lecture 2 – linear systems - web.stanford.edu · lecture 2 – linear systems ... – broad class...

EE392m - Spring 2005Gorinevsky

Control Engineering 2-1

Lecture 2 – Linear Systems

This lecture: EE263 material recap + some controls motivation • Continuous time (physics) • Linear state space model• Transfer functions • Black-box models; frequency domain analysis• Linearization



Modeling and Analysis

This lecture considers• Linear models. More detail on modeling in Lecture 7• Simulation: computing state evolution and output signal• Stability: does the solution diverge after some time? • Approximate linear models

System Model

[internal states]Input signal(control)

Output signal(observation)

state evolution



Linear Models

• Model is a mathematical representations of a system– Models allow simulating the system – Models can be used for conceptual analysis – Models are never exact

• Linear models – Have simple structure – Can be analyzed using powerful mathematical tools – Can be matched against real data using known procedures– Many simple physics models are linear – They are just models, not the real systems



State space model • Generic state space model

– is described by ODEs– e.g., physics-based system

model– state vector x– observation vector y– control vector u

),(

),,(

txgy

tuxfdtdx

=

= state evolution

observation

Example: F16 Longitudinal Model

e-

e--

e-

.q..V.qq

.q..V.-

δ.q...V.-V

δαθ

δααθα

180081 82010952

1015291002110542

17058023282810931

12

34

2

−−+⋅==

⋅−+−⋅=

+−−+⋅=

&

&

&

&

z

x

δe

αθV

eu δ=

θ3.57=y

from Aircraft Control and Simulationby Stevens and Lewis

x1 - velocity V [ft/sec] x2 - angle of attack α [rad] x3 - pitch angle θ [rad] x4 - pitch rate q [rad/sec]δe - elevator deflection [deg]



Linear state space model • Linear Time Invariant (LTI) state space model:

• Can be integrated analytically or numerically (simulation) • Can be well analyzed: stability, response

Cxy

BuAxdtdx

=

+= state evolution

observations


[ ] xy

uxdtdx

⋅=

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⋅−+⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⋅

−⋅−−−⋅−

=−

−

−

−

03.5700

18.00

1015.217.0

08.1082.01095.2100091.0002.11054.248.02.3282.81093.1

3

12

4

2



Integrating linear autonomous system

• Matrix exponential– Can be computed in Matlab

as expm(A)– The definition corresponds

to integrating the ODE by Euler method

Axdtdx =

( ) tt

ttAIAt ∆

→∆∆+= /

0lim)exp(

( ) )()()()( TxtAIttAxTxtTx ∆+=∆+≈∆+

Example: >>A = [-1.93e-2 8.82 -32.2 -0.58;

-2.54e-4 -1.02 0 0.91;0 0 0 1;

2.95e-12 0.82 0 -1.08];

>> expm(A)ans =

0.9802 3.1208 -31.8830 -10.2849-0.0002 0.5004 0.0031 0.3604-0.0000 0.2216 1.0002 0.6709-0.0001 0.3239 0.0007 0.4773

( ) )()exp()()( TxtAnTxtAItnTx n ∆→∆+=∆+



Integrating linear autonomous system

• Initial condition response

0)0(, xxAxdtdx ==

0)exp()( xAttx =

Example: %Take A, B, C from the F16 example>> k = 0.02>> G = A + k*B*C;>> x0 = [0.1700;

-0.0022;0;-0.1800];

>> for j = 1:length(t);

x(:,j)=expm(G*t(j))*x0; end;

0 5 10 15 20

1

2

3

0 5 10 15 20-0.04-0.02

00.02

0 5 10 15 20-0.08-0.06-0.04-0.02

00.02

0 5 10 15 20-0.15-0.1

-0.050

0.05

TIME



Eigenvalues and Stability• Consider eigenvalues of the matrix A

• Suppose A has all different and nonzero eigenvalues, then

• The system solution is exponentially stable if• If A has eigenvalues with multiplicity more than 1, things are

a bit more complicated: Jordan blocks, polynomials in t• Still the condition of exponentially stability is

