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Purpose This paper aims to propose a general and rigorous study on the propagation property of invariant errors for the model conversion of state estimation problems with discrete group affine systems. Design/methodology/approach The evolution and operation properties of error propagation model of discrete group affine physical systems are investigated in detail. The general expressions of the propagation properties are proposed together with the rigorous proof and analysis which provide a deeper insight and are beneficial to the control and estimation of discrete group affine systems. Findings The investigation on the state independency and log-linearity of invariant errors for discrete group affine systems are presented in this work, and it is pivotal for the convergence and stability of estimation and control of physical systems in engineering practice. The general expressions of the propagation properties are proposed together with the rigorous proof and analysis. Practical implications An example application to the attitude dynamics of a rigid body together with the attitude estimation problem is used to illustrate the theoretical results. Originality/value The mathematical proof and analysis of the state independency and log-linearity property are the unique and original contributions of this work.

In this paper, first, general dynamic modeling is proposed for multi-input/output nonlinear systems in the form of affine and non-affine systems. Then, an observer barrier function-based adaptive fuzzy scheme is suggested to estimate unknown functions. Briefly, the main contributions of the proposed scheme are: (1) the proposed modeling can be employed in various classes of nonlinear systems, e.g., Single Input–Single Output (SISO), Single Input–Multi Output (SIMO), Multi Input–Single Output (MISO), and Multi Input–Multi Output (MIMO) systems with square or non-square control gain matrix, (2) by combining an observer error signal and the barrier Lyapunov function, the proposed Observer-based Barrier Lyapunov Function (OBLF) method can be employed to solve the problems of output constraint by preventing the output from violating the constraint, and (3) a non-singular robust adaptive fuzzy approach is presented for various classes of nonlinear systems so that the uncertainties are attenuated by a robust bounded H ∞ -like control term. The proposed scheme guarantees the stability of the closed-loop system based on the Strictly Positive Real (SPR) condition and OBLF theory, but it does not need the SPR conditions to be well known. Finally, to show the usefulness of the proposed technique, the simulation examples are employed for various classes of nonlinear affine and non-affine systems with square or non-square control gains.

The main contribution of this work is to propose a design technique for state-feedback control of continuous-time piecewise-affine (PWA) systems that is robust not only to the system uncertainties, but also to variations of the controller gains. More specifically, this study presents sufficient conditions to synthesise a robust non-fragile PWA controller that exponentially stabilises the closed-loop equilibrium point. Furthermore, these conditions are cast as an optimisation problem subject to a set of linear matrix inequalities (LMIs), which can then be solved efficiently. In addition, a set of LMI conditions are derived to ensure that the control input will always meet a pre-assigned upper bound. Simulation results demonstrate the effectiveness of the proposed approach.

In this paper, a robust control approach is applied for both MIMO/SISO affine/non-affine nonlinear systems based on a modeling error-based adaptive fuzzy observer controller, in the presence of input saturation. In the proposed scheme, non-affine nonlinear systems can be transformed to affine systems and unknown higher-order term of expansion (HOTE) that appears due to the use of this method can be estimated by an adaptive fuzzy technique. Using the modeling error between the system states observer and a serial–parallel estimator model, a modeling error-based adaptive fuzzy observer estimator is proposed that uses the modeling error as the input of fuzzy system to approximate and adaptively compensate the unknown HOTE and also the external disturbance. The proposed scheme is able to hold control performance in the presence of input saturation. An analysis of the controlled system is presented to verify the stability of the system under control. The stability of the closed-loop system is provided based on the strictly positive real condition and Lyapunov theory. The proposed approach is effectual and robust. The simulation results demonstrate the usefulness of the proposed method for both MIMO and SISO systems.

