A frequency domain approach to state estimation - CiteSeerX

Journal of the Franklin Institute 340 (2003) 147–157

Short communication

A frequency domain approach to state estimation H.J. Marquez* Department of Electrical & Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 Received 14 December 2000; received in revised form 11 March 2003; accepted 13 March 2003

Abstract A novel input-to-state approach to the problem of robust state estimation in the presence of model uncertainty as well as plant disturbance and sensor noise is considered. A new observer structure is introduced and shown to have certain advantages over the classical Luenberger observer when studying robustness issues. This structure is then used to solve the robust estimation problem in the HN framework. r 2003 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.

1. Introduction Very often when dealing with state space realizations of dynamical systems in real time, it is necessary to manipulate the state vector ‘‘x’’. Examples include control applications via state feedback, fault detections, and system monitoring. Unfortunately, more often than not x is either difficult, too expensive, or even impossible to measure. In this case, an observer can be used to obtain an estimate x# of the true state x: This estimate can then be used as a substitute for x; as required. This approach, however, is not without problems. Indeed, plant disturbance, sensor noise, and modeling errors will invariably lead to deviations from the true state unless due precautions are taken during the observer design. In the sequel we will refer to the robust observer design problem, defined loosely as the problem of designing observers which are insensitive to these effects. The classical approach to state estimation consists of studying a free or unforced system subject to nonzero initial conditions, and design the observer gain to stabilize

*Tel.: +1-780-492-3334; fax: +1-780-492-1811. E-mail address: [email protected] (H.J. Marquez). 0016-0032/03/$30.00 r 2003 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S0016-0032(03)00017-6

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# thus achieving asymptotic convergence. See, for the error dynamics x* ¼ x  x; example, any of the standard reference on linear systems [1–3] or [4]. Modeling errors are very difficult to incorporate in this setting. Much attention has been focused on this problem in recent years, particularly, in connection with the problem of residual generation in observer-based fault detections. See, for example, [5,6]. A commonly used approach consists of modeling uncertainties and unknown external excitations as unknown inputs. See [5,7–10]. See also the recent survey [11] for an extensive account of available methods for robust estimation along with applications in fault detections. In this paper, we study the robust estimation problem, but we follow a very different approach. Robust observer design has similarities with robust controller design, understood as the design of controllers which maintain their performance in the presence of model uncertainties noise and disturbances, an issue which has been extensively studied in recent years and that has resulted and more that one well developed techniques. See, for example [12–14], and the references therein for an HN approach to robust controller design, and [15] for an L1 approach to the same problem. Much of the success in robust controller design is due to the systematic exploitation of the well-known relationship between state space and input–output realizations of dynamical systems. While computations are invariably performed in state space, modeling errors are better described in the input–output framework, typically in the form of frequency dependent bounds on the amplitude of a perturbation term. The same concepts, however, have not been emphasized in the robust observer design problem, perhaps due to the intrinsic state space nature of the state observation problem. In this paper, we pursue an input–output approach to the state estimation problem. We view systems as mappings from input-to-state and look for small estimation errors in the presence of persistent excitation. This approach will allow us to consider a very general class of perturbations acting on the nominal plant model. In this context, an observer will be seen as a filter designed so that the error dynamics # has some desirable frequency domain characteristics. Thus it soon becomes ðx  xÞ apparent that the structure typically adopted for the state space estimator, often referred to as Luenberger observer,1 is not the most convenient one. To see this, we simply notice that in this structure the filter order is that of the original state space realization of the plant. This is a severe limitation in any filter since it will strongly limit the shape of the frequency characteristics of the resulting filter, and implies among other things that robust state estimation is simply not achievable with loworder models. Notice also that this problem gets more serious given that the frequency shaping of this filter has all of the tradeoffs and limitations encountered in the design of any feedback loop, in particular those related to the location of the plant zeros. See, for example [16], for a thorough discussion of these issues. It is then

1 We emphasize here that we refer to the structure of the state space estimator and not the method used for its design. Thus, the structure of a Luenberger observer is the same observer structure used in virtually all of the literature on state estimation.

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apparent that a modified observer structure with additional degrees of freedom in the design might be desirable in certain cases. In this paper, we introduce a new observer structure, in the future referred to as an input–output observer, which allows additional degrees of freedom in the design. This structure is then used to analyze and solve the robust observer design problem stated earlier. More important, as explained earlier, our input–output approach to the state estimation problem is fundamentally different from the classical solution. Our approach is rather novel and characterized by the fact that (i) can easily handle unstructured uncertainties in the plant, (ii) the solution can be made optimal with respect to worst case exogenous perturbations, and (iii) poses the robust observer design problem in the context of the robust control theory, thus taking advantage of the vast existing literature and software tools available for the design.

