Feedback System Stability

Continuous-time and discrete-time feedback control systems take the following general forms:

The systems have many different transfer functions. To distinguish these, we make the distinction between external and internal signals:

• External signals: $r,d$ – we have no control over these
• Internal signals: $u,e,y$ – these depend on our control design

We consider transfer functions from external signals to internal signals (closed-loop transfer functions):

$$T_{ry},T_{re},T_{ru},T_{dy},T_{de},T_{du}$$

• We consider external-to-internal TFs because our goal is to understand the influence of external factors on the internals of our system

Example:

$$\begin{array}{rl}y& =Pu=P(d+Ce)=P(d+C(r-y))\\ & ⟹y=Pd+PCr-PCy\\ & ⟹(1+PC)y=Pd+PCr\\ & ⟹y=\underset{T_{dy}}{\underbrace{\frac{P}{1+PC}}}d+\underset{T_{ry}}{\underbrace{\frac{PC}{1+PC}}}r\end{array}$$

We can generalize these in matrix form for both continuous and discrete time:

$$\begin{array}{r}\text{Continuous-time:}\left[\begin{array}{c}U\\ E\\ Y\end{array}\right]=\left[\begin{array}{cc}\frac{C}{1+PC}& \frac{1}{1+PC}\\ \frac{1}{1+PC}& \frac{-P}{1+PC}\\ \frac{PC}{1+PC}& \frac{P}{1+PC}\end{array}\right]\left[\begin{array}{c}R\\ D\end{array}\right]\\ \text{Discrete-time:}\left[\begin{array}{c}U\\ E\\ Y\end{array}\right]=\left[\begin{array}{cc}\frac{D}{1+GD}& \frac{1}{1+GD}\\ \frac{1}{1+GD}& \frac{-G}{1+GD}\\ \frac{GD}{1+GD}& \frac{G}{1+GD}\end{array}\right]\left[\begin{array}{c}R\\ D\end{array}\right]\end{array}$$

Well-posed

A feedback system is well-posed if all closed-loop transfer function from external signals to internal signals are real, rational, and proper.

Closed-loop stable / Internally stable

A closed-loop system is closed-loop stable or internally stable if all closed-loop transfer functions from external signals to internal signals $T_{ry},T_{re},T_{ru},T_{dy},T_{de},T_{du}$ are BIBO stable.