控制理論基礎(chǔ)-中英文對(duì)照

1、Course: Fundamentals of Control Theory控制理論基礎(chǔ)CHAPTER 1 INTRODUCTION 第一章 緒論1.1 INTRODUCTION 引言1.2 INTRODUCTION TO CONTROL SYSTEMS 控制系統(tǒng)概述1.3 DEFINITIONS 定義1.4 HISTORICAL BACKGROUND 歷史背景1.5 DIGITAL CONTROL DEVELOPMENT 數(shù)字控制的發(fā)展1.6 MATHEMATICAL BACKGROUND 數(shù)學(xué)背景（基礎(chǔ)）1.7 GENERAL NATURE OF THE ENGINEERING CONTR

2、OL PROBLEM 工程控制問題的一般特點(diǎn)1.8 COMPUTER LITERACY 用于控制系統(tǒng)的計(jì)算機(jī)程序 1.9 OUTLINE OF TEXT 全書內(nèi)容要點(diǎn) 1.1 Introduction 引言 With the rapidly development of the modern science and technologies, man created many miracles. One of them is the invention of automatic control equipments. Today, we cant live on without the hel

3、p of these auto-machines. 隨著現(xiàn)代科技的迅速發(fā)展，人類創(chuàng)造了很多奇跡。其中之一就是自動(dòng)控制設(shè)備。在現(xiàn)代社會(huì)，人類如果離開了這些自動(dòng)機(jī)器將無法生存下去。 Automatic control systems permeate (滲透) life in all advanced societies today. 自動(dòng)控制系統(tǒng)技術(shù)已經(jīng)滲透到了現(xiàn)代社會(huì)生活的方方面面： produce goods required by an increasing world population 生產(chǎn)足夠的商品以滿足不斷增長的人口的需求； travel to the moon and expl

4、ore outer space 到月球旅行和探索外層空間； space vehicles, the space shuttle, space stations 宇宙飛船、航天飛機(jī)、空間站； reconfigurable flight control systems 可重新配置的飛行控制系統(tǒng)。1.2 Introduction to control systems 控制系統(tǒng)概述 There are many equipments with control systems in our daily lives, such as: 在我們的日常生活中，有很多帶控制系統(tǒng)的設(shè)備，例如： Refrigera

5、tors電冰箱 Robots機(jī)器人 air-conditioner空調(diào) auto-washers 全自動(dòng)洗衣機(jī) elevators電梯 pilotless aircraft無人駕駛飛機(jī) men 人類自身Example: toaster 例子：烤箱期望的面包黑度設(shè)定 input: timer 輸入：由定時(shí)器設(shè)定的時(shí)間 output: degree of darkness and crispness 輸出：面包的黑度和松脆程度 The output quantity has no influence on the input quantity, there is no feedback in th

6、is system. 輸出量對(duì)輸入量沒有影響，系統(tǒng)中不存在反饋。Example: DC shunt motor 例子：直流電機(jī)勵(lì)磁電壓源主軸轉(zhuǎn)速電樞電壓勵(lì)磁電流電樞電壓源電壓選擇器直流電機(jī) input: armature voltage 輸入：電樞電壓 output: speed of the shaft 輸出：電機(jī)主軸的轉(zhuǎn)速 DC: Direct Current 直流 AC: Alternating Current 交流 open-loop control systems: Systems in which the output quantity has no effect upon the

7、 input quantity 開環(huán)控制系統(tǒng)：輸出量對(duì)輸入量沒有影響的系統(tǒng) functional block diagram: 功能方框圖指令輸入?yún)⒖歼x擇器動(dòng)態(tài)單元參考輸入輸出 command input: desired darkness of the toast or the desired speed of the motor 命令輸入：面包的期望黑度值或電機(jī)的期望轉(zhuǎn)速 reference-selector block: selection of the value of time on the toaster timer or the value of voltage applied

