Synthesis of Output Feedback Static Regulators by Using the Gradient Flows: LQR and LS-Matching Eigenvalue Assignment

Synthesis of regulators based on the output measured variable is considered more difficult than synthesis based on the state variables and synthesis of dynamic regulators. This problem has not been solved in its general form so far. In this paper, we consider the case when neither the necessary (mp ≥ n)  nor sufficient (mp > n) conditions for the pole placement synthesis at the output are fulfilled, where m is the number of inputs, p the number of outputs, and n the order of the system. At the same time, two approaches to solving this problem are being investigated: LQR and the pole placement approximation using gradient flows. In the first case, Lyapunov algebraic equations are solved and gradient equations are integrated. In this case, an extremum (local or global) is always reached. The second approach uses gradient flows on group Lie

GL(n, R) Ѕ R^mЅp, where a least squares pole assignment (mp < n) is performed with the choice of a part of the desired spectrum of a closed system. Here, optimization is performed before the modes enter a small neighborhood of the desired modes. To obtain gradient equations in both approaches, gradients of objective functions tr(M) from the matrix argument are derived. The properties of the function tr(M) and the rules for finding its gradient from the matrix argument are considered in detail. The obtained gradient equations are verified using a practical example: a 4th-order object with weakly damped dynamics with a large range of parameter uncertainty, where the task is to synthesize a robust static regulator of the minimum (second) order. This problem has been solved by both methods with the fulfillment of the set technical requirements.

1. Brockett R. W. Dynamical Systems that Sort Lists, Diagonalize Matrices and Solve Linear Programming Problems // Lin. Alg. and Appl. 1991. Vol. 146. P. 79—91.

2. Brockett R. W. Least Squares Matching Problems // Lin. Alg. and Appl. 1989. Vol. 12—124. P. 761—777.

3. Helmke U., Moore J. B. Optimization and Dynamical Systems. Berlin: Springe, 1996. 403 p.

4. Absil P. A., Mahony R., Sepulchre R. Optimization Algorithms on Matrix Manifolds. Princeton: Princeton Univ. Press. 2008. 224 p.

5. Jiang D., Wang J. Least Squares Pole Assignment by Memoryless Output Feedback // Systems and Control Lettters.1996. Vol. 39, N. 1. P. 31—42.

6. Schulte-Herbruggen T., Glaser S. Y., Dirr G., Helmke U. Gradient Flows for Optimization and Quantum Control. Foundations and Applications. URL: http:arXiv:0802.4195v1[quantph]28Feb.2008.

7. Mahony R. E., Helmke U., Moore J. B. Gradient Algorithms for Principal Component Analysis // J. Austral. Math. Soc. Ser. В. 1996. Vol. 37. P. 430—450.

8. Yan W. Y., Teo K. L., Moore J. B. А Gradient Flow Approach to Computing LQ Optimal Output Feedback Gains // Proceeding American. Cont.Conf.San. Franc. Calif. 1993. P. 1266—1270.

9. Chan H. C., Lam J., Ho D. W. А Gradient Flow Approach to Robust Pole Assignment in Second-Order Systems // Proceeding 35 Conf. on Dec. and Control. Kobe. Japan. Dec. 1995. P. 4589—4590.

10. Jiang D., Wang J. Augmented Gradient Flows for on-line Robust Pole Assignment via State and Output Feedback // Automatica. 2002. Vol. 38. P. 279—286.

11. Willems J. C., Hesselink W. H. Generic properties of the pole placement problem // Proceedings of the 7th IFAC Congress, Helsinki Finland. 1978. P. 1725—1729.

12. Petersen K. B., Peders en M. S. The Matrix Cookbook. Version: November,15, 2012. 69 p. URL:http://matrixcookbook.com (дата обращения: 03 февраля 2022 г.).

13. Geering Н. P. On Calculating Gradient Matrices // IEEE Trans. Aut.Control. 1976. Vol. 21. P. 615—616.

14. Bellman R. Introduction to Matrix Analysis. NY: McCraw-Hill, 1970. 440 p.

15. Levine W. S., Athans M. On the Determination of the Optimal Constant Output Feedback Gains for Linear Multivariable Systems // IEEE Trans. Aut.Control. 1970. Vol. 15, N. 1. P. 44—48.

16. Hall В. C. Lie Groups, Lie Algebras and Representations. An Elementary Introduction. NY: Springer-Verlag, 2003. 354 p.

17. Curtis M. L. Matrix Groups. 2nd Edition. NY: Springer-Verlag, 1984. 210 p.

18. Trofimov V. V. Introduction in geometry of manifolds with symmetries, Moscow, Publ. house of MSU, 1989, 359 p. (in Russian).

19. Krasnoschechenko V. I. Synthesis Hinf robust regulator of the low order by using of linear matrix inequalities and projective lemmas, Mehatronika, Avtomatizatsija, Upravlenie, 2018, vol. 19, no. 4, pp. 219—231 (in Russian).

20. Krasnoschechenko V. I. Synthesis of dynamic output feedback controller using functions of matrix root-clustering in D-regions, Mehatronika, Avtomatizacija, Upravlenie, 2023, vol. 24, no. 5, pp. 227—238 (in Russian).