0)det();eig(}{ =−= AIAj λλ

1}diag{ −⋅⋅= VVA jλ

0Re <jλ

1}diag{)exp( −⋅⋅= VeVAt tjλ

Example: % take A from % the F16 example>> eig(A)ans =-1.9125 -0.1519 + 0.1143i-0.1519 - 0.1143i0.0970

0Re <jλ



Input-output models

• Black-box models – describe system P as an operator

Px

uinput signal

youtput signal

internal state (hidden)

• Historically (50 years ago)– Black-box models EE– State-space models ME, AA



Linear System (input-output)

• Linearity

• Linear Time-Invariant systems - LTI

)()( 11 ⋅⎯→⎯⋅ yu P

)()( TyTu P −⋅⎯→⎯−⋅

)()()()( 2121 ⋅+⋅⎯→⎯⋅+⋅ byaybuau P

)()( 22 ⋅⎯→⎯⋅ yu P

⎯→⎯Pu

t

y

t



Convolution representation

• Convolution integral

• Impulse response

• Step response: u = 1 for t > 0

∫∞−

−=t

duthty τττ )()()(

)()()()( thtyttu =⇒= δ

∫∫ =−=tt

dhdthtg00

)()()( ττττ )()( tgdtdth =

uhy *=signal processing notation



Impulse Response for State Space Model

• Impulse response for the state x

• System impulse response

{1

)0()( tuBtAxtx ∆+∆≈∆

BtuBtx =∆≈∆ )(

BAttx )exp()( =

Cxy

BuAxdtdx

=

+= state evolution

observation

Example: >> A = [-0.3130 56.7000 0;

-0.0139 -0.4260 0;0 56.7000 0];

>> B = [0.232; 0.0203; 0];>> C = [0, 0, 1];

0 10 20-0.5

00.5

1

0 10 20-0.020

0.020.04

0 10 200

0.5

1

TIME

BAtCth )exp()( =



Formal transfer function• Rational transfer function

= IIR (Infinite Impulse Response) model– Broad class of input-output linear models

• Differentiation operator

• Formal transfer function – rational function of s

• For a causal system m ≤ n

sdtd →

ubdt

udbdt

udbyadt

ydadt

yda nn

n

n

n

mm

m

m

m

11

1

2111

1

21 +−

−

+−

−

+++=+++ KK

usDsNusHy ⋅=⋅=)()()( 11)( ++++= mm

m asasasN K

11)( ++++= nnn bsbsbsD K



Poles, Impulse Response

• Expand

Then

• Quasi-polynomial impulse response – see a textbook

• Example:

usDsNy ⋅=)()( 0)( =sN

0)( =sD- zeros- poles

KMK

M pspssD )()()( 11 −⋅⋅−= K

ups

sNups

sNyKM

K

KM ⋅

−++⋅

−=

)()(

)()(

11

1 K

( ) ( ) tpMK

MK

tpM

M K

K

K ectcectcth 1,1

1,1,11

1,11

1

1)( +−

+− ++++++= KKK

udt

yd =2

2u

sy ⋅= 2

1

00)( xtvth +=

Transfer function:

Impulse response:



Transfer Function for State Space

• Characteristic polynomial

• Poles are the same as eigenvalues of the state-space matrix A• For stability we need Re pk = Re λk < 0

( )( ) BAsICsH

uBAsIy1

1

)( −

−

−=

⋅−=

Poles eigenvalues( ) 0det =− AsI

( ) 0det)( =−= AsIsN

CxyBuAxsx

=+=

• Formal transfer function for a state space model

sdtd →



Laplace transform• Laplace integral transform:

• Laplace transform of the convolution integral yields

• Transfer function:– function of complex variable s– analytical in a right half-plane Re s ≥ a– for a stable system a ≤ 0– for an IIR model

∫∞

=0

)()( dtethsH st)(ˆ)()(ˆ susHsy =

)(ˆ sxsdtdx →

dtetxsxtx st∫∞

=→0

)()(ˆ)(

Re s

Imag s

)(sH

)()()(

sDsNsH =



Frequency decomposition• Sinusoids are eigenfunctions of an LTI system:

LTIPlant

• Frequency domain analysis

usHy )(=

∑∑ =→ tikk

tik

kk eiHuyeu ωω ω )(

tie ω tieiH ωω)(

( ) tititi eiedtdes ωωω ω=→



Frequency domain description

∫∫ −− =⇒= ωωωπ

ωωπ

ω

ω

ω deuiHydeuu ti

y

ti43421)(~

)(~)(21)(~

21

)(~ ω

ω

ue ti

uPacket of sinusoids

)(~ ω

ω

ye tiPacket

of sinusoids

)( ωiH y

• Frequency domain analysis

)(ˆ)()()(~ ωωωωω

ω iudetudetuuis

stti =⎟⎠⎞⎜

⎝⎛==

=

∞

∞−

∞

∞− ∫∫

• Fourier transform – numerical analysis• Laplace transform – complex analysis u(t) = 0, for t < 0



Continuous systems in frequency domain

• Fourier transform

• Inverse Fourier transform

• I/O impulse response model

• Transfer function

• System frequency response)(~)()(~

)()(

)()()(

)(~21)(

)()(~

0

ωωω

ττ

ωωπ

ω

ω

ω

uiHy

dtethsH

dtuthty

dextx

dtetxx

st

t

ti

ti

=

=

−=

=

=

∫

∫

∫

∫

∞∞−

∞

∞−

−

∞

∞− ],[],[ ∞−∞→∞−∞



Frequency domain description• Bode plots:

ti

ti

eiHyeu

ω

ω

ω )(==

• Example:

7.01)(

−=

ssH

Bode Diagram

Frequency (rad/sec)

Phas

e (d

eg)

Mag

nitu

de (d

B)

-5

0

5

10

15

10-2 10-1 100-180

-135

-90

-45

0

• |H| is often measured in dB– [dB] = 20 log10 M

)(arg)()()(ωωϕ

ωωiH

iHM==



Model Approximation

• Model structure – physics, computational • Determine parameters from data • Step/impulse responses are close the input/output

models are close • Example – fit step response• Linearization of nonlinear model



Black-box model from data• Linear black-box model can be determined from the data,

e.g., step response data, or frequency response

• Example problem: fit an IIR model of a given order• This is called model identification • Considered in more detail in Lecture 8

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

TIME

HEA

T FL

UX

STEP RESPONSE



Linear PDE models • Include functions of spatial variables

– electromagnetic fields – mass and heat transfer– fluid dynamics – structural deformations

• Example: sideways heat equation

1

2

2

0)1(;)0(

=∂∂=

==∂∂=

∂∂

xxTy

TuTxTk

tT

yheat flux

x

Toutside=0Tinside=u



Linear PDE System Example • Heat transfer equation,

– boundary temperature input u– heat flux output y

• Impulse response and step response• Transfer function is not rational

0)1()0(

2

2

==∂∂=

∂∂

TTuxTk

tT

1=∂∂=

xxTy

0 20 40 60 80 1000

2

4

6x 10

-2

TIME

HEA

T FL

UX

PULSE RESPONSE

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

TIME

HEA

T FL

UX

STEP RESPONSE

0

0.5

105

1015

0

0.2

0.4

0.6

0.8

1

TIME

TEMPERATURE

COORDINATE

heat flux



Impulse response approximation• Approximating impulse

and step responses by a low order rational transfer function model

• Higher order model can provide very accurate approximation

• Methods:– trial and error– sampled time response fit,

e.g., Matlab’s prony

– identification, Lecture 8– formal model reduction

approaches - advanced

03.034.08.24.008.001.0)( 2

2

+++−=

sssssH

0 20 40 60 80 1000

0.02

0.04

0.06

0 20 40 60 80 1000

0.5

1

TIME



Validity of Model Approximation

• Why can we use an approximate model instead of the ‘real’ model?

