The Hybrid Adaptive Control of T-S Fuzzy System Based on Niche

Abstract

Based on the niche characteristics, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Considering the close degree of Niche as the consequent of adaptive T-S fuzzy control system, the hybrid control law is designed by tracking, continuous supervisory, and adaptive compensation. Adaptive compensator is used in the controller to compensate the approximation error of fuzzy logic system and the effect of the external disturbance. The adaptive law of consequent parameters, which is achieved in this paper, embodies system adaptability as biological individual. It is proved that all signals in the closed-loop system are bounded and tracking error converges to zero by Lyapunov stability theory. The effectiveness of the approach is demonstrated by the simulation results.

1. Introduction

In recent years, fuzzy technique has gained rapid development in complex nonlinear plants [1]. Fuzzy logic offers human reasoning capabilities to capture nonlinearities and uncertainties, which cannot be described by precise mathematical models. Theoretical proofs of fuzzy models as universal approximators have been presented in the last decade [2, 3]. Fuzzy adaptive control methodologies have emerged in recent years as promising ways to approach nonlinear control problems. Fuzzy control, in particular, has had an impact in the control community because of the simple approach it provides to use heuristic control knowledge for nonlinear control problems. Recently, fuzzy systems have aroused a great resurgence of interest from part of the control community on the ground that they may be applied to model ill-defined complex systems. Direct and indirect schemes are two staple configurations for adaptive fuzzy controls. It has been established that indirect and direct controls (DAFC and IAFC) are able to incorporate plant knowledge and control knowledge, respectively, to yield stable and robust control systems [4]. The last two decades or so have witnessed a large quantity of results on indirect [5, 6] or direct [7–11] adaptive control (IAC/DAC) using fuzzy systems. However, the fuzzy adaptive controller proposed in [6, 7] ensure that the tracking error asymptotically convergence to zero (or a neighborhood of the origin) if the minimum approximation error is squared integrable along the state trajectory. And those algorithms are just confined to the linear and nonlinear systems whose state variables are assumed to be available for measurement. In many complicated cases, all state variables are not measurable such that output feedback or observer-based adaptive control techniques have to be applied. In [12–15], observer-based IAC and DAC algorithms are proposed for nonlinear systems, respectively. But [6, 12, 13] proposed adaptive control gain is only applicable to nonlinear systems with unknown constant gain. So a hybrid adaptive fuzzy control is needed.

In order to exploit the relative advantages of indirect and direct adaptive configurations at the same time, some researchers have developed several hybrid IAC and DAC algorithms [16–18] where a weighting factor, which can be adjusted by the tradeoff between knowledge of the plant and knowledge of the control, is adopted to sum together the control efforts from both the IAC and DAC. However, those schemes have their limitations. Above all, they take full-state feedback, which can be unsuitable for nonlinear systems without state variables available. Moreover, the conventional adaptive controller proposed in [19] combines indirect, direct, and variable structure methods; nonetheless its plants are assumed to be linear systems only. The author of [6] has developed a hybrid adaptive fuzzy control (HAFC) for nonlinear systems. However, input gains are required to be a constant 1 and control gain is an unknown constant. The HAFC algorithm of [20] can just be applied to robot manipulators with bounds estimation whereas that of [21] derives an unsupported HAFC scheme from a faulty Lyapunov derivative. On the other hand, a certain observer-based combined direct and indirect adaptive fuzzy neural controller is developed in [22]. Sensor fault may be in any form, even unbounded, which will make the system failure unavoidably. Paper [23] proposed a reliable observer-based controller, which makes the system work well no matter whether sensor faults occur or not.

Among various kinds of fuzzy models, there is a very important class of Takagi-Sugeno (T-S) fuzzy models [24] which have recently gained much popularity because of their special rule consequent structure and success in a functional approximation [6, 25]. In the recent two years, with the stability theory of T-S fuzzy system drawing lots of researchers attention, Yeh et al. [25] proposed stability analysis of interconnected and robustness design of time-delay fuzzy systems using fuzzy Lyapunov method. Moreover, T-S fuzzy model plays an important role in dealing with practical problems, such as oceanic structure [26], Vehicle occupant classification [27], and engineering systems [28], in which T-S fuzzy systems are applied to sensor fault estimation.

Upon all the fuzzy control method mentioned above, the consequent rules only contain the mathematical expressions without the practical significances. In this paper, the self-adaptability of the ecosystem is introduced into the fuzzy control method. Ecosystem is a complex large system, the interaction between individuals within the system, the interaction between the system and the outside world, made the internal dynamics of ecological system extremely complex. Biological evolution in complex ecosystems displays a strong nonlinearity [29], and increasing complexity is an indication of biological evolution, which makes it easier for organisms to adapt to environmental changes, and higher adaptability is the performance of a higher degree of evolution, that is, the so-called “survival of the fittest” [30]. Because of the redundancy and stability of ecological system, individuals acquire their own living ability and evolve to the advantage direction. This is why the ecological system could keep balance. The stability of the system is closely related with individuals’ ecological niche. Through such feature, Wang and Yi-min [31] discussed a method of fuzzy control based on niche model. The basic concept and methods of niche are introduced. Using the similarity measures, the capabilities of the living’s self-learning and self-organization are joined in fuzzy system, gaining the fuzzy system which has the meaning of niche. Considering the high-order nonlinearity, complexity, and biological individuals’ adaptability of ecological systems, we combine biological characteristics with adaptive fuzzy control method and raise a new hybrid T-S fuzzy adaptive control way, which contains great biological significance itself. The consequent of fuzzy system is parameter’s ecological niche close degree functions for T-S model. In condition of immeasurable biological individuals’ ecological factors, in [18, 30], the author designs controller and obtains adaptive law with consequent parameter which takes good tracking effect to changing error target function. Fuzzy T-S system of ecological niche reflects the degree of biological exploiting and usage and develops to its favorable way. During this course, the system moves forward by mutual changes, coordination with environment and themselves. This fuzzy adaptive control of niche reflects that the biology of individual organisms in the new environment has a strong adaptability, strong in the biological and practical significance, but also provided a reference value for the artificial cultivation of rare species. Therefore, fuzzy control background gets its biological meaning in this way.

