Adaptive Control of Advanced G-L Fuzzy Systems with Several Uncertain Terms in Membership-Matrices

Abstract

In this paper, a set of novel adaptive control strategies based on an advanced G-L (proposed by Ge-Li-Tam, called GLT) fuzzy system is proposed. Three main design points can be summarized as follows: (1) the unknown parameters in a nonlinear dynamic system are regarded as extra nonlinear terms and are further packaged into so-called nonlinear terms groups for each equation through the modeling process, which reduces the complexity of the GLT fuzzy system; (2) the error dynamics are further rearranged into two parts, adjustable membership function and uncertain membership function, to aid the design of the controllers; (3) a set of adaptive controllers change with the estimated parameters and the update laws of parameters are provided following the current form of error dynamics. Two identical nonlinear dynamic systems based on a Quantum-CNN system (Q-CNN system) with two added terms are employed for simulations to demonstrate the feasibility as well as the effectiveness of the proposed fuzzy adaptive control scheme, where the tracking error can be eliminated efficiently.

1. Introduction

The concept of fuzzy logic was proposed by L. A. Zadeh in 1965 [1] that has received much attention as a powerful tool for interdisciplinary applications, such as clustering [2,3], fault diagnosis [4,5,6], prediction [7,8] and decision making [9,10,11,12], etc. For example, Khoshdast et al. [3] presented an approach to reduce the mesh-induced error for CFD analysis; Krzywanski et al. [9] applied a set of fuzzy logic-based methods to raise the performance of heat transfer in a super-heater for an industrial CFBC; Khoshdast et al. [10] developed a coupled fuzzy logic design for the simulation of a coal classifier in an industrial environment; and in [11], Akbar et al. proposed a fuzzy logic-based model to elucidate the effect of FSW parameters on the ultimate tensile strength and elongation of pure copper joints.

Among the various kinds of applications in the fuzzy area, fuzzy logic control [13,14,15,16,17,18,19,20], especially fuzzy model-based control, is widely used because the design and analysis of the overall fuzzy system can be systematically performed using the well-established classical linear systems theory. The Takagi–Sugeno fuzzy model proposed in 1985 [21] is widely accepted as a powerful tool for modeling, design and analysis of nonlinear dynamic systems [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. This well known mathematical tool aims to build a fuzzy model of dynamic systems where fuzzy implications and reasoning are applied to describe a fuzzy subspace of inputs and its consequence for a linear input-output form.

According to the modeling concept revealed by the T-S fuzzy model, Ge and Li [37] proposed a novel modeling approach—applying the fuzzy concept to model each equation in nonlinear systems. This new modeling approach was proposed in an attempt to raise the efficiency of fuzzy modeling, simplify the modeling procedure, and reduce the numbers of modeling materials and feedback control inputs during fuzzy control. This novel modeling approach has been cited over 50 times in various research fields and applications; for instance, H∞ control design [38,39,40,41,42], design and control of memristive systems [43,44,45,46], data encryption [47,48,49,50], sampled-data control [51,52,53] and other applications [54,55,56,57,58,59,60]. Moreover, Li et al. [61] extended a new concept of a nonlinear terms group to further improve the effectiveness of the G-L fuzzy model in 2015, called advanced G-L fuzzy modeling strategy (GLT fuzzy system). For all kinds of nonlinear dynamic systems, there will be only two linear subsystems and two membership function matrices in the final output of this advanced fuzzy model. In summary, this proposed modeling approach provides a novel way to model systems with different structures as the same form, and therefore, the fuzzy controllers can be designed in a more efficient way.

Furthermore, in practical engineering, the structure and parameters in a system are usually unobtainable, especially when the parameters are always changing according to the operation of the system. As a result, adaptive control [62,63] has been employed to address this kind of practical issue. Fuzzy logic systems (FLSs) provide a very effective method for coping with uncertainties due to its excellent functional approximation abilities. For example, Liu et al. [64,65] investigated adaptive fuzzy control for MIMO systems with unknown dead-zones, Tong et. al. [66,67] designed a set of state observers for a fuzzy adaptive control system, and Chen et. al. [68] applied a fuzzy neural network to address the adaptive control problem for a class of uncertain nonlinear stochastic systems.

The adaptive control studies mentioned above can be divided into two types: one designs the adaptive control scheme following fuzzy logic theory, and the other derives the stability of adaptive control via transferring the original dynamic systems to a T-S fuzzy system. A novel fuzzy model [37,61] provides an alternative way to model a dynamic system in a more efficient way, and as a consequence, we became interested in how to solve the adaptive control problem by applying a novel fuzzy model. In this paper, a GLT fuzzy system based on an adaptive control strategy is proposed to address the control problem for membership function matrices with mismatched parameters. The concept provided via this set of adaptive control strategies is to consider the unknown parameters in the original dynamic system as extra nonlinear terms and then package them into a nonlinear terms group in each equation. With this approach, the unknown parameters problem is transferred to an uncertain membership function matrices problem. This strategy provides a different perspective for solving the adaptive control problem that is based on a GLT fuzzy model.

