7. Stability and The Lyapunov Method
7.2 Local Stability of Nonlinear Systems
Example 7.1.1 (Asymptotic and BIBO stability 1) A BIBO-stable but not asymptotically stable LTI system
Consider a LTI system with the state-space representation
˙ x=
−2 0 0 3
x+ 2
1
u (7.20)
y= 1 0
(7.21) The system above is not asymptotically stable since one of the eigenvalues of the state matrix is positive, but it is BIBO-stable because the “unstable state” does not appear in the output.
The above system is callednon-detectable, since the unstable state is clearly non-observable.
Example 7.1.2 (Asymptotic and BIBO stability 2) Another BIBO-stable but not asymptotically stable LTI system
Consider an LTI system with the truncated state equation
˙ x=
0−2 2 0
x (7.22)
The eigenvalue analysis shows that λ1,2= +i
that is, the poles of the system are on the stability boundary. This system is a pure oscillator.
7.2 Local Stability of Nonlinear Systems
One of the most widespread approach for analyzing local asymptotic stability of a nonlinear system in the neighborhood of a steady-state operating point is to linearize the system model around the operating point and then perform linear stability analysis. This approach is the subject of this section, where we discuss how to linearize nonlinear state-space models, how to relate local and global analysis results and how local stability depends on the system parameters.
7.2.1 Local Linearization of Nonlinear State-space Models
In order to linearize a state or output equation in a nonlinear state-space model, we first need to find a steady-state operating or reference point to linearize around and then proceed with the linearization.
The Reference Point. The reference point (u0, x0, y0) for the linearization is found by specifying a reference inputu0 for a nonlinear state-space model in its general form:
˙
x=f(x, u), y=h(x, u) (7.23)
and computing the reference value of the state and output signals from the above equations by specifying ˙x= 0 to have a steady-state operating point:
fe(x0, u0) = 0, y0=eh(x0, u0) (7.24) In the special case of input-affine state-space models, we have:
f(x0) + Xm i=1
gi(x0)ui0) = 0, y0=h(x0) (7.25) from the state-space model
˙
x(t) =f(x(t)) + Xm i=1
gi(x(t))ui(t)
y(t) =h(x(t)) (7.26)
Thereafter,centered variables are introduced for all variables to have e
x=uưu0, xe=xưx0, ye=yưy0 (7.27) The Principle of Linearization. The linearization is based on a Taylor- series expansion of a smooth nonlinear functionf around a steady-state ref- erence pointx0.
In the case of a univariate (i.e.whenxis scalar) scalar-valued function y=f(x), the following Taylor series expansion is obtained:
f(x) =f(x0) + df dx x
0
(x−x0) + d2f dx2 x
0
(x−x0)2
2! +. . . (7.28) wherex0 is the reference point and ddxifi denotes thei-th partial derivative of the functionf with respect tox.
The linear approximation is then obtained by neglecting the higher-order terms, which gives:
7.2 Local Stability of Nonlinear Systems 147
f(x)≃f(x0) + df dx
x0
(x−x0) (7.29)
This way, we obtain a linear approximation of the original nonlinear equation y=f(x) in the form:
y=f(x0) + df dx
x0
(x−x0) (7.30)
The above equation can be simplified to obtain (ye+y0) =f(x0, t) + df
dx
x0
(x−x0) (7.31)
because
y0=f(x0) and ey=y−y0 (7.32)
Finally we obtain e
y= df dx
x0
e
x (7.33)
which is the final linearized form expressed in terms of the centered variable ex.
It is easy to extend the above formulae (7.33) for the case of a multivariate vector-valued functiony=f(x1, . . . , xn, t) to obtain
e
y= Jf,xx
0ãxe (7.34)
whereJf,xis the so-called Jacobian matrix of the functionf containing the partial derivatives of the function with respect to the variablexevaluated at the steady-state reference point
Jjif,x
0= ∂fj
∂xi
x
0
(7.35) The Linearized Form of the State-space Model. Now we can apply the principle of linearization to the case of nonlinear state-space models. To do this, the nonlinear functions feand eh in the general state-space model in Equation (7.23) or the nonlinear functionsf, g andhin Equation (7.26) can be linearized separately around the same steady-state reference point (u0, x0, y0).
