First-Order Low-Pass Filter - Princeton University

Filters, Cost Functions, and Controller Structures

Robert Stengel Optimal Control and Estimation MAE 546

Princeton University, 2018

Copyright 2018 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE546.html

http://www.princeton.edu/~stengel/OptConEst.html

minΔu

J = minΔu

12

ΔxT (t)QΔx(t)+ ΔuT (t)RΔu(t)⎡⎣ ⎤⎦dt0

∞

∫subject to

Δ!x(t) = FΔx(t)+GΔu(t)

!  Dynamic systems as low-pass filters!  Frequency response of dynamic systems!  Shaping system response

!  LQ regulators with output vector cost functions!  Implicit model-following!  Cost functions with augmented state vector

1

First-Order Low-Pass Filter

2

Low-Pass FilterLow-pass filter passes low-frequency signals and

attenuates high-frequency signals1st-order example

Δ!x t( ) = −aΔx t( ) + aΔu t( )a = 0.1, 1, or 10

Δx s( ) = as + a( )Δu s( )

Δx jω( ) = ajω + a( )Δu jω( )

Laplace transform, with x(0) = 0

Frequency response, s = jω

3

Response of 1st-Order Low-Pass Filters to Step Input and Initial Condition

Δ!x t( ) = −aΔx t( ) + aΔu t( )a = 0.1, 1, or 10

Smoothing effect on sharp changes4

Frequency Response of Dynamic Systems

5

Response of 1st-Order Low-Pass Filters to Sine-Wave Inputs

Δu t( ) =sin t( )sin 2t( )sin 4t( )

⎧

⎨⎪⎪

⎩⎪⎪

!x t( ) = −x t( ) + u t( )

x t( ) = −10x t( ) +10u t( )

x t( ) = −0.1x t( ) + 0.1u t( )

Input

Output

6

Response of 1st-Order Low-Pass Filters to White Noise

Δ!x t( ) = −aΔx t( ) + aΔu t( )a = 0.1, 1, or 10

7

Relationship of Input Frequencies to Filter Bandwidth

Δu t( ) =sin ωt( )sin 2ωt( )sin 4ωt( )

⎧

⎨⎪⎪

⎩⎪⎪

How do we calculate frequency

response?

BANDWIDTHInput frequency at

which output magnitude = –3dB

8

Bode Plot Asymptotes, Departures, and Phase Angles for 1st-Order Lags

•  General shape of amplitude ratio governed by asymptotes

•  Slope of asymptotes changes by multiples of ±20 dB/dec at poles or zeros of H(jω)

•  Actual AR departs from asymptotes

•  Phase angle of a real, negative pole

–  When ω = 0, ϕ = 0°–  When ω = λ, ϕ =–45°–  When ω -> ∞, ϕ -> –90°

•  AR asymptotes of a real pole–  When ω = 0, slope = 0 dB/

dec–  When ω ≥ λ, slope = –20 dB/

dec

9

H red ( jω ) =

10jω +10( )

H blue( jω ) =100jω +10( )

H green ( jω ) =100

jω +100( )

2nd-Order Low-Pass Filter

Δ!!x t( ) = −2ζω nΔ!x t( )−ω n2Δx t( ) +ω n

2Δu t( )

Laplace transform, with I.C. = 0

Frequency response, s = jω

Δx s( ) = ω n2

s2 + 2ζω ns +ω n2 Δu s( )

Δx jω( ) = ω n2

jω( )2 + 2ζω n jω( ) +ω n2Δu jω( )

!H ( jω )Δu jω( )10

2nd-Order Step Response

11

Increased damping decreases oscillatory over/undershootFurther increase eliminates oscillation

Amplitude Ratio Asymptotes and Departures of Second-Order Bode

Plots (No Zeros)• AR asymptotes of a

pair of complex poles–  When ω = 0, slope

= 0 dB/dec–  When ω ≥ ωn,

slope = –40 dB/dec

• Height of resonant peak depends on damping ratio

12

Phase Angles of Second-Order Bode Plots (No Zeros)

