QUESTION 3 (8 marks) Given an inverted pendulum, --motor Its model is approximated by the following second order ordinary differential equation (ODE), 24(1) di a-sin ( (1))-b. d (1) -c-u(t) dt for which the parameters are a=2, b=1 and c=1. 3.1. Without linearizing the ODE, obtain the model in state space representation, according to the general dx(t) form = f(x(t), u(t)). Propose an adequate state vector, X (I), choosing its components dt from this set of candidate variables: (t), do(1) d(t) d(1) (1 mark) 3.2. Apply Taylor's approximation at X =[0,0] to the model obtained in (3.1), to obtain a linear model of dx (1) the type: =A-X(t)+B-u(t). dearly indicate the values obtained for A and B. dt 3.3. For this inverted pendulum the eigenvalues of the matrixA are the following real numbers: {-1.618, +0.618}. (1 mark) 3.3. a) Is this system stable? Base your answer on the information provided by those eigenvalues. (1 mark) 3.3. b) Give a brief interpretation (even intuitive) about the stability of the physical system. (1 mark) Note: you do not need to obtain the eigenvalues; those are provided. You need to base your answer assuming that those provided eigenvalues are correct. 3.4. Controllability 3.4. a) Show that the controllability matrix, for tha linearized inverted pendulum is [0 +1] +1-1 (1 mark) 3.4. b) What is the condition about the Controllability matrix C, for considering the system to be controllable? (1 mark) 3 Please see over 3.4. c) Based on the controllability matrix C given in (3.4.a), and by applying a basic calculation, infer if that system (whose controllability matrix is that matrix) is controllable. 3.4 d) Can a system which is stable be not controllable? (1 mark) (1 mark)