Finite-time stabilization of non-autonomous uncertain chaotic centrifugal flywheel governor systems with input nonlinearities

Abstract

The centrifugal flywheel governor is a mechanical device that automatically controls the speed of an engine and avoids the damage caused by an abrupt change of load torque. Recent research has discovered that this system exhibits very rich and complex dynamics such as chaos. In this paper, the problem of finite-time stabilization of non-autonomous chaotic centrifugal flywheel governor systems in the presence of model uncertainties, external disturbances, fully unknown parameters and input nonlinearities is studied. Appropriate adaptation laws are designed to undertake the system’s unknown parameters. Using the adaptation laws and finite-time control theory, a robust adaptive controller is derived to stabilize the non-autonomous uncertain centrifugal flywheel governor system with nonlinear control inputs in a given finite time. The finite-time stability and convergence of the closed-loop system are analytically proved. A numerical simulation is given to show the robustness and effectiveness of the proposed finite-time controller and to verify the theoretical results of the paper.

Keywords

Adaptation law centrifugal governor finite-time stabilization input nonlinearity robust adaptive controller

1. Introduction

The centrifugal flywheel governor is one of the most interesting and attractive nonlinear dynamical systems. It is a mechanical device that automatically regulates the speed of an engine and avoids the damage caused by an abrupt change of load torque. In 1788, James Watt designed a conical pendulum governor to control the speed of steam engines. Formerly, centrifugal governors were used to regulate the distance and pressure between millstones in windmills since the 17th century (Hills, 1996). Afterwards, the centrifugal flywheel governors have found useful applications in many practical systems such as the diesel engine, steam engine, gas turbine and so on. For example, the ignition timing of an automotive engine has been controlled by a distributor composed of a spring and centrifugal governor (Abno and Hata, 1981). Recent research has recognized different kinds of centrifugal flywheel governor systems with a rich variety of nonlinear behavior. Furthermore, it has been shown that these systems display a diverse range of dynamic behaviors including regular, quasi-periodic and chaotic motions.

Chaotic dynamical systems are very complex nonlinear systems that display unpredictable and irregular behaviors. The main specific feature of a chaotic system is that a small change in the initial conditions and the system parameters leads to an enormous difference in the long-term behavior of the system. Since the pioneering work by Ott et al. (1990), chaos control of autonomous chaotic systems has received considerable attention in recent years (Park and Kwon, 2005a,b; Hseng et al., 2010; Lee et al., 2010; Kuo, 2007, 2011; Kwon et al., 2011; Aghababa and Heydari, 2012). On the other hand, with the discovery of more and more non-autonomous chaotic systems in engineering sciences and physics, stabilization and synchronization of non-autonomous chaotic systems have attracted the significant interest of many researchers. In this regard, several control techniques have been developed to stabilize/synchronize non-autonomous chaotic systems, which include linear state error feedback control (Wu et al., 2006), impulsive control (Yang and Chua, 2000), adaptive control (Ge and Lee, 2005; Aghababa, 2011; Aghababa and Aghababa, 2011), active control (Lei et al., 2006), sliding mode control (Salarieh and Alasty, 2008), sinusoidal state error feedback control (Cai et al., 2007), variable substitution control (Chen et al., 2009) and fuzzy observer-based method (Yoo et al., 2010).

However, most of the abovementioned control methods have been introduced to guarantee the asymptotic stability of chaotic systems. In other words, the state trajectories of the chaotic systems can approach to zero with infinite settling time. However, from a practical engineering point of view, it is more valuable to stabilize chaotic systems within a given finite time. To achieve faster convergence in control systems, the finite-time control methods are efficient and effective techniques. Finite-time stabilization means the optimality in the settling time. Moreover, the finite-time control techniques have demonstrated better robustness and disturbance rejection properties (Bhat and Bernstein, 2000).

In the past few years, chaos control, anti-control and synchronization of the centrifugal flywheel governor systems have been studied by some researchers. In Chu et al. (2008), the nonlinear dynamics of a non-autonomous centrifugal flywheel governor system and its synchronization have been studied. Regular and chaotic dynamics of both autonomous and non-autonomous rotational machines with a centrifugal governor subjected to external disturbances have been studied in Ge et al. (1999). Zhang et al. (2009) have investigated the complex dynamical behaviors of a class of centrifugal flywheel governor systems and have proposed a parametric open-plus-closed-loop approach for controlling chaos. By numerically integrating the Lagrangian equations of the motion, the bifurcation and chaos of a non-autonomous centrifugal flywheel governor system have been investigated (Zhang et al., 2008). Sotomayor et al. (2008) have studied the Lyapunov stability and the Hopf bifurcation in a system coupling a hexagonal centrifugal governor with a steam engine. Ge and Jhuang (2007) have investigated chaos and its control and synchronization for a fractional order rotational mechanical system with a centrifugal governor. Nonlinear dynamics and the control of chaos for a rotational machine with a hexagonal centrifugal governor and a spring considering the effects of external disturbances have been reported by Ge and Lee (2003).Anti-control and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor have been addressed by Ge and Lee (2005a). Moreover, Ge and Lee (2005b) have studied the chaotic behavior of an autonomous rotational machine system with a centrifugal governor and spring. Considering the effects of time-delays, they have proposed linear and adaptive controllers to stabilize the system.

