The limited memory filter appears to be the only device for preventing divergence in the presence of unbounded perturbation.

Andrew H. Jazwinski [79], p. 255

The term state estimation implies that we want to estimate the state of some process, system, or object using its measurements. Since measurements are usually carried out in the presence of noise, we want an accurate and precise estimator, preferably optimal and unbiased. If the environment or data is uncertain (or both) and the system is being attacked by disturbances, we also want the estimator to be robust. Since the estimator usually extracts state from a noisy observation, it is also called a filter, smoother, or predictor. Thus, a state estimator can be represented by a certain block (hardware or software), the operator of which allows transforming (in some sense) input data into an output estimate. Accordingly, the linear state estimator can be designed to have either infinite impulse response (IIR) or finite impulse response (FIR). Since IIR is a feedback effect and FIR is inherent to transversal structures, the properties of such estimators are very different, although both can be represented in batch forms and by iterative algorithms using recursions. Note that effective recursions are available only for delta‐correlated (white) noise and errors.

In this chapter, we introduce the reader to FIR and IIR state estimates, discuss cost functions and the most critical properties, and provide a brief historical overview of the most notable works in the area. Since IIR‐related recursive Kalman filtering, described in a huge number of outstanding works, serves in a special case of Gaussian noise and diagonal block covariance matrices, our main emphasis will be on the more general FIR approach.

1.1 What Is System State?

When we deal with some stochastic dynamic system or process and want to predict its further behavior, we need to know the system characteristics at the present moment. Thus, we can use the fundamental concept of state variables, a set of which mathematically describes the state of a system. The practical need for this was formulated by Jazwinski in [79] as “…the engineer must know what the system is “doing” at any instant of time” and “…the engineer must know the state of his system.”

Obviously, the set of state variables should be sufficient to predict the future system behavior, which means that the number of state variables should not be less than practically required. But the number of state variables should also not exceed a reasonable set, because redundancy, ironically, reduces the estimation accuracy due to random and numerical errors. Consequently, the number of useful state variables is usually small, as will be seen next.

When tracking and localizing mechanical systems, the coordinates of location and velocities in each of the Cartesian coordinates are typical state variables. In precise satellite navigation systems, the coordinates, velocities, and accelerations in each of the Cartesian coordinates are a set of nine state variables. In electrical and electronic systems, the number of state variables is determined by the order of the differential equation or the number of storage elements, which are inductors and capacitors.

In periodic systems, the amplitude, frequency, and phase of the spectral components are necessary state variables. But in clocks that are driven by oscillators (periodic systems), the standard state variables are the time error, fractional frequency offset, and linear frequency drift rate.

In thermodynamics, a set of state variables consists of independent variables of a state function such as internal energy, enthalpy, and entropy. In ecosystem models, typical state variables are the population sizes of plants, animals, and resources. In complex computer systems, various states can be assigned to represent processes.

In industrial control systems, the number of required state variables depends on the plant program and the installation complexity. Here, a state observer provides an estimate of the set of internal plant states based on measurements of its input and output, and a set of state variables is assigned depending on practical applications.

1.1.2 What Model to Estimate State?

Systems and processes can be both nonlinear or linear. Accordingly, we recognize nonlinear and linear state‐space models. Linear models are represented by linear equations and Gaussian noise. A model is said to be nonlinear if it is represented by nonlinear equations or linear equations with non‐Gaussian random components.

Nonlinear Systems

A physical nonlinear system with random components can be represented in continuous time by the following time‐varying state space model,

(1.1)

(1.2)

where the nonlinear differential equation (1.1) is called the state equation and an algebraic equation 1.2 the observation equation. Here, is the system state vector; , is the input (control) vector; is the state observation vector, is some system error, noise, or disturbance; is an observation error or measurement noise; is a nonlinear system function; and is a nonlinear observation function. Vectors and can be Gaussian or non‐Gaussian, correlated or noncorrelated, additive or multiplicative. For time‐invariant systems, both nonlinear functions become constant.

In discrete time , a nonlinear system can be represented in state space with a time step using either the forward Euler (FE) method or the backward Euler (BE) method. By the FE method, the discrete‐time state equation turns out to be predictive and we have

(1.3)

(1.4)

where is the state, is the input, is the observation, is the system error or disturbance, and is the observation error. The model in (1.3) and (1.4) is basic for digital control systems, because it matches the predicted estimate required for feedback and model predictive control.

By the BE method, the discrete‐time nonlinear state‐space model becomes

(1.5)

(1.6)

to suit the many signal processing problem when prediction is not required. Since the model in (1.5) and (1.6) is not predictive, it usually approximate a nonlinear process more accurately.

Linear Systems

A linear time‐varying (LTV) physical system with random components can be represented in continuous time using the following state space model

(1.7)

(1.8)

where the noise vectors and can be either Gaussian or not, correlated or not. If and are both zero mean, uncorrelated, and white Gaussian with the covariances and , where and are the relevant power spectral densities, then the model in (1.7) and (1.8) is said to be linear. Otherwise, it is nonlinear. Note that all matrices in (1.7) and (1.8) become constant as , , , , when a system is linear time‐invariant (LTI). If the order of the disturbance is less than the order of the system, then , and the model in (1.7) and (1.8) becomes standard for problems considering vectors and as the system and measurement noise, respectively.

By the FE method, the linear discrete‐time state equation also turns out to be predictive, and the state‐space model becomes

(1.9)

(1.10)

where , , , , and are time‐varying matrices. If the discrete noise vectors and are zero mean and white Gaussian with the covariances and , then this model is called linear.

Using the BE method, the corresponding state‐space model takes the form

(1.11)

(1.12)

and we notice again that for LTI systems all matrices in (1.9)–(1.12) become constant.

