第一节系统的稳定性精选文档.ppt

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1、第一节系统的稳定性本讲稿第一页，共四十四页稳定性和代数稳定判据(Stability of the system and the algebra criteria)典型输入作用和时域性能指标(Typical input and time performance index)一阶系统的瞬态响应(Transient response of one order dynamical system)二阶系统的瞬态响应(Transient response of two order dynamical system)稳态误差分析(Steady error analyse)主要内容(Main issues)本

2、讲稿第二页，共四十四页第一节 系统的稳定性和 代数稳定判据Section 1 Stability of the control system and the its algebra evaluation criteria本讲稿第三页，共四十四页1.稳定的基本概念和线性系统稳定的充要条件(Basic concept of system stability and its the sufficient,necessary condition of the linear control system)1)Stabilizing of control system is the most import

3、ant condition for system to run properly.2)In fact,the real system is always affected by the outside or inside disturbances,such as load varying,energy wave,system parameter changing,environment changing etc.3)If the system is unstable,system will departure the initial balance state under any small

4、wave,and will disperse with the time going.4)To analyze the system stability and to make out the plan to ensure the system stable is the basic goal of control theory.稳定的充要条件和属性本讲稿第四页，共四十四页q 稳定的基本概念(Basic concept of stability)：If the system is in the state of balance,it will depart the state under th

5、e effect of outside exciting.when the outside exciting is disappeared,the system will return to the original state as it runs for so long time.The system is stable,or the system is off stability.Or else the system is unstable,or the system is of instability.本讲稿第五页，共四十四页Consider the follow differenti

6、al equation：+polynomial relative with initial value稳定的充要条件和属性Do Laplace transform：where：x(t)input y(t)output;is constants.本讲稿第六页，共四十四页1.The first item is zero state solution,and is related with the response excited by the input.2.The second item is zero input solution,and is related with the respons

7、e excited by the initial value.本讲稿第七页，共四十四页q necessary and sufficient condition of linear system stabilizing is:all the system characteristic roots are of negative real part(Eigenvalue),or all the system characteristic roots are lain on the left half plane of s complex plane.稳定的充要条件和属性本讲稿第八页，共四十四页充要

8、条件说明 if there is a positive real root for a system,it means the system response is dispersive.if there is a pair of positive real part complex root for a system,it means the system response is period dispersive oscillation.both cases are unstable.if there is a zero root for a system,it means the sys

9、tem response is random balance state.if there is a pair of imaginary roots for a system,it means the system response is oscillating state in a constant size.Stable zoneNonstable zoneCriticalStable zoneS plane本讲稿第九页，共四十四页For the one order system,if and only if are positive,the system is stable.If and

10、 only if are positive,the system is stable.For 3 or more order system,it is more difficult to get the roots of a algebra equation.How can we do?充要条件说明Notice：System stability is a quality of linear system,and is just related with the system structure and parameter,but not related with the input signa

11、l.not related with the initial condition.System stability is just related with the polar,not related with the zero.For the 2 order system本讲稿第十页，共四十四页2.Routh-Hurwitz criterion Considering the characteristic equation of the linear system 劳斯判据1).Routh criterion：The necessary and sufficient condition of

12、 the system is as follow:b.All the elements which are lain on the first column of the Routh array,and are composed of the coefficients of the characteristic equation should be positive.a.All the polynomial coefficients of the characteristic equation should be positive;本讲稿第十一页，共四十四页The first two line

13、 elements of the Routh array consist of the coefficients of characteristic equation.The first row elements are composed of the coefficients an,an-2,an-4,.;The second row elements are composed of the coefficients an-1,an-3,an-5,.How to construct the Routh table?本讲稿第十二页，共四十四页劳斯判据The rules to calculate

14、 the 3th row elements is as follow：本讲稿第十三页，共四十四页劳斯判据The rules to calculate the 4th row elements is as follow：本讲稿第十四页，共四十四页According to the above similar method,the remain elements can also be lead.The rules to calculate the 5th row elements is as follow：本讲稿第十五页，共四十四页劳斯判据例子example considering the sys

