New Design of State Space Linear Quadratic Fault-Tolerant Tracking

Article pubs.acs.org/IECR

New Design of State Space Linear Quadratic Fault-Tolerant Tracking Control for Batch Processes with Partial Actuator Failure Ridong Zhang,†,‡ Liangzhi Gan,‡ Jingyi Lu,‡ and Furong Gao*,‡ †

Information and Control Institute, Hangzhou Dianzi University, Hangzhou 310018, People’s Republic of China Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

‡

ABSTRACT: In this paper we present a new state space model based linear quadratic fault-tolerant tracking control scheme for batch processes with unknown disturbances and partial actuator faults. To develop the linear quadratic fault-tolerant control, the batch process is first treated with a new state space representation that incorporates both the state and the output tracking error dynamics. Then relevant concepts of the subsequent linear quadratic control are formulated, where improved closed-loop control performance is achieved even with unknown disturbances and actuator faults compared with traditional linear quadratic control. Application to injection velocity control shows that the proposed scheme achieves the design objective well with performance improvement.

1. INTRODUCTION Batch processing technology has received widespread attention because it is a preferred method of manufacturing low-volume products with high value. The relevant studies can be dated back to the 1930s; however, the progress of both control technology and application has been witnessed only significantly in the past 10 years, driven by the business of manufacturing.1 Batch processes are now facing challenging operation conditions because of the demand of high productivity, which will lead to process system failures. If a fault cannot be detected and then corrected immediately, process operating performance will generally deteriorate or serious safety problems may happen. Therefore, it is important for fault-tolerant control (FTC) to maintain closed-loop control performance even in the presence of faults. Research of studies on FTC has received considerable attention recently.2−5 Due to the complex nature of batch processes and the immature supporting technologies, results focusing on fault detection/diagnosis and control of batch processes are insignificant and limited.6−9 It is also of note that design of traditional analysis, synthesis, and learningtype controllers is difficult because batch process operations are affected by both nonrepetitive time-invariant and nonrepetitive time-varying failures. In practical situations, actuator faults cannot be avoided because friction, saturation, dead zones, etc. exist everywhere and cause difficulty for the actuator to achieve the specified or desired position. This is an unfavorable fact for process controllers because controller design is always based on the assumption of perfect actuator action; thus, serious deterioration of performance may result.10 To deal with these issues, studies on fault diagnosis and control of different kinds of processes have been conducted; for example, see refs 11−15. For batch processes with actuator failures, Wang et al.8 propose a 2D iterative learning reliable controller (ILRC) with linear matrix inequality (LMI) conditions of closed-loop stability. However, iterative learning can be effective if the process is of © 2013 American Chemical Society

repetitive nature, which may not hold in practice where many batch processes are slowly time-varying, causing nonrepetitive behavior. In addition, performance enhancement of many previous cycles cannot be achieved by iterative learning control (ILC) until the final achievement of performance improvement. It is therefore also important to improve performance within cycles. To the best of our knowledge, few results for batch processes with actuator failures have appeared to the present.16 On the basis of the aforementioned fact, an improved linear quadratic fault-tolerant tracking control strategy for batch processes is proposed in this paper. In the proposed strategy, the batch process is transformed into a new state space representation that incorporates both the state and the output tracking error dynamics, and subsequent linear quadratic control is then designed to yield improved control performance. The proposed control ensures improved output tracking performance under unknown disturbances both for normal operation and in the case of admissible faults compared with control via a traditional linear quadratic regulator (TLQG). The proposed method is applied to injection velocity control to show its feasibility and effectiveness. The paper is organized as follows: Section 2 proposes the problem formulation. In section 3, the idea of a traditional linear quadratic regulator is introduced. The main strategy of the proposed method is detailed in section 4, where the batch process is first transformed into a new state space formulation and a corresponding linear quadratic control strategy is designed. In section 5, the effectiveness of the proposed method is demonstrated through injection velocity control. Section 6 concludes the paper. Received: Revised: Accepted: Published: 16294

July 1, 2013 September 12, 2013 October 22, 2013 October 22, 2013 dx.doi.org/10.1021/ie402066p | Ind. Eng. Chem. Res. 2013, 52, 16294−16300

Industrial & Engineering Chemistry Research

Article

2. PROBLEM FORMULATION For simplicity, the batch process under consideration is assumed to be single-input single-output (SISO), and when it is considered to be operated around a set point, this kind of process can be described through linearization as A(z −1) y(z) = B(z −1) u(z) + C(z −1)

e(z) Δ

law such that the output of the batch process tracks a given set point as closely as possible, even under actuator failures.

