Theorem 1.10 If the disturbance system (1.30) is quadratic stable, the closed-loop dynamic fuzzy control system (1.26) is asymptotically stable.

Proof: The perturbed system (1.30) is quadratic stable according to definition (1.10):

For the closed-loop dynamic fuzzy control system (1.26), we have the Lyapunov function

where (P↼,P⇀) is a dynamic fuzzy symmetric positive definite matrix. As the system global representation matrix (C↼,C⇀) is an interval matrix, and (C↼,C⇀)=((C↼0,C⇀0)+(M↼,M⇀)(Σ↼,Σ⇀)(k)(N↼,N⇀),(Σ↼,Σ⇀)T(k)(Σ↼,Σ⇀)(k)≤(I↼,I⇀)n2, we have

and the closed loop dynamic fuzzy control system (1.26) is asymptotically stable.

Lemma 1.1 (A↼,A⇀),(D↼,D⇀),(E↼,E⇀)and(F↼,F⇀) are dynamic fuzzy positive definite matrices and, if (P↼,P⇀)−(ε↼,ε⇀)(D↼,D⇀)(D↼,D⇀)T>(0↼,0⇀) holds for any dynamic fuzzy symmetric matrix (P↼,P⇀) and scalar (ε↼,ε⇀)>(0↼,0⇀) then [35]

Theorem 1.11 If there exist a scalar (α↼,α⇀)>(0↼,0⇀) and dynamic fuzzy symmetric positive definite matrix (P↼,P⇀), the following holds:

and the closed-loop dynamic fuzzy control system (1.26) is asymptotically stable.

Proof: Let the Lyapunov function of the closed-loop dynamic fuzzy control system (1.26) be

where (P↼,P⇀) is a dynamic fuzzy symmetric positive definite matrix. Then,

ΔV[(x↼,x⇀)(k)]=V[(x↼,x⇀)(k+1)]−V[(x↼,x⇀)(k)]=(x↼,x⇀)T(k){[∑i=1m∑j=1m(w↼i,w⇀i)(k)(w↼j,w⇀j)(k)((A↼i,A⇀i)+(B↼i,B⇀i)(K↼j,K⇀j))]T×(P↼,P⇀)[∑i=1m∑j=1m(w↼i,w⇀i)(k)(w↼j,w⇀j)(k)(A↼i+A⇀i)+(B↼i,B↼i)(K↼j,K↼j))]−(P↼,P⇀)}×(x↼,x⇀)​(k)

according to (1.30), and

ΔV[(x↼,x⇀)(k)]=(x↼,x⇀)T(k){(C↼0,C⇀0)+(M↼,M⇀)(∑↼,∑⇀)(k)(N↼,N⇀))T(P↼,P⇀)                                    ((C↼0,C⇀0)+(M↼,M⇀)(∑↼,∑⇀)(k)(N↼,N⇀))−(P↼,P⇀)}(x↼,x⇀)(k).

Using Lemma 1.1, if there exists a scalar (α↼,α⇀)>(0↼,0⇀) and a dynamic fuzzy symmetric positive definite matrix (P↼,P⇀)satisfying(P↼,P⇀)−1−(α↼,α⇀)(M↼,M⇀)(M↼,M⇀T)>(0↼,0⇀), then

ΔV[(x↼,x⇀)(k)]≤(x↼,x⇀)T(k){(C↼0,C⇀0)T((P↼,P⇀)−1−(α↼,α⇀)(M↼,M⇀)(M↼,M⇀)T)−1                                    (C↼0,C⇀0)+(α↼,α⇀)−1(N↼,N⇀)T(N↼,N⇀)−(P↼,P⇀)}(x↼,x⇀)(k)<(0↼,0⇀).

According to the Lyapunov stability theorem, the closed-loop dynamic fuzzy system (1.26) is asymptotically stable.

