Abstract
This paper considers the problem of designing dynamic controllers with a simple structure for a linear time-invariant dynamic system with a scalar control, a vector feedback channel, and a perturbation. A simple controller is a controller whose structure cannot be simplified: an attempt to eliminate any controller element will make the designed system not satisfy the requirements. The robustness of the designed system is ensured using the original index of the controller’s roughness. Among all simple controllers, this index allows identifying the one that provides the system with the maximum robustness. A method for solving the design problem is proposed and an illustrative example is given.
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REFERENCES
V. V. Solodovnikov, “Synthesis of corrective devices of the tracking systems with typical effects,” Avtom. Telemekh. 12, 352–388 (1951).
V. V. Solodovnikov and V. L. Lenskii, “Synthesis of minimum complexity management systems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 2, 56–68 (1966).
V. V. Solodovnikov, V. F. Biryukov, and V. I. Tumarkin, The Principle of Complexity in Control Theory (Nauka, Moscow, 1977) [in Russian].
V. Solodovnikov, “Poorly defined problems of stochastic optimization and their solution with the aid of the principle of complexity,” IFAC Proc. Vols. 10, 649–651 (1977).
V. V. Solodovnikov and V. I. Tumarkin, Complexity Theory and Design of Control Systems (Nauka, Moscow, 1990) [in Russian].
Yu. I. Paraev and V. I. Smagina, “Tasks to simplify the structure of optimal regulators,” Avtom. Telemekh., No. 6, 180–183 (1975).
A. Balestrino and G. Celentano, “CAD of minimal order controllers,” IFAC Proc. Vols. 12 (7), 1–8 (1979).
A. Balestrino and G. Celentano, “Dynamic controllers in linear multivariable systems,” Automatica 17, 631–636 (1981).
A. R. Gaiduk, “Selection of reverse links in the minimum complexity control system,” Avtom. Telemekh., No. 5, 29–37 (1990).
L. H. Kell and S. P. Bhattacharyya, “State-space design of low-order stabilizers,” IEEE Trans. Autom. Control 35, 182–186 (1990).
D. W. Gu, B. W. Choi, and I. Postlethwaite, “Low-order stabilizing controllers,” IEEE Trans. Autom. Control 38, 1713–1717 (1993).
Q. G. Wang, T. H. Lee, and J. B. He, “Low-order stabilizers for linear systems,” Automatica 33, 651–654 (1997).
J. C. Fan and T. Kobayashi, “A simple adaptive PI controller for linear systems with constant disturbances,” IEEE Trans. Autom. Control 43, 733–736 (1998).
P. Grieder, M. Kvasnica, M. Baotić, and M. Morari, “Stabilizing low complexity feedback control of constrained piecewise affine systems,” Automatica 41, 1683–1694 (2005).
H. Sano, “Low order stabilizing controllers for a class of distributed parameter systems,” Automatica 92, 49–55 (2018).
J. Bu and M. Sznaier, “A linear matrix inequality approach to synthesizing low-order suboptimal mixed 𝓁1/H p controllers,” Automatica 36, 957–963 (2000).
K. M. Grigoriadis and R. E. Skelton, “Low-order control design for LMI problems using alternating projection methods,” Automatica 32, 1117–1125 (1996).
S. Wang and J. H. Chow, “Low-order controller design for SISO systems using coprime factors and LMI,” IEEE Trans. Autom. Control 45, 1166–1169 (2000).
S. P. Bhattacharyya, H. Shapellat, and L. Keel, Robust Control: the Parametric Approach (Prentice Hall, Upper Saddle River, NJ, 1995).
O. N. Kiselev and B. T. Polyak, “Synthesis of low-order controllers by criterion H ∞ and by the criterion of maximum robustness,” Avtom. Telemekh., No. 3, 119–130 (1999).
V. I. Goncharov, A. V. Liepin’sh, and V. A. Rudnitskii, “Synthesis of low-order robust controllers,” J. Comput. Syst. Sci. Int. 40, 542 (2001).
E. N. Gryazina, B. T. Polyak, and A. A. Tremba, “Design of the low-order controllers by the H ∞ criterion: A parametric approach,” Autom. Remote Control 68, 456 (2007).
O. S. Kozlov and L. M. Skvortsov, “Synthesis of simple robust controllers,” Autom. Remote Control 76, 1598 (2015).
B. D. O. Anderson and Y. Liu, “Controller reduction: Concepts and approaches,” IEEE Trans. Autom. Control 34, 802–812 (1989).
D. Mustafa and K. Glover, “Controller reduction by H ∞-balanced truncation,” IEEE Trans. Autom. Control 36, 668–683 (1991).
V. V. Dombrovskii, “Reducing the order of linear multidimensional systems with H ∞ restrictions,” Avtom. Telemekh., No. 4, 123–132 (1994).
V. V. Dombrovskii, “Synthesis of dynamic regulators of reduced order at H ∞ restrictions,” Avtom. Telemekh., No. 11, 10–17 (1996).
V. V. Apolonskii and S. V. Tararykin, “Methods for the synthesis of reduced state controllers of linear dynamic systems,” J. Comput. Syst. Sci. Int. 53, 799 (2014).
V. V. Apolonskii, L. G. Kopylova, and S. V. Tararykin, “Reducing controllers of linear dynamic systems based on the analysis of physical features of the plant,” J. Comput. Syst. Sci. Int. 55, 683 (2016).
M. G. Zotov, “Algorithm for synthesizing optimal controllers of given complexity,” J. Comput. Syst. Sci. Int. 56, 343 (2017).
V. A. Mozzhechkov, “Design of simple-structure linear controllers,” Autom. Remote Control 64, 23 (2003).
V. A. Mozzhechkov, Simple Structures in the Control Theory (TulGU, Tula, 2000) [in Russian].
S. P. Bhattacharyya, “Robust control under parametric uncertainty: An overview and recent results,” Ann. Rev. Control 44, 45–77 (2017).
V. V. Voevodin, Linear Algebra (Nauka, Moscow, 1980) [in Russian].
G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations (Prentice Hall, Englewood Cliffs, NJ, 1977).
P. D. Krut’ko, Control of Executive Systems of Robots (Nauka, Moscow, 1991) [in Russian].
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Mozzhechkov, V.A. Design of Simple Robust Controllers for Time-Invariant Dynamic Systems. J. Comput. Syst. Sci. Int. 60, 353–363 (2021). https://doi.org/10.1134/S1064230721030126
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DOI: https://doi.org/10.1134/S1064230721030126