Both the FE‐ and BE‐based discrete‐time state‐space models are employed to design state estimators with the following specifics. The term with matrix is neglected if the order of the disturbance is less than the order of the system, which is required for stability. If noise in (1.9)–(1.12) with is Gaussian and the model is thus linear, then the optimal state estimation is provided using the batch optimal FIR filtering and recursive optimal Kalman filtering. When and/or are non‐Gaussian, then the model becomes nonlinear and other estimators can be more accurate. In some cases, the nonlinear model can be converted to linear, as in the case of colored Gauss‐Markov noise. If and are unknown and bounded only by the norm, then the model in (1.9–1.12). can be used to derive different kinds of estimators called robust.

1.2 Properties of State Estimators

The state estimator performance depends on a number of factors, including cost function, accurate modeling, process suitability, environmental influences, noise distribution and covariance, etc. The linear optimal filtering theory [9] assumes that the best estimate is achieved if the model adequately represents a system, an estimator is of the same order as the model, and both noise and initial values are known. Since such assumptions may not always be met in practice, especially under severe operation conditions, an estimator must be stable and sufficiently robust. In what follows, we will look at the most critical properties of batch state estimators that meet various performance criteria. We will view the real‐time state estimator as a filter that has an observation and control signal in the input and produces an estimate in the output. We will also consider smoothing and predictive state estimation structures. Although we will refer to all the linear and nonlinear state‐space models discussed earlier, the focus will be on discrete‐time systems and estimates.

1.2.1 Structures and Types

In the time domain, the general operator of a linear system is convolution, and a convolution‐based linear state estimator (filter) can be designed to have either IIR or FIR. In continuous time, linear and nonlinear state estimators are electronic systems that implement differential equations and produce output electrical signals proportional to the system state. In this book, we will pay less attention to such estimators.

In discrete time, a discrete convolution‐based state estimator can be designed to perform the following operations:

• Filtering, to produce an estimate at
• Smoothing, to produce an estimate at with a delay lag
• Prediction, to produce an estimate at with a step
• Smoothing filtering, to produce an estimate at taking values from future points
• Predictive filtering, to produce an estimate at over data delayed on points

These operations are performed on the horizon of data points, and there are three procedures most often implemented in digital systems:

• Filtering at over a data horizon , where , to determine the current system state
• One‐step prediction at over to predict future system state
• Predictive filtering at over to organize the receding horizon (RH) state feedback control or model predictive control (MPC)

It is worth noting that if discrete convolution is long, then the computational problem may arise and batch estimation becomes impractical for real‐time applications.

Nonlinear Structures

To design a batch estimator, observations and control signals collected on a horizon , from to , can be united in extended vectors and . Then the nonlinear state estimator can be represented by the time‐varying operator and, as shown in Fig. 1.1, three basic ‐shift state estimators recognized to produce the filtering estimate, if , ‐lag smoothing estimate, if , and ‐step prediction, if :

• FIR state estimator (Fig. 1.1a), in which the initial state estimate and error matrix are variables of
• IIR limited memory state estimator (Fig. 1.1b), in which the initial state is taken beyond the horizon and becomes the input
• RH FIR state estimator (Fig. 1.1c) that processes one‐step delayed inputs and in which and are variables of

Due to different cost functions, the nonlinear operator may or may not require information about the noise statistics, and the initial values may or may not be its variables. For time‐invariant models, the operator is also time‐invariant. Regardless of the properties of , the ‐dependent structures (Fig. 1.1) can give either a filtering estimate, a ‐lag smoothing estimate, or a ‐step prediction.

In the FIR state estimator (Fig. 1.1a) the initial and represent the supposedly known state at the initial point of . Therefore, and are variables of the operator . This estimator has no feedback, and all its transients are limited by the horizon length of points.

In the limited memory state estimator (Fig. 1.1b), the initial state is taken beyond the horizon . Therefore, goes to the input and is provided through estimator state feedback, thanks to which this estimator has an IIR and long‐lasting transients.

The RH FIR state estimator (Fig. 1.1c) works similarly to the FIR estimator (Fig. 1.1a) but processes the one‐step delayed inputs. Since the predicted estimate by appears at the output of this estimator before the next data arrive, it is used in state feedback control. This property of RH FIR filters is highly regarded in the MPC theory [106].

Linear Structures

Due to the properties of homogeneity and additivity [167], data and control signal in linear state estimators can be processed separately by introducing the homogeneous gain and forced gain for LTV systems and constant gains and for LTI systems. The generalized structures of state estimators that serve for LTV systems are shown in Fig. 1.2 and can be easily modified for LTI systems using and .

The ‐shift linear FIR filtering estimate corresponding to the structure shown in Fig. 1.2a can be written as [173]

(1.13)

where the ‐dependent gain is defined for zero input, , and for zero initial conditions. For Gaussian models, the OFIR estimator requires all available information about system and noise, and thus the noise covariances, initial state , and estimation error become variables of its gains and . It has been shown in [229] that iterative computation of the batch OFIR filtering estimate with is provided by Kalman recursions. If such an estimate is subjected to the unbiasedness constraint, then the initial values are removed from the variables. In another extreme, when an estimator is derived to satisfy only the unbiasedness condition, the gains and depend neither on the zero mean noise statistics nor on the initial values. It is also worth noting that if the control signal is tracked exactly, then the forced gain can be expressed via the homogeneous gain, and the latter becomes the fundamental gain of the FIR state estimator.

The batch linear limited memory IIR state estimator appears from Fig. 1.2b by combining the subestimates as

(1.14)

where the initial state taken beyond the horizon is processed with the gain . As will become clear in the sequel, the limited memory filter (LMF) specified by (1.14) with is the batch KF.

