formulation of generalized block backstepping control law of underactuated mechanical systems
TRANSCRIPT
Design of Control Law for Underactuated Mechanical Systems: A Modif ied Backstepping Approach
byShubhobrata Rudra
Inspire Research FellowElectrical Engineering Department
Jadavpur University
1
Contents
• A few words on the Underactuated Mechanical Systems [UMS]
• Motivations and Contributions of the Present Work
• Generalized Formulation of the Proposed Block-Backstepping
Control Law for Generic Underactuated Mechanical System
• Comparative Analysis with well known Olfati Saber’s Method
• Applications of the Proposed Control Law on different
Underactuated Mechanical Systems
• Conclusions2 2
Definition: underactuated mechanical systems are a special class
of mechanical systems that have fewer actuating
inputs (control inputs) than configuration variables
(outputs from the system).
3
Advantages of Underactuated Mechanical System
• Cost Effective Design
• Backup control algorithm for fully-actuated system
Underactuated Mechanical Systems
• Use of fewer control inputs
• Complicated nonlinear structure
• Lack of Controllability
• Most of the system fails to satisfy Brockett’s condition
of feedback linearization
4
Control problems of the UMS
Important Contributions in the Field of Underactuated Control System
Year of
PublicationTitle of the Publication Main Contribution
1976Control of unstable mechanical system Control of
pendulum [35]
Proposed a control algorithm for swing up and stabilization of
Inverted Pendulum
1990Nonlinear controllers for non-integrable systems: the
Acrobot example [30]
Developed a basic control algorithm for nonholonomic
underactuated system
1991
Control of mechanical systems with second-order
nonholonomic constraints: Underactuated
manipulators [97]
First paper that identify second order nonholonomic constraints
and discussed the control aspects of such system
1993An analysis of the kinematics and dynamics of under-
actuated manipulators [8]
Provided a detail analysis of forward and inverse dynamics of
underactuated manipulators
1994Partial feedback linearization of underactuated
mechanical systems [32]
Devised a feedback control law that partially linearizes any
arbitrary underactuated system
1995Control of vehicles with second-order nonholonomic
constraints: Underactuated vehicles [84]
Proposed control algorithm for higher order underactuated
system with second order nonholonomic constraints
5
Important Contributions in the Field of Underactuated Control System
1996
Energy based control of a class of underactuated
mechanical systems [23]
First Proposed a novel approach of controlling the flow of
energy in an UMS
Hybrid feedback laws for
a class of cascade nonlinear control systems [24]
Proposed the concept of a logic based switching controller
to select the suitable control law between various time-
periodic control functions at discrete-time instants.
Control of underactuated mechanical systems using
switching and saturation [125]
Developed a saturation control algorithm for stabilization of
UMS
1997
Non-smooth stabilization of an underactuated
unstable two degrees of freedom mechanical
system [112]
Proposed the formulation of a nonsmooth control law to
address the control problem of nonholonomic system
1998
Controller design for a class of underactuated
nonlinear systems [20]
Proposed a novel state transformation to convert the state
model of 2 DOF systems into normal form and then apply
the concept of integral backstepping on that.
1999
Dynamics and control of a class of underactuated
mechanical systems [10]
Provided a detailed analysis of controllability and
stabilizability of the underactuated mechanical system
Adaptive Variable Structure Set-Point Control of
Underactuated Robots [130]
Devised a model based adaptive control scheme where
the uncertainty bounds only depends on the inertial
parameter of the system 6
Important Contributions in the Field of Underactuated Control System
2000
Nonlinear control of underactuated
mechanical systems with application to
robotics and aerospace vehicles [14]
Developed a novel coordinate transform that converts the state
model of the underactuated system into Byrens-Isidori normal
form (for both 2-DOF and n-DOF system).
Discontinuous feedback control of a
special class of underactuated mechanical
systems [17]
Devised a generalized discontinuous control law for class of
nonholonomic system with 2nd order motion constraints.
Controlled Lagrangians and the
stabilization of mechanical systems I:
Potential shaping [123]
Devised a kinematic symmetry based stabilization algorithm for
UMS. Modification to the Lagrangian of uncontrolled system is
achieved by means of the energy shaping technique.
2001
Controlled Lagrangians and the
stabilization of mechanical systems II:
Potential shaping [124]
Extended version of the previous approach to address the control
problem of the system without symmetry.
Stabilization of underactuated mechanical
systems: A nonregular backstepping
approach [19]
At first the state model of the underactuated system transformed
into a chained system. And then the authors applied cascade
backstepping approach on the chained system.
2002
Stabilization of a class of underactuated
mechanical systems via interconnection
and damping assignment [57]
This is basically a passivity based method. The proposed
algorithm generates a smooth asymptotic stabilizing control law to
ensure the local stability of the UMS. 7
Important Contributions in the Field of Underactuated Control System
2003Position tracking of underactuated
vehicles [118]
Proposed a novel controller that ensures exponential convergence of
Underactuated vehicles to any arbitrary trajectory.
2004
Design of a stable sliding-mode
controller for a class of second-order
underactuated systems [140]
Proposed the concept of hierarchical sliding mode control to address
the control problem of second order underactuated system
2005Global time-varying stabilization of
underactuated surface vessel [85]
Proposed three different time varying control laws to address the
stabilization control problem of USV
2006Optimal sliding mode control for
underactuated systems [18]
Based on nonlinear predictive control the authors proposed a novel
design of optimal linear surfaces for sliding model control of
underactuated nonlinear systems.
2007
Trajectory-tracking and path-following
of underactuated autonomous vehicles
with parametric modeling uncertainty
[91]
Proposed an adaptive supervisory control combined with
backstepping control law to address the control problem of UAV with
uncertain parameter.
2008Sliding mode control of a class of
underactuated systems [73]
The authors proposed a sliding mode control approach to stabilize a
class of underactuated systems that could be represented in
cascaded form.8
Important Contributions in the Field of Underactuated Control System-
2009Control of a class of underactuated mechanical
systems using sliding modes [144]
Developed a higher order sliding mode control algorithm
to address the stabilization control problem of the
underactuated system
2010
Output feedback control of a quadrotor UAV using
neural networks
[148]
Neural network is introduced to learn the uncertain
dynamics of UAV and then apply a output feedback
control algorithm to stabilize the UAV.
2011
Inverse dynamics of underactuated mechanical
systems: A simple case study and experimental
verification [149]
Studied the dynamic inversion problem for underactuated
mechanical systems
2012Controller design for a class of underactuated
mechanical systems [66]
A backstepping-like adaptive controller based on function
approximation technique (FAT) is designed to address the
control problem of UMS
2013Output-feedback stabilization control for a class
of under-actuated mechanical system [150]
An output feedback based backstepping controller is
designed to address the control problem of UMS
(actuated shape variable)
2014
Nonlinear State Feedback Controller design for
Underactuated Mechanical System: a Modified
Block Backstepping Approach [29]
Proposed a novel block-backstepping design approach to
address the control problem of generalized underactuated
systems9
Solutions to the Control Problems of UMS
• By means of some state transformation state model of the
systems could be converted into a new state model that
conforms application of well-known control laws.
• Design a control law for the system in new state space using
advanced control law that will also ensure proper functioning
of the physical system.
• A few popular control algorithms for UMS are feedback
linearization, backstepping, sliding mode control, etc.
10
Important applications of Backstepping Control on Underactuated System
Year of
PublicationTitle of the Publication Main Contributions
1996Nonlinear tracking of underactuated surface vessels
[86]
Proposed a backstepping based output decoupling
controller design for USV
1998
Controller design for a class of underactuated
nonlinear systems [20]
Proposed a novel state transformation to convert the
underactuated system state model into normal form and
then apply the concept of integral backstepping on that.
Global practical stabilization and tracking for an
underactuated ship—A combined averaging and
backstepping approach [88]
Developed a combined averaging and backstepping control
approach to address the tracking and stabilization for USV.
2000Trajectory tracking control design for autonomous
helicopters using a backstepping [135]
Devised a backstepping based controller on the
approximate dynamic model of autonomous helicopter.
2001Stabilization of underactuated mechanical systems:
A non-regular backstepping approach [19]
At first the state model of the underactuated system
transformed into a chained system. And then the authors
have applied cascade backstepping approach on the
chained system.
2002Global tracking control of underactuated ships by
Lyapunov's direct method [111]
Proposed a cascade backstepping approach to address the
control problem of the USV 11
Important applications of Backstepping Control on Underactuated System
2003 Position tracking of underactuated vehicles [118]
Proposed a novel controller that ensures exponential
convergence of Underactuated vehicles to any arbitrary
trajectory.
2004
Global uniform asymptotic stabilization of an
underactuated surface vessel: Experimental
results [83]
Proposed a smooth time varying state feedback to achieve
the global asymptotic stabilization of USV.
