Design of reduced-order observers with precise loop transfer recovery

This paper concerns the design of reduced-order observers for systems in which the number of measurements is more than the number of controls. We develop an algorithm that applies to regular systems that have no transmission zeros. The algorithm uses eigenstructure assignment whereas other approaches use Kalman filter methods. The advantages of this approach are the following: 1) precise loop transfer recovery rather than approximate loop transfer recovery, 2) finite observer gain rather than asymptotic observer gain, and 3) modest computational tools and operations counts. Case studies are presented illustrating these features. T HE problem of designing an observer that can achieve loop transfer recovery (LTR) has received continuous at- tention since Doyle1 presented an example with a Kalman-fil- ter-based observer (linear quadratic Gaussian (LQG)) design lacking robustness even though the full-state feedback (linear quadratic regulator (LQR)) controller had impressive robust- ness properties, namely gain margins of - 6 dB to + oo dB and phase margins of ±60 deg.2 To alleviate this problem, Doyle and Stein3 developed a robustness recovery procedure in which fictitious process noise is added to the input in the design model. The LQR robustness properties are preserved with the loop open at the input since the loop transfer function is re- covered asymptotically as the intensity of the fictitious pro- cess noise is increased. Stein and Athans4 call this procedure LQG/LTR. There have been further developments and appli- cations of LQG/LTR by a number of workers. Madiwale and Williams5 extended the theory to reduced-order observer- based LQG designs for nonsquare, minimum phase, and left invertible plants. Calise and Prasad6 developed an approach for designing a fixed-order compensator and obtained an ap- proximate LTR for nonsquare, minimum phase systems. It is similar to the full-order compensator design of ordinary LQG/ LTR. Fu7 developed the necessary and sufficient condition for exact LTR employing a general feedback structure for model matching control that includes the observer-based state feedback control as a special case. He formulated the prob- lem of model matching by assigning certain stable matrices such that the desired loop transfer function became equal to the associated loop transfer function of the closed-loop input- output transfer function. He claimed that exact LTR is achiev- able under the conditions for strong stabilization of systems. Tsui8'9 introduced a theoretical analysis of an entirely new approach to the problem of loop transfer recovery. His ap- proach was to minimize the observer gain to the system input by observer pole selection. Furthermore, he claimed that this new approach aimed directly at achieving the necessary and sufficient condition of LTR. The purpose of this paper is to present a computational algorithm for the solution of the constrained matrix Sylves- ter equation that arises in Tsui's approach for designing ro- bust reduced-order Luenberger observers.10 Precise LTR is achieved with finite observer gain for regular nonsquare plants with no transmission zeros, for which the number of sensors is greater than the number of controls. Other approaches have been unable to attain these results. The method developed here provides freedom to select eigenvalues and eigenvectors for the observer.