CONTROL SYSTEM’S COURSEWORK PROJECT

OVERVIEW

Josh NeighborM.Sc Mechanical EngineeringUniversity of California San Diego

COURSES WITH PROJECTS HIGHLIGHTED:• Linear Systems Theory• Linear Control Design • Optimal Estimation• Robust and Multivariable Control

LINEAR SYSTEMS THEORY

Tasks: Derivation of Linearized Models

Created state space realization using linearized equations of motion, motor dynamics, and known parameters

Model Analysis Stability analysis of open loop system Analysis of controllability and observability of system

Output feedback control design with observer Observer and controller design to stabilize MiP

Mobile Inverted Pendulum (MiP) figures from Figure 17.6 of Dr. Thomas Bewley’s Numerical Rennaisance, First

Edition, 2014

Project: Linear control in state variable form for a mobile inverted pendulum

(pictured)

LINEAR SYSTEMS THEORY

System model (solid line) with Observer (dashed line) for output feedback controlled system: (1) motor torque, (2) Wheel angle and MiP Angle with respect to normal, (3) Change in wheel angle and MiP angle.

Result:• Successfully modeled system

stabilized by output feedback controller

Notes:• The system was also modeled for a

constant disturbance torque (wind)• Controller and Observer poles

placed arbitrarily to be stable and at desired speed (wanted to see system dynamics well). They could have been placed using more robust techniques.

LINEAR SYSTEMS THEORY

System response to a constant disturbance. States include Torque, change in Theta, Theta, Phi and change in Phi. We see both the change

in Phi and Theta converge to zero while Theta and Phi converge to constants.

Result:• Successfully modeled system that

stabilized the MiP under constant disturbance.

Notes:• The system was also modeled for a

constant disturbance torque (wind)• Determined the Observability and

Controllability properties of the system in order to design functional observer and controller.

LINEAR CONTROL DESIGNProject:

Analysis and controller design for a model of the pitch dynamics of an aircraft.

Tasks:• LQR Control Design to minimize cost function• Stability Margin Analysis• Estimator Design for system with noise and subsequent LQG controller design• Loop Transfer Recovery for noisy system

LINEAR CONTROL DESIGN

Bode Plot for LQR Controller. With no noise on the system, the cost function was minimized while

offering an infinite gain margin

Bode Plot for LQG controller after Loop Transfer Recovery. Some gain margin was gained after the

losses due to the noise on the system. Increases in stability come with losses of optimization for the

objective function

OPTIMAL ESTIMATIONProject:

Modeling of a chemical reaction with a nonlinear dependence. Estimate the concentration of the reagent to drive a feedback controller.

Tasks• Analyze observability properties of time-varying, nonlinear system• Create extended and linearized Kalman filters• Evaluate filter performance

OPTIMAL ESTIMATION

Linearized and Extended Kalman Filters performance evaluated by analysis of autocorrelation of the filter error. More uncorrelated error indicates better performance.

ROBUST AND MULTIVARIABLE CONTROLProject:

State feedback control and observer design for a 2DOF helicopter model

Tasks• State feedback and full order design

for given multivariable transfer function

• Pole placement at given poles using Sylvester’s equation

• Decouple systemResult:• Successful pole placement for all poles at -

1• Decoupled system (pictured)

Multivariable transfer function step response for

the decoupled system

ROBUST AND MULTIVARIABLE CONTROLProject:

Design stabilizing feedback controller that aims at minimizing the effect of additive disturbances on the output of the plant, given constrains on control signals and/or uncertainty present in the plant.

Tasks• Verify nominal disturbance rejection

performance• Verify nominal tracking or measurement

noise sensitivity performance• Derive relevant transfer function M11

used for robust stability condition and plot of maximum/structured singular value of M11 to check stability robustness.

Result:• Successfully used loop shaping to meet the

performance specifications.• Minimized the effect of additive disturbances.

Hard Disk Drive dual state actuator: MIMO 2 input, 1 output, 8 state model.

ROBUST AND MULTIVARIABLE CONTROL

M11 was found using the following equation, as it is

used for the robust stability test. We have an additive

uncertainty on one resonant frequency of the system,

so we use the additive uncertainty equation to then

find M11 of our system.

END Please feel free to contact me with any further questions.

Thank you.