Improved State Space Model Predictive Control Design for Linear

Article pubs.acs.org/IECR

Improved State Space Model Predictive Control Design for Linear Systems with Partial Actuator Failure Jili Tao,* Yong Zhu, and Qinru Fan Ningbo Institute of technology, Zhejiang University, Ningbo 315100, P R China ABSTRACT: This paper presents a new state space model predictive control scheme for linear systems with partial actuator faults. To consider dealing with the uncertainties caused by actuator faults, the process is first treated into a nonminimal representation using state space transformation. Then the subsequent design of model predictive control is introduced to yield improved closed-loop control performance under unknown disturbances and actuator faults. The advantages of the proposed method over existing methods lie in the fact that the proposed control method can regulate both the process output/input changes in the design, which cannot be achieved by traditional state space model predictive control. For performance comparison, a traditional model predictive fault-tolerant control is also designed. Simulations are given to illustrate the feasibility and effectiveness of the proposed scheme.

1. INTRODUCTION Actuator faults are a common phenomenon in process control systems. Because of the actuator faults, the actuator may not react as the controller requires, which causes control performance degradation.1 Since industrial processes are operating under challenging conditions to meet the demand of high productivity and market needs, these processes tend to encounter system failures. The reasons lie in the fact that industrial processes often have a lot of measurement sensors and actuators. It is generally known that significant damage or performance deterioration can be caused if a fault is not detected and corrected immediately. In view of the importance of fault detection and correction, fault-tolerant control (FTC) has been an important research area up to the present. Recently, there have been some studies of FTC on actuator faults. The representatives are as follows. In ref 2, an adaptive fault-tolerant control strategy for backlash actuator faults is proposed. Mosemann et al. proposed a stability criterion for the systems under friction− type actuator faults.3 In ref 4., Ahn et al. presented an adaptive compensation method to cope with the cogging and coulomb friction in the permanent-magnet motors. Xu. et al. proposed an iterative learning control strategy for deadzone faults in the actuators.5 Lagerberg and Egardt have studied the backlash phenomenon of industrial actuator in automotive powertrains.6 Theoretically, Wu et al show the stabilizing feedback control approach for linear systems with actuator saturation.7 Zhang et al proposed a linear quadratic control for batch processes with actuator faults.8 In practice, the design of controller is often based on the assumption that the actuator acts exactly in response to the controller output because the actual fault cannot be known. However, this inevitably causes control performance deterioration since the controller is operating under the model/process mismatch.9−13 Recent results of fault diagnosis and control for different kinds of process can be seen in refs 14−18. However, to efficiently improve control performance under model/process mismatch still remains a challenge. Though iterative learning control has been proposed, it is only confined to the assumption that repetitive nature of disturbance is considered. In practice, nonrepetitive time-varying behavior and disturbances are common, which pose the difficulty for process controllers. © 2014 American Chemical Society

Recently, model predictive control (MPC) has also been thought of as a useful tool to improve control performance.19−21 And since state space design can effectively use the states’ information for controller design, MPC based on state space models has attracted a lot of interest from researchers.22−37 However, the numerical difficulty caused by observers and the constraints in the process pose great difficulty for traditional state space MPC.38,39 In view of this, the idea of incorporating the measured process inputs, outputs and their past measured values into a nonminimal state space model (NMSS) based MPC has been intensively studied. Research on NMSS model based control can be seen in many different areas.40−56 However, there are still issues for MPC to deal with model/process mismatch for desired product quality, together with faults in the control actuators. In this paper, a model predictive control related to NMSS models are designed for linear systems with partial actuator faults. In the controller formulation, the process is first transformed into a nonminimal representation using state space models, then the subsequent MPC is designed to cope with model/process mismatch. The proposed MPC is demonstrated to show improved output tracking performance under unknown disturbances and admissible faults compared with traditional FTC. A typical process in recent literature is studied to show the feasibility and effectiveness of the proposed. The paper is organized as follows. Section 2 shows the problem formulation. In section 3, the idea of traditional FTC using state space model is introduced. Subsequently, the proposed MPC design is detailed in section 4. In section 5, the effectiveness of the proposed is demonstrated through a typical process in recent literature. In section 6, conclusion is given.

