# Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method

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1 .. AERO 422: Active Controls for Aerospace Vehicles Frequency Response- Method Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University.

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3 ... Response to Sinusoidal Input.. u(t) y(t). P Let u(t) = A u sin(ωt) Vary ω from to A linear system s response to sinusoidal inputs is called the system s frequency response AERO 422, Instructor: Raktim Bhattacharya 3 / 52

4 ... Response to Sinusoidal Input.. Example Let P (s) = s+, u(t) = sin(t) y(t) = 2 e t 2 cos(t) + 2 sin(t) = 2 e t + sin(t π }{{} 2 4 ) }{{} natural response forced response Forced response has form A y sin(ωt + ϕ) A y and ϕ are functions of ω AERO 422, Instructor: Raktim Bhattacharya 4 / 52

5 ... Response to Sinusoidal Input.. Generalization In general ω Y (s) = G(s) s 2 + ω 2 = α α n + + α + α s p s p n s + jω s jω = y(t) = α e pt + + α n e p nt + A }{{} y sin(ω + ϕ) }{{} natural forced Forced response has same frequency, different amplitude and phase. AERO 422, Instructor: Raktim Bhattacharya 5 / 52

6 ... Response to Sinusoidal Input.. Generalization (contd.) For a system P (s) and input u(t) = A u sin(ω t), forced response is y(t) = A u M sin(ω t + ϕ), where M(ω ) = P (s) s=jω = P (jω ), magnitude ϕ(ω ) = P (jω ) phase In polar form P (jω ) = Me jϕ. AERO 422, Instructor: Raktim Bhattacharya 6 / 52

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8 ... Fourier Series Expansion.. Given a signal y(t) with periodicity T, y(t) = a 2 + ( 2πnt a n cos T a = 2 T a n = 2 T b n = 2 T n=,2, T T T y(t)dt ( 2πnt y(t) cos T ( 2πnt y(t) sin T ) dt ) dt ) + b n sin ( ) 2πnt T AERO 422, Instructor: Raktim Bhattacharya 8 / 52

9 ... Fourier Series Expansion.. Approximation of step function.2 N=2.2 N= N=8.2 N= N=2.2 N= AERO 422, Instructor: Raktim Bhattacharya 9 / 52

10 .. Fourier Transform... Step function Fourier transform reveals the frequency content of a signal AERO 422, Instructor: Raktim Bhattacharya / 52

11 .. Fourier Transform... Step function frequency content.2.8 y(t) t ŷ(ω) ω AERO 422, Instructor: Raktim Bhattacharya / 52

12 .. Signals & Systems... Input Output. u(t) P y(t) Fourier Series Expansion superposition principle. i u i(t) P i y i(t) Fourier Transform Ụ(jω) Y (jω). P u i (t) = a i sin(ω i t) y iforced (t) = a i M sin(ω i t + ϕ) Y (jω) = P (jω)u(jω) Suffices to study P (jω) P (jω), P (jω) AERO 422, Instructor: Raktim Bhattacharya 2 / 52

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14 .. First Order System... Bode Diagram Freq =. rad/s 5 y(t) y(t) y(t) Freq =. rad/s Freq = 5. rad/s Phase (deg) 45 y(t) Freq =. rad/s P (s) = /(s + ) loglog scale db = log ( ) 2dB = log (/) u(t) = A sin(ω t) y forced (t) = AM sin(ω t + ϕ) AERO 422, Instructor: Raktim Bhattacharya 4 / 52

15 ... Second Order System Bode Diagram y(t) y(t) 2 Freq =. rad/s Freq =. rad/s Freq = 5. rad/s Phase (deg) y(t) y(t) Freq =. rad/s P (s) = /(s 2 +.5s + ) ω n = rad/s u(t) = A sin(ω t) y forced (t) = AM sin(ω t + ϕ) AERO 422, Instructor: Raktim Bhattacharya 5 / 52

16 .. Lead Compensator 5 5 Lead Controller. Phase lead low gain at low frequency high gain at high frequency.. relate it to derivative control 5 6 Phase (deg) AERO 422, Instructor: Raktim Bhattacharya 6 / 52

17 .. Lag Compensator Lag Controller. Phase lag high gain at low frequency low gain at high frequency relate it to integral control.. 5 Phase (deg) AERO 422, Instructor: Raktim Bhattacharya 7 / 52

18 .. S(jω) + T (jω) =. d.. n r + u e y + y. m C P y m Magnitude S(jω) P (s) = C(s) = (s+)(s/2+) S = G er = +P C = +P T = G yr = P C +P C = P +P ω rad/s Bode Diagram S T S+T Magnitude T(jω) ω rad/s AERO 422, Instructor: Raktim Bhattacharya 8 / 52

19 .. All transfer functions... With proportional controller G er G ed G en G yr G yd G yn AERO 422, Instructor: Raktim Bhattacharya 9 / 52

