Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control


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1 Chapter 2 Classical Control System Design
2 Overview Ch Classical control system design Introduction Introduction Steadystate Steadystate errors errors Type Type k k systems systems Integral Integral control control Frequency Frequency response response plots plots Bode Bode plots plots Classical Classical design design techniques techniques Classical Classical design design specifications specifications Lead, Lead, lag, lag, leadlag leadlag compensation compensation GuilleminTruxal GuilleminTruxal method method Quantitative Quantitative Feedback Feedback Theory Theory Root Root locus locus Nyquist Nyquist plots plots M M and and Ncircles Ncircles Nichols Nichols plots plots
3 Steadystate errors1 r + F C P y Tracking behavior: Assume n t rt () = 1() t n! rs ˆ( ) = 1 n 1 s + Response ys $( ) Ls ( ) = ( )$( ) Ls ( ) Fsrs H( s) Tracking error ε ˆ() s = rˆ() s yˆ() s = [1 H()]() s rˆ s
4 Steadystate errors2 Steadystate tracking error ( n) ε = lim ε( t) = lim sεˆ ( s) = lim t s 0 s 0 If F(s)=1 (no prefilter) then 1 1 H( s) = 1 + Ls ( ) 1 H( s) s n ( n) ε = 1 lim 0 n [1 + ( )] s s L s
5 Type k system A feedback system is of type k if Then Lo () s Ls () =, Lo (0) 0 k s ( n) ε = 1 L s lim s 0 n [1 + ( )] s 0 for 0 n< k n s = lim = 1/ Lo (0) for n k s 0 k = s + Lo () s for n> k k
6 Steadystate errors3
7 Integral control1 Integral control: Design the closedloop system such that Type k control: Ls () = L o k s () s Ls () = Lo () s s Results in good steadystate behavior Also: k 1 s k Ss () = = = O( s ) for s Ls ( ) k s + L () s o
8 Integral control2 Type k control: Hence if k Ss () = O( s ) for s 0 n t 1 vt () = 1(), t vs ˆ( ) = n! n s + 1 then the steadystate error is zero if n < k (rejection) k = 1: Integral control: Rejection of constant disturbances k = 2: Type2 control: Rejection of ramp disturbances Etc.
9 Integral control3 Integral control: Lo () s Ls () = = PsCs () () k s The loop has integrating action of order k Natural integrating action is present if the plant transfer function has one or several poles at 0 If no natural integrating action exists then the compensator needs to provide it
10 Integral control4 Pure integral control: Cs () = 1 st i PI control: Cs () = g 1+ 1 st i PID control: Cs () = g std st i ZieglerNichols tuning rules
11 Internal model principle Asymptotic tracking if model of disturbance is included in the compensator Francis, D.A. and Wonham, W.M., (1975) The internal model principle for linear multivariable regulators, Applied Mathematics and Optimization, vol 2, pp
12 Frequency response plots Bode plots Nichols plots Nyquist plots
13 Bode plots1 Bode plot: doubly logarithmic plot of L(jω) versus ω semi logarithmic plot of arg L(jω) versus ω L( jω ) = 2 ωo o o j + o ( jω ) 2 ζ ω ( ω) ω
14 Bode plots2 Helpful technique: By construction of the asymptotic Bode plots of elementary first and secondorder factors of the form The shape of the Bode plot of ( jω z1)( jω z2) L( jω zm ) L( jω ) = k ( j ω p )( j ω p ) L( j ω p ) o o jω + α and ( jω) + 2 ζ ω ( jω) + ω may be sketched m
15 Nyquist plots Nyquist plot: Locus of L(jω) in the complex plane with ω as parameter Contains less information than the Bode plot if ω is not marked along the locus L( jω ) = 2 ωo o o j + o ( jω ) 2 ζ ω ( ω) ω
16 M and Ncircles1 r + L y Closedloop transfer function: H L = = T 1 + L Mcircle: Locus of points z in the complex plane where z = M 1+ z Ncircle: Locus of points z in the complex plane where arg z 1+ z = N
17 M and Ncircles2
18 Nichols plots Nichols plot: Locus of L(jω) with ω as parameter in the log magnitude versus argument plane 2 ωo o o j + o L( jω ) = ( jω) 2 ζ ω ( ω) ω Nichols chart: Nichols plot with M and Nloci included
19 Classical design specifications Time Rise time, delay time, overshoot, settling time, steadystate error of the response to step reference and disturbance inputs; error constants domain domain Frequency Bandwidth, resonance peak, rollon and rolloff of the closedloop frequency response and sensitivity functions; stability margins
20 Classical design techniques Lead, lag, and laglead compensation (loopshaping) (Root locus approach) (GuilleminTruxal design procedure) Quantitative feedback theory QFT (robust loopshaping)
21 Classical design techniques Rules for loopshaping Change openloop L(s) to achieve certain closedloop specs first modify phase then correct gain
22 Lead compensation Lead compensation: Add extra phase in the crossover region to improve the stability margins Typical compensator: Phaseadvance network 1+ jωt C( jω) = α, 0< α < 1 1+ jωα T
23 Lead/lag compensator C( jω) = α 1+ jωt 1+ jωα T
24 Lag compensation Lag compensation: Increase the low frequency gain without affecting the phase in the crossover region Example: PIcontrol: C( jω ) = k 1+ jωt jωt
25 Leadlag compensation Leadlag compensation: Joint use of lag compensation at low frequencies phase lead compensation at crossover Lead, lag, and leadlag compensation are always used in combination with gain adjustment
26 Notch compensation (inverse) Notch filters: suppression of parasitic dynamics additional gain at specific frequencies Special form of general second order filter
27 Notch compensation H = u ε = s ω s ω β + 2β 1 2 s ω s ω Notch filter :ω 1 = ω 2
28 Notch compensation ampl. β β 1 2 fase 0
29 Root locus method1 Important stage of many designs: Fine tuning of gain compensator pole and zero locations Helpful approach: the root locus method (use rltool!)
