This paper designs a double-loop cascade active disturbance rejection control (ADRC) to overcome the external disturbances and parameter uncertainty during hypersonic vehicle flight. The vehicle attitude angle and attitude angular velocity are regulated in outer loop and inner loop, respectively. A stochastic robust approach is employed to further tune the ADRC parameters for better control performances. The Monte Carlo sampling of uncertain parameter is adopted to evaluate the stochastic robust performance. An improved differential evolution algorithm that combines neighborhood field optimization and triangular mutation is employed as the numerical solver. Simulation results show that the ADRC controller with optimized parameters manifests improved robustness as well as good control performances.

The development of hypersonic vehicle (HV) technology has received increasing attention over the past decades. One of the key issues in this area is the attitude control. The difficulty of HV attitude control is incurred from the high nonlinearity, strong coupling, uncertainty, and external disturbance. Due to the high speed and the significant interactions across the aerodynamics, propulsion system, and structural dynamics, HVs are sensitive to uncertainties [

Wang and Stengel [

Han [

In ADRC, the ESO estimates the unmeasured uncertainties and disturbances of the controlled system, such that they can be compensated in the control law design, so as to improve the performance of the control system. Designing an ESO in ADRC for HV control involves tuning a lot of parameters. Although ADRC is inherent robust, the ADRC parameters can influence the ADRC stability and ESO convergence, which is the key issue in its successful applications. Wan [

For flight control, especially HV control, it remains an open problem to choose proper parameters to guarantee the system stability and further dynamic performances. It is worth studying introducing the optimization technologies to improve the ADRC performances for flight control [

The bioinspired optimization, a.k.a. the intelligent optimization technique, has been proved to be effective against complicated problems in HV control area [

This paper mainly investigates the design of a double-loop cascade active disturbance rejection control for hypersonic vehicles, where the angular velocity ADRC and attitude angular ADRC are designed, respectively. During the controller design process, the coefficients of inner and outer ESO and the parameters of related nonlinear controller are the tuned via a stochastic robust optimization approach, which guarantees the good performance of ADRC system against unmeasured uncertainties and disturbances. In order to solve the optimization problem, an improved differential evolution (DE) algorithm that incorporates neighborhood field optimization and triangular mutation is employed as the numerical solver. Simulation results illustrate the effectiveness of the proposed approach.

The structure of this paper is as follows. The dynamic model of hypersonic vehicle is introduced in Section

Before the implement of the controller design, the hypersonic vehicle should be established. A generic hypersonic vehicle (HV) model proposed by the American Langley Institute is hereby employed [

A general ADRC is composed of four parts, which are the scheduling transition process, the ESO, the nonlinear combination, and the disturbance compensation. The main idea of the ADRC is to treat all the parametric uncertainty estimations as unknown disturbances, which will be compensated upon the design of the control input [

When applied to the attitude angle control of HV, the classical second-order single-loop ADRC ignores a large number of uncertainties, and the disturbance observed by the ESO is too complex, resulting in undesirable large chattering in the control output. Hence, a cascade ADRC design is employed, where the inner loop and outer loop are both cascade first-order systems, such that the influence of uncertainty on the control output is reduced. The basic structure of the double-loop ADRC is shown in Figure

Basic structure of the double-loop cascade ADRC.

This structure is based on the idea of the backstepping method. Firstly, the angular velocity of HV in each channel is determined by the ADRC in the outer loop, which will be taken as the set point for the inner loop. Thereafter, according to the ADRC in the inner loop, the control quantity of rudder is determined. Finally, the control input is acquired from reverse recursion.

