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This paper presents the current developments of a novel approach to robust Adaptive Feedforward Control (AFC) of uncertain linear systems affected by harmonic disturbance of known frequency. The features that set the proposed method apart from existing ones are the following: (i) knowledge of the sign of both the real and imaginary parts of the transfer function at the frequency of excitation is not needed; (ii) persistence of excitation is not required; (iii) stability analysis tools based on averaging are avoided, hence the requirement of an exponentially stable equilibrium is circumvented. The methodology reposes upon recent results on adaptive regulation of uncertain linear systems with weak immersions as well as on classic tools in adaptive control. A noticeable drawback of the proposed controller - at this stage of the research - is its relatively high dimensionality, which stems from the necessity of a convexification of a non-convex parameter set.

In this paper, a novel non-asymptotic method for target localization based on the algebra of Volterra linear integral operators is presented aiming at estimating the coordinate of a stationary source by a single mobile agent. The algorithm assumes that the agent is only allowed to obtain the measurement of distance from the source. By properly designing the kernel of the Volterra operators, the influence of initial conditions on the transient phase can be eliminated in order to achieve – ideally – a deadbeat mode of behavior. The stability analysis shows that the algorithm is robust to bounded additive measurement perturbations. Moreover, the bias on the estimate due to time-discretization is characterized. Simulation results show that the proposed algorithm is characterized by fast convergence and good noise immunity.

This paper deals with a novel adaptive observer-based technique for estimating the amplitude, frequency and phase of a single sinusoidal signal from a measurement affected by structured and unstructured disturbances. The structured disturbances are modelled as a time-polynomial so as to represent bias and drift phenomena typically present in applications, whereas the unstructured disturbances are modelled as bounded noise signals. The proposed estimation technique exploits a specific adaptive observer scheme equipped with a switching criterion allowing to properly address in a stable way poor excitation scenarios. The estimator's stability properties are analyzed by Input-to-State Stability arguments. The practical characteristics of the proposed estimation approach are evaluated and compared with other existing tools by extensive simulation trials. Real experimental results are provided as well.

A novel finite-time convergent estimation technique is prop osed for identifying the amplitude, frequency and phase of a biased sinusoidal signal. Resorting to Volterra integral operato rs with suitably designed kernels, the measured signal is pr ocessed yielding a set of auxiliary signals in which the influence of t he unknown initial conditions is removed. A second-order sl iding mode-based adaptation law – fed by the aforementioned auxil iary signals – is designed for finite-time estimation of the f requency, amplitude, and phase. The worst case behavior of the propose d algorithm in presence of the bounded additive disturbance s is fully characterized by Input-to-State Stability argume nts. The effectiveness of the estimation technique is evalua ted and compared with other existing tools via extensive numerical simulations.

This article addresses the observer design problem for simultaneous state and parameter estimation for a class of one-dimensional linear parabolic PDEs. The design is based on the backstepping PDE methodology, Including a modified integral transformation to compensate for the parameter uncertainty. The solution of the resulting Kernel-PDE is recast as a convex optimization problem via a Sum-of-Squares formulation which is solved by polynomial optimization techniques (semidefinite programming). This allows computing – in a fast and direct way - the state and parameter observer gains at every sampling time. In addition to an observer based on the Volterra transformation, an observer design method with two boundary measures via a Fredholm-type transformation is presented. The uniqueness and invertibility of this transformation are proved for polynomial kernels in the space of continuous functions. The effectiveness of this approach is illustrated by numerical simulations.

This article deals with the observer design problem for the simultaneous estimation of the solid Lithium concentration and of the diffusion parameter for a Single Particle Model of Lithium-Ion Batteries. The design is based on the Backstepping PDE methodology, including a modified Volterra transformation to compensate for the diffusivity uncertainty. The resulting coupled/uncoupled Kernel-PDE and Ordinary Differential Equation (ODE) are recast, via a Sum-of-Squares decomposition, in terms of a convex optimization problem and solved by semidefinite programming, allowing, at each fixed time, an efficient computation of the state and parameter observer gains. In addition, based on the Moment approach, a novel scheme of inversion of the nonlinear output mapping of the Single Particle Model is presented. The effectiveness of this approach is illustrated by numerical simulations.