3. TRADITIONAL LINEAR QUADRATIC CONTROL20 The idea of TLQG is illustrated in this section for later comparison with the proposed controller in this paper. The process model is first formulated through the state space model as follows:

(1)

⎧ ⎪ x(k + 1) = Ax(k ) + Bu(k ) ⎨ ⎪ y(k) = Cx(k) ⎩

where u(z) and y(z) are the z-transforms of process input u(k) and output y(k). e(z) is the z-transform of a zero mean white noise e(k). Δ = 1 − z−1 is the difference operator. A(z−1), B(z−1), and C(z−1) are corresponding polynomials as follows:

where x(k), y(k), and u(k) are the process state, output, and input, respectively. {A, B, C} are the system matrices with appropriate dimensions. To eliminate the steady-state tracking error, the process model is treated with the difference variables as

−1 A(z −1) = 1 + Fz + F2z −2 + ... + Fnz −n 1

B(z −1) = H1z −1 + H2z −2 + ... + Hmz −m

(2)

C(z −1) = 1 + c1z −1 + c 2z −2 + ... + crz −r

⎧ ⎪ Δx(k + 1) = AΔx(k ) + BΔu(k ) ⎨ ⎪ Δy(k) = CΔx(k) ⎩

For the above batch process, a controller is designed to produce a control input signal for the actuator to implement. For control input signal u(k), let uF(k) denote the signal from the actuator that has failed. Then, if there is no fault, the following holds: F

u (k) = u(k) −1

y(k + 1) = y(k) + CAΔx(k) + CBΔu(k)

(9)

Thus, by combination of eq 9 with eq 8, an augmented model is derived as

−1

Remark 1. Because polynomials A(z ), B(z ), and C(z ) are constant, eq 1 is a linear description of batch processes. However, batch processes are generally nonlinear in many cases, and batch processes based on a nonlinear model are still an open issue. In ref 17, a linear perturbation model is proposed to describe nonlinear batch processes. Similarly, for nonlinear batch processes, use of the proposed method in this paper is feasible. Remark 2. Time delay can be incorporated into the above process model by letting the coefficients H1 = H2 = ... = Hd = 0. However, the above ideal case cannot always be achieved due to physical limitations. The following failure model is adopted: u F(k) = αu(k)

(8)

It is noted that the following relation holds:

(3) −1

(7)

x I(k + 1) = A Ix I(k) + BIΔu(k)

(10)

where ⎡ Δx(k)⎤ ⎥ x I(k + 1) = ⎢ ⎢⎣ y(k) ⎥⎦

⎡ A 0⎤ AI = ⎢ ⎣ CA 1 ⎥⎦

⎡B⎤ BI = ⎢ ⎥ ⎣ CB ⎦

(11)

To design a linear quadratic controller, the following quadratic performance criterion is considered: kf − 1

(4)

J=

where

∑ [(y(k) − yr (k))T Q̅ (y(k) − yr (k)) k=k0

0 < α̲ ≤ α ≤ α̅

(5)

+ Δu T(k) R̅ Δu(k)] + [y(k f ) − yr (k f )]T Q̅ f

The terms α̲ (α̲ ≤ 1) and α̅ (α̅ ≥ 1) are known scalars. Remark 3. Mainly three situations are considered for actuator failure: the partial failure case, i.e., partial degradation of the actuator, the outage case, and the stuck fault, which makes the output of an actuator stay at a constant value. In the case of the latter two failures, the control system is no longer controllable. Therefore, a partial failure case model described by eq 4 is widely used.18,19 Remark 4. Generally, α > 0 corresponds to the partial failure case and α = 0 the outage case. It is obvious that α > 0 is used in this paper. Meanwhile, the parameter α is unknown but assumed to vary within a known range that is described by eq 5. α̲ = α̅ corresponds to the normal case. Hence, a batch process with actuator failures may be described by A(z −1) y(z) = B(z −1) u F(z) + C(z −1) F

e(z) Δ

[y(k f ) − yr (k f )]

(12)

Here, yr(k), Q̅ > 0, R̅ > 0, and Q̅ f > 0 are the set point signal, the output weighting matrix, the input weighting matrix, and the terminal weighting matrix, respectively. [k0, kf] is the future optimization horizon. For the augmented system described by eq 10, the corresponding linear quadratic controller can be designed to yield tracking performance under both normal and admissible faults.

