A rigid body in planar translation is actuated by two independent thrusters

A rigid body in planar translation is actuated by two independent thrusters. With \(t\) in seconds and state \(\mathbf{x}=(p,v,w)^{\mathrm{T}}\), where \(p\) is position in metres, \(v\) is velocity in metres per second, and \(w\) is a thruster-filter state in metres per second, the model is \(\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}\) with \[ A=\begin{pmatrix}0&1.00&0\\ 0&0&1.00\\ 0&0&-4.00\end{pmatrix},\qquad B=\begin{pmatrix}1.00&0\\ 0&0\\ 0&1.00\end{pmatrix}. \] The first column of \(A\) is dimensionless in the \((1,2)\) sense of mapping \(v\) onto \(\dot p\); the entry \(-4.00\) has units \(\mathrm{s}^{-1}\); the second column of \(B\) has units \(\mathrm{s}^{-1}\) if the second input is taken in metres per second. Take both inputs in a consistent set of units so that the displayed numerical matrices apply as written. Let \(\mathbf{u}=(u_1,u_2)^{\mathrm{T}}\). The controllability matrix is the \(3\times 6\) block \(\mathcal{C}(A,B)=[B\ AB\ A^2B]\). (a) Compute \(AB\) and \(A^2B\). Form \(\mathcal{C}(A,B)\) and determine \(\operatorname{rank}\mathcal{C}(A,B)\). Decide complete controllability. (b) Apply the PBH test at each eigenvalue of \(A\) (compute \(\det(sI-A)\) first). Confirm the conclusion of (a). (c) Now disable the first thruster by replacing \(B\) with its second column \(B'=(0,0,1.00)^{\mathrm{T}}\) only. Recompute \(\operatorname{rank}\mathcal{C}(A,B')\) and the PBH ranks. Decide controllability of the single-input pair \((A,B')\). Identify a one-dimensional subspace of states that cannot be reached from the origin using only \(u_2\).

** Compute the two-input Kalman matrix first; then repeat with only the second column of \(B\), which feeds the last coordinate of the displayed chain.