Pole Assignment for Uncertain Systems via Symbolic Algebra

This paper presents the application of symbolic algebra techniques to the MATHEMATICA implementation of a set of output-feedback pole assignment algorithms, for systems characterised by parametric uncertainty. For multivariable systems there may be more than one feedback matrix solutions leading to the same closed-loop poles based on the same algorithm used. Thus overparameterised solutions are sought by generalising the existing algorithms with extra degrees of freedom retained in the symbolic variables. The general parametric form of output-feedback compensators is developed in terms of the uncertain parameters and symbols representing the extra degrees of freedom. The implementation of three output-feedback pole assignment techniques is presented, with the theory brie y introduced and examples illustrating the effectiveness of the algorithms described. Control Systems Group, Department of Electronic and Electrical Engineering, Loughborough University, Loughborough, LE11 3TU, UK, {x.zheng, a.c.zolotas}@lboro.ac.uk; corresponding author : X. Zheng. Control Systems Centre, School of Electrical and Electronics Engineering, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK, [email protected]. 1 Author final version (International Journal of Control, Vol. 79, No. 11, November 2006, 1431-1446) Notation n Number of states m Number of inputs l Number of outputs A n× n system matrix B n×m input matrix bi i thcolumn of the input matrix C l × n output matrix [A,B] A system defined by A and B matrices [A,B,C] A system defined by A, B and C matrices Φ Controllability matrix of a system In n× n identity matrix i The imaginary number of √ −1 q Order of the output-feedback compensator q r × 1 uncertain parameter vector r Degree of uncertainties At Transpose of matrix A A Inverse of matrix A |A| Determinant of matrix A Adj(A) Adjoint matrix of matrix A ⌊x⌋ The nearest integer lower than or equal tox ⌈x⌉ The nearest integer greater than or equal tox Γ The set of desired closed-loop poles φi Constants defined as φi = ⌊ max (m,l) min (m,l−i+1 ⌋ φ A constant defined as φ = ∑min (m,l) i=1 φi G(s) Transfer function matrix of the open-loop system F (s) Output-feedback compensator matrix SISO Single Input Single Output SIMO Single Input Multi Output MISO Multi Input Single Output MIMO Multi Input Multi Output