Start with a model for a submerging submarine:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system for the controlled model:

Perturb the submarine with a vertical force:

Simulate the closed-loop system with the same disturbance:

Plot the depth of the submarine in the original model:

In the closed-loop system, the submarine changes its density to preserve its depth:

Start with a model for a spacecraft in a circular orbit:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system for the controlled model:

Simulate the original model with an initial velocity:

Simulate the closed-loop system with the same initial velocity:

Plot the deviations from the circular trajectory in the original model:

The closed-loop system brings the spacecraft back to its circular orbit:

Start with a model for an inverted pendulum:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system for the controlled model:

Apply a tangential force to the pendulum and simulate it:

Simulate the closed-loop system with the same disturbance:

Plot the angle of the pendulum in the original model:

The closed-loop system applies a horizontal force and brings the pendulum back to the vertical position:

Start with a model for a continuous stirred-tank reactor:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system:

Simulate the original model with a deficit in the concentration of the reactant from its equilibrium value:

Simulate the closed-loop system with the same initial condition:

Plot the change in reactant concentration in the original model:

The closed-loop system controls the flow rate and brings the concentration back to the equilibrium value:

Start with a model for a ball placed on top of a beam that can rotate around its center of mass:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system:

Simulate the original model placing the ball at the edge of the beam:

Simulate the closed-loop system with the same initial condition:

Plot the position of the ball, measured from the middle of the beam, in the original model:

The closed-loop system applies a torque and brings the ball back to the middle of the beam:

Start with a model for a continuous stirred-tank reactor:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system:

Simulate the original model with a deficit in the concentration of the reactant from its equilibrium value:

Simulate the closed-loop system with the same initial condition:

Plot the change in reactant concentration in the original model:

The closed-loop system controls the flow rate and brings the concentration back to the equilibrium value:

Start with a model for a ball placed on top of a beam that can rotate around its center of mass:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system:

Simulate the original model placing the ball at the edge of the beam:

Simulate the closed-loop system with the same initial condition:

Plot the position of the ball, measured from the middle of the beam, in the original model:

The closed-loop system applies a torque and brings the ball back to the middle of the beam:

Start with a model for a spacecraft in a circular orbit:

Linearize it around an equilibrium point:

Design a controller and observer and put them together in EstimatorRegulator:

Generate the closed-loop system for the controlled model:

Simulate the closed-loop system with an initial velocity:

The closed-loop system brings the spacecraft back to its circular orbit:

Start with a model for an inverted pendulum:

Linearize it around an equilibrium point:

Design a controller and observer and put them together in EstimatorRegulator:

Generate the closed-loop system for the controlled model:

Simulate the closed-loop system with an initial angle away from the equilibrium:

The closed-loop system applies forces to bring the pendulum back to vertical position:

Start with a model for a continuous stirred-tank reactor:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system for the controlled model:

Simulate the original model with a deficit in the concentration of the reactant from its equilibrium value:

The closed-loop system controls the flow rate and brings the concentration back to the equilibrium value:

Start with a model for a ball placed on top of a beam that can rotate around its center of mass and add some rotational damping:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system for the controlled model:

Simulate the closed-loop system placing the ball away from the middle of the beam:

The closed-loop system applies a torque and brings the ball back to the middle of the beam:

Start with a model for an RLC circuit:

Linearize it around an equilibrium point:

Design a controller:

Generate the closed-loop system for the controlled model:

Provide a reference input and plot the output:

Start with a model for a camera stabilizer:

Linearize it around an equilibrium point:

Create a controller:

Generate the closed-loop system for the controlled model:

Perturb the system with a vertical force:

Simulate the closed-loop system with the same disturbance:

Plot the position of the camera in the original model:

In the closed-loop system, the position of the camera does not change as much:

Start with a model for a continuous stirred-tank reactor:

Linearize it around an equilibrium point:

Create a controller:

Generate the closed-loop system for the controlled model:

Simulate the original model with a deficit in the concentration of the reactant from its equilibrium value:

Simulate the closed-loop system with the same initial condition:

Plot the change in reactant concentration in the original model:

The closed-loop system controls the flow rate and brings the concentration back to the equilibrium value:

Start with a model for a submarine:

Linearize it around an equilibrium point:

Design a controller that tracks the depth of the submarine:

Generate the closed-loop system for the controlled model:

Simulate the closed-loop system providing a reference signal for the depth and noise:

Plot the reference and the output:

Plot the density change needed to reproduce the reference:

Start with a model for a ball placed on top of a beam that can rotate around its center of mass:

Linearize it around an equilibrium point:

Design a controller that tracks the position of the ball:

Generate the closed-loop system for the controlled model:

Simulate the closed-loop system providing a reference signal for the position and noise:

Plot the reference and the output:

Plot the input torque needed to reproduce the reference: