State estimation plays a fundamental role in feedback control, system monitoring and system optimization because noisy measurements is the only information available from the system. Several methods have been developed for accomplishing such task (see jazwinski2007stochastic; crassidis2004optimal; among others). All these methods have been developed upon the assumption on the knowledge of noises and model of the system, as well as, the absence of constraints.
In practice, these assumptions are not easily satisfied and research efforts were focused on approaches that do not relay on such requirements (see li1997linear, sayed2001framework, blanchini2008set, among others). For example, an filter is designed minimizing the norm of the mapping between disturbances and estimation error. In li1997linear, el1997robust and hu2009improved an approach that solves a least-square estimation problem is introduced. Both methods are based on the adequate selection of the uncertainty model instead of relying on statistical assumptions on noises. In these approaches, uncertainty models are formulated based on the available information of the system. In the same way, robust estimation algorithms based on as min-max robust filtering, set-valued estimation and guaranteed cost paradigm, have attracted the attention of the research community (see sayed2001framework, zhu2002design).
Building on the success on moving horizon control, moving horizon estimation (MHE) has attracted attention of researchers since the pioneering work of jazwinski1968limited (see also schweppe1973uncertain, rao2001constrained and rao2003constrained). The interest in such estimation methods stems from the possibility of dealing with limited amount of data, instead of using all the information available from the beginning, and the ability to incorporate constraints. In recent years, both theoretical properties of various MHE schemes as well as efficient computational methods for real-time implementation have been studied (see alessandri2005robust, alessandri2008moving, alessandri2012min, garcia2016new, sartipizadeh2016computationally, sanchez2017adaptive). In particular, it is of interest to establish robust stability and estimate convergence properties. In recent years several results have been obtained for different algorithms, advancing from idealistic assumptions (observability and no disturbances) to realistic situations (detectability and bounded disturbances).
For nonlinear observable systems, rao2003constrained established the asymptotic stability of the estimation error for the standard cost function. Furthermore, if the disturbances are asymptotically vanishing the estimation error is robust asymptotically stable and it asymptotically converges to zero (rawlings2009model-rawlings2012optimization). alessandri2008moving and alessandri2010advances proposed an estimation scheme, based on least-square cost function of the estimation residuals, that guaranteed the boundedness of estimation error for observable systems subject to bounded additive disturbances. Finally, for the general case of nonlinear detectable systems subject to bounded disturbances, ji2016robust and muller2017nonlinear showed the robust global asymptotic stability (RGAS) and convergence of estimation error in case of bounded or vanishing disturbances, respectively. In these works, the least-square objective function was modified by adding a max-term. ji2016robust established RGAS for the full information estimator while muller2017nonlinear established RGAS and convergence for the moving horizon estimator. Furthermore, for a particular choice of the weights of the objective function, muller2017nonlinear established these results for the least-squares type objective function.
This paper introduces the RGAS and convergence analysis for the moving horizon estimator based on adaptive arrival cost proposed in sanchez2017adaptive in the practical case of nonlinear detectable systems subject to bounded disturbances. To establish robust stability properties for MHE it is crucial that the prior weighting in the cost function is chosen properly. In various schemes the necessary assumptions in the prior weighting are difficult to verify (rao2003constrained, rawlings2009model), while in others can be verified a prior muller2017nonlinear. In the MHE scheme analysed in this work, the assumption on the prior weighting can be verified a prior by design. Furthermore, the disturbances gains become uniform (i.e., they are valid independent of ), allowing to extend the stability analysis to full information estimators with least-square type cost functions.
The rest of the paper is organized as follows: Section 2 introduces the notation, definitions and properties that will be used through the paper. Section 3 presents the main result and shows its connections with previous stability analysis. Section 4 discusses simple examples, previously used in the literature, with the purpose of illustrating the concepts and also in order to show the difference with others MHE algorithms. Finally, Section 5 presents conclusions.
2 Preliminaries and setup
Let denotes the set of integers in the interval denotes the set of integers greater or equal to . Boldface symbols denote sequences of finite or infinite length, i.e., , respectively. We denote as the finite sequence given at time . By
we denote the Euclidean norm of a vector. Let denote the supreme norm of the sequence . A function is of class if is continuous, strictly increasing and . If is also unbounded, it is of class . A function is of class if is non increasing and . A function is of class if is of class for each fixed , and of class for each fixed .
The following inequalities hold for all
The preceding inequalities hold since is included in the sequence and functions are non-negative strictly increasing functions.
Bounded sequences: A sequence is bounded if is finite. The set of bounded sequences is denoted as for some
Convergent sequences: A bounded infinite sequence is convergent if as . Let denote the set of convergent sequences :
Analogously, is defined for the sequence .
2.2 Problem statement
Let us consider the state estimation problem for nonlinear discrete time systems of the form
where are the state, process noise, measurement and estimation residuals vectors, respectively. The process disturbance and estimation residuals are unknown but assumed to be bounded, i.e, for some . are compact and convex sets with the null vector belongs to them. In the following we assume that is continuous, locally Lipschitz on and is continuous. The solution to the system (2) at time is denoted by , with initial condition and process disturbance sequence . Furthermore, the initial condition is unknown, but a prior knowledge is assumed to be available and its error is assumed to be bounded, i.e., , .
The solution of the estimation problem aims to find at time an estimate of the current state minimizing a performance metric using by the MHE. At each sampling time , given the previous measurements , the following optimization problem is solved
where is the optimal estimated and is the optimal process noise estimate at sample based on measurements available at time . The process noise and are the optimization variables. The stage cost penalizes the estimated process noise sequence and the estimation residuals , while penalizes the prior estimated . The adequate choice of and , and their parameters, allows to ensure the robust stability of the estimator muller2017nonlinear. While the estimation window is not full, , problem (3) can be reformulated and solved as a full information problem
as increases this problem becomes (3) for all .
