Stability analysis of fuzzy control systems

ZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 105 (1999) 33-48

Stability analysis of fuzzy control systems A. K a n d e l * , Y. L u o , Y . - Q . Z h a n g Department of Computer Science and Engineering, University of South Florida. 4202 East Fowler Avenue, Tampa, FL 33620-5350, USA

Received March 1996;receivedin revised form June 1997

Abstract In this paper, we present a general review of the stability issue as related to fuzzy control systems. The concept of stability and the general criterion used for fuzzy control systems are discussed. The intuitive and nonlinear system stability analysis of fuzzy control systems are given. Popov's technique is proposed to test the stability of fuzzy systems. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Stability analysis of fuzzy systems; Popov technique; Lyapunov method; Energetic analysis; Describing

function analysis; Circle criterion; Input/output system stability

1. Introduction Fuzzy control is a nonlinear control based on heuristic control tables, and is actually "another" way of implementing the proportional-integralderivative (PID) control strategy, although not limited to PID. Any nonlinear system can be approximated as accurately as required with some number of fuzzy control rules. The fuzzy controller is also regarded as a universal approximator, and its use offers many advantages [10]. The wider application of fuzzy systems require solid and systematic analysis of system performance. Stability is of particular importance to the application of fuzzy systems - the first and last concern for any system design and a fundamental issue in every control system. Various concepts,

* Corresponding author. Tel.: (813) 974-4232; fax: (813) 9745456; e-mail: [email protected].

such as Lyapunov stability, asymptotic stability, exponential stability, local stability, and global stability or global exponential stability, must be well understood to accurately characterize the rich and complex stability behavior exhibited by nonlinear systems. This paper addresses the stability issue of fuzzy control systems. A brief review of the related system control theories is given where required. In Section 2, a short review of linear and nonlinear system control theory is given. In Section 3, the intuitive and qualitative approach is detailed. In Section 4, Popov's approach is proposed. The final section contains some concluding remarks.

2. Theory of control systems All physical systems are inherently nonlinear in nature [26], differing only in their degree of nonlinearity. Both linear and nonlinear differential

0165-0114/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved. PII: S0165-01 14(97)00234-0

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

34

equations can represent and model the nonlinear system under certain assumptions. Sometimes a nonlinear system is approximated by a set of linearized systems with linear differential equations. The nonlinear system, however, demonstrates far more complex and richer behavior than the linear system. Mathematical limitations and the availability of analytical tools contribute to this situation. Mathematical limitations: Most nonlinear systems cannot be represented by nonlinear differential equations, or these equations are either difficult or impossible to solve, thus making nonlinear system behavior difficult to analyze. Finding an accurate universal mathematical model for fuzzy control systems is difficult, because fuzzy control is nonlinear by nature [10]. Modern control theory has shown that no universal approach exists for nonlinear systems. Analytical tools: Because a nonlinear system does not follow the principle of superposition, effective analytical tools such as frequency domain transformation, Fourier transforms, and Laplace transforms are not applicable. Unfortunately, fuzzy control cannot take full advantage of linear control theory unless a good linear approximation of fuzzy control systems can be found.

2.1. Linear system Control systems are generally described by the differential equation

= f ( x , u), where f is a nonlinear function, u is the system input, and x the state of the system used to describe system behavior. Because those states close to equilibrium are being considered, where the nonlinearity is weak, the function f can be expanded into the Taylor series and only the linear part considered [12]. A common mathematical model [28] of a linear system with n state variables, and m inputs is

Y¢= Ax + Bu, where the dot denotes differentiation with respect to time (t), x is the n x 1 state vector, u is the m x 1 external input vector, A is the n x n coefficient con-

stant matrix of linear system, and B is the n x m input matrix. The linear system contains many important properties, including superposition, frequency domain, and unique equilibrium point, which are all related to stability analysis.

2.2. Nonlinear system As stated earlier, a nonlinear system demonstrates much richer and more complicated properties and system behavior than a linear system. Nonlinear systems are traditionally classified as inherently nonlinear or intentionally nonlinear [26]. The study of nonlinear system behavior is very important when analyzing the stability of fuzzy control systems. Modeling: A commonly used dynamic model of a nonlinear system [9] is given by

= f ( x , t), where x is a state vector of the nonlinear system with n states, f is a nonlinear vector function of the control system and t is the time. The system state x as time t changes from zero to infinity is known as the system trajectory. Multiple equilibrium points: A unique feature of a nonlinear system is its multiple equilibrium points. Motion of stability: The nonlinear control system responds both to initial conditions and to the input to the system. The stability of a nonlinear system depends on its initial condition. Linear control allows the analysis of a system in the time frequency domain. These standard approaches cannot be used for nonlinear systems, however, because an explicit solution of nonlinear system equations is almost impossible to obtain and the common tools used in linear analysis are not applicable.

