Homework 3 - Due: 02/08

Problem 1

Consider the following matrices:

$$A_1=\left[\begin{array}{cc}1/2& -1/2\\ 1/2& -1/2\end{array}\right],A_2=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],A_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& -2& 0\\ -3& 0& -2\end{array}\right].$$

1. Compute $e^{A_{i}t}$ for $i=1,2,3$. Note that $A_2$ is a special case of a matrix whose exponential was computed in class.
2. For $i=1,2,3$, write down the solution of $\dot{x}=A_{i}x$ for a general initial condition.
3. In each case, determine whether the solutions of $\dot{x}=A_{i}x$ decay to 0, stay bounded, or go to $\mathrm{\infty }$ (for various choices of initial conditions).
4. Try to state the general rule which can be used to determine, by looking at the eigenstructure of $A$, whether the solutions of $\dot{x}=Ax$ decay to 0, stay bounded, or go to $\mathrm{\infty }$.

Problem 2

The pictures in this demo page show possible trajectories of a linear system $\dot{x}=Ax$ in the plane, when the eigenvalues ${\lambda }_1$ and ${\lambda }_2$ of $A$ are in one of the following six configurations:

#	Configuration	#	Configuration
(a)	${\lambda }_1<{\lambda }_2<0$	(b)	${\lambda }_1<0<{\lambda }_2$
(c)	$0<{\lambda }_1<{\lambda }_2$	(d)	${\lambda }_1,{\lambda }_2=a\pm ib,a<0$
(e)	${\lambda }_1,{\lambda }_2=a\pm ib,a=0$	(f)	${\lambda }_1,{\lambda }_2=a,a>0$

Match each picture with the corresponding eigenvalue distribution. Justify your answers using your knowledge of the solution of $\dot{x}=Ax$. Plotting in MATLAB or use of the demo applet cannot be used as a justification.

Animations courtesy of Prof. R.B. Israel

Problem 3

This exercise illustrates the phenomeom known as resonance. Consider the system

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]& =\left[\begin{array}{cc}0& \omega \\ -\omega & 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u,y=x_2\end{array}$$

where $\omega >0$, $u$ is the control input, and $y$ is the output. Let the input be $u(t)=\cos \nu t$, $\nu >0$. Using the variation-of-constants formula, compute the response $y(t)$ to this input from the zero initial condition $x(0)=0$, considering separately two cases: $\nu \ne \omega$ and $\nu =\omega$. In each case, determine whether this response is decaying to 0, bounded, or unbounded.

Problem 4

Compute the state transition matrix $\mathrm{\Phi }(t,t_0)$ of $\dot{x}=A(t)x$ with $$A(t)=\left[\begin{array}{cc}-1+\cos t& 0\\ 0& -2+\cos t\end{array}\right]$$

Problem 5

Consider the LTV system $\dot{x}=A(t)x$, where $A(t)$ is periodic with period $T$, i.e., $A(t+T)=A(t)$. Let $\mathrm{\Phi }(t,t_0)$ denote the corresponding state transition matrix. The goal of this exercise is to show that we can simplify the system and characterize its transition matrix with the help of a suitable (time-varying) coordinate transformation.

Being a nonsingular matrix, $\mathrm{\Phi }(T,0)$ can be written as an exponential: $\mathrm{\Phi }(T,0)=e^{RT}$, where $R$ is some (possibly complex-valued) matrix. Define also:

$$P(t):=\mathrm{\Phi }(t,0)e^{-Rt}$$

1. Using the properties of state transition matrices from class, show that $\mathrm{\Phi }(t,t_0)=P(t)e^{R(t-t_0)}{P}^{-1}(t_0).$
2. Deduce from a) that $\overline{x}(t):=P^{-1}(t)x(t)$ satisfies the time-invariant differential equation $\dot{\overline{x}}=R\overline{x}$.
3. Prove that $P(t)$ is periodic with period $T$.
4. Consider the LTV system from Problem 4 . Find new coordinates $\overline{x}(t)$ in which, according to part b), this system should become time-invariant. Confirm this fact by differentiating $\overline{x}(t)$.

Problem 6

Prove the variation-of-constants formula for linear time-varying control systems stated in class: $$x(t)=\varphi (t,t_0)x_0+\underset{t_0}{\overset{t}{\int }}\varphi (t,s)B(s)u(s)ds$$ satisfies the LTV controlled systems’ differential equation.

Hint: Differentiate both sides. See also Class Notes, end of Section 3.7.