september 11, 2015 analysis and control of systems considering signal transmission...
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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
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2Outline• Introduction
– Motivations– Background
• Modeling of Delay Systems and Solution
• Methods for Analysis
• Methods for Control
• Concluding Remarks
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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]
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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
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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
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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 < −
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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
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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
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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= ±
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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= + − +
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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
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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− −≅ = + + + −
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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…
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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= ±
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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
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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
= +
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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−=
+ −
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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)
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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 − = ⇒
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20Outline
• Introduction
• Modeling of Delay Systems and Solution – Modeling– Solution to DDEs
• Methods for Analysis
• Methods for Control
• Concluding Remarks
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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 (?)
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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−
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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]
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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
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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.
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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
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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
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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.)
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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)]
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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.
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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
Real (Sk: roots of the characteristic equation)
0.3598h s=• Spectrum when
(s)
(s)
so
s-1
ξ = cos𝜃𝜃 =)𝑅𝑅𝑒𝑒(𝑠𝑠
𝑅𝑅𝑒𝑒(𝑠𝑠) 2 + 𝐼𝐼𝑚𝑚(𝑠𝑠) 2
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32Stability Using the Estimated Delay
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2-150
-100
-50
0
50
100
150
Real (Sk: roots of the characteristic equation)
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
Real (Sk: roots of the characteristic equation)
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
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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)
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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
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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
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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)
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37Outline
• Introduction
• Derivation of Solution to Delay Differential Eqs.
• Methods for Analysis
• Methods for Control
• Concluding Remarks
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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
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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]
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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.
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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 →
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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
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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
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44Outline
• Introduction
• Derivation of Solution to Delay Differential Eqs.
• Methods for Analysis
• Methods for Control
• Concluding Remarks
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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
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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
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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]
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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
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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.
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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
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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
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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
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53Outline
• Introduction
• Derivation of Solution to Delay Differential Eqs.
• Methods for Analysis
• Methods for Control
• Concluding Remarks
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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
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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
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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