The RH FIR state estimator (Fig. 1.2c) is the FIR estimator (Fig. 1.2a) that produces a ‐shift state estimate over one‐step delayed data and control signal as

(1.15)

By , this estimator becomes the RH FIR filter used in state feedback control and MPC. The theory of this filter has been developed in great detail by W. H. Kwon and his followers [91].

It has to be remarked now that a great deal of nonlinear problems can be solved using linear estimators if we approximate the nonlinear functions between two neighboring discrete points using the Taylor series expansion. State estimators designed in such ways are called extended. Note that other approaches employing the Volterra series and describing functions [167] have received much less attention in state space.

1.2.2 Optimality

The term optimal is commonly applied to estimators of linear stochastic processes, in which case the trace of the error covariance, which is the mean square error (MSE), is convex and the optimal gain is required to keep it to a minimum. It is also used when the problem is not convex and the estimation error is minimized in some other sense.

The estimator optimality is highly dependent on noise distribution and covariance. That is, an estimator must match not only the system model but also the noise structure. Otherwise, it can be improved and thus each type of noise requires its own optimal filter.

Gaussian Noise

If a nonlinear system is represented with a nonlinear stochastic differential equation (SDE) (1.1), where is white Gaussian, then the optimal filtering problem can be solved using the approach originally proposed by Stratonovich [193] and further developed by many other authors. For linear systems represented by SDE (1.7), an optimal filter was derived by Kalman and Bucy in [85], and this is a special case of Stratonovich's solution.

If a discrete‐time system is represented by a stochastic difference equation, then an optimal filter (Fig. 1.1) can be obtained by minimizing the MSE, which is a trace of the error covariance . The optimal filter gain can thus be determined by solving the minimization problem

(1.16)

to guarantee, at , an optimal balance between random errors and bias errors, and as a matter of notation we notice that the optimal estimate is biased. A solution to (1.16) results in the batch ‐shift OFIR filter [176]. Given , the OFIR filtering estimate can be computed iteratively using Kalman recursions [229]. Because the state estimator derived in this way matches the model and noise, then it follows that there is no other estimator for Gaussian processes that performs better than the OFIR filter and the KF algorithm.

In the transform domain, the FIR filter optimality can be achieved for LTI systems using the approach by minimizing the squared Frobenius norm of the noise‐to‐error weighted transfer function averaged over all frequencies [141]. Accordingly, the gain of the OFIR state estimator can be determined by solving the minimization problem

(1.17)

where the weights for are taken from the error covariance, which will be discussed in detail in Chapter . Note that if we solve problem (1.17) for unweighed , as in the early work [109], then gain will be valid for unit‐intensity noise. It is also worth noting that, by Parseval's theorem and under the same conditions, the gains obtained from (1.16) and (1.17) become equivalent. The disadvantage of (1.17) is stationarity. However, (1.17) does not impose any restrictions on the bounded noise, which thus can have any distribution and covariance, which is a distinct advantage.

It follows that the optimality in state estimates of Gaussian processes can be achieved if accurate information on noise covariance and initial values is available. To avoid the requirement of the initial state, an estimator is often derived to be optimal unbiased or maximum likelihood. The same approach is commonly used when the noise is not Gaussian or even unknown, so the estimator acquires the property of unbiased optimality.

1.2.3 Unbiased Optimality (Maximum Likelihood)

We will use the term unbiased optimality to emphasize that the optimal estimate subject to the unbiasedness constraint becomes optimal unbiased or an estimator that involves information about noise is designed to track the most probable process value under the assumed statistical model. In statistics, such an estimator is known as the maximum likelihood (ML) estimator, and we will say that it has the property of unbiased optimality. For example, the ordinary least squares (OLS) estimator maximizes the likelihood of a linear regression model and thus has the property of unbiased optimality. From the standpoint of Bayesian inference, the ML estimator is a special case of the maximum a posteriori probability estimator under the uniform a priori noise or error distribution, in which case the ML estimate coincides with the most probable Bayesian estimate. In frequentist inference, the ML estimator is considered as a special case of the extremum estimator. The ML estimation approach is also implemented in many artificial intelligence algorithms such as machine learning, supervised learning, and artificial neural networks.

For further study, it is important that the various types of state estimators developed using the ML approach do not always have closed‐form engineering solutions in state space (or at least reasonably simple closed‐form solutions) due to challenging nonlinear and nonconvex problems.

Gaussian Noise

As was already mentioned, the property of unbiased optimality can be “inoculated” to an optimal state estimate of Gaussian processes if the unbiasedness condition is obeyed in the derivation. The FIR filter derived in this way is called the a posteriori optimal unbiased FIR (OUFIR) filter [222]. The ‐dependent gain can be determined for the OUFIR filter by solving the minimization problem

(1.18)

and we notice that (1.18) does not guarantee optimality in the MSE sense, which means that the unbiased OUFIR estimate is less accurate than the biased OFIR estimate. A distinct advantage is that a solution to (1.18) ignores the initial state and error covariance. It was also shown in [223] that the ML FIR estimate is equivalent to the OUFIR estimate and the minimum variance unbiased (MVU) FIR estimate [221], and thus the property of unbiased optimality is achieved in the following canonical ML form

(1.19)

where matrix is combined with matrices and , , and weight is a function of the noise covariances and , .

It turns out that the recursive algorithm for the batch a posteriori OUFIR filter [222] is not the KF algorithm that serves the batch a posteriori OFIR filter. The latter means that the KF is optimal and not optimal unbiased.

In the transform domain, the property of unbiased optimality can be achieved by applying the approach to LTI systems if we subject the minimization of the squared Frobenius norm of the noise‐to‐error weighted transfer function averaged over all frequencies to the unbiasedness condition as

(1.20)

and we notice that, by Parseval's theorem and under the same conditions, gains produced by (1.18) and (1.20) become equivalent.

