MEC 560: Advanced control systems¶

Assignment 2¶

Due 10/3/2016 at 4:00 pm.¶

NO LATE SUBMISSIONS ACCEPTED.¶

Please submit all your work in electronic for via blackboard. Use latex or equation editor to typeset your equations. Include all code and graphs in your submission.¶

Question 1: Similarity Transforms¶

Two square-matrices $A $ and $ C $ are called similar if there exists an invertible square matrix $ P $ such that,

$$ C = P^{-1} A P .$$

The transformation $ P^{-1} A P $ is also called similarity or conjugation of $A$. As we saw before, a dynamic system $ \dot{x} = Ax + Bu $ with transformation $ x = P \hat{x} $ transforms as, $ \dot{\hat{x}} = P^{-1} A P + P^{-1} B $.

Show that the controllability of the matrix (rank of the $M_A $) does not change due to similarity transform.
Do the entries of the controllability matrix themselves change?

Question 2: Simulation of dynamic systems¶

Consider the dynamic system defined as,

$$ \ddot{x} +4 \dot{x} + 3 x = u $$

Express the dynamics of the system in state space form and compute the eigen values of the system dynamics matrix. What can you say about the stability properties of the system.
Using $ u = 0 $, and assuming you start at $ x = 1 $ and $ \dot{x} = 0 $, numerically solve the the system dynamics equation using euler difference method, runge-kutta 4 with fixed time-step, and ode45. For fixed time-step, choose $ dt = 0.1 $ and total time of $ 10 s $.
Compare the solutions obtained from euler difference method, runge-kutta 4 with fixed time-step and ode45 to the analytical solution.
Compare the solutions obtained from euler difference method, runge-kutta 4 with fixed time-step and ode45 to the analytical solution, for $ dt = 0.05 $, $ dt = 0.1 $ and $ dt = 0.2 $ and simulate for total time of $ 10 s $. What are the advantages and disadvantages of choosing a smaller or larger $ dt $ values?
Based on results above, which numerical integration scheme will you recommend?

Question 3: Control of dynamic systems¶

a. Control of a second order dynamic system¶

Consider the dynamic system defined as,

$$ \ddot{x} +5 \dot{x}-4 x = u $$

Assume that you start from $ x = 10 $, $\dot{x} = 0 $ and $ u = 0 $.

Express the dynamics of the system in state space form and compute the eigen values of the system dynamics matrix. What can you say about the stability properties of the system.
Compute the controllability matrix of the state-space equation you obtained.
Using $ u = 0 $, numerically solve the the system dynamics equation to verify your predictions (use ode45).
Use pole-placement technique to compute the gain matrix such that the eigen values of the modified system are at -2 and -3.
Use this control law to achieve a desired set point given by $ x = 4 $. What can you say about the control thus generated? Is this control signal feasible?
Use this control law to track a time-dependent signal given by $ x(t) = sin(2t) $. What can you say about the control thus generated? Is this control signal feasible?

b. Adding integrator error state and controller dynamic¶

Modify your controller to add an additional state, the integral of the error between current and desired state.
Use pole-placement technique to compute the gain matrix such that the eigen values of the modified system are at -2,-3 and -4.
Use this control law to achieve a desired set point given by $ x = 4 $. What can you say about the control thus generated? Is this control signal feasible?
Use this control law to track a time-dependent signal given by $ x(t) = sin(2t) $. What can you say about the control thus generated? Is this control signal feasible?

c. Adding integrator error state and controller dynamic¶

Modify your controller to add an additional state, the integral of the error between current and desired state, and include first order actuator dynamics given by $ \dot{u} = -5(u - u_c $.
Use pole-placement technique to compute the gain matrix such that the eigen values of the modified system are at -2,-3,-4 and -5.
Use this control law to achieve a desired set point given by $ x = 4 $. What can you say about the control thus generated? Is this control signal feasible?
Use this control law to track a time-dependent signal given by $ x(t) = sin(2t) $. What can you say about the control thus generated? Is this control signal feasible?

Compare the controllers from parts a,b and c, and make a recommendation of which controller you will choose and why?

Question 4: Project: System dynamics¶

In previous assignment you identified the type of system you would like to design controller for. Find atleast 3 research articles where other researchers have designed control system for same or similar task that you are interested in. What assumptions do they make regarding the dynamics of the system?
Identify the states and controls for your system dynamics.
Obtain differential equations describing the dynamics of the system. Are the system dynamics equation linear or nonlinear? Have other researchers used linearizing methods to design controllers for this task?
Identify the specific tasks you would like your system to perform? Tracking, regulator, set-point.

Question 5: Cart on Pole¶

In previous assignment you derived linearized equations of motion for a pendulum on the cart. Assuming that the pendulum starts at a position where the initial states are $ \theta = 0 $ and $ x = 0 $, i.e. balanced vertically up.

Is the linearized system controllable?
Design a full-state feedback control for the linearized system using pole-placement technique.
(Bonus) Assuming a disturbance of $ F = 0.2sin(t) $ is applied to the system, perform numerical simulations to verify if the stabilizing control obtained from linearized control stabilizes the system.
(Bonus) Repeat the previous part for $ A $ in $ F = A sin(t) $ ranging from 0 to 2, and check for which values of $ A $, the pole-placement technique based on linearized system fails to stabilize the system.

Bonus questions are additional 25% of the points obtained on other questions. Note, while simulating the dynamic system, you need to use the full non-linear dynamic system model. Control is designed based on linearized assumptions, but the system behaves in non-linear manner.