Non-fragile adaptive fuzzy observer for nonlinear systems with uncertainties

Abstract

In this article, a non-fragile adaptive fuzzy observer is proposed for nonlinear systems with uncertain external disturbance and measurement noise. Firstly, the nonlinear system is augmented by an output filtered transformation. The output with measurement disturbance is put into the state equation of the augment system. Then, we introduce fuzzy logic system (FLS) to approximate the measurement disturbance, and construct an augmented non-fragile adaptive fuzzy observer for the augment system. A Lyapunov function is constructed to reveal that the characteristic of estimation errors is uniformly ultimately boundedness (UUB). Finally, two experimental simulations are offered to confirm the validity of the proposed design method.

Keywords

Non-fragile high-gain observer adaptive observer fuzzy logic system

1 Introduction

It is reported in industrial applications that due to round-off errors in calculation and sensor devices aging, observer gains may exist drifts [1]. For example, the application of observer in secure communication has been studied in [2, 3]. The basic principle of secure communication is that introduce a state observer to construct a receiving system synchronized with the chaotic system, and modulate a digital signal to a certain parameter of the transmission system. Then, the signal is demodulated by using the synchronization error at the receiving terminal. In addition, secure communication has been also achieved by designing electronic circuits in [4]. With aging of electronic components, there may exist observer gain perturbation. Recently, some researchers have commenced to discuss about observer design with gain perturbations [5–7], since the concept of non-fragile observer has been formally presented to resist the resilient observer gains [8]. However, the obtained results rely on linear matrix inequality (LMI) technology. The designed approach will be invalidity if the LMI-based sufficient conditions are infeasible.

On the other hand, uncertainties not only exist in observer gain but also in outputs of a system. For instance, there inevitably exists output noise in practical systems such as electrical devices [9] and mechanical systems [10]. The high-gain technology is one of the effective ways to resist unknown measurement uncertainty [11–14]. However, larger high-gain will produce the phenomenon of oscillation [15] and variation [16] in the presence of measurement uncertainty. Recently, filtering technique has been introduced to deal with the measurement disturbance [17]. The references [18, 19] also showed the influence of measurement uncertainties in high-gain observers when low-pass filter were inserted.

With the development of computer technology and artificial intelligence, fuzzy logic system (FLS) was proposed based on fuzziness characteristics of human brain thinking. FLS is especially suitable for nonlinear and time-varying systems [20]. Base on a fuzzifier, a fuzzy inference machine, a de-fuzzifier and a knowledge database, FLS reveals an excellent approximation ability to estimate an unknown disturbance [21]. It has been gradually applied in nonlinear system control field to approximate an unknown function [22–16]. An adaptive technology was also proposed to identify the weights of FLS [27].

It is well known that the adaptive technique devotes to estimate unknown states and parameters simultaneously for linear systems [28–33] and nonlinear systems [34–40]. For a speed sensorless induction motor, an adaptive high-gain observer has been designed in [41]. For a switched nonlinear system, an adaptive fuzzy observer has been established in [42]. Moreover, the author proposed a new adaptive estimation algorithm independent with the positive definite matrix by an easy-to-check persistent excitation condition in [37]. In the latest study [43], an enhanced feedback adaptive observer without persistent excitation has been proposed. However, the adaptive observer revealed an erratic performance under the circumstance of measurement disturbances [44, 45].

Motivate by above investigations, we propose a non-fragile adaptive fuzzy observer for nonlinear systems with measurement disturbance. The definition of non-fragile adaptive fuzzy observer is given, firstly. Secondly, the nonlinear system is augmented by applying an output filter. Then, the measurement noise is transformed to the state equations, and approximated by FLS. Next, the adaptive observer technique is used to estimate the weights of FLS and the unmeasurable states. Finally, by constructing a Lyapunov function, it is shown that the estimation errors are uniformly ultimately boundedness (UUB).

The main difficulties are as follows. (a) How to deal with gain perturbations of high-gain observers. (b) How to attenuate the effect of measurement noise to high-gain observers. The innovations and contributions of our work are as follows. (a) A non-fragile adaptive fuzzy observer is proposed for nonlinear systems with uncertain external disturbance and measurement noise based on an output filter and FLS approximation theory. (b) Sufficient conditions are given such that the estimation errors are UUB. The estimation error bound is adjustable.

This article is organized as follows. The preliminaries and problem description are located in Section 2. In Section 3, we propose the non-fragile adaptive fuzzy observer design method for nonlinear systems with uncertain external disturbance and measurement noise. The experimental simulation results are arranged in Section 4. The last section summarizes this article.

2 Preliminaries and problem description

In this section, we will present the problem statement and some fundamentals. Similar with [43], we are going to consider the following nonlinear system with both states uncertainty and measurement noise d (t),

\begin{matrix} \dot{\bar{x}} (t) = A_{0} \bar{x} (t) + B_{0} u (t) + F_{0} (t, \bar{x}) \\ + \sum_{j = 1}^{p} β_{j} {\bar{Ξ}}_{j} (t), \\ \bar{y} (t) = C_{0} \bar{x} (t) + d (t), \end{matrix}

(1)

where

\bar{x} (t)

\bar{y} (t)

and u (t) are the state variable, the output variable and the control variable, respectively.

