Fault‐tolerant predictive trajectory tracking of an air vehicle based on acceleration control

IET Control Theory & ApplicationsVolume 14, Issue 5 p. 750-762 Research ArticleFree Access Fault-tolerant predictive trajectory tracking of an air vehicle based on acceleration control Seyyed Ali Emami, Seyyed Ali Emami Department of Aerospace Engineering, Sharif University of Technology, Tehran, IranSearch for more papers by this authorAfshin Banazadeh, Corresponding Author Afshin Banazadeh Banazadeh@sharif.edu Department of Aerospace Engineering, Sharif University of Technology, Tehran, IranSearch for more papers by this author Seyyed Ali Emami, Seyyed Ali Emami Department of Aerospace Engineering, Sharif University of Technology, Tehran, IranSearch for more papers by this authorAfshin Banazadeh, Corresponding Author Afshin Banazadeh Banazadeh@sharif.edu Department of Aerospace Engineering, Sharif University of Technology, Tehran, IranSearch for more papers by this author First published: 01 March 2020 https://doi.org/10.1049/iet-cta.2019.0596Citations: 5AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract A novel fault-tolerant model predictive control (MPC)-based trajectory tracking approach for an aerial vehicle is presented in this study. A generalised online sequential extreme learning machine is introduced first to identify the corresponding coefficients of actuator faults. Subsequently, a robust trajectory tracking control is developed based on MPC, where the system constraints can be effectively considered in the designed control scheme. Trajectory tracking control is achieved by controlling only the acceleration of the aerial robot in the MPC structure. This leads to less computational burden and faster closed-loop dynamics. In addition, an effective disturbance observer is employed, which can satisfactorily capture the effects of unmodelled dynamics and external disturbances on the system dynamics. The stability of the proposed control algorithm is proved based on the Lyapunov theory under realistic assumptions. Finally, the proposed control strategy is applied to a quadrotor unmanned aerial vehicle. Both simulation and experimental results are presented to demonstrate the effectiveness of the introduced fault-tolerant control scheme. The obtained results suggest that the designed control approach is capable of stabilising the attitude and providing acceptable performance in tracking a predefined trajectory in a three-dimensional environment in the simultaneous presence of actuator faults and external disturbances. 1 Introduction Model predictive control (MPC) is an advanced control approach which can be satisfactorily employed in the case of different non-linear multi-input multi-output (MIMO) systems. It is capable of considering the system constraints in the design process, effectively. Owing to the acceptable performance of MPC in the case of different complex systems, in recent decades, it has been extensively used in various industrial applications [1–3]. There are a variety of MPC-based trajectory tracking control approaches in the literature [4, 5]. As discussed in [6], in most research, trajectory tracking has been performed through the control of the position of the vehicle in the optimisation problem of MPC. Herdt et al. [7] proposed a new control scheme, where the stable trajectory tracking in a horizontal plane could be achieved through the control of only the velocity components of the vehicle. It has been also proved in the study that the control of the velocity and position components, simultaneously, leads to more accurate tracking. Two linear and non-linear MPC (NMPC) approaches have been designed in [8] for trajectory tracking of a rotary wing micro air vehicle. The MPC formulation has been developed based on the control of both position and velocity components. Also, it has been shown that the closed-loop responses of the linear MPC and NMPC for trajectory tracking under external disturbances are comparable, while the NMPC has demonstrated slightly better disturbance rejection capability. A NMPC has been introduced in [9] for trajectory tracking control of a quadcopter unmanned aerial vehicle (UAV) considering the system constraints in the design steps. Both position and velocity components of the vehicle have been utilised in the optimisation problem of the MPC in order to provide a precise trajectory tracking. In addition, Galffy et al. [10] proposed a path following control approach for a fixed-wing aircraft using a cascaded control design. More precisely, an acceleration-based control has been developed in the inner loop control, where the desired acceleration components have been determined using a feedback linearisation block in the outer loop. However, the stability of the closed-loop system has not been analysed in the study. In this regard, to the best of the authors' knowledge, the problem of MPC-based trajectory tracking control through the control of only the acceleration components, which can lead to faster closed-loop dynamics and significantly less computational cost, has not yet been studied in the literature. On the other hand, the occurrence of actuator faults during the operation of an air vehicle results in reduced performance of the closed-loop system and in some cases, it may lead to closed-loop instability. Different fault-tolerant control (FTC) approaches have been proposed in the literature in order to overcome this problem [11–13]. Aoki et al. [14] employed an NMPC to stabilise the states of a hexacopter in the presence of specific fault scenarios. The proposed approach attempted to solve an optimisation problem in each case to determine the control commands assuming that the faulty actuators are known during the