Abstract: A new method to design a decentralized Fuzzy Adaptive Controller (FAC) for a class of large scale nonaffine nonlinear systems is proposed in this study. It is assumed that functions of the subsystems and their interactions are unknown. To design controller, the lyapunov function is proposed for the system and then unknown parameters of controller and system are derived based on the stability theory. The robustness against uncertainty and external disturbance, the boundedness of the estimation errors, the convergence of the output error to zero and the lyapunov stability of the closed loop system are guaranteed. To use the knowledge of the experts in FAC is another advantage of controller. Robust adaptive control has been used to avoid chattering in adaptation laws. An illustrative example is given to show the promising performance of the proposed method.
INTRODUCTION
In the recent years, controller design for Large Scale Systems (LSS) and effort to extend it has attracted much attention. Research in controllers of LSS is motivated by many emerging applications that employ novel actuation devices for active control of industrial automation, cooperating robotic systems, power systems and aerospace processes. Centralized controller for the LSS is usually impractical due to the requirement of a large amount of information exchanges between subsystems and the lack of computing capacity (Karimi et al., 2007).
The tunable structure of the FAC and using the knowledge of experts in the FAC are reasons to attract many researchers to developed appropriate controllers for nonlinear systems especially for LSS (Ioannou and Sun, 1996).
In the recent year, FAC has been fully studied as follow:
In the first case, the TS fuzzy systems have been used to model nonlinear systems and then TS based controllers have been designed with guaranteed stability (Feng, 2002; Fenga et al., 2002). To model affine nonlinear system and to design stable TS based controllers have been employed by Hsu et al. (2003). Designing of the sliding mode fuzzy adaptive controller for a class of multivariable TS fuzzy systems are presented by Cheng and Chien (2006). The nonaffine nonlinear function are first approximated by the TS fuzzy systems and then stable TS fuzzy controller and observer are designed for the obtained model (Goleaa et al., 2003; Park and Park, 2004). In these studies, modeling and controller has been designed simply, but the systems must be linearizable around some operating points.
In the second case, the linguistic fuzzy systems have been used to design controllers for nonlinear systems.
Ying-Guo and Hua-Guang (1998), Jagannathan (1998), Tong et al. (1999), Tong et al. (2000) and Zhang and Bien (2000) have considered linguistic fuzzy systems to design stable adaptive controller for affine systems based on feedback linearization and furthermore, Tong et al. (2000) and Zhang and Bien (2000) has considered that the zero dynamic is stable. Stable FAC based on sliding mode is designed for affine systems by Labiod et al. (2005). Designing of the FAC for affine chaotic systems are presented by Tang et al. (1999) and Chen et al. (1999). To design stable FAC and linear observer for class of affine nonlinear systems are presented by Ho et al. (2005), Zhang (2006), Tong et al. (2004) and Shaocheng et al. (2005). Fuzzy adaptive sliding mode controller is presented for class of affine nonlinear time delay systems by Yu (2004), Chiang (2005) and Jianga et al. (2005). The output feedback FAC for class of affine nonlinear MIMO systems is suggested by Yiqian et al. (2004). The main incompetency of these studies is those restricted conditions on their nonlinear functions.
Labiod and Guerra (2007) and Tong et al. (2004) are involved stable FAC for class of nonaffine nonlinear systems. The deficiencies of these studies are bad performance of the controller when the controller has not been adjusted. Stable adaptive controller for class of linear LSS is proposed by Pagilla et al. (2007), Ioannou and Ponte (1988), Shi and Singh (1992) and Yousef and Simaan (1991). Chiang and Lu (2007) delt with designing FAC based on sliding mode for class of large scale affine nonlinear systems. (Zhang et al., 2002) presented decentralized sliding mode fuzzy adaptive tracking for a class of affine nonlinear systems in large scale systems. Wu (2002) designed FAC for a class of affine nonlinear time delayed systems. These studies have many restricted conditions on their nonlinear function.
FAC has been never applied to nonaffine nonlinear large scale systems. In this study, the stable decentralized robust adaptive controller has been designed based on fuzzy systems for a class of large scale nonaffine nonlinear systems. The controller is robust against uncertainties, external disturbances and approximation errors.
PROBLEM STATEMENT
Consider the following large scale nonaffine nonlinear system.
| (1) |
where,
The control objective is to design an adaptive fuzzy controller for system (1) such that the system output yi(t) follows a desired trajectory yd(t) while all signals in the closed-loop system remain bounded.
In this study, the following assumptions have been considered concerning the system (1) and the desired trajectory yd(t).
Assumption 1: Without loss of generality, it is assumed that the
nonzero function
| (2) |
where, fdm ∈ R is nonzero, known and constant.