However, the abovementioned works have studied the synchronization and stabilization problem of the centrifugal flywheel governor systems without considering the effects of both unknown parameters and system uncertainties. However, in real world applications the exact values of chaotic systems’ parameters cannot be determined in advance and there are usually some model uncertainties and external disturbances in the dynamics of chaotic systems. Furthermore, when the controller is implemented in real physical systems, the limitations of actuators cause nonlinearity in the control input. On the other hand, since chaotic systems are very sensitive to the variations of any parameters of the system, the presence of the input nonlinearities can lead to unpredictable and undesirable behaviors in chaotic systems. Therefore, consideration of the effects of the input nonlinearities in the controller design for chaotic systems cannot be ignored in the design and implementation.Hence, the problem of the finite-time stabilization of non-autonomous chaotic centrifugal flywheel governor systems in the presence of model uncertainties, external disturbances, fully unknown parameters and input nonlinearities is a significant research issue and remains an open and challenging problem.

Motivated by the above discussions, this paper proposes a robust adaptive controller to stabilize non-autonomous chaotic centrifugal flywheel governor systems in a given finite time. The effects of model uncertainties, external disturbances, unknown parameters and input nonlinearities are fully taken into account. To undertake the unknown parameters, suitable adaptation laws are introduced. On the basis of the adaptation laws and finite-time control technique, a robust adaptive controller is designed to stabilize an uncertain non-autonomous centrifugal flywheel governor system in finite time. The finite-time stability and convergence of the closed-loop system are analytically proved. A numerical simulation is given to illustrate the robustness and applicability of the proposed technique and to validate the theoretical results of the paper.

2. Description of centrifugal flywheel governor system and problem formulation

In this section, first the dynamics of a non-autonomous chaotic centrifugal flywheel governor system is described. Then, the problem of robust finite-time stabilization of the centrifugal flywheel governor with fully unknown parameters, model uncertainties, external disturbances and nonlinear control inputs is formulated.

2.1. Centrifugal flywheel governor system description

A centrifugal flywheel governor system (Beltrami, 1987) is depicted in Figure 1. The motor drives the flywheel to rotate with angular velocity ω. The flywheel is joined with an axis through a gear case. The axis rotates with angular velocity $n ω$ . Rods 1 and 2 of the length l are joined to a hinge at the end of the axis. There is a braced ball of mass m at the end of the rods 1 and 2. The balls are also connected to a sleeve over the axis by rods 3 and 4. A linear spring of stiffness k is attached to the sleeve, covering the upper portion of the axis. The vapor’s flux Q into the engine can be adjusted by the mechanical centrifugal flywheel governor on the sleeve, which is set to make the flywheel rotate at a certain angular velocity $ω_{0}$ . When $Δ ω = ω - ω_{0} \neq 0$ , the balls will move outward and inward, and the sleeve will move up and down (Chu et al., 2008).

Two basic assumptions in modeling the system are made as follows:

The masses of the rods and spring are neglected;

Viscous damping in the rod bearings is presented by a damping constant c.

Based on the notation used in Figure 1, the kinetic energy T_k and potential energy V_p of the system are written as follows:

T_{k} = 2 \times {\frac{1}{2} m [(l \sin ϕ) 2 n 2 ω 2 + l 2 \overset{\cdot}{ϕ} 2]} = m (l 2 n 2 ω 2 \sin 2 ϕ + l 2 \overset{\cdot}{ϕ} 2)

(1)

V_{p} = 2 kl 2 (1 - \cos ϕ) 2 + 2 mgl (1 - \cos ϕ)

(2)

where r is the distance between the rotational axis and the suspension joint and ϕ represents the angle between the rotational axis and the rods.

Therefore, the Lagrangian of the system is achieved as follows:

L = T_{k} - V_{p} = m (l 2 n 2 ω 2 \sin 2 ϕ + l 2 \overset{\cdot}{ϕ} 2) - 2 kl 2 (1 - \cos ϕ) 2 - 2 mgl (1 - \cos ϕ)

(3)

Using Lagrange’s equation, the equation of motion can be obtained for the governor as follows:

2 ml 2 \overset{··}{ϕ} - 2 ml 2 n 2 ω 2 \sin ϕ \cos ϕ + 4 kl 2 (1 - \cos ϕ) \sin ϕ + 2 mgl \sin ϕ = - c \overset{\cdot}{ϕ}

(4)

where c is the damping coefficient.

For the rotational machine, the net torque is the difference between the torque P₁ produced by the engine and the load torque P, which is available for angular acceleration. Hence, on the basis of Newton’s second law, there is

I \frac{d ω}{dt} = P_{1} - P

(5)

where I is the moment of inertia of the machine.

For a given angular velocity ω, it is assumed that P₁ is substituted for the average engine torque ${\bar{P}}_{1}$ . The position of the control valve, which admits the fuel, varies as the angle ϕ is varied. Thus, there is

P_{1} = {\bar{P}}_{1} + α (\cos ϕ - \cos ϕ_{0})

(6)

where

ϕ_{0}

and α are the angle corresponding to

ω_{0}

and a constant of proportionality, respectively.

