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092 - Control 2004 University of Bath UK September 2004...

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ROBUST H2/LQG DESIGN FOR A CROSS COUPLED PROCESS K. Cooney 1 , G. Caffrey 2 , E Coyle 3 Dublin Institute of Technology Ireland E-mail: [email protected] 1 , [email protected] 2 , [email protected] 3 Keywords: LQG, H 2 Control, Modern Paradigm. Abstract This paper examines the application of modern control techniques on a cross-coupled multivariable process model perturbed by a process disturbance. The designs use time and frequency domain weights for performance and robustness criteria while minimising the generalised energy. 1. Introduction Lewis [6] stated that “naturally occurring systems exhibit optimality in their motion and so it makes sense to design man-made control systems in an optimal fashion”. Indeed the notion of optimal control has been around for many years but it is only relatively recently that practical solutions to important control problems have been available. Kalman introduced important techniques in linear quadratic design for both control and estimation problems using Lyapunov equations and state variable descriptions. Control theory entered the modern era and has grown considerably since. Kwakernaak and Siven [5], Saberi et al [9], Lublin and Athans [7] outline the solution to a range of deterministic optimal control problems in the time domain. Most notable are the linear quadratic regulator LQR (state feedback control) and linear quadratic gaussian regulator LQG (observer based compensation) problems, which achieve optimality (and a certain degree of robustness) by minimising a quadratic cost function thus keeping the generalised energy required to regulate the system small. Further algorithms such as loop transfer recovery LTR were developed by Doyle and Stein [3] to improve the robustness of the more practical LQG when compared to the more robust but less practical LQR. Unfortunately, these methods do not tackle robustness issues such as modelling errors and sensitivity to process parameter variation directly as they are time domain approaches. It is possible to adjust the transient response in an iterative search but loop shaping the important singular values of the closed loop system is not the primary focus. A frequency domain approach is obviously more applicable to a robustness problem. How then to achieve both optimal H 2 control and robust stability to parameter changes in the process model. Solutions have been proposed by Doyle et al [2], Francis [4] and Lublin et al [8] to achieve H 2 and H norm specifications. Other design methods, which can be used to achieve this type of control, are based on coprime factors and yoala parameterisation, but this paper will discuss the design of state space, central multivariable controllers from both the LQ approach and the more modern robust design of H 2 and is structured to show the close mathematical relationship between them. The modern paradigm outlined by Boyd and Barrett [1] is well suited to visualise both types of problem and perform the synthesis.
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