Stability Analysis and Design of a Class of Fuzzy Control Systems

Copyright © IFAC System Structure and Control, Bucharest, Romania, 1997 STABILITY ANALYSIS AND DESIGN OF A CLASS OF FUZZY CONTROL SYSTEMS Simona Doboli Radu-Emil Precup Department ofAutomation, "Politehnica" University ofTimisoara Address: Bd. V Parvan no.2, RO-J900 Timisoara, Romania E-mail: [email protected]. [email protected] Abstract: The paper presents a design method of fuzzy controllers (FCs) derived from a stability analysis method (Aracil, et af. , 1989). The stability conditions are expressed by a number of indices which verifies the character of the equilibrium point, its uniqueness and also measures the relative degree of stability. The method is applied to the development of a fuzzy control system meant for the position control of an electrohydraulic servosystem. Copyright © 1998 1FAC Keywords: fuzzy control, stability, nonlinear systems, electro-hydraulic systems, position control 1. INTRODUCTION higher order systems that can be reduced to second order systems (see for example Lohmann, 1994). The method is based on a linearized approximate model of the nonlinear control system in the vicinity of its equilibrium points. The stability conditions are expressed by a number of indices that indicate the character of the equilibrium point. To assure the global stability of the system, another index verifies the uniqueness of the equilibrium point. In (Doboli and Precup, 1996), a new way of applying this method to fuzzy control systems was presented. Also, a new index is introduced for second order systems to measure the relative degree of stability (Doboli and Precup, 1997). The development of fuzzy control systems is usually performed by heuristically means due to the lack of general design methods applicable to a large category of systems. A major problem which follows from the heuristic ally way of designing FCs is the analysis of the structural properties of the control system such as stability, controllability and robustness. From this results the necessity of design methods that ensure the global stability of the fuzzy control system in the development phase. Several approaches have been used for the stability analysis of fuzzy control systems such as: the use of classical stability analysis methods (SAMs) for nonlinear systems (hyperstability theory (Opitz, 1993), Lyapunov theory (Driankov, et al., 1993), Popov's criterion (Opitz, 1993», SAMs for linear systems applied to a linearized model of the nonlinear system (Aracil, et af. , 1989), geometrical methods (Driankov, et al., 1993). The design method is based on the selected SAM. The analytical function of the FC is approximated by expressing membership functions (MFs) with exponential components (Gartner and Astolfi, 1995). Two inequalities result between the parameters of the FC and desired values of the indices defmed by SAM. The pole position can be imposed, so that not only the stable character of the control system in the equilibrium point is ensured, but also a desirable dynamic behaviour of the control system . The SAM used here belongs to the second category and was presented by (Aracil, et al., 1989). Among its advantages are its simplicity, and applicability to a large class of non linear control systems. The only constraint imposed on the controlled plant inputoutput map function is continuity and partially differentiable character (Aracil, et al., 1989). It can be easily applied to SI SO, up to second order systems or The case study includes the development of a fuzzy state controller meant for the position control of an unstable second order electrohydraulic servosystem. Design inequalities are used and the stability of the resulted control system is verified by means of 333 stability of the equilibrium point at the origin is assured. simulation. The paper includes a short presentation of the SAM and of the design method, followed by a case study, interpretations of the results and conclusions. The closed loop system will be globally stable if the origin remains the only stable equilibrium point (Aracil, et al. , 1989). It can be shown by geometrical approach that another solution of the equation (2) exists only when the controller vector field (h' <p(i) compensates the plant component (f(A» (Aracil, et al., 1989). This is possible to happen only in regions where both vectors have the same direction. The subspace where this condition is fulfilled is defined by relation (7): 2. ST ABILITY ANALYSIS METHOD The dynamics of the plant is described by the following state equation: i = f(~) +~ .U (I) where x=[xI' x2, '''' xn]T is the state space vector, 11 a [n, I] dimensional vector, u=q,(i) the control signal expressed through the nonlinear function <pm, i the input vector of the form i=T[w; K] , where T is a constant matrix of dimension [n;n+ 1], not necessarily to be