Stabilization of gyrostat system with dead-zone nonlinearity in control input

Abstract

This paper proposes a robust adaptive finite-time controller for stabilization of a non-autonomous electromechanical gyrostat system in the presence of model uncertainties and external disturbances. The effect of the dead-zone nonlinearity in the control input is also taken into account. Moreover, all parameters of the system are assumed to be fully unknown in advance. To deal with the system's unknown parameters, some adaptation laws are introduced. Subsequently, based on the finite-time control theory, an adaptive robust controller is proposed to stabilize the non-autonomous gyrostat system in a given finite time. Then, Lyapunov's stability theory is applied to prove the finite-time stability of the designed control scheme. Finally, a numerical simulation is given to demonstrate the efficacy and robustness of the proposed finite-time control approach.

Keywords

Adaptive controller dead-zone nonlinearity finite-time stabilization Lyapunov theory non-autonomous gyrostat

1. Introduction

Owing to special attributes such as extraordinary sensitivity to initial conditions, broad Fourier transform spectra and fractal motion in phase space, chaotic systems have found many useful applications in mechanical engineering, secure communication, ecological systems, economical systems, power converters, information processing and chemical reactions. In this line, various control techniques, such as H_∞ control (Ahn, 2009, 2010a,b, 2012; Ahn et al., 2010), fuzzy control (Ahn, 2010c–e), sliding mode control (Aghababa et al., 2011; Pai and Yau, 2011; Aghababa and Aghababa, 2012a,b; Aghababa and Heydari, 2012; Aghababa and Akbari, 2012; Aghababa and Feizi, 2012a,b), finite-time control (Aghababa, 2011, 2012a; Aghababa and Aghababa, 2012c–h), adaptive control (Aghababa and Aghababa, 2012i, 2013), fractional control (Aghababa, 2012b–d) and active control (Njah, 2011), have been proposed to stabilize/synchronize chaotic systems.

The electromechanical gyrostat is a fourth-order non-autonomous satellite system that exhibits chaotic behavior. In recent years, the study of chaotic behavior of the chaotic gyrostat satellites and its control/synchronization has been widely reported in the literature. For example, the nonlinear dynamics of chaotic electromechanical gyrostat systems has been studied by Ge and Lin (2002, 2003), who adopted the delayed feedback control method to control and/or synchronize the chaos of the electromechanical gyrostat system. Aghababa and Aghababa (2012j) and (2011) proposed adaptive control schemes for synchronization and stabilization of chaotic gyrostat systems, respectively. E1-Gohary and Hassan (1999) studied the problem of exponential asymptotic stability of the rotational motion of a gyrostat using servocontrol moments. The problem of stabilizing the relative programmed motion of a satellite-gyrostat has been addressed in E1-Gohary (1998). An approximate solution of the problem of optimal stabilization of rotary motion of a gyrostat system has been derived in Gorshkov (1979). In E1-Gohary (2009) optimal asymptotic controllers were proposed for the steady rotations of a satellite-gyrostat on a circular orbit. El-Gohary (2000a,b) proposed control schemes to guarantee optimal asymptotic stability of the angular and rotational motion of the gyrostat systems. Chaos synchronization of electromechanical gyrostat systems via variable substitution control (Chen et al., 2009) and linear state error feedback control (Chen et al., 2010) has also been reported in the literature.

However, most of the abovementioned works have assumed that the parameters of the gyrostat systems are known in advance, while, in practice, the precise values of the gyrostat system's parameters are usually unknown beforehand. Moreover, in most of the abovementioned works, the stability of the closed-loop system has not been guaranteed in a given finite-time. Whereas, in many engineering applications, finite-time stabilization of the gyrostat system is required. Furthermore, the proposed controllers in the aforementioned studies work only for linear control inputs. However, when a controller is implemented in real physical situations, limitations of actuators result in nonlinearities, such as saturation, backlash and dead-zone, in the control input. On the other hand, because chaotic systems are sensitive to the variations of the system parameters, input nonlinearities can lead to unpredictable and undesirable behaviors in the system dynamics.

Nowadays, it is known that the dead-zone characteristic is frequently encountered in various engineering systems and can be a cause of instability. Moreover, the dead-zone nonlinearity is common in mechanical connections, hydraulic servovalves, piezoelectric translators, and electric servomotors (Shyu et al., 2005). Thereby, the effects of the dead-zone nonlinearities in the control inputs cannot be neglected in controller design and realization for the mechanical gyrostat systems. However, to the best knowledge of the authors, the problem of finite-time stabilization of the non-autonomous electromechanical gyrostat systems with model uncertainties, external disturbances, unknown parameters and dead-zone nonlinearity in the control input has received less attention, and the relevant theoretical advances have seldom been reported in the literature.

The main purpose of this paper is to design an adaptive robust controller to stabilize an uncertain electromechanical gyrostat system with the dead-zone nonlinearity in the control input in a given finite time. All the parameters of the gyrostat system are assumed to be unknown in advance. To estimate the unknown parameters, appropriate adaptation laws are introduced. Subsequently, using the finite-time control idea, an adaptive robust controller is designed to ensure the stabilization of the closed-loop system in a given finite time. The finite-time stability and convergence of the proposed controller are analytically proved. A numerical simulation is given to demonstrate the efficiency and robustness of the proposed finite-time controller.

