Adaptive neural output feedback control for a class of switched non-linear systems with unknown backlash-like hysteresis of the actuator

Abstract

This paper investigates the problem of adaptive neural output feedback control for a class of switched non-linear systems, and the unknown backlash-like hysteresis of the actuator is also taken into consideration. First, neural networks are used to approximate the uncertain functions in the studied system. Second, a state-observer is proposed to estimate the system states. Finally, an adaptive neural output feedback control algorithm based on a backstepping technique is constructed; in addition, dynamic surface control is applied to eliminate the explosion in complexity caused by the backstepping technique. By using Lyapunov stability theory, it is proved that all the signals of the switched system are bounded under the proposed control scheme. The effectiveness of the proposed approach is further confirmed by simulation experiments.

Keywords

Adaptive neural control backlash-like hysteresis dynamic surface control uncertain switched non-linear systems

Introduction

A switched system is a very important hybrid system composed of a series of subsystems and a switching law. Switched systems have been widely used over the past few decades in applications such as aircraft flight control systems, robotic systems, electric power systems, network control systems and others. In Lian et al. (2017), a sampled data state feedback controller was constructed to control the stability of a class of switched system, and the proposed method was verified by a simulation experiment for the F-18 aircraft. In the work of Liu et al. (2007), a type-II fuzzy control algorithm was proposed for biped robotic control. An innovative pulse-width modulation switched algorithm (Williams and Hoft, 1991) has been designed for the programmable logic controller adaptive frequency domain control. An adaptive controller based on Bernoulli random variables has been constructed for a random switched intelligent network system (Yang et al., 2013). Compared with the specific engineering applications, the theoretical research on switched systems is more extensive. For some linear switched systems (Philippe et al., 2016; Santis et al., 2003; Su and Huang, 2012; Wirth, 2005; Valcher and Zorzan, 2016) and non-linear switched systems (Heydari and Balakrishnan, 2014; Li et al., 2017; Liu and Wu, 2017; Su et al., 2016), the researchers put forward the corresponding control methods and solve a series of related problems in which fuzzy logic systems (Li and Tong, 2016; Tong et al., 2015) and neural networks (Cai and Xiang, 2015; Han et al., 2009; Long and Zhao, 2015) are widely used to approximate the uncertain functions in non-linear switched systems.

There are many issues in the study of non-linear switched systems, such as the actuator dead-zone, time-delay, backlash-like hysteresis, and a series of problems. The existence of such problems may lead to system instability or reduced system performance; hence, the study of these issues is necessary. Tong et al. (2015) proposed the adaptive output feedback control algorithm by using the backstepping technique and the average dwell-time method, so as to solve the problems of the non-linear function, the unmeasured state and the actuator dead-zone in the switched system. For switched systems consisting of a finite number of linear delay differential equations (DDEs) (Kim et al., 2006), a common Lyapunov function is used to construct a controller to stabilize the switched system with time delay. Additionally, as a non-linear input, the problem of hysteresis is ubiquitous in a variety of systems, such as aerospace (Viswamurthy and Ganguli, 2007), military (Peroutka et al., 2010) systems. Researchers have done a lot of related work on hysteresis of actuators in the system. Robust adaptive state feedback control algorithms (Su et al., 2000; Zhou et al., 2004) have been proposed for non-linear systems with hysteresis of the actuator. An adaptive output feedback control method based on the fuzzy logic system (Li et al., 2012; Shahnazi et al., 2010) has been proposed, which makes the non-linear system with unknown hysteresis of the actuator stable. In the work of Cui et al. (2017), Wang et al. (2017) and Niu et al. (2016), the authors proposed an adaptive state feedback control algorithm based on neural networks for a class of switched systems with hysteresis of the actuator, in which the neural network is used to approximate the non-linear unknown function in the switched system. However, in practical applications, many system states are not measurable. Even if the system status is measurable, the measured status may great deviate from the true status due to sensor fault, for example, which eventually leads to system instability.

When using backstepping techniques, traditional design methods require repeated differentiation of the virtual controller, and the complexity of the control increases rapidly as the order of the system increases. Then, there is the issue of explosion of complexity. However, the dynamic surface control (DSC) method simplifies the backstepping design for a class of non-linear strict-feedback systems, and further overcomes the problem of explosion of complexity. Wang et al. (2017) and Bai et al. (2017), through the combination of DSC, constructed a state feedback and an output feedback control method for a class of strict-feedback systems.

Inspired by the above research, this article combines the DSC method and the adaptive neural network control method to construct an output feedback controller to solve the tracking problem of a switched system with input hysteresis. First, this article can achieve a similar control effect Cui et al. (2017), Wang et al. (2017) and Niu et al. (2016), but the system states must be measurable in these works. Second, compared with Li and Tong (2016), we consider a class of switched systems with input hysteresis and further avoid the issue of the explosion of complexity caused by the traditional backstepping technique.

The notation used here can be found at the end of the paper.

Problem formulation and preliminaries

Model description

A class of uncertain switched non-linear systems with unknown backlash-like hysteresis of the actuator is given as follows; such uncertain switched non-linear systems can represent a one-link robot system (Li and Tong, 2016), as well as a double-propeller helicopter system, for example.

