Eight Irrationalities of Basic State Space Control System Design

Abstract

In basic state space control system design, the state feedback control Kx(t) is designed assuming the system state x(t) is available. This design has the following eight irrationalities. 1) The design of Kx(t) ignores the fact that a big effort such as an observer is needed to generate the Kx(t) signal. 2) This design ignores the parameters of the realizing observer and the parameter C of the open loop system model (A,B,C). 3) To guarantee the generation of such Kx(t) signal, a state observer is needed to generate all n elements of x(t), while the number of signals of Kx(t) is much less than n. 4) For most open loop systems, their state observers must have the information of open loop system input, thus abandoning the well established output feedback compensator structure of the classical control theory. 5) It is proven that only an output feedback compensator can realize (if it can generate the signal) the loop transfer function of Kx(t)-control, K(sI - A)-1B, which determines the sensitivity/robustness properties of the feedback system. 6) An observer has a dynamic part which determines the observer state and implicitly x(t), and an output part which determines the observer output Kx(t). While the dynamic part is much more important than the output part, the latter (Kx(t)) is designed before the former. 7) The only other existing basic form of control is static output feedback control KyCx(t), which is a special and extremely constrained Kx(t)-control (K must be linear combinations of only m rows of C). At the other extreme, no constraint is attached to the K of existing Kx(t)-control. 8) The order of KyCx(t)-control is fixed at zero, an extreme low, while the order of (state) observers of Kx(t)-control is fixed at least n - m, an extreme high. While each of these eight irrationalities is obvious enough to demand a fundamental adjustment of the basic state space control design of the past forty years, a new and simple design approach accomplishes this adjustment and eliminates these irrationalities all at once