Discrete-Time Steady-State Minimum-Variance Prediction and Filtering

This book describes the classical smoothing, filtering and prediction techniques together with some more recently developed embellishments for improving performance within applications. It aims to present the subject in an accessible way, so that it can serve as a practical guide for undergraduates and newcomers to the field. The material is organised as a ten-lecture course. The foundations are laid in Chapters 1 and 2, which explain minimum-mean-square-error solution construction and asymptotic behaviour. Chapters 3 and 4 introduce continuous-time and discrete-time minimum-variance filtering. Generalisations for missing data, deterministic inputs, correlated noises, direct feedthrough terms, output estimation and equalisation are described. Chapter 5 simplifies the minimum-variance filtering results for steady-state problems. Observability, Riccati equation solution convergence, asymptotic stability and Wiener filter equivalence are discussed. Chapters 6 and 7 cover the subject of continuous-time and discrete-time smoothing. The main fixed-lag, fixed-point and fixed-interval smoother results are derived. It is shown that the minimum-variance fixed-interval smoother attains the best performance. Chapter 8 attends to parameter estimation. As the above-mentioned approaches all rely on knowledge of the underlying model parameters, maximum-likelihood techniques within expectation-maximisation algorithms for joint state and parameter estimation are described. Chapter 9 is concerned with robust techniques that accommodate uncertainties within problem specifications. An extra term within Riccati equations enables designers to trade-off average error and peak error performance. Chapter 10 rounds off the course by applying the afore-mentioned linear techniques to nonlinear estimation problems. It is demonstrated that step-wise linearisations can be used within predictors, filters and smoothers, albeit by forsaking optimal performance guarantees.

where v k is a stationary measurement noise sequence over an interval k  [1, N], R . An objective is to design a filter  that operates on the above measurements and produces an estimate, /

Application of the Time-Varying Filter Recursions
A naïve but entirely valid approach to state estimation is to apply the standard minimumvariance filter recursions of Section 4 for the problem (1) - (3). The predicted and corrected state estimates are given by where L k = / 1 As before, the above Riccati equation is iterated forward at each time k from an initial condition P 0 . A necessary condition for determining whether the states within (1) can be uniquely estimated is observability which is discussed below.

The Discrete-time Observability Matrix
Observability is a fundamental concept in system theory. If a system is unobservable then it will not be possible to recover the states uniquely from the measurements. The pair (A, C) within the discrete-time system (1) -(2) is defined to be completely observable if the initial states, x 0 , can be uniquely determined from the known inputs w k and outputs y k over an interval k  [0, N]. A test for observability is to check whether an observability matrix is of full rank. The discrete-time observability matrix, which is defined in the lemma below, is the same the continuous-time version. The proof is analogous to the presentation in Chapter 3.

Proof:
Since the input w k is assumed to be known, it suffices to consider the unforced system It follows from (8) -(9) that "What happens depends on our way of observing it or the fact that we observe it." Werner Heisenberg www.intechopen.com Smoothing, Filtering and Prediction: Estimating the Past, Present and Future 104 which can be written as From the Cayley-Hamilton Theorem, A k , for k ≥ n, can be expressed as a linear combination of A 0 , A 1 , ..., A n-1 . Thus, with N ≥ n -1, equation (11) uniquely determines x 0 if O N has full rank n.
Thus, if O N is of full rank then its inverse exists and so x 0 can be uniquely recovered as Observability is a property of the deterministic model equations (8) - (9). Conversely, if the observability matrix is not rank n then the system (1) -(2) is termed unobservable and the unobservable states are called unobservable modes.

Discrete-time Observability Gramians
Alternative tests for observability arise by checking the rank of one of the observability gramians that are described below.

Lemma 2: The pair (A, C) is completely observable if the observability gramian
is of full rank.
Proof: It follows from (8) -(9) that From the Cayley-Hamilton Theorem, A k , for k ≥ n, can be expressed as a linear combination of A 0 , A 1 , ..., A n-1 . Thus, with N = n -1, is unique provided that W N is of full rank.
It is shown below that an equivalent observability gramian can be found from the solution of a Lyapunov equation.

