Situación antes de las reformas

Process and Performance-Sparsity Trade-Off

For previously-mentioned 20 × 20 randomly distributed system and the same setup for all parameters except τ , by considering the 26 equidistant values for τ (with step

(a)

(b)

Figure 5.2: (a) Sparsity visualization of K for τ = 0 (b) Sparsity visualization of K for τ = 9. Blue dots represent the non-zero elements.

size 0.5), cardinality percentage

kKk0

kKLQR_k 0

× 100,

and performance loss percentage

J∗− JLQR

JLQR × 100,

are visualized versus time-delay τ in Figures 5.3(a) and 5.3(b), respectively. The superscript LQR is utilized to denote the quantities related to standard traditional LQR design corresponding to A when there is no time-delay (τ =0).

As Figures 5.3(a) and 5.3(b) demonstrate, as time-delay τ increases, cardinality percentage and performance loss percentage get increased. Thus, it is observed that

the larger time-delay we have, the poorer quality of sparsification process we get. To assess the effect of time-delay on performance-sparsity trade-off, considering the same 20 × 20 randomly distributed system and choosing the fixed time-delay from set {0, 2.5, 5, 7.5, 10, 12.5}, we run BRPTS Algorithm for 20 logarithmically spaced sparsity-promoting parameter γ ∈ [10−4, 10−1] which leads to plots depicted by Figure 5.4.

As Figure 5.4 showcases, when time-delay τ gets larger, the performance-sparsity trade-off gets worse. In other words, prescribing a fixed value of cardinality percentage (proportional to number of controller communication links), having a larger time- delay leads to higher performance loss which is not desired.

5.7

Conclusion

Considering the class of uncertain linear time-delay systems, the sparse memoryless LQR design is presented. Utilizing the LMI techniques, deriving the equivalent rank- constrained reformulation, and applying the bi-linear rank penalty technique, sub- optimal sparse memoryless LQR design is achieved. Employing the various numerical experiments, the negative effect of constant time-delay on sparsification process and performance-sparsity trade-off is observed. The improvement of sub-optimality level of utilized technique can be seen as a possible future work.

τ -2 0 2 4 6 8 10 12 14 Cardinality Percentage 0 5 10 15 20 25 (a) τ -2 0 2 4 6 8 10 12 14

Performance Loss Percentage

0 10 20 30 40 50 60 70 80 (b)

Cardinality Percentage

0 20 40 60 80 100

Performance Loss Percentage

0 10 20 30 40 50 60 70 80 90 100 τ = 0 τ = 2.5 τ = 5 τ = 7.5 τ = 10 τ = 12.5

Figure 5.4: Performance-Sparsity trade-off for different values of 6 uniformly selected time-delay τ and 20 logarithmically spaced sparsity-promoting parameter γ ∈ [10−4, 10−1].

Chapter 6

Row-Column Sparse Linear

Quadratic Controller Design via

Bi-Linear Rank Penalty Technique

and Non-Fragility Notion

6.1

Introduction

Sparsity-Promoting control problems can be categorized as two important classes: (i) sparse controller design,

(ii) row/column sparse controller design.

Some examples of the first category can be found in [3, 18, 19, 21, 26, 27, 30, 32, 42, 68, 89]. On the other side, row/column sparse controller design is investigated by [10, 28].

In this chapter, focusing on the second category of sparsity-promoting control problems, i.e., row/column sparse controller design, we consider row-column (r, c)- sparse controller design. In such a design problem, each node will communicate to

at most r other nodes and information of each node will be used by at most c other nodes. While in [10] the definition of row/column sparsity is different from ours and [28]’s, the new contribution of our work compared to [10] is capability of having row and column sparsity at the same time with possibly distinct values for r and c. Also, in comparison with [28], instead of using majorization theory and computationally expensive algorithm, we utilize the non-fragility notion provided by [33] to have a purely utilized bi-linear rank penalty technique and relatively fast algorithm. It is also remarkable that numerical simulations provided by [28], do not satisfy the sparsity constraints for all rows or columns. As we will see in our simulation section, all the sparsity constraints hold for all rows and all columns.

The chapter is arranged as follows: The section 6.2 is dedicated to explain our mathematical notations which are used along the chapter. In Section 6.3, we express the problem which we aim at solving. In Section 6.4, we show how our problem can equivalently be translated to an optimization problem constrained to several linear matrix inequalities and m + n + 1 rank constraints. Section 6.5 provides the vision to our bi-linear rank penalty technique and its details. Our numerical simulations are visualized for the class of randomly-generated systems in Section 6.6. At last, we finish the chapter with drawing some future directions in Section 6.7.

6.2

Mathematical Notations

Throughout the chapter, matrices are denoted by capital letters, and the elements are shown by capital letters with subscripts. The vectors, on the other hand, are represented by lower-case letters, with elements denoted by the same letter with subscripts. The identity matrix of size n × n is denoted by In. The Hadamard matrix

product is denoted by ◦. The number of non-zero elements of a matrix is denoted by k.k0. The `2 norm of a matrix is represented by k.k2. The notationkXkmaxrepresents

the maximum absolute value of all elements of matrix X. The trace operator is denoted by Tr(.) and the rank operator is demonstrated by rank(.). The block diagonal matrix construction operator is shown by diag(.). The operator sign(.) is used to take element-wise sign of a matrix. The element-wise comparison between two matrices is denoted by usual ≥. A matrix is called Hurwitz if its all eigenvalues lie in the open left half of the complex plane. The set of n × 1 real vectors and m × n real matrices are represented by Rn and Rm×n, respectively. A symmetric matrix is called positive definite (positive semi-definite) if all of its eigenvalues are positive (non- negative). The space of positive definite (positive semi-definite) matrices is denoted by Sn

++ (Sn+) and X − Y ∈ Sn+ (X − Y ∈ Sn++) is symbolized with X  Y (X  Y ).

The normal distribution with zero mean and unit variance is represented by N (0, 1) and the expected value is represented by E. The ith _{row and j}th _{column of matrix X}

are shown by X(i, :) and X(:, j), respectively.

Definition 5. For a given  ≥ 0 and matrix X, rank of X is k with tolerance  and denoted by rank(X; ), if exactly k singular values of X are greater than .

6.3

Problem Formulation