Elsevier

Automatica

Volume 45, Issue 11, November 2009, Pages 2651-2658
Automatica

Brief paper
Risk-sensitivity conditions for stochastic uncertain model validation

https://doi.org/10.1016/j.automatica.2009.07.019Get rights and content

Abstract

The paper presents sufficient and necessary conditions that verify the relevance of an assumed linear stochastic system model for problems in which probabilistic characteristics of the plant are not known exactly. The approach is to establish the existence of an admissible probability space on which the output of the candidate stochastic system model is consistent (in a stochastic sense) with the noisy output of the plant.

Introduction

In the system identification theory the mismatch between the modeled and observed system outputs is often attributed to noise, or excitation (e.g., see Ljung and Soderstrom (1983)). However, it was argued in Poolla et al., 1994, Smith and Doyle, 1992 that models of feedback control systems must account for other types of uncertainties such as, e.g., unmodeled dynamics, coprime factor uncertainty or norm-bounded disturbances. This argument motivated a substantial research effort in the area of deterministic uncertain systems model validation; e.g., see Savkin and Petersen (1996). The deterministic model validation problem was formulated in Problem 3.1 in Smith and Doyle (1992) as follows: Given an uncertain system model (consisting of a nominal model and a set of uncertain perturbations) and an experimental datum, does the uncertainty set include an uncertainty element such that the observed datum can be produced exactly? The solution to this problem, as noted in Poolla et al. (1994), Smith and Doyle (1992), is instrumental for eliminating models which are inconsistent with observed measurements. Related issues of model equivalence arising in the problem of uncertain model reduction were considered in Antoulas and Willems, 1993, Beck et al., 1996.

In stochastic control systems, effects of noise may further augment those attributed to the system uncertainty. While a common description of stochastic systems employs the classic Gaussian excitation model to describe dynamics which are difficult to predict accurately, in many applications, dynamics are often driven by non-Gaussian disturbances. Such disturbances often have fixed but unknown probability laws. One way to deal with this uncertainty is to embed the true process into a class of models obtained by perturbing a simpler, e.g., a Gaussian white noise, model (Boel, James, & Petersen, 2002). In the context of stochastic control systems with measurement noise, the following stochastic model validation problem enables one form of such embedding. Given a causal plant and a set of linear stochastic systems consisting of a nominal model and a set of uncertain perturbations, does the uncertainty set include an uncertainty element such that the probability law of the plant’s output, given the disturbance datum, can be produced exactly? In this formulation, our stochastic model validation problem becomes a version of the weak stochastic realization problem concerned with “the existence and the classification of all stochastic systems for which the output process equals a given process in distribution” (van Schuppen, 1989).

In this paper, we establish conditions under which the above problem has a solution. The class of stochastic disturbances under consideration is based on the interpretation of the system uncertainty as resulting from perturbations of reference Brownian motions; this allows one to account for a rich class of system uncertainties including some standard uncertainty models such as norm-bounded parametric disturbances and H-norm bounded unmodeled dynamics (Petersen et al., 2000b, Ugrinovskii and Petersen, 2001). Thanks to its connection to the H control and risk-sensitive control theories (Dai Pra et al., 1996, Fleming and McEneaney, 1995, James, 1992), this approach has been shown to enable tractable solutions to a number of robust control and filtering problems involving uncertain stochastic systems; e.g., see Petersen, James, and Dupuis (2000a), Petersen et al. (2000b), Charalambous and Rezaei (2007). In this paper we complement this recent stochastic robust control and robust filtering theory by addressing issues of model validation in relation to the stochastic uncertainty description used in the above references. Our aim is to pave the way for future research into issues of stochastic model reduction and stochastic systems realization for this class of stochastic uncertain systems, which would parallel the deterministic theory concerned with issues of system equivalence and system realizations (Beck et al., 1996, Petersen, 2007).

