Post on 04-Jul-2020
transcript
ECE7850 Wei Zhang
ECE7850 Lecture 4:
Basics of Stability Analysis
• Basic Stability Concepts
• Lyapunov Stability Theorems
• Converse Lyapunov Functions
• Semidefinite Programming (SDP)
• Basic Polynomial Optimization
• Computational Techniques for Stability Analysis
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ECE7850 Wei Zhang
Basic Stability Concepts
• Consider a time-invariant nonlinear system:
x = f(x) with IC x(0) = x0 (1)
• Assume: f Lipschitz continuous; origin is an isolated equilibrium f(0) = 0
• x = 0 stable in the sense of Lyapunov, if
∀ǫ > 0, ∃δ > 0, s.t. ‖x(0)‖ ≤ δ ⇒ ‖x(t)‖ ≤ ǫ, ∀t ≥ 0
Basic Stability Concepts 2
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• x = 0 asymptotically stable if it is stable and δ can be chosen so that
‖x(0)‖ ≤ δ ⇒ x(t) → 0 as t → ∞
if the above condition holds for all δ, then globally asymptotically stable
• Region of Attraction: RA = {x ∈ Rn : whenever x(0) = x, then x(t) → 0}
• x = 0 exponential stable if there exist positive constants δ, λ, c such that
‖x(t)‖ ≤ c‖x(0)‖e−λt
Basic Stability Concepts 3
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• Does attractive implies stable in Lyapunov sense?
– Answer is NO. e.g.:
x1 = x21 − x2
2
x2 = 2x1x2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−3
−2
−1
0
1
2
3
Basic Stability Concepts 4
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Stability Analysis Using Lyapunov Functions
How to verify stability of a system:
• Trivial answer: explicit solution of ODE x(t) and check stability definitions
• Need to determine stability without explicitly solving the ODE
• Preferably, analysis only depends on the vector field
• The most powerful tool is: Lyapunov function
Stability Analysis Using Lyapunov Functions 5
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• Classes of functions: (Assuming 0 ∈ D ⊆ Rn)
– g : D → R is called positive semidefinite (PSD) on D if g(0) = 0 and g(x) ≥ 0, ∀x ∈ D
– g : D → R is called positive definite (PD) on D if g(0) = 0 and g(x) > 0, ∀x ∈ D \ {0}
– g is negative semidefinite (NSD) if −g is PSD
– g : Rn → R is radically unbounded if V (x) → ∞ as ‖x‖ → ∞
– Cn: n-times continuously differential functions g : Rn → Rm
• Lie derivative of a C1 function V : Rn → R along vector field g is:
LgV (x) ,
∂V
∂x(x)
T
g(x)
Stability Analysis Using Lyapunov Functions 6
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• Theorem 1 (Lyapunov Theorem) Let D ⊂ Rn be a set containing an open neighborhood
of the origin. If there exists a PD function V : D → R such that
LfV is NSD (2)
then the origin is stable. If in addition,
LfV is ND (3)
then the origin is asymptotically stable.
• Remarks:
– A PD C1 function satisfying (2) or (3) will be called a Lyapunov function
– For the latter case, if V is also radially unbounded ⇒ globally asymptotically stable
Stability Analysis Using Lyapunov Functions 7
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• Proof of Lyapunov Theorem:
Stability Analysis Using Lyapunov Functions 8
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• Definition: V : D → R is called an Exponential Lyapunov Function (ELF) on D ⊂ Rn if
∃k1, k2, k3 > 0 such that
k1‖x‖2 ≤ V (x) ≤ k2‖x‖2
LfV (x) ≤ −k3‖x‖2
• Theorem 2 (ELF Theorem) If system (1) has an ELF, then it is exponentially stable.
• Proof:
1. show V (x(t)) ≤ V (x(0))e−(k3/k1)t
2. show x(t) ≤ ce−λt‖x(0)‖
Stability Analysis Using Lyapunov Functions 9
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• Example 1
x1 = −x1 + x2 + x1x2
x2 = x1 − x2 − x21 − x3
2
Try V (x) = ‖x‖2
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• Example 2
x1 = −x1 + x1x2
x2 = −x2
– Fact: The system is GAS (Homework: try V (x) = ln(1 + x21) + x2
2)
– Can we find a simple quadratic Lyapunov function? First try: V (x) = x21 + x2
2
– In fact, the system does not have any (global) polynomial Lyapunov function
Stability Analysis Using Lyapunov Functions 11
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When there is a Lyapunov Function?
• Converse Lyapunov Theorem for Asymptotic Stability
origin asymptotically stable;
f is locally Lipschitz on D
with region of attraction RA
⇒ ∃V s.t.
