lecture 31
TRANSCRIPT
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Lecture – 33
Stability Analysis of Nonlinear Systems Using Lyapunov Theory – I
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Outline
Motivation
Definitions
Lyapunov Stability Theorems
Analysis of LTI System Stability
Instability Theorem
Examples
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References
H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003.
J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991.
H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996.
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Techniques of Nonlinear Control Systems Analysis and Design
Phase plane analysisDifferential geometry (Feedback linearization)
Lyapunov theoryIntelligent techniques: Neural networks, Fuzzy logic, Genetic algorithm etc.Describing functionsOptimization theory (variational optimization, dynamic programming etc.)
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MotivationEigenvalue analysis concept does not hold good for nonlinear systems.Nonlinear systems can have multiple equilibrium points and limit cycles.Stability behaviour of nonlinear systems need not be always global (unlike linear systems).Need of a systematic approach that can be exploited for control design as well.
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Definitions
System Dynamics
Equilibrium Point
( ) : (a locally Lipschitz map)
: an open and connected subset of
n
n
X f X f D
D
= →R
R
( ) 0e eX f X= =
( )eX
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Definitions
Open Set
Connected SetA connected set is a set which cannot be represented as the union of two or more disjointnonempty open subsets. Intuitively, a set with only one piece.
( )A set is open if for every ,
n
r
Ap A B p A
⊂
∈ ∃ ⊂
Space A is connected, B is not.
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Definitions
Stable Equilibrium
Unstable EquilibriumIf the above condition is not satisfied, then the equilibrium point is said to be unstable
0
is stable, provided for each 0, ( ) 0 :(0) ( ) ( )
e
e e
XX X X t X t t
ε δ εδ ε ε
> ∃ >
− < ⇒ − < ∀ ≥
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DefinitionsConvergent Equilibrium
Asymptotically StableIf an equilibrium point is both stable and convergent, then it is said to be asymptotically stable.
( ) ( )If : 0 lime etX X X t Xδ δ
→∞∃ − < ⇒ =
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DefinitionsExponentially Stable
Convention
(without loss of generality)
( ) ( )( )
, 0 : 0 0
whenever 0
te e
e
X t X X X e t
X X
λα λ α
δ
−∃ > − ≤ − ∀ >
− <
The equilibrium point 0eX =
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DefinitionsA function is said to be positive semi definite in
if it satisfies the following conditions:
is said to be positive definite in if condition (ii) is replaced by
is said to be negative definite (semi definite) in if is positive definite.
( )( ) 0 (0) 0( ) 0,i D and Vii V X X D
∈ =
≥ ∀ ∈
( ) 0 {0}V X in D> −
:V D →RD
:V D →R
:V D →R
D
D ( )V X−
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Lyapunov Stability TheoremsTheorem – 1 (Stability)
( )
( )( )( )
Let 0 be an equilibrium point of , : .Let : be a continuously differentiable function such that:( ) 0 0
( ) 0, {0}
( ) 0, {0}Then 0 is "stable".
nX X f X f DV D
i V
ii V X in D
iii V X in DX
= = →
→
=
> −
≤ −
=
RR
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Lyapunov Stability TheoremsTheorem – 2 (Asymptotically stable)
( )
( )( )( )
Let 0 be an equilibrium point of , : .Let : be a continuously differentiable function such that:( ) 0 0
( ) 0, {0}
( ) 0, {0}Then 0 is "asymptotically stable".
nX X f X f DV D
i V
ii V X in D
iii V X in DX
= = →
→
=
> −
< −
=
RR
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Lyapunov Stability TheoremsTheorem – 3 (Globally asymptotically stable)
( )
( )( )( )( )
Let 0 be an equilibrium point of , : .Let : be a continuously differentiable function such that:( ) 0 0
( ) 0, {0}
( ) is "radially unbounded"
( ) 0, {0}Then 0 is "glo
nX X f X f DV D
i V
ii V X in D
iii V X
iv V X in DX
= = →
→
=
> −
< −
=
RR
bally asymptotically stable".
