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Problem 4: (25) A system is given in the state-space form as below, $\dot{x} = \begin{bmatrix} 0 & 1 \ -2 & -1 \end{bmatrix} x + \begin{bmatrix} 0 \ 1 \end{bmatrix} u$ y = $\begin{bmatrix} 1 & 0 \end{bmatrix} x$ (a) Determine whether the given system is controllable and observable. (10) (b) Find the gains $k_1$ and $k_2$ to get 0.5 of the damping ratio and 10 [rad/s] of the undamped natural frequency of the system if the state-feedback control input, u(t) = -[$k_1$ $k_1$]x(t) + r(t), is applied to this system. (15)

          Problem 4: (25)
A system is given in the state-space form as below,
$\dot{x} = \begin{bmatrix} 0 & 1 \ -2 & -1 \end{bmatrix} x + \begin{bmatrix} 0 \ 1 \end{bmatrix} u$
y = $\begin{bmatrix} 1 & 0 \end{bmatrix} x$
(a) Determine whether the given system is controllable and observable. (10)
(b) Find the gains $k_1$ and $k_2$ to get 0.5 of the damping ratio and 10 [rad/s] of the
undamped natural frequency of the system if the state-feedback control input,
u(t) = -[$k_1$ $k_1$]x(t) + r(t), is applied to this system. (15)

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Problem 4: (25)
A system is given in the state-space form as below,
ẋ =
< b m a t r i x >
x +
< b m a t r i x >
u
y = < b m a t r i x >
x
(a) Determine whether the given system is controllable and observable. (10)
(b) Find the gains k1 and k2 to get 0.5 of the damping ratio and 10 [rad/s] of the
undamped natural frequency of the system if the state-feedback control input,
u(t) = -[k1 k1]x(t) + r(t), is applied to this system. (15)

Added by Crystal E.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sufiyan Alam

Step 1

The controllability matrix is given by: C = [B AB A^2B ... A^(n-1)B] where A is the system matrix and B is the input matrix. In this case, the system matrix A is not given, so we cannot determine the controllability of the system. The observability matrix is Show more…

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Problem 4: (25) A system is given in the state-space form as below: y = [1 0]x (a) Determine whether the given system is controllable and observable. (10) (b) Find the gains k1 and k2 to get 0.5 of the damping ratio and 10 [rad/s] of the undamped natural frequency of the system if the state-feedback control input, u(t) = -[k1 k2]x(t) + r(t), is applied to this system. (15)

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