Optimal Multivariable PID Control Based on LMI Approach
In the previous two chapters, design methods for multivariable PID control have been demonstrated based on IMC and dominant pole placement, respectively. They are in line with traditional design methods commonly used for process control and are broadly ac
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In the previous two chapters, design methods for multivariable PID control have been demonstrated based on IMC and dominant pole placement, respectively. They are in line with traditional design methods commonly used for process control and are broadly accepted in practice. On the other hand, most of the developments in the modern control theory have not been applied to PID control due to their complexity or inconvenience of use. It is our wish to bridge the modern control theory and the industrial control practice which could benifit from the advancement in the former. This chapter aims to make an initial step towards solving this problem. The basic idea is to transform the PID control to the equivalent static output feedback (SOF) control by augmenting to the process with some new state variables induced by the PID controller such that the well established results in SOF field can be employed to design a multivariable PID controller for various specifications such as stability, H2 /H∞ performance and maximum output control.
7.1 Introduction The static output feedback plays a very important role in control theory and applications. Recently, it has attracted considerable attention (see e.g. [159, 85, 86, 160, 161, 162, 163] and references therein). Yet, it is still left with some open problems. Unlike the state feedback case, a SOF gain which stabilizes the system is not easy to find. Linear matrix inequality (LMI) [89] is one of the most effective and efficient tools in controller design and a great deal of LMI-based design methods of SOF design have been proposed in the last decade [90, 164, 91, 165, 166, 167, 168, 169, 170, 171, 172]. Among these methods, an iterative linear matrix inequality (ILMI) method was proposed by Cao et al. [90] and later employed to solve some multivariable PID controller design problems [79, 80]. In this context, a new additional matrix-valued variable is introduced so that the involved stability condition becomes conservative (sufficient but far from necessary). The iterative algorithm in [90] tried to find a sequence of the additional variables such that the relevant sufficient condition is close to the necessary and sufficient one. The similar idea is used in the so-called substitutive LMI method in [172]. In both works, the additional matrix variables are updated at the current Q.-G. Wang et al.: PID Control for Multivariable Processes, LNCIS 373, pp. 167–202, 2008. c Springer-Verlag Berlin Heidelberg 2008 springerlink.com
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7 Optimal Multivariable PID Control Based on LMI Approach
iteration step using the decision variables (matrix-valued) obtained in the preceding step. With additional variables, the dimensions of the LMIs become higher. It is possible that the decision variables obtained in the preceding step can be used in the next one directly without introducing the additional variables and the dimensions of the LMIs need not be increased. In addition, we will develop some efficient way to get suitable initial values for some decision variable in the iterative proce
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