The linearized model can then be expressed in terms of centered variables in an LTI form:
e˙
x=Aexe+Beue
ey=Cexe+Deeu (7.36)
where in the general nonlinear state-space model case the matrices can be computed as
Ae= Jf ,xe
0, Be=Jf ,ue
0, Ce= Jeh,x
0, De = Jeh,u
0 (7.37)
while in the input-affine case they are in the form:
Ae= Jf,x
0+Jg,x|0u0, Be=g(x0), Ce= Jh,x
0, De = 0 (7.38) 7.2.2 Relationship Between Local and Global Stability of
Nonlinear Systems
The relationship between local and global stability is far from being trivial. It is intuitively clear that global stability is a stronger notion, that is, a locally stable steady-state point may not be globally stable. This is the case when more than one steady-state point exists for a nonlinear system.
There is a related but important question concerning local stability of nonlinear systems:how and when can we draw conclusions on the local sta- bility of a steady-state point of a nonlinear system from the eigenvalues of its locally linearized state matrix? The following theorems provide answers to this question.
Theorem 7.2.1 (Theorem for proving local asymptotic stability).
Suppose that each eigenvalue of the matrixAein Equation (7.36) has a neg- ative real part. Then x0 is a locally asymptotically stable equilibrium of the original system (7.26).
The following theorem shows how one can relate the eigenvalues of a locally linearized state matrix to the unstable nature of the equilibrium point.
Theorem 7.2.2 (Theorem for proving instability). Suppose that the matrixAein (7.36) has at least one eigenvalue with a positive real part. Then x0 is an unstable equilibrium of the original system (7.26).
The following simple examples illustrate the application of the above results to simple systems.
Example 7.2.1 (Global and local asymptotic stability 1)
We show that Theorem 7.2.1 is only true in one direction. For this purpose, consider the one-dimensional autonomous system
7.2 Local Stability of Nonlinear Systems 149
˙
x=−x3 (7.39)
The only steady-state equilibrium of this system isx0= 0.
The Jacobian matrix is now a constantJf,x=−3x2evaluated at x0= 0, which is again 0.
It’s easy to check that for any initial conditionx(0) x(t) = x(0)
p2tx(0)2+ 1 (7.40)
From this, it follows that
|x(t)| ≤ |x(0)|, ∀t≥0 (7.41) and obviously
t→∞lim x(t) = 0 (7.42)
Therefore 0 is a globally asymptotically stable equilibrium of the system (7.39).
Example 7.2.2 (Global and local asymptotic stability 2 )
In this simple example we show that only stability of the linearized system is not enough even for the stability of the original nonlinear system. For this, consider the one-dimensional autonomous system
˙
x=−x2 (7.43)
It’s easy to see that the linearized system (around the equilibrium statex0= 0) reads
e˙
x= 0 (7.44)
which is stable (but not asymptotically stable). It can be checked that the solution of Equation (7.43) for the initial state x(0) is given by
x(t) = x(0)
tx(0) + 1, ∀t≥0 (7.45)
This means that 0 is an unstable equilibrium state of (7.43) since the solution is not even defined for allt≥0 ifx(0)<0.
Example 7.2.3 (Asymptotic stability, nonlinear case)
Consider the following two-dimensional model (which is the model of a mathematical pendulum with rod lengthl and gravity con- stantg)
˙
x1=x2 (7.46)
˙ x2=−g
l sin(x1) (7.47)
It’s easy to see that this system has exactly two equilibrium points, namelyx01= [0 0]T andx02= [π 0]T. The system matrix of the linear approximation inx02 is given by
Ae= 0 1
g l 0
(7.48) which has an eigenvalue with a positive real part, thereforex02
is an unstable equilibrium of the system according to Theorem 7.2.2. We can’t say anything about the stability ofx01 based on the linear approximation. However, it’s not difficult to prove with other methods thatx01 is a locally stable but not asymptotically stable equilibrium state.
7.2.3 Dependence of Local Stability on System Parameters:
Bifurcation Analysis
Roughly speaking, bifurcations represent the sudden appearance of a quali- tatively different solution for a nonlinear system as some parameter varies.