Laplace Noise

The heavy‐tailed Laplace distribution may better reflect measurement noise associated with harsh environments such as industrial ones. The Laplace noise is observed in underlying signals in radar clutter, ocean acoustic noise, and multiple access interference in wireless system communications [13,86,134]. To deal with heavy‐tailed noise, a special class of ML estimators called ‐estimators was developed in the theory of robust statistics, and it was shown that the nonlinear median filter is an ML estimator of location for Laplace noise distribution [90]. For multivariate Laplace measurement noise, the median approach [13] can be applied in state space if we consider the sum of absolute errors on and determine the ‐dependent gain of the median ML FIR filter by solving the minimization problem in the infinum as

(1.21)

It is worth noting that the nonlinear minimization problem (1.21) cannot be solved by applying the derivative with respect to , and the exact analytic form of the median FIR filter gain is thus unavailable. However, a numerical solution can be found if we solve the minimization problem (1.21) by approximating with a differentiable function.

When the heavy‐tailed measurement noise is represented by a ratio of two independent zero mean Laplacian random variables, it acquires the meridian distribution [13]. Following the meridian strategy, the ‐shit meridian ML FIR filter gain can be determined by solving the minimization problem

(1.22)

where is referred to as the medianity parameter [13], and we notice the same advantages and drawbacks as in the case of the median ML FIR filter.

Cauchy Noise

Another type of heavy‐tailed noise has the Cauchy distribution [128], and the corresponding nonlinear filtering approach based on the ML estimate of location under Cauchy statistics is called myriad filtering [60]. Under the Cauchy distributed measurement noise, the ‐dependent myriad ML FIR filter gain can be determined by solving the nonlinear minimization problem

(1.23)

where is called the linearity parameter [60]. This nonlinear problem also cannot be solved analytically with respect to the filter gain, but approximate numerical solutions can be feasible for implementation.

1.2.4 Suboptimality

The property of suboptimality is inherent to minimax state estimators, where gains do not exist in closed analytic forms but can be determined numerically by solving the discrete algebraic Riccati inequality (DARI) or linear matrix inequality (LMI). The most elaborated estimators of this type minimize the disturbance‐to‐error (‐to‐) transfer function for maximized norm‐bounded random components and are called robust. Using the LMI approach, the FIR filter gain is computed for the smallest possible error‐to‐disturbance ratio . It is important to note that although the LMI problem is convex, the numerical solution does not guarantee the exact optimal value of . Moreover, estimators of this type may fail the robustness test if they are too sensitive to tuning factors [180].

Performance

We have already mentioned that a solution to the problem can be found analytically and, for LTI systems and Gaussian noise, the FIR filter is equivalent to the OFIR filter. For arbitrary positive definite symmetric error matrices, the time‐invariant gain of the suboptimal FIR filter can be found numerically to match the hybrid FIR structure [109].

The suboptimal filter performance can be obtained by considering the error matrix and introducing an auxiliary matrix such that . Then a nonlinear inequality can be represented as a function of in the LMI form, where structure and complexity depend on disturbances and uncertainties. The FIR filter gain can finally be computed numerically by solving the minimization problem

(1.24)

It is worth noting that the LMI approach can be used to determine suboptimal gains for all kinds of FIR state estimators applied to LTI systems affected by bounded errors, model uncertainties, and external disturbances.

Performance

The estimation performance is reached in state space by minimizing the induced norm of for the maximized disturbance in what is known as the ‐to‐ or energy‐to‐energy filter. The approach is developed to minimize the norm , where is the maximum singular value of . In the Bode plot, the norm minimizes the highest peak value of . In the designs of estimators, the induced norm of is commonly minimized using Parseval's theorem [70], where the ratios of the squared norms have meanings the ratios of the energies. Since the norm reflects the worst estimator case, the estimator is called robust.

The optimal FIR filtering problem implies that the FIR filter gain can be found by solving on the following optimization problem,

(1.25)

where and are some proper weights. Unfortunately, closed‐form optimal solutions to (1.25) are available only in some special cases. Therefore, the following suboptimal algorithm can be used to determine the gain numerically,

(1.26)

where a small enough factor indicates a part of the disturbance energy that goes to the estimator error. The solution to (1.26) is commonly found using LMI and the bounded real lemma.

Hybrid Performance

Hybrid suboptimal FIR state estimation structures are developed to improve the robustness by minimizing simultaneously the trace of the average weighted transfer function using the approach and the peak value of using the approach. An example is the FIR filter, where gain obeys both the and constraints and can be determined by solving the following minimization problem

(1.27)

Similarly, a hybrid FIR state estimator can be developed.

Generalized Performance

In generalized filtering, the energy‐to‐peak transfer function is minimized and the generalized performance is achieved by minimizing the peak error for the maximized disturbance energy [213] in what is called the energy‐to‐peak or ‐to‐ estimation algorithm. Because an optimal solution to the generalized filtering problem does not exist in closed form, suboptimal algorithms were elaborated in [188] using LMI and the energy‐to‐peak lemma and then developed by many authors [5,31,144,159].

The energy‐to‐peak filtering approach [213] implies minimizing the infinity error norm¹ [188] over the maximized disturbance norm . Accordingly, the suboptimal FIR filter gain can be determined by solving the optimization problem

(1.28)

using the energy‐to‐peak lemma [188]. In Chapter, we will show that this lemma is universally applicable to BE‐ and FE‐based state‐space models.