2.3. Concept of stability in nonlinear control system Stability is generally regarded as a tendency toward the origin in linear control systems [30]. Since the nonlinear system displays complex behavior, a clear understanding of the concept of stability is needed before beginning an analysis of fuzzy control systems. Because most fuzzy systems can be

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A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

treated as autonomous, discussion will be limited to such systems. Equilibrium points: A nonlinear system may contain more than one isolated equilibrium point. An equilibrium point is the state of a system when the state x(t) = x* and the state of the system will stay at x* for all future time. A linear system has only one equilibrium point; otherwise, the system is unstable and may have many continuous, uninterrupted equilibrium points. Stability in the Lyapunov sense: A system is said to be stable if its trajectories can be made arbitrarily close to the origin when the initial starting state is chosen. Stability in the Lyapunov sense is a widely used definition in the control community, originating from the concept of "energy" proposed by the Russian mathematician Alexander Lyapunov in the 19th century. Instability or limit cycles may occur when the trajectories of a system cannot be made arbitrarily close to the origin. Asymptotic stability: When a system is stable and initial states that are close to the region of origin converge to the origin, the system has asymptotic stability. A system with Lyapunov stability does not guarantee asymptotic stability because asymptotic stability is stricter than Lyapunov stability. Additionally, one needs to know how fast the system converges to the equilibrium point. Exponential stability is used to estimate how fast the system trajectory approaches and converges to the equilibrium point as time goes to infinity. The system has exponential stability when its state vector converges to the origin no less than an exponential function or in an exponential order. The function f(t) is of exponential order if a real, positive constant 6 exists such that the function [26]

e-~tl f(t)l ~ 0 approaches zero as time t approaches infinity. Exponential stability is stricter than asymptotic stability. Thus, exponential stability guarantees both Lyapunov stability and asymptotic stability but not vice versa. Local and 91obal stability: The definitions above describe the local stability of systems with small motion. The local behavior of a system tells

very little about global behavior of a system. Global stability and its relation with local stability needs to be considered when the initial point of a system is far from the equilibrium point. The above definitions can be extended from local stability to global stability [1]. A system is considered asymptotically stable in the global sense if every initial state leads the system to an asymptotically stable state [30]. In other words, local stability leads to global stability under the condition of asymptotic stability for all initial states. The system is thus asymptotically stable, implying that linear system stability is guaranteed globally by using the superposition principle. For fuzzy control systems, global stability or instability can be studied by a local linearization model if the system is linearized "near" the equilibrium point. For a simple fuzzy control system with weak nonlinearity, Lyapunov's linearization method justifies the application of linear control techniques to study the global stability of a fuzzy system. No conclusions may be reached, however, for fuzzy systems with strong nonlinearity by using the Lyapunov linearization model. The nonlinearity of a fuzzy control system may come from either the controlled plant or the nonlinear fuzzy controller.

3. Stability of fuzzy control systems Before proceeding with the study of fuzzy control systems, an assumption is the dynamics of the objective system - the nonfuzzy part of the control system under study - are available or predictable. Fig. 1 depicts the basic configuration of the fuzzy control system. The early design and analysis of the fuzzy control system is based on specific applications, depending largely on the knowledge and expertise of the

control input ~ I l nonfuzzy ~ l Process Under Control Fuzzy Controller

output & state nonfuzzy .,4

Fig. 1. Structure of fuzzy control system.

36

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

I ~ ControlFUZZY~ _ ~ System [ [ I ~

Nonlinear System Use the Theory of ! blonlinear Systems [ Dynamic Fuzzy System Use Intuitive Methods to Study Fuzzy Systems

u .inputS[

ControlPlantG(s)

[I

Y ~ (-~ internal FeedbackGainF(x,t) ]~ output ~

r external input

Fig. 4. Feedback loop with external input [7].

Fig. 2. General analysis of fuzzy control systems.

Circle criteria ~ Hyper-stabilitvapproach ) _ [ Nonl!near 1_ ~ Input output theory ) ;;;':ach [P" ~ Lyapunovstability ) I I -I ~ Phase plane criteria ) Fuzzy I ~ System ~ ----q~ Poleassignment ) Stability I I Rule modification ) L.~ Qualitative ~ Approach ~

Stability indices Energeticstablhty

) )

Fig. 3, Stability analysis of fuzzy control systems.

particular designer. No systematic and general methodologies exist for the design and analysis of such dynamic fuzzy control systems. Fuzzy control systems are essentially nonlinear systems, due to the nature of the nonlinear fuzzy controller employed. Fig. 2 illustrates the approaches to designing fuzzy control systems. As already noted, no universal methodology exists for the design and analysis of a fuzzy control system. The first approach is the intuitive and qualitative approach, corresponding to the dynamic fuzzy system depicted in Fig. 2. The second is the nonlinear theory approach [37] shown in Fig. 3. There are two approaches to general stability theory. The first is Lyapunovs' direct method. The direct method of Lyapunov is a time domain method based on a nonlinear state-space description of the system. Suppose an autonomous system, described by the vector differential equation [26]:

%: = f ( x , t), f(O,t)=O

for allt.

