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The design of a state feedback and robust controller for regulating the tridimensional (3D) movement of autonomous underwater mobile crafts (AUMCs). The controller design is based on the applicatio...

This paper presents a novel method for computing the Nash bargaining equilibrium for finite, ergodic and controllable Markov chains games. To solve the bargaining process we first set the disagreem...

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...

This article concerns a proximal-point algorithm with time penalization. The case where the cost of moving from one position to a better one is penalized by the time taken by the agent for the decision-making is studied and the restriction employing the penalty method is incorporated. It is shown that the method converges monotonically with respect to the minimal weighted norm to a unique minimal point under mild assumptions. The gradient method is employed for solving the objective function, and its convergence is proven. The rate of convergence of the method is also estimated by computing the optimal parameters. The effectiveness of the method is illustrated by a numerical optimization example employing continuous-time Markov chains.

This article deals with the Tikhonov regularization method for the constrained Lagrange approach, taking into account polylinear programming problems. A regularized Lagrange function is str...

This paper presents a novel method for computing the multi-objective problem in the case of a metric state space using the Manhattan distance. The problem is restricted to a class of ergodi...

This paper presents a Stackelberg–Nash game for modeling multiple leaders and followers. The model involves two Nash games restricted by a Stackelberg game. We propose a computational approach to find the equilibrium point based on the extraproximal method for ergodic controlled finite Markov chains. The extraproximal method consists of a two-step iterated procedure: the first step is a prediction and the second is a basic adjustment of the previous step. We formulate the game as coupled nonlinear programming problems using the Lagrange principle. The Tikhonov’s regularization method is used to guarantee the convergence to a unique equilibrium point. Validity of the method is demonstrated applying this framework to model an oligopoly competition.

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.

We study the behaviour of stochastic discrete-time models controlled by an output linear feedback during a tracking process. The controlled system is assumed to be nonlinear satisfying the global ‘quasi-Lipschitz’ condition and subjected to stochastic input and output disturbances. Two gain matrices in a feedback and in an observer define an ‘averaged’ ellipsoid in the tracking-error space where all system's trajectories arrive ‘with probability one’. The selection of the ‘best’ gain matrices is realised numerically by application of the robust attractive ellipsoid method with the linear matrix inequality technique application. The suggested approach is illustrated by designing of a robust tracking controller for a benchmark example in the presence of stochastic noises both in the state dynamics and in the output observations.

There are many examples in science and engineering which are reduced to a set of partial differential equations (PDEs) through a process of mathematical modelling. Nevertheless there exist many sources of uncertainties around the aforementioned mathematical representation. Moreover, to find exact solutions of those PDEs is not a trivial task especially if the PDE is described in two or more dimensions. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set to an arbitrary accuracy. In this article, a strategy based on the differential neural network (DNN) for the non-parametric identification of a mathematical model described by a class of two-dimensional (2D) PDEs is proposed. The adaptive laws for weights ensure the ‘practical stability’ of the DNN-trajectories to the parabolic 2D-PDE states. To verify the qualitative behaviour of the suggested methodology, here a non-parametric modelling problem for a distributed parameter plant is analysed.

In this article, we deal with time-varying matrix estimation in continuous-time stochastic models under a coloured noise. The suggested estimation algorithm is based on the least squares method (LSM) with forgetting factor. The forming filter generating this 'coloured' noise from a standard 'white noise' is assumed to be partially known. An analysis of the convergence zone is presented. A numerical example illustrates the effectiveness of the proposed algorithm.

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...

Time varying parameters identification of stochastic systems is addressed via sliding mode parameter observers. Sliding mode observer is governed by control that compensates a so-called Ito's term, which reflects a stochastic nature of a system. The matrix estimation algorithm based on equivalent control is proposed. A numerical example illustrates the effectiveness of the proposed approach.

Identification problem in a nonlinear system with constant coefficients is addressed via traditional and high-order sliding mode parameter observer based on the least-squares continuous-time technique. Measurement noise effects on the parameter estimation algorithm are analyzed. The effectiveness of the proposed method is verified via the parameter estimation in a second-order system with noisy measurement.

The problem of observability for systems with unknown inputs is revised. The sufficient and necessary conditions are used for the design of an observer for linear systems with bounded unknown inputs. To realize the observation of the state, a second-order sliding-mode observer is suggested to be applied. Such an observer provides a robust estimate of the state vector in a finite time, without filtration. The design is based on the concept of the hierarchical output injection maintaining zero value for output tracking error at each level of the hierarchy. The equivalent control is used to identify the unknown inputs. A numerical example illustrates the effectiveness of the suggested technique.

A finite time multi-persons linear-quadratic differential game with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances, it is demonstrated that the standard feedback Nash strategies bear to a 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, containing three different versions is suggested. They are sliding modes, second order sliding modes and window integration. All of them realize the identification of unknown periodic disturbances during an "identification period" when all participants apply the standard feedback Nash strategies with the, so-called, "shifting signal" generated only by a known external exciting signal. After that period the complete standard strategies with "pre-identification" including the recalculated shifting signal are activated. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.

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...