2005Global time-varying stabilization of
underactuated surface vessel [85]
Proposed three different time varying control laws to address
the stabilization control problem of USV
2006Asymptotic backstepping stabilization of an
underactuated surface vessel [87]
Proposed a natural coordinate transformation that converts
the USV state model into a third order chained system and
then implement a discontinuous feedback control on it.
2007
Trajectory-tracking and path-following of
underactuated autonomous vehicles with
parametric modeling uncertainty [91]
Proposed an adaptive supervisory control combined with
backstepping control law to address the control problem of
UAV with uncertain parameter.
2008
Modeling and Backstepping-based Nonlinear
Control Strategy for a 6 DOF Quadrotor
Helicopter [151]
Proposed a backstepping based pd controller design
approach for 6 DOF helicopter
12
Important applications of Backstepping Control on Underactuated System
2010Global stabilisation and tracking control of
underactuated surface vessels [90]
Proposed an unified backstepping control design approach
to address the stabilization and exponential tracking control
of USV
2011
Backstepping design for cascade systems with
relaxed assumption on Lyapunov functions
[59]
Use of the feedforoward control with backstepping approach
has been proposed
2012Controller design for a class of underactuated
mechanical systems [66]
A backstepping-like adaptive controller based on function
approximation technique (FAT) is designed to address the
control problem of UMS
2013
Output-feedback stabilization control for a
class of under-actuated mechanical system
[150]
An output feedback based backstepping controller is
designed to address the control problem of UMS (actuated
shape variable)
13
A Few Words on Backstepping
• Simplifies the control problem of an n-dimensional
system by treating it as a cascade of n number 1st
order systems.
• Does not cancel useful nonlinearities.
• It requires system representation in controllable
canonical form.
14
Motivations & Contributions of the Present Work
• Motivations:
• An effective way of solving the UMS control problems is to convert the
state model of the UMS into a new state model using some state
transformation.
• However, most of the previously proposed control laws of UMS have
employed complicated state transformation that in turn makes the
control law inapt for practical applications.
• Conversely, some of the application oriented approaches have
resulted in a handy controller for real-time implementation, but have
failed to ensure global asymptotic stability.
• As a matter of fact, scientist and practice engineers are still looking
for a plausible solution to the control problem of the UMS.15
16
Motivations & Contributions of the Present Work
Main Contributions of the Present Work
• A simple algebraic state transformation technique has been proposed to
convert the state model of the UMS into a convenient form that is suitable
for backstepping control law design.
• A thorough analysis has been made to find out the condition of global
diffeomorphism for the proposed algebraic state transformation.
• Block-backstepping based control law has been designed to address the
control problem of the generic underactuated mechanical systems.
• Stability of the internal dynamics has thoroughly been analyzed to ensure
the global asymptotic stability of the control law.
17
Motivations & Contributions of the Present Work
Main Contributions of the Present Work
• Integral action has been incorporated in the stabilizing function to make
the system partially insensitive with respect to parameter variation and
model uncertainty.
• Proposed control law has been applied on several underactuated systems
to corroborate the theoretical claims (Seven 2-DOF systems, and three 3-
DOF systems).
• Performances of the proposed control law during application on different
UMS have been compared with the performances of the Hierarchical
Adaptive Backstepping Sliding Mode Control.
Formulation of the Block Backstepping Control Law
11 12 1
21 22 2
01 2
1 2
q q q q h q,p
q q q q h q,p q
m m
m m B
Lagrangian model of n-DOF underactuated system
Feedback Law by Spong
1 1 1 1
21 11 1 22 21 11 12, ,2
q h q p q q h q p q q q q q uB m m B m m m m
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
Where , , ,
, , ,
, , ,
1
1q
n 2
2q
n ,
1 2q q q
ncol
1 1
11 qn n
m
1 2
12 qn n
m
1 2
21 qn n
m
2 2
22 qn n
m
q p
1,1
h q pn
2
2 ,h q pn
2 2n nB q
2n
18
1
11 1
1
11 12
, ,f q p q h q
q q q
m p
g m m
0X
State Model of n-DOF underactuated system
State Vector: 21 1 2X q p q pT
Desired Configuration Vector:
Control Objective:2
as t E 0
State model is
not in Controllable
Canonical form
19
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
0 E X XError:
Formulation of the Block Backstepping Control Law
• Define new state variable z1
• Derivative of state variable z1
• Define Stabilizing Function for z1
• Define 2nd State variable z2
• Derivative of z1
1 2 1 2z p p f pK D g
1 1 1c K D g 1 1 1 2
α z p f p 1
nij
ij k
k k
gD g p
q
2 2 1z p α
1 1c 1 2 1 1z z z χ
Where , is a constant matrix such
that only when i=j or otherwise.
2 1n nK
0ijk 0ijk
20
Integral action
1 2 1 1 2z q q p pK g ……(i)
……(ii)
..(iii)
…...(iv)
…...(v)
Formulation of the Block Backstepping Control Law
Hence,
1
1 1 2
1 1 1 1
2 1 2
2 2 2 2
1 2 2 2
1 1 1 1
q q p p
p p p
c c
K g Df Df Df g Df
D g D g D g D g g D g
c c
2 1 2 1 1
1 2
2
1 2 1 1
z u z z χ z
f u p p f u u
u p f u u
u z z χ z φ
1 2
2 2
1 2 2 2p p p pI K g Df g Df D g D g g D g
2 2
1 2 1 1 2 1q q p pK Df Df Df D g D 1 2 2φ f p p f p f
where
21
…...(vi)
Formulation of the Block Backstepping Control Law
• Desired Dynamics of the system
• Desired expression of the time derivative of z2
• Actual time derivative of z2
• Expression of the control input:
1 1
2
c
c
1 2 1 1
2 1 2
z z χ z
z z z
22 1 2z z zc
1 1 1 1 2c c c 1 2 1 1 1 2
u z z χ z Φ z z 1 2
1 1 1 1 1 21 c c c 1 1 2u c χ z z φ
1 1 1 1c c 2 1 2 1 1
z u z z χ z φ
22
…...(vii)
…...(viii)
…(ix)
..(x)
Formulation of the Block Backstepping Control Law
• Remark 1: The proposed control law (u) ensures global asymptotic
stability of the reduced order system in (z1, z2 ) coordinate.
• Remark 2: The proposed control law relies on the fact that ψ is invertible.
• Remark 3: The Proposed control law only ensures the global asymptotic stability of reduced order system (order =2n2).
1 1
2
c
c
1 2 1 1
2 1 2
z z z χ
z z z
23
Formulation of the Block Backstepping Control Law
• The proposed control law transforms original system (order 2n) into
reduced order system (2n2).
• Now, 2n = 2n2 + 2n1
• Therefore, the order of the Internal dynamics of the system is 2n1.
Original State Model
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
Reduced order Model
1 1 1
2
c
c
1 2 1
2 1 2
z z z
z z z
24
Formulation of the Block Backstepping Control Law
• Stability of the internal dynamics can be analyzed using the concept of
zero dynamics stability.
• Two state variables q1 and p1 are selected to represent the internal
dynamics of the underactuated system.
• z1 identically equal to zero implies , z1 , first derivative of z1 , second
derivative of z1 are equal to zero. That is
Solution of these three
Equations will give the
expressions of q2 and p2
in terms of q1 and p1`.
Zero Dynamics Analysis
25
01 2 1 1 2
z q q p pK g
1 02 1 2z = p p f pK D g
1z u φ 0
..(xi.a)
..(xi.c)
..(xi.b)
• Zero dynamics:
• Stability of the zero dynamics depends on the choice of controller
parameter K.
• A Lyapunov function Vz may be constructed to analyze the stability of the
zero dynamics system.
• The controller parameter K should be selected in a manner to ensure the
negative definiteness of the
1
1,
1 1
1 1
q p
p f u f Φ F q pg g
1 1
2 2
T T
1 1 1 1q q p pzV
1
1
T
zV T
1 1q p p f g φ
26
Zero Dynamics Analysis
..(xii)
..(xiii)
..(xiv)
zV
27
Pictorial Representation of Proposed Control Law
Fig: 1. Pictorial representation of the proposed control law
• proposed Transformation
28
Condition of Global Diffeomorphism
1 2
1
1
2 1 1 21
2 2 1 1 1
3
4
q q p pz
z p z p fZ X
z
z
K g
c K D g pT
q
p
..(xv)
1 1 1 1
12
2
1
1 1
1 0 0 0
0 1 0 0
2
2
1 2
12 2
1 2 2
q
fp
p qf
fqX p p
p q q
gK K K Kg
gK c c K c Kc g KD g
TD gK c
D g D g Kpp
..(xvi)
• Simplification yields
• Therefore, it can be concluded that invertibility of ψ implies
that the proposed algebraic transformation is a global
diffeomorphism.
