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Stability of Nonlinear Systems

The equilibria of a system can tell us a great deal about the stability of the system. For nonlinear systems, stability is a property of the equilibrium points, and to be stable is to converge to or stay equilibrium.

Definition 36

An equilibrium point
xeR\boldsymbol{x}_e\in \mathbb{R}
is a stable equilibrium point in the sense of Lyapunov if and only if
ϵ>0,δ(t0,ϵ)\forall \epsilon > 0,\exists \delta(t_0, \epsilon)
such that
tt0, x0xe<δ(t0,ϵ)    x(t)xe<ϵ\forall t \geq t_0,\ \|\boldsymbol{x}_0 - \boldsymbol{x}_e\| < \delta(t_0, \epsilon) \implies \|\boldsymbol{x}(t) - \boldsymbol{x}_e\| < \epsilon
Lyapunov Stability essentially says that a finite deviation in the initial condition from equilibrium means the resulting trajectory of the system stay close to equilibrium. Notice that this definition is nearly identical to Theorem 10. That means stability of an equilibrium point is the same as saying the function which returns the solution to a system given its initial condition is continuous at the equilibrium point.

Definition 37

An equilibrium point
xeR\boldsymbol{x}_e\in \mathbb{R}
is an uniformly stable equilibrium point in the sense of Lyapunov if and only if
ϵ>0,δ(ϵ)\forall \epsilon > 0,\exists \delta( \epsilon)
such that
tt0, x0xe<δ(ϵ)    x(t)xe<ϵ\forall t \geq t_0,\ \|\boldsymbol{x}_0 - \boldsymbol{x}_e\| < \delta(\epsilon) \implies \|\boldsymbol{x}(t) - \boldsymbol{x}_e\| < \epsilon
Uniform stability means that the
δ\delta
can be chosen independently of the time the system starts at. Both stability and uniform stability do not imply convergence to the equilibrium point. They only guarantee the solution stays within a particular norm ball. Stricter notions of stabilty add this idea in.

Definition 38

An equilibrium point
xe\boldsymbol{x}_e
is attractive if
t0>0, c(t0)\forall t_0 > 0,\ \exists c(t_0)
such that
x(t0)Bc(xe)    limtx(t,t0,x0)xe=0\boldsymbol{x}(t_0) \in B_c(\boldsymbol{x}_e) \implies \lim_{t\to\infty} \|\boldsymbol{x}(t, t_0, \boldsymbol{x}_0) - \boldsymbol{x}_e\| = 0
Attractive equilibria guarantee that trajectories beginning from initial conditions inside of a ball will converge to the equilibrium. However, attractivity does not imply stability since the trajectory could go arbitarily far from the equilibrium so long as it eventually returns.

Definition 39

An equilibrium point
xe\boldsymbol{x}_e
is asymptotically stable if
xe\boldsymbol{x}_e
is stable in the sense of Lyapunov and attractive.
Asymptotic stability fixes the problem of attractivity where trajectories could go far from the equilibrium, and it fixes the problem with stability where the trajectory may not converge to equilibrium. It means that trajectories starting in a ball around equilibrium will converge to equilibrium without leaving that ball. Because the constant for attractivity may depend on time, defining uniform asymptotic stability requires some modifications to the idea of attractivity.

Definition 40

An equilibrium point is uniformly asympototically stable if
xe\boldsymbol{x}_e
is uniformly stable in the sense of Lyapunov, and
c\exists c
and
γ:R+×RnR+\gamma:\mathbb{R}_+\times\mathbb{R}^n\to\mathbb{R}_+
such that
x0Bc(xe), limτγ(τ,x0)=0,tt0, x(t,t0,x0)xeγ(tt0,x0)\forall \boldsymbol{x}_0\in B_c(\boldsymbol{x}_e),\ \lim_{\tau\to\infty}\gamma(\tau, \boldsymbol{x}_0) = 0, \qquad \forall t\geq t_0,\ \|\boldsymbol{x}(t, t_0, \boldsymbol{x}_0) - \boldsymbol{x}_e\| \leq \gamma(t-t_0, \boldsymbol{x}_0)
The existence of the
γ\gamma
function helps guarantee that the rate of converges to equilibrium does not depend on
t0t_0
since the function
γ\gamma
is independent of
t0t_0
. Suppose that the
γ\gamma
is an exponential function. Then solutions to the system will converge to the equilibrium exponentially fast.

