De La Salle University – Manila, ECE Technical Report, October 1, 2004
ADAPTIVE CONTROL USING FUZZY-PI MEANS Emmanuel A. Gonzalez1 and Leonard U. Ambata2 Department of Electronics and Communications Engineering College of Engineering De La Salle University, Manila 2401 Taft Ave., Malate, Manila 1004, Philippines 1
[email protected] [email protected]
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ABSTRACT
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It has been a practice in control system design to provide certain means of control in any control system by using different techniques such as adding compensators like PI, PD, and PID, with the incorporation of machine intelligence techniques such as fuzzy logic, artificial neural networks, and evolutionary computation.
u PI Controller
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Fig. 1. Unity-gain feedback control system employing a general PI controller with the application of disturbance, w.
PI Controller
This paper presents a design approach in the development of compensators for control systems using the hybrid of the conventional proportional-integral (PI) method and fuzzy logic - “fuzzy-PI”.
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I.
INTRODUCTION
System control is a very challenging area in the field of control systems engineering, which widens the research and development of different control techniques in linear and non-linear, single- and multivariable systems. Different control techniques have been formulated such as conventional PI, PD, and PID control, machine intelligence (e.g. fuzzy logic, neural networks, evolutionary computation) [4]-[5], and the mixture of both with the hybridization of the conventional PI control method and fuzzy logic [1]-[2] as an example.
Fig. 2. Specific block diagram of the feedback control system using a PI compensator.
II. CONVENTIONAL PI CONTROL
of steady-state error of a control system, i.e. bringing down the steady-state error to zero. The addition of a pole makes an effect of minimizing the error of the system. However, one disadvantage of this conventional compensator is its inability to improve the transient response of the system. With a PI controller, it is very difficult to decrease the system’s percent overshoot and the settling time, which on the other hand, can be controlled by a proportional-derivative (PD controller.) This is why researchers use the combination of both PD and PI compensators to develop a proportional-integralderivative (PID) controller for the control of the system’s transient response and steady-state error. However, PID controllers are difficult to implement, in which, the use of PI compensators becomes more practical especially when transient response control is unnecessary.
The reason behind the extensive use of proportionalintegral (PI) controllers is its effectiveness in the control
The conventional PI compensator (see Figs. 1 and 2) has the form of Eq. (1), where u is the control output which is
In contribution to the numerous researches available in circulation, we intend to develop a design methodology for a PI compensator with the incorporation of fuzzy reasoning for adaptive tuning of systems, in which fuzzy logic has been proven in many applications and researches in the field of control engineering [3], [6]-[7].
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De La Salle University – Manila, ECE Technical Report, October 1, 2004
Adaptive PI Controller
K max = K max − 0.1( K max − K min )
Rule Base
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After the determination of the domain of the proportional and integral constants, the tuning of the instantaneous values of the constants takes place. Depending on the value of the error signal, e, the values of the constants adjusts formulating an adaptive control system. With a high value of e, the constants KP and KI changes to ensure that the steady-state error of the system is reduced to minimum if not zero. This makes a single-input-singleoutput (SISO) compensator, being the error signal, e, as the input and the control signal, u, as the output.
KP , KI
Inference Engine e
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Fig. 3. Insertion of an inference system that controls the proportional and integral constants of the PI controller.
fed to the plant, KP and KI are the proportional and integral constants, respectively, and e is the error signal, which is the difference of the output signal to the reference input signal, i.e. e = r – y.
dω ( t ) = K P e ( t ) + K I ∫ e ( t ) dt T dt
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The disadvantage of the developed SISO PI compensator is its inability to react to abrupt changes in the error signal, e, because it is only capable of determining the instantaneous value of the error signal without considering the change of the rise and fall of the error, which in mathematical terms is the derivative of the error signal, denoted as De. To solve this problem, a multipleinput-single-output (MISO) equivalent of the compensator is implemented having the error signal, e, and the rate-of-error, De, as the inputs.
(1)
Eq. (1) shows that the PI controller introduces a pole in the entire feedback system, considering that the system is a unity-feedback type, consequently, making a change in its original root locus. Analytically the pole introduces a change in the control system’s response to step inputs by increasing the system type. The effect is the reduction of steady-state error. On the other hand, the constants KP and KI determine the stability and transient response of the system, in which, these constants rely on their universe of discourses
An adaptive PI controller is shown in Fig 3. The determination of the output control signal, u is done in an inference engine with a rule base having i if-then rules in the form of Ri: IF e AND De, THEN KP AND KI with the inference engine implemented in the PI controller that tunes the compensator constants according the applied rule base.
K P ∈ K P ,min , K P ,max K I ∈ K I ,min , K I ,max
With the rule base, the values of the constants KP and KI are changed according to the value of the error signal, e, and the rate-of-error, De. The structure and determination of the rule base is done using trial-and-error methods and is also done through experimentation.
where the values of the minimum and maximum proportional and integral constants are evaluated through experimentation and using some iterative techniques. This makes the design of the conventional PI controller dependent on the knowledge of the expert. Nevertheless, the evaluation and synthesis of the controller constants is not limited to someone’s expertise, hence, it is still realizable for a new designer without apriori knowledge of the system to develop such controllers.