• In turn, it can be inferred that invertibility of ψ together with
zero dynamics stability ensures that the proposed control law
can be used to stabilize an underactuated system at a desired
equilibrium. 29
Condition of Global Diffeomorphism
….(xvii)detX
T
Comparative Analysis with Olfati’s Method
• Olfati’s method in brief:
11 12 1
21 22 2
, 0
,
1 2
1 2
q q q q h q p
q q q q h q p q
m m
m m B
Lagrangian model of n-DOF underactuated system
Spong’s Feedback Law
1 1 1 1
21 11 1 22 21 11 12, ,2
q h q p q q h q p q q q q q uB m m B m m m m
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
State Model of n-DOF underactuated system
Define change of coordinate
2
2
1 2
2 2
1 1
1
y q γ q
yp
ξ q
ξ p
L
12
11
2q
2
0
γ q dsm s
m s
1
2p p q
TL M V
30
State model in new coordinate
1
11
,
1 1 2
2 1 1 1
1 2
2
y ξ y
y y γ ξ ξ
ξ ξ
ξ u
m
g
1
,1 2q qq
Vg
Define change of coordinate
2
2
1 2
2 2
1 1
1
y q γ q
yp
ξ q
ξ p
L
12
11
2q
2
0
sγ q ds
s
m
m
1
2p p q
TL M V
31
Comparative Analysis with Olfati’s Method
Comparative Analysis with Olfati’s Method
• A short demonstration on Inertia Wheel Pendulum
32
1 1
2 2
1 1
2 2
q
q
p
p
1 1
21 2 0 12
1 1
11 11
2 2
2
sin
q p
m m g mp q u
m m
q p
p u
2 2
11 1 1 2 1 1 2 12 21 22 2
0 1 1 2 1 0
, m m l m L I I m m m I
m m l m L g
where
33
Comparative Analysis with Olfati’s Method
0
1 1
21 2 . 12
1 1
11 11
2 2
2
sin
q p
m m g mp q u
m m
q p
p u
State model of IWP
Olfati’s method Proposed method
12
1 1 2
11
2 11 1 12 2
1
1 2
2 2
my q q
m
Ly m p m p
p
q
p
12
1 2 1 1 2
11
21 2 0
1 2 1 1
11
21 2 0
1 1 1 1 1 1 1
11
2 2 1
sin
sin
mz q k q p p
m
m m gz p k p q
m
m m gc z k p q
m
z p
New set of state variable Definition of error variables
State Model State Model
1
1 11 2
12
2 12 2 0 1 1
11
1 2
2
sin
y m y
my m m g y
m
u
1 2 1 1 1 1
2 1 1 1 2 1 1 1 1
11
12
21 0 0 21 0 0
1 1 1
11 11
1
sin cos
z z c z
z u z c z c z
mk
m
m m g m m gk q q p
m m
34
Comparative Analysis with Olfati’s Method
Olfati’s method Proposed method
State Model State Model
1
1 11 2
12
2 12 2 0 1 1
11
1 2
2
sin
y m y
my m m g y
m
u
1 2 1 1 1 1
2 1 2 1 1
11
12
21 0 0 21 0 0
1 1 1
11 11
1
sin cos
z z c z
z u c z c z
mk
m
m m g m m gk q q p
m m
Expression of Control Law Expression of Control Law
011 12
2 1 3 2 1 1 1
12 11 11
1 0 1 1
212
1 0 1 0 1 1 1 2
11
2 12 1 12 2
1 0 1 0 1 2 1 1
11 11
12
1 0 1 1 1
11
sin
tanh
sin 1 tanh
1 tanh cos
2 tanh sin
mm mu c c y k
m m m
k c c z
mk c c m y c y
m
m y mk c c m c y y
m m
mc m c y y
m
2
1
12 2 12 2
1 1 1 1 2 1
11 11
; m y m
y k km m
1
211
1 1 1 1 2 2
12
21 0 0 21 0 0
1 1 1 1 1 1
11 11
1 1
sin cos
mu k c z c c z
m
m m g m m gc k q q p
m m
35
Application of the Proposed Control Algorithm
• Acrobot (nonholonomic system with actuated shape variable)
• Pendubot (nonholonomic system with un actuated shape variable)
• TORA (Holonomic system with actuated shape variable)
• Furuta pendulum (nonholonomic system with un actuated shape variable)
• Inertia Wheel Pendulum (Flat underactuated system)
• Inverted Pendulum (Holonomic system with unactuated shape variable)
• Overhead crane (Holonomic system with unactuated shape variable)
• USV (nonholonomic system with actuated shape variable)
• VTOL ( Nonminimum phase Flat underactuated system)
• 3DOF redundant manipulator (nonholonomic system, interacting input) 36
Conclusions
37
• Proposed Control law relies on a simple algebraic state transformation.
• Global diffeomorphism of the state transformation ensures controllability.
• It converts the system into a block-strict feedback form that consists of a
reduced order system, and internal dynamic model.
• Stability of the zero dynamics has thoroughly been analyzed.
• Integral actions has been incorporated in the control law.
• Proposed algorithm is quite generalized.
• It performs well in both the simulation as well as in the real-time environments.
• Only drawback is the tedious calculation of the zero dynamics.
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• contd. 38
39
• Define new state variable z1
• Derivative of state variable z1
• Define Stabilizing Function for z1
• Define 2nd State variable z2
• Derivative of z1
1 2 1 1 2z q q p pK g
1 2 1 2z p p f pK D g
11 1 1 2α z p f pc K D g
1
nij
ij k
k k
gD g p
q
2 2 1z p α
11 2 1z z zc
Where , is a constant matrix such
that only when i=j or otherwise.
2 1n nK
0ijk 0ijk
1
2
3
0 0
0 0
0 0
k
K k
k
Detailed Analysis
40
• Derivative of z2
1 1
1 2 1
2
2
2 2 1
1 2
2
z u z z
f u p p f u
u u p
q q p
p
c c
K g Df Df Df g
Df D g D g
2 2 2
1 2D g D g D g
2
2
1
1 1
n nij
ij k l
l k k l
gD g p p
q q
2
2
1
nij
ij k
k k
gD g p
q
Detailed Analysis
• But derivative of p vector will generate components of u vector
1 1
2
p f uP
p u
g
• Structure of matrix 2
2D g
2
1 1 2
111
1 1
2
2
1
1 1
n nn
k k
k kk k
n nn n n
k k
k kk k
ggp p
q q
D g
g gp p
q q
41
Detailed Analysis
2
1 1 2 2
111
1 1 21
2
2
1 2
1 1
2p
n nn
k k
k kk k
n nn n n n
k k
k kk k
ggp p
q q p
D g
g g pp p
q q
Instead of , let us consider 2
2D g 2
2 2pD g
Or,
2
2
1 1 2
2
111
21 2
1 1
2
2
1
21 2
1 1
2p
n nn
k n k
k kk k
n nn n n
k n k
k kk k
ggp p p p
q q
D g
g gp p p p
q q
2 2
2
1 1 1 2 1 2
2
1 111 11
21 2
1 1
2
2
1 1
21 2
1 1
2
p+ + p
p
p+ + p
n n
n
n n
n n n n n n
n
n n
g gg gp p
q q q q
D g
g g g gp p
q q q q
2 2
2
1 1 1 2 1 2
2
1 111 11
21 2
1 1
2
2
1 1
21 2
1 1
2
+ +
p p
+ +
n n
n
n n
n n n n n n
n
n n
g gg gp p
q q q q
D g
g g g gp p
q q q q
42
Detailed Analysis
2 2
2
1 1 1 2 1 2
2
1 111 11
21 2
1 1
2
2
1 1
21 2
1 1
1
2
2
+ +
pp
p
+ +
n n
n
n n
n n n n n n
n
n n
g gg gp p
q q q q
D g
g g g gp p
q q q q
Or,
2 2 2 2
1 1
2 2 2 2
1 1 1 1
1 1
1 1 1 1
2 2 2 2
1 1 1 11 1
2
2 2
2 2 2 2
1 1 1 11 1
1 2p p
n n n n
r r r r
r r r r
r r r rn n n
n n n nn r n r n r n r
r r r r
r r r rn n n
g g g gp p p p
q q q q
D g p
g g g gp p p p
q q q q
2 2
2 2
1 1
1 1
2 2
1 11
12
2 2
2
2 2
1 11
n n
r r
r r
r r n
n nn r n r
r r
r r n
g gp p
q qp
D g pp
g gp p
q q
43
Detailed Analysis
1p f ug
2p u
Since
2 2 2 2
1 1
2 2 2 2
1 1 1 1
1 1
1 1 1 1
2 2 2 2
1 1 1 11 1
2
2
2 2 2 2
1 1 1 11 1
2p f u u
n n n n
r r r r
r r r r
r r r rn n n
n n n nn r n r n r n r
r r r r
r r r rn n n
g g g gp p p p
q q q q
D g g
g g g gp p p p
q q q q
2 2 2 2
1 1
2 2 2 2
1 1 1 1
1 1
1 1 1 1
2 2 2 2
1 1 1 11 1
2
2 2
2 2 2 2
1 1 1 11 1
1 2p p
n n n n
r r r r
r r r r
r r r rn n n
n n n nn r n r n r n r
r r r r
r r r rn n n
g g g gp p p p
q q q q
D g p
g g g gp p p p
q q q q
Or,
2 2 2 2
2 2 2 21 1 22 p p pp f u uD g D g D g g D g
44
Detailed Analysis Hence,
1
1 1 2
1 1
2 1 2
2 2 2 2
1 2 2 2
1 1
2 2 1
1 2
2
2 1
z u z z
f u p p f u u
u p f u u
u z z Φ
q q p p
p p p
c c
K g Df Df Df g Df
D g D g D g D g g D g
c c
1 2
2 2
1 2 2 2p p p pI K g Df g Df D g D g g D g
2 2
1 2 1 1 2 11 2 2f p p f p f q q p pK Df Df Df D g D
where
45
• Desired Dynamics of the second order system
• Desired expression of the time derivative of z2
• Actual time derivative of z2
• Expression of the control input:
1
2
1 2 1
2 1 2
z z z
z z z
c
c
22 1 2z z zc
1 1 21 2 1 1 1 2u z z z χ Φ z zc c c 1 2
1 1 21 1 2u z z Φc c c
Detailed Analysis
1 12 2 1z u z z Φc c
46
47
Stepwise Development of the
Proposed Control Algorithm-2 DOF
0 E X X
1 1
2 2
1 1 1
2 2 2
q,p q
q,p q
q p
q p
p f g u
p f g u
State Model of 2 DOF underactuated system
State Vector: 1 2 1 2
Tq q p pX
State Error:
Control Objective: E 0 as t
State model is
not in Controllable
Canonical form
48 c
• Define new state variable z1
• Derivative of state variable z1
• Define Stabilizing Function for z1
• Define 2nd State variable z2
• Derivative of z1
1 2 1 2 1 1 2z q k q g p g p
1 2 1 2 1 1 1 2 22 1dg p dg p z p k p g f p g f p
1 1 1 1 2 1 1 1 2 22 1dg p dg pc z k p g f p g f p
2 2 1z p
1 2 1 1z z c z
Contd.
