[SOLVED] ECE4530J Homework 2

Problem 1

Consider the trajectory tracking problem with acceleration being the control input:

[𝑥[𝑡 + 1]] = [1 𝛿] [𝑥[𝑡]] + [0] 𝑢[𝑡]. 𝑣[𝑡 + 1] 0 1 𝑣[𝑡] 𝛿

Let 𝑥̅[𝑡] = 𝑣̅𝑡 be the reference trajectory to track.

1. a)  Reformulate the state-space model with the position and speed errors being the state.
2. b)  Using the reformulated model, find a linear controller that stabilizes the system.
3. c)  Find a linear controller that destabilizes the system.

Problem 2

Consider a one-dimensional non-linear system

𝑥̇ = 𝑎1𝑥 + 𝑎2𝑥2 + 𝑏𝑢 + 𝑐.

1. a)  Use Taylor expansion to linearize the RHS of the dynamical equation in the neighborhood of 𝑥 = 0.
2. b)  For the linearized system, design a linear controller 𝜇(𝑥) that stabilizes the linearized system. Hint: a linear system 𝑥̇ = 𝑎̃𝑥 is stable if and only if 𝑅𝑒(𝑎̃) < 0.
3. c)  For the continuous-time system, design a controller 𝜇(𝑥) such that, with 𝑢 = 𝜇(𝑥), the RHS of the dynamical equation is linear. Hint: do not confuse this part with part a).

Problem 3

Consider a two-vehicle platoon with states [𝑥1[𝑡]] , [𝑥2[𝑡]]. Vehicle 1 tracks a pre-specified 𝑣1[𝑡] 𝑣2[𝑡]

trajectory 𝑥̅[𝑡] = 𝑣̅𝑡, 𝑡 = 0,1,2, … Vehicle 2 follows vehicle 1 to keep a spacing of 𝑑 to vehicle

1. The inputs are the engine torques 𝜏1[𝑡] and 𝜏2[𝑡]. For 𝑖 = 1,2, the propelling force is given

by 𝛼𝜏 , while the resistant force is given by 𝛽𝑣2. The vehicle masses are 𝑚 , 𝑚 , respectively. 𝑖𝑖12

1. a)  Formulate Newton’s second law for both vehicles.
2. b)  Formulate the state-space model for both vehicles using absolute position and speed as the

   state.

3. c)  Reformulate the model with the tracking/following errors being the state.
4. d)  Construct a trajectory-tracking policy for vehicle 1, i.e., a function that maps tracking errors

   𝑒1, 𝑦1 to 𝜏1. Explain why the policy will work. No mathematical proof needed.

5. e)  [Bonus] Construct a vehicle-following algorithm for vehicle 2, i.e., a function that maps

   𝑥1, 𝑣1, 𝑥2, 𝑣2 to 𝜏2. Explain why the policy will work. No mathematical proof needed.