Design and Stability Analysis of Single-Input Fuzzy Logic Controller

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 30, NO. 2, APRIL 2000 303 Design and Stability Analysis of Single-Input Fuzzy Logic Controller Byung-Jae Choi, Seong-Woo Kwak, and Byung Kook Kim Abstract In existing fuzzy logic controllers (FLC s), input variables are mostly the error and the change-of-error _ regardless of complexity of controlled plants. Either control input or the change of control input 1 is commonly used as its output variable. A rule table is then constructed on a two-dimensional (2-D) space. This scheme naturally inherits from conventional proportional-derivative (PD) or proportional-integral (PI) controller. Observing that 1) rule tables of most FLC s have skew-symmetric property and 2) the absolute magnitude of the control input or 1 is proportional to the distance from its main diagonal line in the normalized input space, we derive a new variable called the signed distance, which is used as a sole fuzzy input variable in our simple FLC called single-input FLC (SFLC). The SFLC has many advantages: The total number of rules is greatly reduced compared to existing FLC s, and hence, generation and tuning of control rules are much easier. The proposed SFLC is proven to be absolutely stable using Popov criterion. Furthermore, the control performance is nearly the same as that of existing FLC s, which is revealed via computer simulations using two nonlinear plants. Index Terms Absolute stability, fuzzy control logic, Lure-type Lyanupov function, signed distance, stability analysis. I. INTRODUCTION FUZZY logic contollers (FLC s) are one of useful control schemes for plants having difficulties in deriving mathematical models or having performance limitations with conventional linear control schemes. Most works in fuzzy control field use the error and the change-of-error as fuzzy input variables regardless of the complexity of controlled plants. Also, either control input (PD-type) or incremental control input (PI-type) is typically used as a fuzzy output variable representing the rule consequent ( then part of a rule) [1], [2]. These conventional FLC s naturally came from the concept of linear proportional-derivative (PD) or proportional-integral (PI) control scheme. Such FLC s are sufficient for simple secondorder plants since we can place two closed-loop poles arbitrarily. However, in the cases of complex higher order plants, all process states are required as fuzzy input variables for implementing state feedback FLC s. Generally, all state variables must be used to represent contents of the rule antecedent ( if part of a rule). Manuscript received June 19, 1998; revised November 20, 1999. This work was supported in part by the 1999 Taegu University Research Grant. This paper was recommended by Associate Editor A. Kandel. B.-J. Choi is with the Department of Computer and Communication Engineering, Taegu University, Kyungpook 712-714, Korea (e-mail: bjchoi@taegu.ac.kr). S.-W. Kwak and B. K. Kim are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon 305-701, Korea (e-mail: ksw@rtcl.kaist.ac.kr; bkkim@ee.kaist.ac.kr.). Publisher Item Identifier S 1083-4419(00)02968-X. However, it requires a huge number of control rules and much effort to create. This is the reason why many FLC s simply use only the error and the change-of-error as fuzzy input variables. These facts motivates design of simpler FLC s. Many researches were performed to improve the performance of the FLC s and ensure their stability. Tang and Mulholland [3] developed a relation between the scaling factors of fuzzy controller and the control gains of equivalent linear PI controller. They also showed that the FLC can be used as a multiband control. Li and Gatland [4] and [5] proposed a more systematic design method for PD and PI-type FLC s. For PID (proportional-integral-derivative)-type FLC, they also presented a simplified rule generation method using two two-dimensional (2-D) spaces instead of a three-dimensional space. Palm [6], [7] proposed a sliding mode fuzzy controller which generates the absolute value of switching magnitude in the sliding mode control law using the error and the change-of-error. Most of researches use two fuzzy input variables in the rule antecedent regardless of the complexity of the controlled plants. In this paper, we suggest a simple but powerful FLC design method using a sole fuzzy input variable instead of the error and the change-of-error (or all process state variables) to represent the contents of the rule antecedent. For conventional FLC s using the error and the change-of-error as fuzzy input variables, the fuzzy rule table is established on a 2-D space of the phase plane. We can find