Time t is in seconds and θ is in radians

Time \(t\) is in seconds and \(\theta\) is in radians. An inverted-pendulum angle (already divided by the appropriate length scale) is modelled by \[ \ddot\theta-8.00\,\theta=5.00\,u,\qquad |u|\le 1.00, \] with \(u\) dimensionless after scaling. A saturating PD law \[ u=\mathrm{sat}_1(-3.20\,\theta-0.900\,\dot\theta) \] is implemented. State coordinates are \(x_1=\theta\) and \(x_2=\omega:=\dot\theta\). (a) In the linear strip \(|3.20\theta+0.900\omega|\le 1\), write \(\dot x=A x\) and compute the eigenvalues of \(A\). Decide whether the origin is a locally exponentially stable focus or node. (b) In the region \(u=+1\), the dynamics are affine: \(\ddot\theta-8.00\theta=5.00\). Compute the unique constant equilibrium \(\theta_+\) of this affine system, and check whether \(\theta_+\) (with \(\omega=0\)) lies in the region \(u=+1\). Repeat for \(u=-1\), obtaining \(\theta_-\). Classify each consistent saturated equilibrium (saddle, node, focus) by linearizing the affine vector field. (c) Linearize the saddle at \((\theta_+,0)\) and compute its eigendirections. The stable manifold of that saddle (a curve in the plane) is a separatrix of the full piecewise-affine system. At the saddle itself, report the slope \(d\omega/d\theta\) of the stable eigenvector. Likewise at \((\theta_-,0)\). (d) A trajectory starts at \((\theta,\omega)=(0.200\,\mathrm{rad},0)\). Compute \(u(0)\) and decide which piece is active. A second trajectory starts at \((0.700\,\mathrm{rad},0)\). Decide, using only the location of the saddles and the linear-strip boundaries (not a numerical integration), whether this second point can lie in the basin of the origin. If the test is inconclusive, say so and compute the unsaturated command at that point.

** The linear strip is a PD strip about the origin; the saturated pieces are affine inverted-pendulum dynamics with a constant forcing.