Abstract: This paper describes the analysis and synthesis of control systems using a graphical programming language within the LabVIEW virtual instrument software environment. LabVIEW is used to analyze the root locus of control systems in classical control theory, the frequency domain response (Bode plot, Nyquist plot ) of linear time-invariant systems, and the correction of control systems. Practice shows that analyzing and synthesizing automatic control systems in a virtual instrument environment has advantages such as a user-friendly interface and convenient implementation. It also demonstrates the powerful data processing capabilities and convenient graphical programming methods of virtual instrument software, providing convenience and support for background information processing and even the realization of networked instruments. Keywords: Virtual instrument, Automatic control system, LabVIEW LabVIEW is used to analyze root locus, frequency response of linear systems (Bade Graph, Nyquist Diagram), and system compensation in classic control principles. Its advantages, such as user-friendliness and ease of implementation, are well demonstrated in practice through the use of virtual instrument software in control systems. It also reflects LabVIEW's powerful data processing capabilities and convenient graphical programming approaches, benefiting background informatization and the realization of virtual network equipment. Keywords: virtual instrument, automatic control system, LabVIEW 0. Introduction LabVIEW, short for Laboratory Virtual Instrument Engineering Warbench, is a virtual instrument development platform software launched by National Instruments (NI). It features a simple graphical programming environment. It has built-in signal acquisition, measurement analysis, and data display functions and integrates thousands of engineering functions, eliminating the complexity of traditional development tools. From simple instrument control to process control and industrial automation systems, LabVIEW has been widely used. For the analysis and design of control systems, NI has developed corresponding toolkits. Leveraging LabVIEW's convenient graphical programming language, the analysis and synthesis of control systems become much easier. Engineers can focus their main efforts on the control system design itself without spending excessive time on programming syntax and instructions. For beginners in automatic control system theory, LabVIEW provides a simple and easy-to-learn simulation and analysis platform, helping them better understand and learn this course. 1. Root Locus of Control Systems The root locus method does not directly solve the characteristic equation. Instead, it uses a graphical method to represent the relationship between the roots of the characteristic equation and all numerical values ​​of a certain system parameter. When this parameter takes a specific value, the corresponding characteristic roots can be found in the above relationship graph. This makes it easy to observe the influence of the open-loop zero and pole distribution on system performance. ① The effect of adding open-loop zeros on the root locus: a. Adding open-loop zeros changes the number and angle of asymptotes; b. Adding open-loop zeros is equivalent to adding differential action, causing the root locus to shift to the left or bend, thereby improving the relative stability of the system. Increased system damping shortens the transient response time; c. The closer the added open-loop zeros are to the origin, the stronger the differential action, and the better the relative stability of the system. ② Effect of adding open-loop poles on the root locus: a. Adding open-loop poles changes the number and angle of asymptotes; b. Adding open-loop poles is equivalent to increasing the integral action, causing the root locus to shift or bend to the right, thus reducing the relative stability of the system. Decreased system damping lengthens the transient response time; c. The closer the added open-loop poles are to the origin, the stronger the integral action, and the worse the relative stability of the system. ③ Effect of adding open-loop dipoles: Adding a pair of open-loop dipoles has almost no effect on the root locus, but it can improve the steady-state performance of the system. 