Hybrid fuzzy direct/indirect adaptive controller for uncertain nonlinear systems

Abstract

This paper describes a new approach to adaptive control of uncertain nonlinear systems. A fuzzy logic controller is used to combine both direct and indirect methods. Based on the fuzzy neural networks, the plant unknown nonlinear functions are estimated, and then combined to form the indirect control law. In parallel, another fuzzy neural network approximates the direct adaptive control. According to the modelling error and its derivatives, the fuzzy logic controller modulates between direct and indirect adaptive controllers. The global stability of the overall system is shown by constructing a Lyapunov function. The simulation results show that within this scheme, the control objectives can be achieved with a fast convergence and optimal control for different dynamic regimes.

Keywords

Fuzzy systems (FS)fuzzy neural networks (FNN)direct adaptive control indirect adaptive control hybrid adaptive control optimal control Lyapanov stability

Introduction

Adaptive controls can be successfully applied when dynamic nonlinear systems are not exactly modelled or when their parameters are variable. According to the works recorded in Bibi et al. (2018), the use of conventional adaptive control is limited to cases where the unknown dynamics of the system have a linear structure with parameters unknown. However, for strongly nonlinear system, it can lose its stability under high disturbances using linear controllers. To overcome these problems, intelligent nonlinear adaptive controls can be used as a powerful alternative.

Adaptive neural-fuzzy robust control, fuzzy approximation-based adaptive control of nonlinear uncertain state constrained systems with time-varying delays, stochastic adaptive control and fuzzy-based multi-error constraint control for switched nonlinear systems have attracted the attention of many researchers in recent years.

In Sun et al. (2019), a new adaptive neural-fuzzy robust position control scheme for the magnetic suspension system of a low-speed maglev train is developed with stability proof. The result of experimental implementation demonstrated that the impact of the disturbance and parameter perturbations is reduced based on the developed control law. Wei et al. (2019) presented an adaptive fuzzy tracking control for flexible-joint robots with full state constraints. It is also proven that it can be found that the use of tan-type BLF to deal with state constraints can have better performance compared with the log-type BLF (Wei et al., 2019).

In literature, two different approaches have been used for developing the intelligent adaptive control law for uncertain nonlinear systems. The first is a direct adaptive approach in which a set of parameters in the control law is directly modified to form a stable closed-loop system (Bibi et al., 2018). In an indirect approach, unknown nonlinear functions of the dynamic model are first estimated, and then combined to form the overall control law (Bibi et al., 2017). Both methods are well established and can be used to design an efficient robust adaptive controller for the same nonlinear system.

However, the singularity problem (the division-by-zero) can be encountered in the case of indirect adaptive controllers for first order nonlinear systems. In Zhang et al. (2016), some techniques have been reported to overcome this problem and improve the stability of the indirect adaptive schemes. The direct adaptive approach does not pose the singularity problem but requires the boundedness of the gain control derivative by a continuous function to ensure the global stability (Labiod and Boucherit, 2013).

Shun-Feng et al. (2006) proved that inappropriate selection of design parameters for direct adaptive estimator can make the system unstable. It is also reported that the presence of noise leads the direct adaptive control to be unbounded (Shun-Feng et al., 2006). Other technical issues (the stability, the parameter convergence, the performance improvement and the robustness) of either direct or indirect adaptive control brought to light new drawbacks and proposed partial solutions (Bibi et al., 2018; Hojati and Gazor, 2002; Shun-Feng et al., 2006).

One of the major questions that arises is: which approach can achieve control objectives with minimum cost and high performance? It has been proven that the integration of both direct and indirect adaptive control laws can improve performance (Li and Tong, 2016; Sun and Hou, 2017; Wang and Tong, 2018). For this purpose, some research has been reported which makes use of a combined direct and indirect adaptive control to compute the global control law (Li and Tong, 2016; Wang and Tong, 2018). Other schemes are based on hybrid parameter adaptation laws which use both the tracking and the modelling errors to update adaptive parameters (Hojati and Gazor, 2002; Sun and Hou, 2017). In this paper, we develop a hybrid fuzzy direct/indirect adaptive controller (HFDIAC) for uncertain nonlinear systems. A fuzzy logic controller (FLC) is designed to modulate between direct and indirect adaptive controllers based on the modelling error and its derivatives. It means that if the modelling error becomes large, therefore the overall control will be dominated by the direct adaptive control and vice versa. Based on the Lyaponov approach, the stability of the closed loop is performed, where the adaptive algorithms parameters are derived. Within this scheme, the control objectives can be achieved with a fast convergence and optimal control for different dynamic regimes.

This paper is organized as follows. First, the development of ideal control law for a class of uncertain nonlinear systems is described. Next, the direct adaptive control using universal approximators is presented, then the indirect adaptive control is detailed. The following section presents the fuzzy universal approximator used to estimate the direct and the indirect adaptive control. Next, the hybrid fuzzy direct/indirect adaptive controller (HFDIAC) for uncertain nonlinear systems is developed. The penultimate section presents numerical results of the simulated models which validate the proposed scheme of control. The paper ends with concluding remarks and comments.

