6.1.3 Closing Feedback Loop
Feedback interconnections of the types shown in Figure 6.4 are performed with the function FeedbackConnect. The function may just close the loop (Figure 6.4a), or may include a second system (typically a controller) in the feedback (Figure 6.4b). Either negative (default), positive, or mixed feedback can be formed. The connection specifications may be omitted if all corresponding inputs and outputs are to be used. The inputs and outputs of the aggregate are the ones of the first system. The states come from all subsystems.
Figure 6.4. Feedback interconnections.
Closing the feedback loop.
Feedback interconnections with second system.
The type descriptor for FeedbackConnect should be one of the reserved words Positive or Negative.
Consider a second-order system with two inputs and two outputs.
This forms the negative feedback by connecting corresponding outputs and inputs.
We can use built-in Mathematica functions to simplify the result.
This forms the closed loop for the first two inputs and outputs of the transfer function tf.
Now we plug the system ss in the positive feedback loop for ss1 (assuming, for simplicity, that there is no direct transmission term in ss).
This connects ss1 and ss according to Figure 6.5 (under the same simplifying assumption).
Figure 6.5. Example of feedback interconnection.
The connection functions described in this section accept transfer functions in arbitrary (rather than rational polynomial) form as long as no type conversion or subsystem selection has to be made.
This is the serial connection of two systems described by the transfer functions g[s] and h[s].
Here are the same systems connected in parallel.
This is the negative feedback connection.