To measure the peaks of Fig. 100.6, one finds y()=1+(y--5n=0 (100.26) occurring at n n:odd(overshoot) n:even(undershoot) (100.27) Hence ymax=1+ exp (100.28) occurring at (100.29) With these parameters, one finds 1+0.75 1+1.18+1.4 Note that increasing 5 decreases the overshoot and decreases the settling time but increases ta and tr When 5=1, the system has a double pole at-O,, resulting in a critically damped response. This is the point when the response just changes from oscillatory to exponential in form. For a unit step input, the response is given by y(t)=1-em(1+ont)(5=1) (100.30) For the range 5>1, the system is overdamped due to two real system poles. For a unit step input, the (t)=1+ C1-c2(c1 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC To measure the peaks of Fig. 100.6, one finds (100.26) occurring at (100.27) Hence (100.28) occurring at (100.29) With these parameters, one finds and Note that increasing x decreases the % overshoot and decreases the settling time but increases td and tr . When x = 1, the system has a double pole at –wn, resulting in a critically damped response. This is the point when the response just changes from oscillatory to exponential in form. For a unit step input, the response is given by y(t) = 1 – e–wnt(1 + wnt) (x = 1) (100.30) For the range x > 1, the system is overdamped due to two real system poles. For a unit step input, the response is given by y t n n n peak exp – = +- . . .( ) ( ) , , - = - 1 1 1 0 1 1 2 px x t n n n n = - p w x 1 2 : odd (overshoot) even (undershoot): ymax = + exp - 1 1 2 –px x t n max = - p w x 1 2 t t t d n r n s n ª + ª + + ª 1 07 1 11 14 3 2 . . . x w x x w xw y t c cc e c e ct c t n n ( ) ( ) = + - - Ê Ë Á ˆ ¯ ˜ 1 > 11 1 1 1 21 2 1 2 w w x
(100.31) Finally, when 5=0, the response is purely sinusoidal. For a unit step, the response is given by y(t)=l-cos o,t (=0) (100.32) Defining Terms se: The response of a system when the input is an impulse function. Steady-state error: The difference between the desired reference signal and the actual signal in steady-state, e,when time approaches infinity. Steady-state response: That part of the response which remains as time approaches infini Transient response: That part of the response which vanishes as time approaches infinity. Tero-input response: That part of the response due to the initial condition only Zero-state response: That part of the response due to the input only. Related Topics 6. 1 Definitions and Properties.7.1 Introduction. 112.2 A Brief History of CACSD Further Inform J J. D'Azzo and C H. Harpis, Linear Control System Analysis and Design, New York: McGraw-Hill, 1981 R C. Dorf, Modern Control Systems, 5th ed, Reading, Mass. Addison-Wesley, 1989 M.E. El-Hawary, Control Systems Engineering, Reston, Va. Reston, 1984 G H. Hostetter, C J. Savant, Jr, and R. T. Stefani, Design of Feedback Control Systems, Philadelphia: Saunders B.C. Kuo, Automatic Control Systems, Englewood Cliffs, N J. Prentice-Hall, 1987. K Ogata, Modern Control Engineering, Englewood Cliffs, N J. Prentice-Hall, 1970 N K. Sinha, Control Systems, New York: Holt, 1986 100.3 Frequency Response Methods: Bode Diagram Approach Andrew P. Sage Our efforts in this section are concerned with analysis and design of linear control systems by frequency response methods Design generally involves trial-and-error repetition of analysis until a set of design specifications has een met. Thus, analysis methods are most useful in the design process, which is one phase of the systems engineering life cycle [Sage, 1992]. We will discuss one design method based on Bode diagrams. We will discuss the use of both simple series equalizers and composite equalizers as well as the use of minor-loop feedback systems design Figure 100.9 presents a flowchart of the frequency response method design process and indicates the role of analysis in linear systems control design. The flowchart of Fig. 100.9 is applicable to control system design methods in general. There are several iterative loops, generally calling for trial-and-error efforts, that omprise the suggested design process. An experienced designer will often be able, primarily due to successful prior experience, to select a system structure and generic components such that the design specifications can be met with no or perhaps a very few iterations through the iterative loop involving adjustment of equalizer or compensation parameters to best meet specifications If the parameter optimization, or parameter refinement such as to lead to maximum phase margin, approach ows t the specifications cannot be met, we are then assured that no equalizer of the specific form selected will pecifications. The next design step, if needed would consist of modification of the equalizer form or e 2000 by CRC Press LLC