fixed, w the reference input, and f(K) the process function. The only restriction imposed on the f(K) is the continuous and partially differentiable character (Aracil, et at., 1989). The relations in the sequel will be particularized for second order systems. (7) h, b2 where b l and b2 are the components of 11. So, a third stability index is defined on the subspace given by (7): 12=min 1f(!J +lzu l (8) with x EA , where A is the subspace (7) with the exception of a small region around origin (Aracil, et aI. , 1989) and IKI means the Euclidean norm . The magnitude of all three indices indicates the stability margin of the control system . The equilibrium points are found as part of the solutions of the equation: dK/dt=O. (2) It can be assumed that the ongm is among the solutions, which is equivalent to f(Q)=f l (Q)=f2(Q), For second order systems a fourth stability index that measures the relative degree of stability can be introduced. The first two already defined indices I I and 1' 1, ensure only the position of the closed loop system eigenvalues in the negative real part of the complex plane. But, it is known that the distance of the complex eigenvalues to the imaginary axis is related to the damping and overshoot of the system. The last one may be too large to be considered acceptable when the distance is too small (Ogata, 1990). So, by computing the tangent of the angle 8 (Fig. I) and by imposing a maximum value for this angle one can verify that the overshoot and damping of the control system are within certain limits. lms where fl and f2 are the two components of f(A) ' This is not a constraint, because any different solution can be translated into ongm by a coordinate transformation (Voicu, 1986). The stability conditions for the ongm equilibrium point are derived from the stability theorem (Voicu, 1986) applied to a linearized approximation of the control system function. Based on a Lyapunov criteria, if the equilibrium point of the linearized system is asymptotically stable, then it will be also asymptotically stable for the non linear system (Aracil, et al. , 1989). Res The Jacobian matrix of the control system can be expressed as: Fig. I . Interpretation of the eigenvalues position (3) 8F21 8x 1 For example, a second order linear system with 8=75° has an overshoot of approximate 43%, and one with 8=60°, a 16% overshoot. It exists a proportionally dependence between the values of 8 (tan(8» and the overshoot (Ogata, 1990). The maximum admissible overshoot/angle depends on each system. The fourth index can be defined in the following manner: 8F2 x · 0 8x 2 x ·0 where F(K)=f(A)+h'u=[F IW F2a)]T. The characteristic polynomial of J is: pes) = det(s·I - J) = s2_ tr (J) · s+det(J) (4) where tr(J)=all+a22, det(J)=alla22" a 12'a21> and all> a12, a21, a22 are the elements of J computed for XQ=O. f', The conditions for J to have negative real part eigenvalues are equivalent with the following relations (Aracil, et at., 1989): (5) 11= - (all+a22)= - tr(J»O (6) 11'= all 'a22"a12'a21= det(J»O For positive values of both indices {II> II'}, the = tan (9) = (J4 . r,-I~)/l (9) and makes sense only when: 121<41'1 (10) Contrary with the other three indices, a more stable character is obtained for smaller values of [" I' From condition (11): 334 I" 1<tan(8 max )=k (ll) and from relation (10), the following inequalities results: ( _1)'4k + 1 2 ' I'S/~<4 ' 1' (relation (3), (5) and (6» . Only the partial derivatives of the process function are known before the design of Fe. The desired interval of values for the partial derivatives of FC is determined from the system of inequations (relation {I 6» re-written in the following manner: 2 . Rem + m:5 - (b l . tPxl + b 2 . tP x2 ) S 2· ReM + m (12) By expressing I I and 11' in terms of input and output MF parameters, relation (11) represents a good design condition. p l :5 n l . tPxl + n 2 · tPx2 :5P2 (17) where b i are the components of the 12 vector (relation (I», ~xl and $x2 the partial derivatives ofFC function ~W computed in origin: n l = f2x2 . b l - f lx2 . b 2 m = f lxl + f2x2 3. THE DESIGN METHOD Contrary with the SAM which only verifies the stability of an already developed fuzzy control system, the proposed design method ensures the stable character of the resulted control system in the design phase. Now, the position of the closed loop poles is imposed in order to obtain a desired stable behaviour of the control system around the equilibrium point. n P2 2 2 2 2 = (18) ReM (I +tg(p) ) +r r = f lxl · f2x2 - f lx2 -f2xl and f ixj i,j= 1,2 are the partial derivatives of f(2l) computed in origin. By expressing the partial derivatives of FC in terms of its parameters, inequations given in relation (17) determine two conditions that ensure the stability of the control system at the equilibrium point. To compute the partial derivatives of $(2l), an approximation of the nonlinear