# Control-Oriented Model Subject to Actuator Faults and Uncertainties

The control-oriented model of the HGV is established by combining the kinematic model and the dynamic model of the HGV attitude, based on which the so-called composite-loop safety control design can be achieved.

As far as an attitude control system is concerned, *p. a,* and /3 can be gathered into a vector *x* = *[p,* a, *8] ^{T}.* In terms of Eq. (6.3), one can obtain:

where

By defining *X**2* = [p, *q,r] ^{T},* Ecjs. (6.6)-(6.7) can be described as:
where /1 = [/

_{M},

*f*and

_{a}, fs^{T}

By accounting for Eqs. (6.4)-(6.5), one can render: where

Further, Eqs. (6.10)-(6.11) can be formed as:
where /2 = *f _{p}, f_{q}, f_{r}]^{T}, и =* [<5

_{Q},

*S*<5

_{e},_{r}]

^{T}, and

Gain fault and bias fault are the faults commonly occurring on flight actuators. In this work, the actuator fault model including both sorts of faults is generally formed as:

where Л = diag{Ai, Л2, A3} represents the gain fault and *p* = *[p _{a}, p_{e}, p_{r}}^{T}*

denotes the bias fault, respectively. Note that 0 < A, < 1, *i* = 1,2,3. Thus, in the presence of actuator faults, Eq. (6.12) is represented as:

Note that *и* in Eqs. (6.12), (6.14), and (6.15) represents the control input vector under normal conditions, while *up* describes the control input vector in the case of faults.

Assumption 6.1. *It is assumed that the condition* ||I) ^^{1}||_{0}o < 1 *holds.*

Remark 6.1. *[150] There exists a condition *

*2*(A

^{—}

*I) yf** I x> < E

*such that the control signals*

*g2U*

*dominate the fault vector function*

*д-2*(Л —

*I) u.*

By accounting for the actuator faults and model uncertainties, the control- oriented model in vector format is established as:

where *A* arises from Д, and Д2 = <72(Л — *I)u + g**2**p* is the lumped uncertainty induced by actuator faults.

By defining *y* = *x* — aq._{(}/ and *У**2** = У**1**Х**2* — *xi.d-* the following equations can be achieved:

where *x _{l c}/* represents the desired states. Letting a(-) =

*g*,ж

_{2}+ .(/2/2

^{— }

*b(-)*=

^{an<}i Д3 =

*g*1Д21 Eq. (6.17) can be simplified as:

Remark 6.2. Ts *can be seen from Fig. 6.1 and also Eq. (6.8), the input vector of the outer-loop HGV model consists of the roll rate (p), pitch rate (q), and yaw rate (r), while the state vector with respect to the outer-loop is composed by the bank angle (р), АОЛ (a), and sideslip angle (8), respectively. Focusing on the inner-loop HGV model of Eq. (6.12), the deflections of the aileron (S _{a}), elevator (5_{e}), and rudder (6_{r}) are regarded as the inputs, while the roll rate, pitch rate, and yaw rate constitute the state vector. It should be mentioned that* /2

*in Eq. (6.12) is closely related to the states of Eq. (6.8).*

Remark 6.3. *Unpredictable aerodynamics due to hypersonic speed and airframe/ structural dynamics interactions constitute the uncertainty source.*

FIGURE 6.1: Block diagram of the HGV model.

/1 *s can be seen from Eq. (6.7), the aerodynamic coefficients Cl and Cy with uncertainty are contained in* /j. *Hence, } = **A **in Eq. (6.16) is regarded as the model uncertainty. Moreover, the term in Eq. (6.18),* Д3 = *992* (A *— I) и + дУ**2**р, includes the information of actuator faults, without* a priori *knowledge. In the following, fixed-time observers are developed to estimate **A **i and* Д3, *respectively.*

Remark 6.4. *Note that **A*1 = /1 = *[ffi, f _{a}, fp]^{T}. In Eq. (*

*6*

*.*

*6*

*), f*/3

_{/t}. f_{a}, and*are regarded as the impact terms of trajectory on the HGV attitude. The value of V is usually very large over HGV flight envelopes. Furthermore, j3*я» 0

*and*7

*ф*±7t/2.

*Therefore, the assumption that Ai is bounded is reasonable. Focusing on*Д

_{3}=

*giQi*(A —

*I) и*+

*gig^P,*Д3

*is related to system states and control inputs, which are bounded. With respect to HGV fight envelopes, (1*~ 0

*and each element of g*

*-2*

*is composed by bounded control moment coefficients. In consequence,*Д3

*is bounded in flight.*

Remark 6.5. H.s *reported in [151], hydraulic driven actuators are configured in hypersonic vehicles to operate all control surfaces. Flush air data system (FADS) that is often mounted in the upper and lower lifting surfaces has been successfully applied to hypersonic vehicles [152]. FADS, which is dependent on the pressure sensor array measurement of aircraft surface pressure distribution, obtains dynamic pressure, bank angle, AO A, and sideslip angle indirectly through a specific algorithm. In addition, an inertial navigation system (INS) can measure the position, orientation, and velocity of a hypersonic vehicle. Hence, with respect to the studied HGV, the bank angle, AO A, and sideslip angle can be measured by an FADS, while the measurements of the angular rates of roll, pitch, and yaw can be provided by an INS.*

# Problem Statement

The objective is to design a safety control scheme such that:

1. The terms including actuator faults and system uncertainties can he estimated within a fixed amount of time, thus:

where *t„* is the fixed convergence time, Д1 and Д3 are the estimates of Д1 and Д3, respectively.

2. The detrimental impact of HGV actuator faults can be counteracted within a finite amount of time, thus:

*where t_{c} denotes the finite convergence time, /t,/. »/3d* correspond to the reference signals of the bank angle, AOA, and sideslip angle, respectively.

3. The composite-loop design is achieved under multivariable situation, by which separating the HGV dynamics into inner and outer loops is no longer needed.