Prasheel

Sliding Mode Controller – Observer for
Uncertain Dynamical Systems
Presented by,
Prasheel V. Suryawanshi
College of Engineering Pune
Guided by,
Dr. P. D. Shendge
PhD Thesis Defence
March 15, 2016 @ University of Pune
Prasheel V. Suryawanshi PhD Thesis Defence March 15, 2016 1

Outline
1 Introduction
2 Main Contributions
3 Conclusions
4 Examiner Comments

The Start SMC Motivation Proposed Method Objectives Methodology
Part I
Introduction

Where it all Started ?
u y
Control
−
ref
Plant

Uncertain Plant
u y
Control
−
ref
Uncertain Plant
disturbance (d)

Control of Uncertain Plant
u y
Sliding Mode Control
−
ref
Uncertain Plant
disturbance (d)

Control of Uncertain Plant using SMC
u y
−
ref
Uncertain Plant
disturbance (d)

Sliding Mode Control (SMC)
special class of variable structure systems (VSS) that alters
the dynamics of a system with a switching control
concept of changing the structure of controller in response to
changing state of the system, to obtain a desired response
eﬀective strategy for controlling systems with signiﬁcant
uncertainties and unmeasurable disturbances
ROBUSTNESS ???

Need for Investigation
Discontinuous control – chatter
excessive wear and tear of actuators
excitation of unmodeled dynamics

The bounds of uncertainty and disturbances are required
not always easy to ﬁnd
rate of change of uncertainty

The bounds of uncertainty and disturbances are required
not always easy to ﬁnd
rate of change of uncertainty
capabilities to compensate disturbances of only matched type
severe limitations on the applicability of SMC

requirement of full state-vector
all the system states are seldom measurable
sliding cannot be enforced
control may not be implementable

requirement of full state-vector
all the system states are seldom measurable
sliding cannot be enforced
control may not be implementable
accurate model may not be always available
uncertainty, un-modelled dynamics and disturbances
estimation imperative

Proposed Conﬁguration
u
Observer
(State Estimation)
yueq
ˆd
−
ref
−
Uncertain Plant
disturbance (d)
Estimation of
Uncertainty and Disturbance
ˆx

Central Theme
DESIGN AND DEVELOPMENT OF
ROBUST CONTROL ALGORITHMS
FOR
UNCERTAIN DYNAMICAL SYSTEMS

Objectives
Design a boundary layer SMC law using UDE for mitigating
chatter.

Objectives
chatter.
Design a SMC law for nonlinear systems by estimating the
states and

Objectives
chatter.
states and
matched uncertainties using UDE.

Objectives
chatter.
states and
mismatched uncertainties using EID Method.

Objectives
chatter.
states and
uncertainties (matched/ mismatched) using EID and DO.

Objectives
chatter.
states and
uncertainties (matched/ mismatched) using EID and DO.
Design a discrete SMC algorithm with estimation of states
and uncertainties.

Methodology
Redeﬁning the Control Problem
Convergence and Robustness Analysis
Stability Analysis
Performance Evaluation
Application Validation

Approach
Theoretical Formulation
Derivation of control law
Derivation of stability proof and stability analysis
Convergence and Robustness analysis
Numerical Simulation
MATLAB - SIMULINK

Approach
Application Validation
Flexible Joint System
Inverted Pendulum System
Antilock Braking System
Active Steering Control
Industrial Motion Control

Boundary layer SMC using UDE Salient Features Outcomes
Part II
Main Contributions

Main Contributions I
1 Boundary layer SMC law using UDE for mitigating Chatter
Application : Flexible Joint System
2 SMC for a Class of Nonlinear System using UDE
Application : Inverted Pendulum System
3 SMC for Mismatched Uncertain System using EID Method
Application : Anti-lock Braking System

Main Contributions II
4 SMC for Uncertain Nonlinear System using EID and DO
Application : Active Steering Control
5 Discrete SMC Algorithm with Estimation of States and
Uncertainties using UDE
Application : Motion Control System

Boundary layer SMC for Chatter Reduction

Uncertain Plant
˙x = A x + b u + b e(x, t)
e(x, t) is lumped uncertainty comprising of
uncertainties in A and b matrices,
and unmeasurable, external disturbance.

Design of Control
σ = s x
˙σ = s ˙x = s A x + s b u + s b e(x, t)
u = un + ueq
where
ueq = −(sb)−1
(sAx + kσ) k > 0
and
un = −ρ(x, t) sgn(σsb)

Smoothing by sat function
un =



−ρ(x, t) sgn(σsb), |σsb| >
−ρ(x,t)
, |σsb| ≤
where is a small positive number
small retains invariance but may result in chatter
large suppresses chatter but results in loss of invariance

sat function inside the boundary layer
0 2 4 6 8 10
−2
−1
0
1
time (sec)
x1andx2
(a) states
0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
time (sec)
x1andx2
(b) magniﬁed states
Figure: states

sat function inside the boundary layer
0 2 4 6 8 10
0
2
4
time (sec)
σ
Figure: Sliding variable

Variation in σ for diﬀerent values of
Boundary layer RMS value of σ
width – for sinusoidal for sawtooth
0.02 0.0004524 0.02545
0.2 0.03646 0.1908
2 0.5244 0.4639

UDE based Control
σ = s x
˙σ = s ˙x = s A x + s b u + s b e(x, t)
ueq = −(sb)−1
(sAx + kσ) k > 0
˙σ = s b un + s b e − k σ
e = (sb)−1
( ˙σ + kσ) − un

UDE based Control
ê = e Gf (s)
Gf (s) is a low-pass filter with sufficient bandwidth.
e = (sb)−1
( ˙σ + kσ) − un
un = −ê
With a first order low pass filter,
un =
(sb)−1
τ
σ + k σ

Modiﬁed Sliding Plane
σ∗
= σ − σ(0) e−αt
where α is a positive constant
σ∗(0) = 0 for any σ(0)
σ∗ → σ as t → ∞

Comparison of Control
0 2 4 6 8 10
−4,000
−2,000
0
time (sec)
u
(a) Conventional sliding plane
0 2 4 6 8 10
−40
−20
0
20
time (sec)u
(b) Modiﬁed sliding plane
Figure: Comparison of control

SMC law using Modiﬁed Sliding Plane
ueq = −(sb)−1
(s A x + σ(0) α e−α t
+ k σ∗
)
un = −(sb)−1 1
τ
σ∗
+ k σ∗

Boundary layer SMC law using Modiﬁed Sliding Plane
ueq = −(sb)−1
(s A x + σ(0) α e−α t
+ k σ∗
)
un =



−ρ(x, t) sgn(σ s b), |σsb| >
−(sb)−1 1
τ
σ∗
+ k σ∗
, |σsb| ≤

Numerical Example
A =
0 1
−2 −3
b =
0
1
∆A =
0 0
−1 −2
∆b =
0
−0.4
x0 = [1 0]T
k = 5 τ = 0.001 = 0.02
σ = 4 x1 + x2
d(x, t) = 10 sin(t)

States : UDE inside the boundary layer
0 2 4 6 8 10
−2
−1
0
1
time (sec)
x1andx2
(a) 1st
order ﬁlter
0 2 4 6 8 10
−2
−1
0
1
time (sec)x1andx2
(b) 2nd
order ﬁlter
Figure: states

Magnified states : UDE inside the boundary layer
0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
time (sec)
x1andx2
(a) 1st
order filter
0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
time (sec)x1andx2
(b) 2nd
order filter
Figure: magnified states

σ : UDE inside the boundary layer
0 2 4 6 8 10
0
2
4
time (sec)
σ
(a) 1st
order ﬁlter
0 2 4 6 8 10
0
2
4
time (sec)
σ
(b) 2nd
order ﬁlter
Figure: sliding surface

Variation in σ with ﬁrst order UDE
Time constant RMS value of σ
τ for sinusoidal for sawtooth
0.001 0.0001926 0.01641
0.01 0.01697 0.1434
0.1 0.4947 0.5625

Effect of Filter Time-Constant (τ)
0 2 4 6 8 10
−40
−20
0
20
time (sec)
u
(a) first order UDE (τ = 0.1 s)
0 2 4 6 8 10
−40
−20
0
20
time (sec)
u
(b) first order UDE (τ = 0.01 s)
Figure: Effect of filter time-constant