Substituting equation (6) into equation (5), it yields

I \overset{\cdot}{ω} = α \cos ϕ - F

(7)

where

F = P + α \cos ϕ_{0} - {\bar{P}}_{1}

is the equivalent torque of the load.

When the equivalent load torque F varies with the time, it can be represented by $F = P + α \cos ϕ_{0} - {\bar{P}}_{1} + Ia \sin wt$ . Therefore, the motion of the centrifugal flywheel governor with a variable equivalent load torque is obtained using equations (4) and (7) as follows (Chu et al., 2008):

\overset{··}{ϕ} = (e + n 2 ω 2) \sin ϕ \cos ϕ - (e + g / l) \sin ϕ - b \overset{\cdot}{ϕ} \overset{\cdot}{ω} = (α \cos ϕ - F) / I - a \sin wt

(8)

where

e = 2 k / m

and

b = c / 2 ml 2

For the parameters’ values of $n = 3$ , $l = 1.5$ , $a = 0.8$ , $w = 1$ , $e = 0.3$ , $b = 0.4$ , $I = 1.2$ , $α = 0.611$ , $F = 0.3$ and $g = 9.8$ , the non-autonomous centrifugal flywheel governor (8) exhibits chaotic behavior (Chu et al., 2008). The irregular motion of this mechanical centrifugal flywheel governor system with the initial conditions of $ϕ (0) = 0.006$ , $\overset{\cdot}{ϕ} (0) = 0.007$ and $\overset{\cdot}{ω} (0) = 0.15$ is illustrated in Figures 2 and 3. Moreover, the Lyapunov exponents of the system (8) are depicted in Figure 4. It is obvious that since there is a positive Lyapunov exponent, the system (8) is chaotic.

2.2. Problem Formulation

Defining $x_{1} = ϕ$ , $x_{2} = \overset{\cdot}{ϕ}$ and $x_{3} = ω$ , the non-autonomous centrifugal flywheel governor system (8) can be rewritten with model uncertainties, external disturbances, unknown parameters and nonlinear control inputs as follows:

{\overset{\cdot}{x}}_{1} = x_{2} + Δ f_{1} (x, t) + h_{1} (u_{1}) {\overset{\cdot}{x}}_{2} = (e + n 2 x_{3}^{2}) \sin x_{1} \cos x_{1} - (e + g / l) \sin x_{1} - {bx}_{2} + Δ f_{2} (x, t) + h_{2} (u_{2}) {\overset{\cdot}{x}}_{3} = (α \cos x_{1} - F) / I - a \sin wt + Δ f_{3} (x, t) + h_{3} (u_{3})

(9)

where

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

is the state vector of the system,

Δ f_{i} (x, t), i = 1, 2, 3

represents time-varying model uncertainties and external disturbances of the system,

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

is the vector of control inputs and

h_{1} (u_{1}), h_{2} (u_{2})

and

h_{3} (u_{3})

with

h_{i} (0) = 0, i = 1, 2, 3

are continuous nonlinear functions inside the sector

[ρ_{i}, ν_{i}], i = 1, 2, 3, ρ_{i} > 0

, i.e.

ρ_{i} u_{i}^{2} \leq u_{i} h_{i} (u_{i}) \leq ν_{i} u_{i}^{2}, i = 1, 2, 3

(10)

Assumption 1

Since the trajectories of chaotic systems are always bounded (Curran and Chua, 1997), the uncertainty $Δ f_{i} (x, t), i = 1, 2, 3$ is assumed to be bounded by

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

(11)

where

d_{i}, i = 1, 2, 3

is a given positive constant.

Assumption 2

In practical or experimental situations, the system parameters are inevitably disturbed by external inartificial factors, such as environment temperature, voltage fluctuation, mutual interference between components, etc., and their values cannot be exactly known in advance. Therefore, it is assumed that the values of the parameters $n, l, a, w, e, b, I, α, F$ and g are unknown in advance. As a result, the nominal values of the system parameters will not be used in control laws (i.e. the nominal values of the system parameters are only utilized to simulate the dynamics of the system). Instead, some adaptive parameters will be proposed in control laws to estimate the system unknown parameters.

For simplicity, let $θ = [θ_{1}, θ_{2}, θ_{3}, θ_{4}] T = [e, n 2, e + g / l, b] T$ and $ψ = [ψ_{1}, ψ_{2}] T = [α / I, a + F / I] T$ to be the vectors of the unknown parameters of the second and third state equations of the centrifugal flywheel governor system (9). Then, the following assumption is made.

Assumption 3

In order to ensure that the governor system exhibits chaos, the system parameters are assumed to lie in a bounded range. Therefore, it is assumed that the unknown vector parameters θ and ψ are norm bounded, i.e.

‖ θ ‖ \leq Θ, ‖ ψ ‖ \leq Ψ

(12)

where

‖ . ‖

denotes the Euclidean norm in

R n

and Θ and Ψ are known positive constants.