Performance

In some applications, it is required to minimize the peak error with respect to the peak disturbance using the peak‐to‐peak, ‐to‐, or most generally state estimator. The need to exploit such estimators arises when a system operates under impulsive external attacks and this is not enough to minimize the peak estimation error with respect to the disturbance energy. The approach was originally proposed in [208] and then developed by many authors [1,133,144,209]. It suggests to solve the peak‐to‐peak filtering problem by minimizing the norm of the peak disturbance to peak error transfer function on as . Accordingly, the FIR filter gain can be computed numerically by solving the optimization problem

(1.29)

It can be shown that an optimal solution to (1.29) is mathematically untractable, but a suboptimal can be found under the following assumptions. Since at does not exceed the upper bound , the following substitution can be made: , where is a scaling scalar. Likewise, substitute , where is a scaling scalar. A suboptimal can then be determined by solving the optimization problem

(1.30)

where is a scalar and is small enough. In Chapter, we will show that the problem (1.30) can be solved numerically using the peak‐to‐peak lemma and LMI.

1.2.5 Unbiasedness

The property of unbiasedness is inherent to state estimators, which satisfy the unbiasedness condition

(1.31)

Because the operator of averaging removes zero mean noise from the model and zero mean random errors from the estimate, then it follows that any estimator will be unbiased if the test by (1.31) is positive. Estimators, which belong to the family of ML state estimators, produce the optimal unbiased estimate and thus obey (1.31). A purely unbiased estimator can be obtained by ignoring the zero mean noise in the system and the observation. That leads to the ‐shift unbiased FIR (UFIR) filtering estimate [179],

(1.32)

which belongs to the class of OLS and also appears from the canonic ML estimate (1.19) by setting . Note that (1.32) does not pretend to minimize error covariance. Even so, this estimator is also able to minimize the MSE on the optimal horizon of points.

1.2.6 Deadbeat

If there is no noise in the state space model, then the state and its estimate become deterministic and the unbiasedness condition (1.31) degenerates to

(1.33)

which is known as the deadbeat property. Any estimator can be tested for (1.33) if we ignore noise in the model. Inherently, the OFIR estimator [176], ML estimator (1.19), and UFIR estimator (1.32) are deadbeat estimators. However, it cannot be said that KF has this property, because the noise covariances cannot be ignored in its recursions.

1.2.7 Denoising (Noise Power Gain)

As a measure of noise reduction in FIR filters, the noise power gain (NPG) was introduced by W. F. Trench in [198] as the ratio of the output noise variance to the input noise variance ,

(1.34)

The NPG is thus akin to the noise figure in wireless communications. For white Gaussian noise, the NPG is equal to the sum of the squared coefficients of the impulse response of the FIR filter or the squared norm of . Thus, for white Gaussian noise it can be computed as , which does not apply for other types of noise.

In state space, the homogeneous gain represents the impulse response coefficients of the FIR filter for each state. Therefore, in [186] a symmetric square matrix of dimensions was introduced and called the generalized NPG (GNPG),

(1.35)

The value of this measure resides in the fact that it shows which of the estimators is prone to best noise reduction. Indeed, the main diagonal components of are proportional to the NPGs in the system states, and the remaining components to the cross NPGs. Therefore, the estimator whose GNPG has the smallest components generates the least random noise at the output. For example, the UFIR filter has a larger GNPG than the OFIR filter and thus provides poorer noise reduction. It then follows that GNPG (1.35) can be used to compare different types of FIR state estimator in terms of noise reduction, provided that gains of these estimators are known.

1.2.9 Robustness

George E. P. Box was the first to introduce the term robustness in statistics [21]. Years later, P. J. Huber summarized that robustness signifies insensitivity to small deviations from the assumptions [74], and many other definitions of robustness have been adopted over time for specific applications. In this book, we will view robustness as the ability of an estimator to be unresponsive to model errors, uncertainties, and disturbances. This, however, has two different meanings. Solving the identification problem requires an estimator to be robust to all kinds of internal and external errors. In contrast, estimating the disturbed state requires an estimator to be robust only to observation errors. In both cases, the estimator must be robust to temporary uncertainties and impulsive attacks.

Obviously, a robust estimator is not one that is called robust, but which passes the test for robustness. The estimator robustness is highly dependent on the tuning factors and variables such as initial state, noise covariances, error matrices, scaling of norm‐bounded estimators, etc. Since each tuning factor is introduced to improve performance, it follows that the performance can degrade dramatically if tuning is not done properly. Thus, the rule of thumb for robustness to tuning errors can be formulated as follows. An estimator with a small number of tuning factors is more robust than an estimator with a large number of tuning factors.

The effect of tuning errors on the estimation accuracy is sketched in Fig. 1.3 for the worst case, when these errors do not compensate for each other and can occur in opposite directions.

At the tuning phase, the blind estimator, which has no tuning factor, gives the largest errors. But this estimator is truly robust because its output is not affected by tuning. Several tuning factors can dramatically improve accuracy. However, tuning errors will degrade its performance. More tuning factors can further improve accuracy. But this can be accompanied by significant performance degradation due to tuning errors. The reader is referred to [180], where the trade‐off in robustness between the UFIR, Kalman, and game theory filters was investigated.

How can we tune an estimator to make it the most robust? The theory suggests that an estimator becomes robust if it is optimally tuned for maximized disturbance. If this is the case, then any deviation from the tuning point will lead to an increase in errors and a decrease in disturbance, which compensate for each other, and hence robustness.

1.2.10 Computational Complexity

The computational complexity of digital estimators is commonly associated with the computation time. In this regard, the most accurate batch FIR estimators, operating with data on the horizon of points, have the highest computational complexity and largest computation time. An iterative algorithm for computing OFIR estimates using KF recursions has complexity and medium computation time. The KF algorithm has the lowest computational complexity and is the fastest state estimator. But the computation time consumed by iterative FIR algorithms can be significantly reduced using parallel computing.

