Part 9 - General controller structure

Introduction

In this part we’ll see how we define a general feedback loop.

General controller structure

Assume the general structure for a feedback loop.

However, we now have some noise that is added (in theory the sign can be whatever) to our out signal. Additionally, the noise, $V(s)$ will go through a process, $G_v(s)$ .

Along with this noise we have some noise in the feedback loop as well, so-called measurement error. We’ll call the new feedback signal for $Y_m = Y(s) - W(s)$ , where $W(s)$ is the extra noise.

Mathematically we will describe this as:

\begin{align*} E(s) & = R(s) - Y_m(s) \newline & = R(s) - (Y(s) - W(s)) \newline & = R(s) - Y(s) + W(s) \end{align*}

Y(s) = G_v(s)V(s) + F(s)G(s) [R(s) - Y(s) + W(s)] \newline Y(s) = G_v(s)V(s) + F(s)G(s)R(s) - F(s)G(s)Y(s) + F(s)G(s)W(s) \newline Y(s) + F(s)G(s)Y(s) = G_v(s)V(s) + F(s)G(s)R(s) + F(s)G(s)W(s) \newline Y(s)(1 + F(s)G(s)) = G_v(s)V(s) + F(s)G(s)R(s) + F(s)G(s)W(s) \newline Y(s)(1 + L(s)) = G_v(s)V(s) + L(s)R(s) + L(s)W(s) \newline Y(s) = \dfrac{1}{1 + L(s)} G_v(s)V(s) + \dfrac{L(s)}{1 + L(s)} R(s) + \dfrac{L(s)}{1 + L(s)} W(s) \newline

Definition 1 (Sensitivity functions)

Let’s denote $\dfrac{1}{1 + L(s)}$ as $S(s)$ , weird notation, but let’s go with it. Let’s denote $\dfrac{L(s)}{1 + L(s)}$ as $T(s)$ .

We call $S(s)$ for the sensitivity function. $T(s)$ is called for the complementary sensitivity function.

Meaning that:

S(s) + T(s) = 1 \ | \ \text{Not hard to see}

We also have a name for $\dfrac{G_v(s)}{1 + L(s)} = G_v(s) S(s)$ , noise-sensitivity function.

What is the error in $E(s)$ ?

The impact from $R(s)$ of the overall error is found by setting $V(s) = 0, W(s) = 0$

If we wanted to know the impact from $V(s)$ we set the others to 0.

\begin{align*} E(s) & = R(s) - Y(s) \newline & = R(s) - T(s)R(s) & = R(s)(1 - T(s)) & = R(s)S(s) \end{align*}

If we want $y(t)$ to follow $r(t)$ , $S(s)$ needs to be small, which is equivalent to saying $T(s)$ should be “large”.

Let’s do an actual example now.

Example 1

We have a car system, where $y(t)$ is velocity of the car. $F_d(t)$ is a traction force. $u(t)$ is throttle.

In the $s$ -domain we have:

Y(s) = \dfrac{5}{1 + 5s} F_d(s)

F_d(s) = \dfrac{0.1}{1 + s}U(s)

U(s) = \dfrac{(1 + s)F_d(s)}{0.1}

The car has a cruise control - this is a feedback loop with a PI-controller.

F(s) = K_p \cdot \dfrac{1 + T_i s}{T_i s}

We know that the transfer function from throttle $\rarr$ velocity is:

G(s) = \dfrac{Y(s)}{U(s)} = \dfrac{\dfrac{5}{1 + 5s} F_d(s)}{\dfrac{(1 + s)F_d(s)}{0.1}} = \dfrac{0.5}{(1 + 5s)(1 + s)}

The loop transfer function is therefore:

L(s) = F(s)G(s) = K_p \cdot \dfrac{1 + T_i s}{T_i s} \cdot \dfrac{0.5}{(1 + 5s)(1 + s)}

If we choose a time constant, such as $T_i = 5$ , we get:

L(s) = F(s)G(s) = \dfrac{0.1K_p}{s(1 + s)}

Let’s study $S(s)$ and $T(s)$ now.

S(s) = \dfrac{1}{1 + L(s)} = \dfrac{1}{1 + \dfrac{0.1K_p}{s(1 + s)}} = \dfrac{s(1 + s)}{s(1 + s) + 0.1K_p} = \dfrac{s^2 + s}{s^2 + s + 0.1K_p}

T(s) = \dfrac{L(s)}{1 + L(s)} = \dfrac{\dfrac{0.1K_p}{s(1 + s)}}{1 + \dfrac{0.1K_p}{s(1 + s)}} = \dfrac{0.1K_p}{s(1 + s) + 0.1K_p} = \dfrac{0.1K_p}{s^2 + s + 0.1K_p}

We can view this as a second-order system.

\omega_n = \sqrt{0.1K_p}

\zeta = \dfrac{1}{2 \omega_n} = \dfrac{1}{2 \sqrt{0.1K_p}}