Problem formulation

Most dynamic systems are essentially represented by a nonlinear state model, allowing their real dynamic behaviour to be described as accurately as possible. Examples of such systems are mechanical and electromechanical systems (unmanned aerial vehicle, unmanned ground vehicle, magnetic suspension system of low-speed maglev train, robot manipulator, etc.). Consider the standard nonlinear dynamic system:

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ ⋮ \\ {\overset{\cdot}{x}}_{n - 1} = x_{n + 1} \\ {\overset{\cdot}{x}}_{n} = f (x) + g (x) u + d \end{matrix}

(1)

y = x_{1}

(2)

$u$ ∈ R is the input and $y$ ∈ R is the output. $f (.)$ and $g (.)$ are unknown continuous functions and $d$ is the external bounded disturbance. When $g (x) \neq 0$ , the system (1) is controllable for measurable $x_{i}$ in a certain controllability region $U_{c} \in R^{n}$ , we assume that $(0 < g (x) < \infty)$ .

The tracking error is defined as

e_{t} (t) = y_{d} (t) - y (t)

(3)

Where $y_{d} (t) = [y_{d}, {\overset{\cdot}{y}}_{d}, \dots, y_{d}^{(n - 1)}]$ is the desired trajectory vector and $y (t) = [y, \overset{\cdot}{y}, \dots, y^{(n - 1)}]$ is the output vector.

The filtered tracking error $s_{t}$ is defined as

s_{t} = (\frac{d}{dt} + k)^{n - 1} e_{t} (t) = e_{t}^{n - 1} + \dots + k_{1} e_{t} = k^{T} e_{t}

(4)

where e _t $= [e_{t}, {\overset{\cdot}{e}}_{t}, \dots, e_{t}^{n - 1}]^{T}$ and $k = [k_{1}, \dots, k_{n - 1}]^{T}$ .

Let $k = [k_{1}, \dots, k_{n - 1}]^{T}$ be chosen such that all the roots of polynominal $H (s_{t}) = s_{t}^{n - 1} + k_{n - 1} s_{t}^{n - 2} + \dots + k_{1}$ are lie in the left half of the complex plane. The following exact feedback linearization control law which stabilizes the global closed-loop system cannot be implemented, because the nonlinear functions $f (x)$ and $g (x)$ are unknowns.

u = \frac{1}{g (x)} [- f (x) - \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} - d + y_{d}^{(n)}]

(5)

The design and the stability analysis for the uncertain nonlinear system (1) has matured to a stage where theoretically results have become well established. Among these, there are the adaptive model-based nonlinear system controllers (indirect adaptive controllers), which achieve global error convergence even in the presence of uncertainties in the nonlinear system dynamics (Bibi et al., 2017). The design process of this type of controller requires the estimated model of the nonlinear system (1), which is given by

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1} = x_{2} \\ ⋮ \\ {\overset{\cdot}{\hat{x}}}_{n - 1} = x_{n + 1} \\ {\overset{\cdot}{\hat{x}}}_{n} = \hat{f} (x) + \hat{g} (x) u \end{matrix}

(6)

where $\hat{f} (x)$ and $\hat{g} (x)$ are the estimates of the nonlinear functions $f (x)$ and $g (x)$ respectively.

The modelling error is defined as

e_{m} (t) = x_{n} (t) - {\hat{x}}_{n} (t)

(7)

We can also estimate directly the control input $u$ , without using the estimated model (6). in which the direct adaptive controller uses a universal approximator as controller to stabilize the nonlinear system (1) (Bibi et al., 2018).

Before the development of the direct and the indirect control law, it is required to make the following assumptions.

Assumption 1. The external bounded disturbance $d$ is bounded by $D$ , where, $| d | \leq D$ .

Assumption 2. The derivative of the unknown nonlinear function $g (x)$ is assumed bounded by an unknown continuous function $| \overset{\cdot}{g} (x) | \leq D (x)$ .

Assumption 3. The minimum approximation error $ϵ_{d} (x)$ has an upper positive bound . $| ϵ_{d} (x) | \leq η$ .

Assumption 4. The minimum approximation errors $ϵ_{f} (x)$ and $ϵ_{g} (x)$ have an upper bound, where $Ψ = ϵ_{f} (x) + ϵ_{g} (x) u$ , $| Ψ | \leq η$ .

Direct adaptive control using universal approximators

The ideal control law which stabilizes the global closed-loop system and ensures an optimal performance can be obtained as follows:

\begin{matrix} u^{*} = \frac{1}{g (x)} [- f (x) - \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} - d + y_{d}^{(n)}] - g (x) η sgn (s_{t}) \\ + \frac{D (x)}{2 g^{2} (x)} s_{t} \end{matrix}

(8)

The ideal control law given by the equation (8) cannot be implemented, because the nonlinear functions $f (x)$ , $g (x)$ and $D (x)$ are unknowns. Therefore, a fuzzy universal approximator is used to approximate the ideal control law given by the equation (8). The optimal direct control law can be written as

u^{*} = u^{*} (x, θ_{d}^{*}) + ϵ_{d} (x) = Φ_{d} (x) θ_{d}^{* T} + ϵ_{d} (x)

(9)

where $θ_{d}^{* T}$ is the vector of optimal parameter estimations and $ϵ_{d} (x)$ is the approximation error. The estimated direct control law is defined as

\hat{u} (x, θ_{d}) = Φ_{d} (x) θ_{d}^{T}

(10)

The adaptive law of the adjusted parameter $θ_{d}$ is chosen as

{\overset{\cdot}{\hat{θ}}}_{d} = γ_{d} s_{t} Φ_{d} (x)

(11)

Figure 1 shows the overall direct adaptive control scheme. Now, let us give the analytic stability proof of the direct adaptive control scheme, based on the direct Lyaponov approach.