Finite-time stabilization of the non-autonomous centrifugal flywheel governor system means that there exists a constant $T = T (x (0)) > 0$ , such that ${Lim}_{t \to T} ‖ x (t) ‖ = 0$ and $‖ x (t) ‖ \equiv 0$ , if $t \geq T$ .

3. Design of a robust adaptive finite-time controller

To design a robust adaptive finite-time controller, first appropriate adaptation laws are derived to undertake the unknown parameters of the system. Then, suitable robust adaptive control laws are proposed to stabilize the closed-loop system in finite time. Finally, the finite-time convergence and stability of the proposed control scheme are proved.

Lemma 1

Assume $u_{i} (t) = χ_{i} sgn (x_{i}), χ_{i} < 0$ , then the following inequality holds:

x_{i} h_{i} (u_{i}) \leq ρ_{i} χ_{i} | x_{i} |

(13)

where

sgn (.)

is the sign function.

Proof

If u_i in the left hand side of equation (10) is replaced by $u_{i} (t) = χ_{i} sgn (x_{i})$ , then there is

ρ_{i} χ_{i}^{2} sgn (x_{i}) 2 \leq χ_{i} sgn (x_{i}) h_{i} (u_{i})

(14)

Using $sgn (x_{i}) 2 = 1$ , there follows

ρ_{i} χ_{i}^{2} \leq χ_{i} sgn (x_{i}) h_{i} (u_{i})

(15)

Since $χ_{i} < 0$ and with multiplying $| x_{i} | / χ_{i}$ to both sides of the above inequality, there is

x_{i} h_{i} (u_{i}) \leq ρ_{i} χ_{i} | x_{i} |

(16)

This completes the proof.

Lemma 2

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

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

(17)

where

q > 0

0 < ξ < 1

are two constants. Then, for any given t₀,

V (t)

satisfies the following inequality:

V 1 - ξ (t) \leq V 1 - ξ (t_{0}) - q (1 - ξ) (t - t_{0}), t_{0} \leq t \leq t_{1}

(18)

and

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

with t₁ given by

t_{1} = t_{0} + \frac{V 1 - ξ (t_{0})}{q (1 - ξ)}

(19)

In order to guarantee the finite-time stabilization of system (9), suitable control laws are derived. The control laws should be with the following features:

The control laws should overcome input nonlinearities. In order to gain this feature, a switching control law is proposed which includes the sign function. This alternative is inspired of the property proven in Lemma 1.

The control inputs must be robust against system uncertainties. On the other hand, since the system uncertainties are unknown in engineering practical applications, only the norm bounds of uncertainties are used to construct a robust and reliable controller. Therefore, motivated by Assumption 1, the control inputs contain the bounds of the system uncertainties.

The other feature of the control laws is that they should be able to undertake the system’s unknown parameters. This problem can be solved using the adaptive control theory. Consequently, proper adaptive parameters are introduced in control laws to adaptively estimate the system’s unknown parameters.

Finally, the control laws should be designed in a form ensuring the finite-time stability of the closed-loop system. Inspired by the finite time control theory (Lemma 2), the control inputs should include some terms to satisfy the results of Lemma 2. As a result, it is discovered that a constant gain with a state-normalized term multiplied by the sum of the bounds of the unknown parameters and adaptive parameters should be employed.

According to the above comments, suitable robust adaptive finite-time control laws are

u_{1} (t) = - \frac{1}{ρ_{1}} (| x_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) (\frac{| x_{1} |}{‖ x ‖ 2}) + d_{1} + η_{1}) sgn (x_{1}) = χ_{1} sgn (x_{1}) u_{2} (t) = - \frac{1}{ρ_{2}} (| {\overset{\land}{θ}}_{1} \sin x_{1} \cos x_{1} | + x_{3}^{2} | {\overset{\land}{θ}}_{2} \sin x_{1} \cos x_{1} | + | {\overset{\land}{θ}}_{3} \sin x_{1} | + | {\overset{\land}{θ}}_{4} x_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) \times (\frac{| x_{2} |}{‖ x ‖ 2}) + d_{2} + η_{2}) sgn (x_{2}) = χ_{2} sgn (x_{2}) u_{3} (t) = - \frac{1}{ρ_{3}} (| {\overset{\land}{ψ}}_{1} \cos x_{1} | + | {\overset{\land}{ψ}}_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) \times (\frac{| x_{3} |}{‖ x ‖ 2}) + d_{3} + η_{3}) sgn (x_{3}) = χ_{3} sgn (x_{3})

(20)

where

\overset{\land}{θ} = [{\overset{\land}{θ}}_{1}, {\overset{\land}{θ}}_{2}, {\overset{\land}{θ}}_{3}, {\overset{\land}{θ}}_{4}] T

and

\overset{\land}{ψ} = [{\overset{\land}{ψ}}_{1}, {\overset{\land}{ψ}}_{2}] T

are estimations for unknown parameters θ and ψ, respectively, if

‖ x ‖ = 0

then

| x_{i} | / ‖ x ‖ 2 = 0

μ = min {η_{i}}, i = 1, 2, 3

and

η_{i} > 0

is a constant gain.

To tackle the unknown parameters, the following adaptation laws are proposed.