Definition 41

An equilibrium point
xe\boldsymbol{x}_e
is locally exponentially stable if
h,m,α\exists h,m,\alpha
such that
x0Bh(xe), x(t,t0,x0)xemeα(tt0)x(t)xe\forall \boldsymbol{x}_0\in B_h(\boldsymbol{x}_e),\ \|\boldsymbol{x}(t, t_0, \boldsymbol{x}_0) - \boldsymbol{x}_e\| \leq me^{-\alpha(t - t_0)}\|\boldsymbol{x}(t) - \boldsymbol{x}_e\|
are all local definitions because the only need to hold for
x0\boldsymbol{x}_0
inside a ball around the equilibrium. If they hold
x0Rn\forall \boldsymbol{x}_0\in\mathbb{R}^n
, then they become global properties.
Just as we can define stability, we can also define instability.

Definition 42

An equilibrium point
xe\boldsymbol{x}_e
is unstable in the sense of Lyapunov if
ϵ>0,δ>0\exists \epsilon > 0, \forall \delta > 0
such that
x0Bδ(xe)    Tt0,x(T,t0,x0)∉Bϵ(xe)\exists \boldsymbol{x}_0\in B_\delta(\boldsymbol{x}_e) \implies \exists T\geq t_0, x(T, t_0, \boldsymbol{x}_0) \not\in B_\epsilon(\boldsymbol{x}_e)
Instability means that for any
δ\delta-
ball, we can find an
ϵ\epsilon-
ball for which there is at least one initial condition whose corresponding trajectory leaves the
ϵ\epsilon-
ball.

Lyapunov Functions

In order to prove different types of stability, we will construct functions which have particular properties around equilibrium points of the system. The properties of these functions will help determine what type of stable the equilibrium point is.

Definition 43

A class
K\mathcal{K}
function is a function
α:R+R+\alpha: \mathbb{R}_+ \to \mathbb{R}_+
such that
α(0)=0\alpha(0) = 0
and
α(s)\alpha(s)
is strictly monotonically increasing in
ss
.
A subset of the class
K\mathcal{K}
functions grow unbounded as the argument approaches infinity.

Definition 44

A class
KR\mathcal{KR}
function is a class
K\mathcal{K}
function
α\alpha
where
limsα(s)=s\lim_{s\to\infty}\alpha(s) = s
.
Class
KR\mathcal{KR}
functions are “radially unbounded”. We can use class
K\mathcal{K}
and class
KR\mathcal{KR}
to bound “energy-like” functions called Lyapunov Functions.

Definition 45

A function
V(x,t):Rn×R+RV(\boldsymbol{x}, t): \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}
is locally positive definite (LPDF) on a set
GRnG\subset \mathbb{R}^n
containing
xe\boldsymbol{x}_e
if
αK\exists \alpha \in \mathcal{K}
such that
V(x,t)α(xxe)V(\boldsymbol{x}, t) \geq \alpha(\|\boldsymbol{x} - \boldsymbol{x}_e\|)
LPDF functions are locally “energy-like” in the sense that the equilibrium point is assigned the lowest “energy” value, and the larger the deviation from the equilibrium, the higher the value of the “energy”.

Definition 46

A function
V(x,t):Rn×R+RV(\boldsymbol{x}, t): \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}
is positive definite (PDF) if
αKR\exists \alpha \in \mathcal{KR}
such that
xRn, V(x,t)α(xxe)\forall \boldsymbol{x}\in\mathbb{R}^n,\ V(\boldsymbol{x}, t) \geq \alpha(\|\boldsymbol{x} - \boldsymbol{x}_e\|)
Positive definite functions act like “energy functions” everywhere in
Rn\mathbb{R}^n
.

Definition 47

A function
V(x,t):Rn×R+RV(\boldsymbol{x}, t): \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}
is decrescent if
αK\exists \alpha \in \mathcal{K}
such that
xBh(xe), V(x,t)β(xxe)\forall \boldsymbol{x}\in B_h(\boldsymbol{x}_e),\ V(\boldsymbol{x}, t) \leq \beta(\|\boldsymbol{x} - \boldsymbol{x}_e\|)
Descresence means that for a ball around the equilibrium, we can upper bound the the energy.
Note that we can assume
xe=0\boldsymbol{x}_e = 0
without loss of generality for Definition 45, Definition 46, Definition 47 since for a given system, we can always define a linear change of variables that shifts the equilibrium point to the origin.