Figure 1.

Direct adaptive control scheme.

Theorem 1: Applying the control law defined by (10) for the nonlinear system modelled by the nonlinear differential equation (1), and using the adaptation law given by (11). If Assumptions 1, 2 and 3 are satisfied, the following properties are guaranteed:

The boundedness of all signals is ensured in the closed loop.

The filtered tracking error $s_{t} \to 0$ when $t \to \infty$ which implicates the asymptotically convergence of the tracking errors and their derivatives to zero $e_{t}^{(i)} \to 0$ when $t \to \infty$ for $i = 0, \dots n - 1$ .

Proof of Theorem 1: Using the direct adaptive control, the candidate Lyapunov function $V_{d}$ is chosen as

V_{d} = \frac{1}{2 g (x)} s_{t}^{2} + \frac{1}{2 γ_{d}} {\tilde{θ}}_{d}^{T} {\tilde{θ}}_{d}

(12)

The derivative of the candidate Lyapunov function ${\overset{\cdot}{V}}_{d}$ is given by

{\overset{\cdot}{V}}_{d} = \frac{- \overset{\cdot}{g} (x)}{2 g^{2} (x)} s_{t}^{2} + \frac{1}{g (x)} s_{t} {\overset{\cdot}{s}}_{t} - \frac{1}{γ_{d}} {\tilde{θ}}_{d}^{T} {\overset{\cdot}{\hat{θ}}}_{d}

(13)

Let us determine the time derivative of the filtered tracking error ${\overset{\cdot}{s}}_{t}$

\begin{matrix} {\overset{\cdot}{s}}_{t} = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + y^{(n)} - y_{d}^{(n)} \\ = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + (f (x) + g (x) u + d) - y_{d}^{(n)} \\ = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + (f (x) + g (x) u + d) - y_{d}^{(n)} + g (x) u^{*} - g (x) u^{*} \end{matrix}

(14)

Applying (8) and (10), the derivative of the filtered tracking error ${\overset{\cdot}{s}}_{t}$ can be rewritten as

\begin{matrix} {\overset{\cdot}{s}}_{t} = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + f (x) - g (x) (u^{*} - \hat{u}) + d - y_{d}^{(n)} + g (x) u^{*} \\ = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + f (x) - g (x) \tilde{u} + d - y_{d}^{(n)} - f (x) - \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} \\ - d + y_{d} - g (x) η sgn (s_{t}) + \frac{D (x)}{2 g^{2} (x)} s_{t} \\ = - g (x) \tilde{u} - g (x) η sgn (s_{t}) + \frac{D (x)}{2 g^{2} (x)} s_{t} \end{matrix}

(15)

Replacing (11) and (15) in (13), the derivative of the candidate Lyapunov function ${\overset{\cdot}{V}}_{d}$ becomes

\begin{matrix} {\overset{\cdot}{V}}_{d} = \frac{[D (x) - \overset{\cdot}{g} (x)]}{2 g^{2} (x)} s_{t}^{2} + s_{t} Φ_{d} (x) {\tilde{θ}}_{d}^{T} + s_{t} [ϵ_{d} (x) - η sgn (s_{t})] \\ - \frac{1}{γ_{d}} {\tilde{θ}}_{d}^{T} {\overset{\cdot}{\hat{θ}}}_{d} \\ \leq \frac{[D (x) - \overset{\cdot}{g} (x)]}{2 g^{2} (x)} s_{t}^{2} + \frac{1}{γ_{d}} {\tilde{θ}}_{d}^{T} (γ_{d} s_{t} Φ_{d} (x) - {\overset{\cdot}{\hat{θ}}}_{d}) \\ + s_{t} [ϵ_{d} (x) - η sgn (s_{t})] \\ < | s_{t} | | ϵ_{d} (x) | - η | s_{t} | \end{matrix}

(16)

Where the approximation errors are bounded $| ϵ_{d} (x) | < η$ , consequently, the filtered tracking error $s_{t}$ and the approximation error ${\tilde{θ}}_{d}$ of the direct adaptive control are bounded.

Indirect adaptive control using universal approximators

For the same nonlinear system defined by the equation (1), there are other ways to design adaptive controllers, which can also achieve the desired performance. In this section we present the principal steps for developing an indirect adaptive control.