Assumption 2: The desired trajectory and its time derivatives are all smooth and bounded.
Assumption 3: the interconnection term satisfies the following:
| (3) |
where, ξij is an unknown time varying parameters.
Assumption 4: The disturbance in the above equation is bounded by:
| (4) |
Define the tracking error vector as:
| (5) |
Where:
| (6) |
Taking the nith derivative of both sides of the Eq. 6, the following equation can be derived.
| (7) |
Use Eq. 5 to rewrite the above equation as:
| (8) |
where, Ai0 and bi are defined below:
| (9) |
Consider the vector
| (10) |
Let vi be defined as:
| (11) |
By adding and subtracting the term
| (12) |
Using assumption 1, Eq. 11 and the signal vi which is not explicitly dependent on the control input ui, the following inequality is satisfied:
| (13) |
Invoking the implicit function theorem, it is obvious that the nonlinear
algebraic equation fi (xi, ui)–vi
= 0 is locally soluble for the input ui for an arbitrary (xi,
vi). Thus, there exists some ideal controller
| (14) |
As a result of the mean value theorem, there exists a constant λ
in the range of f
<λ1 , such that the nonlinear function f
i
(xi, ui) can be expressed around
| (15) |
Using Eq. 15 and 14 to rewrite Eq. 12 as follow:
| (16) |
However, the implicit function theory only guarantees the existence of
the ideal controller
FUZZY SYSTEMS
Figure 1 shows the basic configuration of the fuzzy systems considered in this study. Here, fuzzy systems can be considered as a multi-input, single-output: x ∈ U ⊂ Rn→y ∈ V ⊂ R. Consider that a multi-output system can be separated into a group of single-output systems.
| Fig. 1: | Configuration of fuzzy system |
The fuzzifier performs a mapping from a crisp input vector x = [x1 ,x2 ….,xn]T to a fuzzy set, where the label of the fuzzy set are such as small, medium, large, etc.
The fuzzy rule base is consisted of a collection of fuzzy IF-THEN rules. Assume that there are M rules and the lth rule is:
| (17) |
where, x = [x1 ,x2 ….,xn]T and y are the crisp input and output of the fuzzy system, respectively. Alj and Bl are fuzzy membership function in Uj and V, respectively.
The fuzzy inference performs a mapping from fuzzy sets in U to fuzzy sets in V, based on the fuzzy IF-THEN rules in the fuzzy rule base.
The defuzzifier maps fuzzy sets in V to a crisp value in V. The configuration of Fig. 1 shows a general framework of fuzzy systems, because many different choices are allowed for each block in Fig. 1 and various combinations of these choices will construct different fuzzy systems (Wang, 1997). Here, the sum-product inference and the center-average defuzzifier are used for fuzzy system. Therefore, the fuzzy system output can be expressed as:
| (18) |
where,
The fuzzy systems in the form of Eq. 18 are proven by Wang and Mendel (1993) to be a universal approximator if their parameters are properly chosen.
Theorem 1: Suppose f(x) is a continuous function on a compact set U (Wang, 1997). Then, for any ∈>0, there exists a fuzzy system like Eq. 18 satisfying:
| (19) |
The output given by Eq. 18 can be rewritten in the following compact form:
| (20) |
where, θ = [y1 y2…yM] is a vector grouping all consequent parameters and is w(x) = [w1(x) w2(x)…wM(x)]T a set of fuzzy basis functions defined as:
| (21) |
The fuzzy system (Eq. 18) is assumed to be well defined so that
FUZZY ADAPTIVE CONTROLLER DESIGN
Here, it has been shown how to develop a fuzzy system to adaptively approximate the unknown ideal controller.
The ideal controller can be represented as:
| (22) |
where,
| (23) |
Denote the estimate of
| (24) |
In which uirob is defined below.
| (25) |
In Eq. 23,
| (26) |
Consider the following update laws.
| (27) |
where,
In following equation, λmin, λmax and svdmax(.) are the minimum, maximum eigenvalue and maximum singular value decomposition, respectively.
Lemma 1: The following inequality holds if
| (28) |
Proof: From assumption 1 and the lemma 1, it is obvious that:
| (29) |
This in turn leads to the following inequality.
| (30) |
After some algebraic manipulations, the following inequality is obtained.
| (31) |
Use Eq. 31 to have the following which completes the proof.
| (32) |
Lemma 2: based on lemma 1 and Eq. 10, the following inequality holds.
| (33) |
Proof: Using Eq. 10 and after some algebraic manipulations, the following inequality is obtained.
| (34) |
Using the Eq. 34, the following equation can be derived.
| (35) |
Use Eq. 29 and 35 to have the following which completes the proof.
| (36) |
Theorem 2: Consider the error dynamical system given in Eq. 26 for the large scale system (1) satisfying assumption 1, interconnection term satisfying assumption 3, the external disturbances satisfying assumption 4 and a desired trajectory satisfying assumption 2, then the controller structure given in Eq. 24, 25 with adaptation laws Eq. 27 makes the tracking error converge asymptotically to a neighborhood of origin and all signals in the closed loop system be bounded.