2. Problem Formulation

Consider the 𝑛th-order nonlinear system of the following form: 𝑥^(𝑛)=𝑓𝑥,𝑥̇,…,𝑥^(𝑛−1)+𝑔𝑥,𝑥̇,…,𝑥^(𝑛−1)𝑢+𝑑(𝑡)𝑦=𝑥, (2.1) where 𝑓 and 𝑔 are unknown but bounded functions, 𝑢,𝑦∈𝑅 are the input, output of the system, respectively, and 𝑦 and 𝑑(𝑡) is the external bounded disturbance. 𝑋=(𝑥₁,…,𝑥_𝑛)^𝑇=(𝑥,𝑥̇,…,𝑥^(𝑛−1))^𝑇∈𝑅^𝑛 is the state vector where not all 𝑥_𝑖 are assumed to be available for measurement. In order for the system (2.1) to be controllable, it is required that 𝑔(𝑋)≠0 for 𝑋 in a certain controllability region 𝑈_𝑐⊂𝑅^𝑛. Since 𝑔(𝑋) is continuous, we assume 0<𝑔(𝑋)≤∞ for 𝑋∈𝑈_𝑐. The control objective is to design a combined controller, tune the correlation parameters of adaptive law, and force the system output 𝑦 to follow a bounded reference signal 𝑦_𝑚(𝑡) under the constraint that all signals involved must be bounded. To begin with, the signal vector 𝑦_𝑚(𝑡), the tracking error vector 𝑒, and estimation error vector 𝑒   are defined as 𝑦_𝑚=(𝑦_𝑚,𝑦_𝑚̇,…,𝑦̇_𝑚^(𝑛−1))^𝑇∈𝑅^𝑛,  𝑒=𝑦_𝑚−𝑋=(𝑒,𝑒,̇…,𝑒^(𝑛−1))^𝑇∈𝑅^𝑛,  𝑒=𝑦_𝑚−𝑋=(𝑒,𝑒,̇…,𝑒^(𝑛−1))^𝑇∈𝑅^𝑛, where 𝑋 and 𝑒 denote the estimates of 𝑋 and 𝑒, respectively. If the functions 𝑓(𝑋) and 𝑔(𝑋) are known and the system is free of external disturbance 𝑑(𝑡), then we can choose the controller 𝑢^∗ to cancel the nonlinearity and design controller. Let 𝑘₀=(𝑘₀₁,…,𝑘_0𝑛)^𝑇∈𝑅^𝑛 be chosen such that all roots of the polynomial 𝑠^𝑛+𝑘_0𝑛𝑠^𝑛−1+⋯+𝑘₀₁ are in the open left half-plane and control law of the certainty equivalent controller is obtained as 𝑢^∗=1𝑔(𝑋)−𝑓(𝑋)+𝑦_𝑚^(𝑛)+𝑘₀^𝑇𝑒. (2.2) Substituting (2.2) to (2.1), we get the main objective of the control is lim_𝑡→∞𝑒(𝑡)=0. However 𝑓(𝑋) and 𝑔(𝑋) are unknown, the ideal controller (2.2) cannot be implemented, and not all system states 𝑋 can be measured. So we have to design an observer to estimate the state vector 𝑋 in the following.