49 c
• Derivative of z2 :
• Expressions of ψ, and ϕ
2 1 2 1 1z u c z c z
21 1 2 2
2 1 1 2 2 1 1 1 1 2
1 2 1 2
2 2 2 1 11 1 2 2 2 1 2 2
1 2 1 2
2
1
dg p
dg p
f f g gg k g g g g g p g p g
p p q q
f f g gg g g g p g p g
p p q q
2 2 2
2 22 2 2 2 2
2 1 1 1 1 2 1 1 2 22 2
1 2 1 21 2
1 1 1 1
1 2 1 2 1 2
1 2 1 2
2
2 2
2
2
1
dg p
dg
g g g g gf k f f p f f p p p p
q q q qq q
f f f fg g p p f f
q q p p
f
2 2 2
2 21 1 2 2 2
2 1 2 1 1 2 22 2
1 2 1 21 2
2pg g g g g
p f f p p p pq q q qq q
Contd.
50 c
• Desired Dynamics of the second order system
• Desired expression of the time derivative of z2
• Actual time derivative of z2
• Expression of the control input:
1 2 1 1
2 1 2 2
z z c z
z z c z
2 1 2 2z z c z
2 1 2 1 1z u c z c z
1 1 2 1 1 1 1 2 2u z c z c z z c z 1 2
1 1 1 2 21u c z c c z
Contd.
51 c
• Remark 1: The proposed control law (u) ensures global asymptotic
stability of the reduced order system in (z1, z2 ) coordinate.
• Remark 2: The proposed control law relies on the fact that ψ is invertible.
• Remark 3: The Proposed control law only ensures the global asymptotic stability of reduced order system (2nd order system).
1 2 1 1
2 1 2 2
z z c z
z z c z
Contd.
52 c
53
54
1 1
2
2 2 1 2 121
11 11
2 2
2
sin 2l
q p
M q p p p mp u
m m
q p
p u
Application on Acrobot
2
2 2 1 2 1 1 2 1 0 1 2 2 0 1 2
2 2 2
1 1 2 1 2 1 2 2 1 2
2 2 2
1 1 2 1 2 1 2 2 1 2
2 2 2
1 1 2 1 2 1 2 2 1 2
sin 2 cos cos
2 cos
2 cos
2 cos
l c c
c c c
c c c
c c c
M q p p p m l m l g q m l g q qf
m l m l l l l q I I
m l m l l l l q I Ig
m l m l l l l q I I
State Model of the Acrobot
Standard 2-DOF state model
1 1
2 2
1
2
,q p q
q p
q p
p f g u
p u
2
1 1 2 1 0 1 2 2 0 1 2 2 1 2 2 2 2 2cos cos , , c c l c cm l m l g q m l g q q M m l l M m l I
2 2 2
1 1 1 2 1 2 1 2 11 1 2 12 2 2, 2 cos , cos c c l lM m l m l l I I m M M q m M M q55
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
1 2 1 1 2z q k q p gp
1 1 1 1 1 1 2c z k p f p dg p
2 2 1z p
1 2 1 2z p k p f p dg p
1 2
1 1 1 1 2 2 1 1 11u c z c c z c
2
1 2 2
1 2f f g
k g g pp p q
23
1 2 22
1 2 1 2
f f f gk f p p f p
q q p q
1 1 2
1 11
sin sinq q qf
q m
2 2
1 11
2 sinlM q pf
p m
2
2 2 2 1 1 2 2
2 11
cos 2 sin 2 sinl lM q p p p q q M q ff
q m
2 1 2
2 11
2 sinlM q p pf
p m
1
0 g
q
2
2 11
1 2 sinlg M qg
q m
2 222 11 2
2 22 11
2 1 2 sin 1 2 cosl lg M q m g M qg
q m
56
Application on Acrobot
Results obtained from simulation
Fig: Acro_1. Angular displacement of the base link (q1)
57
1 2 1 23, 4, 0 and 0q q p p Initial Conditions:
58
Fig: Acro_2. Angular Velocity of the base link (p1)
Results obtained from simulation
59
Results obtained from simulation
Fig: Acro_3. Angular displacement of the upper link (q2)
60
Results obtained from simulation
Fig: Acro_4. Angular Velocity of the upper link (p2)
61
Results obtained from simulation
Fig: Acro_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF Systems
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Algorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
Comparison with HABSMC
63Fig: Acro_6. Comparison of Angular displacement of the base link (q1)
1 2 1 24, 6, 0 and 0q q p p Initial Conditions:
Comparison with HABSMC
64Fig: Acro_7. Comparison of Angular velocity of the base link (p1)
Comparison with HABSMC
Comparison of q2
65
Fig: Acro_8. Comparison of Angular displacement of the upper link (q2)
Comparison with HABSMC
Comparison of p2
66Fig: Acro_9. Comparison of Angular Velocity of the upper link (p2)
Comparison with HABSMC
Comparison of input u
67Fig: Acro_10. Comparison of the input (u)
68
1 1
1
2 2
2
2 2 1 2 12
2
11 11
sin 2l
q p
p u
q p
M q p p p mp u
m m
Application on Pendubot
2
2 2 1 2 1 1 2 1 0 1 2 2 0 1 2
2 2 2
1 1 2 1 2 1 2 2 1 2
2 2 2
1 1 2 1 2 1 2 2 1 2
2 2 2
1 1 2 1 2 1 2 2 1 2
sin 2 cos cos
2 cos
2 cos
2 cos
l c c
c c c
c c c
c c c
M q p p p m l m l g q m l g q qf
m l m l l l l q I I
m l m l l l l q I Ig
m l m l l l l q I I
State Model of the Pendubot
Standard 2-DOF state model
1 1
2 2
1
2 q,p q
q p
q p
p u
p f g u
2
1 1 2 1 1 2 2 1 2 2 1 2 2 2 2 2cos cos , , c c l c cm l m l g q m l g q q M m l l M m l I
2 2 2
1 1 1 2 1 2 1 2 11 1 2 12 2 2, 2 cos , cos c c l lM m l m l l I I m M M q m M M q69
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
2
2 11
1 2 sinlg M qg
q m
2
2 2 2 1 1 2 2
2 11
cos 2 sin 2 sinl lM q p p p q q M q ff
q m
1 1 2 2 1z q k q p gp
1 1 1 1 1 2 1c z k p f p dg p
2 1 1z p
1 1 2 1z p k p f p dg p
1 2
1 1 1 1 2 2 1 1 11u c z c c z c
2
1 2 2
1 2f f g
k g g pp p q
23
1 2 22
1 2 1 2
f f f gk f p p f p
q q p q
1 1 2
1 11
sin sinq q qf
q m
2 2
1 11
2 sinlM q pf
p m
2 1 2
2 11
2 sinlM q p pf
p m
1
0 g
q
2 222 11 2
2 22 11
2 1 2 sin 1 2 cosl lg M q m g M qg
q m
70
Application on Pendubot
Results obtained from simulation
Fig: Pendu_1. Angular displacement of the base link (q1)
71
Initial Conditions:1 2 1 24, 3, 0 and 0q q p p
72
Fig: Pendu_2. Angular Velocity of the base link (p1)
Results obtained from simulation
73
Results obtained from simulation
Fig: Pendu_3. Angular displacement of the upper link (q2)
74
Results obtained from simulation
Fig: Pendu_4. Angular Velocity of the upper link (p2)
75
Results obtained from simulation
Fig: Pendu_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF SystemsAlgorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Comparison with HABSMC
77
Fig: Pendu_6. Comparison of Angular displacement of the base link (q1)
1 2 1 26, 4, 0 and 0q q p p Initial Conditions:
Comparison with HABSMC
78
Fig: Pendu_7. Comparison of Angular velocity of the base link (p1)
Comparison with HABSMC
Comparison of q2
79
Fig: Pendu_8. Comparison of Angular displacement of the upper link (q2)
Comparison with HABSMC
Comparison of p2
80
Fig: Pendu_9. Comparison of Angular Velocity of the upper link (p2)
Comparison with HABSMC
Comparison of input u
81
Fig: Pendu_10. Comparison of the input (u)
82
1 1
1
2 2
2 1
2 2 2 1 3 2
2
tan sincos
q p
p u
q p
kp k q p k q u
q
Application on Furuta Pendulum
21 2
2 1 2
1 2 1
2
2 2 2
2 1 2 2
tan sin
cos
gl lf q p q
L l L
J m lg
m l l q
State Model of the Furuta Pendulum
Standard UMS state model
1 1
2 2
1
2 q,p q
q p
q p
p u
p f g u
2
2 2 2 1 2
1 2 3
2 1 2 1 2 1
, ,
J m l gl l
k k km l l L l L 83
1
2
1
2
q
