# Preliminaries

A brief description of finite-time stability and homogeneity is presented, serving as a foundation of the HGV safety control design.

Consider the system:

where / : D —> R" is continuous on an open neighborhood D of the origin £ = 0. The equilibrium £ = 0 of Eq. (5.1) is finite-time convergent if there are an open neighborhood U C D of the origin and a function : £/{0} —> (0, oo), such that V£0U, the solution trajectories £(f,£0) of Eq. (5.1) starting from the initial point £o 6 t/{0} is well-defined and unique in forward time for t € [0. 7^(£o)). and linit^j^o) £(t, £o) = 0. Here T^(^o) is called the settling time (of the initial state £o)- The equilibrium of Eq. (5.1) is finite-time stable if it is Lyapunov stable and finite-time convergent . When U = D = R". then the origin is in globally finite-time stable equilibrium.

Definition 5.1. Let dilation (ri, • • • , rn) € Rn with r, > 0, г = 1, • ■ ■ , n. Let /(£) = [У1 (C)5' ’ ’ >fn(£)]T he a continuous vector field. f(£) is recognized to be homogeneous of degree dR with respect to dilation (r*i,--- ,rn) if, for any given £ > 0,

System Eq. (5.1) is said to be homogeneous if f(£) is homogeneous.

Lemma 5.1. [132] The continuous system Eq. (5.1) is named globally finitetime stable if it is globally asymptotical stable and locally homogeneous of degree d < 0.

# Mathematical Model of a HGV

## Nonlinear HGV Model

The HGV model is based on the assumption of a rigid vehicle structure, a flat, non-rotating Earth and uniform gravitational field. In the following, the kinematic model and dynamic model of a HGV are described, respectively. The inertial position coordinates are described as:

where x, у, and z represent the positions with respect to x-, y-. and ^-directions of the Earth-fixed reference frame, respectively. V stands for the total velocity, 7 and x denote the flight-path angle and the heading angle, respectively.

The force equations are described as:

where p is the bank angle, g is the gravitational constant, Q is the dynamic pressure, Sr is the reference area, m is the HGV mass, C/J. Co, and Cy are the aerodynamic coefficients with respect to lift, drag, and side force, respectively. The model of attitude is written as:

where a and ,5 denote the angle of attack (AOA) and sideslip angle, respectively.

The model of angular velocities is given as:

where p, q, and r are roll, pitch, and yaw angular rates, respective^. I а, гпа, and п.4 denote the roll, pitch, and yaw moments, while Ixx, Iyy, and Izz represent the moments of inertia.

The aerodynamics forces L. D, and Y are represented as:

where Cl — G Lclt.a n v G'l ,Sa 8a V ^L.S, 8c ■ CpD .cl сан ~~Cp fiaSa d G/;<)fV v Cp^r8r, and Cy = Cy.0 + Cy.s,,8a + Cyr\$c6e + Cyt\$r6r. 8a. Se, and 8r are the so-called control deflections of the aileron, elevator, and rudder, respectively. The rolling, pitching, and yawing moments are:

where b is the span of the HGV, c is the mean aerodynamic chord, xcg is the distance between the centroid and reference moment along x body-axis. The corresponding coefficients are calculated as: C) = C)jif3 + Cps„8a + Сц,,. 8,, + Ci,sr\$r T Ci,r 2V Giер2V? Cm = Cm ciean + Cm>sa8a + Cm>sc8e + Cm>sT8r + Cmл/ 2у • and Cn = Cat.3fl + Cn sa6a -f- Сп \$с8е -+- Cn.<5,8r i Gn pfy + Cn>r^y.

## Actuator Fault Model

When actuation systems work under a normal condition, appropriate aerodynamic forces and moments are produced. The required HGV maneuver can be thereby accomplished with a baseline/nominal controller. If the HGV encounters actuator malfunctions, the nominal controller’s attempts to maintain the expected maneuver may be futile and the flight safety can be jeopardized [30, 133]. Gain fault and bias fault are the faults commonly appearing on flight actuators. The actuator fault model is generally formed as:

where Л = diag{Ai, A2, A3} represents the gain fault, p = [p. />2- Рз}Т denotes the bias fault, and и = [<5a, 8e, Sr]T.

Remark 5.1. It is reported in Ref. [30] that the leakage of hydraulic fluid can be the root cause of the degradation of the actuator effectiveness. Therefore, A = diag{A1; Аз, A3} in Eg. (5.9) is used to describe the effectiveness of the HGV actuators, where 0 < Ai, A2, A3 < 1. In addition, the sensor fault in an actuator system can result in the actuator bias faults. To be more specific, if the amplitude sensor encounters a bias fault, the measured amplitude is the actual amplitude plus the bias value, ds a consequence, the sensed amplitude is forced to be equal to the referenced signal. However, the actual value of the actuator amplitude is deviated from the expected value. Hence, p = [pi,p2, рз]Т is adopted in Eq. (5.9) to describe the bias faults of the aileron, elevator, and rudder, respectively.

## Problem Statement

The purpose is to develop a safety control scheme based on adaptive multivariable integral TSMC such that:

1) The deleterious effects of HGV actuator faults can be compensated within a finite amount of time, thus:

where tf is the finite time, pd, &d, and /3d correspond to the reference commands of the bank angle, AOA, and sideslip angle, respectively;

• 2) A composite-loop design for HGV attitude tracking control under actuator faults can be achieved, without the need of dividing the HGV dynamics into the inner-loop and outer-loop; and
• 3) Multivariable design can be integrated into the safety control.