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Abstract

The tracking control problem for nonlinear system with unknown control coefficient and external disturbances is investigated in this article. By inspiring extended state observer and K filter, a novel extended state filter is first developed for state and disturbance observations. Then, by means of backstepping approach, an adaptive composite output feedback controller is constructed based on the state and disturbance estimations. The stability analysis is presented on the basis of the Lyapunov stability method. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed control approach.

Keywords

Adaptive control anti-disturbance control unknown control coefficient extended state filter

Introduction

Since disturbances widely exist in many practice systems, for example, spacecraft system, power systems, industrial processes, and many other systems, a control system is required to have the ability to handle disturbances to provide satisfactory control performance. A number of control approaches have been proposed to deal with disturbances for nonlinear system, for example, stochastic control,¹ output regulator,^2,3 $H_{\infty}$ control,^4,5 sliding mode control,^6–9 and robust integral of the sign of the error.¹⁰ Those control methods improve the anti-disturbance performance from different aspects. Among them, $H_{\infty}$ control mainly pays attention to stability (or robust stability) analysis of systems in the absence of disturbances, and the robustness is generally achieved at the price of sacrificing their nominal control performances.¹¹ Sliding mode control method provides an efficient way to reject disturbances, but the unexpected chattering restrains its applications in many industrial systems.¹²

Recently, disturbance compensation control schemes (including active disturbance rejection control (ADRC)^13,14 and disturbance-observer-based control (DOBC)¹⁵) have received more and more attention and been regarded as a nice choice to accomplish robustness against uncertainties and disturbance rejection performance.^13–24 The basic idea of disturbance compensation control method is that an observer, including disturbance observer or extended state observer, is constructed to estimate the unknown disturbances and then a feedforward compensator together with a conventional feedback controller is combined to reject unknown disturbances to obtain a satisfactory control performance.^13,15–17 Furthermore, disturbance compensation control methods can provide a flexibility of structure, that is, the observer and the feedback controller can be independently designed and integrated. In consideration of the advantage, disturbance compensation control methods have been applied to various control fields.^{25–31–33,34} The fact that the control direction is unknown is one of sever uncertainties. And many scholars have paid attention to studying the control problem for systems with unknown control direction and obtained many meaningful results.^35–37

Note that in the disturbance compensation approach, when the control coefficient is unknown, the usual approach is to choose a nominal value of control coefficient. Then, the corresponding error and external disturbances are regarded as the lumped disturbances, and the disturbance estimation technique is employed to estimate them. However, if the nominal value is chosen to larger or smaller, it may not obtain good control performance. So, it is necessary to propose some control approaches to enhance the control performance of nonlinear system with unknown control coefficient.

In this article, the tracking control problem for nonlinear system with unknown control coefficient and external disturbances is addressed. First, a novel extended state filter is constructed to estimate system states and external disturbances. Then, using backstepping method, a composite adaptive output feedback controller is developed. The stability analysis is established on the basis of the Lyapunov function theory. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method.

The main contribution of this article is as follows: first, a novel extended state filter is proposed to estimate the unknown states and unknown disturbances, and second, in the disturbance compensation control framework, an adaptive control scheme is developed to solve the control problem for systems with control coefficient unknown.

Problem formulation and preliminaries

In this article, we consider the following uncertain system

y^{(n)} (t) = b u (t) + f (y (t)) + d (t)

(1)

where $y \in R$ is the controlled input, $u (t) \in R$ is the control input, $d (t) \in R$ is the external disturbances, $f (y (t))$ denotes the known nonlinear function, and $b > 0$ is an unknown constant.

Letting $x_{i} (t) = y^{(i - 1)} (t)$ for $i = 1, \dots, n - 1$ , system (1) is rewritten as

\begin{matrix} \overset{\cdot}{x} = Ax + Bbu + B (f (x_{1} (t)) + d (t)) \\ y (t) = x_{1} (t) \end{matrix}

(2)

where

\begin{matrix} x = [x_{1} (t), \dots, x_{n} (t)]^{T} \\ A = {[\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{matrix}]}_{n \times n}, B = {[\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}]}_{n \times 1} \end{matrix}

The control objective is to find an adaptive composite anti-disturbance controller such that

All closed-loop signals are uniformly ultimately bounded (UUB);

y tracks the reference signal $y_{r}$ , where $y_{r}$ and its kth-order derivative $y_{r}^{(k)}$ are supposed to be bounded and continuous.

