Technical communiqueJoint state and parameter estimation based on constrained zonotopes☆
Introduction
Without assuming knowledge of the stochastic properties of unknown variables, set-based state estimation methods are able to provide guaranteed enclosures of the system trajectories in applications affected by bounded uncertainties (Chisci et al., 1996, Scott et al., 2016). Set-based methods have also been widely used in the parameter identification field as an alternative to stochastic methods, since they are able to provide guaranteed enclosures of the model parameters when the uncertain model parameters have unknown stochastic properties. Zonotopes have been used to approximate the parametric set for discrete-time systems with additive uncertainties in Bravo et al. (2006), which was later extended to allow multiplicative uncertainties in Wang et al. (2017). However, both methods are applied only to systems described by regression models, and rely on conservative intersections with strips to refine the parametric set. Intervals have been used in the context of optimal design of experiments in Denis-Vidal et al. (2019), to minimize the conservatism of the parametric enclosure. Moreover, a bisection-based interval algorithm has been used in Rumschinski et al. (2010) to deal with non-convex parameter sets using collections of intervals. Nevertheless, intervals are not able to capture dependencies between variables, which may result in conservative enclosures due to wrapping effect.
In the literature, parameter identification is typically addressed as a separated problem from state estimation, in which a model is identified off-line. Few state estimation strategies in the literature refine online the model parametric uncertainties in order to improve the accuracy of state estimation. Such methodology is referred to as joint state and parameter estimation, which enables the simultaneous estimation of both states and model parameters. It allows for a more efficient update of these variables using available measurement, besides taking into account state-parameter dependencies, rather than dealing with two separated problems. A Kalman filtering (KF) strategy, based on multi-innovation recursive extended least squares algorithm, has been proposed in Cui et al. (2020) to enhance parameter estimation. However, bias issues introduced by KF make such approaches unreliable in case the assumptions on the stochastic properties of the uncertainties are violated. Deterministic approaches include Luenberger-based observers (Zhang et al., 2020) and set-based interval estimation (Raıssi et al., 2004). The latter propose a prediction-update state and parameter estimator suitable for nonlinear continuous-time systems. However, besides not being able to capture the dependencies between states and parameters, the method can lead to high computational complexity due to the use of multiple sets.
The work presented in this note proposes a method for set-based joint state and parameter estimation of discrete-time systems. The strategy extends the algorithms based on constrained zonotopes (CZs) proposed in Rego et al. (2021) and Scott et al. (2016), to include parameter estimation in a unified framework for the first time. In contrast to interval-based methods,1 this framework implemented using CZs allows the estimated enclosures to propagate existing dependencies between states and model parameters. Besides, both the state and parameter enclosures (which are unified in our method) are refined using generalized intersections, unlike in zonotope-based estimation methods. These advantages result in a significant improvement in the accuracy of both state and parameter estimation.
Section snippets
Preliminaries
Consider , , and a real matrix . Let be the Cartesian product, and define the linear mapping, Minkowski sum, and generalized intersection, as respectively. In this note, functions with set-valued arguments will be used to denote the exact image of the set under the function, i.e. . In addition, let be a function of class (i.e., continuously differentiable) and denote its argument. Then, denotes
Linear systems
Consider a linear discrete-time system with unknown-but-bounded disturbances and model parameters, given by where is the system state, is the known input, is the process disturbance, is the measured output, is the measurement disturbance, and are the unknown parameters. In addition, , , , , , , , and . The initial state, model parameters, and
Numerical examples
This section presents numerical results4 for the set-based joint state and parameter estimation method proposed in this note. We compare the results provided by the new framework (denoted by CZ-J for the linear case, and CZMV-J for the nonlinear case) with the CZ methods proposed in Rego et al. (2021) and Scott et al. (2016), denoted by CZ and CZMV, respectively (i.e., with prediction step given by Proposition 1, in which MV stands
Conclusions
This note developed a new method for set-based joint state and parameter estimation of discrete-time systems with unknown-but-bounded model parameters. By extending state estimation methods using CZs to a unified framework, allowing to maintain the dependencies between states and parameters, the accuracy of both state and parameter estimation was significantly improved. Future works will include extending the method developed in Section 3.1 to joint state and parameter estimation of linear
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Work supported by the Brazilian agencies CNPq (INCT Grant No. 465755/2014-3, PQ Grant No. 315695/2020-0), CAPES (Grants no. 001 and 88887.136349/2017-00), FAPEMIG (Grant No. APQ-03090-17), and FAPESP (INCT Grant No. 2014/50851-0), and the Italian Ministry for Research in the framework of the 2017 Program for Research Projects of National Interest (PRIN) (Grant No. 2017YKXYXJ). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André L. Tits.