Feedback Control Design by Lyapunov’s Direct Method Aníbal M. Blanco, José L. Figueroa and J. Alberto Bandoni Planta Piloto de Ingeniería Química, UNS - CONICET, Camino La Carrindanga, Km. 7 8000 Bahía Blanca, ARGENTINA, fax: +54 291 486 1600, E-mail:bandoni@plapiqui.edu.ar The purpose of this work is to introduce a systematic technique for feedback control design. Based on Lyapunov´s stability theory a Non-linear Programming Problem is formulated in order to obtain an optimal closed loop design in some sense. The proposed technique is applied to the feedback control design of a classic stirred tank reactor. 1.INTRODUCTION In this work we deal with the so-called first and second fundamental problems in the theory of automatic control, as introduced by Letov (1961). The first fundamental problem, or stability problem, consists of determining the values of the parameters of the controller, which are required to guarantee stability of a steady state point. The second fundamental problem, or control quality problem, deals with the character of the convergence of the motion in terms of response speed. Both aspects will be considered in the control design formulation, by inclusion of Lyapunov’s stability conditions. Lyapunov´s direct (or second) method, is the most general available tool for assessing stability of non-linear dynamic systems, described by a set of differential equations. It is based on an energetic approach and neither explicit nor numerical solutions of the equations are required. Besides the stability issue, it also admits the consideration of transient response speed in an indirect way. 2. LYAPUNOV´S STABILITY THEORY In the present section, most relevant issues of Lyapunov´s Stability Theory are outlined. See, for example, Vidyasagar (1993) for a complete analysis. 2.1. Lyapunov’s linearization method Consider the free, autonomous system: d dt x fx = ( ), f0 0 () = (1) where x , represents the deviation state vector. We can write ) x ( f x A ) x ( f 1 + = , where A f x x 0 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ∂ ∂ , and ) x ( f 1 is the residual. Then, it can be proved (Vidyasagar, 1993) that 0 is