2. Gyrostat system and finite-time stabilization

In this section, first a brief description of the nonlinear dynamics of the chaotic non-autonomous electromechanical gyrostat system is given. Then, the problem of robust finite-time stabilization of the gyrostat system with fully unknown parameters, uncertainties, disturbances and dead-zone control inputs is formulated.

2.1. Gyrostat system

The gyrostat is a torque-free satellite whose rotors are axis symmetric and constrained to relative rotation about their symmetric axes. The motion of an electromechanical gyrostat system is governed by (Ge and Lin, 2003)

\begin{matrix} \overset{\cdot}{x} = \frac{B - C}{A} yz - \frac{h_{3}}{A} (1 + f \cos ω t) y + \frac{h_{2}}{A} z \\ + \frac{k_{1}}{A} (ω_{r} - x) + \frac{k_{2}}{A} (ω_{r}^{3} - x 3) \\ \overset{\cdot}{y} = \frac{C - A}{B} xz - \frac{h_{1}}{B} z + \frac{h_{3}}{B} (1 + f \cos ω t) x \\ + \frac{k_{3}}{B} (ω_{r} - y) + \frac{k_{4}}{B} (ω_{r}^{3} - y 3) \\ \overset{\cdot}{z} = \frac{A - B}{C} xy + \frac{h_{3}}{C} f ω \sin ω t - \frac{h_{2}}{C} x + \frac{h_{1}}{C} y - \frac{b}{C} z \\ + \frac{k_{5}}{C} (ω_{r} - z) + \frac{k_{6}}{C} (ω_{r}^{3} - z 3) + \frac{K_{T}}{C} I \\ \overset{\cdot}{I} = \frac{K_{a} (ω_{r} - z)}{L} - \frac{K_{b}}{L} z - \frac{R}{L} I \end{matrix}

(1)

where

x = ω_{x}, y = ω_{y}

and

z = ω_{z}

are the angular velocities,

I

is the current of the control-motor and

A = 500, B = 500, C = 1000, h_{1} = h_{2} = 200, h_{3} = 250, f = 12.9, b = 200, ω = 1, k_{i} (i = 1, 2, \dots, 6) = 1, ω_{r} = 0, K_{T} = 300, K_{a} = 50, K_{b} = 1.3, R = 100, L = 2

are all positive constants. More details for the system can be seen in Ge and Lin (2003).

For the abovementioned parameter values and the initial conditions of $(0.1, 0.2, 0.3, 0)$ , the electromechanical gyrostat system (1) exhibits chaotic behavior (Ge and Lin, 2003). The irregular chaotic motion and the strange attractor of this electromechanical gyrostat system are illustrated in Figures 1 and 2.

2.2. Finite-time stabilization problem

Defining $x_{1} = x$ , $x_{2} = y$ , $x_{3} = z$ and $x_{4} = I$ and considering $ω_{r} = 0$ (Ge and Lin, 2003), the nonlinear chaotic electromechanical gyrostat system (1) with model uncertainties, external disturbances, unknown parameters and dead-zone control inputs can be rewritten as follows:

\begin{matrix} {\overset{\cdot}{x}}_{1} = \frac{B - C}{A} x_{2} x_{3} - \frac{h_{3}}{A} (1 + f \cos ω t) x_{2} + \frac{h_{2}}{A} x_{3} - \frac{k_{1}}{A} x_{1} \\ - \frac{k_{2}}{A} x_{1}^{3} + Δ f_{1} (x, t) + ϕ_{1} (u_{1}) \\ {\overset{\cdot}{x}}_{2} = \frac{C - A}{B} x_{1} x_{3} - \frac{h_{1}}{B} x_{3} + \frac{h_{3}}{B} (1 + f \cos ω t) x_{1} - \frac{k_{3}}{B} x_{2} \\ - \frac{k_{4}}{B} x_{2}^{3} + Δ f_{2} (x, t) + ϕ_{2} (u_{2}) \\ {\overset{\cdot}{x}}_{3} = \frac{A - B}{C} x_{1} x_{2} + \frac{h_{3}}{C} f ω \sin ω t - \frac{h_{2}}{C} x_{1} + \frac{h_{1}}{C} x_{2} \\ - \frac{b + k_{5}}{C} x_{3} - \frac{k_{6}}{C} x_{3}^{3} + \frac{K_{T}}{C} x_{4} + Δ f_{3} (x, t) + ϕ_{3} (u_{3}) \\ {\overset{\cdot}{x}}_{4} = - \frac{K_{a} + K_{b}}{L} x_{3} - \frac{R}{L} x_{4} + Δ f_{4} (x, t) + ϕ_{4} (u_{4}) \end{matrix}

(2)

where

x = [x_{1}, x_{2}, x_{3}, x_{4}] T

is the state vector of the system,

Δ f_{i} (x, t), i = 1, 2, 3, 4

denotes unknown time-varying model uncertainties and external disturbances of the system, which is bounded by a known positive constant

a_{i}

, i.e.

