Adolfo - I must admit that I do not understand the problem. For example, what is the meaning of the following phrase:

H infinity norm is less than 1 + delta.

Dear Adolfo,

I think it will be necessary to study these papers in order to identify the problem in question. that why i suggest for you attached files in topics.

Best regards

Lutz von Wangenheim, thank you for your observation. I've edited the question and changed the figure to be more clear.

Mohamed, thank you for the files.

Z.H. Khan, actually I've already choose the target loop shape "Wp(s)", to get good tracking (at low frequency), noise reduction and robust to high frequency unmodeled dynamics (at high frequency), good roll-off at the gain frequency crossover (to get good stability margins).  My question is how to prove that affirmation. I've noticed if ||H(s)| < 1 + delta|,  the L(s) approximate the target loop shape.

The problem of loop transfer recovery (LTR) was raised by John Doyle in the late 1970's. The problem is to make the loop transfer function L(s) of the observer feedback system equal (or "recovered") to a target loop transfer function Lkx(s) = K(sI - A)-1B, where K is the state feedback control gain.

For over half of a century until now and in all other existing literature, this K (or Lkx(s) ) has been separately designed before observer design, and has been designed independent of parameter C of system (A,B,C). This overall design procedure is the well known separation principle.

Unfortunately, satisfactory solution of this specific problem (under separation principle) does not generally exist for the great majority of systems (such as non-minimum-phase systems). A branch of these solutions is at "H infinity setting", and was initiated by J.B. Moore in the late 1980's.

This conclusion was made in my publications "Observer Design -- A Survey" of 2015 and Robust Control System Design, 2nd Ed. of 2004, in my Researchgate Q&A  to "Limitations of Linear Quadratic Regulator in non-minimum phase systems?" few months ago (I notice that you upvoted my answer to another relevant Q&A question), and in my review of an SCL paper "Loop transfer recovery, a review" (I was the only reviewer of that paper) six months ago. Let me quote a paragraph of my review, regarding the significance of the LTR problem.

"This problem is critically important to all feedback control designs because loop transfer function determines the sensitivity/robustness of the corresponding feedback system, because robustness is THE main purpose of feedback control since this control has side effects on performance, and because Kx-control is far better in improving feedback system performance and robustness than all other forms of feedback control. Feedback control theory (state space control theory in particular) has not found many successful applications ever since the 1960’s, mainly because the majority of control community (including the authors of this review paper) only knew unsatisfactory design solutions to this problem."

The clear and basic reason of this non-existence of satisfactory solution to this specific LTR problem and separation principle, was provided in the above listed literature and will be quoted later in this answer.

I suspect that this situation is also the reason of the shifted interest to H infinity design problem, for otherwise the optimal LQG system is already fully achieved. However H infinity problem has its own critical difficulties such as HOW to actually determine and adjust the really optimal n2 parameters of the weighting matrices of the optimality itself, HOW to really and effectively guarantee and adjust actual performance and robustness, and HOW to generally and effectively consider loop phase which is as important as loop gain! 

The design of K (or Lkx(s)) has no such difficulties at all especially in eigenstructure assignment, even if K is constrained by equation K = K [T':C']' (where T is the key observer parameter (observer states converge to Tx(t))) and is designed using static output feedback control (SOFC) methods. This is because Kx-control is based on the BEST information of system state and system structure, and because, specifically,  eigenstructure assignment can generally and effectively optimize a robust stability criterion that is far more generally accurate than other existing criterion s. This is the reason that LTR problem is critically important, and is unavoidable by other design propositions.

Fortunately this problem has found decisively satisfactory solution, with exact LTR for a great majority of systems! The key is to abandon separation principle, enable lower and flexible observer (or compensator) orders, and design K based on (not separated from) parameters T and C using SOFC methods. In this revolutionary synthesized design principle, observer (or T) is designed for guaranteed exact LTR, before the design of corresponding K.

This idea and result were fully developed in the 1990's, fully documented in my literature listed above, and has apparently been missed by almost all control researchers (including those involved in this Q&A question).

The clear and basic reason of this decisively satisfactory solution to this most critical LTR problem, was provided in the above literature and will be quoted at the end of this answer, after the clear and basic reason of the unsatisfactory solution to the existing LTR problem and separation principle:

"The reason of this critical problem is actually also extremely simple and obvious. As Dr. Whidhorne stated in his answers, the LQR design or full state feedback design (only full state feedback control can guarantee the minimization of the LQ criteria) is based on the assumption that the ideal information of full state measurement is available. But this assumption is invalid for all systems if an observer is needed, and is very far from being valid for all non-trivial systems such as non-minimum-phase systems.

We all know this assumption (all systems have the ideal and full state measurement information) is invalid since the very beginning. But all existing designs (including LQG designs and the LTR designs of 1980's) are still fully based on this invalid assumption. All of these existing designs follow separation principle (design full state feedback control and observer separately).

How can a control designed based on this ideal assumption, be fully realized for all systems, almost all them are far from being ideal (or this ideal assumption is far from being valid)?

This is the fundamental reason for the failed realization of robust properties (or failed LTR) of full state feedback control at almost all system conditions. This is understandably the main reason that state space control theory has not found many successful application s since the 1960's."

This is also the clear and basic reason of the above mentioned "decisively satisfactory LTR solution. " This is because by "abandoning separation principle, enabling lower and flexible observer (or compensator) orders, and designing K based on parameters T and C", the above mentioned existing drawbacks, irrationalities, and reasons were all eliminated!

Z.H. Khan,

I've looked the book that you cited, but I could not find the proof explicit. Could you tell me where it is at the book?

Best regards.

The explicit proof is at Theorems 3.3 and 3.4 of my book, on pages 71-75.

The explicit proof can also be found at Subsection 3.1 of my paper "Observer Design -- A Survey".