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The Principle of Automatic Control 自动控制原理

The Principle of Automatic Control 自动控制原理

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The Principle of Automatic Control 自动控制原理

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  1. The Principle of Automatic Control自动控制原理 Lecturers:Prof. Jiang Bin Dr. Lu Ningyun College of Automation Engineering NUAA,2008. Autumn

  2. NUAA-The Principle of Automatic Control Chapter 3 Time-domain analysis of control systems 控制系统时域分析

  3. Review of Chap2 • Modeling of control systems • Differential equation model • How to obtain differential equation model? • Linearization a nonlinear model • Solving differential model is difficult • Laplace and inverse Laplace transform • Formula • Properties • How to use laplace transform to solve differential model • Transfer function

  4. Review of Chap2 • Transfer function • What is Transfer function? • What types of system can be characterized by transfer functions? • How to find transfer function? • polynomial form and zero-pole-gain form • Relationship between poles and responses • Modeling a complex control system by structure diagram or signal-flow graph

  5. Chap 3 Time-domain analysis of control systems 3-1 Time-domain criteria (时域指标) of control systems 3-2 Time response of a first-order system(一阶系统) 3-3 Time response of a prototype 2nd-order system(2阶原型系统) 3-4 Analysis of higher-order system 3-5 Stability of linear control systems 3-6 Steady-state error(稳态误差)

  6. 3-1 Time-domain criteria of control systems

  7. What is time-response(时间响应)? Since time is used as an independent variable in most control system, it is usually of interest to evaluate the output response with respect to time, or simply, the time response. • Typical input signals: • 单位阶跃 1(t) • 单位斜坡 t • 单位加速度 0.5t^2 • 单位脉冲δ(t) • 正弦函数 Asin(wt) 为了便于分析和设计;以及各种控制系统的性能比较,需要基本的输入函数形式

  8. Output c(t) 1 0 t Transient process and steady-state process Time response of a control system is usually divided into transient process and steady-state process Transient process (瞬态过程),transition from initial condition to the final condition 由系统特性,瞬态过程可能衰减(decaying)、发散(divergence)(或振荡(oscillation)等形式--提供系统稳定性(stability)的信息 瞬态过程同时提供响应速度(speed)、阻尼(damping)等信息 Steady-state process(稳态过程) the part of the time response that remains after the transient has died out, 提供了稳态误差(steady-state error)信息

  9. Time-domain criteria • 稳定性 (stability) • 动态性能指标(transient performance) • 响应速度 (speed of response) • Rise time(上升时间),peak time(峰值时间),settling time(调节时间) • 响应形式 (type of response) • Damping (阻尼) ratio, overshoot(超调量),period of oscillation(振荡周期) • 稳态性能指标(steady state performance) • 稳态误差,反映系统控制精度或抗扰动能力(disturbance resistance)

  10. A typical unit-step response curve h(t) 超调量σ% = A 稳态误差 A 峰值时间tp 100% B 上 升 时间tr 调节时间ts t 误差带 B

  11. Rise timetr(上升时间)is defined as the time required for the step response to rise from zero to its first steady-state value, or to rise from 10 to 90 percent of its steady-state value. Delay time td(延迟时间)is defined as the time required for the step response to reach 50 percent of its final value. Peak timetp(峰值时间)is defined as the time required for the step response to reach the first peak value. Setting time ts(调节时间)is defined as the time required for the step response to decrease and stay within a specified percentage (5% or 2%) of its final value. 动态性能指标

  12. Percent maximum overshoot σ%(超调量),响应的最大偏差与终值的查与终值之比的百分数 Note: The maximum overshoot is often used to measure the relative stability of a control system. A system with a large overshoot is usually undesirable. 实际应用中,常用的动态指标: 上升时间,评价系统的响应速度(quickness) 超调量,评价系统的阻尼程度(smoothness) 调节时间,同时反映响应速度和阻尼程度

