On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs

In this paper, a globally optimal filtering framework is developed for unbiased minimum-variance state estimation for systems with unknown inputs that affect both the system state and the output. The resulting optimal filters are globally optimal within the unbiased minimum-variance filtering over all linear unbiased estimators. Globally optimal state estimators with or without output and/or input transformations are derived. Through the global optimality evaluation of this research, the performance degradation of the filter proposed by Darouach, Zasadzinski, and Boutayeb [Darouach, M., Zasadzinski, M., & Boutayeb, M. (2003). Extension of minimum variance estimation for systems with unknown inputs. Automatica, 39, 867-876] is clearly illustrated and the global optimality of the filter proposed by Gillijns and De Moor [Gillijns, S., & De Moor, B. (2007b). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43, 934-937] is further verified. The relationship with the existing literature results is addressed. A unified approach to design a specific globally optimal state estimator that is based on the desired form of the distribution matrix from the unknown input to the output is also presented. A simulation example is given to illustrate the proposed results.     

On optimal output regulation for linear systems

The classical output regulation problem formulation for linear systems has a number of shortcomings; among them a primary one is that it does not take into account the transient response. Although this problem has been studied since the 1970s, a complete picture has not emerged yet. We formulate and study here a number of output regulation problems which incorporate the notion of transient performance into them. By defining a performance index that involves transient response, we define optimal and suboptimal control problems which are constrained by the steady state output regulation requirement. Such constrained optimal and suboptimal control problems for a given system are studied by transforming them to unconstrained optimal and suboptimal control problems for certain auxiliary systems. Several issues such as obtaining expressions for the infimal performance measure, solvability conditions, regulator construction are studied in detail. Both state and measurement feedback controllers are considered.     

Output regulation of linear plants subject to constraints

Output regulation problems for continuous-time linear systems with state and/or input constraints are studied. The problems are formulated in global and semi-global setting by using state or full information feedback. The goal of this paper is to develop solvability conditions for the posed problems. Moreover, appropriate regulators are constructed under the solvability conditions. To state the solvability conditions clearly, a taxonomy of constraints is introduced which delineates the constraints into several categories. Such a taxonomy of constraints provides a classification of linear plants with constraints and identifies what types of output regulation problems are solvable. Results developed here include as a special case the results obtained in the literature for systems with only input constraints. The constraint taxonomy also identifies some intrinsically hard constraints (non-right invertible constraints) for which the solvability conditions of global/semi-global output regulation problems are not clear yet. As a special case of output regulation, we also consider tracking problems with constraints. It is shown that if there exists a state feedback controller with a stabilizing domain of attraction, then one can find a regulator with a tracking domain of attraction arbitrarily close to the stabilizing domain.     

Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough

This paper extends previous work on joint input and state estimation to systems with direct feedthrough of the unknown input to the output. Using linear minimum-variance unbiased estimation, a recursive filter is derived where the estimation of the state and the input are interconnected. The derivation is based on the assumption that no prior knowledge about the dynamical evolution of the unknown input is available. The resulting filter has the structure of the Kalman filter, except that the true value of the input is replaced by an optimal estimate.     

Analysis, design, and performance limitations of optimal filtering in the presence of an additional input with known frequency

In this paper, the inputs are considered to be of two types. The first type of input, as in standard H₂ optimal filtering, is a zero mean wide sense stationary white noise, while the second type is a linear combination of sinusoidal signals each of which has an unknown amplitude and phase but known frequency. The generalized H₂ optimal filtering problem seeks to find a linear stable filter that estimates a desired output such that the H₂ norm of the transfer matrix from the white noise input to the estimation error is minimized subject to the constraint that the mean of the error converges to zero for all initial conditions of the given system and filter and for all possible external sinusoidal signals. The analysis, design, and performance limitations of generalized H₂ optimal filters are presented here.     

Linear dynamic filtering with noisy input and output

State estimation problems for linear time-invariant systems with noisy inputs and outputs are considered. An efficient recursive algorithm for the smoothing problem is presented. The equivalence between the optimal filter and an appropriately modified Kalman filter is established. The optimal estimate of the input signal is derived from the optimal state estimate. The result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem.     