4. NEW STATE SPACE MODEL BASED LINEAR QUADRATIC CONTROL 4.1. New State Space Model. Similar to the description in section 3, the batch process described by eq 1 is first treated with the difference state space model shown in eq 8 as follows:

(6)

Δx(k + 1) = AΔx(k) + BΔu(k)

F

where u (z) is the z-transform of the failed control input u (k). The control objective is to determine a fault-tolerant control

Δy(k + 1) = CΔx(k + 1) 16295

(13)

dx.doi.org/10.1021/ie402066p | Ind. Eng. Chem. Res. 2013, 52, 16294−16300


Article

Although the cost function of TLQG described by eq 12 can also be expressed with the state xI(k) instead of y(k), the elements in the corresponding weighting matrices Q and Qf will be fixed due to the specified transformation. Thus, there is no choice to designate the elements in Q and Qf. 4.2.2. Controller Derivation. The derivation of the corresponding linear quadratic controller follows the general procedure of linear quadratic control design. Here the optimal control law is summarized in the following theorem. Theorem 1. This theorem is for the batch process described by eq 13. The linear quadratic tracking control is given as eq 20 for the performance criterion in eq 19. Hk,kf and gk,kf in eq 20 are obtained from Riccati equations in eqs 21a and 21b with the boundary condition in eq 22.

Unlike the further treatment of the TLQG design, here, the expected output is defined as r(k), and then the output tracking error is therefore formulated as

e(k) = y(k) − r(k)

(14)

By combination of eqs 13 and 14, and noticing that Δr(k + 1) = 0, the formulation of e(k + 1) can be derived as e(k + 1) = e(k) + CAΔx(k) + CBΔu(k)

(15)

It is noted that, by further augmenting eqs 13 and 15 into a new model, the following will result: z(k + 1) = A mz(k) + Bm Δu(k)

(16)

where ⎡ A 0⎤ Am = ⎢ ⎣ CA 1 ⎥⎦

⎡B⎤ Bm = ⎢ ⎥ ⎣ CB ⎦

u(k) = −R−1BTm [I + Hk + 1, k f Bm R−1BTm ]−1 [Hk + 1, k f A mz(k) (17)

+ gk + 1, k ]

⎡ Δx(k)⎤ ⎥ z(k ) = ⎢ ⎢⎣ e(k) ⎥⎦

Hk , k f = A Tm[I + Hk + 1, k f Bm R−1BTm ]−1 Hk + 1, k f Am + Q

(18)

(R + BTm Hk + 1, k f Bm )−1BTm Hk + 1, k f A m + Q gk , k = A Tm[I + Hk + 1, k f Bm R−1BTm ]−1 gk + 1, k − Qzr(k) f

f

(21b)

Hk f , k f = Q f g k , k = − Q f z r (k f ) f

(22)

f

It is noted that, by formulating the process model as eq 16, the actual reference signal is really zero. For a zero reference signal, gk,kf becomes zero so that we have u(k) = −R−1BTm [I + Hk + 1, k f Bm R−1BTm ]−1 Hk + 1, k f A mz(k) (23)

Remark 8. It is shown in eq 18 that the process state variables are the state changes and the output tracking error; thus, the set points for such state variables are actually zero, i.e., zr(k) = 0, showing that only eq 21a needs to be solved, which reduces the computation burden compared with that of TLQG, which has to solve the two Riccati equations in eqs 21a and 21b, because the process state variable includes the output instead of the tracking error. Proof of Theorem 1. According to the minimum principle of Pontryagin, the Hamiltonian corresponding to the proposed process model in eq 16 is

kf − 1

∑ [(z(k) − zr(k))T Q(z(k) − zr(k)) k=k0

+ Δu T(k) RΔu(k)] + [z(k f ) − zr(k f )]T Q f [z(k f ) − zr(k f )]

