Robust filtering with guaranteed energy-to-peak performance — an LMI approach

doi:10.1016/S0005-1098(99)00211-3

Automatica

Volume 36, Issue 6, June 2000, Pages 851-858

https://doi.org/10.1016/S0005-1098(99)00211-3 Get rights and content

Abstract

The problem of robust energy-to-peak filtering for linear systems with convex bounded uncertainties is investigated in this paper. The main purpose is to design a full order stable linear filter that minimizes the worst-case peak value of the filtering error output signal with respect to all bounded energy inputs, in such a way that the filtering error system remains quadratically stable. Necessary and sufficient conditions are formulated in terms of linear Matrix Inequalities — $LMI$ s, for both continuous- and discrete-time cases.

Introduction

Since the Kalman filtering theory has been introduced (Kalman, 1960) much effort has been devoted to the problem of estimating an output error signal (linear combination of the states) through a filter structure such that a guaranteed performance criteria is minimized in an estimation error sense. In this setting, the $H_{2}$ filtering design arises as an efficient strategy whenever the noise input is assumed to have a known power spectral density. In the literature, the $H_{2}$ filtering problem has been faced using Riccati-based approaches (Shaked & de Souza, 1994; Xie & de Souza, 1995), and more recently by means of linear matrix inequalities ( $LMI$ s) (Khargonekar, Rotea & Baeyens, 1996; Palhares & Peres, 1998).

In the case where there exists insufficient statistical information about the noise input, two strategies can be employed. The first one is the well-known $H_{∞}$ filtering design, in which the input is supposed to be an energy signal and the energy-to-energy gain is minimized, or simply bounded by a prescribed value. Many papers have dealt with $H_{∞}$ filter design (as well as with the mixed $H_{2} / H_{∞}$ filtering problem), even when uncertain parameters are taken into account. Basically, the solutions are obtained through Riccati-like equation (Takaba & Katayama, 1996; Park & Kailath, 1997; Xie & de Souza, 1995), or $LMI$ s (Khargonekar et al., 1996; Li & Fu, 1997; Palhares & Peres, 1998; Geromel, Bernussou, Garcia & de Oliveira, 1998; Geromel & de Oliveira, 1998; Palhares & Peres, 1999).

The second approach is the peak-to-peak filtering design, that is, the problem of finding a linear filter which minimizes the worst-case peak value of the filtering error for all bounded peak values of the input signals (Nagpal, Abedor & Poolla, 1996; Vincent, Abedor, Nagpal & Khargonekar, 1996; Voulgaris, 1996). In other words, the maximal peak-to-peak gain of the filtering error system is used as performance criterion (ℓ₁ norm for discrete-time, $L_{1}$ norm for continuous-time systems). In the uncertain case, an algorithm for the guaranteed ℓ₁ norm computation has been proposed in Fialho and Georgiou (1995) (see also the references therein).

On the other hand, the energy-to-peak gain filtering problem has received less attention. This kind of performance criterion has been discussed in Wilson (1989), where it is shown that the energy-to-peak gain can be computed from the controllability Grammian and the state-space representation of the system. In control system design, the problem of finding a controller such that the closed-loop gain from $L_{2}$ to $L_{∞} (ℓ_{2}$ to ℓ_∞) is below a prespecified level is called the generalized $H_{2}$ control problem, since this optimization criterion reduces to the usual $H_{2}$ norm when the controlled output is a scalar (see Rotea (1993) and also Scherer (1995) for details). The objective of the $L_{2}$ – $L_{∞}$ (ℓ₂–ℓ_∞) filter design problem is to minimize the peak value of the estimation error for all possible bounded energy disturbances. In this sense, the energy-to-peak filtering can be viewed as a deterministic formulation of the Kalman filter (see Grigoriadis & Watson, 1997). This strategy has been used for both full and reduced order filter design through $LMI$ s (Grigoriadis & Watson, 1997; Watson & Grigoriadis, 1997) and also as a criterion for model reduction (Grigoriadis, 1997). However, to the authors’ knowledge, only precisely known systems have been addressed.