• Phase angle of a pair of complex negative poles

–  When ω = 0, ϕ = 0°–  When ω = ωn, ϕ =–

90°–  When ω -> ∞, ϕ -> –

180°• Abruptness of phase

shift depends on damping ratio

13

Δ!x(t) = FΔx(t)+GΔu(t)Δy(t) = HxΔx(t)+HuΔu(t)

Time-Domain System Equations

Laplace Transforms of System Equations

sΔx(s)− Δx(0) = FΔx(s)+GΔu(s)

Δx(s) = sI− F[ ]−1 Δx(0)+GΔu(s)[ ]Δy(s) = HxΔx(s)+HuΔu(s)

Transformation of the System Equations

14

Transfer Function Matrix

Transfer Function Matrix relates control input to system output

with Hu = 0 and neglecting initial condition

H (s) = Hx sI− F[ ]−1G (r x m)

Laplace Transform of Output Vector

Δy(s) = HxΔx(s)+HuΔu(s) = Hx sI− F[ ]−1 Δx(0)+GΔu(s)[ ]+HuΔu(s)

= Hx sI− F( )−1G +Hu⎡⎣ ⎤⎦Δu(s)+Hx sI− F[ ]−1Δx(0)

= Control Effect + Initial Condition Effect

15

Scalar Frequency Response from Transfer Function Matrix

Transfer function matrix with s = jω

H (jω ) = Hx jωI− F[ ]−1G (r x m)

Δyi s( )Δuj s( ) = H ij (jω ) = Hxi

jωI− F[ ]−1G j (r x m)

Hxi= ith row of Hx

G j = j th column of G16

Second-Order Transfer Function

H (s) = HxA s( )G =h11 h12h21 h22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

adjs − f11( ) − f12− f21 s − f22( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

dets − f11( ) − f12− f21 s − f22( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

g11 g12g21 f22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

(n = m = r = 2)

Second-order transfer function matrix

r × n( ) n × n( ) n × m( )= r × m( ) = 2 × 2( )

Δ!x t( ) =Δ!x1 t( )Δ!x2 t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

f11 f12f21 f22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx1 t( )Δx2 t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

g11 g12g21 f22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δu1 t( )Δu2 t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δy t( ) =Δy1 t( )Δy2 t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

h11 h12h21 h22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx1 t( )Δx2 t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Second-order dynamic system

17

Scalar Transfer Function from Δuj to Δyi

H ij (s) =

kijnij (s)Δ(s)

=kij s

q + bq−1sq−1 + ...+ b1s + b0( )

sn + an−1sn−1 + ...+ a1s + a0( )

# zeros = q# poles = n

Just one element of the matrix, H(s)Denominator polynomial contains n roots

Each numerator term is a polynomial with q zeros, where q varies from term to term and ≤ n – 1

=kij s− z1( )ij s− z2( )ij ... s− zq( )ij

s−λ1( ) s−λ2( )... s−λn( )18

Scalar Frequency Response Function for Single Input/Output Pair

H ij (jω ) =

kij jω − z1( )ij jω − z2( )ij ... jω − zq( )ijjω − λ1( ) jω − λ2( )... jω − λn( )

Substitute: s = jω

Frequency response is a complex function of input frequency, ω

Real and imaginary parts, or** Amplitude ratio and phase angle **

= a(ω)+ jb(ω)→ AR(ω) e jφ (ω )

19

MATLAB Bode Plot with asymp.mhttp://www.mathworks.com/matlabcentral/

http://www.mathworks.com/matlabcentral/fileexchange/10183-bode-plot-with-asymptotes

2nd-Order Pitch Rate Frequency Response, with zero

asymp.mbode.m

20

• High gain (amplitude) at low frequency

–  Desired response is slowly varying

•  Low gain at high frequency

–  Random errors vary rapidly

• Crossover region is problem-specific

Desirable Open-Loop Frequency Response Characteristics (Bode)