The problem of estimating the n unknown amplitudes, frequencies and phases of the components of a multi-sinusoidal signal is addressed in this paper. The proposed methodology theoretically allows the exact identification of the above unknown parameters within an arbitrarily small finite time in the noise-free scenario. The measured signal is processed by a bank of Volterra integral operators with a suitably designed kernel, that yields a set of auxiliary signals which are computable on-line by causal linear filters. These auxiliary signals are in turn used to estimate the frequencies in an adaptive fashion, while the amplitudes and the phases estimates can be calculated by means of algebraic formulas. The effectiveness of the estimation technique is evaluated and compared with other existing finite-time estimators via numerical simulations.

This paper deals with the problem of estimating the frequencies of the n sinusoidal components of a multi-sinusoidal signal. A distinctive feature of the proposed method is that the frequencies are directly adapted, thus not requiring further steps of eigenvalue extraction or polynomial root-finding to retrieve the frequencies from the characteristic polynomial, as typically done in the literature. The frequency estimation problem is approached by formulating a new state-space realization of the signal generator (oscillatory internal model) which is characterized by a minimal parameterization in the sense that only n parameters are used to assign the spectrum of the generator, i.e., the n frequencies of the components. In contrast with other existing adaptive observer-based methods that provide direct estimates of the frequencies, the proposed technique does not require state augmentation, making the overall dynamic order (internal model's state+parameters) equal to 3n.

This paper deals with a new practical approach to continuous-time parametric identification for dynamic models in Hammerstein form and its application in the context of estimating the dynamics of joint stiffness. The proposed methodology deals the static non-linearities as a linear combination of nonlinear basis functions, while the linear dynamic parts of the system are modeled by a parametric rational transfer function. The proposed identification method is enabled by the algebra of Volterra linear integral operators by a suitable design of kernels admitting finite-dimensional and internally stable state-space realizations. Simulation results show the effectiveness of the proposed continuous-time identification technique.

The paper deals with the design of a state observer for linear time-invariant systems, which converges in finite-time without resorting to high-gain injection. The devised observer is based on modulation integrals as the main enabling tool, and can be implemented as a jump-linear system. The dynamics of the state estimation error is proven to be input-to-state stable with respect to the additive measurement noise on the system's output. Numerical examples are included to show the effectiveness of the proposed method and to provide comparisons with some existing techniques.

This paper deals with backstepping design for boundary PDE control/observer as a convex optimization problem. Both Volterra and Fredholm operators are analysed for a class of parabolic and hyperbolic PDEs. The resulting Kernel-PDEs are formulated in terms of polynomial functions, the parameters of which are optimized using Sum-of-Squares (SOS) techniques and solved via semidefinite programming. Uniqueness and invertibility of the Fredholm-type transformation are proven for polynomial Kernels in the space of real-analytic functions. The inverse kernels are approximated as the optimal solution of a SOS and moment problem. The effectiveness of this approach is illustrated by numerical simulations.

This paper introduces a novel deadbeat frequency estimator for possibly biased noisy sinusoidal signals. The proposed estimation scheme is based on processing the measurements by Volterra integral operators with suitably designed kernels, that allow to obtain auxiliary signals not affected by the unknown initial conditions. These auxiliary signals are exploited to adapt the frequency estimate with a variable structure adaptation law that yields finite-time convergence of the estimation error. The worst case behavior of the proposed algorithm in the presence of bounded additive disturbances is characterized by Input-to-State Stability arguments. Numerical simulations are given to show the effectiveness of the proposed method and to compare it with some other techniques available in the recent literature.

This paper deals with an adaptive observer methodology for estimating the frequency of an unknown periodic signal with non-zero fundamental component corrupted by unstructured and bounded uncertainties. The proposed estimator is characterized by Input-to-State Stability (ISS) with respect to the bounds on the higher order harmonics and on the unstructured disturbance. The influence of each tuning parameter is analyzed through a specified ISS asymptotic bound. The effectiveness of the proposed technique is shown through numerical simulations.

In this paper, a parallel pre-filtering scheme is presented to address the problem of estimating the parameters of a sinusoidal signal from biased and noisy measurements. Extending some recent result on pre-filtering-based frequency estimators, a parallel pre-filtering scheme is proposed to deal with the unknown offset and bounded measurement perturbations, which are typically present in several practical applications. A simple frequency estimator, having parallel second-order pre-filters, is introduced. The behaviour of the proposed algorithm with respect to bounded additive disturbances is characterized by Input-to-State Stability arguments. Numerical examples shows the effectiveness of the proposed technique.