{\begin{matrix} {\overset{\cdot}{x}}_{i} = x_{i + 1} + f_{i, δ (t)} ({\bar{x}}_{i}) + d_{i . δ (t)} (t) \\ {\overset{\cdot}{x}}_{n} = u (v) + f_{n, δ (t)} ({\bar{x}}_{n}) + d_{n . δ (t)} (t) \\ y = x_{1} \end{matrix},

(1)

where ${\bar{x}}_{i} = [x_{1}, x_{2}, \dots, x_{i}]^{T} \in R^{i}, i = 1, 2, \dots, n$ indicates the status of the system; $δ (t) = k : [0, + \infty) \to S = {1, 2, \dots, s}$ denotes the switching signal; $f_{i, k} ({\bar{x}}_{i}) (i = 1, 2, \dots, n$ and $k = 1, 2, \dots, s)$ represent an unknown smooth non-linear function representing the uncertainty of the system model; $d_{i, k} (t) (i = 1, 2, \dots, n$ and $k = 1, 2, \dots, s)$ indicate the external disturbance; y indicates the output of the switched system; $u (v)$ is the output of an unknown backlash-like hysteresis of the actuator; and the specific form of expression is

\frac{du}{dt} = τ | \frac{dv}{dt} | (φ v - u) + η \frac{dv}{dt},

(2)

where v indicates the input of the unknown backlash-like hysteresis of the actuator, $τ$ , $φ$ and $η$ are the unknown constants, and $φ > 0$ indicates the slope of the line, simultaneously $φ > η$ .

Through the analysis of Su et al. (2000, 2003), the solution of system (2) can be defined as:

u (t) = φ v (t) + κ (v),

(3)

κ (v) = [u_{0} - φ v_{0}] e^{- τ (v - vo) sgn \overset{\cdot}{v}} + e^{- τ v sgn \overset{\cdot}{v}} \int_{v_{0}}^{v} [η - φ] e^{τ ς (sgn \overset{\cdot}{v})} d ς,

(4)

where $u_{0}$ and $v_{0}$ are the initial values of u and v. Further, according to Su et al. (2000, 2003), $κ (v)$ is bounded as shown:

| κ (v) | \leq \bar{κ} .

(5)

Applying equation (3), equation (1) can be written as:

{\begin{matrix} {\overset{\cdot}{x}}_{i} = x_{i + 1} + f_{i, δ (t)} ({\bar{x}}_{i}) + d_{i . δ (t)} (t), \\ {\overset{\cdot}{x}}_{n} = φ v (t) + κ (v) + f_{n, δ (t)} ({\bar{x}}_{n}) + d_{n . δ (t)} (t) \\ y = x_{1} \end{matrix}

(6)

Assumption 1. The external disturbance $d_{i, k} (t)$ of the switched system is bounded, and it is assumed that

| d_{i, k} | \leq {\bar{d}}_{i} .

(7)

Assumption 2. The reference signal $y_{d}$ is smooth, bounded and available.

Neural network

In this paper, the radial basis function (RBF) neural network is used to approximate the non-linear functions . For arbitrary $ε > 0$ , there exists a neural network $Φ^{T} ϕ (Z)$ , then one can have

H (Z) = Φ^{T} ϕ (Z) + σ (Z), | σ (Z) | \leq ε,

(8)

where $Z \in Ω_{z} \subset R^{m}$ represents the input vector of the RBF neural network, $ϕ (Z) = [ϕ_{1} (Z), ϕ_{2} (Z), \dots, ϕ_{n} (Z)]^{T}$ indicates the basis function vector, $Φ = [Φ_{1}, Φ_{2}, \dots, Φ_{N}]^{T}$ denotes the weight vector, $N > 1$ is the number of nodes in the neural network. The following Gaussian function can be chosen as a basis function:

ϕ_{i} (Z) = \exp [- \frac{{(Z - μ_{i})}^{T} (Z - μ_{i})}{ψ_{i}^{2}}],

(9)

where $ψ_{i}$ and $μ_{i} = [μ_{i 1}, μ_{i 2}, \dots, μ_{im}]^{T}$ represent the width and centre vector of the Gaussian function, respectively.

Adaptive output feedback control design and stability analysis

State-observer design

Before designing the controller, it is necessary to design the state-observer. In this section, a state-observer is constructed to estimate the unmeasured states of system (1).

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i} = {\hat{x}}_{i + 1} + l_{i} (y - x_{1}) i = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{\hat{x}}}_{n} = u (v) + l_{n} (y - x_{1}) \end{matrix}

(10)

Where $l_{i}$ is a constant that is designed, the above formula can be described in the following form:

\overset{\cdot}{\hat{x}} = A \hat{x} + Ly + Bu,

(11)

where the specific expressions of A, L and B are given by

A = [\begin{matrix} - l_{1} \\ ⋮ & I_{n - 1} \\ - l_{n} & \dots & 0 \end{matrix}],

(12)

L = [l_{1} l_{2} \dots l_{n}]^{T},

(13)

B^{T} = [0 \dots 0 1] .

(14)

Let A be Hurwitz by choosing the suitable L . For any symmetric positive definite matrix Q, there is a matrix $P = P^{T} > 0$ such that equation (15) holds:

A^{T} P + PA = - Q .

(15)

Define the observer error

{\begin{matrix} e_{1} = x_{1} - {\hat{x}}_{1} \\ e_{2} = x_{2} - {\hat{x}}_{2} \\ ⋮ \\ e_{n} = x_{n} - {\hat{x}}_{n} \end{matrix}

(16)

By using systems (1) and (11), one has

\overset{\cdot}{e} = Ae + F_{k} + D_{k},

(17)

where $e = [e_{1} e_{2} \dots e_{n}]^{T}$ , $F_{k} = [f_{1, k} f_{2, k} \dots f_{n, k}]^{T}$ , and $D_{k} = [d_{1, k} d_{2, k} \dots d_{n, k}]^{T}$ .