8、to the motor armature 參考選擇器方框：選擇設(shè)定烤箱定時(shí)器的時(shí)間值或電樞兩端外加電壓值 reference input: output of the reference-selector block 參考輸入：參考選擇器方框的輸出 dynamic unit: the unit which the reference input is applied to 動(dòng)態(tài)單元：施加參考輸入信號(hào)的單元 desired output: output of the dynamic unit 期望輸出：動(dòng)態(tài)單元的輸出 closed-loop control systems: Systems i

9、n which the output has a direct effect upon the input quantity閉環(huán)控制系統(tǒng)：輸出信號(hào)對(duì)輸入信號(hào)有直接影響的系統(tǒng)。 functional block diagram of a single-input single-output (SISO): 單輸入單輸出系統(tǒng)的功能方框圖：指令輸入?yún)⒖歼x擇器參考輸入激勵(lì)信號(hào)前向元件反饋信號(hào)反饋元件系統(tǒng)動(dòng)態(tài)部分輸出 characteristic: the output is fed back and is compared with the input 特點(diǎn)：輸出信號(hào)被反饋到輸入端并與輸入信號(hào)進(jìn)行比較

10、 comparison unit: performs comparison between the reference input and the feedback signal and results in an actuating signal 比較單元：將參考輸入信號(hào)與反饋信號(hào)進(jìn)行比較，并產(chǎn)生一個(gè)執(zhí)行信號(hào)Example: home heating control system 家用取暖控制系統(tǒng)繼電器爐子熱空氣排出口恒溫器期望室內(nèi)溫度設(shè)定實(shí)際室內(nèi)溫度 A bimetallic coil in the thermostat is affected by both the actual roo

11、m temperature (output) and the reference-selector setting. 恒溫器中的雙金屬片既受到實(shí)際室溫（輸出），又受到參考選擇器設(shè)定的室溫雙重影響。 Another example: Refer to Ch. material P1-14 其它的例子：見材料1141.3 Definitions 定義 From the preceding discussion the following definitions are evolved, based in part on the standards of the IEEE, and are used

12、 in this text. 根據(jù)前面的討論，引伸出了下列在本教材中使用的定義，（部分定義基于IEEE標(biāo)準(zhǔn)）。 IEEE: Institute for Electrical and Electronic Engineers 電氣和電子工程師學(xué)會(huì)美 System. A combination of components that act together to perform a function not possible with any of the individual parts. 系統(tǒng)：許多部件的組合，這些部件組合起來能完成一定的功能，而其中任何單獨(dú)的部件不能完成此功能。 Comman

13、d input. The motivating input signal to the system, which is independent of the output of the system and exercises complete control over it (if the system is completely controllable). 命令輸入：施加于系統(tǒng)的使用信號(hào)，該信號(hào)與系統(tǒng)的輸出無關(guān)，并且在整個(gè)過程中對(duì)系統(tǒng)進(jìn)行全面控制(假設(shè)系統(tǒng)是完全可控的)。 Reference selector (reference input element). The unit th

14、at establishes the value of the reference input. The reference selector is calibrated in terms of the desired value of the system output. 參考選擇器(參考輸入環(huán)節(jié))：建立參考輸入信號(hào)的單元。根據(jù)系統(tǒng)輸出的期望值對(duì)參考選擇器進(jìn)行設(shè)定。 Reference input. The reference signal produced by the reference selector, i.e., the command expressed in a form di

15、rectly usable by the system. It is the actual signal input to the control system. 參考輸入：參考選擇器產(chǎn)生的參考信號(hào)，即以系統(tǒng)能直接使用的形式所表示的指令，它是控制系統(tǒng)的實(shí)際輸入信號(hào)。 Disturbance input. A disturbance input signal to the system that has an unwanted effect on the system output. 干擾輸入：系統(tǒng)的干擾輸入信號(hào)，對(duì)系統(tǒng)的輸出產(chǎn)生不期望的影響。 Forward element (system d

