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Ind. Eng. Chem. Res. 2006, 45, 9050-9060
Iterative Learning Fault-Tolerant Control for Batch Processes Youqing Wang,†,‡ Jia Shi,‡ Donghua Zhou,† and Furong Gao*,‡ Department of Automation, Tsinghua UniVersity, Beijing, 100084, People’s Republic of China, and Department of Chemical Engineering, Hong Kong UniVersity of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
This paper develops an iterative learning reliable control (ILRC) scheme for batch processes with unknown disturbances and actuator faults. To develop the iterative learning control design, the batch process is transformed to, and treated as, a two-dimensional Fornasini-Marchsini (2D-FM) model. The relevant concepts of fault-tolerance along two-dimensional (2D) axes are introduced. The proposed control law can guarantee the closed-loop convergency along both the time and the cycle directions to satisfy H∞ performance, even with unknown disturbances and actuator faults. For performance comparison, a traditional fault-tolerant control law is also developed by considering the batch process in each cycle as a continuous process. Conditions for the existence of the proposed ILRC scheme are given as linear matrix inequalities. Applications to injection velocity control show that the proposed ILRC achieve the design objectives well, with performance improvement along both the time and cycle directions. 1. Introduction Studies on batch process control can be dated back to 1930s.1 The last 10 years have witnessed a resurgent interest in batch processing technologies that is driven by the business of manufacturing.2 Batch processes are the preferred manufacturing choice for low-volume and high-value products such as specialty chemicals, pharmaceuticals, consumer products, and bioproducts. Batch processing is widely used in chemical industries, including food, beverage, pharmaceuticals, agricultural chemicals, paints, flavors, polymers, and so on.3 The demand for productivity leads increasingly for the chemical plant to operate under challenging conditions, which consequently exposes the possibility of system failures. A chemical process typically has a large number of measurements and actuators; if a fault is not detected promptly with a proper corrective action, it will degrade the process performance, and in serious cases, result in safety problems for the plant and personnel. Fault detection and diagnosis (FDD) can detect and estimate the faults, whereas fault-tolerant control (FTC) is capable of maintaining the performance of the closed-loop system at an acceptable level in the presence of faults. Because of this, FDD and FTC, using the model-based analysis redundancy, have received significant attention;4-6 however, most of them are designed for continuous processes. Batch processes are also suffering from this problem. FDD studies on batch processes have began to emerge recently,7,8 whereas, to the best knowledge of the authors, there are no reported studies on FTC for batch processes using analysis redundancy. If a batch process within each cycle is viewed and treated as a continuous process, the traditional FTC method can be used directly for batch processes, as shown in section 3 later in this paper. The use of the traditional FTC design has two major problems. First, only asymptotical behavior of the closed-loop system is considered in the traditional FTC design. As each cycle of a batch process has finite time only, the control * Corresponding author: Telephone: +852-2358 7139. Fax: +8522358 0054. E-mail:
[email protected]. † Department of Automation, Tsinghua University. ‡ Department of Chemical Engineering, Hong Kong University of Science and Technology.
performance in such a finite time cannot be guaranteed by the traditional FTC. Second, with the traditional design method, the control performance cannot be improved from cycle to cycle, despite the batch process having a certain repetitive nature. Reliable control is a popular FTC method.9-11 Recently, the study of reliable control has received considerable attention, because of the growing demands on reliability. The main task of this study is to design a fixed controller such that the closedloop system can maintain stability and performance, not only when all components are operational, but also in case of faults. To exploit the repetitive nature of batch processes, iterative learning control (ILC) has been used widely. ILC is motivated to mimic human learning and it was first mathematically formulated by Arimoto, Kawamura, and Miyazaki.12 In ref 13, a robust ILC was proposed for an injection mold process with uncertain initial resetting and disturbances, and the output can track an arbitrary bounded reference. A systematic method for the analysis and design of ILC systems was presented in ref 14, in regard to the frequency domain. A model-based ILC strategy for the tracking control of product quality in batch processes was proposed in ref 15. Shi, Gao, and Wu proposed a general design framework for ILC of batch process: the batch process under ILC is modeled as a two-dimensional (2D) system and the design of a robust ILC for a batch process is transformed to a robust stabilization problem of the 2D system.16-18 Based on the aforementioned framework, in this paper, the batch process is transformed into a 2D Fornasini-Marchsini (2D-FM) model, and ILC and reliable control are integrated to result in iterative learning reliable control (ILRC) for batch processes. The objective of a reliable control scheme is to guarantee outputs and track performance, not only when all components are operational but also in case of admissible faults. An iterative learning scheme improves the tracking performance in the batch direction, taking advantage of the repetitive nature of batch processes. The proposed closed-loop system has good robust H∞ performances to unknown disturbances. New design concepts of FTC in 2D space are introduced. Sufficient conditions for the proposed fault tolerance are expressed as linear matrix inequalities (LMIs). Finally, the feasibility and effectiveness of the proposed method are demonstrated with injection velocity control.
