september 11, 2015 analysis and control of systems considering signal transmission...

September 11, 2015

Analysis and Control of Systems Considering Signal Transmission

DelaysSun Yi

Assistant ProfessorLaboratory for Automation of Mechanical Systems (LAMS)

Dept. Mechanical EngineeringNorth Carolina A&T State University

2Outline• Introduction

– Motivations– Background

• Modeling of Delay Systems and Solution

• Methods for Analysis

• Methods for Control

• Concluding Remarks

3Control for Formation Flying• Ever-increasing interest in ‘small’

satellites, such as micro-, nano- satellites,

• As a substitute for huge, complex and

expensive monolithic satellites,

• Should be capable of maintaining inter-

spacecraft distance accurately to operate

as a virtual satellite with a large

capability,

• Performance degradation that arises from

the round-trip communication delay.

[Smith and Hadaegh, 2007]

4Networks of Agents: Intro• Communications between agents often

experience delays due to signal transmission, intermittent connectivity and link breakage.

• Their effects on system performance are not trivial but not predictable. Just known:– Delays can cause instability and

unexpected behaviors (e.g., out of control).

• For reliable performance of networked multi -agent systems, new control strategy is critically needed.

,( ) ( ) ( )C D GPS dt t t t= − −ρ r r

5Effects of Time Delay on Synchronization

-16 -14 -12 -10 -8 -6 -4 -2 0 2-20

-15

-10

-5

0

5

10

15

20PV Position Control, Eigenvalues vs. Time-Delay

ℜ(λ)

ℑ(λ

)

Unstable

h (time-delay) =0

h =0.1h =0.01

h =0.02h =0.03

h =0.04 h =0.06h =0.05 h =0.07h =0.08

h =0.09

Stable

• As delay increases, the rightmost eigenvalues are ↓

Testbed for control of networked multi-agent systems

Agent 1

Agent 2 Agent 3

6Tele-Operation

from www.nasa.gov

− Time Delay: information transmission between the local and remote environment (between “master” and “slave”).

2( ) 0 ( ) 0 ( ) 0( ) ( )

( ) 0 ( ) 0 ( ) 0s s s s s s s s s

m mm m m m m m m m m

x t A B K x t B R x t T B GF t F t T

x t A B K x t B R x t T B+ −

= + + + − + −

Cs∫Bs ys

As

delay

Rm

Cm∫Bm

Am

ymFm

delay

G2

Ks

Km

Rs

delay

slavem

aster

[Azorin, 04]

* Padé approximation can’t meet requirement,

( )Re 5rightmostλ < −

7Delay Systems Delays are inherent in many systems, e.g.

– In engineering, biology, chemistry, economics, etc. [Niculescu, 2001]– Control loops: sensors, actuators, computational delays.

What is challenging?– Delay operator leads to infinite spectrum due to

– Difficulty in 1) determining stability, 2) designing controllers

e−sh =(−sh)k

k!=1− sh +

12!

(sh)2

k= 0

∞

∑ −13!

(sh)3 + ...

( ) ( )x t ax t=

time delay

( ) ( ) ( )dx t ax t a x t h= + −

Re(s)

Imag(s)

aλ =

Imag(s)

?λ =

Re(s)

Ordinary differential equations Delay differential equations

8Delay Differential Equations (DDEs)• History

– 18th century: Laplace and Condorcet– Fundamental theory: Bellman [1963], Hale and Lunel [1993], etc.– Approximate, numerical, graphical methods

• Considered linear time-invariant systems with a single delay, h:

• This type of equations can represent those systems:– Tele-operation, formation flight, neural network, etc.– Machine tool chatter and wind turbines – Bio (HIV/HBV/HCV) dynamic model – Automotive powertrain systems control due to fluid transport

t

x(t)

-h 0

g(t)

x0

h 2h 3h 4h0

( ) ( ) ( ) ( ), 0( ) ( ), [ ,0)( ) , 0

t t t h t tt t t ht t

+ + − = >= ∈ −= =

dx Ax A x Bux gx x

9Existing MethodsRepresentative current approaches:

– Approximation, e.g., Padé approximation of the delay:

– Prediction-based methods, e.g., Smith predictor [Smith, 1958], finite

spectrum assignment (FSA) [Manitius and Olbrot, 1979]

– Bifurcation analysis, e.g., [Olgac et al., 1997]

– Numerical solutions, e.g., dde23 in Matlab (Runge-Kutta)

– Graphical methods, e.g., Nyquist [Desoer and Wu, 1968]