Proof: Consider the following lyapunov function.
| (37) |
The time derivative of the lyapunov function becomes.
| (38) |
Use Eq. 26, to rewrite above equation as:
| (39) |
Using assumption 1 yields and by assumptions 3 and 4, to rewrite Eq. 39 as follow:
| (40) |
Equation 40 can be rewritten as below:
| (41) |
Using Eq. 27, the above inequality rewrites as:
| (42) |
Use the lemma 1,
Remark 1: The term tanh(.) is a smooth approximation of the discontinuous term sign(.). The sign(.) function is not used in the study due to avoiding chattering in the response.
Remark 2: It is very important to select properly the controller parameters to gain a satisfactory performance. Here, at this stage, number of the rules and the input membership functions are obtained by trial and error.
Remark 3: To guarantee the boundedness of the parameters in the presence of the unavoidable approximation error, the proposed adaptive laws Eq. 27 is modified it by introducing a σ-modification term as follows:
| (43) |
SIMULATION RESULTS
Here, the proposed decentralized fuzzy model reference adaptive controller is applied to a two-inverted pendulum problem (Karimi et al., 2007) in which the pendulums are connected by a spring as shown in Fig. 2. Each pendulum may be positioned by a torque input ui applied by a servomotor and its base. It is assumed that the angular position of pendulum and its angular rate are available and can be used as the controller inputs. The pendulums dynamics are described by the following nonlinear equations.
| (44) |
where, y1, y2 are the angular displacements of the pendulums from vertical position. m1 = 2 kg, m2 = 2.5 kg are the pendulum end masses j1 = 0.5 kg, j2 = 0.62 kg are the moment of inertia, k = 100 N m-1 is spring constant, r = 0.5 m is the height of the pendulum, g = 9.81 m sec-2 shows the gravitational acceleration, l = 00.5 m is the natural length of spring, α1, α2 = 25 are the control input gains and b = 0.4 m presents distance between the pendulum hinges.
| Fig. 2: | Two inverted pendulum connected by a spring |
| Fig. 3: | Performance of the PID controller in first subsystem |
The desired value of the outputs are zero (yid = 0 for i = 1, 2). As discussed earlier, the following primary PI controller are obtained after some trials and errors.
| (45) |
Figure 3 and 4 show the outputs of the system where only the controller defined in Eq. 45 is applied to the system.
Obviously the primary controller by itself is not admissible. Now the proposed controller defined in Eq. 24, 25 has been applied to mentioned system. Initially the PID controller keeps the states of system xi1, xi2 in the range of [–1, 1], [–5, 5]. Let xi = [xi1, xi2]T, z =[xi1, xi2, vi]T and vi are defined over [–45, 45]. For each fuzzy system input, it is defined 6 membership functions over the defined sets. Consider that all of the membership functions are defined by the Gaussian function:
where, c is center of the membership function and δ is its variance.
| Fig. 4: | Performance of the PID controller in second subsystem |
| Fig. 5: | Performance of the proposed controller in first subsystem |
| Fig. 6: | Performance of the proposed controller in second subsystem |
It has been assumed that the initial value of θi1(0),
θi2(0), uir(0), uicom(0) and
| Fig. 7: | Control input u1 |
| Fig. 8: | Control input u2 |
The parameters fdm, fmin and the vector ki = [ki1, ki2,…,ki,ni]T have been chosen so that the lemma 2 holds.
As shown in Fig. 3-6, it is obvious that the performance of the proposed controller is promising. Based on these simulation results, the controller can stabilized the closed loop system. It can decrease error estimation and disturbances effect in the output of the subsystems. Figure 7 and 8 show the total input of each subsystem. It is shown that ripple in the input controller can decrease error estimation and disturbances attenuation.
CONCLUSION
A decentralized fuzzy model reference adaptive output tracking controller is proposed for a class of large scale nonaffine nonlinear systems in this study. Fuzzy systems used to approximate the knowledge of the experts in the controller design procedure. It has been shown that the derived adaptation laws guaranty the Lyapunov’s stability of closed-loop system. Asymptotic convergence of the tracking error to zero is guarantied. Robustness against external disturbances and approximation errors, relaxing the conditions and using knowledge of experts are the merits of the proposed controller.