Fuzzy Logic Control:

integrating qualitative and quantitative design


( Ph.D. Thesis )


H.X. Li


Summary


Research in fuzzy logic control (FLC) has progressed rapidly in recent years, however, there is still no systematic method for analysing and designing this fantastic controller. In industry, FLC is experience based design. Sound knowledge of the process is often needed. The tuning for matching linguistic rules and numerical input/output is normally done by rule adjustment. This qualitative design is entirely heuristic, and thus it is difficult to obtain the systematic design. In the academic area, there is research in artificial neural network (NNW) to possess self-learning capability. Only limited or no initial knowledge is needed. The quantitative training is carried out to generate the rule base. Except for being time-consuming, however, this pure quantitative approach may lose the original linguistic interpretation. Another method of research in developing adaptive fuzzy control is by combining NNW and fuzzy logic. The rule base is built by fuzzy logic and the data base trained by NNW. This approach is still complex and time consuming. The aim of this thesis is to provide some practical and simple methods for designing and tuning the conventional FLC - an error feedback type FLC. These methods integrate the qualitative approach from the knowledge control together with some quantitative approach from the classical control. The basic concept is to design rule base qualitatively and tune the data base quantitatively.

Just like its non-fuzzy counterparts, the conventional FLC has two-term control (FZ-PD, FZ-PI) and three-term control (FZ-PID). A general two-dimensional rule base is designed qualitatively by the phase plane technique based on the general dynamics of the process. The rule base derived in this way not only can keep the important triple-property but also can be easily modified to include other information, like time-delay, etc.. In addition, this general rule base can provide a robust performance because it is from the general process, instead of a particular process or experience from some experts. These results can be extended easily to the high-dimensional rule base. On the other hand, by simply modifying the data base, a two-dimensional rule base can be used also to construct a simplified FZ-PID controller. This type of FZ-PID is fast in computation, simple in structure and easy in the implementation.

As the rule base conveys a general control policy, it should be preserved during the operation. As the data base provides a necessary numerical calibration for the control, it is most suitable for tuning. The concept of fuzzy transfer function is first introduced to describe the influence of input scaling gains to the performance. It is well known that PI/PD/PID control can be designed or tuned easily because they already have systematic theory for analysis, design and tuning. Then, a comparative gain design is proposed to find the initial scaling gains for conventional FLC by using its well-tuned non-fuzzy counterparts, and a comparative tuning method for their fine tuning. These methods require little of the trial and error process, and achieve quite satisfactory performance. FLC designed in this way shows superior performance when compared with their non-fuzzy counterparts for increasingly complex processes.

The linguistically designed rule base can be quantised by using the inference cells (ICs). The rule base can be decomposed into many ICs, from which the mathematical derivation can be carried out to obtain the quantitative model of FLC. This mathematical model shows that conventional FLC with a linear rule base is a nonlinear time-varying control, consisting of two parts: a nonlinear relay term and a nonlinear two-/three- term. It becomes a linear control at the equilibrium state. Further analysis of the variable structure control (VSC) theory shows that the relay term provides more VSC features. The relationship between scaling gains and the performance, the membership functions and performance, and the influences of uncertainties to the design become more clear. The unified method based on VSC theory is proposed for designing and tuning scaling gains of the conventional FLC.

Both qualitative and quantitative analyses show that conventional FLC is actually a dual control bridging two-/three- term control and VSC. Thus, dual design methodologies are suggested for the design of FLC. When the process is more linear, FLC can be designed and tuned by the comparative method to provide a smoother response. When the process is more nonlinear, FLC can be designed and tunedby using VSC theory to suppress more uncertainties.

Applications of fuzzy control on three different processes are demonstrated by using the above theory. These processes include: a thermal process with variable time-delay, a magnetic suspension system, and an inverted pendulum system.

The thesis also discusses the more advanced approaches for fuzzy control, such as: FLC for high-order systems, FLC for delayed processes and adaptive FLC. The high-order FLC should have a high-dimensional rule base, which can be implemented by a high-order switching function from VSC theory. As the data base is meant to achieve matching betweennumerical input/output and linguistic control rules, then properly adjusting data base may reduce the short delay effects. The gradient method from model reference adaptive control (MRAC) is borrowed to design the model reference adaptive fuzzy control (MRAFC). Two different approaches are discussed. Though the derivation is based on the linear model around the equilibrium state, the results can be extended globally to a wide range of systems. The simulations show that MRAFC is more robust than MRAC for systems with large model-plant mismatches, especially for nonlinear or time-varying systems.