On the Lyapunov Design of Systems with Zeros in the Right-Half Plane

The effect of a zero in the right-half plane on the controller derived through the stability theorem of Lyapunov is considered through a simple illustrative example. It is shown that the controller should have the form of a linear saturating function instead of the bang-bang form for the minimum-phase systems. Many industrial processes are known to exhibit a dead-time delay in their txansfer characterist.ics. It is also known t,hat such delay effects can often be approximated by placing a zero of t,he transfer function in t.he right-half plane (RHP). This usudy causes an unsatisfactory design of linear controllers for these plants since one of t,he root loci branches off towards t.he RKP. A design technique which has been successfully employed to derive cont.rollers that ensures system stability is based on the stability theorem of Lyapunov. Several interesting applicat.ions of this approach have been reported in the literature (see, for example, Grayson [ l ] ) . All of the systems considered, however, are of t.he minimum phase t.ype. It has been established that the desired cont.roller for such system is of the bang-bang t,ype. The aim of this correspondence is to point out that a RHP zero causes t.he optimal controller to have a saturating form. To illust.rate the point consider a simple feedback regulator system as shown in Fig. 1. The control signal u.(e) is yet unknown and should be so chosen as to ensure that the controlled system remains asymptotically stable with the error of regulation suitably bounded. (As discwed in [ l ] this choice of u ako minimizes a quadratic cost function.) The plant transfer function G(s) is taken to be G(s) = (s +~a). This corresponds to the system differential equation y + UI~ + ~oy z i bu. Defining t.he state variables x1 = y