Abstract
Aiming at the uncertainty and external disturbance sensitivity of the near space vehicles (NSV), a novel sliding mode controller based on the high-order linear extended state observer (LESO) is designed in this paper. In the proposed sliding mode controller, the double power reaching law is adopted to enhance the state convergence rate, and the high-order LESO is designed to improve the antidisturbance ability. Moreover, the appropriate observer bandwidth and extended order are selected to further reduce or even eliminate the disturbance by analyzing their influences on the observer performance. Finally, the simulation demonstrations are given for the NSV control system with uncertain parameters and external disturbances. The theoretical analyses and simulation results consistently indicate that the proposed high-order LESO with carefully selected extended order and observer bandwidth has better performance than the traditional ones for the nonlinear NSV system with parametric uncertainty and external disturbance.
1. Introduction
Near space vehicle (NSV) has many excellent characteristics, such as fast speed, strong survival and penetration ability, high flight altitude, wide application range, and strong precision strike ability. Despite the great challenges of NSV control considering the nonlinearity, strong coupling and uncertainty of NSV, in addition with their susceptibility to external disturbances [1], the control technology based on modern control theory has been widely studied. Morden control methods such as the robust control, predictive control, sliding mode control, and intelligent control have been applied to hypersonic vehicle control and achieved certain results. On the other hand, the single control method has its own advantages and disadvantages in the NSV control application, which might not meet the multiple requirements of flight control. Therefore, the combination of different control methods to give full play to their respective advantages has become one of the research focus.
In practical applications, uncertainties such as modeling errors and external disturbances are always existent, which make the system output unstable or asymptotically track the desired target. Therefore, different robust control strategies have been proposed. Thereinto, the sliding mode control has been widely used in the flight control system due to its simple structure, fast response, and its insensitivity to external disturbances [2, 3]. In [4], a finite-time attitude control is developed to ensure that the required thrust vector is met exactly at the prescribed time. In [5], a novel recursive singularity free fast terminal sliding mode strategy for finite-time tracking control of nonholonomic systems is proposed. Specially, the sliding mode control with disturbance observation compensation is quite suitable for uncertain nonlinear systems with external disturbances and parameter perturbations [6, 7]. In [8], a novel nonsingular fast terminal sliding mode control method based on disturbance observer is proposed for the stabilization of the uncertain time-varying and nonlinear third-order systems. In [9], the uncertainties are effectively solved by the combination of the sliding mode control and the interval type-2 Takagi-Sugeno-Kang (TSK) fuzzy control. The double power reaching law sliding mode control is adopted in [10] to track the large range command of the climbing stage of the NSV for better performance. In the aspect of anti-interference, the extended state observer can estimate the total disturbance in real time according to the input and output system information and can be eliminated in the feedback control, so that the closed-loop dynamic system has better control performance. Moreover, there is a set of mature empirical formula for the parameter tuning of the observer, which is more convenient for engineering application. In [11], a robust adaptive finite-time fast terminal sliding mode controller is proposed to achieve the desired formation in the presence of model uncertainties and external disturbances. In [12], a novel sliding mode control approach is proposed for the control of a class of underactuated systems which are featured as in cascaded form with external disturbances. Authors in [13] proposed a combination of finite-time robust-tracking theory and composite nonlinear feedback approach for the finite-time and high-performance synchronization of the chaotic systems in the presence of the external disturbances, parametric uncertainties, Lipschitz nonlinearities, and time delays. In [14], an adaptive controller is proposed by employing an event-triggered control and an extended-state-observer, where a simple strategy to tune the observer parameters is provided. Authors in [15] compensate the disturbance by combining the sliding mode control with the autodisturbance rejection control. A novel antisaturation controller is designed in [16] for the air breathing hypersonic vehicle, using the observer to compensate the mismatched disturbance. A state observer with adaptive extension and a continuous sliding mode controller with disturbance observer are proposed and applied to the hypersonic vehicle system in [17, 18], respectively. The nonlinear disturbance observer is applied to the robust flight control of the air breathing hypersonic vehicle in [19], which can significantly improve the robustness and antidisturbance ability of the system. Authors in [20, 21] have pointed out that the high-gain error feedback can guarantee the fast convergence of the observation error and the sufficient estimation accuracy. The performance analyses of the LESO and its higher order form for the second-order system in [22] indicate that the high-order form has faster response speed and better low-frequency characteristics.
