**7. Numerical simulation**

The parameters of nonlinear UH dynamic model used in numerical simulations is obtained from reference [6]. This model is used as the simulation model in this section. The step size of this simulation is 0.02s, which is the same as the control period of the real UH flight platform.

#### **7.1. FDD simulation**

in steady case are also determined by references such as *x <sup>f</sup>* <sup>1</sup> which may be outside the safety region *Ω*pf. Obviously, this reference is unreachable and tracking such reference may lead UH

analyzed before system motion and a new reachable reference is necessary. Compared with

According to Theorem 2, the safety region S is an invariant set. Clearly, a postfault system will be safe if initial states are inside the safety region S *<sup>f</sup>* of the postfault system. In this way, the initial states of the postfault system can be evaluated. Second, the steady states will be analyzed. In steady‐state case, actuators are not expected to be saturated so that the remaining efficiency of actuators can be used for disturbance defence. Hence, the original reference should be inside the reachable set of the postfault system such as ref∈Srf, where Srf is the reachable set of the postfault system; otherwise, the original reference ref is not reachable and tracking the original one may lead UH unsafe. The reason is that the actuator efficiency is reduced in the postfault system and tracking unreachable reference will lead fault‐free actuator saturated which implies that UH cannot respond to control signal correctly. Under this condition, a new optimal reference is required which can be calculated by the trajectory replanning approach. In other words, if the original reference is not reachable after detecting the actuator fault, the ISBP approach should be called to calculate new trajectory and controller

The parameters of nonlinear UH dynamic model used in numerical simulations is obtained from reference [6]. This model is used as the simulation model in this section. The step size of this simulation is 0.02s, which is the same as the control period of the real UH flight platform.

*<sup>f</sup>* 1). Hence, reference reachability should be

unsafe (as shown by the state trajectory *x*(*<sup>k</sup> <sup>f</sup>* <sup>2</sup>)*<sup>x</sup>* ′

128 Recent Progress in Some Aircraft Technologies

*x <sup>f</sup>* <sup>1</sup>, *x <sup>f</sup>* 2 may be more reasonable which is inside *Ω*pf.

**Figure 6.** A sample of safety region of fault‐free and postfault system in 2D state space.

reference based on the postfault dynamic model of UH.

**7. Numerical simulation**

Based on the simplified nonlinear model, the EKF is designed and LNN is used to train weight matrices and offsets of faulty actuators. The parameters of the EKF filter and the training results of the left swashplate actuator by LNN are as follows: *Q* =*diag*(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), *R* =2.5×10−<sup>5</sup> ⋅*diag*(40, 40, 40, 1, 1, 1, 1, 1, 1, 1, 1, 1), *W* = −0.0095, −0.00015, 0.000073, 0.076, 0.01023, −0.01008 , *b* =0.03099

where *Q* and *R* are covariance matrices of the process noise and the measurement noise, respectively. Then, we introduce a multiplicative fault on the left swashplate actuator as follows:

$$\begin{cases} \mathcal{A}\_1 = 1 & 0 \le t < 20 \\ \mathcal{A}\_1 = 0.6 & t \ge 20 \end{cases}$$

The fault detection module based on the filter residuals can get the WSSR curve shown in **Figure 7**. When no fault occurs, the WSSR curve cannot exceed the threshold.

**Figure 7.** The curve of filter WSSR under the left servo fault.

After fault detection, isolation residuals are calculated to isolate this fault. The curves of four isolation residuals are shown in **Figure 8**. According to this figure, the isolation residual of the left swashplate actuator is less than other residuals. That is to say, the left actuator of swash‐ plate is faulty. The fault isolation module cannot provide the right isolation results as soon as the completion of fault detection because of the vibration caused by the measurement noise. There may be misdiagnosis of the fault location within a very short time.

**Figure 8.** Curves of isolation residuals of every actuator under the left servo fault.

The curve of the proportional effectiveness of the left actuator under the fault is shown in **Figure 9**. After fault isolation, the isolation module passes the serial number of the faulty actuator to fault identification module to identify the AHC of this faulty actuator, and the simulation results indicate the size of this multiplicative fault. Because the weight matrices and offsets are trained by the LNN under the faults whose AHCs' parameters are all 0.5 for the online simulations, there may be error between the identification results and the actual size of this fault. By the calculation of average, the fault identification result is equal to the actual fault approximately.

The simulation results show that fault detection and isolation modules could detect and isolate a fault quickly, and the identification result is accurate enough; moreover, this method is with good real‐time performance.

**Figure 9.** The curve of AHC proportional effectiveness under the left servo fault.

#### **7.2. Controller and trajectory (re‐)planning simulation**

After fault detection, isolation residuals are calculated to isolate this fault. The curves of four isolation residuals are shown in **Figure 8**. According to this figure, the isolation residual of the left swashplate actuator is less than other residuals. That is to say, the left actuator of swash‐ plate is faulty. The fault isolation module cannot provide the right isolation results as soon as the completion of fault detection because of the vibration caused by the measurement noise.

There may be misdiagnosis of the fault location within a very short time.

130 Recent Progress in Some Aircraft Technologies

**Figure 8.** Curves of isolation residuals of every actuator under the left servo fault.

fault approximately.

good real‐time performance.

