Abstract
In this paper, the tube model predictive control (Tube-MPC) via notch filter is proposed for flexible air-breathing hypersonic vehicle with actuator fault. Firstly, considering the low-order flexible modes, the polytopic linear parameter varying (LPV) model of flexible air-breathing hypersonic vehicle is established using Jacobian linearization and tensor product. Actuator fault and parameter uncertainty are transformed into a lumped disturbance term. The baseline Tube-MPC trajectory tracking controller is developed by designing robust auxiliary feedback control law and nominal control law. In order to weaken the effect of flexibility, a notch filter is introduced to eliminate the frequency response peak of flexible modes. And the effect of notch filter on control performance and stability is analysed. Secondly, an alternative controller design strategy is proposed to ensure stability of the closed-loop system. The filter and the original nonlinear model are combined together such that the new polytopic LPV model is codesigned. On the basis, the adjustable Tube-MPC controller is developed to improve the control performance and ensure the stability of the system. Finally, the effectiveness of the two controllers is verified by simulations.
1. Introduction
Air-breathing hypersonic vehicle has the advantages of high-speed and low-launch cost in space transportation and potential of fast global strike in military application [1]. Compared with the traditional aircraft, air-breathing hypersonic vehicle has more significant characteristics such as the interaction of aerodynamic-heating-flexible-propulsion, strong time-varying parameters, and modelling uncertainty. Hence the flexible control system faces new challenges [2].
Since the 1980s, many researchers have devoted to modelling flexible air-breathing hypersonic vehicle. There are two categories of approaches: analytical method based on physical principles and computational fluid dynamics (CFD) method. In the former, the longitudinal nonlinear dynamics model derived by Bolender and DOman [3] is widely applied. This model represents complex interaction among aerodynamics, structural dynamics and propulsion system, and has flexible deformation and vibration. In addition, Waszak et al. [4] establish the fuselage/structure coupling model of hypersonic vehicle using the average axis method and decouple the rigid body from the flexible motion equation. Sudalagunta et al. [5] take six independent displacements of the rigid section to model the flexible deformation. In the latter, the purpose of CFD modelling is to analyse aerodynamics by fluid dynamics and calculate aerodynamic coefficients by data fitting. Lisa [6], David [7], and Parker [8] establish flexible hypersonic vehicle models by means of CFD. Regardless of analytical modelling or CFD modelling, the flexible air-breathing hypersonic vehicle model is strongly nonlinear and coupled, which is very unfavourable to control system.
Many control methods have been applied to the control of hypersonic vehicle, such as linear quadratic regulator (LQR) [9], feedback linearization control [10], backstepping control [11], sliding mode control [12], intelligent control [13], robust control [14], and adaptive control [15]. For hypersonic vehicle with high nonlinearity and large flight envelope, linear parameter varying (LPV) control has the advantage of simple design and attracted more attentions. When hypersonic vehicle is flying at high speed and large maneuver, flexibility cannot be ignored. Commonly, flexibility includes two modes: known flexibility and unknown flexibility. In the first case, the flexible modes are taken as known states for controller design. Huang et al. [16] build polytopic LPV model for hypersonic vehicle with known flexibility, and design gain-scheduling switch controller. Zhang et al. [17] investigate a novel switching LPV framework based on mode independent/dependent persistent dwell time (PDT/MPDT) switching signals to deal with the case of coexistence of slow maneuver and frequent rapid maneuver. Wu et al. [18] build the LPV model of flexible hypersonic vehicle by utilizing the curve fit and least-squares method and design the nonfragile output tracking controller. When flexibility is unknown, either the observer is designed to estimate and compensate flexibility, or the robustness of closed-loop system is utilized to suppress flexibility. Xu et al. [19] model the coupling among flexibility, rigid body, and the accessional angle of attack caused by wind as unknown disturbance, and design neural network and disturbance observer to deal with the lumped coupling. Sun et al. [20] develop a robust backstepping cruise tracking control scheme with closed-loop finite-time convergence for flexible hypersonic vehicle, where a fixed-time sliding-mode disturbance observer is designed for flexibility, uncertainties, and external disturbances.
