Modeling and Control Based on Generalized Fuzzy Hyperbolic Model

Transcription

1 5 American Control Conference June 8-5. Portland OR USA WeC7. Modeling and Control Based on Generalized Fuzzy Hyperbolic Model Mingjun Zhang and Huaguang Zhang Abstract In this paper a novel generalized fuzzy hyperbolic model (GFHM) is proposed. First the definition of the GFHM is proposed and the approximation capability of the GFHM is discussed. The GFHM is proved to be an universal approximator by Stone-Weierstrass theorem. Further this fuzzy model is used as identifier for nonlinear dynamic systems and the back-propagation training algorithm is given. Finally the adaptive fuzzy control scheme based on the GFHM is presented which can guarantee that the closed-loop system is globally asymptotically stable. The simulation results show the applicability of the modeling scheme and the effectiveness of the proposed adaptive control scheme. I. INTRODUCTION For many real systems which are highly complex and inherently nonlinear conventional modeling approaches often cannot be applied whereas the fuzzy approach might be a viable alternative. In [] a fuzzy basis function network is used to approximate an unknown nonlinear function. The disadvantage of this fuzzy model is that some important dynamical behavior of the system cannot be represented. On the other hand a closed-loop system in the T-S fuzzy model was presented in [] [3]. The control law must be expressed by a T-S fuzzy model which is hardly used to other kinds of nonlinear systems. Also there are some closely related neural-fuzzy control approaches [4] [5] [6]. Recently Zhang and Quan proposed a new fuzzy hyperbolic model (FHM) in [7]. The fuzzy hyperbolic model is a nonlinear model that is very suitable for expressing nonlinear dynamic properties. However due to the structural characteristic of the FHM it cannot approximate some nonlinear continuous functions to arbitrary accuracies that is it is not an universal approximator. In this paper we extend the result of [7] and propose a generalized fuzzy hyperbolic model (GFHM) that can be shown to be an universal approximator. Moreover we use the GFHM equipped with the back-propagation training algorithm as identifier for nonlinear dynamic systems. Finally an adaptive control law based the GFHM is designed. This work was supported by the National Natural Science Foundation of P. R. China (6747 and 6353) and the Foundation for Doctoral Special Branch by the Ministry of Education of P. R. China (453) and Shenyang City Science Foundation (33--7). Mingjun Zhang is currently working toward the Ph.D degree in the School of Information Science and Engineering Northeastern University Shenyang 4 P. R. China. She is an associate professor with the School of Information Science and Engineering Beihua University Jilin City Jilin 3 P. R. China. zhang-mj966@63.com Huaguang Zhang is with School of Information Science and Engineering Northeastern University Shenyang Liaoning 4 P. R. China. hgzhang@ieee.org II. THE GENERALIZED FUZZY HYPERBOLIC MODEL In this section we first define generalized fuzzy hyperbolic model (GFHM) and then analyze its approximation capabilities. A. The Definition of the GFHM In [7] the membership function of the FHM P x and N x is defined as: µ Px (x) e (x k) µ Nx (x) e (x + k) () where k >.We can see that only two fuzzy sets are used to represent the input variable and that the fuzzy sets cannot cover the whole input space. The model cannot be an universal approximator. Now we transform the input variable x as follows: x i x d i () where i...w(w is a positive