Examples of Control Systems

Example 1: Cruise Control of a Car

• - Force imparted by the engine
• - Mass of the car
• - Friction
• - Slope incline

Governing Equation

Let to get a linear order ordinary differential equation

Say we want to obtain a certain speed

Note that is a constant and refers to and is measured by a sensor in the plant

Example 2: Voltage Control of an RLC Circuit

• input (voltage)
• output (capacitor voltage)

Substituting from eq. 2 into eq. 1 and dividing out by gives

State-space Model

State Vector Defines the state of our dependent variables. It is essentially a vector of all the variables in all our equations that change during the system operation, except the input

Now we write eq. 1 and 2 in matrix form solving for the derivatives

Example 3: Inverted Pendulum

• - Point mass
• - Length of massless rod
• - Angle of pendulum from vertical
• - Torque applied at the pivot

Governing Equation

The specifics of the governing equation isn’t the focus of the course

State-space Model

Example 4: DC Motor

• - Back emf; voltage induced from movement of armature coils in magnetic field
• - Friction
• - Torque produced by current in coils and magnetic field of magnets
• - Moment of inertia
• - Input voltage
• - Back EMF Constant,
• - Torque constant,

The Way I like to think of making State-Space Models

1. List all of the variables
2. Remove the constants
3. Take an integral of remaining constants (e.g. if we have , then we’re left with )
4. Remove the constants again
5. What you’re left with are variables of the state-space model