| Δ f_{i} (x, t) | \leq a_{i}, i = 1, 2, 3, 4

u (t) = [u_{1} (t), u_{2} (t), u_{3} (t), u_{4} (t)] T

is the vector of control inputs and

φ_{i} (u_{i}), i = 1, 2, 3, 4

is a dead-zone nonlinear function described by

\begin{matrix} φ_{i} (u_{i}) \\ = {\begin{matrix} φ_{+ i} (u_{i} (t) - u_{+ i}), & u (t) > u_{+ i} \\ 0, & - u_{- i} \leq u (t) \leq u_{+ i}, \\ φ_{- i} (u_{i} (t) + u_{- i}), & u (t) < - u_{- i} \end{matrix} i = 1, 2, 3, 4 \end{matrix}

(3)

where

φ_{+ i} (\cdot)

and

φ_{- i} (\cdot)

are nonlinear continuous functions of

u_{i} (t)

and

u_{+ i}

and

u_{- i}

are positive constants. In addition, the nonlinear input

φ_{i} (u_{i})

outside of the dead-band is with a gain reduction tolerance

β_{+ i}

and

β_{- i}

, i.e., it satisfies the following property:

\begin{matrix} {\begin{matrix} (u_{i} (t) - u_{+ i}) φ_{i} (u_{i}) \geq β_{+ i} (u_{i} (t) - u_{+ i}) 2, & u (t) > u_{+ i} \\ (u_{i} (t) + u_{- i}) φ_{i} (u_{i}) \geq β_{- i} (u_{i} (t) + u_{- i}) 2, & u (t) < - u_{- i} \end{matrix} \end{matrix}

(4)

where

β_{+ i} \leq φ_{+ i}

and

β_{- i} \leq φ_{- i}

are some constants.

Remark 1

When the external disturbances (either periodic or non-periodic) are added in the original chaotic system, the chaotic system will probably become non-chaotic. Therefore, in this paper, we consider the general problem of stabilizing the electromechanical gyrostat system.

Assumption 1

All the parameters $A, B, C, h_{1}, h_{2}, h_{3}, f, b, ω, k_{i} (i = 1, 2, \dots, 6), K_{T}, K_{a}, K_{b}, R$ and $L$ are fully unknown in advance.

Considering

\begin{matrix} α_{1} = [\frac{B - C}{A}, \frac{h_{3}}{A} (1 + f), \frac{h_{2}}{A}, \frac{k_{1}}{A}, \frac{k_{2}}{A}], \\ α_{2} = [\frac{C - A}{B}, \frac{h_{1}}{B}, \frac{h_{3}}{B} (1 + f), \frac{k_{3}}{B}, \frac{k_{4}}{B}], \\ α_{3} = [\frac{A - B}{C}, \frac{h_{3}}{C} f ω, \frac{h_{2}}{C}, \frac{h_{1}}{C}, \frac{b + k_{5}}{C}, \frac{k_{6}}{C}, \frac{K_{T}}{C}] and \\ α_{4} = [\frac{K_{a} + K_{b}}{L}, \frac{R}{L}] \end{matrix}

as the vectors of the unknown parameters of the first, second, third and fourth equations of the gyrostat system (2), respectively, the following assumption is presented.

Assumption 2

It is assumed that the unknown vector parameter $α_{1}, α_{2}, α_{3}$ and $α_{4}$ are norm bounded, i.e.

α_{i} \leq D_{i}, i = 1, 2, 3, 4

(5)

where ċ denotes the Euclidean norm in

R n

and

D_{i}, i = 1, 2, 3, 4

is a known positive constant.

Finite-time stabilization of the nonlinear electromechanical gyrostat system means that the trajectories of the gyrostat system converge to zero within a finite time. In other words, if there exists a constant $T = T (x (0)) > 0$ , such that

{lim}_{t \to T} x (t) = 0

(6)

and

x (t) \equiv 0

, if

t \geq T

, then the stabilization of the non-autonomous electromechanical gyrostat system (2) is achieved in a finite time.

3. Design of an adaptive robust finite-time controller

In this section, an adaptive robust finite-time controller is designed to stabilize the electromechanical gyrostat system (2) with model uncertainties, external disturbances, unknown parameters and dead-zone control inputs in finite-time. First, a useful necessary lemma is presented below.

Lemma 1

(Wang et al., 2009). Assume that a continuous, positive-definite function $V (t)$ satisfies the following differential inequality:

\overset{\cdot}{V} (t) \leq - cV ξ (t) \forall t \geq 0, V (t_{0}) \geq 0

(7)

where

c > 0

0 < ξ < 1

are two constants. Then,

V (t)

satisfies the following inequality:

V 1 - ξ (t) \leq V 1 - ξ (0) - c (1 - ξ) (t), 0 \leq t \leq T

(8)

and

V (t) \equiv 0 \forall t \geq t_{1}

with

t_{1}

given by

T = \frac{V 1 - ξ (0)}{c (1 - ξ)}

(9)

In order to guarantee the finite-time stabilization of the uncertain electromechanical gyrostat system (2), suitable adaptive robust control inputs and adaptation laws are introduced as follows.

\begin{matrix} {\begin{matrix} u_{i} (t) = - ρ_{i} χ_{i} - u_{- i}, & x_{i} (t) \geq 0 \\ u_{i} (t) = ρ_{i} χ_{i} + u_{+ i}, & x_{i} (t) < 0 \end{matrix} \end{matrix}

