Abstract
The finite-time attitude tracking control for gliding-guided projectile with unmatched and matched disturbance is investigated. An adaptive variable observer is used to provide estimation for the unmeasured state which contains unmatched disturbance. Then, an improved adaptive twisting sliding mode algorithm is proposed to compensate for the matched disturbance dynamically with better transient quality. Finally, a proof of the finite-time convergence of the closed-loop system under the disturbance observer and the adaptive twisting sliding mode-based controller is derived using the Lyapunov technique. This attitude tracking control scheme does not require any information on the bounds of uncertainties. Simulation results demonstrate that the proposed method which is able to acquire the minimum possible values of the control gains guaranteeing the finite-time convergence performs well in chattering attenuation and tracking precision.
1. Introduction
The most significant difference between gliding-guided projectile (GGP) and traditional projectile is that the former one is equipped with wing assembly. On the one hand, the wing assembly makes the projectile be capable of gliding, which enables the gliding-guided projectile to strike the target from a greater distance. On the other hand, the wing assembly provides the projectile with the ability of controlling its flight path, which means that the gliding-guided projectile can achieve precise strikes [1, 2]. Due to these advantages, the gliding-guided projectile has become the current research hotspot. It should be noted that the main way to ensure the stability of a projectile’s airframe is a high body roll-axis spin rate, which is a significant difference from traditional missiles, and is the source of nonlinear coupling between the yaw and pitch channel of GGP. Therefore, the design of GGP, especially the design of the attitude controller, is a challenging task. This is due to the fact that the GGP model suffers from high nonlinearity and strong coupling characteristics. In addition, a GGP is vulnerable to various disturbances during flight, such as the aerodynamic parameter perturbation, the unmodeled dynamics, and strong and sudden wind gusts which change greatly along the latitude and longitude [3].
To compensate for uncertainties and disturbances suffered by the flight control system, a lot of works have been carried out. In [4], the authors propose a pitch attitude and lateral decoupling control based on active disturbance rejection control for aircraft. To obtain robustness, they utilize an extended state observer (ESO). In [5], the finite-time attitude control for a reusable launch vehicle with unmatched disturbance is investigated, and they propose an adaptive multivariable disturbance compensation method to estimate the disturbance. In [6], the adaptive control on the longitudinal dynamics of a hypersonic flight vehicle in the presence of wind effects is investigated. The unknown disturbance due to wind is estimated by neural networks. In [7], they propose a dual closed-loop control framework for attitude control of a quadrotor. In the dual-loop framework they proposed, active disturbance rejection control and proportional-derivative control are used in the inner and outer loops, respectively. To cancel the effect of dynamic disturbance, they use an ESO. In [8], they utilize the prescribed performance control technique to tackle the issue of attitude control of hypersonic flight vehicles (HFVs), which largely improves the transient characteristics of HFVs. In [9], the authors propose an intelligent flight control scheme based on reinforcement learning technique. Besides, many other control methods, such as fuzzy control, neural network control, backstepping control, and model following control, have been applied on this issue [10–20]. It should be noted that various feedback linearizations have been applied to this issue [21–23]. The aims of such methods are to cancel the nonlinear dynamics via mathematical transformation so as to control the system in a linear manner. It is able to provide a full envelope nonlinear flight control system. However, the design techniques need much information about the system; otherwise, the output will not be desired.
Higher-order sliding mode (HOSM) control is another popular method to deal with an uncertain system. A twisting controller (TC) is historically the first 2-SMC algorithm [24] that drives the output and derivative of the system to the origin in finite time in the presence of disturbance whose boundary is known. If the boundary of the disturbance is unknown, it is necessary to design adaptive twisting sliding mode control (ATSMC) methods to satisfy the condition for sliding mode to exist. One efficient scheme is designing algorithms which are able to adjust dynamically the control gains. In [25], this kind of adaptive twisting sliding mode control is integrated into attitude tracking controller for the reentry reusable launch vehicle. In [26], the twisting sliding mode control law is modified with a proposed gain adaptation strategy to improve the attitude tracking performance of quadrotor unmanned aerial vehicles. In [27], the Lyapunov theory is used to design the ATSMC. However, the aforementioned methods fail to search the minimum possible value of the control gain. As we know, the amplitude of chattering which hinders the applications of the sliding mode control is proportional to the magnitude of discontinuous control.
In [28], an adaptive algorithm is designed via equivalent control. Their method is able to search the minimum possible value of control. In their work, the gains can be adjusted adaptively to compensate for the disturbances with lower values without knowledge of the bounds of the disturbances. However, this method may lead to large chattering when the values of the gains are close to the ideal ones. This is due to the fact that the changing rate of the gains still keeps a large value in such circumstances.
Motivated by the aforementioned work, we propose a robust attitude tracking scheme for GGP with chattering attenuation in this paper. The error dynamics of the attitude tracking system suffers from both matched and unmatched disturbances, where the bounds of disturbances are not known. In order to estimate the unmeasured state influenced by unmatched disturbance, an adaptive observer is utilized. Then, an improved equivalent control-based adaptive twisting sliding mode control (IECATSMC) algorithm for [28] is designed and adopted by the controller to compensate for the matched disturbance dynamically. The main contributions of this paper can be summarized as follows: (i)The improved adaption algorithm is able to adaptively adjust the changing rates of the gains on a larger scale. Thus, the transient characteristics of the adaptation strategy in [28] are improved. Since the proposed control scheme is able to acquire the minimum possible values of the control gains, it performs better in chattering attenuation than the fixed gain TSMC and the existing ATSMC, which can be seen from the simulation results given in the latter(ii)A high-performance disturbance estimator proposed in [29] is used in this paper to reconstruct the unmeasurable system state, which extends the application of this estimator. More importantly, by doing so, the finite-time convergence property of the closed-loop system is guaranteed(iii)A proof of the finite-time convergence of the closed-loop system under the observer and the IECATSMC-based controller is derived using the Lyapunov stability theory
The paper is organized as follows: the GGP attitude kinematics model is given in Section 2; then, error dynamics of attitude tracking system is derived. To estimate the unmeasured state in error dynamics, an adaptive observer is utilized in Section 3. The improved adaptive twisting algorithm which is able to search the minimum possible values of the control gains is investigated for the attitude control in Section 4, and a proof of the finite-time convergence for the overall system is derived. The simulation results are provided in Section 5. Finally, concluding remarks are summarized in Section 6.
Notations. Throughout the paper the following notations will be used. For a given vector , denote . For a given scalar variable , denote as the absolute value of .
2. Gliding-Guided Projectile Attitude Kinematics Model
In actual flight, the speed of the GGP is variable due to the varying thrust and drag. However, if we study the dynamics of the projectile which is gliding along a very short flight path, we can safely think that the flight speed and rotational speed of the projectile are constant. Besides, we can also neglect the variation of the dynamic pressure and Mach number as well as other physical parameters. Then, under the assumption that the attack angle and side slip angle are small, the attitude kinematics model of the gliding-guided projectile can be constructed as [30, 31] where and denote kinetic coefficients, where , , , , , , , and .
By accepting that the attitude change speed is more faster than the change speed of the velocity vector direction of the projectile center of mass on the short flight path [30], and without loss of generity, we assume that , . From (2) and the assumption of , , and , we obtain where , , , , , and .
Assume that the steering gear system of the projectile is designed as a second-order one. For the convenience of study, we only consider the steady output of equivalent rudder deflection [32]: where where and stand for equivalent rudder deflect angle control commands for pitching channel and yaw channel, respectively; is the gain of canard; denotes the time constant of steer gear system; denotes the system damping ratio; represents delay time;, and represents delay phase angle.
Taking matched and unmatched disturbances into consideration, from ((1)) and ((2)), we get where besides, , , , and are uncertainties induced by parameter variation and model simplification; stand for external unmeasurable disturbances.
The object is to design a control input so as to drive the attitude angle vector to follow the control command in spite of the internal and external disturbances.
Thus, defining the attitude tracking error as , using (1) and recalling the relationships and , the error dynamics for (5) can be calculated as where where the deviations induced by the uncertainties in and can be seen as internal disturbances.
The following assumption adopted from [29] is needed to the existences of causal control schemes for system (5).
Assumption 1. The disturbances , , , , , and and the first-order derivatives of them are bounded, but the boundaries are unknown.