3. T-S Fuzzy System Based on Niche

3.1. Description of T-S Fuzzy System Based on Niche

Because 𝑓(𝑋) and 𝑔(𝑋) are unknown, T-S fuzzy systems are used to approximate them. The basic configuration of T-S [24] system is expressed as 𝑅^𝑙_𝑓∶if𝑥₁is𝐴^𝑙₁,…,𝑥_𝑛is𝐴^𝑙_𝑛,then𝑓_𝑙(𝑋)=32𝜆−𝜙^𝑙𝑓−𝜆𝜎^𝑙_𝑓,𝑅+𝜎_𝑙𝑔∶if𝑥₁is𝐺^𝑙₁,…,𝑥_𝑛is𝐺^𝑙_𝑛,then𝑔_𝑙(𝑋)=32𝜆−𝜙^𝑙𝑔−𝜆𝜎^𝑙_𝑔,𝑅+𝜎_𝑙𝑢∶if𝑥₁is𝑈^𝑙₁,…,𝑥_𝑛is𝑈^𝑙_𝑛,then𝑢_𝑙(𝑋)=32𝜆−𝜙^𝑙𝑢−𝜆𝜎^𝑙_𝑢).+𝜎(𝑙=1,…,𝑚 (3.1) Here, 𝐴^𝑙_𝑖,𝐺^𝑙_𝑖,𝑈^𝑙_𝑖 are fuzzy sets, 𝜆^𝑙=(𝑥₁,𝑥₂,…,𝑥_𝑛) represents the real ecological factor of ecologic niche, and 𝜆=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛) is the ideal ecological factor. Consequent indicates the difference between real state of ecologic niche and ideal one. we choose Gaussian membership function, which satisfies 𝑥^𝑙_𝑖−𝑥^𝑙_𝑗≫𝛿, where 𝑥^𝑙_𝑖,𝑥^𝑙_𝑗 are the centers and 𝜎 is the variance of the functions. Then difference could be showed by approach functions as 𝐻(𝐴,𝐵)=3/2−𝜙[(𝜆^𝑙_𝑓−𝜆)/(𝜎^𝑙_𝑓+𝜎)], where 𝜙(𝑥)=(1/√2𝜋)∫^𝑥_−∞𝑒^−𝑡2/2𝑑𝑡. From approach functions above, we could know consequent is a zero-order T-S mode. Here, we use 𝑎^𝑙₀ as 3/2−𝜙[(𝜆^𝑙_𝑓−𝜆)/(𝜎^𝑙_𝑓+𝜎)] for convenience. Let 𝑚 be the number of systems with central average defuzzifier, and product inference and singleton fuzzifier can be expressed as𝑓𝑋∣𝜃_𝑓=∑^𝑚_𝑙=1𝑎^𝑙₀∏^𝑛_𝑖=1exp−𝑥_𝑖−𝑥^𝑙_𝑖/𝜎^𝑙_𝑖²𝑋∑^𝑚_𝑙=1∏^𝑛_𝑖=1exp−𝑥_𝑖−𝑥^𝑙_𝑖/𝜎^𝑙_𝑖²=^𝑚_𝑙=1𝜉^𝑙(𝑋)𝜃^𝑙_𝑓=𝜉(𝑋)^𝑇𝜃_𝑓, (3.2) where 𝜉^𝑙(𝑋)=∏^𝑛_𝑖=1exp(−((𝑥_𝑖−𝑥^𝑙_𝑖)/𝜎^𝑙_𝑖)²)𝑋/∑^𝑚_𝑙=1∏^𝑛_𝑖=1exp(−((𝑥_𝑖−𝑥^𝑙_𝑖)/𝜎^𝑙_𝑖)²), 𝜃^𝑙_𝑓=(𝑎^𝑙₀,0,…,0)∈𝑅^1×(𝑛+1), 𝑋=[1,𝑥₁,…,𝑥_𝑛]^𝑇;𝜉(𝑋)=(𝜉¹(𝑋),…,𝜉^𝑚(𝑋))∈𝑅^1×𝑚, 𝜃_𝑓=^𝜃1𝑓⋮_𝜃𝑚𝑓∈𝑅^{𝑚×(𝑛+1)}; 𝑔(𝑋∣𝜃_𝑔)=𝜉(𝑋)^𝑇𝜃_𝑔 and 𝑢_𝐷(𝑋∣𝜃_𝐷)=𝜂(𝑋)^𝑇𝜃_𝐷 can be expressed similarly.

3.2. Niche T-S Model Parameter Optimization Method

Niche fuzzy T-S model based on parameter optimization of backpropagation algorithm known input and output data (𝑥^𝑝₀,𝑦^𝑝₀),𝑥^𝑝₀∈𝑈∈𝑅^𝑛,𝑦^𝑝₀∈𝑉∈𝑅, the task is to determine the form (3.2) Niche T-S model parameters, To fitting error 𝑒^𝑝=(1/2)[𝑓(𝑥^𝑝₀)−𝑦^𝑝₀]² the minimum. Assume 𝑀 is known, by adjusting 𝜆^𝑙_𝑓,𝜎^𝑙_𝑓,𝑥^𝑙_𝑖,𝜎^𝑙_𝑖, and let 𝑒^𝑝 be the minimum. To facilitate the discussion, use 𝑒,̂𝑓,𝑦 to represent 𝑒^𝑝,𝑓(𝑥^𝑝₀), and 𝑦^𝑝₀. 𝑏=∑^𝑚_𝑙=1𝑍^𝑙,𝑍^𝑙=∏^𝑛_𝑖=1exp(−((𝑥_𝑖−𝑥^𝑙_𝑖)/𝜎^𝑙_𝑖)²), 𝑎=∑^𝑚_𝑙=1(𝑎^𝑙₀𝑍^𝑙).

Using gradient descent to determines the parameters [6]    𝜆^𝑙_𝑓𝑞+1=𝜆^𝑙_𝑓𝑞−𝛾𝜕𝑒̂𝜕𝜆^𝑙_𝑓𝑞,𝑙=1,2,…,𝑚,𝑞=0,1,2,…,𝛾 (3.3) determines the steps.