2. PROBLEM FORMULATION For simplicity, a single-input single-output (SISO) process is considered here. When considering the operation around Received: Revised: Accepted: Published: 3578

September 9, 2013 January 16, 2014 February 14, 2014 February 14, 2014 dx.doi.org/10.1021/ie402969r | Ind. Eng. Chem. Res. 2014, 53, 3578−3586

Industrial & Engineering Chemistry Research

Article

To let the control system to be free of steady-state tracking error, the tracking error integral action is introduced as follows:

a set-point, this process can be described through linearization as A(z −1)y(z) = z −dB(z −1)u(z) + C(z −1)

e(k) = r(k) − y(k)

e(z) Δ

(1)

where r(k) is the expected output. A subsequent dynamic model of integral of error is then introduced

where u(z) and y(z) are the z-transforms of control input u(k) and output y(k) of the process. e(z) is the z-transforms of a zero mean white noise e(k). d is the process dead time. Δ is the differenced operator Δ = 1 − z−1. A(z−1), B(z−1), and C(z−1) are the corresponding polynomials with the following forms −1

A(z ) = 1 + Fz 1 −1

B(z ) = H1z

−1

−1

−1

+ F2z

+ H2z

C(z ) = 1 + c1z

−1

−2

+ c 2z

−2

+ ... + Fnz

xe(k + 1) = xe(k) + e(k) = xe(k) + r(k) − y(k)

−n

= xe(k) + r(k) − Cx(k)

+ ... + Hmz −m −2

+ ... + crz

(8)

(9)

with

−r

(2)

xe(0) ≡ 0

For the above process described by eq 1, a controller will be designed for the actuator to implement. Denote uF(k) as the signal from the actuator that has failed. Then, the ideal case for the actuator is as follows:

u F (k) = u(k)

and xe(k) denotes the integral of tracking error. By combination of eq 9 with eq 7, an augmented model is derived as x I(k + 1) = AIx I(k) + BI au(k − d) + C Ir(k)

where

(3)

⎡ x(k) ⎤ ⎥ x I(k) = ⎢ ⎢⎣ xe(k)⎥⎦

However, the above case, eq 3, cannot always be achieved since there exist physical limitations for actuators, causing actuator failure. The failure model in this study is adopted as

u F (k) = αu(k)

(10)

(11)

(4)

where (5) 0 < α ≤ α ≤ α̅ The terms α̲ (α̲ ≤ 1) and α̅ (α̅ ≥ 1) are known scalars. Remark 1. There are mainly three kinds of actuator failures: partial failure case, the outage case, and the stuck fault. It is shown that under the latter two failures the control system can no longer be controllable. Therefore, the partial failure described by eq 4 is widely used.57,58 Remark 2. Generally, α > 0 denote the partial failure case and α = 0 the outage case. Thus α > 0 will be studied in this paper. It can also been seen from eq 5 that the parameter α is assumed to vary within a known range. It is also clearly that α = 1 corresponds to the normal case. Hence, a process under actuator failures can be described as follows:

A(z −1)y(z) = z −dB(z −1)u F (z) + C(z −1) F

e(z) Δ

(6) F

with u (z) denotes the z-transforms of failed control input u (k). The control objective is now to design a controller such that the output of the process tracks the set-point as closely as possible under actuator failures.

3. TRADITIONAL FAULT-TOLERANT CONTROL First, the strategy of traditional FTC is presented here for convenience of later comparisons. The traditional FTC uses a state space model, which can be obtained through directly modeling or transformations as follows: ⎧ ⎪ x(k + 1) = Ax(k ) + Bu(k − d) ⎨ ⎪ y(k) = Cx(k) ⎩

(7)

where x(k), y(k), and u(k) represent the state, output, and input of the process, respectively. d indicates process delay. {A,B,C} are the system matrices with appropriate dimensions.

