VII. CONCLUSIONThis note introduces and characterizes a new type of multiplier, which can be used for robustness analysis in L 1 .In order to highlight the main idea, we have restricted our presentation to systems with a single nonlinearity. An extension of the robustness result to (diagonally structured) multiple nonlinearities is straightforward by using diagonal L1-multipliers along with diagonal scaling. L 1 -multipliers can be employed for controller synthesis by using an iteration which switches between L 1 -controller synthesis and robustness analysis, similar to the synthesis in L2.Our characterization of L 1 -multipliers allows to numerically search for multipliers improving the robust performance conditions. In the interest of more efficient algorithms, the conditions involved have to be studied more thoroughly. REFERENCES[1] V. Balakrishnan, "Linear matrix inequalities in robustness analysis with multipliers," Syst. -trix norms for robust stability and performance with block-structured uncertainty," Int.
A new algorithm for the determination of reachability of a marking in generalized Petri nets is proposed . The algorithm is based on the matrix-equation approach . Given a Petri net, a so called complemente d Petri net is created that consists of the given Petri net and a complementary transition with input an d output places depending on the initial and target markings. Then the reachability task is reduced to the investigation of T-invariants of the complemented Petri net. Using a technique of search for T-invariants , the algorithm determines minimal-support T-invariants, that include the complementary transition, an d then calculates the finite set of their linear combinations . For each T-invariant, the algorithm tries t o create sequentially all reachability paths from the given initial marking to the target marking o r determines that there are no such paths at all . During creation of the reachability paths, the algorithm needs memory only for storing the reachability path being created . If it is sufficient to determine only one firing sequence that transforms the initial marking into the target one, then the algorithm may be terminated after successful creation of the first reachability path . I . INTRODUCTIONThe reachability problem is formulated as follows : for any Petri net PN, with an initial markin g M°, and for some other marking M, determine whether the relation M E R(PN, M°) is true, where R(PN, M°) is the reachability set of PN for its initial marking M° [1], The decidability of the reachability problem has been proved for a number of restricted classes of Petri nets, and there are efficient algorithms for such classes as acyclic Petri nets, marked graphs, and others [2 -4] . In ou r recent paper [5], a novel algorithm for determining the reachability of a marking in acyclic Petri net s is described . Using the matrix representation of a Petri net, the algorithm solves the fundamenta l equation of the Petri net, creates a temporal ordering relation between transition firings and uses thi s relation for determining the shortest sequence of transition firings transforming the given initia l marking into the target one .It has been shown that the reachability problem is decidable for generalized Petri nets as well [6] . The main contribution of the algorithm described in [6] is in proving that the reachability problem fo r generalized Petri nets (and for vector replacement systems in terms of which Petri nets can b e formulated) is decidable . However, the practical use of this algorithm is limited . Actually, the algorithm creates a series of so called regular constrained refined graphs each of which is a generalization of the basic coverability tree . As the author admits, the first refined graph woul d enumerate the whole reachability set of the given Petri net . To the best of our knowledge, no existin g Petri net software tool uses that algorithm because of its high time and space complexity .In practice, two different techniques are used most often to determine the reachability i n general...
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