1. LINEARIZATION MODELING FOR NON-LINEARIZATION MODELING FOR NON-
SMOOTH DYNAMICAL SYSTEMS WITHSMOOTH DYNAMICAL SYSTEMS WITH
APPROXIMATED SCALAR SIGN FUNCTIONAPPROXIMATED SCALAR SIGN FUNCTION
ROY G. PERRY COLLEGE OF ENGINEERING
DEPARTMENT OF ELECTRICALAND COMPUTER ENGINEERING
Dr. Warsame H. Ali
Supported in part by the US Army Research Office under the research cooperative
agreement grant, W911NF-04-2-0054 &Department of Energy NNSA

2.  Introduction
 Proposed Solution
 Approximated Scalar Sign Function
 Optimal Linearization
 Input Delay Compensation via Digital Redesign
 Illustrative Examples
 Conclusions
 Future Work
 References
ICGST 2012 presented July 16, 2012 7/16/2012

3.  Overview…
 Merits:
 Reduction in wiring cost
 Flexibility in reconfiguration and scalability
 Ease of fault diagnosis and maintenance
 Potential applications
 Space and terrestrial exploration
 Automobiles
 Tele-operations
ICGST 2012 presented July 16, 2012 7/16/2012

4.  Overview…
 Time delay issue
 Network transmission induces time delay in the control loop.
 Time delays may degrade the control performance or even make the
system unstable [1].
 Research efforts on delay compensation
 The optimal stochastic method approaches the problem as a linear-
quadratic-Gaussian problem [2].
 The robust control method considers the delays as multiplicative
perturbations on the system [3].
 Linear matrix inequality based sufficient conditions is derived in [4] for
the stability for nonlinear systems.
 Queuing strategies are proposed by many researchers for coping with
the network delay for both linear and nonlinear plant [5,6].
ICGST 2012 presented July 16, 2012 7/16/2012

5.  Overview…
 Sign-function-constrained Non-smooth System
 Refers to the system modeled using sign function or absolute value
function.
 Typical examples include friction, hysteresis, saturation, backlash, etc.
Coulomb friction
( )sign
where
C
C
F F v
F Nµ
=
=
v
F
N
ICGST 2012 presented July 16, 2012 7/16/2012

6.  Overview…
 Sampled-data system
System with continuous-time plant and discrete-time
controller have been becoming the most population
implementation style in current control systems.
 System model in concern
ICGST 2012 presented July 16, 2012 7/16/2012

7. Nonsmooth system
Smooth nonlinear model
Approximation
Linear model
Analog controller
Linearization
Linear control theory
Digital controller
Digital redesign for delay compensation
ICGST 2012 presented July 16, 2012 7/16/2012

8. Nonsmooth system
Smooth nonlinear model
Approximation
Linear model
Analog controller
Linearization
Linear control theory
Digital controller
Digital redesign for delay compensation
ICGST 2012 presented July 16, 2012 7/16/2012

9. ( )
( ) ( )
( ) ( )
1 1
1 1
j j
j j j
z z
sign z
z z
+ − −
=
+ + −
-0.5 0 0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
σ
signj
(σ)
(a)
-10 -5 0 5 10
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
σ
sign
j
(σ)
(b)
j=10
j=20
j=30
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-5
0
5
10
15
20
25
30
35
σ
d(sign
j
(σ))/dσ
j=10
j=20
j=30
( ) ( )lim j
j
sign z sign z
→∞
=
ICGST 2012 presented July 16, 2012 7/16/2012

10. Nonsmooth system
Smooth nonlinear model
Approximation
Linear model
Analog controller
Linearization
Linear control theory
Digital controller
Digital redesign for delay compensation
ICGST 2012 presented July 16, 2012 7/16/2012

11.  Linearization
 Jacobian Linearization [9]
( ) ( )x f x G x u= +& k kx A x B u= +&
( )x f x=& kx A x=&
( ) ( ) ( ) H.O.T.k k kf x f x x x= + ∇ − +
( ) ( ) ( )k k k kf x x f x f x x≈ ∇ + −∇
ICGST 2012 presented July 16, 2012 7/16/2012

12.  Jacobian Linearization [9]…
 Usually gives an affine other than linear model so that extra
efforts are required for controller design and stability analysis.
 The only exception is when the operating point is not only the
equilibrium but the origin, which cannot be guaranteed in a
nonlinear control process.
ICGST 2012 presented July 16, 2012 7/16/2012

13.  Optimal Linearization [Teixeira and Zak,10]
 An optimal linear model can be obtained at any operating
point, which has the exact dynamics of the nonlinear system at
the operating point and minimum approximation error (in the
least square sense) in the vicinity of the operating point.
( )
( ) ( )
( )
2
2
0
0 0
k k k T
k k k
kk
k
f x f x x
f x x for x
xA
f for x
 −∇
∇ + ≠
= 