Control design

In this section, the backstepping technique and the DSC method are combined to propose an adaptive neural output feedback controller.

Define the following coordinate transformation

z_{1} = {\hat{x}}_{1} - y_{d},

(18)

z_{i} = {\hat{x}}_{i} - ω_{i},

(19)

χ_{i} = ω_{i} - α_{i - 1},

(20)

where $ω_{i}$ represents the output of the first-order filter, input $α_{i - 1}$ is the virtual control, and $i = 2, 3, \dots, n$ .

For the following controller design, define the following constant:

θ = max {‖ Φ_{i, k} ‖^{2} : i = 0, 1, 2, \dots, n, and k \in S},

(21)

where $θ$ is an unknown constant, $\hat{θ}$ represents an estimate of $θ$ , and $\tilde{θ} = θ - \hat{θ}$ .

Virtual control is designed as follows:

α_{i} = - (c_{i} + \frac{1}{2 γ_{i}^{2}} \hat{θ} + 1) z_{i} .

(22)

Consider the following adaptive law

\overset{\cdot}{\hat{θ}} = \sum_{i = 1}^{n} \frac{r}{2 γ_{i}^{2}} z_{i}^{2} - ℓ \hat{θ}

(23)

\overset{\cdot}{\hat{W}} = - Γ z_{n} \bar{v} (t) - π \hat{W}

(24)

where $\hat{W}$ is the estimate of $1 / φ$ . For system (6), the control input is designed as follows:

v (t) = \hat{W} \bar{v} (t),

(25)

\bar{v} (t) = - (c_{n} + \frac{1}{2 γ_{n}^{2}} \hat{θ} + 1) z_{n},

(26)

where $c_{i}$ , $γ_{i}$ , r, $ℓ$ , $Γ$ and $π$ are positive parameters.

The specific process of controller design is as follows.

Step 1: Consider a Lyapunov function:

V_{1} = e^{T} Pe + \frac{1}{2} z_{1}^{2} + \frac{1}{2} χ_{2}^{2} + \frac{1}{2 r} {\tilde{θ}}^{2} .

(27)

The time derivative of (27) is as follows:

\begin{matrix} {\overset{\cdot}{V}}_{1} = - e^{T} Qe + 2 e^{T} P F_{k} + 2 e^{T} P D_{k} + z_{1} (z_{2} + χ_{2} + α_{1}) \\ + z_{1} l_{1} e_{1} + χ_{2} {\overset{\cdot}{χ}}_{2} - \frac{1}{r} \tilde{θ} \overset{\cdot}{\hat{θ}} . \end{matrix}

(28)

The RBF neural networks are used to approximate this unknown function, $F_{k} = [f_{i, k} f_{2, k} \dots f_{n, k}]^{T}$ , then the following holds:

\begin{matrix} H_{0 i, k} (Z_{0}) = f_{i, k} ({\bar{x}}_{i}) = Φ_{io, k}^{T} ϕ (Z_{0}) + σ_{i 0, k} (Z_{0}) \\ σ_{i 0, k} (Z_{0}) | \leq ε_{0}, \end{matrix}

(29)

where $Z_{0} = {\hat{\bar{x}}}_{i}$ and ${\hat{\bar{x}}}_{i} = [{\hat{x}}_{1}, {\hat{x}}_{2}, \dots, {\hat{x}}_{i}]^{T}$ , then for this unknown function vector we have

\begin{matrix} H_{0, k} (Z_{0}) = F_{k} (\bar{x}) = Φ_{0, k}^{T} ϕ_{0, k} (Z_{0}) + σ_{0, k} (Z_{0}) \\ | | σ_{0, k} (Z_{0}) | | \leq ε_{0}, \end{matrix}

(30)

where $Φ_{0, k} = [Φ_{10, k} Φ_{20, k} \dots Φ_{n 0, k}]$ and $σ_{0, k} = [σ_{01, k} σ_{02, k} \dots σ_{0 n, k}]^{T}$ .

It is not difficult to obtain $0 < ϕ^{T} (Z_{0}) ϕ (Z_{0}) \leq 1$ , then by equation (21), further based on Young’s inequality, one can obtain:

\begin{matrix} 2 e^{T} P F_{k} = 2 e^{T} P (Φ_{0, k}^{T} ϕ_{0, k} (Z_{0}) + σ_{0, k} (Z_{0})) \\ \leq | | e {| |}^{2} + | | P {| |}^{2} θ + | | P {| |}^{2} ε_{0}^{2} . \end{matrix}

(31)

The following can be obtained from Assumption 1 and Young’s inequality:

2 e^{T} P D_{k} \leq \frac{1}{ρ} | | e | |^{2} + ρ | | P | |^{2} Σ_{i = 1}^{n} {\bar{d}}_{i},

(32)

z_{1} l_{1} e_{1} \leq \frac{ξ}{4} z_{1}^{2} l_{1}^{2} + \frac{1}{ξ} e_{1}^{2},

(33)

z_{1} χ_{2} \leq \frac{z_{1}^{2}}{2} + \frac{χ_{2}^{2}}{2},

(34)

Substituting equations (31), (32), (33) and (34) into equation (28) leads to the following:

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - λ_{min} [Q] | | e {| |}^{2} + | | e {| |}^{2} + | | P {| |}^{2} θ + | | P {| |}^{2} ε_{0}^{2} \\ + \frac{1}{ρ} | | e {| |}^{2} + ρ | | P {| |}^{2} Σ_{i = 1}^{n} {\bar{d}}_{i} + z_{1} (z_{2} + α_{1}) + \frac{z_{1}^{2}}{2} \\ + \frac{χ_{2}^{2}}{2} + \frac{ξ}{4} z_{1}^{2} l_{1}^{2} + \frac{1}{ξ} e_{1}^{2} + χ_{2} {\overset{\cdot}{χ}}_{2} - \frac{1}{r} \tilde{θ} \overset{\cdot}{\hat{θ}} . \end{matrix}

(35)

By using the approximation of neural networks, we can get

\begin{matrix} H_{1} (Z) = \frac{ξ}{4} z_{1} l_{1}^{2} = Φ_{1}^{T} ϕ_{1} (Z) + σ_{1} (Z) \\ | σ_{1} (Z) | < ε_{1} . \end{matrix}

(36)

Because $ω_{2}$ is the output of the first-order filter and $α_{1}$ is the input, we have

ϖ_{2} {\overset{\cdot}{ω}}_{2} + ω_{2} = α_{1}, ω_{2} (0) = α_{1} (0),

(37)

where $ϖ_{2}$ is a positive constant, and applying equations (20) and (37) results in

{\overset{\cdot}{χ}}_{2} = {\overset{\cdot}{ω}}_{2} - {\overset{\cdot}{α}}_{1} = \frac{- χ_{2}}{ϖ_{2}} + {\overset{\cdot}{α}}_{1} .

(38)

By inserting equation (22) into (38), one has

{\overset{\cdot}{χ}}_{2} = {\overset{\cdot}{ω}}_{2} - {\overset{\cdot}{α}}_{1} = \frac{- χ_{2}}{ϖ_{2}} - \frac{d (c_{1} + \frac{1}{2 γ_{1}^{2}} \hat{θ} + 1) z_{1}}{dt} .

(39)

A positive continuous function $ϒ_{2}$ is introduced; then, the following holds:

| {\overset{\cdot}{χ}}_{2} + \frac{χ_{2}}{ϖ_{2}} | \leq ϒ_{2} (z_{1}, z_{2}, θ),

(40)

where $ϒ_{2}$ is an upper-bound function and the maximum value is $m_{2}$ (Wang et al., 2014), so we have

χ_{2} {\overset{\cdot}{χ}}_{2} \leq \frac{- χ_{2}^{2}}{ϖ_{2}} + | χ_{2} | ϒ_{2} \leq \frac{- χ_{2}^{2}}{ϖ_{2}} + χ_{2}^{2} + \frac{1}{4} m_{2}^{2} .

(41)

Substituting equations (23), (36) and (41) into equation (35), we get

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - λ_{min} [Q] | | e {| |}^{2} + | | e {| |}^{2} + | | P {| |}^{2} θ + | | P {| |}^{2} ε_{0}^{2} \\ + \frac{1}{ρ} | | e {| |}^{2} + ρ | | P {| |}^{2} Σ_{i = 1}^{n} {\bar{d}}_{i} + z_{1} z_{2} \\ + z_{1} (- (c_{1} + \frac{1}{2 γ_{1}^{2}} \hat{θ} + 1) z_{1}) + \frac{z_{1}^{2}}{2} \\ + \frac{χ_{2}^{2}}{2} + z_{1} (Φ_{1}^{T} ϕ_{1} (Z) + σ_{1} (Z)) \\ + \frac{1}{ξ} e_{1}^{2} + \frac{- χ_{2}^{2}}{ϖ_{2}} + χ_{2}^{2} + \frac{1}{4} m_{2}^{2} \\ - \frac{1}{r} \tilde{θ} (\sum_{i = 1}^{n} \frac{r}{2 γ_{i}^{2}} z_{i}^{2} - ℓ \hat{θ}) . \end{matrix}

(42)

By Young’s inequality, we can obtain

z_{1} Φ_{1}^{T} ϕ_{1} (Z) \leq \frac{z_{1}^{2}}{2 γ_{1}^{2}} | | Φ_{1} | |^{2} + \frac{γ_{1}^{2}}{2},

(43)

z_{1} σ_{1} (Z) \leq \frac{z_{1}^{2}}{2} + \frac{ε_{1}^{2}}{2} .

(44)

Then, equation (42) can be further written in the following form:

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - q | | e {| |}^{2} + z_{1} z_{2} - c_{1} z_{1}^{2} - (\frac{1}{ϖ_{2}} - \frac{3}{2}) χ_{2}^{2} \\ - \frac{1}{r} \tilde{θ} (\sum_{i = 2}^{n} \frac{r}{2 γ_{i}^{2}} z_{i}^{2}) - \frac{ℓ}{2 r} {\tilde{θ}}^{2} \\ + \frac{1}{ξ} e_{1}^{2} + Δ_{1}, \end{matrix}

(45)

where $q = λ_{min} [Q] - 1 - \frac{1}{ρ}$ and $Δ_{1} = | | P | |^{2} θ + | | P | |^{2} ε_{0}^{2} + ρ | | P | |^{2} Σ_{i = 1}^{n} {\bar{d}}_{i} + \frac{γ_{1}^{2} + ε_{1}^{2}}{2} + \frac{1}{4} m_{2}^{2} + \frac{ℓ}{2 r} θ^{2}$ .

