Sampled-Data System Stability

The map from $\left[\begin{array}{c}r\\ d\end{array}\right]$ to $\left[\begin{array}{c}u\\ e\\ y\end{array}\right]$ is the function that sends $r(t),d(t)$ to the resulting $u(t),e(t),y(t)$.

• This is basically a more generalized version of a transfer function.

Sampled-Data System Closed-Loop Stability

A sampled-data system is closed-loop stable if the map from from $\left[\begin{array}{c}r\\ d\end{array}\right]$ to $\left[\begin{array}{c}u\\ e\\ y\end{array}\right]$ is BIBO stable.

BIBO Stability for Sampled-Data System

The map $\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]$ to $\left[\begin{array}{c}{y}_1\\ ⋮\\ y_n\end{array}\right]$ is BIBO stable if for every collection of bounded signals $x_1(t),\dots ,x_n(t)$, the resulting $y_1(t),\dots ,y_n(t)$ are bounded.

Corresponding Discrete Time System

Can we find the closed-loop stability of a sampled-data system (at all times) by evaluating the stability of a corresponding discrete time system (at the sample points)?

For a given sampled-data system:

We can draw the following corresponding discrete system, which is equal to the sampled-data system at sample points:

Pathological Sampling Time

For a plant $P(s)$, a sampling time $T>0$ is pathological if the number of poles of $G[z]$ is less than the number of poles of $P(s)$.

Example:

$$P(s)=\frac{1}{s-1+j}+\frac{1}{s-1-j}\text{(2 poles)}$$

Then:

$$\begin{array}{r}G[z]=\frac{1}{j-1}\frac{1-e^{(1-j)T}}{z-e^{(1-j)T}}+\frac{1}{-j-1}\frac{1-e^{(1+j)T}}{z-e^{(1+j)T}}\end{array}$$

where we are using $-\frac{1}{\lambda }\frac{1-e^{\lambda T}}{z-e^{\lambda T}}$.

If we choose $T=2\pi$ we will have

$$\begin{array}{rl}G[z]& =\frac{1}{j-1}\frac{1-e^{2\pi }}{z-e^{2\pi }}+\frac{1}{-j-1}\frac{1-e^{2\pi }}{z-e^{2\pi }}\\ & =\frac{2(e^{2\pi }-1)}{z-e^{2\pi }}\text{(1 pole!)}\end{array}$$

Note: for almost all $T>0$, $T$ is non-pathological.

Theorem

For the sampled-data system, is $T>0$ is non-pathological, then the SD system is closed-loop stable if and only if its associated discrete-time system is itself closed-loop stable.

This is a lot easier to check than checking for BIBO stability! We can check the stability of a sampled-data system by looking only at its associated discrete-time system.

Thus, we can say that a SD system is closed-loop stable if and only if roots of the characteristic polynomial $\Delta$ (for the system with $G[z],D[z]$) are in the open unit disk $\mathbb{D}$.

• Schur Characteristic Polynomials