Effect of Filter Order
0 2 4 6 8 10
−40
−20
0
20
time (sec)
u
(a) second order UDE (τ = 0.1 s)
0 2 4 6 8 10
−40
−20
0
20
time (sec)
u
(b) second order UDE (τ = 0.01 s)
Figure: Effect of filter order

Variation in σ
Approximation RMS value of σ
sat function 4.524 × 10−4
ﬁrst order UDE 1.926 × 10−4
second order UDE 6.475 × 10−5

¨θ + F1
˙θ −
Kstiﬀ
Jeq
α = F2Vm
¨θ − F1
˙θ +
Kstiﬀ(Jeq + Jarm)
JeqJarm
α = −F2Vm
where,
F1
∆
=
ηmηg KtKmK2
g + BeqRm
JeqRm
and
F2
∆
=
ηmηg KtKm
JeqRm

Considering the output as y = θ + α,
the dynamics in terms of y and θ is re-written as,
ÿ =
Kstiff
Jeq
F3y −
Kstiff
Jeq
F3θ
¨θ =
Kstiff
Jeq
y −
Kstiff
Jeq
θ − F1
˙θ + F2Vm
where F3
∆
= 1 −
Jeq + Jarm
Jarm

Flexible Joint : state tracking
0 2 4 6 8 10
−1
−0.5
0
time (sec)
x1andxm1
(a) τ = 10ms
0 2 4 6 8 10
−1
−0.8
−0.6
−0.4
−0.2
0
time (sec)
x1andxm1
(b) τ = 1ms

0 2 4 6 8 10
−0.4
−0.2
0
0.2
0.4
time (sec)
x2andxm2
(c) τ = 10ms
0 2 4 6 8 10
−0.4
−0.2
0
0.2
0.4
time (sec)
x2andxm2
(d) τ = 1ms

0 2 4 6 8 10
−0.5
0
0.5
time (sec)
x3andxm3
(e) τ = 10ms
0 2 4 6 8 10
−0.5
0
0.5
time (sec)
x3andxm3
(f) τ = 1ms

0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
x4andxm4
(g) τ = 10ms
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
x4andxm4
(h) τ = 1ms
Figure: state tracking

Salient Features
discontinuous control outside the boundary layer
UDE based control inside
modiﬁed sliding surface to address problem of large initial
control
performance improvement over baseline SMC using ‘sat’
function
overall stability of the system is proved
application to ﬂexible joint system

Outcomes
Journal Publication – 1
International Journal of Dynamics and Control: Springer
DOI: 10.1007/s40435-015-0150-9
Conference Publication – 1
World Congress on Engineering and Computer Science

SMC for a Class of Nonlinear System using UDE Salient Features Outcomes
SMC for a Class of Nonlinear System using UDE

Plant Dynamics
˙x = f (x, t) + g(x, t) u + ∆f (x, t) + ∆g(x, t) u + d(x, t)
x Rn is the state vector
u R1 is the control input
f (x, t) is known nonlinear system vector
g(x, t) is known nonlinear input vector
∆f (x, t) & ∆g(x, t) are uncertainties in plant and input vector
d(x, t) is unmeasurable disturbances

Assumption 1 : Matching Conditions
∆f (x, t) = g(x, t) · ef (x, t)
∆g(x, t) = g(x, t) · eg (x, t)
d(x, t) = g(x, t) · ed (x, t)



where ef , eg and ed are unknown
˙x = f (x, t) + g(x, t) u + g(x, t) · e(x, u, t)
e(x, u, t) = ef (x, t) + eg (x, t) u + ed (x, t)

Assumption 2 : Disturbance
The lumped uncertainty e(x, t) is continuous and satisﬁes,
d(j)e(x, t)
dt(j)
≤ µ for j = 0, 1, 2, . . . , r
where µ is a small positive number

Model Following
˙xm = Amxm + bmum
f (x, t) − Am x = g(x, t) · L
bm = g(x, t) · M
L and M are suitable known matrices of appropriate dimensions

Design of Control
σ = bT
x + z
˙z = −bT Am x − bT bm um
z(0) = −bT x(0)




Design of Control
˙σ = bT
˙x + ˙z
˙σ = bT
g L + bT
g u + bT
g e − bT
g M um
u = ueq + un
ueq = −L + Mum − (bT
g)−1
kσ
k is a positive constant.
˙σ = bT
g un + bT
g e − kσ

Estimation of Uncertainty
ê = e Gf (s)
Gf (s) is a low-pass filter with sufficient bandwidth
e = (bT
g)−1
( ˙σ + kσ) − un
un = −ê
un =
(bT g)−1
τ
σ + k σ

Improvement in Estimation
Gf (s) =
1 + 2τs
τ2s2 + 2τs + 1
(2nd
order IDC)
where τ is a small positive constant
ê1 be the estimate of e(x, u, t) and
ê2 = ˙ê1 be the estimate of ˙e(x, u, t)
ê1 = (bT
g)−1 2
τ
σ +
2τk + 1
τ2
t
0
σ +
k
τ2
t
0
t
0
σ

Estimation of States
˙x = f (x, t) + g(x, t) u + ∆f (x, t) + ∆g(x, t) u + d(x, t)
˙x(t) = A x(t) + B u(t) + B e(x, u, t)
e(x, u, t) = f (x, t)+g(x, t) u+∆f (x, t)+∆g(x, t) u+d(x, t)−A x−B u

Estimation of States
˙x(t) = A x(t) + B u(t) + B e(x, u, t)
˙ˆx(t) = A ˆx(t) + B u(t) + B ˆe(ˆx, u, t) + J(y(t) − ˆy(t))
˙˜x = (A − JC) ˜x + B ˜e
e(x, u, t) = B+
(˙x − Ax − Bu)

Uncertainty Estimation with Estimated States
ê(ˆx, u, t) = Gf (s) B+
(˙ˆx − Aˆx − Bu)
Gf (s) =
1
1 + τs
˙ê =
B+JC
τ
(˜x)
˙ê =
B+J
τ
(y − ˆy)

Stability Analysis
˙˜x(t) = (A − JC)˜x(t) + B˜e(x, u, t)
˙˜e(t) = −
B+JC
τ
˜x(t) + ˙e(t)

Stability Analysis
˙˜x(t) = (A − JC)˜x(t) + B˜e(x, u, t)
˙˜e(t) = −
B+JC
τ
˜x(t) + ˙e(t)
Therefore,
˙˜x
˙˜e
=


(A − JC) B
−
B+JC
τ
0


˜x
˜e
+
0
1
˙e

Stability Analysis
˙˜x(t) = (A − JC)˜x(t) + B˜e(x, u, t)
˙˜e(t) = −
B+JC
τ
˜x(t) + ˙e(t)
Therefore,
˙˜x
˙˜e
=


(A − JC) B
−
B+JC
τ
0


˜x
˜e
+
0
1
˙e
A compact form can be written as,
˙w = Ak w + T ˙e

Stability Analysis
If the observer gains are selected such that all eigen values of Ak
have negative real parts, one can always ﬁnd a positive deﬁnite
matrix P such that,

Stability Analysis
matrix P such that,
PAk + AT
k P = −Q
for a given positive deﬁnite matrix Q
Let λ be the smallest eigen value of Q

Stability Analysis
matrix P such that,
PAk + AT
k P = −Q
for a given positive deﬁnite matrix Q
Let λ be the smallest eigen value of Q
Deﬁning a Lyapunov function as,
V (w) = wT
P w

Stability Analysis
Taking derivative of V (w)
˙V (w) = wT
P ˙w + ˙wT
Pw
= wT
(PAk + AT
K P)w + 2wT
PT ˙e
= −wT
Qw + 2wT
PT ˙e
≤ −λ w 2
+ 2 w PT µ
≤ − w ( w λ − 2 PT µ)

Stability Analysis
˙V (w) = wT
P ˙w + ˙wT
Pw
= wT
(PAk + AT
K P)w + 2wT
PT ˙e
= −wT
Qw + 2wT
PT ˙e
≤ −λ w 2
+ 2 w PT µ
≤ − w ( w λ − 2 PT µ)
Thus, w is ultimately bounded by,
w ≤
2 PT µ
λ

