Simulation Research of Second-Order Control System Based on Matlab/Simulink

1 Second-order control system model

A system that can be described by a second- order differential equation is called a second-order control system. In control engineering practice, second-order control systems are very common, for example, armature-controlled DC motors, mechanical displacement systems composed of RLC networks and spring-mass-damper, etc. In addition, many high-order systems are often studied as second-order control systems under certain conditions. Therefore, it is of great practical significance to discuss and analyze the characteristics of second-order control systems in detail. The mathematical model structure of a typical second-order control system is shown in Figure 1.

Its closed-loop transfer function is:

Where: ζ is the system damping ratio; ωn is the undamped natural oscillation angular frequency, in rad/s. The dynamic characteristics of the second-order control system can be described by the above two parameters. The output response of the second-order control system under the action of a unit step signal can be divided into the following cases:

(1) When ζ = 0, the second-order control system is in a zero-damping state. The system has a pair of conjugate imaginary roots, and the unit step response of the system is an undamped constant-amplitude oscillation curve.

(2) When 0<ζ<1, the second-order control system is in an underdamped state. The poles of the system are conjugate complex numbers and are located in the left half plane of S. The unit step response of the system consists of a steady-state response and a transient response. The steady-state response is 1, and the transient response is an oscillation attenuation process. The oscillation angular frequency is determined by the damping ratio ζ and the undamped natural oscillation angular frequency ωn, and as ζ decreases, its oscillation amplitude increases.

(3) When ζ=1, the second-order control system is in a critical damping state. The system has two identical real poles located in the left half plane of S. The unit step response of the system is a monotonically rising curve with no overshoot and no oscillation, and there is no steady-state error.

(4) When ζ>1, the second-order control system is in an overdamped state. The system has two unequal real poles located in the left half plane of S. The unit step response of the system is similar to that of the critical damping case, which is a monotonically rising curve with no overshoot and no oscillation, but its transition process time is longer than that of the critical damping.

2. Second-order control system simulation design and research

2.1 Second-order control system simulation structure design

Start Matlab 7.04 and enter the Simulink  simulation interface. According to the mathematical model structure of the second-order control system, the Simulink simulation structure of the second-order control system is designed as shown in Figure 2. Double-click each function module, set the corresponding parameters in the parameter dialog box that appears, input a unit step signal during simulation, the start time is 0, change the values ​​of ωn and ζ respectively, click the start command under the simulation menu to simulate, double-click the oscilloscope module to observe the simulation results, obtain the step response curve of the system, and then analyze the influence of ωn and ζ on the dynamic performance of the system.

2.2 Relationship between the unit step response of a second-order control system and the parameter ζ

The Simulink simulation structure of the second-order control system with ωn=10 rad/s unchanged and parameter ζ changed to 0, 0.25, 1 and 2 is shown in Figure 3. The unit step signal is input, and its simulation response curve is shown in Figure 4. From the experimental data analysis in the figure, it can be seen that the response curves are, from top to bottom, undamped equal-amplitude oscillation curves, underdamped oscillation attenuation curves, critical damping and overdamped monotonic rising curves without overshoot. When 0<ζ<1, the step response curve of the underdamped state ζ of the second-order control system is shown in Figure 5. As ζ increases, the overshoot of the unit step response of the system decreases, but the rise time is lengthened and the peak value of the curve is larger. Therefore, considering the two factors of overshoot and rise time, ζ should be selected to be close to the optimal damping ratio of 0.707.

2.3 Relationship between the unit step response of the second-order control system and the parameter ωn

The Simulink simulation structure of the second-order control system is shown in Figure 6 when ζ=0.1 is set unchanged and the parameter ωn is changed to 10 rad/s and 20 rad/s respectively. The unit step signal is input and the simulation response curve is shown in Figure 7. From the experimental data analysis in the figure, it can be seen that when ζ=0.1, as ωn increases, the oscillation period of the system unit response becomes shorter, and its adjustment time is also shortened accordingly; when ζ≥1, the system becomes a critical damping or underdamped system, and there is a similar conclusion at this time. Figure 8 shows the step response curve of the second-order control system when ζ=1 and ωn is 10 rad/s and 20 rad/s respectively.

3 Conclusion

The simulation analysis method based on Matlab/Simulink environment , through the basic modules provided by Simulink toolbox, does not require any hardware, and uses simulation examples to well realize the simulation research of second-order control systems under the action of unit step signals, directly observe and analyze the changes in the output performance of second-order control systems, and verify the correctness of related theories of second-order control systems. It has great practical value in experimental teaching and scientific research of second-order control systems, and fully reflects the intuitive and convenient characteristics of Matlab/Simulink simulation .