Polynomials and polynomial matrices play an important role in linear system theory, from first principles multivariable linear systems are modelled by sets of differential equations in the input u and the output y of the form D(d/dt)y(t) = N(d/dt)u(t). D and N are polynomial matrices in the differential operator d/dt. Polynomial matrix models do not replace state space and frequency domain descriptions but provides a powerful additional tool.

The course shows how to systems may be analyzed and manipulated using this approach based on the mathematical properties of polynomial matrices. Many of the characteristics of polynomial matrices are reviewed, such as canonical and reduced forms, elementary operations and unimodular polynomial matrices, and divisors, primness and factorization.

The closed-loop properties of single-input single-output and multivariable systems may conveniently be analyzed using polynomial matrix theory. It is only a small step from stability analysis to the pole assignment problem. The pole assignment problem is intrinsically linked to linear polynomial matrix equations of the form DX + NY = C in the unknown polynomials or polynomial matrices X and Y. This crucial equation is extensively discussed.

Polynomial methods lend themselves very well to the frequency domain solution of famous and proven control system design methods such as LQG, H2 and deadbeat. These approaches to control system design and their application are thoroughly reviewed.

Whenever appropriate the Polynomial Toolbox 2 for MATLAB will be used for on-line illustrations and demonstrations. Copies of the slides will be made available to the participants.

1. Polynomials and polynomial matrices
   Preview; Polynomial; Polynomial Toolbox 2; Fields and rings; Ring of polynomials; Polynomial fractions; Polynomial equations; Polynomial matrices; Polynomial matrix fractions; Polynomial matrix equations.
2. Computer session 1 - Polynomial Toolbox
   Polynomial Toolbox 2; Handling polynomials and matrices. Handling basic models.
3. Polynomials in control systems
   IO systems and polynomials; Properties of SISO IO systems; feedback SISO systems; Running examples.
4. Discrete-time systems
   D-t systems and polynomials; Feedback design using d; Deadbeat regulation; Asymptotic tracking; Stochastic problems.
5. Continuous-time and MIMO systems
   Basic design tasks for c-t systems; MIMO systems and polynomial matrix fractions; Design for MIMO systems.
6. Computer session 2 - CAD based on polynomial methods
   Practical design using polynomial methods.
7. Future perspectives
   Robust control analysis and design using polynomial methods; Time-delay systems; n-D systems.