Decomposed fuzzy proportional–integral–derivative controllers

Applied Soft Computing 1 (2001) 201–214 Decomposed fuzzy proportional–integral–derivative controllers Marjan Golob∗ Faculty of Electrical Engineering and Computer Science, Institute for Automation, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Accepted 22 August 2001 Abstract In this paper, several types of decomposed proportional–integral–derivative fuzzy logic controllers (PID FLCs) are tested and compared. An important feature of decomposed PID FLCs are their simple structures. In its simplest version, the decomposed PID FLC uses three one-input one-output inferences with three separate rule bases. Behaviours of proportional, integral and derivative PID FLC parts are defined with simple rules in proportional rule base, integral rule base and derivative rule base. The proposed decomposed PID FLC has been compared with several PID FLCs structures. All PID FLCs have been realised by the same hardware and software tools and have been applied as a real-time controller to a simple magnetic suspension system. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy logic control; PID control; Decomposed fuzzy system; Magnetic suspension system 1. Introduction Fuzzy control techniques have been widely used in industrial processes, particularly in situations where conventional control design techniques have been difficult to apply. The main advantage of the fuzzy logic controller (FLC) [12] is that it can be applied to plants that are difficult to get the mathematical model, and the controller can be designed to apply heuristic rules that reflect the experience of human experts. Recently, fuzzy logic and conventional control design methods have been combined to design proportional–integral–derivative fuzzy logic controller (PID FLC), such as [1–5,8,14,15]. PID FLCs have been successfully applied to a variety of practical problems. However, in spite of its practical success, there is not a standard procedure for ∗ Tel.: +386-2-220-7161; fax: +386-2-251-1178; URL: http://www.au.feri.uni-mb.si/∼marjan/. E-mail address: [email protected] (M. Golob). tuning PID FLCs. The design of most PID FLCs is a very time consuming activity involving, knowledge acquisition, definition of the control structure, definition of rules, and tuning a variety of gains and other parameters. If the number of fuzzy inputs increases, a fuzzy controller gets increasingly intractable. The knowledge acquisition suffers increasingly from the engineering knowledge base and computational and memory demands of fuzzy inference systems increase strongly, thus suffering from course of dimensionality. The problem of knowledge acquisition is addressed by learning techniques, while the problem of multidimensionality by decomposing the fuzzy inference system. Next problem of conventional methods to design PID FLCs is a fact that the relation between PID FLC structures, their design parameters and the performance of the control loops is also difficult due to the inherent non-linear nature of controllers with fuzzy logic components. Therefore, design PID FLC with a simple structure, limited number of tuning parameters, and the knowledge base with restricted rules is important 1568-4946/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 1 5 6 8 - 4 9 4 6 ( 0 1 ) 0 0 0 1 9 - 9 202 M. Golob / Applied Soft Computing 1 (2001) 201–214 for the solution of the real world problems, especially the problems with high real-time systems demands. This paper presents the structure and advantages of decomposed PID FLCs as an alternative to the fuzzy logic—PID complete design, a design that require three inputs, which will substantially expand the rule base and make the design more difficult. The decomposed PID FLCs, as we proposed, are based on the inference decomposition and simplification. The fuzzy inference, described by a three-dimensional relational fuzzy equation, has been decomposed on the set of two or three fuzzy inferences, each described by a one- or two-dimensional relational equations. In comparison with the exiting fuzzy PID controllers the proposed decomposed PID FLC has the following features. 1. The decomposed PID FLC keeps the simple structure of the PID controller and enables easy connection between fuzzy parameters and operation of the controller. 2. Most of the actual fuzzy PID controllers are multivariable fuzzy controllers and have specially large knowledge bases. The larger the knowledge base is, the slower is the system. The decomposed PID FLC knowledge base is combined by three separate rule bases with simple rules. The number of rules is reduced. 