System controller

In control theory, a controller is used within a closed-loop system to allow it to enforce a key set of performance characteristics, like stability, transient parameters, or steady-state tracking.

There are a few basic important types of controllers:

• Proportional control, where we only have a $k$ constant term.
• Proportional-integral (PI) control, where we have an additional integral term.

Tracking problem

One core problem is the tracking problem, where we want to design a controller $C(s)$ such that:

• The feedback loop system is BIBO stable.
• For all reference signals $r(t)$, when $d(t)=0$, the asymptotic tracking error $e(\infty )$ is either bounded or converges to zero, i.e., when there’s no disturbance, the system should track the reference system.
• For all disturbance signals $d(t)$ and for all reference signals $r(t)$, the asymptotic tracking error $e(\infty )$ is bounded or is zero.

The feedback system is BIBO stable if for any bounded signals $r(t),d(t)$ the signals $u(t),e(t)$ are bounded as well, where $e(t)=r(t)-y(t)$. If $r(t),e(t)$ are bounded, then $y(t)$ is also bounded.

We can model this in a system of two equations:

$$\left[\begin{array}{c}E(s)\\ U(s)\end{array}\right]=\left[\begin{array}{cc}{F}_{11}(s)& F_{12}(s)\\ F_{21}(s)& F_{22}(s)\end{array}\right]\left[\begin{array}{c}R(s)\\ D(s)\end{array}\right]$$

The system is BIBO stable if and only if all the transfer functions in the matrix above are BIBO stable.

Classes of controllers

There are three types of basic controllers. The first is a proportional controller, where it takes the form:

$$G_C(s)=K$$

The proportional-integral-derivative (PID) controller takes the form:

$$G_C(s)=K_p\left(1+\frac{1}{T_{i}s}+T_{d}s\right)$$