A framework for the automatic synthesis of hybrid fuzzy/numerical controllers
Introduction
Soft computing refers to the integration of artificial intelligence techniques in hybrid frameworks for solving real world problems [1]. In particular, much work is being done to combine different control paradigms in order to obtain hybrid controllers with better performance. For example, in [2], neural networks and dynamic programming are used to control a nonlinear process. In this context, the use of fuzzy logic together with other approaches has been extensively worked out. For example, in [3] hybrid controllers making use of fuzzy logic and neural networks are proposed, while in [4] genetic algorithms are used to generate and/or calibrate fuzzy controllers. Indeed, fuzzy control represents a powerful technique to cope with continuous systems.
In this paper, we focus on the integration of fuzzy and numerical control, starting from previous works based on cell mapping. Cell mapping [5] is based on the discretization of state variables of the system, partitioned into cells, and represents a very efficient computational technique for the global analysis of continuous systems.
Hsu used cell mapping to generate a numerical controller, that is a control table containing state-action pairs [6]. However, the values in the table are discretized and this can introduce steady state errors when dealing with continuous systems. Furthermore, even using a very fine discretization, the table can require a lot of memory for a large number of cells. In contrast, a fuzzy logic controller is continuous and only requires memory for the input fuzzy set definitions and the output function parameters.
Much work is being done to provide methodologies for the automatic calibration of fuzzy logic controllers in order to improve their performance. In particular, cell mapping has been used to generate control tables to improve the quality of fuzzy controllers [7], [8], [9].
However, to the best of our knowledge, no previous works have addressed the problem of simultaneously using a fuzzy controller together with an optimal control table. In this context, a very important issue is to provide a methodology for the automatic generation of such a hybrid controller.
In this paper, we propose a new approach based on the following considerations:
- 1.
Model checking techniques [10], [11] allow one to explore huge state spaces (also up to billions of states) and to use a very fine discretization of the state space, so obtaining a more precise analysis of a system than the one provided by other techniques as cell mapping.
- 2.
Recently, these techniques have been used to automatically synthesize optimal control tables.
- 3.
In previous works, we proposed an automatic compression process for numerical controllers that uses ordered binary decision diagrams (OBDD). The OBDD representation results very compact and very fast to read.
- 4.
Fuzzy controllers still remain very suitable to cope with continuous systems due to their robustness.
We propose to exploit properties of numerical and fuzzy controllers highlighted in 1–4 in order to automatically generate a hybrid controller consisting of
- •
a numerical component: that is the optimal control table synthesized by model-checking-based tools;
- •
a fuzzy component: that is a fuzzy controller generated and/or calibrated using the information in the numerical controller.
As explained in Section 4, the hybrid controller takes advantage of the optimality of numerical controllers and the robustness of fuzzy ones. Finally, making use of the OBDD-compression, the resulting hybrid controller requires a small amount of memory and can be easily implemented in a microprocessor.
The paper is organized as follows. In Sections 2 Fuzzy control, 3 Numerical controllers we introduce fuzzy and numerical controllers, respectively. In Section 4 we describe, step by step, our methodology for the automatic generation of hybrid controllers. Section 5 shows experimental results and presents a comparison between the controllers performance. Section 6 concludes the paper.
Section snippets
Fuzzy control
Fuzzy control is well known as a powerful technique for designing and realizing controllers, especially suitable when a mathematical model is lacking or is too complex to allow an analytical treatment [12].
A fuzzy controller (FC) is based on qualitative fuzzy rules which have the form “if condition then control action”, where the condition represents the rule premise and the control action the rule consequent.
Fuzzy controllers are very effective in handling “uncertain” or “partially known”
Numerical controllers
As mentioned above, a numerical controller is a table, indexed by plant states, whose entries are commands for the plant. These commands are used to set the control variables in order to reach the set point from the corresponding states. Namely, when the controller reads a state from the plant, it looks up the action described in the associated table entry and sends it to the plant.
There is a number of well-established techniques for the synthesis of numerical controllers. Among them, one of
Automatic generation of hybrid controllers
In this section we present the general framework for the automatic generation of hybrid fuzzy/numerical controllers.
Remark 4.1 We recall that, in this paper, we do not concern with the generation of fuzzy controllers, instead we focus on the improvement of a given one and on how to integrate it with a numerical controller, so obtaining the final hybrid controller.
To this aim, we assume an initial TS fuzzy controller to be given. Indeed, there is a lot of work on the design of fuzzy rules, we refer the
Experimental results
To show the effectiveness of our approach, in this section we present experimental results related to two benchmark problems (see, e.g., [28], [29], [30]), the linear DC motor and the nonlinear inverted pendulum. In both cases we compare the performance of initial fuzzy, numerical, updated fuzzy and hybrid controllers.
Conclusions
In this paper, we have presented a methodology for the automatic generation of hybrid fuzzy/numerical controllers.
We have first shown how numerical controllers generated through model checking can be used to consistently improve TS fuzzy controllers. Moreover, we have presented the HCGMurphi framework which allows one to automatically generate a numerical controller starting from the plant description and to integrate it with a given TS fuzzy controller updated by the tool itself. The resulting
Acknowledgements
We are grateful to two anonymous reviewers for their helpful comments which helped in improving the presentation of this paper.
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