Research Papers

Multivariable Proportional-Integral-Derivative Controller Tuning Via Linear Matrix Inequalities Based on Minimizing the Nonconvexity of Linearized Bilinear Matrix Inequalities

[+] Author and Article Information
Farshad Merrikh-Bayat

Department of Electrical and
Computer Engineering,
University of Zanjan,
Zanjan 4537138791, Iran
e-mail: f.bayat@znu.ac.ir

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received October 11, 2017; final manuscript received May 19, 2018; published online June 18, 2018. Assoc. Editor: Yunjun Xu.

J. Dyn. Sys., Meas., Control 140(11), 111012 (Jun 18, 2018) (9 pages) Paper No: DS-17-1516; doi: 10.1115/1.4040420 History: Received October 11, 2017; Revised May 19, 2018

This paper proposes an iterative linear matrix inequality (LMI) method for tuning the parameters of multi-input multi-output (MIMO) proportional–integral–derivative (PID) controllers. The proposed method calculates the parameters of controller such that the singular values (SVs) of the sensitivity function are shaped according to the given weight function. For this purpose, first using bounded real lemma (BRL), this problem is represented as a bilinear matrix inequality (BMI) where one of the matrix variables is the variable of Lyapunov equation and the other is structured and contains the matrices of the state-space representation of the controller. This BMI is solved approximately using a novel iterative procedure which linearizes the BMI around an initial solution to arrive at an LMI. The point around which the BMI is linearized is updated automatically at each iteration and the linearized BMI has the nice property that it is obtained by minimizing the amplitude of nonconvex terms.

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Fig. 1

The MIMO feedback system

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Fig. 2

Representation of the MIMO PID tuning as a weighted sensitivity problem

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Fig. 3

Closed-loop step response from r to yp for the PID controlled system, corresponding to Example 1

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Fig. 4

Singular values of S(s) (dash-dotted: iteration 1, dashed: iteration 3, solid: iteration 5) and Bode magnitude plot of Wp−1(s), corresponding to Example 1

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Fig. 5

The effect of using different sets of LMIs for calculating the initial P on the closed-loop step responses. At each case, the controller is obtained after five iterations.

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Fig. 6

Comparing the proposed method with hinfstruct function of matlab

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Fig. 7

Closed-loop step response from r to yp for the PID controlled system, corresponding to Example 2

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Fig. 8

The effect of using different sets of LMIs for calculating the initial P on the closed-loop step responses. At each case, the controller is obtained after five iterations.

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Fig. 9

Comparing the proposed method with hinfstruct function of matlab




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