In addition, actuator fault possibly occurs [21] due to high-speed and large-maneuver flight. Thereby, online observation and compensation or improvement of controller’s robustness are common strategies to handle actuator fault. The former belongs to active fault-tolerant control (FCT) method and the latter is passive approach. For example, Zhao et al. [22] develop FTC strategy based on direct Lyapunov method and the bilinear matrix inequalities technique for flexible hypersonic vehicle. An et al. [23] regard the flexible dynamics as equivalent disturbances, and design disturbance observer and sliding-mode tracking controller for flexible hypersonic vehicle ith actuator fault. In order to identify the lumped effect involving flexible modes and actuator fault online, Shao et al. [24] propose neural estimator based on hysteresis quantizer to achieve appointed-time tracking of flexible hypersonic vehicle. In passive FTC research, a weighted LPV Tube-MPC is proposed for hypersonic vehicle with three kinds of actuator fault [25]. The passive fault-tolerant tracking problem is investigated for the bounded external disturbance and sensor fault [26].
Model predictive control (MPC) is popular due to its good ability of handling constraint, and LPV model is generally combined for controller design [27, 28]. This control method is often applied to control complex nonlinear systems. For example, based on the LPV model of unmanned airship, a gain-scheduling MPC controller was designed for lateral control [29]. A systematic design procedure for approximate explicit MPC is presented for constrained nonlinear systems described in LPV form [30]. One-step ahead robust MPC for LPV model with bounded disturbance is developed [31]. Similarly, MPC and LPV are also applicable to hypersonic vehicle, which can effectively deal with the nonlinearity and uncertainty of hypersonic vehicle model. For example, a novel parameter-dependent robust MPC (PD-RMPC) algorithm with explicit time-delay compensation is presented for hypersonic vehicle.
Inspired by the previous work, a Tube-MPC control strategy with north filter is proposed for flexible air-breathing hypersonic vehicle with loss of actuator effectiveness fault in the presence of parameter uncertainty. The weak coupling between flexibility and rigid body is ignored, by which the continuous LPV model of vehicle is established by Jacobian linearization. And the corresponding polytopic formulation is built by tensor product. Parameter uncertainty or actuator fault are uniformly transformed into an additional disturbance term. On the basis, baseline Tube-MPC controller is designed, where robust auxiliary feedback control law and nominal control law are calculated, respectively. The nominal control law provides a reasonable reference trajectory for the actual system. The robust auxiliary feedback control law steers the actual trajectory into the mRPI set and makes it approximate the nominal trajectory. In order to reduce the flexible effect, a notch filter is introduced into the short-period mode. Moreover, the effect of notch filter on the control system is analysed. Alternatively, in order to ensure the stability of the closed-loop system, the transfer function model of filter is transformed into state-space equation. The states of filter and filtered pitch rate are both combined with the original nonlinear model. Considering the weak coupling between flexibility and rigid body, the new polytopic LPV model with notch filter is codesigned. To improve control performance and reduce conservatism, an adjustable Tube-MPC controller is developed. The simulation results show that the two types of Tube-MPC design method with notch can both effectively get rid of the elastic effect and provide good tracking performance.
The rest of this paper is organized as follows. Firstly, the nonlinear longitudinal model of hypersonic vehicle is presented. Secondly, the control-oriented polytopic LPV model is established and the baseline Tube-MPC controller design via notch filter is proposed. Then, the adjustable Tube-MPC controller based on the codesigned polytopic LPV model is presented. The effectiveness of these two strategies is verified by simulations. Finally, conclusion is drawn.
2. Flexible Hypersonic Vehicle Model
The flexible hypersonic vehicle is regarded as a pair of cantilever beams and assumed to follow Hook’s law. The longitudinal dynamic equations (Coupling model, CM) are introduced [8], which consists of rigid body motion equations and flexible vibrations.where are altitude, velocity, attack angle, pitch angle, and pitch rate, respectively; are quality, moment of inertia, and gravitational acceleration, respectively. represent the generalized flexible coordinates of the ith flexible mode; are natural frequencies, damping ratios and inertial coupling parameter of the ith flexible mode, respectively. . In addition, L, D, T, and M denote lift, drag, thrust, and pitching moment, respectively, and denotes the generalized force of the ith flexible mode. They are expressed by the following curve-fitting approximation:where is the elevator deflection. is the dynamic pressure. represent reference area of vehicle, thrust to moment coupling coefficient, and mean aerodynamic chord, respectively. and coefficients of the nonlinear model are calculated aswhere is the air density, are the air density at nominal altitude, nominal altitude for air density approximation, and air density decay rate, respectively. is the fuel-to-air ratio; is the elevator coefficient; are thrust parameters.