simulation. In addition, the stability of the closed-loop system has not been studied in the paper. A learning-based fuzzy FTC approach has been designed in [15] for a linear model of a quadrotor UAV. A state feedback control approach has been employed as the baseline controller, where a fuzzy control has been used to determine the appropriate gains of the baseline controller. The structure of the membership functions and the rules of the fuzzy controller, in turn, have been tuned using an online optimisation. A fault-tolerant MPC has been introduced in [16] for a conventional fixed-wing aircraft. The designed control approach in the study includes a target calculation block, an MPC optimiser, and a linear observer. The controller attempted to redistribute the control commands on healthy actuators in the presence of actuator faults using a control allocation scheme. Finally, the proposed control approach has been used to control the lateral mode of a transport aircraft. However, closed-loop stability has been studied only for a linear model of the system. Also, it has been assumed that the actuator faults are known accurately during the flight, which is a questionable assumption. Indeed, in practical applications, the identification of system faults is typically performed using a separate fault detection and isolation (FDI) block. In addition, FDI blocks generally employ some predefined symptoms to detect and isolate an actuator fault, where the symptoms are commonly defined based on the instantaneous prediction error of the model. However, using such an approach, an FDI block may lead to incorrect fault detections in the presence of unexpected external disturbances. This, in turn, can lead to unacceptable performance or even instability of the closed-loop system in the simultaneous presence of actuator faults and external disturbances. Motivated by the above discussion, in this study, a fault-tolerant predictive trajectory tracking control is designed for an aerial vehicle. A generalised online sequential extreme learning machine (GOS-ELM) is developed to effectively identify the dynamics of faulty actuators. Afterwards, an acceleration-based MPC structure is introduced, which can be satisfactorily used to control the attitude and trajectory of an air vehicle in a three-dimensional (3D) environment in the presence of different actuator faults. In addition, the stability of the closed-loop system is proved based on the Lyapunov theory. Furthermore, most of the current trajectory tracking control approaches in the literature require a separate attitude stabiliser in the closed-loop system, which can cause excessive computational cost. As an instance, Shi et al. [17] proposed a finite-time trajectory tracking control for a quadrotor UAV based on the exponential coordinates. An attitude controller has been designed as the inner loop of the closed-loop system, while in the outer loop, the position has been controlled through a homogeneous feedback plus dynamics compensation scheme. Also, an MPC-based trajectory tracking control for a quadrotor UAV has been proposed in [18] in the presence of model uncertainties and external disturbances. Active disturbance rejection control has been used to stabilise the attitude, while the stable trajectory tracking has been provided using an extended state observer to compensate for the external disturbances. However, using the proposed MPC-based trajectory tracking control approach in the current study, there is no need for a separate attitude stabiliser in the closed-loop system and the designed control approach can satisfactorily control both the attitude and trajectory of the aerial robot. Accordingly, the main contributions of the current study are as follows: (a) a novel acceleration -based predictive trajectory tracking control is designed for the non-linear model of a quadrotor UAV in order to provide faster closed-loop dynamics and more accurate trajectory tracking while requiring a less computational cost. (b) A GOS-ELM is presented in the study, which can effectively identify the system dynamics after the fault occurrence using an online multi-objective optimisation. The introduced approach leads to acceptable performance even in the simultaneous presence of actuator faults, model uncertainties, and external disturbances, and there is no need for conventional FDI blocks in the proposed FTC. Also, unlike the conventional FDI blocks, the proposed approach has a general formulation that can be satisfactorily applied to other dynamic systems. In addition, due to the indirect structure of the proposed control approach, the introduced GOS-ELM can be effectively integrated with other conventional control schemes. (c) Different types of system constraints are considered in the control design steps. Indeed, the controllability of a dynamic system can be significantly reduced after the fault occurrence. Accordingly, the satisfaction of system constraints in the presence of actuator faults is a crucial problem in designing an efficient FTC scheme. However, the system constraints are not usually considered in the control design procedure. This can lead to reduced effectiveness of a designed controller in practical applications, where the closed-loop system may even become unstable. Here, all the system constraints can be effectively considered in the control design. In addition, the asymptotic stability of the closed-loop system in the presence of the actuator faults has been carefully analysed considering the system constraints, which suggests the possibility of utilising the proposed control approach in the case of different practical applications. (d) The formulation of the introduced MPC is modified in order to stabilise the attitude of the aerial robot with no need to employ a separate attitude stabiliser. The rest of the paper is organised as follows: the GOS-ELM algorithm is introduced in Section 2. Subsequently, the steps of the acceleration-based MPC is studied in Section 3. Sections 4 and 5 consist of the simulation and experimental results, and finally, Section 6 concludes the paper. 