An LMI approach to peak-to-peak gain minimisation: filtering and control

Mixed-norm H2/L1 controller synthesis via fixed-order dynamic compensation: A Riccati equation approach

“…Furthermore, referring to Schweppe' s work, within the context of L ¥ induced signals, the L 1 norm bound (41) also appears without proof in Nagpal et al (1994) and Sznaier et al (1995).…”

“…However, as in the pure discrete-time z 1 case (Dahleh and Pearson 1987a, McDonald and Pearson 1991, Sta ans 1991, Dahleh and Diaz-Bobillo 1995 the resulting mixed H 2 / z 1 controller order could be arbitrarily high. In recent research Sznaier et al (1995) posed a constrained (non-optimal) mixed H 2 bound/L 1 bound control problem using the linear matrix inequality feasibility framework developed by Nagpal et al (1994). However, since the approach of Sznaier et al (1995) is based on a matrix inequality feasibility problem, there are no optimality guarantees of the resulting mixed-norm H 2 bound/L 1 bound controller.…”

“…Even though recent work on suboptimal L 1 controllers Sznaier 1994, Wang et al 1995) yield rational (® nite dimensional) controllers (by solving a particular discrete-time z 1 problem) that guarantee L 1 performance arbitrarily close to the optimal L 1 performance, the order of these controllers may be very high. In order to minimize controller complexity a new and novel approach to point-wise-in-time peak-to-peak gain minimization was formulated by Nagpal et al (1994). Speci® cally, by approximating the closure of the set of reachable closed-loop system states due to unit-peak system disturbances by an ellipsoid (Schweppe 1973), Nagpal et al (1994 obtain an upper bound to the L 1 norm ( L ¥ equi-induced norm with Euclidean spatial norms as opposed to spectral spatial norms).…”

“…In order to minimize controller complexity a new and novel approach to point-wise-in-time peak-to-peak gain minimization was formulated by Nagpal et al (1994). Speci® cally, by approximating the closure of the set of reachable closed-loop system states due to unit-peak system disturbances by an ellipsoid (Schweppe 1973), Nagpal et al (1994 obtain an upper bound to the L 1 norm ( L ¥ equi-induced norm with Euclidean spatial norms as opposed to spectral spatial norms). In this case the L 1 synthesis problem reduces to the evaluation of a linear matrix inequality feasibility problem.…”

“…Speci® cally, these multiobjective problems are treated by forming a convex combination of the L 1 norm bound proposed by Nagpal et al (1994) and the H 2 closed-loop system norm. The proposed approach is reminiscent of scalarization techniques for Pareto optimization (Haddad and Bernstein 1990).…”