Meaning
Mathematical control theory employs systems of first-order differential equations to represent the internal states, inputs and outputs of a dynamic physical system. A state space model provides a framework for analyzing multi-variable systems where classical transfer functions are difficult to apply. This mathematical representation allows engineers to simulate the behavior of complex production machinery before any physical hardware is built.
Matrix Representation
Linear systems of equations use transition and input matrices to describe how system states evolve over time. When designing a state space model, the state vector must contain enough independent variables to capture the essential dynamics of the system. This model is validated during pilot runs by comparing the simulated state trajectories against the measurements recorded from the physical prototype.
Discrepancies identified during this phase require recalibrating the model matrices before the control software is deployed in production.
Variable Optimization
Identifying the critical variables of a system minimizes the computational effort required for real-time control. In a state space model, optimizing the selection of state variables helps to avoid redundant calculations. This optimization allows the control algorithm to run within the hardware limits of the target embedded controller.
Control Verification
Feedback gain matrices are calculated to ensure the closed-loop system is stable and has the desired response characteristics. Using a state space model to synthesize the feedback controller ensures that all states are driven to their target values without excessive oscillation. This verification step confirms the system’s capability to operate reliably under variable loading conditions.
