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.
Highlights
One of the active problems with major impact which have been studied for years in Control Theory refers to robustness
We present a controller synthesis procedure which manages to find a fixed structure fractional-order controller using the μ-synthesis technique from the Robust Control framework, where the non-convex subproblem involved in the classical D–K iteration is replaced by a swarm optimization algorithm
The process is represented by a Computer Numerical Control (CNC) machine with two orthogonal axis which are operated by two servo DC motors
Summary
One of the active problems with major impact which have been studied for years in Control Theory refers to robustness. Robust control problems use H2 and H∞ norms defined in frequency domain as a performance measure. One possible solution is presented in [1] and is based on Algebraic Riccati Equations (AREs). A more numerically stable approach to solve ARE was developed using Popov triplets in [2], approach recently implemented in an open-source manner in [3], with an iterative refinement method presented in [4] , but an ARE-based solution presents a limitation due to the impossibility of solving singular problems. An alternative way which manages to solve such problems was introduced in [5], where AREs were replaced by Algebraic Riccati Inequalities (ARIs). An open-source solver for Robust Control problems using LMIs is presented in [6]
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