If there exists a scalar function V(x, t) having continuous, first partial derivatives and satisfying the following conditions: 1. V (x, t) is positive definite, 2. 12(x,t) is negative definite, then the equilibrium state at the origin is uniformly asymptotically stable. If, in addition, l?(x, t) ~ oo as llxll ~ oo, then the equilibrium state at the origin is uniformly asymptotically stable. The second approach is input-output stability, which deals with the external representation where relatively small outputs result from the bounded input [10]. Consider the system below (Fig. 4). Each block in the feedback loop is regarded as an operator which transforms input signals into output signals, and stability is defined in terms of the properties of such transformations. Lz-norms on the signals are introduced as [7]

HuH=/(;o for a scalar or vector signal u(t). The feedback system is finite-gain input-output stable if the norms of its internal signals are bounded by some fixed multiple of the external input norm. The Lyapunov function can be obtained to guarantee the global asymptotic stability [7]. More recently proposed has been the hyperstability approach [14]. Although close to 100 papers were reviewed for this paper, particular approaches were selected for presentation here. These approaches have either rich backgrounds in control system theory or strong practical engineering values. Table 1 lists papers involving the most important developments and research in fuzzy control stability. Knowledge of a dynamic nonfuzzy model of the process or plant to be controlled is assumed. Even if the model is known only approximately, however, some conclusions about the stability of the closed

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

37

Table 1 Important research papers on fuzzy control stability 1978 1978 1978 1979 1980 1980 1981 1984

C.V. Negoita [25] W.M. Kickert and E.H. Mamdani 1-18] R.M. Tong [34] M. Braae and D.A. Rutherford [3] R.M. Tong [34] A.A. Kania et al. [16] W. Pedrycz [27] K.S. Ray and D.D. Majumder [29]

1985 1985 1988 1991 1992 1992 1993 1993 1993

W.M. Kickert and E.H. Mamdani [19] J.B. Kiszka et al. [21] M. Sugeno and G.T. Kang [31] T. Yamashita [37] M. Sugeno and K. Tanaka [32] M. Maeda et al. [23] W.C. Kim et al. [20] F. Hara and K. Yamamoto [11] A.Y.W. Cho et al. [4]

1993 1994

K.H. Cho et al. [6] T.A. Johansen [14]

On the stability of fuzzy systems Analysis of a fuzzy logic controller Analysis and control of fuzzy systems using finite discrete relations Selection of parameters for a fuzzy logic controller Some properties of fuzzy feedback systems On stability of formal fuzziness systems An approach to the analysis of fuzzy systems Application of circle criteria for stability analysis of linear SISO and MIMO systems associated with fuzzy logic controllers Analysis of fuzzy logic controller Energetic stability of fuzzy dynamic systems Structure identification of fuzzy model Stability analysis of fuzzy control system applying conventional method Stability analysis and design of fuzzy systems Stability analysis of fuzzy control system by 3D display Stability analysis of linguistic fuzzy model systems in state space A design method of Lyapunov-stable MMG fuzzy controller Design technique of fuzzy controller using pole assignment method and the stability analysis of the system On stability analysis of nonlinear plants with fuzzy logic controller Fuzzy model based control: Stability, robustness, and performance issues

system can still be obtained [10]. In fact, any stability analysis requires some knowledge a b o u t the process dynamics. Furthermore, if the closed-loop system is robust and far from the point of stability loss, it can remain stable even if the model does not exactly represent the actual d y n a m i c behavior of the process under control. 3.1. Intuitive and qualitative approach The study of the internal representation of a dynamic system gives the category a p p r o a c h and the energetistic approach, both of which are qualitative in nature. The category approach: Using the category approach, N e g o i t a [25] first discussed issues relative to fuzzy system stability and suggested a process of synthesis. A fuzzy system specifies interaction rules between different descriptions, with structural stability emerging as a consequence. Stability is treated as a property of the system as a whole related to the c o o r d i n a t i o n of the actions between its parts. Also noted is that every stable system has the p r o p e r t y that, if displaced from a state of equilibrium and released, the subsequent

m o v e m e n t is m a t c h e d in such a way to the initial displacement that the system is b r o u g h t back to the state of equilibrium. The category a p p r o a c h is, in essence, a theoretical analysis. N o practical engineering application has been proposed. Energetic approach: Following a philosophy similar to the L y a p u n o v method, the concept of "energy" m a y be applied to stability analysis of fuzzy systems. Kiszka et al. [21] used an energy function to study both the stability and cause of instability in a fuzzy system. A formulated energy function is called the energy "measure" of a fuzzy d y n a m i c system. U n d e r the guideline that a system is stable unless its total energy decreases monotonically, the researchers investigated the system stability by using the energetic stability algorithm. Observation of the state of a physical system reveals that the states in a stable system tend toward a position in space where m i n i m u m energy is attained. If the system achieves a stable position, then the energy at this point is constant and minimal. If the stored energy in the system m o n o t o n i cally decreases as time increases, and after some

38

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

time attains a minimal value, the system is locally stable. A position in the state space where the system has minimal energy is called a stable point. A stable point is identified as an equilibrium point. In the sense of local stability, attainment of the stable point does not depend upon a starting point (initial condition) of the system. A system whose energy decreases on the average may have multiple equilibrium points (states) and, as such, its energy may not decrease at each instant. Kiszka's definition of fuzzy system energy is more general than that of the energy in a strictly physical sense and is more widely interpreted because it considers certain specific properties of fuzzy relations, such as the degree of fuzziness, spread, cardinality, etc. System stability is observed by studying the change of the energy. The system converges and is stable. Further conclusions about the property of system stability can be reached by considering different initial conditions of the system. The above system is autonomous and independent of the system initial conditions. With some considerations of the physical properties of the system behavior, this approach may be used to study a system with limit cycles. The fuzzy system is said to be unstable if the energy of the fuzzy relation increases with time. The fuzzy system is said to be oscillatory if the energy of the fuzzy relation fluctuates with some period of repeatability with time. One may easily find a fuzzy control system meeting this "measure" but it is either in the marginal stable region or is unstable. When engineering exactness is emphasized, the approach has no significant practical use. Furthermore, the local stability of the fuzzy system does not necessarily guarantee global stability.

