Finite time attitude tracking control of an autonomous airship

Abstract

The problem of station-keeping attitude tracking control for an autonomous airship with system uncertainties and external disturbances is investigated. Adaptive laws are applied to estimate the upper bounds of uncertainties and disturbances, and a nonlinear finite time control scheme is proposed by combing input/output feedback linearization with integral sliding mode technique. Different from the existing works on attitude control of airship, the developed controller can guarantee the yaw, pitch and roll angle trajectories track the desired attitude in finite time in spite of uncertain system uncertainties and external disturbances. Simulation results are provided to illustrate the attitude tracking performance.

Keywords

Autonomous airship finite time attitude control input/output feedback linearization integral sliding mode control adaptive control

Introduction

The study of autonomous airships has gradually become a hot research topic (see, e.g., Li et al., 2009; Won, 2002, and the references therein) in the past years because they suit a wide range of applications, ranging from advertising, aerial photography and aerial inspection platforms. The key benefits of airships over other aircraft are their ability to hover and fly at low speed. However, the autonomous airship is still in its early stages of development, especially at high altitudes. One of the key techniques is the flight control system. Many control approaches have been developed for airships in recent years, including nonlinear control (see Azinheira et al., 2009; Chen et al., 2012; Zheng et al., 2012), adaptive control (see Yang et al., 2013; Zhang et al., 2008), intelligent control (see Hong et al., 2009; Rao et al., 2007), sliding mode control (see Yang et al., 2012) and so on.

As one key focus of research, attitude control is an important component of airship control system and is the precondition for the airship’s trajectory control. Recent decades have witnessed important developments in the design of attitude laws for airship tracking, and many control schemes have been proposed. In Wang and Shan (2006), an attitude tracking control law using input/output feedback linearization and the Lyapunov method was designed. In Yang et al. (2011, 2012), fuzzy sliding mode control methods were proposed for attitude control of an unmanned airship. In Liesk et al. (2013), the problem of attitude, velocity and height control of an unmanned, unstable, fin-less airship was solved by designing a combined backstepping/Lyapunov controller. In Valle et al. (2015), a new linearization-based gain-scheduled controller method was proposed. To improve fault tolerance capability of airship attitude control system, the backstepping (Wu et al., 2012) and Lyapunov methods (Wang et al., 2014) have been employed.

Note that the existing attitude control laws for airships are manly asymptotically stable control laws, which means the convergence is at best exponential. In other words, the desired attitude cannot be tracked in finite time. Obviously, attitude control laws with finite convergence rates are more desirable. Besides the faster convergence rate, the systems under finite time control usually demonstrate high accuracies, better disturbance rejection properties, and better robustness against uncertainties (Li et al., 2009, 2011). Due to the above superiorities, the study on finite time attitude control for diverse types of aircraft (especially for spacecraft) has gained much attention in recent years: see, for example, Jin and Sun (2008), Du et al. (2011), Zou et al. (2011), Zhu et al. (2011), Du and Li (2012), Xiao et al. (2015) and references therein. Due to the fact that the gravity centre does not coincide with the buoyancy centre, the dynamic equations in airships contain attitude variables, which is usually different with other types of aircraft. Hence, the most of existing finite time attitude control methods cannot be applied to airship dynamics directly. So far, to the best of author’s knowledge, there is no finite time control result about attitude tracking for airships, although it is of both theoretical and practical importance.

In this paper, motivated by the above discussion, a finite time attitude control scheme is developed for airships. By using the input/output feedback linearization approach, the control of station-keeping attitude motion for airships is equivalent to stabilization of a linear multivariable system with bounded uncertainties. Adaptive laws are used to estimate the bounds of uncertainties and an integral sliding mode controller is designed such that the states of transformed system can be stabilized in finite time. Simulation results illustrate the effectiveness of proposed attitude control method. The main contributions of this paper can be briefly outlined as follows.

A finite time attitude tracking control law is developed by combing input/output feedback linearization and integral sliding mode technique, which can force the attitude tracking errors to converge to the origin in a finite time. Based on the proposed controller, the problem of finite time attitude tracking control for airships is investigated for the first time.

In a practical airship system, the inertia uncertainties and external disturbances are unavoidable due to complex flight condition. The uncertainties in the inertial matrix and external disturbances are not considered in Yang et al. (2012), Liesk et al. (2013), Valle et al. (2015), Wu et al. (2012), Wang et al. (2014). In Wang and Shan (2006) and Yang et al. (2011), the research works give a control design based on the assumption that the uncertainties are bounded by known upper bounds, which are not easily obtained in practice. In contrast to the previously cited works on attitude control for airships, the adaptive laws are designed and the upper bounds of inertia uncertainties and disturbances are not required.

The rest of this paper is organized as follows. In the section on airship attitude model and problem formulation, the airship model is introduced and the input/output feedback linearization of the station-keeping attitude tracking model followed by controller design is given. In the section on finite time attitude tracking controller design, a finite time controller is designed by combing input/output feedback linearization with integral sliding mode control. The next section simulates the results of an autonomous airship with a derived controller. The last section concludes and considers future work.

