ISSN 0005-1179, Automation and Remote Control, 2017, Vol. 78, No. 8, pp. 1417–1429.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
D.V. Balandin, M.M. Kogan, 2017, published in Avtomatika i Telemekhanika, 2017, No. 8, pp. 76–90.
Pareto Optimal Generalized H
and Vibroprotection Problems
D. V. Balandin
and M. M. Kogan
Lobachevsky Nizhny Novgorod State University, Nizhny Novgorod, Russia
Nizhny Novgorod State University of Architecture and Civil Engineering,
Nizhny Novgorod, Russia
Received October 25, 2016
Abstract—We consider a novel multi-objective control problem where the criteria are general-
-norms of transfer matrices of individual channels from the disturbance input to various
objective outputs. We obtain necessary conditions for Pareto optimality. We show that syn-
thesis of Pareto optimal controls can be done in terms of linear matrix inequalities based on
optimizing Germeier’s convolution, which also turns out to be the generalized H
-norm of the
closed-loop system with output composed of the objective outputs multiplied by scalars. As ap-
plications we consider multi-objective problems of vibration isolation and oscillation suppression
with new types of criteria.
Keywords: multi-objective, Pareto set, Pareto optimal controls, Germeier convolution, general-
-norm, vibroprotection, vibration isolation, oscillation suppression.
Real life control problems are always multi-objective, and this is perfectly true for the problem
of reducing the inﬂuence of external disturbances on various outputs of a controllable object. One
of the ﬁrst publications where multi-objective optimization techniques were used to design a Pareto
optimal control were the works  for a linear-quadratic Gaussian control and , where the Pareto
set of optimal controllers in a multi-objective problem in the form of the H
-norm was characterized
in terms of solutions of Riccati equations under Youla parametrization and a scalar multi-objective
function in the form of a linear convolution of criteria.
Although over the last decade signiﬁcant progress has been achieved in solving optimal control
problems with H
-norms as criteria, which have clear physical interpretations in the
form of suppression levels for deterministic or stochastic disturbances from various classes, it still
presents many obstacles to consider multi-objective problems with these criteria. These obstacles
are primarily related to the complex characterization of the Pareto set and ﬁnding the corresponding
scalar multi-objective function that would deﬁne this set. Besides, the problem is complicated by
the fact that each criterion is characterized by its quadratic Lyapunov function with a matrix
which is a solution of Riccati equations or linear matrix inequalities, and scalar optimization of
the multi-objective function in the form of some convolution leads, in the general case, to bilinear
deterministic inequalities with respect to matrices of these Lyapunov functions and the controller’s
feedback matrix. To solve such a system, researchers usually imposed an additional condition that
all Lyapunov functions are equal, which introduced conservatism into the problem [3–7]. At the
same time, the main question, namely how much the resulting control laws diﬀer from Pareto
optimal ones, remained unanswered.