The organization of the research can be summarized below. The modeling concept for the GLT fuzzy model theory and the fuzzy adaptive control scheme are introduced in Section 2. The modeling process, simulation results and some further discussion are included in Section 3. Finally, the conclusions are given in Section 4.

2. Materials and Methods

2.1. GLT Fuzzy-Model Theory

According to the proposed fuzzy model [61], to model each nonlinear equation and package the nonlinear terms groups, the consequent part of the proposed fuzzy system is designed to be a linear equation, and the system output can be simply described as weight sums of two linear subsystems as given below:

$$\dot{\mathit{Y}}(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\mathit{M}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})$$

(1)

where $\mathit{Y}(\mathit{t})\in {\Re }^{n\times 1}$ is the state of the system, revealed to be a vector form, and the coefficient matrices, ${\mathit{A}}_{\mathit{i}}\in {\Re }^{n\times n}$ and ${\mathit{B}}_{\mathit{i}}\in {\Re }^{n\times 1}$, indicate the coefficient of states and constants transformed via modeling, respectively. Moreover, p = 2, the number of linear subsystems and membership function matrices according to the modeling process, where ${\mathit{M}}_{\mathit{i}}\in {\Re }^{n\times n}$ are matrices including the membership functions generated by the proposed fuzzy if-then rules:

$${\mathit{M}}_{\mathit{i}}=\left[\begin{array}{cccc}{M}_{ji}& 0& 0& 0\\ 0& M_{ji}& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& M_{ji}\end{array}\right],$$

(2)

where j = 1, 2, …, n; i = 1, 2, ${\mathit{M}}_1+{\mathit{M}}_2$ is the identity matrix and ${\displaystyle \sum _{i=1}^2{M}_{ji}=1}$.

2.2. Adaptive Control Scheme

The proposed adaptive control scheme is constructed in this subsection. The unknown parameters are regarded as part of the nonlinear terms, and we package those unknown parameters into nonlinear terms groups, i.e., through this design, the unknown parameters are included in the membership function and no longer appear in the coefficient matrix of linear subsystems of the final output. Equation (3) and (4) are two GLT fuzzy systems with consistent structures, the first is regarded as the drive system, and the second is viewed as a response system with designed fuzzy controllers, where the unknown system parameters are arranged in the membership function matrices in Equation (3):

Master system:

$$\dot{\mathit{Y}}(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\mathit{M}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})$$

(3)

Slave system:

$$\dot{\mathit{X}}(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{X}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})+\mathit{U}(\mathit{t})$$

(4)

where $\mathit{Y}(\mathit{t})\in {\Re }^{n\times 1}$, $\mathit{X}(\mathit{t})\in {\Re }^{n\times 1}$ are the states of the master and slave systems, respectively; ${\mathit{M}}_{\mathit{i}}\in {\Re }^{n\times n}$ and ${\widehat{\mathit{O}}}_{\mathit{i}}\in {\Re }^{n\times n}$ are the membership function matrices with unknown and estimated parameters, respectively. Once again, p = 2, the number of linear subsystems and membership function matrices according to the modeling process. It is important to mention that A_i $\in {\Re }^{n\times n}$ and B_i $\in {\Re }^{n\times n}$ are the coefficient matrices, which are the same in the master and slave system through the design given above. U(t)$\in {\Re }^{n\times 1}$ is the adaptive control input, designed to not share the same membership functions as the plant model.

The error states $\mathit{e}(\mathit{t})=\mathit{Y}(\mathit{t})-\mathit{X}(\mathit{t})$ are defined, $\mathit{e}(\mathit{t})\in {\Re }^{n\times 1}$, then we have the following error dynamics description in Equation (5):

$$\dot{\mathit{e}}=\dot{\mathit{Y}}-\dot{\mathit{X}}={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\mathit{M}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})-{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{X}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})-\mathit{U}(\mathit{t})$$

(5)

In observation of the mathematics structure in Equation (5), the structure of the error system is not simple, where the drive system, response system and the designed fuzzy controllers are included. Also, the membership function matrices M_i in the master system comprise unknown parameters, the estimated parameters are designed in ${\widehat{\mathit{O}}}_{\mathit{i}}$ and ${\mathit{M}}_{\mathit{i}}\ne {\widehat{\mathit{O}}}_{\mathit{i}}$. Thus, the fuzzy controllers cannot be assigned to remove these uncertain terms directly. According to the observation mentioned above, Equation (5) can be further expanded in the following way:

$$\begin{array}{ll}\dot{\mathit{e}}=\dot{\mathit{Y}}-\dot{\mathit{X}}& ={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\mathit{M}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})-{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{X}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})\\ & -{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})+{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})-\mathit{U}(\mathit{t})\end{array}$$

