 2DLQ optimal controls are proposed: one based on a single-cycle quadratic performance index, referred to as SC-2DLQ optimal control; the other based on a multicycle quadratic performance index, referred as MC-2DLQ optimal control. Optimal control algorithms will be derived by using dynamic programming. In a 2D system view, the obtained control laws are essentially optimal time-varying 2D state feedback controls. For SC-2DLQ optimal control, the resulting control law uses real-time state information and all state information of the last cycle over future time, while, for MC2DLQ optimal control, the resulting control law depends on the all historical state information of the current cycle and all state information of the last cycle over future time. To balance the computational load and control performance of the MC2DLQ optimal control scheme, a new quadratic performance index with a receding horizon along the cycle direction is introduced, resulting in a 2DLQ optimal control scheme referred to as the cyclewise receding horizon 2DLQ (CWRH-2DLQ) optimal control scheme. As all of these 2DLQ optimal control schemes are designed based on finite horizon optimization algorithms, it is important to investigate the stability and robustness of the control systems, so methods for stability and robustness analysis are introduced. The results obtained in this paper will be used as the background material for the ILC schemes to be developed in the second paper in this series. The remainder of this paper part is so organized: The mathematical description of a repetitive process with 2D

dynamics is presented in section 2 together with the definitions of single-cycle (SC) and multicycle (MC) 2DLQ optimal control problems. The derivations of control laws and the introduction of the CWRH-2DLQ optimal control scheme are given in section 3. Cyclewise stability analysis is performed for the nominal system in section 4. Section 5 addresses the cyclewise robust stability and disturbance rejection analysis. Examples are presented in section 6 to demonstrate the effectiveness of the proposed schemes. Finally, conclusions are drawn in section 7. Throughout this paper, the following notations are used: Rn represents Euclidean n space with the norm denoted by |‚|. Rn×m is a set of n × m real matrices. For any matrix M ∈ Rn×n, M > 0 (M g 0) means M is a positive (semipositive) definite symmetric (PDS/SPDS) matrix. MT represents the transpose of matrix M. If matrix A ∈ Rn×m represents a linear transformation form space Rm to Rn, then |A| denotes the induced norm. I and 0 respectively denote the identity matrix and the zero matrix with appropriate dimensions depending on the context, and In and 0n×m are used to specify the dimensions of these two matrices. For any vector function f(‚,‚), f([t1:t2],‚) represents the value set {f(t,‚)}t)t1,t1+1,...,t2 and f(|tt21,‚) denotes super vector (fT(t1,‚) fT(t1 + 1,‚) ... fT(t2,‚))T. For a signal w(k), if |w|2e ) N |w(k)|2 < ∞ for any integer N > 0, then w(k) is said to x∑k)0

be in l2e space, denoted by w ∈ l2e. 2. Problem Formulation

2.1. 2D Model. In this paper, the process of interest is of a class of repetitive processes with 2D dynamics, called repetitive 2D processes, described by the following FM-type discretetime 2D state-space equation

∑:

{

p

x(t + 1,k) ) A1(t,k)x(t,k) + A2(t,k)x(t + 1,k - 1) + B(t,k)u(t,k) y(t,k) ) C(t,k)x(t,k)

(3) x(0,k) ) x0,k, x(t,0) ) xt,0, t ) 0, 1, ..., T, k ) 0, 1, ...

where t and k denote the discrete 2D indices. For a repetitive process, t denotes the time and k represents the cycle index. x(t,k) ∈ Rn, u(t,k) ∈ Rm, and y(t,k) ∈ Rl are, respectively, the state, input, and output of the process at time t in the kth cycle, {A1(t,k),A2(t,k),B(t,k),C(t,k)} are the system matrices with appropriate dimensions. Without loss generality, here, they are assumed to be both time- and cycle-dependent variables. The term xt,0, t ) 1, 2, ..., T is the initial state profile of a 2D system, and x0,k is the initial state of each cycle. The last two form the boundary condition of a 2D model (eq 3). Model 3 is a special 2D system with dynamics repeated over a finite time duration. Information available at time t in the kth cycle are {{f([0:t],k), f([0:T],[0:k - 1])}: f ∈ {x,u,y}}. Use of such a 2D model description allows the timewise and cyclewise dynamics to be modeled simultaneously. 2.2. 2DLQ Optimal Control Problems. For engineering application, a basic requirement for controller design is that the control algorithm should be causal. For a repetitive 2D system ∑p, the causality of the control algorithm is defined as follows: Definition 2.1. For 2D system ∑p, a control algorithm is causal if, and only if, the value of the input u(t,k) at time t in the kth cycle is computed from the available system information {{f([0:t],k), f([0:T],[0:k - 1])}, f ∈ {x,u,y}}. Compared with an LQ control problem of a 1D system, the 2DLQ problem is more complicated as control performance needs to be considered in a 2D sense. For 2D model 3, the following two kinds of 2DLQ control problems are important and interesting to be considered:

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4605

Definition 2.2. SC-2DLQ Optimal Control Problem. For given PDS matrices R([0:T - 1]) and SPDS matrices Q([1:T]), design a causal control law at the end of the kth cycle such that the following single-cycle quadratic performance index is minimized

the optimal control strategy for the (k + 1)th cycle is

Jk+1(u([0:T - 1],k + 1)) )

which minimizes the performance index (eq 4) computed by

u*(t,k + 1) ) - K(t,k + 1)

T-1

(xT(t + 1,k + 1)Q(t + 1)x(t + 1,k + 1) + ∑ t)0

Jk+1(u*([0:T-1],k +1)))

u (t,k + 1)R(t)u(t,k + 1)) (4) T

Definition 2.3. MC-2DLQ Optimal Control Problem. For given PDS matrices R([0:T - 1],[1:N]) and SPDS matrices Q([1: T],[1:N]), design a causal control law such that the following multicycle quadratic performance index is minimized

J[1:N](u([0:T - 1],[1:N])) ) N T-1

∑ ∑(x (t + 1,i)Q(t + 1,i)x(t + 1,i) + i)1 t)0 T

uT(t,i)R(t,i)u(t,i)) (5) Here, the positive integer N g1 is defined as the cyclewise optimization horizon. Note that both (4) and (5) define the control performances in a 2D sense and use the time duration of the process as the default timewise optimization horizons. In a cyclewise sense, the SC2DLQ optimal control problem is essentially a one-cycle LQ optimal control problem, while the MC-2DLQ optimal control problem is a multicycle LQ optimal control problem. 3. 2DLQ Optimal Control Schemes The optimal control law can be derivated by the use of the principle of optimality and dynamic programming. The principle of optimality states that an optimal policy has the property that, with whatever initial state and initial decision, the remaining decisions are optimal with respect to the state resulted from the first decision. According to this principle, it is possible to determine the optimal control law by starting from the end point of the optimization horizon and going backward along the dynamic of the process. To conduct this programming, the following one-step optimization result plays an important role. Lemma 3.1. For the following one-step LQ optimization problem

u* ) arg min{J(u|x′) ) xTSx + uTRu:x ) T1x′ + T2u} (6) u

where matrices R > 0 and S g 0, x′ is known, and u is the decision variable, the optimal decision is

u* ) -(R + T2TST2)-1T2TST1x'