  13. Steady-state error(稳态误差) It is defined as the discrepancy between the output and the reference input when the steady state is reached. Detailed in 3-6 稳态性能指标

  14. Followed by… 3-2 Time response of a first-order system(一阶系统) 3-3 Transient response (瞬态响应) of a prototype 2nd-order system(2阶原型系统) Why we emphasize the 1-th order and 2-nd order systems? • Higher-order systems can be considered to be sum ofthe responses of first- and second-order systems

  15. 3-2 Time response of a first-order system(一阶系统)

  16. First-order system Gain 增益 Time constant (时间常数) Example Q: What is RC circuit used for?

  17. Time response 1. 零初始条件下系统的单位阶跃响应

  18. No overshoot, no oscillation

  19. 一阶系统单位阶跃响应的特点: 1.响应曲线具有非振荡特征,故也称为非周期响应。 2. 可用时间常数T度量系统输出量的数值 3. 响应曲线斜率初始值为1/T,可用来实验测定一阶系统参数T 4. 时间参数反映了系统的惯性,T越大系统响应越慢,惯性越大。

  20. Example 1. Consider a first-order system, • calculate the settling time ts when kt=0.1; • calculate the value of Kt that makes ts=0.

  21. Closed-loop TF

  22. 2. 零初始条件下系统的单位斜坡响应

  23. The unit-ramp response is shown as follows: Definition of steady-state error: When t tends to infinity, the difference between actual steady-state value and the expected one : c(t) r(t)=t T T t 0 一阶系统单位斜坡达到稳态时具有和输入相同的斜率,只要在时间上滞后T,这就存在着ess=T的稳态误差。

  24. 课后复习 • 自学中文教材3-2节 • 一阶系统单位脉冲响应 • 一阶系统单位加速度响应

  25. 3-3 Time response of a prototype2-order system

  26. Second-order system Nature Frequency 自然频率 Damping ratio 阻尼比 二阶系统响应特性决定于根的位置,即阻尼比ζ和自然频率ωn这两个参数

  27. j 0 j 0 j 0 j 0 二阶系统单位阶跃响应定性分析 过阻尼 ζ>1 临界阻尼 ζ=1 欠阻尼 0> ζ<1 零阻尼 ζ=0

  28. 二阶系统的阻尼特性 1.当0< ζ <1时,一对负实部的共轭复根, 系统的单位阶跃响应具有衰减振荡特性,称为欠阻尼(underdamped)。 2.当ζ=1时,两个相等的负实根,称为临界阻尼(critically damped) 3.当ζ>1时,特征方程具有两个不相等的负实根,称为过阻尼 (overdamped) 4.当ζ=0时,系统有一对共轭纯虚根,系统单位阶跃响应作等幅振荡,称为无阻尼(undamped)。 5.当ζ < 0时,此时系统特征方程具有一对正实部的共轭复根或正实根,系统的单位阶跃响应具有发散特性,称为negatively damped).

  29. 二阶系统的单位阶跃响应 1. ζ>1 过阻尼系统

  30. 起始速度小,然后上升速度逐渐加大,到达某一值后又减小,响应曲线不同于一阶系统。过阻尼二阶系统的动态性能指标主要是调节时间ts,根据公式求ts的表达式很困难,一般用计算机计算出的曲线确定ts。起始速度小,然后上升速度逐渐加大,到达某一值后又减小,响应曲线不同于一阶系统。过阻尼二阶系统的动态性能指标主要是调节时间ts,根据公式求ts的表达式很困难,一般用计算机计算出的曲线确定ts。

  31. 2. ζ=1 临界阻尼系统 响应曲线的变化率始终为正,t趋于无穷时,变化率趋于0。 临界阻尼系统单位阶跃响应是稳态值为1的无超调单调上升过程(monoganly

  32. 大阻尼二阶系统可近似等效为一阶系统,调节时间可用3T1来估算。WHY大阻尼二阶系统可近似等效为一阶系统,调节时间可用3T1来估算。WHY 过(临界)阻尼二阶系统调节时间特性