Lemma 3: Suppose that the system (8) -(9) is stable, that is, |λ i (A)| < 1, i = 1 to n, then the pair (A, C) is completely observable if the nonnegative symmetric solution of the Lyapunov equation
is of full rank.
Proof: Pre-multiplying C T C = W -A T WA by (A T ) k , post-multiplying by A k and summing from k = 0 to N results in It is noted below that observability is equivalent to asymptotic stability.

Lemma 4 [3]: Under the conditions of Lemma
Proof: It follows from (16) that from which the claim follows.
Another criterion that is encountered in the context of filtering and smoothing is detectability. A linear time-invariant system is said to be detectable when all its modes and in particular its unobservable modes are stable. An observable system is alsodetectable.
(ii) Now suppose that measurements of the first state are not available, that is, are of rank 1, the pair (A, C) is unobservable. This system is detectable because the unobservable mode is stable.

Monotonicity
It will be shown below that the solution (6) monotonically approaches a steady-state asymptote, in which case the gain is also timeinvariant and can be precalculated.

Theorem 1: Riccati Equation Comparison Theorem [4] -[10]: Suppose for a t ≥ 0 and for all k ≥ 0 the two Riccati difference equations
have solutions t k P  ≥ 0 and The above result can be verified by substituting The above theorem is used below to establish Riccati difference equation monotonicity.

Theorem 2 [6], [9], [10], [11]: Under the conditions of Theorem 1, suppose that the solution of the Riccati difference equation (19) has a solution t k
"We follow abstract assumptions to see where they lead, and then decide whether the detailed differences from the real world matter." Clinton Richard Dawkins www.intechopen.com The assumption t k P  ≥ 0 is the initial condition for an induction argument. For the induction step, it follows from ( The above theorem serves to establish conditions under which a Riccati difference equation solution monotonically approaches its steady state solution. This requires a Riccati equation convergence result which is presented below.

Convergence
When the model parameters and second-order noise statistics are constant then the predictor gain is also time-invariant andpre-calculated as where P is the symmetric positive definite solution of the algebraic Riccati equation A real symmetric nonnegative definite solution of the Algebraic Riccati equation (21) is said to be a strong solution if the eigenvalues of (A -KC) lie inside or on the unit circle [4], [5]. If there are no eigenvalues on the unit circle then the strong solution is termed the stabilising solution. The following lemma by Chan, Goodwin and Sin [4] sets out conditions for the existence of 7 solutions for the algebraic Riccati equation (21).

Lemma 5 [4]: Provided that the pair (A, C) is detectable, then i) the strong solution of the algebraic Riccati equation (21) exists and is unique;
ii) if A has no modes on the unit circle then the strong solution coincides with the stabilising solution.
A detailed proof is presented in [4]. If the linear time-invariant system (1) -(2) is stable and completely observable and the solution P k of the Riccati difference equation (6) is suitably initialised, then in the limit as k approaches infinity, P k will asymptotically converge to the solution of the algebraic Riccati equation. This convergence property is formally restated below.

Lemma 6 [4]: Subject to:
iii) (P 0 − P) ≥ 0; "We know very little, and yet it is astonishing that we know so much, and still more astonishing that so little knowledge can give us so much power." Bertrand Arthur William Russell A proof appears in [4]. This important property is used in [6], which is in turn cited within [7] and [8]. Similar results are reported in [5], [13] and [14]. Convergence can occur exponentially fast which is demonstrated by the following numerical example.

Example 2.
Consider an output estimation problem where A = 0.9 and B = C = Q = R = 1. The solution to the algebraic Riccati equation (21) is P = 1.4839. Some calculated solutions of the Riccati difference equation (6) Table. 1. Solutions of (21) for Example 2.