q
p
p
Actuated Configuration
Variable: φ
Unactuated Configuration
Variable: θ
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
1 1 2 2 2 1 z q k q p g p
1 1 1 1 1 2 2 1 2c z k p f p dg p
2 1 1 z p
1 1 2 2 1 2 z p k p f p dg p
1 2
1 1 1 1 2 2 1 1 11u c z c c z c
2
1 2
1 2f g
k g g pp q
23
2 22
2 1 2
f f gk f p f p
q p q
1
0f
q
3 2 1
1
2 sinf
k q pp
1
0 g
q
1 2 2
2
sec tang
k q qq
2
2 21 2 2 22
2
sec tan secg
k q q qq
2 2
2 2 3 2 1
2
sec cosf
k q k q pq
2
0f
p
84
Application on Furuta Pendulum
Results obtained from simulation
Fig: FP_1. Angular displacement of the base (q1)
85
Initial Conditions:1 2 1 23, 6, 0 and 0q q p p
86
Fig: FP_2. Angular Velocity of the base (p1)
Results obtained from simulation
87
Results obtained from simulation
Fig: FP_3. Angular displacement of the upper link (q2)
88
Results obtained from simulation
Fig: FP_4. Angular Velocity of the upper link (p2)
89
Results obtained from simulation
Fig: FP_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF SystemsAlgorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Comparison with HABSMC
91
Fig: FP_6. Comparison of the Angular displacement of base (q1)
1 2 1 24, 4, 0 and 0q q p p Initial Conditions:
Comparison with HABSMC
92
Fig: FP_7. Comparison of the Angular velocity of the base (p1)
Comparison with HABSMC
Comparison of q2
93
Fig: FP_8. Comparison of the Angular displacement of Pendulum (q2)
Comparison with HABSMC
Comparison of p2
94
Fig: FP_9. Comparison of the Angular Velocity of Pendulum(p2)
Comparison with HABSMC
Comparison of input u
95
Fig: FP_10. Comparison of the input (u)
96
1 1
2
1 3 1 2 2 1 2 2
2 2
2
sin cos
q p
p k q k q p k q u
q p
p u
Application on TORA system
2
3 1 2 2 1
2 2
sin
cos
f k q k q p
g k q
State Model of the TORA System
Standard UMS state model
1 1
2 2
1
2
q,p q
q p
q p
p f g u
p u
2 2 1 2 3 1 1 2, k m r m m k k m m
where
97
1
2
1
2
x
q x
q
p v
p
Actuated Configuration
Variable: θ
Unactuated Configuration
Variable: x
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
1 2 1 1 2z q k q p gp
1 1 1 1 1 1 2c z k p f p dg p
2 2 1z p
1 2 1 2z p k p f p dg p
1 2
1 1 1 1 2 2 1 1 11u c z c c z c
2
1 2
1 2f g
k g g pp q
23
1 2 22
1 1 1 2
f f g gk f p f p f p
q p q q
2 2 1
1
2 sinf
k q pp
1
0 g
q
2 2
2
cosg
k qq
2
2 222
sing
k qq
2
0f
p
3
1
fk
q
2
2 2 1
2
cosf
k q pq
98
Application on TORA system
Results obtained from simulation
Fig: TORA_1. Displacement of the cart (q1)
99
Initial Conditions:1 2 1 21, 3, 0 and 0q q p p
q1
[m]
100
Fig: TORA_2. Velocity of the cart (p1)
Results obtained from simulation
101
Results obtained from simulation
Fig: TORA_3. Angular displacement of the rotor(q2)
102
Results obtained from simulation
Fig: TORA_4. Angular Velocity of the rotor(p2)
103
Results obtained from simulation
Fig: TORA_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF SystemsAlgorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Comparison with HABSMC
105
Fig: TORA_6. Comparison of Cart displacement (q1)
1 2 1 20.5, 4, 0 and 0q q p p Initial Conditions:
Comparison with HABSMC
106
Fig: TORA_7. Comparison of the base velocity (p1)
Comparison with HABSMC
Comparison of q2
107
Fig: TORA_8. Comparison of the rotor Angular displacement (q2)
Comparison with HABSMC
Comparison of p2
108
Fig: TORA_9. Comparison of the rotor angular velocity (p2)
Comparison with HABSMC
Comparison of input u
109
Fig: TORA_10. Comparison of the input (u)
110
1 1
21 2 0 12
1 1
11 11
2 2
2
sin
q p
m m g mp q u
m m
q p
p u
Application on IWP
21 2 0
1
11
2
2 2
1 1 2 1 1 2
sinm m g
f qm
Ig
m l m L I I
State Model of the IWP
Standard UMS state model
1 1
2 2
1
2
q,p q
q p
q p
p f g u
p u
2 2
11 1 1 2 1 1 2 12 21 22 2, m m l m L I I m m m I
where
111
1 1
2 2
1 1
2 2
q
q
p
p
Actuated Configuration
Variable: θ2
Unactuated Configuration
Variable: θ1
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
1 2 1 1 2z q k q p gp
1 1 1 1 1 1 2c z k p f p dg p
2 2 1z p
1 2 1 2z p k p f p dg p
1 2
1 1 1 1 2 2 1 1 11u c z c c z c
1 kg
1
1
fk f p
q
1
0f
p
1
0 g
q
2
0g
q
2
0f
p
1
1
cosf
2
2 2 1
2
cosf
k q pq
112
Application on IWP
Results obtained from simulation
Fig: IWP_1. Angular displacement of the pendulum (q1)113
Initial Conditions:1 2 1 26, 3, 0 and 0q q p p
114
Fig: IWP_2. Angular Velocity of the base (p1)
Results obtained from simulation
115
Results obtained from simulation
Fig: IWP_3. Angular displacement of the Wheel (q2)
116
Results obtained from simulation
Fig: IWP_4. Angular Velocity of the Wheel (p2)
117
Results obtained from simulation
Fig: IWP_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF SystemsAlgorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Comparison with HABSMC
119Fig: IWP_6. Comparison of the angular displacement of pendulum (q1)
1 2 1 26, 4, 0 and 0q q p p Initial Conditions:
Comparison with HABSMC
120
Fig: IWP_7. Comparison of the angular velocity of the pendulum (p1)
Comparison with HABSMC
Comparison of q2
121
Fig: IWP_8. Comparison of the angular displacement of wheel (q2)
Comparison with HABSMC
Comparison of p2
122
Fig: IWP_9. Comparison of the angular velocity of wheel (p2)
Comparison with HABSMC
Comparison of input u
123
Fig: IWP_10. Comparison of the input (u)
124
1 1
2 2
2
2 2 2 1
1
2
2 2 2 1
2
sin 2 sin
2
sin 2 sin
2
q p
q p
gl q p q bp aup
d d d d
g q p q bp lup
d d d d
2
2 2 2 1
1
2
2 2 2 1
2
1
2
sin 2 sin
2
sin 2 sin
2
gl q p q bpf
d d d
g q p q bpf
d d d
ag
d
lg
d
125
Applications on Inverted Pendulum
μ=l M+m 2
2sind J l q 2 J
a lM m
State Model of Inverted Pendulum
Standard 2-DOF state model
1 1
2 2
1 1 1
2 2 2
q,p q
q,p q
q p
q p
p f g u
p f g u
1
2
1
2
x
q x
q
p v
p
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
1 2 1 1 2z q k q ldp adp
1 2 1 1 2 1 2z p k p d lp ap d lf af
1 1 1 1 3 3 4 1 2c z k x d lx ax d lf af
2 2 1z p
1 2
1 1 1 2 2 1 11u c z c c z c
2 2
2 2 2 2 2sin 2 2 sin 2 sinl q l q p alp ql a bak l a
d d d d d d
2 2
2 1 2 2 1 2 2 1 2 2 22
3
2 2 2 2 2 1 2 2 2 2 2
3
2 2 1 2 2
2 cos 2 2 sin 2 cos 2
2 sin cos cos 2 sin
+ cos sin 2
f k f lp q lp ap lp q lf af gl p q
lp q f lp q blf agp q ap q f
ap q baf lf q
126
Applications on Inverted Pendulum
Results obtained from Real-time experiment
Fig: IP_1. Displacement of the cart (q1)
