# Raktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries

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1 . AERO 632: of Advance Flight Control System. Preliminaries Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University.

2 Preliminaries Signals & Systems Laplace transforms.. Transfer functions from ordinary linear differential equations System interconnections Block diagram algebra simplification of interconnections General feedback control system interconnection. d n r + u + +. e C P y + + y m y m AERO 632, Instructor: Raktim Bhattacharya 2 / 46

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4 Signals & Systems.. u(t). P y(t) Actuator applies u(t) Sensor provides y(t) Feedback controller takes y(t) and determines u(t) to achieve desired behavior The controller is typically implemented as software, running in a micro controller Imperfections exist in real world sensors have noise actuators have irregularities plant P is not fully known AERO 632, Instructor: Raktim Bhattacharya 4 / 46

5 System Response to u(t).. u(t). P y(t) Given plant P and input u(t), what is y(t)? P is defined in terms of ordinary differential equations y(t) is the forced + initial condition response. Linear Dynamics mẍ + cẋ + kx = u(t) dynamics y(t) = x(t) measurement Nonlinear Dynamics ẍ µ( x 2 )ẋ + x = u(t) dynamics y(t) = x(t) measurement In this class we focus on linear systems AERO 632, Instructor: Raktim Bhattacharya 5 / 46

6 Linear Systems... u(t) P y(t) Dynamics is defined by linear ordinary differential equation Super position principle applies u (t) y (t) u 2 (t) y 2 (t) = (u (t) + u 2 (t)) (y (t) + y 2 (t)) AERO 632, Instructor: Raktim Bhattacharya 6 / 46

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8 Given signal u(t), Laplace transform is defined as Exists when L {u(t)} := u(t)e st dt lim t u(t)e σt =, for some σ >.. Very useful in studying linear dynamical systems and designing controllers AERO 632, Instructor: Raktim Bhattacharya 8 / 46

9 Properties Linear operator Additive L {u (t) + u 2 (t)} = Superposition = (u (t) + u 2 (t)) e st dt u (t)e st dt +.. = L {u (t)} + L {u 2 (t)} L {au(t)} = al {u(t)}, a is a constant u 2 (t)e st dt AERO 632, Instructor: Raktim Bhattacharya 9 / 46

10 Properties (contd.). U(s) := L {u(t)}.. 2. L {au (t) + bu 2 (t)} = al {u (t)} + bl {u 2 (t)} = au (s) + bu 2 (s) 3. t s U(s) u(τ)dτ 4. U (s)u 2 (s) u (t) u 2 (t) Convolution 5. lim su(s) lim s 6. lim su(s) u( + ) Initial value theorem s 7. du(s) tu(t) { ds } du 8. L su(s) su() dt t u(t) Final value theorem 9. L {ü} s 2 U(s) su() u() AERO 632, Instructor: Raktim Bhattacharya / 46

11 Important Signals... L {δ(t)} = δ(t) is impulse function 2. L {(t)} = s (t) is unit step function at t = 3. L {t} = s 2 ω 4. L {sin(ωt} = s 2 + ω 2 s 5. L {cos(ωt} = s 2 + ω 2 AERO 632, Instructor: Raktim Bhattacharya / 46

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13 Spring Mass Damper System.. m u(t) Equation of Motion mẍ + cẋ + kx = u(t) Take L { } on both sides L {mẍ + cẋ + kx} = L {u(t)} ml {ẍ} + cl {ẋ} + kl {x} = L {u(t)} m ( s 2 X(s) sx() ẋ() ) + c (sx(s) x()) + kx(s) = U(s) (ms 2 + cs + k)x(s) = U(s) ẋ() and x() are assumed to be zero AERO 632, Instructor: Raktim Bhattacharya 3 / 46

14 Transfer Function.. m u(t). u(t) P y(t) (ms 2 + cs + k)x(s) = U(s) = X(s) U(s) = ms 2 + cs + k Choose output y(t) = x(t) = Y (s) = X(s). Therefore P (s) := Y (s) U(s) = ms 2 Transfer function + cs + k AERO 632, Instructor: Raktim Bhattacharya 4 / 46

15 Transfer Function (contd.) In general P (s) = N(s) D(s) where N(s) and D(s) are polynomials in s Roots of N(s) are the zeros Roots of D(s) are the poles determine stability.. AERO 632, Instructor: Raktim Bhattacharya 5 / 46

