機(jī)械控制理論基礎(chǔ)講義 2

1、Fundamentals of Mechanical Control Theory機(jī)械控制理論基礎(chǔ)（Bilingual Course 雙語教學(xué)）吳吉平School of Mechanical Engineering機(jī)械工程學(xué)院Teaching Material: Linear Control System Analysis and Design Fourth Edition教材：線性控制系統(tǒng)分析與設(shè)計 第4版References 參考書目1. 楊叔子，機(jī)械工程控制基礎(chǔ)，第四版，武漢，華中理工大學(xué)出版社 2. 緒方盛彥, 現(xiàn)代控制工程, 科學(xué)出版社，1978 3. 李友善, 自動控制原理(上冊)

2、 , 國防工業(yè)出版社 4. 張伯鵬, 控制工程基礎(chǔ), 機(jī)械工業(yè)出版社 5. 陽含和, 機(jī)械控制工程(上冊) , 機(jī)械工業(yè)出版社 6. 姚伯威, 控制工程基礎(chǔ), 電子科技大學(xué)出版社 7. 薛定宇, 控制系統(tǒng)計算機(jī)輔助設(shè)計MATLAB語言及應(yīng)用, 清華大學(xué)出版社 Learning requirement 學(xué)習(xí)要求1. All exercises and experiment reports must be done in English, and all questions must be answered in English. 所有的作業(yè)和實驗報告必須用英文完成，所有的問題必須用英文回答。2.

3、 must be present (must not be absent) 上課不能缺席3. taking notes：Dont merely copy the blackboard, trying to write down what you consider is important. 作筆記：不能僅僅抄黑板，而是要記下你認(rèn)為重要的東西。4. preview 預(yù)習(xí)5. review 復(fù)習(xí)6. do exercises and experiments independently 獨立完成作業(yè)和實驗7. summarizing, concluding 總結(jié)，概括8. Dont read the

4、 book word by word, the emphases should be put on the understanding the concepts, terminologies as well as their logic relations. 不要一個單詞一個單詞地讀書，應(yīng)該將重點放在概念、術(shù)語以及它們之間的邏輯關(guān)系的理解上9. Daring to put forward questions. 勇于提出問題10. If you have any question, please contact me. 如果你有什么問題，請聯(lián)系我Final examination result

5、考試成績評定1. Daily behaves: 30%. Including: present, exercises, experiments.平時成績占30%，包括：到課率，作業(yè)，實驗2. Final examination: 70%. 考試成績占70%Well, we will get to business（言歸正傳）, well begin the first chapter.CHAPTER 1 INTRODUCTION 第一章 緒論1.1 INTRODUCTION 引言1.2 INTRODUCTION TO CONTROL SYSTEMS 控制系統(tǒng)引言1.3 DEFINITIONS

6、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 CONTROL PROBLEM 工程控制問題的一般特點1.8 COMPUTER LITERACY 計算機(jī)文化1.9 OUTLINE OF TEXT 全文概括1.1 Introduction 引言With the rapidly development of the modern science and tec

7、hnologies, man created many miracles. One of them is the invention of automatic control equipments. Today, we cant live on without the help 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。其中之一就是自動控制設(shè)備。今天，如果沒有這些自動機(jī)器的幫助，我們幾乎無法生存。Automatic control systems permeate (滲透) life in all advanced societies toda

8、y.現(xiàn)在，自動控制系統(tǒng)技術(shù)已經(jīng)滲透到所有的高級社會生活。produce goods required by an increasing world population制造滿足日益增長的人口需求的商品travel to the moon and explore outer space到月球旅行和探索外層空間space vehicles, the space shuttle, space stations飛船、航天飛機(jī)、空間站 reconfigurable flight control systems 重組的飛行控制系統(tǒng)1.2 Introduction to control systems 控制

9、系統(tǒng)引言There are many equipments with control systems in our daily lives, such as: 我們的日常生活中有很多帶控制系統(tǒng)的設(shè)備，例如： Refrigerators冰箱Robots機(jī)器人air-conditioner空調(diào)auto-washers 全自動洗衣機(jī)elevators電梯pilotless aircraft無人駕駛飛機(jī)men 人類Example: toaster 例子：烤箱期望的面包黑度設(shè)定input: timer 輸入：時間output: degree of darkness and crispness 輸出：面包

10、的黑度和松脆程度The output quantity has no influence on the input quantity, there is no feedback in this system. 輸出量對輸入量沒有影響，系統(tǒng)中不存在反饋。馬達(dá)速度電樞電壓磁場電流激磁電壓源電樞電壓源電壓選擇器直流馬達(dá)Example: DC shunt motor 例子：直流并勵馬達(dá)input: armature voltage 輸入：電樞電壓output: speed of the shaft 輸出：軸的轉(zhuǎn)速DC: Direct Current 直流AC: Alternating Current