Kalman filtering in extended noise environments

This note introduces an extended environment for Kalman filtering that considers also the presence of additive noise on input observations in order to solve the problem of optimal (minimal variance) estimation of noise-corrupted input and output sequences. This environment includes as subcases both errors-in-variables filtering (optimal estimate of inputs and outputs from noisy observations) and traditional Kalman filtering (optimal estimate of state and output in presence of state and output noise). A Monte Carlo simulation shows that the performance of this extended filtering technique leads to the expected minimal variance estimates.     

Unbiased minimum-variance input and state estimation for linear discrete-time systems

This paper addresses the problem of simultaneously estimating the state and the input of a linear discrete-time system. A recursive filter, optimal in the minimum-variance unbiased sense, is developed where the estimation of the state and the input are interconnected. The input estimate is obtained from the innovation by least-squares estimation and the state estimation problem is transformed into a standard Kalman filtering problem. Necessary and sufficient conditions for the existence of the filter are given and relations to earlier results are discussed.     

A note on interval observer design for unknown input estimation

This paper addresses the problem of interval observer design for unknown input estimation in linear time-invariant systems. Although the problem of unknown input estimation has been widely studied in the literature, the design of joint state and unknown input observers has not been considered within a set-membership context. While conventional interval observers could be used to propagate with some additional conservatism, unknown inputs by considering them as disturbances, the proposed approach allows their estimation. Under the assumption that the measurement noise and the disturbances are bounded, lower and upper bounds for the unmeasured state and unknown inputs are computed. Numerical simulations are presented to show the efficiency of the proposed approach.     

Semi-global stabilization and output regulation of constrained linear plants via measurement feedback

Semi-global stabilization and output regulation of linear systems subject to state and/or input constraints have been studied in our earlier work by using state feedback. For the same problems, observer based measurement feedback control designs are the topics of this paper. High-gain observers are used in the feedback design in order to obtain accurate estimates of the state so that the constraint violation can be avoided. Due to the peaking phenomenon associated with a high-gain observer, a special saturation protection is built in the control laws to avoid possible constraint violation. The results in this paper show that the semi-global stabilization and semi-global output regulation problems for constrained linear systems are solvable via measurement feedback under solvability conditions similar to those in the state feedback.     

飞机颤振模态参数辨识试验的快速滤波算法

研究飞机颤振随机模型中实际输入-输出信号序列的最优滤波估计问题,利用矩阵论中的矩阵因式分解和统计信号处理中的条件期望公式,将由新息过程构成的块Toeplitz矩阵进行三角分解,得到一种有效的递推滤波算法。对于滤波输入-输出信号的估计值,推导该算法下的估计误差和方差表达式。最后用仿真算例验证采用滤波后得到的输入-输出信号估计值作为飞机颤振模态参数辨识试验的观测信号可得到较为准确的传递函数,进而使得模态参数的辨识也更精确。     

On the estimation problem of a class of continuous bioreactors with unknown input

The estimation problem of substrate–biomass reactors with isotonic or nonisotonic growth is addressed. The unmeasured reactor and time-varying feed substrate concentrations must be estimated from the dilution rate input and biomass measurements. First, the solvability conditions (nonlinear-global observability or detectability) are characterized, finding that: (i) the conditions depend on particular motions and growth mechanisms, and (ii) with isotonic (or nonisotonic) growth, the unknown input–state pair is observable in the classical (or nonstandard) single-valued (S) [or bivalued (B)] sense. Then, a robustly convergent S (or B)-observer is designed accordingly. The S-observer yields the estimate of the only unmeasured input–state trajectory pair associated with the measured input–output signal. The B-observer yields the estimates of the two possible unmeasured input–state trajectory pairs associated with the measured input–output signal, each one satisfying the reactor mass balances. The developments and findings are illustrated with representative (Monod and Haldane) examples through analytic assessment and numerical simulation.     

