May 19, 2012

Compressed sensing meets control systems

After I had a lecture on compressed sensing (CS) in 2010 given by Prof. Toshiyuki Tanaka , I had a vague idea to apply CS to control systems.

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:

1. The term "Ax-b" where A is highly structured (not randomized) and x is unknown whether it is sparse or not.
2. The matrix "A" includes model error (e.g., error from linearization).
3. The vector "b" is subject to noise (e.g., quantization noise).
4. Computation should be extremely fast since computational delay may cause instability of the closed-loop system.
5. 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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