Stability Analysis
˙V (w) = wT
P ˙w + ˙wT
Pw
= wT
(PAk + AT
K P)w + 2wT
PT ˙e
= −wT
Qw + 2wT
PT ˙e
≤ −λ w 2
+ 2 w PT µ
≤ − w ( w λ − 2 PT µ)
Thus, w is ultimately bounded by,
w ≤
2 PT µ
λ
This implies,
˜x ≤ λ1 =
2 PT µ
λ
˜e ≤ λ2 =
2 PT µ
λ

Stability Analysis
The dynamics of ˆσ can be written as,
˙ˆσ = −kˆσ + BT
JC˜x

Stability Analysis
JC˜x
Therefore,
ˆσ ˙ˆσ = −kˆσ2
+ BT
JC ˆσ˜x
≤ −k|ˆσ|2
+ |BT
JC| |ˆσ| ˜x
≤ −|ˆσ| k |ˆσ| − |BT
JC| λ1

Stability Analysis
JC˜x
Therefore,
ˆσ ˙ˆσ = −kˆσ2
+ BT
JC ˆσ˜x
≤ −k|ˆσ|2
+ |BT
JC| |ˆσ| ˜x
≤ −|ˆσ| k |ˆσ| − |BT
JC| λ1
Thus, the sliding variable (ˆσ) is ultimately bounded by,
|ˆσ| ≤ λ3 =
|BT JC| λ1
k

Numerical Example
f (x, t) =


0 1
2 x2 sin x1
2
3 + cos x1
cos x1
2
3 + cos x1

 , g(x, t) =


0
1
2
3 + cos x1



Numerical Example
f (x, t) =


0 1
2 x2 sin x1
2
3 + cos x1
cos x1
2
3 + cos x1

 , g(x, t) =


0
1
2
3 + cos x1


∆f (x, t) =
0 0
0.2 sin x2 0.1 cos x2
, ∆g(x, t) =
0
0.1 sin x2

Numerical Example
f (x, t) =


0 1
2 x2 sin x1
2
3 + cos x1
cos x1
2
3 + cos x1

 , g(x, t) =


0
1
2
3 + cos x1


∆f (x, t) =
0 0
0.2 sin x2 0.1 cos x2
, ∆g(x, t) =
0
0.1 sin x2
Am =
0 1
−ω2
n −2ζωn
, Bm =
0
−ω2
n
;

Numerical Example
f (x, t) =


0 1
2 x2 sin x1
2
3 + cos x1
cos x1
2
3 + cos x1

 , g(x, t) =


0
1
2
3 + cos x1


∆f (x, t) =
0 0
0.2 sin x2 0.1 cos x2
, ∆g(x, t) =
0
0.1 sin x2
Am =
0 1
−ω2
n −2ζωn
, Bm =
0
−ω2
n
;
x(0) = 1 0 xm(0) = 0 1
d(x, t) = x2
1 sin (t) + x2 cos (t) + 1

Estimation of States and Uncertainty
0 5 10 15 20 25
−2
−1
0
1
2
·10−2
time (sec)
−xˆx
(a) state estimation error
0 5 10 15 20 25
−2
−1
0
1
2
·10−2
time (sec)
−eˆe
(b) uncertainty estimation error

Estimation of States and Uncertainty
0 5 10 15 20 25
−2
−1
0
1
2
·10−4
time (sec)
σ
Figure: sliding variable

Case 1 : Model Following
0 2 4 6 8 10
−1
0
1
time (sec)
x1andxm1
(a) Plant and Model State x1
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
x2andxm2
(b) Plant and Model State x2

Case 1 : Model Following
0 2 4 6 8 10
−50
0
50
time (sec)
u
(c) control u
0 2 4 6 8 10
−5
0
5
10
time (sec)
eandˆe
(d) uncertainty : actual and estimated
Figure: Model following performance

Case 2 : Increase in the Uncertainty of g(x, t)
0 2 4 6 8 10
−1
0
1
time (sec)
x1andxm1
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)x2andxm2

0 2 4 6 8 10
−100
−50
0
50
time (sec)
u
(c) control u
0 2 4 6 8 10
−2
0
2
4
·10−2
time (sec)σ
(d) sliding surface σ

0 2 4 6 8 10
−20
0
20
40
60
time (sec)
eandˆe
(e) uncertainty : actual and estimated
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
estimationerror(%)
(f) estimation error
Figure: Robust performance

Case 3 : Fast Disturbance
0 2 4 6 8 10
−1
0
1
time (sec)
x1andxm1
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)x2andxm2

0 2 4 6 8 10
−50
0
50
time (sec)
u
(c) control u
0 2 4 6 8 10
−5
0
5
·10−3
time (sec)
σ
(d) sliding surface σ

0 2 4 6 8 10
0
5
10
time (sec)
eandˆe
(e) uncertainty : actual and estimated
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
estimationerror(%)
(f) estimation error
Figure: Robust performance with fast disturbance

Case 4 : Improvement with Higher Order Filter
0 2 4 6 8 10
−5
0
5
·10−3
time (sec)
σ
(a) σ with 1st
order ﬁlter
0 2 4 6 8 10
−2
0
2
4
·10−4
time (sec)
σ
(b) σ with 2nd
order ﬁlter

Case 4 : Improvement with Higher Order Filter
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
estimationerror(%)
(c) ˜e with 1st
order filter
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)estimationerror(%)
(d) ˜e with 2nd
order filter
Figure: Improvement with higher order filter

Application : Inverted Pendulum System
˙x1 = x2
˙x2 =
g sin(x1) −
m l x2
2 cos(x1) sin x1
(mc +m)
l 4
3 − m cos2(x1)
(mc +m)
+
cos(x1)
(mc +m)
l 4
3 − m cos2(x1)
(mc +m)
u(t) + d(t)



[x1 x2] are the state variables – position and velocity of the pole
u(t) is the control input
d(t) is the external disturbance
mc is the mass of cart
m is the mass of pole
l is half length of pole

Plant Dynamics
f (x) =


x(2)
g sin(x1) − m l x2
2 cos(x1) sin x1/(mc + m)
l 4
3 − m cos2(x1)/(mc + m)


g(x) =



0
cos(x1)/(mc + m)
l 4
3 − m cos2(x1)
(mc +m)



d(t) =
0
20 sin(2 π t)

System Parameters
mc = 1 kg
m = 0.1 kg
l = 0.5 m
g = 9.81
initial conditions of plant; x(0) = [π/60 0]
initial conditions of model; xm(0) = [0 0]
uncertainty in the plant is 50%
d(t) = 20 sin(2 π t)
r(t) = 0.2 sin π t +
π
2

Case 1 : State Tracking
0 1 2 3 4 5
−0.2
−0.1
0
0.1
time (sec)
x1andxm1
(a) position x1
0 1 2 3 4 5
−1
0
1
·10−3
time (sec)
σ
(b) sliding variable
Figure: state tracking without disturbance

Case 2 : State Tracking with Disturbance
0 1 2 3 4 5
−0.2
−0.1
0
0.1
time (sec)
x1andxm1
(a) position x1
0 1 2 3 4 5
−2
−1
0
1
2
·10−2
time (sec)
σ
Figure: state tracking with disturbance

Case 3 : Effect of Filter Order
0 5 10 15 20 25
−2
−1
0
1
2
time (sec)
estimationerror(%)
(a) 1st
order filter
0 5 10 15 20 25
−2
−1
0
1
2
·10−2
time (sec)
estimationerror(%)
(b) 2nd
order filter
Figure: Improvement in uncertainty estimation

0 5 10 15 20 25
−2
−1
0
1
2
·10−2
time (sec)
σ
(a) 1st
order ﬁlter
0 5 10 15 20 25
−2
−1
0
1
2
·10−4
time (sec)
σ
(b) 2nd
order ﬁlter
Figure: Improvement in sliding

Salient Features
model-following SMC using UDE
UDE to estimate the lumped uncertainty and improvement in
estimation accuracy with higher-order UDE
system considered – uncertain nonlinear, subjected to time
varying disturbance with state-dependent nonlinearity in the
input vector, in addition to plant matrix
sliding enforced without using discontinuous control and
without requiring knowledge of uncertainties or their bounds
control law made implementable by estimating the states
ultimate boundedness of estimation error (˜e) and sliding
variable (σ) is proved

Outcomes
Journal Publication – 1
Nonlinear Dynamics: Springer
Vol. 78, No. 3, pp. 1921-1932, 2014
Journal Publication – 1 (under review)
Transactions of Institute of Measurement and Control

SMC using EID Salient Features Outcomes
SMC for Mismatched Uncertain System
using EID Method

Deﬁnition of EID
An Equivalent Input Disturbance (EID) is a disturbance on the
control input channel that produces the same eﬀect on the
controlled output as actual disturbances do.
An EID always exists for a controllable and observable plant with
no zeros on the imaginary axis.