State Estimation
The formulation of the steady-state Kalman filter (which is also known as the limiting Kalman filter) follows by allowing k to approach infinity and using the result of Lemma That is, the filter employs fixed gains that are calculated using the solution of the algebraic Riccati equation (21) instead of the Riccati difference equation (6). The filtered state is calculated as where L = ( T T PC CPC + 1 ) R  is the time-invariant filter gain, in which P is the solution of the algebraic Riccati equation (21). The predicted state is given by where the time-invariant predictor gain, K, is calculated from (20).

Asymptotic Stability
The asymptotic stability of the filter (24) -(25) is asserted in two ways. First, recall from Lemma 4 (ii) that if |λ i (A)| < 1, i = 1 to n, and the pair (A, C) is completely observable, then |λ i (A − KC)| < 1, i = 1 to n. That is, since the eigenvalues of the filter's state matrix are within the unit circle, the filter is asymptotically stable. Second, according to the Lyapunov stability theory [1], the unforced system (8) is asymptotically stable if there exists a scalar continuous function V(x), satisfying the following.
Consider the function ( ) x Px where P is a real positive definite symmetric matrix.
Observe that By inspection, the design algebraic Riccati equation (22) is of the form (26) and so the filter is said to be stable in the sense of Lyapunov.

Output Estimation
For output estimation problems, the filter gain, L, is calculated differently. The output estimate is given by where the filter gain is now obtained by L = ( T T CPC CPC + 1 ) R  . The output estimation filter (24) -(25) can be written compactly as from which its transfer function is Smoothing, Filtering and Prediction: Estimating the Past, Present and Future 110

Equivalence of the Wiener and Kalman Filters
As in continuous-time, solving a discrete-time algebraic Riccati equation is equivalent to spectral factorisation and the corresponding Kalman-Yakubovich-Popov Lemma (or Positive Real Lemma) is set out below. A proof of this Lemma makes use of the following identity Lemma 7. Consider the spectral density matrix Then the following statements are equivalent.

(iii) There exists a nonnegative solution P of the algebraic Riccati equation (21).
Proof: Following the approach of [12], to establish equivalence between (i) and (iii), use (21) within (30) to obtain Premultiplying and postmultiplying (32) by , respectively, results in

The Schur complement formula can be used to verify the equivalence of (ii) and (iii).
In Chapter 2, it is shown that the transfer function matrix of the optimal Wiener solution for output estimation is given by where { } + denotes the causal part. This filter produces estimates / k k y from measurements z k .
By inspection of (33) it follows that the spectral factor is The Wiener output estimator (34) involves 1 ( ) z   which can be found using (35) and a special case of the matrix inversion lemma, namely, Thus, the spectral factor inverse is It can be seen from (36) that { } the Wiener filter (34) can be written equivalently which is identical to the transfer function matrix of the Kalman filter for output estimation (29). In Chapter 2, it is shown that the transfer function matrix of the input estimator (or equaliser) for proper, stable, minimum-phase plants is Substituting (35) into (38) gives The above Wiener equaliser transfer function matrices require common poles and zeros to be cancelled. Although the solution (39) is not minimum-order (since some pole-zero cancellations can be made), its structure is instructive. In particular, an estimate of w k can be obtained by operating the plant inverse on Thus, under conditions of diminishing measurement noise, the output estimator will be devoid of dynamics and its maximum magnitude will approach the identity matrix.
"It is not the possession of truth, but the success which attends the seeking after it, that enriches the seeker and brings happiness to him." Max Karl Ernst Ludwig Planck Smoothing, Filtering and Prediction: Estimating the Past, Present and Future 112 Therefore, for proper, stable, minimum-phase plants, the equaliser asymptotically approaches the plant inverse as the measurement noise becomes negligible, that is, Time-invariant output and input estimation are demonstrated below.   The resulting transfer function of the output estimator is H OE (z) = (z + 0.9) 2 (z + 0.9) −2 , which illustrates the low-measurement noise asymptote (41). The minimum-variance input estimator is calculated as where L = QD T Ω -1 . The input estimator transfer function is "Your theory is crazy, but its not crazy enough to be true." Niels Henrik David Bohr www.intechopen.com Smoothing, Filtering and Prediction: Estimating the Past, Present and Future 114 H IE (z) = (z + 0.1) 2 (z + 0.9) −2 , which corresponds to the inverse of the plant and illustrates the asymptote (42). A simulation was generated based on the fragment of Matlab ® script shown in Fig. 1 and some sample trajectories are provided in Fig. 2. It can be seen from the figure that the actual and estimated process noise sequences are superimposed, which demonstrates that an equaliser can be successful when the plant is invertible and the measurement noise is sufficiently low. In general, when measurement noise is not insignificant, the asymptotes (41) -(42) will not apply, as the minimum-variance equaliser solution will involve a trade-off between inverting the plant and filtering the noise. Table 2. Main results for time-invariant output estimation.