(7)

(8)

where P is a PDS matrix defined by

P ) T1T(S - ST2(R + T2TST2)-1T2TS)T1

(9)

Proof is omitted here because it follows directly from the optimization algorithm. The solutions for the 2DLQ optimal control problems of the last section are given as the following theorems: Theorem 3.2. SC-2DLQ Optimal Control Scheme. For the SC-2DLQ optimal control problem proposed in definition 2.2,

)

x(t,k + 1) , t ) 0,1, ..., T - 1 x(|t+1 T ,k) (10)

( )

( )

x0,k+1 T x0,k+1 P(0,k +1) x(|1T,k) x(|1T,k)

(11)

where the time-varying matrices K([0:T - 1],k + 1) and P(0,k + 1) are determined by the following backward recursive algorithm: Algorithm 1

S(T,k + 1) ) Q(T)

(12)

P(t,k + 1) ) A h T(t,k + 1)(S(t + 1,k + 1) S(t + 1,k + 1)B h (t,k + 1)(R(t) + T h (t,k + 1))-1 × B h (t,k + 1)S(t + 1,k + 1)B B h T(t,k + 1)S(t + 1,k + 1))A h (t,k + 1) (13) S(t,k + 1) )

(

)

Q(t) 0 0(T-t)n×(T-t)n + P(t,k + 1) 0

(14)

K(t,k + 1) ) (R(t) + B h T(t,k + 1)S(t + 1,k + 1)B h (t,k + 1))-1 B h T(t,k + 1)S(t + 1,k + 1)A h (t,k + 1) (15) where t ) T - 1, T - 2, ..., 0 and

A h (t,k + 1) )

(

)

A1(t,k + 1) A2(t,k + 1) 0 I(T-t-1)n , 0 0 B(t,k + 1) B h (t,k + 1) ) 0 (T-t-1)n×m

(

)

(16)

Proof. To prove this theorem, dynamic programming will be used. We start from the end point of the (k + 1)th cycle and iteratively move backward along time t. First, introduce an LQ performance index as

J[t:T-1],k+1(u([t:T - 1],k + 1)) ) T-1

(xT(i + 1,k)Q(i + 1)x(i + 1,k) + uT(i,k)R(i)u(i,k)) ∑ i)t

(17)

It is clear that J[0:T-1],k+1(u([0:T - 1],k + 1)) ) Jk+1(u([0:T 1],k + 1)). For t ) T - 1, according to Lemma 3.1, we have

u*(T - 1,k + 1) ) -K(T - 1,k + 1)

which minimizes the performance index computed by

J(u*|x′) ) x′TPx′

(

(

x(T - 1,k + 1) x(T,k)

)

(18)

where

K(T - 1,k + 1) ) (R(T - 1) + BT(T - 1,k + 1)Q(T)B(T - 1,k + 1))-1 BT(T - 1,k +1)Q(T)(A1(T-1,k+1) A2(T - 1,k + 1) ) (19) and the minimal performance index is computed by

JT-1,k+1(u*(T - 1,k + 1)) ) x(T - 1,k + 1) T x(T - 1,k + 1) P(T - 1,k + 1) x(T,k) x(T,k)

(

)

(

)

(20)

4606


(ii) for i ) N,

where

(

)

A1T(T - 1,k + 1) A2T(T - 1,k + 1) (Q(T) - Q(T)B(T - 1,k + 1)(R(T - 1) + BT(T - 1,k + 1)Q(T)B(T - 1,k + 1))-1 × A T(T - 1,k + 1) BT(T - 1,k + 1)Q(T)) 1T A2 (T - 1,k + 1)

P(T - 1,k + 1) )

(

u*(t,N) ) -K(t,N)

)

T

(21)

J[T-2:T-1],k+1(u*([T - 2:T - 1],k + 1)) ) T-1

(xT(t+1,k +1)Q(t +1)x(t +1,k +1) + ∑ u([T - 2:T - 1],k + 1) t)T-2 min

uT(t,k + 1)R(t)u(t,k + 1)) ) (xT(T - 1,k + 1)Q(T - 1)x(T - 1,k + 1) +

J[1:N](u*([0:T - 1],[1:N])) )

u(T-2,k)

u (T - 2,k + 1)R(T - 2)u(T - 2,k + 1) + JT-1,k+1(u*(T - 1,k + 1))) ) x(T - 1,k + 1) T x(T-1,k + 1) S(T -1,k +1) + x(T,k) x(T,k)

((

)

(

) )

uT(T - 2,k + 1)R(T - 2)u(T - 2,k + 1) (22) where

S(T - 1,k + 1) )

(

)

Q(T - 1) 0 + P(T - 1,k + 1) (23) 0n×n 0

which satisfys relationship 14. Note that

( (

)

x(T - 1,k + 1) ) x(T,k)

S(T,i) )

Jk+1(u*([0:T - 1],k + 1)) ) J[0:T-1],k+1(u*([0:T - 1],k + 1)) ) x0,k+1 T x0,k+1 P(0,k + 1) 1 x(|T,k) x(|1T,k)

( )

(25)

(28)

(29)

)

(31)

K(t,i) ) (R(t,i) + B ˜ T(t,i)S(t + 1,i)B ˜ (t,i))-1B ˜ T(t,i)S(t + 1,i)A ˜ (t,i) (32)

( ( )

Itn 0 A ˜ (t,i) ) 0 0

x(|0t ,i) , t ) 0,1,...T - 1 x(|t+1 T ,i - 1)

0 In A1(t,i) 0

0 0 A2(t,i) 0

0 0 0 I(T-t-1)n

0tn×m 0n×m B ˜ (t,i) ) B(t,i) 0(T-t-1)n×m

)

(33)

Proof. This procedure is similar to the proof of theorem 3.2. First, we introduce the following performance index

J[t,T-1],[i:N](u([t:T - 1],i),u([0:T - 1],[i + 1:N])) ) J[t:T-1],i(u([t:T - 1],i)) + J[i+1:N](u([0:T - 1],[i + 1:N])) (34) where the LQ function J[i+1:N](‚) is defined as

and completes the proof. Theorem 3.3. MC-2DLQ Optimal Control Scheme. Assume x(0,[1:N]) ) x0,1. For the MC-2DLQ control problem proposed in definition 2.3, the optimal control strategy over the optimization horizon is defined as follows (i) for i ) 1, 2, ..., N - 1,

