a

b

c

d

The objective of LQR is to design an optimal control signal that minimizes the following loss function

For solving the optimization using dynamic programming, we introduce the following notation,

$V_k=mi{n}_{u(k)dots..\mathrm{.}1(N-1)}{\displaystyle \underset{i=k}{\overset{N-1}{}}}(x^T(i)Q_{1}x(i)+u^T(i)Q_{2}u(i)+2x^T(i)Q_{12}u(i))+x^T(N)Q_{0}x(N)$

Here $v_k$ is the minimum loss from $k$ to $N-$th iteration. Then state whether the following is true or false

minJ$=V_0$

a$.$ True

b$.$ False

a

b

A first$-$order continuous system is represented by the following differential equation:

$x^{}(t)=x(t)+u(t)$

We want to design a feedback controller such that $u(t)=-Kx(t)$ and desired equilibrium condition is:

$x(t)=0$ as $t.$ What should be the value of $K$ for the above system to be stable?

a$.K1$

b$.K1$

c$.K<mn\; id="EditorElement\_289655">1</mn>$

d$.K>1$

a

b

c

d

Consider the same system and stability criteria given in Question $3.$ The performance of the system is

measured by the following loss function:

$J={\displaystyle \underset{0}{\overset{}{}}}[x^2(t)+{}^2(t)]dt$

Given $m>1,$ compute $K$ such that $J$ is minimum. $[$Hint: find out value of $K$ for which $\frac{dJ}{dK}=0].$

a$.K=1+\frac{\sqrt[\phantom{2}]{m^2+m}}{m}$

b$.K=1+\frac{\sqrt[\phantom{2}]{m^2-m}}{m}$

c$.K=1-\frac{\sqrt[\phantom{2}]{m^2+m}}{m}$

d$.K=1-\frac{\sqrt[\phantom{2}]{m^2-m}}{m}$