© 2000 by CRC Press LLC (100.31) Finally, when x = 0, the response is purely sinusoidal. For a unit step, the response is given by y(t) = 1 – cos wnt (x = 0) (100.32) Defining Terms Impulse response: The response of a system when the input is an impulse function. Steady-state error: The difference between the desired reference signal and the actual signal in steady-state, i.e., when time approaches infinity. Steady-state response: That part of the response which remains as time approaches infinity. Transient response: That part of the response which vanishes as time approaches infinity. Zero-input response: That part of the response due to the initial condition only. Zero-state response: That part of the response due to the input only. Related Topics 6.1 Definitions and Properties • 7.1 Introduction • 112.2 A Brief History of CACSD Further Information J.J. D’Azzo and C.H. Harpis, Linear Control System Analysis and Design, New York: McGraw-Hill, 1981. R.C. Dorf, Modern Control Systems, 5th ed., Reading, Mass.: Addison-Wesley, 1989. M.E. El-Hawary, Control Systems Engineering, Reston, Va.: Reston, 1984. G.H. Hostetter, C. J. Savant, Jr., and R. T. Stefani, Design of Feedback Control Systems, Philadelphia: Saunders, 1989. B.C. Kuo, Automatic Control Systems, Englewood Cliffs, N.J.: Prentice-Hall, 1987. K. Ogata, Modern Control Engineering, Englewood Cliffs, N.J.: Prentice-Hall, 1970. N.K. Sinha, Control Systems, New York: Holt, 1986. 100.3 Frequency Response Methods: Bode Diagram Approach Andrew P. Sage Our efforts in this section are concerned with analysis and design of linear control systems by frequency response methods. Design generally involves trial-and-error repetition of analysis until a set of design specifications has been met. Thus, analysis methods are most useful in the design process, which is one phase of the systems engineering life cycle [Sage, 1992]. We will discuss one design method based on Bode diagrams.We will discuss the use of both simple series equalizers and composite equalizers as well as the use of minor-loop feedback in systems design. Figure 100.9 presents a flowchart of the frequency response method design process and indicates the key role of analysis in linear systems control design. The flowchart of Fig. 100.9 is applicable to control system design methods in general. There are several iterative loops, generally calling for trial-and-error efforts, that comprise the suggested design process. An experienced designer will often be able, primarily due to successful prior experience, to select a system structure and generic components such that the design specifications can be met with no or perhaps a very few iterations through the iterative loop involving adjustment of equalizer or compensation parameters to best meet specifications. If the parameter optimization, or parameter refinement such as to lead to maximum phase margin, approach shows the specifications cannot be met, we are then assured that no equalizer of the specific form selected will meet specifications. The next design step, if needed, would consist of modification of the equalizer form or c c 1 2 2 2 = -x x + - 1 = -x - x - 1
I Identify Requirements dentify Specifications Identify Design Performance Criteriah r Modify Design Performance Criteria Select Initial Design Structure and Para Evaluate Performance Criteria Feasible? Is Design Perfomance Acceptable? FIGURE 100.9 System design life cycle for frequency-response-based desig structure and repetition of the analysis process to determine equalizer parameter values to best meet specifications. If specifications still cannot be met, we will usually next modify generic fixed components used in the system. This iterative design and analysis process is again repeated. If no reasonable fixed components can be obtained meet specifications, then structural changes in the proposed system are next contemplated. If no structure can be found that allows satisfaction of specifications, either the client must be requested to relax the frequency response specifications or the project may be rejected as infeasible using present technology. As we might suspect, economics will play a dominant role in this design process. Changes made due to iteration in the inner loops of Fig. 100.9 normally involve little additional costs, whereas those made due to iterations in the outer en involve major cost changes. Frequency Response Analysis Using the Bode Diagram The steady-state response of a stable linear constant-coefficient system has particular significance, as we know from an elementary study of electrical networks and circuits and of dynamics we consider a stable linear system with input-output transfer functio h()=2(s) U(s) We assume a sinusoidal input u( t)=cos ot so that we have for the Laplace transform of the system output 2()=s(s) We expand this ratio of polynomials using the partial-fraction approach and obtain z(s)=F)+.+ s+10s-J0 In this expression, F(s) contains all the poles of H(s). All of these lie in the left half plane since the syst represented by H(s), is assumed to be stable. The coefficients a, and a, are easily determined as e 2000 by CRC Press LLC