15、tem which characteristic equation is:To write down the Routh array as right position According to the necessary and sufficient condition of a stable system,we can get：andTry to determine the system stability.product of 2 inner coefficients minus product of 2 out coefficients is positive.本讲稿第十六页，共四十四

16、页2)Discussion of the special condition of the Routh array and some conclusionb.The system is unstable if all the elements in the first column of Routh array are not zero but not all are positive.劳斯判据特殊情况a.It will not affect the system stability to multiple or divide all elements in a row of the Rout

17、h array with a positive number;c.It also indicate that there are some characteristic roots in the right half plan of complex number plane s.d.The number of the unstable roots is equal to the changed sign number of elements in the first rank of Routh array.本讲稿第十七页，共四十四页example Assume that the system

18、characteristic equation is:-1 3 0（2）1 0 0（）There is 2 sign changes in the first column.Try to find the number of unstable root of the system.Discuss:To list the Routh array;There is a negative number in the first column.The system is unstable.2 unstable characteristic roots are in the right half s p

19、lane.本讲稿第十八页，共四十四页劳斯判据特殊情况 e.If the first element is zero but the others in one line of the Routh table are not all zero,A new method should be considered.Solution：to substitute the element 0 with a very small positive number .At last counting the sign-changed number.After then to calculate the othe

20、r elements on the line or below the line.本讲稿第十九页，共四十四页example Considering the characteristic equation:Let ,then There is 2 sign changes that means 2 unstable roots in the right half plane of complex number plane s.to analyse：Try to find the number of unstable root of the system.Discuss:to list the R

21、outh array;Clearly,there is a negative number in first column.The system is unstable.本讲稿第二十页，共四十四页f.All the numbers in a Routh array row are zero.It means that there is a pair of characteristic roots which are equal in size and opposite in sign.劳斯判据特殊情况There are 3 cases as:a pair of real roots which

22、 are equal in size and opposite in sign;or a pair of conjugate imaginary roots;or 2 pair of conjugate complex number roots which are symmetric to the imaginary axis.本讲稿第二十一页，共四十四页exampleSolution：a.to construct an assistant algebra equation of complex number variable s according to the coefficients o

23、f the last row in which the coefficients are nonzero.b.To differentiate the assistant equation and get the new equationc.To substitute the coefficients of the zero row with the coefficients of the new equation.Notice:the assistant equation must be even order.本讲稿第二十二页，共四十四页example to discuss the stab

24、ility of the below system.1 6 81 6 81 3 03 8 0劳斯判据特殊情况 to simplify it;Analyse:to list the Routh array;to build the assistant equation;to differentiate the above equation to get a new one;to substitute the coefficients with the new coefficients from the simplified equation.to continue the list the Ro

25、uth array to determine the unstable roots.本讲稿第二十三页，共四十四页It seems that the system is stable because the elements are bigger than or equal to zero.Clearly the system is critical stable that means unstable in an engineering meaning.1 6 81 6 81 3 03 8 0To build an assistant equation as below and to solv

26、e it,we may get:本讲稿第二十四页，共四十四页 Each of the host diagonal elements is the coefficients of characteristic polynomial from the second to the last .Problem 1.How to construct the Hurwitz array?Each element of each row below the host diagonal is some coefficients according to subscript increasing.Each el

27、ement of each row above the host diagonal is some coefficients according to subscript decreasing.All the element is 0 when the subscript is bigger than n or smaller than 0.subscript decreasingsubscript increasing本讲稿第二十六页，共四十四页Problem 2.How to construct the host sub-determinant?本讲稿第二十七页，共四十四页3.Applic

28、ation of Routh Hurwitz criterion1)To determine the system stabilityexample if the system characteristic equation is：,try to determine the system stability.Analyse:To list the Routh array as below：2 unstable roots are in the right half plane.The system is unstable.There is 2 sign changes本讲稿第三十页，共四十四页

29、2)To analyze the influence of system parameter changingexample the system block diagram is given as below,try to determine the critical amplifying coefficient.Solution closed loop transfer function is：The characteristic equation is：An important effect of the Routh-Hurwitz criterion is to analyze the

