# American Institute of Mathematical Sciences

2016, 6(4): 491-504. doi: 10.3934/naco.2016022

## 2D system analysis via dual problems and polynomial matrix inequalities

 1 Arzamas Polytechnic Institute, Alekseev Nizhny Novgorod State Technical University, 607220, Arzamas, Russian Federation

Received  September 2015 Revised  November 2016 Published  December 2016

Application of the Lyapunov method to 2D system stability and performance analysis yields algebraic systems that can be interpreted as either sum-of-squares problems for nontrivial matrix polynomials, or parameterized linear matrix inequalities that need to be satisfied for certain ranges of parameter values. In this paper we show that dualizing core inequalities in the latter forms allows converting these systems to conventional optimization problems on sets described by polynomial matrix inequalities. Methods for solving these problems include moment-based methods or the “atomic optimization” method proposed earlier by the author. As a result, we obtain necessary conditions for 2D system stability and lower bounds on system performance. In particular, we demonstrate respective results for discrete-discrete system stability and mixed continuous-discrete system $\mathcal{H}_\infty$ performance. A numerical example is provided.
Citation: Vladimir Pozdyayev. 2D system analysis via dual problems and polynomial matrix inequalities. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 491-504. doi: 10.3934/naco.2016022
##### References:
 [1] P. Agathoklis, E. Jury and M. Mansour, Algebraic necessary and sufficient conditions for the stability of 2-D discrete systems, Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, 40 (1993), 251-258. [2] B. Anderson, P. Agathoklis, E. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems, Circuits and Systems, IEEE Transactions on, 33 (1986), 261-267. doi: 10.1109/TCS.1986.1085912. [3] S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, 1994. doi: 10.1137/1.9781611970777. [4] G. Chesi, LMI techniques for optimization over polynomials in control: A survey, Automatic Control, IEEE Transactions on, 55 (2010), 2500-2510. doi: 10.1109/TAC.2010.2046926. [5] G. Chesi and R. Middleton, Necessary and sufficient LMI conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems, Automatic Control, IEEE Transactions on, 59 (2014), 996-1007. doi: 10.1109/TAC.2014.2299353. [6] B. Cichy, P. Augusta, E. Rogers, K. Galkowski et al., On the control of distributed parameter systems using a multidimensional systems setting, Mechanical Syst. and Signal Processing, 22 (2008), 1566-1581. [7] G. Dullerud and R. D'Andrea, Distributed control of heterogeneous systems, Automatic Control, IEEE Transactions on, 49 (2004), 2113-2128. doi: 10.1109/TAC.2004.838499. [8] D. Henrion and J.-B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback, Automatic Control, IEEE Transactions on, 51 (2006), 192-202. doi: 10.1109/TAC.2005.863494. [9] D. Henrion and J.-B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, in Positive Polynomials in Control (eds. D. Henrion and A. Garulli), Springer Berlin Heidelberg, Berlin, Heidelberg, (2005), 293-310. doi: 10.1007/10997703_15. [10] T. Hinamoto, 2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, 40 (1993), 102-110. [11] V. Kamenetskiy and Y. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions, Systems & Control Letters, 8 (1987), 445-451. doi: 10.1016/0167-6911(87)90085-5. [12] S. Knorn and R. Middleton, Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs, Automatic Control, IEEE Transactions on, 58 (2013), 2579-2590. doi: 10.1109/TAC.2013.2264852. [13] J.-B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization, 11 (2001), 796-817. doi: 10.1137/S1052623400366802. [14] Y. Li, M. Cantoni and E. Weyer, On water-level error propagation in controlled irrigation channels, in Proc. IEEE Conf. Decision Control and Eur. Control Conf., Seville, Spain, (2005), 2101-2106. [15] M. C. D. Oliveira, J. C. Geromel and J. Bernussou, Extended H2 and H∞ norm characterizations and controller parametrizations for discrete-time systems, International Journal of Control, 75 (2002), 666-679. doi: 10.1080/00207170210140212. [16] W. Paszke, E. Rogers and K. Galkowski, H2/H∞ output information-based disturbance attenuation for differential linear repetitive processes, International Journal of Robust and Nonlinear Control, 21 (2011), 1981-1993. doi: 10.1002/rnc.1672. [17] V. Pozdyayev, Atomic optimization. I. Search space transformation and one-dimensional problems, Automation and Remote Control, 74 (2013), 2069-2092. doi: 10.1134/S0005117913120096. [18] V. Pozdyayev, Atomic optimization, II, Multidimensional problems and polynomial matrix inequalities, Automation and Remote Control, 75 (2014), 1155-1171. doi: 10.1134/S0005117914060150. [19] V. Pozdyayev, Necessary conditions for 2D systems' stability, in Preprints, 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, (2015), 800-805. [20] R. Rabenstein and L. Trautmann, Towards a framework for continuous and discrete multidimensional systems, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 73-86. [21] R. P. Roesser, A discrete state-space model for linear image processing, Automatic Control, IEEE Transactions on, 20 (1975), 1-10. [22] E. Rogers, K. Galkowski and D. Owens, Control systems theory and applications for linear repetitive processes, in Lecture Notes in Control and Information Sciences, vol. 349, Springer-Verlag, Berlin, 2007. [23] E. Rogers and D. Owens, Stability analysis for linear repetitive processes, in Lecture Notes in Control and Information Sciences, vol. 175, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0007165. [24] E. Rogers and D. Owens, Kronecker product based stability tests and performance bounds for a class of 2D continuous-discrete linear systems, Linear Algebra and its Applications, 353 (2002), 33-52. doi: 10.1016/S0024-3795(02)00287-2.

