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Technical Brief

Many-Objective Optimal and Robust Design of Proportional-Integral-Derivative Controls With a State Observer

[+] Author and Article Information
Yousef Sardahi

School of Engineering,
University of California,
Merced, CA 95343
e-mail: ysardahi@ucmerced.edu

Jian-Qiao Sun

Professor
School of Engineering,
University of California,
Merced, CA 95343;
e-mail: jqsun@ucmerced.edu

Carlos Hernández

CINVESTAV-IPN,
Computer Science Department,
Mexico City 07360, Mexico
e-mail: chernandez@computacion.cs.cinvestav.mx

Oliver Schütze

CINVESTAV-IPN,
Computer Science Department,
Mexico City 07360, Mexico
e-mail: schuetze@cs.cinvestav.mx

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 26, 2016; final manuscript received August 31, 2016; published online November 9, 2016. Assoc. Editor: Zongxuan Sun.

J. Dyn. Sys., Meas., Control 139(2), 024502 (Nov 09, 2016) (4 pages) Paper No: DS-16-1273; doi: 10.1115/1.4034749 History: Received May 26, 2016; Revised August 31, 2016

This paper presents a many-objective optimization (MaOO) approach to design control and observer gains simultaneously. The many-objective optimization (MaOO) approach finds trade-offs between different nonagreeable design goals. We report the MaOO results of a proportional-integral-derivative (PID) control with a state estimator applied to a second-order oscillator. The overall system is optimized to minimize the peak time, overshoot, tracking error, and control energy, and maximize the rejection of external disturbance and measurement noise, while the relative stability, which is defined by maximum real part of eigenvalues of the closed-loop system, is constrained to be less than or equal to −1. The numerical simulations show promising findings of the proposed method.

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Figures

Grahic Jump Location
Fig. 1

The Pareto set: (a) kp versus ki, (b) kp versus kd, (c) kp versus l1, and (d) kp versus l2. The color code indicates the level of the objective function JIAE. Red denotes the highest value, and dark blue denotes the smallest value (see figure online for color).

Grahic Jump Location
Fig. 2

Projections of the Pareto front: (a) Mp versus tp, (b) Eu versus tp, (c) LTL versus ‖S(jω)‖∞, and (d) Eu versus ‖S(jω)‖∞. The color code indicates the level of the objective function JIAE. Red denotes the highest value, and dark blue denotes the smallest value (see figure online for color).

Grahic Jump Location
Fig. 3

(a) The poles of the closed-loop system A − BK corresponding to the Pareto set in Fig. 1. (b) The estimator poles of the matrix A − LC. The color code indicates the level of the objective function JIAE. Red denotes the highest value, and dark blue denotes the smallest value (see figure online for color).

Grahic Jump Location
Fig. 4

System responses before (blue solid line) and after (red solid line) adding an external disturbance (as in subplots (a) and (b)) or measurement noise (as in subplots (c) and (d)). (a) max(‖S(jω)‖∞), (b) min(‖S(jω)‖∞), (c) max(LTL), and (d) min(LTL) (see figure online for color).

Grahic Jump Location
Fig. 5

System responses with x̂(0)=0 (blue solid line) and with x̂(0)≠0 (red solid line). (a) x̂(0)=[−1−1], and the estimator is very slow, (b) x̂(0)=[−1−1] and the estimator is very fast, (c) x̂(0)=[1 1] and the estimator is very slow, and (d) x̂(0)=[1 1] and the estimator is very fast (see figure online for color).

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