Airship attitude model and problem formulation

The motion of an airship is usually represented by a set of nonlinear kinematics and dynamics equations that describe its evolution in a six degrees-of-freedom (6-DOF) space. The coordinate system is depicted in Figure 1. $P x_{i} y_{i} z_{i}$ is an earth-fixed inertial frame, with the origin on the surface of the earth, the $x_{i}$ -axis points north, the $y_{i}$ -axis points east, and the $z_{i}$ -axis points down. $O x_{b} y_{b} z_{b}$ is the body-fixed frame, with the origin at the centre of the volume, the $x_{b}$ -axis points forward, the $y_{b}$ -axis points right, and the $z_{b}$ -axis points downward. $O x_{v} y_{v} z_{v}$ is the velocity-fixed frame, with the origin at the centre of the volume, the $x_{v}$ -axis points forward, the $y_{v}$ -axis points right, and the $z_{v}$ -axis points downward. Under the established coordinate frames, the attitude motion of an autonomous airship can be described by its angle and angular velocity over time. The vector $r_{G} = {[x_{G}, y_{G}, z_{G}]}^{T}$ denotes the distance from centre of volume to the centre of gravity. The attitude is described by using Euler angles $Ω = {[\begin{matrix} ϑ & ψ & ϕ \end{matrix}]}^{T}$ , where $ϑ$ , $ψ$ , $ϕ$ are pitch, yaw and roll angles, respectively. The angular velocities are described by $w = {[\begin{matrix} p & q & r \end{matrix}]}^{T}$ , where p,q,r are the rolling, pitching and yawing angular velocities, respectively.

The kinematics equations of the airship can be expressed as follows (Li et al., 2009)

[\begin{matrix} \overset{\cdot}{P} \\ \overset{\cdot}{Ω} \end{matrix}] = [\begin{matrix} S_{1} & 0_{3 \times 3} \\ 0_{3 \times 3} & S_{2} \end{matrix}] [\begin{matrix} v \\ w \end{matrix}]

(1)

where $S_{1}$ is the rotation matrix from body-fixed frame to earth-fixed inertial frame

S_{1} = [\begin{matrix} \cos ψ \cos ϑ & \cos ψ \sin ϑ \sin ϕ - \sin ψ \cos ϕ & \cos ψ \sin ϑ \cos ϕ + \sin ψ \sin ϕ \\ \sin ψ \cos ϑ & \sin ψ \sin ϑ \sin ϕ + \cos ψ \cos ϕ & \sin ψ \sin ϑ \cos ϕ - \cos ψ \sin ϕ \\ - \sin ϑ & \cos ϑ \sin φ & \cos ϑ \cos ϕ \end{matrix}]

(2)

and $S_{2}$ denotes the transforming relationship between Euler angles rate and angular velocity

S_{2} = [\begin{matrix} 0 & \cos ϕ & - \sin ϕ \\ 0 & \sec ϑ \sin ϕ & \sec ϑ \cos ϕ \\ 1 & \sin ϕ \tan ϑ & \cos ϕ \tan ϑ \end{matrix}]

(3)

The airship is assumed to be a rigid body, and the aero-elastic effects are ignored. The dynamics equations of the platform can be derived from Newton–Euler formulation, which can be expressed by two vector equations in the earth frame as follows

\frac{d p}{dt} = F

(4)

\frac{d H}{dt} = τ

(5)

where $F$ represents the sum of all external forces, $τ$ represents the sum of all external moments, $p$ is the linear momentum and $H$ is the angular momentum.

Accounting for the added mass and added inertia, the linear momentum and angular momentum of the airship can be stated as

\begin{array}{l} p = (m E_{3 \times 3} + M_{a}) v + w \times r_{G} \\ = (M_{0} + M_{a}) v + w \times r_{G} = M v + w \times r_{G} \end{array}

(6)

H = (I_{0} + I_{a}) w + m r_{G} \times v = Iw + r_{G} \times (m v)

(7)

where m is the mass of airship, $E$ is the $3 \times 3$ identity matrix, $M_{0}$ is the added mass matrix, $I_{0}$ is the inertial matrix, $I_{a}$ is the added inertia matrix, $M$ is the total mass matrix, and $I$ is the total inertial matrix.

Substituting equation (6) into equation (4), the time derivation of the linear momentum in the earth frame is obtained as

\begin{array}{l} \frac{d p}{d t} = \frac{d (M v + w \times r_{G})}{d t} \\ = \frac{d M v}{d t} + m (\frac{d w}{d t} \times r_{G} + w \times \frac{d r_{G}}{d t}) = F \end{array}

(8)

and then transported into the body frame according to the vector differential principle, it is obtained as

M \overset{\cdot}{v} + w \times (M v) + m \overset{\cdot}{w} \times r_{G} + m \overset{\cdot}{w} \times (w \times r_{G}) = F

(9)

Substituting equation (7) into equation (5), we obtain the equation of angular motion

[I \dot{w} + w \times (I w)] + m [r_{G} \times v + w \times (r_{G} \times v)] = τ

(10)

Combining equations (9) and (10), dynamics model of the airship can be stated as

[\begin{matrix} M & - m r_{G}^{\times} \\ m r_{G}^{\times} & I \end{matrix}] [\begin{matrix} \overset{\cdot}{v} \\ \overset{\cdot}{w} \end{matrix}] + [\begin{matrix} w \times (Mv) + m w \times (w \times r_{G}) \\ w \times (Iw) + m w \times (r_{G} \times v) \end{matrix}] = [\begin{matrix} F \\ τ \end{matrix}]

(11)

where $r_{G}^{\times}$ is the symmetry matrix of the vector $r_{G}$ .

From the kinematics and dynamics equation of the airship in equations (1) and (11), we obtain the equations of the station-keeping attitude motion as follows (Sergio et al., 1998)

{\begin{matrix} \overset{\cdot}{ϑ} = - r \sin ϕ + q \cos ϕ \\ \overset{\cdot}{ψ} = \frac{1}{\cos ϑ} (r \cos ϕ + q \sin ϕ) \\ \overset{\cdot}{φ} = p + \tan ϑ (r \cos ϕ + q \sin ϕ) \\ \overset{\cdot}{p} = (I_{1} r - I_{2} p) q + I_{3} z_{G} G \cos ϑ \sin ϕ + I_{4} u_{1} + I_{4} T_{d 1} \\ \overset{\cdot}{q} = I_{5} rp - I_{6} (p^{2} - r^{2}) + I_{7} (u_{2} - z_{G} G \sin ϑ + T_{d 2}) \\ \overset{\cdot}{r} = (I_{2} r + I_{8} p) q + I_{9} z_{G} G \cos ϑ \sin ϕ + I_{4} u_{3} + I_{4} T_{d 3} \end{matrix}