integer) and d i is a constant. We call the input variables after transformation x i x d i (i...w) generalized input variables. We can see that after the linear transformation of x the fuzzy sets may cover the whole input space if w is large enough. Before defining the GFHM we first give the definition of fuzzy rule base for inferring the GFHM. Definition : Given a plant with n input variables x ( x (t)... x n (t)) T and an output variable y we call the fuzzy rule base a type of a generalized fuzzy hyperbolic rule base if it satisfies the following conditions: ) Every fuzzy rule takes the following form: R l : IF ( x d ) is F x and... and ( x d w ) is F xw and ( x d ) is F x and... and ( x n d nwn ) is F xnwn THEN y l c F + + c Fw + c F + + c Fnwn (i m ) (3) where w z (z...n) are the numbers of transforming x z and d zj (z...n j...w z ) are the constants where x z is transformed F xzj are fuzzy sets of x z d zj which include P x (Positive) and N x (Negative) subsets. c Fzj are constants corresponding to F xzj. ) The constants c Fzj (z...n j...w z ) in the THEN part correspond to F xzj in the IF part that is if there is F xzj in the IF part c Fzj must appear in the THEN part. Otherwise c Fzj does not appear in the THEN part /5/$5. 5 AACC 79

2 3) There are s m fuzzy rules in the rule base where m n i w i that is all the possible P x and N x combinations of input variables in the IF part and all the linear combinations of constants in the THEN part. Theorem : For a multi-input single-output system y f( x x... x n ) given a generalized fuzzy hyperbolic rule base as in Definition define the generalized input variables as follows: x x d... x w x d w... x w+ x d x w+w x d w... x m x n d nwn where m n i w i w z (z...n) are the numbers to be transformed about x z d zj (z...n j...w z ) are the constants where x z is transformed. If we define the membership function of the generalized input variables P x and N x as () then we can derive the following model: y c Pi e kixi + c Ni e kixi e kixi e kixi p i + q i i i i P + Q tanh(k x x) (4) where p i (c Pi + c Ni )/ q i (c Pi c Ni )/ P m i p i Q [q...q m ] is constant vector tanh(k x x) is defined by tanh(k x x) [ tanh(k x ) tanh(k m x m ) ] T x [x...x m ] T K x diag[k...k m we call (4) the GFHM. Proof: By applying the product-inference rule the singleton fuzzifier and the center of gravity defuzzifier to the generalized fuzzy hyperbolic rule base we have: where y U/V U (c P + c P + + c Pm )µ P µ P µ Pm + +(c N + c N + + c Nm )µ N µ Nm V µ P µ P µ Pm + µ N µ P µ Pm µ N µ Nm Then y U/V c Pi µ Pi + c Ni µ Ni µ i Pi + µ Ni c Pi e (x i k xi ) + c Ni e (x i + k i ) i e (x i k i ) + e (x i + k i ) (5) From (5) we have c Pi e x i k i + k ix i + c Ni e x i k i k ix i y i e x i k i + k ix i + e x i k i k ix i c Pi e kixi + c Ni e kixi i let p i (c Pi + c Ni )/ q i (c Pi c Ni )/ e kixi e kixi y p i + q i i i P + Q tanh(k x x) which is the same as (4) Remark : The differences between the GFHM and the FHM are: ) The input variables of the GFHM are generalized input variables which are transformed from original input variables. ) After the linear transformation of x we may choose the number of fuzzy rules arbitrarily until the model approximates a nonlinear function at an arbitrary accuracy. 