Quadratic Lyapunov Functions

Definition 48

A Quadratic Lypunov function is of the form
V(x)=xPx,P0V(\boldsymbol{x}) = \boldsymbol{x}^\top P \boldsymbol{x},\quad P \succ 0
Quadratic Lyapunov Functions are one of the simplest types of Lyapunov Functions. Their level sets are ellipses where the major axis is the eigenvector corresponding to
λmin(P)\lambda_{min}(P)
, and the minor axis is the eigenvecctor corresponding to
λmax(P)\lambda_{max}(P)
.

Theorem 14

Consider the sublevel set
Ωc={xV(x)c}\Omega_c = \{ \boldsymbol{x} | V(\boldsymbol{x}) \leq c \}
. Then
rr_*
is the radius of the largest circle contained inside
Ωc\Omega_c
, and
rr^*
is the radius of the largest circle containing
Ωc\Omega_c
.
r=cλmax(P)r=cλmin(P)r_* = \sqrt{\frac{c}{\lambda_{max}(P)}} \qquad r^* = \sqrt{\frac{c}{\lambda_{min}(P)}}

Sum-of-Squares Lyapunov Functions

Definition 49

A polynomial
p(x)p(\boldsymbol{x})
is sum-of-squares (SOS) if
g1,,gr\exists g_1,\cdots,g_r
such that
p(x)=i=1rgi2(x)p(\boldsymbol{x}) = \sum_{i=1}^r g_i^2(\boldsymbol{x})
SOS polynomials have the nice property that they are always non-negative due to being a sum of squared numbers. Since any polynomial can be written in a quadratic form
P(x)=z(x)Qz(x)P(\boldsymbol{x}) = z^\top(\boldsymbol{x}) Q z(\boldsymbol{x})
where
zz
is a vector of monomials, the properties of
QQ
can tell us if
PP
is SOS or not.

Theorem 15

A polynomial is SOS if and only if it can be written as
p(x)=z(x)Qz(x),Q0p(\boldsymbol{x}) = z^\top(\boldsymbol{x}) Q z(\boldsymbol{x}), \quad Q \succeq 0
Note that
QQ
is not necessarily unique, and if we construct a linear operator which maps
QQ
to
PP
, then this linear operator will have a Null Space. Mathematically, consider
L(Q)(x)=z(x)Qz(x).\mathcal{L}(Q)(\boldsymbol{x}) = z^\top(\boldsymbol{x})Qz(\boldsymbol{x}).
This linear operator has a null space spanned by the polynomials
NjN_j
. Given a matrix
Q00Q_0 \succeq 0
such that
p(x)=z(x)Q0z(x)p(\boldsymbol{x}) = z^\top(\boldsymbol{x})Q_0z(\boldsymbol{x})
(i.e
pp
is SOS), it is also true that
p(x)=z(x)(Q0+jλjNj(x))z(x).p(\boldsymbol{x}) = z^\top(\boldsymbol{x})\left( Q_0 + \sum_{j} \lambda_j N_j(\boldsymbol{x}) \right) z(\boldsymbol{x}).
SOS polynomials are helpful in finding Lyapunov functions because we can use SOS Programming to find SOS polynomials which satisfy desired properties. For example, if we want
V(x)V(\boldsymbol{x})
to be PDF, then one constraint in our SOS program will be that
V(x)ϵxx,ϵ>0V(\boldsymbol{x}) - \epsilon \boldsymbol{x}^\top \boldsymbol{x}, \quad \epsilon > 0
is SOS.

Proving Stability

To prove the stability of an equilibrium point for a given nonlinear system, we will construct a Lyapunov function and determine stability from the properties of the Lyapunov functions which we can find. Given properties of
VV
and
dVdt\frac{d^{}V}{dt^{}}
, we can use the Lyapunov Stability Theorems to prove the stability of equilibria.

