Multi-input and Multi-variable systems

Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.

In the absence of disturbances, the cruise control system's block diagram can be simplified to a specific transfer function. This transfer function represents the relationship between the desired speed input and the vehicle's actual speed.

where T_d(s) is the transfer function from the desired speed R(s) to the actual speed Y(s).

Conversely, when the primary input signal (the desired speed) is nullified, the block diagram simplifies to another transfer function, representing the system's response to external disturbances alone.

where T_u(s) is the transfer function from the disturbance D(s) to the actual speed Y(s).

The overall response of the cruise control system is the superposition of the responses to both the desired speed and the disturbance inputs. This can be mathematically represented as:

This superposition principle illustrates how the system adjusts to maintain the desired speed while counteracting disturbances.

In a more complex system, such as an airplane, multiple inputs and outputs must be considered. Inputs might include control signals from the pilot, such as aileron, rudder, and elevator adjustments, while outputs are the aircraft's responses, such as changes in roll, pitch, and yaw. The complexity of such a system necessitates the use of vectors and matrices to represent the multiple inputs and outputs succinctly.

Block diagrams for multi-variable systems like airplanes can be simplified using vector representations. Inputs and outputs are expressed as vectors, and their relationships are captured in a transfer matrix. Feedback loops in these systems can also be described using matrix equations, allowing for a comprehensive representation of the system's dynamics.