© 2000 by CRC Press LLC structure and repetition of the analysis process to determine equalizer parameter values to best meet specifications. If specifications still cannot be met, we will usually next modify generic fixed components used in the system. This iterative design and analysis process is again repeated. If no reasonable fixed components can be obtained to meet specifications, then structural changes in the proposed system are next contemplated. If no structure can be found that allows satisfaction of specifications, either the client must be requested to relax the frequency response specifications or the project may be rejected as infeasible using present technology. As we might suspect, economics will play a dominant role in this design process. Changes made due to iteration in the inner loops of Fig. 100.9 normally involve little additional costs, whereas those made due to iterations in the outer loops will often involve major cost changes. Frequency Response Analysis Using the Bode Diagram The steady-state response of a stable linear constant-coefficient system has particular significance, as we know from an elementary study of electrical networks and circuits and of dynamics. We consider a stable linear system with input-output transfer function We assume a sinusoidal input u(t) = cos wt so that we have for the Laplace transform of the system output We expand this ratio of polynomials using the partial-fraction approach and obtain In this expression, F(s) contains all the poles of H(s). All of these lie in the left half plane since the system, represented by H(s), is assumed to be stable. The coefficients a1 and a2 are easily determined as FIGURE 100.9 System design life cycle for frequency-response-based design. H s Z s U s ( ) ( ) ( ) = Z s sH s s ( ) ( ) = + 2 2 w Zs Fs a s j a s j () () = + + + - 1 2 w w
We can represent the complex transfer function HGo) in either of two forms, HGo=B(O)+ jC(o) H(-jo)=B(o)-jC(o) The inverse Laplace transform of the system transfer function will result in a transient term due to the inverse transform of F(s), which will decay to zero as time progresses. A steady-state component will remain, and this is, from the inverse transform of the system equation, given b z(1)=a12e e combine several of these relations and obtain the result e/or +e")or z(t)=B(0 C(0 This result becomes, using the Euler identity, z(t)=b(o) cosat-C(o) sino =[B2(o)+C2(o)]2cos(0+阝) H(o) cos(ot+阝) where tan B(O)=C(o)/B(o) As we see from this last result there is a very direct relationship between the transfer function of a linear constant-coefficient system, the time response of a system to any known input, and the sinusoidal steady-state response of the system. We can always determine any of these if we are given any one of them. This is a very nportant result. This important conclusion justifies a design procedure for linear systems that is based only as it is possible to determine transient responses, or responses to any given dy-state sinusoidal responses, at least in theory. In practice, this might Bode diagram Design-Series equalizers In this subsection we consider three types of series equalization 1. Gain adjustment, normally attenuation by a constant at all frequencies 2. Increasing the phase lead, or reducing the phase lag, at the crossover frequency by use of a phase lead network 3. Attenuation of the gain at middle and high frequencies such that the crossover frequency will be decreased to a lower value where the phase lag is less, by use of a lag network The Euler identity is erot cos ot+ jsinot e 2000 by CRC Press LLC
© 2000 by CRC Press LLC We can represent the complex transfer function H(jw) in either of two forms, H(jw) = B(w) + jC(w) H(–jw) = B(w) – jC(w) The inverse Laplace transform of the system transfer function will result in a transient term due to the inverse transform of F(s), which will decay to zero as time progresses. A steady-state component will remain, and this is, from the inverse transform of the system equation, given by z(t) = a1e–jwt + a2 jwt We combine several of these relations and obtain the result This result becomes, using the Euler identity, 1 z(t) = B(w) coswt – C(w) sinwt = [B2(w) + C2(w)]1/2 cos(w + b) = *H(jw)* cos(wt + b) where tan b(w) = C(w)/B(w). As we see from this last result, there is a very direct relationship between the transfer function of a linear constant-coefficient system, the time response of a system to any known input, and the sinusoidal steady-state response of the system. We can always determine any of these if we are given any one of them. This is a very important result. This important conclusion justifies a design procedure for linear systems that is based only on sinusoidal steady-state response, as it is possible to determine transient responses, or responses to any given system