Analyse Rocket Altitude Control using the Dead-Zone Non-Linearity

Consider the folowing block diagram :-

Figure 1.1: Rocket Control

The state space variables are :-

$$x_1 = \theta$$ (1.1)

$$x_2 = \dot {\theta}$$ (1.2)

For the Controller , the control action can be represented as :-

$$\delta = - x_1 - kx_2$$ (1.3)

$$\begin{array}{ccccccc} \tau &=& T && && \delta > \delta_1 \\... ...leq \delta_1 \\ &=& -T && && \delta <-\delta_1 \end{array}$$ (1.4)

where $\tau$ is the torque applied. From the above equations, the equations of the switching lines are :-

$$\begin{array}{ccccccc} \delta &=& +-\delta_1 \\ -x_1 - kx_2 &=& \delta_1 \\ -x_1 - kx_2 &=& -\delta_1 \end{array}$$ (1.5)

The Trajectories are as shown in the figure below :- Trajectory1 As compared to the system containing controller without the dead-zone, this system will have lesser fuel consumption but at the same time, it may not give the shortest time required to reach the equillibrium point. Another way of controlling the rocket is by using a controller having memory where the controller non-linearity is positive Hysteresis. (Assignment)