This paper presents a MIMO nonlinear adaptive observer, which is characterized by a robust excitation-based switching strategy. The proposed switching algorithm allows to address the scenario of poor excitation, while a conservative minimum duration of excitation interval for ensuring a progressive improvement is determined. The robustness of the devised method with respect to the bounded unstructured perturbation is studied by a input-to-state stability analysis. Simple simulation results show the effectiveness of the proposed technique.

The objective of this work consists in the offline approximation of possibly discontinuous model predictive control laws for nonlinear discrete-time systems, while enforcing hard constraints on state and input variables. Obtaining an offline approximation of the receding horizon control law may lead to a very significant reduction of the online computational burden with respect to algorithms based on iterated optimization, thus allowing the application to fast dynamics plants. The proposed approximation scheme allows to cope with discontinuous control laws, such as those arising from constrained nonlinear finite horizon optimal control problems. A detailed stability analysis of the closed-loop system driven by the approximated state-feedback controller shows that the devised technique guarantees the input-to-state practical stability with respect to the (non-fading) approximation-induced errors. Two examples are provided to show the effectiveness of the method when the approximator is chosen either as a discontinuous nearest point function or as a smooth neural network.

This work introduces a framework devoted to the design of parametric estimators with very fast convergence properties for continuous-time dynamic systems characterized by bounded relative degree and possibly affected by structured perturbations. More specifically, the design of suitable kernels of non-anticipative linear integral operators gives rise to estimators that are ideally not influenced by the transient effects due to the unknown initial conditions of the hidden states of the system under concern. The analysis of the properties of the kernels guaranteeing such a fast convergence is addressed and two classes of admissible kernel functions are introduced. The operators induced by the proposed kernels admit implementable (i.e., finite-dimensional and internally stable) state-space realizations. Numerical examples are reported to show the effectiveness of the proposed methodology; comparisons with some existing algebraic estimators are addressed as well.

We propose a method for trajectory classification based on a general cluster-based methodology, that can be used both off-line in an unsupervised fashion, both on-line, classifying new trajectories or part of them. We use the Earth Mover's Distance (EMD) and we adapt it in order to employ it as a tool for trajectory clustering. We propose a novel effective method to identify the clusters’ representatives by means of the p—median location problem. This methodology is able to manage different length and noisy trajectories and takes velocity profiles and stops into account. We discuss the experimental results and we compare our approach with other trajectory clustering methods.

This paper deals with the on-line simultaneous estimation of frequencies, amplitudes, and phases of a pair of sinusoidal signals with values of the frequencies that are close to each other. The proposed nonlinear parametric identification method is based on the interconnection of two quadrature-based adaptive filters, originally conceived for the extraction of a single sinusoidal signal. The distinctive feature of quadrature-based filters, inherited by the method under consideration, consists in performing the adaptation in a parametric space having higher dimensions than the number of unknowns. The stability analysis, based on two-time scale averaging arguments, shows the existence of multiple asymptotically stable manifolds corresponding to the simultaneous detection of the two sinusoidal signals.

In this paper, the robust state feedback stabilization of uncertain discrete-time constrained nonlinear systems in which the loop is closed through a packet-based communication network is addressed. In order to cope with model uncertainty, time-varying transmission delays, and packet dropouts (typically affecting the performances of networked control systems), a robust control scheme combining model predictive control with a network delay compensation strategy is proposed in the context of non-acknowledged UDP-like networks. The contribution of the paper is twofold. First, the issue of guaranteeing the recursive feasibility of the optimization problem associated to the receding horizon control law has been addressed, such that the invariance of the feasible region under the networked closed-loop dynamics can be guaranteed. Secondly, by exploiting a novel characterization of regional Input-to-State Stability in terms of time-varying Lyapunov functions, the networked closed-loop system has been proven to be Input-to-State Stable with respect to bounded perturbations.

This work deals with the analysis and the design of robust minimum-time control laws for nonlinear discrete-time systems with possibly non robustly controllable target sets. Given a Lipschitz nonlinear transition map with bounded control inputs, in the paper we show that the reachability properties of the target set can be used to assess the existence of a robust positively controllable set which includes the target in its interior. This result is exploited to formulate a minimum-time control scheme with enhanced robustness properties able to ensure the ultimate boundedness of the state-trajectories in presence of bounded uncertainties even in the case in which the target set is not robustly positively controllable.

