Linear Robust Control System Design via Evolutionary Computation

Modern robust controller design often involves solving a mixed-sensitivity optimisation problem to maintain system response and error signals within pre-specified tolerances despite uncertainties and disturbances.  Although many approaches based upon the solution of a set of Riccati equations have been proposed to derive an analytical solution, these methods are often confined within a narrow problem domain with frequently encountered numerically or physically ill-posed limitations.  Moreover, the resulting controller is often of very high-order and thus hard to be implemented.  These techniques also encountered difficulties in incorporating practical time-domain design constraints, such as actuator saturation and time delays, and cannot easily be applied if the plant to be controlled has imaginary poles or/and zeros.  Due to these limitations and constraints, a conventionally designed controller may lead to system degradation or may not realise the full potential when on-line implementation is performed.

 

Emulating the Darwinian-Wallace principle of “survival-of-the-fittest” in natural selection and genetics, evolutionary algorithms (EAs) have been found to be very effective and efficient in searching a poorly understood, irregular and complex space for optimisation and machine learning.  Such an algorithm evaluates performances of candidate controllers at multiple points simultaneously leading to several globally optimised designs, which is naturally well suited for the robust controller design optimisation in a usually multi-modal multi-dimensional searching space.  An EA-based design methodology that is capable of solving the mixed-sensitivity optimisation to achieve desired closed-loop performance in the face of perturbations and constraints for practical systems as shown in Figure 1 has been developed.

Without loss of generality, a DC servomotor system for velocity control is used to illustrate the EA controller design methodology.  Stability verification is carried out for every candidate controller, such that any design with unstable poles on the right s-plane will be assigned a predefined high cost, without performing the closed-loop simulation so as to reduce the overall computation time.  The excellent closed-loop response of the EA evolved system for a step input of 60 r.p.m. (after a 9:1 step-down gear-box) is shown by Curve 1 in Figure 2.  To validate the robustness of the controller, a 0.2 Hz sine wave disturbance with peak-to-peak amplitude of 0.2 and 10 ms sampling period as shown by Curve 2 of Figure 2 was applied to the system.  The attenuated disturbance at the system output and the response of the motor system that suffered from this disturbance is shown by Curve 3 and Curve 4 in Figure 2, respectively.  The performance clearly reveals that the effect of external disturbance has been attenuated satisfactorily.

 

Figure 2: Response of the step and disturbance inputs.

 

The EA evolved robust controller has been further validated in a physical system involving a DC servo-mechanism for further validation.  The captured closed-loop response of the system at two different operating points, subject to both step-up and step-down command tests is shown in Figure 3.  Note that the eddy-current brake of the system is released at t = 3s and reapplied at t = 8s, to test the system robustness in tolerating plant perturbations.  The response clearly confirms that the EA evolved system is consistent for both the simulated and implemented experimentation, with excellent closed-loop response and good robustness against uncertainties.  The evolutionary design methodology is currently being extended to include multi-objective optimisation for MIMO systems, and to integrate other design specifications such as mixed-norm, controller structures and economical costing considerations. 

 

This work was performed in collaboration with Dr KK Tan.

 

Figure 3: Implemented performance of the EA evolved DC servomotor system, where plant uncertainties occur at the boundaries of the shaded area.