∇ =
( )k kB G x=
ICGST 2012 presented July 16, 2012 7/16/2012

14. Nonsmooth system
Smooth nonlinear model
Approximation
Linear model
Analog controller
Linearization
Linear control theory
Digital controller
Digital redesign for delay compensation
ICGST 2012 presented July 16, 2012 7/16/2012

15.  Digital Redesign
 Digital redesign is a two-step procedure: an analog controller is
firstly designed to meet the stability and performance
requirement, and then redesigned to the digital one.
 Digital redesign is advantageous in terms of available analysis
and design methods since analog control theory is far more
mature than digital control theory.
 Bilinear transformation [11]:
High sampling frequency is required.
( ) ( )2 1 / 1s z T z= − +  
ICGST 2012 presented July 16, 2012 7/16/2012

16.  Prediction-based Digital Redesign [Shieh et. al.12]
 Control period can be enlarged without compromising the
essential control performance.
 Control signals are much smaller in the transient, suitable for
physical implementation.
 This method can be conveniently extended to achieve time
delay compensation.
ICGST 2012 presented July 16, 2012 7/16/2012

17.  Prediction-based Digital Redesign [Shieh et. al.12]…
ICGST 2012 presented July 16, 2012 7/16/2012

18.  Prediction-based Digital Redesign [Shieh et. al.12]…
t
u(t)
o
( )cu t
( )du t
t
u(t)
o
( )cu t
( )du t
( )( )d cu kT u k v T= +  
ICGST 2012 presented July 16, 2012 7/16/2012

19.  Input Delay Compensation via Digital Redesign
( )( )d cu kT u k v T τ= + +  
( ) ( ) ( ) ( )1d d d d d du kT K x kT D u k T E r kT= − − − +  
ICGST 2012 presented July 16, 2012 7/16/2012

20.  Input Delay Compensation via Digital Redesign…
Remarks
 The proposed method is based on the assumption that input delay is known
to the controller. In NCSs where delay may be unknown to the controller,
this method can be applied in a way similar to Networked Predictive
Control (NPC) [13].
 It is noted that for a Single-Input-Single-Output (SISO) system, input delay,
output delay, and their combination can be converted to either input or
output side only through appropriate mathematical transformation [14].
ICGST 2012 presented July 16, 2012 7/16/2012

21. Nonsmooth system
Smooth nonlinear model
Approximation
Linear model
Analog controller
Linearization
Linear control theory
Digital controller
Digital redesign for delay compensation
 Example I
ICGST 2012 presented July 16, 2012 7/16/2012

22.  Example I: Linearization of DC Motor with Coulomb
Friction
m a a m
t L f
V R i L i K
K i T T J B
ω
ω ω
 = + +

= + + +
&
&
/ / 1/
/ / 0
a a m a a
m
t
R L K L i Li
V
K J B J ωω
− −      
= + ÷  ÷  ÷ −     
&
&
( ) ( )f j fT sign sign Kµ ω µ ω ω= × ≈ × ≈
( )
0
0
j k
k
f k
k
sign
for
where K
j for
ω
µ ω
ω
µ ω

≠
= 
 =
( )
/ / 1/
/ / 0
a a m a a
m
t f
R L K L i Li
V
K J B K J ωω
− −      
= +  ÷  ÷  ÷
− +        
&
&
ICGST 2012 presented July 16, 2012 7/16/2012

23.  Example I: Linearization of DC Motor with Coulomb
Friction
0 1 2 3 4 5 6
-30
-20
-10
0
10
20
30
Time (second)
MotorSpeed(rad/s)
non-smooth model
conventional linear model
proposed linear model
ICGST 2012 presented July 16, 2012 7/16/2012

24.  For non-smooth systems, the approximated scalar sign
function is proposed to transform differential equation-based
non-smooth models into smooth models. Then optimal
linearization can be applied to obtain an optimal linear model
at any operating point.
 A state-space based methodology is proposed on the digital
controller design for continuous-time nonlinear systems with
input delay. In addition to delay compensation, the proposed
method has several notable features: prolonged control
period and the reduced control signal.
ICGST 2012 presented July 16, 2012 7/16/2012

25.  Choose a proper linear control design method for
developing a controller based on the obtained linear
model.
 Verify the effectiveness of developed controller by
simulation and experiment.
 Extend the proposed network-induced-delay
compensation method to non-smooth system.
ICGST 2012 presented July 16, 2012 7/16/2012

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ICGST 2012 presented July 16, 2012 7/16/2012