3. Tube-MPC via Notch Filter
For longitudinal trajectory tracking of flexible air-breathing hypersonic vehicle in presence of uncertain parameter and actuator fault, the baseline Tube-MPC controller is designed [32]. Meanwhile, a notch filter is introduced to suppress flexibility. The schematic of the closed-loop control system is shown in Figure 1.
3.1. Control-Oriented Polytopic LPV Modelling
To linearize nonlinear vehicle model, the weak coupling between the rigid body and the flexible modes is ignored (called Decoupling Model, DM). The following longitudinal motion equations are adopted [16]:
In this paper, the polytopic LPV model of DM is built by Jacobian linearization and tensor product.
Firstly, within the flight envelope, altitude and velocity are selected as scheduling variables, denoted as . Then, Jacobian linearization is performed at each equilibrium point. Define and . To compute the equilibrium points of flexible vehicle, let . According to the selected height and velocity , the states and control inputs at the equilibrium point are described as
Define the neighbourhood of equilibrium point as . The first-order Taylor series expansion is performed as
Define new state variables and control inputs as
Then,
Further, model (4) is rewritten aswhere . Obviously, equation (6) is a linear time-varying (LTI) system.
Within the flight envelope, multiple equilibrium points are selected for Jacobian linearization. The corresponding LTI systems are fitted into the following continuous LPV model:where and are matrices with scheduling variables.
The continuous LPV model (7) is transformed into a polytopic LPV system by tensor product [33].
Assuming that m singular values are determined in height and n singular values in velocity, a polytopic LPV system with vertices is obtained. A time-varying matrix is defined as , expressed by a convex combination of vertex systemswhere represents each vertex system, is the weight coefficient of each vertex system being dependent on scheduling variables, and . is the modelling error between the original nonlinear model and the polytopic LPV model.
The above polytopic LPV model is discretized, and there iswhere . are the polytopic vertices, is the convex set, and is the convex hull. , and , .
The Bode diagram of the DM model and the polytopic LPV model are drawn at any point in the flight envelope. As shown in the figure, the solid red line represents the DM system, and the dotted blue line for the polytopic LPV system. It can be seen that the two lines overlap very well. In other words, the established LPV model has high accuracy.
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Due to aeroelastic deformation during flight, moment of inertia, reference area, and some aerodynamic parameters of hypersonic vehicle will change. Considering the uncertain parameters and linearization modelling error, the following model is obtained: is a summarized term resulting from parameter uncertainty and modelling error. It is assumed that is bounded, i.e., .
In some cases, actuator faults may occur during vehicle flying. Loss of effectiveness fault is addressed in this paper and its corresponding faulty model is written aswhere is the control inputs with loss of effectiveness fault occurring on elevator.
Then,where , and is bounded, that is,
In summary, uncertain parameters, modelling error, and actuator failure of flexible hypersonic vehicle can be lumped into a bounded term . Thereby, model (11) is rewritten aswhere and is bounded, that is,
Remark 1. In polytopic LPV modelling, the uncertain term caused by uncertain parameters, the disturbance term caused by actuator faults, and the lumped term are all assumed to be bounded. In controller design, the worst conditions will be considered to obtain the maximum robustness, are taken as maximum or minimum.
3.2. Baseline Tube-MPC Controller Design
For model (14), the nominal system is defined aswhere are nominal states and are nominal control inputs.
The Tube-MPC control law is designed aswhere K is the nominal control gain used to generate a nominal trajectory and is the auxiliary robust feedback gain to reject the lumped term. The control law (16) forces the trajectory of the actual system into mRPI set and the actual trajectory will get close to the nominal.
Firstly, the robust feedback control gain is designed. The control errors between the actual system and the nominal system are defined as . Then, by subtracting (15) from (14), the following error equation can be obtained:
It can be known that is bounded when satisfies the conditions of Hurwitz matrix.
The system without parameter uncertainty is defined as
The auxiliary robust feedback control gain is solved by the following lemma.
Lemma 1. (see [34]) For polytopic LPV systems (18), if there are matrix , so that the following LMI holds:Then the auxiliary robust feedback control gain can ensure the stability of system (18).
To compute mRPI set, the following definition is given.
Definition 1. (see [35]) The set is the mRPI set of the polytopic LPV system (14) if and only if there is for all error states and .
From Definition 1, we know that is always located in the mRPI set. Therefore, holds, where is Minkowski set addition. Due to W being symmetric around the origin, the mRPI set Z has the same symmetric characteristic around the origin. Consequently, the mRPI set of model (14) is defined aswhere is the upper bound.
For , Z satisfies the following equation:Thereby, the constraint (21) can be converted towhere is the identity matrix.