V is continuuos and PD on RA
LfV is ND on RA
V (x) → ∞ as x → ∂RA
• Converse Lyapunov Theorem for Exponential Stability
origin exponentially stable on D;
f is C1⇒ ∃ an ELF V on D
• Proofs are involved especially for the converse theorem for asymptotic stability
• IMPORTANT: proofs of converse theorems often assume the knowledge of system solution.
When there is a Lyapunov Function? 12
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Semi-definite Programming
• Converse Lyapunov function theorems are not constructive
• Basic idea for Lyapunov function synthesis
– Select Lyapunov function structure (e.g. quadratic, polynomial, piecewise quadratic, ...)
– Parameterize Lyapunov function candidates
– Find values of parameters to satisfy Lyapunov conditions
• Many Lyapunov synthesis problems can be formulated as Semidefinite programming (SDP)
problems.
Semi-definite Programming 13
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Convex Cone
• Recall: A set S is convex if x1, x2 ∈ S implies λx1 + (1 − λ)x2 ∈ S, ∀α ∈ [0, 1].
• A set S is a cone if λ > 0, x ∈ S ⇒ λx ∈ S.
• Conic combination of x1 and x2:
x = α1x1 + α2x2 with α1, α2 ≥ 0
• convex cone: (i) a cone that is convex; (ii) equivalently, a set that contains all the conic
combinations of points in the set
Semi-definite Programming 14
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Real Symmetric Matrices:
• Sn: set of real symmetric matrices
• All eigenvalues are real
• There exists a full set of orthogonal eigenvectors
• Spectral decomposition: If A ∈ Sn, then A = QΛQT , where Λ diagonal and Q is unitary.
Semi-definite Programming 15
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Positive Semidefinite Matrices
• A ∈ Sn is called positive semidefinite (p.s.d.), denoted by A � 0, if xT Ax ≥ 0, ∀x ∈ Rn
• A ∈ Sn is called positive definite (p.d.), denoted by A ≻ 0, if xT Ax > 0 for all nonzero x ∈ Rn
• Sn+: set of all p.s.d. (symmetric) matrices
• Sn++: set of all p.d. (symmetric) matrices
• p.s.d. or p.d. matrices can also be defined for non-symmetric matrices. But we focus on
symmetric ones.
e.g.:
1 1
−1 1
Semi-definite Programming 16
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• Other equivalent definitions for symmetric p.s.d. matrices:
– All 2n − 1 principal minors of A are nonnegative
– All eigs of A are nonnegative
– There exists a factorization A = BT B
• Other equivalent definitions for p.d. matrices:
– All n leading principal minors of A are positive
– All eigs of A are strictly positive
– There exists a factorization A = BT B with B square and nonsingular.
Semi-definite Programming 17
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• Useful facts:
– If T nonsingular, A ≻ 0 ⇔ T T AT ≻ 0; and A � 0 ⇔ T TAT � 0;
– Sn+ is a convex cone: positive semidefinite cone
– Inner product on Rm×n: < A, B >, tr(AT B) , A • B.
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– For A, B ∈ Sn+, tr(AB) ≥ 0 (the cone Sn
+ is acute)
– Schur complement lemma: Define M =
A B
BT C
1. M ≻ 0 ⇔
A ≻ 0
C − BT A−1B ≻ 0⇔
C ≻ 0
A − BC−1BT ≻ 0
2. If A ≻ 0, then M � 0 ⇔ C − BT A−1B � 0
3. If C ≻ 0, then M � 0 ⇔ A − BC−1BT � 0
Semi-definite Programming 19
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– Proof of Schur complement lemma:
Semi-definite Programming 20
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Operations that preserve convexity
• intersection of possibly infinite number of convex sets:
– e.g.: polyhedron:
– e.g.: PSD cone:
Semi-definite Programming 21
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• affine mapping f : Rn → Rm (i.e. f(x) = Ax + b)
– f(X) = {f(x) : x ∈ X} is convex whenever X ⊆ Rn is convex
e.g.: Ellipsoid: {‖x ∈ Rn : (x − xc)
T P (x − xc) ≤ 1} or equivalently {xc + Au : ‖u‖2 ≤ 1}
– f−1(Y ) = {x ∈ Rn : f(x) ∈ Y } is convex whenever Y ⊆ R
m is convex
e.g.: {Ax ≤ b} = f−1(Rn+), where R
n+ is nonnegative orthant
Semi-definite Programming 22
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Linear Matrix Inequality
• Given symmetric matrices F0, . . . , Fm ∈ Sn,
F (x) = F0 + x1F1 + · · · + xnFn � 0
is called a Linear Matrix Inequality in x ∈ Rn
• The function F (x) is affine in x
• The constraint set {x ∈ Rn : F (x) � 0} is nonlinear but convex
Semi-definite Programming 23
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• Example 3 Characterize the constraint set: F (x) =
x1 + x2 x2 + 1
x2 + 1 x3
� 0
Semi-definite Programming 24
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• Example 4 Find a Lyapunov function V (x) = xT P x for a linear system x = Ax
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Semidefinite Programming (SDP)
• SDP: optimization problem with linear objective, and LMI and linear equality constraints:
minimize: cT x
subject to: F0 + x1F1 + · · · + xnFn � 0
Ax = b
(4)
• Global optimal solution of SDP can be found efficiently.