16、ynamics). The unit that reacts to an actuating signal to produce a desired output. This unit does the work of controlling the output and thus may be a power amplifier. 前向單元（系統(tǒng)動(dòng)態(tài)環(huán)節(jié)）：在激勵(lì)信號(hào)作用下產(chǎn)生期望輸出的單元。該單元起控制輸出的作用，因此它可能是一個(gè)功率放大器。 Output (controlled variable). The quantity that must be maintained at a pr

17、escribed value, i.e., it must follow the command input without responding to disturbance inputs. 輸出(被控變量)：該輸出量必須維持在所規(guī)定的值上，即跟隨指令輸入。 Open-loop control system. A system in which the output has no effect upon the input signal. 開環(huán)控制系統(tǒng)：輸出信號(hào)對(duì)輸入信號(hào)不產(chǎn)生影響的系統(tǒng)。 Feedback element. The unit that provides the means

18、for feeding back the output quantity, or a function of the output, in order to compare it with the reference input. 反饋單元：該單元將輸出量或輸出量的函數(shù)反饋到輸入端，并與參考輸入進(jìn)行比較。 Actuating signal. The signal that is the difference between the reference input and the feedback signal. It is the input to the control unit that

19、causes the output to have the desired value. 作用信號(hào)：參考輸入和反饋信號(hào)兩者之差。該信號(hào)作用于控制單元節(jié)，使輸出達(dá)到期望值。 Closed-loop control system. A system in which the output has an effect upon the input quantity in such a manner as to maintain the desired output value. 閉環(huán)控制系統(tǒng)：輸出對(duì)輸入產(chǎn)生直接影響，以使輸出值維持期望值上。 The fundamental difference be

20、tween the open- and closed-loop systems is the feedback action. 開環(huán)與閉環(huán)系統(tǒng)兩者之間的主要區(qū)別是反饋?zhàn)饔谩?Servomechanism (often abbreviated as servo). The term is often used to refer to a mechanical system in which the steady-state error is zero for a constant input signal. Sometimes, by generalization, it is used to

21、refer to any feedback control system. 伺服機(jī)構(gòu)（?？s寫為servo）：在定常輸入信號(hào)作用下穩(wěn)態(tài)誤差為零的系統(tǒng)，有時(shí)泛摜任何反饋控制系統(tǒng)。 Regulator. This term is used to refer to systems in which there is a constant steady-state output for a constant signal. The name is derived from the early speed and voltage controls, called speed and voltage reg

22、ulators. 調(diào)節(jié)器：在常值輸入信號(hào)下有常值穩(wěn)態(tài)輸出的系統(tǒng)，從早期的速度和電壓控制系統(tǒng)引伸而來(故又稱速度和電壓調(diào)節(jié)器)。1.4 Historical background 歷史背景汽車前進(jìn)的實(shí)際方向（輸出）汽車前進(jìn)的指定方向（輸入）A pictorial demonstration of an automobile as a feedback control system駕駛汽車作為反饋控制系統(tǒng)的圖示配重Heroes device for opening temple doors打開寺廟大門的古老裝置James Watt's flyball governor 飛球調(diào)速器，即蒸汽機(jī)

23、離心調(diào)速器離心調(diào)速器的應(yīng)用 離心調(diào)速器的工作原理很簡(jiǎn)單，假定機(jī)器在平衡狀態(tài)遠(yuǎn)行時(shí)突然加上負(fù)載，此時(shí)機(jī)器速度下降，調(diào)速器飛球的高度下降，通過杠桿的作用使蒸汽閥門(它在這里是執(zhí)行器) 打開以使更多的蒸汽進(jìn)入機(jī)器從而使速度大體上能復(fù)原。為了使閥門維持在這個(gè)新的開啟狀態(tài)，飛球張開的角度自然和原先(未加負(fù)載時(shí))不同，這意味著在有負(fù)載時(shí)的速度實(shí)際不會(huì)完全和原先的速度相等，亦即負(fù)數(shù)的變動(dòng)產(chǎn)生了轉(zhuǎn)速誤差。如果該誤差在允許范圍內(nèi)，則上述控制已滿足要求，若要求轉(zhuǎn)速不因加上負(fù)載而變化，可以設(shè)法改變閥門的閥桿長度。后來的發(fā)明者設(shè)計(jì)了自動(dòng)更新調(diào)整機(jī)構(gòu)實(shí)現(xiàn)了轉(zhuǎn)速恒定。 load disturbance M rotated

