Multi-input and Multi-variable systems

A multi-input system accepts multiple input signals that influence its behavior. Cruise control exemplifies this by accepting two inputs: the driver's desired speed and external disturbances like road incline. The system adjusts engine throttle to maintain the desired speed while compensating for terrain changes, demonstrating how multiple inputs work together to achieve system objectives.

Block diagrams illustrate multi-input systems by showing each input signal and its path through the system. For cruise control, the diagram includes the desired speed input and disturbance input, each flowing through transfer functions that describe how the system responds. This visual representation clarifies the relation between mathematical equations and block diagrams, making system behavior transparent.

When disturbances are nullified, the transfer function Td(s) describes the relationship between desired speed and actual speed. When the primary input is nullified, Tu(s) represents the system's response to disturbances alone. These individual transfer functions allow engineers to analyze how each input independently affects system output.

The superposition principle states that a multi-input system's overall response equals the sum of individual responses from each input. For cruise control, the total speed adjustment combines the response to desired speed changes and the response to road incline disturbances. This principle simplifies analysis of complex systems with multiple simultaneous inputs.

Multi-variable systems have multiple inputs and outputs that interact dynamically. An airplane has pilot control inputs like aileron and rudder adjustments, producing outputs such as roll, pitch, and yaw changes. This complexity requires vector and matrix representations to capture all input-output relationships and feedback interactions comprehensively.

Vectors represent multiple inputs and outputs as single entities, while transfer matrices capture relationships between them. Feedback loops in multi-variable systems are described using matrix equations that express how outputs depend on inputs and system states. This mathematical framework enables analysis of complex systems like aircraft with numerous interacting variables.

The closed-loop transfer matrix describes the final relationship between outputs and inputs in a feedback system. It is calculated by solving matrix equations that incorporate the system's transfer functions and feedback paths. This matrix provides the complete input-output behavior of multi-variable systems after all feedback interactions are resolved.