127
128
Fig: IP_2. Velocity of the cart (p1)
Results obtained from Real-time experiment
129
Results obtained from Real-time experiment
Fig: IP_3. Angular displacement of the pendulum(q2)
130
Results obtained from Real-time experiment
Fig: IP_4. Angular Velocity of the pendulum(p2)
131
Results obtained from Real-time experiment
Fig: IP_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF SystemsAlgorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Comparison with HABSMC
133
Fig: IP_6. Comparison of Cart displacement (q1)
Comparison with HABSMC
134
Fig: IP_7. Comparison of the cart velocity (p1)
Comparison with HABSMC
Comparison of q2
135
Fig: IP_8. Comparison of the pendulum angular displacement (q2)
Comparison with HABSMC
Comparison of p2
136
Fig: IP_9. Comparison of the pendulum angular velocity (p2)
137
Application on Granty crane
2
2 2 2 1
1
2
2 2 2 1
2
1
2
sin 2 sin
2
sin 2 sin
2
gl q p q bpf
d d d
g q p q bpf
d d d
ag
d
lg
d
μ=l M+m 2
2sind J l q 2 J
a lM m
State Model of Granty Crane
Standard 2-DOF state model
1 1
2 2
1 1 1
2 2 2
q,p q
q,p q
q p
q p
p f g u
p f g u
1 1
2 2
2
2 2 2 1
1
2
2 2 2 1
2
sin 2 sin
2 2
sin 2 sin
2
q p
q p
gl q p q bp aup
d d d
g q p q bp lup
d d d d
138
1
2
1
2
x
q x
q
p v
p
Application on Granty crane
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
• Control Input Required to realize the desired dynamics for z2:
where
and
1 2 1 1 2z q k q ldp adp
1 2 1 1 2 1 2z p k p d lp ap d lf af
1 1 1 1 3 3 4 1 2c z k x d lx ax d lf af
2 2 1z p
1 2
1 1 1 2 2 1 11u c z c c z c
2 2
2 2 2 2 2sin 2 2 sin 2 sinl q l q p alp ql a bak l a
d d d d d d
2 2
2 1 2 2 1 2 2 1 2 2 22
3
2 2 2 2 2 1 2 2 2 2 2
3
2 2 1 2 2
2 cos 2 2 sin 2 cos 2
2 sin cos cos 2 sin
+ cos sin 2
f k f lp q lp ap lp q lf af gl p q
lp q f lp q blf agp q ap q f
ap q baf lf q
139
Results obtained from Real-time experiment
Fig: OC_1. Displacement of the cart (q1)
140
141
Fig: OC_2. Velocity of the cart (p1)
Results obtained from Real-time experiment
142
Results obtained from Real-time experiment
Fig: OC_3. Angular displacement of the payload (q2)
143
Results obtained from Real-time experiment
Fig: OC_4. Angular Velocity of the payload (p2)
144
Results obtained from Real-time experiment
Fig: OC_5. Control Input (u)
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 2 DOF SystemsAlgorithm
i) Design backstepping sliding mode control
law for two subsystems. These two sliding
surfaces are known as first layer sliding
surface.
ii) Similar to MRAC Identify the disturbance
bound for each subsystems.
iii) Design a second layer sliding surface to
establish the coupling between two of the
first layer sliding surface.
iv) Second layer sliding surface yields the
complete expression of input
1 1
1
2 2
2
q p
p f X g X u
q p
p u
Comparison with HABSMC
146
Fig: OC_6. Comparison of Cart displacement (q1)
Comparison with HABSMC
147
Fig: OC_7. Comparison of the cart velocity (p1)
Comparison with HABSMC
Comparison of q2
148
Fig: OC_8. Comparison of the payload angular displacement (q2)
Comparison with HABSMC
Comparison of q2
149
Fig: OC_9. Comparison of the payload angular velocity (p2)
Comparison with HABSMC
Comparison of q2
150
Fig: OC_10. Comparison of the input signal(u)
151
1 3
1 6
2 1 2
2 4 5
4 2 6 2
tan & tan 0
T
T
y
q x
p x
q x x
p x x
cf x x x g x
m
1 4
2 5
3 6
4 1
5 2
6 1 2 4 2 6tan tan
y
x x
x x
x x
x u
x u
cx u x x x x
m
Application on USV
State Model of the USV
Standard UMS state model
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
152
1
2
3
4
5
6
x
y
x x
x
x y
x v
x
x v
Actuated Configuration
Variable: x,θ
Unactuated Configuration
Variable: y
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
Application on USV
K g1 2 1 1 2
z q q p p
K D g1 2 1 2
z p p f p
0K k
1 1c K D g 1 1 1 2
α χ z p f p
3
2 2 2 24 5 6
1 1 2 3
tan tan tan tan0 0k
k k
x x x xD g p x x x
q x x x
2 2z p α
2
2 5sec 0D g x x
153
Dynamics of z2:
1
1 1 2
1 1
2 1 2
2 2 2 2
1 2 2 2
1 1
2 2 1
1 2
2
2 1
z u z z
f u p p f u u
u p f u u
u z z Φ
q q p p
p p p
c c
K g Df Df Df g Df
D g D g D g D g g D g
c c
Application on USV
Dynamics of z2:
1 1
1 2 1
2
2
2 2
2 1
1 2
2
z p α
u z z
f u p p f u
u u p
q q p
p
c c
K g Df Df Df g
Df D g D g
(5.9)
23 3
2
1
1 1
ij
ij k l
l k k l
gD g p p
q q
32
2
1
ij
ij k
k k
gD g p
q
2 2
2 1 2D g D g D g
1 1 2
2 2 2 2
2 2 2 22p f u up p pD g D g D g g D g
154
2
1
2
2 2 1
1 1
1, ,n
mi
p mr i
i r
gD g p m n
q
2 22 2
2 2 4 5 4 5 2 4
1 1 2 2
0 0tan tan0 secp
x xD g x x x x x x
x x x x
2 22 1 4 5
3 3
tan 00p
xD g x x
x x
1 4 2 6
6
tany y
p
c cDf x x x
m x m
2 4 2 6 4 2 6 2
4 5
tan tan tan 0y y
p
c cDf x x x x x x x
m x x m
2
2
2
2 2 1
1 2
1, ,n
mi
p mr i
i r
gD g p m n
q
4 2 6 2tan & tan 0yc
f x x x g xm
Application on USV
Control Input 1 2
1 1 1 2 1 11 c c c c 1 2 1u z z χ Φ
1 2
2 2
1 2 2 2p p p pI K g Df g Df D g D g g D g
155
23 3
2 2 22
1 2 2 5
1 1
tan0 2 tan sec 0ij k l
l k k l
xD g p p x x x
q q
1 4 2 6
3
tan 0y
q
cDf x x x
m x
2
2 4 2 6 4 2 6 4 2
1 2
tan tan 0 secy y
q
c cDf x x x x x x x x
m x x m
Application on USV
2 2
1 2 1 1 2 11 2 2f p p f p f q q p pK Df Df Df D g D
1 4 2 6
6
tany y
p
c cDf x x x
m x m
2
2 1 0pD g
2
2
1
1 1
n nij
ij k l
l k k l
gD g p p
q q
156
Results obtained from simulation
1 2 1 20.8, 0.19 , 0.1 and 1.62 2
q q p p
Initial Condition
Fig: USV_1. Longitudinal Displacement
157
Results obtained from simulation
Fig: USV_ 2. Longitudinal Velocity
158
Fig: USV_3. Angular Orientation of the surface vessel
159
Results obtained from simulation
Fig: USV_4. Angular Velocity of the surface vessel160
Results obtained from simulation
Fig: USV_ 5.Lateral Displacement
161
Results obtained from simulation
Fig: USV_6. Lateral Velocity
162
Results obtained from simulation
163
Results obtained from simulation
Fig: USV_7. Control Input u1
164
Results obtained from simulation
Fig: USV_8. Control Input u2
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 3 DOF Systems
1 1
1 1
21 21
22 22
21 1
22 2
q p
p f X g X u
q p
q p
p u
p u
Algorithm
(i) Follow the same principle of 2-DOF
hierarchical architecture for controller
design for first input (u1)
(ii) Repeat the adaptive backstepping
sliding mode design for input u2 too.
