Simulation Results
HGV Flight Condition and Simulation Scenarios
The initial flight conditions of the HGV are: V(0) = 3000 m/s, II = 30 000 m, p(0) = 2°, a(0) = 2°, ,5(0) = 2°, and p(0) = _{9}(0) = r(0) = 0. The model uncertainties, actuator faults, and sensed signals with white noises are introduced to assess the performance of the developed safety control scheme.
1) The uncertainties corresponding to the roll, pitch, and yaw moments of inertia (I_{xx}, lyy. Fz) are 10% of the nominal values. The maximal 20% mismatch exists in the HGV mass, C;, C_{m}, and C_{n}, respective!)'.
2) Focusing on the fault pattern of the HGV actuators, the gain faults and bias faults are included with consideration of both time-invariant and time- varying cases, as can be found in Eqs. (5.56)-(5.57), respectively.
Actuator time-invariant faults:
Actuator time-varying faults:
3) The white noise with a mean of 0 and covariance of 0.01 is injected into
each measurement channel.
Two scenarios are studied to demonstrate the use of the algorithms for HGV attitude tracking control. (1) Scenario I: the model uncertainty, time- invariant actuator gain and bias faults, and measurement noises are considered; and (2) Scenario //: the model uncertainty, time-varying actuator gain and bias faults, and measurement noises are involved.
The control parameters are selected as: k = 3, fc_{2} = 4, rq = r_{2} =
// = 0.1, and Ф = 0.15. In the adaptive laws: qq = d-. q_{2} ^{=} pp, and y_{3} = dj. ci(0) =c_{2}(0) =c_{3}(0) =0.
For quantitatively evaluating the tracking performance, an index is defined as:
where [2] covers the time frame of the overall simulation. The defined metric is the scalar valued L_{2} norm, as a measure of average tracking performance.
Simulation Analysis of Scenario I
It is highlighted in Fig. 5.2(a) that after the actuator faults take place (t > 9 s), the reference signal can be quickly tracked under the proposed safety control scheme. As can be seen from Figs. 5.2(b) and 5.2(c), the AOA and sideslip angle can converge to the intended values within finite time in the presence or absence of actuator faults. Focusing on Fig. 5.2, the developed safety control scheme allows the HGV to follow the prescribed tracking profiles as closely as possible under the actuator faults and model uncertainties. The defined indices in Eq. (5.58) with respect to the bank angle, AOA, and sideslip angle are 0.4068°, 0.1904°, and 0.1764°, respectively. Based on Fig. 5.3, the amplitude of the actuators becomes larger than that of the normal case, such that the effects induced by the faults can be eliminated. Key observations from Fig. 5.4 are : (1) the estimated values of the parameters (2, and c_{3}) hold at constant values to counteract model uncertainties (0 < t < 9 s) and (2) the estimated values respond appropriately by applying the adaptive laws after the occurrence of the actuator faults.
Simulation Analysis of Scenario II
The performance against actuator time-varying faults is evaluated in Scenario II. From Fig. 5.5, the tracking performance is satisfactory, when the actuator time-varying malfunctions, measurement noises, and model uncertainties simultaneously exist. The HGV states can be steered to the intended values in a timely manner. The defined metrics corresponding to the bank angle, AOA, and sideslip angle are 0.4671°, 0.2025°, and 0.2137°, respectively. As compared to those in Scenario I (see Table 5.1), the performance is
FIGURE 5.2: The curves of the tracking angles in Scenario I.
decreased by 14.82%, 6.36%, 21.15%, respectively. This condition arises due to that the impact of time-varying faults is worse than that of time-invariant ones. The deflections of the actuators and the adaptation process of the control gains are depicted in Figs. 5.6 and 5.7, respectively. The control gains can be promptly updated in response to the time-varying faults. The actuators are appropriately managed to maintain the HGV safety. In summary, the applicability of the developed safety control scheme is further verified through the simulation studies of Scenario II.
FIGURE 5.3: The curves of the deflections in Scenario I.
Concluding Remarks
A safety control architecture, including the multivariable integral TSMC and adaptive approaches suitable for HGV' attitude tracking control, is developed against actuator faults and model uncertainties. The unique advantages of the proposed method he in three aspects.
FIGURE 5.4: Curves of the adaptive gains in Scenario I.
- 1) The finite-time stability of the faulty HGV can be guaranteed so that unacceptable HGV behaviors are not created by actuator gain and bias malfunctions;
- 2) The composite-loop design under actuator faults is achieved on the basis of control-oriented model, without the need of the time-scale separation principle; and
FIGURE 5.5: The curves of the tracking angles in Scenario II.
TABLE 5.1: Performance index.
- |
&p,H |
&p,a |
^{a}p,P |
Scenario I |
0.4068 |
0.1904 |
0.1764 |
Scenario II |
0.4671 |
0.2025 |
0.2137 |
FIGURE 5.6: The curves of the deflections in Scenario II.
3) The multivariable integral TSMC method is presented to enable integration into the HGV safety control design, instead of the decoupled single-input and single-output method.
The simulations of a full nonlinear model of the HGV dynamics show that the investigated scheme can be successfully employed to handle scenarios involving actuator faults and model uncertainties.
FIGURE 5.7: Curves of the adaptive gains in Scenario II.
Notes
This chapter presents a safety control strategy for a hypersonic gliding vehicle (HGV) subject to actuator malfunctions and model uncertainties. The control-oriented model of the HGV is established according to the HGV kinematic and aerodynamic models. A composite-loop design for HGV safety con?trol under actuator faults is subsequently developed, where newly developed multivariable integral terminal sliding-mode control (TSMC) and adaptive techniques are integrated. The simulations show that the HGV can handle the time-invariant, the time-varying actuator faults and model uncertainties well with the proposed safety control design techniques.
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 [130, 131]. One needs to develop two controllers corresponding to the separated loops. However, it is difficult to guarantee the finite-time stability of the overall closed-loop system. In this study, Ecj. (5.28) is integrated by Eqs. (5.13) and (5.20), which provides the basis of the proposed composite-loop design.
In Refs. [127, 136], multivariable TSMC design is discussed for hypersonic vehicles. The sliding manifold [127, 136] is essentially established by a decoupled treatment. Instead, based on sliding manifold of vector expression, the approach developed in this chapter has twofold benefits: (1) the problem related to the decoupled design is avoided; and (2) the multivariable safety control can maintain the globally finite-time stability under actuator malfunctions. These improvements have the potential to enhance the safety of operational HGVs.