From the above recent works, it can be seen that the sliding mode control has good robustness in the NSV control system. The disturbance observation compensation can reduce or even eliminate the disturbance and improve the antidisturbance ability. However, considering the high-order, uncertainty, strong coupling of the NSV control system and other factors in the practical application, the traditional LESO could not meet the performance requirements of NSV control system. Given that, a novel sliding mode controller based on the high-order LESO is proposed in this paper. Based on the traditional LESO method, the high-order LESO is used to estimate and compensate the external disturbance. The sliding mode control method with disturbance compensation is used to improve the stability and anti-interference ability of the system. In this paper, we have proven that compared with the traditional LESO, the high-order LESO has better low-frequency characteristics and anti-disturbance ability for the NSV control system. The main contributions of our work can be concluded as follows. (1)The feedback linearization equivalent model of NSV is established. According to the equivalent model, a sliding mode controller with high-order disturbance compensator is designed to improve the system stability and anti-interference ability. The convergence region of high-order disturbance observer under the condition of lumped disturbance is analyzed, and the system stability is also proved by Lyapunov method(2)The sliding mode controller with high-order LESO compensation is proposed, and the design method of the high-order LESO is also provided(3)The transfer functions from the lumped disturbance to the disturbance estimate and the disturbance estimate error in high-order form are deduced, respectively. Moreover, the influence of the observer bandwidth and the extended order on the observer performance are analyzed specifically in the time domain and frequency domain, respectively(4)The nonlinear NSV model with uncertain parameters and external disturbance has been proven, and the simulation results also show that the high-order LESO has better robustness and antidisturbance ability by selecting appropriate order and observation bandwidth
The remainder of this paper is organized as follows. The theoretical bases are presented in Section 2, including the NSV modeling, the sliding mode controller, and the traditional observer design method. The proposed high-order LESO is detailed in Section 3. The aircraft case simulations and discussions are provided in Section 4, followed by the conclusion in Section 5.
2. Preliminaries
2.1. NSV Model
According to [9] and [23–26], the motion model of NSV longitudinal channel is defined as follows: where , , , , and indicate the velocity, track angle, pitch rate, angle of attack, and altitude, respectively; , , and are severally the throttle setting, engine natural frequency, and damping coefficient, followed by , , , , , , , and as the lift, drag, thrust, mass, gravitational constant, distance from the center of mass to the center of gravity, pitching moment, and the moment of inertia, respectively. Thereinto, the lift, drag, and thrust can be expressed as [9] where , , , and are coefficients for the lift, drag, thrust, and pitch moment, respectively; the control input signals include the throttle setting and elevator deflection angle .
Note that the NSV model in Equation (1) is nonlinear. However, according to the nonlinear theory, the input-output linearization can be adopted first; then, and are derived three times and four times, respectively. Thus, the linearized model can be obtained as follows [9]: where the first, second, and third derivatives of are obtained according to Equation (1) as where , ,, and.
Note that in Equation (3) and Equation (5), the flight state vector is available for measurement. Accordingly, the linearized model can be described as follows:
When the external disturbances are considered, the linearized model can be uniformly described as [27, 28].
where , and and in Equation (6) are substituted by and as the control inputs. This linearized model considering external disturbances of Equation (8) is defined under the hypotheses as follows [28]:
Assumption 1. The external disturbance and its first-order differential are bounded as , , where and are known normal numbers.
Assumption 2. The matrix is nonsingular. In the whole branch envelope, it is reasonable to assume that the matrix is nonsingular because the flight path angle of the aircraft satisfies .
Note that Assumption 1 is only suitable for the traditional state expansion observer. The high-order form of observer will be discussed in the following sections.
2.2. Design of the Sliding Mode Controller
The design process of the sliding mode controller can be described as the following two steps. (1)Select the integral sliding surface aswhere and are the tracking errors for the velocity channel and the altitude channel, respectively. and are the positive constants. The derivatives of Equation (9) can be obtained as
Let , and . By introducing the model Equation (6) into Equation (10), we can get (2)Define the sliding mode approaching rate aswhere , ,,,,,, and. When the system state is far away from the sliding mode, and play a dominant role, while and play a dominant role when the system state is close to the sliding mode. According to Equations (8), (11), and (12), the continuous-time controller can be obtained as follows: where
The sliding mode controller can make the system state converge into the neighborhood of the equilibrium zero point quickly, if there are uncertain external disturbances in the system [29]. To ensure the convergence of system state, the observer can be used to compensate the bounded lumped disturbance caused by the system modeling error and bounded external disturbances.
2.3. Design of Traditional Extended State Disturbance Observer
Considering the following third-order system with external disturbances and uncertainties, the state space form is described as where is the sum of system uncertainties and external disturbances. Equation (15) can be rewritten as follows: where is the derivative of the system lumped disturbance. Then, the traditional extended state observer can be designed as follows [25, 26]: where , , , and are observer states, , , , and are positive real numbers, and is the observation bandwidth.
