2D system analysis via dual problems and polynomial matrix inequalities

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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

Abstract
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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.

Keywords: 2D systems, polynomial inequalities., Lyapunov method, matrix inequalities, nonlinear programming.

Mathematics Subject Classification: Primary: 93D05, 90C26; Secondary: 49M2.

Citation: Vladimir Pozdyayev. 2D system analysis via dual problems and polynomial matrix inequalities. Numerical Algebra, Control & 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, 40 (1993), 251. Google Scholar
[2]	B. Anderson, P. Agathoklis, E. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems,, Circuits and Systems, 33 (1986), 261. doi: 10.1109/TCS.1986.1085912. Google Scholar
[3]	S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Studies in Applied Mathematics, (1994). doi: 10.1137/1.9781611970777. Google Scholar
[4]	G. Chesi, LMI techniques for optimization over polynomials in control: A survey,, Automatic Control, 55 (2010), 2500. doi: 10.1109/TAC.2010.2046926. Google Scholar
[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, 59 (2014), 996. doi: 10.1109/TAC.2014.2299353. Google Scholar
[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. Google Scholar
[7]	G. Dullerud and R. D'Andrea, Distributed control of heterogeneous systems,, Automatic Control, 49 (2004), 2113. doi: 10.1109/TAC.2004.838499. Google Scholar
[8]	D. Henrion and J.-B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback,, Automatic Control, 51 (2006), 192. doi: 10.1109/TAC.2005.863494. Google Scholar
[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)*, (2005), 293. doi: 10.1007/10997703_15. Google Scholar
[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, 40 (1993), 102. Google Scholar
[11]	V. Kamenetskiy and Y. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions,, Systems & Control Letters, 8 (1987), 445. doi: 10.1016/0167-6911(87)90085-5. Google Scholar
[12]	S. Knorn and R. Middleton, Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs,, Automatic Control, 58 (2013), 2579. doi: 10.1109/TAC.2013.2264852. Google Scholar
[13]	J.-B. Lasserre, Global optimization with polynomials and the problem of moments,, SIAM Journal on Optimization, 11 (2001), 796. doi: 10.1137/S1052623400366802. Google Scholar
[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., (2005), 2101. Google Scholar
[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. doi: 10.1080/00207170210140212. Google Scholar
[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. doi: 10.1002/rnc.1672. Google Scholar
[17]	V. Pozdyayev, Atomic optimization. I. Search space transformation and one-dimensional problems,, Automation and Remote Control, 74 (2013), 2069. doi: 10.1134/S0005117913120096. Google Scholar
[18]	V. Pozdyayev, Atomic optimization, II, Multidimensional problems and polynomial matrix inequalities,, Automation and Remote Control, 75 (2014), 1155. doi: 10.1134/S0005117914060150. Google Scholar
[19]	V. Pozdyayev, Necessary conditions for 2D systems' stability,, in Preprints, (2015), 800. Google Scholar
[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. Google Scholar
[21]	R. P. Roesser, A discrete state-space model for linear image processing,, Automatic Control, 20 (1975), 1. Google Scholar
[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, (2007). Google Scholar
[23]	E. Rogers and D. Owens, Stability analysis for linear repetitive processes,, in Lecture Notes in Control and Information Sciences, (1992). doi: 10.1007/BFb0007165. Google Scholar
[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. doi: 10.1016/S0024-3795(02)00287-2. Google Scholar

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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, 40 (1993), 251. Google Scholar
[2]	B. Anderson, P. Agathoklis, E. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems,, Circuits and Systems, 33 (1986), 261. doi: 10.1109/TCS.1986.1085912. Google Scholar
[3]	S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Studies in Applied Mathematics, (1994). doi: 10.1137/1.9781611970777. Google Scholar
[4]	G. Chesi, LMI techniques for optimization over polynomials in control: A survey,, Automatic Control, 55 (2010), 2500. doi: 10.1109/TAC.2010.2046926. Google Scholar
[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, 59 (2014), 996. doi: 10.1109/TAC.2014.2299353. Google Scholar
[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. Google Scholar
[7]	G. Dullerud and R. D'Andrea, Distributed control of heterogeneous systems,, Automatic Control, 49 (2004), 2113. doi: 10.1109/TAC.2004.838499. Google Scholar
[8]	D. Henrion and J.-B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback,, Automatic Control, 51 (2006), 192. doi: 10.1109/TAC.2005.863494. Google Scholar
[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)*, (2005), 293. doi: 10.1007/10997703_15. Google Scholar
[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, 40 (1993), 102. Google Scholar
[11]	V. Kamenetskiy and Y. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions,, Systems & Control Letters, 8 (1987), 445. doi: 10.1016/0167-6911(87)90085-5. Google Scholar
[12]	S. Knorn and R. Middleton, Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs,, Automatic Control, 58 (2013), 2579. doi: 10.1109/TAC.2013.2264852. Google Scholar
[13]	J.-B. Lasserre, Global optimization with polynomials and the problem of moments,, SIAM Journal on Optimization, 11 (2001), 796. doi: 10.1137/S1052623400366802. Google Scholar
[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., (2005), 2101. Google Scholar
[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. doi: 10.1080/00207170210140212. Google Scholar
[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. doi: 10.1002/rnc.1672. Google Scholar
[17]	V. Pozdyayev, Atomic optimization. I. Search space transformation and one-dimensional problems,, Automation and Remote Control, 74 (2013), 2069. doi: 10.1134/S0005117913120096. Google Scholar
[18]	V. Pozdyayev, Atomic optimization, II, Multidimensional problems and polynomial matrix inequalities,, Automation and Remote Control, 75 (2014), 1155. doi: 10.1134/S0005117914060150. Google Scholar
[19]	V. Pozdyayev, Necessary conditions for 2D systems' stability,, in Preprints, (2015), 800. Google Scholar
[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. Google Scholar
[21]	R. P. Roesser, A discrete state-space model for linear image processing,, Automatic Control, 20 (1975), 1. Google Scholar
[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, (2007). Google Scholar
[23]	E. Rogers and D. Owens, Stability analysis for linear repetitive processes,, in Lecture Notes in Control and Information Sciences, (1992). doi: 10.1007/BFb0007165. Google Scholar
[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. doi: 10.1016/S0024-3795(02)00287-2. Google Scholar

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