A new method is proposed to solve the model inversion problem that is part of model based iterative learning control (ILC) for nonlinear systems. The model inversion problem consists of finding the input signal corresponding to a given output signal. This problem is formulated as a nonlinear dynamic optimization problem in time domain and solved efficiently using a constrained Gauss-Newton algorithm. A nonlinear ILC algorithm based on this model inversion approach is validated numerically and experimentally. The considered application is an electric circuit described by a polynomial nonlinear state-space model. The nonlinear ILC algorithm shows fast convergence and accurate tracking control. ispartof: pages:718-723 ispartof: Proc. of the Mediterranean Conference on Control and Automation pages:718-723 ispartof: Mediterranean Conference on Control and Automation location:Thessaloniki, Greece date:24 Jun - 26 Jun 2009 status: published

In this note, a robust model predictive control scheme for constrained discrete-time nonlinear systems affected by bounded disturbances and state-dependent uncertainties is presented. In order to guarantee the robust satisfaction of the state constraints, restricted constraint sets are introduced in the optimization problem, by exploiting the state-dependent nature of the considered class of uncertainties. Moreover, unlike the nominal model predictive control algorithm, a stabilizing state constraint is imposed at the end of the control horizon in place of the usual terminal constraint posed at the end of the prediction horizon. The regional input-to-state stability of the closed-loop system is analyzed. A simulation example shows the effectiveness of the proposed approach.

This paper presents a fault detection and isolation scheme for a class of Lipschitz nonlinear systems with nonlinear and unstructured modeling uncertainty. This significantly extends previous results by considering a more general class of system nonlinearities which are modeled as functions of the system input and partially measurable state variables. A new fault diagnosis method is developed using adaptive estimation techniques. The fault isolability conditions are rigorously established via the use of the so-called fault mismatch functions, which are defined to characterize the mutual difference between pairs of faults.

This note introduces the notion of switching-driving Lyapunov function which can be used to provide a sufficient stabilizability condition for general nonlinear systems. The theory is illustrated by means of the ldquoball-and-platerdquo system, for which locally stabilizing control laws are explicitly derived.

The theory of implicit models, introduced in previous papers, is used here in order to define a new adaptive control algorithm based on either m-step-ahead or infinite horizon LQ optimization and on recursive least squares identification techniques in the presence of systems having an ARMAX structure. The adaptive algorithm is based on the identification of a single ARX implicit model, which is defined as a model capable of representing the system input-output behaviour correctly only in certain closed-loop conditions. It is shown that, by properly structuring the algorithm, a single whitening (i.e. yielding a white residual sequence) possible convergence point exists coinciding with the optimal control law. Simple conditions assuring that a generic convergence point coincides with the above one are also provided, as well as preliminary simulation experience.

This paper deals with the stabilization problem for nonlinear systems: it provides a sufficient condition for the existence of a time-varying switching control scheme which globally asymptotically stabilizes the zero equilibrium. The sufficient condition is proven to hold for a class of non-holonomic systems and the corresponding switching control law is described in detail.

The paper deals with the global stabilization of both the attitude and the angular velocities of an underactuated rigid body. First a stability theorem is proven for a class of systems; subsequently, the equations describing the physics of the rigid body are presented, showing that the rigid body belongs to the considered class of systems, and a sufficient condition for the application of the theorem to the stability of the rigid body equilibrium is pointed out. Finally, some simulation results are reported showing the effectiveness of the proposed methodology.

In this paper, we consider the problem of actively providing an estimate of the state of a stochastic dynamic system over a (possibly long) finite time horizon. The active estimation problem (AEP) is formulated as a stochastic optimal control one, in which the minimization of a suitable uncertainty measure is carried out. Toward this end, the use of the Renyi entropy as an information measure is proposed and motivated. A neural control scheme, based on the application of the extended Ritz method (ERIM) and on the use of a Gaussian sum filter (GSF), is then presented. Simulation results show the effectiveness of the proposed approach.

This paper presents a methodology for actuator fault detection in unknown, input-affine, nonlinear systems using neural networks. Both state feedback and output feedback cases are considered. Neural net tuning algorithms are derived and fault identifiers are developed using the Lyapunov approach. The paper studies properties of the fault dynamics, the dynamics of a fault evolution process. The actuator-fault dynamics are analyzed and a rigorous detectability condition is given for actuator faults relating the actuator desired input signal, neural net-based observer sensitivity, and detectability time. Moreover, the issue of fault propagation through the system dynamics towards the measurable output is addressed and specific conditions under which such faults can be detected are proposed. Simulation results are presented to illustrate the effectiveness of the proposed technique.

