×

zbMATH — the first resource for mathematics

Discrete-time pole placement with stable controller. (English) Zbl 0766.93022
Summary: The classical polynomial approach to pole placement involves the computation of a particular solution of a given Diophantine equation. The solution corresponds to a proper controller whose order is fixed by the number of nonminimum phase zeros of the plant transfer function, by its order, and by the desired closed-loop characteristic polynomial. This controller might be unstable for some desired closed-loop characteristic polynomials, even though the plant can be stabilized by a stable controller. Now, unstable controllers are known to deteriorate closed- loop system performances, compared with stable ones. Hence, we propose a procedure to build, for a specific class of plants, a stable pole placement controller when the classical pole placement controller is unstable. The properties of both stable and unstable controllers are compared via simulations. It appears that, for the considered plants, the stable controller improves at least one of the following properties, compared with its unstable counterpart: step response transient, gain margin, measurement noise rejection. The results obtained with our procedure are also compared with the results obtained via Middleton’s approach, which consists in modifying the desired closed-loop poles so as to obtain a stable and minimum phase controller with the classical pole placement method. Finally, our pole placement design with stable controller is applied to solve a particular simultaneous partial pole placement problem.

MSC:
93B55 Pole and zero placement problems
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
93C62 Digital control/observation systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Åström, K.J.; Wittenmark, B., Self tuning regulators based on pole-zero placement, (), 120-130
[2] Åström, K.J.; Wittenmark, B., Computer controlled systems: theory and design, (1984), Prentice Hall Englewood Cliffs, NJ · Zbl 0217.57903
[3] Åström, K.J.; Wittenmark, B., Adaptive control, (1989), Addison-Wesley · Zbl 0217.57903
[4] Blondel, V.; Gevers, M.; Mortini, R.; Rupp, R., Simultaneous stabilization of three or more plants: conditions on the positive real axis do not suffice, () · Zbl 0825.93682
[5] Dorato, P.; Park, Hong Bae; Li, Y., An algorithm for interpolation with units in H∞, with application to feedback stabilization, Automatica, 25, 427-430, (1989) · Zbl 0684.93057
[6] Doyle, J.C.; Stein, G., Multivariable feedback design: concepts for a classical/modern synthesis, IEEE trans. aut. control, AC-26, 4-15, (1981) · Zbl 0462.93027
[7] Emre, Erol, Simultaneous stabilization with fixed closed-loop characteristic polynomial, IEEE trans. aut. control, AC-28, 103-104, (1983) · Zbl 0499.93049
[8] Freudenberg, J.S.; Looze, D.P., Right half plane poles and zeros and design tradeoffs in feedback systems, IEEE trans. aut. control, AC-30, 555-565, (1985) · Zbl 0562.93022
[9] Ghosh, B.K., Simultaneous partial pole placement: a new approach to multimode system design, IEEE trans. aut. control, AC-31, 440-443, (1986)
[10] Middleton, R.H.; Goodwin, G.C., Digital control and estimation: a unified approach, (1990), Prentice Hall Englewood Cliffs, NJ · Zbl 0636.93051
[11] Middleton, R.H., Trade-offs in linear control system design, Automatica, 27, 281-292, (1991) · Zbl 0753.93024
[12] Mohtadi, C., Bode’s integral theorem for discrete-time systems, (), 57-66 · Zbl 0734.93037
[13] Saeks, R.; Murray, J.; Chua, O.; Karmokolias, C.; Iyer, A., Feedback system design: the single-variate case—part I, Circuits, systems and signal processing, 1, 137-169, (1982) · Zbl 0491.93021
[14] Saeks, R.; Murray, J.; Chua, O.; Karmokolias, C.; Iyer, A., Feedback system design: the single-variate case—part II, Circuits systems and signal processing, 2, 3-34, (1983) · Zbl 0511.93030
[15] Vidyasagar, M., Control system synthesis: a factorization approach, (1987), MIT Press MA · Zbl 0655.93001
[16] Vidyasagar, M., Some results on simultaneous stabilization with multiple domains of stability, Automatica, 23, 535-540, (1987) · Zbl 0624.93055
[17] Vidyasagar, M., A state-space interpretation of simultaneous stabilization, IEEE trans. aut. control, AC-33, 506-508, (1988) · Zbl 0638.93064
[18] Vidyasagar, M.; Viswanadham, N., Algebraic design techniques for reliable stabilization, IEEE trans. aut. control, AC-27, 1085-1095, (1982) · Zbl 0496.93044
[19] Youla, D.C.; Bongiorno, J.J.; Lu, C.N., Single-loop feedback stabilization of linear multivariable dynamical plants, Automatica, 10, 159-173, (1974) · Zbl 0276.93036
[20] Wei, K., Stabilization of a linear plant via a stable compensator having no real unstable zeros, Systems & control letters, 15, 259-264, (1990) · Zbl 0724.93066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.