The curve of the proportional effectiveness of the left actuator under the fault is shown in **Figure 9**. After fault isolation, the isolation module passes the serial number of the faulty actuator to fault identification module to identify the AHC of this faulty actuator, and the simulation results indicate the size of this multiplicative fault. Because the weight matrices and offsets are trained by the LNN under the faults whose AHCs' parameters are all 0.5 for the online simulations, there may be error between the identification results and the actual size of this fault. By the calculation of average, the fault identification result is equal to the actual

The simulation results show that fault detection and isolation modules could detect and isolate a fault quickly, and the identification result is accurate enough; moreover, this method is with A linear UH model (2) with added three‐axis positions is used in this part and related param‐ eters can be found in [13]. The added positions are considered integration of velocities in the body coordinate. Thus, the system states, control inputs, and outputs are

$$\begin{aligned} \mathbf{x}\_h &= \begin{bmatrix} p\_x & p\_y & p\_z & u & v & w & \phi & \theta & \psi & p & q & r & a\_{1s} & b\_{1s} \end{bmatrix} \\ \mathbf{u} &= \begin{bmatrix} \theta\_l & \theta\_r & \theta\_b & \theta\_r & \phi\_r & \phi\_r \end{bmatrix}, \ y\_h = \begin{bmatrix} p\_x & p\_y & p\_z & \psi \end{bmatrix} \end{aligned}$$

Assume that the left swashplate actuator has fault at 6.4s with *λ*<sup>1</sup> =0.6, and the LQR parameter matrices are

$$\begin{aligned} \underline{Q}\_{\text{x}} &= \text{diag}(0.0 \, 0 \, 0 \, 0.01 \, 0.01 \, 0.01 \, 3 \, 1 \, 0.02 \, 0 \, 0 \, 0 \, 0.1 \, 0.1 \, 0.001 \, 0.001 \, 0.003 \, 0.01) \\ \underline{Q}\_{\text{u}} &= \text{diag}(1110.50.1) \end{aligned}$$

The preknown work space is shown in **Figure 10** where the start point is (0,0,0) and the target point is (20,20,20). The external environment constraint considered here is a sphere obstacle whose position is (10,10,10) and radius is 1. The feasible path is given by path nodes, such as (0,0,0), (8,8,13), and (20,20,20), which can guide UH from the start point to the target point with obstacle avoidance.

**Figure 10.** Preknown work space and feasible path.

According to Theorem 3, a series of controllers *K*, invariant sets S, and reachable sets S*<sup>r</sup>* satisfying obstacle avoidance and UH dynamic limits can be computed as shown in **Fig‐ ure 11** where the green ellipsoids are the reachable sets in 3D and the centers of ellipsoids, blue stars, are controller references. According to these references, the UH can run from the start point to the target point and the actual trajectory is shown by the blue curve. Clearly, the trajectory is kept inside the green ellipsoids so that it is also inside the invariant sets.

**Figure 11.** Reachable sets and UH actual trajectory in fault‐free condition.

Compared with fault‐free case as shown in **Figure 11**, **Figure 12** shows the results of the postfault case without the SHC framework. The actuator fault is detected at 6.4s and the related state is marked by red points in the figure. According to the postfault dynamic model, a new reachable set, red ellipsoid, is calculated. It is easy to see that the original reference, black point, is outside the reachable set of the postfault system which implies that the original reference is unreachable. Under this condition, if it does nothing, the UH system may be in danger as shown by the blue curve. The related manipulated variables are shown in **Figure 13** where the blue curves are the actuator outputs and the red dashed lines are the actuator constraints. Clearly, the actuator outputs are saturated at last which leads the UH system out of order.

**Figure 12.** Reachable sets and UH actual trajectory in postfault condition without SHC.

**Figure 10.** Preknown work space and feasible path.

132 Recent Progress in Some Aircraft Technologies

**Figure 11.** Reachable sets and UH actual trajectory in fault‐free condition.

According to Theorem 3, a series of controllers *K*, invariant sets S, and reachable sets S*<sup>r</sup>* satisfying obstacle avoidance and UH dynamic limits can be computed as shown in **Fig‐ ure 11** where the green ellipsoids are the reachable sets in 3D and the centers of ellipsoids, blue stars, are controller references. According to these references, the UH can run from the start point to the target point and the actual trajectory is shown by the blue curve. Clearly, the

trajectory is kept inside the green ellipsoids so that it is also inside the invariant sets.

**Figure 13.** Actuator outputs in postfault condition without SHC.

**Figure 14** shows the results of the postfault system with a SHC framework. After fault detection, Theorem 3 is recalled to calculate the new controllers, invariant sets, and reachable sets to evaluate the performance and guarantee the postfault UH system to be stable. At last, the UH can reach the target point with obstacle avoidance as shown by the actual trajectory.

**Figure 14.** Reachable sets and UH actual trajectory in postfault condition with SHC.

### **8. Conclusions**

In this chapter, a self‐healing control framework is proposed for UH systems. The SHC framework aims at providing a solution to guarantee UHs safety and maximum ability to achieve the desired missions under both fault‐free and postfault conditions. The EKF‐ and LNN‐based FDD approach is used to detect and diagnosis actuator faults modeled by AHCs. Then, the AHC‐based reconfigurable controller design method is proposed to calculate the fault‐tolerant controller and the related safety region against both actuator faults and con‐ straints by solving a set of LMIs. Third, the ISBP approach is presented for planning a feasible trajectory and computing the related controller reference under both external environment constraints and UH dynamic limits at the same time. After fault occurrence, based on the calculated safety region and controller reference, the performance of the postfault UH system can be evaluated, which can provide information whether the fault can be compensated and the original reference can be reached. If the original reference is not reachable, the ISBP approach will be recalled to calculate the new trajectory and reference again according to the postfault dynamic model. Finally, numerical simulations illustrate the effectiveness of the proposed SHC framework.