10.1021/ie060726p CCC: $33.50 © 2006 American Chemical Society Published on Web 11/08/2006
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9051
This paper is organized as follows. The problem formulation is introduced in section 2. In section 3, a traditional reliable controller (TRC) is designed for the batch process. The main results of this paper are presented in section 4: the batch process is transformed to an equivalent 2D-FM system, an ILRC scheme is designed, and an algorithm for performance optimization is proposed. In section 5, the feasibility and effectiveness of the proposed scheme are demonstrated through injection velocity control. Finally, the conclusions of this paper are given in section 6. Throughout this paper, the following notations are used. Rp represents Euclidean p-space with the norm denoted by |•|. Rp×q is the set of all p × q real matrixes; for any matrix M ∈ Rp×p, M > 0 (M < 0) means M is a positive (negative) definite matrix. MT represents the transpose of matrix M. I and 0 respectively denote the identity matrix and the zero matrix with appropriate dimensions. The notation “*” represents the transposed elements in the symmetric position. For a 2D signal w(t,k), if |w(•,k)|T-2e
Remark 2. The failure model described by eq 2 is widely adopted.10,19 The parameter Ri is unknown but assumed to vary within a known range. If Ri ) R j i, then it corresponds to the normal case uFi ) ui. When Ri ) 0, it covers the outage case. If Ri > 0, it corresponds to a partial failure case, i.e., partial degradation of the actuator. Denote
< ∞ for any integers N1,N2 > 0, then w(•,•) ∈ l2D-2e. VΩ(ξ) is defined as VΩ(ξ) ) ˆ ξTΩξ, where ξ is a vector and Ω is a symmetric matrix with appropriate dimensions. 1
2
ΣBPF:
j 2, ..., R j m] R j ) diag[R j 1, R
(4b)
R ) diag[R1, R2, ..., Rm]
(4c)
R ) diag[R1, R2, ..., Rm]
(4d)
{
x(t + 1,k) ) Ax(t,k) + BRu(t,k) + w(t,k) y(t,k) ) Cx(t,k)
}
(5)
The control objective is to determine a fault-tolerant control law such that the output of the process tracks a given trajectory, yr(t), as closely as possible, even with actuator failures. Define
e(t,k) ) ˆ yr(t) - y(t,k)
2. Problem Formulation A batch process, which involves repetitively performing a given task over a period of time (called a batch/cycle), can be described by the following state-space model:
{
x(t + 1,k) ) Ax(t,k) + Bu(t,k) + w(t,k) y(t,k) ) Cx(t,k) for x(0,k) ) x0,k; t ) 0, 1, ...,T; k ) 1, 2, ...
ΣBP:
(4a)
Hence, a batch process with actuator faults may be described by
N ) ˆ x∑t)0 |w(t,k)|2 < ∞ for any integer N > 0, then it is said that w(•,k) ∈ lT-2e; similarly, if |w(•,•)|2D-2e ) ˆ N N ∑k)0 |w(t,k)|2 x∑t)0
uF ) [uF1 , uF2 , ..., uFm]T
}
The following notations and definitions are introduced:
β ) diag[β1, β2, ..., βm]
(7a)
β0 ) diag[β10, β20, ..., βm0]
(7b)
βi )
R j i + Ri 2
(for i ) 1, 2, ..., m)
(8a)
βi0 )
R j i - Ri R j i + Ri
(for i ) 1, 2, ..., m)
(8b)
(1) with
where t denotes time, k denotes the batch index, and x0,k is the time-wise initial state of the kth batch run. (Because of the fact that the initial state of each batch can usually be reset in many applications, it is assumed in this paper that x0,k ≡ x0.) The expressions x(t,k) ∈ Rn, y(t,k) ∈ Rl, and u(t,k) ) [u1(t,k), u2(t,k), ..., um(t,k)]T ∈ Rm represent, respectively, the state, output, and input of the process at time t in the kth batch run. The term w(t,k) represents the disturbance resulted from unmodeled dynamics or external disturbance, and {A,B,C} are the system matrixes with appropriate dimensions. Remark 1. Because {A,B,C} are constant matrixes, eq 1 can only describe linear processes; however, batch processes are nonlinear in many cases. In ref 15, a linear perturbation model is proposed to describe nonlinear batch processes and an ILC strategy is designed for the nonlinear processes based on this linear model. Similarly, the proposed method in this paper might be used in nonlinear batch processes. Designing iterative learning fault-tolerant control law for batch processes based on nonlinear model is an open problem, which is also our future work. For control input ui(t,k) (where i ) 1, 2, ..., m), let uFi (t,k) denote the signal from the actuator that has failed. The following failure model is adopted here:
uFi (t,k) ) Riui(t,k)
(for i ) 1, 2, ..., m)
(2)
ji 0 e R i e Ri e R
(for i ) 1, 2, ..., m)
(3)
where The terms Ri (Ri e 1) and R j i (R j i g 1) are known scalars.