Theorem 16

If
V(x,t)\exists V(\boldsymbol{x}, t)
such that
VV
is LPDF and
dVdt0-\frac{d^{}V}{dt^{}} \geq 0
locally, then
xe\boldsymbol{x}_e
is stable in the sense of Lyapunov.

Theorem 17

If
V(x,t)\exists V(\boldsymbol{x}, t)
such that
VV
is LPDF and decrescent, and
dVdt0-\frac{d^{}V}{dt^{}} \geq 0
locally, then
xe\boldsymbol{x}_e
is uniformly stable in the sense of Lyapunov.

Theorem 18

If
V(x,t)\exists V(\boldsymbol{x}, t)
such that
VV
is LPDF and decrescent, and
dVdt-\frac{d^{}V}{dt^{}}
is LPDF, then
xe\boldsymbol{x}_e
is uniformly asymptotically stable in the sense of Lyapunov.

Theorem 19

If
V(x,t)\exists V(\boldsymbol{x}, t)
such that
VV
is PDF and decrescent, and
dVdt-\frac{d^{}V}{dt^{}}
is LPDF, then
xe\boldsymbol{x}_e
is globally uniformly asymptotically stable in the sense of Lyapunov.

Theorem 20

If
V(x,t)\exists V(\boldsymbol{x}, t)
and
h,α>0h, \alpha > 0
such that V is LPDF is decrescent,
dVdt-\frac{d^{}V}{dt^{}}
is LDPF, and
xBh(xe), dVdtαxxe\forall \boldsymbol{x}\in B_h(\boldsymbol{x}_e),\ \left\lvert\left\lvert\frac{d^{}V}{dt^{}}\right\rvert\right\rvert \leq \alpha \|\boldsymbol{x}-\boldsymbol{x}_e\|
The results of Theorem 16, Theorem 17, Theorem 18, Theorem 19, Theorem 20 are summarized in Table 1.
Table 1: Summary of Lyapunov Stability Theorems
Going down the rows of Table 1 lead to increasingly stricter forms of stability. Descresence appears to add uniformity to the stability, while
dVdt-\frac{d^{}V}{dt^{}}
being LPDF adds asymptotic convergence. However, these conditions are only sufficient, meaning if we cannot find a suitable
VV
, that does not mean that an equilibrium point is not stable.
One very common case where it can be difficult to find appropriate Lyapunov functions is in proving asymptotic stability since it can be hard to find
VV
such that
dVdt-\frac{d^{}V}{dt^{}}
is LPDF. In the case of autonomous systems, we can still prove asymptotic stability without such a
VV
.

Theorem 21 (LaSalle's Invariance Principle)

Consider a smooth function
V:RnRV:\mathbb{R}^n\to\mathbb{R}
with bounded sub-level sets
Ωc={xV(x)c}\Omega_c = \left\{\boldsymbol{x} | V(\boldsymbol{x}) \leq c \right\}
and
xΩc\forall \boldsymbol{x}\in \Omega_c
,
dVdt0\frac{d^{}V}{dt^{}} \leq 0
. Define
S={xdVdt=0}S = \left\{\boldsymbol{x}\bigg\lvert\frac{d^{}V}{dt^{}} = 0\right\}
and let
MM
be the largest invariant set in
SS
, then
x0Ωc, x(t,t0,x0)M as t.\forall \boldsymbol{x}_0\in \Omega_c,\ \boldsymbol{x}(t, t_0, x_0) \to M \text{ as } t\to \infty.
LaSalle’s theorem helps prove general convergence to an invariant set. Since
VV
is always decreasing in the sub-level set
Ωc\Omega_c
, trajectories starting in
Ωc\Omega_c
must eventually reach
SS
. At some point, they will reach the set
MM
in
SS
, and then they will stay there. Thus if the set
MM
is only the equilibrium point, or a set of equilibrium points, then we can show that the system trajectories asymptotically converges to this equilibrium or set of equilibria. Moreover, if
V(x)V(\boldsymbol{x})
is PDF, and
xRn,dVdt0\forall \boldsymbol{x}\in\mathbb{R}^n, \frac{d^{}V}{dt^{}} \leq 0
, then we can show global asymptotic stability as well.
LaSalle’s theorem can be generalized to non-autonomous systems as well, but it is slightly more complicated since the set
SS
may change over time.

