Abstract
This paper presents a multi-objective digital PID controller design method using the parameter space approach of robust control. Absolute stability is treated first by finding the digital PID controller gain parameter space corresponding to closed loop poles being inside the unit circle. Additionally, phase margin, gain margin and a mixed sensitivity bound are treated as frequency domain constraints. Determination of digital PID controller parameter space regions satisfying these constraints is presented. All of these regions are superimposed to obtain a multi-objective digital PID controller gain parameter space solution region. The path following controller design of an automated driving vehicle is used as an example to illustrate the method. This multi-objective parameter space design approach can be extended to other digital controller forms also.
Highlights
The parameter space approach is a part of the parametric approach to robust control [1]–[3]
PID controllers should, be designed directly as digital controllers in order for their designed performance not to be degraded later due to the effect of sampling as very high sampling rates are usually not possible in most applications. It is for this reason that this paper focuses on the design of digital PID controllers instead of treating the easier to handle and well-established continuous time PIDs
The contributions of this paper are: 1) analytical treatment of digital PID controller design, 2) using a multi-objective approach by calculating and superimposing stability constraints, phase margin bounds, gain margin bounds and mixed sensitivity bounds in the same controller parameter space, 3) not being constrained to plants that are first or second order with a delay and, instead, being able to treat plants of any order with time delay, 4) the ability to extend and apply the design approach to other fixed order controllers beyond PID controllers, if needed, 5) guaranteeing a stable design automatically by calculating the stable region of PID gains, 6) being able to incorporate the sampling time into the computation process to evaluate its effect on the solution region obtained
Summary
The parameter space approach is a part of the parametric approach to robust control [1]–[3]. Digital PID controller design approaches that can treat general plants instead of low order fixed forms, that can handle multiobjective constraints, that offer ease of visualization to the designer through graphical representations of stability and performance regions rather than obtaining one possible solution within that PID controller parameter space through optimization are needed. The contributions of this paper are: 1) analytical treatment of digital PID controller design, 2) using a multi-objective approach by calculating and superimposing stability constraints, phase margin bounds, gain margin bounds and mixed sensitivity bounds in the same controller parameter space, 3) not being constrained to plants that are first or second order with a delay and, instead, being able to treat plants of any order with time delay, 4) the ability to extend and apply the design approach to other fixed order controllers beyond PID controllers, if needed, 5) guaranteeing a stable design automatically by calculating the stable region of PID gains, 6) being able to incorporate the sampling time into the computation process to evaluate its effect on the solution region obtained. The stable region, the corresponding two poles of the closed loop transfer function (4) are inside the unit circle, on the unit circle and outside the unit circle, respectively, as expected
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