CS is not so popular in control systems community as in signal processing. Although L1 optimal control is a relatively classical problem (see, e.g., a work by Dahleh and Pearson here), they have not cared about the sparsity-promoting property of L1 optimization.

On the other hand, networked control have recently attracted a lot of attention in control systems community. In networked control, a controller is placed away from a controlled plant and the controller should communicate with the plant over rate-limited networks, such as wireless networks (imagine controlling this for example). In this situation, the data should be compressed to satisfy the rate-limiting constraint.

My idea is to use a technique of CS for compressing data of signals in networked control systems. More precisely, we use L1 optimization for sparse representation of control signals to be transmitted through rate-limited and erasure networks. I discussed the subject with Dr. Daniel E. Quevedo and presented the work at a conference:

M. Nagahara and D. E. Quevedo,

Sparse Representations for Packetized Predictive Networked Control,

*IFAC 18th World Congress*, pp. 84-89, Aug 2011.

and also a journal version

M. Nagahara, D. E. Quevedo, and J. Ostergaard,

Sparse Packetized Predictive Control for Networked Control over Erasure Channels,

*IEEE Transactions on Automatic Control*, Vol. 59, No. 7, Jul 2014.

I believe this is the first-ever paper to apply CS (more precisely, sparse representation or sparse approximation) to networked control systems. However, there remain a couple of difficulties:

- The term "
**Ax-b**" where**A**is highly structured (not randomized) and**x**is unknown whether it is sparse or not. - The matrix "
**A**" includes model error (e.g., error from linearization). - The vector "
**b**" is subject to noise (e.g., quantization noise). - Computation should be extremely fast since computational delay may cause instability of the closed-loop system.
- Only cheap quantization (e.g., a uniform scalar quantizer) can be used.

To see these difficulties, imagine again controlling the helicopter to make it fly stably along a desired trajectory. The problem is very challenging.

For recent papers on CS for control systems, see this entry.

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