Then 𝜕𝑒̂𝜕𝜆^𝑙_𝑓=𝜕𝑒̂𝜕𝑓𝜕𝑓𝜕𝑎𝜕𝑎𝜕𝑎^𝑙₀𝜕𝑎^𝑙₀𝜕𝜆^𝑙_𝑓=⋅1𝑓−𝑦√𝑒2𝜋^{−((𝜆𝑙𝑓−𝜆)/(𝜎𝑙𝑓+𝜎))2/2}⋅1𝜎^𝑙_𝑓⋅1+𝜎𝑏⋅Z^𝑙𝑋. (3.4) Then,𝜆^𝑙_𝑓𝑞+1=𝜆^𝑙_𝑓𝑞⋅1−𝛾𝑓−𝑦√𝑒2𝜋^{−((𝜆𝑙𝑓(𝑞)−𝜆)/(𝜎𝑙𝑓(𝑞)+𝜎))2/2}⋅1𝜎^𝑙_𝑓𝑞⋅1+𝜎𝑏⋅Z^𝑙𝑋. (3.5) Equally, 𝜎^𝑙_𝑓𝑞+1=𝜎^𝑙_𝑓𝑞𝑓⋅1−𝛾𝑦−√𝑒2𝜋^{−((𝜆𝑙𝑓(𝑞)−𝜆)/(𝜎𝑙𝑓(𝑞)+𝜎))2/2}⋅𝜆^𝑙_𝑓𝑞−𝜆𝜎^𝑙_𝑓𝑞+𝜎²⋅1𝑏⋅Z^𝑙𝑋 (3.6)adjustable parameters 𝑥^𝑙_𝑖,𝜎^𝑙_𝑖 with the same way:𝑥^𝑙_𝑖𝑞+1=𝑥^𝑙_𝑖𝑞−𝛾𝜕𝑒̂𝜕𝑥^𝑙_𝑖𝑞=𝑥^𝑙_𝑖𝑞𝑁−𝛾𝑓−𝑦𝑓−𝑏⋅2𝑥_𝑖−𝑥^𝑙_𝑖𝑞𝜎^𝑙_𝑖𝑞²𝑋𝜎^𝑙_𝑖𝑞+1=𝜎^𝑙_𝑖𝑞−𝛾𝜕𝑒̂𝜕𝜎^𝑙_𝑖𝑞=𝜎^𝑙_𝑖𝑞𝑁−𝛾𝑓−𝑦𝑓−𝑏⋅2𝑥_𝑖−𝑥^𝑙_𝑖𝑞²𝜎^𝑙_𝑖𝑞³𝑋, (3.7) where 𝑁=∑^𝑚_𝑙=1(3/2−𝜙((𝜆^𝑙_𝑓−𝜆)/(𝜎^𝑙_𝑓+𝜎))),𝜆^𝑙_𝑔,𝜎^𝑙_𝑔,𝜆^𝑙_𝑢 and 𝜎^𝑙_𝑢 can be optimized directly.

4. Hybrid Controller with Supervisory and Compensation Control Scheme

The overall control law is constructed as 𝑢=𝛼𝑢_𝐼𝑋+(1−𝛼)𝑢_𝐷𝑋∣𝜃+𝑢_𝑠+𝑢_𝑐, (4.1) where 𝑢_𝐼 is an indirect controller, 𝑢_𝐷 is the output of the T-S controller, 𝑢_𝑠 is the supervisory control to force the state within the constraint set, 𝛼∈[0,1] is a weighting factor, and 𝑢_𝑐 is the compensate controller of adaptive control. Since 𝑋 cannot be available and 𝑓(𝑋) and 𝑔(𝑋) are unknown, we replace the functions 𝑓(𝑋),𝑔(𝑋), and error vector  𝑒  by estimation functions 𝑓(𝑋),𝑔(𝑋), and 𝑒. The certainty equivalent controller can be rewritten as 𝑢^∗=1𝑔(𝑋)−𝑓(𝑋)+𝑦_𝑚^(𝑛)+𝑘₀^𝑇𝑒. (4.2) The indirect control law is written as  𝑢_𝐼=(1/𝑔(𝑋))[−𝑓(𝑋)+𝑦_𝑚^(𝑛)+𝑘₀^𝑇𝑒]. Applying (4.2), (4.1) to (2.1), the error dynamic equation is 𝑒̇=𝐴𝑒−𝐵𝑘^𝑇₀𝑒+𝐵𝛼𝑓𝑋−𝑓(𝑋)+𝑔𝑋−𝑔(𝑋)𝑢_𝐼+(1−𝛼)𝑔(𝑋)𝑢^∗−𝑢_𝐷+𝐵𝑔(𝑋)𝑢_𝑠+𝑢_𝑐−𝐵𝑑, (4.3)defining   𝑘_𝑐=(𝑘_𝑐𝑛,…,𝑘_𝑐1) as the observer vector, the observation errors are defined as 𝑒̃=𝑒−𝑒 from (4.2) and [24] and we get 𝑒̇̃=Λ𝑒̃+𝐵𝛼𝑓𝑋−𝑓(𝑋)+𝑔𝑋−𝑔(𝑋)𝑢_𝐼+(1−𝛼)𝑔(𝑋)𝑢^∗−𝑢_𝐷−𝐵𝑔(𝑋)𝑢_𝑠+𝑢_𝑐−𝐵𝑑, (4.4)where    Λ=𝐴−𝑘_𝑐𝐶^𝑇=−𝑘_𝑐𝑛100⋯00−𝑘_{𝑐(𝑛−1)}010⋯00⋮⋮⋮⋮⋱⋮⋮−𝑘_𝑐2000⋯01−𝑘_𝑐1,00⋮0110⋮00.000⋯00,𝐴=0100⋯000010⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯010000⋯00𝐵=,𝐶= (4.5) Let 𝐴=𝐴−𝐵𝑘^𝑇₀ be strictly Hurwitz matrix; so there exists a positive definite symmetric 𝑛×𝑛  matrix 𝑃, 𝑃 which satisfies the Lyapunov equation Λ^𝑇𝑃+𝑃Λ=−𝑄, 𝐴^𝑇𝑃+𝑃𝐴=−𝑄,   where 𝑄 and 𝑄 are arbitrary  𝑛×𝑛 definite symmetric matrix. Let 𝑉_𝑒=(1/2)𝑒^𝑇𝑃𝑒. Since 𝑄 and 𝑘_𝑐 are designer by the designer, we can choose 𝑄 and 𝑘_𝑐, such that 𝑉̇_𝑒≤0. Hence, 𝑉_𝑒 is a bounded function and there exists a constant value 𝑉_𝑒, such that 𝑉_𝑒≤𝑉_𝑒.