(6)

where $\pm {\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})$ are the extra terms added in Equation (6), then we can obtain the following expression:

$$\begin{array}{ll}\dot{\mathit{e}}=\dot{\mathit{Y}}-\dot{\mathit{X}}& ={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}-{\mathit{A}}_{\mathit{i}}\mathit{X}(\mathit{t})+{\mathit{B}}_{\mathit{i}}-{\mathit{B}}_{\mathit{i}})\\ & +{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}({\mathit{M}}_{\mathit{i}}-{\widehat{\mathit{O}}}_{\mathit{i}})({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})-\mathit{U}(\mathit{t})\end{array}$$

(7)

Finally, the error dynamic system can be described in a clearer way in Equation (8):

$$\dot{\mathit{e}}=\dot{\mathit{Y}}-\dot{\mathit{X}}={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{e}(\mathit{t})})+{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}({\mathit{M}}_{\mathit{i}}-{\widehat{\mathit{O}}}_{\mathit{i}})({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})-\mathit{U}(\mathit{t})$$

(8)

Here, the error dynamic system derived in Equation (8) can be classified into two parts given in Equation (9). The first part is developed via the estimated membership function matrices ${\widehat{\mathit{O}}}_{\mathit{i}}$ and the coefficient matrices A_i with error states, which can be controlled directly through designing U₁(t). The second part is composed of the difference of the two membership function matrices M_i,${\widehat{\mathit{O}}}_{\mathit{i}}$, and the linear subsystems of the master system, where the control input in U₁(t) cannot address it directly. Therefore, U₂(t) is needed.

$$\dot{\mathit{e}}=\underset{{\mathit{U}}_1(\mathit{t})}{\underbrace{{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{e}(\mathit{t})})}}+\underset{{\mathit{U}}_2(\mathit{t})}{\underbrace{{\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}({\mathit{M}}_{\mathit{i}}-{\widehat{\mathit{O}}}_{\mathit{i}})({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})}}-\mathit{U}(\mathit{t})$$

(9)

Consequently, the fuzzy controllers $\mathit{U}(\mathit{t})={\mathit{U}}_1(\mathit{t})+{\mathit{U}}_2(\mathit{t})$ can be designed as follows:

$$\left\{\begin{array}{l}{\mathit{U}}_1(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^{\mathit{p}}{\widehat{\mathit{O}}}_{\mathit{i}}(-{\mathit{C}}_{\mathit{i}}{\mathit{F}}_{\mathit{i}}\mathit{e}(\mathit{t})})\\ {\mathit{U}}_2(\mathit{t})={[{\mathit{u}}_1(\mathit{t}),{\mathit{u}}_2(\mathit{t}),{\mathit{u}}_3(\mathit{t})\dots {\mathit{u}}_l(\mathit{t})]}^T\end{array}\right.$$

(10)

where ${\mathit{U}}_1(\mathit{t})\in {\Re }^{n\times 1}$ is designed as the fuzzy control input to pre-process the error dynamic systems in Equation (9); $C_{\mathit{i}}\in {\Re }^{n\times n}$ is a constant matrix; $F_{\mathit{i}}\in {\Re }^{n\times n}$ is a matrix designed to be the control gain satisfying the condition ${\mathit{A}}_{\mathit{i}}-C_{\mathit{i}}{F}_{\mathit{i}}=G$, where G is a negative definite matrix. The fuzzy controllers ${\mathit{U}}_2(\mathit{t})\in {\Re }^{n\times 1}$ should be designed with appropriate parameter update laws, where l is the total number of control inputs in U₂(t).

The Lyapunov function given in Equation (11) is applied,

$$V(\mathit{e}(\mathit{t}),{\tilde{\mathit{a}}}_j)=\frac{1}{2}\left({\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}({\mathit{e}}_{\mathit{i}}{(\mathit{t})}^2)}+{\displaystyle \sum _{j=1}^{\mathit{m}}({\mathit{r}}_j{\tilde{\mathit{a}}}_j^2)}\right)$$

(11)

where ${\tilde{\mathit{a}}}_{\mathit{i}}$ is the difference of parameters in the membership function; m indicates the number of parameters we have estimated; r_j refer to the adaptation gains, where r_j > 0 In this paper, all r_j are set to be r_j = 1. Through designing appropriate fuzzy controllers $\mathit{U}(\mathit{t})$ and the adaptation law of system parameters ${\tilde{\mathit{a}}}_{\mathit{i}}$, we aim to have the following:

$$\dot{\mathit{V}}=-{\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}({\eta }_{\mathit{i}}{\mathit{e}}_{\mathit{i}}{(\mathit{t})}^2)}\le 0$$

(12)

where ${\eta }_i$ is designed to be positive. It is worth to mentioning that the pragmatically asymptotical stability theorem provided by Ge et al. in 1999 [69,70] has proved in a strict way that the error states, as well as the estimated parameters, can approach the uncertain or goal parameters in Equation (12).