21

Examples of Proportional LQ

Regulator Response

22

Example: Open-Loop Stable and Unstable 2nd-Order LTI System Response to Initial Condition

Stable Eigenvalues = -0.5000 + 3.9686i -0.5000 - 3.9686i

Unstable Eigenvalues = 0.2500 + 3.9922i 0.2500 - 3.9922i

FS =0 1

−16 −1⎡

⎣⎢

⎤

⎦⎥

FU = 0 1−16 +0.5

⎡

⎣⎢

⎤

⎦⎥

23

Example: Stabilizing Effect of Linear-Quadratic Regulators for Unstable

2nd-Order System

r = 1Control Gain (C) = 0.2620 1.0857

Riccati Matrix (S) = 2.2001 0.0291 0.0291 0.1206

Closed-Loop Eigenvalues = -6.4061 -2.8656

r = 100Control Gain (C) = 0.0028 0.1726

Riccati Matrix (S) = 30.7261 0.0312 0.0312 1.9183

Closed-Loop Eigenvalues = -0.5269 + 3.9683i -0.5269 - 3.9683i

minΔu

J = minΔu

12

Δx12 + Δx2

2 + rΔu2( )dt0

∞

∫⎡

⎣⎢

⎤

⎦⎥

Δu(t) = − c1 c2⎡⎣

⎤⎦

Δx1(t)Δx2 (t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −c1Δx1(t)− c2Δx2 (t)

For the unstable system

24

Example: Stabilizing/Filtering Effect of LQ Regulators for the Unstable 2nd-

Order Systemr = 1Control Gain (C) = 0.2620 1.0857

Riccati Matrix (S) = 2.2001 0.0291 0.0291 0.1206

Closed-Loop Eigenvalues = -6.4061 -2.8656

r = 100Control Gain (C) = 0.0028 0.1726

Riccati Matrix (S) = 30.7261 0.0312 0.0312 1.9183

Closed-Loop Eigenvalues = -0.5269 + 3.9683i -0.5269 - 3.9683i

25

Example: Open-Loop Response of a Stable 2nd-Order System to Random

Disturbance

Eigenvalues = -1.1459, -7.8541

26

Example: Disturbance Response of Unstable System with Optimal Feedback

27

LQ Regulators with Output Vector Cost Functions

28

Quadratic Weighting of the Output

J = 12

ΔyT (t)QyΔy(t)⎡⎣ ⎤⎦dt0

∞

∫

= 12

HxΔx(t)+HuΔu(t)[ ]T Qy HxΔx(t)+HuΔu(t)[ ]{ }dt0

∞

∫

Δy(t) = HxΔx(t) +HuΔu(t)

29Note: State, not output, is fed back

Quadratic Weighting of the

Output

Δu(t) = − RO +Ro( )−1 MOT +GO

TPSS⎡⎣ ⎤⎦Δx t( ) = −CΔx t( )

Δumin J =

Δumin

12

ΔxT (t) ΔuT (t)⎡⎣

⎤⎦Hx

TQyHx HxTQyHu

HuTQyHx Hu

TQyHu +Ro

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dt

0

∞

∫

!Δumin

12


⎤⎦QO MO

MOT RO +Ro

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dt

0

∞

∫

30

Output vector may not contain control (i.e., Hu = 0)Ad hoc Ro added to assure invertability

State Rate Can Be Expressed as an

Output to be Minimized

minΔu

J = 12

ΔyT (t)QyΔy(t)⎡⎣ ⎤⎦dt0

∞

∫ = 12

Δ!xT (t)QyΔ!x(t)⎡⎣ ⎤⎦dt0

∞

∫

Δu(t) = − RSR +Ro( )−1 MSRT +GSR

TPSS⎡⎣ ⎤⎦Δx t( ) = −CΔx t( )

Δ!x(t) = FΔx(t)+GΔu(t)Δy(t) = HxΔx(t)+HuΔu(t) " FΔx(t)+GΔu(t)

