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NOTE: Quiz only covers LEC 1,2,3

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1 State variables analysis

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A Digital Control System

State space design (modern Design approach)

1.1 Review of continuous-time state space representations

$$ \dot{x}(t) = A x(t) + B u(t) \ state \ equation \\y(t) = C x(t) + D u(t) \ output \ equation \\

x(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{bmatrix}, \quad u(t) = \begin{bmatrix} u_1(t) \\ u_2(t) \\ \vdots \\ u_r(t) \end{bmatrix}, \quad y(t) = \begin{bmatrix} y_1(t) \\ y_2(t) \\ \vdots \\ y_m(t) \end{bmatrix}

$$

For single-input, single-output systems, r = 1 and m = 1.

1.2 Transfer function from the given state space model

$$ s X(s) = A X(s) + B U(s) \\ \\ X(s) = [s I - A]^{-1} B U(s) \\ Y(s) = C X(s) + D U(s) $$

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The denominator of the transfer function is det[sI-A].

Hence, the poles of the system are identical to the Eigenvalues of the matrix A.