3. Simulation Results and Discussion

In this section, a Quantum-CNN system (Q-CNN system) [71] with some extra nonlinear terms is proposed for illustration. This complicated system is presented to show the effectiveness of the proposed modeling and control strategy developed in this article.

$$\left\{\begin{array}{l}{\dot{y}}_1=-2b_1\sqrt{1-y_1^2}\sin y_2\\ {\dot{y}}_2=-w_1(y_1-y_3)+2b_3\frac{y_1}{\sqrt{1-y_1^2}}\cos y_2+{\Delta }_1\\ {\dot{y}}_3=-2b_2\sqrt{1-y_3^2}\sin y_4\\ {\dot{y}}_4=-w_2(y_3-y_1)+2b_4\frac{y_3}{\sqrt{1-y_3^2}}\cos y_4+{\Delta }_2\end{array}\right.$$

(13)

where ${\Delta }_1$ and ${\Delta }_2$ are the extra nonlinear terms defined as ${\Delta }_1=b_5{y}_1{y}_4^2$, ${\Delta }_2=b_6{y}_3{y}_2^3$. When the system parameters b₁ = b₃ = 6.8, b₂ = b₄ = 4.3, w₁ = 4.7, w₂ = 3.9, b₅ = 5 and b₆ = 2 and initial states are set as (y₁₀, y ₂₀, y₃₀, y₄₀) = (0.1, 0.5, 0.1, 0.5), the system shows chaotic behaviors, as shown in Figure 1.

According to the modeling theory given in Section 2.1, Equation (13) can be expressed in the following form:

$$\dot{\mathit{Y}}(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^2{\mathit{M}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{Y}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})$$

(14)

where M_i are membership function matrices, M₁ + M₂ is the identity matrix and A_i, B_i are the coefficient matrices of the GLT fuzzy system:

$$\begin{gathered} M_1=\left[\begin{array}{cccc}{M}_{11}& 0& 0& 0\\ 0& M_{21}& 0& 0\\ 0& 0& M_{31}& 0\\ 0& 0& 0& M_{41}\end{array}\right], M_2=\left[\begin{array}{cccc}{M}_{12}& 0& 0& 0\\ 0& M_{22}& 0& 0\\ 0& 0& M_{32}& 0\\ 0& 0& 0& M_{42}\end{array}\right], \\ {A}_1=\left[\begin{array}{cccc}0& 0& 0& 0\\ W_2-w_1& 0& w_1& 0\\ 0& 0& 0& 0\\ w_2& 0& W_4-w_2& 0\end{array}\right], B_1=\left[\begin{array}{c}-2b_1{W}_1\\ 0\\ -2b_2{W}_3\\ 0\end{array}\right] \\ {A}_2=\left[\begin{array}{cccc}0& 0& 0& 0\\ -W_2-w_1& 0& w_1& 0\\ 0& 0& 0& 0\\ w_2& 0& -W_4-w_2& 0\end{array}\right], B_2=\left[\begin{array}{c}2b_1{W}_1\\ 0\\ 2b_2{W}_3\\ 0\end{array}\right] \end{gathered}$$

where M₁₁, M₁₂, M₂₁, M₂₂, M₃₁, M₃₂, M₄₁, M₄₂ are fuzzy sets and can be described as:

$$\begin{gathered} M_{11}=\frac{1}{2}(1+\frac{N_1}{W_1}), M_{12}=\frac{1}{2}(1+\frac{N_1}{W_1}), M_{21}=\frac{1}{2}(1+\frac{N_2}{W_2}), M_{22}=\frac{1}{2}(1-\frac{N_2}{W_2}), \\ {M}_{31}=\frac{1}{2}(1+\frac{N_3}{W_3}), M_{32}=\frac{1}{2}(1-\frac{N_3}{W_3}), M_{41}=\frac{1}{2}(1+\frac{N_4}{W_4}), M_{42}=\frac{1}{2}(1-\frac{N_4}{W_4}) \end{gathered}$$

Further, W₁ = 1, W₂ = 30, W₃ = 1, W₄ = 15 are the boundary values defined by the ranges for the nonlinear terms groups N₁, N₂, N₃, N₄ separately:

$$\left\{\begin{array}{l}Nonlienar\begin{array}{c}\end{array}Terms\begin{array}{c}\end{array}Group\begin{array}{c}\end{array}1:N_1\in [-W_1,W_1]\begin{array}{c}\end{array}and\begin{array}{c}\end{array}{W}_1>0\\ Nonlienar\begin{array}{c}\end{array}Terms\begin{array}{c}\end{array}Group\begin{array}{c}\end{array}2:N_2\in [-W_2,W_2]\begin{array}{c}\end{array}and\begin{array}{c}\end{array}{W}_2>0\\ Nonlienar\begin{array}{c}\end{array}Terms\begin{array}{c}\end{array}Group\begin{array}{c}\end{array}3:N_3\in [-W_3,W_3]\begin{array}{c}\end{array}and\begin{array}{c}\end{array}{W}_3>0\\ Nonlienar\begin{array}{c}\end{array}Terms\begin{array}{c}\end{array}Group\begin{array}{c}\end{array}4:N_4\in [-W_4,W_4]\begin{array}{c}\end{array}and\begin{array}{c}\end{array}{W}_4>0\end{array}\right.$$