Step i: For each step i ( $2 \leq i < n - 1$ ), define a Lyapunov function:

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

(46)

Differentiating equation (46), the following can be obtained from equations (10), (19) and (20):

{\overset{\cdot}{V}}_{i} = {\overset{\cdot}{V}}_{i - 1} + z_{i} (z_{i + 1} + χ_{i + 1} + α_{i} + l_{i} e_{1} - {\overset{\cdot}{ω}}_{i}) + χ_{i + 1} {\overset{\cdot}{χ}}_{i + 1} .

(47)

Similarly, by using Young’s inequality,

z_{i} l_{i} e_{1} \leq \frac{ξ}{4} z_{i}^{2} l_{i}^{2} + \frac{1}{ξ} e_{1}^{2},

(48)

z_{i} χ_{i + 1} \leq \frac{z_{i}^{2}}{2} + \frac{χ_{i + 1}^{2}}{2} .

(49)

According to equations (48) and (49), one has

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq {\overset{\cdot}{V}}_{i - 1} + z_{i} z_{i + 1} + \frac{z_{i}^{2}}{2} + \frac{χ_{i + 1}^{2}}{2} + \frac{ξ}{4} z_{i}^{2} l_{i}^{2} \\ + \frac{1}{ξ} e_{1}^{2} + z_{i} (α_{i} - {\overset{\cdot}{ω}}_{i}) + χ_{i + 1} {\overset{\cdot}{χ}}_{i + 1} . \end{matrix}

(50)

By using the approximation ability of the neural network, we can get

\begin{matrix} H_{i} (Z) = \frac{ξ}{4} z_{i} l_{i}^{2} - {\overset{\cdot}{ω}}_{i} + z_{i - 1} = Φ_{i}^{T} ϕ_{i} (Z) + σ_{i} (Z) \\ | σ_{i} (Z) | < ε_{i} . \end{matrix}

(51)

Consider $ω_{i + 1}$ as the output of the first-order filter and $α_{i}$ as the input; we can get

ϖ_{i + 1} {\overset{\cdot}{ω}}_{i + 1} + ω_{i + 1} = α_{i}, ω_{i + 1} (0) = α_{i} (0),

(52)

where $ϖ_{i + 1}$ is a positive constant. Similar to the first step, we can get

\begin{matrix} χ_{i + 1} {\overset{\cdot}{χ}}_{i + 1} \leq \frac{- χ_{i + 1}^{2}}{ϖ_{i + 1}} + | χ_{i + 1} | ϒ_{i + 1} \\ \leq \frac{- χ_{i + 1}^{2}}{ϖ_{i + 1}} + χ_{i + 1}^{2} + \frac{1}{4} m_{i + 1}^{2}, \end{matrix}

(53)

where $ϒ_{i + 1}$ is an upper-bound function and the maximum value is $m_{i + 1}$ .

Remark 1. $ϒ_{i + 1}$ $(i = 1, 2, 3, \dots, n - 1)$ is a bounded function. For the specific proof of the process we can refer to Wang et al. (2014, 2017).

Substituting equations (22), (51) and (53) into equation (50), one gets

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq {\overset{\cdot}{V}}_{i - 1} + z_{i} z_{i + 1} + z_{i} (Φ_{i}^{T} ϕ_{i} (Z) + σ_{i} (Z)) + \frac{z_{i}^{2}}{2} + \frac{χ_{i}^{2}}{2} \\ + \frac{1}{ξ} e_{1}^{2} + z_{i} (- (c_{i} + \frac{1}{2 γ_{i}^{2}} \hat{θ} + 1) z_{i}) - z_{i} z_{i - 1} \\ + \frac{- χ_{i + 1}^{2}}{ϖ_{i + 1}} + χ_{i + 1}^{2} + \frac{1}{4} m_{i + 1}^{2} . \end{matrix}

(54)

By Young’s inequality, we have

z_{i} Φ_{i}^{T} ϕ_{i} (Z) \leq \frac{z_{i}^{2}}{2 γ_{i}^{2}} | | Φ_{i} | |^{2} + \frac{γ_{i}^{2}}{2}

(55)

z_{i} σ_{i} (Z) \leq \frac{z_{i}^{2}}{2} + \frac{ε_{i}^{2}}{2} .

(56)

By inserting equations (55) and (56) into equation (54), a further simplification can be obtained as

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - q | | e {| |}^{2} + z_{i} z_{i + 1} - \sum_{j = 1}^{i} c_{j} z_{j}^{2} - \sum_{j = 1}^{i} (\frac{1}{ϖ_{j + 1}} - \frac{3}{2}) χ_{j + 1}^{2} \\ - \frac{1}{r} \tilde{θ} (\sum_{j = i + 1}^{n} \frac{r}{2 γ_{j}^{2}} z_{j}^{2}) - \frac{ℓ}{2 r} {\tilde{θ}}^{2} \\ + \frac{i}{ξ} e_{1}^{2} + Δ_{i}, \end{matrix}

(57)

where $Δ_{i} = | | P | |^{2} θ + | | P | |^{2} ε_{0}^{2} + ρ | | P | |^{2} \sum_{i = 1}^{n} {\bar{d}}_{i} + \sum_{j = 1}^{i} \frac{γ_{j}^{2} + ε_{j}^{2}}{2} + \sum_{j = 1}^{i} \frac{1}{4} m_{j + 1}^{2} + \frac{ℓ}{2 r} θ^{2}$ .

Step n: Continue to consider a Lyapunov function:

V_{n} = V_{n - 1} + \frac{1}{2} z_{n} + \frac{φ}{2 Γ} {\tilde{W}}^{2},

(58)

where $\tilde{W} = W - \hat{W}$ .