5. Hybrid Adaptive Control of Niche

We will develop the hybrid adaptive control such that the closed-loop output 𝑦(𝑡) follows 𝑦_𝑚(𝑡). Let us replace 𝑓(𝑋),𝑔(𝑋), and 𝑢_𝐷(𝑋) by 𝑓(𝑋∣𝜃_𝑓),𝑔(𝑋∣𝜃_𝑔), and 𝑢_𝐷(𝑋∣𝜃_𝐷), respectively. Therefore, the error dynamics (4.3) can be rewritten as 𝑒̇̃=Λ𝑒̃+𝐵𝛼𝑓𝑋∣𝜃_𝑓−𝑓(𝑋)+𝑔𝑋∣𝜃_𝑔−𝑔(𝑋)𝑢_𝐼+(1−𝛼)𝑔(𝑋)𝑢^∗−𝑢_𝐷𝑋∣𝜃_𝐷+𝐵𝑔(𝑋)𝑢_𝑠+𝑢_𝑐−𝐵𝑑. (5.1) Let 𝑉_̃𝑒=(1/2)𝑒^𝑇̃𝑃𝑒̃ then using (5.1), we have 𝑉̇_̃𝑒≤−12𝑒^𝑇̃𝑄𝑒̃+𝑒^𝑇̃𝑃𝐵𝛼𝑓𝑋∣𝜃_𝑓+𝑓(𝑋)+𝑔𝑋∣𝜃_𝑔𝑢_𝐼+𝑔(𝑋)𝑢_𝐼+(1−𝛼)𝑔(𝑋)𝑢^∗+𝑔(𝑋)𝑢_𝐷𝑋∣𝜃_𝐷+𝑒^𝑇̃𝑃𝐵(1−𝛼)𝑔(𝑋)𝛼+𝑢_𝑐−𝑒^𝑇̃𝑃𝐵𝑔(𝑋)𝑢_𝑠. (5.2)

In order to design 𝑢_𝑠 so that 𝑉̇_̃𝑒≤0, we need the following assumption.

Assumption 5.1. We could find three functions as 𝑓^𝑢(𝑋),𝑔^𝑢(𝑋),𝑔_𝑙(𝑋) and get |𝑓(𝑋)|≤𝑓^𝑢(𝑋)≈𝑓^𝑢(𝑋) and 0<𝑔_𝑙(𝑋)≈𝑔_𝑙(𝑋)≤𝑔(𝑋)≤𝑔^𝑢(𝑋)≈𝑔^𝑢(𝑋), in which 𝑋∈𝑈_𝑐 and 𝑓^𝑢(𝑋)≈𝑓^𝑢(𝑋)<∞,𝑔^𝑢(𝑋)≈𝑔^𝑢(𝑋)<∞. This is due to the fact that we can choose 𝑘_𝑐 to let 𝑋≈𝑋. Also external disturbance is bounded. We design |𝑑(𝑡)|≤𝑑_𝑚.

From Assumption 5.1, we choose the supervisory control 𝑢_𝑠 as 𝑢_𝑠=𝑉_̃𝑒𝑉^𝑝sgn𝑒^𝑇̃𝛼𝑃𝐵𝑔_𝑙(𝑋)𝑓𝑋∣𝜃_𝑓+𝑓^𝑢𝑔𝑋∣𝜃_𝑔+𝑔^𝑢𝑢_𝐼+𝑓^𝑢(𝑋)+𝑦_𝑚^(𝑛)+𝑘^𝑇𝑒+𝑉_̃𝑒𝑉^𝑝sgn𝑒^𝑇̃𝑃𝐵(1−𝛼)𝑢_𝑑+1𝑔_𝑙(𝑋)𝑓^𝑢(𝑋)+𝑦_𝑚^(𝑛)+𝑘₀^𝑇𝑒+𝑔^𝑢𝑢_𝑐+𝑑_𝑚. (5.3) And we choose 𝑢_𝑐=𝑐_𝑜sgn(𝑒^𝑇̃𝑃𝐵), and 𝑐_𝑜 is a nonnegative constant. Considering the case 𝑉_̃𝑒>𝑉 and substitute (5.3) into (5.2), we obtain 𝑉̇_̃𝑒≤−(1/2)𝑒^𝑇̃𝑄𝑒̃≤0. Therefore, if the closed-loop system with the fuzzy controller  𝑢  as 𝑢=𝛼𝑔⁻¹𝑋∣𝜃_𝑔−𝑓𝑋∣𝜃_𝑓+𝑦_𝑚^(𝑛)+𝑘₀^𝑇𝑒+(1−𝛼)𝑢_𝐷𝑋∣𝜃_𝐷+𝑢_𝑠+𝑢_𝑐 (5.4)works well in the sense that the error is not too large, if 𝑉_̃𝑒≤𝑉, then the supervisory control 𝑢_𝑠 is zero. If the system tends to diverge, that is, 𝑉_̃𝑒>𝑉, then be 𝑢_𝑠 gins to force 𝑉_̃𝑒≤𝑉.