3. The decomposed PID FLC is suitable for hardware realisation, for instance, with a method for implementing fuzzy controllers on semi-custom VLSI chips, in particular field programmable gate arrays (FPGAs). In our recent work [6] the simple decomposed PID FLC has been compared with discrete PID controller and gain scheduling discrete PID controller. The main question is, do the decomposed PID FLC structure do a good job as the non-simplified PID FLC form? The simple magnetic suspension system is designed to investigate the characteristics of different fuzzy control algorithms and the results in stabilisation, time responses for set point changes, time responses for load disturbances and trajectory tracking need to be interpreted in the context of its physical characteristics. All PID FLCs are realised by the same hardware and software tools and applied to a simple magnetic suspension system [7]. The real-time implementation of the control for the magnetic suspension system allowed to perform several interesting experiments where the fuzzy controller proved to be very efficient. Two types of real-time experiments have been done: experiments for set point changes and experiment for load disturbances. This paper is organised as follows: Section 2 introduces decomposed fuzzy systems and decomposed PID FLCs. Section 3 reviews other decomposed fuzzy PID controllers. The simple electromagnetic suspension system is presented in Section 4. In Section 5 control experiments and results are compared and Section 6 concludes the paper. 2. Decomposed fuzzy systems The development of the PID FLC is followed from the structure of the discrete PID controller. The basic idea of the discrete PID controller is to choose the control law by considering error E(k), change-of-error DE(k) := (E(k)E(k − 1))/T and the numerically approximated integral of error IE(k) := IE(k − 1) + TE(k). The PID control law is uPID (k) = KP E(k) + KD DE(k) + KI IE(k) (1) where KP is the proportional constant, KD the differential constant, KI the integral constant and T the sampling period. For a linear process the control parameters KP , KD and KI are designed in such a way that the closed-loop system is stable. The corresponding analysis can be done by means of the knowledge of process parameters taking into account special performance criteria. In the case of non-linear processes that can be linearised around an operating point, conventional PID controllers also work successfully. However, the PID controller with constant parameters in the whole working area is only robust for some systems. In this case, tuning of the PID parameters may be performed. A fuzzy PID controller starts from the same assumptions which are used for the conventional PID controller. The output of the general fuzzy controller u(k) is given by u(k) = N (E(k), DE(k), IE(k)) (2) where N(·) is a non-linear function determined by the fuzzy parameters. In the case we assume the structure of the fuzzy PID controller has three input variables M. Golob / Applied Soft Computing 1 (2001) 201–214 and one output variable with three base fuzzy sets on each fuzzy variable, we get one rule base with a maximum of 27 rules. To minimise the number of rules the simplification of the fuzzy PID structure is proposed as described in [6]. The basic idea is to decompose the multivariable control rule base into three sets of one-dimensional rule bases for each input. The control algorithm is represented by fuzzy rules. The linguistic description of the general fuzzy PID controller is given by 203 has an analytical solution. With this decomposition, the equality in Eq. (5) becomes an inclusion and the relation reduces to U ′ ⊂ E ′ ◦ {(E(1) ∧ U(1) ) ∨ · · · ∨ (E(m) ∧ U(m) )} ∧ DE ′ ◦ {(DE(1) ∧ U(1) ) ∨ · · · ∨ (DE(m) ∧ U(m) )} ∧IE ′ ◦ {(IE(1) ∧ U(1) ) ∨ · · · ∨ (IE(m) ∧ U(m) )} (6) IF E ′ = E(1) AND DE ′ = DE(1) AND IE ′ = IE(1) THEN U ′ = U(1) .. . IF E ′ = E(i) AND DE ′ = DE(i) AND IE ′ = IE(i) THEN U ′ = U(i) .. . IF E ′ = E(m) AND DE ′ = DE(m) AND IE ′ = IE(m) THEN U ′ = U(m) where E′ , DE′ and IE′ are fuzzy input variables (error, change-of-error and integral of error), E(i) , DE(i) , IE(i) the ith corresponding base fuzzy sets, and U′ is the fuzzy control output with corresponding base fuzzy sets U(i) . The number of rules is denoted by m. The fuzzy relation R of the