European Journal of Control (2001)7:58-66 © 2001 EUCA European Journal of Control On Stability of Dynamical Controllers Using Pole Assignment G. P. Liu I, S. Dal ey 2,* and G. R. Duan 3 ,t 'School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK; 2ALSTOM Power Technology Centre, Cambridge Road, Leicester LE8 6LH, UK; 3Department of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Belfast BT9 5AH, UK This paper is concerned with the stability of dynamical controllers using pole assignment. Three types of dynamical controllers are considered: polynomial out- put-feedback dynamical controllers, state-space out- put-feedback dynamical controllers and observer based state-feedback controllers. The design formulations of stable dynamical controllers are presented using polynomial, output-feedback, state-feedback and pole assignment techniques. They ensure that the closed- loop poles of the systems are assigned in the desired set and also the dynamical controllers are stable. Keywords: Dynamical controller; Pole assignment; Stability 1. Introduction As advanced control design techniques become more readily accepted in industrial process control, it is necessary to pay more attention to the properties of the controllers being implemented. The main problem which exists in practice is the stability of the stabi- lising controllers. Some design strategies, such as, H OO control and LQG, may produce controllers which are unstable, but which stabilise the closed- loop systems. Despite the fact that controller stability is often overlooked in the design strategy, it is of fundamental importance since the practical Correspondence and offprints request to: G.P. Liu, School of Mechanical, Materials, Manufacturing Engineering and Manage- ment, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Tel.: +441159514003; Fax: +441159514000; Email: guoping.liu@nottingham.ac.uk 'Fax: +44 1162015464; Email: steve.daley@power.alstom.com tEmail: g.r.duan@qub.ac.uk implementation of an unstable controller is extremely difficult. The undesirable forward path instability which results from the unstable controller might be- come important if saturated signals drive the control- ler unit for a period of time. Thus, although the unstable stabilising controllers may be optimal in some sense, it is not practical. The literature for the construction of stable stabi- lising controllers (also called strong stabilisation con- trollers) has grown in H oo and H 2 /LQG optimal control areas [3,14,25]. The stable stabilising control- ler problem was first addressed for single-input sin- gle-output (SISO) systems by [27]. Some attempts to provide such a stable controller using the Nevan- linna-Pick interpolation technique were given for SISO systems in [1,4,11], and for multi-input multi- output systems (MIMO) [23,24]. Some modifications in the Riccati equations that will result in a stable controller have been suggested in [8,9]. A number of techniques for dynamical controller design using pole assignment have been developed in recent years [2,19]. However, the resulting controllers, although internally stabilising the system, may them- selves not be stable [18]. As a result, some design for- mulations which exploit design freedom to guarantee both internal stability and controller stability are sought. This paper therefore addresses stable dynamical controller design using three pole assignment techni- ques, which are state-feedback, output-feedback and polynomial pole assignment. The stability of three types of dynamical controllers, which are observer based state-feedback controllers, state-space output- feedback dynamical controllers and polynomial Received 25 November 1998. Accepted in revised form 10 February 2001. Recommendedby M. Sebek and A. 1 sidori