2. Preliminaries and notation In the sequel R represents the field of real numbers, Rn the set of n-tuples of real numbers, and Rnp the set of real matrices of order n by p. Similarly, R[s] represents the set of proper rational functions of the form pðsÞ ¼ nðsÞ=dðsÞ; where nðsÞ and dðsÞ are polynomials in the variable s with the degree of n being at most equal to the degree of d. R[s]p  q represents the set of matrices with elements in R[s]. We will say that PðsÞAR½spq is an stability matrix if and only if none of the entries of P have poles in the right-half of the complex plane. We will consider a system S, defined via a minimal state space realization of the form x’ ¼ Ax þ Bu; y ¼ Cx;

AARnn ; BARnp ;

CARqn ;

ð1Þ ð2Þ

where for simplicity, we have assumed that D ¼ 0 in Eq. (2). Alternatively, S will be seen as a mapping from input-to-state and represented in transfer functions form as follows: P0 ðsÞ ¼ ðsI  AÞ1 B;

ð3Þ

H0 ðsÞ ¼ CP0 ðsÞ ¼ CðsI  AÞ1 B:

ð4Þ

These two mappings will be represented using a black box model, as shown in Fig. 1. A key role in this representation is played by the state vector x: The standard approach used to estimate the state, consists of employing the following structure: # x’# ¼ Ax# þ Bu þ L½y  C x: ð5Þ Eq. (5) defines what is usually referred to as the Luenberger Observer. It is well known that the observer error dynamics is given by x’* ¼ x’  x’# ¼ ðA  LCÞx*

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Fig. 1. Black box model of the systems P0 and H0 .

Fig. 2. Uncertainty representation.

thus Eq. (5) is sometimes called an asymptotic observer, since x# asymptotically converges to the true state x; provided that the eigenvalues of the matrix (A2LC) lie in the left-half of the complex plane. This analysis, however, is affected by problems neglected in our analysis: first it ignores sensor noise and plant disturbance. Second, it assumes perfect modeling. As a consequence of these two effects, the residual # will not be zero, thus posing the question of how to design the observer to ðx  xÞ minimize the estimation error. Problems of this type are typical in the so-called robust control theory, and a solution usually involves three steps: (i) Include some form of uncertainty description associated with the model which quantifies the accuracy of the predictions by the model. (ii) Estimate the characteristics of the disturbance acting on the plant, as well as those of the measurement noise. (iii) Incorporate the assumed bounds in (i) and (ii) directly into the design process. Robust estimation, however, has certain features not encountered in robust control. These differences will be discussed throughout the rest of the article. In robust control, frequency dependent bounds on the input–output map have been used successfully in a variety of applications. In this paper, we apply the same concept to the input-to-state description. We will adopt the following formulation: PðsÞ ¼ ðsI  AÞ1 B þ D ¼ P0 ðsÞ þ D;

ð6Þ

HðsÞ ¼ CPðsÞ ¼ CP0 ðsÞ þ CD ¼ H0 ðsÞ þ CD

ð7Þ

which can be represented as shown in Fig. 2.

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Thus the difference between the true state x and that predicted by the model P0 is assumed to be within the bound D: The perturbation D is unknown, but satisfies a frequency dependent bound of the form jDðjoÞjplðjoÞ

8oAR:

ð8Þ

We will also assume that the disturbance D satisfies: Z N jdðtÞj2 dtoN; N Z N 2 # jdðjoÞj2 jWðjoÞj doo1

ð9Þ

N

# i.e. dAL2 and satisfies the inequality Eq. (9) for some function WðÞ; where qq # WðsÞAR½s is analytic and has no zeros in the right-half plane. Similar assumptions apply to the noise n. These assumptions are typical in the HN framework.

3. An input–output observer With the assumptions described above, the observer structure Eq. (5) is perhaps not the most suitable candidate. Indeed, our interest is in looking for the optimal robust state estimator, in a sense to be defined. Define the following structure, in the future referred to as an input–output observer: # ¼ P0 ðsÞ½uðsÞ þ GðsÞeðsÞ; xðsÞ

ð10Þ

# eðsÞ ¼ yðsÞ þ nðsÞ þ dðsÞ  yðsÞ;

ð11Þ

# ¼ C x# yðsÞ

ð12Þ

which can be represented as shown in Fig. 3. Before discussing the properties of this observer notice that we have used transfer functions, rather than a state space realization of the plant S. Here P0 ðsÞ ¼

Fig. 3. Input–output observer.