(6)
From eqs 4 and 8, an unknown matrix R0 exists, such that
R ) (I + R0)β
(9)
|R0| e β0 e I
(10)
and
where
ˆ diag[R01, R02, ..., R0m] R0 ) and
ˆ diag[|R01|, |R02|, ..., |R0m|] |R0| ) 3. Traditional Reliable Control (TRC) This traditional reliable controller is designed here for comparison with a controller that will be introduced later. It is well-known that a controller that contains a tracking error integral action can effectively eliminate the steady-state tracking error. To obtain a robust tracking controller with state and tracking error integral feedback, we introduce the following dynamic model:
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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006
ΣE: xe(t + 1,k) ) xe(t,k) + e(t,k)
regarded as an optimization variable, the reliable tracking controller design problem can be redefined as
) xe(t,k) + yr(t) - y(t,k) ) xe(t,k) + yr(t) - Cx(t,k) where
(20)
X,Y,
The problem described by expression 20 can be solved using the LMI optimization toolbox in MATLAB.
xe(0,k) ≡ 0 and xe(t,k) denotes the tracking error integral. The combination of the extended model ΣE with the batch process model ΣBPF leads to an augmented model:
ΣBP-E: X1(t + 1,k) ) A1X1(t,k) + B1Ru(t,k) + W1(t,k) (12)
4. Iterative Learning Reliable Control (ILRC) 4.1. Equivalent Two-Dimensional (2D) Representation. For the batch process that is described by eq 1, introduce an ILC law with the form
ΣILC: u(t,k) ) u(t,k - 1) + r(t,k) (for u(t,0) ) 0, t ) 0, 1, 2, ..., T) (21)
where
X1(t,k) )
min γ (subject to expression 17)
(11)
[ ] x(t,k) xe(t,k)
(13)
(14)
where u(t,0) is the initial profile of iteration and r(t,k) ∈ Rm is called the updating law of the ILC. The objective for ILC design is to determine the updating law r(t,k) such that y(t,k) tracks yr(t). Define
Consider the augmented system described by eq 12 with the following state-feedback tracking controller:
δt(f(t,k)) ) f(t,k) - f(t - 1,k), δk(f(t,k)) ) f(t,k) - f(t,k - 1) (22)
and
A1 )
[ ]
[]
[ ]
w(t,k) A 0 B , B1 ) , W1(t,k) ) yr(t) -C I 0
u(t,k) ) K1X1(t,k) ) K11x(t,k) + K12xe(t,k)
(15)
where
K1 ) [K11 K12 ] ∈ Rm×(n+l)
δk(x(t + 1,k)) ) Aδk(x(t,k)) + BRr(t,k) + δk(w(t,k))
ΣBP-E-C: X1(t + 1,k) ) (A1 + B1RK1)X1(t,k) + W1(t,k)
]
Ω 0 -Ω ΩAT1 + YTβBT1 Ω Tβ 2 T 0 I * -Ω + B1β0 B1 0 * * -I 0 0 0 and γ > 0 such that the following LMI holds:
X12(t,k) e γVΩ ∑ t)0
(23)
e(t + 1,k) ) e(t + 1,k - 1) - Cδk(x(t + 1,k))
The closed-loop augmented system is given by
[
It can be derived from the model that is described by eq 1, the ILC law that is described by eq 21, and eq 6 that
(19)
The proof is giVen in Appendix A.1. Notice that xe(0,k) ≡ 0; if x(0,k) ) 0, we get |X1(•,k)|T-2e e γ|W1(•,k)|T-2e. Hence, γ denotes the H∞ norm |TX1W1|∞ of the transfer function from W1 to X1. Because γ can be further
(24)
From eqs 23, 24, and 11, we obtain an augmented model:
{
Σ2D-BP: X2(t + 1,k) ) A21X2(t,k) + A22X2(t + 1,k - 1) + B2Rr(t,k) + C2δk(w(t,k)) Z(t,k) ) ˆ e(t,k) ) GX2(t,k)
}
(25)
where
[ ]
δk(x(t,k)) X2(t,k) ) e(t,k) xe(t,k)
(26)
A 0 0 A21 ) -CA 0 0 0 I I
(27a)
[
[ ] [ ]
]
0 0 0 A22 ) 0 I 0 0 0 0
(27b)
B B2 ) -CB 0
(27c)
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9053
[]
I C2 ) C 0
(27d)
G ) [0 I 0 ]
(27e)
Model Σ2D-BP is a typical 2D-FM model20 with uncertain perturbations. Because this model equivalently represents the dynamical behavior of the tracking error of the system that is described by eq 5, it is called the equivalent 2D tracking error model of system 5. Therefore, it is clear that the design of the updating law r(t,k) for system 5 is equivalent to the design of a fault-tolerant control law for the equivalent 2D tracking error model Σ2D-BP. Design a control law as follows:
[
r(t,k) ) K2
]
[
X2(t,k) X (t,k) ) ˆ [K21 K22 ] 2 X2(t + 1,k - 1) X2(t + 1,k - 1)
) K21X2(t,k) + K22X2(t + 1,k - 1)
]
(28)
The closed-loop 2D-FM system then is given by
{
Σ2D-BP-C: X2(t + 1,k) ) (A21 + B2RK21)X2(t,k) + (A22 + B2RK22)X2(t + 1,k - 1) + C2δk(w(t,k)) (29) Z(t,k) ) ˆ e(t,k) ) GX2(t,k) For 2D system Σ2D-BP-C, the state evolves along two axes, which are called the T-axis and the K-axis, respectively. Obviously, the boundary conditions of the 2D system should be of two dimensions, which are denoted by
{
(for t ) 1, 2, ...) X2(t,0) ) Xt,0 2 0,k X2(0,k) ) X2 (for k ) 1, 2, ...)
(30)
0,k the T-boundary. Here, we call Xt,0 2 the K-boundary and X2 To design the fault-tolerant control law and analyze the robust stability of the closed-loop system, some definitions are introduced. Definition 1. For any bounded boundary conditions {Xt,0 2 , 0,k X2 } and any admissible actuator faults that satisfy eq 3, if the state of the 2D closed-loop system Σ2D-BP-C satisfies
lim X2(t,k) ) 0
t,kf∞
(31)
then the control law that is defined by eq 28 is called a 2Dfault-tolerant control law for system Σ2D-BP and the closedloop system Σ2D-BP-C is called a 2D-fault-tolerant system. The state of system Σ2D-BP-C evolves along two axes and 2D fault tolerance only describes the fault tolerance of the system in the 2D sense. To investigate the fault tolerance for the system along the two axes separately, the following concepts are introduced. Definition 2. Assume that the K-boundary Xt,0 2 is zero. For , any integer N > 0, and any admissible any T-boundary X0,k 2 actuator faults that satisfy eq 3, if the state of the 2D closedloop system Σ2D-BP-C satisfies N
∑|X2(t,k)| ) 0 tf∞ k)1
lim
(32)
then the control law that is defined by eq 28 is called a T-faulttolerant control law and the 2D system Σ2D-BP-C is called a T-fault-tolerant system. Definition 3. Assume that the T-boundary X0,k 2 is zero. For any T-boundary Xt,0 2 , any integer N > 0, and any admissible actuator faults that satisfy eq 3, if the state of the 2D closedloop system Σ2D-BP-C satisfies N
lim
∑|X2(t,k)| ) 0
(33)
kf∞ t)1
then the control law that is defined by eq 28 is called a K-faulttolerant control law and the 2D system Σ2D-BP-C is called a K-fault-tolerant system. Remark 3. In the 2D sense, traditional fault-tolerant control is a T-fault-tolerant control law, not a K-fault-tolerant control law. Hence, the control performances cannot improve from cycle to cycle. Batch processes have a good repetitive nature; hence, the K-fault tolerance should be emphasized. To describe the sensitivity of the controlled output to the disturbance, the following definition is extended from the robust control theory of a one-dimensional (1D) system. Definition 4.21 For a scalar γ > 0, the 2D-FM system Σ2D-BP-C is considered to have robust H∞ performance γ if it is asymptotical stable and, for zero boundary conditions and any disturbance δk(w(t,k)) ∈ l2D-2e, the controlled output of the system satisfies
|Z|2D-2e < γ |δk(w)|2D-2e
(34)
The robust H∞ performance γ indicates the maximum sensitivity of the controlled output to the disturbance. A smaller γ value indicates better disturbance rejection performance. For controller design, γ should be minimized. Lemma 1. The 2D closed-loop system Σ2D-BP-C is 2D-faulttolerant if there are a function V(•) and a scalar F > 1 that satisfy the following conditions: (a) V(x) g 0 for ∀ x ∈ Rn, and V(x) ) 0 T x ) 0; (b) V(x) f ∞ as |x| f ∞; (c) For any boundary conditions and any admissible actuator faults that satisfy eq 3,