Basic Concept
˙xo = Axo + Bu + Bd d
yo = C xo

Basic Concept
˙x = Ax + Bu + Bde
y = Cx

Nonlinear Plant Dynamics
˙x(t) = A x(t) + f (x, t) + B u(t) + g(x, t) u(t) + Bd d(x, t)
y(t) = C x(t)



where
x(t) Rn is the state vector
u(t) R1 is the control input
f (x, t) is the uncertain nonlinear plant vector
g(x, t) is the uncertain nonlinear input vector
d(x, t) Rn is external unmeasurable disturbance
A and B are the plant and input matrices representing the nominal
plant

The uncertain nonlinear plant f (x, t) and input vector g(x, t) u
can be considered as state and input dependent disturbance and is
lumped with external unmeasurable disturbances Bd d.

The uncertain nonlinear plant f (x, t) and input vector g(x, t) u
can be considered as state and input dependent disturbance and is
lumped with external unmeasurable disturbances Bd d.
The plant is now given by,
˙x(t) = A x(t) + B u(t) + Bde(x, u, t)
y(t) = C x(t)



where de = f (x, t) + g(x, t) u + Bd d(x, t)

Assumption
The EID de(x, t) is continuous and satisﬁes,
d(j)de(x, u, t)
dt(j)
≤ µ for j = 0, 1, 2, . . . , r
where µ is a small positive number.

Servo Control with EID
u
˙ˆx = Aˆx + Bueq + LC(x − ˆx)
ˆy = Cˆx
yueq
(1+τs)k
−(τs)k
(1+τs)k
ˆd
SMC
−
ref
−
d
-
σ = y − yd
˙x = Ax + Bu + Bdd + f + gu
y = Cx

Estimation of EID
The control u can be written as,
u = ueq + un
where ueq is the nominal control designed using sliding mode
un = −ˆd is the control for the uncertain part
with ˆd as the estimate of de

Estimation of EID
de = B+
(˙x − Ax − Bu)

Estimation of EID
de = B+
(˙x − Ax − Bu)
The observer is given as,
˙ˆx = Aˆx + Bueq + LC (x − ˆx)

Estimation of EID
de = B+
(˙x − Ax − Bu)
˙ˆx = Aˆx + Bueq + LC (x − ˆx)
Let, ˜x = x − ˆx
Therefore, ˙˜x = ˙x − ˙ˆx

Estimation of EID
The observer error dynamics can be determined as
˙˜x = A(x − ˆx) + B(u − ueq) + Bde − LC (x − ˆx)
= (A − LC)˜x + B˜d
where ˜d = de − ˆd

Estimation of EID
The observer error dynamics can be determined as
˙˜x = A(x − ˆx) + B(u − ueq) + Bde − LC (x − ˆx)
= (A − LC)˜x + B˜d
where ˜d = de − ˆd
The EID (de) can now be estimated with observed states as,
d∗
= B+
(˙ˆx − Aˆx − Bu)
= B+
(Aˆx + Bueq + LC ˜x − Aˆx − Bu)
= B+
LC ˜x + ˆd

Estimation of EID
The estimate ˆd is obtained by passing d∗ through a low-pass Gf (s)
ˆd = Gf (s) d∗

Estimation of EID
ˆd = Gf (s) d∗
For a ﬁrst order ﬁlter,
ˆd =
d∗
1 + τs

Estimation of EID
ˆd = Gf (s) d∗
For a ﬁrst order ﬁlter,
ˆd =
d∗
1 + τs
ˆd + τ ˙ˆd = d∗
ˆd + τ ˙ˆd = B+
LC ˜x + ˆd
˙ˆd =
B+LC
τ
˜x
˙de −
˙ˆd = −
B+LC
τ
˜x + ˙de

Estimation of EID
The estimation error dynamics can be determined as,
˙˜d = −
B+LC
τ
˜x + ˙de

Estimation of EID
˙˜d = −
B+LC
τ
˜x + ˙de
˙˜x
˙˜d
=
(A − LC) B
−B+LC
τ 0
˜x
˜d
+
0
1
˙de

Estimation of EID
˙˜d = −
B+LC
τ
˜x + ˙de
˙˜x
˙˜d
=
(A − LC) B
−B+LC
τ 0
˜x
˜d
+
0
1
˙de
The error dynamics can be further simpliﬁed as,
˙˜x
˙˜d
=
A B
01×2 0
−
L
B+L
τ
C 0
˜x
˜d
+
02×1
1
˙de

Estimation of EID
A compact form of can be written as,
˙˜e = (H − K G) ˜e + E ˙de
where,
˜e =
˜x
˜d
H =
A B
01×2 0
K =
L
B+L
τ
G = C 0 E = 0 0 1
T

Improvement in Estimation with Higher Order Filter
The EID (de) and its derivative ( ˙de) can be estimated using a
second order ﬁlter of the form,
Gf (s) =
1 + 2τs
τ2s2 + 2τs + 1
where τ is a small positive constant.

second order ﬁlter of the form,
Gf (s) =
1 + 2τs
τ2s2 + 2τs + 1
where τ is a small positive constant.
Therefore,
ˆd1 =
1 + 2τs
τ2s2 + 2τs + 1
B+
LC ˜x + ˆd1

τ2 ¨ˆd1 + 2τ ˙ˆd1 + ˆd1 = B+
LC˜x + 2τB+
LC ˙˜x + ˆd1 + 2τ ˙ˆd1
τ2 ¨ˆd1 = B+
LC˜x + 2τB+
LC ˙˜x
¨ˆd1 =
1
τ2
B+
LC ˜x +
2
τ
B+
LC ˙˜x
¨ˆd1 = ˙ˆd2 +
2
τ
B+
LC ˙˜x

τ2 ¨ˆd1 + 2τ ˙ˆd1 + ˆd1 = B+
LC˜x + 2τB+
LC ˙˜x + ˆd1 + 2τ ˙ˆd1
τ2 ¨ˆd1 = B+
LC˜x + 2τB+
LC ˙˜x
¨ˆd1 =
1
τ2
B+
LC ˜x +
2
τ
B+
LC ˙˜x
¨ˆd1 = ˙ˆd2 +
2
τ
B+
LC ˙˜x
Therefore,
˙ˆd1 =
2
τ
B+
LC ˜x + ˆd2
˙ˆd2 =
1
τ2
B+
LC ˜x

The estimation error is written as,
˙˜d1 = −
2
τ
B+
LC ˜x + ˜d2
˙˜d2 = −
1
τ2
B+
LC ˜x + ¨de

The estimation error is written as,
˙˜d1 = −
2
τ
B+
LC ˜x + ˜d2
˙˜d2 = −
1
τ2
B+
LC ˜x + ¨de
The error dynamics can now be written as,



˙˜x
˙˜d1
˙˜d2


 =

 A B
02×2 02×1
−


L
2
τ B+L
1
τ2 B+L

 C 01×2




˜x
˜d1
˜d2


+


0
01×1
1

 ¨de

A compact form can be written as,
˙˜e = (H − K G) ˜e + E ¨de
where,
˜e =


˜x
˜d1
˜d2

 H =
A B
02×2 02×1
K =


L
2
τ B+L
1
τ2 B+L


G = C 01×2 E = 0 0 0 1
T

Generalization for kth Order Filter
˙˜d1 = −
k
τ
B+
LC ˜x + ˜d2
˙˜d2 = −
k
τ2
B+
LC ˜x + ˜d3
...
˙˜dk−1 = −
k
τk−1
B+
LC ˜x + ˜dk
˙˜dk = −
1
τk
B+
LC ˜x + de
(k)

Application : Antilock Braking System (ABS)

ABS : Dynamic Model
The equation of motion for upper wheel
J1 ˙z1 = Fnr1sµ(λ) − q1z1 − s1M10 − s1Tb
The equation of motion for lower wheel
J2 ˙z2 = −Fnr2sµ(λ) − q2z2 − s2M20
The normal force Fn can be derived by calculating the sum of
torques at A
Fn =
Mg + s1Tb + s1M10 + q1z1
L(sin ϕ − sµ(λ) cos ϕ)