24、 speed n centrifugal framework expansion sliding cover of comparing framework valve (through the level of the transition framework) steam quantity Q n (Ch1-9)炮彈擊中飛機(jī)的前置位置提前角雷達(dá)天線雷達(dá)和計(jì)算機(jī)回路射擊角防空火炮激勵(lì)信號(hào)驅(qū)動(dòng)電機(jī)飛機(jī) 負(fù)載擾動(dòng)下降M則（軸）轉(zhuǎn)速上升離心力機(jī)構(gòu)張開度比較機(jī)構(gòu)滑套閥門（通過比較機(jī)構(gòu)杠桿）蒸汽量Q 轉(zhuǎn)速反饋Antiaircraft radar-tracking control system 防空雷達(dá)跟

25、蹤控制系統(tǒng) 1868: J C Maxwell, On speed regulating machine, Proceeding of the Royal Society of Landon, Vol. 16 1868年，麥克斯韋爾在英國皇家學(xué)會(huì)學(xué)報(bào)上發(fā)表“論速度調(diào)節(jié)器”論文 1884: E J Routh, Stability criterion of linear system 1884年，勞斯提出線性系統(tǒng)的穩(wěn)定性判據(jù) 1923: Heaviside, arithmetic operators method 1923年：海維賽德提出代數(shù)算子法（海維賽德展開式） 1930s: H Nyqui

26、st, Nyquist Stability criterion 19世紀(jì)30年代，奈奎斯特提出奈奎斯特穩(wěn)定性判據(jù) H W Bode, negative feedback amplifiers 伯德提出負(fù)反饋放大器概念 1945: publication of servomechanism “伺服機(jī)構(gòu)”一書出版 1948: publication of Theory of controls “控制理論”一書出版 1950: W R Evans, Root-Locus method 伊文斯提出根軌跡法 conventional control theory: developed through t

27、he late 1950s. Which is effectively applied to many control-design problems, especially to SISO systems. 經(jīng)典控制理論：是20世紀(jì)50年代末以前出現(xiàn)的，已經(jīng)有效地應(yīng)用于許多控制設(shè)計(jì)問題中，特別是對(duì)于單輸入單輸出信號(hào)系統(tǒng)。 modern control theory: developed for the design of more complicated systems and for multiple-input multiple-output (MIMO) systems 現(xiàn)代控制理論：

28、為了設(shè)計(jì)更復(fù)雜的系統(tǒng)和多輸入多輸出系統(tǒng)而發(fā)展起來的控制理論。 The development of control concepts in the engineering field has been extended to the realm of human and biomedical engineering 控制概念已經(jīng)從工程界延伸到人類和生物醫(yī)學(xué)工程領(lǐng)域。1.5 Digital control development 數(shù)字控制的發(fā)展 aerospace vehicles: 宇航飛行器 control surfaces: 操縱面 stability augmentation syste

29、m: 增穩(wěn)系統(tǒng), SAS control stick: 駕駛桿 airborne digital processors: 機(jī)載數(shù)字處理器 1.6 Mathematical Background 數(shù)學(xué)背景 differential equations: a classical means, used in the solution of control systems in the early studies. The approach is tedious and does not readily indicate what changes should be made to improve

