1、Ch6 The Stability of Linear Feedback Systems,The concept of stability The Routh-Hurwitz stability criterion The relative stability,1,6.1 The concept of stability,A stable system is a dynamic system with a bounded output to a bounded input (BIBO).,The issue of ensuring the stability of a closed-loop

2、feedback system is central to control system design. An unstable closed-loop system is generally of no practical value.,absolute stability, relative stability,2,Absolute stability: We can say that a closed-loop feedback system is either stable or it is not stable. This type of stable/not stable char

3、acterization is referred to as absolute stability.,Relative stability: Given that a closed-loop system is stable, we can further characterize the degree of stability. This is referred to as relative stability.,3,4,5,6.2 The Routh-Hurwitz stability criterion,6,where,7,A necessary and sufficient condi

4、tion for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.,8,A necessary condition: All the coefficients of the polynomial must have the same sign and be nonzero if all the roots are in left-hand plane (LHP).,The characteristic equation is

5、 written as,9,Hurwitz and Routh published independently a method of investigating the stability of a linear system. The number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array.,Routh-Hurwitz stability criterion,10,CASE1 No ele

6、ment in the first column is zero.,CASE2 Zero in the first column while some other elements of row containing a zero in the first column are nonzero.,CASE3 Zeros in the first column,and other elements of the row containing the zero are also zero.,11,Consider the characteristic polynomial,The Routh ar

7、ray is,12,Case 3,Consider the characteristic polynomial,The Routh array is,The auxiliary polynomial,13,14,Design example： welding control,15,6.3 The relative stability,The relative stability of a system can be defined as the property that is measured by the relative real part of each root or pair of

8、 roots. Axis shift and examples,16,17,Consider control system,Determine the range of K satisfying the stability and all poles -1.,18,The Routh array is,19,The Routh array is,Let , we obtain,20,Design example： Tracked vehicle turning control,21,22,Summary,In this chapter, we have considered the conce

9、pt of the stability of a feedback control system. A definition of a stable system in terms of a bounded system response was outlined and related to the location of the poles of the system transfer function in the s-plane. The Routh-Hurwitz stability criterion was introduced, and several examples wer

10、e considered. The relative stability of a feedback control system was transfer function in the s-plane.,23,Assignment,E6.1 E6.4 E6.5 E6.8,24,Ch7 The Root Locus Method,Main content:,The Root Locus Concept The Root Locus Procedure Parameter Design by the Root Locus method Sensitivity and the Root Locu

11、s Three-term(PID) Controllers The Root Locus using MATLAB,25,7.1 The Root Locus Concept,The response of a closed-loop feedback system can be adjusted to achieve the desired performance by judicious selection of one or more parameters. The locus of roots in the s-plane can be determined by a graphica

12、l method. The root locus method was introduced by Evans in 1984 and has been developed and utilized extensively in control engineering practice.,The root locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter is changed.,26,Closed-loop control s

13、ystem with a variable parameter K,27,unity feedback control system, the gain K is a variable parameter,28,29,30,31,7.2 The Root Locus Procedure,Step1: Write the characteristic equation as 1+F(s)=0 And rearrange the equation. If necessary, so that the polynomial in the form poles and zeros follows: 1

14、+KP(s)=0 Step2 Factor P(s), if necessary, and write the polynomial in the form of poles and zeros as follows:,32,Step3 Locate the poles and zeros on the s-plane with selected symbols. The locus of the roots of the characteristic equation 1+KP(s)=0 begins at the poles of p(s) and ends at the zeros of

15、 p(s) as K increases from 0 to infinity. with n poles and M zeros and nM. Step4 The root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros. Step5 Determine the number of separate loci, SL, the number of separate loci is equal to the numb

16、er of poles.,33,Example7.1 Second-order system,34,Step6 The root loci must be symmetrical with respect to the horizontal real axis with angles. Step7 The root loci proceed to the zeros at infinity along asymptotes centered at and with angles . These linear asymptotes are centered at a point on the r

17、eal axis given by The angle of the asymptotes with respect to the real axis is,35,Example7.2 Fourth-order system,36,37,Step8 Determine the point at which the locus crosses the imaginary axis (if it does so), using the Routh-Hurwitz criterion. The actual point at which the root locus crosses the imag

18、inary axis is readily evaluated by utilizing the Routh-Hurwitz Criterion. Step9 Determine the breakaway point on the real axis (if any). Let or Step10 The angle of locus departure from a pole is The angle of locus arrival from a zero is,38,39,40,41,42,Step11 Determine the root locations that satisfy

19、 the phase criterion at root . The phase criterion is q=1,2. Step12 Determine the parameter value at a specific root using the magnitude requirement. The magnitude requirement at is,43,Example7.4 Fourth-order system,44,45,7.3 Parameter Design by the Root Locus method,This method of parameter design

20、uses the root locus approach to select the values of the parameters,The effect of the coefficient a1 may be ascertained from the root locus equation,46,47,48,49,50,51,7.4 Sensitivity and the Root Locus,The root sensitivity of a system T(s) can be defined as,the sensitivity of a system performance to

21、 specific parameter changes,we have,52,53,54,55,56,7.5 Three-term(PID) Controllers,The controller provides a proportional term, an integration term, and a derivative term,57,58,59,60,61,62,Summary,In this chapter, we have investigated the movement of the characteristic roots on the s-plane as the sy

22、stem parameters are varied by utilizing the root locus method. The root locus method, a graphical technique, can be used to obtain an approximate sketch in order to analyze the initial design of a system and determine suitable alterations of the system structure and the parameter values. Furthermore

23、, we extended the root locus method for the design of several parameters for a closed-loop control system. Then the sensitivity of the characteristic roots was investigated for undesired parameter variations by defining a root sensitivity measure.,63,Assignment,E7.4 E7.8,64,Ch8 Frequency Response Me

24、thods,Basic concept of frequency response Frequency response plots Drawing the Bode diagram Performance specification in the frequency domain,65,8.1 Basic concept of frequency response,The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input sign

25、al. The resulting output signal for a linear system, is also a sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle.,66,Let input,The Laplace transformation,The output,undetermined coefficient,67,68,is complex vector,69,Frequency Characteristics,Transf

26、er function and Laplace transform Frequency characteristics and Fourier transform,70,Frequency characteristic, Transfer function and differential equation are equivalent in representation of system.,71,Frequency characteristic and Transfer function,72,Computation of frequency response,73,8.2 Frequen

27、cy response plots,Polar plot Bode diagram Nichols chart Frequency response plots of typical elements,74,75,frequency response of an RC filter,76,77,78,The primary advantage of the logarithmic plot is the conversion of multiplicative factor into additive by virtue of the definition of logarithmic gai

28、n,79,Bode diagram of an RC filter,80,81,Nichols chart,0o,180o,-180o,w,0,-20dB,20dB,82,Frequency response plots of typical elements,Gain Pole at origin Zero at origin,83,Pole on the real axis (jwT+1) Zero on the real axis (jwT+1) Two complex poles Two complex zeros,84,85,86,87,88,Bode diagram of a twin-T network,89,90,8.3 Drawing the Bode diagram,91,92,93,94,95,96,Drawing Bode diagram : (1) (2) Draw the asymptotic approximation of L() in the low frequency range； (3) Change the slope at the break frequency; (4) This appro