Uncertainty in system identification: learning from the theory of risk ⋆ Patricio E. Valenzuela ∗ Cristian R. Rojas ∗ H˚akan Hjalmarsson ∗ ∗ Department of Automatic Control, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden (e-mail: {pva, crro, hjalmars}@kth.se). Abstract: This article addresses the issue of measuring uncertainty in optimization problems arising in system identification. The issue of uncertainty has been studied in the theory of risk, where the results are mainly employed in finance applications. Here we explore how the results in the literature of theory of risk can be used to obtain a systematic approach to uncertainty in system identification. For concreteness, the discussion is illustrated by an application to input design, but it can be extended to other areas of the field. Keywords: System identification, input design, theory of risk, uncertainty. 1. INTRODUCTION Uncertainty is an issue common to many research areas. By uncertainty we understand the lack of knowledge to fully describe a phenomenon. The lack of knowledge causes severe difficulties when we are interested in determining the best decision with limited information. Examples of this problem can be found in control design, where the con- troller must be designed with limited information about the plant dynamics (Zhou and Doyle, 1998; Zhou et al., 1996), and in portfolio optimization, where the returns are maximized subject to limited information about the future evolution of the assets price (Bertsimas and Thiele, 2006; Black and Litterman, 1992; Krokhmal et al., 2002; Perold, 1984; Postek et al., 2014). In the same line, many problems arising in system iden- tification are solved with limited information. In system identification, the uncertainty can be understood as the lack of knowledge about the true dynamics of the process to be modeled. The uncertainty associated with the pro- cess dynamics is of importance in applications where the optimal decision depends on the true model description. This is the case in input design, where the optimal input sequence depends on true process dynamics (Ljung, 1999). Approaches to solve this issue have been presented in the literature, which can be classified in two classes: (i) sequential or adaptive procedures, where a new design is obtained based on the current estimates of the process dynamics (Gerencs´er and Hjalmarsson, 2005; Gerencs´er et al., 2009; Lindqvist and Hjalmarsson, 2001; Pronzato, 2000); and (ii) robust procedures, where the design is obtained by including the uncertainty in the optimization problem (Jansson and Hjalmarsson, 2005; Pronzato and Walter, 1985; Rojas et al., 2007). In this article, we are interested in addressing the uncertainty in input design by using the robust approach. The robust approach to uncertainty in system identifi- cation has been analyzed in the literature, and several techniques have been proposed (Larsson et al., 2012; ⋆ This work was supported by the Swedish Research Council un- der contracts 621-2011-5890 and 621-2009-4017, and by the Euro- pean Research Council under the advanced grant LEARN, contract 267381. M˚artensson and Hjalmarsson, 2006; Rojas et al., 2012). The main idea behind these results is the inclusion of a mapping from the space of functions with uncertainty to a scalar value. The scalar value takes into account the uncer- tainty associated with the process dynamics. Some exam- ples of the mappings employed are the expected value, and the supremum over the set of possible descriptions of the process dynamics. However, there is no analysis of how well the mappings address the issue of uncertainty in system identification. By how well we mean if the mapping is either a weak or a conservative measure of the uncertainty in the optimization problem. The problem of properly measuring the uncertainty has been addressed in the theory of risk measures. In the theory of risk, the uncertainty is understood as the risk associated with the portfolios (Cram´er, 1930). To properly measure the risk associated with the portfolios, the notion of coherent measure of risk has been introduced (Artzner et al., 1999). Some of the requirements for a functional to be a coherent measure of risk are the convexity and monotonicity, which imply that the resulting optimization problem is convex if the original problem with uncertainty is convex. In addition, a coherent measure of risk en- courages diversification, i.e., it is always better to invest in several assets rather than in a single one, which is a property usually required by investors to reduce the risk associated with the portfolios. In this article we explore the connection between uncer- tainty in the theory of risk and uncertainty in system identification. In particular, we discuss how the notion of a coherent measure of risk can be employed to obtain a systematic approach to uncertainty in system identifica- tion. In addition, we introduce a coherent measure of risk that can be useful in system identification: the conditional value at risk (CVaR) (Rockafellar and Uryasev, 2000). The usefulness of coherent risk measures will be illustrated by its application to input design. However, we emphasize that the applicability of this approach is not only limited to this topic, it can also help to address the uncertainty issue in other areas (e.g., a theory of risk approach has already been used in Markov control processes and model predictive control, see Chow and Pavone (2013, 2014); Shen et al. (2014)).