Thus, the obvious reason for getting recursive forms is to reduce complexity when it is critical for real‐time applications. But in trying to find suitable recursions, one realizes that they are available only for Gaussian processes and diagonal block error matrices. To illustrate this issue, let us consider the block measurement noise covariance matrix . For delta‐correlated noise, this matrix is diagonal and can readily be represented recursively as

(1.36)

that, for OFIR filter with diagonal , leads to Kalman recursions [229]. It can be shown that recursions for nondiagonal and are complex, which can increase the computational complexity so that the original batch estimate can become more efficient. This observation explains the fact that the optimal recursions derived by Kalman for white Gaussian noise and serving the OFIR filter with diagonal and and the unbiased recursions derived by Shmaliy for the UFIR filter by ignoring zero mean noise appear to be the only low‐complexity engineering schemes used in practice.

1.3 More About FIR State Estimators

Now, it might be worthwhile to outline the most general features of FIR state estimators to make the rest of this book easier to read.

First, recall that the FIR state estimator is a batch estimator (Fig. 1.1 and Fig. 1.2) and, as such, is generally more accurate than a recursive computation algorithm. This is because the batch structure operates with full block noise covariance and error matrices, while suitable recursive forms are available when such matrices are diagonal. An exception is white Gaussian processes, where the block noise covariance matrices are diagonal and the Kalman recursions exactly compute the batch OFIR estimates.

Next, the folk belief that the batch is computationally less efficient than the recursive form is true only for Gaussian processes. For non‐Gaussian processes, low‐complexity recursive forms are commonly not available. Furthermore, in the modern world of fast computers and large memory, computational complexity no longer matters much, unlike in Kalman's days. Consequently, there is growing interest in batch FIR state estimators that produce the highest accuracy with the highest robustness.

Jazwinski was the first to draw attention in [79] to another fact that finite horizon (FH) control (read: RH FIR control) is more robust to temporary uncertainties than infinite horizon control (read: KF‐based control). His argument was that the FH estimator does not project old errors beyond the horizon to the current point, and therefore the LMF is preferable. The general idea behind the FH approach was then formulated by Schweppe [161] as an old estimate updated not over all data on but over an FH of most recent observations. It was later rephrased by Maybeck [126] that it is preferable to rerun the growing memory filter over the data FH for each . In the sequel, FH estimators were developed using KF as LMFs for several control theories. In the general formulation of the FH estimation problem, it is required to minimize the cost function on at , , taking into account the input signal . In the RH FIR control or MPC [106], this goal is achieved by predicting the state at over most recent past data taken from . The most elaborated RH minimum variance FIR (MVF) filter was designed in [105] to be bias‐constrained or ML (not optimal).

Finally, it should be noted that the FIR approach recognizes the difference between the linear FE‐based state model used in control and the BE‐based used when prediction is not required. For example, the a posteriori ‐shift OFIR filter [176] uses the BE‐based state model, and the MVF filter [105] is based on the FE‐based state model. This difference is essential when developing batch estimators but is poorly recognized in recursive schemes.

1.4 Historical Overview and Most Noticeable Works

The history of the development of methods of batch FIR state estimation related to filtering, smoothing, and prediction is rooted in a wide area of estimation theory. Probably the first and rather awkward batch method was used by J. Kepler² in 1601 [1.1] to estimate the orbit of Mars from 12 observations. The next noticeable step was made in 1795 by C. F. Gauss³ [1.2], who contributed significant theoretical advances used to predict the planetary orbits with reasonable accuracy for a more elegant batch estimation approach, now referred to as the least squires.

In the 1930s, A. N. Kolmogorov⁴ developed the least squires approach for discrete stationary random processes from purely mathematical consideration [1.3]. Soon after, in the early 1940s, N. Wiener⁵ reconsidered the problem from the engineering perspective, extended the least squires approach to continuous‐time stationary random processes, and derived an optimal filter, now known as the Wiener filter [1.4]. The optimal Wiener filter was then applied to a wide range of applications, including weather forecasting, economics, and communications. Further, Wiener investigated and solved several smoothing problems including working in the frequency domain with signals characterized by their power spectral densities and estimators derived in terms of their transfer functions.

Since the class of random stationary processes is relatively narrow, further progress has been achieved for dynamic systems governed by white Gaussian noise. This made it possible to conveniently simulate nonstationary processes in state space, both in continuous and discrete time, and subsequently to elegantly solve the optimal state estimation problem. The most significant works on the optimal state estimation of linear nonstationary stochastic processes with white Gaussian noise were published in 1960–1961 by R. E. Kalman.⁶ The approach is based on the use of Bayes⁷ formula, a priori and a posteriori probability distributions, and likelihood function. Despite the fact that Bayes' rule cannot be applied directly to the filtering problem, the optimality conditions imposed on it lead to an optimal filtering algorithm. The Kalman approach in discrete time [1.5] and the Kalman‐Bucy approach in continuous time [1.6] are known now as Kalman filtering, and the corresponding algorithms are referred to as the Kalman and Kalman‐Bucy filters, respectively. Moreover, since this approach covers both stationary and nonstationary stochastic processes, the linear optimal filtering theory is called simply linear optimal Kalman filtering. About the time the Kalman filtering theory appeared, fundamental results on filtering of non‐Gaussian nonlinear random processes were presented in 1955–1965 by R. L. Stratonovich⁸ [1.7], whose nonlinear approach generalizes Kalman filtering in the linear case.

Soon after the optimal KF became a powerful tool for state estimation in linear systems, many authors also showed that its operation conditions are not always met in practice. Accordingly, the KF algorithm has been modified for colored Gauss‐Markov noise, extended to nonlinear cases using Taylor series and unscented transform, reorganized to solve various smoothing problems, and robustified. These and other developments of the KF algorithm can be found in D. Simon's book [1.8] and many other outstanding works.

In 1968, A. H. Jazwinski has made a fundamental observation [1.9], which, decades later, arouses great interest in state estimation using FIR structures. He stated [1.10] that the LMF appears to be the only device for preventing divergence in the presence of unbounded perturbation in the system and concluded that FH control is more robust than infinite horizon control. It then turned out that the initial conditions for optimal LMF operation could not always be set properly, and the approach was not widely adopted, although the basic idea was latter used in FIR filtering.

