## Abstract

Unmanned helicopters (UHs) develop quickly because of their ability to hover and low speed flight. Facing different work conditions, UHs require the ability to safely operate under both external environment constraints, such as obstacles, and their own dynamic limits, especially after faults occurrence. To guarantee the postfault UH system safety and maximum ability, a self‐healing control (SHC) framework is presented in this chapter which is composed of fault detection and diagnosis (FDD), fault‐tolerant control (FTC), trajectory (re‐)planning, and evaluation strategy. More specifically, actuator faults and saturation constraints are considered at the same time. Because of the existence of actuator constraints, usable actuator efficiency would be reduced after actuator fault occurrence. Thus, the performance of the postfault UH system should be evaluated to judge whether the original trajectory and reference is reachable, and the SHC would plan a new trajectory to guarantee the safety of the postfault system under environment constraints. At last, the effectiveness of proposed SHC framework is illustrated by numerical simulations.

### Keywords

- fault detection and diagnosis
- fault‐tolerant control
- invariant set
- self‐healing control framework
- trajectory (re‐)planning

## 1. Introduction

Safe operation and reliability are important performances of unmanned aerial vehicles (UAVs). Unmanned helicopters (UHs) as a kind of UAVs develop quickly because of their ability to hover and low speed flight. Nevertheless, traditional control and planning strategies cannot guarantee their safety and reliability in the face of malfunctions such as sensor faults, actuator faults, and component faults. Furthermore, taking physical limits such as actuator constraints and state constraints into account, the situation would be more serious. In order to ensure the reliability and safety of UHs, fault detection and diagnosis (FDD) and fault‐tolerant control (FTC) approaches become focus of research and many related research results have been presented [1]. However, physical limits are not well considered in conventional FTC approaches which may affect their efficiency. On the other hand, besides UH's dynamic limits, external environment constraints, such as obstacles, also affect UH's safety. However, traditional planning methods cannot consider environment constraints and vehicles’ limits at the same time which makes these methods helpless in the face of faults. In this chapter, a self‐healing control (SHC) framework is proposed against actuator faults and constraints of single‐rotor UHs under external environment constraints. The SHC framework is shown in Figure 1 which involves FDD module, reconfigurable controller, trajectory (re‐)planning module, and evaluator module.

The tasks of the above modules are as follows:

FDD module: Estimate the actuator healthy coefficient (AHC) [2] in real time. AHCs can indicate healthy conditions of actuators and actuator fault information is the basis of controller reconfiguration.

Reconfigurable controller: This module realizes the same functions as conventional FTC approaches and considers actuator constraints at the same time. After fault occurrence, the fault‐free controller will be configured as a postfault controller according to AHCs provided by FDD module to guarantee the stability of the postfault UH system.

Trajectory (re‐)planning module: Compute realizable trajectory based on desired path and UH dynamic model under environment constraints. This module calculates a feasible trajectory under both fault‐free and postfault conditions. Furthermore, controller references are computed directly in this module which can be used by reconfigurable controller.

Evaluator module: Evaluate system performance of UHs according to AHCs, UH dynamic model, and controller and trajectory information. If the original controller and trajectory are not feasible, this module will ask related modules to reconfigure controller and replan trajectory.

The remaining part of this chapter is organized as follows: single‐rotor UH and AHC models are simply introduced in Section 2; Section 3 investigates FDD approach against actuator faults based on extended Kalman filter (EKF) and linear neural network (LNN). In Section 4, the postfault controller is designed to guarantee the stability of UH system under both actuator faults and constraints; Section 5 presents invariant‐set based planning (ISBP) approach that can compute controller reference according to desired path and UH dynamic model. Evaluation strategy is introduced in Section 6 and numerical simulations are shown in Section 7. Finally, Section 8 summarizes the chapter with conclusions.

## 2. Model of UH and AHCs

### 2.1. UH model

The 6‐DOF dynamics of UH is given by the following Newton–Euler equations:

where

Aerodynamic forces and moments can be calculated by

where

Note that

In order to guarantee the accuracy of AHC estimation, nonlinear model is used to design FDD approach. However, the above nonlinear model is too complex to design EKF filter so that a simplified nonlinear discrete‐time model is used [6]:

where

On the other hand, the linear dynamic model is used to design controller and trajectory planning which is shown as follows:

where

### 2.2. AHC model

Suppose

where *i*th actuator's AHC. *i*th actuator is fault‐free; otherwise, the *i*th actuator has fault. Thus, the fault condition of actuators can be represented by AHCs.

## 3. EKF‐ and LNN‐based FDD approach

Because of the serious nonlinear coupling between all control channels, we assume that an actuator has fault whereas others are still well, and the actuator for rotate speed control,

### 3.1. Fault identification module

The task of fault identification is to determine the size of the fault and the fault time after fault isolation. The size of a fault is indicated by the proportional effectiveness of AHC, so we should estimate the actual manipulated variable vector

As shown in Eq. (1), in a typical nonlinear discrete‐time system, the input vector is obtained from actuators, and the current state vector

For the calculation, processes of forces and torques of the simplified nonlinear model have been linearized, Eq. (3) is a linear equation set, and the values of

Typically, Eq. (3) is an overdetermined linear equation set and could not be solved directly [8]. After knowing which actuator is faulty, many output estimations of this actuator can be obtained by the faultless manipulated variables of other actuators according to Eq. (3). Multiply the sample column vectors formed by these estimations by a weight matrix and correct this result with an offset. Then, the manipulated variable estimation of this faulty actuator can be obtained.

