| Summary: | 博士 === 國立成功大學 === 化學工程學系碩博士班 === 90 === This dissertation is concerned with the time-domain simulation and design of SISO (Single-Input Single-Output) feedback control systems. The major effort is devoted to developing fast and accurate numerical methods for computing time responses of closed-loop systems. These methods greatly facilate the design of fractional-order feedback control systems. Effort is also devoted to enhancing the performance of feedback SISO control systems with the use of non-conventional controllers such as fractional-order PID controllers, I-P (Integral-Proportional) controllers with Smith predictors, and PID-deadtime controllers. The latter two types of control schemes are constructed mainly for the lag-dominant processes containing longer delay.
For achieving fast and accurate time-domain simulations for systems with fractional-order or having pure and/or distributed time delays, we represent the transient time response of a closed-loop control system in terms of B-spline series. More precisely, it regards to the inversion of Laplace transforms of irrational type with the B-spline series expansion approach. The B-spline series representation for a time function $f(t)$ with $t ge 0$ contains first several terms in boundary splines and the remaining terms in interior B-splines. It is shown that by matching the initial conditions of the response exactly, the coefficients associated with the interior B-splines can be accurately obtained by a computationally efficient FFT-based algorithm.
The study of feedback fractional-order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional-order models, and that fractional-order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer-order controllers. Owing to the lack of effective analytic methods for the time-domain analysis and simulation of linear feedback fractional-order systems, we suggest two reliable and accurate numerical methods for inverting fractional-order Laplace transforms. One is based on computing Bromwich's integral with a numerical integration scheme capable of accuracy control, and the other is based on expanding the time response function in a B-spline series. In order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the Gr"unwald--Letniknov approximation method and Podlubny's analytic formulas, which are in a form of double infinite series, the time-domain simulations of the feedback control of a fractional-order process with a PD$^mu$-controller and a fractional-order band-limited lead compensator are worked out. The simulation results indicate that a convergence problem indeed occurs in using Podlubny's infinite series expressions, and that the problem could not be overcome by a series acceleration scheme.
In this work, we are also concerned with the design of a fractional-order PID controller which involves noninteger-order integrator and differentiator. The three controller gains and two real orders of the fractional-order PID controller are determined to minimize an integral square error (ISE) performance index while satisfying the specified gain and phase margins. We formulate the design problem as a constrained parametric optimization and apply a differential evolution algorithm to search globally the optimal controller parameters. Due to the problem lack of analytic time-domain analysis methods for fractional-order systems, an effective numerical method is utilized to compute the ISE performance index as well as to check the stability in the frequency domain. Design examples are given to compare the performance of the optimal fractional-order PID controller with the optimal integer-order PID controller in controlling integer-order as well as fractional-order processes.
For lag-dominant process, the features gained by using a I-P controller with a Smith predictor are: 1) the dead time is compensated by the Smith predictor and 2) sluggish and overshoot can be improved owing to rearranging the controller scheme from P-I to I-P structure. A delay-free model is obtained for inducing a matched Smith predictor so that the time responses of system and controller output can be solved exactly, and the specifications in time and frequency domains of the feedback system can also be formulated in sets of analytical equations. Due to this benefit, a spherical solution tracing method is applied to computing the $K_p$-$tau_i$ manifold corresponding to each performance specification with great accuracy and efficiency. Thus, the feasible domain in which the I-P controllers synthesized always meet the desired constrains is enclosed by these mainfolds, and the high gain problem in controller design caused by classic Smith predictors is avoided in turn.
Since the potential of PID-deadtime controllers in control of process existing long or distributed delay, in our work, the optimal-ISE PID-deadtime controllers are designed for integral and first-order unstable plus time delay plants. Once the dead time of the compensator is designated to be the delay induced by process, $theta_p$, the quadratic performance index, such as ISE, and its derivatives of the feedback control system can be represented in terms of controller parameters analytically via a residue computation method. As a spherical method is invoked to meet the constraints of minimum ISE, we can trace the three $K_p$-$theta_p$, $tau_i$-$theta_p$ and $tau_d$-$theta_p$ manifolds simultaneously. These manifolds represent the optimal settings of the PID deadtime controller with respect to the process deadtime. The tuning rules for servo and regulatory control are also correlated and presented. As compared with the optimal-ISE PID controllers, the designed PID-deadtime controllers exhibit a significant performance promotion for integral and unstable process control.
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