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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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

B. M. ChenA. SaberiP. 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. Google Scholar

[6]

J. C. Doyle, B. A. Francis and A. R. Tanenbaum, Feedback Control Theory, Mac Millan, New York, 1992. Google Scholar

[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. Google Scholar

[8]

A. Saberi, P. Sannuti and B. M. Chen, $H_2$ -Optimal Control, Prentice-Hall, New Jersey, 1995.Google Scholar

[9]

G. S. WangB. 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. Google Scholar

[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. Google Scholar

[11]

E. I. Veremey, Efficient spectral approach to SISO problems of $H_2$ -optimal synthesis, Appl. Math. Sciences, 9 (2015), 3897-3909. Google Scholar

[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.Google Scholar

show all references

References:
[1]

B. Anderson and J. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, New Jersey, 1989. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

B. M. ChenA. SaberiP. 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. Google Scholar

[6]

J. C. Doyle, B. A. Francis and A. R. Tanenbaum, Feedback Control Theory, Mac Millan, New York, 1992. Google Scholar

[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. Google Scholar

[8]

A. Saberi, P. Sannuti and B. M. Chen, $H_2$ -Optimal Control, Prentice-Hall, New Jersey, 1995.Google Scholar

[9]

G. S. WangB. 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. Google Scholar

[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. Google Scholar

[11]

E. I. Veremey, Efficient spectral approach to SISO problems of $H_2$ -optimal synthesis, Appl. Math. Sciences, 9 (2015), 3897-3909. Google Scholar

[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.Google Scholar

Figure 1.  Fragment of the surface $J=J(h_{0}, h_{1} )$
Figure 2.  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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