13控制原理課件-wanghui chapter3

1、2022/9/26Chapter 3-11CHAPTER 3Time Domain Analysis of Control System and Its Characteristics By Hui Wang2022/9/26Chapter 3-12Outline of this chapterIntroductionSolution of linear differential equationsDynamic of first and second order system Time response specifications(indices)Rouths stability crit

2、erionSteady-state error analyzingSolution of the state Equation2022/9/26Chapter 3-13tTest signalOutput responsetNew steady state (if the system is stable) Transient response IntroductionMathematical modelStable?Satisfied?.Control Systemx(t)y(t)if so, the system is unstable 2022/9/26Chapter 3-14 Intr

3、oductionBecause control systems are inherently dynamic, their performance is usually specified in terms of both the transient response and the steady state responseThe transient response is the response that fades with time The steady state response is the one that persists after transients have ela

4、psed (i.e, when the system reaches steady state)Design specifications: normally include several time-response indices (pl. of index) for a specified input command as well as a desired steady-state accuracy.2022/9/26Chapter 3-15 IntroductionThe ability to adjust the transient and steady-state respons

5、e of a feedback control system is a beneficial e of the design of control systems. One of the first steps in the design process is to specify the measures of performance for the controlled system.In the first, we review how to get the general solution of a linear differential equation and its compon

6、ents.Then, we will introduce the common time-domain specifications such as percent overshoot, settling time, time to peak, time to rise, and steady-state tracking error. To observe responses of the systems, we will use selected input signals such as the step and ramp to test the response of the cont

7、rol system. 2022/9/26Chapter 3-16 IntroductionThe correlation between the system performance and the location of the system transfer function poles and zeros in the s-plane is discussed. We will develop valuable relationships between the performance specifications and the natural frequency and dampi

8、ng ratio for second-order systems.Relying on the notion of dominant poles, we can extrapolate the ideas associated with second-order systems to those of higher order. A general format for the complementary solution in terms of the state transition matrix (STM) is introduced too. Based on Rouths stab

9、ility criterion, we will discuss the stability of control systems2022/9/26Chapter 3-17Solution of linear differential equations Solution of linear differential equations Standard inputs to control system Steady-state responses Transient response2022/9/26Chapter 3-18 The solution of differential equa

10、tionThere are two methods to obtain the solution (time response) of a linear differential equation. One is solving differential equation directly, getting the particular solution part, and complementary solution part respectively. Then add these two parts got the whole solution. (see P66-73). Anothe

11、r method is using Laplace transform (see P97- 106).-review it! 2022/9/26Chapter 3-19The solution of differential equation: LT method (see P97-106) Popularly, a nth system expressed by differential equation in time domain To get the system time response, it is required to perform the inverse-transfor

12、m operation.where usually nm.Take Laplace transfome(zero initial condition), it will be注意：此式與P98(4.32)式中的符號與系數(shù)標(biāo)注不同！If initial condition not zero, can we get the time response? How to do?2022/9/26Chapter 3-110 Usually, the inverse-transform operation can be performed by direct reference to transform

13、tables or by use a digital computer program.The key is how to get the partial-fraction expansions?There are four classes, depending on the denominator X(s)(see 4.8).The solution of differential equation: LT method (see P97-106)2022/9/26Chapter 3-111 The solution of differential equation: classical m

14、ethodThe general solution (response) of a linear differential equation:Particular integralSteady-state component of the solution: which has the same form as input Transient component of the solution: which is the solution of the corresponding homogeneous equation The form of the transient response d

15、epends only on the roots of the characteristic equation. complementary function+2022/9/26Chapter 3-112Different from linear system, for nonlinear differential equations, the form of the response also depends on the initial or boundary conditions, the roots of the characteristic equation, and the ins

16、tantaneous value of the steady-state component.Complete solution (response) of a linear differential equation: The solution of differential equation: classical method2022/9/26Chapter 3-113 Ex.3-1 As Fig. shown system, find the output response when input is a unit step function. R (s)C(s)+-Solution：s

17、tep 1: find close-loop G(s)step 2: find C(s)step 3: to get c(t)The solution of differential equation: LT method (see P97-106)Transient responseSteady state 2022/9/26Chapter 3-114Classical Method: see P76-77, 3.8Example: Two methods finding solution of differential equationThe differential equation i

