Hybrid Adaptive Bionic Fuzzy Control Method

The bionic fuzzy system embeds the biologically active adaptive strategy into the traditional T-S fuzzy system, which increases the active adaptability. On the basis of the researches, an identificationmodel is added to the system, and a hybrid bionic adaptive fuzzy control method is proposed in this paper, which makes the system have biological adaptability and strong anti-interference ability. The adaptive law contains two items: the first one is the general term for adjusting system parameters by using the current state and the second one is a compensation item for the adjustment of system parameters based on the development trend. Lyapunov synthesis method is used to analyze the stability and convergence of system. The design method of fuzzy controller, adaptive laws, and parameter constraints are given. Finally, the effectiveness of the method is verified by simulation of inverted pendulummodel.


Introduction
Due to the high complexity of dynamic environment and system structure, the research of nonlinear systems is difficult to achieve accurate description.At the same time, higher requirements on the stability and effectiveness of the controlled system needed to meet certain performance index.By combining the adaptive control strategy and fuzzy control method, the stability and anti-interference of the system can be improved due to the active adaptability of the organism [1,2].The niche of individual organism is introduced into the design of fuzzy systems, and adaptive behavior and individual feedback are used to establish the fuzzy control model based on ecological niche [3].The design method of genetic algorithm and double nested fuzzy control method are given to make optimized controllable niche structure.A tracking feedback control method based on optimum niche is proposed, which is applied in the intelligent greenhouse system to realize the fuzzy control design for optimal ecological system [4].In [5,6], the self-organization and selflearning of niche are added to the fuzzy system, and a new fuzzy control method based on niche model is given.At the same time, the universal approximation of system was demonstrated, which achieved good results by using the system function as approximation.In [7], the T-S adaptive fuzzy control method is established by using the niche proximity with normal distribution characteristics as the consequent of fuzzy rules.The research [8] gives  ∞ tracking performance designs in both indirect and direct adaptive fuzzy systems.
On the basis of the study, in this paper, combining the identification model [9,10] with bionic fuzzy system and mixing the model error with tracking error, the hybrid bionic fuzzy control method is built.Stability and convergence analyzed by Lyapunov synthesis method [11][12][13], adaptive law, and constraints of the system parameters are given.The two items of bionic adaptive law, respectively, represent the current state and development trend of systems, which makes the system have good active adaptability.Finally, the strong anti-interference and better stability of highly complexity nonlinear system are verified by the simulation [14][15][16][17][18].
A hybrid adaptive controller is constructed as follows [12]: is equivalent controller,   is output controller,   is supervisory controller,   is adaptive compensation controller, and  ∈ [0, 1] is weighted factor.

Niche Ecostate-Ecorole Theory Function.
Considering the ecosystem with  biological unit, the size of the niche of the  biological unit is defined as Using the center-average defuzzifier, product inference engine, singleton fuzzifier, and Gauss membership function, Adaptive fuzzy system based on biological adaptive strategy is as follows: where Similarly, where  1 ,  1 ,  1 are on behalf of the current state of the biological individuals and  2 ,  2 ,  2 represent development trend of the biological individuals in the future.

Bionic Hybrid Controller
A nonlinear system is introduced in parallel with system (1): Define model error as From system 3 and system (1), it can be obtained that The model error of the original system and the identification model combined as replace f(), ĝ(), and   (); the minimum approximation error is Tracking error can be written as Following design   satisfies V ≤ 0, and design supervision controller as Choose the compensation controller as 0 is positive number; then the fuzzy controller of the closed loop system is If  ≤ , the supervisory controller is 0; if the system tends to separate,  ≥ , using   to ensure  ≤ .

The Design of Adaptive Laws
Adjusting optimal parameters of hybrid fuzzy system based on Bionic  * 1 , is a positive definite matrix and satisfies the Lyapunov equation, and  is an arbitrary positive definite matrix.
The bionic adaptive laws of the identification system are as follows: where  1 ,  1 ,  1 represent the current state and  2 ,  2 ,  2 represent the development trend of systems, which makes the systems have good active adaptability.

Analysis of System Stability and Convergence
It will affect the stability when the system is disturbed.In this paper, a hybrid adaptive fuzzy controller is designed, and the adaptive law is given.The following analysis is conducted for the performance of fuzzy controller.To ensure which satisfies the projection operator Otherwise,  3), ( 8), (15), and (16).The adaptive law is given by ( 21), and adaptive fuzzy control system must have the following characteristics: ( (2) The boundary of tracking error and the model error is given by the minimum approximation error set  = (  +   )  (); then , and ‖  2 ‖ ≤  2 have been proved in Theorem 1, and the following is to prove that || ≤   .
While designing the controller   , set  < ,  is a given constant, set  min is the minimum eigenvalue of , and due to  = (1/2)  , it can be obtained that Proof.Combine the adaptive laws (21), ( 22), ( 24), (25), and (26) into the following Lyapunov function: Then due to that, and there is so it can be obtained that

Simulation
Example 1.The inverted pendulum problem is studied by the hybrid adaptive fuzzy control method:  The membership function is Set the initial conditions (0) = (−/9, /12)  , the original system, and control system with internal disturbance (system state input  is substituted by (1 +  1 ),  1 = 0.5) and external disturbance ( 2 = 0.01).
From Figures 1-3, when the system is disturbed, the hybrid bionic adaptive controller proposed in this paper makes the system have more anti-interference and good convergence.In addition, this paper makes a comparison when the output trajectory has both internal and external disturbances; it can be seen that the fuzzy system has achieved good results.From Figure 4(a), when   = 1, the system has good convergence and biological adaptability than system controlled by traditional method.From Figure 4   by comparing the output trajectories, it can be obtained that the bionic hybrid fuzzy control has better result.
From Figure 5, to the uncertain system with disturbance, the hybrid bionic adaptive controller proposed makes the system have more anti-interference and good convergence.From Figure 6, when   = 1, the system has good Mathematical Problems in convergence and biological adaptability than system controlled by traditional method.

Conclusion
In this paper, the hybrid fuzzy control method combined model error with tracking error to design the adaptive laws.Compared with traditional method, it has faster convergence rate.In addition, the biological characteristics are embedded in the bionic fuzzy system, which improved the condition that traditional adaptive law just considers the current state of system.The method proposed in this paper makes system have advantages of anti-interference and strong self-adaptability.

Figure 3 :
Figure 3:  1 and  2 with both internal and external disturbances.

Figure 4 x
Figure 4

x 1 Figure 6 :
Figure 6:  1 with both internal and external disturbances (when   chooses different values).
indicates the state of the  biological unit,   is the dimensional conversion coefficient, and   indicates the rate of change of the  biological unit,  = 1, 2, . . ., .If the biological individual has a good state and good development trend, then the niche of the biological individual is relatively large,   ∈ [0, 1].When the niche   of the  biological unit is greater, it indicates the greater niche role relatively in the biological unit in the system in study.