(10)

\begin{matrix} \overset{\cdot}{\overset{\land}{}} α_{1} (t) = [x_{1} x_{2} x_{3}, | x_{1} | | x_{2} |, x_{1} x_{3}, - x_{1}^{2}, - x_{1}^{4}], {\overset{\land}{α}}_{1} (0) = {\overset{\land}{α}}_{10} \\ \overset{\cdot}{\overset{\land}{}} α_{2} (t) = [x_{1} x_{2} x_{3}, - x_{2} x_{3}, | x_{1} | | x_{2} |, - x_{2}^{2}, - x_{2}^{4}], {\overset{\land}{α}}_{2} (0) = {\overset{\land}{α}}_{20} \\ \overset{\cdot}{\overset{\land}{}} α_{3} (t) = [x_{1} x_{2} x_{3}, | x_{3} |, - x_{1} x_{3}, x_{2} x_{3}, - x_{3}^{2}, - x_{3}^{4}, x_{3} x_{4}], \\ {\overset{\land}{α}}_{3} (0) = {\overset{\land}{α}}_{30} \\ \overset{\cdot}{\overset{\land}{}} α_{4} (t) = [- x_{3} x_{4}, - x_{4}^{2}], {\overset{\land}{α}}_{4} (0) = {\overset{\land}{α}}_{40} \end{matrix}

(11)

where

ρ_{i} = 1 / β_{i}

β_{i} = min {β_{+ i}, β_{- i}}

χ_{i} = | {\overset{\land}{α}}_{i} (t) | | f_{i} (x) | + {μ D}_{D} (\frac{| x_{i} |}{x 2}) + a_{i} + η_{i} > 0

{\overset{\land}{α}}_{i} (t), i = 1, 2, 3, 4

is an estimation for the unknown vector parameter

α_{i}, i = 1, 2, 3, 4

f_{1} (x) = [x_{2} x_{3}, | x_{2} | sgn (x_{1}), x_{3}, - x_{1}, - x_{1}^{3}] T

f_{2} (x) = [x_{1} x_{3}, - x_{3}, | x_{1} | sgn (x_{2}), - x_{2}, - x_{2}^{3}] T

f_{3} (x) = [x_{1} x_{2}, sgn (x_{3}), - x_{1}, x_{2}, - x_{3}, - x_{3}^{3}, x_{4}] T

f_{4} (x) = [- x_{3}, - x_{4}] T

μ = min {η_{i}}, i = 1, 2, 3, 4

D_{D} = \sum_{i = 1}^{4} [{\overset{\land}{α}}_{i} (t) + D_{i}]

η_{i} > 0, i = 1, 2, 3, 4

is a constant gain,

sgn (.)

is the signum function,

{\overset{\land}{α}}_{i 0}, i = 1, 2, 3, 4

is the initial value of the adaptation parameters

{\overset{\land}{α}}_{i} (t), i = 1, 2, 3, 4

\frac{| x_{i} |}{x 2} = 0

, if

x = 0, i = 1, 2, 3, 4

and the absolute value for a vector such as

Z = [z_{1}, z_{2}, \dots, z_{n}] T

is considered as

| Z | = [| z_{1} |, | z_{2} |, \dots, | z_{n} |] T

Theorem 1

Consider the non-autonomous electromechanical gyrostat system (2) with model uncertainties, external disturbances, unknown parameters, dead-zone nonlinear control inputs controlled by the signals (10) with the adaptation laws (11). Then the system trajectories converge to zero in the finite time

T = \sqrt{2} / μ (1 / 2 (x (0) 2 + \sum_{i = 1}^{4} {\overset{\land}{α}}_{i} (0) - α_{i}^{2})) 1 / 2 .

Proof

Select a positive definite continuous function in the form of

V (t) = 1 / 2 (x (t) 2 + \sum_{i = 1}^{4} {\overset{\land}{α}}_{i} (t) - α_{i}^{2})

(12)

Taking the derivative of

V (t)

with respect to time, one has

\overset{\cdot}{V} (t) = \sum_{i = 1}^{4} [x_{i} (t) {\overset{\cdot}{x}}_{i} (t) + ({\overset{\land}{α}}_{i} (t) - α_{i}) T {\overset{\cdot}{\overset{\land}{}} α}_{i} (t)]

(13)

Inserting

{\overset{\cdot}{x}}_{i} (t)

from equation (2) into the above equation, yields

\begin{matrix} \bar{\overset{\cdot}{}} V (t) = x_{1} (\frac{B - C}{A} x_{2} x_{3} - \frac{h_{3}}{A} (1 + f \cos ω t) x_{2} + \frac{h_{2}}{A} x_{3} \\ - \frac{k_{1}}{A} x_{1} - \frac{k_{2}}{A} x_{1}^{3} + Δ f_{1} {(x, t) + ϕ}_{1} {(u}_{1})) + x_{2} (\frac{C - A}{B} x_{1} x_{3} \\ - \frac{h_{1}}{B} x_{3} + \frac{h_{3}}{B} (1 + f \cos ω {t) x}_{1} - \frac{k_{3}}{B} x_{2} - \frac{k_{4}}{B} x_{2}^{3} \\ + Δ f_{2} {(x, t) + ϕ}_{2} {(u}_{2})) + x_{3} (\frac{A - B}{C} x_{1} x_{2} + \frac{h_{3}}{C} f ω \sin ω t \\ - \frac{h_{2}}{C} x_{1} + \frac{h_{1}}{C} x_{2} - \frac{{b + k}_{5}}{C} x_{3} - \frac{k_{6}}{C} x_{3}^{3} + \frac{k_{T}}{C} x_{4} \\ + Δ f_{3} {(x, t) + ϕ}_{3} {(u}_{3})) + x_{4} (- \frac{K_{a} + K_{b}}{L} x_{3} - \frac{R}{L} x_{4} \\ + Δ f_{4} {(x, t) + ϕ}_{4} {(u}_{4})) + \sum_{i = 1}^{4} [({\overset{\land}{α}}_{i} (t) - α_{i}) T {\overset{\cdot}{\overset{\land}{}} α}_{i} (t)] \end{matrix}