(12)

where $I_{1} = \frac{I_{xz}^{2} - I_{z} (I_{y} - I_{z})}{- I_{x} I_{z} + I_{xz}^{2}}$ , $I_{2} = \frac{I_{xz} (I_{x} + I_{z} - I_{y})}{- I_{x} I_{z} + I_{xz}^{2}}$ , $I_{3} = \frac{I_{z}}{- I_{x} I_{z} + I_{xz}^{2}}$ , $I_{4} = \frac{1}{- I_{x} I_{z} + I_{xz}^{2}}$ , $I_{5} = \frac{I_{z} - I_{x}}{I_{y}}$ , $I_{6} = \frac{I_{xz}}{I_{y}}$ , $I_{7} = \frac{1}{I_{y}}$ , $I_{8} = \frac{- I_{xz}^{2} - I_{x} (I_{x} - I_{y})}{- I_{x} I_{z} + I_{xz}^{2}}$ and $I_{9} = \frac{I_{xz}}{- I_{x} I_{z} + I_{xz}^{2}}$ ; $I_{xz}$ is the product of inertia about the plane of $x_{b} O z_{b}$ ; $I_{x}$ , $I_{y}$ , $I_{z}$ are moments of inertia about $O x_{b}$ , $O y_{b}$ and $O z_{b}$ , respectively; $I_{x}$ , $I_{y}$ , $I_{z}$ , $I_{xz}$ include added mass; G is the weight of airship; $T_{d 1}$ , $T_{d 2}$ , $T_{d 3}$ are the disturbances torques; $u_{1}$ , $u_{2}$ , $u_{3}$ are the input moments of rolling, pitching and yawing, respectively.

Consider the inertia uncertainties of station-keeping attitude control system in (12), we express $I_{x}$ , $I_{y}$ , $I_{z}$ and $I_{xz}$ as

I_{x} = I_{x}^{*} + Δ I_{x}, I_{y} = I_{y}^{*} + Δ I_{y}, I_{z} = I_{z}^{*} + Δ I_{z}, I_{xz} = I_{xz}^{*} + Δ I_{xz}

where $I_{x}^{*}$ , $I_{y}^{*}$ , $I_{z}^{*}$ , $I_{xz}^{*}$ are known nominal values of inertia; $Δ I_{x}$ , $Δ I_{y}$ , $Δ I_{z}$ , $Δ I_{xz}$ are the uncertain parts of inertia.

Set the state variables as $x (t) = {[\begin{matrix} ϑ & ψ & ϕ & p & q & r \end{matrix}]}^{T}$ , the output as $y (t) = h (x) = {[\begin{matrix} ϑ & ψ & ϕ \end{matrix}]}^{T}$ , then the expressions of the controlled system are as follows

{\begin{matrix} \overset{\cdot}{x} (t) = f (x) + Δ f (x) + [g (x) + Δ g (x)] u \\ y (t) = h (x) \end{matrix}

(13)

where

\begin{array}{l} f (x) = [\begin{matrix} - r \sin ϕ + q \cos ϕ \\ \frac{1}{\cos ϑ} (r \cos ϕ + q \sin ϕ) \\ p + t g ϑ (r \cos ϕ + q \sin ϕ) \\ (I_{1}^{*} r - I_{2}^{*} p) q + I_{3}^{*} z_{G} G \cos ϑ \sin ϕ \\ I_{5}^{*} r p - I_{6}^{*} (p^{2} - r^{2}) - I_{7}^{*} z_{G} G \sin ϑ \\ (I_{2}^{*} r + I_{8}^{*} p) q + I_{9}^{*} z_{G} G \cos ϑ \sin ϕ \end{matrix}], \\ g (x) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ I_{4}^{*} & 0 & 0 \\ 0 & I_{7}^{*} & 0 \\ 0 & 0 & I_{4}^{*} \end{matrix}] \end{array}

Δ f (x) = [\begin{matrix} 0 \\ 0 \\ 0 \\ (I_{1} - I_{1}^{*}) r q - (I_{2} - I_{2}^{*}) p q + (I_{3} - I_{3}^{*}) z_{G} G \cos ϑ \sin ϕ + I_{4} T_{d 1} \\ (I_{5} - I_{5}^{*}) r p - (I_{6} - I_{6}^{*}) (p^{2} - r^{2}) - (I_{7} - I_{7}^{*}) z_{G} G \sin ϑ + I_{7} T_{d 2} \\ (I_{2} - I_{2}^{*}) r q + (I_{8} - I_{8}^{*}) p q + (I_{9} - I_{9}^{*}) z_{G} G \cos ϑ \sin ϕ + I_{4} T_{d 3} \end{matrix}]

Δ g (x) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ I_{4} - I_{4}^{*} & 0 & 0 \\ 0 & I_{7} - I_{7}^{*} & 0 \\ 0 & 0 & I_{4} - I_{4}^{*} \end{matrix}]

where $I_{1}^{*} = \frac{I_{xz}^{* 2} - I_{z}^{*} (I_{y}^{*} - I_{z}^{*})}{- I_{x}^{*} I_{z}^{*} + I_{xz}^{* 2}}$ , $I_{2}^{*} = \frac{I_{xz}^{*} (I_{x}^{*} + I_{z}^{*} - I_{y}^{*})}{- I_{x}^{*} I_{z}^{*} + I_{xz}^{* 2}}$ , $I_{3}^{*} = \frac{I_{z}^{*}}{- I_{x}^{*} I_{z}^{*} + I_{xz}^{* 2}}$ , $I_{4}^{*} = \frac{1}{- I_{x}^{*} I_{z}^{*} + I_{xz}^{* 2}}$ , $I_{5}^{*} = \frac{I_{z}^{*} - I_{x}^{*}}{I_{y}^{*}}$ , $I_{6}^{*} = \frac{I_{xz}^{*}}{I_{y}^{*}}$ , $I_{7}^{*} = \frac{1}{I_{y}^{*}}$ , $I_{8}^{*} = \frac{- I_{xz}^{* 2} - I_{x}^{*} (I_{x}^{*} - I_{y}^{*})}{- I_{x}^{*} I_{z}^{*} + I_{xz}^{* 2}}$ and $I_{9}^{*} = \frac{I_{xz}^{*}}{- I_{x}^{*} I_{z}^{*} + I_{xz}^{* 2}}$