that the 2-D rule table has the skew-symmetric property and the absolute magnitude of the control input is proportional to the distance from its main diagonal line on the normalized input space. This property also holds in case of the PID-type FLC which uses the error, the sum-of-error, and the change-of-error as fuzzy input variables [1]. Similarly, we find that the absolute magnitude of the control input is proportional to the distance from its main diagonal surface. This skew-symmetric property is satisfied in most -dimensional FLC s which use the error and its time derivative terms, namely, and as fuzzy input variables. This property allows us to suggest a new variable called the signed distance, which is the distance to an actual state from the main diagonal line (or hyperplane) and is positive or negative according to the position of the actual state. The derived signed distance is then used as the sole fuzzy input variable of a simple FLC called single-input FLC (SFLC). As a result, the number of fuzzy rules is greatly reduced compared to the case of the conventional FLC s, but its control performance is almost the same as conventional FLC s. We also prove that the proposed SFLC is absolutely stable using Popov criterion. This paper is constructed as follows. In Section II, we propose a simple design method for the FLC. We here derive the 1083 4419/00$10.00 2000 IEEE

304 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 30, NO. 2, APRIL 2000 signed distance, which is used as a sole fuzzy input variable of the SFLC. Stability analysis of the proposed SFLC is described in Section III by showing that the SFLC is absolutely stable. In Section IV, we compare the control performances of the proposed SFLC with existing FLC s via computer simulations. Concluding remarks are presented in Section V. TABLE I RULE TABLE FOR THE CONVENTIONAL FLC II. SINGLE-INPUT FLC The SFLC is designed for FLC s with skew-symmetric property in the control rule table. First, we consider a second-order system, and then extend to th order case. A. Notations Let the controlled process be a system with or nonlinear) state equation th order (linear (1) with (2) are partially known continuous functions representing system dynamics and unknown external disturbances, is the process state vector, and and are the input and output of the system, respectively. We frequently omit the time parameter for simplicity. The control problem is to force to track a given bounded reference input signal. Let be the tracking error vector as follows: The rule form for the conventional (PD-type) FLC using two fuzzy input variables of the error and the change-of-error is as follows: If is and is then is, and are the linguistic values taken by the process state variables, and, respectively. Here the number of control rules is. In case of complex higher order plants, fuzzy input variables generally require all process states. Then the number of rules is numerous and generation and tuning of rules is very difficult. Hence, PD or PI-type FLC is used in many applications regardless of the complexity of the controlled plants. B. Design of SFLC We first consider a control rule table of conventional FLC s with the control rule form. When every five linguistic values for error, change-of-error, and control input are used, a typical rule table is shown in Table I with 25 rules. (3) Fig. 1. Rule table with infinitesimal quantization levels. In Table I, subscripts 2, 1, 0, 1, and 2 denote fuzzy linguistic values of negative big (NB), negative small (NS), zero (ZR), positive small (PS), and positive big (PB), respectively. Similar to Table I, most rule tables have skew-symmetric property, namely,. Note that the boundaries of for the same control input shapes. Also, the magnitude of the control input is approximately proportional to the distance from the main diagonal line. If the quantization level of the independent variables is halved, then the boundaries of the control regions become staircase shapes with double number of stairs and half pitches. Furthermore, as they become infinitesimal, the boundaries become straight lines as shown in Fig. 1. Then, the control law describes the multilevel relay controller with five bands. Also, note that the absolute magnitude of the control input is proportional to the distance from the following straight line called the switching line: Note that the control inputs above and below the switching line have opposite signs. Now we introduce a new variable called the signed distance. Let be the intersection point of the switching line and the line perpendicular to the switching line from an operating point, as illustrated in Fig. 2. The distance between and, can be expressed as (4) (5)

