Robust Identification of Uncertain Dynamical Systems where Adaptation is Impossible James T. Lo and Devasis Bassu Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 21250, U.S.A. e-mail: jameslo@umbc.edu Abstract - This paper shows that training with the risk- averting error criterion yields a robust system identifier in the presence of an uncertain environmental parameter that is impos- sible to adapt to. Numerical results comparing least-squares and risk-averting identifiers illustrate the efficacy of the proposed method. I. Introduction Robust control and signal processing have been intensively studied for averting excessively large or disastrous errors in the past 20 years [1], [7], [6], [2]. There have been three situations for which the robust pro- cessing has been used. First, an environmental parameter for the processor (i.e. controller and signal processor) is observ- able, but an adaptive processor is difficult to design. Second, the environmental parameter is unobservable. Third, the oper- ating environment involves a fine feature or a dynamics under- represented in the measurements or is very complex (e.g. non- linearity) so that an analytic closed-form solution is difficult if not impossible. The adaptive neural networks with long- and short-term memories described in [3], [4] is simple, systematic, general and effective for adaptation to an observable parameter and even allows adaptation to unobservable parameter and elim- inates the need for robust processing as long as the parame- ter stays constant long enough for a proper adjustment of the short-term memory. It is shown in [5] that fine features and under-represented dynamics can be treated effectively by ro- bust neural processors. This leaves open only the problem of processing in the presence of an uncertain environmental pa- rameter that is impossible to adapt to. We note that robust control and signal processing have been studied mainly for linear environment, and robust processors This work was supported in part by the National Science Foundation un- der Grant No. ECS0114619 and the Army Research Office under Contract DAAD19-99-C-0031. The contents of this paper do not necessarily reflect the position or the policy of the Government. Devasis Bassu is also with the Applied Research, Telcordia Technologies, 445 South Street, Morristown, NJ 07960. (e-mail: dbassu@telcordia.com). for the same have been developed. However, the same cannot be said about the nonlinear environments. Moreover, robust- ness in the literature usually refers to that with respect to the pessimistic H ∞ and minimax criteria. Although risk-sensitive criterion has been used, the idea of different degrees of robust- ness has not been found in the control and signal processing literatures by the present authors. The purpose of this paper is to show that the foregoing re- maining problem can be easily but effectively resolved by the use of a risk-averting error criterion that is slightly but signif- icantly different from the ordinary risk-sensitive criterion. An adaptive training method based on the risk-averting criteria to train robust neural networks is used here. A description of the method can be found in a companion paper also presented at the IJCNN’02 by the authors. A fundamental problem used for illustration is the identifi- cation of a dynamical system, called a plant, y t = f (y t-1 ,...,y t-p ,u t ,...,u t-q ,ξ t ,c) A (1) in the presence of an environmental parameter c that changes so fast that adaptation to it is impossible, where u (t) and y (t) are the system input and output of the dynamical system at time t; and ξ (t) is a random driver that has unbiase effect on the system output. The primary objective is to compare the performances of the neural networks (NNs) trained as identifiers of (1), called neural identifier, with respect to the RA (risk-averting) error criterion J λ,p and the risk-neutral error criterion J 0,p (i.e. L p ) given below: J λ,p (w) =  ω∈S T  t=1 exp λ |e (t, w, ω)| p (2) J 0,p (w) =  ω∈S T  t=1 |e (t, w, ω)| p (3) e (t, w, ω) := y (t, ω) - ˆ f (t, w, ω) (4) where ˆ f (t, w, ω) denotes the output of the neural identi- fier with weights w subject to the input u (t, ω); u (t, ω) 0-7803-7278-6/02/$10.00 ©2002 IEEE