20 ... Piper Dakota Control System.. ed with root locus method System Transfer function from δ e (elevator angle) to θ (pitch angle) is P (s) = θ(s) δ e (s) = 6(s + 2.5)(s +.7) (s 2 + 5s + 4)(s 2 +.3s +.6) Control Objective an autopilot so that the step response to elevator input has t r < and M p < % = ω n >.8 rad/s and ζ >.6 2 nd order Controller C(s) =.5 s + 3 ( +.5/s) s + 25 AERO 422, Instructor: Raktim Bhattacharya 2 / 52

21 ... Piper Dakota Control System.. Time Response Ref to Control Ref to Error Ref to Output.5.5 Elevator angle (deg).5 Error (deg).5 Pitch Angle (deg) Time (seconds).5 2 Time (seconds) 2 Time (seconds) Dist to Control Dist to Error Dist to Output 5 Elevator angle (deg).5 Error (deg) Pitch angle (deg) Time (seconds) Time (seconds) 2 4 Time (seconds) AERO 422, Instructor: Raktim Bhattacharya 2 / 52

22 ... Piper Dakota Control System.. Frequency Response G er G ed G en G yr G yd G yn AERO 422, Instructor: Raktim Bhattacharya 22 / 52

23 ... Piper Dakota Control System.. Frequency Response (contd.) G ur G ud AERO 422, Instructor: Raktim Bhattacharya 23 / 52

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25 ... Approximate.. Useful for & Analysis Let open-loop transfer function be Write in Bode form KG(s) = K (s z )(s z 2 ) (s p )(s p 2 ) KG(jω) = K (jωτ + )(jωτ 2 + ) (jωτ a + )(jωτ b + ) K is the DC gain of the system. Example G(s) = (s + ) (s + 2)(s + 3) = G(jω) = jω + (jω + 2)(jω + 3) = 6 jω + (jω/2 + )(jω/3 + ) AERO 422, Instructor: Raktim Bhattacharya 25 / 52

26 ... Approximate.. contd. Transfer function in Bode Form KG(jω) = K (jωτ + )(jωτ 2 + ) (jωτ a + )(jωτ b + ) Three cases. K (jω) n pole, zero at origin 2. (jω + ) ± real pole, zero 3. [ ( jω ω n ) 2 + 2ζ jω ω n + ] ± complex pole, zero AERO 422, Instructor: Raktim Bhattacharya 26 / 52

27 ..... Case: K (jω) n pole, zero at origin Gain log K (jω) n = log K + n log jw = log K + n log w Phase K (jω) n = K + n jω = + n Gain Phase (a) Gain (b) Phase AERO 422, Instructor: Raktim Bhattacharya 27 / 52

28 ..... Case:2 (jωτ + ) ± real pole, zero Gain (jωτ + ) = Frequency ω = /τ is the break point {, ωτ <<, jωτ, ωτ >>. 2 Gain AERO 422, Instructor: Raktim Bhattacharya 28 / 52

29 ... Case:2 (jωτ + ) ± real pole, zero (contd.).. Phase, ωτ <<, = jωτ + = jωτ, ωτ >>, jωτ = 9 ωτ, jωτ + = Phase AERO 422, Instructor: Raktim Bhattacharya 29 / 52

30 .. Example... G(s) = 2(s +.5) s(s + )(s + 5) Bode Diagram Bode Diagram Phase (deg) Phase (deg) AERO 422, Instructor: Raktim Bhattacharya 3 / 52

31 Errors

32 .. Closed-loop system... r + e u y. C P y Closed-loop transfer function G er = + P C = K (jω) n (jωτ + )(jωτ 2 + ) (jωτ a + )(jωτ b + ) Steady-state gain lim sger(s) s s lim Ger(jω) ω P C = 2(s +.5) s(s + )(s + 5) Typically analysis is done with open-loop system Bode Diagram AERO 422, Instructor: Raktim Bhattacharya 32 / 52

33 .. Open-loop system... r + e u y. C P y Open-loop transfer function 2(s +.5) P C = s(s + )(s + 5) = K (jω) n (jωτ + )(jωτ 2 + ) (jωτ a + )(jωτ b + ) Steady-state error step e ss = Steady-state error ramp + K p, K p := K. 4 2 Bode Diagram e ss = K v 2 4 System type is the slope of the low frequency asymptote K v is the value of low frequency asymptote at ω = rad/s AERO 422, Instructor: Raktim Bhattacharya 33 / 52

34 Analysis

35 .. r + e u y. C P y Given open-loop data C(s) = K, P (s) = s(s+) 2 Imaginary Axis (seconds ) Root Locus. All points on root locus satisfy + P (s)c(s) = P (s)c(s) = = P (s)c(s) = and P (s)c(s) = 8 At neutral stability point s = jω, P (jω)c(jω) = P (jω)c(jω) = Stable for K < 2 Real Axis (seconds ) AERO 422, Instructor: Raktim Bhattacharya 35 / 52