3. Finite-Time Adaptive Observer for
As we see from (6), the error state contains unknown disturbance ; therefore, it cannot be measured directly. To addresss this problem, an adaptive finite-time variable observer is used for (5) in this section.
Assumption 2. Suppose that the disturbance and , where and are unknown.
For system (5), the controller based on twisting algorithm can be designed as where are control gains. It should be noted that the contoller (16) is impractical because of the existence of lumped disturbance , whose upper bound is unknown. Meanwhile, as mentioned before, the error state is not measurable, so an adaptive finite-time observer is needed. Inspired by [29, 33], an adaptive finite-time observer for (16) is formulated as where the auxiliary sliding variable and the adaptive gains and are where represents the arbitrary small threshold value, and , , and are positive constants. It can be concluded that can be estimated by the term in finite time regardless of the unknown bounded uncertainty [29].
4. Main Results
If and in (16) satisfy and , the system represented by (5) is stable [34]. However, to make traditional twisting sliding mode control strategies meet the demand is a challenging thing because the upper bound of disturbance is often unknown especially in the case of gliding-guided projectile. Therefore, adaptive methods which can adjust the values of control gains have been proposed [27]. It should be noted that such methods usually cause large gains, which may result in an undesirable aggressive chattering phenomenon. In this section, an improved adaptive twisting sliding mode control method is developed to search the minimum possible values of control with better transient quality.
Using estimated in (17), the actual adaptive twisting controller is then designed as where the estimation with represents the estimation error of . Substituting (25) into (5), the closed-loop system is changed into where .
4.1. Improved Adaptive Twisting Sliding Mode Control Algorithm Design
To simplify the adaptive controller design, the ideal estimate of will be exploited, and the following representation for (26) can be considered: where the state variables , and the control gains with to be designed. represents lumped uncertainty, which satisfies and , where the constants and are unknown.
Before we proceed, the following lemma is needed to be introduced.
Lemma 3. Considering the system defined as (29), if the gain satisfies , then the states and will converge to origin in finite time.
The proof of Lemma 3 is given in Appendix A.
Thus, the problem to be solved in the following part becomes to design a gain adaptive algorithm, which ensures the condition holds without any knowledge about the disturbance .
Theorem 4. For the uncertain systems (27), the dynamics of the control gains are designed as , where is defined as where , , , , and are positive constants, and they satisfy , . satisfies where is called the equivalent control [23] function, which is an average value ranging from (-1, 1). is a positive constant and satisfies . Then, the states will converge to origin in finite time.
Proof. The analysis can be divided into three phases.
Phase 1. the gain satisfies .
The system (28) is on reaching phase, so we have
Recall that , thus so the value of will keep increasing until it is large enough, which means can be achieved.
Assuming , we can have here, is used. Therefore, by solving the inequation , we can obtain , which means the condition can be achieved at most after .
Phase 2. the gain satisfies .
According to Lemma 3, the states and will converge to the origin in finite time. Furthermore, the convergence time satisfies , where the definitions of and can be seen in Appendix A.
Phase 3. the system enters the sliding mode.
First, Let us assume that , which means , and the time derivative of exist. A Lyapunov function candidate is defined as . Since system (27) has entered the sliding mode in phase 3, the following equation holds where is defined as equivalent control input. Taking the time derivative of , assuming that , which means that and using (28), (29), and (33) yield
If , we will have
Therefore, is achievable, at least after . After the adaption process for is over , we obtain which means . Another case is that , then increases until , and it will be maintained at that level. The theorem is proven.
If is very close to , the minimal possible gain will be found for the current value of disturbance. As a result, the amplitude of chattering will be reduced.
Remark 5. The main feature of the improved adaptive algorithm is that it can adaptively adjust the changing rate of the gain , which is realized by a regulatory factor . The factor enables the changing rate of to be larger or smaller than . In general, when becomes larger than , which means is far away from its ideal value, can also increase and become larger than . Hence, will tend more quickly toward its ideal value. On the other hand, once enters a small region , which means has been already very close to its ideal value, will become smaller than . As a result, the amplitude of gain oscillation will be reduced so that the condition can be safely ensured.
