A unitary approach on adaptive control synthesis IOAN URSU, ADRIAN TOADER Elie Carafoli National Institute for Aerospace Research Bdul Iuliu Maniu 220, Bucharest 061126 ROMANIA iursu@incas.ro , atoader@incas.ro , http://www.incas.ro/english/people/iursu/iursu.htm Abstract. The paper proposes a unitary approach of adaptive output feedback control for non-affine uncertain systems, about which the positive knowledge refers to the relative degree r. Given a reference model, the objective is to design a controller that forces the measured system output to track the reference model output with bounded error. The components of the so called pseudocontrol, thought on a superposition effects principle, are the following: 1) the output of reference model, 2) the output of a Kalman type stabilizing compensator of the pair of systems composed by a) an output dynamics of a set of integrators of order tantamount to the assumed known relative degree r of the controlled system and b) an internal model, of order r – 1, oriented to the tracking error decreasing in the presence of step input signals, and 3) the adaptive control designed to approximately cancel the error of approximate dynamic inversion by virtue of whom the real control is hereby determined from pseudocontrol. A single hidden layer neural network is used to counteract this dynamic inversion error. The common approach of pseudocontrol design based on tracking error dynamics estimation is evaded. A proof of stable working of this intelligent type controller is sketched. The mathematical model for the longitudinal channel of a hovering VTOL-type aircraft is used as framework of synthesis and validation by numerical simulations. Keywords: Uncertain systems, Adaptive control, Dynamic inversion, Neural network, Kalman synthesis, VTOL-type aircraft, Numerical simulation. 1. Introduction A way in treating the control of uncertain systems is the adaptive control. Research in this field is of particular importance, taking into account the emerging applications such as modern fighter and civilian aircrafts, unmanned aerial vehicles (UAV), flexible structures, robotics, flow physics, combustion processes and so on. Indeed, modelling for all these applications suffers of uncertainty, both in parameters and dynamics. To highlight the framework of the paper, let the dynamics of an observable [1] nonlinear single-input- single-output (SISO) non affine system be given by the equations ( 29 ( 29 x= f x,u ,y=g x & (1) where n x x D ∈ ⊂ R is the state vector, , uy ∈ R (for sake of simplicity) are the input signal (control), respectively, the output signal (measurement), and f , g are uncertain functions, sufficiently smooth; moreover, n need not be necessary prescribed! For this real or virtual system, e. g. a helicopter or its mathematical model, various problems are stated in control theory. Let’s consider such a problem: design (more specific, synthesize) a control law ( 29 y u , which uses the available measurement y , so that the measured and controlled output y to follow asymptotically a prescribed reference signal ( 29 r rm y t C ∈ (the class of continuous functions with continuous derivatives until de order r ). In fact, this is the problem of trajectory tracking for an airplane or rocket, for example. In addition, the control law u is subjected to saturating restrictions, M u u < . A premise of solving the problem is the ability of the artificial intelligence techniques – of neural networks (NNs), for example – in compensating the lack of system knowledge, in other works, in compensating the uncertainties in its modeling. For systems such as (1), let consider a relative degree r n < and the fulfilment of feedback linearization conditions stated in [1]. This means that a certain state coordinates transformation involving Lie derivatives ( 29 i f Lh will operate, and the system will be rewritten in the normal form ( ( 29 ( 29 ( 29 0 1 1 1, ..., 1 ,, , , , : i i r r r r f f i r g u y g u L g + ϕ= ϕ,ξ , ξ =ξ , ..., = - ξ= ξχ =ξ ξ, χ = & & & (2) χ is the state vector associated with the zero dynamics ( 29 0 0, f χ= χ & . These considerations are not hazardous, MATHEMATICAL METHODS, COMPUTATIONAL TECHNIQUES, INTELLIGENT SYSTEMS ISSN: 1790-2769 71 ISBN: 978-960-474-188-5