To analyze this problem, we’ll break down each part systematically:

Part (i): Finding $(A, b)$

The differential equation given is:

$(s^2 + p_1 s + p_0) [y(t)] = k_p u(t)$

This can be rewritten as a state-space form by defining the state vector $x(t) = \begin{pmatrix} y(t) \\ \dot{y}(t) \end{pmatrix}$. To express this in the form:

$\dot{x}(t) = Ax(t) + bu(t)$

we'll identify the matrices $A$ and $b$ from the structure of the differential equation. Generally, this involves arranging the equation to isolate the second derivative term and setting up the system in a canonical form.

Part (ii): Choosing a Model System

For the model system

$\dot{x}_m(t) = A_m x_m(t) + b_m r(t)$

where $A_m$ and $b_m$ are chosen to make the system exponentially stable. Stability implies that the eigenvalues of $A_m$ have negative real parts. You would then ensure the matching conditions $A_m = A + b k_1^{*T}$ and $b_m = b k_2^*$ are satisfied for some gains $k_1^*$ and $k_2^*$.

Part (iii): Control Law and Stability Analysis

The control law

$u(t) = k_1^{*T} x(t) + k_2^{*} r(t)$

should make $e(t) = x(t) - x_m(t)$ converge to zero as $t \to \infty$, indicating the system follows the reference model. You’ll prove this by analyzing the error dynamics $e(t)$ and showing that it converges to zero under the given conditions.

Would you like detailed calculations for each part, including how to derive $A$, $b$, $A_m$, and $b_m$ specifically?

Related Questions

1. How would you determine the stability of the system using the eigenvalues of $A$ or $A_m$?
2. What are the general steps to derive a state-space representation from a second-order differential equation?
3. How do you choose the gains $k_1^*$ and $k_2^*$ to satisfy the matching conditions?
4. Why is it necessary for $A_m$ to be exponentially stable in a model-following control system?
5. What conditions are required for the error $e(t)$ to asymptotically approach zero?

Tip

Always check the controllability of the system before proceeding with control design to ensure that all states can be controlled as intended.