11、交流open-loop control systems: Systems in which the output quantity has no effect upon the input quantity開環(huán)控制系統(tǒng)：輸出量不影響輸入量的系統(tǒng)functional block diagram: 功能框圖輸出輸入?yún)⒖紕討B(tài)單元參考選擇器輸入指令command input: desired darkness of the toast or the desired speed of the motor 指令輸入：（面包的）期望黑度或者電動機(jī)的期望轉(zhuǎn)速reference-selector block:

12、selection of the value of time on the toaster timer or the value of voltage applied to the motor armature 參考選擇器框：烤箱上定時器的時間值或電樞兩端外加電壓值的選擇reference input: output of the reference-selector block 參考輸入：參考選擇器框的輸出dynamic unit: the unit which the reference input is applied to 動態(tài)單元：參考輸入應(yīng)用的單元desired output: o

13、utput of the dynamic unit 期望輸出：動態(tài)單元的輸出closed-loop control systems: Systems in which the output has a direct effect upon the input quantity閉環(huán)控制系統(tǒng)：輸出影響到輸入的系統(tǒng)。functional block diagram of a single-input single-output (SISO):單輸入單輸出系統(tǒng)的功能框圖：指令輸入?yún)⒖歼x擇器參考輸入激勵信號前向元件反饋信號反饋元件系統(tǒng)動態(tài)部分輸出characteristic: the output is

14、 fed back and is compared with the input 特性：輸出是反饋并與輸入比較comparison unit: performs comparison between the reference input and the feedback signal and results in an actuating signal 比較單元：執(zhí)行參考輸入與反饋信號之間的比較，并產(chǎn)生一個激勵信號Example: home heating control system 家用取暖控制系統(tǒng)期望室內(nèi)溫度設(shè)定熱空氣排出口實際室內(nèi)溫度繼電器恒溫器爐子A bimetallic coil

15、 in the thermostat is affected by both the actual room temperature (output) and the reference-selector setting. 恒溫器中的雙金屬片既受到實際室溫（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

16、 in part on the standards of the IEEE, and are used in this text. 根據(jù)前面的討論，引伸出了下列使用在本文中的定義，部分是基于IEEE標(biāo)準(zhǔn)1。IEEE: Institute for Electrical and Electronic Engineers 電氣和電子工程師學(xué)會美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

17、的組合，這些部件組合起來能完成一定的功能，而其中任何單獨的部件不能完成此功能。Command 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). 指令輸入 加到系統(tǒng)的激勵輸入信號，該信號與系統(tǒng)的輸出無關(guān)，并且在整個過程中對系統(tǒng)進(jìn)行全面控制(假設(shè)系統(tǒng)是完全可控的)。Reference se

18、lector(reference input element). The unit that 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)) 建立參考輸入值的單元。根據(jù)系統(tǒng)輸出的期望值對參考選擇器進(jìn)行設(shè)定。Reference input. The reference signal produced by the reference selector

19、, i.e., the command expressed in a form directly 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，即以系統(tǒng)能直接使用形式表示的指令，它是控制系統(tǒng)的實際輸入信號。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)的干擾輸入信號對系統(tǒng)

20、的輸出產(chǎn)生一個不需要的影響。Forward element (system dynamics). 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. 前向環(huán)節(jié)(系統(tǒng)動態(tài)環(huán)節(jié)) 它在激勵信號作用下產(chǎn)生期望輸出。該環(huán)節(jié)起控制輸出的作用，因此它可能是一個功率放大器。Output (controlled variable). The quan

21、tity that must be maintained at a prescribed value, i.e., it must follow the command input without responding to disturbance inputs. 輸出(被控變量) 這個量必須維持在所規(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) 輸出不影響輸入的系統(tǒng)。Feedback element. The uni

22、t that provides the means for feeding back the output quantity, or a function of the output, in order to compare it with the reference input. 反饋環(huán)節(jié) 該環(huán)節(jié)將輸出量或輸出的函數(shù)反饋到輸入端，使之與參考輸入進(jìn)行比較。Actuating signal. The signal that is the difference between the reference input and the feedback signal. It is the input

23、to the control unit that causes the output to have the desired value. 激勵信號 它是參考輸入和反饋信號兩者之差的信號。它激勵控制環(huán)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) 輸出對輸入有影響的系統(tǒng)，這種影響使輸出維持期望值。The

24、 fundamental difference between 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)別是反饋作用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 genera