V(x(t,k)) < F-1 ∑ V(x(t,k)), ∑ t+k)T +K +i+1 t+k)T +K +i 0
0
T0eteT0+i K0ekeK0+i
0
0
T0eteT0+i K0ekeK0+i
∀ T0 > 0, K0 > 0, i > 0 (35)
where the maximum value of F that satisfies eq 35 is called the 2D conVergence index (2D-CI) of the system. Proof. In ref 18, it has been proVen that system Σ2D-BP-C is 2D robust asymptotically stable under conditions (a), (b), and (c). From the definition of 2D robust asymptotical stability20 and Definition 1, we determine that system Σ2D-BP-C is 2Dfault-tolerant for all admissible actuator faults that satisfy eq 3. Similar results can be obtained for T-fault tolerance and K-fault tolerance. Lemma 2. The 2D closed-loop system Σ2D-BP-C is T-faulttolerant if there are a function V(•) and a scalar λ1 > 1 that satisfy the following conditions: (a) V(x) g 0 for ∀ x ∈ Rn, and V(x) ) 0 T x ) 0; (b) V(x) f ∞ as |x| f ∞; (c) For any T-boundary X0,k 2 , any integer N > 0, a zero K-boun, and any admissible actuator faults that satisfy eq 3, dary Xt,0 2
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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 N
N
∑V(x(t + 1,k)) < λ1-1k)1 ∑V(x(t,k)), k)1
∀t>0
(36)
where the maximum value λ1 that satisfies eq 36 is called the T conVergence index (T-CI) of the system. Lemma 3. The 2D closed-loop system Σ2D-BP-C is K-faulttolerant if there are a function V(•) and a scalar λ2 > 1 that satisfy the following conditions: (a) V(x) g 0 for ∀ x ∈ Rn, and V(x) ) 0 T x ) 0; (b) V(x) f ∞ as |x| f ∞; any integer N > 0, a zero (c) For any K-boundary T-boundary X0,k , and any admissible actuator faults that satisfy 2 eq 3, Xt,0 2 ,
N
N
V(x(t,k + 1)) < λ2 ∑V(x(t,k)), ∑ t)1 t)1 -1
∀k>0
(37)
where the maximum value λ2 that satisfies eq 37 is called the K conVergence index (K-CI) of the system. Remark 4. The convergence indexes 2D-CI, T-CI, and K-CI quantify the 2D-fault tolerance, T-fault tolerance, and K-fault tolerance, respectively. A larger 2D-CI (T-CI, K-CI) value indicates better 2D-fault tolerance (T-fault tolerance, K-fault tolerance) and faster convergence along the 2D-axis (T-axis, K-axis). Note that 2D-CI (T-CI, K-CI) is only a possible lower bound of the real convergence rate along the 2D-axis (T-axis, K-axis). 4.2. Reliable Controller Design and System Structure. In this section, we will design a reliable updating law r(t,k) such that the closed-loop system Σ2D-BP-C is 2D-fault-tolerant with satisfied convergence indices. The results are as follows. Theorem 2. Assume δk(w(t,k)) ≡ 0. For giVen scalars λ1,λ2 > 1, if there exist positiVe definite matrixes Ω,S1,S2 ∈ R(n+2l)×(n+2l), matrixes Y1,Y2 ∈ Rm×(n+2l), and a scalar > 0, such that the following LMIs hold:
[
]
-S1
0 ΩAT21 + YT1 βBT2 YT1 β * -S2 ΩAT22 + YT2 βBT2 YT2 β 1, if there exist positiVe definite matrixes Ω,S1,S2 ∈ R(n+2l)×(n+2l), matrixes Y1,Y2 ∈ Rm×(n+2l), and scalars ,γ > 0 such that the following matrix inequalities hold:
]
0 ΩAT21 + YT1 βBT2 YT1 β ΩGT 0 0 * -S2 ΩAT22 + YT2 βBT2 YT2 β 0 2 T C 0 * * -Ω + B2β0 B2 0 2 1 are specified, the LMIs given as eqs 38 and 41 can be solved using the LMI optimization toolbox in MATLAB. The design objective is such that λ1,λ2 are as large as possible. It is reasonable to regard the two indices as the design parameters to be optimized. There are many methods to optimize λ1,λ2; for example, we can introduce a cost function CF(λ1,λ2) of λ1 and λ2, and then solve the following problem:
max
X,S1,S2,Y1,Y2,,γ
CF(λ1,λ2)
(subject to eqs 38 and 39, or eqs 41 and 42) (45) There are many choices for the cost function, e.g. CF(λ1,λ2) ) |(λ1 λ2)T|.
