The goal of all of this is to build to an unique theorem we can build to, that summarizes everything we have learned so far.

Example

A = [0,1,0;0,0,0;0,0,-1;]
The eigenvalues of a are [0,0,-1]
We have with multiplicity
and , = 1

The sum of the multiplicities must be the size of the matrix A.

The study of minimal multiplicity

Geometric multiplicity is the dimension of the eigenspace.
We define the minimal polynomial of the matrix A, , the least common multiple of the denominators of the matrix
The characteristic polynomial of A, will be defined as

The roots of are also the roots of , equal to .
,the minimal polynomial multiplicity, will be the multiplicity of as root of .

If we have zero real part eigenvalues, if our minimal multiplicity is = 1, we have stability. If multiplicity 2, we have instability

The minimal multiplicity tells us how many natural modes there are corresponding to an eigenvalue of zero.

This is true for all LTI systems.

!! Summary

An LTI System is internally stable if the zero-input state response is bounded for any initial state

An LTI system is internally stable if and only if all the natural modes are bounded.

An LTI system is internally stable if and only if all the system eigenvalues have non-positive real part and the ones with null real part have unitary() minimal multiplicity , which is the root of the minimal polynomial , corresponding to the least common multiple of the denominators of the matrix

Unitary - equal to 1(one unit)

An LTI system is asymptotically stable if and only if:
The zero-input state response is convergent to 0 as
or
all its natural modes are convergent
or
all the system eigenvalues have strictly negative real part.

UNNECESSARY. YOU CAN CHECK FOR INSTABILITY BY CHECKING FOR LACK OF STABILITY.

Conversely, an LTI system is unstable if is divergent
or
any natural mode diverges
or
some eigenvalue with strictly positive real part or all system eigenvalues have nonpositive real part and the ones with null real part have minimal polynomial multiplicity larger than 1

In MATLAB we determine minimal multiplicity either explicitly, by computing the matrix

s = tf('s')
zpk(minreal(inv(s*eye(dim(A))-A),1e-3))

Or, we can simply use the command minpoly. It computes the minimal polynomial of the matrix A.

P = minpoly(A) returns the vector containing the coefficients of the minimal polynomial of the matrix A.

roots(minpoly(A)) - returns roots of minimal polynomial.