The nonlinear term groups N₁, N₂, N₃ and N₄ are designed as follows, and the time series are given in Figure 2, where parameters b₃ and b₄ have been packaged into nonlinear term groups, because b₃ and b₄ are designed to be the unknown parameters in the following section of fuzzy adaptive control. This step provides the generated GLT fuzzy system two linear subsystems without any uncertainty. Chaotic behavior of a GLT Fuzzy Q-CNN system with extra nonlinear terms is given in Figure 3.

$$\left\{\begin{array}{l}{N}_1=\sqrt{1-y_1^2}\sin y_2\\ N_2=2b_3\cos y_2/\sqrt{1-y_1^2}+b_5{y}_4^2\\ N_3=\sqrt{1-y_3^2}\sin y_4\\ N_4=2b_4\cos y_4/\sqrt{1-y_3^2}+b_6{y}_2^3\end{array}\right.$$

Fuzzy Adaptive Control of GLT Systems

In this subsection, two identical Q-CNN GLT fuzzy systems are illustrated for further discussion. The goal of this section is to control the slave fuzzy system with estimated parameters to achieve a drive system with several unknown parameters via the proposed fuzzy adaptive control strategy devised in Section 2.2.

• CASE I: Parameter b₃ is unknown

Consider the fuzzy system in Equation (14) is the master system, where b₃ is designed as unknown parameters, which leads the nonlinear terms group N₂ to be uncertainties, and the membership functions M₂₁, M₂₂ become uncertain membership functions as well. The slave Q-CNN fuzzy system with the designed fuzzy controllers U(t) and the estimated parameters ${\widehat{b}}_3$ are given below:

$$\dot{\mathit{X}}(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^2{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{X}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})+\mathit{U}(\mathit{t})$$

(15)

where $\dot{\mathit{X}}(\mathit{t})$$={\left[\begin{array}{cccc}{\dot{x}}_1& {\dot{x}}_2& {\dot{x}}_3& {\dot{x}}_4\end{array}\right]}^T$. When initial conditions are set as (x₁₀, x₂₀, x₃₀, x₄₀) = (5, −5, 5, −5), the coefficient matrices A_i and B_i are the same to those in the master system in Equation (14), the boundary values (W₁, W₂, W₃, W₄) = (1, 30, 1, 15) and the membership function matrices ${\widehat{\mathit{O}}}_{\mathit{i}}$ with estimated parameters can be organized as follows:

$${\widehat{O}}_1=\left[\begin{array}{cccc}{O}_{11}& 0& 0& 0\\ 0& {\widehat{O}}_{21}& 0& 0\\ 0& 0& O_{31}& 0\\ 0& 0& 0& O_{41}\end{array}\right],{\widehat{O}}_2=\left[\begin{array}{cccc}{O}_{12}& 0& 0& 0\\ 0& {\widehat{O}}_{22}& 0& 0\\ 0& 0& O_{32}& 0\\ 0& 0& 0& O_{42}\end{array}\right]$$

where ${\widehat{\mathit{O}}}_1+{\widehat{\mathit{O}}}_2$ is the identity matrix and each element is proposed as follows:

$$\begin{gathered} O_{11}=\frac{1}{2}(1+\frac{N_1}{W_1}), O_{12}=\frac{1}{2}(1+\frac{N_1}{W_1}), {\widehat{O}}_{21}=\frac{1}{2}(1+\frac{{\widehat{N}}_2}{W_2}), {\widehat{O}}_{22}=\frac{1}{2}(1-\frac{{\widehat{N}}_2}{W_2}), \\ {O}_{31}=\frac{1}{2}(1+\frac{N_3}{W_3}), O_{32}=\frac{1}{2}(1-\frac{N_3}{W_3}), O_{41}=\frac{1}{2}(1+\frac{N_4}{W_4}), O_{42}=\frac{1}{2}(1-\frac{N_4}{W_4}) \end{gathered}$$

The nonlinear terms groups N₁, N₂, N₃, N₄ in ${\widehat{\mathit{O}}}_{\mathit{i}}$ are as follows:

$$\left\{\begin{array}{l}{N}_1=\sqrt{1-x_1^2}\sin x_2\\ {\widehat{N}}_2=2{\widehat{b}}_3\cos x_2/\sqrt{1-x_1^2}+b_5{x}_4^2\\ N_3=\sqrt{1-x_3^2}\sin x_4\\ N_4=2b_4\cos x_4/\sqrt{1-x_3^2}+b_6{x}_2^3\end{array}\right.$$

It can be observed that the membership functions ${\widehat{O}}_{21}$ and ${\widehat{O}}_{22}$ are composed of the nonlinear terms groups ${\widehat{N}}_2$ comprising the estimated parameter ${\widehat{b}}_3$, i.e., ${\widehat{N}}_2$ is the estimated group and ${\widehat{O}}_{21}$ and ${\widehat{O}}_{22}$ are the estimated membership functions. The initial value of the estimated parameter is set as ${\widehat{b}}_{30}=6$.