ABS : Dynamic Model
Deﬁning
S(λ) =
sµ(λ)
L(sin ϕ − sµ(λ) cos ϕ)
µ(λ) can be approximated by the following formula
µ(λ) =
w4λP
a + λP
+ w3λ3
+ w2λ2
+ w1λ

ABS : Dynamic Model
The slip is given as,
λ =
r2z2 − r1z1
r2z2
The actuator dynamics is given as,
˙Tb = c31(b(u) − Tb)
b(u) =
b1u + b2, u ≥ u0
0, u < u0

ABS : Dynamic Model
Considering λ = x1 and Tb = x2, the dynamics can be written as,
˙x1 = f (λ, z2) + g(λ, z2)x2
˙x2 = −c31x2 + c31b1u + c31b2
f (λ, z2) = − (S(λ)c11 + c13)(1 − λ) + (S(λ)c12 + c14)
r1
r2z2
+
(1 − λ)
z2
(S(λ)c21(1 − λ)
r2
r1
+ c23)z2 + S(λ)c22 + c24
g(λ, z2) = −
r1
r2z2
S(λ)c15 + c16 −
r2
r1
S(λ)c25(1 − λ) s1

Simulation Results
Case 1: Nominal plant
Case 2: Uncertainty in mass (m)
Case 3: Uncertainty in road gradient (θ)
Case 4: Diﬀerent friction models
observer poles : [ −20 − 30 − 50 ]
controller gains : kl = 10 and ks = 2
reference slip : λd = 0.2
initial value of z1 and z2 : 1700 rpm

Case 1 : Nominal Plant
0 0.2 0.4 0.6 0.8 1
0
5 · 10−2
0.1
0.15
0.2
time (sec)
λandλd
(a) slip ratio
0 0.2 0.4 0.6 0.8 1
0
0.5
1
time (sec)u(V)
(b) control

0 0.2 0.4 0.6 0.8 1
500
1,000
1,500
time (sec)
z1andz2(rpm)
(c) angular velocity
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
time (sec)Tb(Nm)
(d) braking torque

0 0.2 0.4 0.6 0.8 1
−0.8
−0.6
−0.4
−0.2
0
time (sec)
ˆd
(e) estimate of EID
0 0.2 0.4 0.6 0.8 1
−0.8
−0.6
−0.4
−0.2
0
time (sec)
σ
(f) sliding variable

0 0.2 0.4 0.6 0.8 1
0
5 · 10−2
0.1
0.15
0.2
time (sec)
λandλd
(g) λ (solid) and λd (dotted)
0 0.2 0.4 0.6 0.8 1
0
5 · 10−2
0.1
0.15
0.2
time (sec)λandˆλ
(h) λ (solid) and ˆλ (dotted)
Figure: Control performance for nominal plant

Case 2 : Uncertainty in Mass (m)
0 0.2 0.4 0.6 0.8 1
0
5 · 10−2
0.1
0.15
0.2
time (sec)
λandλd
(a) slip ratio
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
time (sec)
u(V)
(b) control
Figure: Control performance with mass uncertainty
nominal (solid), +30% (dashed) and –30% (dashed-dot)

Uncertainty in ˜λ u σ ˜y
mass (%) (λ − λref ) (y − ˆy)
+20 0.0455 0.7296 0.1115 0.0016
–20 0.0440 0.6924 0.1094 0.0013
+30 0.0459 0.7394 0.1121 0.0017
–30 0.0436 0.6834 0.1091 0.0013
+40 0.0462 0.7493 0.1127 0.0017
–40 0.0433 0.6745 0.1088 0.0012

Case 3 : Uncertainty in Road Gradient (θ)
Road inclination ˜λ u σ ˜y
angle (deg.) (λ − λref ) (y − ˆy)
+10 0.0446 0.7064 0.1103 0.0015
+20 0.0445 0.6940 0.1101 0.0014
+30 0.0442 0.6735 0.1097 0.0014
+40 0.0439 0.6462 0.1092 0.0014
+50 0.0434 0.6122 0.1086 0.0013

Case 4 : Diﬀerent Friction Models (µ)
0 0.2 0.4 0.6 0.8 1
0
5 · 10−2
0.1
0.15
0.2
time (sec)
λandλd
(a) slip ratio
0 0.2 0.4 0.6 0.8 1
0
0.5
1
time (sec)
u(V)
(b) control
Figure: Control performance with variation in µ(λ)
nominal (solid), +25% (dashed) and –25% (dashed-dot)

Case 4 : Diﬀerent Friction Models (µ)
0 0.2 0.4 0.6 0.8 1
0
5 · 10−2
0.1
0.15
0.2
time (sec)
λandλd
(a) slip ratio
0 0.2 0.4 0.6 0.8 1
0
0.5
1
time (sec)
u(V)
(b) control
Figure: Control performance for diﬀerent friction models
Inteco (solid), Magic formula (dashed) and Burckhardt formula (dashed-dot)

Experimental Results

0 1 2 3
0
0.2
0.4
0.6
0.8
1
time (sec)
λandλd
(a) slip ratio
0 1 2 3
0
0.2
0.4
0.6
0.8
1
time (sec)λandˆλ
(b) control

0 1 2 3
−1
0
1
time (sec)
u(V)
(c) λ (solid) and λd (dotted)
0 1 2 3
0
500
1,000
1,500
time (sec)
z1andz2(rpm)
(d) λ (solid) and ˆλ (dotted)
Figure: Experimental results for nominal plant

0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
time (sec)
λandλd
(a) slip ratio
0 1 2 3 4
−1
0
1
time (sec)
u(V)
(b) control
Figure: Control performance with mass uncertainty (+40%)

Case 3 : Diﬀerent Road Surface
0 1 2 3
0
0.2
0.4
0.6
0.8
1
time (sec)
λandλd
(a) slip ratio
0 1 2 3
−1
0
1
time (sec)
u(V)
(b) control
Figure: Control performance for dry road surface

0 1 2 3
0
0.2
0.4
0.6
0.8
1
time (sec)
λandλd
(a) slip ratio
0 1 2 3
−2
−1
0
1
time (sec)
u(V)
(b) control
Figure: Control performance for wet road surface

0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
time (sec)
λandλd
(a) slip ratio
0 2 4 6 8 10 12
−2
−1
0
time (sec)
u(V)
(b) control
Figure: Control performance for oily road surface

Summary of Experimental Results
Case ˜λ u σ stopping
(λ − λref ) distance (m)
nominal plant 0.1852 0.3505 0.0315 24.26
mass (+20%) 0.1733 0.3708 0.0397 25.39
mass (+30%) 0.1741 0.3708 0.0389 27.89
mass (+40%) 0.1435 0.2969 0.0180 28.59
dry road 0.1756 0.3619 0.0354 25.68
wet road 0.1635 0.3314 0.0308 26.89
oily road 0.1654 0.4933 0.0094 95.17

Salient Features
SMC combined with EID and extended to nonlinear system
with mismatched uncertainties
SMC for nominal control mitigates the need of integral action
used in conventional EID based control
SMC law augmented with estimate of EID; obtained using
observed states and a low-pass filter
effect of higher-order filter for improving the accuracy of
estimation with generalization to kth-order filter
application to anti-lock braking systems (ABS) control and
testing for different friction models
Experimental validation

Outcomes
IEEE Transactions on Industrial Electronics
IEEE Transactions on Control System Technology

SMC using EID and DO Salient Features Outcomes
SMC for Uncertain Nonlinear System
using EID and DO

Uncertain Plant
˙x(t) = A x(t) + B u(t) + Bde(x, u, t)
y(t) = C x(t)



where de comprises of nonlinearities, uncertainties and disturbance
in either or all channels

Estimation of EID
u yueq
ˆde
SMC
−
ref
−
d
-
Sliding
Plant
State Observer
Disturbance
Observer
Surface

Estimation of EID
de = B+
(˙x − Ax − Bu)
˙ˆx = Aˆx + Bueq + L (y − ˆy)
Therefore,
˙˜x = (A − LC)˜x + B ˜de
where ˜de = de − ˆde

Estimation of EID with DO
ˆde = p + Jˆx
˙p = −J(Aˆx + Bu + B ˆde)
The dynamics of de can be written as,
˙ˆde = JLC˜x
Therefore, the estimation error dynamics can be written as,
˙˜de = −JLC˜x + ˙de