• Will the analysis hold? • The input-output maps of two systems are ‘close’ if the

convolution kernels (impulse responses) are ‘close’

• The closed-loop stability impact of the modeling error– Control robustness– Will be discussed in Lecture 9

∫∞−

−=t

uthty τττ )()()(



Nonlinear map linearization• Nonlinear - detailed model • Linear - conceptual design model • Differentiation, secant method

• Example: static map linearization

)()( 0uuufufy −

∆∆≈=



Linearization Example: RTP• RTP – Rapid Thermal

Processing• Major semiconductor

manufacturing process

uinput heaters

heated partT

output furnace

)()( 244

1 FF TTcTTcbudtdT −−−−=

T – part temperatureu – IR heater powerTF – furnace temperature

• Stefan-Boltzmann law nonlinearity• TF is assumed to be constant



RTP, cont’d

Linearize around a steady state point

b = 1000, c1 = 1.1⋅10-10, c2 = 0.8, TF = 300

buTfdtdT += )(

300 400 500 600 700 800 900 1000 1100 1200 1300-1200

-1000

-800

-600

-400

-200

0

TEMPERATURE

f(T)

)()()( 244

1 FF TTcTTcTf −−−−=

∆−∆+≈′= )()()( **

*TfTfTfa

dTTaTfL +−= )()( *

)( *Tfd =buTf

dtdT

L += )(



RTP, cont’d

Simulate performance:

kxudbuaxx

−=++=&

)( bkap +−=Linear system with a pole

T* = 1000, a = -1.7425, b = 1000, k = 0.01

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5300

400

500

600

700

800

900

1000

TIME

T

Linear model, d = 0

Non-linear model

*TTx −=

7425.11−=p

p1



Nonlinear state space model linearization

• Linearize the r.h.s. map in a state-space model

BvAqq

uuufxx

xfuxfx

vq

+=

−∆∆+−

∆∆≈=

&

4342143421& )()(),( 00

{]0......0[

),(),(

#

0000

j

jj

j

jj

dsd

uxfusxfxf

=

−+=⎥⎦

⎤⎢⎣⎡

∆∆

),(0 00 uxf=• Linearize around an equilibrium• Secant method

• This is how Simulink computes linearization



• State vector xx1 - velocity V [ft/sec] x2 - angle of attack α [rad] x3 - pitch angle θ [rad] x4 - pitch rate q [rad/sec]• Control input

u - elevator deflection δe [deg].

z’

x’

δe

α θV

z

x

eu δ=

θ3.57=y

q

IM

q

qFFmV

FFm

V

y

y

zx

zx

=

=

++−=

+=

θ

ααα

αα

&

&

&

&

)cossin(1

)sincos(1

),(cos),(

sin)(

,

,

,

etmy

etzz

txx

RCMmgrCF

TmgrCF

δαθδα

θα

=+=

+−=⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

θ

αq

V

x

),( uxfdtdx =


For more detail see: Aircraft Control and Simulation by Stevens and Lewis



Nonlinear Model of F16

• Aircraft models are understood by groups of people

• Could take many man-years worth of effort

• Aerodynamics model is based on empirical data

• f(x,u) available as a computational function can be used without a deep understanding of the model

• The nonlinear model can be used for simulation, or linearized for analysis

)(

),(

xgy

uxfdtdx

=

= state evolution

observation



Linearized Longitudinal Model of F16• Assume trim condition

• Linearize the nonlinear function f(x,u) by a finite difference method (secant method). Step = [1 0.001 0.01 0.001]

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⋅−=

∆∆=

−

18.00

1015.217.0

3

ufB

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

−⋅−−−⋅−

=∆∆=

−

−

−

08.1082.01095.2100091.0002.11054.248.02.3282.81093.1

12

4

2

xfA

- velocity V [ft/sec] - angle of attack α [rad] - pitch rate q [rad/sec]- pitch angle θ [rad⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0393.00

0393.0500

0

0

0

0

0

θ

αq

V

x

• These are the matrices we considered in the linear F16 model example



Simulation-based validation• Simulate with nonlinear model, compare with linear model

results

• Doublet response



LTI models - summary

• ODE model• State space linear model • Linear system can be described by impulse response or

step response• Linear system can be described by frequency response =

Fourier transform of the impulse response• Linear model approximations can be obtained from more

complex models – Approximation of a linear model response– Linearization of a nonlinear model

lecture 2 – linear systems - web.stanford.edu · lecture 2 – linear systems ... – broad class...

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