Special case of output weighting

minΔu

J = minΔu

12


⎤⎦FTQyF FTQyG

GTQyF GTQyG +Ro

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dt

0

∞

∫

! minΔu

12


⎤⎦QSR MSR

MSRT RSR +Ro

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dt

0

∞

∫

31

Implicit Model-Following LQ Regulator

Δu(t) = −CIMFΔx t( )

Δx(t) = FΔx(t) +GΔu(t)

Simulator aircraft dynamics

ΔxM (t) = FMΔxM (t)Ideal aircraft dynamics

Feedback control law to provide dynamic similarity

Another special case of output weighting32

Princeton Variable-Response Research Aircraft

F-16


Assume state dimensions are the sameΔ!x(t) = FΔx(t)+GΔu(t)Δ!xM (t) = FMΔxM (t)

If simulation is successful,ΔxM (t) ≈ Δx(t) and Δ!xM (t) ≈ FMΔx(t)

33


J = 1

2Δx(t)− ΔxM (t)[ ]T QM Δx(t)− ΔxM (t)[ ]{ }dt

0

∞

∫

Δu(t) = − R IMF +Ro( )−1 M IMFT +G IMF

TPSS⎡⎣ ⎤⎦Δx t( ) = −CΔx t( )

Cost function penalizes difference between actual and ideal model dynamics

Therefore, ideal model is implicit in the optimizing feedback control law

minΔu

J = minΔu

12

Δx(t) Δu(t)⎡⎣

⎤⎦T F − FM( )T QM F − FM( ) F − FM( )T QMG

GTQM F − FM( ) GTQMG +Ro

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dt

0

∞

∫

! minΔu

12

Δx(t) Δu(t)⎡⎣

⎤⎦T QIMF M IMF

M IMFT R IMF +Ro

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪dt

0

∞

∫

34

Cost Functions with Augmented State Vector

35

Integral Compensation Can Reduce Steady-State Errors


Δ!ξ t( ) = H IΔx(t)

!  Selector matrix, HI, can reduce or mix integrals in feedback

!  Sources of Steady-State Error!  Constant disturbance!  Errors in system dynamic model

36

Δ!x(t)

Δ!ξ t( )⎡

⎣⎢⎢

⎤

⎦⎥⎥=

F GH I 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx(t)Δξ t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥+ G

0⎡

⎣⎢

⎤

⎦⎥Δu(t)

LQ Proportional-Integral (PI) Control

Δu(t) = −CBΔx t( )−CI H IΔx τ( )dτ0

t

∫! −CBΔx t( )−CIΔξ t( )

where the integral state is

Δξ t( ) ! H IΔx τ( )dτ0

t

∫dim H I( ) = m × n

define Δχ t( ) !Δx t( )Δξ t( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

37

Integral State is Added to the Cost Function and the Dynamic Model

minΔu

J = minΔu

12

ΔxT (t)QxΔx(t)+ ΔξT t( )QξΔξ t( ) + ΔuT (t)RΔu(t)⎡⎣ ⎤⎦dt0

∞

∫

= minΔu

12

ΔχT (t)Qx 00 Qξ

⎡

⎣⎢⎢

⎤

⎦⎥⎥Δχ(t)+ ΔuT (t)RΔu(t)

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥dt

0

∞

∫

subject to Δ!χ(t) = FχΔχ(t)+GχΔu(t)

Δu(t) = −R−1 GT 0⎡⎣ ⎤⎦PSSΔχ(t) = −CχΔχ(t)

= −CBΔx t( )−CIΔξ t( )38

Integral State is Added to the Cost Function and the Dynamic Model

Δu(t) = −CχΔχ(t)

= −CBΔx t( )−CIΔξ t( )Δu(s) = −CχΔχ(s)

= −CBΔx s( )−CIΔξ s( )

= −CBΔx s( )−CIHxΔx s( )

s

Δu(s) = −CBs +CIHx[ ]

sΔx s( )