(14)

Knowing that all the parameters of the gyrostat system are positive (Ge and Lin, 2003), one can obtain

\begin{matrix} \overset{\cdot}{V} (t) \leq x_{1} (\frac{B - C}{A} x_{2} x_{3} + \frac{h_{3}}{A} (1 + f) | x_{2} | {sgn (x}_{1}) + \frac{h_{2}}{A} x_{3} \\ - \frac{k_{1}}{A} x_{1} - \frac{k_{2}}{A} x_{1}^{3} + | Δ f_{1} (x, t) | {sgn (x}_{1} {) + ϕ}_{1} {(u}_{1})) \\ + x_{2} (\frac{C - A}{B} x_{1} x_{3} - \frac{h_{1}}{B} x_{3} + \frac{h_{3}}{B} (1 + f) | x_{1} | {sgn (x}_{2}) \\ - \frac{k_{3}}{B} x_{2} - \frac{k_{4}}{B} x_{2}^{3} + | Δ f_{2} (x, t) | {sgn (x}_{2} {) + ϕ}_{2} {(u}_{2})) \\ + x_{3} (\frac{A - B}{C} x_{1} x_{2} + \frac{h_{3}}{C} {f ω sgn (x}_{3}) - \frac{h_{2}}{C} x_{1} + \frac{h_{1}}{C} x_{2} \\ - \frac{{b + k}_{5}}{C} x_{3} - \frac{k_{6}}{C} x_{3}^{3} + \frac{K_{T}}{C} x_{4} + | Δ f_{3} (x, t) | {sgn (x}_{3}) {+ ϕ}_{3} {(u}_{3})) + x_{4} (- \frac{K_{a} + K_{b}}{L} x_{3} - \frac{R}{L} x_{4} \\ + | Δ f_{4} (x, t) | {sgn (x}_{4} {) + ϕ}_{4} {(u}_{4})) + \sum_{i = 1}^{4} [({\overset{\land}{α}}_{i} (t) - α_{i}) T \overset{\cdot}{\overset{\land}{}} α_{i} (t)] \end{matrix}

(15)

Since

\begin{matrix} α_{1}^{T} (t) \overset{\cdot}{\overset{\land}{}} α_{1} (t) = \frac{B - C}{A} x_{1} x_{2} x_{3} + \frac{h_{3}}{A} (1 + f) | x_{1} | | x_{2} | + \frac{h_{2}}{A} x_{1} x_{3} \\ - \frac{k_{1}}{A} x_{1}^{2} - \frac{k_{2}}{A} x_{1}^{4}, \end{matrix}

\begin{matrix} α_{2}^{T} (t) \overset{\cdot}{\overset{\land}{}} α_{2} (t) = \frac{C - A}{B} x_{1} x_{2} x_{3} - \frac{h_{1}}{B} x_{2} x_{3} \\ + \frac{h_{3}}{B} (1 + f) | x_{1} | | x_{2} | - \frac{k_{3}}{B} x_{2}^{2} - \frac{k_{4}}{B} x_{2}^{4}, \end{matrix}

\begin{matrix} α_{3}^{T} (t) \overset{\cdot}{\overset{\land}{}} α_{3} (t) = \frac{A - B}{C} x_{1} x_{2} x_{3} + \frac{h_{3}}{C} {f ω x}_{3} - \frac{h_{2}}{C} x_{1} x_{3} \\ + \frac{h_{1}}{C} x_{2} x_{3} - \frac{b + k_{5}}{C} x_{3}^{2} - \frac{k_{6}}{C} x_{3}^{4} + \frac{K_{T}}{C} x_{3} x_{4} \end{matrix}

and

α_{4}^{T} (t) \overset{\cdot}{\overset{\land}{}} α_{4} (t) = - \frac{K_{a} + K_{b}}{L} x_{3} x_{4} - \frac{R}{L} x_{4}^{2}

and

sgn (x_{i}) = | x_{i} | / x_{i}

we have

\overset{\cdot}{V} (t) \leq \sum_{i = 1}^{4} [x_{i} ϕ_{i} (u_{i}) + a_{i} | x_{i} | + {\overset{\land}{α}}_{i}^{T} (t) \overset{\cdot}{\overset{\land}{}} α_{i} (t)]

(16)

when

x_{i} (t) < 0

, surveying (4) and (10), it is apparent that

u (t) > u_{+}

and

\begin{matrix} (u_{i} (t) - u_{+ i}) φ_{i} (u_{i}) = ρ_{i} χ_{i} φ_{i} (u_{i}) \geq β_{+ i} (ρ_{i} χ_{i}) 2 \\ \geq β_{i} (ρ_{i} χ_{i}) 2 = ρ_{i} χ_{i}^{2} \end{matrix}

(17)