– Lyapunov methods, e.g., LMI, ARE [Niculescu, 2001]

– Great number of monographs devoted to this field of active research:

[Richard, 2003; Yi et al., 2010]

12

12

sh

hs

e hs−

−≈

+

s iv= ±

10

10

Padé Approximation

( ) ( ) ( )i nfinite dimensional

finite dimensional

2

2( ) 0 ( ) 02

2 ( ) 2 2 0

(2 ) 2( ) 2 0

shd d

d

d d

shs a k a e s a k ash

s sh a k sh a sh

s h s ah kh a h a k a

− −− + − = → − + − =

+

→ + − + + − − =

→ + − − + − + − =

The Padé approximation is1

21

2

sh

hs

e hs−

−=

+

Then,

DDE becomes a simple 2nd order ODE

e−sh =(−sh)k

k!=1− sh +

12!

(sh)2

k= 0

∞

∑ −13!

(sh)3 + ... ( )time-delay

( ) ( ) ( )dx t ax t a x t h kx t= + − +

11Padé Approximation• With the parameters 1, 2, 1da a h= = − =

2

2

(2 ) 2( ) 2 0

( 1 ) 2 2 0d ds h s ah kh a h a k a

s s k k+ − − + − + − =

→ + − − + − =

• For the value of k= -1.1, the above 2nd order equation has two stable poles, but the gain is applied to the original system and causes to instability.

But the system was still unstable

-2 -1.5 -1 -0.5 0 0.5-15

-10

-5

0

5

10

15

Real (λ)

Imag

inay

( λ)

eigenvalues

Unstable

12

12

Rational Function Approximation• Higher order Padé:

• Battle & Miralles (2000)

• Inappropriate to be used in designing feedback control strategies, because the approximation methods cannot be free from errors.

( )

( )

2

2

12 12

12 12

hs

hshs

ehshs

−− +

≅

+ +

4 4 2 2 2 3 3 2 3 3( ) , ( ) 6 6 24( )

hs p hse p hs hs h s h s h sp hs

π π π π− −≅ = + + + −

13

13

Rational Function Approximation• “Approximating the delay term by means of

rational function by truncating infinite series, such as Padé approximation, constitutes a limitation in analysis and may lead to unstable behaviors of the true system.”

e.g.) [1] J. P. Richard, "Time-delay systems: an overview of some recent advances and open problems," Automatica, vol. 39, pp. 1667-1964, 2003.[2] G. J. Silva and Datta, A. Bhattacharyya, S.P., "Controller design via pade approximation can lead to instability," in Proceedings of the 40th IEEE Conference on Decision and Control 2001, pp. 4733-4737. [3] PID controllers for time-delay systems /Guillermo J. Silva, Aniruddha Datta, S.P. Bhattacharyya.. Boston : Birkhäuser, c2005…

14Bifurcation Analysis

Originally stable

Stability may change.

Re

Imag

Re

Imag

e−sh =(−sh)k

k!=1− sh +

12!

(sh)2

k= 0

∞

∑ −13!

(sh)3 + ... s iv= ±

15

15

Bifurcation analysis• Given a system of delay differential equation with stability for

τ=0, and with the characteristic equation

• If the Eq. has a pure imaginary root, iv,

• Separated into its real and imaginary parts

• So,

• If there is a solution, v, then stability may change.

∑∑=

−

=

=+M

i

ii

N

i

ii bea

110λλ λτ

0)()( 21 =+ − τiveivPivP

0))sin()))(cos(()(()()( 2211 =−+++ ττ vivviQvRviQvR

0)cos()()sin()()(0)sin()()cos()()(

221

221

=+−=++

ττττ

vvQvvRvQvvQvvRvR

16Smith Predictor

0 1 2 3 4 50

0.5

1

1.5

Time, t

Res

pons

e, y

1M

pM

KGsτ

=+

ye u

-

+r

1M

pM

KGsτ

=+

she−ye u

-

+r

* Two responses show disagreement due to

delay

desired

real with time-delay

where 0.5; 0.2; 0.5; 2.5; 1M n Mh Kτ ζ ω= = = = =

IS p

KG Ks

= +

IS p

KG Ks

= +

17Smith Predictor: Basic Concept

y v

Smith Predictor

Gr(s) GP(s)

(1-e-sT)GP(s)

Gc

e-sh+ yu+- -

yGr(s) Gp(s)+

yu

-

equivalent

e-sh

* Delay is moved outside the feedback loop.