30、 influence of some system parameters varying such as the open-loop system amplifying coefficient K.We can use the criterion to determine the maximum critical amplifying coefficient.本讲稿第三十二页，共四十四页To write out the Routh array as below：According to the necessary and sufficient condition:all the coeffic

31、ients must be bigger than 0.the elements lain on the first column of the Routh array should be positive.Then we may get:The critical amplifying coefficient is .The characteristic equation is：本讲稿第三十三页，共四十四页3)To determine the relative system stability (stabilization abundance)As we know,we can use the

32、 Routh-Hurwitz criterion to determine whether a control system is stable or unstable.It is a absolute stability.if we want to know the relative stability of a control system,or how can it be determined?Usually,the distance between the characteristic root p of the maximum real part and the imaginary

33、axis is used virtually to express the system stabilization abundance.Clearly,if p is lain on the imaginary axis,that means the system stabilization abundance is 0.How do we determine the system stabilization abundance?本讲稿第三十四页，共四十四页To draw a vertical line on the complex plane s which is parallel to

34、the imaginary axis,and if all the characteristic roots are on the left of the line,the system is called of stabilization abundance.The bigger the is,the more stable the system is.Problem:How to find the in a control system?Let ,and substitute the complex variable s with in the characteristic equatio

35、n,then lead a new characteristic equation with a new complex variable z.Use Routh-Hurwitz criterion to analyze the system stability according to the new characteristic equation.If the new system is stable,the original system is called of stabilization abundance.本讲稿第三十五页，共四十四页example a system charact

36、eristic equation is ,There is a pair of imaginary roots.The new system is critical stable.And the original system is of 1 abundance stability.How about the relative system stability?solution clearly,And The system is stable.Let ,substitute the s with z-1,the new equation is:or本讲稿第三十六页，共四十四页Usually,t

37、he real part of the cogent complex root represents the attenuation speed of system response,whereas the imaginary part of the cogent complex root represents the oscillating of system response.is the angle between the polar and the negative real axis.The smaller the angle is,the better the system qua

38、lity is.Another form to discuss the relative stability,the relative stability is worst.本讲稿第三十七页，共四十四页3.Essential unstable system and the plan to improve1)What is the essential unstable system?It is the system whose performance can not be improved just by adjusting the system parameters.结构不稳定系统及其改进措施

39、-杠杆和放大器的传递函数执行电机的传递函数进水阀门的传递函数控制对象水箱的传递函数2)Example:liquid height control system本讲稿第三十八页，共四十四页结构不稳定系统及其改进措施Closed loop transfer function：let：Characteristic equation：or：clearly：Routh array：No matter how to change the parameter K and T,can the system not be stable?What should we do to make the system t

40、o be stable?The system is unstable.本讲稿第三十九页，共四十四页结构不稳定系统及其改进措施cause:There are 2 integral components in the forward access.The numerator of the closed loop transfer function is a constant,The closed-loop characteristic equation lack of some item.-2 methods:To change the integral component;To introduc

41、e a open loop zero.本讲稿第四十页，共四十四页结构不稳定系统及其改进措施 to change the integral property feedback is used in the component.The advantage is to improve the system stability,but the disadvantage is to cut down the steady accuracy.本讲稿第四十一页，共四十四页结构不稳定系统及其改进措施 introduce the open loop zero speed feedback proportiona

42、l+differential control本讲稿第四十二页，共四十四页结构不稳定系统及其改进措施Closed loop transfer function：Characteristic equation：Clearly：Routh array：Necessary and sufficient condition：ie.ie.To introduce the proportional+differential control,is to increase the lack coefficient in the characteristic equation.If the parameter i

43、s properly adjusted(),the system is stable.本讲稿第四十三页，共四十四页summaryqNecessary and sufficient condition of linear systemqRouth-Hurwitz criterionqThe application of Routh-Hurwitz criterion to determine the system stability to analyze the effect of system parameter changingto analyze the relative system stabilityqEssential unstable system and the plan to improveassignments：3-1(1),(3),3-2(1),(3),(5),3-3本讲稿第四十四页，共四十四页