Parameterized bilinear matrix inequality techniques for gain‐scheduling proportional integral derivative control design

Abstract

Proportional-integral-derivative (PID) structured controller is the most popular class of industrial control but still could not be appropriately exploited in gain-scheduling control systems. To gain the practicability and tractability of gain-scheduling control systems, this paper addresses the  ∞ gain-scheduling PID control. The design of such a controller is based on parameterized bilinear matrix inequalities, which are then solved via a bilinear matrix inequality optimization problem of nonconvex optimization. Several computational procedures are developed for its computation. The merit of the developed algorithms is shown through the benchmark examples. K E Y W O R D SH ∞ proportional-integral-derivative control, bilinear matrix inequality, gain-scheduling control system, nonconvex optimization techniques, parameterized bilinear matrix inequality INTRODUCTIONGain-scheduling systems, which are easily implemented online, have proved as one of the most practical tools for studying complex nonlinear systems. 1 Treating nonlinear systems as gain-scheduling systems allows the application of advanced gain-scheduling control techniques in tackling state feedback and output feedback stabilization of nonlinear systems. [2][3][4][5] Until now, most gain-scheduling controllers are assumed structure-free and full-rank to admit computationally tractable parameterized linear matrix inequality (PLMI)-based or linear matrix inequality (LMI)-based formulations. [4][5][6][7][8][9] Meanwhile, proportional-integral-derivative structured (PID) controller is an indispensable component of industrial control thanks to its simple but versatile structure. PID control theory is still the subject of recent research. 10-19 However, the main focus is restricted to linear time-invariant systems in the frequency domain. PID controller for Takagi-Sugeno fuzzy systems, 20 which are particular gain-scheduling systems, was considered in Referencs 21-24. Reference 25 proposed an LMI-based iterative algorithm for a proportional-integral (PI) controller in Takagi-Sugeno systems under the specific structure of both system and controller. A bilinear matrix inequality (BMI) approach was proposed in Reference 26 to solve the H ∞ gain-scheduling problem in the discrete-time case. This approach, however, is restricted to more favorable state-feedback controllers and PID output-feedback controllers are not considered. Gain-scheduled PID controller design for uncertain linear parameter varying (LPV) systems was studied in Reference 27, where solvability conditions for robust controller synthesis are formulated in terms of BMIs. Note that such characterizations rely on guaranteed-cost performance design of linear quadratic-type and involve matrix terms of degree-3 in the decision variables which seems extremely challenging from a computational viewpoint. Additionally, matrix inequalities were solved using the general 3886

A mesh‐based partitioning algorithm for decreasing conservatism in solving bilinear matrix inequality problems

“…Since then, the researchers have listed several problems resulting in a BMI problem 4,5 . For example, the applications of the BMI problems in designing control systems 6,7 such as guaranteed cost control, 8,9 static output feedback controller for spectral abscissa optimization, 10 uncertain fractional order system, 11,12 $$$ {H}_2 $$$ and $$$ {H}_{\infty } $$$ optimization, 13 observer‐based robust controller design, 14 model predictive control, 15 fuzzy controller design, 16 Self Optimizing Control, 17 non‐quadratic controller design, 18 and power flow problem in power systems 19 are reported in the literature.…”

Novel robust gain scheduled PID controller design usingDRregions

“…Stabilizing H inf controller for discrete‐time switched system including dwell time conditions for each mode is proposed in [25]. Gain‐scheduling PID controller design based on quadratic stability and special technique to simplify the corresponding BMI solution is presented in [26]. In [27], an improved modeling method which combines partial linearization and function substitution is proposed to build an LPV model for a nonlinear unmanned aerial vehicle.…”