The error and error dynamic systems of the master and slave GLT fuzzy systems provided in Equations (14) and (15) are defined following the form revealed in Section 2.2, where the system dimension n = 4. In order to analyze the stability of the error dynamic system, an appropriate candidate Lyapunov function should be defined. We chose the candidate Lyapunov function with square form as follows:

$$V(\mathit{e}(\mathit{t}),{\tilde{a}}_{\mathit{i}})=\frac{1}{2}\left(\mathit{e}{(\mathit{t})}^2+{\displaystyle \sum _{\mathit{i}=1}^n(F_{\mathit{i}}{\tilde{a}}_{\mathit{i}}^2)}\right)=\frac{1}{2}(e_1^2+e_2^2+e_3^2+e_4^2+{\tilde{b}}_3^2)$$

(16)

where ${\tilde{b}}_3=b_3-{\widehat{b}}_3$ is the error of unknown parameter b₃ and estimated parameter ${\widehat{b}}_3$. The derivatives of the Lyapunov function in Equation (16) can be described as follows:

$$\dot{V}(\mathit{t})=e_1{\dot{e}}_1+e_2{\dot{e}}_2+e_3{\dot{e}}_3+e_4{\dot{e}}_4+{\tilde{b}}_3{\dot{\tilde{b}}}_3$$

(17)

According to the adaptive control scheme in Section 2.2, we can design the fuzzy controllers $\mathit{U}(\mathit{t})={\mathit{U}}_1(\mathit{t})+{\mathit{U}}_2(\mathit{t})$ shown in Equation (18) and the update laws of parameters shown in Equation (19):

$$\left\{\begin{array}{l}{\mathit{U}}_1(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^2{\widehat{\mathit{O}}}_{\mathit{i}}(-{\mathit{B}}_{\mathit{i}}{F}_{\mathit{i}}\mathit{e}(\mathit{t})})\\ {\mathit{U}}_2(\mathit{t})={[{\mathit{u}}_1(\mathit{t}),{\mathit{u}}_2(\mathit{t}),{\mathit{u}}_3(\mathit{t}),{\mathit{u}}_4(\mathit{t})]}^T\end{array}\right.$$

(18)

where ${\mathit{B}}_1$, ${\mathit{B}}_2$ are set as the identity matrix,

$$\begin{gathered} F_1=\left[\begin{array}{cccc}1& 0& 0& 0\\ W_2-w_1& 1& w_1& 0\\ 0& 0& 1& 0\\ w_2& 0& W_4-w_2& 1\end{array}\right], F_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ -W_2-w_1& 1& w_1& 0\\ 0& 0& 1& 0\\ w_2& 0& -W_4-w_2& 1\end{array}\right] \\ {u}_1(t)=(M_{12}-O_{12})\times 4b_1{W}_1, u_2(t)=b_5{y}_1(y_4^2-x_4^2)-2{\widehat{b}}_3{y}_1(\frac{\cos x_2}{\sqrt{1-x_1^2}}-\frac{\cos y_2}{\sqrt{1-y_1^2}}) \\ {u}_3(t)=(M_{32}-O_{32})\times 4b_2{W}_3, u_4(t)=(M_{41}-O_{41})\times 2y_3{W}_4 \end{gathered}$$

The Update Laws of Parameters

$${\dot{\widehat{b}}}_3=-{\dot{\tilde{b}}}_3=\frac{2y_1\cos y_2}{\sqrt{1-y_1^2}}{\mathit{e}}_2$$

(19)

Some interesting designs can be further discussed in this case. In Equation (18), ${\mathit{u}}_1(\mathit{t})$, ${\mathit{u}}_3(\mathit{t})$ are controllers designed in the form of parallel distributed compensation (PDC). Since there are no uncertain parameters in the first and third equations, the membership functions in the master and slave systems can be used to design the corresponding controllers. In addition, ${\mathit{u}}_2(\mathit{t})$ is a controller designed to let the estimated parameter in the slave system approach the original parameter in the master system. Since the membership functions $M_{21},M_{22}$ comprise unknown parameters b₃, controller ${\mathit{u}}_2(\mathit{t})$ should be designed without uncertain membership functions.

According to the design in Equations (18) and (19), we have the following derivative of the Lyapunov function, which is a negative semi-definite function of error states:

$$\dot{V}(\mathit{t})=-e_1^2-e_2^2-e_3^2-e_4^2\le 0$$

(20)

The derivative of the Lyapunov function in Equation (20) is a negative definite function of e. In the sense of Lyapunov stability theory, the Lyapunov asymptotic stability theorem is not satisfied. We cannot obtain the common origin of error dynamics and the parameter dynamics are asymptotically stable. Through the pragmatic asymptotic stability theorem [60,61] proposed by Prof. Ge, Yu and Chen, D is a 5-manifold, n = 5 and the number of error state variables p = 4. When e1 = e2 = e3 = e4 = 0, ${\widehat{b}}_3$ takes arbitrary values, $\dot{V}=0$, so X is of 4 dimensions and m = n − p = 5 − 4 = 1, m + 1 < n is satisfied. According to the pragmatic asymptotic stability theorem, error vector e approaches zero and the estimated parameters also approach the uncertain parameters. The equilibrium point is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymptotically stable. The simulation results are shown in Figure 4.

