From Classical to State-Feedback-Based Controllers

A utomatic control teaching can be fragmented, comprising different streams that seem to be disconnec t ed or even competing. The appeal of new theories and concepts may encourage students to forget that the key issue in the discipline is the control design problem. If two or more different approaches provide a good solution to that problem, then a strong connection should exist between them. If such connections can be established, this should help students better understand the underlying concepts in the control design problem. In this article, we consider continuous-time systems and explore the connections between single-input, single-output (SISO) linear classical controllers (i.e., those defined by transfer functions) and the control design approach based on feedback of an estimated system state. The aim of this article is to provide insight for automatic control teaching and control system design. The transition from estimated state feedback to classical control is well known; however, to the authors’ knowledge, the reverse transition has not been previously articulated for the general case. As is well known, the combination of an observer together with feedback of the observed state can be reduced to an equivalent classical control loop in which the controller is expressed in transfer function form [1]. For instance, assume that the state-space plant model is given by the four-tuple (Ao , Bo , Co , 0), a full-order state observer is built with gain J, and the observed state is fed back with gain K. Hence, the equivalent classical controller has a transfer function C s ( ) and an associated state-space model given by the four-tuple (Ac, Bc, Cc, Dc), where