A_{0} = (\begin{matrix} 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \\ 0 & 0 & \dots & 0 \end{matrix}),

B_{0} = (\begin{matrix} 0 \\ ⋮ \\ 0 \\ 1 \end{matrix})

and

C_{0} = (\begin{matrix} 1 & 0 & \dots & 0 \end{matrix})

, β_j is the unknown parameter,

{\bar{Ξ}}_{j} (t) = (\begin{matrix} φ_{1, j} (t) \\ ⋮ \\ φ_{n, j} (t) \end{matrix})

and φ_i,j (t) (i = 1, ⋯ , n, j = 1, ⋯ , p) is continuous function with an upper bound φ_max. The measurement uncertainty d (t) is a continuous nonlinear function. The nonlinear function

F_{0} (t, \bar{x}) = (f_{1} (t, {\bar{x}}_{1}^{ι}), f_{2} (t, {\bar{x}}_{2}^{ι}), \dots, f_{n} (t, {\bar{x}}_{n}^{ι}))^{T} \in ℝ^{n}

and

f_{i} (t, {\bar{x}}_{i}^{ι}) \in ℝ

are continuous nonlinear functions and satisfy

\begin{array}{l} | f_{i} (t, {\bar{x}}_{i}^{ι}) - f_{i} (t, {\hat{\bar{x}}}_{i}^{ι}) | \\ \leq ϱ (| {\bar{x}}_{1} (t) - {\hat{\bar{x}}}_{1} (t) | + \dots + | {\bar{x}}_{i} (t) - {\hat{\bar{x}}}_{i} (t) |), \end{array}

where ${\bar{x}}_{i}^{ι} = ({\bar{x}}_{1}, {\bar{x}}_{2}, \dots, {\bar{x}}_{i})$ , (i = 1, ⋯ , n) and ϱ > 0 is a constant.

Lemma 1.[46]: A continuous function $F (υ)$ on a compact set ϒ ⊂ R^q can be estimated by a FLS,

\begin{matrix} \hat{F} (υ) = {\hat{O}}^{T} φ (υ) + ξ, \end{matrix}

where υ = (υ₁, υ₂, ⋯ , υ_q) ^T ∈ ϒ is the input vector,

\hat{O} \in R^{p}

is the weight vector and ξ is the approximation error. The basis continuous function vector

φ (υ) = (φ_{1} (υ), φ_{2} (υ), \dots, φ_{p} (υ))^{T} \in ℝ^{p}

. Moreover, there exists a positive constant

\bar{ξ}

to let

| ξ | \leq \bar{ξ}

. The ideal weight O^* can be figured out by

O^{*} = \arg \min_{\hat{O} \in H_{F}} {\sup_{υ \in Υ} | \hat{F} (υ | \hat{O}) - F (υ) |},

where

H_{F} = {\hat{O} : ‖ \hat{O} ‖ \leq \bar{H}}

and

\bar{H}

is a constant. Then,

F (υ) = O^{* T} φ (υ) + ξ^{*}, | ξ^{*} | \leq \bar{ξ},

where ξ^* is the optimal approximation error.

From Lemma 1, the measurement noise d (t) can be approximated by,

\begin{matrix} d (t) = \sum_{i = 1}^{m} α_{i}^{*} φ_{i} (t) + d^{*} (t), \end{matrix}

(2)

where φ_i (t) is continuous basis function with the upper bound φ_max,

α_{i}^{*}

is the optimal coefficient and d^* (t) is the optimal approximation error with an upper bound

\bar{d}

. Our aim is to design a non-fragile adaptive fuzzy observer for the nonlinear system (1) with the measurement noise (2).

Lemma 2.[47] Let $A_{a} = (\begin{matrix} - a_{0} η (t) & v & 0 & \dots & 0 \\ - a_{1} η (t) & 0 & v^{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - a_{n} η (t) & 0 & 0 & \dots & 0 \end{matrix})$ and $P_{0} = (\begin{matrix} 1 & 0 & \dots & 0 \\ - b_{1} & 1 & \dots & 0 \\ 0 & - b_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix})$ , where η (t) is an unknown continuous and bounded function with ∞ > η_max ≥ η (t) ≥ η_min > 0, η_max and η_min are two constants. The parameters a_i, (i = 1, ⋯ , n) and b_i, (i = 1, ⋯ , n) can be calculated by,

\begin{matrix} v & = \frac{1}{2} v_{0}^{2}, \\ b_{i} & = \sum_{j = i}^{n} v_{0} v^{n - j}, \\ a_{i} & = b_{i} a_{i - 1}, \end{matrix}

where v₀ > 0 and v ≤ 1. The parameters a₀ and b₀ satisfy the following conditions,

\begin{matrix} a_{0} \geq 0.5 (0.75 v_{0} + \frac{b_{1}^{2}}{b_{0}} nv v_{0} + 2 v b_{1}) η_{min}^{- 1}, \end{matrix}

b_{0} = {\begin{matrix} 0.75, n = 1, \\ \max {b_{1}, \dots, b_{n}}, n \geq 2, \end{matrix}

and

\begin{matrix} b_{0} \leq \min \\ {\frac{0.75}{n - 1}, \frac{0.25}{(n - i) v + i - 1}, i = 1, \dots, n - 1} . \end{matrix}

Then,

A_{a}^{T} P_{1} + P_{1} A_{a} \leq - I,

(3)

where

P_{1} = \frac{P_{0}^{T} P_{0}}{0.25 v v_{0}^{n} λ_{min} (P_{0}^{T} P_{0})}

Note that the specific matrix P₁ is independent of η (t).

