(Solved): 1. An observer feedback controller is applied to a plant that incorporates a disturbance input and ...

1. An observer feedback controller is applied to a plant that incorporates a disturbance input and measurement noise: $\begin{array}{l}\stackrel{?}{x}(t)=Ax(t)+B_{u}u(t)+B_{w}w(t)\\ y(t)=Cx(t)+v(t)\end{array}$ Generate the state equation for the closed-loop system, using the following state: $\left[\begin{array}{l}x(t)\\ \hat{x}(t)\end{array}\right]$. Note that this state equation should have both the disturbance, $w(t)$, and the measurement noise, $v(t)$, as inputs. 2. The elevation control system for an antenna is designed to drive the elevation error angle to zero. The antenna is described by the following differential equation: $J\ddot{?}(t)+G\stackrel{?}{?}(t)=u(t)+w(t),$ where $?$ is the elevation angle error in radians, $J=1000\mathrm{kg}?{\mathrm{m}}^2$ is the moment of inertial of the antenna, $G=5\mathrm{kg}?{\mathrm{m}}^2/\mathrm{s}$ is the friction, $u$ is the control torque, and $w$ is the disturbance torque. The control torque is generated from the elevation error and the elevation error rate: $u(t)=?2000?(t)?800\stackrel{?}{?}(t)$ 1. Generate the state equations of this system with the disturbance torque as the input. Simulate the system's response to both impulse and step disturbances. 2. What are the poles of the system? 3. What are the natural modes of the system (without the controller)?