minreal

Minimal realization or pole-zero cancellation.

Syntax

• [Am, Bm, Cm, Dm] = minreal(A, B, C, D)
• [Am, Bm, Cm, Dm] = minreal(A, B, C, D, tol)
• sysOut = minreal(sysIn)
• sysOut = minreal(sysIn, tol)

Input argument

• A (n x n) - Represents the system's state-transition matrix. It describes how the system's internal state evolves over time.
• B (n x m) - Describes the input-to-state mapping. It shows how control inputs affect the change in the system's state.
• C (p x n) - Represents the state-to-output mapping. It shows how the system's state variables are related to the system's outputs.
• D (p x m) - Describes the direct feedthrough from inputs to outputs. In many systems, this matrix is zero because there is no direct feedthrough.
• tol - scalar real (tolerance).
• sysIn - LTI model.

Output argument

• Am, Bm, Cm, Dm - a minimal realization of the state-space system A, B, C, D.
• sysOut - a minimal realization of LTI input.

Description

minreal function reduces state-space models by eliminating uncontrollable or unobservable states.

In transfer functions or zero-pole-gain models, it cancels pole-zero pairs. The resulting model maintains the same response characteristics as the original model but with minimal order.

When using sysOut = minreal(sysIn, tol), you can customize the tolerance for state elimination or pole-zero cancellation.

The default tolerance is set to sqrt(eps), and increasing this value prompts more aggressive cancellations, potentially further simplifying the model.

Example

sysIn = ss([1 0;0 -2], [-1;0], [2 1], 0, 3.2);
sysOut = minreal(sysIn)

History

Author

Allan CORNET