1 1

shc p r psh

shc p r p

G G e G Ge

G G e G G

−−

− =+ +

( )1

rc sh

r p r p

GG sG G G G e−=

+ −

18Smith Predictor: Basic concept (II)

1 1

sTc p r psT

sTc p r p

G G e G Ge

G G e G G

−−

− =+ +

( )1

rc sT

r p r p

GG sG G G G e−=

+ −

y v

( )

( )

( ) 1 ( )

( )

( ) ( )

sT sTp p

p

sT sTp

sT

y t T

y v e G s u e G s u

G s u

G s e u s e

y e

− −

−

+

+ = + −

=

= ⋅

= ⋅

* Why it is a “predictor”:

Smith Predictor

Gr(s) Gp(s)

(1-e-sT)Gr(s)

Gc

e-sT+ yu+- -

(→ preceded response)

19Discretizing• Discretizing method

–– Can handle delay terms in a easy way,– If time step is small (for accuracy) and delay time, h, is large,

then the dimension of the delayed system becomes high (n=100, 1000,….).

– Need processes:

System in t-domain

System in z-domain

Design feedback

controller in z-domain

System in t-domain

Also, if delays are uncertain, or vary over time, analysis using discretization may not be possible.

{ } 1Delay : ( 1) rational function of .Z t zz

δ − = ⇒

20Outline

• Introduction

• Modeling of Delay Systems and Solution – Modeling– Solution to DDEs




21StrategyOrdinary differential equations Delay differential equations

• Solution

• Stability

• Controllability & Observability

• Design of feedback control (w/ observer)

• Transient response & robust control

?

( )0. ., ( ) te g t e= Ax x • Solution (?)

22

• Definition:

• Infinite number of branches

• Already embedded in MATLAB, Maple, Mathematica, etc.• Contributions: Lambert [1758], Euler [1779], Corless et al.,

[1996], Asl and Ulsoy [2003]…• Used to study jet fuel, combustion, enzyme, molecular forces.

The Lambert W Function

(k = 0)

( 0)k ≠

Principal branch

( )( ) kW xkW x e x=

( ) ( ) ( )( )( )( )( )

( )( )

( )

( ) ( )

0 1

ln lnln ln ln

ln

1where, 1 (Stirling Cycle Numbers)1!

ln ln 2

m

kk k k lm l m

l m k

llm

k

xW x x x c

x

l mc

lm

x x ikπ

∞ ∞

+= =

= − +

+ = − +

= +

∑∑

( ) 1

01

( )!

nn

n

nW x x

n

+∞

=

−= ∑

( - , , -1, 0, 1, , )k = ∞ ∞

-2 -1 0 1 2 3-4

-3

-2

-1

0

1

x

W(x

)

0W

1W−

23

( ) ( )

( ) ( )( )

( )( )

1

ahd

h s a ahd

W a heah ahd d

ahd

ahd

h s a e a he

W a he e a he

s a h W a he

s W a he ah

+

−

+ = −

− = −

+ = −

= − −

Solution of DDEs (Free Scalar)

Roots in terms of the parameters, a, ad , h !

Using the definition of the “Lambert W function “

• Scalar (first-order) delay differential equation (DDE)

0

( ) ( ) ( ) 0, for 0( ) ( ), for [ ,0)( ) , for 0

dx t ax t a x t h tx t g t t hx t x t

+ + − = >

= ∈ −= =

• Obtain a transcendental characteristic equation:

( )( ) W xW x e x=

0shds a a e−+ + =

( )( )multiplying h s ahe +

* [Asl & Ulsoy, ASME JDSMC, 2003]

24Example: Free Scalar DDEs

[ )0

with preshape function ( ) 1, for -1,01, for 0

g t tx t

= ∈

= =

( ) ks t Ik

kx t e C

∞

=−∞

= ∑

-6 -5 -4 -3 -2 -1 0-100

-50

0

50

100

Real (Sk: roots of the characteristic equation)

