Multi-input and Multi-variable systems

Cruise control in cars, a multi-input system, adjusts to the driver's speed preference and compensates for uphill disturbances.

The block diagram for this system includes the desired speed and the disturbance as inputs.

When the disturbance is nullified, the block diagram simplifies to a specific transfer function. When the primary input signal is nullified, the block diagram yields another transfer function.

The system's overall response is the sum of the responses due to the disturbance and input signals.

An airplane is a multi-variable system with multiple inputs and outputs, like the flight controls and the plane's movement.

Block diagrams can illustrate each input and output, while a simplified version employs vectors to represent multiple inputs and outputs.

These systems can also be depicted using feedback, where the relationships between the different parts of the system are expressed in a matrix form.

By solving these equations, the closed-loop transfer matrix gives the final relationship between the output and input.

Consider specific matrices for the transfer functions. Upon substitution, the closed-loop transfer function matrix can be calculated.