Improvement with Extended DO
second order DO as,
ˆde = p1 + J1ˆx
˙p1 = −J1(Aˆx + Bu + B ˆde) + ˆ˙de
ˆ˙de = p2 + J2ˆx
˙p2 = −J2(Aˆx + Bu + B ˆde)

Improvement with Extended DO
second order DO as,
ˆde = p1 + J1ˆx
˙p1 = −J1(Aˆx + Bu + B ˆde) + ˆ˙de
ˆ˙de = p2 + J2ˆx
˙p2 = −J2(Aˆx + Bu + B ˆde)
The error dynamics can be written as,
˙˜de = −J1LC˜x + ˜˙de
˙˜˙de = −J2LC˜x + ¨de

Design of Control
The sliding surface,
σ = y − yd
σ = Cˆx − yd

Design of Control
The sliding surface,
σ = y − yd
σ = Cˆx − yd
The modiﬁed sliding surface,
σ∗
= σ − σ(0) e−αt
where α is a user chosen positive constant

Design of Control
˙σ∗
= CAˆx + CBueq + CLC˜x + ασ(0)e−αt
− ˙yd

Design of Control
˙σ∗
− ˙yd
ueq = −(CB)−1
CAˆx + ασ(0)e−αt
− ˙yd + kl σ∗
+ ks sat(σ∗
)

Design of Control
˙σ∗
− ˙yd
ueq = −(CB)−1
− ˙yd + kl σ∗
+ ks sat(σ∗
)
un = − ˆde

Design of Control
˙σ∗
− ˙yd
ueq = −(CB)−1
− ˙yd + kl σ∗
+ ks sat(σ∗
)
un = − ˆde
Therefore,
˙σ∗
= −kl σ∗
− ks sat(σ∗
) + CLC˜x

Active Steering : Bicycle Model
lr
lf
Y
β
αr
δ
αf
vx
vy
Fyr
Fyf
Fw
lw
r
X
v
Fxf
Fxr
CG
W
yaw angle

Dynamic Model I
m( ˙vy + vx
˙ψ) = Fyf cos δ + Fyr
J ¨ψ = Fyf cos δ lf − Fyr lr
where δ is the front wheel steer angle
β and ˙ψ are vehicle side slip angle and yaw rate
Fyf and Fyr are the lateral forces in the front and rear tire
lf and lr are distances from vehicle’s center of mass to front and
rear axis
vx and vy are longitudinal velocity and lateral velocity

Dynamic Model II
The lateral forces of both the axes can be expressed as,
Fyf = −µ Cf αf , Fyr = −µ Cr αf
where Cf , Cr are the front and rear tire cornering stiﬀness
αf , αr are the slip angles of the front and rear tire
µ describes the road surface condition
Using simple geometrical relations,
αf = δ − β +
lf
vx
˙ψ
αr = −β +
lr
vx
˙ψ

Dynamic Model III
The sideslip angle (β) at CG is deﬁned as angle between vehicle
longitudinal axis and local direction of travel given by,
β = arctan
vy
vx
For small angles, β can be approximated as
β ≈
vy
vx
The lateral acceleration ay at the vehicle CG is given by
ay = vx ( ˙β + ˙ψ)

Dynamic Model IV
m( ˙vy + vx r) = Fyf + Fyr + Fyf cos δ − Fyf
J ¨ψ = Fyf lf − Fyr lr + Fyf cos δf − Fyf lf
Let the non linear terms in alongwith uncertainties in lateral forces
due to cornering stiﬀness being dependent on friction coeﬃcient
and vehicle mass variations be taken into disturbance without
modifying the system equations as follows :
m( ˙vy + vx r) =Fyf + Fyr + d1
J ¨ψ =Fyf lf − Fyr lr + d2

Dynamic Model V
Rearranging the terms
˙β =
Fyf + Fyr
mvx
+ d1 and ¨ψ =
Fyf lf − Fyr lr
J
+ d2
where
d1 =
Fyf cos δ − Fyf
mvx
+
∆Fyf cos δ + ∆Fyr
∆mvx
d2 =
Fyf cos δlf − Fyf lf
J
+
∆Fyf lf cos δ − ∆Fyr lr
J
Writing the system matrices in state space form
˙x = Ax + Bu + Bd d
y = Cx + Du

Dynamic Model VI
where x = [β, ˙ψ]T , u = δ, y = [ay , ˙ψ]T , d = [d1, d2]T
A =
−Cf −Cr
mvx
−lf Cf +lr Cr
mv2
x
− 1
−lf Cf +lr Cr
J
−l2
f Cf −l2
r Cr
Jvx
, B =
Cf
mvx
lf Cf
Izz
C =
−Cf −Cr
m
−lf Cf +lf Cr
mvx
0 1
, D =
Cf
m
0
, Bd =
1 0
0 1

Modeling
βd = 0
˙ψd =
vx
l + Kuv2
x
δf
where l = lf + lr is the total wheelbase length of the vehicle and
Ku =
m(lr Cr − lf Cf )
lCf Cr
is the vehicle stability parameter describing steering characteristics
of the vehicle

Simulation
Parameters Value Unit
Vehicle mass (M) 1430 kg
Yaw moment inertia (J) 2430 kg m2
Distance from CG to front axle (lf ) 1.1 m
Distance from CG to rear axle (lr ) 1.4 m
Front cornering stiﬀness (Cf ) 40000 N/rad
Rear cornering stiﬀness (Cr ) 42503 N/rad
Front tire relaxation length (af ) 0.75 m
Rear tire relaxation length (ar ) 0.75 m

Results
Case 1: Uncertainty in road adhesion coeﬃcient (µ)
Case 2: Uncertainty in mass and inertia (m and J)
Case 3: Variable longitudinal velocity vx with wind
disturbance Fw
Case 4: Eﬀect of second-order DO
observer poles : [ −10 − 20 − 30 ]
controller gains : kl = 20 and ks = 2

Case 1 : Uncertainty in Road Adhesion Coeﬃcient (µ)
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
time (sec)
˙ψand˙ψd
(a) yaw rate
0 2 4 6 8 10
−1.5
−1
−0.5
0
time (sec)
u
(b) control
Figure: Single lane change maneuver (SLC)

0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
˙ψand˙ψd
(a) yaw rate
0 2 4 6 8 10
−1
0
1
time (sec)u
(b) control
Figure: Double lane change maneuver (DLC)

0 2 4 6 8 10
−1
0
1
·10−2
time (sec)
˙ψ−˙ψd
(a) output tracking error
0 2 4 6 8 10
−1
0
1
2
·10−3
time (sec)
˙ψ−
ˆ˙ψ
(b) output estimation error
Figure: Tracking and estimation accuracy for DLC maneuver

Case 2 : Uncertainty in Mass and Inertia (m and J)
0 2 4 6 8 10
−4
−2
0
2
4
time (sec)
˙ψand˙ψd
(a) yaw rate
0 2 4 6 8 10
0
2
4
6
time (sec)u
(b) control
Figure: Control performance for DLC maneuver

Case 3 : Variable Longitudinal Velocity with Wind Disturbance
0 2 4 6 8 10
5
10
15
20
time (sec)
vx
(a) longitudinal velocity
0 2 4 6 8 10
0
2
4
time (sec)Fw(N)
(b) wind disturbance

Case 3 : Variable Longitudinal Velocity with Wind Disturbance
0 2 4 6 8 10
−1
0
1
time (sec)
˙ψand˙ψd
(c) yaw rate
0 2 4 6 8 10
−2
0
2
4
6
time (sec)
u (d) control
Figure: Control performance for DLC maneuver

Case 4 : Eﬀect of Second-order DO
0 2 4 6 8 10
0
0.1
0.2
0.3
time (sec)
˙ψ−˙ψd
(a) output tracking error (SLC)
0 2 4 6 8 10
−5
0
5
·10−2
time (sec)
˙ψ−˙ψd
(b) output tracking error (DLC)
Figure: Comparative performance

Summary of Performance : DLC Maneuver
Uncertainty (β-βd ) × 10−3 ( ˙ψ- ˙ψd ) × 10−3 uRMS
µ = 0.3 111 9.28 0.3748
µ = 0.5 116 9.37 0.3821
µ = 0.8 119 9.68 0.3892
∆m, ∆J = +20% 73.9 15.02 4.1237
∆m, ∆J = −20% 74.6 12.78 2.2343
varying vx 49.5 23.74 4.1172
2nd order DO 12.89 3.26 0.1638