)

)

0Tn×Tn 0 + P(0,i + 1) Q(T,i) 0

(

)

According to Lemma 3.1, it is clear that the control law designed by algorithm 1 is the optimal control that minimizes the control performance index J[T-2:T-1],k+1(u). The above procedure can be repeated backward along the time until t ) 0, which leads to

(

(

0tn×tn 0 0 Q(t,i) 0 + P(t,i) S(t,i) ) 0 0(T-t)n×(T-t)n 0 0

B(T - 2,k + 1) u(T - 2,k + 1) (24) 0n×m

u*(t,i) ) -K(t,i)

T x0,1 x0,1 P(0,1) 1 x(|T,0) x(|1T,0)

where the initial matrix P(0,N) is determined in step 1, t ) T - 1, T - 2, ..., 0, and

)( )

( )

( ) ( )

P(t,i) ) A ˜ T(t,i)(S(t + 1,i) - S(t + 1,i)B ˜ (t,i)(R(t,i) + T -1 T ˜ (t,i)) B ˜ (t,i)S(t + 1,i))A ˜ (t,i) (30) B ˜ (t,i)S(t + 1,i)B

A1(T - 2,k + 1) A2(T - 2,k + 1) 0 x(T - 2,k + 1) + In x(|T-1 0 0 T ,k)

(

(27)

where the time-varying matrices K([0:T - 1],[1:N]) and P(0,1) are determined by the following backward recursive algorithm: Algorithm 2 (i) Step 1. For i ) N, the matrices K([0:T - 1],N) and P(0,N) are determined by algorithm 1 proposed in theorem 3.2 where the index k + 1 is replaced by N and matrices Q(t) and R(t) are replaced by Q(t,N) and R(t,N), respectively. (ii) Step 2. For i ) N - 1, N - 2, ..., 1, the matrices K([0:T - 1],i) and P(0,i) are determined by the following backward recursive algorithm:

u(T-2,k+1) T

min

)

x(t,N) , t ) 0,1,...T - 1 x(|t+1 T ,N - 1)

leading to the minimal performance index (eq 5) computed by

It is easy to see that eqs 19 and 21 are the case of algorithm 1 when t ) T - 1. Now, using (20) for t ) T - 2 gives

min

(

(26)

J[i+1:N](u([0,T - 1],[i + 1,N])) ) N T-1

(xT(t + 1,j)Q(t + 1,j)x(t + 1,j) + uT(t,j)R(t,j)u(t,j)) ∑ ∑ j)i+1 t)0 (35) and J[t:T-1],i(‚) is defined by (17). For i ) N, it results from theorem 3.2 that control law 27 leads to the minimal performance index computed by


JN(u*([0:T - 1],N)) ) J[0:T-1],N(u*([0:T - 1],N)) ) T x0,N x0,N P(0,N) 1 x(|T,N - 1) x(|1T,N - 1)

(

)

(

)

(36)

On the basis of the assumption that x0,N-1 ) x0,N, the above identity can be rewritten as

Definition 3.1. CWRH-2DLQ Optimal Control Problem. For the given PDS matrices R([0:T - 1],[1:N]) and SPDS matrices Q([1:T],[1:N]), design a causal control law at the end of the kth cycle such that the following quadratic performance index is minimized

J[k+1:k+N](u([0:T - 1],[k + 1:k + N])) ) N T-1

JN(u*([0:T - 1],N)) ) xT(|0T,N - 1)P(0,N)x(|0T,N - 1) (37)

(xT(t + 1,k + i)Q(t + 1,i)x(t + 1,k + i) + ∑ ∑ i)1 t)0 uT(t,k + i)R(t,i)u(t,k + i)) (40)

Then, for i ) N - 1, according to dynamic programming and relationships (eq 37), we have

JT-1,[N-1:N](u*(T - 1,N - 1),u*([0:T - 1],N)) ) min

u(T-1,N-1) T

(xT(T,N - 1)Q(T,N - 1)x(T,N - 1) +

u (T - 1,N - 1)R(T - 1,N - 1)u(T - 1,N - 1) + JN(u*([0:T - 1],N))) ) min

u(T-1,N-1) T

(xT(|0T,N - 1)S(T,N - 1)x(|0T,N - 1) +

u (T - 1,N - 1)R(T - 1,N - 1)u(T - 1,N - 1)) (38)

where S(T,N - 1) is defined by (29). It also follows from dynamic model 3 and definition 33 that

(

)

0 x(|T-1 ,N - 1) + x(T,N - 2) B ˜ (T - 1,N - 1)u(T - 1,N - 1) (39)

˜ (T - 1,N - 1) x(|0T,N - 1) ) A

Now, using lemma 3.1 again, one can derivate that the control laws {u*(T - 1,N - 1),u*([0:T - 1],N)} determined by algorithm 2 are the optimal controls in terms of the performance index JT-1,[N-1:N]. This procedure can be repeated backward along the dynamic of the process until t ) 0 and i ) 1, which leads to performance index 5 being minimized. We give following remarks on theorems 3.2 and 3.3. Remark 3.1. Note that, for algorithm 1, the sizes of timevarying matrices {P,S,K} increase with a decrease of time t. For algorithm 2, the sizes of matrices {P,S,K}, however, remain constant for i ) 1, 2, ..., N - 1. Remark 3.2. From a 2D system viewpoint, the optimal control laws proposed in theorems 3.2 and 3.3 both are timevarying 2D state feedback controls. For the SC-2DLQ optimal control scheme as well as the last cycle of the MC-2DLQ optimal control scheme, the control lawsssee (10) and (27)s are computed from the real-time state information and all state information of last cycle over the remaining time, while, for the other cycles, the MC-2DLQ optimal control scheme depends on all available state information of the current cycle and all state information of the last cycle over the remaining time. Both algorithm 1 and algorithm 2 are causal for the 2D process ∑p. Remark 3.3. It is easy to see from the proof of theorem 3.3 that the constant initial condition can guarantee the algorithm causality. It is worth pointing out that this is not a prerequisite if the initial states of each cycle can be predetermined and used for the computation of the optimal control. Remark 3.4. As mentioned before, the MC-2DLQ optimal control scheme is essentially a finite LQ optimal control in a cycle sense. Theoretically, a larger N can be chosen for better performance. However, large values of N increase the computational load of the design, especially for a process with a large time duration. One way to balance the computation load and control performance is to use a quadratic performance index with a receding horizon, leading to the following cyclewise receding horizon (CWRH) 2DLQ optimal control problem.