36 ... P (jω)c(jω) < at P (jω)c(jω) = 8.. Bode Diagram 5 5 Phase (deg) K=. K=2 K= AERO 422, Instructor: Raktim Bhattacharya 36 / 52

37 .. Gain Margin... Open loop Bode Diagram 5 5 Phase (deg) K=. K=2 K= Gain Margin (GM): factor by which gain can be increased at P (jω)c(jω) = 8 AERO 422, Instructor: Raktim Bhattacharya 37 / 52

38 .. Phase Margin... Open loop Bode Diagram 5 5 Phase (deg) K=. K=2 K= Phase Margin (PM): amount by which phase exceeds 8 at P (jω)c(jω) = AERO 422, Instructor: Raktim Bhattacharya 38 / 52

39 .. Nyquist Plot... Relates open-loop frequency response to number of unstable closed-loop poles Residue theorem in complex analysis Plot P (jω)c(jω) in the complex plain Number of encirclements of equals Z P of + P (s)c(s) AERO 422, Instructor: Raktim Bhattacharya 39 / 52

40 .. Nyquist Plot... contd. Write P (s)c(s) = KG(s) = K N(s) D(s) = + P (s)c(s) = D(s) + KN(s) D(s) Poles of + P (s)c(s) = Poles of G(s) none of them on RHP Number of encirclements = number of zeros of + P (s)c(s) on RHP number of poles of closed-loop system AERO 422, Instructor: Raktim Bhattacharya 4 / 52

41 .. Nyquist Plot... Example: P (s)c(s) = K s(s+) 2 Nyquist Diagram 3 2 K= K=2 K= Imaginary Axis Real Axis AERO 422, Instructor: Raktim Bhattacharya 4 / 52

42 .. Nyquist Plot Determining Gain. Given P (s)c(s) = K, what is K for stability? s(s+) 2 Encirclement of /K + G(s) =.. Nyquist Diagram 2.5 Imaginary Axis Real Axis AERO 422, Instructor: Raktim Bhattacharya 42 / 52

43 .. Nyquist Plot... Gain and Phase Margin Nyquist plot of P (s)c(s) Nyquist Diagram 2.5 Imaginary Axis Real Axis AERO 422, Instructor: Raktim Bhattacharya 43 / 52

44 Frequency Domain

45 ... Using of P (jω)c(jω) Loop Shaping.. Develop conditions on the Bode plot of the open loop transfer function Sensitivity +P C Steady-state errors: slope and magnitude at lim ω Robust to sensor noise Disturbance rejection Controller roll off = not excite high-frequency modes of plant Robust to plant uncertainty Look at Bode plot of L(jω) := P (jω)c(jw) AERO 422, Instructor: Raktim Bhattacharya 45 / 52

46 ... Frquency Domain Specifications Constraints on the shape of L(jω).. P (j!)c(j!) Steady-state error boundary slope! c Sensor noise, plant uncertainty! Choose C(jω) to ensure L(jω) does not violate the constraints Slope at ω c ensures P M 9 stable if P M > = Sensor noise, disturbance Plant uncertainty P C > 8 AERO 422, Instructor: Raktim Bhattacharya 46 / 52

47 .. Plant Uncertainty... P (jω) = P (jω)( + P (jω)) Bode Diagram 5 5 True Model Unc+ Unc P (j!)c(j!) Steady-state error boundary slope!c Sensor noise, plant uncertainty! slope 5 2 P (j!)c(j!) Steady-state error boundary!c Sensor noise, disturbance Plant uncertainty! AERO 422, Instructor: Raktim Bhattacharya 47 / 52

48 ... Sensor Characteristics.. Noise spectrum P (j!)c(j!) Steady-state error boundary slope!c Sensor noise, plant uncertainty! P (j!)c(j!) slope!c Steady-state error boundary Sensor noise, disturbance Plant uncertainty! G yn = P C + P C AERO 422, Instructor: Raktim Bhattacharya 48 / 52

49 .. Reference Tracking... Bandlimited else conflicts with noise rejection Spectrum of r(t) X(f) P (j!)c(j!) Steady-state error boundary slope!c Sensor noise, plant uncertainty! Frequency (Hz) X(f) Spectrum of n(t) slope Sensor noise, disturbance Plant uncertainty!!c Steady-state error boundary Frequency (Hz) P (j!)c(j!) G yr = G yn = P C +P C P C + P C AERO 422, Instructor: Raktim Bhattacharya 49 / 52

50 ... Disturbance Rejecton.. Bandlimited else conflicts with noise rejection Spectrum of d(t) X(f) P (j!)c(j!) Steady-state error boundary slope!c Sensor noise, plant uncertainty! Frequency (Hz) X(f) Spectrum of n(t) slope Sensor noise, disturbance Plant uncertainty!!c Steady-state error boundary Frequency (Hz) P (j!)c(j!) G yd = G yn = P C +P C P + P C AERO 422, Instructor: Raktim Bhattacharya 5 / 52

51 ..... AERO 422, Instructor: Raktim Bhattacharya 5 / 52

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