(21a)

= A TmHk + 1, k f A m − A TmHk + 1, k f Bm

In eq 17, 0 is a zero vector with appropriate dimensions. Remark 5. Equation 16 is the newly formulated state space model. Unlike the traditional state space model that is used for the TLQG design, the proposed model will facilitate the control system design to regulate both the output tracking error and the process state changes, which will have more degrees than the TLQG method and lead to improved control performance. Remark 6. Note that although the proposed model is in a state space form, input−output process models can also be first transformed into the proposed model and then for MPC design. This can be done using numerous transformations in control theory. However, if measured process input and output variables are to be used, one possible way is to directly adopt these variables as state variables. For example, for the input− output process that is described by eqs 1 and 2, we can first choose a process state space variable as xm(k) = [y(k) ... y(k − n + 1) u(k − 1) ... u(k − m + 1)]T; then a corresponding state space model can be formulated in the form of eq 13. The subsequent design then follows the steps in eqs 14−18. 4.2. Linear Quadratic Fault-Tolerant Control. 4.2.1. Cost Function. The cost function is as follows: J=

(20)

f

and

(19)

Hk = [(z(k) − zr(k))T Q(z(k) − zr(k)) + Δu T(k)

Here, zr(k), Q > 0, R > 0, and Qf > 0 are the set point signal, the output weighting matrix, the input weighting matrix, and the terminal weighting matrix, respectively. [k0, kf] is the future optimization horizon. Remark 7. It can be clearly seen that the new state space model facilitates the controller design to regulate the dynamics of both the output tracking error and the process state variables. From eq 18, it is seen that the state z(k) includes both the original process state variables and the output tracking error, showing that the elements in Q and Qf can be tuned to regulate both the process states and the tracking error.

RΔu(k)] + pkT+ 1 [A mz(k) + Bm Δu(k)]

(24)

where pk+1 is the Lagrange multiplier and k ∈ [k0, kf]. The necessary conditions for getting the optimal control law are pk =

∂Hk = 2Q(z(k) − zr(k)) + A Tmpk + 1 ∂z(k)

(25a)

pk = 2Q f (z(k f ) − zr(k f )) f

and ∂Hk/∂Δu(k) = 0, which yields the following: 16296


Industrial & Engineering Chemistry Research ∂Hk = 2RΔu(k) + BTm pk + 1 = 0 ∂Δu(k)

Article

(25b)

It is noted that ∂ Hk/∂Δu (k) = 2R is positive definite and Hk is a quadratic form in Δu(k); thus, the solution of eqs 25a and 25b will be an optimal control law to minimize Hk, which is 2

2

1 Δu(k) = − R−1BTm pk + 1 2

(26)

By assuming that pk = 2Hk , k f z(k) + 2gk , k

(27)

f

and from eq 25a, the boundary conditions are then given in eq 22. Substituting eqs 16 and 26 into eq 27, we have pk + 1 = 2Hk + 1, k f z(k) + 2gk + 1, k

f

⎛ ⎞ 1 = 2Hk + 1, k f ⎜A mz(k) − Bm R−1BTm pk + 1 ⎟ ⎝ ⎠ 2 + 2gk + 1, k

(28)

f

By solving for pk+1, eq 28 will be further written as pk + 1 = [I + Hk + 1, k f Bm R−1BTm ]−1 [2Hk + 1, k f A mz(k) + 2gk + 1, k ]

(29)

f

Thus, the optimal control law in eq 20 will be derived by substituting eq 29 into eq 26. It can also be noted that, by substituting eq 29 into eq 25a, we have pk = 2Q(z(k) − zr(k)) + A Tmpk + 1 = 2[A Tm(I + Hk + 1, k f Bm R−1BTm )−1Hk + 1, k f A m + Q]z(k) + 2 [A Tm(I + Hk + 1, k f Bm R−1BTm )−1gk + 1, k − Qzr(k)]

Figure 1. (a) Output responses under case 1. (b) Input signals under case 1.