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9055
The problem that is described in eq 45 is not easy to solve. A easier method is to assume that λ ) ˆ λ1 ) λ2 and solve the following problem:
max
X,S1,S2,Y1,Y2,,γ
Using fixed-step or variable-step search algorithms, the problem that is described by eq 46 can be solved. For the LMI that is described by eq 41, γ > 0 can be further regarded as an optimization variable, and the design objective is such that γ is as small as possible. Solve eq 46 and get λ*, then let λ1 ) λ2 ) λ* in the LMI that is described by eq 42. Solve the following problem:
min
X,S1,S2,Y1,Y2,
γ
(subject to eqs 41 and 42)
(47)
λ* K21 K22
Algorithm 1 solve the problem described by eq 46 and get λ* let λ1 ) λ2 ) λ* and solve the problem described by eq 47 using eq 43, the control gains can be obtained
5. Illustration Injection molding, which is a batch process, consists of three main stages: filling, packing, and cooling.13 For the filling stage, the injection velocity, which is a key process variable, should be controlled to follow a given profile to ensure product quality. In this section, the proposed ILRC scheme is applied to injection velocity control to demonstrate the feasibility and effectiveness of the proposed design. The injection velocity response to the proportional valve has been identified as an autoregressive model:16
P(z) )
1.69z + 1.419 z2 - 1.582z + 0.5916
(48)
The state-space representation of the model that is described by eq 48 is introduced as
[
]
[]
1.582 -0.5916 1 x(t,k) + u(t,k) + w(t,k) 1 0 0 y(t,k) ) [1.69 1.419 ]x(t,k), 0 e t e 200, k ) 1, 2, ... (49)
x(t + 1,k) )
value 1.058 [-2.1577 0.6573 0.0278 0.0278] [0 0 0.2947 0]
To evaluate the tracking performance, introduce the following performance index:
x∑ 200
DT(k) ) ˆ
e2(t,k)
(51)
t)1
The smaller DT(k) value indicates the better tracking performance in the kth cycle. Solving the problem that is described by eq 20, we obtain that
K1 ) [-2.7527 0.1785 0.3375 ]
The problem that is described by eq 47 can be solved using the LMI optimization toolbox in MATLAB. In summary, the following algorithm can be used to optimize λ1, λ2, and γ.