Lemma 3. Define $D = (\begin{matrix} v^{0} & 0 & 0 & \dots & 0 \\ 0 & v^{1} & 0 & \dots & 0 \\ 0 & 0 & v^{1 + 2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & v^{\frac{n (n + 1)}{2}} \end{matrix})$ and $A_{1} = (\begin{matrix} - {\bar{a}}_{0} η (t) & 1 & 0 & \dots & 0 \\ - {\bar{a}}_{1} η (t) & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - {\bar{a}}_{n} η (t) & 0 & 0 & \dots & 0 \end{matrix})$ , where ${\bar{a}}_{i} = v^{\frac{i (i + 1)}{2}} a_{i}$ . Let P = D^-1P₁D^-1. Then,

\begin{matrix} A_{1}^{T} P + P A_{1} \leq - I . \end{matrix}

(4)

Proof. From (3), it follows that,

\begin{matrix} A_{a}^{T} P_{1} + P_{1} A_{a} \\ = {DA}_{1}^{T} D^{- 1} P_{1} + P_{1} D^{- 1} A_{1} D \\ = {DA}_{1}^{T} D^{- 1} DPD + {DPDD}^{- 1} A_{1} D \\ = D (A_{1}^{T} P + P A_{1}) D \\ \leq - I . \end{matrix}

Note that λ_min (D^-1) =1. Then,

\begin{matrix} A_{1}^{T} P + P A_{1} \leq - I . \end{matrix}

The proof is completed.

3 The non-fragile adaptive fuzzy observer design strategies

Generally, the output error $\bar{y} (t) - \hat{y} (t)$ is used to design observers. However, for the nonlinear system with output disturbance, the observer gains will couple together with the output disturbance. In order to avoid the influence of the output disturbance on observer gains, we introduce an output filter to transform the output disturbance into the state equation.

Insert the following filtering action to the output with disturbance. Then, according to Lemma 1, we have

\begin{matrix} {\dot{x}}_{0} (t) = - {Lk}_{0} θ (t) x_{0} (t) + y (t) \\ = - {Lk}_{0} θ (t) x_{0} (t) + {\bar{x}}_{1} (t) \\ + \sum_{i = 1}^{m} φ_{i} (t) α_{i} + d^{*} (t), \end{matrix}

(5)

where L and k₀ are two positive constants, α_i is the unknown parameter and θ (t) ∈ (θ_min, θ_max) is the gain perturbation.

Remark 1. In order to explain the sensitivity of the observer gain, take the follow linear system as an example,

\begin{matrix} {\dot{\bar{x}}}^{*} (t) = A_{1}^{*} {\bar{x}}^{*} (t) + B_{1}^{*} u^{*} (t), \\ {\bar{y}}^{*} (t) = C_{1}^{*} {\bar{x}}^{*} (t), \end{matrix}

where

{\bar{x}}^{*} (t)

{\bar{y}}^{*} (t)

and u^* (t) are the state variable, the output variable and the control variable for the linear system, respectively.

A_{1}^{*} = (\begin{matrix} 0 & 0.5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix})

B_{1}^{*} = (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix})

and

C_{1}^{*} = (\begin{matrix} 1 & 0 & 0 & 0 \end{matrix})

. By introducing the output filter,

{\dot{x}}_{0}^{*} (t) = - x_{0}^{*} (t) + {\bar{y}}^{*} (t) = - x_{0}^{*} (t) + {\bar{x}}_{1}^{*} (t),

an augmented linear system can be obtained,

\begin{matrix} {\dot{x}}^{*} (t) = A^{*} x^{*} (t) + B^{*} u^{*} (t), \\ y^{*} (t) = C^{*} x^{*} (t), \end{matrix}

where y^* (t) is the output of the augmented linear system,

x^{*} (t) = (\begin{matrix} x_{0}^{*} (t) \\ {\bar{x}}^{*} (t) \end{matrix})

A^{*} = (\begin{matrix} - 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix})

B^{*} = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix})

and

C^{*} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \end{matrix})

. Then, design the following observer as,

\begin{matrix} {\dot{\hat{x}}}^{*} (t) = A^{*} {\hat{x}}^{*} (t) + B^{*} u^{*} (t) + κ C^{*} (x^{*} (t) - {\hat{x}}^{*} (t)) \\ {\hat{y}}^{*} (t) = C^{*} {\hat{x}}^{*} (t), \end{matrix}

where the gain vector

κ = (\begin{matrix} 6.49995 \\ 2371.84 \\ 15214.6 \\ 61223.4 \\ 2675.63 \end{matrix})

. Let

e (t) = x^{*} (t) - {\hat{x}}^{*} (t)

. The error system can be obtained as,

\dot{e} (t) = (A^{*} - κ C^{*}) e (t) .

The eigenvalue vector of A^* - κC^* is presented as follows,

(\begin{matrix} - 0.0385376546014988 + 48.6699287123693 i \\ - 0.0385376546014988 - 48.6699287123693 i \\ - 5.99488174522310 \\ - 0.37816911834361 \\ - 0.04982382722791 \end{matrix}) .

Obviously, all the eigenvalues have negative real parts, which means the error system is stable. However, after introducing a small perturbation Δκ = (-0.1 - 0.1 - 0.1 - 0.1 - 0.1) ^T, the eigenvalue vector of A^* - (κ + Δκ) C^* will be

(\begin{matrix} 0.0107712822402293 + 48.6754280153126 i \\ 0.0107712822402293 - 48.6754280153126 i \\ - 5.99349779786267 \\ - 0.37817304544277 \\ - 0.04982172117263 \end{matrix}) .

It reveals the error system is unstable. Due to

\frac{‖ Δ_{κ} ‖}{‖ κ ‖} = 0.0000061356

, we can conclude the error system may be fragile with an observer gain perturbation rate less than 10^-6.