fuzzy controller rule base is expressed as follows: m R = ∨ {E(i) ∧ DE(i) ∧ IE(i) ∧ U(i) } i=1 Now we define the fuzzy relations RE , RDE and RIE as m RE = ∨ {(E(i) ∧ U(i) )}, i=1 m RDE = ∨ {(DE(i) ∧ U(i) )}, i=1 m RIE = ∨ {(IE(i) ∧ U(i) )} (4) i=1 (7) and the following is obtained from Eq. (6): where ∨ is the aggregation operator and ∧ is the implication operator. For each rule in Eq. (3) a fuzzy relation R(i) has to be constructed. To obtain the fuzzy controller relation R, fuzzy relations R(i) are aggregated. The dimension of the relation matrix R is dim[R] = dim[E] × dim[DE] × dim[IE] × dim[U ]. To obtain the new fuzzy controller output U′ , given the current fuzzy inputs E′ , DE′ and IE′ the compositional rule of inference is used: U ′ = E ′ ◦ DE ′ ◦ IE ′ ◦ R (3) (5) where ◦ is the composition operator of fuzzy relations. Because of the multidimensionality of the fuzzy relation R (Eq. (4)) the composition rule of inference (Eq. (5)) is difficult to perform and analytical solutions usually cannot be obtained. To overcome this difficulty, it is proposed [6] to break up the inference of a multidimensional rule base into three rule bases for which the inference is easier to perform or U ′ ⊂ E ′ ◦ RE ∧ DE ′ ◦ RDE ∧ IE ′ ◦ RIE (8) Because of the inference break-up, the code optimisation can be applied to software implementation. The main advantage of the decomposition and simplification is a reduction in the number of linguistic rules. In the case when we assume the structure of the decomposed fuzzy PID controller has three input variables and one output variable with three base fuzzy sets on each fuzzy variable, we get three rule bases with a maximum of three rules in each of the rule base. Due to the simplification, the output of the decomposed PID FLC is a little more fuzzified than the output of the general PID FLC. The output of the decomposed fuzzy controller u(k) is given by u(k + 1) = defuzz{RE ◦ fuzz(e(k)) ∧RDE ◦ fuzz(de(k)) ∧RIE ◦ fuzz(ie(k))} (9) 204 M. Golob / Applied Soft Computing 1 (2001) 201–214 Fig. 1. The structure of the decomposed PID FLC realised using Eq. (9). where fuzz(·) is the fuzzification operator and defuzz(·) is the defuzzification operator. The structure of the decomposed PID FLC realised using Eq. (9) is shown in Fig. 1. The decomposed PID FLC, presented in Fig. 1 and described using Eq. (9), is contained in three decomposed fuzzy inferences. Linguistic output variables are aggregated and the result is defuzzified. In [6] we proposed a decomposition of entire fuzzy systems, including the defuzzification. The aggregation operator has been substituted with a sum of defuzzified outputs. The output of such a decomposed PID FLC is given by u(k + 1) = defuzz{RE ◦ fuzz(e(k))} + defuzz{RDE ◦ fuzz(de(k))} + defuzz{RIE ◦ fuzz(ie(k))} (10) and the structure is shown in Fig. 2, where decomposed PID FLC is used as a controller in a discrete feedback control system. 3. Other decomposed fuzzy PID controllers Several FLC configurations have been presented in the literature. 3.1. PD + I FLC The PD + I FLC was proposed by Li and Ng [9]. The control law of the PD + I FLC is constructed by a sum of fuzzy PD action and fuzzy integral action. The structure of a simple control system with PD + I FLC is shown in Fig. 3. The characteristic of the PD + I FLC is a combination of a two-dimensional rule base for the PD control and a one-dimensional rule base for the incremental integral control. 3.2. PI FLC + conventional D controller The PI FLC + D was proposed by Li and Gatland [10] and Qin [13]. The control law consists of the Fig. 2. Control system with decomposed PID FLC. M. Golob / Applied Soft Computing 1 (2001) 201–214 205 Fig. 3. Control system with PD + I FLC. Fig. 4. Control system with PI FLC + D. PI FLC and derivative control action of the process output: uPI FLC+D (k) = uPI FLC (k) + uD (k) (11) The derivative control action has the following continuous time transfer function: UD (s) Td s + 1 = (12) FD (s) = Y (s) (Td /Kd )s + 1 and after “zero-order hold” conversion method the FD (z) discrete time transfer function is realised. The structure of a simple control system with PI FLC + D is shown in Fig. 4. This structure has the benefit of implementing derivative control on the output, avoiding derivative kicks for step set point changes. 