Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems

In Chapters 8 and 9 of this book we have designed dynamic controllers such that the closed-loop systems display the desired transient response and steady state characteristics. The design techniques presented in those chapters have sometimes been limited to trial and error methods while searching for controllers that meet the best given specifications. Furthermore, we have seen that in some cases it has been impossible to satisfy all the desired specifications, due to contradictory requirements, and to find the corresponding controller. Controller design can also be done through rigorous mathematical optimization techniques. One of these, which originated in the sixties (Kalman, 1960)—called modern optimal control theory in this book—is a time domain technique. During the sixties and seventies, the main contributor to modern optimal control theory was Michael Athans, a professor at the Massachusetts Institute of Technology (Athans and Falb, 1966). Another optimal control theory, known as , is a trend of the eighties and nineties. optimal control theory started with the work of Zames (1981). It combines both the time and frequency domain optimization techniques to give a unified answer, which is optimal from both the time domain and frequency domain points of view (Francis, 1987). Similarly to optimal control theory, the so-called optimal control theory optimizes systems in both time and frequency domains (Doyle et al., 1989, 1992; Saberi et al., 1995) and is the trend of the nineties. Since optimal control theory is mathematically quite involved, in this section we will present results only for the modern optimal linear control theory due to Kalman. It is worth mentioning that