Conclusion
In the linear time-invariant case, it is assumed that the signal model and observations can be described by x k+1 = Ax k + Bw k , y k = Cx k , and z k = y k + v k , respectively, where the matrices A, B, C, Q and R are constant. The Kalman filter for this problem is listed in Table 2. If the pair (A, C) is completely observable, the solution of the corresponding Riccati difference equation monotonically converges to the unique solution of the algebraic Riccati equation that appears in the table.
The implementation cost is lower than for time-varying problems because the gains can be calculated before running the filter. If |λ i (A)| < 1, i = 1 to n, and the pair (A, C) is completely "Clear thinking requires courage rather than intelligence." Thomas Stephen Szasz

ASSUMPTIONS
MAIN RESULTS

Signals and system
and C are known. observable, then |λ i (A -KC)| < 1, that is, the steady-state filter is asymptotically stable. The output estimator has the transfer function 1 ( ) ( )( ) Problem 5. Assuming that a system G has the realisation x k+1 = A k x k + B k w k , y k = C k x k + D k w k , expand ΔΔ H (z) = GQG(z) + R to obtain Δ(z) and the optimal output estimation filter.

Glossary
In addition to the terms listed in Section 2.6, the notation has been used herein.
A, B, C, D A linear time-invariant system is assumed to have the realisation x k+1 = Ax k + Bw k and y k = Cx k + Dw k in which A, B, C, D are constant state space matrices of appropriate dimension. This book describes the classical smoothing, filtering and prediction techniques together with some more recently developed embellishments for improving performance within applications. It aims to present the subject in an accessible way, so that it can serve as a practical guide for undergraduates and newcomers to the field. The material is organised as a ten-lecture course. The foundations are laid in Chapters 1 and 2, which explain minimum-mean-square-error solution construction and asymptotic behaviour. Chapters 3 and 4 introduce continuous-time and discrete-time minimum-variance filtering. Generalisations for missing data, deterministic inputs, correlated noises, direct feedthrough terms, output estimation and equalisation are described. Chapter 5 simplifies the minimum-variance filtering results for steady-state problems. Observability, Riccati equation solution convergence, asymptotic stability and Wiener filter equivalence are discussed. Chapters 6 and 7 cover the subject of continuous-time and discrete-time smoothing. The main fixed-lag, fixedpoint and fixed-interval smoother results are derived. It is shown that the minimum-variance fixed-interval smoother attains the best performance. Chapter 8 attends to parameter estimation. As the above-mentioned approaches all rely on knowledge of the underlying model parameters, maximum-likelihood techniques within expectation-maximisation algorithms for joint state and parameter estimation are described. Chapter 9 is concerned with robust techniques that accommodate uncertainties within problem specifications. An extra term within Riccati equations enables designers to trade-off average error and peak error performance. Chapter 10 rounds off the course by applying the afore-mentioned linear techniques to nonlinear estimation problems. It is demonstrated that step-wise linearisations can be used within predictors, filters and smoothers, albeit by forsaking optimal performance guarantees.

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