<BODY bgcolor="#aaeebb"> <CENTER> <H3><u>Another Example of the Well Known Witsenhausen Problem (1968)</u></H3> </CENTER> <H4>Extremely simple but exceedingly hard:</H4> <P> <CENTER> (optimal solution known to exist but remains unsolved after 28 years)<BR> <H2> <i>x</i><sub>1</sub> = <i>x</i><sub>o</sub> + <i>u</i><sub>1</sub>; <i>z</i><sub>1</sub> = <i>x</i><sub>o</sub>; <i>x</i><sub>o</sub> ~ <i>N</i>(0,sigma<sup>2</sup>);<br> <i>x</i><sub>2</sub> = <i>x</i><sub>1</sub> + <i>u</i><sub>2</sub>; <i>z</i><sub>2</sub> = <i>x</i><sub>1</sub> + <i>v</i>; <i>v</i> ~ <i>N</i>(0,1);<br> <br> <i>J</i> = <i>x</i><sub>2</sub><sup>2</sup> + <i>k u</i><sub>1</sub><sup>2</sup>; <i>k</i> &gt; 0 </H2> </CENTER> <H4>Tradeoff between two decision makers with</H4> <P> <CENTER> good information but costly control<BR> vs.<BR> poor information with inexpensive control </CENTER> <P> <H4>For <i>k</i> = 0.2, sigma = 5</H4> <UL> <LI><i>J</i> based on best linear control law = 0.9 <LI><i>J</i> from best known nonlinear control law (Witsenhausen) = 0.404 <LI>best answer based on our Approach = 0.19<br> (<I>M. Deng &amp; Y.C. Ho, paper submitted to AUTOMATICA 1995</I>) </UL> <BR> <tt> <A HREF="Slide02.html" target="_parent">To Next Slide</A> / <A HREF="../OOTOC.html" target="_parent">Table of Content</A> </tt> <BR> <hr> <hr> <br> <P> This is a two step control problem with the simplest linear system. The initial condition <i>x</i><sub>o</sub> is a standard Gaussian random variable <i>N</i>(0,sigma<sup>2</sup>) which is perfectly observed by decision maker DM1 (Decision Maker 1). It is possible for DM1 to directly cancel <i>x</i><sub>o</sub> via control <i>u</i><sub>1</sub>. However it is costly for DM1 to act (the <i>k u</i><sub>1</sub><sup>2</sup> term in the performance criterion). DM2 acts after DM1. DM2 has no cost to act but s/he does not know what DM1 knows. Instead DM2 knows a noisy observation of <i>x</i><sub>1</sub> = <i>x</i><sub>o</sub> + <i>u</i><sub>1</sub>, the result of DM1 acting on <i>x</i><sub>o</sub>, through <i>z</i><sub>2</sub> = <i>x</i><sub>1</sub> + <i>v</i>, <i>v</i> ~ <i>N</i>(0,1). Thus, we have a tradeoff between a DM with perfect information but costly control and another DM with noisy information but inexpensive control. The classical version of the problem in which the DM who acts later always knows what the previous DM knows, the so-called perfect memory information structure is trivial to solve. But the twist here is that the problem is decentralized. DM2 does not know <i>z</i><sub>1</sub> the information of DM1. It is also known that if one of DM uses a linear strategy, then the optimal strategy of the other DM must also be linear. In fact, this linear strategy pair is also easy to solve and yields the performance of <i>J</i> = 0.9 (for <i>k</i> = 0.2 and sigma = 5). But surprisingly, a nonlinear strategy exist which out-performs the best linear strategy. The basic idea here is that of signaling. DM1 can use his/her control variable <i>u</i><sub>1</sub> not so much to reduce <i>x</i><sub>o</sub> but to signal to DM2 his knowledge of <i>x</i><sub>1</sub> so that s/he can cancel it with no cost. One such control found by Witsenhausen is <BR> <BR> The action of DM1 converts <i>x</i><sub>o</sub> to &#177;sigma. Then DM2 only needs to figure out whether or not <i>x</i><sub>1</sub> is positive or negative. For large sigma, this can be done with negligible error and thus insure the canceling of <i>x</i><sub>1</sub>. The performance of such a strategy is <i>J</i> = 0.404, considerably better than the best linear strategy. However, the optimal nonlinear strategy is not known even after 28 years. The problem is also known to be NP-hard.<BR> <BR> </BODY> </HTML>