In order to enhance robustness, it is expected that the invariant set becomes as small as possible. Define , where is the maximum value of each column. By minimizing , the boundary of is solved, and the best robustness is achieved. Farkas lemma is introduced to solve mRPI set .
Lemma 2 (see [36]). For , if there is satisfying , then the following two statements are equivalent.(1), such that ,(2) such that .The optimization problem for the mRPI set is formulated as follows:where , is the n-dimensional unit vector, and is the interference distribution matrix. Then, , and Z can be obtained by (20).
Consider the control input constraints of flexible hypersonic vehiclewhere represents the maximum of control inputs.
Based on the auxiliary robust feedback control gain F and the mRPI set Z, the control inputs of the nominal system should satisfywhere is the Pontryagin set difference, and there is .
The nominal control law at the vertex of each convex hull is designed to provide an appropriate nominal trajectory. For system (15), the following infinite time-domain quadratic programming performance index is given:where .
If there are matrix , symmetric matrix , and a positive scalar , such that all polytopic vertices satisfying the following optimization problem, the nominal control law can stabilize system.where is the maximum of the jth nominal input.
The quadratic programming criterion “min-max” (26) is converted to minimization of the upper bound , that is, a new minimum optimization problem (32) is formulated [37]. LMI constraint (33) guarantees that the nominal states stay in the invariant set with respect to the nominal control law , i.e., , and . Equation (34) indicates that the Lyapunov function of the nominal system (15) is decreasing [37]. Equation (35) is derived from the input constraint .
In conclusion, the nominal control law can guarantee the stability of the nominal system (15).
The real control law is written aswhere is the control input at the equilibrium point.
3.3. Notch Filter Design
The Tube-MPC controller designed above can guarantee the stability of the rigid body states and flexible states of hypersonic vehicle with actuator faults. In order to further handle flexibility and reduce the coupling originating from flexible modes, a notch filter is introduced in this paper. The notch filter can reduce the flexible vibration as much as possible by eliminating the peak of flexible mode frequency response [38, 39].
The frequency response of an ideal notch filter is described aswhere is the notch frequency.
A notch filter should have the following two characteristics.(1)The notch is deep enough and has a certain width. All zeros of transfer function of notch filter should be on the unit circle.(2)Non-notch frequency signals are unaffected. The zero-pole of transfer function of notch filter must match.
The transfer function of notch filter is [38]where are frequency and are damping ratio.
In equation (33), the numerator transmission function can determine the frequency of the notch centre. The denominator transmission function is used for the physical implementation of notch filter.
From CM (1), it can be seen that the coupling between rigid body states and the flexible modes is mainly reflected in pitch rate q. Therefore, a notch filter is introduced into the feedback of channel. The open-loop Bode diagram is presented as Figure 3, to analyse the effect of notch filter on flexible hypersonic vehicle.
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From the figure, it can be seen that the peak caused by the flexible mode is suppressed after the notch filter is connected into the flexible hypersonic vehicle. The different parameter alternation of notch filter has the similar effect as [40]. In conclusion, the reasonable choice of notch filter will not destroy the stability of the original system.
For the effects of the first-order and second-order flexible modes on hypersonic vehicle, we linearize CM (1) at equilibrium point based on small disturbance linearization, and the open-loop pole distribution diagram is drawn in Figure 4.
From Figure 4, it can be seen that CM (1) has a total of 7 motion modes, including five motion modes of rigid body and two aeroelastic modes. Two flexible modes correspond to two pairs of underdamped conjugate complex roots, whose natural frequencies are almost consistent with the natural frequencies of structural flexibility. Since the two pairs of underdamped conjugate complex roots are close to the imaginary axis, both of the two flexible modes have a great influence on CM (1). Therefore, the notch filter is designed for the first- and second-order flexible modes, respectively. Only single one flexible mode is focused. However, by means of reasonable selection of notch filter parameters, it is possible to reduce the other flexible mode while suppressing the focused one. Take the first-order flexible mode as an example, as shown in Figure 5. The first-order flexible mode is the centre of notch filter and aimed to be cancelled. Meanwhile, the second-order flexible mode is partially decreased at the edge of notch filter.
After the notch filter is introduced, the system stability of flexible hypersonic vehicle can still be guaranteed by control law (31). The reasons are as follows.
First of all, the principle of notch filter is that the non-notch frequency is not affected, so only the first flexible mode is filtered out. As previously analysed, the appropriate parameter selection of notch filter will not destroy the stability of the original system, and will even increase the phase margin of the system.