where integer N > 0 is called the cyclewise optimization horizon. The only difference between definition 2.3 and definition 3.1 is that the performance index of the latter adopts a cyclewise receding horizon. So, it is easy to understand the following solution for the CWRH-2DLQ optimal control problem. Theorem 3.4. CWRH-2DLQ Optimal Control Scheme. For the (k + 1)th cycle, assume x(0,[k + 1:k + N]) ) x0,k+1. Then, for the CWRH-2DLQ control problem proposed in definition 3.1, the optimal control law is

(

)

x(|0t ,k + 1) , u*(t,k + 1) ) -K(t,k + 1) x(|t+1 T ,k) t ) 0, 1, ..., T - 1 (41) where the gain matrices K([0:T - 1],k + 1) are determined by the following backward recursive algorithm: Algorithm 3 (i) Step 1. For i ) N, the matrices K([0:T - 1],k + N) is determined by algorithm 1 proposed in theorem 3.2 where the index k + 1 is replaced by k + N and matrices Q(t) and R(t) are replaced by Q(t,N) and R(t,N), respectively. (ii) Step 2. For i ) N - 1, N - 2, ..., 1, the matrix K([0:T - 1],k + i) are computed by the following backward recursive algorithm:

S(T,k + i) )

(

)

0Tn×Tn 0 + P(0,k + i + 1) (42) Q(T,i) 0

P(t,k + i) ) A ˜ T(t,k + i)(S(t + 1,k + i) - S(t + 1,k + i)B ˜ (t,k + i)(R(t,i) + T B ˜ (t,k + i)S(t + 1,k + i)B ˜ (t,k + i))-1 T B ˜ (t,k + i)S(t + 1,k + i))A ˜ (t,k + i) (43) 0tn×tn 0 0 Q(t,i) 0 + P(t,k + i) (44) S(t,k + i) ) 0 0(T-t)n×(T-t)n 0 0

(

)

K(t,k + i)) (R(t,i)+ B ˜ T(t,k + i)S(t + 1,k + i)B ˜ (t,k + i))-1 × B ˜ T(t,k + i)S(t + 1,k + i)A ˜ (t,k + i) (45) where the initial matrix P(0,k + N) is computed in step 1, t ) T - 1, T - 2, ..., 0, and

(

Itn 0 A ˜ (t,k + i) ) 0 0

0 In A1(t,k + i) 0

( )

0tn×m 0n×m B ˜ (t,i) ) B(t,k + i) 0(T-t-1)n×m

0 0 A2(t,k + i) 0

0 0 0 I(T-t-1)n

)

(46)

4608


Although the optimal control is designed by algorithm 3 for all cycles within the cyclewise optimization horizon, the only optimal control result of the current cycle is applied. For the next cycle, the cyclewise optimization horizon of the performance index will be moved forward, and based on the new performance index, new system parameters, and all information obtained up to now, the optimal control will be redesigned. Similar to the receding horizon LQ optimal control of a 1D system, the CWRH-2DLQ optimal control scheme gives a good compromise between the computational burden and the control performance along the cycle. In the following part of this paper, the discussions will focus on the SC-2DLQ optimal control scheme and the CWRH-2DLQ optimal control scheme. Remark 3.5. For the proposed 2DLQ optimal control schemes, the matrices {P,S,K} are independent from the input, output, and state of the process. For algorithms 1 and 3, if system matrices {A1,A2,B} are both time- and cycle-varying, then {P,S,K} will be both time- and cycle-varying and need to be computed from cycle to cycle; if {A1,A2,B} are cycle-independent, then the matrices {P,S,K} will be cycle-independent and need to be computed only once. Remark 3.6. For the proposed optimization algorithms, the major computational load results from the numerical inversion of matrix R(t,‚) + BT(t,‚)S(t + 1,)B(t,‚). Fortunately, this matrix is always an m × m PDS matrix when R is taken as a PDS matrix. Therefore, the computational load depends proportionally on the value of T and N. Furthermore, it also follows from relations 15, 32, and 45 that eqs 13, 30, and 43 can be written respectively as

P(t,k + 1) ) (A h (t,k + 1) - B h (t,k + 1)K(t,k + 1))TS(t + 1,k + 1)(A h (t,k + 1) B h (t,k + 1)K(t,k + 1)) + KT(t,k + 1)R(t)K(t,k + 1) (47) P(t,i) ) (A ˜ (t,i) - B ˜ (t,i)K(t,i))TS(t + 1,i)(A ˜ (t,i) - B ˜ (t,i)K(t,i)) + KT(t,i)R(t,i)K(t,i)

(48)

P(t,k + i) ) (A ˜ (t,k + i) - B ˜ (t,k + i)K(t,k + i))TS(t + 1,k + i)(A ˜ (t,k + i) - B ˜ (t,k + i)K(t,k + i)) + KT(t,k + i)R(t,i)K(t,k + i) (49) These equations imply that matrix P(‚,‚) is a PDS matrix if Q and R are SPDS and PDS, respectively. This property of matrix P(‚,‚) is useful for the cyclewise stability analysis that will be addressed in the next section. 4. Cyclewise Stability Analysis Stability is important when analyzing dynamic systems. All 2DLQ optimal control schemes proposed in the last section are based on the performance indices with finite horizons along both time and cycle. As the time duration of the process in every cycle is finite, the cyclewise stability is important to investigate. Cyclewise stability analysis has a close analogy to the convergence analysis of the ILC algorithm, which will be discussed in the second paper of this series. For simplicity, in this section, we will discuss the cyclewise stability analysis of the SC-2DLQ optimal control scheme and the CWRH-2DLQ optimal control scheme for the repetitive 2D processes described by the following 2D model.

∑p1:

{

x(t + 1,k) ) A1(t)x(t,k) + A2(t)x(t + 1,k - 1) + B(t)u(t,k) y(t,k) ) C(t)x(t,k)

(50)

x(0,k) ) x0,k, x(t,0) ) xt,0, t ) 0, 1, ..., T, k ) 0, 1, ... Clearly, the system matrices {A1,A2,B,C} are cycle-independent. As mentioned before, the SC-2DLQ optimal control law and the CWRH-2DLQ optimal control law will be cycle-independent and have the following forms: (i) for the SC-2DLQ optimal control scheme,

u*(t,k) ) -K(t)

(

)

(

x(t,k) x(t,k) : ) -(K1(t) l K2,t+1(t) ‚‚‚ K2,T(t)) x(|t+1 x(|t+1 ,k 1) T T ,k - 1)

)

(51)

(ii) for the CWRH-2DLQ optimal control scheme,

u*(t,k) ) -K(t)

(

)

