# Control-Oriented Model

The establishment of the control-oriented model is presented, which provides a basis of the composite-loop design of the HGV safety control.

For the safety flight control of the HGV, define *X* = [p, a, *0 ^{T}* and

*x*

*-2*= [p,

*q.*r]

^{7}. In accordance with Eq. (5.5), one can obtain:

where .....

By recalling the definitions *Xi = [p,a,/3] ^{T}* and

*x*

*-2*= [p,

*q, r*Eqs. (5.11)- (5.12) can be therefore expressed as:

^{T}.

where /, = [/,,, *f _{a}, fp]^{T},* and

Moreover, combining the angular rate dynamics of Eq. (5.6) and aerodynamic moments of Ecj. (5.8) gives:

By defining *f _{p} = QbS_{r} (с^р/З* +

*Ci*(^)

_{>r}*+ Ci*

_{iP}*/Ixx + {{I*

_{yy}~hz)qr)/*I _{xx},* Ecj. (5.15) can be recast in the form:

Similarly, the pitch angular rate dynamics is represented as:

As long as letting /, = *(QcS _{r}(C_{m>c}i_{ean} + C_{mq}($y)))/I_{yy}* + (

*Qx*

_{cg}S_{r}{CD,*clean*sincv-f-

*Cb^iean*

*cos a))/Iyy*+ ((

*I*

_{zz}- I_{xx})pr)/I_{yy}, Eq. (5.17) can be simplified as:

The yaw angular rate dynamics is described as:

where *f _{r} — QbS_{r}(C_{n}^f3 С_{пр}(^у) -- C_{nr}(^y))^I_{zz}* +

*QxcgS*

_{r}CY,pfi / Izz, Iyy)PQ/hz-According to Eqs. (5.16), (5.18), and (5.19), one can obtain:

where *f _{2} = [f_{p},f_{q},fr]^{T}, U =* [

*S*d

_{e},_{r}]

^{r}.

Combining Eqs. (5.13) and (5.20) gives:

Differentiating Ecj. (5.13) and recalling Eq. (5.20) achieve:

The control-oriented model with actuator anomalies and model uncertainties is built as follows. Firstly, Eq. (5.22) can be expressed as:

where *F(xi, x _{2}) = *

*9*

*iX*and G(aq,

_{2}+gxf_{2}*x*In addition,

_{2}) = gg_{2}.*F(x i,x*contains two terms:

_{2})where *F„* and *Ap* denote the nominal portion and the uncertain portion of *F,* respectively. G(*Xi,x _{2})* can be specified in a manner similar to

*F(x*[. ж

_{2}):

The nominal term of G is G_{n}, which solely relies on the known portions of *g*i and *g _{2}.* With respect to the studied HGV, det(—sec/3. One can obtain that

*g*is invertible if

*/3*does not equal to ±7r/2. Focusing on the known portion of

*g-*

*2*

*,*it can be regarded as control allocation matrix which is invertible in HGV flight envelopes. Therefore, the nominal portion

*G*is invertible.

_{n}Consequently, Eq. (5.23) can be further written as:

By accounting for the gain and the bias faults in actuators as Eq. (5.9), one can render:

where *I* is a 3 x 3 identity matrix.

Assumption 5.1. *It is assumed that the boundedness of f _{jt}, f_{a}, and fg is associated with the norm of system states. It can be further assumed that:*

*where* £_{1;} £_{2}> ^{a7l(}l ^{£}з ^{ar(}- positive scalars.

Remark 5.2. *f =* [/,,. *f _{n}. fg]^{T} can be seen as an impact term of trajectory on the HGV attitude. Since the attitude dynamics is much faster than the translation motion, the values of* /

_{д},

*f*yls

_{a}, and fg are usually small,*can be seen from (5.29), the lumped additive uncertainty term fi*+

*Ai?*+ (

*G*/,,

_{n}+ Ag)p is dependent on the system states.*is not only greatly dependent on*7,

*g, a, (5, and V, but also on p, q, and r. Essentially, the HGV angles as well as the HGV velocity are bounded in typical HGV flight envelopes. Therefore, the bound of*/,,

*is closely related to the norm of HG V states if fi ~*0, 7

*ф*90°,

*and V ф*0.

*The similar assumption can be applied to the boundedness of f*

_{a}and fg.Remark 5.3. *The second inequality in (5.29) is essential such that the control signal G _{n}u dominates the uncertain vector function* (ДсЛ + G

_{n}(A —

*I))u, which is induced by the actuator anomalies and the control input matrix uncertainty. This condition in turn ensures that the actuators configured are capable of addressing the HG V uncertainty and fault issues. In addition, G*Ag solely rely on HGV angles including a and fi which are bounded in typical flight envelopes, instead of the angular velocities p, q, andr. Hence, it is assumed that the second term of (5.29) is bounded.

_{n }andRemark 5.4. *Time-scale separation of the independent inner-loop and outer- loop designs, stemming from Eq. (5.22), is typically enforced in most of the design approaches. It is difficult to guarantee the finite-time stability of the overall closed-loop system. By contrast, Eq. (5.28) is integrated by Eqs. (5.13) and (5.20), which provides the basis of the proposed composite-loop design.*

# Safety Control System Design of a HGV against Faults and Uncertainties

The system depicted in Fig. 5.1 is composed by the guidance and control units. Generally, the guidance system generates the commands of the bank angle, AOA, and sideslip angle. The safety control scheme that is the main focus of this study outputs the actuator commands necessary to track the desired attitude and to handle actuator malfunctions. Two problems are addressed in the following. The first is the composite-loop design of safety control problem: construct the HGV safety control law against actuator malfunctions and model uncertainties, using the multivariable integral TSMC technique. The second is the problem of selecting the control parameters within the developed safety control scheme: determine the control parameters by exploring the adaptive tuning method.

FIGURE 5.1: Schematic illustration of the studied HGV safety control.