In previous works, the robust stability of MHE has been achieved by modifying the standard least-square cost function through the inclusion of a –term (ji2016robust; muller2017nonlinear) or by a suitable choice of the cost’s function parameters (muller2017nonlinear). Another mechanism to solve this problem is combining a suitable choice of the stage cost with a time–varying prior weight of the form
whose parameters are recursively updated using the information available at time (sanchez2017adaptive, 2017). The prior weighting is defined in this way to avoid the introduction of artificial cycling in the estimation process (see rawlings2009model). In this approach, the prior weight matrix is given by
where and , where
denotes the process noise variance. The prior knowledge of the windowis updated using a smoothed estimate (findeisen1997moving)
The optimization problem (3) can be reformulated in terms of the initial condition and the estimated process noises and the residuals along the entire trajectory as follows
This formulation of problem (3) allows to explicitly see the effect of past data on the current state estimate . In this formulation it is easy to see the exponential averaging of these data. Allowing change in time, the past data has different affects on the current estimates depending on .
Before proceeding to the development of the main results, we state the main properties and assumptions about the prior weighting .
The updating mechanism (5) is a time-varying filter whose inputs are and the initial condition . It generates recursively a real-time estimation of by updating with an exponential time-averaging of . The updating mechanism (5) only use data and it does not rely on a model of the system. The sequence is positive definite, it is decreasing in norm and it is bounded. The proof of these properties follows similar steps as in sanchez2017adaptive.
The prior weighting is a continuous function lower bounded by and upper bounded by such that:
for all and
where and .
The system (2) is incrementally input/output-to-state stable if there exist functions and such that for every two initial states , , and any two disturbances sequences the following holds for all :
This definition combines the concepts of output-to-state-stability (OSS) and input-to-state-stability (ISS). As stated in sontag1997output, the notion of IOSS represents a natural combination of the ideas of strong observability and ISS, and it was called detectability in sontag1989some and strong unboundedness observability in jiang1994small. In addition, the existence of an observer for the system (2), which is incrementally input-output-to-state stable (i-IOSS) instead of IOSS (see Remark 24 in sontag1997output), is assumed. Note that , since These assumptions will help us to bound the functions involved in the definition of i-IOSS and to relate them with the terms of the MHE cost function (stage cost and prior weight).
In the following sections the updating mechanism (5) and the assumption of i-IOSS sontag2008input will be used to prove robust stability of the proposed MHE in the presence of bounded disturbances and convergence to the true state in the case of convergent disturbances. Some assumptions about functions related to system (2) and Definition 1 will be helpful in the sequel.
The function and satisfies the following inequality
for some , and and .
The stage cost is a continuous function bounded by such that the following inequalities are satisfied
Functions and from Definition 1 are related with the bounds of stage cost and through the following inequalities
In this work, we claim that the proposed estimator holds the property of being robust global asymptotic stable, which is defined as follows.
Consider the system described (2) subject to disturbances and for , with prior estimate for . The moving horizon state estimator given by equation (3) with adaptive prior weight is robustly globally asymptotically stable (RGAS) if there exists functions and , such that for all , all , the following is satisfied for all
3 Robust stability of moving horizon estimation under bounded disturbances
We are ready to derive the main result: RGAS of the proposed moving horizon estimator with a large enough estimation horizon for nonlinear detectable systems under bounded disturbances. Furthermore, a function exist such that (14) is valid with this and for all estimation horizon .
Consider an i-IOSS system (2) with disturbances , . Assume that the arrival cost weight matrix of the MHE problem is updated using the adaptive algorithm (5). Moreover, Assumptions 1, 2 and 3 are fulfilled and initial condition is unknown, but a prior estimate is available. Then, the MHE estimator (3) is .
Proof. The optimal cost of problem (3) is given by
Due optimality, the following inequalities hold
then, taking into account the lower and upper bounds we have
Analogously, bounds for and can be found
Next, let us consider some sample . Assuming that system (2) is i-IOSS with and for all . Since we obtain
In order to get a finite upper bound for the estimation error, the three terms in the right hand side of equation (17) must be upper bounded. The first term can be written
Taking in account that is a symmetric positive definite matrix for all , then , where
denotes the maximal eigenvalue of matrix. Denoting as the minimal eigenvalue of matrix and taking in account that , the maximum conditioning number of matrix can be defined as , then can be bounded by
The first term in the right side of this equation is bounded due the assumption that , while the second term are finite constants. To extend the validness of (18) to the full estimation horizon, an extension of the function at the beginning of the estimation, , is required.
The second term in the right hand side of equation (17), can be bounded by the following inequality
Recalling Assumption 3, the reader can verify the following inequality
In an equivalent manner, a bound for the third term in the right hand side of equation (17) can be found
Defining the functions and for all and as follows
equation (21) can be written as follows
To guarantee the validity of previous results on the entire time horizon we must extend the definition of . Because of , and for , it is sufficient to define for some to extend the definition of . We would like to determinate the decreasing rate for the function samplings time in the future. In order to do that, let define the constants
The minimum horizon length required to accomplish a decreasing rate will be given by
Adopting an estimator with a window length greater or equal to such that
the effects of the initial conditions will vanish with a decreasing rate . As , the estimation will entry to the bounded set defined by the noises of the system
This set define the minimum size region of error space that the error can achieve by removing the effect of errors in initial conditions (). Equation (27) establish a trade off between speed of convergence and window length, which is related with the size of .
For any MHE with adaptive arrival cost and window length two situations can be considered
The estimator removed the effects of on such that , and
The estimator has not removed the effects of on such that ,
This equation implies the fact that the estimation error .