The time derivatives of equation (58) are as follows:

{\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + z_{n} {\overset{\cdot}{z}}_{n} - \frac{φ}{Γ} \tilde{W} \overset{\cdot}{\hat{W}} .

(59)

According to equations (3), (10), (18), (24), (25) and (26), the derivative of $V_{n}$ can be written as

\begin{matrix} {\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + z_{n} (- (c_{n} + \frac{1}{2 γ_{n}^{2}} \hat{θ} + 1) z_{n}) \\ + z_{n} (κ (v) + l_{n} e_{1} - {\overset{\cdot}{ω}}_{n}) + \frac{φ π}{Γ} \tilde{W} \hat{W} . \end{matrix}

(60)

Because of Young’s inequality, we have

z_{n} l_{n} e_{1} \leq \frac{ξ}{4} z_{n}^{2} l_{n}^{2} + \frac{1}{ξ} e_{1}^{2},

(61)

z_{n} κ (v) \leq \frac{1}{2} z_{n}^{2} + \frac{{\bar{κ}}^{2}}{2} .

(62)

Inserting equations (61) and (62) into equation (60) gives the following:

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq {\overset{\cdot}{V}}_{n - 1} + z_{n} (- (c_{n} + \frac{1}{2 γ_{n}^{2}} \hat{θ} + 1) z_{n}) + z_{n} {\overset{\cdot}{ω}}_{n} \\ + \frac{ξ}{4} z_{n}^{2} l_{n}^{2} + \frac{1}{ξ} e_{1}^{2} + \frac{1}{2} z_{n}^{2} + \frac{{\bar{κ}}^{2}}{2} + \frac{φ π}{Γ} \tilde{W} \hat{W} . \end{matrix}

(63)

If we continue to use the approximation function of the neural network, we can get the following:

\begin{matrix} H_{n} (Z) = \frac{ξ}{4} z_{n} l_{n}^{2} - {\overset{\cdot}{ω}}_{n} + z_{n - 1} = Φ_{n}^{T} ϕ_{n} (Z) + σ_{n} (Z) \\ | σ_{n} (Z) | < ε_{n} . \end{matrix}

(64)

Applying equation (64), one gets

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq {\overset{\cdot}{V}}_{n - 1} + z_{n} (- (c_{n} + \frac{1}{2 γ_{n}^{2}} \hat{θ} + 1) z_{n}) \\ + z_{n} (Φ_{n}^{T} ϕ_{n} (Z) + σ_{n} (Z)) \\ + \frac{1}{ξ} e_{1}^{2} + \frac{1}{2} z_{n}^{2} + \frac{{\bar{κ}}^{2}}{2} - z_{n} z_{n - 1} \\ + \frac{φ π}{Γ} \tilde{W} \hat{W} . \end{matrix}

(65)

By using Young’s inequality, one gets

z_{n} Φ_{n}^{T} ϕ_{n} (Z) \leq \frac{z_{n}^{2}}{2 γ_{n}^{2}} | | Φ_{n} | |^{2} + \frac{γ_{n}^{2}}{2},

(66)

z_{n} σ_{n} (Z) \leq \frac{z_{n}^{2}}{2} + \frac{ε_{n}^{2}}{2} .

(67)

Inserting equations (66) and (67) into equation (65) leads to the following:

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - Λ | | e {| |}^{2} - \sum_{i = 1}^{n} c_{i} z_{i}^{2} - \sum_{i = 1}^{n - 1} (\frac{1}{ϖ_{i + 1}} - \frac{3}{2}) χ_{i + 1}^{2} \\ - \frac{ℓ}{2 r} {\tilde{θ}}^{2} - \frac{φ π}{2 Γ} {\tilde{W}}^{2} + Δ_{n}, \end{matrix}

(68)

where $Λ = q - \frac{n}{ξ}$ and $Δ_{n} = | | P | |^{2} θ + | | P | |^{2} ε_{0}^{2} + ρ | | P | |^{2} \sum_{i = 1}^{n} {\bar{d}}_{i} + \sum_{i = 1}^{n} \frac{γ_{n}^{2} + ε_{n}^{2}}{2} + \sum_{i = 1}^{n - 1} \frac{1}{4} m_{i + 1}^{2} + \frac{ℓ}{r} θ^{2} + \frac{φ π}{Γ} W^{2} + \frac{{\bar{κ}}^{2}}{2}$ .

Remark 2. In the controller design process $c_{i}$ , $γ_{i}$ , r, $ℓ$ , $Γ$ , $π$ and $l_{i}$ are the positive parameters to be designed. If we select the appropriate parameters, the tracking error will converge to a small neighbourhood of the origin.

Remark 3. This paper considers the unknown backlash-like hysteresis problem of an uncertain switched non-linear system. In addition, the problem of time delay is also widespread in uncertain switched non-linear systems. For this problem, we can refer to the methods proposed by Karimi et al. (2005, 2009), and further solve the time-delay problem in an uncertain switched non-linear system.

Remark 4. For an uncertain switched non-linear system like aircraft flight control systems, robotic systems, electric power systems and network control systems, during actual system operation the system may be unstable because of actuator failure or sensor failure. However, fault-tolerant control can automatically compensate for the effects of faults to maintain system stability and restore the system’s pre-failure performance as much as possible, therefore making the system stable and reliable. Based on other research in the literature (Kandukuri et al., 2016; Karimi, 2010; Kommuri et al., 2016; Sakthivel et al., 2016), fault-tolerant control of an uncertain switched non-linear system should be studied and discussed in future works.