According to (5), the following augmented system from (1) can be obtained,

\begin{matrix} \dot{x} (t) = A_{2} x (t) + B_{1} u (t) + F (t, x) + Ξ (t) γ \\ + B_{2} d^{*} (t), \\ y (t) = C_{1} x (t), \end{matrix}

(6)

where y (t) is the output of the augmented system (6),

x (t) = (\begin{matrix} x_{0} (t) \\ \bar{x} (t) \end{matrix})

B_{1} = (\begin{matrix} 0 \\ B_{0} \end{matrix})

B_{2} = (\begin{matrix} 1 \\ 0_{n \times 1} \end{matrix})

F (t, x) = (\begin{matrix} 0 \\ F_{0} (t, \bar{x}) \end{matrix})

C_{1} = (\begin{matrix} C_{0} 0 \end{matrix})

A_{2} = (\begin{matrix} - {Lk}_{0} θ (t) & 1 & \dots & 0 \\ ‖ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 \\ 0 & 0 & \dots & 0 \end{matrix})

, γ = (α₁ ⋯ α_m β₁ ⋯ β_p) ^T and

Ξ (t) = (\begin{matrix} φ_{1} (t) & \dots & φ_{m} (t) & 0 & \dots & 0 \\ 0 & \dots & 0 & φ_{1, 1} (t) & \dots & φ_{1, p} (t) \\ ⋮ & ⋮ & ⋮ \\ ‖ 0 & \dots & 0 & φ_{n, 1} (t) & \dots & φ_{n, p} (t) \end{matrix})

The time-varying matrix Γ (t) is obtained by linearly filtering Ξ (t) through

\dot{Γ} (t) = LA Γ (t) + L Ξ (t), Γ (0) = 0,

(7)

where

A = (\begin{matrix} - k_{0} θ (t) & 1 & \dots & 0 \\ - k_{1} θ (t) & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ - k_{n} θ (t) & 0 & \dots & 0 \end{matrix})

and k₀, k₁, ⋯ , k_n are some positive constants to be designed.

Lemma 4. The time-varying matrix Γ (t) satisfies

\begin{matrix} ‖ Γ (t) ‖_{2} \leq \frac{2 λ_{\max}^{\frac{3}{2}} (P) (m + (n \times p))^{\frac{1}{2}} φ_{\max}}{λ_{\min}^{\frac{1}{2}} (P)}, \end{matrix}

(8)

where P is given in Lemma 3.

Proof. Define the Lyapunov function as $L (t) = tr {Γ^{T} (t) P Γ (t)}$ . From Lemma 3, we can deduce

\begin{matrix} \dot{L} (t) = tr {{\dot{Γ}}^{T} (t) P Γ (t) + Γ^{T} (t) P \dot{Γ} (t)} \\ \leq Ltr {Γ^{T} (t) (A^{T} P + PA) Γ (t)} \\ + 2 Ltr {Γ^{T} (t) P Ξ (t)} \\ \leq - Ltr {Γ^{T} (t) Γ (t)} + \frac{1}{2 λ_{\max} (P)} L L (t) \\ + 2 L λ_{\max}^{2} (P) tr {Ξ^{T} (t) Ξ (t)} \\ \leq - \frac{1}{λ_{\max} (P)} L L (t) + \frac{1}{2 λ_{\max} (P)} L L (t) \\ + 2 L λ_{\max}^{2} (P) (m + (n \times p)) φ_{\max}^{2} \\ \leq - \frac{1}{2 λ_{\max} (P)} L L (t) \\ + 2 L λ_{\max}^{2} (P) (m + (n \times p)) φ_{\max}^{2} . \end{matrix}

(9)

It is easy to find out that

\begin{matrix} L (t) \leq \exp (- \frac{1}{2 λ_{\max} (P)} L (t - t_{0})) L (t_{0}) \\ + 4 λ_{\max}^{3} (P) (m + (n \times p)) φ_{\max}^{2} . \end{matrix}

Moreover,

\begin{matrix} ‖ Γ (t) ‖_{2}^{2} \leq ‖ Γ (t) ‖_{F}^{2} \\ \leq \frac{1}{λ_{\min} (P)} L (t) \\ \leq \frac{4 λ_{\max}^{3} (P) (m + (n \times p)) φ_{\max}^{2}}{λ_{\min} (P)} . \end{matrix}

Therefore, the inequation (8) can be obtained.

Propose the following observer for the augmented system (6),

\begin{matrix} \dot{\hat{x}} (t) = A_{2} \hat{x} (t) + Ω (y (t) - \hat{y} (t)) + \hat{F} (t, \hat{x}) \\ + B_{1} u (t) + L^{- 1} Γ (t) \dot{\hat{γ}} (t), \\ \dot{Γ} (t) = LA Γ (t) + L Ξ (t), \\ \dot{\hat{γ}} (t) = Λ Γ^{T} (t) C_{1}^{T} (y (t) - \hat{y} (t)), \\ \hat{y} (t) = C_{1} \hat{x} (t) . \end{matrix}

(10)

where

\hat{γ} = ({\hat{α}}_{1} (t) \dots {\hat{α}}_{m} (t) {\hat{β}}_{1} (t) \dots {\hat{β}}_{p} (t))^{T}

Ω = (\begin{matrix} 0 \\ L^{2} k_{1} θ (t) \\ ⋮ \\ L^{n + 1} k_{n} θ (t) \end{matrix})

and Λ is a positive definite diagonal matrix.

