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This paper proposes an online constrained extremum-seeking approach for an unknown convex function with unknown constraints within a class of uncertain dynamical systems with an available output di...

In this work, the sliding mode control based on the super-twist observer is presented. The parameters of the controller as well as the observer are admitted to be time-varying and depending on available current measurements. In view of that, the considered controller is referred to as an adaptive one. It is shown that the deviations of the generated state estimates from real state values together with a distance of the closed-loop system trajectories to a desired sliding surface reach a μ-zone around the origin in finite time. The application of the suggested controller is illustrated for the orientation of a tethered satellite system in a required position.

Two-dimensional nonlinear systems with parametrical interval uncertainty are studied. Differential geometric extremal deviations method is developed. Its basic elements are integral funnels (IFs) and their boundaries. Extreme matrix-valued functions determining the branches of the boundaries of IFs are synthesised. Typical phase portraits of considered uncertain systems with different oscillatory properties are presented. Analytical criteria of robust stability for different oscillatory classes of uncertain systems are formulated.

In this paper we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantised. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimisation problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented in order to illustrate the applicability of the proposed design method.

This article deals with a class of continuous nonlinear control systems in the presence of sampled outputs. The dynamical models under consideration are described by ordinary differential equations with additive bounded uncertainties. The linear feedback control design proposed in this article is based on an extended version of the classical invariant ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves the celebrated ‘descriptor techniques’ from the extended Lyapunov methodology. Finally, the implementability of the proposed control design scheme is illustrated by a computational example. A brief discussion on the principal numerical issues is also included.

In this study the tracking problem for a class of non-linear uncertain systems is analyzed. The considered class of non-linear systems are restricted by those verifying the global Lipschitz condition on non-linearities making them linear-like. A new sliding mode neurocontroller is suggested to solve this problem. The controller desing includes the on-line state estimates construction and the corresponding tracking control based on sliding mode approach and the reconstructed dynamics generated by a special non-linear observer. A special sliding mode technique during the “off-line training” to estimate the right-hand side of the given dynamics in finite-time was applied. This procedure allows use of these estimates for the best (in LQ-sense) nominal weights selection in the neuro observer designed. A switching (sign) type term is incorporated in the observer structure to correct the current state estimates using just the on-line measurable output. This observer is supplied with a new learning procedure with...

The problem of the realization of integral sliding mode controllers based only on output information is discussed. The implementation of an output integral sliding mode controller ensures insensitivity of the state trajectory with respect to the matched uncertainties from the initial time moment. In the case when the number of inputs is more than or equal to the number of outputs, the closed loop system, describing the output integral sliding mode dynamics, is shown to lose observability. For the case when the number of inputs is less than the number of outputs, a hierarchical sliding mode observer is proposed. The realization of the proposed observer requires a filtration to obtain the equivalent output injections. Assigning the first order low-pass filter parameter small enough (during this filter realization), the convergence time and the observation error can be made arbitrarily small. The results obtained are illustrated by simulations.

A finite time multi-persons linear-quadratic differential game (LQDG) with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances and uncertainties, the standard feedback Nash strategies are shown to yield to an e-(or quasi) Nash equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, is suggested. It is defined an “adaptation period” where all participants apply the standard feedback Nash strategies with the, so-called, “shifting signal” generated only by a known external exciting signal. Then, during the adaptation, the readjustment (or correction) of the control strategies is realized to estimate the effect of unknown periodic disturbances by the corresponding correction of the shifting vector. After that adaptation period, the complete standard strategies including the recalculated shifting signal are activated allo...

This paper develops a version of the robust maximum principle applied to the minimax Mayer problem formulated for stochastic differential equations with a control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each possible value of the parameter. The cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term. A terminal condition, given by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of multi-model systems given by a stochastic differential equation with parameters from a given finite set are constructed. This problem belongs to the class of minimax stochastic optimization problems. The p...

The robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control-dependent diffusion term. The parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each value of the uncertain parameter from a given finite set. The cost function is considered on a finite horizon (contains the mathematical expectation of both an integral and a terminal term) and on an infinite one (a time-averaged losses function). These problems belong to the class of minimax stochastic optimization problems. It is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of Ric...

This paper presents the version of the robust maximum principle in the context of multi-model control formulated as the minimax Bolza problem. The cost function contains a terminal term as well as an integral one. A fixed horizon and terminal set are considered. The necessary conditions of the optimality are derived for the class of uncertain systems given by an ordinary differential equation with parameters from a given finite set. This problem consists in the control design providing a good behaviour for a given class of multi-model system. It is shown that the design of the minimax optimal controller is reduced to a finite-dimensional optimization problem given at the corresponding simplex set containing the weight parameters to be found. The robust optimal control may be interpreted as a mixture (with the optimal weights) of the controls which are optimal for each fixed parameter value. The proof is based on the recent results obtained for minimax Mayer problem (Boltyanski and Poznyak 1999a). The mini...

In this paper the problem of simultaneous robust state and parameter estimation for a class of MIMO non-linear systems under mixed uncertainties (unmodelled dynamics as well as observation noises) it tackled. A switching gain robust 'observer-identifier' is introduced to obtain the corresponding estimates. This is achieved by applying an observer to the so-called nominal extended system, obtained from the original system without any uncertainties and considering the parameters as additional constant states. As it is shown, in general the extended system can lose the global observability property, supposed by valid for the original non-extended system, and a special procedure is needed to provide a good estimation process in this situation. The suggested adaptive observer has the Luenberger type observer structure with switching matrix gain that guarantees a good enough upper bound for the identification error performance index. The Van der Monde generalized transformation is introduced to derive this boun...

To identify on-line a quite general class of non-linear systems, this paper proposes a new stable learning law of the multilayer dynamic neural networks. A Lyapunov-like analysis is used to derive this stable learning procedure for the hidden layer as well as for the output layer. An algebraic Riccati equation is considered to construct a bound for the identification error. The suggested learning algorithm is similar to the well-known backpropagation rule of the multilayer perceptrons but with an additional term which assure the stability property of the identification error.

The problem of robust boundary control for a class of infinite dimensional systems under mixed uncertainties is addressed. A strong solution of the Dirichlet boundary problem corresponding to the perturbed evolution operator is introduced. The Lyapunov function approach is used for proving that there is a controller which stabilizes this class of systems under the presence of smooth enough internal and external perturbations and guarantees some tolerance level for the joint cost functional. The derived control consists of two parts: a compensating one and a linear feedback controller with a gain operator which is a positive inverse solution of a corresponding operator Riccati equation. A heating boundary control process is given as an illustration of the suggested approach