It is easy to see that the uncertainties in system (13) satisfy the following matching condition

Δ f (x) = g (x) d_{f} (x), Δ g (x) = g (x) d_{g} (x)

(14)

where

d_{f} (x) = [\begin{matrix} 0 \\ 0 \\ 0 \\ [(I_{1} - I_{1}^{*}) r q - (I_{2} - I_{2}^{*}) p q + (I_{3} - I_{3}^{*}) z_{G} G \cos ϑ \sin ϕ + I_{4} T_{d 1}] / I_{4}^{*} \\ [(I_{5} - I_{5}^{*}) r p - (I_{6} - I_{6}^{*}) (p^{2} - r^{2}) - (I_{7} - I_{7}^{*}) z_{G} G \sin ϑ + I_{7} T_{d 2}] / I_{7}^{*} \\ [(I_{2} - I_{2}^{*}) r q + (I_{8} - I_{8}^{*}) p q + (I_{9} - I_{9}^{*}) z_{G} G \cos ϑ \sin ϕ + I_{4} T_{d 3}] / I_{4}^{*} \end{matrix}]

d_{g} (x) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ (I_{4} - I_{4}^{*}) / I_{4}^{*} & 0 & 0 \\ 0 & (I_{7} - I_{7}^{*}) / I_{7}^{*} & 0 \\ 0 & 0 & (I_{4} - I_{4}^{*}) / I_{4}^{*} \end{matrix}]

The vector field $f (x)$ , along with the functions $h_{i}$ , is assumed to be smooth in their arguments, allowing arbitrary derivatives of each to be calculated without concern for their existence. Calculate the relative degree of system in (13), we have

L_{gi} L_{f}^{0} h_{j} (x) = 0, i = 1, 2, 3, j = 1, 2, 3

(15)

\begin{array}{l} B (x) = [\begin{matrix} L_{g 1} L_{f}^{1} h_{1} (x) & L_{g 2} L_{f}^{1} h_{1} (x) & L_{g 3} L_{f}^{1} h_{1} (x) \\ L_{g 1} L_{f}^{1} h_{2} (x) & L_{g 2} L_{f}^{1} h_{2} (x) & L_{g 3} L_{f}^{1} h_{2} (x) \\ L_{g 1} L_{f}^{1} h_{3} (x) & L_{g 2} L_{f}^{1} h_{3} (x) & L_{g 3} L_{f}^{1} h_{3} (x) \end{matrix}] \\ = [\begin{matrix} 0 & \cos ϕ I_{7}^{*} & - \sin ϕ I_{4}^{*} \\ 0 & \frac{\sin ϕ}{\cos ϑ} I_{7}^{*} & \frac{\cos ϕ}{\cos ϑ} I_{4}^{*} \\ I_{4}^{*} & \tan ϑ I_{7}^{*} & \tan ϑ I_{4}^{*} \end{matrix}] \end{array}

(16)

Remark 1. Note that the matrix $B (x)$ is nonsingular at $x (t) = x^{0} = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T}$ and the relative degree $r = 6$ . Thus, according to differential geometry theory (Isidori, 1989), the system (13) has full vector relative degree and the input/output map can be rendered a set of decoupled chains of integrators using feedback law.

The attitude tracking error is defined as $e (t) = h (x) - h_{d} (x) = {[\begin{matrix} ϑ - ϑ_{d} & ψ - ψ_{d} & ϕ - ϕ_{d} \end{matrix}]}^{T}$ , where $ϑ_{d}$ , $ψ_{d}$ and $ϕ_{d}$ are desired attitudes to be tracked. Differentiate $e (t)$ two times, we obtain

\overset{\cdot\cdot}{e} (t) = - {\overset{\cdot\cdot}{h}}_{d} (x) + A (x) + B (x) d_{f} (x) + B (x) (I_{3} + d_{g} (x)) u

(17)

where

A (x) = {[\begin{matrix} L_{f}^{2} e_{1} (t) & L_{f}^{2} e_{2} (t) & L_{f}^{2} e_{3} (t) \end{matrix}]}^{T}

\begin{matrix} L_{f}^{2} e_{1} (t) = (- r \cos ϕ - q \sin ϕ) (p + \tan ϑ (r \cos ϕ + q \sin ϕ)) \\ + \cos ϕ (I_{5}^{*} rp - I_{6}^{*} (p^{2} - r^{2}) - I_{7}^{*} z_{G} G \sin ϑ) \\ - \sin ϕ (I_{2}^{*} rp + I_{8}^{*} pq + I_{9}^{*} z_{G} G \cos ϑ \sin ϕ), \end{matrix}

\begin{matrix} L_{f}^{2} e_{2} (t) = & \frac{\sin ϑ}{\cos^{2} ϑ} (r \cos ϕ + q \sin ϕ) (- r \sin ϕ + q \cos ϕ) \\ + \frac{1}{\cos ϑ} (- r \sin ϕ + q \cos ϕ) (p + \tan ϑ (r \cos ϕ + q \sin ϕ)) \\ + \frac{\sin ϑ}{\cos^{2} ϑ} (I_{5}^{*} rp - I_{6}^{*} (p^{2} - r^{2}) - I_{7}^{*} z_{G} G \sin ϑ) \\ + \frac{\cos ϕ}{\cos ϑ} (I_{2}^{*} rp + I_{8}^{*} pq + I_{9}^{*} z_{G} G \cos ϑ \sin ϕ), \end{matrix}

