Distributed stabilisation of spatially invariant systems: positive polynomial approach
- 301 Downloads
The paper gives a computationally feasible characterisation of spatially distributed controllers stabilising a linear spatially invariant system, that is, a system described by linear partial differential equations with coefficients independent on time and location. With one spatial and one temporal variable such a system can be modelled by a 2-D transfer function. Stabilising distributed feedback controllers are then parametrised as a solution to the Diophantine equation ax + by = c for a given stable bi-variate polynomial c. The paper is built on the relationship between stability of a 2-D polynomial and positiveness of a related polynomial matrix on the unit circle. Such matrices are usually bilinear in the coefficients of the original polynomials. For low-order discrete-time systems it is shown that a linearising factorisation of the polynomial Schur-Cohn matrix exists. For higher order plants and/or controllers such factorisation is not possible as the solution set is non-convex and one has to resort to some relaxation. For continuous-time systems, an analogue factorisation of the polynomial Hermite-Fujiwara matrix is not known. However, for low-order systems and/or controller, positivity conditions on the closed-loop polynomial coefficients can be invoked. Then the computational framework of linear matrix inequalities can be used to describe the stability regions in the parameter space using a convex constraint.
KeywordsMultidimensional systems Algebraic approach Control design Positiveness Convex optimisation
Unable to display preview. Download preview PDF.
- Augusta, P., Hurák, Z., & Rogers, E. (2007). An algebraic approach to the control of statially distributed systems—the 2-D systems case with a physical application. In: Preprints of the 3rd IFAC symposium on system, structure and control. IFAC.Google Scholar
- Bose, N. (Ed.). (1985). Multidimensional systems theory: Progress, directions and open problems in multidimensional systems. D. Riedel Publishing Company, iSBN 90-277-1764-8.Google Scholar
- Cichy, B., Gakowski, K., & Rogers, E. (to be published). Iterative learning control for spatio-temporal dynamics using crank-nicholson discretization. Multidimensional Systems and Signal Processing.Google Scholar
- D’Andrea, R., & Dullerud, G. E. (2003). Distributed control design for spatially interconnected systems. IEEE Transactions on Automatic Control 48, 9.Google Scholar
- Justice, J. H., & Shanks, J. L. (1973). Stability criterion for N-dimensional digital filters. IEEE Transaction on automatic control, pp 284–286.Google Scholar
- Kamen, E. W. (1978) Lectures on algebraic systems theory: Linear systems over rings. Contractor report 316, NASA.Google Scholar
- Krstic, M., & Smyshlyaev, A. (2008). Boundary control of PDEs: A course on backstepping designs. : SIAM.Google Scholar
- Löfberg, J. (2004). Yalmip: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, http://control.ee.ethz.ch/~joloef/yalmip.php.
- Mastorakis, N. E. (1997) A new stability test for 2-D systems. In: Proceedings of the 5th IEEE Mediterranean Conference on control and systems (MED ’97).Google Scholar
- Rami, M. A., & Henrion, D. (2010). Inner approximation of conically constrained sets with stability aplication. to be published.Google Scholar
- Rogers E., Gałkowski K., Owens D. H. (2007) Control systems theory and applications for linear repetitive processes, Lecture notes in control and information sciences, vol 349. Springer, BerlinGoogle Scholar
- Rouchaleau, Y. (1972) Linear, discrete time, finite dimensional, dynamical systems over some classes of commutative rings.Google Scholar
- Šebek, M. (1994). Multi-dimensional systems: Control via polynomial techniques. Prague, Czech Republic: Dr.Sc. thesis, Academy of Sciences of the Czech Republic.Google Scholar
- Serban, I., & Najim, M. (2007). A new multidimensional Schur-Cohn type stability criterion. In: Proceedings of the 2007 American Control Conference.Google Scholar
- Šiljak, D. (1975). Stability criteria for two-variable polynomials. IEEE Transaction on Circuits and Systems 22(3).Google Scholar
- Sontag E. (1976) Linear systems over commutative rings: A survey. Ricerche di Automatica 7: 1–34Google Scholar