The control performance of fractional order PDU controller and conventional PD controller in inverted pendulum system is compared and analyzed.
Taking the linear inverted pendulum as the controlled object, the state space equation of the whole system is established by Newton-Euler method. A dual-loop PD control scheme is proposed. The two controlled variables of the inverted pendulum are closed-loop controlled together. Conventional PD controller and fractional order PDU controller are designed respectively. The system is simulated in MATLAB. The simulation results show that the two control strategies can achieve the stability control of the inverted pendulum, but the fractional order PDU controller makes the whole system have smaller overshoot, smaller oscillation and shorter adjustment time, which has better control effect than the conventional PD controller. The linear inverted pendulum is an unstable and multivariable non-linear system.
The control of the linear inverted pendulum requires the knowledge of computer, control engineering, sensors and other fields. Inverted pendulum is widely used in actual control experiment test. It can analyze the robustness, stability and controllability of the system. The principle of inverted pendulum is no longer at the preliminary stage of experimental research, and it is currently used in various projects. Therefore, the study of inverted pendulum has both strong theoretical value and strong practical value.
In this paper, the inverted pendulum is taken as the specific research object. Based on the theory of conventional PID control strategy and fractional order PID control strategy, the inverted pendulum model creation, controller design and simulation are completed by using MATLAB simulation. Finally, two kinds of PID control methods are compared and analyzed. Although the inverted pendulum itself is an unstable system, when the air resistance and various friction components are ignored, it is a classical rigid body system of motion [1-2].
The dynamic equation of the system can be created in the inertial coordinate system by applying the knowledge system of classical mechanics theory.
Therefore, the exact mathematical model of the inverted pendulum can be established by Newton-Euler method. PID controller is the most widely used and mature technology controller [3]. Despite the emergence of various new controllers, the PID controller still plays a leading role because of its simple structure and strong robustness. The PID controller is composed of proportional unit P, integral unit I and differential unit D.
It achieves good control effect by adjusting three parameters of Kp, Ki and Kd. The mathematical model shows that the single inverted pendulum is a controlled object with single input and double output. Therefore, this paper proposes a dual-loop PD control scheme, which can adjust the two controlled variables of the system in a closed loop at the same time and complete the stability control of the inverted pendulum. Its model is the same as that of the state feedback control system, so six parameters of the PD controller can be obtained by pole placement method. The three system variables Kp, Ki and Kd of the conventional PID controller are the same as those of the fractional order PI lambda D_ controller.
At the same time, it has two more adjustable variables, lambda and mu. Because the values of lambda and mu can be taken as fractions instead of just one or other integers in the conventional PID controller, the adjustable region of the variables of fractional order PI lambda D_ controller is wider and the controlled system can be adjusted more flexibly in order to obtain better control results. In the same way, thermostatic element the dual-loop fractional-order PDU control strategy is also adopted here. With the same expected poles as above, the parameters of the controller are obtained by particle swarm optimization algorithm [4], =-15.5326, =-22.6401, = 137.5812, = 25.9820, a= 0.8952, and beta= 1,0384. Let the initial value of the car displacement X be 0, and the initial value of the pendulum angle be 0.05 rad. The simulation results are shown in Fig. 1. It can be concluded that both of the two control strategies can make the car displacement stationary at zero and maintain stable state, but the fractional order PDU controller can make the car displacement smaller overshoot, less system oscillation and shorter adjustment time.
The two control strategies can also make the angle of the pendulum rod stationary at zero position and keep inverted stable state, but the fractional order PDU controller can make the angle of the pendulum rod smaller overshoot, smaller oscillation and faster response. In this paper, the inverted pendulum is taken as the controlled object, and the state space equation of the whole system is established by Newton-Euler method. A dual-loop PD control scheme is proposed.
The two controlled variables of the inverted pendulum are closed-loop controlled together. Conventional PD controller and fractional order PDU controller are designed respectively. The system is simulated in MATLAB. The simulation results show that the two control strategies can achieve the stability control of inverted pendulum, but the fractional order PDU controller makes the whole system have smaller overshoot, smaller oscillation and shorter adjustment time, which has better control effect than the conventional PD controller.