Liu Hsu and Ramon R. Costa
Department of Electrical Engineering
COPPE/Federal University of Rio de Janeiro
C.P. 68504, 21941-972, Rio de Janeiro, RJ, Brazil
Abstract
One basic difficulty of the classical design of stable multivariable MRAC, or MIMO (Multi-Input-Multi-Output) MRAC, is the requirement of a symmetry condition for the high frequency gain matrix. This has been noticed for quite long time. The symmetry condition is not generic once arbitrarily small parametric perturbations can destroy it. Some solutions to overcome this important restriction have been proposed: (a) static compensation to symmetrize the high frequency gain K_p by means of a multiplier matrix; (b) factorization method which assumes the existence of a Symmetric-Diagonal-Upper triangular (SDU) factorization of K_p to obtain a new parameterization of the control law; (c) the Immersion and Invariance approach (I & I) which requires the knowledge of a less restrictive multiplier than in (a); (d) more recently, a multiplier derived from a more general concept of passivity, named W-strict passivity (WSP). While the usual strict passivity condition is associated with the SPR condition, WSP is associated with the corresponding WSPR condition. Solution (a) is not generally applicable to systems with parametric uncertainties in K_p since the symmetry condition is not generic. Compared to the so called SPR-Lyapunov design, which leads to a simple unnormalized gradient update law, solutions (b) and (c) overcome this difficulty under weak assumptions about K_p, but require either additional overparameterization of the control law or additional filtering and auxiliary control signals. Solution (d), discussed in more detail in what follows, is less general than (b) or (c) but has reduced complexity, consisting of a straightforward extension of the SISO (Single Input-Single Output) SPR-Lyapunov design. A simple necessary and sufficient condition to render the plant WSPR under static output feedback was established and can be briefly stated as K_p having real eingenvalues and diagonal Jordan form. Departing from some nominal assignment of〖 K〗_p, a constant matrix multiplier can be easily obtained so that WSPR can be achieved in a robust way. Also, it was verified that the violation of such condition can lead to system instability. One advantage of using the WSPR approach to design MIMO MRAC is that the overparametrization introduced by the factorization method is avoided. The adaptation law is a direct extension of the SISO Lyapunov design. The WSPR approach was recently applied to design a Multivariable Binary MRAC which exhibits good transient and robustness properties based on fast adaptation and parameter projection. WSPR, similarly to SPR systems must have uniform relative degree one. Preliminary extensions to plants with arbitrary relative degree were also developed.
Biographies
Liu Hsu was born in Hunan, China. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Instituto Tecnológico de Aeronáutica (ITA), São José dos Campos, Brazil, in 1968 and 1970, respectively, and the Docteur d’Etat ès Sciences Physiques degree from the Université Paul Sabatier, Toulouse, France in 1974. His doctoral research was developed at the Laboratorie d’Automatique et d’Analyse des Systèmes — CNRS, Toulouse. In 1975, he joined the Graduate School and Research in Engineering of the Federal University of Rio de Janeiro. His current research interests include adaptive control systems, variable structure systems, nonlinear systems dynamics and their applications to industrial process control, nanopositioning, industrial robotics, and terrestrial and underwater robotics. He is a member of the Sociedade Brasileira de Automática (SBA) and a Senior Member of the IEEE Control Systems Society. Liu Hsu is a member of the Brazilian Academy of Sciences. He has served as an associate editor of the scientific journal of the SBA “Controle & Automação”, until 2010, of the Annals of the Brazilian Academy of Sciences and of the IET-Control Theory and Applications. He is a member of the IEEE CSS Technical Committee on Variable Structure and Sliding Mode Control. In 2005, he was nominated to the Brazilian National Order of Scientific Merit.
Ramon R. Costa was born in São Paulo, Brazil in 1956. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Federal Engineering School of Itajubá, Brazil, in 1979 and 1982, respectively, and the D.Sc. degree in electrical engineering from the Federal University of Rio de Janeiro, Brazil, in 1990. Currently, he is an Associate Professor in the Department of Electrical Engineering of COPPE at the Federal University of Rio de Janeiro. In the period 1999–2000 he was a visiting scholar in the Center for Control Engineering and Computation at the University of California, Santa Barbara (UCSB). His research interests include adaptive control, sliding mode control, and robotics. He is a member of the IEEE Control Systems Society and the Sociedade and Brasileira de Automática.
