Question 3 of 8: State Space, Controllability/Observability, Steady-State Error
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 98-Phys-B5 Systems & Control, National Examination
May 2016 — a three-hour closed-book examination with one double-sided handwritten
formula/notes sheet permitted and an approved calculator. Questions 1 and 2 are compulsory;
candidates choose three of the remaining six (3–8). Every question is nonetheless
answered in full below so the paper remains a complete study resource. All eight questions
carry equal value (20 marks each).
Reference texts. N. S. Nise, Control Systems Engineering, 7th ed.
(transient-response specifications, root locus, Routh–Hurwitz, frequency-response and
lead/lag design, state-space representation, steady-state errors); K. Ogata, Modern
Control Engineering, 5th ed. (root-locus and frequency-response compensator design,
controllability/observability, Nyquist stability).
Question 3: State Space, Controllability/Observability, Steady-State Error (20 marks)
Given. State-space process model $A=\begin{bmatrix}0&1\\-1&-1\end{bmatrix}$,
$B=\begin{bmatrix}0\\1\end{bmatrix}$, $C=\begin{bmatrix}1&1\end{bmatrix}$, $D=1$; forward
Proportional gain $K_p$; feedback path $H(s)=1/(s+1)$ (non-unity feedback).
Find. (1) the process transfer function and the loop transfer function
$K_pG(s)H(s)$; (2) controllability and observability of $(A,B,C,D)$; (3) the range of $K_p$ for
closed-loop stability and for $e_{ss(step)}\%\le5\%$.
Approach. Convert the state-space model to a transfer function via
$G(s)=C(sI-A)^{-1}B+D$; test controllability/observability with the standard rank tests;
assemble the loop gain $K_pG(s)H(s)$, form the characteristic equation, and apply
Routh–Hurwitz for stability and the final-value theorem (on the actuating error $E(s)$
shown at the summing junction) for the error spec.
Part 1) — process transfer function and loop gain.
$(sI-A)^{-1}=\dfrac{1}{s^2+s+1}\begin{bmatrix}s+1&1\\-1&s\end{bmatrix}$, so
$$G_{proc}(s)=C(sI-A)^{-1}B+D=\frac{s+1}{s^2+s+1}+1=\boxed{\frac{s^2+2s+2}{s^2+s+1}}.$$
The Open Loop transfer function of the diagram (forward path times the non-unity feedback
path, the quantity that enters $1+K_pG_{proc}H=0$) is
$$K_pG_{proc}(s)H(s)=\boxed{\frac{K_p(s^2+2s+2)}{(s+1)(s^2+s+1)}}.$$
Part 2) — controllability and observability of the open-loop process.
$$\mathcal C=[B\ \ AB]=\begin{bmatrix}0&1\\1&-1\end{bmatrix},\quad\det\mathcal C=-1\ne0
\ \Rightarrow\ \boxed{\text{controllable}}.$$
$$\mathcal O=\begin{bmatrix}C\\CA\end{bmatrix}=\begin{bmatrix}1&1\\-1&0\end{bmatrix},\quad
\det\mathcal O=1\ne0\ \Rightarrow\ \boxed{\text{observable}}.$$
Both matrices are full rank, so the state-space realization is a minimal one and every pole of
$G_{proc}(s)$ is reachable and visible at the output.
Part 3a) — stability range. Clearing denominators,
$$Q(s)=(s^2+s+1)(s+1)+K_p(s^2+2s+2)=s^3+(2+K_p)s^2+(2+2K_p)s+(1+2K_p)=0.$$
Routh's $s^1$ row is $b_1=\dfrac{(2+K_p)(2+2K_p)-(1+2K_p)}{2+K_p}=\dfrac{2K_p^2+4K_p+3}{2+K_p}$;
the numerator $2K_p^2+4K_p+3$ has a negative discriminant ($16-24<0$), so it is positive for
EVERY real $K_p$. Stability then needs only $2+K_p>0$ and $1+2K_p>0$, i.e.
$\boxed{K_p>-\tfrac12}$ — for any physically realizable positive proportional gain, the
closed loop is unconditionally stable.
Part 3b) — steady-state error $\le5\%$. The summing junction's own
signal is $E(s)=R(s)-H(s)Y(s)$, so $E/R=1/(1+K_pG_{proc}H)$ and
$e_{ss(step)}=1/(1+K_pG_{proc}(0)H(0))$. With $G_{proc}(0)=2$ and $H(0)=1$, the loop DC gain is
$2K_p$, so
$$e_{ss(step)}=\frac{1}{1+2K_p}\le0.05\ \Longrightarrow\ 1+2K_p\ge20\ \Longrightarrow\ \boxed{K_p\ge9.5.}$$
Since Part 3a already showed the loop is stable for every $K_p>-0.5$, this error bound is the
BINDING constraint: $\boxed{K_p\ge9.5}$ simultaneously satisfies both requirements.
