How the Software Treats Loop Openings

Simulink^® Control Design™ software linearizes models using a block-by-block approach. The software individually linearizes each block in your Simulink model and produces the linearization of the overall system by combining the individual block linearizations. For more information, see Exact Linearization Algorithm.

To obtain an open-loop transfer function from a model, you specify a loop opening. Loop openings affect only how the software recombines the individual linearized blocks. In other words, the software ignores loop openings when determining the input signal levels for each block, which affects how nonlinear blocks are linearized.

For example, in the following model, to compute the response from e₂ to y₂ without the effects of the outer loop, you open the outer loop by placing a loop opening analysis point at y₁.

Here, k₁, k₂, g₁, and g₂ are nonlinear blocks.

The software linearizes each individual block at the specified operating point, creating the linearized blocks K₁, K₂, G₁, and G₂. At this stage, the software does not break the signal flow at y₁. Therefore, the block linearizations include the effects of the inner-loop and outer-loop feedback signals.

To compute the transfer function from e₂ to y₂, the software enforces the loop opening at y₁, injects an input signal at e₂, and measures the output at y₂.

Here, K₁, K₂, G₁, and G₂ are the linearized blocks.

The resulting linearized transfer function is (I+G₂K₂)^-1G₂K₂.