Linear Matrix Inequalities in Stability Problems: Retrospective and Theoretical Aspects

Some aspects of the development of the theory of linear matrix inequalities are considered. A number of results obtained at the initial stage of the development of this theory, both in the development of numerical methods and in obtaining analytical conditions for their solvability, are highlighted. The main attention is focused on the system of linear matrix inequalities arising in solving the absolute stabi lity problem. E. S. Pyatnitskiy and his followers showed that the solvability of this system is a criterion for the existence of a quadratic Lyapunov function and a sufficient condition for absolute stability. The prerequisites leading to this result are considered here. The use of the considered system of inequalities for studying the stability of hybrid systems described by differential inclusions and switching systems is shown. An analysis is given of citing some works of Pyatnitskiy’s school on the theory of stability and the theory of systems of linear matrix inequalities, from which the relevance of the results of these works at the present time follows.

In developing numerical methods, it was first shown in the work of Pyatnitskiy and Skorodinskiy that the solvability problem for a system of linear matrix inequalities reduces to a convex programming problem. An interesting gradient algorithm for finding solutions to such a system is also presented. In analyzing analytical conditions of solvability, an unsolvability criterion for the system of our interest obtained by Kamenetskiy and Pyatnitskiy is noted. In modern terms, this result can be considered as a description of an admissible set in the dual semidefinite programming problem. A similar result is given in the famous book by S. Boyd et al. The paper shows that the result of Boyd et al. is a simple corollary of the unsolvability criterion. Here the unsolvability criterion is generalized and refined.

1. Chaikovskii M. M., Kurdyukov A. P. Algebraic Riccati Equations and Linear Matrix Inequalities for Discrete-Time Systems, Moscow, Inst. Probl. Upravlen. RAN, 2005 (in Russian).

2. Balandin D. V., Kogan M. M. Synthesis of Control Laws Based on Linear Matrix Inequalities, Moscow, Fizmatlit, 2007 (in Russian).

3. Polyak B. T., Shcherbakov P. S. Robust Stability and Control, Moscow, Nauka, 2002 (in Russian).

4. Polyak B. T., Khlebnikov M. V., Shcherbakov P. S. Control of Linear Systems: Technique of linear matrix inequalities, Moscow, LENAND, 2014 (in Russian).

5. Emel’ja nova Ju. P., Pakshin P. V., Pakshina N. A. Matrix Equations and Inequalities of Second Order: Study Guide, Nizhny Novgorod, Publishing house of Nizhny Novgorod State Techn. Univ., 2013.

6. Boyd S., El Ghaoui L., Feron E., Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. SIAM. Philadelphia, 1994.

7. Boyd S., El Ghaoui L., Feron E., Balakrishnan V. History of Linear Matrix Inequalities in Control Theory, Procceding of the American Control Conference, Baltimore, Maryland, 1994, pp. 31—34.

8. Kamenetskiy V. A., Pyatnitskiy Ye. S. An Iterative Method of Lyapunov Function Construction for Differential Inclusions, Systems and Control Letters, 1987, vol. 8, pp. 445—451.

9. Gorbunov A. V., Kamenetskiy V. A. LMI, Absolute Stability and Hybrid Systems, Stability and oscillations of nonlinear control systems: Proceedings of the XIII International Conference, Moscow, Publishing house of IPU RAS, 2016, pp. 86—87 (in Russian).

10. Gelig A. Kh., Leonov G. A., Yakubovich V. A. Stability of Nonlinear Systems with a Nonunique Equilibrium State, Moscow, Nauka, 1978 (in Russian).

11. Boyd S., Yang Q. Structured and Simultaneous Lyapunov Functions for System Stability Problems, Internat. J. Control, 1989, vol. 49, no. 6, pp. 2215—2240.

12. Kamenetskii V. A., Absolute Stability and Absolute Instability of Control Systems with Several Nonlinear Nonstationary Elements, Autom. Remote Control, 1983, vol. 44, no. 12, pp. 1543—1552 (in Russian).

13. Pyatnitskii E. S., Skorodinskii V. I. Numerical Method of Construction of Lyapunov Functions and Absolute Stability Criteria in the Form of Numerical Procedures, Autom. Remote Control, 1983, vol. 44, no. 11, pp. 1427—1437 (in Russian).

14. Bellman R., Fan K. On Systems of Linear Inequalities in Hermitian Matrix Variables, In V. L. Klee, editor, Convexity, volume 7 of Proccedings of Simposia in Pure Mathematics, pp. 1—11, American Math. Society, 1963.

15. Pyatnitskiy Ye. S., Skorodinskiy V. I. Numerical methods of Liapunov function construction and their application to the absolute stability problem, Systems and Control Letters, 1982, vol. 2, pp. 130—135.