Definition 1. By the output filter (5), the nonlinear system (1) can be augmented to the system (6). We construct the observer (10) with the gain perturbation θ (t) ∈ (θ_min, θ_max). If there exists two positive real numbers t₁ and d₁ satisfying

\begin{matrix} | {\bar{x}}_{j} (t) - {\hat{x}}_{j} (t) | \leq d_{1}, j = 1, \dots, n, \forall t > t_{1}, \end{matrix}

(11)

which indicates the estimation errors satisfying UUB, then we call the system (10) is a non-fragile adaptive fuzzy observer of the nonlinear system (1).

Make the following coordinate transformation,

\begin{matrix} {\tilde{x}}_{i} (t) = \frac{x_{i} (t) - {\hat{x}}_{i} (t)}{L^{i}}, i = 0, 1, \dots, n, \\ \tilde{γ} (t) = L (γ - \hat{γ} (t)) . \end{matrix}

According to (6) and (10), it is easy to find out

\begin{matrix} \dot{\tilde{x}} (t) = LA \tilde{x} (t) + \tilde{F} + L Ξ (t) \tilde{γ} (t) + Γ (t) \dot{\tilde{γ}} (t) \\ + B_{2} d^{*} (t), \end{matrix}

(12)

where

\tilde{F} = (\begin{matrix} 0 \\ \frac{1}{L} (F_{1} (t, x) - {\hat{F}}_{1} (t, \hat{x})) \\ ⋮ \\ \frac{1}{L^{n}} (F_{n} (t, x) - {\hat{F}}_{n} (t, \hat{x})) \end{matrix})

Let $z (t) = \tilde{x} (t) - Γ (t) \tilde{γ} (t)$ . From (7), the error system becomes

\begin{matrix} \dot{z} (t) = LAz (t) + \tilde{F} + B_{2} d^{*} (t) + (LA Γ (t) \\ + L Ξ (t) - \dot{Γ} (t)) \tilde{γ} (t) \\ = LAz (t) + \tilde{F} + B_{2} d^{*} (t) . \end{matrix}

(13)

Next, we give the following results of the non-fragile adaptive fuzzy observer.

Theorem 1.If the high-gain L satisfies that $L > 4 ϱ n ‖ P ‖_{2} + 2 ‖ P ‖_{2}^{2} ‖ B_{2} ‖_{2}^{2} + 2 ‖ C_{1}^{T} C_{1} ‖_{2}$ , then the system (10) is a non-fragile adaptive fuzzy observer for the nonlinear system (1).Proof. Construct the following Lyapunov function $V (t) = z^{T} (t) Pz (t) + {\tilde{γ}}^{T} (t) Λ^{- 1} \tilde{γ} (t)$ . Then,

\begin{matrix} \dot{V} (t) = {\dot{z}}^{T} (t) Pz (t) + z^{T} (t) P \dot{z} (t) \\ + {\dot{\tilde{γ}}}^{T} (t) Λ^{- 1} \tilde{γ} (t) + {\tilde{γ}}^{T} (t) Λ^{- 1} \dot{\tilde{γ}} (t) \\ = {\dot{z}}^{T} (t) Pz (t) + z^{T} (t) P \dot{z} (t) \\ - 2 {\tilde{γ}}^{T} (t) Λ^{- 1} Λ Γ^{T} (t) C_{1}^{T} (y (t) - \hat{y} (t)) \\ \leq {\dot{z}}^{T} (t) Pz (t) + z^{T} (t) P \dot{z} (t) \\ - 2 {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} \tilde{x} (t) \\ \leq {Lz}^{T} (t) (A^{T} P + PA) z (t) + 2 z^{T} (t) P \tilde{F} \\ + 2 z^{T} (t) {PB}_{2} d^{*} \\ - 2 {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} (z (t) + Γ (t) \tilde{γ} (t)) . \end{matrix}

Due to

\begin{matrix} ‖ \tilde{F} ‖_{2} \leq \sqrt{ϱ^{2} (| z_{1} (t) |)^{2} \dots + ϱ^{2} (| \frac{z_{1} (t)}{L^{n - 1}} | + \dots + | z_{n} (t) |)^{2}} \\ \leq ϱ \sqrt{n^{2} (z_{1}^{2} (t) + \dots + z_{n}^{2} (t))} \\ \leq ϱ n ‖ z (t) ‖_{2}, \end{matrix}

and

\begin{matrix} - 2 {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} (z (t) + Γ (t) \tilde{γ} (t)) \\ = - 2 {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} z (t) \\ - 2 {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} Γ (t) \tilde{γ} (t) \\ \leq - {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} Γ (t) \tilde{γ} (t) \\ - {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} Γ (t) \tilde{γ} (t) \\ - {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} z (t) - z^{T} (t) C_{1}^{T} C_{1} Γ (t) \tilde{γ} (t) \\ - z^{T} (t) C_{1}^{T} C_{1} z (t) + z^{T} (t) C_{1}^{T} C_{1} z (t) \\ \leq - {\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} C_{1} Γ (t) \tilde{γ} (t) + z^{T} (t) C_{1}^{T} C_{1} z (t) \\ - ({\tilde{γ}}^{T} (t) Γ^{T} (t) C_{1}^{T} + z^{T} (t) C_{1}^{T}) (C_{1} Γ (t) \tilde{γ} (t) + C_{1} z (t)) \\ \leq ‖ C_{1}^{T} C_{1} ‖_{2} ‖ z (t) ‖_{2}^{2}, \end{matrix}

and Lemma 3, we can obtain

\begin{matrix} \dot{V} (t) \leq - L ‖ z (t) ‖_{2}^{2} + 2 ‖ P ‖_{2} ϱ n ‖ z (t) ‖_{2}^{2} \\ + ‖ P ‖_{2}^{2} ‖ B_{2} ‖_{2}^{2} ‖ z (t) ‖_{2}^{2} + ‖ C_{1}^{T} C_{1} ‖_{2} ‖ z (t) ‖_{2}^{2} + {\bar{d}}^{2} \\ \leq - \frac{1}{2} L ‖ z (t) ‖_{2}^{2} + {\bar{d}}^{2}, \end{matrix}

where $\frac{1}{2} L \geq 2 ϱ n ‖ P ‖_{2} + ‖ P ‖_{2}^{2} ‖ B_{2} ‖_{2}^{2} + ‖ C_{1}^{T} C_{1} ‖_{2}$ .