The estimated indirect control law is presented as

u_{I} = \frac{1}{\hat{g} (x, θ_{g})} [- \hat{f} (x, θ_{f}) - \sum_{i = 1}^{n - 1} k_{i} e^{i} + y_{d}^{(n)} - (η + D) sgn (s_{t})]

(17)

The unknown nonlinear functions $f (x)$ and $g (x)$ are presented as

f (x) = f^{*} (x, θ_{f}^{*}) + ϵ_{f} (x) = Φ_{f} (x) θ_{f}^{* T} + ϵ_{f} (x)

(18)

g (x) = g^{*} (x, θ_{g}^{*}) + ϵ_{g} (x) = Φ_{g} (x) θ_{g}^{* T} + ϵ_{g} (x)

(19)

where $θ_{f}^{* T}$ and $θ_{g}^{* T}$ are the vectors of optimal parameter estimations. $ϵ_{f} (x)$ and $ϵ_{f} (x)$ are the approximation errors of $f (x)$ and $g (x)$ respectively.

Using a fuzzy universal approximator, the estimation of unknown nonlinear functions $f (x)$ and $g (x)$ is defined as

\hat{f} (x, θ_{f}) = Φ_{f} (x) θ_{f}^{T}

(20)

\hat{g} (x, θ_{g}) = Φ_{g} (x) θ_{g}^{T}

(21)

The adaptive laws of the adjusted parameter $θ_{f}$ and $θ_{g}$ are chosen as

{\overset{\cdot}{\hat{θ}}}_{f} = γ_{f} s_{t} Φ_{f} (x)

(22)

{\overset{\cdot}{\hat{θ}}}_{g} = γ_{g} s_{t} Φ_{g} (x) u

(23)

The indirect adaptive control scheme is illustrated in Figure 2. Now, let us prove the stability of the closed-loop system using the indirect adaptive control defined by (17).

Figure 2.

Indirect adaptive control scheme.

Theorem 2: Applying the control law defined by (17) for the nonlinear system modelled by the nonlinear differential equation (1), and using the adaptation law given by (22-23). If Assumptions 1 and 4 are satisfied, the following properties are guaranteed:

The boundedness of all signals is ensured in the closed loop.

Proof of Theorem 2: Using the indirect adaptive control, the candidate Lyapunov function $V_{I}$ is chosen as

V_{I} = \frac{1}{2} s_{t}^{2} + \frac{1}{2 γ_{f}} {\tilde{θ}}_{f}^{T} {\tilde{θ}}_{f} + \frac{1}{2 γ_{g}} {\tilde{θ}}_{g}^{T} {\tilde{θ}}_{g}

(24)

The derivative of the candidate Lyapunov function ${\overset{\cdot}{V}}_{I}$ is given by

{\overset{\cdot}{V}}_{I} = s_{t} {\overset{\cdot}{s}}_{t} - \frac{1}{γ_{f}} {\tilde{θ}}_{f}^{T} {\overset{\cdot}{\hat{θ}}}_{f} - \frac{1}{γ_{g}} {\tilde{θ}}_{g}^{T} {\overset{\cdot}{\hat{θ}}}_{g}

(25)

Based on the indirect adaptive control given by (17), the time derivative of the filtered tracking error ${\overset{\cdot}{s}}_{t}$ can be expressed as

\begin{matrix} {\overset{\cdot}{s}}_{t} = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + y^{(n)} - y_{d}^{(n)} \\ = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + (f (x) + g (x) u + d) - y_{d}^{(n)} \\ = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + f (x) + g (x) u + d - y_{d}^{(n)} - \hat{g} (x, θ_{g}) u \\ + \hat{g} (x, θ_{g}) u \\ = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + f (x) + (g (x) - \hat{g} (x, θ_{g})) u - y_{d}^{(n)} \\ + \hat{g} (x, θ_{g}) u + d \end{matrix}

(26)

Introducing (17) in (26), the derivative of the filtered tracking error ${\overset{\cdot}{s}}_{t}$ can be rewritten as

\begin{matrix} {\overset{\cdot}{s}}_{t} = \sum_{i = 1}^{n - 1} k_{i} e_{t}^{i} + f (x) + (g (x) - \hat{g} (x, θ_{g})) u - y_{d}^{(n)} \\ - \hat{f} (x, θ_{f}) - \sum_{i = 1}^{n - 1} k_{i} e^{i} + y_{d}^{(n)} - (η + D) sgn (s_{t}) + d \\ = (f (x) - \hat{f} (x, θ_{f})) + (g (x) - \hat{g} (x, θ_{g})) u - (η + D) sgn (s_{t}) + d \\ = \tilde{f} (x) + \tilde{g} (x) u - (η + D) sgn (s_{t}) + d \end{matrix}

(27)

Let us determine the approximation errors $\tilde{f} (x)$ and $\tilde{g} (x)$ of the nonlinear functions $\tilde{f} (x)$ and $g (x)$ . The subtraction of (18) from (20) and (19) from (21) are given by

\begin{matrix} f (x) - \hat{f} (x, θ_{f}) = f^{*} (x, θ_{f}^{* T}) - \hat{f} (x, θ_{f}) + ϵ_{f} (x) \\ = Φ_{f} (x) {\tilde{θ}}_{f}^{T} + ϵ_{f} (x) \end{matrix}