Secondly, the model parameter uncertainty, including and other 10 parameters, is considered while designing Tube-MPC controller. These uncertainties are caused by many factors such as flexibility, fuel consumption, and flight environment change. With the Tube-MPC controller, the actual states of system are limited within the mRPI set and are constantly approximating the nominal trajectory. The filtering of first flexible mode can actually be regarded as the reduction of the model parameter uncertainty. In this paper, the mRPI set is designed on the basis of maximum parameter uncertainty. Therefore, the system states still stay in mRPI set after filtering. At the same time, compared with the system without notch filter, the position of system states with notch filter in the mRPI set is closer to the nominal trajectory, as shown in Figure 6. Therefore, the control law (31) can still ensure the stability of flexible hypersonic vehicle. Moreover, the performance of system with notch filter is better than without notch filter.
4. An Adjustable Controller Design Scheme
In this paper, an alternatively adjustable controller based on the codesigned polytopic LPV model is proposed to ensure the stability of the system. Filter states and output are introduced into the original model to formulate an extended states coupling model (ECM). The new polytopic LPV model is codesigned using the same linearization method for the weak coupling between flexibility and rigid body. To improve the control performance, the adjustable Tube-MPC controller is designed. The schematic of the new control system is shown in Figure 7.
4.1. ECM Modelling
Convert the transfer function of notch filter into state space equation, as follows:where are states of notch filter, respectively; is output of filter, that is, the filtered pitch rate; are the parameters of corresponding matrix, respectively.
In order to ensure the stability of the control system, the filter states are introduced into the original CM model, and is replaced by . Consequently, an ECM involving notch filter is presented as follows:
4.2. Codesigned Polytopic LPV Modelling
Considering the weak coupling between flexible modes and rigid body, the ECM model is transformed into a control-oriented linearization model by using the same linearization method as above, where the notch filter is included.
In the flight envelope, the speed and altitude are also taken as the scheduling variables, and the corresponding system states and control inputs are defined as . The states at the equilibrium point can be obtained as
The corresponding LTI model can be built by Taylor expansionwhere
Similarly, select different equilibrium points in the flight envelope for linearization and fitting to formulate the corresponding continuous LPV model:where and are matrices over scheduling variables.
Then, the discrete polytopic LPV model can be obtained by tensor product
The corresponding nominal system has the following formwhere, are nominal system states and control inputs.
The comparison of Bode diagram is also given in Figure 8 to show the accuracy of the codesigned polytopic LPV model, in which the blue line represents the codesigned model and the red line for the ECM model. It can be seen that the new model can also conform to the frequency characteristics of the original model.
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4.3. Adjustable Tube-MPC Controller Design
The baseline Tube-MPC controller is designed in the worst case of fault. However, in the actual flight, the worst case may not happen. In order to balance the control performance and robustness of system, an adjustable Tube-MPC control method is introduced [25].
Considering the case of incomplete actuator fault, the faulty model is formed as follows:where . denotes incomplete disturbance set, that is, . is the adjustable Tube-MPC control inputs, written as the following weighted formula:where is the adjustable factor. and are adjustable robust feedback gain and nominal control gain, respectively.
Firstly, the design method of is the same as the baseline controller by solving (19). It should be noted that the matrix in (19) should be expanded to the corresponding dimension.
Next, the invariant set is designed. In order to achieve better control performance, it is not necessary to choose the mRPI set for the extreme faults. The size of is positively correlated with the size of disturbance set . Considering the nonworst case actuator fault, a smaller mRPI set can be selected, where is called incomplete disturbance mRPI (idmRPI) set. In this paper, an adjustable factor is used to regulate the size of mRPI set corresponding to incomplete fault. In general, should be a diagonal matrix representing the scaling degree of each dimension in . For simplicity, this paper considers that each dimension has the same scaling degree, so can be taken as a value from 0 to 1. Therefore, the relationship between and can be expressed as .
Finally, the adjustable nominal control law is designed. The following optimization problems are presented:where is a matrix. is a symmetric matrix. is a positive scalar. is the maximum of the jth adjustable nominal inputs satisfying the following constraint:
Solving the above optimization problems, the adjustable nominal control law can stabilize the system.
From (46), it can be seen that expands the constraint range of nominal control input. When is decreased, the nominal system constraint is relaxed such that the control performance becomes better and the robustness is worse. On the contrary, the nominal system constraints become strict such that the control performance decreases but the robustness of the system is improved. By adjusting , the stability of the closed-loop system can be further guaranteed.
