Exercise 1. Asymptotic stability and Lyapunov stability.

For each of the systems given below, determine whether it is Lyapunov stable, whether it is asymptotic stable.

(a) (5 points)

(b) (5 points)

Exercise 2. Stabilizability

Decompose the state equation

to a controllable form. Is the reduced state equation observable, stabilizable, detectable?

Exercise 3. Stability

Figure 1: The VTOL. aircraft

The following is a planar model of a Vertical Take-off and Landing (VTOL) aircraft such as Lock-heed’s F35 Joint Strike fighter around hover (cf. Figure 1):

where x, y are the position of the center of mass of the aircraft in the vertical plane and θ is the roll angle of the aircraft. u1 and u2 are the thrust forces (control inputs). The thrust is generated by a powerful fan and is vectored into two forces u1 and u2. J is the moment of inertia, and is a small coupling constant. Determine the stability of the linearized model around the equilibrium solution

x˜(t),y˜(t),θ^˜(t) = 0,u˜₁(t) = mg; ˜u₂(t) = 0.

The linearized model should be time invariant. The state z = [θ,x,˙ y,˙ θ^˙]^T, u = [u₁,u₂]^T Exercise 4. Lyapunov’s direct method (10 points)

An LTI system is described by the equations

Use Lyapunov’s direct method to determine the range of variable a for which the system is asymptotically stable. Consider the Lyapunov function,

Exercise 5. Stability of Non-Linear Systems (20 points) Consider the following system:

x˙1 = x2 − x1x22 x˙2 = −x31

Is the system stable:

• Based on Lyapunov’s Indirect method? (5 points)

[Hint: is the approximated linearized system stable?]

• Based on Lyapunov’s Direct method? (5 points) Consider the Lyapunov function:
• Plot the Phase Portrait plot of the original system and linearized system in a. (5 points). Submit the code to Gradescope.
• Generate a 3D plot showing the variation of V^˙ with respect to x₁ and x₂. (5 points) [Hint: Use Axes3D python library]. Submit the code to Gradescope.

Note: For (c) and (d), include the code along with the plot in the pdf to be submitted. No need to submit .py file.

Exercise 6. BIBO Stability (10 points)

For each of the systems given below, determine whether it is BIBO stable.

(a) (5 points)

(b) (5 points)

Exercise 7. BIBO Stability (Manual-grading: 15 points)

Figure 2: A simple heat exchanger

Consider a simplified model for a heat exchanger shown in Figure 2, in which f_C and f_H are the flows (assumed constant) of cold water and hot water, T_H and T_C represent the temperatures in the hot and cold compartments, respectively, T_Hi and T_Ci denote the temperature of the hot and cold inflow, respectively, and V_H and V_C are the volumes of hot and cold water. The temperatures in both compartments evolve according to:

) (1)

) (2)

Let the inputs to the system be u₁ = T_Ci, u₂ = T_Hi, the outputs are y1 = T_C and y2 = T_H, and assume that f_C = f_H = 0.1(m³/min), β = 0.2(m³/min) and V_H = V_C = 1(m³).

1. Write the state space and output equations for this system. (5 points)
2. In the absence of any input, determine y₁(t) and y₂(t). (5 points)
3. Is the system BIBO stable? Show why or why not. (5 points)