{
parameter
λ
(subject to eqs 38 and 39, or eqs 41 and 42) (46)
step1 step2 step3
Table 1. Design Parameters in Case 1
where w(t,k) is introduced to denote the unknown disturbance. The initial value of the states is assumed to be x(0,k) ≡ [0 0]T for all k. Assume that there exists an unknown actuator fault R; however, we know that 0.8 ) R eR e R j ) 1. Using eq 8, we obtain that β ) 0.9, β0 ) 0.1. In this paper, the set-point profile takes the following form:
yr(t) ) 15
(for 1 e t < 100)
(50a)
yr(t) ) 30
(for 100 et e200)
(50b)
and γ* ) 5.1581. This control gain will be used in the following simulations. The proposed schemes are illustrated with different cases of fault and disturbance. The faults are assumed to occur after Cycle 11 in all cases. Case 1. Constant Fault and Repetitive Disturbance. In this case, it is assumed that R ≡ 7 (this information is not used in the controller design) and
w(t,k) ) 0.1 × [sin(t) sin(t) ]T Hence, δk(w(t,k)) ≡ 0. Use a fixed-step search algorithm with a search step of 0.001 to solve the problem that is described by eq 46. The design results are shown in Table 1. Using these design parameters, the simulation results are shown in Figures 2 and 3. Figure 2 gives the output responses of TRC and ILRC. Because the disturbance is repetitive, before fault occurs, the closed-loop process under TRC is completely repetitive without any improvement from cycle to cycle, as shown in Figure 2a; after fault occurs, the tracking performance declines, as shown in Figure 2b. Under ILRC, the output responses for Cycles 1, 10, 11, and 20 are given in Figures 2c, 2d, 2e, and 2f, respectively. Figure 2 indicates that the tracking performance of ILRC improves in the cycle direction. Because of the fault, the tracking performance experiences degradation after Cycle 11; however, the tracking performance can achieve a perfect level again some cycles later. This can be illustrated clearly in Figure 3, where the values of DT for different cycles are presented. It is clear that the tracking performance of ILRC outperforms TRC after Cycle 2. Case 2. Time-Varying Fault and Nonrepetitive Disturbance. It is assumed that
R ) 0.9 + 0.1 sin(t) and
w(t,k) ) sin(t + φ(k)) × [0.1 0.1 ]T in this case, where φ(k) is assumed to be a random variable that is uniformly distributed within an interval [0,2π]. Obviously, δk(w(t,k)) * 0. Using Algorithm 1, the design parameters are obtained in Table 2. The control performance comparison is illustrated in Figure 4. Again, it shows that the proposed ILRC can improve the control performance, even with a certain degree of timevarying fault and nonrepetitive disturbance.
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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006
Figure 2. Output responses in case 1: (a, b) under TRC; (c, f) under ILRC. Table 2. Design Parameters in Case 2 parameter λ* γ* K21 K22
value 1.058 5 [-2.1592 0.6573 0.0264 0.0264] [0 0 0.2957 0]
In the following case, ILRC will be compared with a typical ILC controller. Case 3. Comparison between ILRC and Typical ILC. The fault and disturbance are same as that in case 1. There are many methods to design ILC. In this simulation, the controller with extended integral feedback proposed in ref 17 is taken as an example. Choosing λ ) 0.5 and using Algorithm 4.1 in ref 17, we obtain that
K ) [-2.2349 0.5927 0.2478 0.4376 ] Figure 3. Tracking performances (DT) in case 1.
In summary, TRC can guarantee control performance improvement only along the time direction, and, thus, TRC is a T-fault-tolerant control law; however, ILRC has control performance improvement not only along the time direction but also along the cycle direction, so ILRC is both T-fault-tolerant control law and K-fault-tolerant control law.
The output responses for Cycles 10 and 20 are given in Figures 5a and b, respectively. Figure 5 indicates that the control performance has an obvious degradation after a fault occurs. This phenomenon can be observed much clearer in Figure 6, which shows the performance decrease in the cycle direction after a fault occurs. Therefore, the ILC proposed in ref 17 is unable to accommodate the fault.
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9057
Figure 4. Tracking performances (DT) in case 2.
Figure 6. Tracking performances (DT) of typical ILC.
have control performance improvement not only along the time direction but also along the cycle direction. The results on injection velocity control have clearly illustrated the feasibility and effectiveness of the proposed design. Acknowledgment This work was supported in part by Hong Kong Research Grant Council (under Project No. 601104), NSFC (through Grant Nos. 60574084, 60434020), the Control Science and Engineering of Zhejiang Provincial “Key in Key” Disciplines (No. Zz050203), and the National 973 Program of China (through Grant No. 2002CB312200). Appendix The following matrix inequalities are needed for the following proof. Lemma A.1.22 Assume X and Y are matrixes or vectors with appropriate dimensions. For any positive definite matrix R > 0 with appropriate dimensions, the following inequality holds:
XY + YTXT e XRXT + YTR-1Y
(52)
Choose R ) I, where > 0; then,
XY + YTXT e XXT + -1YTY
(53)
Lemma A.2 (Schur Complement).22 Assume that W, L, and V are given matrixes with appropriate dimensions, where W and V are positive definite matrixes; then,
LTVL - W < 0 if and only if Figure 5. Output responses under typical iterative learning control (ILC): (a) output of cycle 10 under typical ILC and (b) output of cycle 20 under typical ILC.