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ðsI2AÞ21 B represents the input-to-state plant model, C is the read-out matrix of the state space realization (1) and (2), and GðsÞ is a filter, or generalized observer gain to be designed. Notice that with this formulation the input-to-state model, and thus the input–output observer, retains the coordinate dependent structure of the state space realization. Define now the feedback systems S1 as shown in Fig. 4: Thus for the system S1 we can define the mapping H1 ðsÞ as follows: ! ! ! ! ðI þ CP0 GÞ1 ðI þ CP0 GÞ1 CP0 u1 u1 e1 ¼ H1 ðsÞ ¼ : ð13Þ e2 u2 u2 ðI þ GCP0 Þ1 G ðI þ GCP0 Þ1 Definition 3.1. We say that the input–output observer of Eqs. (10)–(12) is internally stable if H1 is an stability matrix. Clearly, the system S1 of Fig. 4 correspond to the feedback loop which defines the observer structure (10)–(12). It is useful to relate this input–output description to the (minimal) state space realizations of H0 and G: Assume that the observer blocks H0 and G are defined via state space realizations F1 and F2 ; respectively, where (with the notation of Fig. 4), F1 : x’# ¼ Ax# þ Be2 ¼ Ax# þ Bðu2 þ y1 Þ;

ð14Þ

# y# ¼ C x;

ð15Þ

# F2 : x’ 1 ¼ A1 x1 þ B1 e1 ¼ A1 x1 þ B1 ðu1  yÞ;

ð16Þ

y1 ¼ C1 x1 :

ð17Þ

Theorem 3.1. Assuming that H0 and G are defined via controllable and observable state space realizations F1 and F2 of form (14)–(17), then the input–output observer is internally stable if and only if the matrix ! A BC1 A% ¼ ð18Þ B1 C A1 has all of its eigenvalues in the open right-half plane.

Fig. 4. The system S1 :

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# x1 ÞT ; a little algebra shows that Proof. Defining x% ¼ ðx; ! ! ! ! ! A BC1 x# 0 B u1 x’# ¼ þ x’% ¼ x1 B1 0 B1 C A1 u2 x’ 1 ! ! ! 0 C1 y1 x# ¼ y2 C 0 x1 or x’% ¼ A% x% þ B% u; %

ð19Þ

y% ¼ C% x: %

ð20Þ

It is clear that if A# has no eigenvalues in the closed RHP, then H1 is an stability matrix. For the converse notice that with F1 and F2 controllable and observable, then the state space realization (19) and (20) of the feedback system S1 is also controllable and observable (see, for example, [1], p. 446). Thus the result follows by the well-known equivalence between input–output and minimal state space realizations of linear time-invariant systems. & Corollary 3.1. With H0 and G as in Theorem 3.1, x# given by the input–output observer ‘‘asymptotically converges to the true state x’’ if and only if the system S is internally stable. Proof. An straightforward application of Theorem 3.1.

&

Thus, designing an input–output observer reduces to stabilizing the feedback loop S, a problem that can be solved using any technique available. In the sequel we will proceed to cast the observer design problem in the HN setting.

4. Error analysis To analyze the error dynamics, it is convenient to introduce an additional input to the feedback system of Fig. 4, as shown in Fig. 5. This system will be denoted S2 :

Fig. 5. The system S2 :

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As with the system S1 ; we define H2 ðsÞ as follows: 0 1 e1 B C @ e2 A ¼ H2 ðsÞu e3

0 B ¼B @

ðI þ CP0 GÞ1

ðI þ CP0 GÞ1 CP0

ðI þ GCP0 Þ1 G

ðI þ GCP0 Þ1

ðI þ P0 GCÞ1 P0 þ G

ðI þ P0 GCÞ1 P0

ðI þ CP0 GÞ1 C

10

u1

1

CB C ðI þ GCP0 Þ1 GC C A @ u2 A 1 u3 ðI þ P0 GCÞ

ð21Þ H2ij :

We will denote the entries of this matrix by The following lemma says that the addition of the input u3 does not affect the internal stability of the system S1 : We omit the proof since it is straightforward. Lemma 4.1. The feedback system S1 is internally stable if and only if the system S2 is internally stable. We can now analyze the observer error. We proceed as follows: y# ¼ CP0 ðu þ GeÞ # ¼ CP0 u þ CP0 Gy þ CP0 Gðn þ dÞ  CP0 G y; y# ¼ ðI þ CP0 GÞ1 ½CP0 u þ CP0 Gy þ CP0 Gðn þ dÞ:

ð22Þ

# we have that Defining y* ¼ y  y; y* ¼ y  ðI þ CP0 GÞ1 CP0 u  ðI þ CP0 GÞ1 CP0 Gy  ðI þ CP0 GÞ1 CP0 Gðn þ dÞ ¼ ðI þ CP0 GÞ1 y  ðI þ CP0 GÞ1 CP0 u  ðI þ CP0 GÞ1 CP0 Gðn þ dÞ and taking into account that y ¼ CP0 u þ CDu; we have that y* ¼ ðI þ CP0 GÞ1 CDu  ðI þ CP0 GÞ1 CP0 Gðn þ dÞ:

ð23Þ

We also have that x ¼ P0 u þ Du; x# ¼ P0 u þ PGe; # we have that where e ¼ y þ ðn þ dÞ  y# ¼ y* þ ðn þ dÞ: Defining x* ¼ x  x; x* ¼ x  x# ¼ Du  P0 G y*  P0 Gðn þ dÞ and substituting Eq. (23), after some straightforward simplifications we have that x* ¼ ðI þ P0 GCÞ1 CDu  ðI þ P0 GCÞ1 P0 Gðn þ dÞ:

ð24Þ

Eq. (24) can be used to discuss the properties of the observer as well as discuss possible design strategies and constraints.

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(i) Notice first that Eq. (24) can be re-written as follows: x* ¼ H233 CDu  H231 ðn þ dÞ and since H231 and H233 are stable matrices by assumption we have that bounded inputs [u, n and d] will lead to bounded estimation errors. (ii) Eq. (24) implies that in the absence of noise and disturbances (n þ d ¼ 0) and without modeling errors (D ¼ 0), then x* is identically zero, provided that the input–output observer is internally stable. This conclusion is consistent with our input–output analysis, which ignores the effect of possible nonzero initial conditions. (iii) Initial conditions can be included as follows: assume that n, d, and D are identically zero in Fig. 3 and consider minimal state space realizations of H0 and G: With the notation of Fig. 3, we have:The plant x’ ¼ Ax þ Bu;

ð25Þ

y ¼ Cx:

ð26Þ

Observer block H0 : x’# ¼ Ax# þ Bðz þ uÞ;

ð27Þ

# y# ¼ C x:

ð28Þ

Observer block G: # x’ 1 ¼ A1 x1 þ B1 ðy  yÞ;

ð29Þ

z ¼ C1 x1 :

ð30Þ

It follows that: x’* ¼ x’#  x’ ¼ ðAx# þ Bu þ BC1 x1 Þ  ðAx þ BuÞ ¼ Aðx#  xÞ þ BC1 x1 ; x’* ¼ Ax* þ BC1 x1 : Also # x’ 1 ¼ A1 x1 þ B1 Cx  B1 C x; or x’ 1 ¼ A1 x1  B1 Cðx#  xÞ; ) x’ 1 ¼ A1 x1  B1 C x: * Defining now

x ¼

x* x1

!

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H.J. Marquez / Journal of the Franklin Institute 340 (2003) 147–157

we have that x’ ¼

x’*

!

x’ 1

¼

A

BC1

B1 C

A1

!

x* x1

!

% ¼ Ax

and since the eigenvalues of A# are in the left-half plane, it follows that x# asymptotically converges to x: Thus, the input–output observer can also be seen as an asymptotic observer, although we have chosen not to emphasize this aspect. (iv) Small estimation errors can be achieved by finding a compensator GðsÞ which (a) internally stabilizes the observer, and (b) minimizes the error in Eq. (24) with respect to uncertainties of form (8), and noise and plant disturbances of known frequency content of form (9). This is a standard problem which can be cast in the context of the HN framework, and whose solution is well known. Moreover, there exist reliable commercial software readily available, such as Ref. [18]. A typical approach, first proposed in Ref. [17] consists of minimizing the following norm:



W ðI þ P GCÞ1 C

0

1 jjFjjN ¼

ð31Þ

:

W2 ðI þ P0 GCÞ1 P0 G

N

In the robust control literature this is referred to as the mixed sensitivity problem and it is analogous to the present case. (v) Speed of response can be directly addressed in the design by appropriate selection of the weighting functions W1 and W2 introduced above. 5. Concluding remarks This paper contains two important contributions. First, we have formulated the state estimation problem in an input–output framework. The significance of this approach is that the input output approach is more sound in the study of robustness problems, which is the main objective of this paper. Second, we have introduced the input–output observer structure. Using this observer, the synthesis of a robust state estimator becomes a standard feedback design, which can be solved via any of the well-established techniques for feedback design. Our approach to the state estimation problem is significant not only in the study of robustness problems. Indeed, the input–output observer suggests that an observer is simply the input-to-state model of the plant, stabilized via output feedback through the compensator G: We strongly believe that the same approach can be used to solve the nonlinear state estimation problem, an important problem that remains largely unsolved. Research in this direction is currently under way.

Acknowledgements The author gratefully acknowledges the financial support provided by the Natural Sciences and Engineering Research Council of Canada.

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