Multiplying both side of the above inequality by

x_{i} (t) < 0 (x_{i} = - | x_{i} |)

, we have

x_{i} φ_{i} (u_{i}) \leq - β_{i} ρ_{i} χ_{i} | x_{i} |

(18)

when

x_{i} (t) > 0

, surveying (4) and (10), it is clear that

u (t) < - u_{-}

and

\begin{matrix} (u_{i} (t) + u_{- i}) φ_{i} (u_{i}) = - ρ_{i} χ_{i} φ_{i} (u_{i}) \\ \geq β_{- i} (ρ_{i} χ_{i}) 2 \geq β_{i} (ρ_{i} χ_{i}) 2 = ρ_{i} χ_{i}^{2} \end{matrix}

(19)

Multiplying both sides of the above inequality by

x_{i} (t) > 0 (x_{i} = | x_{i} |)

, one has

x_{i} φ_{i} (u_{i}) \leq - χ_{i} | x_{i} |

(20)

From equations (16), (18) and (20) and by using

χ_{i} = | {\overset{\land}{α}}_{i} (t) | | f_{i} (x) | + {μ D}_{D} (\frac{| x_{i} |}{x 2}) + a_{i} + η_{i}

, we have

\begin{matrix} \overset{\cdot}{V} (t) \leq \sum_{i = 1}^{4} [- | x_{i} | (| {\overset{\land}{α}}_{i} (t) | | f_{i} (x) | + {μ D}_{D} (\frac{| x_{i} |}{x 2}) + a_{i} + η_{i}) \\ + a_{i} | x_{i} | + {\overset{\land}{α}}_{i}^{T} (t) \overset{\cdot}{\overset{\land}{}} α_{i} (t)] \end{matrix}

(21)

It is obvious that

\begin{matrix} \overset{\cdot}{V} (t) \leq \sum_{i = 1}^{4} [- | x_{i} | (| {\overset{\land}{α}}_{i} (t) | | f_{i} (x) | + {μ D}_{D} (\frac{| x_{i} |}{x 2}) + η_{i}) \\ + | {\overset{\land}{α}}_{i} (t) | T | \overset{\cdot}{\overset{\land}{}} α_{i} (t) |] \end{matrix}

(22)

Since

| x_{i} | | {\overset{\land}{α}}_{i} (t) | | f_{i} (x) | = | {\overset{\land}{α}}_{i} (t) | T | \overset{\cdot}{\overset{\land}{}} α_{i} (t) |, i = 1, 2, 3, 4

, one can obtain

\overset{\cdot}{V} (t) \leq - \sum_{i = 1}^{4} [| x_{i} | ({μ D}_{D} (\frac{| x_{i} |}{x 2}) + η_{i})]

(23)

Based on the well-known mathematical fact that for

c_{1}, c_{2}, \dots, c_{n} \in R

, the inequality

| c_{1} | + | c_{2} | + \dots + | c_{n} | \geq \sqrt{c_{1}^{2} + c_{2}^{2} + \dots + c_{n}^{2}}

holds and using

\sum_{i = 1}^{4} [| x_{i} | (\frac{| x_{i} |}{x 2})] = 1

, one has

\overset{\cdot}{V} (t) \leq - μ x (t) - {μ D}_{D}

(24)

Using

D_{D} = \sum_{i = 1}^{4} [{\overset{\land}{α}}_{i} (t) + D_{i}]

and

{\overset{\land}{α}}_{i} (t) - α_{i} (t) \leq {\overset{\land}{α}}_{i} (t) + α_{i} (t) \leq {\overset{\land}{α}}_{i} (t) + D_{i}, i = 1, 2, 3, 4

, one has

\begin{matrix} \overset{\cdot}{V} (t) \leq - μ (x (t) + \sum_{i = 1}^{4} {\overset{\land}{α}}_{i} (t) - α_{i} (t)) \\ \leq - \sqrt{2} μ (1 / 2 (x (t) 2 + \sum_{i = 1}^{4} {\overset{\land}{α}}_{i} (t) - α_{i} (t) 2)) 1 / 2 \\ = - \sqrt{2} μ V \frac{1}{2} \end{matrix}

(25)

Hence, according to Lemma 1, the system state trajectories

x (t)

will converge to zero in the finite time

T = \sqrt{2} / μ (1 / 2 (x (0) 2 + \sum_{i = 1} 4 {\overset{\land}{α}}_{i} (0) - α_{i}^{2})) 1 / 2

. Therefore, the non-autonomous electromechanical gyrostat system (2) with dead-zone input nonlinearities will be stable in finite time. This completes the proof.

