How Robust is Robust Control in Discrete Time?
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How Robust is Robust Control in Discrete Time? Marco P. Tucci1 Accepted: 16 July 2020 © The Author(s) 2020
Abstract By applying robust control, the decision maker wants to make good decisions when his model is only a good approximation of the true one. Such decisions are said to be robust to model misspecification. In this paper it is shown that, in many situations relevant in economics, a decision maker applying robust control implicitly assumes that today’s worst-case adverse shock is serially uncorrelated with tomorrow’s worst-case adverse shock. Then, further investigation is needed to see how strong is the ‘immunization against uncertainty’ provided by these popular frameworks. Keywords Linear quadratic tracking problem · Optimal control · Robust optimization · Time-varying parameters JEL Classification C61 · C63 · D81 · D91 · E52 · E61
1 Introduction A characteristic “feature of most robust control theory”, observes Bernhard (2002, p. 19), “is that the a priori information on the unknown model errors (or signals) is nonprobabilistic in nature, but rather is in terms of sets of possible realizations. Typically, though not always, the errors are bounded in some way…. As a consequence, robust control aims at synthesizing control mechanisms that control in a satisfactory fashion (e.g., stabilize, or bound, an output) a family of models”.1 Then “standard control theory tells a decision maker how to make optimal decisions when his model is correct (whereas) robust control theory tells him how to make good decisions when 1 Robust control has been a very popular area of research in economics in the last two decades and shows no sign of fatigue. See, e.g., Giannoni (2002, 2007), Hansen and Sargent (2001, 2003, 2007a, 2007b), Hansen et al. (1999, 2002), Onatski and Stock (2002), Rustem (1992, 1994, 1998), Rustem and Howe (2002) and Tetlow and von zur Muehlen (2001a, b). However the use of the minimax approach in control theory goes back to the 60’s as pointed out in Basar and Bernhard (1991, pp. 1–4).
* Marco P. Tucci [email protected] 1
Dipartimento di Economia Politica e Statistica, Università di Siena, Piazza S. Francesco, 7, 53100 Siena, Italy
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his model approximates a correct one” (Hansen and Sargent 2007a, p. 25). In other words, by applying robust control the decision maker makes good decisions when it is statistically difficult to distinguish between his approximating model and the correct one using a time series of moderate size. “Such decisions are said to be robust to misspecification of the approximating model” (Hansen and Sargent 2007a, p. 27).2 Tucci (2006, p. 538) argues that “the true model in Hansen and Sargent (2007a) … is observationally equivalent to a model with a time-varying intercept.” In the sense that, unless some prior information is available, it is impossible to distinguish between the two models by simply observing the output. Then he goes on showing that, when the same worst-case adverse shock and objective functional are used in both procedure
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