Validation of the Control Quality of Characteristic Field Based Fuzzy Controllers

Transcription

1 Validation of the Control Quality of Characteristic Field Based Fuzzy Controllers R. Hampel Institute of Process Automation and Measuring Technique (IPM) University of Applied Sciences Zittau Theodor-Korner-Allee Zittau, Germany N. Chaker Institute of Process Automation and Measuring Technique (IPM) University of Applied Sciences Zittau Theodor-Korner-Allee Zittau, Germany Abstract Rule based systems for Fuzzy Control, Fuzzy Supervising and Fuzzy Diagnosis are non-linear multi-input, multi-output systems. The result of the signal processing within the system is a high dimensional characteristic field. For optimizing the behaviour of the system, we have a high number of fiee degrees. The reconstruction of the knowledge base from the characteristic field as starting point is impossible. With this background, the paper describes the quality of a two dimensional Fuzzy Controller characteristic field in connection with the necessary deformation for the compensation of non-linear effects. We will demonstrate that one should define two restrictions for the characteristic field: continuity without local extrema and differentiability. Under such conditions we need only two fiee degrees for optimizing the controller behaviour using the characteristic field deformation. A related example will be presented. Keywords: Fuzzy Control, Control Quality, Multi-dimensional Fuzzy Systems, Cascading, Optimization. 1 Introduction Rule based Fuzzy Controller are advantageously applied in such areas where the analytic description of processes is difficult and the knowledge basis is incomplete. That applies especially in the process boundaries nearby fault conditions and in the case of deviations fiom nominal operating point. For keeping a given control quality, requirements regarding dynamics should be fulfilled by the Fuzzy Controller. A concept allowing the representation of knowledge using characteristic fields implemented in a Fuzzy Controller will be presented, and criteria for validating the control quality will be derived as well. In this case, the following aspects will be particularly considered. - Reduction of the number of variable parameters for the parameterization of a Fuzzy Controller, - Function in the process boundaries. By the example of a PI-like Fuzzy Controller in a simplified closed-loop control system, the influence of the characteristic field deformation on the control quality will be presented. For deducing the advantages of the concept, it is meaningful to transform MISO-systems into cascaded structures. In this case, it should be guaranteed that the knowledge basis for designing the controller will not be distorted. The problem will be analysed with the help of a controller with three input variables. The results are transferable to multi-dimensional controllers. For demonstrating the concept performance, the control of a combustion process using fuzzy logic will be described.

2 2 Knowledge Representation Using Fuzzy Logic Based Characteristic Fields Classical PID Controllers with different complements (adaptation, disturbance and setpoint feedforward) have a high acceptance in process automation. Hence, we aim at a comparison between the quality of Fuzzy Controllers and classical PID Controllers [l-121. The Fuzzy Controller in the process has the same interface as the classical PLD Controller. Independently from the chosen structure, the dynamic is determined by differentiation and integration of the system deviation outside the Fuzzy Controller module. With it, the Fuzzy-PID- Controller is provided with the input variables. The dimension still increases if additional disturbance and setpoint variables are feedforwarded [9]. As a result, the system becomes rather complicated. The strategy developed at the IPM aims at the decription of multi input - single output (MISO) systems by structures with two-dimensional Fuzzy Controllers that can be optimized and parameterized individually [3-61. Figure 2 exemplarily shows a Fuzzy-PI-Controller for which the basic rule has the form IF X1 AND X2 THEN Y X1 - integral of system deviation X2 - system deviation Y - manipulated variable. The characteristic field for the PI-Controller presents two areas - non-deformed characteristic field according to the classical non-adjustable PI-Controller, - deformed characteristic field according to the non-linear Fuzzy-PI-Controller. For the parameterization and optimization of the Fuzzy Controller characteristic field there exist many degrees of freedom. The following quality criteria for the shape and deformation of the characteristic field are used: - Differentiability (smooth characteristic by changeable transmission coefficients) - Slight waviness - Ability to deformation (maximal necessary deviation from the linear characteristic field). Differentiability and waviness are essentially determined by the combination operatormembership function. Well suited are: - Operator: Sum-Prod for h-sets - Operator: Max-Min for Gaussian-sets. For these cases it is sufficient to vary the number and distribution of the fuzzy sets for the deformation of the characteristic field. The optimization results from deforming the characteristic field. The optimal behaviour of the Fuzzy Controller is copied in the deformation of the characteristic field. 3 Analysis of Control Quality The analysis of control quality is exemplarily carried out on a simplified process with a first order time delay and dead time. The input-output behaviour of the Fuzzy PI controller represented in Figure 1 shows an optimal characteristic field (Fig. 2) when using appropriate operators for the inference. The change of operator for the conjunction (bounded difference) leads to a non-optimal deformation of the characteristic field as schown in Figure 3. The control behaviour using the bounded difference operator shows an oscillatory course (Fig. 4) accompanied with a high overshoot. This unfavourable behaviour depends on the working point and disturbance height. The main result of this investigation is that the optimal deformation of the characteristic field is continuous and without local minima and maxima. The optimal deformation of the Fuzzy Controller characteristic field can be realized using suitable operators and membershipfunctions. The deduction of these results is made possible because of the clear representation of the control law as a characteristic field. This applies only to systems with two input variables. In the case of multidimensional systems it is necessary to transform them into a cascade of two-dimensional systems in order to keep the transparence of their representation and the advantages of an analysis in a twodimensional system. bl Controller Figure 1 : Fuzzy PI Controller Structure