3.2. Nonlinear theory approach If the fuzzy control system can be modeled as a linear or nonlinear system, the stability analysis can be performed by using the classical control theory tools. Again, correct modeling of the corresponding practical fuzzy control system is assumed. Traditional analytical tools for dealing with stability analysis of nonlinear control systems in-

clude describing function, phase plane, and Lyapunov stability. Each of these methods has been used in analyzing fuzzy control system stability. Because analytical solutions of nonlinear differential equations are difficult, if not impossible, to obtain, the Lyapunov method is particularly important in determining fuzzy control system stability. Almost all other methods are built on the basis of Lyapunov theory. Lyapunov's linearization studies motion of nonlinear systems around equilibrium points. Mainly of theoretical value, it justifies using linear control for design and analysis of physical systems with weak nonlinearities. At one time, the method was referred to as Lyapunov's first method and used as an example of the direct method (Lyapunov's second method) application. Actually, the method is the method of exponents, currently used in chaos analysis. Lyapunov's direct method, based on Lyapunov functions, is not restricted to small motions. In principle, it is applicable to all dynamic systems, fuzzy and nonlinear. This method suffers, however, due to the difficulty in finding a Lyapunov function for a given system. No general practical approach for finding a Lyapunov function exists, so trialand-error, experience, and intuition are needed to search for the appropriate Lyapunov function. The Krasovski method (gradient method) can also be used to find a Lyapunov function [22]. Physical insight into a particular application or theorem on the existence of Lyapunov functions may suggest a good direction for finding the Lyapunov function. A number of instability theorems may also prove helpful if the linearized control systems (whether non-autonomous or autonomous) are marginally stable. Sugeno and Tanaka [32] derived a stability theorem in terms of the Lyapunov second method. By assuming a continuous polynomial membership function, they modeled the fuzzy logic controller to a Sugeno model. Most nonlinear systems, such as fuzzy control systems, are continuous in nature and hard to meaningfully discretize, while digitally controlled systems may be treated as continuous-time systems in analysis if high sampling rates are implemented.

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48 NM

NS

ZO

PS

PM

NM

NB

NB

NM

NS

ZO

NS

NB

NM

NS

ZO

PS

ZO

NM

NS

ZO

PS

..........

PS

NS ..........

PM

ZO

4. . . . . . . . . . . . . .

ZO ~ ..............

PS

-I- .............

PS .. .............

PM

I ..............

PM t ..............

PB

39

PM .E . . . . . . . . . . . . .

PB , .............

PB

Fig. 5. Fuzzy inference rule.

In the above discrete model, P is a positivedefinite matrix and A~ the corresponding matrix of the fuzzy subsystem. This is a sufficient condition for fuzzy control stability. If one of the subsystems is unstable, the closed fuzzy control system may still be stable. For the existence of the positive matrix P, every Ai'Aj must be a stable matrix. Sugeno and Tanaka [32] adopted a strategy where at each discrete time step a linear controller was designed based on a fuzzy model linearization at the current operating point. The assumption was that the operating point changed slowly and in the near range of the equilibrium point compared to the control-loop dynamics. Sugeno's stability theorem is more of approximate stability of the nonlinear system under the condition that the fuzzy system can be approximated by a linear function. All locally approximated linear systems are stable, and, as pointed out, local stability does not guarantee global asymptotic stability. It is also difficult to find the common positive-definite matrix P as no effective algorithm exists for finding matrix P. Example. Consider the fuzzy controller with the rule table shown in Fig. 5. The system trajectory is the curve of the system state history. It maps to the space partition of the fuzzy rule where the linguistic trajectory is formed [10]. This mapping is depicted in Fig. 6. In Figs. 5 and 6: PB is the linguistic partition positive big of a fuzzy membership function. PM is the linguistic partition positive medium of a fuzzy membership function. PS is the linguistic partition positive small of a fuzzy membership function.

Fig. 6. M a p p i n g of system trajectory.