6. Design of Adaptive Law

In order to adjust the parameters in the fuzzy logic system, we have to derive adaptive laws. Hence, the optimal parameters estimates 𝜃^∗_𝑓,𝜃^∗_𝑔, and 𝜃^∗_𝐷 are defined as 𝜃^∗_𝑓=argmin_{𝜃𝑓∈Ω𝑓}sup_{𝑋∈Ω𝑋,𝑋∈Ω𝑋}𝑓𝑋∣𝜃_𝑓(𝑋,𝜃−𝑓)_∗𝑔=argmin_{𝜃𝑔∈Ω𝑔}sup_{𝑋∈Ω𝑋,𝑋∈Ω𝑋}𝑔𝑋∣𝜃_𝑔(𝑋,𝜃−𝑔)_∗𝐷=argmin_{𝜃𝐷∈Ω𝐷}sup_{𝑋∈Ω𝑋,𝑋∈Ω𝑋}𝑢^∗(𝑋)−𝑢𝑋∣𝜃_𝐷, (6.1) where Ω_𝑓,Ω_𝑔,Ω_𝐷,Ω_𝑋, and Ω_𝑋, are compact sets of suitable bounds on 𝜃_𝑓,𝜃_𝑔,𝜃_𝐷,𝑋, and 𝑋, respectively, and they are defined as Ω_𝑓=𝜃_𝑓∶𝜃_𝑓≤𝑀_𝑓,Ω_𝑔=𝜃_𝑔∶𝜃_𝑔≤𝑀_𝑔,Ω_𝐷=𝜃_𝐷∶𝜃_𝐷≤𝑀_𝐷,Ω_𝑋=𝑋∶𝑋≤𝑀_𝑋,Ω_𝑋=𝑋∶‖𝑋‖≤𝑀_𝑋, (6.2) where 𝑀_𝑓,𝑀_𝑔,𝑀_𝐷,𝑀_𝑋 and 𝑀_𝑋, are positive constants. Define the minimum approximation errors as 𝜔=𝛼𝑓𝑋∣𝜃^∗_𝑓−𝑓(𝑋)+𝑔𝑋∣𝜃^∗_𝑔−𝑔(𝑋)𝑢_𝐼+(1−𝛼)𝑔(𝑋)𝑢^∗(𝑋)−𝑢𝑋∣𝜃^∗_𝐷−𝑑(𝑡). (6.3)

Now consider the Lyapunov function 1𝑉=2𝑒^𝑇̃1𝑃𝑒̃+2Γ_𝑓𝜃_𝑓−𝜃^∗_𝑓^𝑇𝜃_𝑓−𝜃^∗_𝑓+𝛼2Γ_𝑔𝜃_𝑔−𝜃^∗_𝑔^𝑇𝜃_𝑔−𝜃^∗_𝑔+(1−𝛼)2Γ_𝐷𝜃_𝐷−𝜃^∗_𝐷^𝑇𝜃_𝐷−𝜃^∗_𝐷, (6.4) where  Γ_𝑓=^𝜏𝑓0⋱_{𝜏𝑓𝑛(𝑛+1)(𝑛+1)} and Γ_𝑔 and Γ_𝐷 is expressed similarly. We get the adaptive law as 𝜃̇_𝑓=−Γ_𝑓𝜉𝑋𝐵^𝑇𝑃𝑒,̃𝜃̇_𝑔=−Γ_𝑔𝜉𝑋𝐵^𝑇𝑃𝑒𝑢̃_𝐼,𝜃̇_𝐷=−Γ_𝐷𝜂𝑋𝑔(𝑋)𝐵^𝑇𝑃𝑒.̃ (6.5)So    𝜃̇_𝑓^𝑇=𝑎^𝑙₀̇𝑎^𝑙₁̇⋮𝑎^𝑙_𝑛̇=𝜉𝑋𝐵^𝑇𝜏𝑃𝑒̃^𝑓00⋮0. (6.6) Furthermore, the adaptive law of niche factors is derived as follows. We derived the adaptive law of real ecologic factors, see Formula (6.7), (6.8), which represented the real niche always develop towards the ideal one. It reflected the compensation of the control system to the external disturbance: 𝜆̇_𝑓𝑞+1=−1√𝑒2𝜋^{−((𝜆𝑓(𝑞)+𝜆)/(𝜎𝑓(𝑞)+𝜎))2/2}1𝜎_𝑓𝑞𝜏+𝜎^𝑓0𝜉𝑋𝐵^𝑇𝜎̇𝑃𝑒,_𝑓̃=1𝑞+1√𝑒2𝜋^{−((𝜆𝑓(𝑞)+𝜆)/(𝜎𝑓(𝑞)+𝜎))2/2}𝜆_𝑓𝑞+𝜆𝜎_𝑓𝑞+𝜎²𝜏_𝑓0𝜉𝑋𝐵^𝑇𝑃𝑒.̃ (6.7) The same way we have 𝜆̇_𝑔𝑞+1=−1√𝑒2𝜋^{−((𝜆𝑔(𝑞)+𝜆)/(𝜎𝑔(𝑞)+𝜎))2/2}1𝜎_𝑔𝑞𝜏+𝜎^𝑔0𝐵^𝑇𝑃𝑒𝑢̃_𝐼,𝜎̇_𝑔=1𝑞+1√𝑒2𝜋^{−((𝜆𝑔(𝑞)+𝜆)/(𝜎𝑔(𝑞)+𝜎))2/2}𝜆_𝑔𝑞+𝜆𝜎_𝑔𝑞+𝜎²𝜏_𝑔0𝐵^𝑇𝑃𝑒𝑢̃_𝐼,𝜆̇_𝐷𝑞+1=−1√𝑒2𝜋^{−((𝜆𝐷(𝑞)+𝜆)/(𝜎𝐷(𝑞)+𝜎))2/2}1𝜎_𝐷𝑞𝜏+𝜎^𝐷0𝜂𝑋𝑔(𝑋)𝐵^𝑇𝜎̇𝑃𝑒,_𝐷̃=1𝑞+1√𝑒2𝜋^{−((𝜆𝐷(𝑞)+𝜆)/(𝜎𝐷(𝑞)+𝜎))2/2}𝜆_𝐷𝑞+𝜆𝜎_𝐷𝑞+𝜎²𝜏_𝐷0𝜂𝑋𝑔(𝑋)𝐵^𝑇𝑃𝑒.̃ (6.8)