function of the FC is necessary. Considering triangular shapes for the MFs of the FC inputs, singleton shapes for the MFs of the output, the SUM-PROD inference method, and a simplified algorithm for singletons as defuzzification technique, the FC output will have the following expression (Opitz, 1993): Ims Res Fig.2 The desired region of the closed loop poles An acceptable choice of the dominant closed loop poles is a pair of complex conjugated values of the following form: PI 2 = Re±j Im, (l3) where Re<O and Im>O are the values corresponding to the real and respectively the imaginary axis. The relation between real and imaginary parts is given by relation (l4): tg([3)=lm/IRel= - ImlRe (l4) From relations (l3) and (l4) it can be seen that only two parameters determine the pole position, for example Re and tg([3). " ' W) , [,~ I"U ",,]{k nu] (19) where N represents the number of the rules, <Xij represents the degree of fulfillment of rule antecedent: <Xij=/lj{i I )/ljCi 2), i 1> i2 are the ~C i~pts, and Ujj the singleton value of the output lIngUIstic terms (L Ts). For symmetric MFs the denominator from relation (19) is equal with 1 (Gartner and Astolfi, 1995). An analytical function of the FC is found by approximating the triangular and trapezoidal MFs with exponential functions (Gartner and Astolfi, 1995). The advantage offered by such an approximation is the continuous and differentiable function obtained, contrary with the original expression of FC function which is only continuous. The expressions corresponding to a symmetric triangle (/It-(x)), a left and right trapezoidal (/ll(x), /lr(x» MF shapes are given in relation (20): The connection between the position of the poles and the first two indices from SAM (I I and 1'1, relations (5) and 6» is as follows : Im = (J4 ' 1'1-I0I2 (l5) Re = -/ 112 By choosing a maximum and minimum value for the real part of the poles (denoted by ReM and Rem respectively), a desirable interval of values for I I and 1'1 related to the position of the poles can be written down: 2 Rem (1 + tg(P) 2) S l' I S Re~ = f lxl . b - f .b 2xl 2 l PI=Rem (l+tg(P) )+r The choice of pole position choice depends on the controlled plant. The area of the complex plane in which the poles are advisable to be found is on the negative real part region with angles between ([3) -600 and 600 degrees (Fig. 2, the shaded area): -2 · Rem S I1 S -2· ReM 2 Il~ (x) = 21 (1 + exp (kl . (x -/~ f)) III (x) = 1/ ( 1 + exp ( 2· kl . (16) (1 + tg(P) 2) x-c-(lI2»)) I Ilr (x) = 1/ ( 1 + exp ( -2· kl . On the other hand, I I and 1'1 can be expressed also in terms of partial derivatives of the plant component (fW) and of the FC function (~i) computed in origin (20) x-c+ (112»)) I where k I is a constant directly proportional with the sharpness of the MFs, and ct-, c, I have the meaning from Fig. 3: 335 conduct to a desired control system behaviour of the closed loop system and then the input MFs parameters will be determined from relation (21). Another alternative could be to perform an optimization of the FC function (relation (17) and (21 )), in order to minimize a cost function (Makkonen and Koivo, 1994), from which also the rule base parameters will be determined. Fig. 3 Auxiliary to relation (20). To exemplify the stability/design conditions from relation (17), it is considered a two input-one output FC, with 3 MFs for each input, and 5 singletons for the output. The number of FC parameters in this case is equal with 4 as shown in Fig. 4 {I\> 12, uI' u2} . With the presented design conditions only the stability of the control system at the equilibrium point is assured. The global stability will be assured only after the second index 12 is computed (relation (8) and shows the uniqueness of the equilibrium point. The index I" I was included in the design conditions at the moment when the position of the closed loop system poles was imposed according to the dynamic of the plant. 4. CASE STUDY The example deals with fuzzy control system meant for a second order, double integrator, electrohydraulic servosystem (ESS). The previously presented SAM was preferred because does not impose restrictions on the plant poles placement as other methods do. The system in question has two poles on the origin. -- u_ The ESS considered here has the following state space model [4]: FigA The input and output MFs The rule base contains 9 rules presented in Table 1, (22) Table 1: The rule base of FC i l \i2 NE ZE PO PO uII ul2 u\3 ZE u21 u22 u23 NE u31 u32 u33 where the constants have the values a=14 .05, b'=26.04 experimentally determined (Preitl and Onea, 1981). The controlled output is y, the position of the piston, and u is the control signal, given by Fe. It can be seen that the origin is an equilibrium point. The FC is a state feedback non linear controller with two inputs: il=xb i2= x2-w, (23) where w is the reference input, and