Figure 1. (a) Output responses under case 1. (b) Input signals under case 1. 3579

dx.doi.org/10.1021/ie402969r | Ind. Eng. Chem. Res. 2014, 53, 3578−3586


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Figure 3. (a) Output responses under case 3. (b) Input signals under case 3.


⎡−F1 −F2 ⎢ 0 ⎢ 1 ⎢ 0 1 ⎢ ⋮ ⎢ ⋮ Am = ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢ 0 0 ⎢ ⋮ ⎢ ⋮ ⎢⎣ 0 0

with ⎡ A 0⎤ ⎡ B⎤ ⎡0⎤ ,B = ,C = AI = ⎢ ⎣−C I ⎥⎦ I ⎢⎣ 0 ⎥⎦ I ⎢⎣1 ⎥⎦

(12)

The next step is then to design a FTC for the augmented system described by eq 10.

4. STATE SPACE MODEL PREDICTIVE CONTROL 4.1. Nonminimal, Input−Output State Space Model. For the process that is described by eq 1, introduce a new state variable as

Bm = [H1 0 0···0 10 0]T

Cm = [1 0 0···0 0 0 0]

Δxm(k) = [Δy(k)Δy(k − 1)···Δy(k − n + 1)Δu(k − 1)

(15)

Define the expected output as r(k), and the output tracking error is therefore formulated as

Δu(k − 2)···Δu(k − m + 1)]T (13)

e(k) = y(k) − r(k)

Then the corresponding state space model is derived as

(16)

Then the dynamics of e(k + 1) can be derived as

Δxm(k + 1) = A mΔxm(k) + BmΔu(k) Δy(k + 1) = CmΔxm(k + 1)

··· −Fn − 1 −Fn H2 ··· Hm − 1 Hm ⎤ ⎥ ··· 0 0 0 ··· 0 0 ⎥ ··· 0 0 0 ··· 0 0 ⎥ ⎥ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⎥ ··· 1 0 0 ··· 0 0 ⎥ ⎥ ··· 0 0 0 ··· 0 0 ⎥ ··· 0 0 1 ··· 0 0 ⎥ ⎥ ··· ⋮ ⋮ ··· ⋮ ⋮ ⋮ ⎥ ··· 0 0 0 ··· 1 0 ⎥⎦

e(k + 1) = e(k) + CmA mΔxm(k) + CmBmΔu(k)

(14)

− Δr(k + 1)

where 3580

(17)



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P

J=

j=1

The combination of the proposed process model eq 14 and the error formulation eq 17 lead to an augmented model as z(k + 1) = Az(k) + BΔu(k) + C Δr(k + 1)

(18)

s. t. Δu(k + j) = 0

(21)

j≥M

where P and M are the prediction horizon and the control horizon, respectively. Qj is the symmetrical weight matrix with appropriate dimension, Lj ≥ 0 is the weight factor of control input increments. And

(19)

and ⎡ Δx (k)⎤ m ⎥ z(k ) = ⎢ ⎢⎣ e(k) ⎥⎦

j=1

× LjΔu(k + j − 1)

where ⎡ Am ⎡ Bm ⎤ 0⎤ ⎡0⎤ ⎥; B = ⎢ ⎥; C = ⎢ ⎥ A=⎢ ⎣−1⎦ ⎢⎣CmA m 1 ⎥⎦ ⎢⎣CmBm ⎥⎦

M

∑ zT(k + j)Q jz(k + j) + ∑ ΔuT (k + j − 1)

Q j = diag{qjy1, qjy2 , ···, qjyn , qju1, qju2 , ..., qju(m − 1), qje}1 ≤ j ≤ P (22)