If $\frac{1}{2} L ‖ z (t) ‖_{2}^{2} \geq 2 {\bar{d}}^{2}$ , the following inequality can be deduced

\begin{matrix} \int_{t_{0}}^{t} \dot{V} (t) dt = V (t) - V (t_{0}) \leq - {\bar{d}}^{2} (t - t_{0}) . \end{matrix}

Let $t_{1} = t_{0} + \frac{V (t_{0})}{{\bar{d}}^{2}} - \frac{4 λ_{\min} (P)}{L}$ . When t > t₁, there exists

\begin{matrix} V (t) \leq V (t_{0}) - {\bar{d}}^{2} (t - t_{0}) \leq λ_{\min} (P) \frac{4 {\bar{d}}^{2}}{L} . \end{matrix}

Obviously, it reveals that

\begin{matrix} ‖ \tilde{γ} (t) ‖_{2} \leq 2 \sqrt{\frac{1}{L}} \bar{d} \frac{λ_{\min} (P)}{λ_{\min} (Λ^{- 1})}, \end{matrix}

and

\begin{matrix} ‖ z (t) ‖_{2} = ‖ \tilde{x} (t) - Γ (t) \tilde{γ} (t) ‖_{2} \leq 2 \sqrt{\frac{1}{L}} \bar{d} . \end{matrix}

(14)

Combining with (8), (9) and (14), for j = 1, ⋯ , n, we have

\begin{matrix} | {\bar{x}}_{j} (t) - {\hat{x}}_{j} (t) | \\ \leq L^{j} (‖ z (t) ‖_{2} + ‖ Γ (t) \tilde{γ} (t) ‖_{2}) \\ \leq 2 L^{j - \frac{1}{2}} \bar{d} (1 + χ), \end{matrix}

(15)

where

χ = \frac{2 λ_{\min} (P) λ_{\max}^{\frac{3}{2}} (P) (m + (n \times p))^{\frac{1}{2}} φ_{\max}}{λ_{\min} (Λ^{- 1}) λ_{\min}^{\frac{1}{2}} (P)}

Obviously, the estimation error is UUB. The proof is completed.

Remark 2. Note that χ and the high-gain L are independent of the optimal approximation error of FLS. Because the FLS is able to approximate a continuous function with an arbitrary accuracy, $\bar{d}$ can be arbitrarily small as long as the basis functions are abundant enough. Therefore, the estimation error bound can be adjusted to an arbitrarily small value.

4 Simulations

In this section, two numerical simulations based on practical systems are presented to illustrate the correctness and validity of our observer design strategies.

Example 1. Consider a DC motor driven fan speed control system with measurement disturbance d (t) described in [49]. The dynamic equation is shown as

{\begin{matrix} J_{1} \dot{s} (t) = κ_{1} I (t) - τ (t), \\ J_{2} \dot{I} (t) = u (t) - κ_{2} s (t) - κ_{3} I (t) + d_{e} (t), \\ y (t) = s (t) + d (t), \end{matrix}

where s (t) , I (t) , τ (t) are the fan speed, armature current and load torque, respectively. The input u (t) =5 is the armature voltage. d_e (t) is the uncertain external disturbance. J₁, J₂, κ₁, κ₂, κ₃ are the coefficients of the system.

Let ${\bar{x}}_{1} (t) = s (t)$ , ${\bar{x}}_{2} (t) = \frac{κ_{1}}{J_{1}} I (t)$ . The fan speed control system becomes

{\begin{matrix} {\dot{\bar{x}}}_{1} (t) = {\bar{x}}_{2} (t) - \frac{1}{J_{1}} τ (t), \\ {\dot{\bar{x}}}_{2} (t) = - \frac{κ_{1} κ_{2}}{J_{1} J_{2}} {\bar{x}}_{1} (t) - \frac{κ_{3}}{J_{2}} {\bar{x}}_{2} (t) + \frac{κ_{1}}{J_{1} J_{2}} u (t) \\ + \frac{1}{J_{2}} d_{e} (t), \\ \bar{y} (t) = {\bar{x}}_{1} (t) + d (t) . \end{matrix}

(16)

Then, insert an output filter, introduce a FLS to estimate d (t) and set the coefficients J₁ = J₂ = κ₁ = κ₂ = κ₃ = 1, τ (t) = x₁ (t) sin(x₁). It becomes

{\begin{matrix} {\dot{x}}_{0} (t) = - {Lk}_{0} θ (t) x_{0} (t) + x_{1} (t) \\ + \sum_{i = 1}^{4} φ_{i} (t) α_{i} + d^{*} (t), \\ {\dot{x}}_{1} (t) = x_{2} (t) - x_{1} (t) sin (x_{1}) + d_{e} (t), \\ {\dot{x}}_{2} (t) = - x_{1} (t) - x_{2} (t) + u (t), \\ y (t) = x_{0} (t) . \end{matrix}

