Stability of Nonlinear Systems
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Last updated
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.
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.
Uniform stability means that the 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.
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.
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.
Just as we can define stability, we can also define instability.
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.
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”.
Descresence means that for a ball around the equilibrium, we can upper bound the the energy.
is SOS.
The results of Theorem 16, Theorem 17, Theorem 18, Theorem 19, Theorem 20 are summarized in Table 1.
The state transition matrix is useful in determining properties of the system.
The Lyapunov Lemma has extensions to the time-varying case.
It turns out that uniform asymptotic stability of the linearization of a system corresponds to uniform, asymptotic stability of the nonlinear system.
For asymptotically stable and exponential stable equilibria, it makes sense to wonder which initial conditions will cause trajectories to converge to the equilibrium.
An equilibrium point is uniformly asympototically stable if is uniformly stable in the sense of Lyapunov, and and such that
The existence of the function helps guarantee that the rate of converges to equilibrium does not depend on since the function is independent of . Suppose that the is an exponential function. Then solutions to the system will converge to the equilibrium exponentially fast.
An equilibrium point is locally exponentially stable if such that
are all local definitions because the only need to hold for inside a ball around the equilibrium. If they hold , then they become global properties.
An equilibrium point is unstable in the sense of Lyapunov if such that
Instability means that for any ball, we can find an ball for which there is at least one initial condition whose corresponding trajectory leaves the ball.
A class function is a function such that and is strictly monotonically increasing in .
A subset of the class functions grow unbounded as the argument approaches infinity.
A class function is a class function where .
Class functions are “radially unbounded”. We can use class and class to bound “energy-like” functions called Lyapunov Functions.
A function is locally positive definite (LPDF) on a set containing if such that
A function is positive definite (PDF) if such that
Positive definite functions act like “energy functions” everywhere in .
A function is decrescent if such that
Note that we can assume 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 are one of the simplest types of Lyapunov Functions. Their level sets are ellipses where the major axis is the eigenvector corresponding to , and the minor axis is the eigenvecctor corresponding to .
Consider the sublevel set . Then is the radius of the largest circle contained inside , and is the radius of the largest circle containing .
A polynomial is sum-of-squares (SOS) if such that
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 where is a vector of monomials, the properties of can tell us if is SOS or not.
Note that is not necessarily unique, and if we construct a linear operator which maps to , then this linear operator will have a Null Space. Mathematically, consider
This linear operator has a null space spanned by the polynomials . Given a matrix such that (i.e is SOS), it is also true that
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 to be PDF, then one constraint in our SOS program will be that
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 and , we can use the Lyapunov Stability Theorems to prove the stability of equilibria.
If such that is LPDF and locally, then is stable in the sense of Lyapunov.
If such that is LPDF and decrescent, and locally, then is uniformly stable in the sense of Lyapunov.
If such that is LPDF and decrescent, and is LPDF, then is uniformly asymptotically stable in the sense of Lyapunov.
If such that is PDF and decrescent, and is LPDF, then is globally uniformly asymptotically stable in the sense of Lyapunov.
If and such that V is LPDF is decrescent, is LDPF, and
Going down the rows of Table 1 lead to increasingly stricter forms of stability. Descresence appears to add uniformity to the stability, while being LPDF adds asymptotic convergence. However, these conditions are only sufficient, meaning if we cannot find a suitable , 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 such that is LPDF. In the case of autonomous systems, we can still prove asymptotic stability without such a .
Consider a smooth function with bounded sub-level sets and , . Define and let be the largest invariant set in , then
LaSalle’s theorem helps prove general convergence to an invariant set. Since is always decreasing in the sub-level set , trajectories starting in must eventually reach . At some point, they will reach the set in , and then they will stay there. Thus if the set 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 is PDF, and , 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 may change over time.
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 . The linearization at the equilibrium is given by
The function is the higher-order terms of the linearization. The linearization is a time-varying system. Consider the time-varying linear system
The state transition matrix of a time-varying linear system is a matrix satisfying
the system is stable at the origin at .
the system is uniformly stable at the origin at .
the system is asymptotically stable.
such that the system is uniformly asymptotically stable.
exponential stability.
If the system was Time-Invariant, then the system would be stable so long as the eigenvalues of were in the open left-half of the complex plane. In fact, we could use to construct positive definite matrices.
For a matrix , its eigenvalues satisfy if and only if , there exists a solution to the equation
In general, we can use the Lyapunov Equation to count how many eigenvalues of are stable.
For and given , if there are no eigenvalues on the axis, then the solution to has as many positive eigenvalues as has eigenvalues in the complex left half plane.
If is bounded and for some , the solution to is bounded, then the origin is a asymptotically stable equilibrium point.
For a nonlinear system whose higher-order terms of the linearization are given by , if
and if is a uniformly asymptotic stable equilibrium point of where is the Jacobian at the , then is a uniformly asymptotic stable equilibrium point of
An equilibrium point is unstable in the sense of Lyapunov if which is decrescent, the Lie derivative is LPDF, , and in the neighborhood of such that .
If is an equilibrium point of a time-invariant system , then the Region of Attraction of is
Suppose that we have a Lyapunov function and a region such that and in . Define a sublevel set of the Lypunov function which is a subset of . We know that if , then the trajectory will stay inside and converge to the equilibrium point. Thus we can use the largest that is compact and contained in as an estimate of the region of attraction.
When we have a Quadratic Lyapunov Function, we can set to be the largest circle which satisfies the conditions on , and the corresponding contained inside 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 which we used to prove asymptotic stability. Then if there exists an which satisfies the following SOS program, then the sublevel set is an estimate of the Region of Attraction.