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##### References:
 [1] P. Agathoklis, E. Jury and M. Mansour, Algebraic necessary and sufficient conditions for the stability of 2-D discrete systems, Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, 40 (1993), 251-258. [2] B. Anderson, P. Agathoklis, E. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems, Circuits and Systems, IEEE Transactions on, 33 (1986), 261-267. doi: 10.1109/TCS.1986.1085912. [3] S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, 1994. doi: 10.1137/1.9781611970777. [4] G. Chesi, LMI techniques for optimization over polynomials in control: A survey, Automatic Control, IEEE Transactions on, 55 (2010), 2500-2510. doi: 10.1109/TAC.2010.2046926. [5] G. Chesi and R. Middleton, Necessary and sufficient LMI conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems, Automatic Control, IEEE Transactions on, 59 (2014), 996-1007. doi: 10.1109/TAC.2014.2299353. [6] B. Cichy, P. Augusta, E. Rogers, K. Galkowski et al., On the control of distributed parameter systems using a multidimensional systems setting, Mechanical Syst. and Signal Processing, 22 (2008), 1566-1581. [7] G. Dullerud and R. D'Andrea, Distributed control of heterogeneous systems, Automatic Control, IEEE Transactions on, 49 (2004), 2113-2128. doi: 10.1109/TAC.2004.838499. [8] D. Henrion and J.-B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback, Automatic Control, IEEE Transactions on, 51 (2006), 192-202. doi: 10.1109/TAC.2005.863494. [9] D. Henrion and J.-B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, in Positive Polynomials in Control (eds. D. Henrion and A. Garulli), Springer Berlin Heidelberg, Berlin, Heidelberg, (2005), 293-310. doi: 10.1007/10997703_15. [10] T. Hinamoto, 2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, 40 (1993), 102-110. [11] V. Kamenetskiy and Y. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions, Systems & Control Letters, 8 (1987), 445-451. doi: 10.1016/0167-6911(87)90085-5. [12] S. Knorn and R. Middleton, Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs, Automatic Control, IEEE Transactions on, 58 (2013), 2579-2590. doi: 10.1109/TAC.2013.2264852. [13] J.-B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization, 11 (2001), 796-817. doi: 10.1137/S1052623400366802. [14] Y. Li, M. Cantoni and E. Weyer, On water-level error propagation in controlled irrigation channels, in Proc. IEEE Conf. Decision Control and Eur. Control Conf., Seville, Spain, (2005), 2101-2106. [15] M. C. D. Oliveira, J. C. Geromel and J. Bernussou, Extended H2 and H∞ norm characterizations and controller parametrizations for discrete-time systems, International Journal of Control, 75 (2002), 666-679. doi: 10.1080/00207170210140212. [16] W. Paszke, E. Rogers and K. Galkowski, H2/H∞ output information-based disturbance attenuation for differential linear repetitive processes, International Journal of Robust and Nonlinear Control, 21 (2011), 1981-1993. doi: 10.1002/rnc.1672. [17] V. Pozdyayev, Atomic optimization. I. Search space transformation and one-dimensional problems, Automation and Remote Control, 74 (2013), 2069-2092. doi: 10.1134/S0005117913120096. [18] V. Pozdyayev, Atomic optimization, II, Multidimensional problems and polynomial matrix inequalities, Automation and Remote Control, 75 (2014), 1155-1171. doi: 10.1134/S0005117914060150. [19] V. Pozdyayev, Necessary conditions for 2D systems' stability, in Preprints, 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, (2015), 800-805. [20] R. Rabenstein and L. Trautmann, Towards a framework for continuous and discrete multidimensional systems, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 73-86. [21] R. P. Roesser, A discrete state-space model for linear image processing, Automatic Control, IEEE Transactions on, 20 (1975), 1-10. [22] E. Rogers, K. Galkowski and D. Owens, Control systems theory and applications for linear repetitive processes, in Lecture Notes in Control and Information Sciences, vol. 349, Springer-Verlag, Berlin, 2007. [23] E. Rogers and D. Owens, Stability analysis for linear repetitive processes, in Lecture Notes in Control and Information Sciences, vol. 175, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0007165. [24] E. Rogers and D. Owens, Kronecker product based stability tests and performance bounds for a class of 2D continuous-discrete linear systems, Linear Algebra and its Applications, 353 (2002), 33-52. doi: 10.1016/S0024-3795(02)00287-2.
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