one output, the control signal u. The block diagram of the control system is presented in Fig. 5: where the values for uij ij=I ..3 belong to the set {-u2, -ub 0, ub U2}. The only fixed value is that from the middle of the table which corresponds to the steadystate regime and therefore is equal to ZE. y=x~ Based on relations (19) and (20), the expressions of the partial derivatives of C/><D corresponding to state variables XI' x2, computed in origin are as follows : ~J\ k\ =2.k\ . (u\2- u32)/(l\ · e ) (21) Combining relations (17), (21) and the connection between vectors i and ~, two design conditions result, from which a number of parameters of FC can be determined. The rest of them must be selected before. One way to do this, is by choosing the rule base parameters Uij' such that the resulted rule base will Fig. 5. Block diagram of the control system. The input and output MFs have the shapes presented in Fig. 4. The expressions of the elements included in 336 relation (18) particularized for ESS are as follows: m = ° nl = ° n2 in the subspace (relation (30» , but outside a small region around the origin. This means that the minimum value of 12 depends only on the choice of the bounds of this region, and therefore cannot be properly defmed. In any case, 12 has defmitely a strictly positive value, and ensures the uniqueness of the origin equilibrium point. So, the global stability of the control system at the equilibrium point at origin is assured. The analysis of the stability of the resulted control system is made by means of numerical simulation. To verify the stability of the fuzzy system, the dynamic behaviour of the free system was simulated, when the system started from different, arbitrarily chosen initial states. The state trajectories are presented in Fig. 6 and show that the control system obtained with this method is stable. = -a ' b' 2 2 PI = Rem (I + tg(P) ) +r (24) P2 = ReM'- (I + tg(p)2) + r r = ° The partial derivatives of the plant function are zero, except for f2x )=a. By choosing the minimum and maximum real part of the pair of complex poles Rem= -I and ReM= -7, and the angle P=600, the numeric values of PI and P2 (relation 24) are: PI=16 P2=196 (25) From relations (17) and (25) the following inequalities result: 4::; -~il ::; 16::; -~ 14 i2 :; (26) 196 5.-------..---------. and ~i2 have the expressions from relation where ~il (21) . Based on relation (23), ~il =~xl and ~i2 = ~x2' It is and ~i2 must be negative. An obvious that both ~il example of a rule base that fulfils this condition is presented in Table 2: Table 2: The rule base ofFC i l \i2 NE ZE PO PO ZE NS NS ZE PB ZE NS NE PS PS ZE -5 L -________L-._ _ _ _ _ _- - - ' -5 0 5 Fig. 6 State trajectories of the free control system It can be observed that the state state trajectories are moving towards the equilibrium point from origin on the direction of the zero control signal right which corresponds to the zero diagonal from the rule base given in Table 2 . From relations (24), (26) and (21), the expressions of the partial derivatives ofFC function result as follows: u I ~ i l = -2· kl . --k-I II . e ~il The design conditions transform into (28): u u2 I 1,059::; - ::; 7,417 u2 = -2 · kl . --k-I (27) II . e 0,3308::; -::; 4,052 5. CONCLUSIONS The following remarks can de drawn from the design method and its application to the case study presented in this paper: (28) II 12 By means of simulation the folk-wing values for FC parameters {I), 12, U), u2} have resulted: 12 =3 u l =9 u2 =12(29) 11 =6 - the design method of a Fe for the presented class of nonlinear systems assures the stability of the control system at the equilibrium point; that also verify the condiiions (28). To ensure the global stability of the control system, index 12 must be computed (relations (7) and (8» . The auxiliary subspace defmed by relation (7) is: xl=O. (30) On the x2 axis the servosystem component is zero (f~=[O a'xd T), and the FCs component Or~(i) increases monotonically from the maximum negative to the maximum positive (see the line from the table rule where il=O). This means that the second stability index 12 (relation (8», has the minimum value, zero, only in origin: 12=minI'~(D· (31) - the method offers a good alternative for design of fuzzy control systems, especially for its simplicity; - it does not impose constraints on the position of plant poles, as other methods do; the only constraint is related to the continuous and partially differentiable character of the plant component f(20 from the state space model (I); - it can be applied to higher order systems; the only problem may appear in the difficult computation of partial derivatives of the FC and in the expression of 1I' But, as stated before, 12 is defmed for points included 337 6. REFERENCES Aracil, J., A. Ollero, A. Garcia-Cerezo (1989). 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