Imag

(Sk: r

oots

of t

he c

hara

cter

istic

equ

atio

n)

k=0 (principal branch)k=1

k=-1

k=2k=3

k=-2k=-3

ex) a =1, ad = -0.5, h =1

0

( ) ( ) ( ) 0, for 0( ) ( ), for [ ,0)( ) , for 0

dx t ax t a x t h tx t g t t hx t x t

+ + − = >= ∈ −= =

( )1 ahk k dS W a he a

h= − −

Poles Analytical Solutions

25Quadrotor Dynamics

T4 T1 XB

XV

T

ω4 ω1

OB,V

d

4 1

T3 T2

φBθB

ω3

φV2

ω2

YB

YV

mg

ZB,V

3 2

XI

ZIYI

θV1

φB,V

OI

�̈�𝑧(𝑡𝑡) = 𝑔𝑔 −4𝑎𝑎𝜔𝜔2 t

𝑚𝑚

𝑇𝑇 = �𝑖𝑖=1

𝑖𝑖=4

𝑇𝑇𝑖𝑖 , 𝑇𝑇𝑖𝑖= 𝑎𝑎𝜔𝜔𝑖𝑖2 𝑎𝑎 > 0

• Total upward thrust, T, on the vehicle is

• Function of rotor speed, ω.• The thrust constant, a, depends on the air

density, the cube of the blade radius, etc. • Taking the equation of motion only in the z-

direction:

• Nonlinearity, noise, connection, disturbance, etc.

26Control Overview• Used ‘AR Drone Simulink Development Kit V1.1’ from

Mathworks.com.• Cascade control is beneficial only if the dynamics of the inner loop

are fast compared to those of the outer loop.

• For autonomous aerial robots, typical delay is around 0.4 ± 0.2𝑠𝑠. Oscillations can emerge due to delays [Vasarhelyi, 2014].

• Large delays (> 0.20𝑠𝑠) causes increased torque dramatically [Ailon, 2014].

1𝑠𝑠

−+

𝑧𝑧(𝑡𝑡)�̇�𝑧𝑎𝑎𝑎𝑎𝑎𝑎(𝑡𝑡)

𝑧𝑧𝑑𝑑𝑑𝑑𝑑𝑑(𝑡𝑡)[−1 1]

𝐀𝐀𝐀𝐀.𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃 𝟐𝟐.𝟎𝟎 𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰 𝐰𝐰𝐰𝐰𝐃𝐃 𝐀𝐀𝐀𝐀𝐀𝐀𝐃𝐃𝐃𝐃𝐀𝐀𝐰𝐰𝐀𝐀𝐀𝐀𝐰𝐰𝐃𝐃𝐀𝐀 𝟏𝟏𝟏𝟏𝐰𝐰 𝐎𝐎𝐀𝐀𝐃𝐃𝐃𝐃 𝐒𝐒𝐒𝐒𝟏𝟏𝐃𝐃𝐰𝐰𝐀𝐀

�̇�𝑧𝑟𝑟𝑑𝑑𝑟𝑟(𝑡𝑡)

𝐀𝐀𝐀𝐀.𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃 𝟐𝟐.𝟎𝟎+ 𝐏𝐏𝐏𝐏𝐃𝐃 𝟏𝟏𝟏𝟏𝐰𝐰 𝐎𝐎𝐃𝐃𝐀𝐀𝐃𝐃𝐃𝐃 𝐒𝐒𝐒𝐒𝟏𝟏𝐰𝐰𝐃𝐃𝐀𝐀

Controller

MATLAB/Simulink Program

𝑒𝑒(𝑡𝑡)

Kp

P_Gain

1s

Plant ScopeReference Height

Time DelayBlock

simout

To Workspace

Control Input Saturation Block

𝑧𝑧𝑧𝑧𝑑𝑑𝑑𝑑𝑑𝑑

27Delay Observation • If there is no delay, increase in the gain,

Kp, does not destabilize the system.• In receiving data,

• In order to design effective controllers (stability boundary), knowing the delay is helpful.

6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, t (seconds)

Alti

tude

, z (m

)

Kp=4

6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, t (seconds)

Alti

tude

, z (m

)

Kp=5

28Time-Delay Estimation• Delays are introduced by

– Computation loads, actuation, and signal transmission/processing.

• Estimation is not straightforward– Even if considerable efforts have been made, there is no common approach.

[Belkoura et. al. (2009)]– “Most of the existing methods for transfer function identification do not consider

the process delay (or dead-time) or just assume knowledge of the delay” [Ljung(1985); Sagara & Zhao (1990) ]

– “None of the existing linear filter methods directly estimate the time delay except for some very special type of excitation” [Ahmed et al. (2006)]

• The delay knowledge can benefit [Belkoura and Richard (2006)]

– Many of control techniques (e.g., Smith predictors, Finite Spectrum Assignment, etc.)

