Consider the $($scalar$)$ transfer function

T$($s$)=$ pmsm $+$pm$1$sm$1++$p$1$s$+$p$0,(5)$

sn $+$qn$1$sn$1++$q$1$s$+$q$0$

with pm $=0$ and m $<$ n$.$ The coefficients pi and qi are real $($and m denotes the degree of the numerator polynomial rather than the number of inputs$).$ The relative degree r of a transfer function is defined as the integer r $=$ n $$ m$.$

$($a$)$ Let $($A$,$ B$,$ C$,0)$ be a realization of T $($s$)$ with the matrix pair $($A$,$ B$)$ controllable. Show that T $($s$)$ has relative degree r if and only if

CAi$1$B $=0$ for all i in $\{1,2,...,$ r $1\}$ and CAr$1$B $=0$ Hint. Exploit the controllability canonical form and recall Theorem $6\mathrm{.}9.$

In the remainder of this problem, we assume r $=$ n and let $($A$,$ B$,$ C$,0)$ be the realization of T $($s$)$ in controllability canonical form. Denote, as usual, the corresponding linear system by

where y$($k$)$ is the kth$-$order time derivative of y$,$ i$.$e$.,$ y$($k$)($t$)=$ dky$($t$).$

dtk

$($c$)$ Derive the dynamics $(7)$ directly from the transfer function $(5).$

x $($t$)=$ Ax$($t$)+$ Bu$($t$),$ y$($t$)=$ Cx$($t$).$

$(6)$

$($b$)$ Show that the dynamics $(6)$ can be written as the higher$-$order differential equation

y$($n$)($t$)+$ qn$1$y$($n$1)($t$)++$ q$1$y$(1)($t$)+$ q$0$y$($t$)=$ p$0$u$($t$),(7)$

Hint. You may use the fact that the Laplace transform is injective, i$.$e$.,$ L$($f$1)=$ L$($f$2)$ implies f$1=$ f$2.$

$2$

We would like to solve a tracking control problem in which the output y asymptotically tracks a given reference trajectory yref : $[0,\backslash$infty $)->$ R$,$ i$.$e$.,$

lim y$($t$)$ yref $($t$)=0.(8)$ t$->\backslash$infty

To do so$,$ we define the tracking error e$($t$)=$ y$($t$)$ yref $($t$).$

Now, assuming that yref is sufficiently differentiable and taking p$0=1$ for simplicity, consider

a controller of the form

k$=0$

Then, the tracking problem can more formally be stated as: find a matrix

F$=$f$0$ f$1$ fn$1$

such that, for each initial condition x$0$ in Rn$,$ the dynamics $(6)$ with x$(0)=$ x$0$ and input $(9)$

satisfy $(8).$

$($d$)$ Define the state z as

n$1$ n$1$

u$($t$)=$ y$($n$)($t$)+$ X qky$($k$)($t$)+$ X fke$($k$)($t$).(9)$

ref ref k$=0$

$$z$1$ e $$z$2$e$(1)$

z$=.=...$

zn e$($n$1)$

and show that the dynamics $(6)$ with the controller $(9)$ can be written as

z $($t$)=$ A $+$ BF z$($t$),$ e$($t$)=$ Cz$($t$).$

$($e$)$ Using the result from $($d$),$ prove that $(8)$ holds for any initial condition x$0$ if and only if $\backslash$sigma $($A $+$ BF$)$ C$.$

Hint. For the only if part, first show that the matrix pair $($A $+$ BF$,$ C$)$ is observable. Then, take some inspiration from the proof of Theorem $6\mathrm{.}16.$