ZO is the linguistic partition zero of a fuzzy membership function. NS is the linguistic partition negative small of a fuzzy membership function. NM is the linguistic partition negative medium of a fuzzy membership function. NB is the linguistic partition negative big of a fuzzy membership function. Fig. 7 depicts that the partition space of the rule set is not covered adequately. The highlighted area is the linguistic trajectory of the fuzzy controller, which is the rule-firing sequence of the rule base. Aracil et al. [-2] interpreted the phase plane geometrically. This technique depends on analyzing the vector interpretation of both the control plant and the fuzzy controller. Consider the closed-loop system modeled by the equations 2 =f(x) +

bu,

u = ¢(x),

wheref(x) is a nonlinear function of the controlled system, qS(x) represents the fuzzy controller and x and b are vectors of dimension n, u is the scalar control input. Closed-loop behavior will depend on the nature off(x) and q~(x). The fuzzy rules used are: RI: If XI is N and X2 is P, then u is NB R2: If X1 is Z and X2 is P, then u is N M R3: If X1 is P and X2 is P, then u is NS R4: If X1 is N and X2 is Z, then u is PM R5: If X1 is Z and X2 is Z, then u is Z

40

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

system trajectory

X2 NB

NB

IN: ,l i

zo

NB

NM

iN ;

PS

~

Xl 'S

PM

PB

M

PB

PB

Fig. 7. Operation covering only the X-axis.

R1

i R2

R5

"" "~"~'~}" XI

R7

~

R8

R9

Fig. 8. Phase plane portrait.

R6: If X1 is P and X2 is Z, then u is NM R7: If X1 is N and X2 is N, then u is PS RS: If X1 is Z and X2 is N, then u is PM R9: If X1 is P and X2 is N, then u is PB PB, PM, PS, ZO, NS, NM and NB represent linguistic values of the fuzzy controller output, such as positive big, positive medium, positive small, zero, negative small, negative medium and negative big. P, Z and N represent linguistic values of the fuzzy controller input state and the change of input, such as positive, zero and negative. Fig. 8 shows the system phase plane portrait. With f(x) and ¢(x) being monotonic functions, the system is stable, providing 1. the nonlinear plant 2 = f ( x ) of the open system is stable, 2. vector field b'O(x) moves toward ¢(x) = 0. In Fig. 8, the vector field leads the system trajectories toward the curve ¢(xl,x2) = 0. The nonlinear plant vector in the vector field will be dominant when the trajectories are close enough to the curve

~b(X1,X2) = 0. Since the precondition states the nonlinear plant of the open system is stable, the system trajectories will converge to the equilibrium point• In the case of a limit cycle, no dominant element exists, only the combination of the nonlinear control plant and the nonlinear fuzzy controller. The decisive vector of the closed-loop system must be found, as the system trajectory and the stability depend on it. This, then, is basically a linguistic trajectory approach, and, in essence, the qualitative approach. Inspecting the system trajectories will provide information on system stability or instability. Other typical nonlinear system dynamics, such as multiple equilibrium points, motion of stability, bifurcation and chaos, also can be studied qualitatively. Corresponding changes of the rules are possible as well to meet the requirements of system performance. Describing function approach (DFA): Many relationships among physical quantities are not linear, although they are often approximated by linear equations mainly for mathematical simplicity. This simplification may be satisfactory so long as the resulting solutions agree with experimental results• A most important characteristic of nonlinear systems is the dependence of the system response on the type and magnitude of the input. For example, a nonlinear system may respond completely differently to step inputs of various magnitudes. The describing function or sinusoidal describing function of a nonlinear element is defined as the complex ratio of the fundamental harmonic component of the output to the input. If no energy-storage element is included in the nonlinear element, then the describing function is a function only of the amplitude of the input. If energy-storage elements are included, however, then the describing function is a function of both amplitude and frequency of the input. To determine system stability, both the - 1/N(x) and the G(jw) local need to be plotted. N(x) represents the nonlinear controller and G(jw) represents the process under control. Linear stability analysis, using the Nyquist criterion, requires the study of the encirclements of the ( - 1, 0) point on the real axis by the G(w) locus. In the describing

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

- 1I N ( x )

o

Re g

G(fi¢.) W

Fig. 9. Stable system.

Re y

W

mm#f

G:jw:l.::(%.'" Fig. 10. Unstable system.

function analysis of nonlinear fuzzy control systems, the conventional frequency response analysis is modified so that the entire -1IN(x) locus becomes the locus of critical points. Thus, the relative location of the - 1/N(x) locus and G(jw) locus will provide the stability information. If the - 1/N locus is not enclosed by the G(jw) locus, then the system is stable, or no limit cycle exists at the steady state, as shown in Fig. 9. If the - 1/N(x) locus is enclosed by the G(jw) locus, the system is unstable and, when subjected to any disturbance, the system output either will increase until breakdown occurs or will increase to some limiting values, as drawn in Fig. 10. When two loci intersect, the - 1IN(x) and the G(jw), the system may have a limit cycle. The system may be stable or unstable, depending on the