input, from a knowledge of steady-state sinusoidal responses, at least in theory. In practice, this might be rather difficult computationally without some form of automated assistance. Bode Diagram Design-Series Equalizers In this subsection we consider three types of series equalization: 1. Gain adjustment, normally attenuation by a constant at all frequencies 2. Increasing the phase lead, or reducing the phase lag, at the crossover frequency by use of a phase lead network 3. Attenuation of the gain at middle and high frequencies such that the crossover frequency will be decreased to a lower value where the phase lag is less, by use of a lag network 1 The Euler identity is ejwt = cos wt + jsinwt. a H j a H j 1 2 2 2 = - = ( ) ( ) w w z t B e e C e e j j t j t j t j t ( ) = ( ) ( ) Ê + Ë Á ˆ ¯ ˜ - Ê - Ë Á ˆ ¯ ˜ - - w w w w w w 2 2
Fre FIGURE 100.10 Phase shift and gain curves for a simple lead network. In the subsection that follows this, we will first consider use of a composite or lag-lead network near crossover to attenuate gain only to reduce the crossover frequency to a value where the phase lag is less. Then we will consider more complex composite equalizers and state some general guidelines for Bode diagram design. Here, we will use Bode diagram frequency domain design techniques to develop a design procedure for each of three elementary types of series equalization Gain reduction Many linear control systems can be made sufficiently stable merely by reduction of the open-loop system gain to a sufficiently low value. This approach ignores all performance specifications, however, except that of phase margin(PM)and is, therefore, usually not a satisfactory approach. It is a very simple one, however, and serves to illustrate the approach to be taken in more complex cases. The following steps constitute an appropriate Bode diagram design procedure for compensat 1. Determine the required PM and the corresponding phase shift P=-I+ PM. 2. Determine the frequency @. at which the phase shift is such as to yield the phase shift at crossover 3. Adjust the gain such that the actual crossover frequency occurs at the value computed in step 2 Phase. Lead Cor In compensation using a phase-lead network, we increase the phase lead at the crossover frequency such that we meet a performance specification concerning phase shift. A phase-lead-compensating network transfer function is -+a)(+a) Figure 100 10 illustrates the gain versus frequency and phase versus frequency curves for a simple lead network with the transfer function of the foregoing equation. The maximum phase lead obtainable from a phase-lead network depends upon the ratio o2/o, that is used in designing the network. From the expression for the phase shift of the transfer function for this system, which is given by e 2000 by CRC Press LLC
© 2000 by CRC Press LLC In the subsection that follows this, we will first consider use of a composite orlag-lead network near crossover to attenuate gain only to reduce the crossover frequency to a value where the phase lag is less. Then we will consider more complex composite equalizers and state some general guidelines for Bode diagram design. Here, we will use Bode diagram frequency domain design techniques to develop a design procedure for each of three elementary types of series equalization. Gain Reduction Many linear control systems can be made sufficiently stable merely by reduction of the open-loop system gain to a sufficiently low value. This approach ignores all performance specifications, however, except that of phase margin (PM) and is, therefore, usually not a satisfactory approach. It is a very simple one, however, and serves to illustrate the approach to be taken in more complex cases. The following steps constitute an appropriate Bode diagram design procedure for compensation by gain adjustment: 1. Determine the required PM and the corresponding phase shift bc = –p + PM. 2. Determine the frequency wc at which the phase shift is such as to yield the phase shift at crossover required to give the desired PM. 3. Adjust the gain such that the actual crossover frequency occurs at the value computed in step 2. Phase-Lead Compensation In compensation using a phase-lead network, we increase the phase lead at the crossover frequency such that we meet a performance specification concerning phase shift. A phase-lead-compensating network transfer function is Figure 100.10 illustrates the gain versus frequency and phase versus frequency curves for a simple lead network with the transfer function of the foregoing equation. The maximum phase lead obtainable from a phase-lead network depends upon the ratio w2/w1 that is used in designing the network. From the expression for the phase shift of the transfer function for this system, which is given by FIGURE 100.10 Phase shift and gain curves for a simple lead network. G s s s c ( ) = + Ê Ë Á ˆ ¯ ˜ + Ê Ë Á ˆ ¯ ˜ 1 1 < 1 2 1 2 w w w w