Introduction
In the previous two chapters, the classical control theory was discussed and the concept of transfer function was used to describe the dynamic behavior of various systems. In practice, the transfer function of higher order than three or four becomes tedious and it is better to use state variable control theory. It was stated that with PID control, three parameters can be adjusted to design control systems. With three parameters, all roots of characteristic equation can not be adjusted to achieve desirable transient and steady state behavior. In practice, it is only possible to make one or two roots to become dominant in response and a compromise between the transient response and steady state error has to be made. The derivative term always amplifies the noise in the practical systems and is not recommended. Instead a lead-lag network produces a better response.
State variable feedback control theory makes use of matrices to describe control systems. This makes it possible to write the governing differential equation in compact form. The use of state variable feedback makes it possible to adjust the location of all roots to desirable position in the s-plane. The practical limitations of measuring all state variables, and the nonlinearities of transducers and saturation of amplifiers limits as where the roots can be located in the s-plane.
Observer may be used to predict state variables from the measurement of a single or few state variables. This makes it possible to control practical systems with many state variables.
In practice, if high performance is not required it is better to model servo systems with second or third order transfer function by making simplifying assumptions. When high performance is required, many factors such as inductance and the compliance of the transmission mechanism must be considered. These considerations make the model very complicated and state variable control theory is recommended as the concept of transfer function becomes very tedious.
In this chapter, the concept of state variables is described from the transfer function but in the later chapters it will be shown that state variables can be defined right from the beginning when the governing equations for each element is written and there is no need to find the overall transfer function.The state variables are not unique and different forms of state variables can be defined. In modeling high-order system, some arrangements have to be made so that the state variables are measurable for feedback. Effort must be made to mini- mize the state variables that are not measurable and have to be predicted from an observer.