Consider a controlled plant ψ(t)=At(u()|0t,ϖ()|0t),t[0,T],ȳ(t)=0tψ(t)dt+ϑ(t), described by a causal input–output mapping A, taking values in a set of Rq-valued continuous functions ψ(t),t[0,T], and a measurement Eq. (2). In (1), (2), ϖ, ϑ represent an exogenous disturbance and a measurement noise signal which are assumed to belong to the spaces of Rr, Rq-valued continuous functions, such that ϖ(0)=0, ϑ(0)=0, respectively. These spaces are denoted C0([0,T],Rr), C0([0,T],Rq). In the sequel, ϖ, ϑ will be interpreted as samples of random processes w(t), θ̄(t) defined on a certain probability space, θ̄ being a Wiener process. Also, u() is an input control signal. We will present a rigorous description of its properties in Section 2. At this stage, we note that the paper focuses on the case where a feedback is allowed in which samples of the input u are causal functions of ϖ. This leads us to assume that u is a random process adapted to the natural filtration generated by the exogenous disturbance; also, see Ugrinovskii and Petersen (2000).

As mentioned above, exogenous disturbances arising in control problems can often be modeled as samples of a random process w whose true probabilistic model is unknown. It may however be possible to represent dynamics of the plant (1) as if they were obtained by perturbing a nominal, possibly linear, system driven by a Gaussian noise while using the same controller u. Also, since the output of the plant (1) can only be measured with some error, it is necessary to augment this perturbed system model with a noisy linear measurement process. This prompts us to consider the following class of candidate linear models to represent the system (1), (2)dx(t)=(A(t)x(t)+B1(t)u(t))dt+B2(t)dw(t),dy(t)=C2(t)x(t)+dv(t),y(0)=0,x(0)=x0. Here w and v are random processes representing the process and measurement noises, such that Eq. (3) can be understood in the Ito sense, and x0Rn.

One can note that the above class of candidate models parallels the Markov representation of a stochastic system arising in the stochastic theory of minimal realization (Akaike, 1974). However, in our case the probability space on which the system (3) is defined and probability distribution of noise processes w, v are not fixed. In particular, the processes generated by the systems (1) and (3) do not have to be Markov.

The model validation problem under consideration is to establish the existence of an admissible probability space on which the process θ(t)=y(t)0tψ(s)ds is a Wiener process with respect to the filtration generated by y, given w. Then, on such a probability space the output y of the system (3) will have the same probability law, conditioned on the disturbance w, as that of ȳ on the underlying probability space of the system (1), (2), i.e., y can be seen as a weak stochastic realization of ȳ (van Schuppen, 1989).

In the light of the robust stochastic control theory mentioned earlier (Boel et al., 2002, Petersen et al., 2000b), we seek to achieve the coupling (4) between ψ and y by perturbing the nominal system which is the special case of the system (3) corresponding to the case where w and v are Wiener processes. Perturbations will be restricted to a given class Ξd; this distinguishes our problem from the general weak stochastic realization problem formulated in van Schuppen (1989). The set of admissible perturbations of the nominal system will be defined rigorously in Section 2.2.

The problem formulation, which seeks to represent the connection between the input and output of a stochastic system in terms of a linear uncertain system, is analogous to the deterministic formulation of the uncertain model validation problem considered in Poolla et al. (1994), Savkin and Petersen (1996), Smith and Doyle (1992). The main difference between the two problems is that our approach directly accounts for stochastic perturbations and noises in the system. Also, the uncertainty affecting probability laws of the system noises is directly included in our model, whereby uncertain systems driven by non-white noises are accounted for.

The main result of the paper is a condition which guarantees the existence of an admissible probability space from the given class Ξd on which (4) holds. Theorem 5 formulates this condition in terms of a risk-sensitive performance cost defined on dynamics of the nominal system (3). The condition is sufficient, and, under some additional technical assumption, necessary. In somewhat lose terms, the cost functional characterizes risk-sensitive performance exhibited by the system (3) when its output y ‘attempts to track’ ψ using the control law u. Our result compares this cost with a given bound on the energy in admissible disturbances. Such a bound is one of the parameters that define admissible magnitude of uncertain disturbances in the system; see Definition 1. In Section 4, we present examples which demonstrate how the proposed conditions can be verified numerically.