The batch FIR estimator can be viewed as a Gauss least squares estimator, although its roots can rather be found in Wiener's filtering theory. In [1.11], N. Levinson used Wiener's MSE criterion to design a discrete‐time filter and predictor in recursive form on the FH of past data. The FIR modification of the Wiener filter was made by L. A. Zadeh⁹ and J. R. Ragazzini in [1.12]. In [1.13], K. R. Johnson extended the Zadeh‐Ragazzini results to discrete time. The first RH UFIR filter based on discrete convolution was obtained by P. Heinonen and Y. Neuvo in [1.14], and the UFIR filter was derived by Y. S. Shmaliy in [1.16]. A general approach to FIR estimation using convolution and Hankel operator norms was developed by D. Wilson in [1.17].

In state space, the theory of various RH FIR structures and recursive forms was first developed in the works of W. H. Kwon et al. [1.18,1.19]. The theory of ‐shift OFIR filtering was developed by Y. S. Shmaliy [1.20,1.21]. In the batch form, the first RH UFIR filter was shown by P. S. Kim and M. E. Lee in [1.22], and the ‐shift UFIR filter was derived by Y. S. Shmaliy in [1.20]. The UFIR filter was then represented with an iterative algorithm using recursions by Y. S. Shmaliy in [1.23] to become a robust alternative to KF [1.24]. Several , , and hybrid FIR structures were first proposed by W. H. Kwon et al. in [1.19, 1.25].

In most recent years, significant progress in FIR state estimation has been achieved in works of Y. S. Shmaliy, S. Zhao, and C. K. Ahn. The first‐ and second‐order extended UFIR filters were proposed for nonlinear models in [1.26]. The optimal horizon for the UFIR filter is justified in [1.27]. In [1.28], it was shown that recursions for OUFIR and ML FIR filters are not Kalman recursions, and therefore KF is optimal, not optimal unbiased, as previously thought. It was shown in [1.29] that all types of bias‐constrained OFIR filters belong to the class of ML state estimators. In [1.30], it was proved that the error covariance of the UFIR filter can be computed iteratively. In [1.31], an improved frequency efficient RH FIR filtering algorithm was proposed. The UFIR filtering approach has been extended in [1.32] to time‐stamped delayed and missing data. In [1.33], a new FIR state estimation approach was developed under disturbances, data errors, and uncertain initial conditions. It was shown in [1.34] that the batch OFIR filtering estimate can be computed using Kalman recursions, and therefore the OFIR and Kalman approaches are not two different methods as previously thought. Chapter and Chapter of this book contribute to the current view of FIR state estimation under disturbances and for uncertain systems, and other important publications can be found in the references.

