Ind. Eng. Chem. Fundam. zyxwvuts 1985, 24, zyxwvut 221-235 Closed-Loop Properties from Steady-State Gain Information 221 Pierre Grosdldler and Manfred Morarl Department of Chemical Engineering, California Institute of Technoicgy, Pasadena, California 9 1 725 Bradley R. HoHt Department of Chemical Engineering, University of Wisconsin, Madison, Wisconsin 53706 Closed-loop properties of open-loop stable multivariable systems are explored when the controllers include integral action. The studied properties comprise closed-loop stability, sensor and actuator failure tolerance, feasibility of decentralized control structures, and robustness with respect to modeling errors. All the results are based on steady-state gain information zyxwvut on&. The relationship between the new theorems and the Relative Gain Array (RGA) is elucidated. Introduction Modeling uncertainties and constantly changing oper- ating conditions make it very difficult to develop reliable dynamic models for chemical processes. Often, only steady-state gain information is available. In multi-input multi-output (MIMO) systems, these data may be repre- sented as a matrix of steady-state gains G(0). Since this matrix zyxwvutsrqpo G(0) is often the only information available on the system, any method that will allow the extraction of zyxwvuts useful feedback properties from it is clearly of great practical importance. The steady-state gain matrix can, for exam- ple, be used to evaluate a measure of steady-state inter- actions between controlled and manipulated variables. This measure of interactions, the Relative Gain Array (RGA), has found widespread acceptance both in industry and in academia since its introduction almost 20 years ago (Bristol, 1966; Shinskey, 1967). This popularity is chiefly a result of the array’s simplicity and empirically confirmed reliability. However, in spite of wide acceptance and abundant studies on practical applications (McAvoy, 1983), the RGA remains an empirical tool with little or no rig- orous theoretical basis. In this article, novel analysis techniques are presented to show that very important closed-loop properties can easily be extracted from the steady-state gain matrix. These techniques have been developed for open-loop stable systems subject to feedback by controllers which include integral action. The properties comprise closed-loop sta- bility, sensor and actuator failure tolerance, feasibility of decentralized control structures, and robustness with re- spect to modeling errors. It will be shown that some of the results can be rigorously expressed in terms of the RGA. This will show that the RGA has, in fact, sound theoretical justifications and is much more than a simple measure of interactions. The approach in this paper is as follows. We shall first provide a definition of the RGA and state some of its properties. It will subsequently be shown how the RGA can be used to predict the closed-loop instability of mul- tivariable control systems. Next, a new analysis technique will be presented which is able to extract much more stability information from the steady-state gain matrix ‘Department of Chemical Engineering, University of Wash- ington, Seattle, WA 98185. 0 196-43 13/85/ 1024-022 1 $0 1.5010 than the RGA. Next, the relationship between Right Half Plane (RHP) zeros and the RGA will be briefly examined. Finally, a new theorem will show how the RGA zyx can be zy used to predict the sensitivity of a multivariable system to modeling errors. Many of the results presented in this paper were con- jectured previously by other researchers. However, as we shall show, some of these conjectures are incorrect, some are partially correct, and some are correct but the argu- ments used to prove them are incorrect. To avoid any misunderstanding based on past misconceptions all the definitions and properties of the RGA known to date will be restated. Throughout this article, it is assumed that we are dealing with square (n X n), open-loop stable and nonsingular transfer matrices. These will be denoted with the bold-face letter G(s) while their individual elements will be denoted byg,(s). Moreover, GI+) will denote the matrix G(s) with its ith row and jth column removed. Steady-state values of these variables will be denoted by the same characters without the “s”, i.e. G(0) = G and gJ0) = g , . Inputs and outputs (manipulated and controlled variables) are de- noted by u, and yI, respectively, when dealing with scalars and by u and y when dealing with vectors of variables. In all cases, it is assumed that these variables represent de- viations from the steady state. Finally, the open right half complex plane will be denoted by C+ and the open left half complex plane by C-. Definition and Algebraic Properties of the RGA Definition of the RGA as an Interaction Measure. In this section, to demonstrate the basic assumptions un- derlying the RGA, its derivation will be carried out for 2 X 2 systems in extensive detail. Consider the transfer matrix G(s) with elements g,(s), inputs u, and outputs yl; (i, j = 1, 2). In the absence of any controller on the system, the transfer function between input u1 and output y1 is simply (Figure 1) where OL indicates “open loop”. Now, suppose that it is desired to control output y2 with input u2 by installiig a controller with the transfer function gC2(s) (Figure 2). In the presence of this controller, the transfer function between u1 and y1 changes. It becomes the sum of the open-loop transfer function g,,(s) and the 0 1985 American Chemical Society