Linear System Theory Examples Paper 6 Controllability, observability and control design Due date: December 19, 2014 Exercise 1. (Controllabilty and observability, 40%) Consider the linear time invariant system w 1 0 x(t) ˙ = x(t) + u(t) 0 0 1 (1.1) y(t) = [1 0] x(t) w 1 0 where w ∈ [1, 2] is a constant parameter. Define A(w) = ,B = and C = [1 0]. 0 0 1 1. Determine for which values of w, the system in (1.1) is controllable and observable. 2. Consider the linear observer xˆ˙ (t) = A(w)ˆ x(t) + Bu(t) + L(y(t) − yˆ(t)) yˆ(t) = C xˆ(t). Find the set of all observer gain matrices L ∈ R2×1 such that, for all possible w ∈ [1, 2], the observation error e(t) = x(t) − xˆ(t) tends to zero asymptotically. Exercise 2. (Controller design, 60%) Consider the following nonlinear system to model the dynamics of an inverted pendulum: ¨ = sin(θ(t)) + u(t). θ(t) 1. Provide a nonlinear state-space model of the system, choosing the angular position ˙ as the state variables. θ(t) and the angular speed ω(t) = θ(t) 2. Show that θ = ω = 0 is an equilibrium point; linearize the system around it. Show that the resulting linear system is unstable. What does this imply about the nonlinear system? 3. Show that the linearized system is controllable. Design a linear, time invariant state feedback u(t) = F x(t) that stabilizes the (linearized) system placing all the closed-loop eigenvalues at −1. 1 4. Use Matlab to simulate the controlled closed-loop system. Estimate numerically the set of initial conditions such that the state of the closed-loop system converges to the origin asymptotically. Provide some plots to justify your claim. Bonus Exercise, 10% Consider the linear time invariant system x(t) ˙ = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t), where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ Rp . Consider the state feedback control u(t) = −Kx(t) + v(t). Show that if the original system is controllable, then the closed-loop system (with state variable x and control input variable v) is also controllable. 2