(30)

proportional valve opening u(k) for this process has been identified as an autoregressive model as follows:

f

By comparing eq 30 with eq 27, we can see that eqs 21a and 21b hold. This completes the proof.

G (z ) =

5. ILLUSTRATION In this section, an injection molding batch process is considered. This process consists of three phases: filling, packing, and cooling.21 To maintain product quality during the filling stage, the injection velocity is very important and should be controlled to follow a given set point. The filling stage in the injection molding machine is as follows. High pressure in the hydraulic cylinder exerts on the injection screw to force the plastic to melt and be pushed into the mold cavity. This stage stops until the mold is completely or almost completely filled for the packing stage to begin to pack the additional material into the mold cavity. During this stage, the injection velocity should be controlled with high precision because it is associated with the mechanical strength, deformation, and accuracy. This variable is operated through a proportional valve opening to regulate the flow of hydraulic oil. Generally, it is necessary to regulate the opening of the proportional valve to control the injection velocity. Thus, the output for this process is the injection velocity denoted as y(k), and the input variable is the opening of the proportional valve denoted as u(k). In this study the typical injection process that has been studied in refs 8− 22 is further studied. According to the previous studies, the injection velocity y(k) response to the

y(z) 1.69z + 1.419 = 2 u(z) z − 1.582z + 0.5916

(31)

Here y(z) and u(z) are the z-transforms of the process output y(k) and the input u(k), respectively. Note that the valve opening ranges from 0 to 1. However, there may be some valve faults in the valve opening, which is denoted by an unknown actuator fault α. In this illustration, the set point profile takes the following form: r(k) = 10 (for 1 ≤ k < 50) r(k) = 20 (for 51 ≤ k < 100)

(32)

To derive the state space model of the batch process, the state variable is first chosen as follows: Δx(k) = [Δy(k) Δy(k − 1) Δu(k − 1)]T

(33)

Thus, the corresponding state space model is formulated as eq 7 with ⎡1.582 − 0.5916 1.419 ⎤ ⎢ ⎥ A=⎢ 1 0 0 ⎥ ⎢⎣ 0 0 0 ⎥⎦

⎡1.69 ⎤ ⎢ ⎥ B=⎢ 0 ⎥ ⎢⎣ 1 ⎥⎦

(34)

The subsequent controllers of TLQG and the proposed method can then be designed according to the previous 16297



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TLQG and the proposed strategy. Note that the smaller the value of α, the more severe the fault. The following three cases are thus studied: case 1, α = 0.8; case 2; α = 0.6, case 3, α = 0.2. The nonrepetitive unknown disturbance added to the process output is a random white noise sequence with a standard deviation of 0.2. The simulation results are shown in Figures 1−3, where Figures 1a−3a show the output responses of the proposed method and TLQG for the three cases, respectively. The performances of the proposed method and TLQG both deteriorate as the fault becomes severe; however, the proposed strategy yields improved output tracking performance where the set point can be tracked as quickly as possible. There is also oscillation and large overshoot in the performance of TLQG. From Figures 1b−3b, it can be seen that TLQG shows drastic input signals in the control system compared with the proposed method, which will indeed have an impact on performance and product quality. It is demonstrated that, in the case of such constant faults, the proposed method can give an improved admissible control result. 5.2. Time-Varying Fault and Nonrepetitive Unknown Disturbance. In this section, time-varying values of α are also studied to further test the performance of the proposed method. The following three cases are investigated: case 4, α = 0.6 + 0.4 sin(k); case 5, α = 0.6 + 0.1 sin(k); case 6, α = 0.6 + 0.01 sin(k). The nonrepetitive unknown disturbance remains the same as that in the previous section. The comparisons of closed-loop control system performance are illustrated in Figures 4−6. It is also shown that the

sections. The two methods are illustrated for different cases of faults and unknown disturbance. The future optimization horizons for the two methods are the same, [1, 20]. The weighting parameters on the output tracking error and the control input of the two methods are also the same, 1 and 0.1, respectively. This shows that, for TLQG, Q̅ = Q̅ f = 1 and R̅ = 0.1, and for the proposed method, the last diagonal element in Q and Qf is 1 and R = 0.1. Note that, due to the merits of the proposed method, there are options to choose other diagonal elements in Q and Qf. In this case, Q and Qf are chosen as ⎡2 ⎢ 0 Q = Qf = ⎢ ⎢0 ⎢⎣ 0