To fulfill the design of the double-loop cascade controller, 1 is divided into inner and outer loop control systems. The inner loop system controls the angular velocity of HV, and the outer loop control system controls the attitude angle of HV. Control systems can be simplified into

The pitch channel is adopted as an example to demonstrate the controller design process. Firstly, an outer loop ADRC is designed in a continuous manner. Since the system is a first-order system, the outer loop ADRC controller only needs position information for feedback. Therefore, the transition process only needs the transition signal of the given signal and does not need the differential signal of the given signal, such that the FAL function can be adopted [

The extended state observer along with the nonlinear feedback and disturbance compensation are thereby designed:

In this way, the outer loop ADRC controller for pitch channel is obtained from the transition process of the outer loop, the extended state observer, nonlinear feedback, and disturbance compensation, and the dynamic analysis is carried out for the above parts:

Equations (

It should be noted that the transition process is eliminated in the ADRC of the inner loop to reduce unnecessary chattering. Further explanations of the ADRC design can be referred to [

Combining the abovementioned ADRC for the outer and inner loops, the cascade ADRC structure is determined, as shown in Figure

Cascade ADRC structure.

Maintaining system stability is a basic requirement for automatic control. The cascade ADRC can provide good antidisturbance performance. However, the flight control is such a complicated problem that a series of dynamic performances are also taken into account. As aforementioned, in order to further improve the control performance of ADRC, a stochastic robust method is employed to find the optimal ADRC parameters that satisfy as many dynamic performance requirements as possible.

The stochastic robust method is composed of stochastic robust analysis (SRA) and stochastic robust design (SRD), as shown in Figure

The stochastic robust optimization method.

Based on uncertainty analysis of the HV in flight, the uncertainty parameter vector of the system is tabulated as Table

Uncertain parameters and their distribution range.

Uncertainty parameter | Uncertainty range |
---|---|

In order to make the optimization result more general, the step responses of the attack angle and the roll angle are employed. The parameters of the ESO are excluded at the current stage, as our major concern is to improve the transient performances of the controller. Including the ESO parameters can result in a much larger optimization problem, with less benefits to the investigated problem. The objective function is designed according to the step responses. For the investigated time-varying nonlinear flight system, the stability performance is judged from the integration of time and absolute error (ITAE) criterion:

There are a number of performance requirements to be satisfied in HV control. Let

For convenience of derivation, assuming that all involved performance indicators are to be minimized.

An example of the rising type judgment, where

A weighted sum approach is employed to obtain the following objective function:

Step response of angle of attack:

_{1}: Index of Angle of Attack ITAE

_{2}: Index of Side-Slip Angle ITAE

_{3}: Index of Roll Angle ITAE

Step response of roll angle:

The weights

If the parameter dimension to be optimized increases, the optimization of parameters will become larger. Hence, minimizing the parameters to be optimized is beneficial to obtain better control effects. According to the cascade ADRC structure, the controller parameters to be optimized are illustrated in Table

ADRC parameters to be optimized.

Structure | Parameters to be optimized | ||
---|---|---|---|

Transient Process | |||

Nonlinear | |||

Combination | |||

Disturbance | |||

Compensation | |||

Channels | Pitch | Yaw | Rolling |

The complexity of the stochastic robust optimization results in demands in both efficient convergence and effective global search ability from the numerical solver. Furthermore, the objective function in (

The NFO is inspired by the real world ecosystems, where a group of animals such as bees and birds are able to communicate and learn from their neighbors within limited perceptual range. Such a communication mechanism is introduced in the NFO method. During the search process, individuals learn from the local environment, instead of the globally “best” ones as in standard evolutionary algorithm. Specifically, an individual keeps following its superior neighbors and diverging from inferior neighbors [

The original triangular mutation adopts the following mutation rule:

The triangular mutation rule can be further improved via incorporating the NFO method. Let

The mutation rule is thereby modified as follows:

In order to keep the balance between the global search and local search, a strategy selection mechanism is added, where the improved DE algorithm and standard DE algorithm are combined. The selection is made based on a nonlinear decreasing probability rule:

The pseudocode of the DE algorithm that incorporates the combined mutation strategy is proposed in Algorithm

Definition:

Set mutation probability

Create a randomly initialized population {

Let

Locate

Select

Select

Generate three random values

Compute the convex combination vector weights

Obtain the combination vector:

Generate the random values

Compute the learning rates

The triangular mutation vector

The conventional mutation vector

Repair

Generate

if

if

if

else

The proposed algorithm combines a newly proposed mutation rule based on the convex combination vector of the triplet as defined by the three vectors and the difference vector between the best and the worst individuals among the three randomly selected vectors from the neighborhood field. The combined mutation strategy manifests advantages in accuracy and convergence speed on the investigated stochastic robust optimization problem. It has yet to verify its superiority on generic issues, e.g., the benchmark problems.

Firstly, the transition process parameters are set as follows:

The uncertainty parameter vector is shown in (

Given

In order to verify the robustness of the cascaded ADRC controller optimized by the stochastic robust method, the aerodynamic parameters are deviated in real time during the simulation. The adopted biased forms for the defection are as follows:

According to above, there are four simulation cases. Firstly, the cascade ADRC simulation with baseline parameters is performed, namely, case (a), where uncertainties are ignored. This is to verify the effectiveness of the cascade ADRC design. For comparison, the uncertainties as characterized by 25 are introduced to perform the ADRC simulation with the baseline parameters, namely, case (b). In this way, the influence of the uncertainties is illustrated. Then, the stochastic robust optimization result, i.e., the optimal parameter

There are three categories of control curves, as the output of the simulations. The first category is the attitude angle-tracking curves, including the attack angle-tracking curve, side-slip angle-tracking curve, and rolling angle-tracking curve. The results from the four cases are depicted in Figure

Attitude angle-tracking curves in the four cases: (a) attitude angle-tracking curve with baseline without uncertainties, (b) attitude angle-tracking curve with baseline under uncertainties, (c) attitude angle-tracking curve with optimal parameters without uncertainties, and (d) attitude angle-tracking curve with optimal parameters under uncertainties.

From Figure

The angular velocity curves in the four cases are also depicted, as shown in Figure

Angular velocity curves in the four cases: (a) angular velocity curve with baseline without uncertainties, (b) angular velocity curve with baseline under uncertainties, (c) angular velocity curve with optimal parameters without uncertainties, and (d) angular velocity curve with optimal parameters under uncertainties.

Steering curves of the servo in the four cases: (a) rudder deflection with baseline without uncertainties, (b) rudder deflection with baseline under uncertainties, (c) rudder deflection with optimal parameters without uncertainties, and (d) rudder deflection with optimal parameters under uncertainties.

According to the above figures, some advantages of the proposed cascade ADRC strategy and the stochastic robust optimization based controller parameter tuning in HVs control can be revealed. Firstly, the stability of the cascade ADRC strategy for HVs in complicated aerodynamic environments is verified. Secondly, the transient performances, even under the influence of aerodynamic parametric uncertainties, can be improved via the parameter optimization. While guaranteeing the convergence, overshoots are removed, oscillations are reduced, and control curves are smoothen with the optimal parameters. In this way, the design objective is achieved.

In this paper, a cascade ADRC is designed for high-speed, strong-coupling, fast time-varying, and strong-nonlinear high-speed HV. In order to obtain better dynamic performances and further enhance ADRC robustness, a stochastic robust optimization method is employed to further tune the ADRC parameters. The robust optimization objective function is designed to characterize the ability of the ADRC parameters to satisfy as many performance requirements as possible. An adaptive DE algorithm based on triangular mutation is employed to solve the complicated robust optimization problem. In simulations, the control performances from the baseline ADRC parameters and the robust optimization results are thoroughly compared. The control curves reveal that significant robustness improvement can be achieved via further parameter tuning in ADRC, such that the effectiveness and necessity of incorporating robust optimization into the ADRC controller design is identified and verified.

All relevant data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Nature Science Foundation of China (grant nos. 61803162, 61873319, and 61903146).