According to the above analyses, the state space model of the velocity channel can be expressed as where
The disturbance observer can be described as where , ,, is selected by the bandwidth-based configuration method. More generally, in traditional LESO for any order system, the adjustable parameter only has the observation bandwidth .
Similarly, the altitude channel can be described as where
and the adjustable parameter only has the observation bandwidth .
Combined Equation (20) with Equation (21), the aircraft disturbance observer can be described as where
3. Proposed High-Order Linear Extended State Observer
3.1. Design of High-Order Linear Extended State Observer
For the sake of generality, the following -th order system with external disturbances and uncertainties are considered. The state space form can be described as where its extended first-order state observer (traditional LESO) is given as
The following hypothesis proposed in [22] is given as
Assumption 3. The derivatives of the lumped disturbance of the system exist and are bounded as .
Accordingly, the form of state equation and extended -th order state observer can be presented as
where is the lumped disturbance , , , , are the first, second, , and -th order derivatives of the lumped disturbance, respectively. is the estimate of lumped disturbance , and , , , are severally the estimates of the first, second, , and -th order derivatives of the lumped disturbance. The parameter is selected by the bandwidth-based configuration method, and is the observation bandwidth, and indicates the traditional LESO.
3.2. Convergence Analysis
In this section, the convergence analysis is presented. First, the definition of the expansion of -th order state observer for the -th order system is given below
The differential form of can be described as where is the observation bandwidth.
For clear illustration, the following hypothesis and theorem proposed in [22] are provided first.
Assumption 4. There are certain constants of , , and the positive definite continuous differentiable functions and : , which make where is the Euclid norm.
Theorem 5. If Assumption 3 and Assumption 4 are satisfied, then for the system Equation (28), the following conclusion is true.
According to Assumption 3 and Assumption 4, it can be concluded that
Considering the relationship between and , there is
Then, according to Gronwall-Bellman inequality, we can obtain
Finally, the inequality of can be concluded from the relationship of and as
where .
Accordingly, when the perturbation parameters are small enough, the estimated state of the system Equation (28) is sufficiently close to the state of the system Equation (27). The estimation error converges to and decreases with the increase of the expansion order.
3.3. Stability Analysis
Theorem 6. For the nonlinear dynamic model of NSV control system show in Equation (1), the continuous time controller of Equation (13) is chosen as the disturbance compensation controller, considering the uncertainty and external disturbance. The system is asymptotically stable when the following conditions are satisfied. The Lyapunov function is defined as where , , , is the lumped disturbance, is the estimate of lumped disturbance, , ,,, The inequality (40) can be described as the following two forms: From inequality (42), we can obtain when From inequality (43), we can obtain when In conclusion, the states and converge to the following regions in the finite time.
4. Results and Discussion
In this simulation parts, we first analyze the influence of the extended order and observation bandwidth on the observer performance; then, we further verify the performance of proposed observer in the NSV control system.
4.1. Parameter Analysis
According to Equation (27) and Equation (28), we can deduce the transfer function from lumped disturbances to disturbance estimate and disturbance estimate error as follows: where is the lumped disturbance estimate error, is the system order, is the expansion order, is the observation bandwidth, and .
Furthermore, the performance analysis of the expansion order and the observation bandwidth is also provided in the time domain and the frequency domain over the third-order velocity channel and the fourth-order altitude channel, respectively. According to Equation (49), the high-order LESO has the following characteristics.
As can be seen from Figure 1, when the observation bandwidth is a constant, the higher the expansion order, the faster the response speed, but the overshoot will increase. When the expansion order is a constant, the bigger the observation bandwidth, the faster the response speed, and the overshoot remains unchanged. It can be seen from the overall effect that the higher the system order, the greater the overshoot. Therefore, it is particularly important to choose the appropriate observation bandwidth and expansion order.
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Figure 2 shows the low-frequency characteristics of high-order LESO lumped disturbance estimates, where the frequency range is . Moreover, the reference frequencies are selected as 0.2, 0.4, and 0.8, respectively. The amplitude frequency characteristics and phase frequency characteristics are shown in Tables 1 and 2, respectively. Compared with the traditional LESO, when the expansion order is , the amplitude errors increase slightly, but the phase errors decrease greatly. When the expansion order is , the amplitude errors and phase errors are smaller than and . When the expansion order is a constant, the larger the observation bandwidth, the smaller of magnitude errors and phase errors. It has been shown in Tables 1 and 2 that the estimation error is also related to the system order. Both of the magnitude errors and the phase errors will increase as the system order becomes higher. Generally, in the low-frequency band, the estimation errors of high-order LESO are smaller than those of the traditional LESO.
