J. Cent. South Univ. Technol. (2007)05-0701-07

DOI: 10.1007/s11771-007-0134-9                                                                                   

Robustly stable model predictive control based on

parallel support vector machines with linear kernel

BAO Zhe-jing(包哲静)¹, ZHONG Wei-min(钟伟民)², PI Dao-ying(皮道映)¹, SUN You-xian(孙优贤)¹

(1. State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China;

2. State Key Laboratory of Chemical Engineering, East China University of Science and Technology,

Shanghai 200237, China)         

Abstract：Robustly stable multi-step-ahead model predictive control (MPC) based on parallel support vector machines (SVMs) with linear kernel was proposed. First, an analytical solution of optimal control laws of parallel SVMs based MPC was derived, and then the necessary and sufficient stability condition for MPC closed loop was given according to SVM model, and finally a method of judging the discrepancy between SVM model and the actual plant was presented, and consequently the constraint sets, which can guarantee that the stability condition is still robust for model/plant mismatch within some given bounds, were obtained by applying small-gain theorem. Simulation experiments show the proposed stability condition and robust constraint sets can provide a convenient way of adjusting controller parameters to ensure a closed-loop with larger stable margin.

Key words: parallel support vector machines; model predictive control; stability; robustness

1 Introduction

The use of support vector machine (SVM) in model predictive control (MPC) has received considerable attention. Several papers have appeared on SVM based MPC^[1-5], all of which concentrate only on the issue related to constructing SVM model, combining SVM model with MPC and deriving the optimal solution^[6]. Some papers applied the gradient descent based method to search the control laws, which cannot be ensured to be globally optimal^[1-3], and others achieved an analytical solution of control laws^[4-5]. However no attention is paid to the stability of the closed-loop system.

In MPC, while the performance of the plant is optimized over the prediction horizon repeatedly, each optimization does not care about what happens beyond the prediction horizon, and so the plant can be put into a state that it will eventually be impossible to stabilize, even though the controlled plant itself is stable^[7-8].

In this paper, the structure block diagram of parallel SVMs based MPC closed loop was obtained. Due to linear kernel, classical theoretical results can be applied to analyze the stability of SVMs-MPC, and then according to SVM model the stability condition was derived. A method of judging the discrepancy between SVM model and the actual plant was proposed, and further the constraint sets for the discrepancy were given to ensure that the real closed-loop system is robustly stable.

2 Parallel SVMs with linear kernel based MPC algorithm

The structure of parallel SVMs based multi-step-ahead MPC is shown in Fig.1, in which H_p (prediction horizon) parallel SVMs predictive models constitute multi-step-ahead predictor, y_s(k+j) is the desired set-point value of output y at time k+j, is the j-step-ahead predictive output at time

k after feedback correction,is the H_p-step-ahead predictive output at time k-H_p after feedback correction, z^-i and z^-j are the delay operators. All sub-models do not influence each other, and then an accumulation of prediction errors can not occur.

Assume an unknown process can be described as follows:

       

      (1)

where  n≥1 and m≥1 are maximum lags in the outputs and inputs, respectively, and .

According to Eqn.(1) and ,

we have

      (2)

Repeat the procedure recursively and the -step-ahead predictor is yielded:

         (3)

SVM with linear kernel can regress the unknown linear and weakly nonlinear system well. Using SVM regression technique, the model with linear kernel can be then represented as^[9-10]:

     (4)

where  n_svj is the number of support vector of model;  is the input vector of support vector corresponding samples of the model  and 

 

Now we can rewrite Eqn. (4) as follows:

         (5)

where ,

 

and

.

In MPC, the quadratic objective function is defined as^[11]

       (6)

where ,,and the weighting matrices Q and R are given by and  .

