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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? 

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### 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.

Consider the system (s^2 + p1s + p0)[y(t)] = kp u(t) with y(0) = ẏ(0) = 0 and kp = z0 ≠ 0, where y(t), ẏ(t) are available for measurement. Let x(t) = (y(t), ẏ(t))^T be the system state variable. (i) Find (A, b) such that ẋ(t) = Ax(t) + bu(t); (ii) choose a model system ẋm(t) = Amxm(t) + bmr(t) such that the equilibrium state xm = 0 of ẋm(t) = Amxm(t) is exponentially stable and Am and bm are independent of the system parameters kp, p1, and p0; but the matching equations Am = A + bk1*^T, bm = bk2* are satisfied by some k1* ∈ R^2 and k2* ∈ R which are dependent on kp, p1, and p0; and (iii) show that if kp, p1, and p0 are known, then the control law u(t) = k1*^T x(t) + k2* r(t) results in ė = Ame for any bounded r(t), where e(t) = x(t) − xm(t), so that lim t→∞ e(t) = 0.

Explore the solution to finding state-space matrices for a given control system and designing a model-following control law to achieve exponential stability. This advanced problem involves key concepts in control systems, including state-space representation, stability analysis, and error dynamics. Suitable for graduate-level students.

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