高階系統的瞬態響應

一般的高階機電系統可以分解成若干一階慣性環節和二階振盪環節的疊加。其瞬態響應即是由這些一階慣性環節和二階振盪環節的響應函式疊加組成。對於一般單輸入——單輸出的線性定常系統，其傳遞函式可表示為

經拉氏反變換，得

可見，一般高階系統瞬態響應是由一些一階慣性環節和二階振盪環節的響應函式疊加組成的。當所有極點均具有負實部時，系統穩定。

在高階系統中，凡距虛軸近的閉環極點，指數函式（包括振盪函式的振幅）衰減就慢，而其在動態過程中所佔的分量也較大。如果某一極點遠離虛軸，這一極點對應的動態響應分量就小，衰減得也快。如果一個極點附近還有閉環極點，它們的作用將會近似相互抵消。如果把那些對動態響應影響不大的項忽略掉，高階系統就可以用一個較低階的系統來近似描述。

在高階系統中，若按求解微分方程得到響應曲線的辦法去分析系統的特性，將是十分困難的。在工程中，常有低階近似的方法來分析高階系統。閉環主導極點的概念就是在這種情況下提出的。

若系統距虛軸最近的閉環極點周圍無閉環極點，而其餘的閉環極點距虛軸很遠。我們稱這個極點為閉環主導極點。高階系統的效能就可以根據這個閉環主導極點來近似估算。工程上往往將系統設計成衰減振盪的動態特性，所以閉環主導極點通常都選擇為共軛複數極點。

高階系統的動態效能

在控制系統的實踐中，通常要求控制系統既具有較快的響應速度又具有一定的阻尼程度，此外，還要求減少死區、間隙和庫倫摩擦等非線性因素對系統性能的影響，因此高階系統的增益常常調整到使系統具有一對閉環共軛主導極點。這時，可以用二階系統的動態效能指標來估算高階系統的動態效能。

翻譯成英文：

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。

參考資料：清華大學控制工程基礎PPT

英文翻譯：Google翻譯

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