Stability

Resources: Chapter 5 page=170

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Two types of stability

BIBO Stability

Deals with the zero-state response ($x(0)=0$). Does a bounded input lead to a bounded response?

An input is bounded if there exists a constant $u_m$ such that

$$|u(t)|\le u_m<\infty$$

An output is bounded if there exists a finite $M$ such that

$${\int }_0^{\infty }|g(\tau )|d\tau \le M$$

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We care about the second part of the general solution

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SISO system

$$\begin{array}{rl}y(t)& ={\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau \\ |y(t)|& \le {\int }_0^t|g(\tau )|u(t-\tau )|d\tau \le {\int }_0^t|g(\tau )|d\tau \cdot u_m\\ |y(t)|& \le {\int }_0^t|g(\tau )|d\tau \le M\end{array}$$

Where $u_m$ is the largest input signal (a constant), such that we get the following inequality, and $M$ is a constant smaller than $\infty$.

A SISO system with proper rational transfer function $\hat{g}(s)$ is BIBO stable if and only if every pole of $\hat{g}(s)$ has a negative real part

• This means that a pole in zero is NOT BIBO STABLE

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Internal Stability

Internal stability deals with the zero-input response ($\dot{x}=Ax$). Does the system converge towards its equilibrium point (or in a neighborhood around it)? Think a pendelum and an inverted pendelum

Definitions

The dynamics of the system can be written as

$$\dot{x}=Ax\to x(t)=e^{At}{x}_0$$

For example: If $A$ is the zero matrix, we get that $\dot{x}=x_0$ meaning that the system is marginally stable

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Methods of finding stability

Lyapunov Theorem

A method of checking the asymptotic stability of $\dot{x}=Ax$.

Theorem 5.5 page=187

The matrix $A$ is CT (?) stable if and only if for any given symmetric Positive-Definite Matrix $N$, the Lyapunov Equation

$$A^{T}M+MA=-N$$

has a unique symmetric solution $M$ and $M$ is as Positive-Definite Matrix

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