Ind. Eng. Chem. Res. 2006, 45, 7151-7162
7151
PROCESS DESIGN AND CONTROL Algebraic Solution to H2 Control Problems. I. The Scalar Case Weidong Zhang,* Linlin Ou, and Danying Gu Department of Automation, Shanghai Jiaotong UniVersity, Shanghai 200030, People’s Republic of China
A frequently encountered problem in industrial control systems is how to control a plant with significant time delay. Dealing with this problem serves then as a starting point of most design methods, regardless of their configurations. In this paper, we give a treatment for the optimal design problem of control systems with time delay in a quadratic cost setting. This is accomplished by exploiting a simple, yet effective, parametrization of all stabilizing controllers, which allows us to compute the optimal controller for both stable and unstable plants with time delays. Differing from most of other methods, the proposed design procedure is conducted analytically, within the framework of algebra theory. The robustness of the closedloop system is also discussed. A simple quantitative tuning procedure is developed, which permits the tradeoff between conflict indices. Examples are given to illustrate the proposed method. 1. Introduction The Linear Quadratic Gaussian (LQG) optimal control laws remain some of the most popular classes of advanced control design methods. In current terminology, this class of LQG optimal problem is often referred to as H2 optimization. The history of H2 optimization dates back to Wiener filtering1 and its early applications for control.2 Optimal control in the state space era contributed to the LQ and LQG problems. The H2 optimization itself is a spinoff of the robust control period, although it is seldom recognized as a robust control tool.3 This problem has been examined thoroughly in many texts, for example, Anderson and Moore4 and Kailath.5 A recent mathematical treatment is presented in Zhou et al.6 The best-known solution to the H2 optimization problem is a state space solution. A systematic introduction of this method can be found in Zhou et al.6 On the basis of the Internal Model Control (IMC) structure, Morari and Zafiriou7 provided an elegant solution. Another solution to the H2 optimization problem is the polynomial solution,3,8 which is based on factorization over polynomial matrixes. Note that the above-mentioned solutions to the H2 optimization problem are not entirely equivalent. Because of the different mathematical tools applied, the problems are solved at different levels of generality under different assumptions.3 The aforementioned methods have an emphasis on rational plants, although the time delay has been considered in some early papers. Time delay often appears in many real control systems. The stability issue and the performance of control systems with time delay are, therefore, both of theoretical and practical importance. The subject has been of interest to academia and practitioners for several decades. For recent progress, one can refer to Gu et al.9 and Niculescu and Gu.10 The aim of this paper is to develop a design procedure for singleinput/single-output (SISO) plants with a time delay, based on Zhang,11,12 and discuss several special design problems. In this paper, attention is given to both theoretical and * To whom correspondence should be addressed. Tel.: +86.21. 34204019. Fax: +86.21.54260762. E-mail:
[email protected].
practical aspects. Basically, we discuss the controller design problem in the framework of algebra theory. There are three steps in our design: (1) Parametrize all stabilizing controllers. (2) Analytically design the optimal controller. (3) Develop a simple tuning method for quantitative performance and robustness. For SISO design, the proposed method is, in fact, internally related to the well-known IMC design of Morari and Zafiriou.7 However, an alternative design procedure with quantitative tuning rules is developed, with an emphasis on systems with time delay and several special design problems, such as optimal disturbance rejection and optimal control of plants with state delays, are studied. In Part II,13 we will show that the proposed method can be extended to the multi-input/multi-output (MIMO) case. For MIMO design, the proposed method is thoroughly different from the IMC results. Our solution offers implementations that are suitable for a wide and flexible class of useful design applications. A typical design example is as follows. Given an uncertain plant,
G(s) )
Ke-θs τs + 1
where K = 4.635-5.665, θ = 2.52-3.08, and τ = 1.62-1.98: (1) Design a controller such that the rise time is as fast as possible and the overshoot is