We propose that -step-ahead predictive model at time k should be corrected by the discrepancy between the latest plant output y(k) and the latest -step-ahead predictor output . So it holds

 (7)

From Eqn. (5), it can be derived that:

      (8)                                                   

Then we can get

     (9)

where

     

  (10)

So the following holds:

           (11)

where  and .

The necessary condition for Eqn.(6) to have a minimum is to set , in which

.

Therefore the optimal control law is

            (12)

where .

Fig.1 Diagram of parallel SVMs based MPC

3 Stability analysis of parallel SVMs with linear kernel based MPC

We can represent Eqn. (10) as  .

where  and

.

So vector g can be rewritten as

       (13)

From Eqn. (5), SVM model has the form

 

(14)                            

where , and.

Eqn. (14) can be rewritten in z-transform form as

  (15)

Define s(z^-1)=p₁(z^-1)+q₁(z^-1) and t(z^-1)=z-n₁(z^-1).

Theorem 1 According to SVM model of the controlled plant, parallel SVMs based MPC closed-loop system is stable if and only if all the roots of polynomial  are inside the unit circle, where

         (16)

Proof Since the controlled plant is unknown, we substitute SVM mode for the plant. Consequently, the structure of closed-loop system is shown in Fig.2.

Fig.2 Structural diagram of closed-loop system

As shown in Fig.2, the SVMs-MPC closed loop has two inputs, one is y_s(k) and the other is b₁. Bias  can be thought as a measurement noise. The transfer functions from y_s(k) and b₁ to output are respectively:

 and

        

where  is the transfer function from input 1(y_s(k)) to output; is the transfer function from input 2(b₁) to output.

The poles of transfer function from bias b₁ to output y(k) are the same as those of transfer function from y_s(k) to output y(k). From classical theory, the necessary and sufficient stability condition of the MPC closed-loop system is derived, that is, all the roots of polynomial G(z)=0 are inside the unit circle, where

Remark 1 From the stability condition, we can know that the stability is relevant to MPC parameters H_p, H_u, Q, R and SVM model. For a specific controlled plant, SVM model is determined, so stability can be achieved by adjusting MPC parameters.

4 Robust analysis of parallel SVMs with linear kernel based MPC

Usually there exists modeling error or system perturbation, but the above mentioned stability condition is derived according to SVM model. Consequently for a closed-loop system composed of the controller and the actual plant, it is necessary to give the constraint sets, which can guarantee that the stability condition is still robust.

Without lose of generalization, Δ is employed to denote the discrepancy between SVM model and the actual plant, and then an additive uncertainty parameter is appended to the transfer function of SVM model, that is

               (17)

where P_actual is the actual transfer function of the controlled plant, Δ is assumed to be stable and .

Theorem 2 The closed-loop system, composed of the controller designed by the stability condition in Theorem 1 and the actual controlled plant with uncertainty, will remain robustly stable if

                         (18)

                                           

where  denotes the largest singular value and .

Proof By considering the system uncertainty as in Eqn.(17) and omitting inputs, the structure of SVMs-MPC and the plant with additive uncertainty is depicted in Fig.3, where  and the transfer function of the block within the broken line is , and this transfer function is

of course stable, resulting from designing a stable controller according to the condition in Theorem 1. Thus the loop gain transfer function is . It then follows the small-gain theorem that the combination of the controller and the actual plant will remain robustly stable

if .

Fig.3 Simplified diagram of combination of controller and actual plant with uncertainty

Remark 2 From a practical point of view, Theorem 2 provides a method to judge whether the stability condition developed in Theorem 1 can ensure the stability of the actual closed-loop system, which can be illustrated as follows.

Although the actual plant is unknown, the discrepancy between SVM model and the actual plant can be measured through experiments. Suppose that a certain input u acts on the actual plant and SVM model respectively, and e denotes the output discrepancy. There exists  from Eqn. (17). Now, assume a unit step-function input is applied, and then , i.e. . It holds true that ≤. From the difference between model output and plant output,  can be easily obtained, and then the upper bound of  is derived. For a SVMs-MPC closed-loop system,  can be calculated. So from the constraint in Theorem 2, it can be determined whether the stability condition developed in Theorem 1 can guarantee that the actual closed loop is still stable.

