We want to use an observer in a textile machine to

Chapter 12, Problem 40

(choose chapter or problem)

The use of feedback control to vary the pitch angle in the blades of a variable speed wind turbine allows power generation optimization under variable wind conditions (Liu, 2008). At a specific operating point, it is possible to linearize turbine models. For example, the model of a three-blade turbine with a 15 m radius working in 12 m/s wind-speed and generating 220 V can be expressed as

\(\dot{\mathbf{x}}=\left[\begin{array}{ccccc} -5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ -10.5229 & -1066.67 & -3.38028 & 23.5107 & 0 \\ 0 & 993.804 & 3.125 & -23.5107 & 0 \\ 0 & 0 & 0 & 10 & -10 \end{array}\right] \mathbf{x}\)

\(+\left[\begin{array}{l} 5 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] u\)

\(y=\left[\begin{array}{lllll} 0 & 0 & 0 & 1.223 x 10^5 & 0 \end{array}\right] x\)

where the state variable vector is given by

\(\mathbf{x}=\left[\begin{array}{lllll} \beta & \xi & \dot{\xi} & \omega_g & \omega_{g m} \end{array}\right]\)

Here, \(\beta\) =  pitch angle of the wind turbine blades, \(\xi\) =  relative angle of the secondary shaft, \(\omega_g\) =  generator speed, and \(\omega_{g m}\) =  generator measurement speed. The system input is u, the pitch angle reference, and the output is y, the active power generated.

a. Find a state feedback vector gain such that the system responds with a 10% overshoot and a settling time of 2 seconds for a step input.

b. Use MATLAB to verify the operation of the system under state feedback.