Applying𝑢_𝑠we have 𝑉̇≤−(1/2)𝑒^𝑇̃𝑄𝑒̃+𝑒^𝑇̃𝑃𝐵𝜔.

Design parameters of, vector adaptive law of 𝜃_𝑓,𝜃_𝑔,𝜃_𝐷, stability, and performance analysis are similar to those of [18]; here we omit it.

7. Simulation

Form combined fuzzy adaptive control of two-dimensional predation system [32] as 𝑑𝑥₁𝑑𝑡=𝑥₁𝑎−𝑐𝑥₁−𝑏𝑥₁²−ℎ(𝑥)−𝑥₂𝛼𝑥²₁1+𝛽𝑥²₁𝑑𝑥₂𝑑𝑡=𝑥₂−𝑞+𝑘𝛼𝑥²₁1+𝛽𝑥²₁, (7.1) where  𝑥₁(𝑡) means total number of food at the time of 𝑡 and 𝑥₂(𝑡) denotes the total number of predators; 𝑎,𝑏,𝑐,𝑞,𝑘,𝛼,𝛽 are the regular numbering ecology, and 𝑘 is transforming factor,  𝑞 represents death ratio of predators. ℎ(𝑥) is the function of Holling’s functional responses, and 𝑥₂(𝛼𝑥²₁/(1+𝛽𝑥²₁)) is the third kind of Holling’s functional responses. The demonstration of the two-dimensional predator system without the controller is shown in Figure 1.

Figure 1 

Two-dimension predator system without the controller.

In order to reach an ideal ecologic balance in this two-dimension predators system, we get a way to control it, where ℎ(𝑥)=0.0012𝑥³₁,𝑎=1.0,𝑏=0.0014,𝑐=0.06,𝑞=0.2,𝑘=0.08,𝛼=5.0,𝛽=1.2.