(28)

\begin{matrix} g (x) - \hat{g} (x, θ_{g}) = g^{*} (x, θ_{g}^{* T}) - \hat{g} (x, θ_{g}) + ϵ_{g} (x) \\ = Φ_{g} (x) {\tilde{θ}}_{g}^{T} + ϵ_{g} (x) \end{matrix}

(29)

where ${\tilde{θ}}_{f} = θ_{f}^{*} - {\hat{θ}}_{f}$ and ${\tilde{θ}}_{g} = θ_{g}^{*} - {\hat{θ}}_{g}$ . When (28) and (29) are incorporated in (27), the filtered tracking error ${\overset{\cdot}{s}}_{t}$ can be written as

\begin{matrix} {\overset{\cdot}{s}}_{t} = Φ_{f} (x) {\tilde{θ}}_{f}^{T} + Φ_{g} (x) {\tilde{θ}}_{g}^{T} u + ϵ_{f} (x) + ϵ_{g} (x) u \\ - (η + D) sgn (s_{t}) + d \end{matrix}

(30)

Replacing (30) in (25), the derivative of the candidate Lyapunov function ${\overset{\cdot}{V}}_{I}$ becomes

\begin{matrix} {\overset{\cdot}{V}}_{I} = s_{t} Φ_{f} (x) {\tilde{θ}}_{f}^{T} + s_{t} Φ_{g} (x) {\tilde{θ}}_{g}^{T} u + s_{t} ϵ_{f} (x) + s_{t} ϵ_{g} (x) u \\ - (η + D) sgn (s_{t}) + d - \frac{1}{γ_{f}} {\tilde{θ}}_{f}^{T} {\overset{\cdot}{\hat{θ}}}_{f} - \frac{1}{γ_{g}} {\tilde{θ}}_{g}^{T} {\overset{\cdot}{\hat{θ}}}_{g} \\ \leq \frac{1}{γ_{f}} {\tilde{θ}}_{f}^{T} (γ_{f} s_{t} Φ_{f} (x) - {\overset{\cdot}{\hat{θ}}}_{f}) + \frac{1}{γ_{g}} {\tilde{θ}}_{g}^{T} (γ_{g} s_{t} Φ_{g} (x) u - {\overset{\cdot}{\hat{θ}}}_{g}) \\ + s_{t} Ψ - (η + D) sgn (s_{t}) + d \\ < - η | s_{t} | + s_{t} Ψ \end{matrix}

(31)

In the last subsections, it was shown that we can design both direct and indirect adaptive controllers for the nonlinear system (1). The stability is also proved for direct and indirect adaptive schemes. But we cannot specify which strategy of control ensures the fast convergence and the best performance.

Fuzzy universal approximator

The fuzzy neural networks (FNN) approximator is selected in the design of the fuzzy hybrid direct/indirect adaptive control. In this hybrid scheme, a FNN approximator is used to approximate the direct adaptive control $\hat{u} (x, θ_{d})$ , and two other FNN approximators are used to approximate the unknown nonlinear functions $f (x)$ and $g (x)$ in the case of indirect adaptive control.

The rule base of FNN consists of $N$ rules, has the following form:

$R^{k} :$ if $x_{1}$ is $F_{1}^{j 1}$ and $x_{2}$ is $F_{2}^{j 2}$ … and $x_{n}$ is $F_{n}^{jn}$ then $y = θ_{k}$ . $k = 1, \dots, N$

where $F_{i}^{ji}$ ⊂ $U_{ji}$ , $i = 1, 2, . . ., n$ , $j = 1, 2, . . ., m$ , $k = 1, 2, . . ., N$ are the antecedent type 1 membership functions, and $θ_{k}$ ⊂ W is the consequent type 1 membership function. By using the singleton fuzzifier, product inference engine and centre-average defuzzifier, the final output of the type 1 fuzzy system is given as follows:

Y (x) = \frac{Σ_{k = 1}^{N} μ_{k} (x) θ_{k}}{Σ_{k = 1}^{N} μ_{k} (x)}

(32)

where

μ_{k} (x) = Π_{i = 1}^{n} μ_{F_{i}^{k} (x_{i})}

The output of type 1 fuzzy system given by (32) can be rewritten in the following compact form:

Y (x) = Φ (x) * {\underline{θ}}^{T}

(33)

where $\underline{θ} = [θ_{1} \dots θ_{k}]$ is a vector of all consequent parameters, and $Φ (x) = [Φ_{1} (x) \dots Φ_{k} (x)]$ is a set of fuzzy basis functions defined as

Φ_{k} (x) = \frac{μ_{k} (x)}{Σ_{k = 1}^{N} μ_{k} (x)}

(34)

The fuzzy system is assumed to be well-defined so that $\sum_{k = 1}^{N} μ_{k} (x) \neq 0$ for all $x \in U$ . Figure 3 shows a schematic representation, in the form of a network, of a fuzzy universal approximator.

Figure 3.

Fuzzy universal approximator architecture.