29Methods for Estimation of Delays• Finite-dimensional Chebyshev spectral continuous time

approximation (CTA) [Torkamani and Butcher (2013)]• Graphical methods [Mamat and Fleming (1995); Rangaiah and

Krishnaswamy (1996); Ahmed et al. (2006)]• Integral equation approach [Wang and Zhang (2001)]: integrate

signals • Discrete and Continuous (“more familiar to practicing control

engineers” [Ahmed et al. (2006)]) • A cost function for a set of time delays in a certain range [Rao and

Sivakumar (1976); Saha and Rao (1983)]• Approximation: Padé approximation, the Laguerre expansion, etc.

→ additional parameters, errors • Frequency-domain maximum likelihood [Pintelon and Biesen

(1990)]

30Altitude Response: Percent Overshoot

Flight1 2 3 4 5

Mo (%) 2.300* 2.290 2.300 2.270 <1

( )( ) ( )2 2

Re0.7684

Re Im

s

s sζ = = −

+

𝑠𝑠 =1𝑇𝑇𝑑𝑑𝑊𝑊(𝑇𝑇𝑑𝑑𝑎𝑎1𝑒𝑒−𝑎𝑎𝑜𝑜𝑇𝑇𝑑𝑑) + 𝑎𝑎𝑜𝑜

0.3598 0.36dT s= ≈

𝑀𝑀𝑜𝑜 = 100𝑒𝑒−𝜁𝜁𝜋𝜋

�√(1−𝜁𝜁2

*When compared to Simulink, Td=0.37s.

31Locus of Rightmost Roots

• As the time delay increases

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5-3

-2

-1

0

1

2

3

Real(s)

Imag

(s)

h =0.3sh =0.4s

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2-150

-100

-50

0

50

100

150


0.3598h s=• Spectrum when

(s)

(s)

so

s-1

ξ = cos𝜃𝜃 =)𝑅𝑅𝑒𝑒(𝑠𝑠

𝑅𝑅𝑒𝑒(𝑠𝑠) 2 + 𝐼𝐼𝑚𝑚(𝑠𝑠) 2

32Stability Using the Estimated Delay

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2-150

-100

-50

0

50

100

150


Imag

(Sk: r

oots

of t

he c

hara

cter

istic

equ

atio

n)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0-150

-100

-50

0

50

100

150


Imag

(Sk: r

oots

of t

he c

hara

cter

istic

equ

atio

n)

Kp=4

Kp=5

TransportDelay

Step ScopeSaturation

1s

Integrator

k

Gain

0.3598dT s=

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, t (seconds)

Alti

tude

, z, m

Kp=3

Kp=4

Kp=5

6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

1.4

Time t (seconds)

Alti

tude

, z (m

)

Kp=4

6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

1.4

Time t (seconds)

Alti

tude

, z (m

)

Kp=5

33Non-homogeneous DDEs

-1 0 1 2 3 4 5 6 7 8 9-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Numerical Integration

Lambert function methodwith 7 branches

Forced Soln. Form[Bellman et al. 1963]

Free solution[Asl & Ulsoy, JDSMC 2003]

*[Yi et. al., CDC 2006]

0

( ) ( , ) ( )t

x t t bu dξ ξ ξ= Ψ∫

) ( , ) ( , )

( , ) ( , ) ) ( , ) 1) ( , ) 0 for

d

a t a t t - h t

a t a t T t - hb t tc t t

ξ ξ ξξ

ξ ξ ξ

ξ ξ

∂Ψ = Ψ ≤ <

∂= Ψ + Ψ + <

Ψ =Ψ = >

0

( ) ( ) ( ) ( ) , 0( ) ( ), [ ,0)( ) , 0

dx t ax t a x t h bu t tx t g t t hx t x t

+ + − = >

= ∈ −= =

( )

0free solution forced solution

(Asl and Ulsoy, 2003) (Yi et al., 2006)

( ) ( )k k

tS t S tI N

k kk k

x t e C e C bu dξ ξ ξ∞ ∞

−

=−∞ =−∞

= +∑ ∑∫

Solution formCondition

External force

*u(t)=cos(t)

34Generalized to Systems of DDE’s

0

( ) ( ) ( ) ( ), 0( ) ( ), [ ,0)( ) , 0

t t t h t tt t t ht t

+ + − = >= ∈ −= =

dx Ax A x Bux gx x

1

2

( )( )

* ( ) , ,

( )

n n n n

n

x tx t

t

x t

× ×

= ∈ ∈

dx A A

1 ( )( )

ahk d

k

W a he a tIh

kx t e C

∞ − −

=−∞

= ∑1 ( )