Bode Plot of Control Law: Integrator plus numerator

matrix of zeros

39

Actuator Dynamics and Proportional-Filter LQ

Regulators

40

Actuator Dynamics May Impact System Response

41

Force multiplication by

large piston

Electro/mechanical

transduction via small piston

Hydraulic Actuator Electric Motor

Actuator Dynamics May Affect System Response

Δ!x(t)Δ !u(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= F G

0 −K⎡

⎣⎢

⎤

⎦⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

0Iv

⎡

⎣⎢⎢

⎤

⎦⎥⎥Δv(t)

Control variable is actuator forcing function

Augment state dynamics to include actuator dynamics

42


Δ !u(t) = −KΔu(t)+ Δv(t)

LQ Regulator with Actuator Dynamics

minΔv

J = minΔv

12

ΔxT (t)QxΔx(t)+ ΔuT (t)RuΔu(t)+ ΔvT (t)RvΔv(t)⎡⎣ ⎤⎦dt0

∞

∫

= minΔv

12

ΔχT (t)Qx 00 Ru

⎡

⎣⎢⎢

⎤

⎦⎥⎥Δχ(t)+ ΔvT (t)RvΔv(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥dt

0

∞

∫

Re-define state and control vectors

Δχ(t) !Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥; Δv t( ) = Control

Fχ !F G0 −K

⎡

⎣⎢

⎤

⎦⎥; Gχ !

0Iv

⎡

⎣⎢⎢

⎤

⎦⎥⎥

43 subject to Δ!χ(t) = FχΔχ(t)+GχΔv(t)


Δv(t) = −CχΔχ(t) = −Rv−1 0 Iv⎡⎣

⎤⎦TPSS

Δx t( )Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= −CBΔx t( )−CAΔu(t)

Δv(s) = −CBΔx s( )−CAΔu(s)44

Control Velocity


Control Displacement

Δ !u(t) = −KΔu(t)−CAΔu(t)−CBΔx t( )sΔu(s) = −KΔu(s)−CAΔu(s)−CBΔx s( )

sI+ K +CA( )⎡⎣ ⎤⎦Δu(s) = −CBΔx s( )Δu(s) = − sI+ K +CA( )⎡⎣ ⎤⎦

−1CBΔx s( )

45

LQ Regulator with Artificial Actuator Dynamics

Δx(t)Δ u(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= F G

0 0⎡

⎣⎢

⎤

⎦⎥

Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥+ 0

I⎡

⎣⎢

⎤

⎦⎥Δv(t)

Δv(t) = Δ !uInt (t) = −CBΔx t( )−CAΔu(t)

CA introduces artificial actuator dynamics, e.g., low-pass filter

LQ control variable is derivative of actual system control

46

Proportional-Filter (PF) LQ Regulator

Δχ(t) =Δx(t)Δu(t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥; Fχ =

F G0 0

⎡

⎣⎢

⎤

⎦⎥; Gχ =

0I

⎡

⎣⎢

⎤

⎦⎥

Δv(t) = Δ !uIntegrator (t) = −CχΔχ(t)

CA provides low-pass filtering effect on the control input

Δu(s) = − sI+CA[ ]−1CBΔx s( )

Optimal LQ Regulator

47

Proportional LQ Regulator: High-Frequency Control in Response to

High-Frequency Disturbances2nd-Order System Response with Perfect Actuator

Good disturbance rejection, but high bandwidth may be unrealistic48

Proportional-Filter LQ Regulator Reduces High-Frequency Control

Signals 2nd-Order System Response

... at the expense of decreased disturbance rejection 49

Next Time: Linear-Quadratic Control

System Design

50

Supplemental Material

51

Princeton Variable-Response Research Aircraft (VRA)

52

Aircraft That Simulate Other Aircraft•  Closed-loop control•  Variable-stability research aircraft, e.g., TIFS, Learjet, NT-33A,

and Princeton Variable-Response Research Aircraft (Navion)

USAF/Calspan TIFS

CalSpan Learjet

Princeton VRA

USAF/Calspan NT-33A

53

First-Order Low-Pass Filter - Princeton University

Documents

Transcript of First-Order Low-Pass Filter - Princeton University