Hybrid fuzzy direct/indirect adaptive controller (HFDIAC) for uncertain nonlinear systems

Using a FNN approximator, the direct and indirect adaptive control are computed simultaneously as

{\begin{matrix} u_{I} = \frac{1}{\hat{g} (x, θ_{g})} [- \hat{f} (x, θ_{f}) - Σ_{i = 1}^{n - 1} k_{i} e^{i} + y_{d}^{(n)}] \\ \hat{u} (x, θ_{d}) = Φ_{d} (x) θ_{d}^{T} \end{matrix}

(35)

where

{\begin{matrix} \hat{f} (x, θ_{f}) = Φ_{f} (x) θ_{f}^{T} \\ \hat{g} (x, θ_{g}) = Φ_{g} (x) θ_{g}^{T} \end{matrix}

(36)

The adaptive laws of the adjusted parameters $θ_{f}$ , $θ_{g}$ and $θ_{d}$ are chosen as

{\begin{matrix} {\overset{\cdot}{\hat{θ}}}_{f} = γ_{f} s_{t} Φ_{f} (x) \\ {\overset{\cdot}{\hat{θ}}}_{g} = γ_{g} s_{t} Φ_{g} (x) u \\ {\overset{\cdot}{\hat{θ}}}_{d} = γ_{d} s_{t} Φ_{d} (x) \end{matrix}

(37)

The robust control component is chosen as

u_{r} = - (η + D) sgn (s_{t})

(38)

A block diagram of a closed-loop system with a fuzzy logic controller (FLC) is shown in Figure 4. The FLC controller uses the modelling error ${\overset{\cdot}{e}}_{m} (t)$ and the rate of change in the modelling error $e_{m} (t)$ as its inputs. The inference engine processes the inputs using a knowledge base. If the plant knowledge is more important and reliable (small $e_{m} (t)$ ) than the direct control knowledge, the overall control will be dominated by the indirect control $u_{I}$ . The output of the inference engine is converted into crisp values, which is a fraction of both direct $\hat{u} (x, θ_{d})$ and indirect $u_{I}$ adaptive controls.

Figure 4.

Schematic diagram of hybrid fuzzy direct/indirect adaptive controller.

The interaction of the inputs and outputs into IF–THEN rules can be presented as the following rules matrix:

$e_{m} (t)$ ${\overset{\cdot}{e}}_{m} (t)$	Negative	Zero	Positive
Negative	$\hat{u} (x, θ_{d})$	$u_{I}$	$\hat{u} (x, θ_{d})$
Zero	$u_{I}$	$u_{I}$	$\hat{u} (x, θ_{d})$
Positive	$u_{I}$	$u_{I}$	$\hat{u} (x, θ_{d})$

We can use the product inference rule and the centre-average defuzzifier (equation (32)) to get the hybrid control law as

\begin{matrix} u_{h} = \frac{μ_{2} (x) + μ_{4} (x) + μ_{5} (x) + μ_{7} (x) + μ_{8} (x)}{Σ_{k = 1}^{9} μ_{k} (x)} u_{I} + \\ \frac{μ_{1} (x) + μ_{3} (x) + μ_{6} (x) + μ_{9} (x)}{Σ_{k = 1}^{9} μ_{k} (x)} \hat{u} (x, θ_{d}) \end{matrix}

(39)

The overall global control law is given by

u = α u_{I} + (1 - α) \hat{u} (x, θ_{d}) + u_{r}

(40)

where

\begin{matrix} α = \frac{μ_{2} (x) + μ_{4} (x) + μ_{5} (x) + μ_{7} (x) + μ_{8} (x)}{Σ_{k = 1}^{9} μ_{k} (x)} \\ 1 - α = \frac{μ_{1} (x) + μ_{3} (x) + μ_{6} (x) + μ_{9} (x)}{Σ_{k = 1}^{9} μ_{k} (x)} \end{matrix}

(41)

The candidate Lyapunov function $V$ is chosen as

V = \frac{1}{2} s_{t}^{2} + \frac{α}{2 γ_{f}} {\tilde{θ}}_{f}^{T} {\tilde{θ}}_{f} + \frac{α}{2 γ_{g}} {\tilde{θ}}_{g}^{T} {\tilde{θ}}_{g} + \frac{(1 - α)}{2 γ_{d}} {\tilde{θ}}_{d}^{T} {\tilde{θ}}_{d}

(42)

Replacing adaptive laws (37) and the ${\overset{\cdot}{s}}_{t}$ derivative of $s_{t}$ , the derivative of the candidate Lyapunov function $\overset{\cdot}{V}$ becomes

\overset{\cdot}{V} < - η | s_{t} | + | s_{t} | Ψ < 0

(43)

Where the approximation errors are bounded $η < Ψ$ , consequently, the filtered tracking error $s_{t}$ is also bounded.

Numerical results

In order to assess the proposed structure, various simulation studies are carried out for single-input single-output (SISO) and multi-input multi-output (MIMO) nonlinear systems; a direct performance comparison is performed based on the plant models taken from literature.