41

relative direction of the two curves at the intersecting point. The describing function approach is an extension of linear techniques used when studying nonlinear systems. Therefore, typical applications are to systems with low degrees of nonlinearity. Using describing functions to analyze high-degree nonlinear systems may cause serious errors, thus limiting its applicability to the analysis and synthesis of lowdegree nonlinear systems. The describing function approach is an approximate method for determining the stability of unforced nonlinear control systems, and analysis results over the years have proven that this approach is sometimes incorrect. The describing function approach is mainly used for predicting limit cycles in nonlinear control systems. Kickert and Mamdani [18] used the describing function approach to evaluate fuzzy system stability. These researchers showed that, under certain restrictive assumptions, fuzzy systems could be treated as a multidimensional, multilevel relay. In this framework, frequency domain analysis can be carried out on the fuzzy system stability. More specifically, by applying fuzzy logic to the linguistic rule-based controller, Kickert and Mamdani first pointed out that the fuzzy controller is identical to a multilevel relay, with the pre-condition that the membership function be symmetrical around the maximum membership value. This gave the one-to-one mapping from input space to output space for the fuzzy controller. The mean of the maxima technique was used in the defuzzification process, assuring uniqueness of mapping, which is critical for further analysis. The fuzzy controller could then be reduced to a typical nonlinear control element of the multilevel relay. The multilevel relay was further assumed to be symmetrical around the origin. Therefore, the analytical approaches in the nonlinear theory may be applied to the stability study of the fuzzy control system. Kickert and Mamdani proposed the describing function approach with a system model drawn in Fig. 11. The fuzzy rule corresponding to the above fuzzy input gives the appropriate implied fuzzy output set. The fuzzy controller rule matrix is shown in Fig. 12. The meaning of PB, PM, PS, ZO, NB, NM, and NS are the same as defined in Fig. 6.

42

A. Kandel et aL / Fuzzy Sets and Systems 105 (1999) 33-48

Nonlinear ~ - I element Feedback

Reference y _ ]

.

Linear element ,i

zero small medium

large

t_,utput

Fig. 11. Feedback nonlinear closed-loop system.

u(y output 0

Change in error NM NS ZO

NB

PS

PM

PB

Yl

Y2

Y3

Y4 "-Y

Fig. 13. Defuzzified output of fuzzy controller.

PB NM

PM PS

zc

NB

Y PS PM

ZO PS

NM ZO i NS

NM

O

Y4

II

I

I I

NS

PM

ZO

I

NS

I

Y3

I

II

I

NI~

I I

Nl:

PB

PM

I

Y2 I t |

input

Fig. 12. Matrix representation of fuzzy rules. u

Yl The input range is divided into regions and the output range is reduced to points; according to the region where the input falls, one output is chosen. The fuzzy controller output using the maxima defuzzification is drawn in Fig. 13. Fig. 14 shows a natural analogy of the multilevel relay formed from the above analysis. Assuming a low-pass linear system, taking sinusoidal inputs, the multilevel relay is approximated and subdivided by the following nonlinear element, which has a dead zone relay, shown in Fig. 15. The nonlinear describing function method may be applied to the above approximated system. The describing function of the nonlinear elements is first obtained and the analysis then focuses on the selfexcited oscillations or limit cycles. The stability conclusions of the describing function can be applied to the reduced fuzzy control system. The typical relay switch nonlinear function with a dead zone is shown in Fig. 16. The describing function DF(x) is

xI

I

X3

x2

"- X

Fig. 14. Multilevel relay input-output.

Applying the above formula to the nonlinear dead zone elements of the fuzzy controller with two relays, the following describing function results by subtracting the DFs of two relays with dead zones: D F ( x ) = 4yi2 (x//~l - x~ - x / x ~ -

x~+ 1),

~x

where y~ is the dead zone level as drawn in Fig. 16. Hence, a general multilevel relay with N input levels x l .... , XN and N corresponding output levels Yl,-.. ,YN has a describing function defined by DF(x) = ~

4 fN-1

( i~=1 Y , ( x / ~

-- x2 -- x~ x21 -- x2+ 1)

In the describing function analysis, only the fundamental harmonic component of the output is assumed significant. Such an assumption is often valid since the higher harmonics in the output of a nonlinear element are often of smaller amplitude than those of the first fundamental component. As

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

output

yi~

Y L. . . . .

input

-x i -Xi+ 1 xi

Xi+l

"-X

-Yi Fig. 15. Nonlinearelementof dead zone.

+MtI

-X I

I

+x 1

- -

r

x

-M

Fig. 16. Typicaldead zone. mentioned previously, the describing function is defined to represent both the energy-storage element and non-storage element, and thus is a function of both the amplitude and frequency of input. In linear systems, it is only a function of the input amplitude, which is why the sinusoidal input is used. Under the assumption of the low-pass filter property and the sinusoidal input, the describing function approach is used to analyze the oscillations and stability of the control system. If the linear control part has the transfer function H(s), the sufficient condition of oscillations after adding the nonlinear fuzzy controller [10], is given by

43

of the fuzzy controller. If the controlled plant is nonlinear, the frequency domain stability analysis of the nonlinear system may not apply. Another problem with the describing function approach is restrictions on the order of the system, which also limits its application to practical system analysis. The describing function approach is basically an approximation method. Besides the multilevel relay modeling of the fuzzy controller, many other modeling techniques are available. These include the on-off nonlinear model (two-position nonlinearity), the nonlinear waveforms model, and the nonlinear saturation model. If the maxima defuzzification method is not used in the fuzzy controller, other nonlinear models might work better than the multilevel relay. Kickert and Mamdani's approach is highly restricted to the maxima defuzzification procedure of the fuzzy controller and may give incorrect results if other defuzzification procedures are used. Many other researchers have approached the problem of fuzzy system stability. Kandel and Langholz [15] introduced the concept of linguistic stability versus numerical stability, where the neurocomputations of a degree of stability were applied. Using the Lyapunov approach, Hara and Yamamoto [l l] studied the stability of MMG fuzzy systems and derived the conditions in terms of scale factors Kp, Kv, and K u for ensuring the fuzzy system stability. The pole assignment method has been used to analyze the stability of the total fuzzy system rather than checking for the existence of the common P matrix (Sugeno's common matrix).

