# 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£_{0} € *U,* 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, • • • , r_{n}) € *R ^{n} 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 d*R with respect to dilation*(r*i,---

*,r*0,

_{n}) if, for any given £ >

*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, *S _{r}* is the reference area,

*m*is the HGV mass,

*C/*and

_{J}. Co,*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 *I _{xx}, I_{yy},* and

*I*represent the moments of inertia.

_{zz }The aerodynamics forces *L. D,* and *Y* are represented as:

where Cl — *G L _{c}l_{t}._{a} n* v G'l ,S

_{a}*8*

*V*

_{a}*^*

*L.S,*8

_{c}

*■ Cp*—

*D .cl сан ~~Cp fi*d G/;<)

_{a}Sa_{f}V v

*Cp^*

_{r}*8*

*and*

_{r},*Cy*= Cy

*.0*

*+ Cy.s,,*

*8*

*+*

_{a}*Cy*

_{r}$_{c}*6*

*+*

_{e}*Cy*

_{t}$_{r}*6*

_{r}.*8*

*and*

_{a}. S_{e},*8*

*are the so-called control deflections of the aileron, elevator, and rudder, respectively. The rolling, pitching, and yawing moments are:*

_{r}

where *b* is the span of the HGV, *c* is the mean aerodynamic chord, *x _{cg}* is the distance between the centroid and reference moment along

*x*body-axis. The corresponding coefficients are calculated as:

*C) = C)jif3*+

*Cps„*

*8*

*+*

_{a}*Сц,,.*

*8*

*,,*+

*Ci,s*T

_{r}$r*Ci,r*

*2*

*V Giер*

*2*

*V*?

*C*

_{m}^{=}

*C*+

_{m c}i_{ean}*C*

_{m>}s_{a}*8*

*+*

_{a}*C*

_{m>}s_{c}*8*

*+*

_{e}*C*

_{m>}s_{T}*8*

*+*

_{r}*Cmл/*

*2*

*у •*and

*C*

_{n}= Cat.*3*

*fl*+

*C*

_{n}s_{a}*6*

*-f-*

_{a}*С*

_{п}$_{с}*8*

*-+- Cn.<5,*

_{е}*8*

*i*

_{r}*G*+

_{n}pfy*C*

_{n>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 =*[<5

_{a},

*8*

_{e}, S_{r}]^{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{A_{1;} Аз, A_{3}} *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,p**2**, 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.