25、lization, it is used to refer to any feedback control system. 伺服機(jī)構(gòu) 指的是在常值輸入信號下穩(wěn)態(tài)誤差為零的機(jī)械系統(tǒng)。有時這一術(shù)語也可用于任何反饋控制系統(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

26、speed and voltage regulators. 調(diào)節(jié)器 指的是在常值輸入信號下有常值穩(wěn)態(tài)輸出的系統(tǒng)。這名稱是從早期的速度和電壓控制系統(tǒng)引伸(也稱速度和電壓調(diào)節(jié)器)得來的。1.4 Historical background 歷史背景汽車前進(jìn)的實際方向（輸出）汽車前進(jìn)的規(guī)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. 打開廟門的古老裝置James Watts flyb

27、all governor飛球調(diào)速器, 即蒸汽機(jī)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單，假定機(jī)器在平衡狀態(tài)遠(yuǎn)行時突然加上負(fù)載，此時機(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)不同，這意味著在有負(fù)載時的速度實際不會完全和原先的速度相等，亦即負(fù)數(shù)的變動產(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)計了自動更新調(diào)整機(jī)構(gòu)實現(xiàn)了轉(zhuǎn)速恒定。load

28、disturbance M rotated speed n centrifugal framework expansion sliding cover of comparing framework valve (through the level of the transition framework) steam quantity Q n (Ch1-9)負(fù)載擾動下降，則（軸）轉(zhuǎn)速上升，離心力上升，炮彈擊中飛機(jī)的實際位置提前角雷達(dá)天線雷達(dá)和計算機(jī)線路開火角功率放大器防空火炮反饋激勵信號驅(qū)動馬達(dá)Antiaircraft radar-tracking control system 防空雷達(dá)跟蹤控制

29、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，速度調(diào)節(jié)器（飛球控制器），英國皇家學(xué)會學(xué)報1884: E J Routh, Stability criterion of linear system 1884年，勞斯，線性系統(tǒng)的穩(wěn)定性判據(jù)1923: Heaviside, arithmetic operators method 1923年：海維賽德，拉氏逆變換定理(海維賽德展開式)1930s: H Nyquist, Nyq

30、uist Stability criterion 19世紀(jì)30年代，奈奎斯特穩(wěn)定性判據(jù)H W Bode, negative feedback amplifiers 波德，負(fù)反饋放大器1945: publication of servomechanism 伺服機(jī)構(gòu)的出版1948: publication of Theory of controls 控制理論的出版1950: W R Evans, Root-Locus method 伊文斯根軌跡理論conventional control theory: developed through the late 1950s. Which is effe

31、ctively 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)計問題中，特別是對于單輸入單輸出信號系統(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：為了設(shè)計更復(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

32、。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 system: 增穩(wěn)系統(tǒng), SAS control stick: 駕駛桿

33、 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 system performance. 微分方程：一種經(jīng)典方法，

34、在早期研究中用于控制系統(tǒng)的求解。很煩瑣，而且不能直接看出為了改善系統(tǒng)的性能應(yīng)采取什么措施。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，以便簡化求解。Nyquists method: deals with the application of steady-state frequency-response techniques to feedback amplifier d

35、esign, extended by Black, Bode, Hall, Harris, etc. 奈魁斯特方法，涉及到穩(wěn)態(tài)頻率響應(yīng)計算法在反饋放大器設(shè)計中的應(yīng)用。root-locus theory: presented by Evans, permits the graphical evaluation of both the time and the frequency response. 根軌跡理論：伊文斯提出，可以對時間和頻率響應(yīng)進(jìn)行圖解計算。Mathematical models: a set of mathematical equations describing a physi

36、cal 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 the designer to spot similarities based upon prev

37、ious experience. 數(shù)學(xué)模型：用一組數(shù)學(xué)方程描述一個控制系統(tǒng)，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)計人員根據(jù)以前的經(jīng)驗找出各系統(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 or complexity of a control-system problem, a desi

38、gner 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)該都不是絕對的。根據(jù)一個控制系統(tǒng)問題的已知因素和復(fù)雜程度，設(shè)計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。根據(jù)每個人在反饋控制設(shè)計中的經(jīng)驗，可以最大程度地利用各種方法的優(yōu)點。1.7 General nature of th

39、e 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，一個控制問題可以分成如下步驟：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. 由于性能指標(biāo)的原因而導(dǎo)致控制問題的存在。3. A set of differe

40、ntial 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 計算機(jī)文化（普及）FREQR: used

