This correspondence revisits the results reported in a recent Automatica paper (Wang et al., 2016). We show that the claim of exponential stability of the extremum-seeking control system presented in Theorem 1 is not correct. An alternative stability analysis using the Lyapunov–Malkin theorem is considered. A counterexample is also presented.

We consider a novel method to design H-infinity observers for a class of uncertain nonlinear systems subject to unknown inputs. First, the main system dynamics are rewritten as an augmented system with state vector including both the state vector of the main system and the unknown inputs. Then, we design a H-infinity reduced-order observer to estimate both state variables and unknown inputs simultaneously. Based on a Lyapunov functional, we derive a sufficient condition for existence of the designed observer which requires solving a nonlinear matrix inequality. To facilitate the observer design, the achieved condition is formulated in terms of a set of linear matrix inequalities (LMI). By extending the proposed method to a multiobjective optimization problem, the maximum bound of the uncertainty and the minimum value of the disturbance attenuation level are found. Finally, the proposed observer is illustrated with an example.

Real plants are, in general, time-varying and uncertain. Yet most industrial control loops are designed using linear time-invariant (LTI) plant models. The present work formalizes the problem of best approximate LTI modelling of BIBO-stable linear time-varying (LTV) systems in a way that is compatible with the notion of induced l∞ norm used in robust l1 control. This setup is closely associated with certain identification methods and controlmotivated model validation/invalidation procedures that can be efficiently implemented with special-purpose linear programming (LP) techniques. Results are given for approximate modelling of BIBO-stable LTV and LTI systems using LTI models, especially BIBO-stable fixed-pole state-space model parametrizations. It is shown that such parametrizations are satisfactory from an approximate modelling point of view, and can be used in output-error-type LP identification techniques and in LP model validation procedures. Various useful constraints on model parameters etc. can be included, as long as they are linear in the unknown parameters, and both insensitive (statistically robust) and sensitive criteria can be used in identification and model validation. Simulation examples are included to illustrate that such techniques indeed give good results and can be used to solve problems of realistic size.

The problem of developing robust thresholds for fault detection is addressed. An inequality for the solution of a linear system with uncertain parameters is provided and is shown to be a valuable tool for developing dynamic threshold generators for fault detection. Such threshold generators are desirable for achieving robustness against model uncertainty in combination with sensitivity to small faults. The usefulness of the inequality is illustrated by developing an algorithm for detection of sensor faults in a turbofan engine. The proposed algorithm consists of a state observer with integral action. A dynamic threshold generator is derived under the assumption of parametric uncertainty in the process model. Successful simulations with measurement data show that the algorithm is capable of detecting faults without generating false alarms.

A nonlinear observer, with the feedback gain weighted by the sensitivity of the output with respect to the state, is developed for systems with nonlinear output map. The observer can be obtained from the extended Kalman filter by a special choice of time-varying weighting matrices. It is shown that the estimation error dynamics are asymptotically stable and a region of attraction is derived. The observer is applied to the top blown converter process for estimating the content of impurities in the liquid metal. Using plant data from the converter at SSAB Oxelösund AB, the observer is shown to provide accurate estimates of the carbon content.

Two finite-spectrum-assignment controllers for continuous linear dynamic systems with multiple time delays in input signal are introduced. The controllers are shown to comprise least-squares state estimators. Disturbance attenuation properties stipulated by the choice of shift operator in the observers are studied.

The well-known parity space method for fault detection and isolation is generalized to a continuous-time case. As in the discrete-time case, it is shown that the parity space equations can be implemented using a number of time delays, and are equivalent to a continuous deadbeat observer. Algorithms for isolation of sensor and actuator faults are derived and illustrated by a numerical example. (

This paper introduces the reader to several recent developments in worst-case identification motivated by various issues of modelling of systems from data for the purpose of robust control design. Many aspects of identification in H∞ and ℓ1 are covered including algorithms, convergence and divergence results, worst-case estimation of uncertainty models, model validation and control relevancy issues.

We consider worst-case analysis of system identification by means of the linear algorithms such as least-squares. We provide estimates for worst-case and average errors, showing that worst-case robust convergence cannot occur in the l1 identification problem. The case of periodic inputs is also analysed. Finally a pseudorandomness assumption is introduced that allows more powerful convergence results in a deterministic framework.

Optimal damping of harmonic disturbances of known frequencies is studied for sampled-data systems. A sampled-data output feedback controller is designed to minimize the intersample variations of the controlled variable. The set of all stabilizing optimal controllers is obtained in terms of the Youla parameterization and a set of interpolation conditions at the disturbance frequencies, which ensure that the stationary cost is minimized.

Approximation of stable linear dynamical systems by means of so-called Laguerre and Kautz functions, which are the Laplace transforms of a class of orthonormal exponentials, is studied. Since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. Assuming that the transfer function of a system is bounded and analytic outside a given disc, it is shown that Laguerre basis functions are optimal in a mini-max sense. This result is extended to the "two-parameter" Kautz functions which can have complex poles, while the poles of Laguerre functions are restricted to the real axis. By conformal mapping techniques the "two-parameter" Kautz approximation problem is recast as two Laguerre approximation problems. Thus, the well-developed theory of Laguerre functions can be applied to analyze Kautz approximations. Unilateral shifts are used to further develop the connection between Laguerre functions and Kautz functions. Results on H2 and H∞ approximation using Kautz models are given. Furthermore, the weighted L2 Kautz approximation problem is shown to be equivalent to solving a block Toeplitz matrix equation