2. Frequency Response of Control Systems Given the open-loop transfer function expression of a unity feedback system: From the analysis of Bode plots in automatic control theory, we know that the system consists of a proportional element, a first-order differential element, an integral element, and a first-order inertial element. The initial amplitude of the plot is , where K is the corresponding gain, and the corner frequencies are 10 and 100, respectively. Due to the presence of the integral element, the initial slope of the amplitude-frequency response is -20dB, and the initial phase angle is -90°. When w=10, the asymptote slope becomes -40dB, and the phase angle becomes -180°. Due to the presence of the differential element, the slope becomes -20dB, and the corresponding phase angle becomes -90°. The Bode plot of the system obtained under LabVIEW conditions is shown in Figure 1. [align=center] Figure 1 Bode plot of the system under LabVIEW conditions[/align] We know that as long as all the closed-loop poles of the system are located in the left half-plane of s, the closed-loop system is stable. When running the LabVIEW program, all its zeros and poles are set in the left half-plane of s, and its gain is varied within the range of 0.001-10. It can be observed that no matter how the gain changes, its Nyquist curve will not approach the point (-1, j0), as shown in Figure 2. The transfer function of the control system is changed to a type I system. From the previous analysis of typical components, we know that a type I system has an integral element. Whenever the signal passes through an integral element, the phase will lag by 90°. The starting point of the Nyquist curve is related to the type of the system. When ω=0, G(j0) = ∞-90°; when ω→∞, G(j0) = 0-90° (nm). With the transfer function set to ∞, the open-loop Nyquist curve starts at ∞-90°, and since nm=0, the Nyquist curve terminates at 00°, as shown in Figure 3. The system is stable. [align=center] Figure 2 All closed-loop poles are located in the right half-plane of s[/align] [align=center] Figure 3 Nyquist plot of type I system[/align] 3. Correction of control system The main task of PID controller design is to quickly determine the proportional coefficient K[sub]c[/sub], integral coefficient K[sub]i[/sub], and derivative coefficient K[sub]d[/sub] for a given controlled object so that the system meets the corresponding index. For a stable system, it is desirable for the system to have a certain stability margin. The PID correction under LabVIEW conditions is used to correct the system that is critically stable or whose stability margin does not meet the requirements. For example, Figure 4 shows the phase frequency of the uncorrected system, Figure 5 shows the phase frequency of the correction device, and Figure 6 shows the phase frequency of the corrected system[2]. The phase margin of the uncorrected system (as shown in Figure 4) and the transfer function show that although the system is stable, its phase margin is not large. As long as the parameters change, the system may become unstable. [align=center]Figure 4 Phase frequency of the uncorrected system[/align] Furthermore, the transfer function shows a zero in the right half-plane of the s-plane, indicating the system is stable under a certain gain. The correction device primarily functions as a PI controller in the low-frequency range and a PD controller in the mid-frequency range. Therefore, the PID controller can improve the steady-state performance and dynamic performance of the system. The transfer function of the correction device is... Figure 6 shows that the phase margin of the corrected system increases significantly, and the steady-state performance is significantly improved. The transfer function of the corrected system is... [align=center]Figure 5 Phase frequency of the correction device Figure 6 Phase frequency of the corrected system[/align] Conclusion Using a virtual instrument platform for the analysis and synthesis of control systems provides a unique and feasible method for the theoretical analysis and design of control systems. Practice has proven that LabVIEW can be used not only for classical control theory research but also for modern control theory, for example, to realize the conversion between transfer functions and state-space expressions, the determination of the observability and controllability of linear time-invariant systems, system state feedback, and arbitrary pole configuration, etc. References [1] Zhou Qiuzhan, Qian Zhihong. Virtual Instruments and LabVIEW7 Express Programming [M]. Beijing: Beijing University of Aeronautics and Astronautics Press, September 2004, P1～P100 [2] Hu Renxi, Wang Henghai, Qi Dongming et al. LabVIEW8.2.1 [M]. Beijing: Machinery Industry Press, January 2008 [3] Liu Junhua (ed.). Virtual Instrument Design Based on LabVIEW [M]. Beijing: Electronic Industry Press, May 2003, P24～P60 [4] Yang Leping, Li Haitao, Xiao Xiangsheng et al. LabVIEW Programming Application [M]. Beijing: Electronic Industry Press, May 2001, P24～P60 [5] Zou Bomin. Automatic Control Theory [M]. Beijing: Machinery Industry Press, January 2006 [6] NI Corporation. PID Control Toolset User Manual (Abstract). 2003.11.