Abstract: Adjustments and fluctuations in various economic variables profoundly impact the dynamic evolution of the product market and currency market in a real-world economic system, causing constant fluctuations, oscillations, imbalances, and other non-equilibrium states. Due to the imbalance and fluctuations in the product market and currency market, the macro-economy is affected with a significant negative impact, while stabilizing the product market and currency market fundamentally promotes the stable and rapid development of the macro-economy. To this end, this study establishes a continuous-time IS-LM dynamic model under the standard assumptions of contemporary economics, studies the dynamic structural characteristics of the continuous-time IS-LM model, designs its control law analysis, conducts simulation analysis, and then obtains the following meaningful conclusion:
   First, the continuous-time IS-LM model dynamic system has reachability and controllability. This proposition describes the relationship between the input variables and state variables of the continuous-time IS-LM dynamic system, indicating that there is always a control variable that enables the initial state of the continuous-time IS-LM dynamic system to be transferred to the required state within a limited time. That is, through policy regulation, the operating state of the continuous-time IS-LM dynamic system can be changed so that it can achieve the economic operating situation desired by the decision-maker.
   Second, the observability index of the continuous-time IS-LM dynamic system is 2, which means it has complete observability. This proposition describes the relationship between the output variables and state variables of the continuous-time IS-LM dynamic system, indicating that the initial state of the continuous-time IS-LM dynamic system can be uniquely determined by the system output in a limited time period, which is the continuous-time IS-LM dynamic system. The system can determine the operating status of the system by observing its output variables.
    Third, for the continuous-time IS-LM dynamic system, a pure gain feedback controller is designed through this study to make the closed-loop system asymptotically stable and achieve output regulation. This proposition shows that by designing an appropriate controller, the continuous-time IS-LM dynamic system can be made asymptotically stable, and the system can have the expected dynamic performance, thereby achieving the control objectives of the continuous-time IS-LM dynamic system. The simulation analysis shows that the feedback control law designed in this study is effective in controlling the continuous-time IS-LM model dynamic system.
    Fourth, IS-LM is a vital framework tool in modern economic theory to describe the movement of real economic systems. Under the standard assumptions of modern economics, the controllability of the IS-LM model has been rigorously proven for the first time. This proposition shows that the authorities can design corresponding fiscal and monetary policies based on the macroeconomic development trend to achieve effective adjustment and control of the evolutionary state of the IS-LM system, that is, the macroeconomic system is controllable. This provides theoretical support for the government to use fiscal and monetary policies to implement macro-control to smooth out the negative impact of economic fluctuations and imbalances on economic development.

Key words: continuous time IS-LM model, feedback control law, controllability, reachability, observability

摘要： 在现代经济学的标准假设下,建立连续时间的IS-LM动态模型,分析IS-LM动态模型的系统结构特征,设计连续时间的IS-LM动态模型的控制律,得到控制律的解析解并进行仿真分析。研究表明:连续时间的IS-LM动态模型具有能控性、能达性和完全能观性等系统结构特征;连续时间IS-LM动态模型系统可以任意配置其极点,使模型系统渐进稳定,达到输出调节并具有满意的动态性能,即在现代经济学的标准假设下,证实了IS-LM动态模型的可控性。根据模型假设及模型参数的经济意义设计仿真实验,仿真实验结果验证了所设计反馈控制律对连续时间IS-LM动态模型控制的有效性。因而,针对复杂多变的经济形势,科学设计经济政策,能有效实现经济调控的目标。

关键词: 连续时间的IS-LM模型, 反馈控制, 能控性, 能达性, 能观性