高阶系统的瞬态响应

　　一般的高阶机电系统可以分解成若干一阶惯性环节和二阶振荡环节的叠加。其瞬态响应即是由这些一阶惯性环节和二阶振荡环节的响应函数叠加组成。 对于一般单输入——单输出的线性定常系统，其传递函数可表示为
　　经拉氏反变换，得
　　可见，一般高阶系统瞬态响应是由一些一阶惯性环节和二阶振荡环节的响应函数叠加组成的。当所有极点均具有负实部时，系统稳定。
　　在高阶系统中，凡距虚轴近的闭环极点，指数函数（包括振荡函数的振幅）衰减就慢，而其在动态过程中所占的分量也较大。如果某一极点远离虚轴，这一极点对应的动态响应分量就小，衰减得也快。如果一个极点附近还有闭环极点，它们的作用将会近似相互抵消。如果把那些对动态响应影响不大的项忽略掉，高阶系统就可以用一个较低阶的系统来近似描述。
　　在高阶系统中，若按求解微分方程得到响应曲线的办法去分析系统的特性，将是十分困难的。在工程中，常有低阶近似的方法来分析高阶系统。闭环主导极点的概念就是在这种情况下提出的。
　　若系统距虚轴最近的闭环极点周围无闭环极点，而其余的闭环极点距虚轴很远。我们称这个极点为闭环主导极点。高阶系统的性能就可以根据这个闭环主导极点来近似估算。工程上往往将系统设计成衰减振荡的动态特性，所以闭环主导极点通常都选择为共轭复数极点。
　　高阶系统的动态性能
　　在控制系统的实践中，通常要求控制系统既具有较快的响应速度又具有一定的阻尼程度，此外，还要求减少死区、间隙和库伦摩擦等非线性因素对系统性能的影响，因此高阶系统的增益常常调整到使系统具有一对闭环共轭主导极点。这时，可以用二阶系统的动态性能指标来估算高阶系统的动态性能。
　　翻译成英文：
　　Transient response of high-order systems
　　A general high-order electromechanical system can be decomposed into a superposition of several first-order inertial links and second-order oscillation links.
　　The transient response is composed of the superposition of the response functions of these first-order inertial links and second-order oscillation links. For a general single-input-single-output linear time-invariant system, the transfer function can be expressed as
　　After inverse Laplace transformation, we get
　　It can be seen that the transient response of a general high-order system is composed of the superposition of the response functions of some first-order inertial links and second-order oscillation links. When all poles have negative real parts, the system is stable.
　　In high-order systems, where the closed-loop poles are close to the imaginary axis, the exponential function (including the amplitude of the oscillation function) decays slowly, and its component in the dynamic process is also larger.
　　If a pole is far away from the imaginary axis, the dynamic response component corresponding to this pole is small and decays quickly. If there are closed-loop poles near a pole, their effects will approximately cancel each other out.
　　If the terms that have little effect on the dynamic response are ignored, the higher-order system can be approximated by a lower-order system.
　　In a high-order system, it will be very difficult to analyze the characteristics of the system by solving the differential equation to obtain the response curve. In engineering, there are often low-order approximations to analyze high-order systems. The concept of closed-loop dominant pole is put forward under this situation.
　　If the system has no closed-loop poles around the closed-loop pole closest to the imaginary axis, and the remaining closed-loop poles are far away from the imaginary axis. We call this pole the closed-loop dominant pole.
　　The performance of high-order systems can be approximated based on this closed-loop dominant pole. In engineering, the system is often designed to dampen the dynamic characteristics of oscillation, so the closed-loop dominant pole is usually selected as the conjugate complex pole.
　　Dynamic performance of high-end systems
　　In the practice of the control system, the control system is usually required to have both a faster response speed and a certain degree of damping. In addition, it is also required to reduce the influence of non-linear factors such as dead zone, gap and Coulomb friction on the system performance.
　　The gain of the system is often adjusted to make the system have a pair of closed-loop conjugate dominant poles. At this time, the dynamic performance index of the second-order system can be used to estimate the dynamic performance of the high-order system.
　　参考资料：清华大学控制工程基础PPT
　　英文翻译：Google翻译
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