ece7850 lecture 4: basics of stability analysiszhang/teaching/hybridsystem...stability analysis...

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

1

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


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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


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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)


ECE7850 Wei Zhang

• 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


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• Proof of Lyapunov Theorem:


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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)‖


ECE7850 Wei Zhang

• 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


ECE7850 Wei Zhang

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

ECE7850 Wei Zhang

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


ECE7850 Wei Zhang

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.


ECE7850 Wei Zhang

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


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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.


ECE7850 Wei Zhang

• 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.


ECE7850 Wei Zhang

– 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


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– Proof of Schur complement lemma:


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Operations that preserve convexity

• intersection of possibly infinite number of convex sets:

– e.g.: polyhedron:

– e.g.: PSD cone:


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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


ECE7850 Wei Zhang

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


ECE7850 Wei Zhang

• Example 3 Characterize the constraint set: F (x) =

x1 + x2 x2 + 1

x2 + 1 x3

� 0


ECE7850 Wei Zhang

• Example 4 Find a Lyapunov function V (x) = xT P x for a linear system x = Ax


ECE7850 Wei Zhang

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)


ECE7850 Wei Zhang

• 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

∗)


ECE7850 Wei Zhang

• 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?


ECE7850 Wei Zhang

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:


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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


ECE7850 Wei Zhang

• 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


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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

ECE7850 Wei Zhang

• 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)


ECE7850 Wei Zhang

Application to Stability Analysis

• Example 7 x = Ax + g(x) with ‖g(x)‖2 ≤ β‖x‖2


ECE7850 Wei Zhang

• Example 8 Find Lyapunov function for

x1 = −x2 − 32x2

1 − 12x3

1

x2 = 3x1 − x2


ece7850 lecture 4: basics of stability analysiszhang/teaching/hybridsystem...stability analysis...

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