41、for frequency domain analysis 頻率域分析方法PARTL: used for partial-fraction expansion of transfer functions and for obtaining a time response 傳遞函數(shù)的部分分式展開和獲得時間響應(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 analysis and design of control

42、 systems and for the simulink simulation 矩陣實驗室，用于控制系統(tǒng)分析與設(shè)計及仿真模擬。1.9 Outline of text 全文概述The text is divided into three parts: 本文分為以下三個部分：PART1: Chap 24, provides the mathematical foundation for modeling physical systems and obtaining time solutions using classical or Laplace transform methods. 第1部分：

43、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Appendix A: Table of Laplace transf

44、orm pairs 附錄A：拉氏變換對表。Appendix B: Interactive CAD programs附錄B：CAD程序Well learn the following chapters: chap16, 8 我們將學(xué)習(xí)以下章節(jié)：chap16, 8Chap 2 Writing System Equation 系統(tǒng)方程的建立2.1 Introduction (Complementary: The principle of superposition about linear system) 引言（補(bǔ)充：線性系統(tǒng)的疊加原理）2.2 Electric circuits and compo

45、nents 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)動機(jī)械系統(tǒng)2.7 Analogous circuits模擬電路2.16 Summary 總結(jié)2.1 Introduction引言In order to research and analysis a system, it is necessary not only to know qualitatively the working principles and its characteristics, b

46、ut 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 be determined in order to an

47、alyze a dynamic system.為了研究和分析一個系統(tǒng)，不僅要定性地了解其工作法則和特性，而且更重要的是定量地描述系統(tǒng)的動態(tài)特性，以揭示系統(tǒng)的結(jié)構(gòu)、參數(shù)以及動態(tài)特性及其它們之間的關(guān)系。所以在分析動態(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 the system by solving diff

48、erential equations on given forces and initial conditions. 在時域內(nèi)描述控制系統(tǒng)動態(tài)特性的是微分方程。我們可以根據(jù)激勵（外力，輸入）和初始條件解微分方程的方法獲得系統(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 between the system input

49、 and output is independent of time. 常系數(shù)線性微分方程（常系數(shù)線性或線性時不變微分方程）：系統(tǒng)輸入與輸出之間的關(guān)系與時間無關(guān)。This chapter presents methods for writing the differential and state equations for a variety of electrical, mechanical, thermal, and hydraulic systems. 這一章介紹對于各種電的、機(jī)械的、熱的和液壓的系統(tǒng)如何建立微分方程和狀態(tài)方程的方法。The basic physical laws ar

50、e given for each system, and the associated parameters are defined. 對于每個系統(tǒng)給出基本的物理定理，規(guī)定有關(guān)的參數(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 that can be represented by linear equati

51、ons over their useful operating range. 其結(jié)果就是一個微分方程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元件所起的作用。 Each function is represented by a block系統(tǒng)的每一部份功能均由一個方框表示 Eac

52、h block is labeled with the name of the component 每一個方框上都注明了一個元件的名字 The blocks are appropriately interconnected by line segments with arrows (Arrows are used to show the direction of the flow of information.) 方框用帶箭頭的線段相連（箭頭表示信息流動的方向）The use of a block diagram provides a simple means by which the fun

53、ctional 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. 方框圖的應(yīng)用提供了一個簡單的方法，通過這個方法可以說明各個部分功能之間的關(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 diagram shows clearly that apparently di

54、fferent physical systems can be analyzed by the same techniques. 從簡單的函數(shù)方框圖中可以清楚地看出，顯然不同的物理系統(tǒng)可以用同樣的方法進(jìn)行分析。In general, variables that are functions of time are represented by lowercase letters. 通常，時間變量用小寫字母表示。To simplify the writing of differential equations, the D operator notation is used. The symbo

55、ls D and 1/D are defined by: 為了簡化微分方程的書寫，使用算子符號D。符號D和1/D分別定義如下： （2.1） （2.2）where Y0 represents the value of the integral at time t=0, that is, the initial value of the integral. 式中表示在時刻的積分值，即積分的初始值。The principle of superposition about linear system 線性疊加系統(tǒng)原理It is assumed that the differential equatio

56、n describing the dynamic relationship between the input x(t) and the output y(t) is as following: 假設(shè)微分方程描述的系統(tǒng)輸入x(t)和輸出y(t)之間的動態(tài)關(guān)系如下： 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 all 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. 如果ai (i = 0,1,2,.,n) 和 bj (j = 0,1,2,.,m)是常數(shù)，也就是說，他們與輸入x(t)和輸出y(t)的變化無關(guān)，該方程就稱為線性時不變方程，相應(yīng)的系統(tǒng)稱為線性時不變系統(tǒng)。a