Industrial Applications of Genetic Algorithms:Genetic Algorithms for H<SUB>2</SUB> Controller Synthesis

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## Chapter 3
Genetic Algorithms for H2 Controller Synthesis

Keith J. Nicolosi

Masters Student, Aerospace Engineering
The University of Alabama;Tuscaloosa, Alabama
email: knicolos@eng.ua.edu

ABSTRACT

Modern control theory has evolved over the past decade, with H2, H∞, and H2/H∞ methods gaining recognition in controller synthesis problems. In the event of any plant changes or disturbances, these methods provide a procedure for multivariable controller tuning. There has been substantial research on the development of homotopy algorithms for solving the H2, H∞, and fixed-order mixed H2/ H∞ compensator synthesis problems [1]. In order for the homotopy algorithm to perform well, the initial gain matrix used in the algorithm must be very close to the optimal gain matrix, such that the initial gain matrix stabilizes the closed-loop system. Also, the homotopy algorithm requires the use of gradient information, which presents a problem with the discontinuities and vast multimodal, noisy search spaces of controller synthesis. As a remedy for this problem, a genetic algorithm is presented for the synthesis of H2compensators to evaluate the efficiency of the algorithm. The results of this study will be used to determine if a genetic algorithm can achieve the desired performance and robustness characteristics associated with the H2compensator synthesis problem. These characteristics include the requirement of closed-loop stability and minimization of the controller’s cost function.

INTRODUCTION

The last decade has witnessed a tremendous increase in the use of vibration isolation systems. These systems have been especially prominent in efforts to protect sensitive devices from vibrations apparent in the environment in which the devices operate. Some examples of the use of these vibration isolation systems include isolating delicate experiments from external vibrations, preventing the transmission of vibrations from rotating machinery in a satellite structure to the scientific sensors, and isolating automobile or aircraft frames from engine vibrations. Current interest in the development of vibration control has motivated research in modern and robust control theory. This is particularly true in those areas related to space structures.

Space shuttle missions have brought about an awareness of the need for autonomous or active vibration isolation for acceleration sensitive payloads.

These active vibration isolation systems will be vital for the success of the microgravity science experiments that will be performed on the International Space Station Alpha. A recent demonstration of active vibration isolation was performed by NASA using the STABLE (Suspension of Transient Acceleration By LEvitation) system. This application for microgravity payloads was the first actively controlled vibration isolation system to be flight tested [2].

Active vibration isolation is also needed for some precision spacecraft that have stringent pointing requirements. These spacecraft are usually flexible, light weight, low stiffness structures, and the stringent pointing requirements result in many closely spaced, lightly damped vibration modes. With the use of an autonomous, robust vibration isolation control system, the spacecraft has the ability not only to reduce the unwanted effects of pointing jitter or excessive motion, but also to re-design, or tune, the control system. These self-tuning spacecraft are able to virtually re-design their control system in the event of any plant changes or disturbances.

The need for this type of autonomous vibration isolation controller tuning was evident with a recent event that occurred to the Hubble Space Telescope [3]. Once the Hubble Space Telescope was in orbit, it began experiencing pointing disturbances due to thermal induced vibration on the solar arrays. A re-design of the control design model was performed with the use of on-orbit data. However, the re-design of the model took important operational time, as well as a substantial investment of manpower, both of which could have been saved with a more efficient on-orbit re-design system. Other events such as deep-space missions stand to benefit from robust re-design systems in the event of disturbances or changes to the spacecraft during its long journey. One approach to acquire this type of self-tuning for multivariable controllers is known as perturbation compensation, where the parameters of a dynamic compensator are tuned to account for plant changes and to provide levels of robustness and performance.

This type of self-tuning system would be advantageous for a robust, active vibration isolation controller. A control design methodology that is well suited for this type of controller synthesis is H2 controller synthesis. The objective of the H2 problem is to minimize the H2 norm on the closed loop transfer function from disturbance inputs to performance outputs, while guaranteeing closed-loop stability [1]. The re-design of the dynamic compensator for a H2 controller is a difficult task. Numerous approaches for H2 controller synthesis have been developed. However, some methods have several limitations that restrict them from optimal performance. Some of these limitations include long run-times, a dependence on stable, initial guesses which are close to the optimal design, and a dependence on derivative information. To circumvent this difficulty, a genetic algorithm (GA) will be developed for this formulation.

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