( )h

k

k

he tIh

kt e

∞ − −

=−∞

= ∑A

dW A Ax C

( )Because h hhe e e +× ≠ × ⇒ ≠d dA A AAd dA A A A

1 ( )( )

k kk

h tt I Ih

k kk k

t e e ∞ ∞ − −

=−∞ =−∞

= =∑ ∑dW A Q A

Sx C C ( )( )* satisfies ' ( ) 'k kh hk k kh e h− −− = −dW A Q A

d dQ W A Q A

( )

0free forced

( ) ( )k k

tt tI N

k kk k

t e e dξ ξ ξ∞ ∞

−

=−∞ =−∞

= +∑ ∑∫S Sx C C Bu

35Systems of DDEs: Example

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Time, t

Res

pons

e, x

(t) numericalwith k=0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Time, t

Res

pons

e, x

(t) numerical

with k=-1, 0, 1

1 1 1

2 2 2

( ) ( ) ( 1)1 3 1.66 0.697( ) ( ) ( 1)2 5 0.93 0.330

x t x t x tx t x t x t

−− − − = + −− −

1

0

1

1.3663 3.9491,

3.2931 9.3999

1.7327 0.0000,

6.5863 0.0000

1.3663 3.9491,

3.2931 9.3999

I

I

I

ii

ii

ii

−

+ = +

− + = − +

− = −

C

C

C

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Time, t

Res

pons

e, x

(t) numerical

with k=-3, -2, -1, 0, 1, 2, 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Time, t

Res

pons

e, x

(t) numericalwith k=0

with k=-1, 0, 1

with k=-3, -2, -1, 0, 1, 2, 3

0.3499 4.9801 1.6253 0.1459 0.3055 1.4150 0.3499 4.9801 1.6253 0.14592.4174 0.1308 5.1048 4.5592 2.1317 3.3015 2.4174 0.1308 5.1048 4.5592

1 0 1( )i i i i

t t ti i i iI I It e e e

− − − + − − + − − + − − − − − +

−= + + + +x C C C

k = 0 (principal branch)

k = 1k = -1

36Analogy to Systems of ODEsODEs DDEs

Scalar case

Matrix-Vector case

0

( ) ( ) ( ), 0( ) , 0

x t ax t bu t tx t x t

+ = >= =

( )0

0

( ) ( )t

at a tx t e x e bu dξ ξ ξ− − −= + ∫

0

( ) ( ) ( ) 0( ) 0t t t tt t

+ = >= =

x Ax Bux x

( )0

0

( ) ( )t

t tt e e dξ ξ ξ− − −= + ∫A Ax x Bu

0

( ) ( ) ( ) ( ) , 0( ) ( ), for [ ,0); ( ) , 0

dx t ax t a x t h bu t tx t g t t h x t x t

+ + − = >= ∈ − = =

( )

0

( ) ( )

1where, ( )

k k

tS t S tI N

k kk k

ahk k d

x t e C e C bu d

S W a he ah

ξ ξ ξ∞ ∞

−

=−∞ =−∞

= +

= − −

∑ ∑∫

0

( ) ( ) ( ) ( ), 0( ) ( ), for [ ,0); ( ) , 0t t t h t tt t t h t t

+ + − = >

= ∈ − = =dx Ax A x Bu

x g x x

( )

0

( ) ( )

1where, ( )

k k

tt tI N

k kk k

k k k

t e e d

hh

ξ ξ ξ∞ ∞

−

=−∞ =−∞

= +

= − −

∑ ∑∫S S

d

x C C Bu

S W A Q A

* Analogy between solution forms enables extension of control methods for systems of ODEs to DDEs.

(Using Lambert Wfunction)

37Outline

• Introduction

• Derivation of Solution to Delay Differential Eqs.




38Stability Analysis

Unstable Stable

DDEs have an infinite eigenspectrum: Rightmost eigenvalues among them?

Proven for scalar case, and some special systems of DDEs, but not for general cases.

for - , , -1, 0, 1, , k k = ∞ ∞S

[ ]0

If has no repeated zero eigenvalues, thenmax Re{eigenvalues of } Re{all other eigenvalues of }k≥

dAS S

39Controllability & Observability Conditions for controllability & observability of DDEs

• Using the developed solution form, and previous results by [Weiss, 1967]• Balanced realization based on Gramians

full rank:

linearly independent

rows:

full rank:

linearly independent

columns:

*[Yi et al., IEEE TAC 2008]

40Networks of Agents: Lab Work

Chris Thomas (Senior, ME) has been supported for autonomous control of AR Drone.