4. The Popov approach -1 H(j~o) = D F " With the known D F function and the linear control plant, calculating the prediction of oscillation is easy. The describing function approach is limited to the SISO system. The system is assumed to be a low-pass system. The MIMO system cannot be represented in the describing function approach. The basic idea is to construct an analytical function

In most control applications, the linear control element is known. This paper explores system stability when using a fuzzy controller and what design limits affect fuzzy controller system stability in general. V.M. Popov proposed a very valuable nonlinear stability method in 1959, based on the model in Fig. 8 of Section 3 [-8], which is a frequency criterion for the asymptotic stability of a SISO nonlinear closed system. Unlike the phase plane

44

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

kl

Re

Fig. 17. System model.

N[x(t)]

y(t)

Fig. 19. Popov line of wl in G(jw) Nyquist plot.

_/

0

x(t)

k x(t) Fig. 18. Nonlinear element function.

technique, it can be applied to a high-order system. It differs from the describing function approach in that it gives an accurate but conservative stability prediction provided the linear element and nonlinear property are known. The stability prediction can be accomplished with some calculation and drawing. Kickert and Mamdani [18] implied that the concept of Popov stability techniques can be applied in analyzing fuzzy control systems. Ray and Majumder [29] gave the circle criteria and also mentioned the potential of Popov criteria but failed to give the Popov techniques. The system under study is modeled as shown in Fig. 17. It is an SISO system with a time-invariant linear part G(s) and a time-invariant nonlinear part N i x ( t ) ] as its components. The system time-invariant nonlinear function is defined as y(t) = N [x(t)].

It may take different forms, including fuzzy controller nonlinearity. The nonlinear element should satisfy the following conditions:

Nix(t)]

0~<--~
N [0] = O.

x¢0,

k is an arbitrary integer and may take the value of infinity. The restriction of the above formula is shown in the graph below (Fig. 18). The curve of the nonlinear element always lies between the x-axis and y(t) = kx(t). The time-invariant linear element is asymptotically stable and has all the negative poles. The sufficient condition for asymptotic stability of the nonlinear system is a number q satisfying Re[(1 + j w q ) G ( j w ) ] + k -1 > 6 > 0 ,

w~>0,

where 6 is an arbitrary small positive constant and Re represent the real part of the G(jw). The G is the linear frequency function. A simplified equation is (Im represents the imaginary part of G(jw)): ReG(jw) > - k -1 + q w I m G ( j o ) ) ,

w >~ O.

Changing the inequality to an equality results in a straight line on the complex plane of G(jw) that passes the point ( - l / k , 0 ) with slope of 1/qw, as shown in Fig. 19. In order to satisfy the inequality equation above, the linear element G(jw) is always on the righthand side and below the Popov line for all frequencies w greater than zero. Because the Popov line is related to the frequency w, the entire line must be drawn to match the changing frequency, which is somewhat impractical. The adjusted frequency F(jw) is used, where F(jw) ,~ Re G(jw) + jw Im G(jw). Replacing the G(jw) with the F(jw) in the inequality gives 1

ReF(jw)>-~+qImF(jw),

w/>0.

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A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

1•

k-

[m

kt'N~ / ~

l

Im

Re F = - 1 / 6 0

Im//P°p°v

slope/ I) ~-Re sl°pe~) ]~Re q-l/~'x F(jw q-1//~ F(jw)

line

• , argtg2

Re

'~w=0

~

Fig. 20. F(jw) Nyquist plot.

Input

Fig. 22. F(jw) Nyquist plot.

Output controller Oumutfeedback

Yl ,.-1 y3 o y2-

Fig. 21. Nonlinear fuzzycontrol system. The Popov line on the F(jw) complex plane passes the point ( - 1/k, 0) with the slope of 1/q. It is not related to the frequency, as shown in Fig. 20. The sufficient condition is the same, in that the F(jw) is on the right side of the Popov line. The following is an example showing how to apply the Popov approach to fuzzy control design to get a stable nonlinear system (Fig. 21). Example. Given the nonlinear fuzzy control system as shown in Fig. 21, we want to find line y(t) = kx(t) for the fuzzy controller and study the effect of the fuzzy controller on the nonlinear system stability. The linear transfer function G(s) is G(s) =

-

1

yl -x3 I -x2 I -xl I

m

n

C¢i xl -yl -y2

J

I

x2

)x3

input v x

•-y3 Fig. 23. Nonlinear fuzzycontroller input output function. y~ y3]-~

/ /

y(t)=kx(t) :

y2; _

yl -x3, -x2 -xl. al~Yo, )j i

t:

,

'

'

input

~

-ylxl x2 x3 -y2 -y3

v

x

(s + 1)(s + 2)(s + 3)"

The linear element is stable because the roots 1, - 2, and - 3 all lie on the left half of the plane. The F(jw) is obtained through the mapping

6(1 - - w 2 ) + jwZ(w2 -- 11) F(jw) = (w 2 + l~-w~- + 4 ~ i + - 6 . The F(jw) Nyquist plot is drawn in Fig. 22. The nonlinear element y(t) = kx(t) must satisfy the restrictions on k to get a stable nonlinear system. With the computer calculation, it is easy to get the restrictions on k for a stable nonlinear system. As discussed in Section 3.2.5, the fuzzy control can be approximated by a multi-relay, as shown in Fig. 23.