In order to control design and stability analysis, an assumption condition is presented.

Assumption 1

The derivative of the disturbances is bounded, that is, $| \overset{\cdot}{d} (t) | \leq \bar{h}$ .

Controller design and stability analysis

Observer design

Similar to ADRC scheme, an extended state variable is added

x_{n + 1} (t) = d (t)

(3)

Then, the extended system equation is described as

\begin{matrix} \overset{\cdot}{\bar{x}} = \bar{A} \bar{x} + \bar{B} bu + \bar{B} f (x_{1} (t)) + Eh (t) \\ y = \bar{C} \bar{x} \end{matrix}

(4)

where

\begin{matrix} \bar{A} = [\begin{matrix} A & {(b_{d})}_{n \times 1} \\ 0_{1 \times n} & 0_{1 \times 1} \end{matrix}], b_{d} = B, \bar{B} = [\begin{matrix} B \\ 0_{1 \times 1} \end{matrix}] \\ \bar{C} = [\begin{matrix} 1 & 0_{1 \times n} \end{matrix}], E = [\begin{matrix} 0_{n \times 1} \\ 1 \end{matrix}], h (t) = \overset{\cdot}{d} (t) \end{matrix}

An extended state filter is constructed

\begin{matrix} \overset{\cdot}{ξ} = (\bar{A} - L \bar{C}) ξ + \bar{B} f + L \bar{C} \bar{x} \\ \overset{\cdot}{η} = (\bar{A} - L \bar{C}) η + \bar{B} u \end{matrix}

(5)

where $ξ = [ξ_{1}, \dots, ξ_{n + 1}]^{T}$ , $η = [η_{1}, \dots, η_{n + 1}]^{T}$ , and $L = [L_{1}, L_{2}, \dots, L_{n + 1}]^{T}$ are the observer parameters, which are determined later.

The estimate state is defined as

\hat{x} = ξ + b η

(6)

The estimation error is defined as

ε = [ε_{1}, \dots, ε_{n + 1}]^{T} = \bar{x} - \hat{x}

(7)

Then, its derivative is

\overset{\cdot}{ε} = (\bar{A} - L \bar{C}) ε + Eh (t)

(8)

Controller design

In this part, the controller is constructed by employing the observer-based adaptive backstepping technique. The recursive design is used in the subsystem

\begin{matrix} \overset{\cdot}{y} = x_{2} \\ {\overset{\cdot}{η}}_{2} = - L_{2} η_{1} + η_{3} \\ ⋮ \\ {\overset{\cdot}{η}}_{n} = - L_{n} η_{1} + u + η_{n + 1} \end{matrix}

(9)

Step 1. Define the tracking error

z_{1} = y - y_{r}

(10)

Then, the derivative of equation (10) is presented as

{\overset{\cdot}{z}}_{1} = x_{2} - {\overset{\cdot}{y}}_{r} = ε_{2} + ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}

(11)

The Lyapunov function for system (11) is defined as

V_{1} = ε^{T} P ε + \frac{1}{2} z_{1}^{2} + \frac{{\tilde{p}}^{2}}{2 p γ}

(12)

Computing the derivative of equation (12), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} & = 2 ε^{T} P ((\bar{A} - L \bar{C}) ε + Eh (t)) \\ + z_{1} (ε_{2} + ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}) - \frac{\tilde{p}}{p γ} \overset{\cdot}{\hat{p}} \\ = ε^{T} (P (\bar{A} - L \bar{C}) + (\bar{A} - L \bar{C})^{T} P) + 2 ε^{T} PEh (t) + z_{1} ε_{2} \\ + z_{1} (ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}) - \frac{\tilde{p}}{p γ} \overset{\cdot}{\hat{p}} \\ = ε^{T} (P (\bar{A} - L \bar{C}) + (\bar{A} - L \bar{C})^{T} P) + 2 ε^{T} PEh (t) + z_{1} ε_{2} \\ + z_{1} (ξ_{2} - {\overset{\cdot}{y}}_{r}) + b z_{1} α_{1} + b z_{1} z_{2} - \frac{\tilde{p}}{p γ} \overset{\cdot}{\hat{p}} \end{matrix}