The problem of the asymptotic stabilization of a five dimensional nonholonomic systems, namely the "ball and plate" or "rolling sphere" system, is discussed and solved by means of a hybrid control law relying on a suitable finite state machine. A control law is associated to each state of the machine and, by using a simple switching strategy, the origin is proven to be globally asymptotically stable in the sense of Lyapunov. Moreover, a particular function is proven to be a Lyapunov function for the considered hybrid system. The chosen control law takes naturally into account the presence of possible control saturations. Simulations are presented showing the effectiveness of the proposed control scheme.

Receding-horizon state estimation problems are addressed for a class of nonlinear discrete-time systems. We assume the system dynamics and measurement equations to be corrupted by additive, bounded noises. The statistics of such disturbances and of the initial state are unknown. We use a generalized least-squares approach that consists in minimizing a quadratic estimation cost function defined on a sliding window composed of a finite number of time stages. New results of convergence for an upper bound on the estimation error are presented that simplify the design of the estimator. The estimator is constructed either by solving an optimization problem on line or by approximating off line the optimal estimation function that solves the problem. In this last case, the approximation can be carried out under suitable assumptions via a minimax optimization.

The objective of this paper is the design of a stabilizing switching control scheme for a class of nonlinear systems. Such systems are relevant to nonlinear plants represented by means of a finite set of nonlinear discrete-time models. A finite set of receding-horizon control laws is defined for each of the nonlinear discrete-time models representing the plant. A rigorous stability analysis is carried out yielding theoretical constraints to be satisfied by the switching strategies to guarantee stability properties. Simulation results on a case study of practical relevance are also presented showing the effectiveness of the multi-model switching control scheme.

The objective of the paper is the design of a stabilizing switching control scheme for a class of nonlinear systems. Such systems are characterized by means of a finite set of nonlinear discrete-time models and for each model a finite set of receding-horizon nonlinear control laws is defined. This highlights a major feature of the considered class of switched systems with respect to previous works, namely, the possibility of switching both between different system models and between different controllers. Some practical stability concepts are then introduced and compared with classical stability definitions. The analysis of the different stability properties is carried out yielding theoretical constraints to be satisfied by the switching strategies in order to guarantee stable modes of behavior of the multi-model switched system. Some simulation results are finally reported showing the effectiveness of the proposed control scheme.

The problem of constructing a receding-horizon estimator for nonlinear discrete-time systems affected by disturbances has been addressed. The noises are assumed to be bounded, additive, and acting on both state and measurement equations. The estimator is designed according to a sliding-window strategy, i.e., so that it minimizes a receding-horizon estimation cost function. The stability of the resulting filter has been investigated and an upper bound on the estimation error has been found. Such a filter can be suitably approximated by parametrized nonlinear approximators as, for example, neural networks. A min-max algorithm turns out to be well-suited to selecting these parameters, as it allows one to guarantee the stability of the error dynamics of the approximate receding-horizon filter. This estimator is designed off line in such a way as to be able to process any possible information pattern. This enables it to generate state estimates almost instantly with a small on-line computational burden.

A multiple model based nonlinear control scheme for nonlinear discrete-time systems is considered. A set of optimal receding-horizon feedback control functions is defined, based on a description of the nonlinear plant by means of interpolation of a set of local models, using a family of nonlinear validity functions. To obtain the control law to be applied to the plant, interpolation of the local control actions is done, using the same validity function family. This choice allows to carry out the stability analysis of the equilibrium of the multiple model control scheme.

Functional optimization problems can be solved analytically only if special assumptions are verified; otherwise, approximations are needed. The approximate method that we propose is based on two steps. First, the decision functions are constrained to take on the structure of linear combinations of basis functions containing free parameters to be optimized (hence, this step can be considered as an extension to the Ritz method, for which fixed basis functions are used). Then, the functional optimization problem can be approximated by nonlinear programming problems. Linear combinations of basis functions are called approximating networks when they benefit from suitable density properties. We term such networks nonlinear (linear) approximating networks if their basis functions contain (do not contain) free parameters. For certain classes of d-variable functions to be approximated, nonlinear approximating networks may require a number of parameters increasing moderately with d, whereas linear approximating networks may be ruled out by the curse of dimensionality. Since the cost functions of the resulting nonlinear programming problems include complex averaging operations, we minimize such functions by stochastic approximation algorithms. As important special cases, we consider stochastic optimal control and estimation problems. Numerical examples show the effectiveness of the method in solving optimization problems stated in high-dimensional setting, involving for instance several tens of state variables.

