3 Short review of ODEs4 Revisiting the Integral Controller5 Transfer functions6 Frequency Responses of Linear Systems7 Saturation and Antiwindup Strategies8 Effect of time delays9 More on ODEs10 Distributions11 DC Motors12 Robustness Margins13 Control of Second-Order System19 Jacobian Linearizations, equilibrium points20 Linear Systems and Time-Invariance21 Matrix Exponential22 Eigenvalues, eigenvectors, stability23 Jordan Form24 Linearized analysis of open-loop and clsoed-loop broom balancing25 Stabilization by State-Feedback26 State-Feedback with Integral Control27 Transfer functions, ODEs, Convolution and State Space Models: Stability28 Relation between Transfer function and Frequency Response29 Root Locus30 Nyquist Criterion: Preliminaries31 Nyquist Analysis: Examples13 Control of Second-Order SystemIn this section, we analyze PD and PID control of a plant typical in mechanicalpositioning systems. We also propose a possible design method. The nominalmodel for the plant isP (s) =As(s + p)where A and p are fixed parameters.13.1 PD controlFirst, consider PD control, specifically proportional control, with inner loop rate-feedback. This is shown below (its just the PID diagram, with the integral actionremoved)KPKDA1s+p1s- i - - i - - - i?- - -?¾6?¾¾6− −urydInner-loop Rate-Feedback×In terms of plant and controller parameters, the loop gain (at breaking pointmarked by ×) isL(s) =A(KDs + KP)s(s + p)In other words, from a stability point of view, the system is just unity-gain, negativefeedback around L.KPKDA1s+p1s- - i - - - i?- - -?¾6?¾¾6y126The closed-loop transfer function from R and D to Y isY (s) =AKPs2+ (AKD+ p)s + AKPR(s) +1s2+ (AKD+ p)s + AKPD(s)The characteristic equation isCE : s2+ (AKD+ p)s + AKPClearly, with two controller parameters, and a 2nd order closed-loop system, thepoles can be freely assigned. Using the (ξ, ωn) parametrization, we set the charac-teristic equation to bes2+ 2ξωns + ω2ngiving design equationsKP:=ω2nA, KD:=2ξωn− pAIn terms of the (ξ, ωn) parametrization, the loop gain and transfer functions areL(s) =(2ξωn− p)s + ω2ns(s + p)Y (s) =ω2ns2+ 2ξωn+ ω2nR(s) +1s2+ 2ξωn+ ω2nD(s) (66)Although this is a 2nd order system, and most quantities can be computed an-alytically, the formulae that arise are rather messy, and interpretation ends uprequiring plotting. Hence, we skip the analytic calculations, and simply numeri-cally compute and plot interesting properties for different values of ωn, p and ξ.Normalization is the key to displaying the data in a cohesive and minimal fashion.For now, take p = 0 (you should take the time to write a MatLab script filethat duplicates these results for arbitrary p). In this case, it is possible to writeeverything in terms of normalized frequency, all relative to ωn. This simultaneouslyleads to a normalization in time (recall homework 8). Hence frequency responsesare plotted G(jω) versusωωn, and time responses plotted y(t) versus ωnt. Weconsider a few typical values for ξ.127The plots below are:• Magnitude/Phase plots of Loop transfer function. These are normalizedin frequency, and show L(jω) versusωωn. From these, we can read off thecrossover frequencies and margins.xi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 10−210−110010110210−210−1100101102103104Normalized Frequency (w/wn)MagnitudeOpen−Loop Transfer Function, PDxi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 10−210−1100101102−180−170−160−150−140−130−120−110−100−90Normalized Frequency (w/wn)Phase (degrees)Open−Loop Transfer Function, PD128• Magnitude/Phase plots of closed-loop R → Y transfer function. These arenormalized in frequency, and show GR→Y(jω) versusωωn.xi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 10−210−110010110210−410−310−210−1100101Normalized Frequency (w/wn)MagnitudeR−>Y Frequency Response, PDxi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 10−210−1100101102−180−160−140−120−100−80−60−40−200Normalized Frequency (w/wn)Phase (degrees)R−>Y Frequency Response, PD129• Magnitude plot of closed-loop D → Y These are normalized in frequencyand magnitude,, and show ω2nGD→Y(jω) versusωωn.xi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 10−210−110010110210−410−310−210−1100101Normalized Frequency (w/wn)Normalized Magnitude, | wn^2 G |Normalized Disturbance Response, PD• Unit step d → y responses. These are normalized both in time, and inresponse. Hence the plot is ω2ny(t) versus ωnt.xi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 0 1 2 3 4 5 6 7 800.20.40.60.811.2Normalized Time (wn*t)Normalized Response, wn^2 yNormalized Disturbance Response, PD130• Unit step r → y responses. These are normalized in time, and show y(t)versus ωnt.xi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 0 1 2 3 4 5 6 7 800.20.40.60.811.2Normalized Time (wn*t)YR−>Y Step Response, PD• Magnitude plot of closed-loop R → E. These are normalized in frequency,and show GR→E(jω) versusωωn.xi = 0.5 xi = 0.707xi = 0.95 xi = 1.3 10−210−110010110210−210−1100101Normalized Frequency (w/wn)MagnitudeR−>E (Sensitivity), PDSome things to notice.131• The r → y response has the canonical 2nd order response we have come toknow and love.• The steady-state disturbance rejection properties are dependent on ωn. Asωnincreases, the effect of a disturbance d on the output y is decreased.Hence, in order to improve the disturbance rejection characteristics, we needto pick larger ωn.• Depending on ξ, the gain-crossover frequency is between about 1.3ωnand2.5ωn. So, using this controller architecture, the gain crossover frequencymust increase when the steady-state disturbance rejection is improved. Thephase margin varies between 53◦and 83◦.• There is no phase-crossover frequency, so as defined, the gain margin is infi-nite.• The closer that the complex frequency response remains to 1 (over a largefrequency range), the better the r → y response. The term “bandwidth” isoften used to mean the largest frequency ωBsuch that for all ω satisfying0 ≤ ω ≤ ωB,|1 − GR→Y(jω)| ≤ 0.3Be careful with the word “bandwidth.” Make sure whoever you are talkingto agrees on exactly what you both mean. Sometimes people use it to meanthe gain crossover frequency. Generally, the higher the bandwidth, the fasterthe response, and better the disturbance rejection. Of course, its hard toexplicitly assess time-domain properties from a single number about a fre-quency response, so use
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