Robust $H_{\infty }$ control for a generic linear rational expectations model of economy

Abstract

Large time-delay and small disturbance attenuation are very important for macroeconomic system. This paper is concerned with the problem of robust $H_{\infty }$ control with large time-delay and small disturbance attenuation for a generic linear rational expectations model of economy with uncertainties, time-varying delay and random shocks. The norm bounded uncertainties are used to describe the uncertainties of economic system. The concept of two levels of conservatism and the approach of Parameters Weak Coupling Linear Matrix Inequalities (PWCLMIs) are developed in this paper. The result is presented in terms of PWCLMIs in this note. Large time-delay and small disturbance attenuation are achieved without increasing conservatism of result. Furthermore, according to the robust $H_{\infty }$ result, one will obtain various results readily by employing different games. An example is given to show the benefit of the presented approach.

Introduction

“Best policies can be evaluated, in theory at least, given an economy. But macroeconomists have only model economies at their disposal and necessarily these economies are abstractions. A concern then is that the model economy used to evaluate policy will provide poor guidance in practice. This leads to the search for policy that performs well for a broad class of economies. This is what robust control theory is all about.” The Nobel Prize-winning economist, Edward C. Prescott, wrote these sentences in the endorsements of book [1].

Robust control for economy has received attention since the early 1960s. In [2], ambiguity preferences of static environment are axiomatized as multiple priors, and decision-making with multiple priors can be represented as max–min expected utility. The static environment of [2] is extended to a dynamic context in [3], where the set of priors is updated over time and the dynamic consistent central axiom leads to a recursive structure for utility. The links between robust control and ambiguity aversion are formally established in [4], which shows that the model set of robust control can be thought of as a particular specification of the set of priors presented in [2], and once the least favorable prior is chosen, behavior could be rationalized as Bayesian with that prior. According to the literature [5], in the economics literature, the most prominent and influential approach to robust control is due to Hansen and Sargent (and their co-authors), which is summarized in their monograph [1]. Hansen–Sargent approach starts with a nominal model and uses entropy as a distance measure to calibrate the model uncertainty set. The principal tools used to solve Hansen–Sargent robust control problems are state-space methods [1], [6]. It needs to note that, all approaches mentioned above adopt a bounded “worst-case” strategy, or can be described as an $H_{\infty }$ problem.

Many of the ideas and inspiration for robust control in economics come from control theory [5]. With the development of robust control for economy, the robust control in control theory is developed very fast. Uncertainties, stochastic disturbances, time-varying or invariant delays, nonlinearities, which always appear in economic systems (see e.g. [7], [8], [9], [10], [11] and references wherein), are investigated sensitively in control theory. Robust stability of uncertain stochastic neural networks with time-delay is studied in [12], [13]. Robust absolute stability for a class of time-delay uncertain singular systems with sector-bounded nonlinearity is studied in [14]. Robust stability for a class of Lur’e singular system with state time-delays is studied in [15]. Robust $H_{\infty }$ output feedback control for uncertain stochastic systems with time-varying delays is studied in [16]. Robust $H_{\infty }$ control for uncertain singular time-delay systems is studied in [17]. Robust exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching is studied in [18]. Linear Matrix Inequality (LMI) approach is adopted in above works due to this approach can be readily checked by exploiting the standard Matlab LMI toolbox, and free-weighting matrices are introduced in some of the above works to reduce the conservatism of results. Unfortunately, although the upper bounds of delays in above works are fit for processing control in engineering, they are not large enough for economic systems. Because the time-delays of economic systems maybe from days to decades. For example, the period of American pork price oscillation is 4 years [19], [20], the average and range length of Kondratiev waves is 50 and from approximately 40–60 years [21], respectively.

Robust $H_{\infty }$ control condition with very large upper bound of time-delay and small disturbance attenuation for a class uncertain stochastic time-varying delay system has been presented by the authors in [22], however, we have not discussed the essence of conservatism fully.

Furthermore, because the LMI approach appeared in very recently, there are few literatures study the robust problem for economic system via LMI approach. One of the authors investigates the condition of stability for the economic discrete-time singular dynamic input–output model in [23]. Furthermore, a state feedback control condition for the economic discrete-time singular dynamic input–output model is presented in [24]. The free-weighting matrix technology has not been introduced into above literatures.

In this paper, we deal with the robust $H_{\infty }$ control with large time-delay and small disturbance attenuation problem for a generic linear rational expectations model of economy with uncertainties, time-varying delay and random disturbances. The norm bounded uncertainties are adopted to illustrate the uncertainties of economic model. The concept of two levels of conservatism of stability and control sufficient conditions is developed. This concept covers the previous concepts of conservatism. The approach of Parameters Weak Coupling Linear Matrix Inequalities (PWCLMIs) is developed. Robust $H_{\infty }$ control sufficient condition is obtained in terms of PWCLMIs, and two levels conservatism of the condition are low. So large time-delay and small disturbance attenuation can be achieved in this note. Furthermore, by using two-person zero-sum game, the $H_{\infty }$ control result of system is obtained too. An example is given to demonstrate the effectiveness and merit of presented method.

Notations: Throughout this paper, the superscript “T” denotes matrix transposition and the notation $X⩾Y$ (respectively, $X>Y$) where X and Y are symmetric matrices, means that X − Y is positive semidefinite (respectively, positive definite). $l_2[0\text{,}\infty )$ is the space of square integrable vector. The shorthand $\mathit{diag}\{M_1\text{,}{M}_2\text{,}\dots \text{,}{M}_n\}$ denotes a block-diagonal matrix with diagonal blocks being the matrices $M_1\text{,}{M}_2\text{,}\dots \text{,}{M}_n$. The asterisk * in a matrix is used to denote term that is induced by symmetry.