1 Control and Maintenance of the Vertical Position of an Inverted Pendulum Group Members: Bashar Fawal Robert Kiwan Arman Matti Shadi Nassrallah Dean Santana

2 Physical System Diagram Parameter Notation: u = applied force acting upon the cart l = length of the rod M = mass of the mobile cart m = mass of the set point on the other end of the rod θ = deviation angle from the vertical reference mg = gravitational force acting upon the set point

3 Problem Statement To maintain the upright vertical position of a rod in a basic mechanical system relative to a mobile cart at all times. A critical problem that arises from the start is the fact that we’re dealing with a nonlinear system. Linearizing a state-space model of this system becomes a top priority in the control of the variables involved.

4 Plan of Attack The simulation process will be carried out in terms of three cases: Case 1: Comparison between the nonlinear and linear systems using classical feedback control. Case 2: Implementation of a state feedback control scheme which allows for a better system response and control at the desired positions. Case 3: Using state feedback control in light of the presence of an external disturbance.

5 Case 1 Observation: The effective differences arising from plotting the open loop step response of both the nonlinear and linear systems becomes clear when monitoring the behavior of the graph at the end of a one second interval. Classical Feedback Control Schemes: The use of feedback to correct the rod position and the cart position respectively produced unsatisfactory results as it relates to stability. The use of a dual feedback setup which accounts for the error between the actual and desired rod and cart positions produced better results.

6 Open Loop Step Response

7 Classical Feedback Control

8 Further Case Studies Case 2: Due to the futility in obtaining desired stability using classical control schemes, a state feedback control method and a state observer will be applied. Similar to chapters 11 and 12, certain conditions such as overshoot tolerance, settling time, and rise time will be examined by designing accordingly the values for ξ, ω n, K, and properly assigning the pole placements for stability purposes.

9 Further Case Studies Case 3: By maintaining the state feedback presence from Case 2, wind gusts compound the external force acting upon the system. Using the same approach as the previous case, the wind affect can be limited to a negligible force by assuring a relative degree of stability for both the cart and rod positions as time increases.

10 Analysis and Expectations The classical approach produced mediocre results characterized by heavy oscillation and instability. The use of state feedback resulted in a more stable system which was able to withstand deviations from the target points and revert to a more controlled setting. Disturbances of a reasonable nature can be effectively handled and minimized through the use of state feedback control methods resulting in the vertical position of the pendulum being maintainable.