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June  2017, 7(2): 121-138. doi: 10.3934/naco.2017009

## SISO H-Optimal synthesis with initially specified structure of control law

 Faculty of Applied Mathematics and Control Processes, Saint-Petersburg State University, Saint Petersburg, 199034, Russia

* Corresponding author: Evgeny I. Veremey

Received  June 2016 Revised  May 2017 Published  June 2017

Fund Project: This article was written based on a study partially supported by the Russian Foundation for Basic Research (RFBR), research project No. 14-07-00083a.

The paper is devoted to particular cases of H-optimization problems for LTI systems with scalar control and external disturbance. The essence of these problems is to find an output feedback optimal controller having initially given structure to attenuate disturbances action with respect to controlled variable and control. An admissible set of controllers can be additionally restricted by the requirement to assign given poles spectrum of the closed-loop system. Specific features of the posed problems are considered and three simple numerical methods of synthesis are proposed to design correspondent H-optimal controllers. To show the simplicity and effectiveness of the proposed approach and the benefits of developed methods, illustrative examples are enclosed to the paper.

Citation: Evgeny I. Veremey, Vladimir V. Eremeev. SISO H-Optimal synthesis with initially specified structure of control law. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 121-138. doi: 10.3934/naco.2017009
##### References:
 [1] B. Anderson and J. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, New Jersey, 1989. [2] S. Bhattacharyya, A. Datta and L. Keel, Linear Control Theory: Structure, Robustness and Optimization, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2009. doi: 10.1201/9781420019612. [3] Ya. M. Bokova and E. I. Veremei, Numerical aspects of spectral method of $H_∞$ -optimal synthesis, Journal of Automat. and Inform. Sciences, 28 (1996), 1-12. [4] S. Boyd, S. L. El. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [5] B. M. Chen, A. Saberi, P. Sanutti and Ya. Shamash, Construction and parameterization of all static and dynamic $H_2$ -optimal state feedback solutions, optimal fixed modes, and fixed decoupling zeros, IEEE Trans. Automat. Contr., 38 (1993), 248-261.  doi: 10.1109/9.250513. [6] J. C. Doyle, B. A. Francis and A. R. Tanenbaum, Feedback Control Theory, Mac Millan, New York, 1992. [7] H. E. Musch and M. Steiner, Tuning advanced PID controllers via direct $H_∞$-norm minimization, Proc. of European Control Conference, Brussel, Belgium, 1-4 July, (1997), 179. doi: 10.1016/j.ejcon.2015.04.008. [8] A. Saberi, P. Sannuti and B. M. Chen, $H_2$ -Optimal Control, Prentice-Hall, New Jersey, 1995. [9] G. S. Wang, B. Liang and G. R. Duan, $H_2$ -optimal control with regional pole assignment via state feedback, International Journal of Control, Automation, and Systems, 4 (2006), 653-659. [10] E. I. Veremey, Algorithms for solving a class of problems of $H_∞$ -optimization of control systems, Journal of Comput. and Syst. Sci. Int., 50 (2011), 403-412.  doi: 10.1134/S1064230711010187. [11] E. I. Veremey, Efficient spectral approach to SISO problems of $H_2$ -optimal synthesis, Appl. Math. Sciences, 9 (2015), 3897-3909. [12] E. I. Veremey, M. V. Sotnikova, V. V. Eremeev and M. V. Korovkin, Modal parametric optimization of control laws with special structure, Proc. of 14th Intern. Conf. on Control, Automation and Systems, Oct. 22-25, KINTEX, Gyeonggi-do, Korea, (2014), 1278–1283.

show all references

##### References:
 [1] B. Anderson and J. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, New Jersey, 1989. [2] S. Bhattacharyya, A. Datta and L. Keel, Linear Control Theory: Structure, Robustness and Optimization, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2009. doi: 10.1201/9781420019612. [3] Ya. M. Bokova and E. I. Veremei, Numerical aspects of spectral method of $H_∞$ -optimal synthesis, Journal of Automat. and Inform. Sciences, 28 (1996), 1-12. [4] S. Boyd, S. L. El. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [5] B. M. Chen, A. Saberi, P. Sanutti and Ya. Shamash, Construction and parameterization of all static and dynamic $H_2$ -optimal state feedback solutions, optimal fixed modes, and fixed decoupling zeros, IEEE Trans. Automat. Contr., 38 (1993), 248-261.  doi: 10.1109/9.250513. [6] J. C. Doyle, B. A. Francis and A. R. Tanenbaum, Feedback Control Theory, Mac Millan, New York, 1992. [7] H. E. Musch and M. Steiner, Tuning advanced PID controllers via direct $H_∞$-norm minimization, Proc. of European Control Conference, Brussel, Belgium, 1-4 July, (1997), 179. doi: 10.1016/j.ejcon.2015.04.008. [8] A. Saberi, P. Sannuti and B. M. Chen, $H_2$ -Optimal Control, Prentice-Hall, New Jersey, 1995. [9] G. S. Wang, B. Liang and G. R. Duan, $H_2$ -optimal control with regional pole assignment via state feedback, International Journal of Control, Automation, and Systems, 4 (2006), 653-659. [10] E. I. Veremey, Algorithms for solving a class of problems of $H_∞$ -optimization of control systems, Journal of Comput. and Syst. Sci. Int., 50 (2011), 403-412.  doi: 10.1134/S1064230711010187. [11] E. I. Veremey, Efficient spectral approach to SISO problems of $H_2$ -optimal synthesis, Appl. Math. Sciences, 9 (2015), 3897-3909. [12] E. I. Veremey, M. V. Sotnikova, V. V. Eremeev and M. V. Korovkin, Modal parametric optimization of control laws with special structure, Proc. of 14th Intern. Conf. on Control, Automation and Systems, Oct. 22-25, KINTEX, Gyeonggi-do, Korea, (2014), 1278–1283.
Fragment of the surface $J=J(h_{0}, h_{1} )$
The crossed sections of the surface $J=J(h_{0}, h_{1} )$ for the fixed value of the parameter $h_1$
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