Sensor integrity monitor

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Sensor integrity monitor. A suite of m sensors yields measurement y R™ of some vector of parameters TER". When the system is operating normally (which we hope is almost always the case) we have y = Ar, where m > n. If the system or sensors fail, or become faulty, then we no longer have the relation y = Ar. We can exploit the redundancy in our measurements to help us identify whether such a fault has occured. We'll call a measurement y consistent if it has the form Ar for some r ER". If the system is operating normally then our measurement will, of course, be consistent. If the system becomes faulty, we hope that the resulting measurement y will become inconsistent, i.e. not consistent. (If we are really unlucky, the system will fail in such a way that y is still consistent. Then we're out of luck.) A matrix B E R is called an integrity monitor if the following holds • By=0 for any y which is consistent. • By *for any y which is inconsistent If we find such a matrix B, we can quickly check whether y is consistent: we can send an alarm if By # 0. Note that the first requirement says that every consistent y does not trip the alarm; the second requirement states that every inconsistent y does trip the alarm. Finally, the problem. Find an integrity monitor B for the matrix 1 2 1 A= -2 1 -1 -2 1 3 -1 -2 1 0 1 1 23 Your B should have the smallest k (i.e., number of rows) as possible. As usual, you have to explain what you're doing, as well as giving us your explicit matrix B. You must also verify that the matrix you choose satisfies the requirements. Hints: You might find one or more of the Matlab commands orth, null, or qr useful. Then again, you might not: there are many ways to find such a B. • When checking that your B works, don't expect to have By exactly zero for a consistent y; because of roundoff errors in computer arithmetic, it will be really, really small. That's OK. Be very careful typing in the matrix A. It's not just a random matrix.