# Controlling a system with delayed measurements

Assume I have a rather simple system I want to control, but all sensor measurements exhibit considerable time delay, i.e.:

$z_t = h(x_{(t-d)}) \neq h(x_t)$

With my limited knowledge about control, I could imagine the following setup:

• One observer estimates the delayed state $x_{(t-d)}$ using control input and (delayed) measurements.
• A second observer uses the delayed observer's estimate and predicts the current state $x_t$ using the last control inputs between delayed measurement and current time.
• The second observer's estimate is used to control the system.

Can I do any better than that? What is the standard approch to this problem? And is there any literature or research about this topic?

• I think the approach is correct and something tells me the second step is done with a Markov chain. – Shahbaz Jun 13 '14 at 14:46
• How long is the delay relative to your sample rate? There is an interesting paper at scholarworks.sjsu.edu/cgi/… that analyzes behavior of PID loops in systems with high time delay and describes a technique for tuning for these systems. – Jason C Jun 17 '14 at 20:17
• Thanks for the various pointers. In addition to what was posted in the answers and comments, I also found this paper helpful. – sebsch Jun 23 '14 at 13:17

## 3 Answers

Delayed measurements are a very common occurrence in LBL sonar positioning, commonly used by AUVs. Modified Kalman filters are a common solution. I searched for "LBL", "delay", and "EKF". Here are a few reasonable papers:

The most straightforward approach is to use a Kalman filter with a memory of recent state history. While waiting for measurements you do the standard time update. When a new measurement arrives, you restore the state and error covariance to the appropriate point in time, apply the measurement update, and then apply time updates to get to the current point in time.

Kalman framework can be adapted to deal with joint state and delay estimation: see this paper as an example.

Don't forget to consider the Smith Predictor, which will help you design the linear regulator as the system was without delays.