Comparison with HABSMC
1 2 1 21, 0.3 , 0.1 and 2.43 3
q q p p
Initial Condition
Fig: USV_9. Performance Comparison of the Longitudinal Displacement
166
Comparison with HABSMC
Fig: USV_10. Performance Comparison of the Longitudinal Velocity
167
Comparison with HABSMC
Fig: USV_11. Performance Comparison of the Angular Orientation of the USV168
Comparison with HABSMC
Fig: USV_12. Performance Comparison of the Angular Velocity of the USV
169
Comparison with HABSMC
Fig: USV_13. Performance Comparison of the Lateral Displacement
170
Comparison with HABSMC
Fig: USV_14. Performance Comparison of the of Lateral Velocity
171
Comparison with HABSMC
Fig: USV_15. Performance Comparison of the Control Input u1
172
Comparison with HABSMC
Fig: USV_16. Performance Comparison of the Control Input u2
173
174
1 5
1 6
2 1 3
2 2 4
5 5 5
sin & cos sin
T
T
q x
p x
q x x
p x x
f g x g x x
1 2
2 1
3 4
4 2
5 6
6 1 5 2 5 5cos sin sin
x x
x u
x x
x u
x x
x u x u x g x
Application on VTOL
State Model of the VTOL
Standard UMS state model
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
175
1
2
3
4
5
6
y
x
x y
x v
x
x
x x
x v
Actuated Configuration
Variable: y,θ
Unactuated Configuration
Variable: x
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
Contd.
K g1 2 1 1 2
z q q p p
K D g1 2 1 2
z p p f p
0K k
1 1c K D g 1 1 1 2
α z χ p f p
3 3
3 33 6 3 6
1 1
cos sinsin cosk k
k kk k
x xD g p p x x x x
q q
2 2z p α
176
Application on VTOL
Dynamics of z2:
1 1
1 2 1
2
2
2 2
2 1
1 2
2
z p α
u z z
f u p p f u
u u p
q q p
p
c c
K g Df Df Df g
Df D g D g
(5.9)
23 3
2
1
1 1
ij
ij k l
l k k l
gD g p p
q q
32
2
1
ij
ij k
k k
gD g p
q
2 2
2 1 2D g D g D g
1 1 2
2 2 2 2
2 2 2 22p f u up p pD g D g D g g D g
Dynamics of z2:
1
1 1 2
1 1
2 1 2
2 2 2 2
1 2 2 2
1 1
2 2 1
1 2
2
2 1
z u z z
f u p p f u u
u p f u u
u z z Φ
q q p p
p p p
c c
K g Df Df Df g Df
D g D g D g D g g D g
c c
177
2
1
2
2 2 1
1 1
1, ,n
mi
p mr i
i r
gD g p m n
q
5 5 5 52
2 2 4 2 4
1 1 2 2
cos sin cos sin0 0p
x x x xD g x x x x
x x x x
2 5 52 1 2 4 5 2 5 4
5 5
cos sinsin cosp
x xD g x x x x x x
x x
1 5
6
9.81 sin 0pDf xx
2 5 5
2 4
9.81 sin sin 0 0pDf x xx x
2
2
2
2 2 1
1 2
1, ,n
mi
p mr i
i r
gD g p m n
q
5 5 59.81 sin & cos sinf x g x x
Application on VTOL
Control Input 1 2
1 1 1 2 1 11 c c c c 1 2 1u z z χ Φ
1 2
2 2
1 2 2 2p p p pI K g Df g Df D g D g g D g
5 6 5 6sin cosD g x x x x
178
2 23 3 3 3
2 25 5
1 5 5 6
1 1 1 1
cos sincos sinij k l k l
l k l kk l k l
x xD g p p p p x x x
q q q q
5
1 5
5
sin9.81 9.81 cosq
xDf x
x
2 5 5
1 3
9.81 sin sin 0 0qDf x xx x
Application on VTOL
2 2
1 2 1 1 2 11 2 2f p p f p f q q p pK Df Df Df D g D
1 5
6
9.81 sin 0pDf xx
2
2
1
1 1
n nij
ij k l
l k k l
gD g p p
q q
2
2 1 5 2 5 4sin cospD g x x x x
179
Results obtained from simulation
1 2 1 22, 4 , 3 and 1 03
q q p p
Initial Condition
Fig: VTOL_1. X axis Displacement of VTOL
180
Results obtained from simulation
Fig: VTOL_ 2. X axis Velocity of VTOL
181
Fig: VTOL_3. Y axis displacement of VTOL
182
Results obtained from simulation
Fig: VTOL_4. Y axis Velocity of the VTOL
183
Results obtained from simulation
Fig: VTOL_ 5. Angular Displacement of VTOL
184
Results obtained from simulation
Fig: VTOL_6. Angular Velocity of the VTOL
185
Results obtained from simulation
186
Results obtained from simulation
Fig: VTOL_7. Control Input v1
187
Results obtained from simulation
Fig: VTOL_8. Control Input v2
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 3 DOF SystemsAlgorithm
(i) Follow the same principle of 2-DOF
hierarchical architecture for controller
design for first input (u1)
(ii) Repeat the adaptive backstepping
sliding mode design for input u2 too.