Here, the controller design section ends.

Stability analysis

Theorem 1. For the closed-loop non-linear switched system (6) with the designed state-observer (10), virtual control (22), adaptive laws (23)–(24), control inputs (25) and appropriate controller parameters, then all the signals in the closed-loop system are bounded under arbitrary switching.

Proof. It is not difficult to get the following results from the above controller design section.

{\overset{\cdot}{V}}_{n} \leq - C V_{n} + Δ_{n},

(69)

where $C = min {\frac{Λ}{λ_{max} [P]}, 2 c_{i}, 2 (\frac{1}{ϖ_{i + 1}} - \frac{3}{2}), ℓ, φ π}$ .

By the comparison principle, one gets

V_{n} (t) \leq (V_{n} (0) - \frac{Δ_{n}}{C}) e^{- Ct} + \frac{Δ_{n}}{C}, t \geq 0 .

(70)

The above equation shows that all signals of the closed-loop system (6) are bounded; then, the following formula holds:

lim_{t \to \infty} | z_{1} | \leq \sqrt{\frac{2 Δ_{n}}{C}} .

(71)

The proof is completed here.

Simulation results

In this part, two sets of simulation studies are implemented, which further verify the effectiveness of the proposed method.

Example 1

A two-dimensional switched system is given as follows:

T_{1} = {\begin{matrix} x_{1} = x_{2} + f_{1, 1} (x_{1}) + d_{1, 1} (t) \\ x_{2} = u (v) + f_{2, 1} (x_{1}, x_{2}) + d_{2, 1} (t) \\ y = x_{1} \end{matrix},

(72)

T_{2} = {\begin{matrix} x_{1} = x_{2} + f_{1, 2} (x_{1}) + d_{1, 2} (t) \\ x_{2} = u (v) + f_{2, 2} (x_{1}, x_{2}) + d_{2, 2} (t) \\ y = x_{1} \end{matrix},

(73)

where $f_{1, 1} (x_{1}) = \sin (x_{1})$ , $f_{2, 1} (x_{1}, x_{2}) = x_{1}^{2} x_{2}$ , $d_{1, 1} (t) = 0.1 \cos (e^{- t})$ , $d_{2, 1} (t) = 0.1 \sin (e^{- t^{2}})$ , $f_{1, 2} = \cos (x_{1})$ , $f_{2, 2} (x_{1}, x_{2}) = x_{1} x_{2}^{2}$ , $d_{1, 2} (t) = 0.1 \sin (t)$ , $d_{2, 2} (t) = 0.1 / (t^{2} + 1)$ , and u shows the output of the following backlash-like hysteresis of actuator:

\frac{du}{dt} = τ | \frac{dv}{dt} | (φ v - u) + η \frac{dv}{dt},

(74)

where $τ = 1$ , $φ = 4$ , and $η = 0.5$ .

Our control goal is to use the actual controller (25), the virtual controller (22), the state-observer (10) and the adaptive laws (23)–(24), so that the output y of the switched system can track the reference signal $y_{d} = \sin (t)$ , where parameters can be selected as: $c_{1} = 5$ , $c_{2} = 5$ , $γ_{1} = 0.25$ , $γ_{2} = 0.25$ , $r = 13$ , $ℓ = 0.05$ , $Γ = 1$ , $π = 0.02$ , $l_{1} = 55$ and $l_{2} = 55$ . The initial state of the switched system and state-observer are $x_{1} (0) = 0.1$ , $x_{2} (0) = 0.2$ , ${\hat{x}}_{1} (0) = - 0.1$ and ${\hat{x}}_{2} (0) = - 0.1$ .

Remark 5. In order to ensure that all signals in the closed-loop system are bounded, in the selection of parameters, the design requirements of the controller and the observer must be satisfied first. In addition, the gain of the controller and the parameter of the observer are related to the uncertainties and disturbances in the uncertain switched non-linear systems, so we cannot simply select a larger value because that will cause serious chattering and consume energy. For these reasons, whether numerical simulation or specific experiment, we will tune the parameter value according to the corresponding result.

The simulation results are as shown in Figures 1 –8.

The trajectory shown in Figure 1 is the switched system output y and reference signal $y_{d}$ ; Figure 2 shows the error between the system output and the reference signal. Figure 3 expresses the trajectory of two adaptive laws designed in this paper. Figures 4 and 5 represent the output and input of unknown backlash-like hysteresis, respectively. The switching signal in Example 1 is selected as shown in Figure 6.

From Figures 1 –6 we can see that all closed-loop signals in the uncertain switched non-linear system are bounded. Not only that, we can obtain from Figures 1 and 2 that under the neural network output feedback control method proposed in this paper, when there is a problem of unknown backlash-like hysteresis of the actuator in the uncertain switched non-linear system, high-precision tracking performance can still be maintained. In addition, a set of comparative simulation results are given. For the two-dimensional uncertain switched non-linear system with unknown backlash-like hysteresis of the actuator in Example 1, the tracking control is performed using the control method of Wang et al. (2017), and the tracking effect and tracking error are shown in Figures 7 and 8. Furthermore, by comparing Figures 1–2 and Figures 7–8, the superiority of the proposed control method of this paper can be seen.

Example 2

As shown in Figure 9, the instrument used in the experiments is a double-propeller helicopter, the Boeing CH-47. When the helicopter is at different heights, over different terrains or encounters different climates, the dynamics model of the helicopter system will experience greater changes. Based on this idea, we use a 3-DOF (three degrees of freedom) experimental helicopter to simulate the CH- 47’s system dynamics model switch.

