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1、第一节系统的稳定性第1页，本讲稿共44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)第2
2、页，本讲稿共44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第3页，本讲稿共44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 important
3、 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 wav
4、e,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第4页，本讲稿共44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 the ef
5、fect 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.第5页，本讲稿共44页Consider the follow differential eq
6、uation：+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.第6页，本讲稿共44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 response exci
7、ted by the initial value.第7页，本讲稿共44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第8页，本讲稿共44页充要条件说明 if
8、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 system resp
9、onse 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第9页，本讲稿共44页For the one order system,if and only if are positive,the system is stable.If and only if
10、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 signal.not rel
11、ated with the initial condition.System stability is just related with the polar,not related with the zero.For the 2 order system第10页，本讲稿共44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 the syst
12、em 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;第11页，本讲稿共44页The first two line elements
13、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?第12页，本讲稿共44页劳斯判据The rules to calculate the 3th ro
14、w elements is as follow：第13页，本讲稿共44页劳斯判据The rules to calculate the 4th row elements is as follow：第14页，本讲稿共44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：第15页，本讲稿共44页劳斯判据例子example considering the system which char
15、acteristic 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.第16页，本讲稿共44页2)Discussion o
16、f 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 Routh array with a
17、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.第17页，本讲稿共44页example Assume that the system characteristic e
18、quation 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 plane.第18页，本讲稿共44
19、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 other elements on the
20、 line or below the line.第19页，本讲稿共44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 Routh array;Clearly
21、,there is a negative number in first column.The system is unstable.第20页，本讲稿共44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 are equal in size
22、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.第21页，本讲稿共44页exampleSolution：a.to construct an assistant algebra equation of complex number variable s according to the coefficients of the last row in whi
23、ch 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.第22页，本讲稿共44页example to discuss the stability of the below syst
24、em.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 Routh array to determine
25、the unstable roots.第23页，本讲稿共44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 solve it,we may get:第24页，本讲稿共
26、44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 element of each row above the
27、 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第26页，本讲稿共44页Problem 2.How to construct the host sub-determinant?第27页，本讲稿共44页3.Application of Routh Hurwitz criterio
28、n1)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第30页，本讲稿共44页2)To analyze the influence of sy
29、stem 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 influence of some system parame
30、ters varying such as the open-loop system amplifying coefficient K.We can use the criterion to determine the maximum critical amplifying coefficient.第32页，本讲稿共44页To write out the Routh array as below：According to the necessary and sufficient condition:all the coefficients must be bigger than 0.the el
31、ements 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：第33页，本讲稿共44页3)To determine the relative system stability (stabilization abundance)As we know,we can use the Routh-Hurwitz criterion to determin
32、e 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 axis is used virtually to express th
33、e 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?第34页，本讲稿共44页To draw a vertical line on the complex plane s which is parallel to the imaginary axis,and if all the char
34、acteristic 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 equation,then lead a new characteristic equat
35、ion 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.第35页，本讲稿共44页example a system characteristic equation is ,There is a pair of
36、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第36页，本讲稿共44页Usually,the real part of the cogent complex root re
37、presents 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 quality is.Another form to discuss the relati
38、ve stability,the relative stability is worst.第37页，本讲稿共44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-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)E
39、xample:liquid height control system第38页，本讲稿共44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 to be stable?The system is unstable.第39页，本讲稿共44
40、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 introduce a open loop zero.第40页，本讲稿共44页结构不稳定系统及其改进措施 to
41、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.第41页，本讲稿共44页结构不稳定系统及其改进措施 introduce the open loop zero speed feedback proportional+differential control第42页，本讲稿共44页结构不稳定系统及其改进措施Clos
42、ed 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 is properly adjusted(),the system is stable.第43页，本讲稿共4
43、4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第44页，本讲稿共44页