2 Development of online identification algorithm of actuator faults The overall process of the proposed control approach is graphically depicted in Fig. 1. Accordingly, an indirect adaptive control structure will be introduced, where the corresponding parameters of actuator faults are identified using a novel online optimisation problem and subsequently, an acceleration-based MPC is designed in order to control the system trajectory. The set-point acceleration of the vehicle, which is employed in the MPC structure, is determined using some auxiliary control loops and finally, a disturbance observer will be designed to compensate for the effects of model uncertainties and external disturbance on the system dynamics. Fig. 1Open in figure viewerPowerPoint Proposed FTC scheme for precise trajectory tracking of an aerial robot 2.1 Online sequential extreme learning machine (OS-ELM) Extreme learning machine (ELM) is generally considered as a neural network with a single hidden layer, where the weights and biases of the hidden layer are chosen as constant random numbers and only the weights of the output layer are updated during the training process [19]. More precisely, consider a feed-forward neural network with a single hidden layer as follows: (1)where , , , , and N represent the network input, the weight vector connecting the input layer to the i th hidden node, the bias of the i th hidden node, the weight connecting the i th hidden node to the output node, and the number of hidden nodes, respectively. Also, is the activation function of each node and denotes the inner product of two vectors. Accordingly, and are random constant numbers in ELM and is updated in the training phase. This, in turn, leads to a linear learning approach with a significantly less computational cost which can be used in the case of online learning approaches, satisfactorily. However, in the case of online identification problems, the measured data are obtained one-by-one and so all the training data are not available at first. An effective approach for online training of ELM (which has been developed based on the recursive least square (RLS) optimisation) is known as the OS-ELM [20]. A variety of OS-ELM-based training approaches have been introduced in the literature for different practical applications [21–23]. The basic OS-ELM has been proposed assuming that all the system data have equal weight in the optimisation problem. However, this assumption is not valid in the case of time-varying systems. In such cases, the last system inputs–outputs should have larger weights in the optimisation problem in comparison with the initial system data. To meet this requirement, the forgetting factor has been introduced in the structure of the OS-ELM approach [24]. On the other hand, if a non-linear system can be modelled as a linear-in-parameters (LIP) model, then the problem of identifying the unknown parameters of the model can be converted to the well-known problem of training an ELM. To be more precise, consider a LIP model as follows: (2)where and represent the model output and a known vector, which is a function of the system parameters and model inputs, . Also, denotes the vector of unknown parameters of the model. Consequently, can be considered as the output of an ELM, where denotes the vector of the weights of the output layer which should be trained. Now, if a set of system inputs-outputs with K distinct samples is initially available, then we have (3)where (4) (5) However, assuming , becomes a non-square matrix. In such cases, the optimal value of can be obtained using the least square optimisation as follows: (6)where (7) (8)and is a constant forgetting factor. Afterwards, vector is updated at each time step after obtaining each new data sample based on the RLS optimisation. The details of the OS-ELM approach can be found in [25]. Accordingly, the OS-ELM approach can be employed satisfactorily for online identification of the unknown parameters of a multi-input single-output (MISO) LIP model. A GOS-ELM approach is introduced in the following subsection, which can be used effectively to identify the unknown parameters of MIMO models. 2.2 Generalised OS-ELM (GOS-ELM) Consider the non-linear dynamic model of an air vehicle as follows: (9)where and are known non-linear functions of the system states, . Also, and denote the system inputs and outputs, respectively. In the case of multiplicative actuator faults, the system inputs are obtained as follows: (10)where and represent the input command and the vector of unknown fault parameters, respectively (as will be discussed in Section 4.4, additive actuator faults can also be estimated by the proposed approach with no need to change the formulation of the introduced algorithm). Thus, we have , where denotes the i th component of . Also, represents a diagonal matrix where . Substituting (10) in (9), we have (11)Accordingly, if a set of distinct system inputs–outputs are available at first, the optimal value of can be determined using the least square optimisation as follows: (12)where (13) (14) (15) (16) (17) Now, at each step, vector should be updated after obtaining each new set of system inputs–outputs. For this purpose, there is a need to develop a recursive formulation for calculating vector . Note that in (14) is a matrix. In the basic OS-ELM approach, a recursive formulation has been developed for a