Higher Order System

In the systems we have considered so far in 6.01, the controller has commanded a velocity based on an input. In the case of our lab robots, the motors have so much torque that they can accelerate to a new commanded velocity faster than the time between steps of the control system.

This is not always the case in life; in this problem, we will consider a controller which has an output not in the form of a velocity, but a force and investigate how to account for these cases where mass and acceleration cannot be ignored or approximated as instantaneous.

Below is the system diagram of a hover-rocket (a rocket that hovers at certain height). The hover rocket possesses a rocket engine which outputs a force depending on the input signal given to it.

In this system, X_i is the desired vertical location of the rocket (in meters), and X_o is the rocket's actual location. The variable M represents the mass of the hover-rocket (in kg), and the variable T represents the length of a timestep (in seconds). The controller gain is k.

Part 1: Unit Analysis

What are the units of the quantity represented by the signal at the labeled points in the diagram?

What is represented by the signal labeled A? -- distance velocity acceleration pure (unitless) force time mass

What is represented by the signal labeled B? -- distance velocity acceleration pure (unitless) force time mass

What is represented by the signal labeled C? -- distance velocity acceleration pure (unitless) force time mass

What is represented by the signal labeled D? -- distance velocity acceleration pure (unitless) force time mass

What is represented by the signal labeled E? -- distance velocity acceleration pure (unitless) force time mass earth fire wind water heart

What are the units of the controller? -- distance force per mass acceleration per distance force per distance pure (unitless) force per time

Part 2: System Functional

Derive the system functional for the hover rocket and enter the coefficients for the top and bottom polynomials in terms of k, M, and T as python lists, starting with the zeroth-order coefficients.

System Functional:
Numerator coeffs (in {\cal R}):
Denominator coeffs (in {\cal R}):