\overset{\cdot}{\overset{\land}{}} θ_{1} (t) = x_{2} \sin x_{1} \cos x_{1}, {\overset{\land}{θ}}_{1} (0) = {\overset{\land}{θ}}_{10} \overset{\cdot}{\overset{\land}{}} θ_{2} (t) = x_{2} x_{3}^{2} \sin x_{1} \cos x_{1}, {\overset{\land}{θ}}_{2} (0) = {\overset{\land}{θ}}_{20} \overset{\cdot}{\overset{\land}{}} θ_{3} (t) = - x_{2} \sin x_{1}, {\overset{\land}{θ}}_{3} (0) = {\overset{\land}{θ}}_{30} \overset{\cdot}{\overset{\land}{}} θ_{4} (t) = - x_{2}^{2}, {\overset{\land}{θ}}_{4} (0) = {\overset{\land}{θ}}_{40} \overset{\cdot}{\overset{\land}{}} ψ_{1} (t) = x_{3} \cos x_{1}, {\overset{\land}{ψ}}_{1} (0) = {\overset{\land}{ψ}}_{10} \overset{\cdot}{\overset{\land}{}} ψ_{2} (t) = | x_{3} |, {\overset{\land}{ψ}}_{2} (0) = {\overset{\land}{ψ}}_{20}

(21)

where

{\overset{\land}{θ}}_{10}, {\overset{\land}{θ}}_{20}, {\overset{\land}{θ}}_{30}, {\overset{\land}{θ}}_{40}, {\overset{\land}{ψ}}_{10}

and

{\overset{\land}{ψ}}_{20}

are the initial values of the adaptation parameters

{\overset{\land}{θ}}_{1}, {\overset{\land}{θ}}_{2}, {\overset{\land}{θ}}_{3}, {\overset{\land}{θ}}_{4}, {\overset{\land}{ψ}}_{1}

and

{\overset{\land}{ψ}}_{2}

, respectively.

Theorem 1

Consider the uncertain non-autonomous centrifugal flywheel governor system (9). If this system is controlled by control inputs (20) with adaptation laws (21), then the system trajectories will converge to zero in a finite time T, determined by

T = \frac{\sqrt{2}}{μ} (\frac{1}{2} (‖ x (0) ‖ 2 + ‖ \overset{\land}{θ} (0) - θ ‖ 2 + ‖ \overset{\land}{ψ} (0) - ψ ‖ 2)) 1 / 2

(22)

Proof

Choose a positive definite function in the form of

V (t) = \frac{1}{2} (‖ x ‖ 2 + ‖ \overset{\land}{θ} - θ ‖ 2 + ‖ \overset{\land}{ψ} - ψ ‖ 2)

(23)

Taking the time derivative of $V (t)$ , there is

\overset{\cdot}{V} (t) = x_{1} {\overset{\cdot}{x}}_{1} + x_{2} {\overset{\cdot}{x}}_{2} + x_{3} {\overset{\cdot}{x}}_{3} + (\overset{\land}{θ} - θ) T \overset{\cdot}{\overset{\land}{}} θ + (\overset{\land}{ψ} - ψ) T \overset{\cdot}{\overset{\land}{}} ψ

(24)

Inserting ${\overset{\cdot}{x}}_{1}$ , ${\overset{\cdot}{x}}_{2}$ and ${\overset{\cdot}{x}}_{3}$ from equation (9) into the above equation, this yields

\overset{\cdot}{V} (t) = x_{1} [x_{2} + Δ f_{1} (x, t) + h_{1} (u_{1})] + x_{2} [(e + n 2 x_{3}^{2}) \times \sin x_{1} \cos x_{1} - (e + g / l) \sin x_{1} - {bx}_{2} + Δ f_{2} (x, t) + h_{2} (u_{2})] + x_{3} [(α \cos x_{1} - F) / I - a \sin wt + Δ f_{3} (x, t) + h_{3} (u_{3})] + (\overset{\land}{θ} - θ) T \overset{\cdot}{\overset{\land}{}} θ + (\overset{\land}{ψ} - ψ) T \overset{\cdot}{\overset{\land}{}} ψ

(25)

Knowing that the parameters of the centrifugal flywheel governor system are all positive (Chu et al., 2008), the following is obtained

\overset{\cdot}{V} (t) \leq x_{1} [x_{2} + h_{1} (u_{1})] + | x_{1} | | Δ f_{1} (x, t) | + x_{2} [(e + n 2 x_{3}^{2}) \sin x_{1} \cos x_{1} - (e + g / l) \sin x_{1} - {bx}_{2} + h_{2} (u_{2})] + | x_{2} | | Δ f_{2} (x, t) | + x_{3} [α \cos x_{1} / I + h_{3} (u_{3})] + | x_{3} | (F / I + a) + | x_{3} | | Δ f_{3} (x, t) | + (\overset{\land}{θ} - θ) T \overset{\cdot}{\overset{\land}{}} θ + (\overset{\land}{ψ} - ψ) T \overset{\cdot}{\overset{\land}{}} ψ

(26)