1 1
1 1
21 21
22 22
21 1
22 2
q p
p f X g X u
q p
q p
p u
p u
Comparison with HABSMC
1 2 1 23, 3 , 2.5 and 1.5 04
q q p p
Initial Condition
Fig: VTOL_9. Performance Comparison of the x-axis Displacement
189
Comparison with HABSMC
Fig: VTOL_10. Performance Comparison of the x-axis Velocity
190
Comparison with HABSMC
Fig: VTOL_11. Performance Comparison of the Y axis displacement of the VTOL
191
Comparison with HABSMC
Fig: VTOL_12. Performance Comparison of the Y axis Velocity of the VTOL
192
Comparison with HABSMC
Fig: VTOL_13. Performance Comparison of the angular orientation of VTOL
193
Comparison with HABSMC
Fig: VTOL_14. Performance Comparison of the of angular velocity of VTOL
194
Comparison with HABSMC
Fig: VTOL_15. Performance Comparison of the Control Input v1
195
Comparison with HABSMC
Fig: VTOL_16. Performance Comparison of the Control Input v2
196
197
1 3
1 6
2 1 2
2 4 5
2
0 & tan 0
T
T
q x
p x
q x x
p x x
f g x
1 4
2 5
3 6
4 1
5 2
6 1 2
1 2 3
tan
, ,
x x
x x
x x
x u
x u
x u x
x x x x y
Application on 3-DOF Redundant Manipulator
State Model of the Manipulator
Standard n-DOF state model
,
1 1
2 2
1
2
q p
q p
p f q p q u
p u
g
198
• Definition of z1:
• Dynamics of z1:
• Stabilizing Function:
• Definition of z2 :
Application on 3-DOF Redundant Manipulator
K g1 2 1 1 2
z q q p p
K D g1 2 1 2
z p p f p
0K k
1 1 1c K D g 1 1 2
α z p f p
3
222 5
1
tan0 sec 0k
k k
xD g p x x
q
2 2z p α
199
Application on 3-DOF Redundant Manipulator
Dynamics of z2:
1 1 1 1
1 2 1
2
2
q q p
p
c c
K g Df Df Df g
Df D g D g
2 2
2 1 1 1
1 2
2
z p α
u z z χ z
f u p p f u
u u p
(5.9)
23 3
2
1
1 1
ij
ij k l
l k k l
gD g p p
q q
32
2
1
ij
ij k
k k
gD g p
q
2 2
2 1 2D g D g D g
1 1 2
2 2 2 2
2 2 2 22p f u up p pD g D g D g g D g
Dynamics of z2:
1
1 1 2
1 1
2 1 2
2 2 2 2
1 2 2 2
1 1
q q p p
p p p
c c
K g Df Df Df g Df
D g D g D g D g g D g
c c
2 2 1
1 2
2
2 1
z u z z
f u p p f u u
u p f u u
u z z Φ
200
2
1
2
2 2 1
1 1
1, ,n
mi
p mr i
i r
gD g p m n
q
2 22 2
2 4 5 4 5 2 4
1 1 2 2
0 0tan tan0 secp
x xD g x x x x x x
x x x x
2 22 1 4 5
3
tan0p
xD g x x
x
1 0pDf 2 0 0pDf
2
2
2
2 2 1
1 2
1, ,n
mi
p mr i
i r
gD g p m n
q
20 & tan 0f g x
Application on 3-DOF Redundant Manipulator
Control Input 1 2
1 1 1 2 1 11 c c c c 1 2 1u z z χ Φ
1 2
2 2
1 2 2 2p p p pI K g Df g Df D g D g g D g
3
222 5
1
tan0 sec 0k
k k
xD g p x x
q
201
23 3
2 2 22
1 2 2 5
1 1
tan0 2 tan sec 0ij k l
l k k l
xD g p p x x x
q q
1 0qDf
2 0 0qDf
Application on 3-DOF Redundant Manipulator
2 2
1 2 1 1 2 11 2 2f p p f p f q q p pK Df Df Df D g D
1 0pDf
2
2
1
1 1
n nij
ij k l
l k k l
gD g p p
q q
2 21 4 5
3
tan0p
xD g x x
x
202
Results obtained from simulation
1 2 1 20.755, 1 , 0.5 and 0.5 04
q q p p
Initial Condition
Fig: 3-DOF_1. X axis Displacement of 3-DOF
203
Results obtained from simulation
Fig: 3-DOF_ 2. X axis Velocity of 3-DOF
204
Fig: 3-DOF_3. Y axis displacement of 3-DOF
205
Results obtained from simulation
Fig: 3-DOF_4. Y axis Velocity of the 3-DOF
206
Results obtained from simulation
Fig: 3-DOF_ 5. Angular Displacement of 3-DOF
207
Results obtained from simulation
Fig: 3-DOF_6. Angular Velocity of the 3-DOF
208
Results obtained from simulation
209
Results obtained from simulation
Fig: 3-DOF_7. Control Input u1
210
Results obtained from simulation
Fig: 3-DOF_8. Control Input u2
Hierarchical Adaptive Backstepping Sliding
Mode Control [HABSMC]
• 3 DOF SystemsAlgorithm
(i) Follow the same principle of 2-DOF
hierarchical architecture for controller
design for first input (u1)
(ii) Repeat the adaptive backstepping
sliding mode design for input u2 too.
1 1
1 1
21 21
22 22
21 1
22 2
q p
p f X g X u
q p
q p
p u
p u
Comparison with HABSMC
1 2 1 20.5, 0.755 , 0.5 and 0.45 06
q q p p
Initial Condition
Fig: 3-DOF_9. Performance Comparison of the x-axis Displacement
212
Comparison with HABSMC
Fig: 3-DOF_10. Performance Comparison of the x-axis Velocity
213
Comparison with HABSMC
Fig: 3-DOF_11. Performance Comparison of the Y axis displacement of the 3-DOF
214
Comparison with HABSMC
Fig: 3-DOF_12. Performance Comparison of the Y axis Velocity of the 3-DOF
215
Comparison with HABSMC
Fig: 3-DOF_13. Performance Comparison of the angular orientation of 3-DOF
216
Comparison with HABSMC
Fig: 3-DOF_14. Performance Comparison of the of angular velocity of 3-DOF
217
Comparison with HABSMC
Fig: 3-DOF_15. Performance Comparison of the Control Input u1
218
Comparison with HABSMC
Fig: 3-DOF_16. Performance Comparison of the Control Input u2
219
220
What is Backstepping?
Stabilization Problem of Dynamical System
Design objective is to construct a control input u which ensures the
regulation of the state variables x(t) and z(t), for all x(0) and z(0).
Equilibrium point: x=0, z=0
Design objective can be achieved by making the above mentioned
equilibrium a GAS.
221
Contd.
Block Diagram of the system:
222
Contd.
First step of the design is to construct a control input for the scalar subsystem
z can be considered as a control input to the scalar subsystem
Construction of CLF for the scalar subsystem
Control Law:
But z is only a state variable, it is not the control input.
223
Contd.
Only one can conclude the desired value of z as
Definition of Error variable e:
z is termed as the Virtual Control
Desired Value of z, αs(x) is termed as stabilizing function.
System Dynamics in ( x, e) Coordinate:
224
Modified Block Diagram
Contd.
Feedback Control Law αsBackstepping
Signal -αs225
So the signal αs(x) serve the purpose of feedback control law inside the block and “backstep” -αs(x) through an integrator.
Contd.
Feedback loop with + αs(x) Backstepping of Signal -αs(x)
Through integrator 226
Construction of CLF for the overall 2nd order system:
Derivative of Va
A simple choice of Control Input u is:
With this control input derivative of CLF becomes:
Contd.
227
Consider the scalar nonlinear system
Control Law( using Feedback Linearization):
Resultant System:
Edurado D. Sontag Proposed a formula to avoid the Cancellation of these
useful nonlinearities.
Why Backstepping?
is it essential to cancel out the
term ?
Not at
all!!!!
This is an Useful Nonlinearity, it has an Stabilizing
effect on the system.
228
Sontag's Formula:
Control Law (Sontag’s Formula):
Control Law (using Backstepping):
Contd.
For large values of x, the
control law becomes
u≈sinx
So this control law avoids the cancellation of
useful nonlinearities!
For higher values of x
But this formula leads a
complicated control input
for intermediate
values of x
0 0
0
42
gx
Vfor
gx
Vfor
gx
V
gx
Vf
x
Vf
x
V
u
back229
230
Basic Concept of Zero Dynamics
• Let us consider a simple 3rd order system transfer function
• State Model Representation
• Three consecutive differentiation of y yields explicit relation
between input and output
• Relative Degree of the system is THREE.231
3 2
1
6 11 6
Y s
U s s s s
1 1
2 2
3 3
1
0 1 0 0
0 0 1 0
6 11 6 1
x x
x x u
x x
y x
1 2 36 11 6y x x x u
• Relative degree indicates the excess number of poles over the
number of zeros.
• Order of the internal dynamics 3-3=0.
• Now append one zero to that same transfer function
• State Model
232
Basic Concept of Zero Dynamics
3 2
4
6 11 6
Y s s
U s s s s
1 1
2 2
3 3
1 2
0 1 0 0
0 0 1 0
6 11 6 1
4
x x
x x u
x x
y x x
• Two consecutive differentiation of y yields explicit relation
between input and output
• Relative degree of the system is 2.
• Order of the internal dynamics is 1.
• Now use of the following feedback law
yields
233
Basic Concept of Zero Dynamics
1 2 36 11 2y x x x u
1 2 36 11 2u x x x v
y v
• Therefore, if we consider the output y as the state of the system
and signal v as the input to the system, then
• Basically this is a second order state model, so we can conclude
that application input , converts the
original system into a 2nd order state model!!!!!
• Actually, it converts the system into a cascade combination of
reduced order system and internal dynamics.
234
Basic Concept of Zero Dynamics
1 2
1 2
2
, z y z y
z z
z v
1 2 36 11 2u x x x v
• Internal dynamics can be represented as
• Stability of internal dynamics can be assessed by the location
of the zero at s plane!!!!
• Consider a nonlinear system
• First order differentiation of y yields
235
Basic Concept of Zero Dynamics
1 14x x y
1 1 2 1
2 1 2 2
2
2 3
x x x x
x x x x u
y x
1 2 22 3y x x x u
• Now application of input yields
• Internal dynamics
• Zero Dynamics can be found using the concept of output
zeroing input.
• Output identically equal to zero implies
• Now that implies
• Again
• Therefore, zero dynamics equation is
236
Basic Concept of Zero Dynamics
1 1
2
x x
x u
1 2 22 3u x x x v y v
1 1 1x x x y
0, 0 y and y
20 0 y x
20 0 0y x u
1 1x x
237
Contd.
238
Contd.
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Contd.
Can’t reach any position Outside the track
Holonomic System
`Nonholonomic systemWith velocity constraint
No velocity componentBut the position is reachable
`Nonholonomic systemWith acceleration
constraint
Position is reachable
May have a sidewise velocity component too
But no sidewiseacceleration
251