(13)

where $α_{1}$ is a virtual control law and $z_{2} = η_{2} - α_{1} .$ The virtual control law $α_{1}$ is designed as

α_{1} = \hat{p} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r})

(14)

Substituting equation (14) into equation (13), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = & ε^{T} (P (\bar{A} - L \bar{C}) + (\bar{A} - L \bar{C})^{T} P) + 2 ε^{T} PEh (t) + z_{1} ε_{2} \\ + z_{1} (ξ_{2} - {\overset{\cdot}{y}}_{r}) + \frac{\hat{p}}{p} z_{1} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r}) + b z_{1} z_{2} - \frac{\tilde{p}}{p γ} \overset{\cdot}{\hat{p}} \\ = & ε^{T} (P (\bar{A} - L \bar{C}) + (\bar{A} - L \bar{C})^{T} P) + 2 ε^{T} PEh (t) + z_{1} ε_{2} \\ + z_{1} (ξ_{2} - {\overset{\cdot}{y}}_{r}) + \frac{p - \tilde{p}}{p} z_{1} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r}) + b z_{1} z_{2} - \frac{\tilde{p}}{p γ} \overset{\cdot}{\hat{p}} \\ = & ε^{T} (P (\bar{A} - L \bar{C}) + (\bar{A} - L \bar{C})^{T} P) + 2 ε^{T} PEh (t) + z_{1} ε_{2} - k_{1} z_{1}^{2} \\ - \frac{\tilde{p}}{p} z_{1} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r}) + b z_{1} z_{2} - \frac{\tilde{p}}{p γ} \overset{\cdot}{\hat{p}} \end{matrix}

(15)

According to Young’s inequality, we have

2 ε^{T} PEh (t) \leq (ε^{T} PE)^{2} + h (t)^{2}

(16)

Note that $| h (t) | = | \overset{\cdot}{d} |$ and combining with assumption 1, we obtain

2 ε^{T} PEh (t) \leq (ε^{T} PE)^{2} + h (t)^{2} \leq ε^{T} PE E^{T} P ε + {\bar{h}}^{2}

(17)

Similarly, we can derive

z_{1} ε_{2} \leq \frac{1}{4} ε_{2}^{2} + z_{1}^{2} \leq \frac{1}{4} ε^{T} ε + z_{1}^{2}

(18)

and define

π_{1} = - γ z_{1} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r})

(19)

Combining equations (15), (17), (18), and (19) yields

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{1}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) + b z_{1} z_{2} + \bar{h} \end{matrix}

(20)

Step 2. We consider the following dynamic system

\begin{matrix} {\overset{\cdot}{z}}_{2} = - L_{2} η_{1} + η_{3} - \frac{\partial α_{1}}{\partial z_{1}} (ε_{2} + ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}) \\ - \frac{\partial α_{1}}{\partial ξ_{1}} (- L_{1} ξ_{1} + ξ_{2} + L_{1} x_{1}) - \frac{\partial α_{1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} - \frac{\partial α_{1}}{\partial {\overset{\cdot}{y}}_{r}} {\overset{\cdot\cdot}{y}}_{r} \\ = - L_{2} η_{1} + α_{2} + z_{3} - \frac{\partial α_{1}}{\partial z_{1}} (ε_{2} + ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}) \\ - \frac{\partial α_{1}}{\partial ξ_{2}} (- L_{2} ξ_{1} + ξ_{3} + L_{2} x_{1}) - \frac{\partial α_{1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} - \frac{\partial α_{1}}{\partial {\overset{\cdot}{y}}_{r}} {\overset{\cdot\cdot}{y}}_{r} \end{matrix}

(21)

where $z_{3} = η_{3} - α_{2}$ .