30 Root locus method2 Ls () N() s ( s z1)( s z2) L( s zm ) = = k D() s ( s p )( s p ) L( s p ) 1 2 n L Closedloop characteristic polynomial χ () s = D() s + N() s = ( s p )( s p ) L( s p ) + k( s z )( s z ) L( s z ) 1 2 n 1 2 Root locus method: Determine the loci of the roots of χ as the gain k varies m
31 Root locus method3 χ () s = ( s p )( s p ) L( s p ) + k( s z )( s z ) L( s z ) 1 2 n 1 2 Rules: For k = 0 the roots are the openloop poles p i For k a number m of the roots approach the openloop zeros z i. The remaining roots approach The directions of the asymptotes of those roots that approach are given by the angles 2i + 1, i 0,1,, n m 1 n m π = L m
32 Root locus method4 The asymptotes intersect on the real axis in the point (sum of openloop poles) (sum of openloop zeros) n m Those sections of the real axis located to the left of an odd total number of openloop poles and zeros on this axis belong to a locus The loci are symmetric with respect to the real axis...
33 Root locus method5 Ls () = k ss ( + 2) Ls () = ks ( + 2) ss ( + 1) Ls () = k ss ( + 1)( s+ 2)
34 GuilleminTruxal method1 r + C P y Closedloop transfer function: PC H = 1 + PC Procedure: Specify H Solve the compensator from C 1 H = P 1 H
35 GuilleminTruxal method2 Example: Choose H() s = m m 1 ams + am 1s + L+ a0 n n 1 m m 1 + n 1 + L+ m + m 1 + L+ 0 s a s a s a s a This guarantees the system to be of type m + 1 How to choose the denominator polynomial? Wellknown options: Butterworth polynomials Optimal ITAE polynomials
36 Butterworth and ITAE polynomials Butterworth polynomials Choose the n lefthalf plane poles on the unit circle so that together with their righthalf plane mirror images they are uniformly distributed along the unit circle ITAE polynomials Place the poles so that () tet dt 0 is minimal, where e is the tracking error for a step input
37 Butterworth and ITAE m = 0
38 GuilleminTruxal method3 Disadvantages of the method: Difficult to translate the specs into an unambiguous choice of H. Often experimentation with other design methods is needed to establish what may be achieved. In any case preparatory analysis is required to determine the order of the compensator and to make sure that it is proper The method often results in undesired polezero cancellation between the plant and the compensator
39 Quantitative feedback theory QFT1 Ingredients of QFT: For a number of selected frequencies, represent the uncertainty regions of the plant frequency response in the Nichols chart Specify tolerance bounds on the magnitude of T Shape the loop gain so that the tolerance bounds are never violated
40 QFT2 Example: Plant Ps () = s 2 g (1 + sθ ) Nominal parameter values: g = 1, θ = 0 Parameter uncertainties: 0.5 g 2, 0 θ 0.2 Tentative compensator: k + std Cs () =, k= 1, Td = 1.414, To = st o
41 QFT3 Responses of the nominal design Specs on T Frequency [rad/s] Tolerance band [db]
42 Uncertainty regions Uncertainty regions for the nominal design The specs are not satisfied Additional requirement: The critical area may not be entered
43 QFT4 Design method: Manipulate the compensator frequency reponse so that the loop gain satisfies the tolerance bounds avoids the critical region Preparatory step 1: For each selected frequency, determine the performance boundary Preparatory step 2: For each selectedfrequency, determine the robustness boundary
44 Performance and robustness boundaries Nominal plant frequency response Robustness boundaries Performance boundaries
45 QFT5 Design step: Modify the loop gain such that for each selected frequency the corresponding point on the loop gain plot lies above and to the right of the corresponding boundary For the case at hand this may be accomplished by a lead compensator of the form 1+ st Cs () = 1 + st Step 1: Set T 2 = 0, vary T 1 Step 2: Keep T 1 fixed, vary T 2 1 2
46 QFT6 Eventual design: T 1 = 3 T 2 = 0.02
47 QFT7 Responses of the redesigned system
48 Prefilter design1 2½degreeoffreedom configuration Closedloop transfer function H = NF D cl F o r F o e C o F X Y X + + u P z For the present case: Dcl ( s) = 0.02 ( s ) ( s )( s ) N() s = 1
49 Prefilter design2 Use the polynomial F to cancel the (slow) pole at , and let 2 ωo Fo ( s) =, ωo = 1, ζo = s + 2ζ ω s+ ω 2 2 Perturbed responses o o o
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