\begin{matrix} L_{f}^{2} e_{3} (t) = & (1 + \tan^{2} ϑ) (r \cos ϕ + q \sin ϕ) (- r \sin ϕ + q \cos ϕ) \\ + \tan ϑ (- r \sin ϕ + q \cos ϕ) (p + \tan ϑ (r \cos ϕ + q \sin ϕ)) \\ + I_{1}^{*} rq - I_{2}^{*} pq + I_{9}^{*} z_{G} G \cos ϑ \sin ϕ \\ + \tan ϑ \sin ϕ (I_{5}^{*} rp - I_{6}^{*} (p^{2} - r^{2}) - I_{7}^{*} z_{G} G \sin ϑ) \\ + \tan ϑ \cos ϕ (I_{2}^{*} rp + I_{8}^{*} pq + I_{9}^{*} z_{G} G \cos ϑ \sin ϕ) . \end{matrix}

The linearizing control law of system (13) is chosen as

u (t) = B^{- 1} (x) [- A (x) + v (t)]

(18)

where $v (t) \in R^{3}$ is the auxiliary control input. This feedback control law partially cancels the system nonlinearity.

Define

e_{i 1} (t) = e_{i} (t), e_{i 2} (t) = {\overset{\cdot}{e}}_{i} (t), i = 1, 2, 3

(19)

From (13), (17) and (18), the attitude tracking error system can be expressed as

{\begin{matrix} {\overset{\cdot}{e}}_{i 1} (t) = e_{i 2} (t), i = 1, 2, 3 \\ {[\begin{matrix} {\overset{\cdot}{e}}_{12} (t) & {\overset{\cdot}{e}}_{22} (t) & {\overset{\cdot}{e}}_{32} (t) \end{matrix}]}^{T} = - {\overset{\cdot\cdot}{h}}_{d} (x) + B (x) d_{f} (x) \\ - B (x) d_{g} (x) B^{- 1} (x) A (x) + (I_{3} + B (x) d_{g} (x) B^{- 1} (x)) v (t) \end{matrix}

(20)

Remark 2. By using input/output feedback linearization, the nonlinear station-keeping attitude-tracking model of airship is transformed into a linear system. The benefit of this approach is that, due to a feedback-linearizing controller in the inner loop of the cascade, the controller design method for linear system can be used in the outer loop.

For the convenience to controller design, the following assumption is introduced.

Assumption 1. There exist constants $ρ_{1} > 0$ and $ρ_{2} > 0$ , such that, for all $x (t) \in R^{6}$ , the uncertain functions in (20) satisfy the following inequalities

{\begin{matrix} ‖ B (x) d_{f} (x) - B (x) d_{g} (x) B^{- 1} (x) A (x) ‖ \leq ρ_{1} \\ ‖ B (x) d_{g} (x) B^{- 1} (x) ‖ \leq ρ_{2} \end{matrix}

(21)

Finite time attitude tracking controller design

In this section, a nonlinear attitude control scheme is proposed, which finite time stabilizes the system (20) in the presence of system uncertainty. Basically, there are two steps for the controller design. The first step is to design an appropriate switching surface function so that the sliding mode dynamics has stability and finite time tracking capability. The second step is to synthesize a suitable adaptive integral sliding mode control law to globally drive the system state trajectories onto the predefined switching surface in a finite time and maintain them there for the subsequent time.

To stabilize the system (20) in a finite time, the following integral sliding mode control law is proposed

u (t) = B^{- 1} (x) [- A (x) + v_{nom} (t) + v_{disc} (t)]

(22)

where $v_{disc} (t)$ is the discontinuous part of the control law which rejects the uncertainties of system (20) and ensures that control objectives are fulfilled in a finite time. $v_{nom} (t)$ is the nominal controller to be designed which stabilizes the sliding mode dynamics in finite time.

In this paper, the following integral-type switching surface function is proposed for station-keeping attitude tracking error system (20)

\begin{array}{l} s (t) = {[\begin{matrix} e_{12} (t) & e_{22} (t) & e_{32} (t) \end{matrix}]}^{T} \\ - {[\begin{matrix} e_{12} (0) & e_{22} (0) & e_{32} (0) \end{matrix}]}^{T} - \int_{0}^{t} v_{n o m} (t) \end{array}

(23)

According to SMC theory (Utkin, 1992), when the system reach onto the sliding surface, it follows that $s (t) = 0$ and $\overset{\cdot}{s} (t) = 0$ . By $\overset{\cdot}{s} (t) = 0$ , we get the equivalent control as

\begin{matrix} v_{disc}^{eq} (t) = {(I_{3} + B (x) d_{g} (x) B^{- 1} (x))}^{- 1} \\ ({\overset{\cdot\cdot}{h}}_{d} (x) - B (x) d_{f} (x) + B (x) d_{g} (x) B^{- 1} (x) A (x) - B (x) d_{g} (x) B^{- 1} (x) v (t)) \end{matrix}

(24)

By substituting (24) into (17), the sliding mode dynamics of station-keeping attitude tracking error system can be represented by the following independent integrator chains

{\begin{matrix} {\overset{\cdot}{e}}_{i 1} (t) = e_{i 2} (t), i = 1, 2, 3 \\ {[\begin{matrix} {\overset{\cdot}{e}}_{12} (t) & {\overset{\cdot}{e}}_{22} (t) & {\overset{\cdot}{e}}_{32} (t) \end{matrix}]}^{T} = v_{nom} (t) \end{matrix}

(25)

The nominal controller $v_{nom} (t)$ can be designed by the following theorem.