Section snippets

Uncertain stochastic systems

Let T>0 be a constant which will denote the finite time horizon considered throughout the paper. Consider a complete probability space (Ω,F,P). On this probability space, consider mutually independent standard Wiener processes w and y with values in Rr, Rq, respectively. As in Dai Pra et al. (1996), the space Ω is thought of as the noise space C0([0,T],Rr)×C0([0,T],Rq), and the probability measure P is defined as the product measure P,w×P,y, where P,w, P,y are standard Wiener measures on C

The structure of a solution to the stochastic model validation problem

Along with the set of admissible uncertain models Ξd, consider the plant (1) driven by sample paths of the disturbance process w. As was observed above, for each admissible probability measure QΞd, its restriction to Fw has a finite relative entropy with respect to P,w. Therefore, if a solution to our model validation problem exists, it must necessarily have this property. This motivates us to start the search for a solution with associating with (1) a collection of probability measures Q̄w

Examples

The examples below demonstrate how the results presented in the previous section can be used in the derivation of tractable model validation conditions.

Conclusions

The paper has presented a sufficient condition for realizability of a given process triple u,ψ,y, in which ψ and y are coupled by a Brownian motion, using perturbations of a linear stochastic system. The class of admissible perturbations comprises uncertainties satisfying a relative entropy constraint. Under an additional technical assumption, the proposed condition is also necessary.

Our result establishes a connection between the realizability of a plant of the form (1) with noisy output, on

Acknowledgments

Discussions with Ian Petersen, Jason Ford are gratefully acknowledged. The author thanks Igor Vladimirov for his comments on the final version of the paper.

Valery A. Ugrinovskii was born in Ukraine in 1960. He received the undergraduate degree in Applied Mathematics in 1982 and a PhD in Physics and Mathematics in 1990, both from the State University of Nizhny Novgorod, Russia. From 1982 to 1995 he held research positions at the Radiophysical Research Institute, Nizhny Novgorod. From 1995 to 1996 he was a Postdoctoral Fellow at the University of Haifa. In 1996 he has joined the School of Engineering and Information Technology, the University of New

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      However, for nonparametric models: “the situation is significantly more complicated” (Lee & Poolla, 1996) and to the best of our knowledge, has not been addressed in the literature. Recently, in the spirit of weak stochastic realization problem (van Schuppen, 1989), Ugrinovskii (2009) investigated the conditions for which the output of a stochastic nonlinear system can be realized through perturbation of a nominal stochastic linear system. In practice, one often encounters the situation where a model is either proposed from physics-based reasoning or a reduced order model is derived for computational convenience.

    Valery A. Ugrinovskii was born in Ukraine in 1960. He received the undergraduate degree in Applied Mathematics in 1982 and a PhD in Physics and Mathematics in 1990, both from the State University of Nizhny Novgorod, Russia. From 1982 to 1995 he held research positions at the Radiophysical Research Institute, Nizhny Novgorod. From 1995 to 1996 he was a Postdoctoral Fellow at the University of Haifa. In 1996 he has joined the School of Engineering and Information Technology, the University of New South Wales at the Australian Defence Force Academy, Canberra, and currently he is an Associate Professor in the School. He is the coauthor of the research monograph Robust Control Design usingHMethods, Springer, London, 2000, with Ian R. Petersen and Andrey V. Savkin. His current research interests include decentralized and distributed control, stochastic control and filtering theory, robust control and switching control.

    This work was supported by the Australian Research Council and the University of New South Wales. The material in this paper was partially presented at the 17th IFAC World Congress. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen.

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