4. Numerical simulations

In this section, numerical simulations are presented to verify the efficacy and applicability of the proposed finite-time controller in the stabilization of uncertain non-autonomous electromechanical gyrostat systems with dead-zone input nonlinearities. The simulations are carried out using MATLAB software. The ode45 solver is applied to solve nonlinear equations. The parameters $A = 500, B = 500, C = 1000, h_{1} = h_{2} = 200, h_{3} = 250, f = 12.9, b = 200, ω = 1, k_{i} (i = 1, 2, \dots, 6) = 1, ω_{r} = 0, K_{T} = 300, K_{a} = 50, K_{b} = 1.3, R = 100, L = 2$ are selected in the simulation. Subsequently, the following model uncertainties and external disturbances are considered in the simulation.

\begin{matrix} Δ f_{1} (x, t) = - 0.15 \sin (2 x_{1}) + 0.3 \cos (3 t) \\ Δ f_{2} (x, t) = 0.2 \sin (5 x_{2}) - 0.3 \sin (3 t) \\ Δ f_{3} (x, t) = - 0.25 \cos (x_{3}) - 0.25 \cos (2 t) \\ Δ f_{4} (x, t) = 0.25 \sin (3 x_{4}) + 0.15 \sin (4 t) \end{matrix}

(26)

As a result, one can obtain

a_{1} = 0.45, a_{2} = 0.5, a_{3} = 0.5

and

a_{4} = 0.4

D_{1}, D_{2}, D_{3}

and

D_{4}

are chosen equal to

10, 10, 5

and

65

, respectively. The initial values of the system are chosen as

x_{1} (0) = 0.1

x_{2} (0) = 0.2

x_{3} (0) = 0.3

and

x_{4} (0) = 0

. Also, the initial values of the adaptation vector parameters

{\overset{\land}{α}}_{i}, i = 1, 2, 3, 4

are all set to

0

. The constant gains

η_{i}, i = 1, 2, 3, 4

are all chosen equal to

1

. The dead-zone nonlinearity is defined as

\begin{matrix} φ_{i} (u_{i}) \\ = {\begin{matrix} (2 - \cos (u_{i}) (u_{i} (t) - 3), & u (t) > 3 \\ 0, & 2 \leq u (t) \leq 3 \\ 1 + 0.5 \sin (u_{i}) (u_{i} (t) - 2), & u (t) < 2 \end{matrix}, i = 1, 2, 3, 4 \end{matrix}

(27)

The time response of the state trajectories of the uncertain non-autonomous electromechanical gyrostat system (2) are shown in Figure 3, where the control inputs are applied at t = 5 s. It is observed that the system trajectories converge to zero in finite time, after the controller is turned on. This means that applying the proposed adaptive robust finite-time controller (10)–(11), the uncertain non-autonomous electromechanical gyrostat system (2) with model uncertainties, external disturbances, fully unknown parameters and dead-zone input nonlinearities is stabilized within a finite time. The time histories of the adaptation vector parameters

{\overset{\land}{α}}_{1}, {\overset{\land}{α}}_{2}, {\overset{\land}{α}}_{3}

and

{\overset{\land}{α}}_{4}

appear in Figures 4–7, respectively. One can see that all adaptation parameters converge to some constants.

5. Conclusions

The present paper studies the problem of finite-time stabilization of uncertain non-autonomous gyrostat systems with dead-zone input nonlinearity. It is assumed that the parameters of the gyrostat system are fully unknown in advance and the system is perturbed by unknown model uncertainties and external disturbances. By means of some adaptation laws and the finite-time control theory, a finite-time adaptive robust controller is designed to stabilize the uncertain non-autonomous gyrostat systems in a given finite time. The finite-time robust stability and convergence of the introduced strategy are mathematically proved. Numerical simulations reveal that the proposed controller stabilizes the uncertain non-autonomous gyrostat systems as quickly as possible, even when the system parameters are fully unknown in advance, unknown model uncertainties and external disturbances are present, and the control inputs include dead-zone nonlinearities.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors

References

Aghababa MP (2011) A novel adaptive finite-time controller for synchronizing chaotic gyros with nonlinear inputs. Chinese Physics B 20: 090505.

Aghababa MP (2012a) Design of an adaptive finite-time controller for synchronization of two identical/different non-autonomous chaotic flywheel governor systems. Chinese Physics B 21: 030502.

Aghababa MP (2012b) Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory. Journal of Computational and Nonlinear Dynamics 7: 021010.

Aghababa

(2012c) Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller. Communications in Nonlinear Science and Numerical Simulation 17: 2670–2681.

Aghababa

(2012d) Finite-time chaos control and synchronization of fractional-order chaotic (hyperchaotic) systems via fractional nonsingular terminal sliding mode technique. Nonlinear Dynamics 69: 247–261.

Aghababa

(2011) Adaptive finite-time stabilization of uncertain non-autonomous chaotic electromechanical gyrostat systems with unknown parameters. Mechanics Research Communications 38: 500–505.

Aghababa

(2012a) Chaos suppression of a class of unknown uncertain chaotic systems via single input. Communications in Nonlinear Science and Numerical Simulation 17: 3533–3538.

Aghababa MP and Aghababa HP (2012b) A novel finite-time sliding mode controller applied to synchronize chaotic systems with input nonlinearity. Arabian Journal for Science and Engineering 00: 1–11 (accessed 11 December 2012). DOI 10.1007/s13369-012-0459-z.

Aghababa

Aghabab

(2012c) Synchronization of mechanical horizontal platform systems in finite time. Applied Mathematical Modelling 36: 4579–4591.

10.

Aghababa

(2012d) Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs. Applied Mathematics and Mechanics 33: 155–164.

11.

Aghababa

(2012e) Chaos suppression of rotational machine systems via finite-time control method. Nonlinear Dynamics 69: 1881–1888.

12.

Aghababa

(2012f) Chaos suppression of uncertain gyros in a given finite time. Chinese Physics B 21: 110505–110505.

13.

Aghababa

(2012g) Finite-time stabilization of a non-autonomous chaotic rotating mechanical system. Journal of the Franklin Institute 349: 2875–2888.