(17)

The non-fragile adaptive fuzzy observer is constructed as,

{\begin{matrix} {\dot{\hat{x}}}_{0} (t) = - {Lk}_{0} θ (t) {\hat{x}}_{0} (t) + {\hat{x}}_{1} (t) \\ + L^{- 1} \sum_{i = 1}^{4} Γ_{0, i} (t) {\dot{\hat{α}}}_{i}, \\ {\dot{\hat{x}}}_{1} (t) = {\hat{x}}_{2} (t) - {\hat{x}}_{1} (t) sin ({\hat{x}}_{1}) \\ + L^{2} k_{1} θ (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{x}}}_{2} (t) = - {\hat{x}}_{1} (t) - {\hat{x}}_{2} (t) + u (t) \\ + L^{3} k_{2} θ (t) (y (t) - \hat{y} (t)), \\ {\dot{Γ}}_{0, 1} (t) = - {Lk}_{0} θ (t) Γ_{0, 1} (t) + L φ_{1} (t), \\ {\dot{Γ}}_{0, 2} (t) = - {Lk}_{0} θ (t) Γ_{0, 2} (t) + L φ_{2} (t), \\ {\dot{Γ}}_{0, 3} (t) = - {Lk}_{0} θ (t) Γ_{0, 3} (t) + L φ_{3} (t), \\ {\dot{Γ}}_{0, 4} (t) = - {Lk}_{0} θ (t) Γ_{0, 4} (t) + L φ_{4} (t), \\ {\dot{\hat{α}}}_{1} (t) = 2 Γ_{0, 1} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{α}}}_{2} (t) = 4 Γ_{0, 2} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{α}}}_{3} (t) = 6 Γ_{0, 3} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{α}}}_{4} (t) = 8 Γ_{0, 4} (t) (y (t) - \hat{y} (t)), \\ \hat{y} (t) = {\hat{x}}_{0} (t) . \end{matrix}

(18)

The initial conditions are given by x (0) = (0, 0, 0) ^T, $\hat{x} (0) = (1, 2, 3)^{T}$ , $\hat{α} (0) = (0, 0, 0, 0)^{T}$ and Γ_0,1 (0) = Γ_0,2 (0) = Γ_0,3 (0) = Γ_0,4 (0) =0. Set L = 10, v₀ = 0.6, $φ_{1} (t) = \frac{1}{1 + e^{- h (t)}} + 3.5, φ_{2} (t) = \frac{1}{1 + e^{- h (t)}} - 1.9, φ_{3} (t) = \frac{1}{1 + e^{- h (t)}} - 2.6, φ_{4} (t) = \frac{1}{1 + e^{- h (t)}} + 1.2$ , where $h (t) = \frac{150 (cos (5 t) - 5)}{1 + 8 t} + sin (5 t)$ . The observer gain vector can be obtained as k = (5, 3.54, 2.124). Set the measurement disturbance d (t) =0.5 sin(8t), the gain perturbation θ (t) =0.9 + 0.2 cos(2t), and the uncertain external disturbance d_e (t) =0. We plot the trajectories of the estimation errors of the non-fragile adaptive fuzzy observer in Fig. 1.

Fig. 1

The trajectories of estimation errors.

In order to demonstrate our observer have better performance than the following high-gain observer of the system (16),

{\begin{matrix} {\dot{\hat{x}}}_{1} (t) = {\hat{x}}_{2} (t) - {\hat{x}}_{1} (t) sin ({\hat{x}}_{1}) \\ + {Lk}_{1} θ (t) (\bar{y} (t) - \hat{y} (t)), \\ {\dot{\hat{x}}}_{2} (t) = - {\hat{x}}_{1} (t) - {\hat{x}}_{2} (t) + u (t) \\ + L^{2} k_{2} θ (t) (\bar{y} (t) - \hat{y} (t)), \\ \hat{y} (t) = {\hat{x}}_{1} (t), \end{matrix}

(19)

where

{\hat{x}}_{1} (t)

{\hat{x}}_{2} (t)

and

\hat{y} (t)

are the states and output of the observer, respectively. Set the same gain perturbation θ (t) =0.9 + 0.2 cos(2t), and the observer gains k₁ = 3.54, k₂ = 2.124, and L = 10. The initial conditions are given by

{\hat{x}}_{1} (0) = 2

{\hat{x}}_{2} (0) = 3

, and

{\bar{x}}_{1} (0) = 0

{\bar{x}}_{2} (0) = 0

. We display the comparison results in Figs. 2 and 3, respectively. It is observed that our observer design method has smaller errors.

Fig. 2

The comparison trajectories of estimation error of x₁.

Fig. 3

The comparison trajectories of estimation error of x₂.