Indirect Method of Lyapunov

It turns out that we can also prove the stability of systems by looking at the linearization around the equilibrium. Without loss of generality, suppose
xe=0\boldsymbol{x}_e = 0
. The linearization at the equilibrium is given by
dxdt=f(x,t)=f(0,t)+fxx=0x+f1(x,t)A(t)x.\frac{d^{}\boldsymbol{x}}{dt^{}} = f(x,t) = f(0, t) + \frac{\partial f}{\partial \boldsymbol{x}}\bigg\lvert_{\boldsymbol{x} = 0}\boldsymbol{x} + f_1(\boldsymbol{x},t) \approx A(t)\boldsymbol{x}.
The function
f1(x,t)f_1(\boldsymbol{x}, t)
is the higher-order terms of the linearization. The linearization is a time-varying system. Consider the time-varying linear system
dxdt=A(t)x, x(t0)=x0.\frac{d^{}\boldsymbol{x}}{dt^{}} = A(t)\boldsymbol{x},\ \boldsymbol{x}(t_0) = \boldsymbol{x}_0.

Definition 50

The state transition matrix
Φ(t,t0)\Phi(t, t_0)
of a time-varying linear system is a matrix satisfying
x(t)=Φ(t,t0)x0, dΦdt=A(t)Φ(t,t0), Φ(t0,t0)=I\boldsymbol{x}(t) = \Phi(t, t_0)\boldsymbol{x}_0,\ \frac{d^{}\Phi}{dt^{}} = A(t)\Phi(t, t_0),\ \Phi(t_0, t_0) = I
The state transition matrix is useful in determining properties of the system.
  1. 1.
    suptt0Φ(t,t0)=m(t0)<    \sup_{t\geq t_0} \|\Phi(t, t_0)\| = m(t_0) < \infty \implies
    the system is stable at the origin at
    t0t_0
    .
  2. 2.
    supt00suptt0Φ(t,t0)=m<    \sup_{t_0\geq 0}\sup_{t\geq t_0} \|\Phi(t, t_0)\| = m < \infty \implies
    the system is uniformly stable at the origin at
    t0t_0
    .
  3. 3.
    limtΦ(t,t0)=0    \lim_{t\to\infty}\|\Phi(t, t_0)\| = 0 \implies
    the system is asymptotically stable.
  4. 4.
    t0,ϵ>0,T\forall t_0,\epsilon>0,\exists T
    such that
    tt0+T, Φ(t,t0)<ϵ    \forall t\geq t_0 + T,\ \|\Phi(t, t_0)\| < \epsilon \implies
    the system is uniformly asymptotically stable.
  5. 5.
    Φ(t,t0)Meλ(tt0)    \|\Phi(t, t_0)\| \leq Me^{-\lambda(t-t_0)} \implies
    exponential stability.
If the system was Time-Invariant, then the system would be stable so long as the eigenvalues of
AA
were in the open left-half of the complex plane. In fact, we could use
AA
to construct positive definite matrices.

Theorem 22 (Lyapunov Lemma)

For a matrix
ARn×nA\in \mathbb{R}^{n\times n}
, its eigenvalues
λi\lambda_i
satisfy
Re(λi)<0\mathbb{R}e(\lambda_i) < 0
if and only if
Q0\forall Q \succ 0
, there exists a solution
P0P\succ 0
to the equation
ATP+PA=Q.A^TP + PA = -Q.
In general, we can use the Lyapunov Equation to count how many eigenvalues of
AA
are stable.

Theorem 23 (Tausskey Lemma)

For
ARn×nA\in\mathbb{R}^{n\times n}
and given
Q0Q \succ 0
, if there are no eigenvalues on the
jωj\omega
axis, then the solution
PP
to
ATP+PA=QA^TP + PA = -Q
has as many positive eigenvalues as
AA
has eigenvalues in the complex left half plane.
The Lyapunov Lemma has extensions to the time-varying case.

Theorem 24 (Time-Varying Lyapunov Lemma)

If
A()A(\cdot)
is bounded and for some
Q(t)αIQ(t) \succeq \alpha I
, the solution
P(t)P(t)
to
A(t)TP(t)+P(t)A(t)=Q(t)A(t)^TP(t) + P(t)A(t) = -Q(t)
is bounded, then the origin is a asymptotically stable equilibrium point.
It turns out that uniform asymptotic stability of the linearization of a system corresponds to uniform, asymptotic stability of the nonlinear system.