14.

Aghababa MP and Aghababa HP (2012h) Finite-time stabilization of non-autonomous uncertain chaotic centrifugal flywheel governor systems with input nonlinearities. Journal of Vibration and Control 00: 1–11 (accessed 2 November 2012). DOI: 10.1177/1077546312463715.

15.

Aghababa

(2012i) A general nonlinear adaptive control scheme for finite-time synchronization of chaotic systems with uncertain parameters and nonlinear inputs. Nonlinear Dynamics 69: 1903–1914.

16.

Aghababa

(2012j) Synchronization of nonlinear chaotic electromechanical gyrostat systems with uncertainties. Nonlinear Dynamics 67: 2689–2701.

17.

Aghababa MP and Aghababa HP (2013) Adaptive finite-time synchronization of non-autonomous chaotic systems with uncertainty. Journal of Computational and Nonlinear Dynamics 8: 031006.

18.

Aghababa

Akbari

(2012) A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances. Applied Mathematics and Computation 218: 5757–5768.

19.

Aghababa

Feizi

(2012a) Nonsingular terminal sliding mode approach applied to synchronize chaotic systems with unknown parameters and nonlinear inputs. Chinese Physics B 21: 060506–060506.

20.

Aghababa

Feizi

(2012b) Design of a sliding mode controller for synchronizing chaotic systems with parameter and model uncertainties and external disturbances. Transactions of the Institute of Measurement and Control 34: 990–997.

21.

Aghababa

Heydari

(2012) Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities. Applied Mathematical Modelling 36: 1639–1652.

22.

Aghababa

Khanmohammadi

Alizadeh

(2011) Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Applied Mathematical Modelling 35: 3080–3091.

23.

Ahn

(2009) An H_∞ approach to anti-synchronization for chaotic systems. Physics Letters A 373: 1729–1733.

24.

Ahn

(2010a) Neural network H_∞ chaos synchronization. Nonlinear Dynamics 60: 295–302.

25.

Ahn

(2010b) Output feedback H_∞ synchronization for delayed chaotic neural networks. Nonlinear Dynamics 59: 319–327.

26.

Ahn

(2010c) T-S Fuzzy H_∞ synchronization for chaotic systems via delayed output feedback control. Nonlinear Dynamics 59: 535–543.

27.

Ahn

(2010d) Fuzzy delayed output feedback synchronization for time-delayed chaotic systems. Nonlinear Analysis: Hybrid Systems 4: 16–24.

28.

Ahn CK (2010e) A T-S fuzzy model based adaptive exponential synchronization method for uncertain delayed chaotic systems: an LMI approach. Journal of Inequalities and Applications 168962.

29.

Ahn

(2012) An answer to the open problem of L₂-L_∞ synchronization for time-delayed chaotic systems. European Physical Journal Plus 127: 22–22.

30.

Ahn

Kang

Joo

(2010) Adaptive H_∞ synchronization for uncertain chaotic systems with external disturbance. Communications in Nonlinear Science and Numerical Simulation 15: 2168–2177.

31.

Chen

Gui

(2010) Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control. Applied Mathematical Modelling 34: 4161–4170.

32.

Chen

Liu

(2009) Global chaos synchronization of electro-mechanical gyrostat systems via variable substitution control. Chaos, Solitons and Fractals 42: 1197–1205.

33.

E1-Gohary

(1998) On the stability of relative programmed motion of satellite-gyrostat. Mechanics Research Communications 25: 371–379.

34.

E1-Gohary

(2000a) The optimal control of an angular motion of a gyrostat containing fluid. International Journal of Non-Linear Mechanics 35: 979–985.

35.

E1-Gohary

(2000b) Optimal control of a rotational motion of a gyrostat on circular orbit. Mechanics Research Communications 27: 59–67.

36.

E1-Gohary

(2009) Chaos and optimal control of steady-state rotation of a satellite-gyrostat on a circular orbit. Chaos, Solitons and Fractals 42: 2842–2851.

37.

E1-Gohary

Hassan

(1999) On the exponential stability of the permanent rotational motion of a gyrostat. Mechanics Research Communications 26: 479–488.

38.

Lin

(2002) Chaos, chaos control and synchronization of a gyrostat system. Journal of Sound and Vibration 251: 519–542.

39.

Lin

(2003) Chaos, chaos control and synchronization of electro-mechanical gyrostat system. Journal of Sound and Vibration 259: 585–603.

40.

Gorshkov

(1979) On optimal stabilization of gyrostat rotary motion. Journal of Applied Mathematics and Mechanics 43: 838–845.

41.

Njah

(2011) Synchronization via active control of parametrically and externally excited Φ⁶ Van der Pol and Duffing oscillators and application to secure communications. Journal of Vibration and Control 17: 493–504.

42.

Pai

Yau

(2011) Generalized projective synchronization for the horizontal platform systems via an integral-type sliding mode control. Journal of Vibration and Control 17: 11–17.

43.

Shyu

Liu

Hsu

(2005) Design of large-scale time-delayed systems with dead-zone input via variable structure control. Automatica 41: 1239–1246.

44.

Wang

Han

Xie

Zhang

(2009) Finite-time synchronization of uncertain unified chaotic systems based on CLF. Nonlinear Analysis: Real World Applications 10: 2842–2849.