Testbed for control of networked multi-agent systems

Ground Robots by Dr. Robot for Autonomous Navigation

Agent 1

Agent 2 Agent 3

Myrielle Allen-Prince (MS, ME) has been supported for networks of drones.

41Motors: PV plus Integral Control• Stable responses with good

convergence.

• But non-zero steady state

errors in results on the

previous slide.

• Not in simulation or theory.

• Due to backlash, friction, and

inductance in circuits.

• Need integral control →

42

42

PV-MRAC

Effects of Disturbance Rejection: Payload

o Stability bounds: 150g o 100g: both performed wello 150g: Adaptive: safe operation & landing PV: crash landing

o Instability: 155g

Designed PV Vs PV-MRAC

Implementation on DroneVideo drone

43

• Software Package Development (being extended to multi-agent systems)– Real-time Control of Aerial and Ground Robots,– Graphic User Interface (GUI) for Simulation and Algorithm

Implementation.

Networks of Agents: Simulation

44Outline

• Introduction





45Eigenvalue Assignment Step 1. Select desired rightmost eigenvaluesStep 2. “Linear” feedback controller

Step 3. With the new coefficients,

Step 4. Equate the selected eigenvalues to those of the matrix S0 as (i.e., k = 0 branch only)

( ) ( ) ( ) ( )t t t h t= + − +dx Ax A x Bu

unknown unknown

( ) ( ) ( )t t t h= + −du K x K x

ˆ ˆ

( ) ( ) ( ) ( ) ( )t t t h= + + + −

d

d dA A

x A BK x A BK x

, , for 1, ,i desired i nλ =

← closed-loop system

0 0ˆ ˆ( )

0 0ˆ ˆ( ) h hh e h+ =dW A Q A

d dW A Q A

0 0 01 ˆ ˆ( )hh

= +dS W A Q A

0 ,( ) , for 1, ,i i desired i nλ λ= =S

46Illustrative Example Example from [Yi et al, Journal of Vibration and Control]

Without feedback control– Rightmost eigenvalues: 0.1098 (unstable); –1.1183

Desired: – 2 & – 4

Resulting linear feedback:

-60 -50 -40 -30 -20 -10 0 10 -150

-100

-50

0

50

100

150

Real (λ)

Imag

inay

(λ)

-5 -4 -3 -2 -1 0 1 2 -3

-2

-1

0

1

2

3

Real (λ)

Imag

inay

(λ)

0 0 1 1 0( ) ( ) ( 0.1) ( )

0 1 0 0.9 1t t t t

− − = + − + −

x x x u

without control

[ ] [ ]( ) 0.1687 3.6111 ( ) 1.6231 0.9291 ( )t t t h= − − + − −

dK K

u x x

with control

unknown unknown

( ) ( ) ( )t t t h= + −du K x K xwith

47Transient Response

0.2σ− = −

1.0

0.5

0.2

dω =

↑

↑

Re

Imag Responses with different imaginary parts

Rule of thumb Assigning the dominant

eigenvalues (real and imaginary parts) with a linear controller.

Possible to meet time-domain specifications of DDEs following design guidelines for ODEs.

0 5 10 15 20 25 300

0.5

1

1.5

2

Time, t

Res

pons

e, x

(t)

-0.2±1.0i

-0.2±0.5i-0.2±0.2i

tr ts Mp tp

σ ↑ ↓ ↓ ↓ fixed

ωd ↑ ↓ fixed ↑ ↓

ωn ↑ ↓ ↓ fixed ↓

* [Yi et al., ASME JDSMC]

48

-4 -3 -2 -1 00.2

0.4

0.6

0.8

1

Rightmost Eigenvalues

Stab

ility

Rad

ii, r

Robust Control

Combined with the ‘stability radius’ concept with algorithms in [Hu and Davison, 2003].

As the eigenvalue moves left, then the stability radius increases, which means, more “robust”.