In this paper, an $LMI$ solution to the guaranteed energy-to-peak filtering problem is proposed for both continuous-time and discrete-time systems. Using as starting point the state-space energy-to-peak gain computation from Wilson (1989) and standard output feedback control results of Scherer, Gahinet & Chilali (1997), necessary (in the sense that the filtering error system is quadratically stable) and sufficient conditions for the guaranteed $L_{2}$ – $L_{∞}$ full order filtering design are provided in terms of $LMI$ s for continuous-time systems with uncertainties in convex bounded domains. It is important to stress that several algebraic manipulations and appropriate change of variables are needed in order to obtain a convex formulation for the robust filtering problem. A similar strategy is used to provide equivalent results for discrete-time systems.

The contributions can be summarized as follows. The paper presents an $LMI$ formulation for linear filter design, which can be immediately extended to handle the problem of robust filtering for linear systems with polytope-type uncertainties. In this context, the energy-to-peak gain is introduced as optimization criterion, allowing the guaranteed cost robust filter design to be performed through convex programming entirely based on $LMI$ s. Continuous-time as well as discrete-time uncertain linear systems with polytopic uncertainty are addressed; in both cases, the problems to be solved are convex optimization problems.

The paper is organized as follows: in the next section the robust filtering problem with guaranteed energy-to-peak performance is stated. In Section 3, the robust $L_{2}$ – $L_{∞}$ (continuous-time) and ℓ₂–ℓ_∞ (discrete-time) guaranteed filtering designs are addressed; the existence and parametrization of a robust filter is established in terms of necessary and sufficient $LMI$ s conditions. Numerical examples and final remarks conclude the paper.

The notation used in this paper is as follows: δ(t) indicates $x ̇ (t)$ for continuous-time systems and x(t+1) for discrete-time systems; L₂ denotes both $L_{2}$ (continuous-time) and ℓ₂ (discrete-time), as well as L_∞ is used for both $L_{∞}$ (continuous-time) and ℓ_∞ (discrete-time) spaces. The boldface characters $I$ and $0$ denote, respectively, the identity and the null matrices of convenient sizes.

As discussed in Wilson (1989), $L_{p}^{n} [0,∞)$ denotes the Lebesgue space of measurable functions f from [0,∞) to $R^{n}$ which satisfy $|| f ||_{L_{p,r}} ≜ (∫ 0 ∞ || f (t) ||^{p}_{r} d t)^{1/p} <∞ for 1≤p<∞, sup t || f (t) ||_{r} <∞ for p=∞,$ where the usual vector r-norm on $R^{n}$ , i.e., ||·||_r is defined as $|| f ||_{r} ≜ ∑ i=0 n | f_{i} (t)|^{r}^{1/r} for 1≤r<∞, max i∈[1,n] | f_{i} (t)| for r=∞.$ Now, considering a discrete-time setting, $Z^{n}$ denotes the space of $R^{n}$ -valued sequences defined on the time set ${0, 1, 2, …}$ ; ℓ_p denotes the set of all sequences ξ in $Z^{n}$ which satisfy $||ξ||_{ℓ_{p,r}} ≜ ∑ t=0 ∞ ||ξ(t)||^{p}_{r}^{1/p} <∞ for 1≤p<∞, sup t ||ξ(t)||_{r} <∞ for p=∞.$

In Wilson (1989), explicit formulas for the induced norm of the bounded linear operator $G : L_{2,r}^{n_{1}} → L_{∞,r}^{n_{2}}$ for r=2 and ∞ can be found. In this paper, only the Euclidean norm (r=2) is considered; to avoid cumbersome notation, $L_{p} (ℓ_{p})$ is used instead of $L_{p,2} (ℓ_{p,2})$ .

Section snippets

Preliminaries

Consider the linear time-invariant system given by $(S) δx(t)=Ax(t)+Bw(t), x(0)=x_{o} y(t)=Cx(t)+Dw(t), z(t)=Lx(t),$ where $x(t) : R → R^{n}$ is the state vector, $y(t) : R → R^{r}$ is the measurement output vector, $w(t) : R → R^{m}$ is the noise signal vector (including process and measurement noises) and $z(t) : R → R^{p}$ is the signal to be estimated. The initial state condition x_o is considered to be known and, without loss of generality, assumed to be zero (see Khargonekar et al., 1996).

The system matrices are assumed to be unknown

L₂–L_∞ guaranteed filtering design

The continuos- and discrete-time state-space characterizations of the energy-to-peak gain presented next can be viewed as the starting point for the main results of this section. First, consider $(A,B,C,D,L)∈ D$ arbitrary but fixed.