30、system performance. 微分方程：在早期研究中用于控制系統(tǒng)求解的一種經(jīng)典方法。使用該法進(jìn)行求解的過程很煩瑣，而且不能分析為改善系統(tǒng)性能應(yīng)采取的措施。 Laplace transform: transforms differential equations to algebraic equations in order to simplify the above method 拉普拉斯變換：將微分方程變換為代數(shù)方程，以簡(jiǎn)化求解過程。 Nyquist's method: deals with the application of steady-state frequency

31、-response techniques to feedback amplifier design, extended by Black, Bode, Hall, Harris, etc. 奈魁斯特方法，將穩(wěn)態(tài)頻率響應(yīng)計(jì)算法應(yīng)用于在反饋放大器的設(shè)計(jì)中。 root-locus theory: presented by Evans, permits the graphical evaluation of both the time and the frequency response 根軌跡理論：由伊文斯提出的可以對(duì)時(shí)間和頻率響應(yīng)進(jìn)行圖解計(jì)算的理論。 Mathematical models: a

32、set of mathematical equations describing a physical system. Once the Mathematical models are set up, the subsequent method of analysis is independent of the nature of the physical system; i.e., it does not matter whether the system is electrical, mechanical, or something else. This technique helps t

33、he designer to spot similarities based upon previous experience 數(shù)學(xué)模型：一組用于描述物理系統(tǒng)的數(shù)學(xué)方程。一旦數(shù)學(xué)模型建立起來，后面的分析就與物理系統(tǒng)的結(jié)構(gòu)特性無關(guān)，即不管系統(tǒng)是電的、機(jī)械的、或者其它什么系統(tǒng)。這種方法有助于設(shè)計(jì)人員根據(jù)以前的經(jīng)驗(yàn)找出各系統(tǒng)之間的相似性。 No single design method is intended to be used to the exclusion of the others. Depending upon the known factors and the simplicity o

34、r complexity of a control-system problem, a designer may use one method exclusively or a combination of methods. With experience in the design of feedback control systems comes the ability to use the advantages of each method 每種方法的應(yīng)該都不是絕對(duì)的。根據(jù)一個(gè)控制系統(tǒng)問題的已知因素和復(fù)雜程度，設(shè)計(jì)者可以只使用其中一種方法，或?qū)追N方法結(jié)合起來使用。1.7 General

35、 nature of the engineering control problem 工程控制問題的一般性質(zhì) In general, a control problem can be divided into the following steps: 一般而言，一個(gè)控制問題的分析可以按如下步驟進(jìn)行： 1. A set of performance specifications is established. 建立一組性能指標(biāo)。 2. The performance specifications establish the control problem. 根據(jù)性能指標(biāo)描述控制問題。 3. A

36、set of differential equations that describe the physical system is formulated. 列出一組描述物理系統(tǒng)的微分方程式。 4. The differential equations are solved by using conventional control theory aided by available or specially written computer programs. 借助于現(xiàn)成的或?qū)ｉT寫出的計(jì)算機(jī)程序，使用經(jīng)典控制理論的方法分析解決不同的問題1.8 Computer literacy 用于控制系統(tǒng)

37、的計(jì)算機(jī)程序 FREQR: used for frequency domain analysis 用于頻域分析 PARTL: used for partial-fraction expansion of transfer functions and for obtaining a time response 用于傳遞函數(shù)的部分分式展開并獲得其時(shí)間響應(yīng) ROOTL: used for obtaining root-locus data and plots 用于獲得根軌跡數(shù)據(jù)和圖形 MATLAB: MATrix LABoratory, which can be use for the analys

38、is and design of control systems and for the simulink simulation 矩陣實(shí)驗(yàn)室，用于控制系統(tǒng)分析與設(shè)計(jì)及仿真模擬。1.9 Outline of text 全文概述 The text is divided into three parts: 本教材分為以下三個(gè)部分： PART1: Chap 24, provides the mathematical foundation for modeling physical systems and obtaining time solutions using classical or Lapla