This chapter deals with the problem of designing a feedback control law that drives a discrete-time dynamic system (in general, nonlinear) so as to minimize a given cost function (in general, nonquadratic). The control horizon lasts a finite number N of decision stages. The model of the dynamic system is assumed to be perfectly known. Clearly, so general non-L Q Q ptimal control problems are very difficult to solve. The proposed approximate solution is based on the following assumption: the control law is assigned a given structure in which a finite number of parameters have to be determined in order to minimize the cost function (the chosen structure is that of a multilayer feedforward neural network). Such an assumption enables us to approximate the original functional optimization problem by a nonlinear programming one. The optimal control problem is then extended from the finite to the infinite control horizon, for which a receding-horizon optimal control scheme is presented. A stabilizing regulator is derived without imposing, as is usually required by this class of control schemes, that either the origin (i.e., the equilibrium point of the controlled plant) or a suitable neighborhood of the origin be reached within a finite time. Stability is achieved by adding a proper terminal penalty function to the process cost. Also the receding-horizon regulator is approximated by means of a feedforward neural network (only one network is needed instead of a chain of N networks, as in the finite-horizon case). Simulation results show the effectiveness of the proposed approach for both finite- and infinite-horizon optimal control problems.

Fine-form detection and discrimination of an object in contact with a skin-like tactile sensor is a basic feature in machine taction for perceptual, grasping and manipulation tasks. The inversion of tactile data in the form of normal and shear stress components in order to recover the contact shape and contact radius in the class of axisymmetry indenters gives rise to a nonlinear inverse problem which must be conceptually solved by using regularization techniques. Radial basis functions networks are used to solve this problem owing to their direct connection with regularization and approximate theory. Simulation results show the effectiveness of the proposed approach even with large noise added to the data. >

The problem of asymptotic stabilization for a class of nonholonomic systems is studied and solved by means of a hybrid control law which makes use of a (deterministic) finite state machine. It is shown that, by using a simple switching control scheme, the origin is a globally asymptotically stable equilibrium in the sense of Lyapunov. The control law can take into account the presence of input saturation. Simulation results are reported showing the performance of the proposed control scheme.

The paper deals with an adaptive observer methodology for estimating the parameters of an unknown sinusoidal signal from a measurement perturbed by structured and unstructured uncertainties. The proposed technique makes it possible to handle measurement signals affected by structured uncertainties like, for example, bias and drifts which are typically present in applications. The stability of the estimator with respect to bounded additive disturbances is addressed by Input-to-State Stability arguments. The effectiveness of the proposed technique is shown through numerical simulations where comparisons with some recently proposed algorithms are also provided.

An approximation procedure termed "extended Ritz method" is presented for the solution of functional optimization problems. The properties of powerful nonlinear approximators, such as neural networks, are exploited to face highly nonlinear optimization problems in high-dimensional settings, with the possibility of avoiding the so-called "curse of dimensionality." As an example, a nonlinear control problem involving several tens of state variables is faced.

The present paper concerns the design of approximate off-line model predictive control laws for nonlinear discrete-time systems subject to hard constraints on state and input variables. The possibility to obtain an approximate receding horizon control law by performing off-line optimization, leads to a dramatic reduction of the real-time computational complexity with respect to on-line algorithms, and allows the application of the developed control technique to plants with fast dynamics, that require small sampling periods. The main feature of the proposed approximation scheme consists in the possibility to cope with possibly discontinuous state-feedback control laws, while guaranteeing the fulfillment of hard constraints on state and input variables despite the perturbations due to the use of an approximate controller. Finally, the resulting closed-loop system is shown to be input-to-state-stable with respect to the approximation-induced perturbations.

In this work, the off-line approximation of state- feedback nonlinear model predictive control laws by means of smooth functions of the state is addressed. The idea is to investigate how the approximation errors affect the stability of the closed-loop system, in order to derive suitable bounds which have to be fulfilled by the approximating function. This analysis allows to conveniently set up the characteristic parameters of some techniques such as Neural Networks which can be used to implement the control law, in order to render the system Input-to-State Practically Stable while satisfying, in addition, hard constraints on the trajectories; both the amount of data storage and the computational time result strongly reduced with respect to Nearest Neighbor or Set Membership approaches, which have been recently proposed to obtain effective off-line approximation of nonlinear MPC. The provided simulations confirm the validity of the method.