5 Simulation results

Two examples were selected to demonstrate the results developed so far.

5.1 Example 1

An unknown linear plant is:

    (19)

Obviously the plant itself is stable. Assume that the initial condition is x(0)=[0, 0]. SVM parameters are chosen as n=2, m=3, C=1×10⁴ and ε=1×10^-4. 200 pairs of samples are generated by applying random inputs uniformly distributed in the interval [-1,1].

The output responses are depicted in Fig.4 and the corresponding closed-loop poles are listed in Table 1, with different MPC parameters.

A unit step input is applied to the actual plant and SVM model respectively, and then we get . For the stable case H_p=H_u=Q=1, R=0.6, the robust stability constraint is . From time 60 to 500 due to system perturbation, the transfer function of the actual plant alters to be  and  respectively, where  and , and then the output response is examined. As shown in Fig.5, the plant still remains stable under perturbation Δ₂, whereas the plant becomes unstable under perturbation Δ₁.

Fig.4 Output responses with different MPC parameters

(a) H_p=H_u=Q=1, R=0; (b) H_p=H_u=Q=1, R=0.5; (c) H_p=H_u=Q=1, R=1; (d) H_p=2, H_u=1, Q=diag{1,1}, R=0

Table 1 Closed-loop poles with different MPC parameters

Fig.5 Output responses under perturbation

(a) ; (b)

5.2 Example 2

An unknown weakly nonlinear plant:

  (20)                        

SVM parameters are n=2, m=3, C=1×10⁴ and ε=  1×10^-4. 200 pairs of samples are generated by applying random inputs uniformly distributed in the interval [-1,1].

A unit step input is applied to the actual plant and SVM model respectively, and then we get  . MPC parameters are chosen as H_p=1, H_u=1, Q=1, R=0.001 and H_p=1, H_u=1, Q=1, R=1. The two closed loops are all stable.  of the former is 0.8063 and that of the latter is 0.8633. So both cases can satisfy the robust constraint and the stability margin of the latter is larger than that of the former. From time 60 to 500, due to system perturbation, the transfer function of the

actual plant alters to be  and  respectively, where Δ₁=0.78/z and Δ₂=0.83/z. The output responses are depicted in Fig.6. And we can see that under perturbation Δ₁ both closed loops still remain stable, but the control performance of the latter is better than that of the former, whereas under Δ₂ the former becomes unstable and the latter is still stable. Fig.6 illustrates that the stability margin of the latter is larger than that of the former.

6 Conclusions

1) Finite horizon model predictive controller based on parallel SVMs may result in closed-loop unstable system.

Fig.6 Output responses under perturbations

(a) Δ₁=0.78/_z;

(b) Δ₁=0.78/z;

(c) Δ₂=0.83/z;

(d) Δ₂=0. 38/z

2) Multi-step-ahead MPC strategy based on parallel SVMs with linear kernel was proposed, and then the necessary and sufficient stability condition of MPC closed loop according to SVM model was derived, and a method of judging the discrepancy between SVM model and the plant was presented.

3) The constraint sets for model/plant mismatch to ensure the real close-loop system are robustly stable. Simulation results show that the stability condition and the robust constraint sets can provide a convenient way to achieve parallel SVMs based MPC closed-loop system with larger stable margin.

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Foundation item: Project(2002CB312200) supported by the National Key Fundamental Research and Development Program of China; project(60574019) supported by the National Natural Science Foundation of China

Received date: 2006-12-20; Accepted date: 2007-03-23

Corresponding author: BAO Zhe-jing, Doctoral candidate; Tel: +86-571-87953761; E-mail: zjbao@iipc.zju.edu.cn

 (Edited by YANG Hua)