CHOI et al.: DESIGN AND STABILITY ANALYSIS OF SINGLE-INPUT FLC 305 TABLE II RULE TABLE FOR THE SFLC Fig. 2. Derivation of the signed distance. Equation (5) can be rewritten in general for any (6) the process state variable in the th rule. In this case, the rule table is established on -dimensional space of and. The number of rules is huge, which makes very difficult to generate reasonable control rules. Similar to the 2-D rule table of Table I, the -dimensional one for also satisfies skew-symmetry property and the absolute magnitude of the control input is proportional to the distance from its main diagonal hyperplane (instead of the diagonal line in the 2-D table). Then the switching line is changed to the following switching hyperplane : (10) Then, the signed distance as follows: is defined for a general point Also, in (7) is summarized as a general signed distance as follows: (11) for for (8) Since the sign of the control input is negative for and positive for and its absolute magnitude is proportional to the distance from the line, we can conclude that Then, a fuzzy rule table can be established on an one-dimensional (1-D) space of instead of the 2-D space of the phase plane for FLC s with skew-symmetric rule table. That is, the control action can be determined by only. So, we call it a SFLC (Single-input FLC). The rule form for the SFLC is given as follows: If is then is is the linguistic value of the signed distance in the th rule. Then, the rule table can be established on an 1-D space like Table II. Hence, the number of rules is greatly reduced compared to the case of the conventional FLC s. Furthermore, we can easily add or modify rules for fine control. C. Extension to General Case The general -input FLC has rules of the following form: If is is and is then is is the number of fuzzy sets for each fuzzy input variable and is the linguistic value taken by (7) (9) That is, represents the signed distance from the operating point to the switching hyperplane of (10). Then the rule table is equivalent to Table II except instead of. From (11), we can see that the general signed distance contains knowledge of all process states as well as the error and the change-of-error. D. Advantages of SFLC The proposed SFLC has many advantage. 1) It requires only one input variable, signed distance, regardless of the complexity of the controlled plants. 2) The control rule table is constructed on an 1-D space. 3) The number of tuning parameters is greatly decreased. Hence, tuning of rules, membership functions, and scaling factors is much easier than conventional FLC s using two or more input variables. 4) The computational complexity is mitigated because the number of control rules is considerably decreased. Although extra computation for the signed distance is required, its effect is slighted by the reduced control rules. 5) The single input variable, signed distance, implies knowledge of all state variables. Hence, in the case of more than third-order controlled plants, the control performance can be superior to the case of conventional two-input FLC s. 6) It is equivalent to the sliding mode control with a boundary layer [8]. The fact implies the closed-loop stability of the SFLC. As a result, the SFLC provides a simpler method for design of FLC s while achieving the desired control performance. III. STABILITY ANALYSIS OF SFLC In this section, we analyze the stability in the case that the SFLC operates as the general nonlinear controller. That is, we assume that the relationship between input and output of the

306 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 30, NO. 2, APRIL 2000 TABLE III RULE TABLE FOR THE GENERALIZED SFLC SFLC is nonlinear. The system (1) can be rewritten as (12) by expanding the system into a Taylor series about [9] (12) Here stands for higher order terms in and and denote the nominal operating point and the nominal input, respectively. In addition, may include uncertainties or disturbances. Then, (12) is called as the perturbed Lure system [9] Next we consider the proposed SFLC. As mentioned in Section II-B., the generalized rule form for the SFLC is as follows: IF is THEN is is the linguistic value of the general signed distance in the th rule. The control inputs are determined by the following Table III. From Table III, we can see that the output of the SFLC is symmetric with respect to zero and bounded by a linear gain. That is, the control input is expressed as follows: (13) can be depicted by Fig. 3. That is, is a nonlinear function that belongs to a sector, is a positive constant. As is expressed by the following equation: Fig. 3. Behavior of (D ). Fig. 4. Block diagram of the system with the SFLC. The block diagram of the SFLC is illustrated in Fig. 4. As shown in Fig. 3, is a time-invariant nonlinearity that satisfies the following sector condition (16) globally (16) Then, the following proposition holds, which states that the proposed SFLC is absolutely stable. Proposition 1: Consider the system (15), is Hurwitz, is a minimal realization of, and the nonlinearity is bounded as follows: (14) (17) (18) and is a time-invariant nonlinearity that satisfies the sector condition (16) globally. Then the system is absolutely stable if there is with not an eigenvalue of such that and Re (19) Then, the system with the SFLC is as follows: Re Im (20) Proof: Consider the following Lure-type Lyapunov function [10]. (15) (21)