Define the following Lyapunov function

V_{2} = V_{1} + \frac{1}{2} z_{2}^{2} + \frac{1}{2} {\tilde{θ}}^{2}

(22)

Computing the derivative of equation (22), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} = & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{1}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ + b z_{1} z_{2} + {\bar{h}}^{2} + z_{2} z_{3} + z_{2} \\ (- L_{2} η_{1} + α_{2} + z_{3} - \frac{\partial α_{1}}{\partial z_{1}} (ε_{2} + ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}) \\ - \frac{\partial α_{1}}{\partial ξ_{2}} (- L_{2} ξ_{1} + ξ_{3} + L_{2} x_{1}) - \frac{\partial α_{1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} - \frac{\partial α_{1}}{\partial {\overset{\cdot}{y}}_{r}} {\overset{\cdot\cdot}{y}}_{r}) - \tilde{θ} \overset{\cdot}{\hat{θ}} \end{matrix}

(23)

The virtual control law $α_{2}$ is designed as

\begin{matrix} α_{2} = & - (k_{2} + {(\frac{\partial α_{1}}{\partial z_{1}})}^{2}) z_{2} + L_{2} η_{1} + \frac{\partial α_{1}}{\partial z_{1}} (ξ_{2} - {\overset{\cdot}{y}}_{r}) \\ + \frac{\partial α_{1}}{\partial ξ_{2}} (- L_{2} ξ_{1} + ξ_{3} + L_{2} x_{1}) + \frac{\partial α_{1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} + \frac{\partial α_{1}}{\partial {\overset{\cdot}{y}}_{r}} {\overset{\cdot\cdot}{y}}_{r} \\ - \hat{θ} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} z_{2} \end{matrix}

(24)

Substituting equation (24) into equation (23) leads to

\begin{matrix} {\overset{\cdot}{V}}_{2} = & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{1}{4} I) ε - (k_{1} + 1) z_{1}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ + {\bar{h}}^{2} - (k_{2} + {(\frac{\partial α_{1}}{\partial z_{1}})}^{2}) z_{2}^{2} + \frac{\partial α_{1}}{\partial z_{1}} z_{2} ε_{2} + b z_{2} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2}) - \hat{θ} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} z_{2}^{2} + z_{2} z_{3} - \tilde{θ} \overset{\cdot}{\hat{θ}} \\ = & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{1}{4} I) ε - (k_{1} + 1) z_{1}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ + {\bar{h}}^{2} - (k_{2} + {(\frac{\partial α_{1}}{\partial z_{1}})}^{2}) z_{2}^{2} + \frac{\partial α_{1}}{\partial z_{1}} z_{2} ε_{2} + b z_{2} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2}) - \hat{b} z_{2} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2}) + z_{2} z_{3} - \tilde{θ} \overset{\cdot}{\hat{θ}} \\ = & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{1}{4} I) ε - (k_{1} + 1) z_{1}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ + {\bar{h}}^{2} - (k_{2} + {(\frac{\partial α_{1}}{\partial z_{1}})}^{2}) z_{2}^{2} + \frac{\partial α_{1}}{\partial z_{1}} z_{2} ε_{2} + b z_{2} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2}) - \hat{θ} z_{2}^{2} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} + z_{2} z_{3} - \tilde{θ} \overset{\cdot}{\hat{θ}} \\ = & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{1}{4} I) ε - (k_{1} + 1) z_{1}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ + {\bar{h}}^{2} - (k_{2} + {(\frac{\partial α_{1}}{\partial z_{1}})}^{2}) z_{2}^{2} + \frac{\partial α_{1}}{\partial z_{1}} z_{2} ε_{2} + b z_{2} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2}) - \hat{θ} z_{2}^{2} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} + z_{2} z_{3} - \tilde{θ} \overset{\cdot}{\hat{θ}} \end{matrix}

(25)

