1 Introduction
For a long time, the output tracking problem has been an important topic in the synthesis of control theory [1-3], and it is a type of control problem that is widely present in engineering practice. For example, the signal tracking of a moving body by a radar antenna [4-5], the tracking of a target object by a missile [6-7], and the trajectory tracking of a mobile robot [8] are all typical examples of output tracking control problems.
In actual tracking systems, there are many uncertainties. For example, in servo systems, due to the presence of friction, the traditional PID control will result in phenomena such as "flat top" and "dead zone" in the tracking results; the traditional sliding mode variable structure control method [9] does not have the robustness of sliding mode in approaching mode; Utkin [10] and Laghrouche [11] adopted the integral sliding mode strategy to ensure that the entire dynamic response process is robust.
This paper investigates the design of an optimal sliding mode tracking controller for a flight simulator turntable servo system when the reference signal is the output of the external system. First, the original system and the external reference system are combined into an augmented system. An integral sliding mode control strategy is used to robustly design the optimal regulator, ensuring complete robustness to uncertainties in satisfying the matching conditions. Based on the augmented system, a sliding mode control law is designed to guarantee that the finite-time arrival condition of the sliding mode is met. Finally, this method is applied to the control of the flight simulator turntable servo system and compared with a traditional optimal regulator. Simulation results demonstrate the effectiveness and superiority of the former.
2 Mathematical Model of Flight Simulation Turntable System
The flight simulation turntable servo system discussed in this paper is a three-axis servo system, consisting of a base and three motion control frames. The outer, middle, and inner frames respectively simulate the yaw, pitch, and roll angles of an aircraft. Under normal conditions, the flight simulation turntable model can be simplified to a linear second-order system. However, at low speeds, it exhibits strong friction, at which point the controlled object becomes nonlinear, making it difficult to achieve high-precision control using traditional control methods.
The system uses a DC motor and ignores the armature inductance. Its structure is shown in Figure 1.
Figure 1. Structure of the turntable servo system
Based on the structure of the servo system, the position state equation of the flight simulator turntable can be described as follows:
3. Design of the Optimal Sliding Mode Tracking Controller
3.1 Design of the optimal tracking sliding surface
The mathematical model of the flight simulator turntable system described above can be classified into a class of uncertain linear systems as shown in the following equation.
Comparing equations (14) and (11), we can see that the ideal sliding mode equation of the uncertain system (8) is the same as the dynamic equation of the optimal control closed-loop system. Therefore, the sliding motion is also asymptotically stable, and the ideal sliding motion is completely robust to system uncertainties. We call (12) the globally robust optimal sliding surface.
3.2 Design of Sliding Mode Control Law
To ensure that the sliding mode reaches the condition, the following variable structure control law is selected:
The above equation shows that the sliding mode control law selected according to equation (15) satisfies the existence condition of the sliding mode and the finite time arrival condition. That is to say, the state trajectory of the system starting from any point can reach the sliding surface (12) in a finite time and then remain on it.
4 System Simulation
Construct the augmented system according to equation (8), where
The nominal system and the uncertain system were controlled by optimal control and global robust optimal sliding mode control, respectively. The simulation results are shown in Figure 2 and Figure 3.
As can be seen from Figures 2 and 3, when there is no external friction in the system, the tracking error response curves of the system under optimal tracking control and optimal sliding mode tracking control are basically the same. However, when the influence of friction on the system is considered, the error response curve under optimal control deviates from its original position, while the tracking performance of the system under the optimal sliding mode tracking controller is almost unaffected.
5. Conclusion
This paper takes the flight simulator turntable servo system as the research object, combines the original system and the external reference system into an augmented system, and uses an integral sliding mode control strategy to robustly design the optimal regulator. Based on the augmented system, a sliding mode control law is designed to ensure that the finite-time arrival condition of the sliding mode is met. Simulation results demonstrate the effectiveness and superiority of the optimal sliding mode tracking control.