By Assumption 1 and using

θ T \overset{\cdot}{\overset{\land}{}} θ = {ex}_{2} \sin x_{1} \cos x_{1} + n 2 x_{2} x_{3}^{2} \sin x_{1} \cos x_{1} - (e + g / l) x_{2} \sin x_{1} - {bx}_{2}^{2}

and

ψ T \overset{\cdot}{\overset{\land}{}} ψ = (α / I) x_{3} \cos x_{1} + (F / I + a) | x_{3} |

, there is

\overset{\cdot}{V} (t) \leq x_{1} [x_{2} + h_{1} (u_{1})] + d_{1} | x_{1} | + x_{2} h_{2} (u_{2}) + d_{2} | x_{2} | + x_{3} h_{3} (u_{3}) + d_{3} | x_{3} | + \overset{\land}{θ} T \overset{\cdot}{\overset{\land}{}} θ + \overset{\land}{ψ} T \overset{\cdot}{\overset{\land}{}} ψ

(27)

Based on Lemma 1 and the control inputs (20), the following can be obtained

\overset{\cdot}{V} (t) \leq | x_{1} x_{2} | + ρ_{1} | x_{1} | [- \frac{1}{ρ_{1}} (| x_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) \times (\frac{| x_{1} |}{‖ x ‖ 2}) + d_{1} + η_{1})] + d_{1} | x_{1} | + ρ_{2} | x_{2} | \times [- \frac{1}{ρ_{2}} (| {\overset{\land}{θ}}_{1} \sin x_{1} \cos x_{1} | + x_{3}^{2} | {\overset{\land}{θ}}_{2} \sin x_{1} \cos x_{1} | + | {\overset{\land}{θ}}_{3} \sin x_{1} | + | {\overset{\land}{θ}}_{4} x_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) \times (\frac{| x_{2} |}{‖ x ‖ 2}) + d_{2} + η_{2})] + d_{2} | x_{2} | + ρ_{3} | x_{3} | \times [- \frac{1}{ρ_{3}} (| {\overset{\land}{ψ}}_{1} \cos x_{1} | + | {\overset{\land}{ψ}}_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) (\frac{| x_{3} |}{‖ x ‖ 2}) + d_{3} + η_{3})] + d_{3} | x_{3} | + \overset{\land}{θ} T \overset{\cdot}{\overset{\land}{}} θ + \overset{\land}{ψ} T \overset{\cdot}{\overset{\land}{}} ψ

(28)

It is clear that

\overset{\cdot}{V} (t) \leq - | x_{1} | [μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) (\frac{| x_{1} |}{‖ x ‖ 2})] - | x_{1} | η_{1} - | x_{2} | [| {\overset{\land}{θ}}_{1} \sin x_{1} \cos x_{1} | + x_{3}^{2} | {\overset{\land}{θ}}_{2} \sin x_{1} \cos x_{1} | + | {\overset{\land}{θ}}_{3} \sin x_{1} | + | {\overset{\land}{θ}}_{4} x_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) (\frac{| x_{2} |}{‖ x ‖ 2})] - | x_{2} | η_{2} - | x_{3} | [| {\overset{\land}{ψ}}_{1} \cos x_{1} | + | {\overset{\land}{ψ}}_{2} | + μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖) (\frac{| x_{3} |}{‖ x ‖ 2})] - | x_{3} | η_{3} + | \overset{\land}{θ} | T | \overset{\cdot}{\overset{\land}{}} θ | + | \overset{\land}{ψ} | T | \overset{\cdot}{\overset{\land}{}} ψ |

(29)

where the absolute value for a vector

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

is defined as

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

Using $\frac{x_{1}^{2}}{‖ x ‖ 2} + \frac{x_{2}^{2}}{‖ x ‖ 2} + \frac{x_{3}^{2}}{‖ x ‖ 2} = 1$ ,

| \overset{\land}{θ} | T | \overset{\cdot}{\overset{\land}{}} θ | = | {\overset{\land}{θ}}_{1} | | x_{2} \sin x_{1} \cos x_{1} | + | {\overset{\land}{θ}}_{2} | | x_{2} x_{3}^{2} \sin x_{1} \cos x_{1} | + | {\overset{\land}{θ}}_{3} | | x_{2} \sin x_{1} | + | {\overset{\land}{θ}}_{4} | x_{2}^{2}

and

| \overset{\land}{ψ} | T | \overset{\cdot}{\overset{\land}{}} ψ | = | {\overset{\land}{ψ}}_{1} | | x_{3} \cos x_{1} | + | {\overset{\land}{ψ}}_{2} | | x_{3} |

, there is

\overset{\cdot}{V} (t) \leq - η_{1} | x_{1} | - η_{2} | x_{2} | - η_{3} | x_{3} | - μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖)

(30)

Using the classic triangle inequality, there follows

\overset{\cdot}{V} (t) \leq - μ ‖ x ‖ - μ (Θ + ‖ \overset{\land}{θ} ‖ + Ψ + ‖ \overset{\land}{ψ} ‖)

(31)

By Assumption 2 and since $‖ \overset{\land}{θ} - θ ‖ \leq ‖ \overset{\land}{θ} ‖ + ‖ θ ‖ \leq ‖ \overset{\land}{θ} ‖ + Θ$ and $‖ \overset{\land}{ψ} - ψ ‖ \leq ‖ \overset{\land}{ψ} ‖ + ‖ ψ ‖ \leq ‖ \overset{\land}{ψ} ‖ + Ψ$ are always satisfied, there is