39、ce transform methods. 第1部分：24章，討論物理系統(tǒng)建模的數(shù)學(xué)基礎(chǔ)，使用經(jīng)典的或拉氏變換方法獲得時(shí)域解。 PART2 : Chap515, represents material that is usually covered in control theory and control system design. 第2部分：515章，介紹有關(guān)控制系統(tǒng)理論與設(shè)計(jì)的一般方法。 PART3: Chap1618, consists of advanced undergraduate or graduate topics. 第3部分：1618，本科畢業(yè)生或研究生提高用 Appen

40、dix A: Table of Laplace transform pairs 附錄A：拉普拉斯變換對(duì)表。 Appendix B: Interactive CAD programs附錄B：CAD程序 Well learn the following chapters: chap16, 89, 11 我們將學(xué)習(xí)以下章節(jié)：chap16, 89, 11Chap 2 Writing System Equation 列寫系統(tǒng)方程2.1 Introduction (Complementary: The principle of superposition about linear system) 引言（補(bǔ)

41、充：線性系統(tǒng)的疊加原理）2.2 Electric circuits and components 電路及其元件2.5 Transfer function and block diagram傳遞函數(shù)和方框圖2.6 Mechanical translation systems直線運(yùn)動(dòng)機(jī)械系統(tǒng)2.7 Analogous circuits相似電路2.1 Introduction引言 In order to research and analysis a system, it is necessary not only to know qualitatively the working principl

42、es and its characteristics, but also more important to quantitatively describe dynamic properties of the system, to reveal the relationship between the structure, parameters as well as dynamic properties of the system (Ch2-1). So an accurate mathematical model that describes a system completely must

43、 be determined in order to analyze a dynamic system. 研究與分析一個(gè)系統(tǒng)，不僅要定性地了解系統(tǒng)的工作原理及其特性，而且更重要的是定量地描述系統(tǒng)的動(dòng)態(tài)性能，揭示系統(tǒng)的結(jié)構(gòu)、參數(shù)與動(dòng)態(tài)特性能之間的關(guān)系。所以在分析動(dòng)態(tài)系統(tǒng)之前，一定要確定完全能描述該系統(tǒng)的精確的數(shù)學(xué)模型。 The differential equations of control systems describe the dynamic performances of the systems in time domain. We can get output response of

44、 the system by solving differential equations on given forces and initial conditions. 控制系統(tǒng)的微方程在時(shí)域內(nèi)描述系統(tǒng)的動(dòng)態(tài)性能，在給定外作用及初始條件下，求解微分方程可以得到系統(tǒng)的輸出響應(yīng)。 linear differential equations with constant coefficients (linear constant-coefficient differential equations or linear time-invariant, LTI): the relationship be

45、tween the system input and output is independent of time. 常系數(shù)線性微分方程（線性常系數(shù)微分方程或線性時(shí)不變微分方程LTI）：系統(tǒng)輸入與輸出之間的關(guān)系與時(shí)間無關(guān)。 This chapter presents methods for writing the differential and state equations for a variety of electrical, mechanical, thermal, and hydraulic systems. 本章介紹關(guān)于各種電的、機(jī)械的、熱的和液壓的控制系統(tǒng)的微分方程和狀態(tài)方程的列

46、寫方法。 The basic physical laws are given for each system, and the associated parameters are defined. 對(duì)于每種類型的系統(tǒng)，將給出其相應(yīng)的基本物理定理，并定義相關(guān)參數(shù)。 The result is a differential equation, or a set of differential equations, that describes the system. The equations derived are limited to linear systems or to systems

47、that can be represented by linear equations over their useful operating range. 其結(jié)果就是用一個(gè)或一組微分方程來描述該系統(tǒng)。這里所得出的方程僅限于線性系統(tǒng)或在它們整個(gè)工作范圍內(nèi)可以由線性方程描述的系統(tǒng)。 block diagram: represents the flow of information and the functions performed by each component in the system. 方框圖：描述系統(tǒng)中的信息流向和每個(gè)元件所起的作用。 Each function is repre

