Decentralized Controller Design for Large-Scale Uncertain Discrete-Time Systems with Non-Block-Diagonal Output Matrix

Round 1

Reviewer 1 Report

The paper deals with the decentralization of large-scale systems into subsystems that are easier to control. This raises a number of new problems, such as the communication of these subsystems and stability issues in the Lyapunov sense, as well as classical stability solutions using eigenvalue calculus. In contrast to earlier works, this time the authors focus on discrete systems. The quality of the work also lies in the proofs of mathematical theorems that document the correctness of the proposed approaches.

However, some parts of the text would require further explanation or additions:      

- What is the advantage of transforming the system to a form with a block-diagonal output matrix? 

- How many subsystems should the decentralized structure contain?

- Page 5, line 171: "The controller can be, for instance, a PI, PID or other type." What is used to select the type of controller?

- The state description in Equation (1) is reduced, the term Du(t) is missing in the expression of the vector y versus the general expression.

- In the notation for discrete systems, the discrete time $kT$ should be included instead of the continuous time $t$, where T is the sampling period.  

- Page 6, Equation (9). The summation variable is missing at the bottom of the sum, probably $t=t_0$ or $t=0$ should be here.

Formal comments:

The equation notation does not distinguish between scalar and vector variables, which makes the text difficult to read. For example, in Equation (1) the vector symbols x, u, y should be in bold.

Page 1, line 27:" methods based on the LMI-BMI approach" - Although there is a list of abbreviations with their meanings at the end of the text, it would be useful to include the meaning at the first occurrence of the abbreviation.

p. 4, l. 140: "i – th" - "i-th"

Author Response

Thank you for the revision. The answers are included in the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper deal with Decentralized Controller Design for Large-Scale Uncertain discrete-time Linear Systems with no block diagonal output matrix

I have the following observations:

1.      The paper it’s well written

2.      I suggest you give more references from last 5 years

3.      Please correct at line 274    …  [?]

4.      Please correct the matrices from example   .05    .1

5.      A comparison with other algorithms in terms of "numerical efficiency" would be interesting.

Author Response

Thank you for the revision. The answers are included in the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

1. The paper addresses the issue of decentralised control design for linear discrete-time systems with an uncertain dynamics. The subject matter of the paper is interesting and fits primarily within the field of control theory for linear systems. However, it is worth noting that this topic is not widely known, and it seems that one could attempt to illustrate the decomposition described in Section 2.1 with a simple diagram. Similarly, it would be useful to illustrate the issue of decentralised control and show the couplings.

2. The considered paper extends the results discussed in [27]. For this reason, it is worth explaining more precisely what is new in the reviewed manuscript. I conclude that the main result is the transfer of the method shown in [27] to the discrete time domain. Is this in fact the case?

3. The control problem given under Definition 2 does not explicitly refer to the content presented in subsection 2.2. Perhaps it would be better to introduce a new subsection.

4. Lines 98-100: The sentence concerning Schur stability sounds strange. In particular, I refer to the statement: "if bar{A}(zeta) is Schur stable ... , in other words, bar{A}(zeta) is Schur stable".

5. In what sense, and if at all, does the form of the matrix E_S introduced in Definition 1 relate to the form of the matrix E defined in the formula under line 188? If E is not related to E_S, it would be better to introduce another name for one of the matrices.

6. The passage contained in lines 164-170 presents a more detailed control problem. Unfortunately, it is under-exposed - in fact, it presents the expectations that the designed decentralised control is supposed to provide.

7. Lines 216-221: I feel that these descriptions are not consistent. The first two points seem to be about open system stability, while the last one is about control design and stability and refers to a closed system.

8. Lines 296-304: This description appears, as it were, in isolation from the earlier section considered in section 4.2. One gets the impression of a significant leap of thought. In essence, this is an algorithm for proceeding and as such should be better exposed.

9. The example shown in section 5, while valuable, requires further comment.
- It is not clear to me whether Case A considered for a nominal system takes into account any uncertainties. If not, does the consideration shown in this paper add anything new to the controller synthesis method?
- Why does the formula under line 324 only show matrix A for i = 1?
- Can Case B be illustrated with an elementary block diagram? Are the designations Am1 and Am2 correct? Should not it be Am_1 and Am_2?

The definition of the matrix \bar{A}_i in equation (4) is missing the sign \bar over the term A_{iMM}.

a) The definition of the matrix \bar{A}_i in equation (4) is missing the sign \bar over the term A_{iMM}.

b) Errors in punctuation are worth noting. In particular, the rules of punctuation are not followed for mathematical formulas, e.g. commas and full stops are missing at the end of equations.

c) Line 201: The definition of the He operator is hardly visible. In analogy with functions, it should be written in a straight rather than italic font - cf. equations (14), (24), (25) and equations under (26).
d) Line 151: unnecessary indentation of the line and no comma at the end of formula (8).
e) Line 334: the symbol Sc should be written in a font analogous to that in the definition (16).
f) Lines 390-398: There are text typesetting errors - Latex's "newline" command appears overtly.
g) Lines 336 and 338: scientific notation with e is not very readable (too much spacing)
h) Lines 343, 346, 366: A different font is used than in lines 336 and 338.

Author Response

Thank you for the revision. The answers are included in the attachment.

Author Response File: Author Response.pdf