Added Sept. 2010

Kim Listmann wrote:
> Dear Prof. Skogestad,
>
> I am a phd cand. in systems & control here at TU Darmstadt, Germany.
> First of all I'd like to congratulate you for your your excellent book
> on multivariable feedback control. Indeed it is one of the best, and I
> am considering to finally buy it instead borrowing it all the time ;).
>
> However, I have a question concerning the chapter on LMIs (chp. 12).
> You provide some helpful tricks to reformulate definiteness equations
> into LMIs and one such possibility is to use Finsler's Lemma. I am
> concerned with a control problem where this Lemma is of utmost
> importance (in my eyes) and I can't find enough material to be really
> safe with what I derived so far.
>
> Q1. My question is the following: In the Example 12.5 on page 484, you
> apply Finsler's Lemma to a state feedback problem resulting in Eqn.
> (12.81). Then it is stated that if a sigma is found for the first
> equation
>
> AQ + QA' - sigma BB' < 0,
>
> we can always find one for the second
>
> AQ + QA' - sigma I < 0.
>
> But both sigma should be the same, shouldn't they? Furthermore, what
> happens if sigma < 0 (which is allowed following Finsler original
> text). Then the situation is vice versa, as every sigma fulfilling the
> last equation, will fulfill the first one. Is that right? I am confused.
>
> Q2. Moreover, consider the output feedback case, then the second equation
> becomes
>
> PA + A'P - sigma C'C < 0, P = Q^-1
>
> Would the problem to find a Q and by that a P, then still be convex?
> 
> Q3. Can you provide some further references highlighting some aspects of
> Finsler's Lemma (if such contributions do exist)?
>
> Sorry for this interruption and thanks in advance for any help you can
> provide.
>
> Best regards
>
> kdl

Dear Kim
 
Very sorry about the delay in replying but Sigurd's email arrived at holiday time and then it slipped through the net.
 
You are on the right track, but for LMIs, experience of working with the algorithms is important. For this reason, I have asked  a colleague of mine who is a real expert in LMIs for his comments.
 
Answers to your questions are given below.
 
Good luck with your PhD studies.
 
With best wishes
ian
 
************************************************************
Q1:

What we're saying in Example 12.5 is that the *existence* of a state-feedback matrix F is guaranteed if there *exists*
a \sigma (just real, positive or negative) such that:

AQ + QA' - sigma BB' < 0
AQ + QA' - sigma I < 0

both hold (for the same sigma).

First, LET US ASSUME THAT THERE EXISTS A SIGMA SUCH THAT THE
FIRST INEQUALITY HOLDS (If not, we immediately know that the state feedback
problem is infeasible i.e. uncontrollable modes are present)

So, let us assume that we can find a sigma_1>0 such that the first inequality holds i.e.

AQ + QA' - sigma_1 BB' < 0

1. Now assume  I >= BB', then with sigma=sigma_1, both inequalities hold - nothing else needs to be done

2. Now, conversely assume that I < BB', then, as I is full rank, we can always find a sigma_2 such
that sigma_2 I > sigma_1 BB' where, obviously, sigma_2 > sigma_1.

Thus we have

AQ+QA' - \sigma_2 I < AQ+QA' - sigma_1 BB' < 0

So with sigma=sigma 2, the second inequality holds. Now, all we do is set sigma=sigma_2 and
because sigma_2 > sigma_1 we have

AQ+QA' - sigma_2 BB' < AQ+QA' - sigma_1 BB' < 0

Hence the first inequality holds also.

Therefore, IF THERE EXISTS A SIGMA SUCH THAT THE FIRST INEQUALITY HOLDS, WE CAN
ALWAYS FIND A SIGMA SUCH THAT BOTH INEQUALITIES HOLD

The main issue here is that BB' is (typically) not full rank so the converse does not hold in general.



Q2:  PA + A'P - sigma C'C < 0, P = Q^-1

This is convex providing no other (non-convex) constraints are given in terms of Q. In other words:

PA + A'P - sigma C'C < 0
P > 0

is a convex feasibility problem (from which Q=P^-1 can be found.

but

PA + A'P - sigma C'C < 0
Q > sigma

is not convex because constraints appear on both P *and* Q (which can't be "convexified")



Q3: For control purposes, the LMI book by Boyd et al (in the references of MFC:A&D) is a
very nice reference for Finsler's Lemma and LMIs more generally. It is available for free download
from Boyd's webpage.