Summary of Performance : SLC Maneuver
Uncertainty (β-βd ) × 10−3 ( ˙ψ- ˙ψd ) × 10−3 uRMS
µ = 0.3 77.03 7.38 1.584
µ = 0.5 81.6 7.87 1.585
µ = 0.8 83.4 8.45 1.586
∆m, ∆J = +20% 37.12 8.76 1.569
∆m, ∆J = −20% 39.43 7.37 1.573
varying vx 46.95 19.7 2.7342
2nd order DO 6.58 1.7 1.6218

Salient Features
SMC combined with EID and extended to nonlinear system
with matched as well as mismatched uncertainties
EID estimated with DO and the accuracy of estimation is
improved with an extended-DO, which is further generalized
to nth-order DO
The ultimate boundedness is guaranteed; and the bounds can
be lowered by appropriate choice of design parameters
Application to yaw-rate tracking in an active steering control
problem for diﬀerent maneuvers with varying types of
uncertainties and disturbances

Outcomes
IEEE Transactions on Industrial Informatics

Discrete SMC Salient Features Outcomes
Discrete SMC Algorithm with Estimation of States
and Uncertainties using UDE

Discrete Plant with δ operator
An uncertain plant in continuous domain can be written as,
˙x = A x + B u + B e(x, t)
The δ operator is deﬁned as,
δxk =
xk+1 − xk
T
T = 0
The discretized plant is given as,
δxk = Axk + Buk + Bek

Sliding Condition I
For sliding to occur
s2
k+1 < s2
k
which implies
s2
k+1 − s2
k < 0
(sk+1 − sk)(sk+1 − sk + 2sk) < 0
Deﬁning
∆sk = (sk+1 − sk)
The sliding condition can be written as
∆sk(∆sk + 2sk) < 0

Sliding Condition II
or
∆s2
k < −2sk∆sk
For sliding to occur, if sk > 0, then ∆sk < 0
−2sk < ∆sk
Thus, for sk > 0
−2sk < ∆sk < 0
Similarly, if sk < 0, then ∆sk > 0
∆sk < −2sk
Thus for sk < 0
0 < ∆sk < −2sk

Sliding Condition III
or
2sk < −∆sk < 0
−2s2
k < sk∆sk < 0
Since δsk = ∆sk
T the sliding condition can be written as,
−2s2
k
T
< skδsk < 0
The condition collapses into the familiar
σ ˙σ < 0
when the sampling time T → 0

Design of Control
δxk = Axk + Buk + Bek
yk = Cxk
δxmk
= Amxmk
+ Bmumk
ymk
= Cmxmk
δˆxk = Aˆxk + Buk + Bˆek + J(yk − ˆyk)
ˆyk = Cˆxk

Error Dynamics
δ˜xk = (A − JC)˜xk + B˜ek
where, ˜ek = ek − êk
êk = ekGf (γ) = ek
1 − e−T/τ
1 + Tγ − e−T/τ
where êk is the estimation of uncertainty and Gf (γ) is discrete first
order low pass filter
δ˜ek = −
1
T
[B+
JC˜xk(1 − e−T/τ
)] + δek

Observer - Controller Combination
δ˜xk
δ˜ek
=



A − JC B
−B+JC
T
(1 − e−T/τ
) 0



˜xk
˜ek
+
0
1
δek
The observer controller combination can be stabilized by
appropriate choice of J, T and τ

Application : Motion Control
Figure: Industrial Plant Emulator - ECP 220

Application : Motion Control I
Jr
¨θ + Cr
˙θ = Td
Jr is reﬂected inertia at drive
Cr is reﬂected damping to drive
Td is the desired torque
The desired torque can be achieved by selecting appropriate
control voltage (u) and hardware gain (khw )
Jr
¨θ + Cr
˙θ = khw u

Application : Motion Control II
The plant dynamics can be modeled in state space notation as,
˙x1
˙x2
=
0 1
0 −Cr
Jr
x1
x2
+
0
khw
Jr
u
y = 1 0
x1
x2
[x1 x2]T are the states - position (θ) and velocity ( ˙θ)
u is the control signal in volts and
y is the output position in degrees

Application : Motion Control III
Cr = C1 + C2 (gr)−2
gr = 6
npd
npl
Jr = Jd + Jp (grprime)−2
+ Jl (gr)−2
Jd = Jdd + mwd (rwd )2
+ Jwd0
Jp = Jpd + Jpl + Jpbl
grprime =
npd
12
Jl = Jdl + mwl (rwl )2
+ Jwl0
Jwl0 =
1
2
mwl (rwl0 )2

Simulation Results
Am =
0 1
−ω2
n −2ζωn
, bm =
0
−ω2
n
x(0) = [0 1]T
xm(0) = [0 0]T
Model parameters : ζ = 1 and ωn = 5
Control gain is K = 2
Observer poles : [−10 − 20 − 30]
Reference : square wave of amplitude 1 and frequency 0.3 rad/sec

0 0.5 1 1.5 2
−0.5
0
0.5
1
time (sec)
x−xm
(a) model tracking error
0 0.5 1 1.5 2
−0.5
0
0.5
1
time (sec)x−ˆx
(b) state tracking error
Figure: Estimation of states and uncertainty
10ms (solid) and 20 ms (dotted)

0 0.5 1 1.5 2
−1
0
1
time (sec)
x1andxm1
(a) plant and model state-1
0 0.5 1 1.5 2
−2
0
2
4
time (sec)
x2andxm2
(b) plant and model state-2

0 0.5 1 1.5 2
−20
0
20
40
time (sec)
u
(c) control
0 0.5 1 1.5 2
−1
−0.5
0
time (sec)
σ
(d) sliding variable
Figure: Model following performance

Case 2 : Effect of Sampling Time
0 0.2 0.4 0.6 0.8 1
−10
−5
0
5
time (sec)
ê
(a) estimate of uncertainty
0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
time (sec)
σ
Figure: Effect of different sampling time
10ms (solid) and 20 ms (dotted)

0 0.2 0.4 0.6 0.8 1
−20
0
20
time (sec)
ê
(a) estimate of uncertainty
0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
time (sec)
σ
Figure: Effect of filter order
second order (solid) and first order (dotted)

(a) Top view (b) Front view
Figure: Industrial Plant Emulator - ECP 220

Case 1: Nominal Plant
Case 2: Addition of Backlash
Case 3: Addition of Coulomb Friction
Case 4: Uneven Load on Drive Motor

0 10 20 30 40
0
10
20
30
40
time (sec)
driveposition(deg)
(a) drive position
0 10 20 30 40
−2
0
2
·10−3
time (sec)
controlinput(Volt)
(b) control
Figure: Control performance for nominal plant

Case 2 : Addition of Backlash
0 10 20 30 40
0
10
20
30
time (sec)
driveposition(deg)
(a) drive position
0 10 20 30 40
−2
0
2
4
·10−3
time (sec)
controlinput(Volt)
(b) control
Figure: Control performance for plant with backlash

Case 3 : Addition of Coulomb Friction
0 10 20 30 40
0
10
20
30
time (sec)
driveposition(deg)
(a) drive position
0 10 20 30 40
−2
0
2
·10−3
time (sec)
controlinput(Volt)
(b) control
Figure: Control performance for plant with coulomb friction

Case 4 : Uneven Load on Drive Motor
0 10 20 30 40
0
10
20
30
40
time (sec)
driveposition(deg)
(a) drive position
0 10 20 30 40
−4
−2
0
2
4
·10−3
time (sec)
controlinput(Volt)
(b) control
Figure: Control performance with uneven load on drive motor

Salient Features
concept of discrete sliding mode using δ-operator
the δ-operator formulates an uniﬁed design thereby resolving
the dichotomy between results in continuous and discrete time
control law
design robustiﬁed by a discrete IDO that estimates states,
uncertainty and disturbance
overall stability proved in the usual way
application to an industrial motion case-study with
experimental validation

Outcomes
International Journal of Robust and Nonlinear Control

ConclusionsPublicationsThe Road Ahead
Conclusions
Part III

Conclusions Publications The Road Ahead
Summary I
1 The method of UDE aids in mitigating the effect of chatter in
conventional sliding mode. The use of UDE gives a better
trade-off inside the boundary layer and this trade-off is
improved by using a higher order UDE.
2 A modified sliding surface handles the problem of large initial
control associated with UDE. The UDE based control enforces
sliding, without using discontinuous control and without
requiring any knowledge of uncertainties or their bounds.
3 The control law is made implementable by simultaneous
estimation of states and uncertainties.

