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Controllability is a key concept in control system design, determining whether a system’s states can be driven to desired values using input signals. Calculating the controllability matrix helps engineers assess this property for a given system modeled in state space form.
State Space Representation
A system in state space form is represented by the equations:
˙x(t) = Ax(t) + Bu(t)
where x(t) is the state vector, A is the system matrix, and B is the input matrix.
Controllability Matrix Calculation
The controllability matrix, 𝓒, is constructed by concatenating the matrices:
[B, AB, A²B, …, Aⁿ⁻¹B]
where n is the number of states in the system. This matrix indicates whether the states can be controlled through the input.
Steps to Calculate
Follow these steps to compute the controllability matrix:
- Identify matrices A and B from the system model.
- Calculate the matrix AB by multiplying A with B.
- Compute higher powers of A (A², A³, etc.) and multiply each by B.
- Concatenate all matrices horizontally to form 𝓒.
Controllability Check
The system is controllable if the controllability matrix 𝓒 has full rank, equal to the number of states. This can be verified by calculating the rank of 𝓒.