3) c Pi and c Ni are unnecessary to be opposite number to each other and we can choose them arbitrarily. From the above description we can see that the GFHM is the generalization of the FHM. When we take c Pi opposite to c Ni we can obtain the FHM. B. Approximation Capability of the GFHM In the following we will show that the GFHM can uniformly approximate any nonlinear function over U to any degree of accuracy if U is compact that is GFHM is an universal approximator. Lemma (Stone-Weierstrass theorem): [8] Let Z be a set of real continuous functions on a compact set U. If) Z is an algebra i.e. the set Z is closed under addition multiplication and scalar multiplication; ) Z separates points on U i.e. for every x y U x y there exists f Z such that f(x) f(y); and 3) Z vanishes at no point on U i.e. for each x U there exists f Z such that f(x) then the uniform closure of Z consists of all real continuous functions on U; i.e. (Z d ) is dense in (C[U]d ). Next we prove that the generalized fuzzy hyperbolic system is an universal approximator. 7

3 Theorem : For any given real continuous g(x) on the compact set U R n and arbitrary ε > there exists f(x) Y such that sup g(x) f(x) <ε (6) x U Proof: First we prove that (Yd ) is algebra. Let f f Y. We can write them as We have f (x) f (x) m i m i c P i Ni e k i x i x i + e k i x i (7) Ni e k i x i x i + e k i x i (8) f (x)+f (x) m c P i Ni e k i x i e k x i i + e k x i i i + m i +m z Ni e k i x i x i + e k i x i c Pz e kzxz + c Nz e kzxz e kzxz + e kzxz (9) It is easy to see that (9) has the same form as (4) that is f + f Y. In the same way we can get f (x) f (x) m c P i Ni e k i x i e k x i i + e k x i i i m Ni e k i x i i e k x i i + e k x i i m c P i e k i x i e k i Pi c N i x i e k i x i i i (e k x i i + e k x i i )(e k x i i + e k x i i ) + c N i e k i x i e k i Ni c N i e k i x i e k i x i (e k x i i + e k x i i )(e k x i i + e k x i i ) (c P i + c P i ) x ie k i x i+(c P i + c N i ) x ie k i x i (e k x i i + e k x i i )(e k x i i + e k x i i ) (c N i + c P i )e k i x ie k i x i+(c N i + c N i )e k i x ie k i x i (e k x i i + e k x i i )(e k x i i + e k x i i ) m m i i m m + i i m i i c P i N i e k i x i e k x i i + e k x i i + c P i N i e k i x i e k x i i + e k x i i +m c Pz e kzxz + c Nz e kzxz e kzxz + e kzxz z () where c P i c N i c P i c N i satisfy the following equations: c P i + c P i c P i c P i + c N i c P i c N i c N i + c P i c N i c N i + c N i c N i c N i. () It is easy to see that () is also in the same form as (4) hence f f Y. Finally for any constant c R wehave cf(x) c Pi e kixi + c Ni e kixi c i c P i e kixi + c N i e kixi i () which is again in the same form as (4) hence cf Y. Therefore (Yd ) is an algebra. Next we prove that (Yd ) separates points on U. We prove this by constructing a required f Y such that f(x ) f(y ) for arbitrarily given x y U with x y. Let x (x x x n) T y (yy yn) T.If x i y i choose input variable as x i x i x i + y i (3) ki x i y i (4) That is x i k i x i x x i + k i x i y. Then from (5) we can get n f(x c Pi e (x i x i ) + c Ni e (x i y i ) ) i e (x i x i ) + e (x i y i ) n c Pi + c Ni e (x i y i ) i +e (x i (5) y i ) n f(y c Pi e (y i x i ) + c Ni e (y i y i ) ) i e (y i x i ) + e (y i y i ) n c Pi e (y i x i ) + c Ni i e (y i (6) x i ) + Let c Pic Ni. We can derive f(x ) f(y ) n n i +e (x i e (x i y i ) y i ) i +e (x i y i ) n i e (x i y i ) + n i e (x i y i ) (7) 7

4 Since x y there must exist some i such that x i y i. Hence we have n i e (x i y i ) thus f(x ) f(y ). Therefore (Yd ) separates points on U. Finally we prove that (Yd ) vanishes at no points of U. By observing () and (4) we simply choose all c Pi > c Ni > (i...m); that is any f Y with c Pi > c Ni > ; serves as the required f. From(4) it is obvious that Y is a set of real continuous functions on U. The universal approximation theorem is therefore a direct consequence of the Stone-Weierstrass Theorem. Remark : There are some distinguishing characteristics in the GFHM: ) The GFHM is nonlinear model. Unlike the T-S fuzzy model which is the combination of local linear models the GFHM is a global nonlinear model. ) The GFHM is a fuzzy model that can easily be derived from known linguistic information. 