Note that

\begin{matrix} \frac{\partial α_{1}}{\partial z_{1}} z_{2} ε_{2} \leq {(\frac{\partial α_{1}}{\partial z_{1}})}^{2} z_{2}^{2} + \frac{1}{4} ε_{2}^{2} \leq {(\frac{\partial α_{1}}{\partial z_{1}})}^{2} z_{2}^{2} + \frac{1}{4} ε^{T} ε \\ b z_{2} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2}) \leq b^{2} z_{2}^{2} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} + \frac{1}{4} = θ z_{2}^{2} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} + \frac{1}{4} \end{matrix}

(26)

Define $π_{2} = z_{2}^{2} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2}$ . Combining equations (25) and (26), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{2}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - k_{2} z_{2}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{2}) + z_{2} z_{3} + {\bar{h}}^{2} + \frac{1}{4} . \end{matrix}

(27)

Step i $(3 \leq i \leq n)$ . Assume that we have designed virtual control laws $α_{j}$ and tuning functions $π_{j}$ , $j = 1, 2, \dots, i$ . The time derivative of the following Lyapunov function

V_{i - 1} = V_{i - 2} + \frac{1}{2} z_{i - 1}^{2}

(28)

satisfies

\begin{matrix} {\overset{\cdot}{V}}_{i - 1} & \leq ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{i - 1}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{i - 1} k_{j} z_{j}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ + z_{i - 1} z_{i} - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{i - 1}) + {\bar{h}}^{2} + \frac{i - 1}{4} \end{matrix}

(29)

where $z_{i} = η_{i} - α_{i - 1}$ . We consider the following Lyapunov function

V_{i} = V_{i - 1} + \frac{1}{2} z_{i}^{2}

(30)

Computing the derivative (30), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{i} & \leq ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{i - 1}{4} I) ε - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{i - 1}) \\ - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{i - 1} k_{j} z_{j}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) + {\bar{h}}^{2} + \frac{i - 1}{4} + z_{i - 1} z_{i} \\ + z_{i} (- L_{i} η_{1} + α_{i} + z_{i + 1} - \frac{\partial α_{i - 1}}{\partial z_{1}} (ε_{2} + ξ_{2} + b η_{2} - {\overset{\cdot}{y}}_{r}) - \sum_{j = 2}^{i - 1} \frac{\partial α_{i - 1}}{\partial η_{j}} (- L_{j} η_{1} + η_{j + 1}) \\ - \sum_{l = 1}^{i} \frac{\partial α_{i - 1}}{\partial ξ_{l}} (- L_{l} ξ_{1} + ξ_{l + 1} + L_{l} x_{1}) - \frac{\partial α_{i - 1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} - \frac{\partial α_{i - 1}}{\partial \hat{θ}} \overset{\cdot}{\hat{θ}} - \sum_{k = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial y_{r}^{(k)}} y_{r}^{(k + 1)}) \end{matrix}

(31)

The virtual control law $α_{i}$ is designed as

\begin{matrix} α_{i} = & - (k_{i} + {(\frac{\partial α_{i - 1}}{\partial z_{1}})}^{2}) z_{i} - z_{i - 1} + L_{i} η_{1} \\ + \frac{\partial α_{i - 1}}{\partial z_{1}} (ξ_{2} - {\overset{\cdot}{y}}_{r}) + \sum_{j = 2}^{i - 1} \frac{\partial α_{i - 1}}{\partial η_{j}} (- L_{j} η_{1} + η_{j + 1}) \\ + \sum_{l = 1}^{i} \frac{\partial α_{i - 1}}{\partial ξ_{l}} (- L_{l} ξ_{1} + ξ_{l + 1} + L_{l} x_{1}) \\ + \frac{\partial α_{i - 1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} + \sum_{k = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial y_{r}^{(k)}} y_{r}^{(k + 1)} + \frac{\partial α_{i - 1}}{\partial \hat{θ}} \overset{\cdot}{\hat{θ}} \\ - {(\frac{\partial α_{i - 1}}{\partial z_{1}} η_{2})}^{2} z_{i} \hat{θ} \end{matrix}

(32)