16. Pyatnitskii E. S. Absolute Stability of Nonstationary Nonlinear Systems, Autom. and Remote Control, 1970, vol. 31, no. 1, pp. 1—9 (in Russian).

17. Kamenetskii V. A., Pyatnitskii E. S. Gradient Method of Constructing Lyapunov Functions in Problems of Absolute Stability, Autom. Remote Control, 1987, vol. 48, no. 1, pp. 1—9 (in Russian).

18. Alimov Yu. I. On the Application of Lyapunov’s Direct Method to Differential Equations with Ambiguous Right Sides, Autom. Remote Control, 1961, vol. 22, no. 7, pp. 713—725 (in Russian).

19. Liberzon D. Switching in Systems and Control, Birkhäuser, Boston, MA, 2003.

20. Shorten R., Wirth F., Mason O., Wulf K., King C. Stability Сriteria for Switched and Hybrid Systems, SIAM Rev, 2007, no. 4, pp. 545—592.

21. Lin H., Antsaklis P. J. Stability and Stabilizability of Switched Linear Systems: a Survey of Recent Results, IEEE Trans. Automat. Contr., 2009, no. 2, pp. 308—322.

22. Molchanov A. P., Pyatnitskii E. S. Lyapunov Functions that Specify Necessary and Sufficient Conditions of Absolute Stability of Nonlinear Nonstationary Control Systems. I, II, III, Autom. Remote Control, 1986, vol. 47. no. 3, pp. 344—354; no. 4, pp. 443—451; no. 5, pp. 620—630 (in Russian).

23. Molchanov A. P., Pyatnitskiy E. S. Criteria of Asymptotic Stability of Differential and Difference Inclusions Encountered in Control Theory, Systems and Control Letters, 1989, vol. 13, pp. 59—64.

24. Pyatnitskiy E. S., Rapoport L. B. Criteria of Asymptotic Stability of Differential Inclusions and Periodic Motions of Timevarying Nonlinear Control Systems, IEEE Trans. Circuits Syst., I, 1996, vol. 43, no. 3, pp. 219—229.

25. Pyatnitskii E. S. Selected Works: in 3 vol., Moscow, Fizmatlit, 2005 (in Russian).

26. Filippov A. F. Stability Conditions in Homogeneous Systems with Arbitrary Regime Switching, Autom. Remote Control, 1980, vol. 41, no. 8, pp. 1078—1085.

27. Laffey T. J., Smigoc H. Common Lyapunov Solutions for Two Matrices whose Difference has Rank One, Linear Algebra and its Applications, 2009, vol. 431, pp. 228—240.

28. Pozdyaev V. V. On an Analytical Solution of Systems of Matrix Inequalities Dual to Lyapunov Inequality Systems, UBS, 2010, vol.28, pp. 58—74 (in Russian).

29. Alekseev V. M., Tikhomirov I. M., Fomin S. I. Optimal Control, Moscow, Nauka, 1979 (in Russian).

30. Griggs W. M., King C. K., Shorten R. N., Mason O., Wulff K. Quadratic Lyapunov Functions for Systems with Statedependent Switching, Linear Algebra and its Applications, 2010, vol. 433, pp. 52—63.

31. Balakrishnan V., Vandenberghe L. Semidefinite Programming Duality and Linear time-invariant systems, IEEE Trans. Automat. Control, 2003, vol. 48, no. 1, pp. 30—41.

32. Vandenberghe L., Boyd S. Semidefinite programming, SIAM Rev., 1996, vol. 38, no. 1, pp. 49—95.

33. Berman A., Ben-Israel A. More on Linear Inequalities with Applications to Matrix Theory, Journal of Mathematical Analysis and Applications, 1971, vol. 33, pp. 482—496.

34. Fradkov A. L. Duality Theorems in Some Nonconvex Extremal Problems, Siberian Math. J., 1973, vol. 14, no. 2, pp. 357—383 (in Russian).

35. Ostrowski A., Schneider М. Some Theorems on the Inertia of General Matrices, J. Math. Anal. Appl., 1962, vol. 4, pp. 72—84.

36. Kamenetskii V. A. Gradient Method for Constructing Lyapunov Functions for Nonlinear Dynamical Systems / Optimization in complex systems. Ser. " Questions of Cybernetics". Ed. by P. P. Parkhomenko, Moscow, Academy of Sciences of the USSR, 1988, pp. 55—72 (in Russian).

37. Dubovitskii A. Ya., Milyutin A. A. Extremum Problems in the Presence of Restrictions, U. S. S. R. Comput. Math. Math. Phys., 1965, vol. 5, no. 3, pp. 1—80 (in Russian).

38. Pshenichny B. N. Convex Analysis and Extremal Problems, Moscow, Nauka, 1980 (in Russian).