3) Just like the T-S fuzzy model we can design a neural network model to identify the model parameters. III. FUZZY NONLINEAR IDENTIFIER Since the GFHM is both universal approximator and the neural network model we can use the GFHM equipped with the back-propagation training algorithm as identifier for unknown nonlinear function. Our task is to determine a GFHM ŷ(k +) in the form of (4) such that J [y(k +) ŷ(k + )] (8) is minimized. BP training algorithm of the GFHM is as follows: P (k +) P (k) η J ŷ(k +) ŷ P(k) P (k)+η[y(k +) ŷ(k + )] (9) q j (k +) q j (k) η J ŷ(k +) ŷ q j (k) q j (k)+η[y(k +) ŷ(k + )] tanh[k j (k)(x z d j (k))] (j m) () k j (k +)k j (k) η J ŷ(k +) ŷ k j (k) k j (k)+η[y(k +) ŷ(k + )]q j (k) [x z d j (k)] sec h [k j (k)(x z d j (k))] () d j (k +)d j (k) η J ŷ(k +) ŷ d j (k) d j (k) η[y(k +) ŷ(k + )]q j (k) k j (k)sech [k j (k)(x z d j (k))] () where η is the rate of training. Remark 3: In some sense the identification complexity of the GFHM is much less than of the T-S fuzzy model. When the number of input variables and fuzzy subsets are n and m i respectively the number of the unknown parameters of the T-S fuzzy model is n i m i + n i m i(n +) while the number of the unknown parameters of the GFHM is +(3/) n i m i. It is clear that as the number of input variable or (and) fuzzy subsets increases the increase of the number of the unknown parameters of the T-S fuzzy model is geometric while that of the GFHM is linear. Therefore when the number of input variables and fuzzy subsets in the GFHM are equal to those of in the T-S model the degree of approximate accuracy of the GFHM is higher than that of the T-S fuzzy model. IV. CONTROL SCHEME Having derived the GFHM for the plant the next step is to design a controller that achieves some control objective. Consider the discrete nonlinear system: y(k +) f(y(k) y(k n + )) + CU(k) f( X(k)) + CU(k) (3) where f( ) is an unknown nonlinear function X(k) [y(k) y(k n + )] T and u(k) R y(k) R are the input and output variable of the system respectively U(k) [u(k) u(k n + )] T C [c c c m ] c c i R(i m ). The aim of control is to determine a controller u(k) such that the output y(k +) of the closed-loop system follows the output y m (k +) of the following reference model: y m (k +)A X m (k)+br(k) (4) where A [a...a n ] a i R (i...n )b R Xm (k) [y m (k)...y m (k n + )] T r(k) R is the input of reference model. That is we want e(k + ) y m (k +) y(k +) to converge to zero as k goes to infinity. Since f( ) is unknown the controller cannot easily be implemented. To solve this problem we identify it by the GFHM. That is we replace the f( ) by ˆf( ) that is in the form of (4). f(k) ˆf( X(k)) f( X(k)) is defined as the identification error. Assumption : ) There exist positive constants M f M e such that f(k) M f e(k) M e ; ) There exists a positive constant M max such that M max M f /M e <. Theorem 3: Consider the plant described by the difference equation (3) assumption is satisfied if the adaptive controller is chosen as: u(k) c [ ˆf( X(k)) + A X m (k)+br(k) CŨ(k)+ρe(k) ] (5) where C [c c m ] Ũ(k) [u(k ) u(k m + )] T ρ is feedback gain which is chosen to satisfy 7

