L04 - Modal Analysis of LTI dynamical Systems - Internal Stability

Stability is the most important property of a dynamical system. Stability prevents system damage.

Stability means that physical variables we are interested in remain bounded(do not exceed a certain range of values) given arbitrary initial conditions and bounded inputs.

(Example)
A bridge is a system that usually undergoes oscillations. It's designed to withstand them. This oscillation must of course be bounded, otherwise it will collapse or create issues.
An example was the tacoma bridge which collapsed in 1940.

Stability is a synonym of boundedness.

System stability -> Boundedness of state/output response.

Internal Stability

Refers to a system's inherent stability. This means that the zero-input state response is always bounded, within a certain range of values. We focus on the state.

Problem

Under which conditions is bounded for all ?

If for all is bounded, we say the system is internally stable
This is connected to the state-space model. We can't define stability for the tf model of the system because the tf model cannot be studied for systems with no input.
We need only know the matrix A.

So how do we study this for all values?

Again, we have the matrix , which is a matrix containing ratios.

This can be differently written as
The roots of the characteristic polynomial of A are the eigenvalues of A.

The purpose is to write in terms of simple fractions.

What happens when is complex?

We have for each term its conjugate pair.
Complex eigenvalues come in conjugate pairs!

When our solution is obtained, we invert and come back to the time domain, getting
In the case of a complex lambda,
or

The key point is that is a linear combination of specific functions of time:
If the poles are complex,
These are all the possible functions that can come up in the state response.

Such functions are called natural modes

The natural modes are the functions that is a linear combination of.

Example:

Natural Modes(NMs):

Let's go back. So, we want to study the boundedness of

We know that the is a linear combination of these natural modes. All we need to do therefore is study the boundedness of the natural modes.

Classification of Natural Modes (Modal Analysis)

$\mathbb{C}$ - **A natural mode is convergent if** $\lim_{t \to \infty} |m(t)| = 0$

- **A natural mode is bounded if

All convergent natural modes are a subset of bounded modes.

- **A natural mode is divergent if

Case 1:

If , convergent.
If , bounded.
If , divergent.

Case 2:

If convergent.
If divergent.
If divergent.

Sines and Cosines are always bounded.

Case 3:

If , convergent.
If , bounded.
If , divergent.

Case 4:

If , convergent.
If , divergent.
If , divergent.

Recap, part 1

A system is internally stable if is bounded for all possible initial conditions.
is bounded only if all its natural modes are bounded. If not, then is unbounded(divergent).
If we can prove that only one, at least one natural mode is divergent, the system is internally unstable
The natural modes are the functions that is a linear combination of.
The classification of the natural modes, therefore the internal stability depends on , the eigenvalues of the matrix A.
Asymptotic stability is when zero-input state response converges to zero, always. More powerful than internal stability.

THE MOST IMPORTANT PARTS

If we have all eigenvalues of negative real part, we have internal stability (asymptotic)
If some eigenvalue has real part larger than zero, we have internal instability
If the eigenvalues have zero real part, we will need to analyze the minimal multiplicity, which we will define in the next part.
If a material is in the lecture appendix, it will not be in the exam

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