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# Control theory

In engineering, control theory deals with the behaviour of dynamical systems over time. The desired output of a system is called the reference variable. When one or more output variables of a system need to show a certain behaviour over time, a controller[?] tries to manipulate the inputs of the system to realize this behaviour at the output of the system.

Take for example cruise control[?]. In this case, the system is a car. The goal of the cruise control is to keep it at a constant speed. So, the output variable of the system is the speed of the car. The primary means to control the speed of the car is the gas pedal.

A simple way to implement cruise control is to lock the position of the gas pedal the moment the driver engages cruise control. This is fine if the car is driving on perfectly flat terrain. On hilly terrain, the car will accelerate when going downhill and slow down when going uphill; something its driver may find highly undesirable.

This type of controller is called an open-loop[?] controller because there is no direct connection between the output of the system and its input. One of the main disadvantages of this type of controller is the sensitivity to the dynamics of the system under control.

Classical Control To avoid the problems of the open-loop controller, control theory introduces feedback. The output of the system $y$ is fed back to the reference value $r$. The controller $C$ then takes the difference between the reference and the output, the error $e$, to change the inputs $u$ to the system under control $P$. This is shown in the figure. This kind of controller is a closed-loop controller[?] or feedback controller[?].

A simple feedback control loop

If we assume the controller $C$ and the plant $P$ are linear, time-invariant and all single input, single output[?], we can analyse the system above by using the Laplace transform on the variables. This gives us the following relations:

$Y(s) = P(s) U(s)$
$U(s) = C(s) E(s)$
$E(s) = R(s) - Y(s)$
Solving for $Y(s)$ in function of $R(s)$, we obtain:
$Y(s) = \frac{PC}{1 + PC}R(s)$

The term $\frac{PC}{1+PC}$ is referred to as the transfer function of the system.

If we can ensure $PC >\!\!> 1$, then $Y(s) \approx R(s)$.

This means we control the output by simply setting the reference.

State space representation To get a coherent model for systems with multiple inputs and multiple outputs, we need a way to record every relation between any input variable and any output variable. With $n$ inputs and $m$ outputs, we have to write down $m n$ Laplace transforms to encode all the information about a system. A more compact representation of a system is its state space representation using $p$ internal states:

$\dot{\mathbf{x}}(t)  = A \mathbf{x}(t) + B \mathbf{u}(t)$

$\mathbf{y}(t)  = C \mathbf{x}(t) + D \mathbf{u}(t)$

where
$\operatorname{dim}[A] = p \times p$
$\operatorname{dim}[B] = p \times n$
$\operatorname{dim}[C] = m \times p$
$\operatorname{dim}[D] = m \times n$
$\dot{\mathbf{x}}(t)  = {d\mathbf{x}(t) \over dt}$.

For simplicity, $D$ is often chosen to be the zero matrix.

The same representation Laplace transformed is:

$s X(s) = A X(s) + B U(s)$
$Y(s) = C X(s) + D U(s)$

The characteristic polynomial of the state space representation is:

$a(s) = \operatorname{det}(sI - A)$

Controllability and observability Controllability is a measure for the ability to use a certain input to control an output of a system. In the cruise control example, an additional input to control the speed of the car is the clutch pedal. By varying the amount of power transferred from the engine to the wheels, you could control the speed of the car. However, most drivers will use this method only for very low speeds. The controllability of the car at high speeds is better with the gas pedal.

Observability[?] is a measure for how well internal states can be observed on the external outputs of a system. The observability and controllability of a system are mathematical equivalent.