5 M max <ρ< M max + then the closed-loop system is globally asymptotically stable. Proof: From (3) and (5) we can get y(k +)f( X(k)) + c u(k)+ CŨ(k) f( X(k)) ˆf( X(k)) + A X m (k)+br(k)+ρe(k) (6) Combining (4) and (6) we have e(k +) ˆf( X(k)) f( X(k)) ρe(k) f(k) ρe(k) (7) We choose the Lyapunov function candidate then V (k) e (k +) e (k) V (k) e (k) (8) [ f (k) ρ f(k)e(k)+ρ e (k) e (k)] e (k)[ρ ρ f(k) e(k) Define M f(k)/e(k) then + f (k) e (k) ] V (k) e (k)(ρ ρm + M ) e (k)[ρ (M )][ρ (M + )] (9) From Assumption we have max(m) M max < min(m) M min M max Let M max < ρ < M max +weget(ρ M + )(ρ M ) < then V (k) < holds. By Lyapunov s stability theorem of discrete-time systems the closed-loop system is globally asymptotically stable. V. SIMULATION RESULTS Consider the discrete nonlinear system: y(k +)f[y(k)y(k )] + CU(k) (3) where C [.8 ] U(k) [ u(k) u(k ) ] T f[y(k)y(k )] y(k)y(k )[y(k)+.5] +y (k)+y (k ) is an unknown nonlinear function. The aim of control is to determine a controller u(k) such that the output y(k +) of the closed-loop system follows the reference model output y m (k +) given by: y m (k +).6y m (k)+.y m (k ) + r(k) A X m (k)+br(k) where A [.6. ] b Xm (k) [ ym (k) y m (k ) ] T r(k) sin(πk/5). To apply Theorem 3 to this system we design adaptive controller u(k) ˆf( X(k)) + A X m (k)+br(k) CŨ(k)+ρe(k) (3) where C [.8] Ũ(k) [u(k )] ρ.5 the output of identifier is as follows ŷ(k +) P + Q tanh(k x x) P + Q tanh[k x ( x z d)] (3) After off-line identification we can derive the following initial parameters of the GFHM: x x x x 3 d P.7966 d d d 3 y(k) d y(k) d y(k ) d Q [ q q q 3 ] [ ] k x diag [k k k 3 ]diag [ ] η..we chose the above initial parameters of the GFHM and the adaptive controller in the form of (3) and the result of simulation is shown in Fig. and Fig.. We see that the control was almost perfect k k (time steps) Fig.. The nonlinear function f( x(k)) (solid line) and the result of identifier ˆf( x(k)) (dotted line) and the identification error f(k) 73

6 k k (time steps) Fig.. The output y(k) (solid line) of the close-loop system and the reference trajectory y m(k) (triangle-downward line) and trajectory error e(k) VI. CONCLUSIONS In this paper a new generalized fuzzy hyperbolic model (GFHM) is proposed. First the GFHM is proved to be an universal approximator and can be as an identifier for nonlinear dynamic systems. We compare between the numbers of the unknown parameters of the T-S fuzzy model and that of the GFHM. As the number of input variables or (and) fuzzy subsets increases the increase of the number of the unknown parameters of the T-S fuzzy model is geometric while that of the GFHM is linear. The back-propagation training algorithm of the GFHM is given. Finally the adaptive fuzzy controller is presented which can guarantee that the closed-loop system is globally asymptotically stable. REFERENCES [] L.X. Wang and J.M. Mendel Fuzzy basis functions universal approximation and orthogonal least-squares learning IEEE Trans. Nerual Networks vol pp [] K. Tanaka and M. Sugeno Stability analysis and design of fuzzy control systems Fuzzy Sets Syst. vol pp [3] Kiriakos Kiriakidis et al. Quadratic stability analysis of the Takagi- Sugeno fuzzy model Fuzzy Sets Syst. vol pp.-4. [4] Jeffrey T. Spooner and Kevin M. Passino Stable adaptive control using fuzzy systems and neural networks IEEE Trans. Fuzzy Syst. vol pp [5] D. Driakov Advances in fuzzy control Physica Verlag New York 998 pp [6] H. G. Zhang and Z. Bien Adaptive fuzzy control of MIMO nonlinear systems Int. Journal of Fuzzy Sets and Systems vol.5 no. pp.9-4. [7] H. G. Zhang and Y. B. Quan Modeling identification and control of a class of non-linear system IEEE Trans. on Fuzzy Systems vol.9 pp [8] W.Rudin Principles of mathematical analysis New York.McGraw- Hill Inc