Remark 6. The function can be derived by low pass filter with a small time constant , respectively. The output satisfies
4.2. Stability Analysis for the Closed-Loop System
In this subsection, the stability of the gliding guided projectile closed-loop system (5) is analyzed.
Theorem 7. or the attitude system (5), the adaptive twisting controller is designed as (25), with the adaptive gains defined as , where is obtained from Theorem 4; then, the states will converge to the origin in finite time.
The proof of Theorem 7 is given in Appendix B.
In order to illustrate the entire control architecture in this paper, the schematic diagram is shown in Figure 1.
5. Simulation and Results
To verify the effectiveness of the adaptive algorithm-based controller, the attitude motion of a certain type of GGP is taken as the simulation project. The parameters of the GGP are shown in Table 1. The initial values for the plant are . Reference commands are taken as deg, deg.
To obtain the equivalent control information, we use a low-pass filter formulated as (37) in the simulation, and the parameter is set as . Simulations are performed in MATLAB/simulink environment, where sample time is set as 0.001 s, and fourth-order Runge-Kutta method is chosen as the numerical integration method.
5.1. Comparison
In the simulation, severe disturbances are considered. , are the uncertainty in , , respectively. and are the uncertainty in and , respectively. The external disturbances are
For performance comparison, the ATSMC proposed in [34] and CTSMC proposed in [24] are also simulated under the same conditions. The gain adaptive schemes are formulated as (i)ATSMC [34]: the gain adaptive scheme is defined as
The term can be computed by (ii)CTSMC [24]: the gain adaptive scheme is defined as
The parameters of the three methods are listed in Table 2. Besides, to validate the effectiveness of the improved algorithm, the method named ECATSMC whose gain adaptive law is defined as [28]: is also simulated, and its control parameters are the same with those of IECATSMC.
Figure 2 shows the curves of the estimation error of and the adaptive gains of the observer . From Figure 2(a), we can see that the estimation error can converge to the origin in a short time. Besides, it is shown in Figure 2(b) that the adaptive gains of the observer keep increasing when the estimation error is large; then, they remain constant.
(a)
(b)
Figure 3 denotes the tracking effect of yaw and pitch commands of the four algorithms. As we can see from Figure 3(a), they both can track the reference commands precisely. After the control begins, CTSMC responses fastest in tracking the commands, IEATSMC takes the second place, and ECATSMC takes the third place, ATSMC responds a little slower than the other three methods. What is more, we can also see from Figure 3(a) that IECATSMC and ECATSMC perform better in terms of tracking accuracy. CTSMC performs a little worse. The tracking effects for pitch command are shown in Figure 3(b). As we see in the picture, ECATSMC responds a little slower than the other three methods. Despite this, ECATSMC can track the command more precisely than CTSMC and ATSMC. Besides, we can also observe that the curve of pith angle under ATSMC deviates from the reference command curve around s. This is due to the fact that the intensive disturbance is imposed around that time. On the contrary, the tracking effects of the other three methods do not degrade.
(a)
(b)
Figure 4 shows the curves of the adaptive gains. We can see from the picture that the gains of IECATSMC and ECATSMC are slightly larger than in almost the whole control process. It should be noted that this approximation can be easily adjusted by choosing the value of . We can also see that ATSMC can increase the gain to satisfy when is larger than . However, in other cases, the gain does not change. Besides, Figure 4 shows that the control gain under CTSMC keep constant, although it satisfies , its value is difficult to be set in practical applications given the upper boundary of disturbance is unknown.
Figure 5 shows the comparison between IECATSMC and ECATSMC and the curve of the exponential term. From Figure 5(a), we can see that at the beginning of the control process, the exponential term automatively increases to a large value to accelerate the changing rate of the gain . Actually, when there is a sudden change occurring in , the exponential term can always adjust its value in time. As a result, IECATSMC reponses faster than ECATSMC, which is demonstrated in Figure 5(b). What is more, when the disturbance changes gently, the exponential term can adjust its value around 1. The result is that IECATSMC has a smaller amplitude of oscillation than ECATSMC, which enables the former one to seldomly violate the condition of .
