The state variable model of a dynamic system comprises a set of first-order differential equations that express the system behavior as time derivatives of the selected variables. The controller design for state state variable model involves selecting feedback gains for each (or a selection) of the state variables. The state feedback controller offers greater flexibility in system design compared to the output feedback controllers designed with transfer function models.
The continuous-time state variable model of a linear time-invariant (LTI) system can be converted to a discrtete-time LTI model by assuming a piece-wise constant input to the system. The conversion is easily done in MATLAB by using the ‘c2d’ command. The assumed sampling frequency should be five to ten times higher than the system bandwidth.
The control system design methods used with continuous-time systems can be extended to digital controller design of sampled-data systems that include sample and hold elements. The time-domain description of a sampled-data system comprisess a set of difference equations that can be easily solved by iteration.
The pole placement design using full state feedback can be similarly performed on discrete state variable models. The desired characteristic polynomial in the discrete case should have its roots inside the unit circle to ensure the stability of the closed-loop system. A discrete system model addionally allow the design of a deadbeat controller that places all roots of the closed-loop characteristic polynomial at the origin. The deadbeat controller ensures that the system response reaches steady state in exactly \(n\) iterations.
Compared to the root locus design that allows selective placement of the closed-loop poles of the characteristic polynomial, arbitrary pole placement is made possible through the use of state feedback design in the case of state variable models. In the following, we assume that the system to be controlled is of single-input single-output (SISO) type.