Figure 10 shows a 3-DOF experimental helicopter produced by the company Quanser. In this part of the simulation, the elevation axis is used to simulate the Boeing CH-47 helicopter ascending and descending. Therefore, the elevation axis of the 3-DOF helicopter is considered, and the dynamic equation of the elevation axis is given below (Meza-Sánchez et al., 2015):

J_{Θ} \overset{\cdot\cdot}{Θ} = - f_{Ψ} \overset{\cdot}{Θ} - k_{f} \sin (Θ) + \cos (Ψ) (V_{f} + V_{b}) + w_{Θ},

(75)

where $Θ$ and $Ψ$ indicate the elevation angle and pitch angle of the 3-DOF helicopter, respectively, $J_{Θ} = 0.91$ kg · $m^{2}$ is a moment of inertia of elevation axis, $f_{Θ}$ indicates the viscous friction level constant, $k_{f}$ represents the force constant of the motor/propeller combination, $V_{f}$ and $V_{b}$ indicate the voltage of the front motor and the back motor, and $w_{Θ}$ represents the external perturbation to the 3-DOF helicopter system.

In order to facilitate the analysis, we carry out the variable conversion $x_{1} = Θ$ , $x_{2} = \overset{\cdot}{Θ}$ . Further assuming that the switched function exists in the 3-DOF helicopter system, and additionally that in the helicopter system exists a common phenomenon of backlash-like hysteresis of the actuator, so equation (75) can be rewritten as follows:

{\begin{matrix} x_{1} = x_{2} + f_{1, δ (t)} (x_{1}) + d_{1, δ (t)} (t) \\ x_{2} = u (v) + f_{2, δ (t)} (x_{1}, x_{2}) + d_{2, δ (t)} (t) \\ y = x_{1} \end{matrix},

(76)

where $f_{1, 1} (x_{1}) = f_{1, 2} (x_{1}) = 0$ , $d_{1, 1} (t) = d_{1, 2} (t) = 0$ , $f_{2, 1} (x_{1}, x_{2}) = (- f_{Θ, 1} x_{2} - k_{f, 1} \sin (x_{1})) / J_{Θ} + 0.1 \sin (x_{1}) \cos (x_{2})$ , $f_{2, 2} (x_{1}, x_{2}) = (- f_{Θ, 2} x_{2} - k_{f, 2} \sin (x_{1})) / J_{Θ} + 0.2 \cos (x_{1} x_{2})$ and $f_{Θ, 1} = 0.4 N \cdot m / V$ , $f_{Θ, 2} = 0.3 N \cdot m / V$ , $k_{f, 1} = 0.5 N / V$ , $k_{f, 2} = 0.3 N / V$ , $d_{2, 1} (t) = 0.1 \cos (e^{- t})$ , $d_{2, 2} (t) = 0.1 / (1 + t^{2})$ , $u (v) = \cos (Ψ) (V_{f} + V_{b}) / J_{Θ}$ . As only the 3-DOF helicopter’s elevation axis is considered and the pitch axis remains horizontal, $u (v) = (V_{f} + V_{b}) / J_{Θ}$ . The controller, virtual control, state-observer and the adaptive laws expressions are as shown in equations (28), (25), (10), (26) and (27), where the parameters in the design process are selected as: $c_{1} = 23$ , $c_{2} = 20$ , $γ_{1} = 0.25$ , $γ_{2} = 0.25$ , $r = 16$ , $ℓ = 0.02$ , $Γ = 1$ , $π = 0.02$ , $l_{1} = 36$ and $l_{2} = 30$ . The reference angle $y_{d} = 0.3 \sin (2 t) (rad)$ is used, and the initial state of the elevation angle of the 3-DOF helicopter is 0.4 rad. The switching signal of this simulation experiment is the same as the switching signal of Example 1.

The tracking performance of the 3-DOF helicopter elevation angle is shown in Figure 11, and the tracking error is given in Figure 12. The two curves of Figure 13 depict two adaptive laws. Figure 14 demonstrates the sum of the voltage inputs of the front motor and back motor in the 3-DOF helicopter system.

It can be seen from the two simulation examples that there are uncertainties and unknown backlash-like hysteresis of the actuator in the uncertain switched non-linear system. In addition, the state of the system is also unmeasured. However, the ideal and high-precision control effect can still be achieved using the output feedback control method based on the neural network proposed in this paper.

Conclusions

An adaptive output feedback control scheme based on a neural network is proposed, which solves the problems of external disturbance, unknown non-linear function, and unknown backlash-like hysteresis of the actuator in a switched non-linear system. In the design of the control scheme, the state-observer is constructed to estimate the unmeasured state of the switched system, and the DSC technique is used to avoid the problem of explosion of complexity caused by the backstepping technique. Finally, the effectiveness of the strategy is further proved by two simulation experiments.

Footnotes

Notation

In this paper, $R^{i}$ denotes the set of i-dimensional real vectors. $sgn (*)$ is the sign function, and $\exp (*)$ indicates the exponential function. We always represent $error \tilde{•} = • - \hat{•}$ , where $\hat{•} is$ the estimate of •. ||★|| denotes the standard Euclidean norm or the induced matrix 2-norm of the matrix. $λ_{min} (★)$ indicates the smallest eigenvalue of the matrix ★.

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported in part by the National Natural Science Foundation of China under Grant 61773143 and 61627901 and in part by the Natural Science Foundation of Heilongjiang Province of China under Grant 2017F009.

ORCID iD

Xuebo Yang

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