1.4.3Design of dynamic fuzzy learning controller

Designing the dynamic fuzzy learning controller has the following considerations:

(1)Determine the input variables and output variables (i.e. the control variables) of the dynamic fuzzy learning controller;

(2)Design the control rules of the dynamic fuzzy learning controller;

(3)Choose the domain of the input variables and output variables of the dynamic fuzzy learning controller;

(4)Prepare the application of the dynamic fuzzy learning controller design algorithm;

(5)Choose a reasonable dynamic fuzzy learning controller design algorithm sampling time.

This section discusses the following.

1.4.3.1Input variables and output variables of dynamic fuzzy learning controller

Choosing which variables constitute the information for the dynamic fuzzy learning controller is a problem worthy of further study. Because this control system services the DFMLS, its input data come from the DFMLS itself.

In a manual control process, the amount of information that people can obtain is based on three factors:

(1)Error;

(2)The variation of error; and

(3)The rate of change in error.

Tab. 1.5: Dynamic fuzzy control rules table.

Therefore, there are three input variables to the dynamic fuzzy learning controller, namely, the error of the output data from the DFMLS, the variation of the error, and the change in the error variation. The output variables of the dynamic fuzzy learning controller generally determine the variation of the control variable.

1.4.3.2Design of control rules for dynamic fuzzy learning controller

The control rules of the dynamic fuzzy learning controller can be described in the following linguistic form:

(1)If (A↼,A⇀)then(B↼,B⇀);

(2)If (A↼,A⇀)then(B↼,B⇀);else(C↼,C⇀);

(3)If (A↼,A⇀)and(B↼,B⇀),then(C↼,C⇀).

These rules can be expressed as: if (A↼,A⇀)then if(B↼,B⇀)then(C↼,C⇀).

The corresponding control strategies of the operator that may be encountered in the operation process are summarized in Tab. 1.5.

The basic idea of establishing a dynamic fuzzy learning control rule table is as follows:

(1)When the error is negative (large), then the error has an increasing trend. To eliminate the existing negative large errors and prevent the error from becoming larger, the control volume change is positive (large);

(2)When the error is negative and the change in error is positive, the system itself has a tendency to reduce the error. Thus, to eliminate the error as soon as possible, a smaller control amount should be used. As can be seen from Tab. 1.5, when the error is negative (large) and the change in error is positive (small), the control variable is taken as positive (middle). When the error is positive (large) or positive (middle), the control volume should not be increased, otherwise an overshoot may produce a positive error. Hence, the change in the control quantity is set to the O level;

(3)When the error is negative (middle), the change in the control quantity should eliminate the error as soon as possible. Based on this principle, the variation of the control variable is the same as when the error is negative (large).

(4)When the error is negative (small), the system is close to the steady state. If the change in error is negative, select a positive (middle) control variable to suppress the change in error in the negative direction. If the error variation is positive, the system itself has a tendency to eliminate negative (small) errors; thus, select the control variable to be positive (small);

(5)The situation when the error is positive is similar to that when the error is negative with the corresponding change of symbol.

Therefore, the principle of selecting the control variable can be summarized as follows: when the error is large, select the control to eliminate the error as soon as possible; when the error is small, the choice of control should prevent an overshoot to ensure the stability of the system.

1.4.3.3Design algorithm of dynamic fuzzy learning controller

To design a stable dynamic fuzzy learning controller according to Theorem (1.28) [36], the design algorithm is as follows:

(1)For the closed-loop dynamic fuzzy control system (1.26), the object parameters (A↼,A⇀)then if(B↼,B⇀)then(C↼,C⇀). and their corresponding membership function are known, whereas the dynamic fuzzy learning controller parameter (A↼l,A⇀l),(B↼l,B⇀l) and its membership function (K↼l,K⇀l) are to be designed. Usually, the membership function (μ↼,μ⇀)(K↼l,K⇀l) is selected and the parameter (K↼l,K⇀l) is designed according to Theorem (1.28).