Then 𝑑𝑥₁𝑑𝑡=𝑥₁1+1.2𝑥²₁1−0.06𝑥₁−0.0014𝑥²₁−0.0012𝑥³₁−𝑥₁𝑥₂𝑑𝑥₂𝑑𝑡=𝑥₂0.16𝑥²₁−0.2+𝑢𝑦=𝑥₁. (7.2) In order to establish the direct relation between output  𝑦  and controller  𝑢, we need the derivation of  𝑦. After derivating twice, we get 𝑦̈=𝑥̈₁=𝑥₁1+1.2𝑥²₁0.8−0.06𝑥₁−0.0014𝑥²₁−0.0012𝑥³₁−𝑥₁𝑥₂²−𝑥²₁𝑥₂0.8−0.06𝑥₁+1.3586𝑥²₁−0.0732𝑥³₁−0.00168𝑥⁴₁−0.00144𝑥⁵₁−𝑥₁𝑥₂+2.3972𝑥²₁−0.2196𝑥³₁−0.00672𝑥⁴₁−0.0072𝑥⁵₁−0.06𝑥₁−𝑥²₁𝑢. (7.3) Here, we command 𝑓(𝑋)=𝑥₁1+1.2𝑥²₁0.8−0.06𝑥₁−0.0014𝑥²₁−0.0012𝑥³₁−𝑥₁𝑥₂²−𝑥²₁𝑥₂0.8−0.06𝑥₁+1.3586𝑥²₁−0.0732𝑥³₁−0.00168𝑥⁴₁−0.00144𝑥⁵₁−𝑥₁𝑥₂+2.3972𝑥²₁−0.2196𝑥³₁−0.00672𝑥⁴₁−0.0072𝑥⁵₁−0.06𝑥₁𝑔(𝑋)=−𝑥²₁. (7.4) Then (7.2) and (7.3) could be shown as 𝑦̈=𝑓(𝑋)+𝑔(𝑋)𝑢. (7.5) If hoping to apply adaptive fuzzy controlling system here, we have to firstly confirm the boundary of 𝑓^𝑢,𝑔^𝑢,  and  𝑔_𝑙. From [29], we know that   0.2<|𝑥₁|<5.5,1.5<|𝑥₂|<5, then 𝑓𝑥₁,𝑥₂≤2.7205𝑒+0.03≈2.7205𝑒+0.03=𝑓^𝑢𝑥₁,𝑥₂0<𝑔_𝑙𝑥₁,𝑥₂≈𝑔_𝑙𝑥₁,𝑥₂=0.04≤𝑔𝑥₁,𝑥₂≤30.25=𝑔^𝑢𝑥₁,𝑥₂≈𝑔^𝑢𝑥₁,𝑥₂. (7.6) According to the above, we find out that the scope of 𝑓(𝑥₁,𝑥₂)  is larger than  𝑔(𝑥₁,𝑥₂). Therefore, we choose 𝜏_𝑓0=50,𝜏_𝑔0=1,𝜏_𝐷0=15. Then we select 𝑀_𝑓=7.7,𝑀_𝑔=31, 𝑀_𝐷=31,  𝜀=0.5,  |𝑢|≤5. The proposal method of this paper does not need to predefine the reference signal, but to reach the balance of each individual self-adaptively. Conveniently, we suppose 𝑘_𝑐=[89184] (in this way, 𝑠²+𝑘₁𝑠+𝑘₂ could be stable) and 𝑘₀=[44],   we select 𝑄=¹⁰¹³₁₃₂₈, from Λ_𝑐^𝑇𝑃+𝑃Λ_𝑐=−𝑄, then we solve 𝑃=²⁹⁻¹⁴₋₁₄₇. 𝑄 as 40252530, and 𝐴=⁰¹₋₄₋₄ so we have 𝑃=¹⁵⁵₅₅; when letting 𝜆_min=2.93, we could get 𝑉=2𝜆_min=5.86. In this way, we get the ideal =0,𝜎=1. The adaptive law comes out as follows: 𝜆̇_𝑓𝑞+1=−50√𝑒2𝜋^{−(𝜆𝑓(𝑞)/(𝜎𝑓(𝑞)+1))2/2}1𝜎_𝑓𝑞𝜉+1𝑋𝐵^𝑇𝜎̇𝑃𝑒_𝑓̃𝑞+1=50√𝑒2𝜋^{−(𝜆𝑓(𝑞)/(𝜎𝑓(𝑞)+1))2/2}𝜆_𝑓𝑞𝜎_𝑓𝑞+1²𝜉𝑋𝐵^𝑇𝜆̇𝑃𝑒_𝑔̃1𝑞+1=−√𝑒2𝜋^{−(𝜆𝑔(𝑞)/(𝜎𝑔(𝑞)+1))2/2}1𝜎_𝑔𝑞𝐵+1^𝑇𝑃𝑒𝑢̃_𝐼𝜎̇_𝑔=1𝑞+1√𝑒2𝜋^{−((𝜆𝑔(𝑞)+0)/(𝜎𝑔(𝑞)+1))2/2}𝜆_𝑔𝑞𝜎_𝑔𝑞+1²𝐵^𝑇𝑃𝑒𝑢̃_𝐼𝜆̇_𝐷𝑞+1=−15√𝑒2𝜋^{−(𝜆𝐷(𝑞)/(𝜎𝐷(𝑞)+1))2/2}1𝜎_𝐷𝑞𝜂+1𝑋𝑔(𝑋)𝐵^𝑇𝜎̇𝑃𝑒_𝐷̃𝑞+1=15√𝑒2𝜋^{−(𝜆𝐷(𝑞)/(𝜎𝐷(𝑞)+1))2/2}𝜆_𝐷𝑞𝜎_𝐷𝑞+1²𝜂𝑋𝑔(𝑋)𝐵^𝑇𝑃𝑒.̃ (7.7) We choose 𝑥₁(0)=0.3,𝑥₂(0)=1.0, then the trend of the prey and predator numbers with T-S controller show as in Figures 2 and 3. The trend of the T-S controller is shown in Figure 4.

Figure 2 

The trend of the prey numbers with T-S controller.

Figure 3 

The trend of the predator numbers with T-S controller.

Figure 4 

The trend of the T-S controller.

From the simulation figures, we can see that the prey and predator numbers reach a stable status in a short period of time under the control of the proposed method in this paper. The creature individuals show a characteristic of self-adaptation according to outside changes.

From the figure of the trend of the T-S controller, it also reaches an ideal status to maintain the overall balance of the two-dimension predator system. The ecological system is optimized with the use of this control method.

8. Conclusion

For the ecological niche, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Let the close degree of Niche which contains parameters as the consequent of adaptive T-S fuzzy control system, then designs the hybrid control law by tracking, continuous supervisory and adaptive compensation. Using gradient descent to optimize the parameters, we get the adaptive law of consequent parameters, embodying biological individual’s ability of adaptability. Based on Lyapunov stability theory, it is proved that all signals in the closed-loop system are bounded and tracking error converges to zero. This paper shows that the fuzzy methods provide good results in practical engineering problems. The performance of the developed approach is illustrated by simulation, on two-dimension predation system model.

Acknowledgment

Work supported by national science foundation (11072090).