Synthesis of Controllers for Modal Shaping in Linear Parameter-Varying Systems via the Implicit Model Following Formulation Paulo C. Pellanda † Pierre Apkarian ‡ † IME, Electrical Engineering Dept., Rio de Janeiro, Brazil - E-mail: pellanda@ime.eb.br ‡ ONERA-CERT, Control System Dept., Toulouse, France - E-mail: apkarian@cert.fr Abstract The control synthesis problem involving Implicit Model Following (IMF) is considered in the context of Linear Parameter-Varying (LPV) and H 2 /H ∞ theories. The well-known quadratic or nominal H 2 IMF problem is first extended to encompass LPV system models with a Linear Fractional Transformation (LFT) structure. This problem is then embedded in the framework of LPV theory. Conditions for dealing with additional mixed H 2 /H ∞ criteria are discussed. The solvabil- ity conditions are provided with little conservatism by a previous multi-channel LFT/LPV result in discrete time. Finally, an illustrative example is used to validate this new formulation. Also, we demonstrate through this example that the IMF formulation is an effective technique to achieve a desired transient behavior for LPV systems. 1 Introduction While most standard methods for robust control de- sign of Linear Time-Invariant (LTI) systems focus on frequency domain specifications, in a number of appli- cations many performance specifications are explicitly stated in time domain in terms of qualities of tran- sient responses and internal state decoupling. It is well-known from classical control theory that the main properties of the time responses can be reflected in the frequency domain. Therefore, the performance objec- tives are often taken into account by choosing an appro- priate synthesis structure and tuning frequency weight- ing functions, filters and/or dynamic scalings. Hence, the application of robust control design methods can lead to a large amount of trial-and-error before obtain- ing satisfactory conventional specifications in terms of time-domain properties. Some robust synthesis methodologies, as those based on H ∞ model matching schemes and on robust pole placement approaches, handle time-domain specifica- tions in a more explicit way. See, for instance, the references [11, 10, 6] and [3]. However, extra diffi- culties appear when non-stationary or nonlinear sys- tems are considered, since they cannot be appropri- ately represented in the frequency domain. The pole notion no longer holds for these systems and some re- quired transient properties are met only for slowly vary- ing conditions. Moreover, because of excessive con- servatism, these techniques are often restrictive in the multi-objective control and Linear Matrix Inequalities (LMI) [2, 5] contexts. Another drawback of the H ∞ model matching methods is that they generally pro- duce high order controllers. In reference [8], the authors present an alternative ap- proach to deal with the control problem involving as- signment of closed-loop modal shapes. The LTI IMF results of [7] are extended to the dynamic feedback case and reformulated in the H 2 context. In this method, time domain specifications are readily reflected in a quadratic criterion that penalizes the error between a desired dynamic behavior and that of the closed-loop system. The purpose of this paper is to study the problem of achieving precise and robust time-domain specifi- cations on specific states of non-stationary LPV sys- tems using IMF and a multi-channel LFT/LPV control method. 2 Problem Statement Consider a continuous-time LPV plant with LFT struc- ture    ˙ x(t) z Δ (t) z(t) y(t)    =    A B Δ B 1 B 2 C Δ D ΔΔ D Δ1 D Δ2 C 1 D Δ1 D 11 D 12 C 2 D 2Δ D 21 D 22       x(t) w Δ (t) w(t) u(t)    w Δ (t) = Δ(t) z Δ (t), (1) where A ∈ R n×n , Δ(t) ∈ R N×N , D 12 ∈ R p1×m2 and D 21 ∈ R p2×m1 define the problem dimension. The no- tation for signals is standard: x for the state vector, w for exogenous inputs, z for controlled or performance variables, u for the control signal, and y for the mea- 0-7803-7896-2/03/$17.00 ©2003 IEEE 5161 Proceedings of the American Control Conference Denver, Colorado June 4-6, 2003