Example 1

Consider the inverted pendulum system:

f (x) = \frac{gsin (x_{1}) - (m_{p} l x_{1}^{2} \cos (x_{1}) \sin (x_{1})) / (m_{p} + m_{c})}{l (4 / 3 - m_{p} co s^{2} (x_{1})) / (m_{p} + m_{c}))}

(44)

g (x) = \frac{\cos (x_{1}) / (m_{p} + m_{c}))}{l (4 / 3 - m_{p} co s^{2} (x_{1})) / (m_{p} + m_{c}))}

(45)

where $x_{1} = θ$ and $x_{2} = \overset{\cdot}{θ}$ denote the angular position and velocity. $[g m_{c} m_{p} l] = [9.81 1 0.1 0.5]$ $= m / s^{2}$ . Three FNNs are used to approximate the nonlinear functions $f (x)$ , $g (x)$ and $\hat{u} (x, θ_{d})$ . $x = [x_{1}, x_{2}]$ is the vector input of the two networks. For each input $x_{i}$ , we define type 3 Gaussian membership functions, where $c = [c_{1}, c_{2}, c_{3}] = [- 0.25, 0, 0.5]$ , $= [σ_{1}, σ_{2}, σ_{3}]$ and $σ_{1} = σ_{2} = σ_{3} = \frac{π}{8}$ . The developed scheme is used to control the output of the system $y = θ$ to track the reference trajectory $y_{d} = θ_{d} = 0.2 \sin (t)$ in the presence of the external disturbance $d (t) = 0.1 \sin (t)$ as highlighted in Figure 5.

Figure 5.

HFDIAC tracking performances.

Figure 6.

Two-link rigid robot manipulator.

Example 2

Now, a case of controlling a two-link rigid robot manipulator is given to illustrate the effectiveness of the proposed control scheme for MIMO nonlinear systems.

The dynamic equations of the two-link rigid robot manipulator system are

[\begin{matrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{matrix}] [\begin{matrix} {\overset{\cdot\cdot}{q}}_{1} \\ {\overset{\cdot\cdot}{q}}_{2} \end{matrix}] + [\begin{matrix} - h {\overset{\cdot}{q}}_{2} & - h ({\overset{\cdot}{q}}_{1} + {\overset{\cdot}{q}}_{2}) \\ h {\overset{\cdot}{q}}_{1} & 0 \end{matrix}] [\begin{matrix} {\overset{\cdot}{q}}_{1} \\ {\overset{\cdot}{q}}_{2} \end{matrix}] = [\begin{matrix} u_{1} \\ u_{2} \end{matrix}]

(46)

where

{\begin{matrix} P_{11} = a_{1} + 2 a_{3} \cos (q_{2}) + 2 a_{3} \sin (q_{2}) \\ P_{12} = P_{21} = a_{2} + a_{3} \cos (q_{2}) + a_{4} \sin (q_{2}) \\ P_{22} = a_{2}, h = a_{3} \sin (q_{2}) - a_{4} \cos (q_{2}) \end{matrix}

with

{\begin{matrix} a_{1} = I_{1} + m_{1} l_{c 1}^{2} + I_{2} + m_{2} l_{c 2}^{2} + m_{2} l_{1}^{2}, a_{2} = I_{2} + m_{2} l_{c 2}^{2}, \\ a_{3} = m_{1} I_{1} l_{c 2} \cos (δ_{2}), a_{4} = m_{1} I_{1} l_{c 2} \sin (δ_{2}) \end{matrix}

In the simulation, the following parameter values are used $[m_{1} m_{2} I_{1} I_{2} l_{1} l_{2} l_{c 1} l_{c 2} δ_{2}] = [1 2 1 0.25 1 1.2 0.5 0.6 30 °]$ .

The system (44) can be written as

\overset{\cdot\cdot}{y} = F (x) + G (x) u

(47)

where $y = [q_{1}; q_{2}]^{T}$ , $u = [u_{1}; u_{2}]^{T}$ , $x = [q_{1}; {\overset{\cdot}{q}}_{1}; q_{2}; {\overset{\cdot}{q}}_{2}]^{T}$ and

{\begin{matrix} F (x) = [\begin{matrix} F_{1} (x) \\ F_{2} (x) \end{matrix}] = - P^{- 1} [\begin{matrix} - h {\overset{\cdot}{q}}_{2} & - h ({\overset{\cdot}{q}}_{1} + {\overset{\cdot}{q}}_{2}) \\ h {\overset{\cdot}{q}}_{1} & 0 \end{matrix}] [\begin{matrix} {\overset{\cdot}{q}}_{1} \\ {\overset{\cdot}{q}}_{2} \end{matrix}] \\ G (x) = [\begin{matrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{matrix}] = P^{- 1} = {[\begin{matrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{matrix}]}^{- 1} \end{matrix}

The matrix $G (x) = P^{- 1}$ is defined positive. The control goal is to make the robot manipulator follow the desired reference in the presence of the external disturbance $d (t) = 0.2 \sin (t)$ , the desired trajectories are $y_{d 1} = \sin (t)$ and $y_{d 2} = \cos (t)$ . Three FNNs are used to approximate the nonlinear functions $F (x)$ , $G (x)$ and $\hat{u} (x, θ_{d})$ . $x = [x_{1}, x_{2}]$ is the vector input of the networks used to approximate $F (x)$ and ${\hat{u}}_{1} (x, θ_{d})$ . $x = [x_{3}, x_{4}]$ is the vector input of the networks used to approximate $G (x)$ and ${\hat{u}}_{2} (x, θ_{d})$ . The FLC controller uses the modelling error $e_{m} (t) = y - \hat{y}$ and the rate of change in the modelling error ${\overset{\cdot}{e}}_{m} (t) = \overset{\cdot}{y} - \overset{\cdot}{\hat{y}}$ as its inputs. For each input $x_{i}$ , we define three Gaussian membership functions as