  33. Im Re 3. 0< ζ<1 欠阻尼系统 the eigenvalues of the closed-loop system are:

  34. When r(t)=1(t), the system output is

  35. Transient part, damping sinusoidal term Steady value=1,indicating that the steady state error=0 for unit step input signal Note: 欠阻尼二阶系统的单位阶跃响应曲线是按指数规律衰减到稳定值的,衰减速度取决于特征值实部-ξwn的大小,而衰减振荡的频率,取决于特征根虚部wd的大小。

  36. 4.ζ=0 零阻尼系统 等幅振荡曲线,振荡频率为wn 因此,wn称为无阻尼振荡频率。

  37. Effects of damping ratio (for a given ) Overshoot rise time The oscillation is smaller The speed of the response is slower We are confronted with a necessary compromise between the speed of response and the allowable overshoot.

  38. Followed by… Note: In control engineering, except those systems that do not allow any oscillation, usually a control system is desirable with - moderate damping (allowing some overshoot) - quick response speed - short settling time Therefore, a second-order control system is usually designed as an underdamped system.

  39. underdamped system 欠阻尼二阶系统 Im Re

  40. Performance analysis Unit-step response: 1. Rise Time tr is the time needed for the response to reach the steady-state value for the first time, so n=1. For a given wn, ζ ↓ ,tr ↓ ; For a given ζ,wn↑,tr↓ .

  41. 2 . Peak time =0 tpis the time needed for the response to reach the maximum value for the first time, so n=1. For a given wn, ζ ↓ ,tp ↓ ; For a given ζ,wn↑,tp↓ .

  42. Im Re 3. Overshoot Overshoot is a function of damping ratioζ, independent of wn.

  43. ζ增大,σ%减小,通常为了获得良好的平稳性和快速性,阻尼比ζ取在0.4-0.8之间,相应的超调量25%-2.5%。ζ增大,σ%减小,通常为了获得良好的平稳性和快速性,阻尼比ζ取在0.4-0.8之间,相应的超调量25%-2.5%。 Damping ratioξ

  44. 根据定义: 不易求出ts,但可得出wnts与ζ的关系曲线: 4 调节时间

  45. 调节时间不连续的示意图 ζ值的微小变化可引起调节时间ts显著的变化。

  46. 当ζ=0.68(5%误差带)或ζ=0.76(2%误差带),调节时间ts最短。所以通常的控制系统都设计成欠阻尼的。当ζ=0.68(5%误差带)或ζ=0.76(2%误差带),调节时间ts最短。所以通常的控制系统都设计成欠阻尼的。 曲线的不连续性,是由于ζ值的微小变化可引起调节时间显著变化而造成的。 Approximated calculation:常用阻尼正弦振荡的包络线衰减到误差带之内所需时间来确定ts。 当ζ<=0.8时,常把 这一项 去掉。写成 即

  47. 两边取对数,得: 可近似表示为: Note: 在设计系统时,ζ通常由要求的最大超调量决定,而调节时间则由无阻尼振荡频率wn来决定。 调节时间与闭环极点实部数值成反比,即极点离虚轴越远,调节时间越短。

  48. Oscillation times N Definition of N:在调节时间内,响应曲线穿越其稳态值次数的一半。 Td为阻尼振荡的周期。 5 振荡次数

  49. 标准二阶系统的性能指标 指标之间有矛盾,响应速度和阻尼程度不能同时达到最好结果 如何选取ζ和wn来满足系统设计要求: (1) 当wn一定,要减小tr和tp,必须减少ζ值,要减少ts则应增大ζwn值,而且z值有一定范围,不能过大。 (2) 增大wn,能使tr,tp和ts都减少。 (3) 最大超调量σ%只由ζ决定, ζ越小, σ%越大。所以,一般根据σ%的要求选择ζ值,在实际系统中, ζ值一般在0.5~0.8之间.