Multimodel analysis and controller design for nonlinear processes

doi:10.1016/j.compchemeng.2004.08.005

Computers & Chemical Engineering

Volume 28, Issue 12, 15 November 2004, Pages 2667-2675

https://doi.org/10.1016/j.compchemeng.2004.08.005 Get rights and content

Abstract

Multimodel analysis and controller design for nonlinear processes via gap metric is discussed. It is shown that the loop-shaping $H_{\infty}$ approach can integrate the procedure of selecting operating points and the local controller design. The local controllers can guarantee not only stability but also performance specified by the pre- and/or post-compensators. Thus, at each operating points, local controllers can have similar performance, and the global performance of the system can be predicted.

Introduction

Multimodel approach is a popular method for nonlinear control systems design. The essence of this approach is to represent a nonlinear system as a combination of linear models. Local controllers can be designed using well-known linear design techniques, such as LQG, $H_{\infty}$ , etc. The concept is simple, yet useful in practical controller design, see, for example, Murray-Smith and Johansen (1997); Galan, Palazoglu, and Romagnoli (2000); Rodriguez, Romagnoli, and Goodwin (2003). However, the question of how many models are sufficient in design and where the models should be selected is not yet solved.

Recently, Galan, Romagnoli, Palazoglu, and Arkun (2003) suggested using the gap metric as a guideline for selecting local models. The idea is that the ‘distance’ between two selected models should not be larger than a prescribed level. Since local controllers can be designed to robustly stabilize all the models within the prescribed level, models selected in this method can guarantee the global stability of the closed-loop system as long as the controller switching is done ‘slowly’. The method is practical in that a detailed nonlinear system model is not needed. However, stability is not the only requirement of a control system. Performance, such as disturbance rejection, dynamic response, etc. was not considered in the paper.

McNichols and Fadali (2003) proposed to use interval mathematics and a classical synthesis design approach to determine a near minimal set of the design points. The algorithm made use of the closed-loop poles, since the performance of the closed-loop system is dependent on them. However, the approach is restricted to systems with ‘interval’ transfer functions, i.e. whose coefficients are within known intervals and vary slowly as a function of some external scheduling variable.

The measure of nonlinearity (Guay, 1996; Helbig, Marquardt, & Allgower, 2000) provides another viewpoint on selecting operating points for multimodel controller design. The idea is that we should select operating points near which the system is most ‘nonlinear’, so the local linear models can ‘cover’ the nonlinearity of the system. For the regime where the system is not so nonlinear, one linear model is sufficient. Unfortunately, the nonlinearity measure is usually not easy to compute, and a detailed nonlinear model is assumed to be available, which is often not possible to obtain in practice.

In this paper, we will extend the method in Galan et al. (2003) to accommodate performance requirements. Motivated by the loop-shaping $H_{\infty}$ approach (McFarlane & Glover, 1990), we use the gap metric for ‘compensated’ models as a guide for selecting operating points. With loop-shaping $H_{\infty}$ design, the local controllers can guarantee not only stability but also performance specified by the pre- and/or post-compensators. Thus, at each operating point, local controllers can have similar performance, and global closed-loop performance can be predicted.

The advantages of using a weighted gap in multimodel control is two fold. First, performance can be incorporated in selecting operating points. Second, the maximum robust stability margin for the shaped plant at a given operating point can be obtained without the knowledge of the local controllers. Thus, operating point selection and local controller design can be integrated in multimodel design procedure.

The remaining part of the paper is arranged as follows. In Section 2, the gap metric is briefly reviewed and the theoretical background for its application in operating point selection is pointed out. In Section 3, performance weights are considered so that not only stability but also performance can be guaranteed in operating point selection and local controller design. Section 4 discusses how to select the performance weights in multimodel approach, and Section 5 illustrates the proposed method using three examples. Conclusions are given in Section 6.

Section snippets

Gap metric

In this section, we review the theory of the gap metric. Details can be referred to El-Sakkary (1985); Georgiou and Smith (1990) and Zhou and Doyle (1998).

Let $P (s)$ be a $p \times m$ rational transfer matrix and let P have the following normalized right coprime factorization $P = N M^{- 1}, with M^{\sim} M + N^{\sim} N = I,$ where ${(\cdot)}^{\sim}$ denotes complex conjugate, i.e. $M {(s)}^{\sim} : = M {(- s)}^{{T}}$ . The graph of P is the subspace of $H_{2}$ (the standard Hardy space in the right half of the complex plane) given by $G (P) = [\begin{matrix} M \\ N \end{matrix}] H_{2} .$ The gap between two linear

Gap metric for shaped plants

The use of the gap metric in selecting the operating points in multimodel controller design has two main shortcomings:

(1)
The gap metric is only related to robust stability, that is, the local controller $K$ can only guarantee that it can stabilize the models at operating points close to the given operating point. However, stability is not the only issue in control system design. Performance should also be guaranteed in selecting operating points.
(2)
Local controller must be available to compute $b_{P, K}$ ,

Compensator selection and multimodel controller design

We choose the pre- and/or post-compensators to reflect the performance requirements. This is done as follows.

The desired open-loop shape normally means high gain at low frequencies, roll-off rates of approximately 20 dB/decade at the desired bandwidth(s), and higher rates at high frequencies (Zhou & Doyle, 1998), as illustrated in Fig. 1.

To obtain a desired loop shape, we choose pre- and post-compensators following the following guidelines (Skogestad & Postlethwaite, 1996):

(1)
The post-compensator

Cases studied

Three examples will be given to illustrate the proposed method.

Conclusions

Multimodel analysis and controller design for nonlinear processes were studied using the gap metric. Operating point selection was extended to accommodate performance requirements. With the loop-shaping $H_{\infty}$ approach, the procedure of selecting operating points and local controller design can be integrated. The local controllers can guarantee not only stability but also performance specified by the pre- and/or post-compensators. Thus, at each operating point, local controllers can have similar

Acknowledgement

W. Tan was sponsored by Specialized Research Fund for the Doctoral Program of Higher Education (20020079007), China.

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Multimodel analysis and controller design for nonlinear processes

Abstract

Introduction

Section snippets

Gap metric

Gap metric for shaped plants

Compensator selection and multimodel controller design

Cases studied

Conclusions

Acknowledgement

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Journal of Process Control

Chemical Engineering Science

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Chemical Engineering Science

The gap metric: robustness of stabilization of feedback systems

IEEE Transactions in Automatic Controls

Gap metric concept and implications for multilinear model-based controller design

Industrial Engineering Chemical Research