• Equivalent formulation (Standard Prime Form):
minimize: fp(X) = C • X
subject to: Ai • X = bi, i = 1, . . . , m
X � 0
(5)
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• Dual form:
maximize: fd(y) = bTy
subject to:∑m
i=1 yiAi � C
• Weak duality: fp(X) ≥ fd(y) for any primal and dual feasible X and y
• Strong duality holds under Slater’s condition: fp(X∗) = fd(y
∗)
Semi-definite Programming 27
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• Example 5 LMIs AT P + P A + I � 0, P � 0 indicate that the Lyapunov function V (x) = xT P x
for linear system x = Ax proves the bound:∫ ∞0 ‖x(τ )‖2dτ ≤ x(0)T P x(0). Suppose x(0) is
fixed. How to find the best possible such bound?
Semi-definite Programming 28
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Basic Polynomial Optimization
• Rn,d: Set of polynomials (with real coefficients) in n variables of degree d:
f(x) =∑
α∈Icαxα1
1 · · · xαn
n
where d = maxα∈I∑n
i αi.
• Pn,d = {f ∈ Rn,d : f(x) ≥ 0, ∀x ∈ Rn}: set of p.s.d. polynomials
• Σn,d = {f ∈ Rn,d : f =∑
i g2i , for some gi ∈ Rn,d}: Sum of Squares (SOS)
• Σn,d ⊂ Pn,d
• checking f ∈ Pn,d is NP-hard
• checking f ∈ Σn,d is a SDP problem
Basic Polynomial Optimization 29
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Representation of Polynomials
• monomial bases Zd(x):
• Linear representation:
Basic Polynomial Optimization 30
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• Quadratic representation (Gram matrix representation):
• Quadratic representation is not unique:
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• Hilbert showed in 1888: Pn,d = Σn,d iff
– d = 2 quadratic polynomials
– n = 1 univariate polynomials
– n = 2, d = 4, quartic polynomials in two variables.
• SOS decomposition for f ∈ Pn,d: if ∃g1, . . . , gs such that f =∑
i g2i
Basic Polynomial Optimization 32
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• Theorem 3 Let Zd(x) be the monomial basis of degree ≤ d. Then f(x) ∈ Σn,d iff there exists
Q such that
Q � 0
f(x) = Zd(x)TQZd(x)
– this is SDP problem
– comparing terms gives affine constraints on the elements of Q
Basic Polynomial Optimization 33
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• Example 6 f(x) = 4x41 + 4x3
1x2 − 7x21x
22 − 2x1x
32 + 10x4
2
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Numerical Construction of Lyapunov Functions
• Important conditions for many stability problems:
g0(x) ≥ 0 on {x ∈ Rn|g1(x) ≥ 0, . . . , gk(x) ≥ 0}
• Conservative but useful condition: ∃ SOS si(x) s.t.
g0(x) −∑
isi(x)gi(x) ≥ 0, ∀x ∈ R
n
This is Generalized S-Procedure
Numerical Construction of Lyapunov Functions 35
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• Important special case: gi(x) = xTGix, i = 0, 1, ... are quadratic polynomials:
– Original condition: ∀x ∈ Rn, xTG1x ≥ 0, . . . , xTGkx ≥ 0 ⇒ xT G0x ≥ 0
– Sufficient condition (S-procedure): ∃α1, . . . , αk ≥ 0 with
G0 � α1G1 + · · · + αkGk
– S-Procedure is lossless if k = 1 and ∃x, xTG1x > 0 (constraint qualification)
Numerical Construction of Lyapunov Functions 36
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Application to Stability Analysis
• Example 7 x = Ax + g(x) with ‖g(x)‖2 ≤ β‖x‖2
Numerical Construction of Lyapunov Functions 37
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• Example 8 Find Lyapunov function for
x1 = −x2 − 32x2
1 − 12x3
1
x2 = 3x1 − x2
Numerical Construction of Lyapunov Functions 38