3.3. P FLC + conventional ID controller The P FLC + ID was proposed by Li [11]. The control law consists of the P FLC and conventional Fig. 5. Control system with P FLC + ID. 206 M. Golob / Applied Soft Computing 1 (2001) 201–214 Fig. 6. Control system with PD + PI FLC. integral–derivative controller. The incremental form of control law is 4. Fuzzy control for the simple electromagnetic suspension system duP FLC+ID (k + 1) The basic principle of a simple electromagnetic suspension system is shown in Fig. 7. The direction of the magnetic force F applied by the electromagnet is opposite to that of gravity G and maintains the suspended steel ball levitated. The magnetic force F is a non-linear function of the electromagnet current I, the electromagnet characteristics, and the air gap X between the steel ball and the electromagnet. Described process is unstable and we shall find a feedback configuration and realise the real-time discrete controller so that the closed loop of the resulting systems is stable. The non-linear characteristic and the strong real-time applications demands are the reasons why the electromagnetic suspension = KP duP FLC (k) + KI Te(k) y(k) − 2y(k − 1) + y(k − 2) − KD T (13) where T is the sampling time, and KI and KD are the same parameters as in the conventional PID controller. The signal duP FLC (k) is the output of the incremental fuzzy logic controller, and KP the proportional coefficient. The structure of the control system with P FLC + ID is shown in Fig. 5. The most important part in the fuzzy P FLC + ID is the fuzzy P part because it is responsible for improving overshoot and rise time. The conventional I part is responsible for reducing steady-state error, and the conventional D part is responsible for the stability of the system and for flatness of the response. 3.4. PD FLC + PI FLC The PD + PI FLC is presented by Kwok et al. [8] and Li and Gatland [10], and it consists of a PD FLC in parallel with a PI FLC. The basic control diagram is shown in Fig. 6. The fuzzy PD + fuzzy PI controller (PD + PI FLC) consists of a PD FLC in parallel with a PI FLC. Its knowledge base is a combination of two-dimensional rule base for the PD control and a two-dimensional rule base for the PI control. Fig. 7. The basic principle of a simple electromagnetic suspension system. M. Golob / Applied Soft Computing 1 (2001) 201–214 207 Fig. 8. Scheme of the real-time application of the simple suspension system. system is suitable for fuzzy controllers’ performance evaluation. The functional diagram for the real-time application to the simple suspension system is shown in Fig. 8. Our fuzzy logic controller programmed on a personal computer (PC-486 with clock of 33 MHz) is extended by a OMRON FB-30AT fuzzy board (with the fuzzy processor FP 3000). An eight-channel analog-to-digital (A/D) converter and a two-channel digital-to-analog (D/A) converter with 12-bit resolution is realised on the plug-in PC board. The first channel of the A/D converter is used to measure the input to the control system, namely, the ball position, which is the output of the optical position measurement system. The fuzzy controller software was implemented in ANSI-C. A real-time sampling frequency of 1 kHz was attained. Measurement of ball position with a simple optical contact-less position measurement system has been realised. The transmitter Dt (LED NLPB-500), with emission angle of 15◦ , is used to generate blue light. The homogenous light wave is produced by an optical system of two optical lenses (L1, L2). The light detector (Dr) provides an output voltage proportional to the position of the ball in the light wave. The influence of ambient light is compensated for by the differential structure of the light detector. The realisation of the model is presented in Fig. 9. Values for nominal parameters, electrical and mechanical time constants are shown in Table 1. Fig. 9. The realisation of the model. 208 M. Golob / Applied Soft Computing 1 (2001) 201–214 Table 1 Nominal system parameters Mass of the steel ball, m (kg) Maximum air gap, D (mm) Number of coils, n Coil resistance, R () Electromagnet inductance, L (mH) 0.147 25 1200 2.8 520 Fig. 11. Output membership functions. 