Substituting (32) into (31), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} & \leq ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{i - 1}{4} I) ε - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{i - 1} k_{j} z_{j}^{2} + z_{i} z_{i + 1} \\ - (k_{i} + {(\frac{\partial α_{i - 1}}{\partial z_{1}})}^{2}) z_{i}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) + {\bar{h}}^{2} + \frac{i - 1}{4} - z_{i} \frac{\partial α_{i - 1}}{\partial z_{1}} ε_{2} \\ - {(\frac{\partial α_{i - 1}}{\partial z_{1}} η_{2})}^{2} z_{i}^{2} \hat{θ} - b z_{i} \frac{\partial α_{i - 1}}{\partial z_{1}} η_{2} - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{i - 1}) + \frac{\partial α_{i - 1}}{\partial \hat{θ}} \overset{\cdot}{\hat{θ}} \end{matrix}

(33)

Note that

\begin{matrix} - z_{i} \frac{\partial α_{i - 1}}{\partial z_{1}} ε_{2} \leq {(\frac{\partial α_{i - 1}}{\partial z_{1}})}^{2} z_{i}^{2} + \frac{1}{4} ε^{T} ε \\ - b z_{i} \frac{\partial α_{i - 1}}{\partial z_{1}} η_{2} \leq {(\frac{\partial α_{i - 1}}{\partial z_{1}} η_{2})}^{2} z_{i}^{2} θ + \frac{1}{4} \\ π_{i} = π_{i - 1} + {(\frac{\partial α_{i - 1}}{\partial z_{1}} η_{2})}^{2} z_{i}^{2} \end{matrix}

(34)

Combining (33) and (34), gives rise to

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq & ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{i}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{i} k_{j} z_{j}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \\ - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{i}) + z_{i} z_{i + 1} + {\bar{h}}^{2} + \frac{i}{4} \end{matrix}

(35)

When $i = n$ , the actual controller u is design as

\begin{matrix} u = & - (k_{n} + {(\frac{\partial α_{n - 1}}{\partial z_{1}})}^{2}) z_{n} - z_{n - 1} \\ + L_{n} η_{1} - η_{n + 1} + \frac{\partial α_{n - 1}}{\partial z_{1}} (ξ_{2} - {\overset{\cdot}{y}}_{r}) \\ + \sum_{j = 2}^{n - 1} \frac{\partial α_{n - 1}}{\partial η_{j}} (- L_{j} η_{1} + η_{j + 1}) \\ + \sum_{l = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial ξ_{l}} (- L_{l} ξ_{1} + ξ_{l + 1} + L_{l} x_{1}) \\ + \frac{\partial α_{n - 1}}{\partial ξ_{n}} (- L_{n} ξ_{1} + ξ_{n + 1} + L_{l} x_{1} + f) \\ + \frac{\partial α_{n - 1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} + \sum_{k = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial y_{r}^{(k)}} y_{r}^{(k + 1)} \\ + \frac{\partial α_{n - 1}}{\partial \hat{θ}} \overset{\cdot}{\hat{θ}} - (\frac{\partial α_{n - 1}}{\partial z_{1}} η_{2})^{2} z_{n} \hat{θ} \end{matrix}

(36)

π_{n} = π_{n - 1} + {(\frac{\partial α_{n - 1}}{\partial z_{1}} η_{2})}^{2} z_{n}^{2}

(37)

The Lyapunov function is chosen as a

V_{n} = V_{n - 1} + \frac{1}{2} z_{n}^{2}

(38)

Then we have

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{n}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{n} k_{j} z_{j}^{2} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{n}) + {\bar{h}}^{2} + \frac{n}{4} \end{matrix}

(39)

The adaptive law is constructed as

\begin{matrix} \overset{\cdot}{\hat{p}} = {Proj}_{\hat{p}} (π_{1} - γ σ \hat{p}) \\ \overset{\cdot}{\hat{θ}} = {Proj}_{\hat{θ}} (π_{n} - σ \hat{θ}) \end{matrix}

(40)

where $Proj$ is a projection operator, which is used to guarantee the boundedness of $\hat{θ}$ and $\hat{p}$ , and the modified one is presented by

{Proj}_{\hat{p}} (•) = {\begin{matrix} 0, & if • = 0 and • < 0 \\ •, & otherwise \end{matrix}

(41)