Lemma 3.1

Letγ>0 be given and assume that the filtering error system is stable. TheL₂–L_∞gain of $(S_{F})$ is limited byγ, i.e., $sup 0≠w∈L_{2} || z ̃ ||_{L_{∞}} ||w||_{L_{2}} <γ$ if and only if there exists $P=P′> 0, P∈ R^{2n×2n}$ such that $C ̃ P C ̃ ′<γ^{2} I,$ $Θ (P)< 0,$ where $Θ (P)≜ A ̃ P+P A ̃ ′+ B ̃ B ̃ ′$ for continuous-time systems or $Θ (P)≜ A ̃ P A ̃$

Examples

Example 1

Consider the following second-order resonant system (borrowed from Grigoriadis and Watson (1997)): $x ̇ (t)= 011 −11 −2.2+α x(t)+ 00 1+β 0 w(t),$ $y(t)=[0 1]x(t)+[0 1+β]w(t),$ $z(t)=[1 0]x(t).$ As pointed out in Grigoriadis and Watson (1997), for α=0 and β=0, this system model corresponds to a vibrating system with natural frequency ω_n=11 rad/s and damping ratio ζ=0.1. The position measurement y(t) is corrupted by noise and the objective is to estimate a velocity signal z(t).

Assuming that the input disturbance is of

Conclusions

The robust filtering problem with guaranteed energy-to-peak performance for continuous- and discrete-time uncertain linear systems in convex bounded domains has been addressed through an $LMI$ approach. Necessary and sufficient conditions are obtained for full order filtering, allowing the problem to be solved through convex optimization procedures (global convergence, efficient algorithms).

Acknowledgements

This research has been supported in part by grants from “Fundação de Amparo à Pesquisa do Estado de Minas Gerais” — FAPEMIG, TEC 1027/98, “Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico” — CNPq and “Fundação de Amparo à Pesquisa do Estado de São Paulo” — FAPESP, Brazil. The authors thank the reviewers for their valuable suggestions on the improvement of the text.

Reinaldo Martinez Palhares was born in Anápolis, Goiás, Brazil, in 1969. He received the B.Sc. degree in electrical engineering from the Federal University of Goiás, UFG, in 1992, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Campinas, UNICAMP, in 1995 and 1998, respectively. In 1998, he joined the Graduate Program in Electrical Engineering of the Pontifical Catholic University of Minas Gerais, as an associated researcher/professor supported by “Fundação de

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^☆: The original version of this paper was presented at the 14th World Congress which was held in Beijing, China during July, 1999. This paper was recommended for publication in revised form by Associate Editor P-O. Gutman under the direction of Editor T. Basar.

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Brief PaperRobust filtering with guaranteed energy-to-peak performance — an LMI approach☆

Abstract

Introduction

Section snippets

Preliminaries

L2–L∞ guaranteed filtering design

Examples

Conclusions

Acknowledgements

Linear Algebra and its Applications

Automatica

Automatica

Conditions for positive and nonnegative definiteness in terms of pseudoinverses

SIAM Journal on Applied Mathematics

H∞ design with pole placement constraints: An LMI Approach

IEEE Transactions on Automatic Control

L2 and L2–L∞ model reduction via linear matrix inequalities

International Journal of Control

Reduced-order H∞ and L2–L∞ filtering via linear matrix inequalities

IEEE Transactions on Aerospace and Electronic Systems

A new approach to linear filtering and prediction problems

Transactions of ASME — Journal of Basic Engineering

Mixed H2/H∞ filtering

International Journal of Robust and Nonlinear Control

A linear matrix inequality approach to robust H∞ filtering

IEEE Transactions on Signal Processing

A linear matrix inequality approach to peak-to-peak gain minimization

International Journal of Robust and Nonlinear Control

Optimal filtering schemes for linear discrete-time systems — an LMI approach

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Brief Paper
Robust filtering with guaranteed energy-to-peak performance — an $LMI$ approach☆

L₂–L_∞ guaranteed filtering design

$H_{∞}$ design with pole placement constraints: An $LMI$ Approach

$L_{2}$ and $L_{2}$ – $L_{∞}$ model reduction via linear matrix inequalities

Reduced-order $H_{∞}$ and $L_{2}$ – $L_{∞}$ filtering via linear matrix inequalities

Mixed $H_{2} / H_{∞}$ filtering

A linear matrix inequality approach to robust $H_{∞}$ filtering

Optimal filtering schemes for linear discrete-time systems — an $LMI$ approach