F_{i}^{ji} = \exp (- \frac{1}{2} (\frac{x_{i} + c_{j}}{σ_{j}}))

(48)

where $i = 1, \dots, 4$ , $j = 1, \dots, 3$ , $c = [c_{1}, c_{2}, c_{3}] = [0, 0.25, 0.5]$ , $= [σ_{1}, σ_{2}, σ_{3}]$ and $σ_{1} = σ_{2} = σ_{3} = \frac{π}{12}$ . The design parameters are selected as $k_{1} = 3$ , the sampling time $t = 0.001$ sec and the adaptive gains $= [γ_{f}, γ_{g}, γ_{d},] = [24, 24, 20.4]$ . The robot initial conditions are $x (0) = [- 0.25 0 0.25 0]^{T}$ .

Comparative simulation is conducted to examine the performance of the developed controller and compared with other control schemes proposed in the literature (Bibi et al., 2017, 2018). For this purpose, the mean absolute error, the mean of squares error and the control effort given by the equations (49), (50) and (51) respectively are fixed as performance criteria, in order to evaluate the efficiency of each controller (direct adaptive control, indirect adaptive control and the proposed controller). The numerical values are summarized in Table 1.

Table 1.

Comparison of the obtained performances.

CriterionThe controller	Direct	Indirect	Hybrid
MAE link 1	0.1369	0.2955	0.0603
MSE link 1	0.0370	0.1447	0.0093
MAE link 2	0.0963	0.1151	0.0665
MSE link 2	0.0191	0.0218	0.0138
Control effort E link 1	$1.14 10^{8}$	$1.18 10^{7}$	$0.45 10^{7}$
Control effort E link 2	$1.8 10^{7}$	$1.7 10^{6}$	$9.28 10^{5}$

The results of the simulation are shown in Figures 7 –12. The position tracking errors are illustrated in Figures 13 –15.

MAE = \frac{| Σ_{i = 1}^{k} y_{d} (k + 1) - y (k + 1) |}{k}

(49)

SSE = \frac{Σ_{i = 1}^{k} (y_{d} (k + 1) - y (k + {1))}^{2}}{k}

(50)

E = \sum_{i = 1}^{k} (u)^{2}

(51)

Figure 7.

Direct adaptive tracking performances for link 1.

Figure 8.

Indirect adaptive tracking performances for link 1.

Figure 9.

Hybrid adaptive tracking performances for link 1.

Figure 10.

Direct adaptive tracking performances for link 2.

Figure 11.

Indirect adaptive tracking performances for link 2.

Figure 12.

Hybrid adaptive tracking performances for link 2.

Figure 13.

Position tracking errors using direct adaptive controller.

Figure 14.

Position tracking errors using indirect adaptive controller.

Figure 15.

Position tracking errors using hybrid adaptive controller.

From the simulation figures and the statistical Table 1, we can see that the three controllers could ensure the tracking of the reference trajectory and also achieve the global stability. However, we notice also that there is a disparity in the control effort and the degree of accuracy for each controller. Figures 9 and 15 show that the hybrid fuzzy direct/indirect adaptive controller (HFDIAC) provides good results compared with the direct and indirect adaptive controller.

In summary, we studied various approaches to the simulation of both direct and indirect adaptive control for uncertain nonlinear systems reported in the literature using fuzzy neural networks as a universal approximator.

When comparing results, it was found that the results obtained using the developed hybrid fuzzy direct/indirect adaptive control (HFDIAC) responded very well compared with those obtained using the approaches reported in the literature. We also found that there are a number of ways to design either a direct or indirect adaptive controller for the same uncertain nonlinear system. In general, it is typically advantageous to use the HFDIAC based on a mechanism of adaptation, which uses both the tracking and the modelling errors; to account for more uncertainty may often result in a more robust closed-loop system.

Finally, when comparing the overall performance of all three controllers in the presence of external disturbances, the HFDIAC performs best under these conditions, followed by the direct controller and finally the indirect controller

Conclusion

In this paper, a new hybrid fuzzy direct/indirect adaptive controller (HFDIAC) for uncertain nonlinear systems is proposed. Based on the modelling error and its derivatives, the HFDIAC modulates between direct and indirect adaptive controllers in order to ensure fast convergence, optimal control for each time and closed-loop stability. The main contribution of the proposed scheme over the existing approaches is that it can combine the advantages of both direct and indirect controllers and optimize the control effort. In addition, our controller is applicable without any prior knowledge of the dynamic system. Numerical simulations for single-input single-output (SISO) and multi-input multi-output (MIMO) nonlinear systems demonstrate the controller performance in achieving control objectives in the presence of external disturbance.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Youssouf Bibi

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