Output Feedback Multiobjective Cumulant Control with Structural Applications Ronald W. Diersing, Michael K. Sain, Khanh D. Pham, and Chang H. Won Abstract— Recently, a cumulant generalization of H2/H∞ control has been given for the state feedback problem. This paper extends those results to the output feedback case. The Nash game approach to the H2/H∞ problem is used. Sufficient conditions for two problems are determined. In the first problem the one player, the control, has only partial state information, while the other player, the disturbance, has full state information. In the other problem both players only have information based upon estimates of the state. Coupled Riccati equations for both cases are given, along with equilibrium solutions. The results are also applied to the first generation structural benchmark for buildings under seismic excitation. I. I NTRODUCTION Cumulants have been used in control with encouraging results [9], [10], [11], [13]. Their use has been particularly interesting for structural vibration problems. Furthermore, cumulants have also been used in game theory [3], [9]. In [3], H 2 /H ∞ was generalized by the use of cumulants and the Nash game. These problems were developed for the case in which both players had full state feedback information. However, the full information of the state is not always available for feedback. Actually, quite often, only a set of output signals, which do not give full information about the states, is available. Here, we assume that the players do not have full state feedback information. We will first give a definition of the problem. In previous work on full state feedback, [3], the problem developed from a nonlinear system with non- quadratic costs. Here, however, the linear system, quadratic cost assumption will be taken from the beginning. With the system defined, the first problem that will be discussed is one in which the control wishes to minimize a linear combination of k cumulants of its cost function, based upon state estimates, while the disturbance has full state information available for its decision. Then, the case in which both players only have information from state estimates will be examined. Finally, output feedback 3-cumulant multi- objective control will be applied to the first generation structural benchmark. R. Diersing is a postdoctoral fellow with the Department of Electrical and Computer Engineering, Temple University, Philadelphia, PA 19122. This work was supported in part by the Center of Applied Mathematics at the University of Notre Dame and the Arthur J. Schmitt Fellowship. rdiersin@temple.edu M. Sain is Freimann Professor of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 avemaria@nd.edu K. Pham is with Dynamics and Controls, Air Force Research Laboratory, Kirtland AFB, NM 87117 khanh.pham@kirtland.af.mil C. Won is Assistant Professor, Department of Electrical and Computer Engineering, Temple University, Philadelphia, PA 19122 cwon@temple.edu II. PROBLEM DEFINITION In the previous work [3], the problem began in a nonlinear framework with non-quadratic costs. However, in the output feedback case, the linear system and quadratic cost case will be considered from the onset. The linear system in question is given by dx(t)=(A(t)x(t)+ B(t)u(t)) dt + D(t)w(t)dt + E(t)dξ (t) (1) with x(t 0 )= x 0 being known. The matrices A, B, D, and E are n×n, n×m, n×p, and n×q respectively, with continuous entries on the interval [t 0 ,t f ]. The process noise ξ is a Wiener process on (Ω, F ,P ). Also, ξ has an autocorrelation function of E{[ξ (t) - ξ (τ )][ξ (t) - ξ (τ )] ′ } = W |t - τ |. The control will have only measurements available, not the full state information that is available to the other player, the disturbance w. The measurements y are given by dy(t)= C(t)x(t)dt + dv(t) (2) where C is a n × r matrix with continuous entries on the in- terval [t 0 ,t f ] and v is a Wiener process with autocorrelation E{[v(t) - v(τ )][v(t) - v(τ )] ′ } =Ξ|t - τ |. The crosscorrelation between ξ and v is given as E{[ξ (t) - ξ (τ )][v(t) - v(τ )] ′ } =Γ|t - τ |, where W - ΓΞ −1 Γ ′ > 0. The cost functions are J 1 (t 0 ,x 0 ; u, w)=  t f t0 ||z 1 (t)|| 2 dt J 2 (t 0 ,x 0 ; u, w)=  t f t0 ( δ 2 ||w(t)|| 2 - ||z 2 (t)|| 2 ) dt (3) where δ is a positive constant and the minimization of the mean of J 2 by w corresponds with constraining the H ∞ norm, ||T z2w || ∞ ≤ δ, see [2], [8], also for the state feedback cumulant case see [3]. Also, z 1 ,z 2 are regulated outputs given by z 1 (t)=C 1 (t)x(t)+ D 1 (t)u(t) z 2 (t)=C 2 (t)x(t)+ D 2 (t)u(t). (4) Furthermore for i =1, 2, the matrices C i ,D i satisfy C ′ i C i = Q i ,C ′ i D i =0,D ′ i D i = R i . It will be assumed that R 1 is a positive definite matrix.