Although bounding methods for estimation and control date back to many decades, research has intensified since the early nineties. The methods developed satisfy practical needs of engineering for various reasons:

  • (1)
    It is often incorrect to assume statistics on measurement errors as that can result in estimated statistical distributions of single point estimates that are misleading.
  • (2)
    Input–output signal uncertainty is mixed with model parameter uncertainties.
  • (3)
    A model with certain given accuracy bounds is often to be fitted to measurement data.
  • (4)
    Sensitivity analysis and estimability of model parameters are needed to validate engineering models.

The introductory paper of this special issue by Kieffer and Walter presents an assessment of various known and new methods for guaranteed estimation of nonlinear system models. Techniques of interval analysis, guaranteed global optimization, set inversion, contractors, Mequation imageller's theorem for continuous time inclusions and a novel use of sensitivity functions are considered. The paper concludes with an interesting finding that guaranteed parameter optimization turns out to be more difficult than guaranteed parameter bonding. The paper by Cerone, Piga and Regruto introduces a new procedure to evaluate parameter bounds of linear dynamic systems for set-membership errors-in-variables problems. Linear-matrix-inequalities relaxation techniques are used to find approximate solutions to the polynomial problems involved in an outer bound approximation of the feasible parameter set. This is found to be less conservative than prior results by bounding techniques.

Single point estimates of state with associated estimates of a statistical distribution can be inappropriate when modelling errors, input uncertainties and measurement noise have unknown statistics, apart from known bounds on their size. The paper by Meselem and Ramdani introduces a generic method for designing interval observers for a broad class of uncertain nonlinear systems, based on nonlinear hybridization and practical stability analysis. The technique is based on inner/outer approximations of the reachable set of states and also on a suitable choice of the observation gain matrix to guarantee the convergence of the observation error under the Lipschitz continuity of the nonlinear part. The paper on Hamiltonian techniques for set membership state estimation, by Kurzhanski, introduces solutions to guaranteed state filtering by using HJB equations and value functions. Value functions are information states for this problem and allow the computation of set-valued guaranteed estimates of the state vector. Upper and lower estimates are obtained for the value function that enable the calculation of external and internal bounds for the state vector. For linear systems, these set estimates belong to specific families of ellipsoids for which explicit equations are derived.

Lhomeau, Jaulin and Hardouin's paper on capture basin approximation introduces new interval-based methods to characterize inner and outer approximations of the capture basin that is a set of states for a nonlinear system from where a target can be reached by allowed inputs. The paper by Ceccarelli, Di Marco, Garulli, Giannitrapani and Vicino uses a set theoretic approach to the path planning problem to measure the localization uncertainty of a robot relative to landmarks. The aim is to find a path minimizing the overall position uncertainty along the path. Note that these two papers are directly applicable in the fields of robotics and autonomous systems; in fact almost all papers of this special issue are applicable to formal verification of hybrid systems.

Finally, estimability is any local property of a parameter that makes a statement on the accuracy of parameter estimation based on measured experimental data. Reynet and Jaulin's paper presents a four-part procedure on a new interval-based method to characterize estimability. It first involves computations of a joint error set, then output set computation, output set inversion and finally uncertainty evaluation. Passive location estimation in a wireless network is used to illustrate the method.

Overall this special issue provides a representative and systematic cross-section of important recent developments in bounding methods for parameter and state estimation.