3 Figure 2: Optimal Characteristic Field of the PI Fuzzy Controller e : control deviation ei : integral of control deviation Y : control variable Figure 3: Non-Optimal Characteristic Field of the Fuzzy PI Controller 4 Cascading High-dimensional Fuzzy Controllers The cascading of multi-dimensional Fuzzy systems consists of a structure transformation into a cascade of two-dimensional systems. The essential restriction of this structure transformation is the keeping of the global transfer behaviour of the fuzzy controller with the aim to reduce computing time and improve the clearness of knowledge. Figure 5 serves as demonstration of the results. XI acd X2 are dominating input variables while X3 is a non-dominating input variable. Its influence on the Controller will be clear through the adaptation rules IF X3 = L displacment of YM in L direction IF X3 = H displacment of YM in H direction The comparison of the rule matrix of the complete three-dimensional Controller (Fig. 6) with the cascaded Controller matrix (Fig. 7 and 8) shows a mismatch at two positions (highlighted in Fig. 6 and 8). By adding two sets to the virtual auxiliary variables Yvl (Fig. 9 and lo), complete consistency between the multi-dimensional and cascaded structure can be obtained. Figure 4: Control Behaviour with Deformed (XB) and Non-deformed (Xp) Fuzzy Characteristic Field Xp, Yp : controlled resp. control variable corresponding to Fig. 2 XB, YB : controlled resp. control variable corresponding to Fig. 3 W : set point variable Z : disturbance variable t : time Figure 5: Cascading a 3-dimensional Fuzzy Controller Figure 6: Complete rule matrix for a 3-dimensional Fuzzy Controller

4 X2 L L N k~ L N H H N H H Figure 7: Fuzzy Controller FC 1 of the cascaded controller structure Figure 8: Fuzzy Controller FC2 of the cascaded controller structure Figure 9: Rule matrix FC 1 for the improved 3-dimensional cascaded Fuzzy Controller (5 Sets for Yvl) Figure 10: Rule matrix FC2 for the improved 3-dimensional cascaded Fuzzy Controller (5 Sets for YVI) 5 Example of Application Combustion processes are characterized by high time delay and transport dead time. For reducing the concentration of pollutants like NOx, SO2 and CO, it is necessary to reduce the delay time and to involve the parameter distribution into the control algorithm. By means of flame pyrometry, it is possible to generate more and better information about the flame for diagnosis of the combustion state. The real flame temperature and optical density of the flue gas are information without time delay. Figure 11 shows the structure used for the combustion control as combination of a classical PI- Controller and two Fuzzy Controllers. The Fuzzy Controller FC 1 realizes the compensation of disturbances as short term controller. The knowledge for the rule basis was accumulated with the help of experiments on a test facility. The input variables are volume flow rate of air V real flue gas temperature T optical density of flue gas D. The output variable is volume flow rate increment of air AV. For the strategic control of the combustion process, the second Fuzzy Controller FC2 is used. The well known knowledge basis for static combustion processes depending on the kind of fuel is accumulated in this Controller. The Fuzzy Controller FCl has a hierarchical structure (Fig. 12). The controllers FCll and FCI2 determine on the basis of the input variables Vk, D, and T the auxiliary variables Y1 and Y2 for the controller FCI3 which generates the control variable Y for changing the air flow rate AV. The Fuzy Controller FC2 represented in Figure 11 is structured in a cascade (Fig. 13). Depending on the input variables h an SO2-concentration, the virtual output variable YI is generated in the controller FC2] and weighted with the input variable NOx-concentration in FC22 to the second virtual variable Y2 which is influenced by the COconcentration in FC23 to generate the control variable Y as changing rate of the set point AS. As example, the optimal characteristic fields of the Fuzzy Controller FC 1 are represented in Figure 14. Figure 15 shows the system behaviour during a disturbance of fuel mass flow rate. Without the Fuzzy Controller results a high overshoot of the air excess number and as consequence higher CO and NOX emission. The strategy and controller structure are also able to control the parameter distribution in the combustion process.

5 Figure 1 1 : Controller structure of the combustion process FC : Fuzzy Controller Figure 12: Structure of the Fuzzy Controller FC 1. Figure 13: Cascaded Structure of the Fuzzy Controller FC2 a) Characteristic Field of FC1l. b) Characteristic Field of FC12. c) Characteristic Field of FCI3. Figure 14: Optimal Characteristic Fields of the Fuzzy Controller FC 1.

6 I 0, m m ) m. m l O. m m l l l 'CI!(.I,om VInI 1.m so: 1wm.1 Figure 15: Process response to a fuel mass flow rate change of 17 % Classical Controller - Fuzzy Controller 3L - Air excess number B - Relative fuel mass flow rate D - Optiscal density CO CO-Conzentration in flue gas T - Flame temperature NOx - NOx-Conzentration in flue gas V - Air volume flow rate SOz - SO2-Conzentration in flue gas 6 Conclusion The paper presented the main results of the investigations in the field of Fuzzy Logic at the IPM. The Fuzzy Controller substitutes the classical one with the same interface to the process. The response characteristic of a Fuzzy Controller can be described by characteristic fields with only two input variables each. Consequently, high dimensional Fuzzy Controllers can be transformed into cadscaded twodimensional ones. Thereby, the knowledge representation of Fuzzy Controllers is made clearer and the optimization simpler. The control quality and the system stability analysis is made possible through the representation as characteristic field. The optimal setting of the Fuzzy Controller is reflected in the deformation of the non-linear characteristic field. The control structure of a combustion process using Fuzzy Logic for processing optical signals was described and the experimental results in comparison with the classical controller structure were presented. The Fuzzy Controller structure is able to control the parameter distribution in the combustion process, and, consequently, to contribute to a minimal pollutants emission.

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