\overset{\cdot}{V} (t) \leq - μ (‖ x ‖ + ‖ \overset{\land}{θ} - θ ‖ + ‖ \overset{\land}{ψ} - ψ ‖) \leq - \sqrt{2} μ (\frac{1}{2} (‖ x ‖ 2 + ‖ \overset{\land}{θ} - θ ‖ 2 + ‖ \overset{\land}{ψ} - ψ ‖ 2)) 1 / 2 = - \sqrt{2} μ V 1 / 2

(32)

Therefore, according to Lemma 2, the system trajectories will converge to zero, in the finite time $T = \frac{\sqrt{2}}{μ} (\frac{1}{2} (‖ x (0) ‖ 2 + ‖ \overset{\land}{θ} (0) - θ ‖ 2 + ‖ \overset{\land}{ψ} (0) - ψ ‖ 2)) 1 / 2$ . Hence the non-autonomous centrifugal flywheel governor system (9) in the presence of model uncertainties, external disturbances, unknown parameters and input nonlinearities will be stable in finite time and the proof is achieved completely.

Remark 1

Since the control inputs (21) include the terms $| x_{i} | / ‖ x ‖ 2$ and $sgn (x_{i}) = | x_{i} | / x_{i}, i = 1, 2, 3$ , the undesirable chattering (high frequency oscillations with finite amplitude) may occur. In order to evade the chattering, $| x_{i} | / ‖ x ‖ 2$ and $sgn (x_{i}) = | x_{i} | / x_{i}, i = 1, 2, 3$ are replaced by $| x_{i} | / (‖ x ‖ 2 + ɛ)$ and $| x_{i} | / (x_{i} + ɛ), i = 1, 2, 3$ , respectively; where $ɛ > 0$ is a sufficiently small constant.

4. Numerical simulations

In this section, numerical simulations are presented to validate the robustness and applicability of the proposed finite-time controller in the stabilization of the uncertain non-autonomous centrifugal flywheel governor system with nonlinear control inputs. The simulations are carried out using MATLAB software. The ode45 solver is used to solve nonlinear equations. The parameters $n = 3$ , $l = 1.5$ , $a = 0.8$ , $w = 1$ , $e = 0.3$ , $b = 0.4$ , $I = 1.2$ , $α = 0.611$ , $F = 0.3$ and $g = 9.8$ are selected in the simulation to simulate the dynamics of the system. All the input nonlinearities are considered to be $h_{i} (u_{i}) = [5 + 3 \cos (t)] u_{i}, i = 1, 2, 3$ . As a result, it can be concluded that $ρ_{i} = 2, i = 1, 2, 3$ . Subsequently, the following uncertainties and disturbances are considered in the simulation.

Δ f_{1} (x, t) = 0.3 \cos (4 x_{1}) - 0.3 \sin (t) Δ f_{2} (x, t) = - 0.25 \sin (2 x_{2}) + 0.35 \cos (2 t) Δ f_{3} (x, t) = - 0.35 \cos (x_{3}) + 0.25 \sin (5 t)

(33)

As a result, it can be obtained that $d_{1} = d_{2} = d_{3} = 0.45$ satisfies the conditions of Assumption 1. Θ and Ψ are chosen equal to 10 and two, respectively. The initial values of the centrifugal flywheel governor system are chosen as $x_{1} (0) = 0.006$ , $x_{2} (0) = 0.007$ and $x_{3} (0) = 0.15$ . In addition, the initial values of both adaptation vector parameters $\overset{\land}{θ}$ and $\overset{\land}{ψ}$ are all set to one. Constant gains $η_{1}$ , $η_{2}$ and $η_{3}$ are all chosen equal to one.

State trajectories of the uncertain chaotic non-autonomous centrifugal flywheel governor system (9) controlled with the controller of equations (20) and (21) are illustrated in Figure 5, where the control inputs are applied at t = 5s. It can be seen that the system trajectories converge to zero quickly. This means that on applying the proposed robust adaptive controller, the uncertain chaotic non-autonomous centrifugal flywheel governor system (9) with model uncertainties, external disturbances, unknown parameters and nonlinear control inputs has been stabilized in a finite time and the closed-loop system is no longer chaotic, and, therefore, there is no strange attractor. The time responses of the adaptation vector parameters $\overset{\land}{θ}$ and $\overset{\land}{ψ}$ are displayed in Figures 6 and 7, respectively. Obviously, all adaptation parameters converge to some constants. Moreover, Figure 8 illustrates the time histories of the applied control inputs. It is apparent that the control inputs have no chattering and oscillation, implying that they are feasible in practical applications.