We speak about a bifurcation when the topological structure of the phase portrait of a dynamical system changes when a parameter value is slightly changed.
Definition 7.2.1 (Topological equivalence)
Let f :M 7→Rn andg :N 7→Rn be class Cr mappings where M, N ⊂Rn are open sets, and consider the autonomous systems
7.2 Local Stability of Nonlinear Systems 151
˙
x=f(x) (7.49)
˙
x=g(x) (7.50)
with flows existing for all t ∈ R. The two differential equations (7.49) and (7.50) are topologically equivalent if there exists a homeomorphism
h : M 7→ N that transforms the trajectories of Equation (7.49) into the trajectories of Equation (7.50) keeping the direction.
If Equation (7.49) and Equation (7.50) are topologically equivalent, thenh maps equilibrium points onto equilibrium points and periodic trajectories onto periodic trajectories (by possibly changing the period).
In order to define bifurcation values, consider a parameter-dependent un- forced nonlinear system
˙
x=f(x, ), x∈M, ∈V (7.51)
whereM ⊂Rn and V ⊂Rl are open sets andf : M ×V 7→Rn is a class Cr function. Notice that Equation (7.51) can be also be interpreted as a parameter-independent system ˙x=f(x, ), ˙= 0. We assume that (7.51) has a solution on the entire setR, andf(0,0) = 0.
Definition 7.2.2 (Bifurcation value)
The parameter= 0∈V is called a bifurcation value if in any neighborhood of 0 there exist∈V parameter values such that the two autonomous systems
˙
x=f(x,0)andx˙ =f(x, )arenot topologically equivalent.
Besides characterizing bifurcation points, Theorem 7.2.3 below can also be used for investigating local stability based on the eigenvalues of the linearized state matrix with multiple zero eigenvalues.
Letλ1, . . . λsbe those eigenvalues of the linearized system matrix Aethat have zero real parts, where
Ae=Dxf(0,0) = ∂fi
∂xj
(0,0) n
i,j=1
(7.52) Furthermore, letAehave exactlym eigenvalues with negative real parts and k=n−s−meigenvalues with positive real parts.
Theorem 7.2.3 (Center manifold theorem).In a neighborhood of 0 with of a sufficiently small norm, the system (7.51) is topologically equivalent with the following system:
˙
x=F(x, ) :=Hx+g(x, )
˙
y=−y (7.53)
˙
z=z (7.54)
wherex(t)∈Rs, y(t)∈Rm,z(t)∈Rk and H is a quadratic matrix of size swith eigenvaluesλ1. . . λs. Furthermore,g is a class Cr function for which g(0,0) = 0andDxg(0,0) = 0.
From the construction of Equation (7.53), it follows that in a neighborhood of 0, the bifurcations of Equation (7.51) can be described using only the equation
˙
x=F(x, ) (7.55)
Equation (7.55) is called thereduced differential equation of Equation (7.53) on thelocal center manifold Mlocc ={(x, y, z)|y= 0, z= 0}.
Often, Equation (7.55) can be transformed into a simpler (e.g. polyno- mial) form using a nonlinear parameter-dependent coordinate transformation which does not alter the topological structure of the phase portrait in a neigh- borhood of the investigated equilibrium point. This transformed form is called a normal form. The normal form is not unique, different normal forms can describe the same bifurcation equivalently.
Definition 7.2.3 (Fold bifurcation)
Letf :R×Rbe a one-parameter C2 class map satisfying
f(0,0) = 0 (7.56)
∂f
∂x
=0,x=0
= 0 (7.57)
∂2f
∂x2
=0,x=0
>0 (7.58)
∂f
∂
=0,x=0
>0 (7.59)
there then exist intervals(1,0),(0, 2)andà >0 such that:
• if ∈ (1,0), then f(ã, ) has two equilibrium points in (−à, à) with the positive one being unstable and the negative one stable, and
• if∈(0, 2), thenf(ã, )has no equilibrium points in(−à, à).
This type of bifurcation is known as a fold bifurcation (also called a saddle- node bifurcation or tangent bifurcation).