This paper deals with the design and analysis of a modified version of the Bush-Mosteller reinforcement scheme applied by partners in a zero-sum repeated game with random pay-offs. The suggested study is based on the learning automata paradigm and a limiting average reward criterion is tackled to analyse the arising Nash equilibrium. No information concerning the distribution of the pay-off is a priori available. The novelty of the suggested adaptive strategy is related to the incorporation of a 'normalization procedure' into the standard Bush-Mosteller scheme to provide a possibility to operate not only with binary but also with any bounded rewards of a stochastic nature. The analysis of the convergence (adaptation) as well as the convergence rate (rate of adaptation) are presented and the optimal design parameters of this adaptive procedure are derived. The obtained adaptation rate turns out to be of o(n 1/3 ).

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 robust stability problem for a wide class of closed-loop nonlinear systems is considered. A nonlinear feedback is assumed to be a function of current state estimates obtained by a nonlinear high-gain observer. The problem is solved in the presence of essential parametric uncertainties as well as external perturbation noise (mixed uncertainties). A two Riccati-equation approach is applied. We demonstrate that the highgain observer under consideration provides sufficiently good state-space estimates which are bounded 'on average'. The same property is valid for the trajectories of the closed-loop nonlinear system. The observation error bound is derived and turns out to be a linear combination of a priori given uncertainties levels of external input and output perturbations. If no uncertainties are considered in the given model description, this bound is zero and corresponds to the asymptotic globally stability property for the observation error as well as for the state-space trajectories. The simulation...

We suggest an approach to provide time-varying parameter estimates in ARMA (Auto Regression Moving Average) models of a stochastic nature based on the use of the recursive version of the Instrumental Variable Method (IVM) with a Matrix Forgetting Factor (MFF). We demonstrate that there exists the best selection of MFF minimizing the error strip bound. This optimal MFF depends in a complex manner on a group of unknown parameters. An adaptation procedure is suggested to obtain asymptotically this optimal value using only the available measurements. The adaptation procedure is based on one Gaussian smoothing technique. The combination of IVM with adaptive MFF is a tool for estimating the entries of a non-stationary parameter matrix involved in the ARMA model. An asymptotic analysis of the error matrix is presented. Simulation results demonstrate the effectiveness of the suggested approach.

This study suggests a new approach to provide time-varying parameter estimates in ARMA (Auto Regression Moving Average) models of stochastic nature based on the use of the recursive version of Instrumental Variable Method (IVM) with a Matrix Forgetting Factor (MFF). This combination is a tool for estimating the entries of a nonstationary parameter matrix involved in the ARMA model. An asymptotic analysis of the error matrix is presented. Simulation results demonstrate the effectiveness of the suggested approach.

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

The selection of a number of nodes in artificial neural nets containing stochastic noise perturbations in the outputs and inputs of each node is examined. The suggested approach is based on a reinforcement learning technique. To solve this optimization problem we introduce a special performance index in such a way that the best number of nodes corresponds to the minimum point of the suggested criterion. This criterion presents a linear combination of a residual minimization functional and some ‘generalized variance’ of the involved disturbances of random nature. A large value of the noise variance leads to a different optimal number of neurons in a neural networks because of the ‘interference’ effect. The optimal point is obtained by the learning procedure based on the Bush-Mosteller reinforcement scheme. This numerical method is commonly used in Intelligent Control Theory. Simulation modelling results are presented to illustrate the effectiveness of the suggested approach

The behaviour of a stochastic automaton operating in an S-model environment is described. The environment response takes an arbitrary value in the closed segment [0, 1] (continuous response). The learning automaton uses a reinforcement scheme to update its action probabilities on the basis of the reaction of the environment. The complete set of actions is divided into a collection of non-empty subsets. The action set is changing from instant to instant. Each action set is selected according to a given probability distribution. Convergence and convergence rate results are presented. These results have been derived using quasimartingales theory.

This paper concerns the adaptive selection of the optimal order of linear regression models using a variable-structure stochastic learning automaton. The Alaike criterion is derived for stationary and non-stationary cases, and it is shown that the optimal order minimizes a loss function corresponding to the evaluation of this criterion. The order of the regression model belongs to a finite set. Each order value is associated with an action of the automaton. The Bush-Mosteller reinforcement scheme with normalized automaton input is used to adjust the probability distribution. Simulation results illustrate the feasibility and performance of this model order selection approach

Multilevel hierarchical systems of learning stochastic automata are investigated in this paper. The system under consideration consists of several levels of automata with different number of outputs. Each automaton is a variable-structure stochastic automaton. A learning scheme which is based on the Bush-Mosteller reinforcement scheme is used to adjust the probabilities associated with the actions of the automata of the hierarchical learning system. Boolean and continuous automaton inputs have been considered. Convergence and convergence rate analysis are presented. The optimization of multimodal functions using this multilevel system of automata is also described. Simulation results indicate the effectiveness of such multilevel learning systems.

This paper deals with the design and the analysis of a new reinforcement scheme for learning automata and its application for multimodal functions optimization. This reinforcement scheme generalizes the well known Bush-Mosteller scheme with decreasing gain for learning automata with continuous inputs. The theoretical analysis is based on martingale theory. The conditions associated with the convergence of this scheme to the optimal pure strategy are stated, and the order of convergence rate is estimated. The variation domains of the variables of the function to be optimized are discretized into subsets which are associated to the outputs of the learning automaton. The values of the function on these subsets are used to construct the continuous automaton inputs. Simulation results show the feasibility and the good performance of this optimization technique.

Algorithms of the method of least squares that are intended for estimating the parameters of non-linear autoregression processes are considered. A class of nonlinear difference equations is isolated which describes autoregression processes for which MLS estimates of parameters in these equations are found to be strongly consistent.