According to equation (41), we obtain

\begin{matrix} - \frac{\tilde{p}}{p γ} (\overset{\cdot}{\hat{p}} - π_{1}) \leq σ \frac{\tilde{p} \hat{p}}{p} = σ \frac{\tilde{p} (p - \tilde{p})}{p} \leq - \frac{σ {\tilde{p}}^{2}}{2 p} + \frac{σ p}{2} \\ - \tilde{θ} (\overset{\cdot}{\hat{θ}} - π_{n}) \leq σ \tilde{θ} \hat{θ} \leq - \frac{σ}{2} {\tilde{θ}}^{2} + \frac{σ}{2} θ^{2} \end{matrix}

(42)

Considering equation (39)–(42), we have

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq ε^{T} (P (\bar{A} - L \bar{C}) + {(\bar{A} - L \bar{C})}^{T} P + PE E^{T} P + \frac{n}{4} I) ε \\ - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{n} k_{j} z_{j}^{2} - \frac{σ {\tilde{p}}^{2}}{2 p} - \frac{σ}{2} {\tilde{θ}}^{2} \\ + \frac{σ}{2} θ^{2} + \frac{σ p}{2} + {\bar{h}}^{2} + \frac{n}{4} \end{matrix}

(43)

Theorem 1

Consider system (1). The following inequalities hold

P (\bar{A} - L \bar{C}) + (\bar{A} - L \bar{C})^{T} P + PE E^{T} P + \frac{n}{4} I = - Q

(44)

Under controller (36) and adaptive laws (40), all signals of the closed-loop system are UUB.

Proof

Combining equations (43) and (44), we have

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq - ε^{T} Q ε - (k_{1} + 1) z_{1}^{2} - \sum_{j = 2}^{n} k_{j} z_{j}^{2} - \frac{σ {\tilde{p}}^{2}}{2 p} - \frac{σ}{2} {\tilde{θ}}^{2} + δ \\ \leq - μ V_{n} + δ \end{matrix}

(45)

where $μ = min {\frac{λ_{min} (Q)}{λ_{max} (P)}, 2 (k_{1} + 1), 2 k_{i}, σ γ, σ}$ and $δ = \frac{σ}{2} θ^{2} + \frac{σ p}{2} + {\bar{h}}^{2} + \frac{n}{4}$ .

Solving the inequality yields

0 \leq V_{n} (t) \leq \frac{δ}{μ} + (V_{n} (0) - \frac{δ}{μ}) e^{- μ t}

(46)

This implies that $V_{n} (t)$ is eventually bounded by $δ / α$ .

From equation (46), we obtain $z_{1}, z_{2}, \dots, z_{n}$ , $ε$ , $\hat{p}$ , and $\hat{θ}$ which are uniformly bounded. Also, $z_{1}$ is bounded, which indicates that $y = x_{1}$ is bounded. Then, with the help of the boundedness of $x_{1}$ , it is not difficult to obtain that $ξ_{1},$ $ξ_{2},$ and $ξ_{n + 1}$ are bounded. Note that $\bar{x} = ε + ξ + b η$ . Therefore, the boundedness of $x_{1}$ , $ε_{1}$ , and $ξ_{1}$ implies that $η_{1}$ is bounded. Furthermore, we can conclude that $x_{2}, \dots, x_{n}$ , $η_{2}, \dots, η_{n}$ , $α_{1}, \dots, α_{n - 1}$ , and u are bounded. Thus, all signals of the closed-loop system are UUB. Moreover, we can obtain that the track error $x_{1} - x_{d}$ converges to a small neighborhood of the origin. □

A numerical example

Consider the following second-order systems

\overset{\cdot\cdot}{y} = bu (t) + f (y (t)) + d (t)

(47)

where $y \in R$ is the controlled output, $u (t) \in R$ is the control input, $d (t) \in R$ is the external disturbances, $f (y (t))$ denotes the known nonlinear function, and $b > 0$ is an unknown constant.

The external disturbances and nonlinear function are presented as $d (t) = 0.2 \sin (t)$ and $f = 1.7 \sin (y (t)) + 0.4 \cos (y (t))$ . The parameter b is given as $b = 2$ . The reference signal is $y_{r} = \sin (t)$ .