(2)The parameter (K¯l,K¯l) is selected so that the approximate linear subsystem is stable. According to (1.4), the approximate linear subsystem is

(x↼,x⇀)(k+1)=(w↼i,w⇀i)(w↼j,w⇀j)(Q↼ij,Q⇀ij)(x↼,x⇀)(k)      (Q↼ij,Q⇀ij)=(A↼i,A⇀i)+(B↼i,B⇀i)(K↼j,K⇀j).

(3)Find the dynamic fuzzy positive definite matrix (P↼i,P⇀i)such that(Q↼ij,Q⇀ij)T(P↼i,P⇀i)(Q↼ij,Q⇀ij)−(P↼i−Pi)<(0↼,0⇀). For a given i∗ ∈ {1, 2, . . ., m}, if there exists some (P↼i∗,P⇀i∗)such that (Q↼ij,Q⇀ij)(P↼i∗,P⇀i∗)(Q↼ij,Q⇀ij)−(P↼i∗,P⇀i∗)<(0↼,0⇀), then select (P↼,P⇀)=(P↼i,P⇀i); otherwise, return to (2), and redesign the parameter (K↼l,K⇀l)until(P↼,P⇀)=(P↼i,P⇀i).

1.4.4Simulation examples

Example 1.3 Consider an inverted pendulum system:

(x↼1,x⇀1)=(x↼2,x⇀2)(x↼2,x⇀2)=gsin(x↼1,x⇀1)−(aml(x↼2,x⇀2)2sin(2x↼1,2x⇀1))/2−acos(x↼1,x⇀1)u4l/3−aml cos2(x↼1,x⇀1),

where (x↼1,x⇀1) is the angle (in radians) between the pendulum and the horizontal direction, (x↼2,x⇀2) is the angular velocity (rad/s) of the pendulum, g = 9.8ms2 is the acceleration due to gravity, m is the mass of the pendulum, M is the mass of the trolley, 2 l is the pendulum length, and u is the horizontal force exerted on the trolley. The parameters are as follows: m = 2.0kg, M = 8.0kg, 2l = 1.0m, a = 1/(m + M).

The following dynamic fuzzy model is adopted [37]:

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where

(A↼1,A⇀1)=[(0↼,0⇀)(1↼,1⇀)(g↼4l/3−aml,g⇀4l/3−aml)(0↼,0⇀)](B↼1,B⇀1)=[(0↼,0⇀)(a↼4l/3−aml,a⇀4l/3−aml)],(A↼2,A⇀2)=[(0↼,0⇀)(1↼,1⇀)(2g↼π(4l/3−amlβ2),2g⇀π(4l/3−amlβ2))(0↼,0⇀)],(B↼2,B⇀2)=[(0↼,0⇀)(aβ←(4l/3−amlβ2),aβ←4l/3−aml)],     β=cos(88∘),

and the membership functions of the DFSs (G↼1,G⇀1)and(G↼2,G⇀2) are

[(μ↼1,μ⇀1)(x)]=((1−11+exp(−7(x−π4)))11+exp(−7(x−π4)),←,→),(μ↼2,μ⇀2)(x)=(1↼,1⇀)−(μ↼1,μ⇀1)(x).

We obtain the following interval representation of the coefficient matrix:

(A↼,A⇀)max=[(0↼,0⇀)(1↼,1⇀)(17.29↼,17.29⇀)(0↼,0⇀)],(A↼,A⇀)min][(0↼,0⇀)(1↼,1⇀)(9.3←6,9.3←6)(0↼,0⇀)],

(B↼,B⇀)max=[(0↼,0⇀)(0.18↼,0.18⇀)],(B↼,B⇀)min=[(0↼,0⇀)(0.005↼,0.005⇀)],and       (C↼,C⇀)max=[(0↼,0⇀)(1.0002↼,1.0002⇀)(−22.46↼,−22.46⇀)(−8.68↼,−8.68⇀)],        (C↼,C⇀)min=[(0↼,0⇀)(1↼,1⇀)(−30.08↼,−30.08⇀)(−10.72↼,−10.72⇀)].

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