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\begin{document}

\null\vspace{-1.75cm}

{\Large
{\bf ECE 486 \hfill Assignment \#  4}

\vspace{.15in}
Issued: February 15 \hfill Due: February 22, 2024}
\rule{6in}{.01in}

\vspace*{.1cm}
 

\head{Reading Assignment:} 
\textbf{FPE},  Sections 3.3-3.6, 4.1-4.3

\head{Problems:} (unless otherwise noted, you can use a calculator/computer to arrive at numerical answers)

\begin{arabnum}
\setcounter{arabnum}{0}  



\item Determine whether or not the following polynomials have any RHP roots:
\begin{romannum}
	\item $s^4 + 10s^3 + 15s^2 + 20s + 1$
	\item $s^6 + 2s^5 - 3s^4 + s^3 + 2s + 3$
	\item $s^8 + 4s^7 + s^6 + 2s^5 + 3s^4 - s^3 + 2s + 3$
	\item $s^4 + 10s^3 + 12s^2 + 20s + 1$
\end{romannum}
(\emph{Computer use not allowed.})

\item
Consider the following feedback system, where $K$ is a constant gain and
$$
G(s) = \dfrac{1}{s^3 + 3s^2 + s + 1}:
$$
\centerline{\includegraphics[width=0.4\textwidth]{unity_feedback_no_disturbances}}

Using the Routh--Hurwitz criterion, show that the system is stable for $-1 < K < 2$ and unstable for $K \ge 2$. (This illustrates the destabilizing effect of feedback when the gain is too high.)

\item
Consider the following open-loop system, consisting of a stable first-order plant
$$
G(s) = \frac{1}{s+p}
$$
(thus, $p > 0$) and a scalar-gain controller. There is also a disturbance $W$ that affects the input to the controller, where $C$ is a fixed constant.

\bigskip

\centerline{\includegraphics[width=0.4\textwidth]{open_loop_with_disturbance}}

\newpage

\begin{romannum}
	\item Choose the value of controller gain $K$ to guarantee perfect tracking of a constant reference.
	\item Show that the resulting system is unable to reject constant disturbances and compute the resulting DC gain from $W$ to $Y$.
\end{romannum}

\item

In class, we have introduced the notion of system type and described how it relates to  reference tracking capabilities (or lack thereof) of closed-loop feedback. In this problem, we will explore the notion of system type with respect to \textit{disturbances}.

Consider the following unity-feedback configuration that consists of a controller $K(s)$ and a plant $P(s)$, and also includes an additive disturbance $W$:

\bigskip

\centerline{\includegraphics[width=0.5\textwidth]{system_type_disturbances}}

Recall that the forward-loop transfer function $K(s)P(s)$ has system type $n$ with respect to reference input if it has a pole of order $n$ at the origin, or, equivalently, if
$$
\lim_{s \to 0} \left[s^n K(s)P(s)\right] \neq 0.
$$
Assuming the closed-loop system is stable, this means that $n$ is the lowest degree of a polynomial that cannot be tracked in feedback with zero steady-state error. 
We will now focus on the control objective of disturbance rejection.
\begin{romannum}
	\item Show that the system type with respect to reference inputs is equal to $n$ whenever
	$$
	\lim_{s \to 0} \frac{1-T_{r\to y}(s)}{s^n} = {\rm const} \neq 0,
	$$
	where $T_{r\to y}$ is the transfer function from the reference $R$ to the output $Y$.
	\item Write down the transfer function $T_{w \to y}$ from the disturbance input $W$ to the output $Y$.
	\item We say that the above system has type $k$ with respect to disturbance inputs if
	$$
	\lim_{s \to 0} \frac{T_{w \to y}(s)}{s^k} = {\rm const} \neq 0.
	$$
	Show that this is the case when $T_{w \to y}$ has a \textit{zero} of order $k$ at the origin, i.e., if $T_{w \to y}(s) = s^k \tfrac{A(s)}{B(s)}$, where $A$ and $B$ are polynomials with real coefficients, such that $A(0) \neq 0$ and $B(0) \neq 0$.
	\item Show that the system of type $k$ with respect to disturbances can acehive perfect steady-state rejection of polynomial disturbances of degree $m < k$, but not when $m \ge k$.
	\item Finally, consider the example we have discussed in class, where
	$$
	P(s) = \frac{1}{s^2+1}
	$$
	and determine the system type with respect to disturbances for P-control $K(s) = K_{\rm P}$, PD-control $K(s) = K_{\rm P} + K_{\rm D}s$, and PID-control $K(s) = K_{\rm P} + K_{\rm D}s + \tfrac{K_{\rm I}}{s}$.
\end{romannum}


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