According to the controller design in section “Controller design and stability analysis,” a composite controller for system (47) is presented as follows

\begin{matrix} u = & - (k_{2} + {(\frac{\partial α_{1}}{\partial z_{1}})}^{2}) z_{2} + L_{2} η_{1} + \frac{\partial α_{1}}{\partial z_{1}} (ξ_{2} - {\overset{\cdot}{y}}_{r}) \\ + \frac{\partial α_{1}}{\partial ξ_{2}} (- L_{2} ξ_{1} + ξ_{3} + L_{2} x_{1}) \\ + \frac{\partial α_{1}}{\partial \hat{p}} \overset{\cdot}{\hat{p}} + \frac{\partial α_{1}}{\partial {\overset{\cdot}{y}}_{r}} {\overset{\cdot\cdot}{y}}_{r} - \hat{θ} (z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})^{2} z_{2} \end{matrix}

(48)

\begin{matrix} \overset{\cdot}{\hat{p}} = {Proj}_{\hat{p}} (- γ z_{1} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r}) - γ σ \hat{p}) \\ \overset{\cdot}{\hat{θ}} = {Proj}_{\hat{θ}} (z_{2}^{2} {(z_{1} - \frac{\partial α_{1}}{\partial z_{1}} η_{2})}^{2} - σ \hat{θ}) \end{matrix}

(49)

\begin{matrix} \overset{\cdot}{ξ} = (\bar{A} - L \bar{C}) ξ + \bar{B} f + L \bar{C} \bar{x} \\ \overset{\cdot}{η} = (\bar{A} - L \bar{C}) η + \bar{B} u \end{matrix}

(50)

where $z_{1} = y - x_{r},$ $α_{1} = = \hat{p} (- k_{1} z_{1} - ξ_{2} + {\overset{\cdot}{y}}_{r})$ , $z_{2} = η_{2} - α_{1},$ $ξ = [ξ_{1}, ξ_{2}, ξ_{3}]^{T}$ , and $η = [η_{1}, η_{2}, η_{3}]^{T}$

\bar{A} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}], \bar{B} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], \bar{C} = {[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]}^{T}, L = [\begin{matrix} L_{1} \\ L_{2} \\ L_{3} \end{matrix}]

The controller parameters are listed as follows

\begin{array}{l} k_{1} = 5, k_{2} = 10, γ = 4, σ = 5, \\ L_{1} = 30, L_{2} = 300, L_{3} = 1000 \end{array}

The initial state of the system is $x (0) = [0.5 - 0.2]^{T}$ . The simulation time is set to $50 s$ . The simulation figures are listed as follows. Figure 1 describes the output curve of the closed-loop system under the proposed controller. In this figure, we can see that the proposed method can guarantee that the tracking performance is satisfactory. The disturbances and disturbance estimation are presented in Figure 2. From this figure, we know that the proposed observer can effectively estimate the unknown disturbances.

Figure 1.

Curves of system output y, reference signal $y_{r}$ , and tracking error.

Figure 2.

Curves of disturbances d and disturbance estimation $\hat{d}$ .

Conclusion

The problem of output tracking control for a class of system with unknown control coefficient and external disturbances has been investigated. In order to obtain the disturbance estimation value, an extended state filter has been proposed. Then, an output feedback controller has been developed on the basis of backstepping method. Using Lyapunov function technique, the closed-loop system stability has been analyzed. Finally, a numerical example has been provided to show the effectiveness of the proposed algorithm.

Footnotes

Academic Editor: Chenguang Yang

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China (nos 61403227, 61320106010, and 61627810), the Taishan Scholar Project of Shandong Province, and the Project supported by the Zhe-jiang Open Foundation of the Most Important Subjects.

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Adaptive anti-disturbance tracking control for nonlinear system with unknown control coefficient

Abstract

Keywords

Introduction

Problem formulation and preliminaries

Assumption 1

Controller design and stability analysis

Observer design

Controller design

Theorem 1

Proof

A numerical example

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References