48、sented by a block系統(tǒng)的每一部份功能均由一個(gè)方框表示 Each block is labeled with the name of the component 在每一個(gè)方框內(nèi)標(biāo)注其所代表的元件的名稱 The blocks are appropriately interconnected by line segments with arrows (Arrows are used to show the direction of the flow of information.) 方框之間用帶箭頭的線段相連（箭頭表示信息流動(dòng)的方向） The use of a block diagr

49、am provides a simple means by which the functional relationship of the various components can be shown and reveals the operation of the system more readily than does observation of the physical system itself. 方框圖提供了一種簡(jiǎn)單的方法，通過該法可以說明各個(gè)元件功能之間的關(guān)系，用它來說明系統(tǒng)的工作情況比考察物理系統(tǒng)本身要容易得多。 The simple functional block d

50、iagram shows clearly that apparently different physical systems can be analyzed by the same techniques. 從簡(jiǎn)單的功能方框圖中可以清楚地看出，明顯不同的物理系統(tǒng)可以用相同的方法進(jìn)行分析。 In general, variables that are functions of time are represented by lowercase letters. 一般情況下，時(shí)域變量用小寫字母表示。 To simplify the writing of differential equations

51、, the D operator notation is used. The symbols D and 1/D are defined by: 為了簡(jiǎn)化微分方程的書寫，引入算子符號(hào)D。符號(hào)D和1/D的定義分別如下： (2.1) (2.2) where Y0 represents the value of the integral at time t=0, that is, the initial value of the integral. 式中Y0表示在時(shí)刻t=0的積分值，即積分的初始值。The principle of superposition about linear system線

52、性系統(tǒng)的疊加原理 It is assumed that the differential equation describing the dynamic relationship between the input x(t) and the output y(t) is as following: 假設(shè)表達(dá)系統(tǒng)輸入x(t)和輸出y(t)之間動(dòng)態(tài)關(guān)系的微分方為：anDny(t)+an-1Dn-1y(t)+.+a1Dy(t)+a0y(t)=bmDmx(t)+.+b1Dx(t)+b0x(t) (1) If ai (i = 0,1,2,.,n) and bj (j = 0,1,2,.,m) are a

53、ll constants, that is, they are not the functions as well as derivatives of y(t) and x(t), the equation is called linear time-invariant, the corresponding system is called linear time-invariant system. 若ai (i = 0,1,2,.,n) 、bj (j = 0,1,2,.,m)均為常數(shù)：即它們不是y(t)和x(t)及它們的導(dǎo)數(shù)的函數(shù)，由此方程是線性時(shí)不變的，相應(yīng)的系統(tǒng)稱為線性時(shí)不變系統(tǒng)。anD

54、ny(t)+an-1Dn-1y(t)+.+a1Dy(t)+a0y(t)=bmDmx(t)+.+b1Dx(t)+b0x(t) (2) If ai and bj is the function of time t, the equation is called linear time-variant, the corresponding system is called linear time-variant system. 若ai和 bj是時(shí)間t的函數(shù)，則該方程為線性時(shí)變的，相應(yīng)的系統(tǒng)稱為線性時(shí)變系統(tǒng)。 (3) If one of the ai and bj is dependent of y(

55、t) and x(t) as well as their derivatives or there exists other forms of t in the equation, the equation is non-linear, the corresponding system is called non-linear system. For examples: 若ai和 bj中只要有一個(gè)依賴于輸入x(t)和輸出y(t)或它們的導(dǎo)數(shù)，或者在微分方程中出現(xiàn)時(shí)間t的其他函數(shù)形式，則該方程為非線性，相應(yīng)的系統(tǒng)稱為非線性系統(tǒng)。如： D2y(t)+2Dy(t)+4y(t)=x(t) linear D2y(t)+Dy