Inverse filtering and deconvolution

This paper studies the so‐called inverse filtering and deconvolution problem from different angles. To start with, both exact and almost deconvolution problems are formulated, and the necessary and sufficient conditions for their solvability are investigated. Exact and almost deconvolution problems seek filters that can estimate the unknown inputs of the given plant or system either exactly or almostly whatever may be the unintended or disturbance inputs such as measurement noise, external disturbances, and model uncertainties that act on the system. As such they require strong solvability conditions. To alleviate this, several optimal and suboptimal deconvolution problems are formulated and studied. These problems seek filters that can estimate the unknown inputs of the given system either exactly, almostly or optimally in the absence of unintended (disturbance) inputs, and on the other hand, in the presence of unintended (disturbance) inputs, they seek that the influence of such disturbances on the estimation error be as small as possible in a certain norm (H₂ or H_∞) sense. Both continuous‐ and discrete‐time systems are considered. For discrete‐time systems, the counter parts of all the above problems when an ??‐step delay in estimation is present are introduced and studied. Next, we focus on the exact and almost deconvolution but this time when the uncertainties in plant dynamics can be structurally modeled by a Δ‐block as a feedback element to the nominally known plant dynamics. This is done either in the presence or absence of external disturbances. Copyright © 2001 John Wiley & Sons, Ltd.     

Connections between H 2 optimal filters and unknown input observers – performance limitations of H 2 optimal filtering

For linear time invariant continuous-time systems with either unknown or white noise input, two well-known filtering problems are considered. These are the unknown input observer problem and the Kalman filtering problem. Most of the available literature on Kalman filtering considers the so-called regular filtering problem. We consider here the general singular filtering problem. We show that such a Kalman filtering problem for a given system can be transformed to the unknown input observer problem for an auxiliary system constructed from the data of the given system. Such transformations between these two filtering problems enable us to study various properties of Kalman filtering, including existence and uniqueness of Kalman filters, computation of performance indices of Kalman filtering, and performance limitations of Kalman filtering as related to the structural properties of the given system.     

A framework for globally optimal state estimation for systems with unknown inputs (I): transformation approach

In this paper, a globally optimal state estimation is addressed in light of the conventional Luenberger observer‐type filter. This paper is the first part of a comprehensive extension of an original work by Hsieh, with the main aim being to develop a transformation‐based filtering framework for global unbiased minimum‐variance state estimation (GUMVSE) for systems with unknown inputs that affect both the system and the output. The main contributions of this paper are (i) a complete optimal solution for the GUMVSE is addressed, where both the globally optimal state filter and predictor are presented, and (ii) additional insights for implementing the globally optimal state filter are highlighted via the proposed decorrelation constraint. Compared with existing results, the proposed globally optimal filter has the most general filter form among all transformation‐based globally optimal filters in the sense that it does not use any specific unknown input transformation matrix in the derivation. A simulation example is given to illustrate the usefulness of the proposed results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On methods of treating DC levels in an adaptive digital Smithpredictor

Two popular methods of treating unknown or slowly varying DC levels in the input and output measurements used for parameter estimation in an adaptive digital Smith predictor are studied. They are the high-pass filtering method and the method of estimating an additional constant. The performance of the former is found to be inferior in that it leads to a larger output variance under a stochastic environment. This is caused by the higher variation in model output due to static gain variations in the estimated explicit model     

Estimation of time delay and lag relationship of quadratic systems

This paper is concerned with the identification of a quadratic non-linear system. The identification includes the estimation of the lag (delay) relationship of the input process, the lead lag (time delay) of the input/output system and the unknown parameters. The estimation procedures are based on the second-order covariances, cross-covariances and spectral densities. The estimation methods described here are illustrated with numerical examples.     

Hierarchical self-tracking control of two-machine system

The paper considers the problem of controlling an interconnected power system with unknown model parameters, and presents a method called self-tracking control. In this method it is assumed that the desired performance of each machine under a given initial condition is prespecified. The controllers are of a priori fixed structure and use local output feedback. Both the identification of unknown model parameters and the evaluation of controller parameters are achieved through a common recursive estimation technique. A hierarchical structure for implementing the above for interconnected machines is proposed. Results of simulation studies for a two-machine system are included.     

Multiple sliding mode observers and unknown input estimations for Lipschitz nonlinear systems

Multiple sliding mode observers for state and unknown input estimations of a class of MIMO nonlinear systems are systematically developed in this paper. A new nonlinear transformation is formulated to divide the original system into two interconnected subsystems. The unknown inputs are assumed to be bounded and not necessarily Lipschitz, and do not require any matching condition. Under structural assumptions for the unknown input distribution matrix, the sliding mode terms of the nonlinear observer are designed to track their respective unknown inputs. Also, the unknown inputs can be reconstructed from the multiple sliding mode structurally. The conditions for asymptotic stability of estimation error dynamics are derived. Finally, simulation results are given to demonstrate the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.