(a)
(b)
Figure 6 denotes the curves of equivalent canard angle commands and equivalent canard angles. Because the results of IECATSMC and ECATSMC are very similar, here, we only show the results of the former one for a clear comparison with the other two methods. Because the values of equivalent canard angle commands are closely related with those of the control gains, we can see from Figures 6(a) and 6(b) that IECATSMC greatly attenuates the amplitude of chattering. This is also reflected in Figures 6(c) and 6(d).
(a)
(b)
(c)
(d)
Figure 7 shows how the gain is adjusted in the three phases in a clearer manner. As we see from the picture, the gains and continue to increase after control begins, and is satisfied after . The gains reach their maximum on . After that, the equivalent control input equals the disturbance; thus, the gains keep decreasing until their curves follow the change of . Therefore, Figure 6 verifies the proof.
5.2. Monte Carlo Experiments
In order to further validate the robustness of the proposed control scheme, Monte Carlo experiments are performed 100 times. The external and internal disturbances used in this subsection are the same as that in Section 5.1, and the standard derivations of the errors of the aerodynamic parameters are set to be of the nominal values.
It can be observed from Figure 8 that the corresponding curves are kept within small envelops, and all of them can track the desired commands accurately.
(a)
(b)
6. Conclusion
In this paper, a robust attitude tracking control scheme is proposed for the gliding-guided projectile. The proposed control method does not require any information on the bounds of disturbances. Simulation results show that the proposed method is not only more efficient in chattering attenuation than existing algorithms but also more precise in tracking performance. Besides, in order to make the proposed attitude tracking controller be more robust, actuator-fault situation should be taken into consideration. Therefore, in our future work, we will extend the proposed control scheme with a fault-tolerant approach.
Appendix
A. Proof of Lemma 3
Consider the following Lyapunov function candidate [33]: and the above function can be rewritten as where , . To make be positive, is chosen to satisfy . By utilizing the relationship , we can have where , and the matrix
Then, (A.3) can be rewritten as
On the other hand,
Next, taking the derivative of , and substituting (27) into it, yields
Recall that and using the relationship , we can obtain where , and the matrix is
If , holds, then the matrix is positive definite, which implies is negative definite, and the following inequality holds
By using the following well-known inequality,
(A.8) can be rewritten as where . Let us denote the upper bound of as , the lower bound of as , then (A.10) can be modified as
Therefore, the states and will converge to origin in finite time at most .
B. Proof of Theorem 7
The proof can be divided into two steps. Firstly, it will be demonstrated that the closed-loop system (26) is bounded with the existence of estimation error . Next, the trajectories of the closed loop (26) without estimation error will be analyzed. Since the uncertainty is bounded due to the boundedness of , there exists a positive constant , which satisfies .
Choose a Lyapunov function candidate as . According to the analysis in phases 1 and 2 in Theorem 4, the gain keeps increasing until , and the flowing inequality can hold where
Note that the estimation error is bounded, assume , where is a positive constant. Then, we obtain where , and the matrix , which is positive definite according to the value of . From (A.5), we can verify that where . Therefore,
Using (B.1) along with the above inequality, one has
Therefore, it is easy to verify that will be confined in a region which is a little larger than . Thus, is bounded, which means the states and are bounded. In Section 3, we have shown that the estimation error for the state will converge to zero in finite time. Without loss of generality, the convergence time is set as . So, holds, when . Thus, the closed-loop system (26) is reduced to which is identical to (27). By following the analysis in Theorem 4, we can conclude that the attitude tracking error will converge to zero at most . What is more, the minimum possible value of the gain can be kept in phase 3, which attenuates the amplitude of chattering.
Nomenclature
| , : | Angle of attack, side slip angle |
| , : | Inclination angle, and deflection angle of flight trajectory |
| , : | Pitch angle and yaw angle |
| , : | Deflection angle of canards |
| : | Mass of the projectile |
| : | Speed of the projectile |
| : | Dynamic pressure |
| : | Projectile diameter |
| , : | Equatorial damping coefficient and polar damping coefficient [10] |
| , : | Characteristic length and characteristic area |
| : | The derivation of the static moment |
| : | The derivation of equatorial damping moment |
| : | The derivation of control moment generated by canard |
| , : | Derivations of the lift induced by projectile body and canard |
| , : | Magnus force and the derivation of Magnus moment. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.