1. [1.1] Kepler J (1602) De Fundamentis Astrologiae Certioribus. Typis Schumanianis, Pragae Bohemorum
2. [1.2] Plackett RL (1972) Studies in the History of Probability and Statistics. XXIX: The discovery of the Method of least squares. Biometrika 59(2):239–251
3. [1.3] Publications of A. N. Kolmogorov (1989) Annals of Probability 17(3):945–964
4. [1.4] Wiener N (1942) Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications. MIT Press, Boston MA
5. [1.5] Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans. ASME (J. Basic Eng.) D82(1):35–45
6. [1.6] Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. Trans. ASME (J. Basic Eng.) D83(1):95–108
7. [1.7] Stratonovich RL Topics in the Theory of Random Noise, Vol. 1 (1963), Vol. 2 (1967). Gordon and Breach, NY
8. [1.8] Simon D (2006) Optimal State Estimaiton: Kalman, , and Nonlinear Approaches. John Wiley & Sons, Hoboken, NJ
9. [1.9] Jazwinski AH (1968) Limited memory optimal filtering. IEEE Trans. Automat. Contr. 13(5):558–563
10. [1.10] Jazwinski AH (1970) Stochastic Processes and Filtering Theory. Academic Press
11. [1.11] Levinson N. (1947) The Wiener RMS error criterion in filter design and prediction. J. Math. Phys. 25:261–278.
12. [1.12] Zadeh LA, Ragazzini JR (1950) An Extension of Wiener's Theory of Prediction. J. Appl. Phys. 21:645–655
13. [1.13] Johnson KR (1956) Optimum, linear, discrete filtering of signals containing a nonrandom component. IRE Trans. Inform. Theory 2(2):49–55
14. [1.14] Friedlander B, Morf M, Kailath T, Ljung L (1978) Extended Levinson‐ and Chandra‐Sekhar‐type equations for a general discrete time linear estimation problem. IEEE Trans. Automat. Contr. 23(4):653–659
15. [1.15] Heinonen P, Neuvo Y (1988) FIR‐median hybrid filter with predictive FIR substructures, IEEE Trans. Acoust. Speech Signal Process. 36(6):892–899
16. [1.16] Shmaliy YS (2006) An unbiased FIR filter for TIE model of a local clock in applications to GPS‐based timekeeping, IEEE Trans. on Ultrason. Ferroelec. Freq. Contr. 53(5):862–8706
17. [1.17] Wilson D (1989) Convolution and Hankel operator norms for linear systems, IEEE Trans. Autom. Contr. 34(1):94–97
18. [1.18] Kwon OK, Kwon WH, Lee KS (1989) FIR filters and recursive forms for discretetime state‐space models. Automatica 25(5):715–728
19. [1.19] Kwon WH, Han S (2005) Receding Horizon Control: Model Predictive Control for State Models. Springer, London
20. [1.20] Shmaliy YS (2010) Linear optimal FIR estimation of discrete time‐invariant state space models, IEEE Trans. Signal Process. 58(6):3086–3096
21. [1.21] Shmaliy YS, Ibarra‐Manzano OG, Time‐variant linear optimal finite impulse response estimator for discrete state‐space models, Int. J. Adapt. Contr. Signal Process. 26(2):95–104
22. [1.22] Kim PS, and Lee ME (2007) A new FIR filter for state estimation and its applications, J. Comput. Sci. Technol. 22(5):779–784
23. [1.23] Shmaliy YS (2011) An iterative Kalman‐like algorithm ignoring noise and initial conditions, IEEE Trans. Signal Process. 59(6):2465–2473
24. [1.24] Shmaliy YS, Zhao S, Ahn CK (2017) Unbiased FIR filtering: an iterative alternative to Kalman filtering ignoring noise and initial conditions, IEEE Contr. Syst. Mag. 37(5):70–89
25. [1.25] Lee YS, Han SH, Kwon WH (2006) FIR filters for discrete‐time state space models. Int. J. Control, Autom. Syst. 4(5):645–652
26. [1.26] Shmaliy YS (2012) Suboptimal FIR filtering of nonlinear models in additive white Gaussian noise, IEEE Trans. Signal Process. 60(10):5519–5527
27. [1.27] Ramirez‐Echeverria Sarr FA, Shmaliy YS (2014) Optimal memory for discrete‐time FIR filters in state space, IEEE Trans. Signal Process 62(3):557–561
28. [1.28] Zhao S, Shmaliy YS, Liu (2016) Fast Kalman‐like optimal unbiased FIR filtering with applications, IEEE Trans. Signal Process. 64(9):2284–2297
29. [1.29] Zhao S, Shmaliy YS, Unified maximum likelihood form for bias constrained FIR filters, IEEE Signal Process. Lett. 23(12):1848–1852
30. [1.30] Zhao S, Shmaliy YS, Liu F (2017) On the iterative computation of error matrix in unbiased FIR filtering, IEEE Signal Process. Lett. 24(5):555–558
31. [1.31] Ahn CK, Zhao S, Shmaliy YS (2017) Frequency efficient receding horizon FIR filtering in discrete‐time state‐space, IEEE Trans. Circ. Syst. I: Regular Papers 64(11):2945–2953
32. [1.32] Uribe‐Murcia K, Shmaliy YS, Ahn CK, Zhao S (2020) Unbiased FIR filtering for time‐stamped discretely delayed and missing data. IEEE Trans. Automat. Contr. 65(5):2155–2162
33. [1.33] Ortega‐Contreras J, Eli Pale‐Ramon E, Shmaliy YS, Xu Y (2021) A novel approach to FIR prediction under disturbances and measurement errors, IEEE Signal Process. Lett. 28:150–154
34. [1.34] Zhao S, Shmaliy YS, Liu F (2021) Optimal FIR filter for discrete‐time LTV systems and fast iterative algorithm, IEEE Trans. Circ. Syst. II, Express Briefs

1.5 Summary

Unlike the classical estimation theory, which was developed to estimate parameters of the approximating function based on measurements with random components, the state estimation theory deals with the internal state of a system. Hence, the state estimator can be thought of as an input‐to‐output structure (hardware or software) that has an IIR or FIR. The set of state variables should be sufficient to predict the future behavior of a system and should not be redundant to avoid excess errors. Physical systems are represented in state space via nonlinear or linear differential equations with random terms. In discrete‐time state‐space, noisy systems are represented via difference equations, which appear from the corresponding differential equations by applying the FE method or BE method.

The theory of linear systems with Gaussian random components is most developed. If a model is nonlinear, then the most common way is to linearize it using the Taylor series expansion and apply methods of linear state estimation. Regardless of structure, the state estimator can be designed to be a filter, ‐lag smoother, ‐step predictor, smoothing filter, or predictive filter.

The term optimal most often refers to estimators of linear stochastic processes, in which case an optimal balance in the MSE sense is obtained between regular bias and random errors. It also applies to systems with random components when the estimation error is minimized in some other sense. In general, each noise requires its own optimal filter: OFIR filter and KF for Gaussian noise, median filter for Laplace noise, myriad filter for Cauchy noise, etc. If the derivation of the optimal estimator obeys the unbiasedness constraint, then the estimate has the property of unbiased optimality. All linear bias‐constrained FIR state estimators belong to the class of ML estimators and have the canonic ML FIR form (1.19).

The property of suboptimality is inherent to norm‐bounded filters, the gains of which cannot be obtained analytically and are determined numerically using LMI. The following suboptimal FIR state estimators having a disturbance‐to‐error transfer function can be designed using LMI: by minimizing the squared Frobenius norm , by minimizing the induced norm of in the ‐to‐ or energy‐to‐energy filter, by combining the and filtering algorithms, generalized by minimizing the energy‐to‐peak in the ‐to‐ structure, and by minimizing the ‐to‐ or peak‐to‐peak .

Unbiasedness is achieved when the state estimator satisfies the unbiasedness condition (1.31). This condition results in a batch UFIR filter whose canonic form (1.32) can be computed using recursions that are not Kalman recursions. An estimator is a deadbeat estimator if it satisfies the condition (1.33) by neglecting noise in the model. All FIR state estimators are BIBO stable, which is not the case for IIR state estimators such as LMF and KF. The state estimator with the lowest GNPG is the most noise canceling. The state estimator with the least number of tuning factors and the sensitivity to these factors is the most robust. In this sense, the UFIR filter, which requires only one tuning factor, the horizon length, is the most robust of all other linear estimators.

The KF algorithm has low computational complexity . Iterative FIR algorithms operating on a horizon of points have complexity and batch FIR state estimators . A computationally complex estimator usually requires a lot of memory to obtain an estimate.

1.6 Problems