• CASE II: Parameters b₃ and b₄ are unknown

When the fuzzy system in Equation (14) is the master system, where b₃ and b₄ are designed as unknown parameters, this leads the nonlinear terms groups N₂ and N₄ to be uncertainties, and the membership functions M₂₁, M₂₂, M₄₁, M₄₂ become uncertain membership functions as well. The slave Q-CNN fuzzy system with the designed fuzzy controllers U(t) and the estimated parameters are given below:

$$\dot{\mathit{X}}(\mathit{t})={\displaystyle \sum _{\mathit{i}=1}^2{\widehat{\mathit{O}}}_{\mathit{i}}({\mathit{A}}_{\mathit{i}}\mathit{X}(\mathit{t})}+{\mathit{B}}_{\mathit{i}})+\mathit{U}(\mathit{t})$$

(21)

where all the system design in Equation (21) is the same as the design in Equation (15), except the initial conditions are set as (x₁₀, x₂₀, x₃₀, x₄₀) = (0, −10, 10, −10), the membership function matrices ${\widehat{\mathit{O}}}_{\mathit{i}}$, which can be organized as follow:

$${\widehat{O}}_1=\left[\begin{array}{cccc}{O}_{11}& 0& 0& 0\\ 0& {\widehat{O}}_{21}& 0& 0\\ 0& 0& O_{31}& 0\\ 0& 0& 0& {\widehat{O}}_{41}\end{array}\right], {\widehat{O}}_2=\left[\begin{array}{cccc}{O}_{12}& 0& 0& 0\\ 0& {\widehat{O}}_{22}& 0& 0\\ 0& 0& O_{32}& 0\\ 0& 0& 0& {\widehat{O}}_{42}\end{array}\right]$$

where ${\widehat{\mathit{O}}}_1+{\widehat{\mathit{O}}}_2$ is the identity matrix, and the main differences for ${\widehat{\mathit{O}}}_{\mathit{i}}$ from those in Equation (15) are ${\widehat{O}}_{41}=(1+{\widehat{N}}_4/W_4)/2$ and ${\widehat{O}}_{42}=(1-{\widehat{N}}_4/W_4)/2$, which includes the estimated parameter ${\widehat{b}}_4$, where ${\widehat{N}}_4=2{\widehat{b}}_4\cos x_4/\sqrt{1-x_3^2}+b_6{x}_2^3$. In this case, ${\widehat{O}}_{21}$, ${\widehat{O}}_{22}$, ${\widehat{O}}_{41}$, ${\widehat{O}}_{42}$ are estimated membership functions. The initial values of the estimated parameters are set as ${\widehat{b}}_{30}=1$ and ${\widehat{b}}_{40}=1$.

The error and error dynamic systems of the master and slave GLT fuzzy systems provided in Equations (14) and (21) are defined following the form revealed in Section 2.2, where the system dimension n = 4. In order to analyze the stability of the error dynamic system, an appropriate candidate Lyapunov function should be defined. We chose the candidate Lyapunov function with square form as follows:

$$\mathit{V}(\mathit{e}(\mathit{t}),{\tilde{a}}_{\mathit{i}})=\frac{1}{2}\left(\mathit{e}{(\mathit{t})}^2+{\displaystyle \sum _{\mathit{i}=1}^n({\mathit{F}}_{\mathit{i}}{\tilde{\mathit{a}}}_{\mathit{i}}^2)}\right)=\frac{1}{2}(e_1^2+e_2^2+e_3^2+e_4^2+{\tilde{b}}_3^2+{\tilde{b}}_4^2)$$

(22)

where ${\tilde{b}}_3=b_3-{\widehat{b}}_3$ and ${\tilde{b}}_4=b_4-{\widehat{b}}_4$ are the errors for the unknown parameters and estimated parameters, respectively. Then the derivatives of the Lyapunov function in Equation (22) can be described as:

$$\dot{\mathit{V}}(\mathit{t})=e_1{\dot{e}}_1+e_2{\dot{e}}_2+e_3{\dot{e}}_3+e_4{\dot{e}}_4+{\tilde{b}}_3{\dot{\tilde{b}}}_3+{\tilde{b}}_4{\dot{\tilde{b}}}_4$$

(23)