18、s :The differential equation is :First: find steady state output i2ss, then i2t , . Laplace transform Method:LTinverse LT transformThe results are same.2022/9/26Chapter 3-115The forms of control systems input may be: an analytical expressionas a specific curverandom in shape Standard inputs to contr

19、ol systemThe pattern used in a machining operation: cutting toolFor example, camera platform used in a photographic airplane.More important is to have a basis of comparison for various systems by standardized input.2022/9/26Chapter 3-116 Standard inputs to control systemSinusoidal functionPower seri

20、es function Step functionRamp functionParabolic functionImpulse functionSingularity function奇異函數(shù)2022/9/26Chapter 3-117 Standard inputs to control systemThese singularity functions can be obtained from one another by successive differentiation or integration. tArea=1Impulse( ) Integration2022/9/26Cha

21、pter 3-118 Standard inputs to control systemImpulse response function (or 權(quán)函數(shù)）LTtArea=1Impulse( ) G(s)x(t)y(t)t Impulse 2022/9/26Chapter 3-119 Standard inputs to control systemStepWhere ,when R=1, unit stepRampParabolicWhere, R=1/2, unit.2022/9/26Chapter 3-120 Standard inputs to control systemSinuso

22、idal functionLTWhich is test input signal for frequency analysis, and usually simulating periodical inputs. 2022/9/26Chapter 3-121 Steady-state response: sinusoidal inputThe equation to be solved is of the formBy using Eulers identity (恒等式）The input quantity r is assumed to be (3.1)(3.2)(3.3)(3.4)Th

23、e response c(t) must be of the formThe nth derivative of c(t)ss is (3.5)phasor2022/9/26Chapter 3-122 Steady-state response: sinusoidal input(3.5)(3.7)Canceling from both sides of the equation: Solving CRule of thumb?(3.8)The time response is:Example see P67. A new word: frequency transfer function G

24、(jw) 2022/9/26Chapter 3-123 Steady-state response: polynomial inputThe coefficients b0, b1, , bq are evaluated by equating the coefficients of like powers of t on both sides of the equation.(3.12)Polynomial input:(3.13)Assume the solution of the same form with input When w=0, q=k. 2022/9/26Chapter 3

25、-124 Steady-state response:step-function inputStep-function input:When w=0, q=k=0. 0(3.17)2022/9/26Chapter 3-125 Steady-state response:ramp-function inputWhen w=0, q=k=1 Ramp-function input:t0:t1:Parabolic input see P69-702022/9/26Chapter 3-126Transient response: classical method(see P70,3.5)Conside

26、ring the a homogeneous equation of a general differential equationAssuming a solution has the formCannot be zero for all timeCharacteristic equationIf all the roots are simple2022/9/26Chapter 3-127Transient response: classical method(see P70,3.5)For each itemIf all the roots are simple, the solution

27、 is of form:LTThe pole of C(s)t m could be plotted in s plane. If all m0, then the system will be unstable. - m- msReIm2022/9/26Chapter 3-128Transient response: classical method(see P70,3.5)If there is a root mq of multiplicity p, the transient response includes corresponding terms of the form Since

28、 the coefficients of the transient solution must be determined from the initial conditions, there must be v+w known initial conditions. If there are complex roots mk , mk+1(they always occur in pairs that are complex conjugates) Then ct=?2022/9/26Chapter 3-129Transient response: classical method(see

29、 P70,3.5)damping coefficient（衰減系數(shù))damped nature frequency(衰減振蕩頻率,有阻尼振蕩頻率)The transient response includes corresponding terms of the form By Euly identityWhereand2022/9/26Chapter 3-130Transient response: classical method(see P70,3.5)This term is called an exponentially damped sinusoid, if is negative

30、, it is shown as Fig. right. The time response always remains between the two branches of the envelope(包絡(luò)線）. If there are complex roots in characteristic equation, the transient response includes terms of the form Only is negative, the system is stable, if is positive, an unstable system, undesirabl

31、e case. （欠阻尼）（過阻尼）2022/9/26Chapter 3-131The roots of this factor are: Damping Ratio and Undamped Natural Frequency n (see P72-73) When the characteristic equation has a pair of complex-conjugate roots, it has a quadratic factor of the form effective damping constant of the system critical value of damping constant zeroifDefine damping ratio2022/9/26Chapter 3-132A underdamped (0 1) second-order system are generally analyzed in terms of and n . Standard formsSystem transient is: Damping Ratio and Undamped Natural Frequen