Stability of LTI Systems

Internal Stability

Stability of $\dot{x} = Ax$ with no input, where $x(t) = e^{At}x(0)$.

• Asymptotically stable iff every eigenvalue has $\mathrm{Re}(\lambda) < 0$, i.e. $A$ is Hurwitz.
• Marginally stable iff every eigenvalue has $\mathrm{Re}(\lambda) \le 0$ and each eigenvalue on the imaginary axis has equal algebraic and geometric multiplicity (no Jordan block of size larger than 1).
• Unstable otherwise.

Input-to-State Stability

The state stays bounded for bounded inputs and goes to zero when the input goes to zero. For LTI systems, ISS $\iff$ $A$ is Hurwitz.

BIBO Stability

Every bounded input produces a bounded output.

• $G$ is BIBO stable iff its impulse response is absolutely integrable, $\int_0^\infty |g(t)|\, dt < \infty$.
• Equivalently, all poles of $G(s)$, after pole-zero cancellation, have negative real parts.

Routh-Hurwitz Criterion

For $a_n s^n + a_{n-1}s^{n-1} + \cdots + a_0$, the first two rows of the table are the alternating coefficients,

$$\begin{array}{c|cccc} s^n & a_n & a_{n-2} & a_{n-4} & \cdots \\ s^{n-1} & a_{n-1} & a_{n-3} & a_{n-5} & \cdots \end{array}$$

and each further entry is

$$R_{i,j} = -\frac{1}{R_{i-1,1}} \det\begin{bmatrix} R_{i-2,1} & R_{i-2,j+1} \\ R_{i-1,1} & R_{i-1,j+1} \end{bmatrix}$$

• The number of sign changes in the first column equals the number of roots in the open right half-plane.
• If a first-column entry is 0 but its row is not all zeros, replace it by $\varepsilon$ and let $\varepsilon \to 0^+$ at the end.
• If a whole row is zero, replace it with the coefficients of the derivative of the auxiliary polynomial formed from the row above.