5. Conclusions

The problem of finite-time stabilization of a non-autonomous centrifugal flywheel governor system is studied in this paper. It is assumed that the parameters of the centrifugal flywheel governor system are fully unknown in advance. In addition, unknown model uncertainties and external disturbances are imposed to the system dynamics. The effects of the nonlinear control inputs are also taken into account. To tackle the system unknown parameters, proper adaptation laws are designed. Using the finite-time control theory and adaptation parameters, a finite-time robust adaptive controller is proposed. The finite-time stability and convergence of the proposed controller are mathematically proved. Numerical simulations reveal that the proposed controller works well for the finite-time stabilization of the uncertain chaotic non-autonomous centrifugal flywheel governor system, even when the system parameters are fully unknown, unknown uncertainties and external disturbances are present in the system dynamics and control inputs contain nonlinearities. It is worth noting that the results of this paper are of practical utility to designers of rotational machines.

Footnotes

Funding

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

References

Abno

Hata

(1981) Microcomputer engine control. Microprocessors and Microsystems 5: 451–457.

Aghababa

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

Aghababa

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

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.

Beltrami

(1987) Mathematics for Dynamic Modeling, Boston, MA: Academic Press.

Bhat

Bernstein

(2000) Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 38: 751–766.

Cai

Chen

(2007) Synchronization criteria for non-autonomous chaotic systems via sinusoidal state error feedback control. Physica Scripta 75: 379–387.

Chen

Liu

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

Chu

Y-D

Zhang

Chang

Luo

(2008) Chaos and chaos synchronization for a non-autonomous rotational machine systems. Nonlinear Analysis RWA 9: 1378–1393.

10.

Curran

Chua

(1997) Absolute stability theory and the synchronization problem. International Journal of Bifurcation and Chaos 7: 1357–1382.

11.

Jhuang

(2007) Chaos control and synchronization of a fractional order rotational mechanical system with a centrifugal governor. Chaos, Solitons and Fractals 33: 270–289.

12.

Lee

(2003) Non-linear dynamics and control of chaos for a rotational machine with a hexagonal centrifugal governor with a spring. Journal of Sound and Vibration 262: 845–864.

13.

Lee

(2005a) Anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor. Journal of Sound and Vibration 282: 635–648.

14.

Lee

(2005b) Control anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay. Chaos, Solitons and Fractals 23: 1855–1864.

15.

Lee

(2005) Chaos synchronization and parameter identification for gyroscope system. Applied Mathematics and Computation 163: 667–682.

16.

Yang

Chen

(1999) Regular and chaotic dynamics of a rotational machine with a centrifugal governor. International Journal of Engineering and Science 37: 921–943.

17.

Hills

(1996) Power from the Wind: a History of Windmill Technology, Cambridge, UK: Cambridge University Press.

18.

Hseng

Huang

(2010) MIMO adaptive fuzzy terminal sliding-mode controller for robotic manipulators. Information Sciences 180: 4641–4660.

19.

Kuo

(2007) Design of an adaptive fuzzy sliding-mode controller for chaos synchronization. International Journal of Nonlinear Science and Numerical Simulation 8: 631–636.

20.

Kuo

(2011) design of fuzzy sliding-mode synchronization controller for two different chaos systems. Computers and Mathematics with Applications 61: 2090–2095.

21.

Kwon

Park

Lee

(2011) Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dynamics 63: 239–252.

22.

Lee

Choi

Park

Won

(2010) Synchronization for chaotic Lur'e systems with sector restricted nonlinearities via delayed feedback control. Nonlinear Dynamics 59: 277–288.

23.

Lei

Shen

Fang

(2006) Global synchronization of two parametrically excited systems using active control. Chaos, Solitons and Fractals 28: 428–436.

24.

Ott

Grebogi

Yorke

(1990) Using chaos to direct trajectories to targets. Physics Review Letters 65: 3215–2158.

25.

Park

Kwon

(2005a) A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos, Solitons and Fractals 23: 495–501.

26.

Park

Kwon

(2005b) LMI optimization approach to stabilization of time-delay chaotic systems. Chaos, Solitons and Fractals 23: 445–450.

27.

Salarieh

Alasty

(2008) Chaos synchronization of nonlinear gyros in presence of stochastic excitation via sliding mode control. Journal of Sound and Vibration 313: 760–771.

28.

Sotomayor

Mello D de

Braga

(2008) Stability and Hopf bifurcation in an hexagonal governor system. Nonlinear Analysis: Real World Applications 9: 889–898.

29.

Wang

Han

Xie

Zhang

(2009) Finite-time chaos control via nonsingular terminal sliding mode control. Communications in Nonlinear Science and Numerical Simulation 14: 2728–2733.

30.

Cai

Wang

(2006) Master–slave chaos synchronization criteria for the horizontal platform systems via linear state error feedback control. Journal of Sound and Vibration 295: 378–387.

31.

Yang

Chua

(2000) Practical stability of impulsive synchronization between two non-autonomous chaotic systems. International Journal of Bifurcation and Chaos 15: 859–867.

32.

Yoo

Won

(2010) Synchronization of two different non-autonomous chaotic systems using fuzzy disturbance observer. Physics Letters A 374: 1354–1361.

33.

Zhang

J-N

Chu

Chang

(2008) Bifurcation and chaos of a non-autonomous rotational machine systems. Simulation Modelling Practice and Theory 16: 1588–1605.

34.

Zhang

Chu

Chang

(2009) Hopf bifurcations Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system. Chaos, Solitons and Fractals 39: 2150–2168.