Fig. 24. Multirelayinput-output function.

The fuzzy controller uses the maxima defuzzification. The Popov restricting line for stability is drawn in Fig. 24. If other defuzzification methods are used, the Popov stability line can still be drawn according to the specific case study shown in Fig. 25. The nonlinearity may not be multilevel relay, but may be any nonlinear function. The stability algorithm is designed based on the above analysis. First, F(jw) is calculated by transformation of linear transfer function G(jw). The Nyquist plot of the corresponding F(jw) function is

46

A, Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

y(t)

'i;t

will w o r k for h i g h e r - o r d e r systems. W i t h m i n o r changes, P o p o v ' s a p p r o a c h m a y also be e x t e n d e d to fuzzy c o n t r o l systems having a pole at the origin.

5. Conclusion Fig. 25. Nonlinear input-output function.

d r a w n . T h e P o p o v line can then be o b t a i n e d directly from the F ( j w ) plot, which satisfies the a s y m p totic stability of the n o n l i n e a r c o n t r o l system. If the n o n l i n e a r e l e m e n t satisfies the P o p o v restricting line r e q u i r e m e n t , the system is stable. P o p o v ' s a p p r o a c h c a n be used in a l m o s t all n o n linear fuzzy c o n t r o l systems w h e r e the linear control e l e m e n t is k n o w n . P o p o v ' s restricting line gives the u p p e r line for the fuzzy c o n t r o l l e r design a n d

In this p a p e r , the use of P o p o v ' s a p p r o a c h is p r o p o s e d for stability testing of fuzzy c o n t r o l systems, with the a l g o r i t h m for its a p p l i c a t i o n given. P o p o v ' s technique can be a p p l i e d to systems of b o t h low a n d high o r d e r a n d is a p r a c t i c a l m e t h o d for real w o r l d engineering design. W i t h s o m e m o d i fications, P o p o v ' s a p p r o a c h m a y be e x t e n d e d to m o r e c o m p l e x systems. Stability analysis is clearly i m p o r t a n t . H o w e v e r , the lack of satisfactory f o r m a l techniques for studying the stability of fuzzy c o n t r o l systems has been c o n s i d e r e d a m a j o r d r a w b a c k of fuzzy a p p l i c a t i o n .

Table 2 Comparison of different approaches Approach

Proposed by

Advantages and limitations

Energetic approach

J.B. Kiszka et al. [21]

Lyapunov method

M. Sugeno and K. Tanaka [32]

Circle criteria

K.S. Ray and D.D. Majumder [29]

Phase plane

M. Braae and D.A. Rutherford [3]

Describing function approach

W.M. Kickert and E.H. Mamdani [19]

Popov approach

In this paper

It is basically a qualitative approach. The approach is problematic in that it is more intuition than mathematics It is a very general approach with the fuzzy controller being modeled as Sugeno model. It is only a sufficient condition for fuzzy control stability. If one of the subsystems is unstable, the closed-loop fuzzy control system may still be stable. It is difficult to find the common positive-definite matrix P as no effective algorithm exists for finding matrix P It is an extension of the classic circle criteria. The approach is restricted to a system that can be modeled as sector bound nonlinearity. Membership has to be symmetrical It is a very simple graphical approach. Inspecting the system trajectories will provide information on system stability or instability. It is restricted to systems with an order less than two Describing function approach is an approximate method for determining the stability of unforced nonlinear control systems, and analysis results over the years have proven that this approach is sometimes inaccurate or incorrect. It applies to systems with low degree of nonlinearity It is a very simple graphical method. Stability can be obtained by studying the Popov line. It is restricted to the system whose control process is known

A. Kandel et al. / Fuzzy Sets and Systems 105 (1999) 33-48

The examination presented here shows that applicable general techniques still do not exist for stability analysis of all fuzzy systems. The study and research of fuzzy system stability require two fundamental components. First is fuzzy modeling. All analyses of fuzzy system stability are based on the particular model of a fuzzy system, with the success of stability analysis depending largely on the modeling of the fuzzy controller. Second is nonlinear system stability theory. Due to the incompleteness of nonlinear stability theory, no powerful analytical method seems to exist for studying fuzzy system stability. Without proper solutions to these two problems, a comprehensive fuzzy system theory cannot be implemented. Fuzzy control system instability should be emphasized in future studies. Indeed, system instability is inseparable from the study of system stability. In many cases, knowledge of system instability can prove critical when stability analysis is unavailable. Finally, the comparison of different approaches is given (Table 2).

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