4.1. A realisation of the fuzzy inference system The fuzzy inference system is performed using the minimum operator and the composition is done using the maximum operator (Mamdani type of the inference engine [12]). The centre of gravity defuzzification method is implemented. All fuzzy inputs are divided into three base membership functions: negative (N), zero (Z) and positive (P). Trapezoid and triangular membership functions (Fig. 10) with 50% overlap are applied to the error fuzzy input E and the derivative of error fuzzy input DE of the fuzzy controller. The output of the controller is the output voltage (ranges of 0–10 V is scaled to integer range 0–4095) from the D/A converter. Singleton memberships functions are applied to fuzzy outputs. Singleton output membership functions are shown in Fig. 11. The knowledge base of the decomposed PID FLC realised using Eq. (10) is composed by three inference rule bases: the proportional rule base, the differential rule base and the integral rule base. Three rules of the rule base for the proportional part of the fuzzy controller are described in Table 2, where E is the fuzzy input variable as result of a fuzzification interface, which has the effect to transforming crisp Table 2 The rule base of the proportional part of the fuzzy PID controller E UP N N Z Z P P variable e(k) into fuzzy sets, and UP is the fuzzy output variable of the proportional fuzzy subsystem. The differential part and the integral part of the fuzzy logic controller are realised with the same type of one-term simple rules. However, the knowledge base of the decomposed fuzzy controller is defined by nine rules. The knowledge base of the PD + I FLC shown in Fig. 3 is composed by two inference rule bases: the two-dimensional proportional–differential rule base and the one-dimensional integral rule base. In this case, the controllers knowledge base is defined by 12 rules (32 + 31). Nine two-term complex rules of the two-dimensional rule base are described in Table 3. According to the structure presented in Fig. 6, the knowledge base of the PD + PI FLC consist of two two-dimensional inference rule bases: the proportional–differential rule base and the proportional–integral rule base. In the case, the PD rule base and the PI rule base contain different rules, the controller knowledge base is defined by 18 rules (32 + 32). Obviously, so long as we take the unified Table 3 The two-dimensional rule base UPD Fig. 10. Input membership functions. DE N Z P E N Z P N N Z N Z P Z P P M. Golob / Applied Soft Computing 1 (2001) 201–214 quantisation method and the appropriate scaling factors, both PI and PD control rules can be realised through unified fuzzy control rule base. It means that a knowledge base can be implemented by using one rule base with nine two-dimensional rules. The advantage of this approach is in simplifying the control rule base significantly. On the other hand, the disadvantage of this approach is that the controller parameters are coupled with each other and have to be regulated in combination. 5. Control experiments and performance comparisons The real-time implementation of the control for the magnetic suspension system allowed to perform several interesting experiments where the fuzzy controller proved to be very efficient. Two types of real-time experiments have been done: experiments for set point changes and experiment for load disturbances. The magnetic suspension system is designed to investigate the characteristics of fuzzy control algorithms and the results in stabilisation, time responses for set point changes, time responses for load disturbances and trajectory tracking need to be interpreted in the context of its physical characteristics. For example, the maximum possible travel of the suspended ball is 16 mm (from 4 to 20 mm). The sampling period T was 1 ms, 209 and 1700 data samples were collected. The adjustment of the scale factors, rule base and membership functions of each controller was accomplished by fine tuning and heuristic corrections linked to the knowledge of the process to be controlled. In all experiments the rule base of fuzzy P controller consisted of three rules and rule bases of the fuzzy PD and the fuzzy PI controllers consisted of nine rules. Time responses at two operational points (14–16 and 6–8 mm) of the magnetic suspension system with PD + I FLC controller are shown in Fig. 12. The simple fuzzy integral controller has been added to fuzzy PD structure. It can be seen that steady-state error has been eliminated. Different set point time responses at high operation point and low operation point are consequence of the non-linear characteristic of the magnetic suspension system. Fig. 13 shows time responses of the magnetic suspension system with decomposed PID FLC in closed loop. Decomposed PID FLC parameters have been optimised to assure stable behaviour at all operating points. Note that the knowledge base of the PD+I FLC is consisted of nine complex rules in PD rule base and three simple rules in I rule base. However, the knowledge base of the decomposed PID FLC is consisted of nine simple rules in three separate rule bases. The increased overshoot present in step responses is the result of the decomposition of the PD rule base and the separate I rule base. The responses to step set point Fig. 12. The real-time PD + I FLC control responses of the magnetic suspension system to step reference changes. 