III. FUZZY-PI CONTROL The hybridization of machine intelligence and conventional control techniques bring up a new approach in designing an intelligent adaptive control system. One example is the application of fuzzy logic to the tuning of conventional PI controllers, which is the focus of this research.
When the compensator constants exceed the maximum allowable values, the control system may come into an unstable state. Preventing this from happening is done by adding an allowance to the evaluated maximum value of the domain of the constants, taking into consideration at least 10% of the length of the domain minus the maximum value, as depicted in Eq. (2).
The fuzzy logic system comprises of four parts, the fuzzification map, defuzzification map, inference engine, and the rule base. The rule base of a fuzzy logic system is
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De La Salle University – Manila, ECE Technical Report, October 1, 2004
Adaptive Fuzzy Logic PI Controller Fuzzy Logic System Rule Base Scaling Factors e, De
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Fig. 4. The fuzzy PI control system with gains placed before fuzzification.
similar to the one used in the previous section, however, the values used are linguistic in nature unlike in the previous section where the values are numerical. An example of this type of ruling is
normalization. Published works in different normalization algorithms can be used [2], [12]. However, normalization heuristics can be developed based solely on trial-and-error methods.
R1: IF e is high AND De is medium, THEN KP is small AND KI is medium
The use of a normalized domain requires input normalization, which maps the physical values of the processes state variables into a normalized domain [2]. The scaling factors are very much important similar to that of the gains of a conventional controller. However, they can also provide instability to the system which can cause oscillation problems and deteriorated damping effect [8]. Gain scheduling and adjustment algorithms have been developed through trial and error [2], [9]-[11].
where the linguistic values of low, medium, and high are arbitrarily set by a membership function formulated by the designer. The inference engine’s basic function is to compute the overall value of the control output variable based on the individual contributions of each rule in the rule base. Each individual contribution represents the control output value as computed by a single rule.
IV. CONCLUSION
The fuzzy logic controller becomes a MIMO system with the error, e, and the rate-of-error, De, as the inputs, and the constants KP and KI as the outputs. The membership functions of the inputs and outputs are incorporated in the fuzzification and defuzzification engine, respectively. The membership of each linguistic variable is based on intuitive perception and are tested, evaluated and changed through trial-and-error methods.
The hybridization of fuzzy logic into a proportionalintegral controller for controlling the steady-state error of a control system is described in this paper. It has been shown that, in order to control the plant, the constants KP and KI of the compensator must be varied according to the fuzzy logic system. Scaling factors are also incorporated to ensure a normalized input vector.
The fuzzy PI controller is shown in Fig. 4. Scaling factors are placed before the fuzzifier to normalize the inputs.
REFERENCES [1] F. Barrero, A. Gonźalez, A. Torralba, E. Galván, and L. G. Franquelo, “Speed control of induction motors using a novel fuzzy sliding-mode structure,” IEEE Trans. Fuzzy Syst., vol. 10, no. 3, pp. 375-383, Jun. 2002.
eˆ = e ⋅ Ge Deˆ = De ⋅ GDe eˆ and Deˆ are the normalized error and rate-of-error inputs. Ge GDe are the scaling factors used for 3
De La Salle University – Manila, ECE Technical Report, October 1, 2004
[2] Y. Zhao and E. G. Collins, Jr., “Fuzzy PI control design of an industrial weigh belt feeder,” IEEE Trans. Fuzzy Syst., vol. 11, no. 3, pp. 311-319, Jun. 2003. [3] D. P. Filev and R. R. Yager, “On the analysis of fuzzy logic controllers,” Fuzzy Sets and Syst., vol. 68, pp. 39-66, 1994. [4] E. Kim, “A discrete-time fuzzy disturbance observer and its application to control,” IEEE Trans. Fuzzy Syst., vol. 11, no. 3, pp. 399-410, Jun. 2003. [5] L. X. Want, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 146-155, May 1993. [6] S.-J. Wu and C.-T. Lin, “Optimal fuzzy control design: local concept approach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 2, pp. 171-185, Apr. 2000. [7] M. Figueiredo, F. Gomide, A. Rocha, and R. Yager, “Comparison of Yager’s level set method for fuzzy logic control with Mamdani’s and Larsen’s methods,” IEEE Trans. Fuzzy Syst., vol. 1, no. 2, May 1993. [8] Z. Z. Zhao, M. Tomizuka, and S. Isaka, “Fuzzy gain scheduling of PID controller,” IEEE Trans. Syst. Man, Cybern., vol. 23, pp. 1392-1398, Oct. 1993. [9] D. Driankov, H. Hellendoorn, and M.Reinfrank, An Introduction to Fuzzy Control. Belrin, Germany: Springer-Verlag, 1993. [10] W. C. Daugherty, B. Rathakrishnan, and J. Yen, “Performance evaluation of a self-tuning fuzzy controller,” in Proc. IEEE Int. Conf. Fuzzy Systems, San Diego, CA, Mar. 1992., pp. 389-397. [11] M. Maedo and S. Murakami, “A self-tuning fuzzy controller,” Fuzzy Sets Syst., vol. 51, no. 1, pp. 29-40, 1992. [12] A. V. Patel and B. M. Mohan, “Analytical structures and analysis of the simpled fuzzy PI controllers,” Automatica, vol. 38, 2002, pp. 981-993.
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