CHOI et al.: DESIGN AND STABILITY ANALYSIS OF SINGLE-INPUT FLC 307 The derivative of by along the trajectories of the system is given (22) From the sector condition, we can see that (23) Fig. 5. The magnetic-levitation system. Thus Choose such that (24) (25) From the given condition (19) we see that there are symmetric positive-definite matrices and, a vector, and a positive constant such that [10] Fig. 6. The fuzzy sets for simulations. A. Regulation Problem We consider a regulation problem for a magnetic-levitation system. Fig. 5 shows the controlled plant. As shown in the figure, the steel ball is suspended in the air by the electromagnetic force generated by the electromagnet. The purpose of the control is to keep the metal ball suspended at the nominal equilibrium position by controlling the current in the magnet. The process dynamics is expressed as (29) Therefore (26) (27) From (17) and (18), the inequality (27) can be summarized as follows: which is negative definite. (28) IV. SIMULATION EXAMPLES Now we reveal control performances of the proposed SFLC via computer simulations. To ensure the performance of the proposed algorithm, we use two controlled processes: Magneticlevitation system and inverted pendulum one. The former is used to show the regulation performance and the latter the tracking performance. Here, we use the same fuzzy sets, inferencing and defuzzification methods for the simulation of both processes. and are the gravitational force on the steel ball and the force generated by the electromagnet, respectively. We use 0.01 and 0.1 kg as and, respectively. Fig. 6 represents the fuzzy sets for error, change-of-error, control input, and signed distance. We use the product inference and the center-average defuzzification. Figs. 7 and 8 show the simulation results of regulation performances and control inputs, respectively. Here, and are the cases of the conventional FLC and the proposed SFLC, respectively. As shown in Figs. 7 and 8, the control performances are almost the same even though the proposed SFLC has only five control rules compared to 25 rules of the conventional FLC. Thus, we can say that the SFLC is better than the conventional FLC. B. Tracking Problem Next we consider a tracking problem for an inverted pendulum system. Fig. 9 shows the plant composed of a pole and a cart. The cart moves on the rail tracks in horizontal direction. The control objective is to balance the pole starting from an arbitrary condition by supplying a suitable force to the cart. For simplicity, we do not consider the position of the cart. The plant dynamics is then expressed as (30)

308 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 30, NO. 2, APRIL 2000 Fig. 9. The inverted pendulum system. Fig. 7. Comparison of regulation performances. (Solid line: actual position, Dashed line: desired position). Conventional FLC. Proposed SFLC. Fig. 10. Comparison of tracking performances. Conventional FLC. Proposed SFLC. Fig. 8. SFLC. Comparison of control inputs. Conventional FLC. Proposed (31) (32) is an acceleration due to gravity ( m/s ), and is the applied force. ( kg) and ( kg) are masses and ( m) is the pole length. Like the cases of the regulation problem, Fig. 6 is used as the fuzzy sets for error, change-of-error, control input, and signed distance. We also use the product inference and the center-average defuzzification. Figs. 10, 11, and 12 show the simulation results of tracking performances, control inputs, and tracking errors, respectively. Fig. 11. SFLC. Comparison of control inputs. Conventional FLC. Proposed Here, and are the cases of the conventional FLC and the proposed SFLC, respectively. From the results, we can see that the control performances are almost the same. Hence, we can state that the SFLC satisfies the following advantages that were described in Section II-D: The number of control rules is only five instead of 25 of the conventional FLC. Because it uses a

