Automatica, Vol.23, No. 5, pp. 601-610, 1987 Printedin GreatBritain. 0005-1098/87 $3.00+ 0.00 Pergamon JournalsLtd. ~) 1987InternationalFederationof Automatic Control Robust Pole Assignment** Y. C. SOH,, R. J. EVANS,, I. R. PETERSEN§ and R. E. BETZ, The interval matrix inequalities provide a neat characterization of pole placement controllers that place closed-loop poles within desired intervals for plants with unknown-but-bounded parameter uncertainties. Key Words--Pole placement; robust control; stability; uncertain interval systems (non-standard); interval matrix inequalities (non-standard). Abstract--This paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots, and applies them to the problem of robust pole-placement. We formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed-loop poles within desired intervals for plants with unknown-but- bounded parameter uncertainties. The algorithms are computationally reasonable and provide a useful addition to currently existing control CAD tools. 1. INTRODUCTION POLE-PLACEMENT is now becoming an accepted technique for controller design (Astrbm and Wittenmark, 1984). There remains, however, some difficulty with the robustness of controllers designed using this method. The problem arises when selecting the desired closed-loop poles. It is well known in classical design procedures that in order to design a robust controller the choice of closed-loop poles depends critically on the plant transfer function (Horowitz, 1963). An arbitrary choice of stable closed-loop poles can lead to a very poor controller design for certain plants. Some assistance in the choice of desired closed-loop poles is given in ~strbm and Wittenmark (1984) and /~strbm (1980) using plant knowledge and properties of the known disturbances. However, the robustness issues are still hidden, making good pole-placement design difficult in practice. In this paper we present techniques which are aimed at improving this situation. We use new results concerning * Received 1 January 1986; revised 30 September 1986; revised 9 March 1987. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor J. Ackermann under the direction of Editor H. Kwakernaak. ¢ Work supported by the Australian Research Grants Scheme. Technical Report No. EE8525. Department of Electrical and Computer Engineering, University of Newcastle, New South Wales 2308, Australia. § Department of Electrical and Electronic Engineering, Australian Defence Forces Academy, Northcott Drive, Campbell, A.C.T. 2600, Australia. 601 polynomials with interval roots, and extend results on interval matrix inequalities (Evans et al., 1983) to derive procedures for designing pole-placement controllers with a prescribed degree of robustness. Our approach is different from sensitivity-based design methods (Berger, 1984; Biernacki, 1986; Kautsky et al., 1985; Kantor and Andres, 1983; Soh and Berger, 1984). These methods seek to find controllers which minimize (in some sense) the sensitivity of achieved closed-loop poles to small perturbations in plant parameters. Our approach also differs from the recent interval polynomial-based approaches initiated by the remarkable result of Kharitonov (1978); see for example Evans and Xie (1985), Soh and Evans (1985a, b). The idea in this case is to model the plant as an interval matrix or interval polynomial and then deter- mine a controller which will stabilize the class of plants. The Kharitonov result gives simple conditions for checking the stability of a class of polynomials with interval coefficients. These results are then used to design controllers which stabilise a class of state-space plants described by interval matrices (Soh and Evans, 1985a,b). The difficulty with this method, however, is the considerable computational complexity which arises because of the complex relationship between the state controller coefficients and the plant closed-loop characteristic polynomials. An alternative to the above approach is to use Lyapunov methods to check the stability of interval matrices (Yedavalli, 1985, 1986). In concept our approach is vaguely similar to the multi-model approaches (Ackerman, 1985; Ghosh, 1986) which seek via computer-aided techniques, to place closed-loop poles inside specified regions for a specified class of plants described by certain parameter variations. There is also some sense in which our maximally robust design is related to the minimax frequency domain methods (Zames and Francis, 1983;