Theorem 1. (Bhat and Berstein, 1998) Let the constants $γ_{i} \in (0, 1)$ , $i = 1, 2, 3$ , the sliding mode dynamics (25) is stabilized at the origin in finite time under the following control law

\begin{array}{l} v_{n o m, i} (t) = - sign (e_{i 2} (t)) {| e_{i 2} (t) |}^{γ_{i}} - sign (ϕ_{γ_{i}} (e_{i 1} (t), e_{i 2} (t))) \\ {| ϕ_{γ_{i}} (e_{i 1} (t), e_{i 2} (t)) |}^{\frac{γ_{i}}{2 - γ_{i}}}, i = 1, 2, 3 \end{array}

(26)

where

ϕ_{γ_{i}} (e_{i 1} (t), e_{i 2} (t)) = e_{i 1} (t) + \frac{1}{2 - ε_{i}} sign (e_{i 2} (t)) {| e_{i 2} (t) |}^{γ_{i}}

In the following, the integral sliding mode control law in (22) is synthesized, by which the trajectories of the transformed attitude control system of airship in (20) on the designed sliding surface $s (t) = 0$ in a finite time and maintain there for all subsequent time.

The discontinuous part of control law $v_{disc} (t)$ is defined as follows

v_{disc} (t) = - Γ sign (s (t))

(27)

where

Γ \geq \frac{{\hat{ρ}}_{2} ‖ v_{nom} (t) ‖ + {\hat{ρ}}_{1} + ‖ {\overset{\cdot\cdot}{h}}_{d} (x) ‖ + δ}{1 + {\hat{ρ}}_{2}}

with $δ > 0$ . The adaptation update laws given as

{\overset{\cdot}{\hat{ρ}}}_{1} (t) = r_{1} ‖ s (t) ‖

(28)

{\overset{\cdot}{\hat{ρ}}}_{2} (t) = r_{2} (‖ v_{nom} (t) ‖ - Γ) ‖ s (t) ‖

(29)

where $r_{1} > 0$ and $r_{2} > 0$ are the design parameters, ${\hat{ρ}}_{1}$ and ${\hat{ρ}}_{2}$ are the estimated bounds, respectively.

Theorem 2. Suppose the sliding surface function is given as (23) and the adaptive ISMC law proposed in (22) with $v_{nom} (t)$ and $v_{disc} (t)$ are given in (26) and (27) and the adaptation laws defined in (28) and (29). Then, the station-keeping attitude tracking errors of system (20) are guaranteed with finite time convergence stability.

Proof. Consider the following Lyapunov function

V (t) = \frac{1}{2} s^{T} (t) s (t) + \frac{1}{2 r_{1}} {\tilde{ρ}}_{1}^{2} + \frac{1}{2 r_{2}} {\tilde{ρ}}_{2}^{2}

(30)

where ${\tilde{ρ}}_{1} = ρ_{1} - {\hat{ρ}}_{1}$ and ${\tilde{ρ}}_{2} = ρ_{2} - {\hat{ρ}}_{2}$

Take the derivative of $V (t)$ along the system trajectories, we have

\begin{matrix} \overset{\cdot}{V} (t) = & s^{T} (t) (- {\overset{\cdot\cdot}{h}}_{d} (x) + B (x) d_{f} (x) - B (x) d_{g} (x) B^{- 1} (x) A (x) \\ + (I_{3} + B (x) d_{g} (x) B^{- 1} (x)) v_{disc} (t) + B (x) d_{g} (x) B^{- 1} (x) v_{nom} (t)) \\ - \frac{1}{r_{1}} {\tilde{ρ}}_{1} {\overset{\cdot}{\hat{ρ}}}_{1} - \frac{1}{r_{2}} {\tilde{ρ}}_{2} {\overset{\cdot}{\hat{ρ}}}_{2} \\ = s^{T} (t) (- {\overset{\cdot\cdot}{h}}_{d} (x) + B (x) d_{f} (x) - B (x) d_{g} (x) B^{- 1} (x) A (x) \\ - Γ (t) (I_{3} + B (x) d_{g} (x) B^{- 1} (x)) sign (s (t)) + B (x) d_{g} (x) B^{- 1} (x) v_{nom} (t)) \\ - \frac{1}{r_{1}} {\tilde{ρ}}_{1} {\overset{\cdot}{\hat{ρ}}}_{1} - \frac{1}{r_{2}} {\tilde{ρ}}_{2} {\overset{\cdot}{\hat{ρ}}}_{2} . \end{matrix}

(31)

Noting Assumption 1, we obtain

\begin{matrix} \overset{\cdot}{V} (t) \leq - Γ ‖ s (t) ‖ - ρ_{2} Γ ‖ s (t) ‖ + ρ_{2} ‖ v_{nom} (t) ‖ ‖ s (t) ‖ \\ + ρ_{1} ‖ s (t) ‖ + ‖ {\overset{\cdot\cdot}{h}}_{d} (x) ‖ ‖ s (t) ‖ - \frac{1}{r_{1}} {\tilde{ρ}}_{1} {\overset{\cdot}{\hat{ρ}}}_{1} - \frac{1}{r_{2}} {\tilde{ρ}}_{2} {\overset{\cdot}{\hat{ρ}}}_{2} \\ \leq - (ρ_{2} + 1) Γ ‖ s (t) ‖ + ρ_{2} ‖ v_{nom} (t) ‖ ‖ s (t) ‖ + ρ_{1} ‖ s (t) ‖ \\ + ‖ {\overset{\cdot\cdot}{h}}_{d} (x) ‖ ‖ s (t) ‖ - \frac{1}{r_{1}} {\tilde{ρ}}_{1} {\overset{\cdot}{\hat{ρ}}}_{1} - \frac{1}{r_{2}} {\tilde{ρ}}_{2} {\overset{\cdot}{\hat{ρ}}}_{2} . \end{matrix}