To obtain the above weight matrices and offsets, consider a linear neuron as a simple LNN, and its input–output relation is expressed as follows:

where

### 3.2. Fault isolation module

The task of fault isolation is to determine the location and type of the fault after fault detection. The structure of this module is shown in Figure 2 to record the completion time and pass the serial number of the faulty actuator to fault identification module.

Four fault identification modules are in parallel in this module, and we assume that there are different actuator faults in different identification modules. Therefore, every fault identification module can obtain the proportional effectiveness of AHC called isolation AHC. Make use of the isolation AHC of the *i*th actuator and let these values of other actuators be both 1 to calculate the manipulated variables called isolation‐manipulated variables according to control variable vector *i*th actuator as follows:

where *i*th actuator can be considered to be faulty.

### 3.3. Fault detection module

The basis of the above two modules is that the fault is detected quickly and accurately. Known from the EKF process, the filter obtains the state vector pre‐estimation

Perform the weighted sum of squared residuals (WSSR) operation [9]:

where the weight matrix *i*th state variable to make the residual that can indicate faults work.

According to the threshold value

## 4. Reconfigurable controller design

In this section, AHC‐based antiwindup controller design method is introduced which can be achieved by solving a set of linear matrix inequalities (LMIs). At the same time, the related invariant‐set‐based safety region is also calculated. The definition of invariant set and the controller design under actuator constraints are given as follows.

### 4.1. Invariant set

Considering a linear system

the (positively) invariant set can be given by the following definition.

**Definition 1 [10]:** The set

Based on Definition 1, the definition of robustly controlled invariant set for the following system is provided:

**Definition 2 [10]:** The set

An invariant set

**Theorem 1 [11]:** The set

For convenience,

### 4.2. Controller design under actuator constraints

Consider linear discrete UH model, Eq. (2), with actuator constraints:

Where

Where

In order to achieve set‐point tracking offset free and compensate actuator saturation, an integrator with a saturation compensator is introduced as follows:

where

where

Note that, if

Based on linear quadratic regulator (LQR) theory, state‐feedback controller

where

**Lemma 1 ([12]):** Define the following polyhedral set with matrix

where *i*th row of matrices

is verified for any positive‐definite matrix

Based on this lemma, saturation compensator can be designed in the following theorem.

**Theorem 2:** Given LQR parameter matrices

where *i*th row of matrices

According to Theorem 2, the safety region

On the other hand, the related admissible set of reference

## 5. ISBP approach

This section proposes an ISBP approach that can plan a feasible trajectory based on the desired path nodes under both external environment constraints and their own dynamic limits [14]. In order to consider environment constraints and dynamic constraints at the same time, invariant set is used as a bridge to connect the two kinds of constraints because the invariant set is calculated based on the Lyapunov function that is linked to dynamic model, and on the other hand, the invariant set has certain geometrical shapes, such as ellipsoid, so that it can easily be represented in work space with obstacles.

Note that, in this section, we have assumed that the heading of UH is kept to be 0, pitch angle and roll angle of UH are kept small so that the position of UH in the world coordinate system can be considered as the integration of velocities in the body coordinate system.

### 5.1. Reachable set

Considering System (4) with controller

**Definition 3 [15]:** The reachable set

where

### 5.2. ISBP method

For the convenience of discussion, a 2D work space is used in the following whereas the proposed ISBP approach can be expanded in 3D condition directly. Consider a preknown environment as shown in Figure 3, where the initial location is

The target of the proposed ISBP approach is to calculate feasible controller reference inputs to move the UH from the initial location to the goal location based on the feasible path and satisfy both external environment constraints and UH's limits.

Consider a simple example in the preknown environment as shown in Figure 4(a) where the initial location is

Clearly, the work space is state space; thus, the location in work space is equality to the state such as the initial state, which is equal to L0. Thus, an invariant set

Suppose the intersection of reachable set

### 5.3. Calculation of the invariant set and the reachable set

According to Theorem 2, the invariant set

However, in order to guarantee controlled plant obstacle avoidance, the state trajectory should be kept outside all of spheres that represent obstacles. For achieving it, the invariant set

According to the above analysis, the following theorem is proposed to calculate the maximum invariant set

**Theorem 3:** Given

where

Clearly, Theorem 2 is included in Theorem 3 because the invariant set is a bridge to connect environment and UH dynamic constraints. Thus, by the ISBP approach, the controller design and controller reference calculation are achieved simultaneously when the preknown work space, the desired path nodes, and the UH dynamic model are given.

The relationship between the related sets and the obstacles are shown in Figure 5 assuming that all sets are projected to a 2D plane.

In other words, based on the calculated invariant set

## 6. Evaluation strategy

The open‐loop model (2) of UH is unstable and the actuator outputs are limited. Thus, the only regional stability of UH can be guaranteed and the stability region is determined by the system structure and actuator constraints [16]. In other words, if the states of UH are inside the stability region, the UH would be of safety; otherwise, the UH may be in danger. Obviously, after actuator fault occurrence, actuator efficiency will be reduced. Hence, the safety region of the postfault system will be different with the fault‐free case as shown in Figure 6(a). Suppose that the safety region of the fault‐free system with a fault‐free controller is

According to Theorem 2, the safety region

## 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

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:

where

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.

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 swashplate 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.

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.

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

A linear UH model (2) with added three‐axis positions is used in this part and related parameters 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

Assume that the left swashplate actuator has fault at 6.4s with

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.

According to Theorem 3, a series of controllers

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 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.

## 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 constraints 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.

## Acknowledgements

This work was supported by Key Program of National Natural Science Foundation of China under Grant 61433016 and National Natural Science Foundation of China under Grant 61503369.