4.4. The Real Control Law Is Written as
where is the control input at the equilibrium point.
The design steps of the adjustable design scheme are as follows.
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5. Simulations
5.1. Simulation Conditions
Two cases are simulated to validate the effectiveness of the proposed method.
Case 1. In this case, the Tube-MPC controller via notch filter is verified. Select as the flight envelope. And a polytopic LPV model with 20 vertices is built. The initial flight conditions are . The commands are step signals: for height and for speed. The input constraints are . The maximum allowable error of longitudinal attitude angle is . It is assumed that the loss of actuator effectiveness fault with 10% occurs at 100s, that is, . The uncertain parameters exist, given as , , , , , , , , , .
Case 2. In this case, the adjustable controller design scheme is verified. In the same flight envelope, the codesigned polytopic LPV model with 6 vertices is built. The command rises from , to , at a constant speed after 20 s. The adjustment factor . Parameter uncertainty and relevant parameters of fault are the same as Case 1.
5.2. Simulation Result and Analysis
To show the advantages of the controller designed in this paper, the traditional robust MPC (RMPC) is compared. It is worth noting that the traditional RMPC cannot achieve satisfactory result in the presence of flexibility. So the traditional RMPC is just used to control the DM (4) without flexibility.
Figure 9 shows the height and velocity trajectory tracking results in the presence of parameter uncertainty and actuator fault of loss of effectiveness. After fault happens, the flexible hypersonic vehicle can still track the reference trajectory stably using Tube-MPC. However, the accurate altitude tracking is not achieved by the traditional RMPC. Figure 10 shows attack angle, pitch angle, and pitch rate vibrate when fault occurs but soon stabilize at equilibrium. Compared with the traditional RMPC, the vibration is smaller. Compared with Tube-MPC without notch filter, the state change of Tube-MPC with notch filter is gentler and smoother after fault occurs. Figure 11 shows the curves of the generalized flexible coordinates and their derivatives. They vibrate after fault occurs, but soon return to stability. Similarly, using the traditional RMPC control, they vibrate a lot more. The jitter of the system under Tube-MPC with notch filter is gentler and smoother. Figure 12 shows the curves of the control inputs. It can be seen that control inputs satisfy the input constraints in presence of fault.
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The notch filter in the above simulation is designed for the first-order elastic mode. To show its power further, the second-order flexible mode is addressed separately by notch filter, shown in Figure 13. It can be known that the notch filter has a slight inhibition on the derivative of the first-order flexible mode, and a significant impact on the derivative of the second-order flexible mode.
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Simulation results show that the Tube-MPC with notch filter can realize accurate and stable trajectory tracking in presence of uncertain parameters and actuator loss of effectiveness fault. The dynamic error of the longitudinal attitude angles is within . By introducing notch filter, the flexible modes can be rejected.
As can be seen from Figures 14 and 15, the adjustable control scheme not only ensures the stability of the aircraft, but also improves the rapidity of speed response. In Figure 16, the adjustable controller can get rid of the influence of elastic characteristics. As can be seen from Figure 17, in case of failure, the control inputs of the adjustable method can still be maintained within a reasonable range.
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6. Conclusion
In this paper, Tube-MPC via notch filter trajectory tracking controller are designed for flexible air-breathing hypersonic vehicle with parameter uncertainty and actuator fault. Firstly, the polytopic LPV model is established by Jacobian linearization and tensor product. The parameter uncertainty and actuator fault are uniformly transformed into the lumped term. The Tube-MPC controller is designed, where the actual trajectory is limited to the mRPI set and is able to approximate the nominal trajectory. The notch filter is introduced to eliminate flexibility. To ensure the stability of the closed-loop system, the adjustable control scheme is proposed. The filter and the original nonlinear model are combined to formulate the ECM model. Based on the codesigned polytopic LPV model, the adjustable Tube-MPC controller is designed to balance control performance and robustness. The simulation results show the effectiveness of the two controllers. Online identification of flexibility of hypersonic vehicle will be considered in future research.
Data Availability
The data of structural and aerodynamic parameters of flexible air-breathing hypersonic vehicle used to support the findings of this study are included in “Control-Oriented Modeling of an Air-Breathing Hypersonic Vehicle” (DOI: 10.2514/1.27830), which are cited within the text as reference [8]. The simulation conditions’ data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported by Aeronautical Science Foundation of China (20180748002) and Tianjin University Innovation Foundation (2022XSU-0003).