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Analysis and Synthesis of Positive Systems Under ℓ1 and L1 by Xiaoming Chen

By Xiaoming Chen

This thesis introduces novel and important effects concerning the research and synthesis of confident structures, specifically less than l1 and L1 functionality. It describes balance research, controller synthesis, and bounding positivity-preserving observer and filtering layout for numerous either discrete and non-stop confident systems.

It in this case derives computationally effective strategies in keeping with linear programming when it comes to matrix inequalities, in addition to a couple of analytical ideas acquired for designated instances. The thesis applies more than a few novel techniques and basic suggestions to the extra research of optimistic structures, therefore contributing considerably to the speculation of confident structures, a “hot subject” within the box of keep an eye on.

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With A = [ai j ] ≥≥ 0 and C = [ch j ] ≥≥ 0, these constraints on k j can be written explicitly as ai j + bi k j ≥ 0, ch j + dh k j ≥ 0, i = 1, 2, . . , n, h = 1, 2, . . , r. It is easy to see that if bi = 0 or dh = 0, the corresponding constraint is slack. ,r ai j ch j , bi dh . 37). Hence, the first part of the result has been established. 4 Analytical Method for Special Case 29 When ρ(A + bK ∗ ) < 1, the system is asymptotically stable and (I − (A + bK ∗ ))−1 = ∞ (A + bK ∗ )i . i=0 For any K ≥≥ K ∗ satisfying ρ(A + bK ) < 1, we have (C + d K )(I − (A + bK ))−1 Bw + Dw = ∞ (C + d K )(A + bK )i Bw + Dw .

Moreover, it follows from Step 1 that if one cannot find such a matrix K 1 , then it can be concluded immediately that there does not exist a solution to Problem PPL1CD. 6 Note that problem in Step 1 is an LMI problem, which has a polynomial time complexity [6]. For LMI problems, the number N (ε) of flops needed to compute an ε-accurate solution is bounded by O(MN 3log(V/ε)), where M is the total row size of the LMI system, N is the total number of scalar decision variables, V is a data-dependent scaling factor, and ε is the relative accuracy set for the algorithm.

Step 2. For fixed K i , solve the following optimization problem for pi and γi . OP2: Minimize γi subject to the following constraints: 26 2 1 -Induced Controller Design for Positive Systems 1T (C + D K ) + piT (A + B K i ) − piT piT 0, Bw + 1 Dw − γi 1 0, pi ≥≥ 0. T T ∗ /γi∗ ≤ Denote γi∗ , pi as the solution to the optimization problem. If γi∗ − γi−1 ε1 , where ε1 is a prescribed bound, then K = K i , p = pi . STOP. • Step 3. For fixed pi , solve the following optimization problem for K i . OP2: Minimize γi subject to the following constraints: A + B K i ≥≥ 0, C + D K i ≥≥ 0, 1 (C + D K ) + T piT (A + B K i ) − piT 0.

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