Set-Based Methodology for White Noise Modeling"

This paper provides a new framework for analyzing white noise disturbances in linear systems: rather than the usual stochastic approach, noise signals are described as elements in sets and their effect is analyzed from a worst-case perspective. The paper studies how these sets must be chosen in order to have adequate properties for system response in the worst-case, statistics consistent with the stochastic point of view, and simple descriptions tha t allow for tractable worst-case analysis. The methodology is demonstrated by considering i ts implications in two problems: rejection of white noise signals in the presence of system uncertainty, and worst-case system identification. A general feature of mathematical models in engineering science is the presence of modeling errors, which arise due to poorly understood or highly unpredictable phenomena, or from simplifications deliberately introduced for the sake of model tractability. Essentially two approaches are available t o assess the consequences of this error: one is to model the uncertainty in terms of a set of allowable perturbations and perform worst-case analysis over this set; the other is to assign the additional structure of a probability measure to the error, and perform analysis in the average. Vncertainty is often the dominant issue in models used for control system design. These models involve substantial approximations (linearizations. unmodeled dynamics) and uncertain parameter values, all of which lead to systematic modeling errors for which the only natural characterization is based on sets. Also, the issue of stability provides an incentive to take the worst-case point of view. This has been the strategy of robust control theory, which has developed mathematical tools for the evaluation of stability and performance in the worst case over sets of systems. In this theory, the methodology based on sets is also applied to disturbance signals (another source of uncertainty), by modeling them in terms of a ball in some signal space (e.g. C2, Cm), which motivates the Em or C1 criteria for worst-case disturbance rejection. The main motivation for these disturbance models is mathematical convenience, since these performance measures can be directly combined with set descriptions of system uncertainty to analyze robust performance (see, e.g. [15]). This approach for disturbance modeling is pessimistic, however, since it ignores a substantial amount of information about empirical disturbances. It is often the case that these exhibit broadband spectral characteristics (white noise, or some filtered version), especially when they describe the cumulative macroscopic effect of very high dimensional fluctuations a t the microscopic level. The statistics of these phenomena have been very accurately modeled by the theory of stochastic processes. The systematic study of the properties of dynamical systems under stochastic noise, pursued by stochastic control theory, often leads to tractable results, the most notable being the classical 'H2 (LQG) problem. The main limitation to its applicability is that noise is rarely the prevailing source of uncertainty, and the others do not fit easily into a stochastic description (1201 contains some work in this direction). The robust performance question one would really want to address in many practical cases is the effect of white noise over sets of systems (the "Robust E2" problem). Many authors (see 125, 71 and references therein) have addressed this problem in terms of a direct combination of the worst-case and stochastic frameworks, and have succeeded in obtaining upper bounds for system performance. At this time. however. this approach is not developed to a competitive level with other performance measures in robust control. In particular, it is difficult to assess the conservatism of these bounds since they involve a combination of worst-case and average case analysis. Another example of the difficulty of combining these frameworks is the relation between robust control and mainstream system identification (as in [ I l l ) , since the latter relies in the stochastic paradigm for noise. Recent efforts in pursuing this unification in the worst-case setting have once again used a pessimistic view of disturbances, resulting in worst-case identification problems with weak consistency properties ([9, 261) and high computational complexity ([6, 211). In this paper we propose a new methodology for white noise modeling, aimed a t resolving these difficulties. l h e starting p o ~ n t is the following question: how does one decide whether a signal can be accurately modeled as a stochastic white noise trajectory? Deciding this from experimental data leads to a statistical hypothesis test on a finite length signal. In other words, one will accept a signal as white noise if it belongs to a certain set. The main idea of our formulation is to take this set as the definition of white noise, and carry out the subsequent analysis in a worst-case setting. For this methodology to be successful, these sets should: e Exclude non-white signals (e.g. sinusoids) which are responsible for the conservatism of the 3-1, and L1 performance measures. a Include likely instances of white noise. Here stochastic noise will be used as a guidance for the choice