(

x(|0t ,k) x(|0t ,k) : ) -(K1,0(t) ‚‚‚ K1,t(t) l K2,t+1(t) ‚‚‚ K2,T(t)) t+1 x(|T ,k - 1) x(|t+1 T ,k - 1)

)

(52)

Substituting (51) and (52) into model ∑p1, one has the closed-loop system as follows: (i) for the SC-2DLQ optimal control scheme, T

x(t + 1,k) ) (A1(t) - B(t)K1(t))x(t,k) + (A2(t) - B(t)K2,t+1(t))x(t + 1,k - 1) -

∑

i)t+2

(ii) for the CWRH-2DLQ optimal control scheme,

B(t)K2,i(t)x(i,k - 1)

(53)

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4609 t-1

x(t + 1,k) ) (A1(t) - B(t)K1,t(t))x(t,k) -

B(t)K1,i(t)x(i,k) + (A2(t) - B(t)K2,t+1(t))x(t + 1,k - 1) ∑ i)0 T

∑ B(t)K2,i(t)x(i,k - 1)

(54)

i)t+2

If the dynamic of closed-loop system along the cycle direction is taken into account only, system (53) and (54) can be uniformly described by the following linear time-invariant (LTI) model

Γx(|1T,k + 1) ) Θx(|1T,k) + Πx(0,k + 1)

(55)

where the constant matrices {Γ,Θ,Π} are defined as follows: (i) for the SC-2DLQ optimal control scheme,

(

(

I 0 B(1)K1(1) - A1(1) I B(2)K (2) - A1(2) 0 1 l l Γ) 0 0 0 0

0 0 I l 0 0

‚‚‚ 0 ‚‚‚ 0 ‚‚‚ 0 ‚ l ‚‚ ‚‚‚ I ‚‚‚ B(T - 1)K1(T - 1) - A1(T - 1)

A2(0) - B(0)K2,1(0) -B(0)K2,2(0) -B(0)K2,3(0) A (1) B(1)K (1) -B(1)K2,3(1) 0 2 2,2 A2(2) - B(2)K2,3(2) 0 0 Θ) l l l 0 0 0

(

(

A1(0) - B(0)K1(0) 0 Π) l 0

0 0 0 l 0 I

)

-B(0)K2,T(0) ‚‚‚ -B(1)K2,T(1) ‚‚‚ -B(2)K2,T(2) ‚‚‚ ‚ l ‚‚ ‚‚‚ A2(T - 1) - B(T - 1)K2,T(T - 1)

)

(56)

)

(57)

(58)

(ii) for the CWRH-2DLQ optimal control scheme

I 0 0 B(1)K1,1(1) - A1(1) I 0 B(2)K1,2(2) - A1(2) B(2)K1,1(2) I l l l Γ) B(T - 2)K1,1(T - 2) B(T - 2)K1,2(T - 2) B(T - 2)K1,3(T - 2) B(T - 1)K1,1(T - 1) B(T - 1)K1,2(T - 1) B(T - 1)K1,3(T - 1)

(

‚‚‚ 0 ‚‚‚ 0 ‚‚‚ 0 ‚ l ‚‚ ‚‚‚ I ‚‚‚ B(T - 1)K1,T-1(T - 1) - A1(T - 1)

A2(0) - B(0)K2,1(0) -B(0)K2,2(0) -B(0)K2,3(0) -B(1)K2,3(1) A2(1) - B(1)K2,2(1) 0 A2(2) - B(2)K2,3(2) 0 0 Θ) l l l 0 0 0

(

A1(0) - B(0)K1,0(0) -B(1)K1,0(1) -B(2)K1,0(2) Π) l -B(T - 1)K1,0(T - 1)

)

-B(0)K2,T(0) ‚‚‚ -B(1)K2,T(1) ‚‚‚ -B(2)K2,T(2) ‚‚‚ ‚ l ‚‚ ‚‚‚ A2(T - 1) - B(T - 1)K2,T(T - 1)

)

0 0 0 l 0 I

)

(59)

(60)

(61)

Now, it is clear that cyclewise stability of the closed-loop system is equivalent to stability of LTI system 55, for which we have following results Theorem 4.1. For LTI system 55, if

|Γ-1Θ| < λ < 1

(62)

4610


then the following conclusions are drawn: (i) if x(0,k) ) 0, k ) 1, 2, ..., then

|x(|1T,k)| < λkM0

(63)

(ii) if |x(0,k)| < M1, k ) 1, 2, ..., then

|x(|1T,k)| < λkM0 +

λ |Γ-1Π|M1 1-λ

(64)

Then, the closed-loop system is asymptotically stable along the cycle if the following inequality is held

P22(0,1) < P22(0,2)

(iii) if |x(0,k + 1) - x(0,k)| < M2, k ) 1, 2, ..., then

|x(|1T,k + 1) - x(|1T,k)| < λkM0 +

λ |Γ-1Π|M2 1-λ

(65)

where M0,1,2 are positive real scalars. Proof. Both (56) and (59) indicate that Γ is an invertible lower triangle matrix. The closed form solution of model 55 is given by

x(|1T,k) ) (Γ-1Θ)kx(|1T,0) +

k-1

(Γ-1Θ)iΓ-1Πx(0,k - i) ∑ i)0

(66)

For linear transformation y ) Ax, the inequality |y| e |A||x| and (62) immediately yield the following relations

|x(|1T,k)| e |Γ-1Θ|k|x(|1T,0)| +

k-1

|Γ-1Θ|i|Γ-1Π||x(0,k - i)| < ∑ i)0 λk|x(|1T,0)| +

λ

|Γ ∑ i)0

-1

Θ| |Γ Π||x(0,k - i + 1) - x(0,k - i)| < i

V(x(|1T,k)) ) xT(|1T,k)P22(0,1)x(|1T,k)

(71)

V(‚) is a positive definite as P(0,1) is a PDS matrix. If x0,k ≡ 0, according to (28) and the proof of theorem 3.3, we have the following relation

xT(|1T,k)P22(0,1)x(|1T,k) ) T 0 0 P(0,1) ) 1 x(|T,k) x(|1T,k) J[k+1:k+N](u*([0:T - 1],[k + 1:k + N])) g J[k+2:k+N](u*([0:T - 1],[k + 2:k + N])) )

( )

(

)

( ) (

)

T 0 0 P(0,2) ) 1 x(|T,k + 1) x(|1T,k + 1)

xT(|1T,k

+ 1)P22(0,2)x(|1T,k + 1) (72)

1-λ

|Γ-1Θ|k(|Γ-1Θ - I||x(|1T,0)| + |Γ-1Π||x(0,1)|) + -1

Proof. Define a quadratic function

|Γ-1Π|M1 (67)

|x(|1T,k + 1) - x(|1T,k)| e k-1

(70)