(32)

With the control law proposed in (22) with $v_{nom} (t)$ and $v_{disc} (t)$ given in (26)–(27) and the adaptation laws defined in (28)–(29), we obtain

\begin{matrix} \overset{\cdot}{V} (t) & \leq - Γ ‖ s (t) ‖ - ρ_{2} Γ ‖ s (t) ‖ + ρ_{2} ‖ v_{nom} (t) ‖ ‖ s (t) ‖ \\ + ρ_{1} ‖ s (t) ‖ + ‖ {\overset{\cdot\cdot}{h}}_{d} (x) ‖ ‖ s (t) ‖ - \frac{1}{r_{1}} {\tilde{ρ}}_{1} {\overset{\cdot}{\hat{ρ}}}_{1} - \frac{1}{r_{2}} {\tilde{ρ}}_{2} {\overset{\cdot}{\hat{ρ}}}_{2} \\ \leq - (ρ_{2} + 1) Γ ‖ s (t) ‖ + ρ_{2} ‖ v_{nom} (t) ‖ ‖ s (t) ‖ + ρ_{1} ‖ s (t) ‖ \\ + ‖ {\overset{\cdot\cdot}{h}}_{d} (x) ‖ ‖ s (t) ‖ - {\tilde{ρ}}_{1} ‖ s (t) ‖ - {\tilde{ρ}}_{2} ‖ v_{nom} (t) ‖ ‖ s (t) ‖ + {\tilde{ρ}}_{2} Γ ‖ s (t) ‖ \\ \leq - ({\hat{ρ}}_{2} + 1) Γ ‖ s (t) ‖ + {\hat{ρ}}_{2} ‖ v_{nom} (t) ‖ ‖ s (t) ‖ \\ + {\hat{ρ}}_{1} ‖ s (t) ‖ + ‖ {\overset{\cdot\cdot}{h}}_{d} (x) ‖ ‖ s (t) ‖ \\ \leq - δ ‖ s (t) ‖ \\ \leq - δ \sqrt{2} \sqrt{V (t)} \end{matrix}

(33)

Since $s (0) = 0$ and $\overset{\cdot}{V} (t) \leq - δ \sqrt{2} \sqrt{V (t)}$ , the system (9) will always be kept on the sliding surface $s (t) = 0$ . As a result, by using Theorem 1, the station-keeping attitude tracking errors of system of airship (20) can converge to zero in finite time. This completes the proof.

Remark 3. It is noted that the finite time controller proposed in (22) does not rely on the upper bound of the disturbance. Instead, the bounds are obtained by designing suitable adaptive laws. Thus, it is more applicable than the existing results.

Remark 4. Control law in (22) is discontinuous when crossing the sliding surface $s (t) = 0$ , which may lead to undesirable chattering. This problem can be alleviated by replacing $sign (s (t))$ with $s (t) / (ε + ‖ s (t) ‖)$ , where $ε > 0$ is a small scalar.

Simulation results

To illustrate the effectiveness of proposed controller design method, the case of ZY-1 airship (Wang et al. (2010)) will be considered, which was manufactured at the School of Aeronautics and Astronautics of Shanghai Jiao Tong University, China, in 2009 (see Figure 2). The geometric configuration of the airship is given in Table 1.

Table 1.

Model parameters of the ZY-1 airship.

Parameter	Value	Unit
G	9003.75	$N$
$I_{x}$	7039.607	$kg m^{2}$
$I_{y}$	18075.59	$kg m^{2}$
$I_{z}$	11347.63	$kg m^{2}$
$I_{xz}$	2198	$kg m^{2}$
$z_{G}$	2.529	$m$

The control objective is to make the output of airship attitude nonlinear system (1) track the following desired attitude angles

\begin{matrix} y_{d} (t) = h_{d} (x) = {[\begin{matrix} ϑ_{d} & ψ_{d} & ϕ_{d} \end{matrix}]}^{T} \\ = [1 - e^{- 0.4 t} (\sin (0.4 t) + \cos (0.4 t))] {[\begin{matrix} 10 & 15 & 20 \end{matrix}]}^{T} \deg \end{matrix}

The initial values of the attitude angles are assumed to be ${[\begin{matrix} ϑ_{d} & ψ_{d} & ϕ_{d} \end{matrix}]}^{T} = {[\begin{matrix} 10 & 5 & - 10 \end{matrix}]}^{T} \deg$ and angular velocities are assumed to be ${[\begin{matrix} p & q & r \end{matrix}]}^{T} = {[\begin{matrix} 3 & 3 & 3 \end{matrix}]}^{T} \deg$ . The airship model is assumed to have the uncertainty of the order 10% on the inertia parameters, i.e. $Δ I = 0.05 * {[\begin{matrix} I_{x} & I_{y} & I_{z} & I_{xz} \end{matrix}]}^{T}$ . The following external disturbances are added to system: $T_{d 1} = 5.2 \sin (t)$ , $T_{d 2} = 3.6 \cos (1.4 t)$ , $T_{d 3} = 0.32 \cos (1.8 t + 2)$ .

The design parameters $r_{1}$ and $r_{2}$ are related to convergence rate about estimation of the bound parameters ${\hat{ρ}}_{1}$ and ${\hat{ρ}}_{2}$ , and the values are chosen as $r_{1} = 1$ and $r_{2} = 1$ . To implement the finite time controller (FTC) proposed in this paper, the control parameters in (22) are chosen as $γ_{1} = γ_{2} = γ_{3} = \frac{1}{3}$ , $δ = 2$ and $ε = 0.05$ .

To illustrate the dynamic performance of proposed controller, the widely-used PID controller is applied to make a comparison. The parameters of PID controller adopted in this paper are $P = 8.9$ , $I = 0.15$ and $D = 6.5$ . The simulation results about attitude tracking errors are shown in Figures 3 –5. Figs. 6 –8 illustrate the responses of three control input moments. Obviously, the attitude tracking system under the finite time controller (22) has a faster convergence rate in the presence of system parameter uncertainties and external disturbances.