CHOI et al.: DESIGN AND STABILITY ANALYSIS OF SINGLE-INPUT FLC 309 Fig. 12. SFLC. Comparison of tracking errors. Conventional FLC. Proposed [4] H.-X. Li and H. B. Gatland, A new methodology for designing a fuzzy logic controller, IEEE Trans. Syst., Man, Cybern., vol. 25, pp. 505 512, 1995. [5], Conventional fuzzy control and its enhancement, IEEE Trans. Syst., Man, Cybern. B, vol. 26, pp. 791 797, 1996. [6] R. Palm, Sliding mode fuzzy control, in Int. Control Fuzzy Syst., 1992, pp. 519 526. [7], Robust control by fuzzy sliding mode, Automatica, vol. 30, no. 9, pp. 1429 1437, 1994. [8] B.-J. Choi, S.-W. Kwak, and B. K. Kim, Design of a single-input fuzzy logic controller and its properties, Fuzzy Sets Syst., vol. 106, no. 8, pp. 299 308, 1999. [9] C.-C. Fuh and P.-C. Tung, Robust stability analysis of fuzzy control systems, Fuzzy Sets Syst., vol. 88, pp. 289 298, 1997. [10] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1996. [11] S. Galichet and L. Foulloy, Fuzzy controllers: Synthesis and equivalences, IEEE Trans. Fuzzy Syst., vol. 3, no. 2, pp. 140 148, 1995. [12] H. L. Malki, H. Li, and G. Chen, New design and stability analysis of fuzzy PD control systems, IEEE Trans. Fuzzy Syst., vol. 2, pp. 245 254, 1994. [13] C. J. Harris, C. G. Moore, and M. Brown, Intelligent Control: Aspects of Fuzzy Logic and Neural Nets. New York: World Scientific, 1993. [14] J. J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. sole input variable, the computational complexity is mitigated and tuning of rules, membership functions, and some scaling factors is much easier than that of the conventional FLC. V. CONCLUDING REMARKS We proposed a simple FLC called the SFLC. We observed that the rule table for conventional PD or PI-type FLC has skew symmetric property, and absolute magnitude of control input is proportional to the distance from its main diagonal line. These properties were also satisfied in the general case of -input FLC s using the error and its time derivative terms as fuzzy input variables. These facts allowed us to derive a new variable called the signed distance, which is the distance with a sign to the actual state from the switching hyperplane. The signed distance was used as a sole input variable of the proposed SFLC, which has many advantages. The number of fuzzy rules was greatly reduced compared to conventional FLC s, and also computational complexity was mitigated. Generation, modification, and tuning of control rules were much easier. Furthermore, the control performance was nearly the same as that of the conventional skewsymmetric FLC s, which has been confirmed through computer simulations using arbitrary two nonlinear plants. The proposed SFLC was also proven to be absolutely stable. Consequently, it is possible to design the FLC very simply while obtaining the desired control performance by using the proposed SFLC. REFERENCES [1] D. Driankov, H. Hellendoorn, and M. Rainfrank, An Introduction to Fuzzy Control, Berlin: Springer-Verlag, 1993. [2] J. Lee, On methods for improving performance of PI-type fuzzy logic controllers, IEEE Trans. Fuzzy Syst., vol. 1, pp. 298 301, 1993. [3] K. L. Tang and R. J. Mulholland, Comparing fuzzy logic with classical controller design, IEEE Trans. Syst., Man, Cybern., vol. SMC-17, pp. 1085 1087, 1987. Byung-Jae Choi was born in Korea on August 20, 1965. He received the B.S. degree in electronics from Kyungpook National University, Kyungpook, Korea, in 1987, and the M.S. degree in nuclear engineering and the Ph.D. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1989 and 1998, respectively. He is currently a Professor at Taegu University, Kyungpook. His research interests are in the areas of intelligent control, variable structure control, and process control. Seong-Woo Kwak was born in Koje, Korea, on March 10, 1970. He received the B.S. and M.S. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1993 and 1995, respectively. He is currently pursuing the Ph.D. degree at Real-Time Control Laboratory, the Department of Electrical Engineering, KAIST. His research interests are in the areas of fault tolerant system, real-time system, intelligent control, and fuzzy system. Byung Kook Kim was born in Korea on October 5, 1952. He received the B.S. degree from Seoul National University, Seoul, Korea, in 1975, and the M.S. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1977 and 1981, respectively. From 1981 to 1986, he was a Chief Researcher at Woo-Jin Instrument Co. Ltd. From 1982 to 1984, and from 1996 to 1997, he was a Research Fellow in the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor. He joined the Department of Electrical Engineering at KAIST in 1986 as an Assistant Professor, he is currently a Professor. His research interests are in the areas of robotics, process control, reliable and intelligent control, fault tolerant system, real-time system, and robot vision.