MISO system. More precisely, in the case of a MISO system, is a column vector rather than a 2D matrix. Thus, the basic OS-ELM approach cannot be employed in the case of a MIMO system. In the following, a recursive formulation is proposed based on the OS-ELM approach, which can be used effectively in the case of MIMO systems. Using (15), we have (18) Also, rewriting (12) for and , we have (19) (20) Thus, using (18), (19) can be reformulated as follows: (21) Finally, combining (21) and (20), the following recursive formulation can be obtained for : (22) On the other hand, inverting (18) and using some well-known matrix identities [24], can be calculated in a recursive manner as follows: (23) Consequently, the unknown vector can be trained online after obtaining each new data set using the aforementioned recursive formulation. The proposed GOS-ELM approach can be summarised as Algorithm 1. Algorithm 1 (GOS-ELM). Set . Calculate using (11). Calculate using (14). Update using (23). Update using (22). Set and go to step 1. Note that, like other conventional optimisation algorithms, the proposed GOS-ELM has an initialisation phase as given in (12). Subsequently, the value of unknown vector is updated after obtaining each new data sample (at each step) using Algorithm 1. Thus, a set of distinct data samples are used only in the initialisation phase, while in the online training phase, a single data sample is required at each time step, which is a practical procedure. Also, concerning the conditions of the convergence of the proposed GOS-ELM to the real values of fault parameters, it should be noted that, theoretically, the proposed approach can be considered as a multi-objective RLS optimisation problem (indeed, the main contribution of the current study in this part is to generalise the OS-ELM algorithm to a multi-objective optimisation problem). Thus, the convergence of the proposed approach can be guaranteed using the standard persistent excitation (PE) condition. As discussed in [26], this condition can be satisfied even in the closed-loop control by injecting some useful excitations to the closed-loop system [27]. Also, a useful modification has been introduced in [28] to update ELMs in the closed-loop structure. The proposed approach does not require the PE condition for the convergence of the parameters to their ideal values. However, addressing this issue is beyond the scope of this study. Accordingly, the unknown vector can be effectively identified during the operation of the air vehicle. As a consequence, the following assumption is considered true in the rest of the paper. Assumption 1.Using the proposed GOS-ELM approach and by appropriate training, the multiplicative (and additive) fault parameters of the non-linear dynamic model (9) can be estimated with a predefined accuracy.Notice that the introduced GOS-ELM approach has been developed for a general non-linear model (in the affine form). Thus, it can be employed satisfactorily in the case of different non-linear systems to identify the corresponding coefficients of actuator faults. As a result, the proposed GOS-ELM approach can be considered as an effective alternative to conventional FDI blocks with less complexity and more general formulation. In the following subsection, a fault-tolerant acceleration-based MPC is proposed based on the introduced identification algorithm in order to provide a robust trajectory tracking control approach for an air vehicle. 3 Acceleration-based MPC 3.1 Non-linear dynamic model of the system The non-linear dynamic model of a conventional quadrotor UAV (using the small angle approximation) can be formulated as follows (see Fig. 2) [29–31]: (24) (25) (26) (27) (28) (29)where , , , g, and () represent, respectively, the diagonal terms of the moment of inertia, the gravity acceleration, and the disturbance components including the effects of external disturbances and model uncertainties. Furthermore, we have (30) (31)where , , and denote the components of the vehicle position in the inertial reference frame. Also, , , and represent the roll angle, pitch angle, and yaw angle, respectively. In addition, we have (32)where l and () represent the arm length of the quadrotor and the thrust force generated by each rotor, respectively. Also, , where and indicate the drag factor and thrust coefficient, respectively [32]. Fig. 2Open in figure viewerPowerPoint Schematic view of a quadrotor UAV 3.2 MPC formulation To develop an acceleration-based control, the system inputs and outputs can be considered as follows: (33) (34)where and represent the system inputs and outputs. In MPC, at each time step, the future outputs of the system are predicted within a predefined prediction horizon, . These predictions are derived in terms of the future control signal within a predefined control horizon, . A set of forthcoming control signals is then calculated by solving an optimisation problem involving the system constraints [33, 34]. Typically, only the first control signal in the optimal sequence is applied to the system and the entire calculation is repeated in the next time step [35]. The optimisation problem of MPC for a MIMO system can be formulated as follows: (35)subject to (36) (37) (38)where , , and represent the predicted system outputs, the set-point outputs, and the increment of the system inputs, respectively. Also, , and represent respectively, the element-wise absolute value, the element-wise inequality, and a constant vector with positive elements, which will be determined in Theorem 1. As will be proved in Theorem 1, the stability of the introduced MPC can be ensured using the proposed terminal constraint (38). In addition, and are diagonal positive definite matrices, which regulate the relative importance of the tracking error and the control effort. 