λk(|Γ-1Θ - I||x(|1T,0)| + |Γ-1Π||x(0,1)|) + λ |Γ-1Π|M2 (68) 1-λ Set M0 ) |x(|1T,0)| or M0 ) |Γ-1 Θ - I||x(|1T,0)| + |Γ-1 Π||x(0,1)|, leading to the conclusions i,ii, and iii. Remark 4.1. Equation 66 provides a clear indication of the state responses of the closed-loop system to the nonzero cyclewise initial state profile and nonzero initial states of each cycle. If condition 62 is satisfied, the state response to the nonzero initial state profile will exponentially converge to zero along the cycle direction. When k is large enough, the nonzero state responses are mainly driven by the nonzero initial condition of each cycle. Inequality 64 indicates that the bounded initial states generate a bounded state response, and inequality 65 indicates further that the fixed nonzero initial states lead to a steady-state response in a cyclewise sense. Theorem 4.1 provides stability analysis directly based on the cyclewise model of the closed-loop system. For the CWRH2DLQ optimal control scheme (N g 2), the PDS matrices P(0,i), i ) 1, ..., N, which are cycle-independent, can also be used to analyze the cyclewise stability of the closed-loop system. The result is given as follows. Theorem 4.2. Consider the CWRH-2DLQ optimal control scheme (N g 2) for 2D system ∑p1with zero initial conditions and a nonzero initial state profile and decompose the matrices P(0,i), i ) 1, 2, ..., N yielded by algorithm 3 as follows:

combining with inequality (70) gives the result

∆V(x(|1T,k)) ) xT(|1T,k + 1)P22(0,1)x(|1T,k + 1) - xT(|1T,k)P22(0,1)x(|1T,k) < xT(|1T,k + 1)P22(0,2)x(|1T,k + 1) - xT(|1T,k)P22(0,1)x(|1T,k) e 0 (73) which implies that V(‚) is a Lyapunov function and the closedloop system is asymptotically stable along the cycle Remark 4.2. Whether inequality (70) can be obtained or not is determined by the parameters of the model and the quadratic performance index, i.e., matrices Q and R and N. From (49), it can be seen that a monotonic increase of matrices R along the k axis easily satisfys condition (70). 5. Robustness Analysis It is of interest to know the robust stability of a control system when there are model-plant mismatches or model uncertainties, as the designs of control systems are usually based on simplified models. Another important issue for robustness analysis is to investigate the sensitivity of a control system to an external disturbance, which is usually unavoidable in application. For the SC-2DLQ optimal control scheme and the CWRH2DLQ optimal control scheme, cyclewise dynamic model 55 should be regarded as the nominal model with an uncertain initial state, x(0,k), to be viewed as the external disturbance. It is also noted from definitions 56-61 that matrices {Γ,Θ,Π} have linear relations with system matrices {A1,A2,B} Therefore, if the parameter uncertainties and external disturbances of the process are taken into account together, the following model with parameter uncertainties can be used to model the cyclewise dynamics of the control system.


∑cp1:

{

(Γ + ∆1)x(k + 1) ) (Θ + ∆2)x(k) + (Π + ∆3)d(k) z(k) ) Φx(k) (74)

where x(k) is the super state, using the state sequence of the process in the kth cycle as the component, d(k) represents an unknown external disturbance, including the uncertain initial condition of each cycle as the integrant, z(k) is the controlled output, {Γ,Θ,Π,Φ} are the nominal system parameters with appropriate dimensions determined by the parameters of the nominal model and control law, and {∆i}i)1,2,3 are unknown matrices, representing the model parameter uncertainties with the forms as

∆i ) EiδiFi, i ) 1,2,3

(75)

where {Ei,Fi}i ) 1,2,3 are known constant matrices indicating the structures of the uncertainties and {δi}i)1,2,3 are unknown matrices satisfying the following norm-bounded conditions

δiTδi < I, i ) 1,2,3

(76)

Clearly, cyclewise robustness analysis of the original 2D control system is equivalent to the robustness analysis of control system ∑cp1. Before giving the results, the following definitions are necessary Definition 5.1. Assume d(k) ) 0. For any nonzero initial condition x(0) and all admissible parameter uncertainties satisfying conditions (76), if the state response of system ∑cp1 satisfies

limx(k) ) 0

(77)

kf∞

then system ∑cp1 is robust asymptotically stable. Definition 5.2. If system ∑cp1 is robust asymptotically stable and, for a zero initial state and any disturbance d ∈ l2e, there exists a real scalar γ > 0 such that

|z|2e e γ|d|2e

)

P + 1F1TF1 - ΓTQΓ ΓTQE1 0, system ∑cp1 has robust H∞ performance γ under assumption 5.1 if there exist PDS matrices P and Q with the appropriate dimensions and

)

P + 1F1TF1 - ΓTQΓ ΓTQE1 0

(

)

Subject to

(78)

then the system ∑cp1 is said to have robust H∞ performance γ. It is also noted from (56) and (59) that Γ is a unit lower triangular matrix. Without a loss of generality, the following assumption is required in our discussion. Assumption 5.1. For any admissible parameter uncertainty ∆1 satisfying (75) and (76), matrix (Γ + ∆1) is nonsingular. The linear matrix inequality (LMI) technique29 plays an important role in the following results. The proofs are provided in the Appendix. Theorem 5.1. Under assumption 5.1, system ∑cp1 is robust asymptotically stable if there exist PDS matrices P and Q with the appropriate dimensions and positive scalars i > 0, i ) 1, 2 such that the following LMI conditions hold

(

positive scalars i > 0, i ) 1, 2, 3 such that the following LMI conditions hold

-ΓTQΓ 0 ΘTQ ΦT 0 -γI ΠTQ 0 < 0 QΘ QΠ -Q 0 Φ 0 0 -γI

(83)

where Φ is a specified measurement matrix for state-space eq 55. 6. Illustration In this section, the SC-2DLQ optimal control scheme and the CWRH-2DLQ optimal control scheme will be tested with a 2D simulation process to illustrate the effectiveness of the proposed control schemes and the performance of these two schemes is compared. Consider an unstable repetitive 2D process described by the following 2D model

{

x(t + 1,k) )

z(t,k) )

()

( (

1.2 1 1.2 1

) )

0 x(t,k) + 0.91 0 1 x(t + 1,k - 1) + u(t,k) 0.91 1

1 x(t,k) 1

()

(84)

t ) 0, 1, ...., T, k ) 0, 1, 2, ... where the nonzero 2D boundary conditions are assumed to be

4612


{

( ) ( )

100 100 10 x(0,k) ) 10

x(t,0) )

t ) 1, 2, ..., T (85) k ) 1, 2, ...