Theorem 25 (Indirect Theorem of Lyapunov)

For a nonlinear system whose higher-order terms of the linearization are given by
f(x,t)f(\boldsymbol{x},t)
, if
limx0supt0f1(x,t)x=0\lim_{\|\boldsymbol{x}\|\to 0}\sup_{t\geq 0} \frac{\|f_1(\boldsymbol{x},t)\|}{\|\boldsymbol{x}\|} = 0
and if
xe\boldsymbol{x}_e
is a uniformly asymptotic stable equilibrium point of
dzdt=A(t)z\frac{d^{}\boldsymbol{z}}{dt^{}}=A(t)\boldsymbol{z}
where
A(t)A(t)
is the Jacobian at the
xe\boldsymbol{x}_e
, then
xe\boldsymbol{x}_e
is a uniformly asymptotic stable equilibrium point of
dxdt=f(x,t)\frac{d^{}\boldsymbol{x}}{dt^{}} = f(\boldsymbol{x},t)

Proving Instability

Theorem 26

An equilibrium point
xe\boldsymbol{x}_e
is unstable in the sense of Lyapunov if
V(x,t)\exists V(\boldsymbol{x},t)
which is decrescent, the Lie derivative
dVdt\frac{d^{}V}{dt^{}}
is LPDF,
V(xe,t)V(\boldsymbol{x}_e, t)
, and
x\exists \boldsymbol{x}
in the neighborhood of
xe\boldsymbol{x}_e
such that
V(x0,t)>0V(\boldsymbol{x}_0, t) > 0
.

Region of Attraction

For asymptotically stable and exponential stable equilibria, it makes sense to wonder which initial conditions will cause trajectories to converge to the equilibrium.

Definition 51

If
xe\boldsymbol{x}_e
is an equilibrium point of a time-invariant system
dxdt=f(x)\frac{d^{}\boldsymbol{x}}{dt^{}} = f(\boldsymbol{x})
, then the Region of Attraction of
xe\boldsymbol{x}_e
is
RA(xe)={x0Rnlimtx(t,t0)=xe}\mathcal{R}_A(\boldsymbol{x}_e) = \{ \boldsymbol{x}_0 \in \mathbb{R}^n | \lim_{t\to\infty} \boldsymbol{x}(t, t_0) = \boldsymbol{x}_e \}
Suppose that we have a Lyapunov function
V(x)V(\boldsymbol{x})
and a region
DD
such that
V(x)>0V(\boldsymbol{x}) > 0
and
dVdt<0\frac{d^{}V}{dt^{}} < 0
in
DD
. Define a sublevel set of the Lypunov function
Ωc\Omega_c
which is a subset of
DD
. We know that if
x0Ωc\boldsymbol{x}_0\in\Omega_c
, then the trajectory will stay inside
Ωc\Omega_c
and converge to the equilibrium point. Thus we can use the largest
Ωc\Omega_c
that is compact and contained in
DD
as an estimate of the region of attraction.
When we have a Quadratic Lyapunov Function, we can set
DD
to be the largest circle which satisfies the conditions on
VV
, and the corresponding
Ωc\Omega_c
contained inside
DD
will be the estimate of the Region of Attraction.
We can find even better approximations of the region of attraction using SOS programming. Suppose we have a
VV
which we used to prove asymptotic stability. Then if there exists an
ss
which satisfies the following SOS program, then the sublevel set
Ωc\Omega_c
is an estimate of the Region of Attraction.
maxc,scs.ts(x) is SOS,(dVdt+ϵxx)+s(x)(cV(x)) is SOS.\begin{aligned} \max_{c, s} &\quad c\\ \text{s.t} &\quad s(\boldsymbol{x}) \text{ is SOS,}\\ &\quad -\left(\frac{d^{}V}{dt^{}} + \epsilon \boldsymbol{x}^\top \boldsymbol{x}\right) + s(\boldsymbol{x})(c - V(\boldsymbol{x})) \text{ is SOS.}\end{aligned}