Conclusion

The finite time attitude-tracking problem for an autonomous airship with system uncertainties and external disturbances has been investigated. Combing input/output feedback linearization with integral sliding mode technique, a robust nonlinear attitude controller has been proposed. In addition, adaptive laws are designed to estimate the upper bounds of the uncertainties. Simulation and comparison results have been given to illustrate the effectiveness of proposed method. Future work includes improving the fault tolerant capability of the proposed controller.

Footnotes

Declaration of conflicting 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 work was supported in part by the China Postdoctoral Science Foundation (grant number 2016M590360), in part by the National Natural Science Foundation of China (grant number 11272205).

References

Azinheira

Moutinho

Paiva

ECD

(2009) A backstepping controller for path-tracking of an underactuated autonomous airship. International Journal of Robust and Nonlinear Control 19: 418–444.

Bhat

Berstein

(1998) Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Transactions on Automatic Control 43(5): 678–682.

Chen

Zhou

Yan

et al . (2012) Composite control strategy of stratospheric airships with moving masses. Journal of Aircraft 49(3): 794–801.

(2012) Finite-time attitude stabilization for a spacecraft using homogeneous method, Journal of Guidance, Control, and Dynamics 35(3): 740–748.

Qian

(2011) Finite-time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Transactions on Automatic Control 56(1): 2711–2717.

Hong

Choi

Kim

(2009) Applications of adaptive neural network control to an unmanned airship. International Journal of Control, Automation and Systems 7(6): 911–917.

Isidori

(1989) Nonlinear Control Systems. New York: Springer.

Jin

Sun

(2008) Robust controller design with finite time convergence for rigid spacecraft attitude tracking control. Aerospace Science and Technology 12: 324–330.

Ding

(2009) Global set stabilization of the spacecraft attitude using finite-time control technique, International Journal of Control 82(5): 822–836,.

10.

Lin

(2011) Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica 47(8): 1706–1712.

11.

Nahon

Sharf

(2009) Dynamics modeling and simulation of flexible airships. AIAA Journal 47(3): 592–605.

12.

Liesk

Nahon

Boulet

(2013) Design and experimental validation of a nonlinear low-level controller for an unmanned fin-less airship. IEEE Transactions on Control System Technology 21(1): 149–161.

13.

Rao

Gong

Luo

(2007) Robotic airship mission path-following control based on ANN and human operator’s skill. Transactions of the Institute of Measurement and Control 29(1): 5–15.

14.

Sergio

Varella

Josue

(1998) Airship dynamic modeling for autonomous operation. In: Proceedings of the 1998 IEEE international conference on robotics and automation, Leuven, Belgium, 16–20 May 1998, pp.3462–3467.

15.

Utkin

(1992) Sliding Modes in Control Optimization. Berlin: Springer-Verlag.

16.

Valle

Menegaldo

Simoes

(2015) Smoothly gain-scheduled control of a tri-turbofan airship. Journal of Guidance, Control and Dynamics 38(1): 53–61.

17.

Wang

Shan

(2006a) Study of robust control for stratosphere airship attitude. Journal of System Simulation 18(5).

18.

Wang

Shan

(2006b) Airship attitude tracking system. Applied Mathematics and Mechanics 27(7): 919–926.

19.

Wang

Duan

et al . (2010) Experiment investigations on aerodynamic characteristics of the ZHIYUAN-1 Airship, Journal of Aircraft 47(4): 1463–1468.

20.

Wang

Zhou

Wang

et al . (2014) Reliable robust sampled-data H_∞ output tracking control with application to flight control. Information Technology and Control 43(2): 174–181.

21.

Won

(2002) Regional navigation system using geosynchronous satellites and stratospheric airships. IEEE Transactions on Aerospace and Electronics Systems 38(1): 271–278.

22.

Lin

Zhang

(2012) Fault-tolerant backstepping attitude control for autonomous airship with sensor failure, Proecedia Engineering, vol.29, pp.2022–2027,.

23.

Xiao

Zhang

(2015) Finite-time attitude tracking of spacecraft with fault-tolerant capability. IEEE Transactions on Control Systems Technology 23(4): 1338–1350.

24.

Yang

Zheng

(2011) Variable structure attitude control for an UAV with parameter uncertainty and external disturbance. Procedia Engineering (15): 408–415.

25.

Yang

Zheng

(2012a) Trajectory tracking for an autonomous airship using fuzzy adaptive sliding mode control. Journal of Zhejiang University-Science C (Computers & Electronics) 13(7): 534–543.

26.

Yang

Zheng

(2012b) Attitude control for a station keeping airship using feedback linearization and fuzzy sliding mode control. International Journal of Innovative Computing, Information and Control 8(12): 8299–8310.

27.

Yang

Zheng

(2013) Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. Advances in Space Research 51: 1157–1167.

28.

Zhang

et al . (2008a) Adaptive stabilization and trajectory tracking of airship with neutral buoyancy. Acta Automatica Sinica 34(11): 1437–1440.

29.

Zhang

et al . (2008b) Stabilization and trajectory tracking of autonomous airship’s planar motion. Journal of Systems Engineering and Electronics 19(5): 974–981,.

30.

Zheng

Huo

(2012) Trajectory tracking control for underactuated stratospheric airship. Advances in Space Research 50: 906–917.

31.

Zhu

Xia

(2011) Attitude stabilization of rigid spacecraft with finite-time convergence. International Journal of Robust and Nonlinear Control 21: 686–702.

32.

Zou

Kumar

Hou

et al . (2011) Finite-time attitude tracking control for spacecraft using terminal sliding mode and Chebyshev neural network. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics 41(4): 950–963.