Homework 10 - Due: 04/16

Problem 1

Find the solution of the vector equation $$\left[\begin{array}{ccc}3& 2& 1\\ 5& 2& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]$$

which minimizes $x_1^2+x_2^2+x_3^2$.

Problem 2

In class, we relied on a theorem from Luenberger’s OVSM to establish that the control law $$u(t)=B^T(t){\varphi }^T(t_0,t)\eta ,\eta =W^{-1}(t_0,t_1)(\varphi (t_0,t_1)x_1-x_0)$$ (appropriately specialized for the LTI case) sending $x_0$ at time $t_0$ to $x_1$ at time $t_1$ minimizes the energy: $$J(u)=\underset{t_0}{\overset{t_1}{\int }}{u}^2(t)dt$$ The above functional can also be interpreted as the squared $L_2$ norm of a function $u(t)$. Use the result of Problem 4 from Homework 4 to provide a proof establishing that the same control law minimizes the $L_2$ norm for LTV systems as well: $$\dot{x}=A(t)x+B(t)u$$

Problem 3

For a system

$$\dot{x}=A(t)x(t)$$

we define the adjoint dynamics to be

$$\dot{p}=-A^T(t)p(t)$$

1. Show that this definition preserves (i.e. holds constant) the inner product between solutions $x(t)$ and $p(t)$. Here the inner product is the standard inner product in ${\mathbb{R}}^n$.

2. Derive the adjoint dynamics $\dot{P}$ for the matrix differential equation below which preserves the inner product $⟨M,N⟩=\mathrm{tr}(M^{T}N)=\mathrm{tr}(N^{T}M)$ between matrices $M,N\in {\mathbb{R}}^{n\times m}$.

$$\dot{X}=A_1(t)X(t)+X(t)A_2(t)$$

Problem 4

Consider a dynamical system $$\dot{x}=Ax+bu,y=cx+u$$ Show that the sytem $$\dot{x}=(A-bc)x+by,u=-cx+y$$ is an “inverse” to the original system by composing their transfer functions.

Hint: Use the Woodbury Matrix Identity.