0 1 0 0

0 0 0 0

0⎤ ⎥ 0⎥ 0⎥ ⎥ 1⎦

(35)

However, for TLQG, this is impossible because its process model and the cost function cannot be integrated to achieve this. Note that even if the cost function of TLQG is transformed to be expressed with the state xI(k) instead of y(k), as pointed out earlier in Remark 6, the elements in the corresponding weighting matrices Q and Qf will be fixed due to the specified transformation and cannot be changed. Thus, there is no choice to designate the elements in Q and Qf freely as the proposed method does. 5.1. Constant Fault and Nonrepetitive Unknown Disturbance. In this section we illustrate the constant fault with nonrepetitive unknown disturbance case, where different constant values of α are studied to test the performance of 16298



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proposed method can still improve control performance when time-varying actuator faults are encountered.

Foundation Funded Project (Grants 2013T60590 and 2012M511368).

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6. CONCLUSION In this work, an improved linear quadratic fault-tolerant control is designed for batch processes with unknown disturbances and partial actuator faults. The process model has been transformed to an augmented model such that output tracking error and process state changes can both be regulated. Comparisons with traditional linear quadratic fault-tolerant control show that control performance has been improved, both under the constant-time and time-varying actuator faults. The illustration of injection velocity control shows the feasibility and effectiveness.

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REFERENCES

(1) Korovessi, E., Linninger, A. A. Batch Processes; CRC/Taylor and Francis: Boca Raton, FL, 2006. (2) Zhou, D. H.; Frank, P. M. Fault diagnosis and fault tolerant control. IEEE Trans. Aerosp. Electron. Syst. 1998, 34, 420−427. (3) Bao, J.; Zhang, W. Z.; Lee, P. L. Decentralized fault-tolerant control system design for unstable processes. Chem. Eng. Sci. 2003, 58, 5045−5054. (4) Wang, Y. Q.; Zhou, D. H.; Gao, F. R. Robust fault-tolerant control of a class of non-minimum phase nonlinear processes. J. Process Control 2007, 17, 523−537. (5) Zhang, X.; Parisini, T.; Polycarpou, M. M. Adaptive fault-tolerant control of nonlinear uncertain systems: an information-based diagnostic approach. IEEE Trans. Autom. Control 2004, 49, 1259− 1274. (6) Lu, N.; Gao, F. R.; Yang, Y.; Wang, F. PCA-based modeling and online monitoring strategy for uneven-length batch processes. Ind. Eng. Chem. Res. 2004, 43, 3343−3352. (7) Scenna, N. J. Some aspects of fault diagnosis in batch processes. Reliab. Eng. Syst. Saf. 2000, 70, 95−110. (8) Wang, Y. Q.; Zhou, D.; Gao, F. Iterative learning fault-tolerant control for batch processes. Ind. Eng. Chem. Res. 2006, 45, 9050−9060. (9) Wang, Y. Q.; Zhou, D.; Gao, F. Active fault-tolerant control of nonlinear batch processes with sensor faults. Ind. Eng. Chem. Res. 2007, 46, 9158−9169. (10) Tsang, T. T. C.; Clarke, D. W. Generalized predictive control with input constraints. IEE Proc.-D: Control Theory Appl. 1988, 135, 451−460.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +852-2358-7139. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Part of this project was supported by the State Key Program of the National Natural Science Foundation of China (Grant 61134007), National Natural Science Foundation of China (Grants 61273101 and 61104058), Hong Kong, Macao and Taiwan Science & Technology Cooperation Program of China (Grant 2013DFH10120), and China Postdoctoral Science 16299



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(21) Gao, F.; Yang, Y.; Shao, C. Robust iterative learning control with applications to injection molding process. Chem. Eng. Sci. 2001, 56, 7025−7034. (22) Shi, J.; Gao, F. R.; 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.

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New Design of State Space Linear Quadratic Fault-Tolerant Tracking

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