Comparing uncertainty and stability radius, one can choose the appropriate positions of the rightmost eigenvalues for robust stability.

stay stable with uncertainties

can be destabilized

A B-4 -3 -2 -1 0 1

-1

-0.5

0

0.5

1

Real(λ)

Imag

( λ)

-4 -3 -2 -1 0-1

-0.5

0

0.5

1

Real(λ)

Imag

( λ)

* [Yi et al., ASME JDSMC]

ISC

Stable, but not robustly stable

Stable, and

robustlystable

49Linear Feedback & Observer: Concept

Feedback Controller

( ) ( )( )

늿 늿 ( )

늿 늿 ( )

= + +

= + + − +

− = − + − − −

⇒ = − +

d d

d d

d d d

d d

x Ax A x Bu

x Ax A x L y Cx Bu

x x A x x A x x L y Cx

e A LC e A e

Observer (state estimator)

( )

= + +

= − +

= − + +

d d

d d

x Ax A x Buu Kx rx A BK x A x Br

stabilize by choosing K.

stabilize by choosing L.

50Diesel Engine: Feedback Control• Diesel Engine: linearized system equation with a transport time delay at N=1500 RPM

• This system has an unstable eigenvalue at 0.9225 (> 0) without feedback

• With feedback control:

• Desired position: -10

)()( tt Kxu =

*eigenvalues by k=0

[ ]

27 3.6 6 0 0 0 0.26 0( ) 9.6 12.5 0 ( ) 21 0 0 ( 0.06) 0.9 0.8 ( )

0 9 5 0 0 0 0 0.18

( ) 0 1 0 ( )

ht t t t

t t

− = − + − + − − −

=dA A B

C

x x x u

y x

51Diesel Engine: State Observer Observer:

[ ]1

2

3

27 3.6 6 0 0 09.6 12.5 0 , 21 0 0 , 0 1 0 ,0 9 5 0 0 0

LLL

− = − = = = −

dA A C L Unstable(one positive real eigenvalue)

1

2

3

27 3.6 6 0 0 0늿 9.6 12.5 0 , 21 0 0

0 9 5 0 0 0

LL

L

− − = − = − − = = − −

d dA A LC A A Goal:Find stabilizing “L”

6.47299.5671

16.0959

=

L

*1.5 times faster than feedback-150 -100 -50 0

-500

0

500

Real(s)

Imag

inar

y(s)

eigenvalues for k=0

-17 -16.5 -16 -15.5 -15 -14.5 -14-10

-5

0

5

10

52Response of Controlled System

1

2

* Reference inputs: square wave with amplitude 0.5: sine wave with amplitude 20 and frequency 0.7( )

rr Hz

0 0.5 1 1.5 2-0.5

0

0.5

Time, t

inta

ke m

anifo

ld p

ress

ure

x1

estimated x1

0 0.5 1 1.5 2-0.5

0

0.5

Time, t

exha

ust p

ress

ure

x2

estimated x2

0 0.5 1 1.5 2-0.5

00.5

1

Time, t

com

pres

sor p

ower

x3

estimated x3

Feedback Control with Observer:

*[Yi et al., JFI 2010]

0 2 4 6 8 10-500

0

500

1000

1500

2000

Time, t

Stat

e V

aria

bles

intake manifold pressure (p1)

exhaust pressure (p2)

compressssor power (Pc)

↑ was unstable

53Outline

• Introduction





54Summary and ContributionsGiven a system of LTI DDEs with single delay h

1. Derive the solution (free & forced): [DCDIS 2007]

3. Check conditions for controllability and observability: [IEEE TAC 2008]

4. Design feedback controller via eigenvalue assignment: [JVC (in press)]

* For systems of ODEs, the above approach is standard.

* Using the Lambert W function-based approach we have extended this to DDEs.

2. Determine stability of the system: [MBE 2007]

5. Transient response & Robust stability : [ASME JDSMC (in press)]

6. Observer-based feedback control: [JFI 2010]

Applications in engineering and biology

55Advantages• Stability analysis

– Accuracy compared to approximation approaches– ‘How stable’ compared to bifurcation analysis

• Feedback control w/ observer– Robustness compared to Smith predictor a– Ease of implementation compared to nonlinear methods (e.g.,

FSA, which has integrals in controllers)

• Controllability/observabiltiy– Quantitative information: ‘How observable/controllable’, and

‘balanced realization’ compared to algebraic conditions

56Concluding Remarks & Future Work

• Still open problems, and long way to go.• Trying to find an effective method, which is more intuitive

and similar to ODEs – Focused on networked systems.– More challenges: noise, time-variance, disturbances, and nonlinearity.

• Used – Time-domain responses and – Analytical solutions in terms of Lambert W function.

• Extended stability analysis and control design. • Multiple Drones• Nonlinear controller: MRAC and Gain scheduling • Path following• Higher-order systems’ delay estimation

september 11, 2015 analysis and control of systems considering signal transmission...

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