Summary II
4 The UDE is able to compensate nonlinear uncertainty even in
input vector to enable a robust SMC law. The efficacy of this
design is confirmed for an inverted pendulum control.
5 The method of EID enables SMC to compensate mismatched
disturbance. The integral action in conventional EID based
control is avoided by the use of SMC for nominal control.
6 The method of EID is able to compensate state-dependent
uncertainties. A higher-order filter facilitates improvement in
the performance of EID based control.

Summary III
7 The EID combined with DO can robustify the control of
uncertain nonlinear systems. An extended DO is successful in
improving the estimation accuracy of EID further by giving
the estimate of derivatives of disturbance as well.
8 The efficacy of SMC with EID in applications for anti-lock
braking system and active steering control is proved. The
results prove that the system is robust to different friction
models.
9 The use of δ-operator in conjunction with a new sliding
condition enables complete and seamless unification of control
law and UDE.

Summary IV
10 The UDE enables reduction of quasi-sliding band for a given
sampling period. The sliding width is significantly reduced by
using a second-order filter.
11 The efficacy of discrete SMC is proved for a motion control
application.
12 The sliding variable, uncertainty estimation error and state
estimation error are ultimately bounded in all cases. It is
proved that the bounds can be lowered by appropriate choice
of control parameters.

Overall Summary
Robust SMC strategy that mitigates the restrictions imposed
in conventional SMC design

Overall Summary
system considered – nonlinear with matched as well as
mismatched uncertainty

Overall Summary
estimation of states, uncertainty and disturbance

Overall Summary
estimation methods – UDE, DO and EID

Overall Summary
ultimate boundedness of σ, ˜e and ˜x is assured in all cases

Overall Summary
ultimate boundedness of σ, ˜e and ˜x is assured in all cases
validation in motion and automotive control applications

Publications I
1 Prasheel V. Suryawanshi, P. D. Shendge, S. B. Phadke
“Robust Sliding Mode Control for a Class of Nonlinear
Systems using Inertial Delay Control”, in Nonlinear Dynamics:
Springer, Vol. 78, No. 3, pp. 1921-1932, 2014.
2 Prasheel V. Suryawanshi, P. D. Shendge, S. B. Phadke “A
Boundary Layer Sliding Mode Control Design for Chatter
Reduction using Uncertainty and Disturbance Estimator”, in
International Journal of Dynamics and Control: Springer,
DOI: 10.1007/s40435-015-0150-9.

Publications II
3 P. D. Shendge, Prasheel V. Suryawanshi, “Sliding Mode
Control for Flexible Joint using Uncertainty and Disturbance
Estimation”, in Proc. of the World Congress on Engineering
and Computer Science, Vol. I, pp. 216-221, San Francisco,
USA, October 2011

Papers under Communication I
“Improving the performance of Equivalent Input Disturbance
based Control with a Higher-Order Filter”, in IEEE
Transactions on Industrial Electronics.
“Sliding Mode Control for Mismatched Uncertain Systems
using Equivalent Input Disturbance Method”, in IEEE
Transactions on Control System Technology.
“Yaw-rate Control in Active Steering using Equivalent Input
Disturbance Method”, in IEEE Transactions on Industrial
Informatics.

Papers under Communication II
4 Prasheel V. Suryawanshi, P. D. Shendge, S. B. Phadke “A
Robust Observer for Control of Uncertain Nonlinear Systems
using Uncertainty and Disturbance Estimator”, in
Transactions of Institute of Measurement and Control.
“Discrete Uncertainty and Disturbance Estimator using Delta
operator with Application in Sliding Mode Control”, in
International Journal of Robust and Nonlinear Control.

The Road Ahead
simplicity of control design
supplemented by uncertainty estimation
attractive proposition
applicability to a wide range of systems

The Road Ahead
simplicity of control design
supplemented by uncertainty estimation
attractive proposition
applicability to a wide range of systems
uncertainty estimation based control techniques

Recommendations for Future Work
control of under-actuated and non-minimum phase systems.
extension to terminal sliding-mode and higher-order sliding
mode control.

mode control.
use of nonlinear sliding surface and nonlinear DO.

mode control.
applications in various verticals like power converters,
fuel-cells, electro-pneumatic systems, electric drives, smart
structures, process control, transportation systems.

mode control.
applications in various verticals like power converters,
fuel-cells, electro-pneumatic systems, electric drives, smart
structures, process control, transportation systems.
redesigning control to consider unmodeled lags, quantization
eﬀects and actuator saturation.
implementation on various digital platforms.

Examiner – 1Examiner – 2Examiner – 3
Examiner Comments
Part IV

Examiner – 1 Examiner – 2 Examiner – 3
Examiner – 1
Page no:10, Second Para, line 4, “lyapunov” should be
replaced by “Lyapunov”.
The correction is done (Page No. 10).

Examiner – 1
On page 34, mention the transformation matrix T.
The correction is done. The complete transformation is
derived (Page No. 36).

Examiner – 1
On page 34, mention the transformation matrix T.
The correction is done. The complete transformation is
derived (Page No. 36).
On page 35, Eq. (2.55) after transformation, matrix A and B
should be renamed.
The correction is done. The notations in equation (2.54),
Page No. 34 is changed.

Examiner – 1
Notation used for Transformation matrix T on page 34, and
on page 49, Eq. (3.59) is same. It needs to be changed. Use
unique symbol for same variable throughout the thesis.
The correction is done. T on page 34 is removed. The thesis
is checked for consistency in symbols and notations.

Examiner – 1
On page 65, Eq. (4.1) input u is scalar not multidimensional.

Examiner – 1
On page 65, Eq. (4.1) input u is scalar not multidimensional.
Page no: 91, Eq. (5.4) deﬁne B+.

Examiner – 1
Add reference to “MATLAB” and “Simulink” software as
these are used in the simulations.

Examiner – 2
List of symbols may be included along with list of ﬁgures and
tables.
The list of Acronyms is included. The symbols are described
at their ﬁrst instance.

Examiner – 2
tables.
Experimental validation of few strategies in chapter 3 to 6.
The concern is addressed. Experimental validation done on
Inteco ABS (chapter 4) and ECP Motion Control (chapter 6)
setup.

Examiner – 2
tables.
Experimental validation of few strategies in chapter 3 to 6.
setup.
In the graphs, units for variables (wherever applicable) should
be indicated.
The correction is done (Page No. 60, 61, 83, 84, 85, 86, 87,
89, 90, 91, 137, 138, 139).

Examiner – 2
Citing the reference of Talole and Phadke for the form of 2nd
order UDE ﬁlter at appropriate places.
The correction is done (Page No. 23, 47).

Examiner – 2
Dimensions of matrices while formulating a problem and
expression of control law in numerical simulations.
The correction is done (Page No. 19, 66, 120).

Examiner – 2
Dimensions of matrices while formulating a problem and
expression of control law in numerical simulations.
The correction is done (Page No. 19, 66, 120).
Disturbance estimation graphs should have the plot of actual
disturbance, for better understanding of proposed strategy.
The graphs are included wherever required (Page No. 54, 55,
56). In case of EID based control, the estimate of EID and
actual disturbance have no correlation.

Examiner – 2
Minor additions at some places is suggested and minor typos
at a couple of instances have been highlighted.
The corrections are done at relevant places in all the chapters.

Examiner – 3
Minor typographic correction: In the second line of page 129
there are repeat references to Figure 6.3a.
The correction is done. The second reference is changed to
6.3b on Page No. 135.

Examiner – 3
Minor typographic correction: In the second line of page 129
there are repeat references to Figure 6.3a.
The correction is done. The second reference is changed to
6.3b on Page No. 135.
greater application of hardware test platforms.
setup.

A Small DOT can STOP
A BIG Sentence,
but FEW MORE DOTS ...
can give a CONTINUITY.
EVERY ENDING CAN BE A NEW
BEGINNING

Prasheel

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