## Control Basics: PID, LQR, MPC

There are many methods to implement control including basic strategies such as a proportional-integral-derivative (PID) controller or more advanced methods such as model predictive techniques. The purpose of this section is to provide a tutorial overview of potential strategies for control of nonlinear systems with linear models.

A following section relates methods to implement dynamic control with nonlinear models.

#### Exercise

**Objective:** Design a model predictive controller with a custom objective function that satisfies a specific problem criterion. Simulate and optimize a pendulum system with an adjustable overhead cart. *Estimated time: 3 hours.*

A pendulum is described by the following dynamic equations:

where *epsilon* is *m _{2}/(m_{1}+m_{2})*,

*y*is the position of the overhead cart,

*v*is the velocity of the overhead cart,

*theta*is the angle of the pendulum relative to the cart, and

*q*is the rate of angle change

^{2}.

The objective of the controller is to adjust the force on the cart to move the pendulum mass to a new final position. Ensure that initial and final velocities and angles of the pendulum are zero. The position of the pendulum mass is initially at -1 and it is desired to move it to the new position of 0 within 6.2 seconds. Demonstrate controller performance with changes in the pendulum position and that the final pendulum mass remains at the final position without oscillation.

#### Solution

#### References

- Bryson, A.E., Dynamic Optimization, Addison-Wesley, 1999.