6. Conclusion By integrating an iterative learning control with a reliable control, an iterative learning reliable control (ILRC) scheme has been proposed for a batch process. The process has been transformed to a two-dimensional Fornasini-Marchsini (2DFM) model, based on which relevant concepts on fault-tolerant control design for a batch process have been presented. Compared to traditional reliable control, the proposed ILRC can
[
]
(55)
[
]
(56)
-W LT 0 such that
[
(57)
]
From eq 16, we haVe
-T4 (T1 + T2βT3)T T3Tβ * -P-1 + T2β02TT2 0 < 0 * * -I
Proof. From eq 9, we haVe
(58)
(T1 + T2RT3) P(T1 + T2RT3) - T4 ) T
(T1 + T2βT3 + T2R0βT3)TP(T1 + T2βT3 + T2R0βT3) - T4 Using Lemma A.2, we obtain the following expression for the inequality giVen in eq 57:
[
-T4
(T1 + T2βT3 + T2R0βT3)T
(T1 + T2βT3 + T2R0βT3)
-P-1
]
0 are positiVe definite matrixes, all functions Vp(•), VQ1(•), and VQ2(•) satisfy conditions (a) and (b) in Lemma 1, 2, and 3, respectiVely. Because δk(w(t,k)) ≡ 0, we haVe
] []
[ ]
TT3 β βT [ 3 0] 0 From eqs 64 and 66, we obtain
0 + T β02[0 TT2 ] + 2
[
[ ] ]
VP(X2(t + 1,k)) < VQ1(X2(t,k)) + VQ2(X2(t + 1,k - 1)) (67)
T
T3 β βT [ 3 0] 0 (60)
[
]
-T4 + -1TT3 β2T3 (T1 + T2βT3)T 0 such that
(59)
From eq 10 and Lemma A.1, we can determine that, for any > 0,
-T4
VP(X1(t + 1,k)) - VP(X1(t,k)) + γ-1X12(t,k) - γW12(t,k) < 0 (63)
holds for all admissible actuator faults that satisfy eq 3, where P ) Ω-1 > 0. Hence,
Without loss of generality, suppose F ) min{λ1,λ2} ) λ1. It results from eq 65 that Q1 < F-1P - Q2, which leads to
VP(X2(t + 1,k)) < F-1VP(X2(t,k)) - VQ2(X2(t,k)) + VQ2(X2(t + 1,k - 1)) (68) Thus, for any integers T0, K0, i > 0, the following inequalities hold:
VP(X2(T0 + 1,K0 + i)) < F-1VP(X2(T0,K0 + i)) VQ2(X2(T0,K0 + i)) + VQ2(X2(T0 + 1,K0 + i - 1)) VP(X2(T0 + 2,K0 + i - 1)) < F-1VP(X2(T0 + 1,K0 + i - 1)) - VQ2(X2(T0 + 1,K0 + i - 1)) + VQ2(X2(T0 + 2,K0 + i - 2)) l VP(X2(T0 + i,K0 + 1)) < F-1VP(X2(T0 + i - 1,K0 + 1)) VQ2(X2(T0 + i - 1,K0 + 1)) + VQ2(X2(T0 + i,K0)) The sum of these inequalities leads to the following result:
∑
t+k)T0+K0+i+1 T0eteT0+i K0ekeK0+i -1 0
0
T0eteT0+i K0ekeK0+i
< F-1
VP(X2(t,k)) - VQ2(X2(T0,K0 + i)) +
VQ2(X2(T0 + i,K0)) - F-1VP(X2(T0 + i,K0))
∑ VP(X2(t,k)) - VQ (X2(T0,K0 + i)) t+k)T +K +i 0
2
VQ1(X2(T0 + i,K0))
0
T0eteT0+i K0ekeK0+i
∑
< F-1
t+k)T0+K0+i T0eteT0+i K0ekeK0+i
VP(X2(t,k))
[
P-1AT21 + P-1KT21βBT2 P-1KT21β -P-1Q1P-1 0 -1 -1 -P Q2P P-1AT22 + P-1KT22βBT2 P-1KT22β * -P-1 + B2β02BT2 0 * * * * * -I
VP(X2(t,k))
0, such that
[
]
(A21 + B2RK21)T (A22 + B2RK22)T P[A21 + B2RK21 A22 + B2RK22 C2 ] CT2
[
]
Q1 - γ-1GTG 0 0 Q2 0 < 0 (73) 0 0 0 γI
l λ1VQ1(X2(t + 1,1)) + λ2VQ2(X2(t + 1,1))
1, we obtain
]
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9059
(74)
holds for all admissible actuator faults that satisfy eq 3. Define
J(t,k) ) ∆V(t,k) + γ-1VGTG(X2(t,k)) - γVI(δk(w(t,k)))
(75)
N
λ1
∑VQ (X2(t + 1,k))