Remark 4. The variables q j y 1 ,q j y 2 ,...,q j y n and qj u1,qj u2,...,qj u(m‑1)are associated with the regulation of the process states (including output changes and the input changes), whileqj e is associated with the regulation of the process output tracking errors. This shows that the proposed MPC can actively tune the control parameters to improve control performance. Remark 5. Because qjy1 is the weight for Δy(k) and qjy2 is for Δy(k−1), etc., we should care more on qjy1 than qjy2 because changes of qjy1 will impact the closed-loop response more significantly than those caused by qjy2.The same applies for qjy2

(20)

In eq 19,0 is a zero vector with appropriate dimension. Remark 3. Eq 18 is the derived new state space model which will later show that this treatment facilitates the controller design to regulate both the process output error and the state changes, leading to improved control performance. Time delay d can be incorporated in the above model by letting the coefficients H1 = H2 = ... =Hd = 0. 4.2. Predictive Controller Design. 4.2.1. Cost Function. 3581



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Figure 7. (a) Output responses under case 3 (prediction horizon 10 and control horizon 4). (b) Input signals under case 3 (prediction horizon 10 and control horizon 4).


and qjy3 . Thus if we want to regulate the closed-loop performance significantly, the choice of qjy1 ≫ qjy2 ≫ qjy3 ≫ ... ≫ qjyn should be taken. For qju1, qju2, ..., qju(m−1), we can let them be zeros because the second term in the cost function eq 21 does the job of penalizing on control inputs. 4.2.2. Controller Derivation. Define the following vectors

with ⎡ B 0 ⎡A⎤ ⎢ AB B ⎢ 2⎥ ⎢ ⎢A ⎥ F = ⎢ ⎥ , Φ = ⎢ A2 B AB ⎢ ⋮ ⎢ ⎥ ⋮ ⋮ ⎢ ⎢⎣ AP ⎥⎦ ⎢ P−1 ⎣ A B AP − 2 B ⎡ C 0 0 ⎢ 0 C ⎢ AC 2 ⎢ S= AC AC C ⎢ ⋮ ⋮ ⎢ ⋮ ⎢ P−1 P 2 P − ⎣ A C A C A − 3C

⎡ z(k + 1) ⎤ ⎡ ⎤ Δu(k) ⎢ ⎥ ⎢ ⎥ ⎢ z(k + 2) ⎥ ⎢ Δu(k + 1) ⎥ Z=⎢ ⎥ , ΔU = ⎢ ⎥, ⋮ ⋮ ⎢ ⎥ ⎢ ⎥ ⎢ z(k + P)⎥ ⎢ Δu(k + M − 1)⎥ ⎣ ⎦ ⎣ ⎦ ⎡ Δr(k + 1) ⎤ ⎢ ⎥ ⎢ Δr(k + 2) ⎥ ΔR = ⎢ ⎥ ⋮ ⎢ ⎥ ⎢ Δr(k + P)⎥ ⎣ ⎦

0 0

(25)

From eq 21, the cost function can be further expressed as the following vector form

(23)

J = ZTQZ + ΔUTLΔU

Through eq 18, The prediction of the process output can be formulated as Z = Fz(k) + ΦΔU + SΔR

⎤ ⎥ ⎥ ⎥, ⋱ 0 ⎥ ⋮ ⋮ ⎥ P−M−1 ⎥ ··· A B⎦ 0 0⎤ ⎥ 0 0⎥ 0 0⎥ ⎥ ⋱ ⋮⎥ ⎥ ··· C ⎦ ··· ···

(26)

where Q = block diag{Q1,Q2,...,QP} and L = block diag{L1,L2,...,LM}.