of a typical set, but not for average case analysis. a Have simple enough descriptions to allow for tractable worst-case analysis. The paper is organized as follows: some notation is established in Section 2. In Section 3, the case of signals over a finite horizon is considered, and set descriptions of white noise are given both from the time and the frequency domain points of view. These sets are analyzed in terms of the ~vorst-case system response and in relation to stochastic noise. Section 4 contains the infinite horizon version. In Section 5, the application of this framework both to Robust 7 i 2 analysis and to worst-case system identification is outlined. Space limitations preclude an extensive development of these directions; the objective here is to show the potential of this methodology. The conclusions are given in Section 6. and some proofs are covered in the Appendix. 2 Assumptions and Notation We will consider discrete time, causal, linear time invariant (LTI) stable systems of the form H ( X ) = CEO h(t)Xt, where X is the shift operator. Most of the results will be presented for single input/single output (SISO) systems; for the multivariable case see Section 4.3. In the SISO case we will assume that h(t) E 11; this implies that the summation converges for each r , defining the autocorrelation sequence of H , and furthermore that r h ( r ) is itself an El sequence, i.e. Cy=-, Irh(r)l < w. The frequency response (Fourier transform of h(t) E 11) is denoted by f l (e ju) , and is a continuous function of w . The Fourier transform of r h ( r ) is the power spectrum sh(;) := 1 H ( ~ J ~ ) / ~ . Also, the X2 norm of the system is given by For some of the frequency domain bounds obtained in this paper, we will further assume that sh(w) is a function of bounded variation (in BV[O, 2 ~ 1 ) . This means (see [22]) tha t where the supremum is over partitions P = {wl,. . . , w,) of the interval [O, 2n]. TV(sh) is the total variation of s h . The time domain condition C / T rh ( r ) l < w is sufficient for sh(w) E BV[O, 2 ~ 1 . 3 The Finite Horizon Case A reasonable starting point for white noise modeling is the case of a scalar valued. finite horizon, discrete time sequence ~ ( 0 ) : . . . , x ( N 1) of length N . The infinite horizon version will be considered in Section 4. which also covers the extension t o vector-valued signals. To analyze the response of a system with memory over this finite horizon, some convention must be made on the "past" values of the input signals. The two simplest choices are either to assume the system is initially at rest, or that i t is in periodic steady state of period N . We will adopt the l , + + , , L a L bcl, since it leads to a more tractable spectral thecry: the sequence x(O), . . . , x ( N 1) will be identified with the periodic signal x(t) of period N . This procedure is justified for analyzing stable systems with time constants which are small compared to N , so that the system is not sensitive t o long range correlations in the input signals; this will be a standing assumption in this section. The discrete Fourier transform (DFT) X(k) , k = 0 . . . N 1 of the sequence x(t) is defined by the relations n7-1 "-1 ' 2 ~ kt X(k) = x( t )e JF ; r ( t ) = x ( k ) e j S k t t=O N k=O The (circular) autocorrelation sequence of x (correlogram) is given by and the sequence power spectrum (periodogram) by sx(k) = jX(k)I2, k = 0 . . . N 1 The sequences r,(r) and s,(k) form a DFT pair. For an N-periodic signal z(t) , we will use as 2 norm the energy over the period, / I x = rx(0) = c:=:' sx(k). The following relations follow immediately from the definitions. Lemma P Let H be a SISO stable system (h(t) E ll). If u(t) is an 17--periodic input signal to H, and y = f lu is the corresponding steady state (periodic) output, then 3.1 White Noise Descriptions in the Time Domain S;le wish to characterize white signals among sequences of length N: when faced with the problem of deciding whether an empirical signal is a sample of white noise, a statistician will perform a hypothesis test in terms of some statistic. A common choice (see [2, 111) is the sample correlogram. which should approximate the expected correlation for white noise (a delta function). In other words a scalar signal is x ( t ) categorized as white if r X ( r ) / r x ( 0 ) is small for T in a certain range (e.g. 1 < T < T). For example1, one can choose t o specify that the correlogram (normalized to r x ( 0 ) = I), must fall inside a band around zero, of width y , as depicted in Figure 1. Figure 1: Correlogram of a pseudorandom sequence From the classical statistical peint of view, the choice of y is associated t o a level of significance of the test, which in turn depends on some stochastic model. But regardless of the reasoning behind this choice, ultimately the "whiteness" of the signal is decided in terms of whether it belongs or not to a parametrized set. This motivates the following: Definition P The set of signals of length iV which are white in the t ime domain sense (accuracy 7 , up to lag T ) is defined b y V V ~ V , ~ . T := {x E Rn : Ir,(r)l 5 yr,(O), T = 1 , . . . ,TI (8) The response of an LTI system t o signals in such sets will now be analyzed from a worst-case perspective. The worst gain of the system under signals in W*v,,,~ (a seminorm on systems) will be denoted s