Example 2. In this example, we consider the nonlinear system (17) containing the uncertain external disturbance d_e (t) =0.5 sin(5t) +1.5 sin(10t) while keeping other conditions fixed. Set $\hat{x} (0) = (3, 2, 1)^{T}$ , $\hat{γ} (0) = (0, 0, 0, 0, 0, 0)^{T}$ and Γ_i,j (0) =0, (i = 0, 1, 2, j = 1, 2, 3, 4, 5, 6). Then, by selecting $φ_{1, 1} (t) = \frac{1}{1 + e^{- h (t)}} + 3, φ_{1, 2} (t) = \frac{1}{1 + e^{- h (t)}} - 1, φ_{2, 1} (t) = \frac{1}{1 + e^{- h (t)}} - 2, φ_{2, 2} (t) = \frac{1}{1 + e^{- h (t)}} + 1$ , the observer can be established as,

\begin{matrix} {\dot{\hat{x}}}_{0} (t) = - {Lk}_{0} θ (t) {\hat{x}}_{0} (t) + {\hat{x}}_{1} (t) + L^{- 1} \sum_{i = 1}^{6} Γ_{0, i} (t) {\dot{\hat{γ}}}_{i}, \\ {\dot{\hat{x}}}_{1} (t) = {\hat{x}}_{2} (t) - {\hat{x}}_{1} (t) sin ({\hat{x}}_{1}) \\ + L^{2} k_{1} θ (t) (y (t) - \hat{y} (t)) + L^{- 1} \sum_{i = 1}^{6} Γ_{1, i} (t) {\dot{\hat{γ}}}_{i}, \\ {\dot{\hat{x}}}_{2} (t) = - {\hat{x}}_{1} (t) - {\hat{x}}_{2} (t) + u (t) \\ + L^{3} k_{2} θ (t) (y (t) - \hat{y} (t)) + L^{- 1} \sum_{i = 1}^{6} Γ_{2, i} (t) {\dot{\hat{γ}}}_{i}, \\ {\dot{Γ}}_{0, 1} (t) = - {Lk}_{0} θ (t) Γ_{0, 1} (t) + L Γ_{1, 1} (t) + L φ_{1} (t), \\ {\dot{Γ}}_{0, 2} (t) = - {Lk}_{0} θ (t) Γ_{0, 2} (t) + L Γ_{1, 2} (t) + L φ_{2} (t), \\ {\dot{Γ}}_{0, 3} (t) = - {Lk}_{0} θ (t) Γ_{0, 3} (t) + L Γ_{1, 3} (t) + L φ_{3} (t), \\ {\dot{Γ}}_{0, 4} (t) = - {Lk}_{0} θ (t) Γ_{0, 4} (t) + L Γ_{1, 4} (t) + L φ_{4} (t), \\ {\dot{Γ}}_{0, 5} (t) = - {Lk}_{0} θ (t) Γ_{0, 5} (t) + L Γ_{1, 5} (t), \end{matrix}

\begin{matrix} {\dot{Γ}}_{0, 6} (t) = - {Lk}_{0} θ (t) Γ_{0, 6} (t) + L Γ_{1, 6} (t), \\ {\dot{Γ}}_{1, 1} (t) = - {Lk}_{0} θ (t) Γ_{0, 1} (t) + L Γ_{2, 1} (t), \\ {\dot{Γ}}_{1, 2} (t) = - {Lk}_{0} θ (t) Γ_{0, 2} (t) + L Γ_{2, 2} (t), \\ {\dot{Γ}}_{1, 3} (t) = - {Lk}_{0} θ (t) Γ_{0, 3} (t) + L Γ_{2, 3} (t), \\ {\dot{Γ}}_{1, 4} (t) = - {Lk}_{0} θ (t) Γ_{0, 4} (t) + L Γ_{2, 4} (t), \\ {\dot{Γ}}_{1, 5} (t) = - {Lk}_{0} θ (t) Γ_{0, 5} (t) + L Γ_{2, 5} (t) + L φ_{1, 1} (t), \\ {\dot{Γ}}_{1, 6} (t) = - {Lk}_{0} θ (t) Γ_{0, 6} (t) + L Γ_{2, 6} (t) + L φ_{1, 2} (t), \\ {\dot{Γ}}_{2, 1} (t) = - {Lk}_{0} θ (t) Γ_{0, 1} (t), \\ {\dot{Γ}}_{2, 2} (t) = - {Lk}_{0} θ (t) Γ_{0, 2} (t), \\ {\dot{Γ}}_{2, 3} (t) = - {Lk}_{0} θ (t) Γ_{0, 3} (t), \\ {\dot{Γ}}_{2, 4} (t) = - {Lk}_{0} θ (t) Γ_{0, 4} (t), \\ {\dot{Γ}}_{2, 5} (t) = - {Lk}_{0} θ (t) Γ_{0, 5} (t) + L φ_{2, 1} (t), \\ {\dot{Γ}}_{2, 6} (t) = - {Lk}_{0} θ (t) Γ_{0, 6} (t) + L φ_{2, 2} (t), \end{matrix}

\begin{matrix} {\dot{\hat{γ}}}_{1} (t) = 2 Γ_{0, 1} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{γ}}}_{2} (t) = 4 Γ_{0, 2} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{γ}}}_{3} (t) = 6 Γ_{0, 3} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{γ}}}_{4} (t) = 8 Γ_{0, 4} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{γ}}}_{5} (t) = 10 Γ_{0, 5} (t) (y (t) - \hat{y} (t)), \\ {\dot{\hat{γ}}}_{6} (t) = 12 Γ_{0, 6} (t) (y (t) - \hat{y} (t)), \\ \hat{y} (t) = {\hat{x}}_{0} (t) . \end{matrix}

The simulation results are shown in Fig. 4, which confirms the effectiveness of the designed non-fragile adaptive fuzzy observer.

Fig. 4

The trajectories of estimation errors for nonlinear system with uncertain external disturbance.

5 Conclusion

We presented the definition and design method of the non-fragile adaptive fuzzy observer for nonlinear systems with uncertainties. By designing a Lyapunov function, the estimation error was proved UUB. In addition, the estimation error bound could be an arbitrarily small value by adjusting some parameters. In the future, the obtained results can be extended to nonlinear adaptive controller with uncertainties.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (62273200).

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