3.3 MPC with non-linear prediction and linearisation along the trajectory (MPC-NPLT) Consider again the non-linear dynamic model (9). Introduction of (39) (40) (41) (42)provides the possibility of formulating the prediction model as follows: (43)where the non-linear function can be obtained using the dynamic equation (9) repeatedly for the future time instants in the prediction horizon. The use of the obtained prediction model (43) in the optimisation equation (35) leads to a non-linear optimisation problem. However, solving a non-linear optimisation problem at each time step is not practical in most cases. Different approaches have been introduced in the literature in order to provide a practical MPC scheme. Three different approaches for approximating such non-linear optimisation problems have been presented in [36, 37]. An appropriate choice of the introduced techniques is as follows: Remark 1.An effective approach called MPC-NPLT has been proposed in [37]. In this approach, the future trajectory of the system is approximated at each time step using the constant system inputs within the prediction horizon (). Subsequently, the non-linear model of the system is linearised along the obtained trajectory. Finally, the optimisation problem is solved using the obtained linear prediction model to determine the optimal control commands at each step. It is notable that the entire process can be further repeated until reaching the desired linearisation accuracy. This can be performed by approximating the future trajectory of the system using the newly determined system inputs. Accordingly, it has been demonstrated in [36] that the introduced MPC-NPLT approach can lead to quite acceptable performance, which is close to that of the ideal NMPC. Using the introduced MPC-NPLT approach, the non-linear optimisation problem of MPC converts to a well-known quadratic programming problem, which can be satisfactorily solved at each time step. The approximated prediction model can be formulated as follows: (44)where represents the effect of the past inputs on the future outputs. In addition, each row of the matrix G consists of the step responses of the corresponding output of that row obtained for different system inputs. Matrix and vector should be calculated at each time step. Furthermore, vector represents the total disturbance, which consists of the effects of non-linear terms and external disturbances on the system outputs. Accordingly, at each time step, the corresponding coefficients of actuator faults are updated using the proposed GOS-ELM approach and subsequently, the updated value of vector is used to generate the prediction model of the system based on the introduced MPC-NPLT algorithm. 3.4 Disturbance observer Typically, the disturbance term is assumed constant over the entire prediction horizon [1, 38]. However, this assumption may lead to inaccurate responses for long prediction horizons. A more effective disturbance observer has been introduced in [5], where instead of assuming a constant disturbance, it has been assumed that the trend of the total disturbance in the future time steps is the same as that in the last time steps. This is a more realistic assumption. More precisely, the total disturbance at can be estimated based on the measured system data up to as follows: (45)where (46) (47)which are computed at . Accordingly, disturbance vector can be obtained as follows: (48) Using Assumption 1 and according to the presented discussion in the previous subsection, in the rest of the paper, the following assumption is considered true. Assumption 2.The outputs of the introduced prediction model (44) can be considered equal to the outputs of the ideal non-linear model (43) within the prediction horizon (the consequences of violating Assumption 2 and the possibility of facilitating this condition to ensure the closed-loop stability will be discussed in Section 3.6).Consequently, the GOS-ELM approach is applied to the nominal model of the system (without considering the external disturbances in the model) and the effects of external disturbances on the model are estimated separately using the introduced disturbance observer. As will be seen in the results section, this approach leads to the acceptable performance of the closed-loop system even in the simultaneous presence of actuator faults and external disturbances.Finally, using the introduced vectors (39)–(42), the optimisation problem (35) can be reformulated as follows: (49)subject to (50) (51) (52)where (53)Consequently, a convex optimisation problem should be solved at each time step to determine the appropriate control commands. The corresponding control commands of the current step are applied to the non-linear model and the entire process is repeated in the future step. 3.5 Trajectory tracking control As mentioned earlier, the current MPC-based trajectory tracking approaches in the literature have been typically designed based on the control of the position or velocity components of the vehicle (or both of them) [6–8]. However, the dynamic equations of an aerial robot, which is obtained using the Newton–Euler equations, are normally written in terms of the acceleration components of the vehicle. Thus, if the MPC can be designed based on only the acceleration components of the vehicle, the prediction model of the system, which should be computed at each step, can be obtained in a straightforward manner using the basic dynamic equations of the system. This, in turn, leads to significantly less computational costs and more accurate predictions. Here, it will be demonstrated that a stable trajectory tracking can be achieved through the control of only the acceleration