210 M. Golob / Applied Soft Computing 1 (2001) 201–214 Fig. 13. The decomposed PID FLC control responses of the magnetic suspension system to step reference changes. changes of the magnetic suspension system, using the PD + PI FLC, is shown in Fig. 14. The overshoot is reduced. The knowledge base consisted of 18 complex rules in two separate rule bases. Next two hybrid controllers, we have applied on magnetic suspension system, are based on combination of the fuzzy control approach and conventional PID control approach. Time responses of the magnetic suspension system with P FLC + ID controller are shown in Fig. 15. Fig. 16 shows time responses of the magnetic suspension system with PI FLC + D controller. Knowledge bases of both hybrid controllers have been realised by rule base with nine complex rules. The benefit of implementing the derivative control on the output, avoiding derivative kicks for step Fig. 14. The real-time PD + PI FLC control responses of the magnetic suspension system to step reference changes. M. Golob / Applied Soft Computing 1 (2001) 201–214 211 Fig. 15. The real-time P FLC + ID control responses of the magnetic suspension system to step reference changes. set point changes, is obviously noted. Increased overshoots present in step responses at higher operating point (14 mm) are results of the conventional derivative control action. It is important to remark the degradation of the performance by the lower operating point (6 mm). Conventional realisation of the I and D controllers are less robust then suitable fuzzy realisation. The mean square error (MSE) performance index have been implemented to assess the quantitative measure of all controllers performances: 1
(ysp (k) − y(k))2 N N MSE = (14) k=1 Table 4 presents the results of performance indices for all FLCs obtained from set point change experiments. Fig. 16. The real-time PI FLC + D control responses of the magnetic suspension system to step reference changes. 212 M. Golob / Applied Soft Computing 1 (2001) 201–214 Table 4 The performance indices for all FLCs obtained from set point change experiments No. of simple rules No. of complex rules Overshoot (mm) Positive Negative Set point = 14 mm Decomposed PID FLC PD + I FLC PD + PI FLC P FLC + ID PI FLC + D 9 3 – – – – 9 18 9 9 2.0 2.0 0.0 0.8 0.2 0.7 0.8 0.1 0.6 0.0 0.329 0.397 0.206 0.222 0.264 Set point = 6 mm Decomposed PID FLC PD + I FLC PD + PI FLC P FLC + ID PI FLC + D 9 3 – – – – 9 18 9 9 2.0 3.0 0.4 1.4 0.8 2.0 2.3 0.2 1.5 0.7 0.885 1.030 0.222 0.348 0.327 In the second experiment a fixed set point of 12 mm has been used. In order to verify the capacity of the FLC control algorithms for disturbance rejection the disturbance magnitude 2 V has been added to the control signal from sample 100 onwards and has been kept till sample 900. Fig. 17 shows the time responses of the load disturbances applied to the magnetic suspension system with diverse PID FLC in closed loop. As is shown in Fig. 18, the decomposed PID FLC time response under load disturbances is compara- MSE ble to the PD + I FLC and the PD + PI FLC time responses. The lower curve denotes the control output (V) of the decomposed PID FLC. Time responses of the load disturbances applied to the magnetic suspension system with the PI FLC + D controller and the P FLC + ID controller (hybrid FLC) are shown in Fig. 18. Table 5 presents the results of performance indices for all FLCs obtained from load disturbances experiments. Fig. 17. The time responses of magnetic suspension FLC control systems to load disturbances. M. Golob / Applied Soft Computing 1 (2001) 201–214 213 Fig. 18. The time responses of the magnetic suspension hybrid fuzzy control systems to load disturbances. Table 5 The performance indices for all FLCs obtained from load disturbance experiments Decomposed PID FLC PD + I FLC PD + PI FLC P FLC + ID PI FLC + D No. of simple rules No. of complex rules Overshoot (mm) Positive Negative 9 3 – – – – 9 18 9 9 2.0 2.1 1.7 1.1 3.2 2.0 2.4 1.9 0.8 2.2 