μ-Synthesis for Fractional-Order Robust Controllers

Abstract

The current article presents a design procedure for obtaining robust multiple-input and multiple-output (MIMO) fractional-order controllers using a μ-synthesis design procedure with D–K iteration. μ-synthesis uses the generalized Robust Control framework in order to find a controller which meets the stability and performance criteria for a family of plants. Because this control problem is NP-hard, it is usually solved using an approximation, the most common being the D–K iteration algorithm, but, this approximation leads to high-order controllers, which are not practically feasible. If a desired structure is imposed to the controller, the corresponding K step is a non-convex problem. The novelty of the paper consists in an artificial bee colony swarm optimization approach to compute the nearly optimal controller parameters. Further, a mixed-sensitivity μ-synthesis control problem is solved with the proposed approach for a two-axis Computer Numerical Control (CNC) machine benchmark problem. The resulting controller using the described algorithm manages to ensure, with mathematical guarantee, both robust stability and robust performance, while the high-order controller obtained with the classical μ-synthesis approach in MATLAB does not offer this.

μ-Synthesis FO-PID for Twin Rotor Aerodynamic System

“…In order to compare the current iteration of FO-PID µ-synthesis with previous methods, the fixed-structure part of the µ-synthesis mixed-sensitivity loop shaping control problem ( 25) is solved using the artificial bee colony (ABC) approach presented in [17]. The hyperparameters used for this experiment are: the swarm dimension N = 50, the maximum number of ABC cycles 50, the maximum number of cycles with no improvements 10, the limit for the abandonment counter 10, the maximum number of D-K iterations 10, the maximum window length for assessing lack of progress 4, while the parameters for the cost functions are α = 1 and β = 10 5 .…”

“…In this section, the mathematical background for the proposed controller design method in terms of Robust Control Framework in Section 2.1 and Fractional-Order Control Framework in Section 2.2 is firstly presented, while in Section 2.3 the method for optimizing the controller parameters using a different approach as against the procedure presented in [17] is described.…”

“…As already known, the last optimization problem can be solved using the so-called D-K iteration [9,17]. This iterative procedure starts with a fixed D (usually considered the unitary system) and alternatively computes the controller K, by solving the H ∞ control problem with fixed D, and the D-scale factor, by solving the Parrot problem, as defined in [6], for each point from a frequency set Ω = {ω l = ω 1 < • • • < ω N = ω u } followed an approximation of the obtained solutions with a minimum phase system.…”

“…The above problem is non-convex in terms of the free tuning parameters of the controller K ∈ K. However, the problem (17) can also be solved using the D-K iteration approach, where the K step from ( 8) is replaced with the following K K step:…”

“…In [16] crone control methodologies were presented, along with LMI formulation for the H ∞ fractional-order control problem. An artificial bee colony optimization for a MIMO FO-PID controller design by solving the mixed-sensitivity µ-synthesis control problem is presented in [17].…”