In this paper we extend the investigation of quasi-affine systems, which were originally introduced by Ron and Shen [J. Funct. Anal. 148 (1997), 408–447] for integer, expansive dilations, to the class of rational, expansive dilations. We show that an affine system is a frame if, and only if, the corresponding family of quasi-affine systems are frames with uniform frame bounds. We also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Finally, we uncover some fundamental differences between the integer and rational settings by exhibiting an example of a quasi-affine frame such that its affine counterpart is not a frame.

This paper deals with state feedback control synthesis for a boiler-turbine system using a set invariance property and a piecewise affine modelling procedure. The nonlinear model of the considered system is linearized around different equilibrium points. After that, the obtained piecewise affine model is exploited to design feedback gains that guarantee compliance with state and control constraints, as well as asymptotic stability of the closed-loop system. The feedback gains are computed by solving a linear programming problem, which can be achieved by numerous effective solvers. Finally, the validity and efficacy of the proposed approach are demonstrated through a numerical example.

In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions $f$ which are $C^2$ except for discontinuities along $C^2$ curves. More specifically, if $f_N^S$ is the $N$-term reconstruction of $f$ obtained by using the $N$ largest coefficients in the shearlet representation, then the asymptotic approximation error decays as $\\norm{f-f_N^S}_2^2 \\asymp N^{-2} (\\log N)^3, N \\to \\infty,$ which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate $N^{-1}$ associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.

In this paper, the problem of distributed adaptive neural control is addressed for a class of uncertain non-affine nonlinear multi-agent systems with unknown control directions under switching directed topologies. Via mean-value theorem, non-affine follower agents’ dynamics are transformed to the structures so that control design becomes feasible. Then, radial basis function neural networks are used to approximate the unknown nonlinear functions. Due to the utilization of a Nussbaum gain function technique, the singularity problem and requirement to prior knowledge about signs of derivative of control gains are removed. On the base of dynamic surface control design and minimal learning parameter approach, a simplified approach to design distributed controller for uncertain nonlinear multi-agent systems is developed. As a result, the problems of explosion of complexity and dimensionality curse are counteracted, simultaneously. By the theoretical analysis, it is proved that the closed-loop network system is cooperatively semi-globally uniformly ultimately bounded. Meanwhile, convergence of distributed tracking errors to adjustable neighborhood of the origin is also proved. Finally, simulation examples and a comparative example are shown to verify and clarify efficiency of the proposed control approach.

This paper presents a new approach to compensate for the current imbalance of an interleaved DC–DC buck converter (IBC), in which the current sensors are not involved in the operation of the converter when it is connected to an invariable load. The current sensors are only used during the offline identification process that builds the universal fuzzy model of the converter’s steady states. Model building involves an upstream identification phase, followed by further dimensionality reduction of the model and error minimization. The method presented here discusses the mathematical complexity of the analytical modelling of hybrid systems and opposes it with a complexity-reduced identification by learning from data. An offline rendered model of the stable and steady states of the IBC is used as a mapping of the required inverter output current to n-fold asymmetric duty cycles, which are distributed among the IBC phases to allow arbitrarily accurate load sharing. The mapping is carried out in the mathematically normalized space of variables or in the physical sense RMS values, achieving the desired robustness in a noisy environment and stability. The final and canonical feedback control is built from the standard and optimized PI controller, which is compensated by the identified IBC model correction. The only measured feedback of the whole controller is the output voltage. Even when applied to the simulation model (physical MATLAB platform) of a two-phase IBC with the built-in system asymmetry, the presented methodology is also applicable to the n-phase IBC without loss of generality.

Sliding dynamics is a peculiar phenomenon to discontinuous dynamical systems, while homoclinic orbits play a role in studying the global dynamics of dynamical systems. This paper provides a method to ensure the existence of sliding homoclinic orbits of three-dimensional piecewise affine systems. In addition, sliding cycles are obtained by bifurcations of the systems with sliding homoclinic orbits to saddles. Two examples with simulations of sliding homoclinic orbits and sliding cycles are provided to illustrate the effectiveness of the results.