State-Space Control
State-space methods describe a system by its internal state vector, enabling systematic control of coupled, multivariable dynamics beyond single-loop PID.
Describing a System by Its State
State-space control represents a dynamic system by a vector of state variables that captures everything needed to predict its future given the inputs. The dynamics are written as matrix equations: the state's rate of change depends on the current state and inputs, and the outputs depend on the state and inputs. This compact form handles systems with many coupled inputs and outputs that single-loop methods cannot address cleanly.
The Standard Form
A linear system is written with four matrices relating state, input, and output. The A matrix describes how the state evolves on its own, B how inputs drive it, C how outputs are read from the state, and D any direct feedthrough. Analysis of A reveals stability and natural modes; controllability and observability tests, computed from these matrices, determine whether the system can be steered and whether its state can be inferred from measurements.
State Feedback and Estimation
- State feedback: command the input from the full state to place the closed-loop dynamics where desired
- Observers and Kalman filters: estimate unmeasured states from available sensors
- Optimal control (LQR): choose feedback gains to minimize a weighted cost of error and effort
- Model-predictive control: optimize commands over a horizon subject to constraints
Why It Suits Complex Plants
When variables interact strongly, adjusting one loop disturbs others, and independent PID loops fight each other. State-space design accounts for the coupling directly, producing coordinated commands. It also handles constraints and optimality explicitly through model-predictive control, at the cost of more computation per cycle, which the timing budget must accommodate.
Application to Plasma Control
Controlling a magnetically confined plasma is inherently multivariable: coil currents, plasma position, shape, and current interact. Advanced tokamak control, of the kind relevant to the Hyperion breeder at the design and simulation stage, uses state-space and model-based methods so that many actuators and diagnostics are coordinated rather than tuned as isolated loops. The estimator reconstructs quantities that cannot be measured directly, and the controller commands all coils together to hold the desired equilibrium.