The parameters of the performances indices are chosen as follows (i) for the SC-2DLQ optimal control scheme

Q(t) )

(

)

100 0 , R(t - 1) ) 1, t ) 1, 2, ..., T 0 100

(86)

(ii) for the CWRH-2DLQ optimal control scheme

Q(t,i) ) 2i-1

(

)

100 0 , R(t - 1,i) ) 2i-1, 0 100 t ) 1, 2, ..., T, i ) 1, 2, ..., N (87)

In the first scenario, we set T ) 100 and assume there is no model mismatch. The system output responses for the two control schemes are shown in Figure 1, which indicates that

both control systems are stable. To compare the control performance of the two control schemes along the cycle, the sum of square error (SSE) of each cycle is calculated and plotted in Figure 2. It is clear that the SC-2DLQ optimal control scheme provides better control performance in the first cycle and smaller control error after the process enters the steady state in a cycle sense, while the CWRH-2DLQ optimal control scheme has faster convergence along the cycle. To investigate the stabilities of the two control systems, the cyclewise models of the closedloop systems are built. The eigenvalue distributions of two control models are shown in Figure 3, which indicates that both control systems are cyclewise asymptotically stable with a good stability margin. The faster convergence rate of the CWRH2DLQ control scheme lies in the fact that the eigenvalues of the resulting closed-loop system are closer to the origin. To test and compare the robustness of these two schemes, in the second scenario, the system parameters {A1,A2,B} are randomly perturbed from 20% to 180% of the nominal values. The output responses of two control schemes are shown in Figure 4, which

Figure 1. Responses of nominal closed-loop systems: (a) SC-2DLQ optimal control scheme; (b) CWRH-2DLQ optimal control scheme.


clearly indicates that the CWRH-2DLQ optimal control scheme is more robust. Corollary 5.1 can be applied to estimate the minimal upper bound of the H∞ performance. For T ) 50, the results are given in Table 1, which shows that the CWRH-2DLQ optimal control scheme can guarantee better disturbance rejection performance. 7. Conclusions The objective of this paper is to develop 2D optimal control schemes that can be used to design ILC systems in the paper of this series. In this paper, the SC-2DLQ optimal control problem and the MC-2DLQ optimal control problem have been formulated and solved for a class of repetitive processes with 2D dynamics. The resulting optimal control laws are essentially time-varying 2D state feedback controls depending on the state information of the current and last cycles. On the basis of the Figure 2. Control errors.

Figure 3. Eigenvalue distributions of cyclewise closed-loop systems: (a) SC-2DLQ optimal control scheme; (b) CWRH-2DLQ optimal control scheme.

4614


Figure 4. Responses of closed-loop systems with uncertain parameter perturbations: (a) SC-2DLQ optimal control scheme; (b) CWRH-2DLQ optimal control scheme. Table 1. Minimum Upper Bounds of H∞ Performance of Cyclewise Closed-Loop Systems SC-2DLQ optimal control scheme

CWRH-2DLQ optimal control scheme

1.74

1.55

γ*

multicycle quadratic performance index with receding horizon, the CWRH-2DLQ optimal control scheme has been proposed to balance the control performance and computational load of the MC-2DLQ optimal control scheme. The cyclewise stability and robustness analysis have been given based on the cyclewise dynamics of the resulting closed-loop system. Simulations show the effectiveness of the all proposed 2DLQ optimal control schemes. On the basis of these results, new ILC schemes will be developed in the second paper. Acknowledgment This work is supported in part by the Hong Kong Research Grant Council under Project No. 601104.

Appendix The following matrix inequalities are needed for the proofs of theorems 5.1 and 5.2. Lemma A.1.29 Assume A,E,F,Q ) QT are matrices with the appropriate dimensions. For all matrix ∆ satisfying ∆T∆ < I, there exists a PDS matrix P > 0 such that

(A + E∆F)TP(A + E∆F) - Q < 0

(A.1)

if, and only if, there is a scale > 0 and PDS matrix P > 0 such that the following LMI is satisfied

(

)

-Q + FTF ATP 0 PA -P PE < 0 0 ETP -I

(A.2)

Lemma A.229 (Schur Complement). Assume W, L, and V are matrices with the appropriate dimensions, where W and V


P - ΓTQΓ + 1-1ΓTQE1E1TQΓ + 1F1T δ1Tδ1F1 < 0 (A.14)

are PDS matrices, then

LTVL - W < 0 if, and only if,

-W LT < 0, L -V-1

)

(A.4)

(

)

From condition 76 and lemma A.2, it is seen that condition 79 implies condition A.10, which completes the proof. B. Proof of Theorem 5.2. Since condition 82 implies inequality 80, it results from theorem 5.1 that system ∑cp1 is robust stable. Now, it remains to show that system ∑cp1 has robust H∞ performance γ. Introduce a quadratic function

(A.5)

J(k) ) ∆V(x(k)) + γ-1zT(k)z(k) - γdT(k)d(k)

(

or

(A.3)

-V-1 L 0 such that

(Γ + ∆1)-TP(Γ + ∆1)-1 - Q < 0

then a sufficient condition leading to J(k) e 0 is that the following matrix inequality is satisfied

(Θ + ∆2 Π + ∆3 )TQ(Θ + ∆2 Π + ∆3 ) P - γ-1ΦTΦ 0 < 0 (A.19) 0 γI

(

)

(

( )( )) ( ( )( )) (

δ2 0 F2 0 T Q (Θ Π ) + 0 δ3 0 F 3 δ 0 F2 0 P - γ-1ΦTΦ 0 < 0 (E2 E3 ) 2 δ F3 0 0 γI 3 0 (A.20)

(Θ Π ) + (E2 E3 )

)

From lemma A.1 and the proof of theorem 5.1, it is seen that conditions 81 and 82 result in J(k) e 0, leading to N

k)0

P - ΓTQΓ - F1T δ1T E1TQΓ - ΓTQE1δ1F1 -

xT(N + 1)Px(N + 1) - xT(0)Px(0) +

F1 δ1 E1 QE1δ1F ) T

(A.18)

∑J(k) )

P - (Γ + ∆1) Q(Γ + ∆1) ) T

T

( ) )( )

i.e.

(A.11)

According to lemma A.1, the above inequality is held if, and only if, condition 80 is satisfied. It is remains to prove that condition 79 implies inequality A.10. On the basis of assumption 5.1, inequality A.10 is equivalent to

P - (Γ + ∆1)TQ(Γ + ∆1) < 0

(A.16)

On the basis of assumption 5.1 and eq 74, (A.15) can be rewritten in a matrix form

Hence

It is shown that V(‚) is a Lyapunov function if the following matrix inequality is held for all admissible parameter uncertainties satisfying norm-bounded condition 76

(A.15)

T

γ-1

P - ΓTQΓ - F1T δ1T E1TQE1δ1F + 1-1ΓTQE1E1TQΓ + 1F1T δ1Tδ1F1 - (1-(1/2)ΓTQE1 + 1(1/2)F1T δ1T) × (1-(1/2)E1TQΓ + 1(1/2)δ1F1) (A.13) where scalar 1 > 0. As Q is a PDS matrix, a sufficient condition for ensuring (A.12) is there exists a scalar 1 > 0 such that

N

∑ k)0

N

zT(k)z(k) - γ

dT(k)d(k) e 0 ∑ k)0

(A.21)

If x(0) ) 0, it is resulted from P > 0 that

γ-1

N

N

∑zT(k)z(k) - γk)0 ∑dT(k)d(k) e 0 k)0

(A.22)

4616


i.e., |z|2e e γ|d|2e, indicating that system ∑cp1 has robust H∞ performance γ. Literature Cited (1) Kaczorek, T. Two-dimensional linear system: Springer: Berlin, 1985. (2) Hinamoto, T. 2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model. IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 1993, 40 (2), 102-110. (3) Lu, W.-S. Some new results on stability and stability robustness of Fornasini-Marchesini state-space 2-D digital filters. Proceedings of IEEE Southeastcon′94. CreatiVe Technology Transfer-A Global Affair, 1994, 2125. (4) Kanellakis, A. New stability results for 2-D discrete systems based on the Fornasini-Marchesini state space model. IEE Proc. Circuits, DeVices Syst. 1994, 141 (5), 427-432. (5) Hinamoto, T. Stability of 2-D discrete systems described by the Fornasini-Marchesini second model. IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 1997, 44 (3), 254-257. (6) Du, C.; Xie, L. LMI approach to output feedback stabilization of 2-D discrete systems. Int. J. Control 1999, 72 (2), 97-106. (7) Guan, X.; Long, C.; Duan, G. Robust optimal guaranteed cost control for 2D discrete systems. IEE Proc. Control Theory Appl. 2001, 148 (5), 355-361. (8) Xie, L.; Du, C.; Soh, Y. C.; Zhang, C. H∞ and robust control of 2-D systems in FM second model. Multidimensional Syst. Signal Process. 2002, 13, 265-287. (9) Du, C.; Xie, L.; Zhang, C. H∞ control and robust stabilization of two-dimensional systems in Roesser models. Automatica 2001, 37, 205211. (10) Lam, J.; Xu, S.; Zou, Y.; Lin, Z.; Galkowski, K. Robust Output feedback stabilization for two-dimensional continuous systems in Roesser form. Appl. Math. Lett. 2004, 17, 1331-1341. (11) Bisiacco, M. New results in 2D optimal control theory. Multidimensional Syst. Signal Process. 1995, 6, 189-222. (12) Sˇ ebek, M.; Kraus, F. J. Stochastic LQ-optimal control for 2-D systems. Multidimensional Syst. Signal Process. 1995, 6, 275-285. (13) Rostan, M.; Lee, E. B. Optimal control (quadratic performance) for linear two-dimensional systems. Proceedings of the 28th Conference on Decision and Control, Tampa, Florida, 1989; pp 307-309. (14) Jagannathan, M.; Syrmos, V. L. A linear-quadratic optimal regulator for two-dimensional systems. Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, 1996; pp 4172-4177. (15) Tsai, J. S.-H.; Li, J. S.; Shieh, L.-S. Disretized quadratic optimal control for continuous-time two-dimensional systems. Trans. Circuits Syst.I: Fundam. Theory Appl. 2002, 49 (1), 116-125. (16) Liu, W.-S.; Antoniou, A. Two-dimensional digital filters; Marcel Dekker: New York, Basel, Hong Kong, 1992.

(17) Huang, T. S. Two-dimensional digital signal processing; SpringerVerlag: Berlin, New York, 1981. (18) Rogers, E.; Owens, D. H. Stability analysis for linear repetitiVe processes; Springer-Verlag: Berlin, Heidelberg, 1992. (19) Galkowski, K.; Rogers, E.; Owens, D. H. New 2D models and a transition matrix for discrete linear repetitive processes. Int. J. Control 1999, 72 (15), 1365-1380. (20) Galkowski, K.; Rogers, E.; Xu, S.; Lam, J.; Owens, D. H. LMIs a fundamental tool in anaysis and controller design for discrete linear repetitive processes. Trans. Circuits Syst.-I: Fundam. Theory Appl. 2002, 49 (6), 768-778. (21) Galkowski, K.; Paszke, W.; Rogers, E.; Xu, S.; Lam, J.; Owens, D. H. Stability and control of differential linear repetitive processes using an LMI Setting. IEEE Trans. Circuits Syst.-II: Analog Digital Signal Process. 2003, 50 (9), 662-666. (22) Galkowski, K.; Paszke, W.; Sulikowski, B.; Rogers, E.; Xu, S.; Lam, J.; Owens, D. H. Stability and control of a physical class of 2D continuous-discrete linear systems using an LMI setting. Proceedings of the American Control Conference, Denver, Colorado, 2003; pp 5058-5063. (23) Geng, Z.; Carroll, R.; Xie, J. Two-dimensional model and algorithm analysis for a class of iterative learning control systems. Int. J. Control 1990, 52 (4), 833-862. (24) Kurek, J. E.; Zaremba, M. B. Iterative learning control synthesis based on 2-D system theory. IEEE Trans. Autom. Control 1993, 38 (1), 121-125. (25) Shi, J.; Gao, F.; Wu, T.-J. Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system. J. Process Control 2005, 15, 907-924. (26) Rogers, E.; Owens, D. H. 2D systems theory and applications - a maturing area. Int. Conf. Control 1994, 1, 63-69. (27) Rogers, E.; Galkowski, K.; Owens, D. H.; Amann, N. 2D systems theory - a common approach to problems in circuits, control and signal processing. UKACC IEE International Conference on Control′96, 1996; pp 965-970. (28) Yamada, M.; Xu, L.; Saito, O. Iterative control of three link manipulator using 2D system control approach. SICE Annual Conference in Fukui, Japan, August, 2003; pp 1880-1885. (29) Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V. Linear matrix inequalities in system and control theory; SIAM: Philadelphia, PA, 1994. (30) Gahinet, P.; Nemirovskii, A.; Laub, A. J.; Chilali, M. The LMI control toolbox. In Proceedings of the 33rd Conference on Decision and Control, Lake Buena, Vista FL, 1994; pp 2038-2041.

ReceiVed for reView November 22, 2005 ReVised manuscript receiVed March 23, 2006 Accepted March 29, 2006 IE051297I

From Two-Dimensional Linear Quadratic Optimal Control to Iterative

Recommend Documents