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Lyapunov Stability TheoremsTheorem – 3 (Exponentially stable)
( )( )
1 2 3
1 2
3
Suppose all conditions for asymptotic stability are satisfied.In addition to it, suppose constants , , , :
( )
( )Then the origin 0 is "exponentially stable".Moreover, if
p p
p
k k k p
i k X V X k X
ii V X k XX
∃
≤ ≤
≤ −
= these conditions hold globally, then the
origin 0 is "globally exponentially stable".X =
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Example:Pendulum Without Friction
System dynamics
Lyapunov function
1 2,x xθ θ ( )21
12 / sinxx
g l xx⎡ ⎤⎡ ⎤
= ⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦
( )2
2 22 1
121 (1 cos )2
V KE PE
m l mgh
ml x mg x
ω
= +
= +
= + −
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Pendulum Without Friction
( ) ( ) ( )
( ) ( )
( )
1 21 2
21 2 2 1
2 1 2 1
sin sin
sin sin 0
0 (nsdf )
T
T
T
V X V f X
V V f X f Xx x
gmgl x ml x x xl
mglx x mglx x
V X
= ∇
⎡ ⎤∂ ∂= ⎡ ⎤⎢ ⎥ ⎣ ⎦∂ ∂⎣ ⎦
⎡ ⎤⎡ ⎤= −⎣ ⎦ ⎢ ⎥⎣ ⎦= − =
≤
Hence, it is a “stable” system.
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Pendulum With Friction
Modify the previous example by adding thefriction force
Defining the same state variables as above
θkl
θθ klmgma −−= sin
1 2
2 1 2sin
x xg kx x xl m
=
= − −
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Pendulum With Friction( ) ( ) ( )
( ) ( )
( ) ( )
1 21 2
21 2 2 1 2
2 22
sin sin
0 nsdf
T
T
T
V X V f X
V V f X f Xx x
g kmgl x ml x x x xl m
kl x
V X
= ∇
⎡ ⎤∂ ∂= ⎡ ⎤⎢ ⎥ ⎣ ⎦∂ ∂⎣ ⎦
⎡ ⎤⎡ ⎤= − −⎣ ⎦ ⎢ ⎥⎣ ⎦= −
≤
Hence, it is also just a “stable” system. (A frustrating result..!)
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Analysis of Linear Time Invariant System
System dynamics:
Lyapunov function:
Derivative analysis:
, n nX AX A ×= ∈R
( )
T T
T T T
T T
V X PX X PXX A PX X PAX
X A P PA X
= +
= +
= +
( ) ( ), 0 pdfTV X X PX P= >
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Analysis of Linear Time Invariant System
For stability, we aim for
By comparing
For a non-trivial solution
( )T T TX A P PA X X QX+ = −
( )0TV X QX Q= − >
0TPA A P Q+ + =
(Lyapunov Equation)
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Analysis of Linear Time Invariant System
( ) The eigenvalues of a matrix satisfy Re 0 if and only if for any given symmetric matrix , a unique
matrix satisfying the Lyapuno
n ni iA
pdf Q
pdf P
λ λ×∈ <
∃
Theorem :
v equation.
Proof: Please see Marquez book, pp.98-99.
0
and are related to each other by the following relationship:
However, the above equation is seldom used to compute . Instead is directly solved from the Lyapunov equation.
TA t At
P Q
P e Qe dt
P P
∞
= ∫
Note :
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Analysis of Linear Time Invariant Systems
Choose an arbitrary symmetric positive definite matrix
Solve for the matrix form the Lyapunov equation and verify whether it is positive definite
Result: If is positive definite, then and hence the origin is “asymptotically stable”.
( )Q Q I=
P
P ( ) 0V X <
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Lyapunov’s Indirect Theorem
( )( )
( )
Let the linearized system about 0 be .
The theorem says that if all the eigenvalues 1, ,
of the matrix satisfy Re 0 (i.e. the linearized systemis exponentially stable), then for t
i
i
X X A X
i n
A
λ
λ
= Δ = Δ
=
<
…
he nonlinear system theorigin is locally exponentially stable.
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Instability theoremConsider the autonomous dynamical system and assume
is an equilibrium point. Let have thefollowing properties:
Under these conditions, is unstable
( )
( )
0 0
( ) (0) 0( ) ,arbitrarily close to 0, such that 0
( ) 0 , where the set is defined as follows{ : and 0}
n
i Vii X X V X
iii V X U UU X D X V Xε
=
∃ ∈ = >
> ∀ ∈
= ∈ ≤ >
R
0X=
0X=
:V D →R
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Summary
Motivation
Notions of Stability
Lyapunov Stability Theorems
Stability Analysis of LTI Systems
Instability Theorem
Examples
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References
H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003.
J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991.
H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996.
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