6. Conclusions An approach of a fuzzy controller based on the decomposition of the multivariable rule base into simple rule bases has been presented and compared to several PID-type FLCs and hybrid-type FLCs. The control output of the decomposed PID FLC is a little more fuzzified then the output of composed PID FLCs, e.g. PD + PI FLC. This is due to the simplification of the multivariable structure. A loss of accuracy must be accepted. The real-time model of the magnetic suspension system has been realised and the comparative analysis of FLC controllers properties has been done. Compared to traditional PID-type FLCs the decomposed PID FLC is comparatively robust and has satisfactory transient performances. Advantages of a decomposed PID FLC structure are: decreasing num- MSE 0.749 1.018 0.334 0.063 0.654 ber of rules; decomposition of multivariable control rules into three sets of one-dimensional rules for each input variable, simplified the evolution of the rule base; a direct relationship between a fuzzy control and a conventional control; easy connection between fuzzy parameters and operation of the controller. An interesting possibility is a design of the fuzzy inference processor which performs composition of a fuzzy set and a two-dimensional fuzzy relation. The PID FLC using decomposed inference method can realise high-speed control. Acknowledgements The author would like to thank the anonymous referees for their helpful comments that have led to the better presentation of this paper. 214 M. Golob / Applied Soft Computing 1 (2001) 201–214 References [1] T. Brehm, Hybrid fuzzy logic PID controller, in: Proceedings of the 3rd IEEE Conference on Fuzzy Systems, Vol. 3, Orlando, FL, 1994, pp. 1682–1687. [2] C.T. Chao, C.C. Teng, A PD-like self tuning fuzzy controller without steady-state error, Fuzzy Set and Systems 87 (1997) 141–154. [3] M. Chen, D.A. Linkens, A hybrid neuro-fuzzy PID controller, Fuzzy Set and Systems 99 (1998) 27–36. [4] L.S. Coelho, A.A.R. Coelho, An experimental and comparative study of fuzzy PID controller structures, in: R. Roy, T. Furuhashi, P.K. Chawdhry (Eds.), Advances in Soft Computing: Engineering Design and Manufacturing, Springer, London, 1999, pp. 147–159. [5] S. Daley, K.F. Gill, Comparison of a fuzzy logic controller with a P + D control law, Transactions of the ASME Journal of Dynamics, Systems and Measurement Control 111 (1989) 128–137. [6] M. Golob, Decomposition of a fuzzy controller based on the inference break-up method, in: R. Roy, T. Furuhashi, P.K. Chawdhry (Eds.), Advances in Soft Computing: Engineering Design and Manufacturing, Springer, London, 1999, pp. 215–227. [7] M. Golob, B. Tovornik, Fuzzy ball position control in the magnetic field, in: Proceedings of the World Multiconference on Systems, Vol. 4, Cybernetics and Informatics, Orlando, FL, 1998, pp. 31–36. [8] D.P. Kwok, P. Tan, C.K. Li, P. Wang, Linguistic PID controllers, in: Proceedings of the 11th IFAC World Congress, Vol. 7, Tallin, Estonia, 1990, pp. 192–197. [9] Y. Li, K.C. Ng, Reduced rule base and direct implementation of fuzzy logic control, in: Proceedings of the 13th IFAC World Congress, San Francisco, CA, 1997, pp. 85–90. [10] H.X. Li, H.B. Gatland, Enhanced methods of fuzzy logic control, in: Proceedings of the FUZZ-IEEE/IFES’95, Vol. 1, Yokohama, Japan, 1995, pp. 331–336. [11] W. Li, Design of a hybrid fuzzy logic proportional plus conventional integral–derivative controller, IEEE Transactions on Fuzzy Systems 6 (1998) 449–463. [12] T.J. Procyk, H. Mamdani, A Linguistic self-organizing process controller, Automatica 15 (1979) 15–30. [13] S.J. Qin, Auto-tuned fuzzy logic controller, in: Proceedings of the American Control Conference, Baltimore, USA, 1994, pp. 2465–2469. [14] K.L. Tang, R.J. Mulholland, Comparing fuzzy logic with classical controller design, IEEE Transactions on Systems, Man and Cybernetics 17 (1987) 1085–1087. [15] L. Zheng, A practical guide to tune of proportional and integral (PI) like fuzzy controller, in: Proceedings of the 1st IEEE International Conference on Fuzzy Systems, San Diego, 1992, pp. 633–640. Marjan Golob was born in 1962 in Postojna, Slovenia. He received his BE (1987), MSc (1991) and PhD (2000) degrees in electrical engineering from University of Maribor. Currently, he is on the faculty staff in the Laboratory for Process Automation, University of Maribor. His research interests include process automation, intelligent control, and fuzzy logic systems. He is a member of IEEE.