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According to Equation (50), it can be seen from Figures 3 and 4 that, at the same observer bandwidth, increasing the order of the extended state can significantly enhance the disturbance suppression in the low-frequency band, which is conducive to better tracking the system state and disturbance. However, in the intermediate-frequency band, the high-order LESO has larger peak amplitudes, which would lead to bigger overshoots or oscillations of the step response, which is not helpful to the system stability. Furthermore, the observation bandwidth only affects the rapidity of high-order LESO, and the overshoot is not affected by the bandwidth, but only related to the expansion order.
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In general, both increasing the observation bandwidth and the expansion order can improve the low-frequency performance of observer. However, the higher the expansion order, the bigger the peak amplitude of intermediate frequency, which is prone to lead a larger overshoot and even oscillation; the lager the observation bandwidth, the worse the intermediate-frequency and high-frequency characteristics. Furthermore, the higher the order of the system, the greater the estimation errors. Therefore, for high-order system, we use high-order LESO by choosing appropriate observation bandwidth and expansion order, which has better low-frequency characteristics than traditional LESO.
4.2. Numerical Simulations
In order to verify the effectiveness of the proposed method, the mathematical model of the NSV is built on the MATLAB/Simulink platform, and the designed controller is also simulated and verified with MATLAB.
In this paper, based on the flight conditions of NSV cruise state, the uncertain parameters are adopted as additional variables. The main parameters are set as follows: , , , , , and . The parametric uncertainty is defined as
The expansion orders of , , and in addition with observation bandwidth of , , and are adopted in the simulations, respectively. Thereinto, indicates the traditional LESO, while and are the recommended LESO in this paper. Under the premise of system stability, the parameters of controller and observer are basically unchanged. Moreover, due to the coupling between the velocity and altitude channels, the observer output needs to be decoupled. The step signal for velocity channel and altitude channel are given as and , respectively. The external disturbance and are introduced in the 30th second.
The simulation results are shown in Figures 5–8, where the curves with parameter of indicate the algorithm in [27], the others with parameter of and represent our proposed algorithm, and the compared nonlinear extended state observer (NESO) is proposed in [30]. Figure 5 shows the time response of compared NSV observers over the velocity channel and altitude channel. It can be seen in Figure 5 that both the controllers with LESO and NESO with appropriate parameters can make the system stable. As to the overall effect, the steady-state accuracy of the proposed high-order LESO is obviously better than traditional LESO. When the expansion order parameter is selected as , the steady-state accuracy of proposed observer is similar to that of NESO. Furthermore, when the expansion order parameter is selected as , the steady-state accuracy of proposed observer is better than NESO. Figure 6 shows the time responses of the observer errors over the velocity channel and altitude channel, where the disturbance estimation errors of the proposed high-order LESO are obviously smaller than traditional LESO and NESO. Figure 7 shows the time responses of the actuator on throttle setting and elevator deflection. It can be concluded from Figure 7 that with the increase of the order and observer bandwidth, the control signal will generate peak amplitude and high-frequency oscillations, which are deleterious to the actuator. However, the curve of NESO is smoother than LESO. The higher the system order and the lager the observer bandwidth, the more serious the situation is. Figure 8 shows the time responses of the attack angle and the track angle. The smaller the observer error is, the smaller the change of attitude angle is, and the better the aircraft stability is. The proposed high-order LESO can improve the system antidisturbance ability by selecting an appropriate order and observer bandwidth. Therefore, for the NSV nonlinear model with uncertain parameters and external disturbance, the simulation results show that the proposed high-order LESO with extended order of 3 and observer bandwidth of 10 is recommendable as the disturbance compensator, which is better than the traditional LESO and the NESO.
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5. Conclusions
In this paper, a sliding mode controller with high-order LESO compensation is designed according to the NSV characteristics, which provides a new way to solve uncertain systems with external disturbances. The proposed high-order LESO is suitable for any order system. The transfer function from the lumped disturbance to the disturbance estimates and disturbance estimates error of any order system with any extended order is derived. It can be found that a carefully selected extended order and observer bandwidth can achieve remarkable observer performance gains, more stable system response, and higher steady-state accuracy. However, there are still some other problems need to be solved in the future work, such as the peak phenomenon of intermediate frequency, the high-frequency antidisturbance ability, the different order coupling, and the design of finite-time sliding mode controller. The future work will be focused on how to design the switching control law between the LESO and the NESO, in order to solve the stability problem of multimode switching for near space variable wing vehicle.
Data Availability
The experimental 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 are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to express their thanks for the support from the Natural Science Foundation of China (No. 61673209, No. 61966010, and No. 81960324).