16 Response to u(t) Given input signal u(t) and transfer function P (s). Determine output response y(t)... Laplace transform U(s) := L {u(t)} 2. Determine Y (s) := P (s)u(s) 3. Laplace inverse y(t) := L {Y (s)} = L {P (s)u(s)}. u(t) P y(t) AERO 632, Instructor: Raktim Bhattacharya 6 / 46

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18 Block Diagram Representation of s Series Parallel Feedback A simple example A complex example.. AERO 632, Instructor: Raktim Bhattacharya 8 / 46

19 Series Connection ụ. G G 2 y.. AERO 632, Instructor: Raktim Bhattacharya 9 / 46

20 Parallel Connection.. G 2 ụ. G. + y AERO 632, Instructor: Raktim Bhattacharya 2 / 46

21 Feedback Connection r. + G y.. AERO 632, Instructor: Raktim Bhattacharya 2 / 46

22 Simple Example.. r. + G G 2 G 3 + y G 4 AERO 632, Instructor: Raktim Bhattacharya 22 / 46

23 Complex Example.. r. + G G 2 G y G 4 AERO 632, Instructor: Raktim Bhattacharya 23 / 46

24 Frequency Response

25 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 632, Instructor: Raktim Bhattacharya 25 / 46

26 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 632, Instructor: Raktim Bhattacharya 26 / 46

27 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 632, Instructor: Raktim Bhattacharya 27 / 46

28 Response to Sinusoidal Input Generalization (contd.) For a system P (s) and input forced response is where u(t) = A u sin(ω t), y(t) = A u M sin(ω t + ϕ), M(ω ) = P (s) s=jω = P (jω ), magnitude.. ϕ(ω ) = P (jω ) phase In polar form P (jω ) = Me jϕ. AERO 632, Instructor: Raktim Bhattacharya 28 / 46

29 Fourier Analysis

30 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 632, Instructor: Raktim Bhattacharya 3 / 46

31 Fourier Series Expansion Approximation of step function...2 N=2.2 N= N=8.2 N= N=2.2 N= AERO 632, Instructor: Raktim Bhattacharya 3 / 46

32 Fourier Transform Step function.. Fourier transform reveals the frequency content of a signal AERO 632, Instructor: Raktim Bhattacharya 32 / 46

33 Fourier Transform Step function frequency content y(t) t ŷ(ω) ω AERO 632, Instructor: Raktim Bhattacharya 33 / 46

34 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 632, Instructor: Raktim Bhattacharya 34 / 46

35 Bode Plot

36 First Order System.. 5 Bode Diagram y(t) Freq =. rad/s Magnitude (db) y(t) y(t) Freq =. rad/s Freq = 5. rad/s Phase (deg) 45 y(t) Freq =. rad/s Frequency (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 632, Instructor: Raktim Bhattacharya 36 / 46

37 Second Order System.. Magnitude (db) 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 8 Frequency (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 632, Instructor: Raktim Bhattacharya 37 / 46

38 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) Magnitude (db) ω rad/s Frequency (rad/s) AERO 632, Instructor: Raktim Bhattacharya 38 / 46

39 All transfer functions With proportional controller.. G er G ed G en Magnitude (db) 2 Magnitude (db) Magnitude (db) Frequency (rad/s) 8 2 Frequency (rad/s) Frequency (rad/s) G yr G yd G yn 2 2 Magnitude (db) 2 4 Magnitude (db) Magnitude (db) Frequency (rad/s) 8 2 Frequency (rad/s) 6 2 Frequency (rad/s) AERO 632, Instructor: Raktim Bhattacharya 39 / 46

40 Controller Considerations

41 Using Bode Plot 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 632, Instructor: Raktim Bhattacharya 4 / 46

42 Frequency 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 632, Instructor: Raktim Bhattacharya 42 / 46

43 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! Magnitude (db) slope 5 2 P (j!)c(j!) Steady-state error boundary!c Sensor noise, disturbance Plant uncertainty! Frequency (rad/s) AERO 632, Instructor: Raktim Bhattacharya 43 / 46

44 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 632, Instructor: Raktim Bhattacharya 44 / 46

45 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 632, Instructor: Raktim Bhattacharya 45 / 46

46 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 632, Instructor: Raktim Bhattacharya 46 / 46

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