Let the plant in the drive system with an elastically

Chapter 12, Problem 42

(choose chapter or problem)

We want to use an observer in a textile machine to estimate the state variables. The 2-input, 1-output system's model is \(\dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B u} ; y=\mathbf{C x}\), where (Cardona, 2010)

\(\mathbf{A}=\left[\begin{array}{ccc} 0 & 1 & 0 \\ -52.6532 & -4.9353 & -2768.1557 \\ -0.001213 & 0 & -0.06106 \end{array}\right]\)

\(\mathbf{B}=\left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \\ 0.001613 & -0.001812 \end{array}\right]\)

\(\mathbf{C}=\left[\begin{array}{lll} 1 & 0 & 0 \end{array}\right]\)

a. Find the system's observability matrix \(\mathbf{O}_{\mathbf{M z}}\) and show that the system is observable.

b. Find the original system's characteristic equation and use it to find an observable canonical representation of the system.

c. Find the observable canonical system's observability matrix \(\mathbf{O}_{\mathrm{Mx}}\) and then find the transformation matrix \(\mathbf{P}=\mathbf{O}_{\mathbf{M z}}^{-1} \mathbf{O}_{\mathbf{M x}}\).

d. Use the observable canonical representation to find an observer gain matrix \(\mathbf{L}_{\mathbf{x}}=\left[\begin{array}{llll}l_{1 x} & l_{2 x} & l_{3 x} & l_{4 x}\end{array}\right]^T\) so that the observer characteristic polynomial is \(D(s)=s^3+30 s^2+316 s+1160\).

e. Find the corresponding observer gain matrix \(\mathbf{L}_{\mathbf{z}}=\mathbf{P L}_{\mathbf{x}}\).