Update content - 2026-10-07

content/zettels/bumpless_transfer.md

@@ -7,7 +7,35 @@ subcategory = "Fundamentals"
 +++
 
 Tags
-:
+: [Anti-Windup Control]({{< relref "anti_windup_control.md" >}})
+
+Bumpless transfer consists in switching between two controllers without creating a discontinuity (a "bump") in the control signal \\(u\\).
+
+A bump occurs because the inactive controller has internal states (integrators, filters) that are not consistent with the signal currently applied to the plant.
+At the switch, its output differs from the active one, and \\(u\\) jumps.
+
+The principle is to keep the inactive controller "warm": its states must be consistent with the applied signal \\(u\_a\\) before the switch.
+Anti-windup is a special case of this, with \\(u\_a = \text{sat}(u)\\).
+
+An example of bumpless transfer between manual and PID control is provided by MathWorks: [Bumpless Control Transfer Between Manual and PID Control](https://www.mathworks.com/help/simulink/slref/bumpless-control-transfer-between-manual-and-pid-control.html).
+
+
+## Controllers with an explicit integrator (PID) {#controllers-with-an-explicit-integrator--pid}
+
+The integrator of the offline controller is driven to track the applied signal using back-calculation:
+
+\begin{equation}
+  \dot{x}\_i = K\_i e + \frac{1}{T\_t} \left( u\_a - u \right)
+\end{equation}
+
+This requires the integrator to be explicit, which is one of the reasons why PID controllers are convenient.
+If the PID is the last element of the chain, its output is \\(u\\) and the tracking is exact.
+
+
+## Controllers made of biquads {#controllers-made-of-biquads}
+
+Filters without integrator (notch, low-pass, lead, lag) have fast dynamics and do not need any tracking.
+The offline chain is simply run in parallel, fed with the same error signal \\(e\\), and its states converge by themselves.
 
 
 ## Bibliography {#bibliography}