According to the adaptive control scheme in Section 2.2, the fuzzy controllers and the update laws of parameters can be designed in the following form, where ${\mathit{B}}_1$, ${\mathit{B}}_2$, are set as the identity matrix:

$$F_1=\left[\begin{array}{cccc}1& 0& 0& 0\\ W_2-w_1& 1& w_1& 0\\ 0& 0& 1& 0\\ w_2& 0& W_4-w_2& 1\end{array}\right], F_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ -W_2-w_1& 1& w_1& 0\\ 0& 0& 1& 0\\ w_2& 0& -W_4-w_2& 1\end{array}\right]$$

(24)

$$\begin{gathered} u_1(t)=(M_{12}-O_{12})\times 4b_1{W}_1, u_2(t)=b_5{y}_1(y_4^2-x_4^2)-2{\widehat{b}}_3{y}_1(\frac{\cos x_2}{\sqrt{1-x_1^2}}-\frac{\cos y_2}{\sqrt{1-y_1^2}}) \\ {u}_3(t)=(M_{32}-O_{32})\times 4b_2{W}_3, u_4(t)=b_6{y}_3(y_2^3-x_2^3)-2{\widehat{b}}_4{y}_3(\frac{\cos x_4}{\sqrt{1-x_3^2}}-\frac{\cos y_4}{\sqrt{1-y_3^2}}) \end{gathered}$$

The Update Laws of Parameters:

$$\left\{\begin{array}{l}{\dot{\widehat{b}}}_3=-{\dot{\tilde{b}}}_3=\frac{2y_1\cos y_2}{\sqrt{1-y_1^2}}{e}_2\\ {\dot{\widehat{b}}}_4=-{\dot{\tilde{b}}}_4=\frac{2y_3\cos y_4}{\sqrt{1-y_3^2}}{e}_4\end{array}\right.$$

(25)

In Equation (24), ${\mathit{u}}_1(\mathit{t})$, ${\mathit{u}}_3(\mathit{t})$ are controllers designed in the form of parallel distributed compensation (PDC). Since there are no uncertain parameters in the first and third equations, the membership functions in the master and slave systems can be used to design the corresponding controllers. In addition, ${\mathit{u}}_2(\mathit{t})$, ${\mathit{u}}_4(\mathit{t})$ are the controllers designed to let the estimated parameters in the slave system approach the original parameters in the master system. Since those membership functions $M_{21},M_{22},M_{41},M_{42}$ comprise unknown parameters b₃, controllers ${\mathit{u}}_2(\mathit{t})$, ${\mathit{u}}_4(\mathit{t})$ should be designed without uncertain membership functions.

With the design of the fuzzy controllers and the adaptation law of parameters mentioned in Equations (24) and (25), we have the following derivative of the Lyapunov function, which is a negative semi-definite function of error states:

$$\dot{\mathit{V}}(\mathit{t})=-e_1^2-e_2^2-e_3^2-e_4^2\le 0$$

(26)

The derivative of the Lyapunov function in Equation (26) is a negative definite function of e. In the sense of Lyapunov stability theory, the Lyapunov asymptotic stability theorem is not satisfied. We cannot obtain the common origin of error dynamics and the parameter dynamics are asymptotically stable. Through the pragmatic asymptotical stability theorem [60,61] proposed by Prof. Ge, Yu and Chen, D is a 6-manifold, n = 6 and the number of error state variables p = 4. When e1 = e2 = e3 = e4 = 0, ${\widehat{b}}_3$ and ${\widehat{b}}_4$ take arbitrary values, $\dot{V}=0$, so X is of 4 dimensions, and m = n − p = 6 − 4 = 2, m + 1 < n is satisfied. According to the pragmatic asymptotic stability theorem, error vector e approaches zero and the estimated parameters also approach the uncertain parameters. The equilibrium point is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymptotically stable. The simulation results are shown in Figure 5 and Figure 6.

Further discussion: comparing the feedback gains F₁ and F₂ designed in CASES I and II. We can figure out by packaging all unknown parameters into nonlinear terms groups that there are no uncertain terms in the two linear subsystems of the final output fuzzy systems. The uncertainty issue has been technologically transferred into the corresponding membership functions, so that the feedback gains in CASES I and II are developed in the same way. Through this design procedure, adaptive control can be achieved with a set of fixed feedback gains, appropriately designed fuzzy controllers and update laws of parameters.

4. Conclusions

In this paper, an extensive application of the GLT fuzzy system for adaptive control is proposed. Simulation results reveal that the proposed fuzzy adaptive control scheme is feasible, and the control goal as well as parameters identification can be achieved effectively. In fact, by using the fuzzy adaptive control scheme, the GTL fuzzy model played an important role in transforming complicated systems with unknown parameters into two linear subsystems by blending their uncertain membership function matrices. As a consequence, fuzzy controllers and parameter update laws can be constructed in a more convenient way. Furthermore, for those equations without unknown parameters, the first stage controller provides influential control results; and for those equations with parameters that need to be identified, the second stage controller and update laws are prerequisite. Consequently, the proposed adaptive control strategy based on the GLT fuzzy system provides an efficient way for designing controllers as well as achieving tracking goals.