(24) 3582



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y(k) = 0.897y(k − 1) + 0.2933u(k − 1) + 0.1486

The optimal control vector through minimization of eq 26 is derived as ΔU = −(ΦT Q Φ + L)−1ΦT Q (Fz(k) + SΔR )

× u(k − 2) +

(27)

r(k) = 15 (for 1 ≤ k < 50)

K s = (ΦTQ Φ + L)−1ΦTQF

r(k) = 30 (for 51 ≤ k < 100) (28)

(29)

where ks and kR are the first rows of the matrices Ks and KR, respectively. Thus control input is u(k) = u(k − 1) + Δu(k)

(32)

The proposed method is tested under different cases that include both actuator faults and unknown disturbances. Note that traditional fault-tolerant model predictive control (TFTMPC) based on eq 10 in section 3 is also tested here for comparison. To illustrate a fair comparison, the common control parameters of the two methods are the same with P = 20, M = 4, Lj = 0.001. The weighting elements on the output error are both chosen as 1. It is seen from eq 22 that the proposed method has another degree of weighting on the process output/input changes, and in this study the weights for the output change qjy1 and the input change qju1 are chosen as 3,0 and 1,0, respectively, to verify its performance. 5.1. Constant Fault and Nonrepetitive Unknown Disturbance. In this case, constant faults are studied to test

The incremental control law can be formulated as Δu(k) = −ksz(k) − kR ΔR

(31)

In this illustration, the trajectory is chosen as

By defining

KR = (ΦTQ Φ + L)−1ΦTQS

e(k) Δ

(30)

5. ILLUSTRATION To demonstrate the feasibility and effectiveness of the proposed method, a simple example is taken from recent literature, which is a 24-plate bubble-cup distillation column process given by Luyben59 as follows. 3583



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5.3. Control Parameters Effects on the Closed-Loop Responses. In this section, we will also briefly illustrate the effect of the control parameters (the prediction horizon and the control horizon) on the control performance. Here, we take the cases 3 and 4 from the illustration. Generally, it is just enough that the prediction horizon is chosen a value to guarantee closeloop stability. The longer the prediction horizon, the sluggish the response of the control system will be. Thus it is not very meaningful to increase the prediction horizon further. For the control horizon and prediction horizon that are originally 4 and 20, respectively, we did further simulations of the following situations: situation 1, the prediction horizon and the control horizon are 10 and 4, respectively; situation 2, the prediction horizon and the control horizon are 10 and 1, respectively. The results for cases 3 and 4 are illustrated in Figures 7−10. It can be seen from these figures that although the control parameters are changed this time, the responses of the proposed are still better. This is because the proposed MPC can have more degrees of regulating the closed-loop dynamics, and thus control performance is improved.

6. CONCLUSION In this article, a state space design of model predictive control is proposed for linear systems with partial actuator faults. The design of the proposed method requires that a new state space formulation from the input-output process model is first done, and then a MPC controller can be formulated through this state space model. Comparisons with traditional model predictive fault-tolerant control demonstrated that improved control performance can be achieved by the proposed strategy for both constant and time-varying actuator faults.

■

AUTHOR INFORMATION

Corresponding Author

*Tel: +86-574-88130021. E-mail: [email protected].


Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Part of this project was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. Q13F030023).

the performance of the proposed method. The following three cases are studied with a random white noise sequence of standard deviation 0.2 added to the process output as unknown disturbance: case 1, α = 0.8; case 2, α = 0.6; case 3, α = 0.25. Figures 1−3 show the simulation results. From Figures 1a−3a, it is seen that as the fault becomes severe, performance of both methods deteriorates. However, since the proposed method can have another degree of weighting on the process output/input changes, the output responses are still acceptable. There are large oscillations in the responses of TFTMPC, which can also be verified because the input signals of TFTMPC are very drastic shown in Figures 1a−3b. 5.2. Time-Varying Fault and Nonrepetitive Unknown Disturbance. In this section, different time-varying faults are studied to further value the performance of the proposed scheme. The following three cases are considered: case 4, α = 0.6 + 0.4 sin (k); case 5, α = 0.6 + 0.2 sin (k); case 6, α = 0.6 + 0.1 sin (k). The noise sequence is the same as that in section 5.1. The comparisons results are illustrated in Figures 4−6. By observation of the output, input, and error signals of the two methods, it is verified again that the proposed method yields improved control performance.

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Improved State Space Model Predictive Control Design for Linear

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