**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. *u*1 and *u*2 are the thrust forces (control inputs). The thrust is generated by a powerful fan and is vectored into two forces *u*1 and *u*2. *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*; ˜*u*_{2}(*t*) = 0.

The linearized model should be time invariant. The state *z *= [*θ,x,*˙ *y,*˙ *θ*^{˙}]* ^{T}*,

*u*= [

*u*

_{1}

*,u*

_{2}]

^{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 = *x*2 − *x*1*x*22 *x*˙2 = −*x*31

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*x*_{1 }and*x*_{2}.**(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*are the flows (assumed constant) of cold water and hot water,

_{H }*T*and

_{H }*T*represent the temperatures in the hot and cold compartments, respectively,

_{C }*T*and

_{Hi }*T*denote the temperature of the hot and cold inflow, respectively, and

_{Ci }*V*and

_{H }*V*are the volumes of hot and cold water. The temperatures in both compartments evolve according to:

_{C }) (1)

) (2)

Let the inputs to the system be *u*_{1 }= *T _{Ci}*,

*u*

_{2 }=

*T*, the outputs are

_{Hi}*y*1 =

*T*and

_{C }*y*2 =

*T*, and assume that

_{H}*f*=

_{C }*f*= 0

_{H }*.*1(

*m*

^{3}

*/min*),

*β*= 0

*.*2(

*m*

^{3}

*/min*) and

*V*=

_{H }*V*= 1(

_{C }*m*

^{3}).

- Write the state space and output equations for this system.
**(5 points)** - In the absence of any input, determine
*y*_{1}(*t*) and*y*_{2}(*t*).**(5 points)** - Is the system BIBO stable? Show why or why not.
**(5 points)**

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