System Identification

Backpropagation is now the most widely used tool in the field of artificial neural networks. At the core of backpropagation is a method for calculating derivatives exactly and efficiently in any large system made up of elementary subsystems or calculations which are represented by known, differentiable functions; thus, backpropagation has many applications which do not involve neural networks as such. This paper first reviews basic backpropagation, a simple method which is now being widely used in areas like pattern recognition and fault diagnosis. Next, it presents the basic equations for backpropagation through time, and discusses applications to areas like pattern recognition involving dynamic systems, systems identification, and control. Finally, it describes further extensions of this method, to deal with systems other than neural networks, systems involving simultaneous equations or true recurrent networks, and other practical issues which arise with this method. Pseudocode is provided to clarify the algorithms. The chain rule for ordered derivatives-the theorem which underlies backpropagation-is briefly discussed.

A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes, as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a user's perspective. It focuses on what are the common features in the di erent approaches, the choices that have to be made and what considerations are relevant for a successful system identi cation application of these techniques.

A compendium of recent theoretical results associated with using higher-order statistics in signal processing and system theory is provided, and the utility of applying higher-order statistics to practical problems is demonstrated. Most of the results are given for one-dimensional processes, but some extensions to vector processes and multichannel systems are discussed. The topics covered include cumulant-polyspectra formulas; impulse response formulas; autoregressive (AR) coefficients; relationships between second-order and higher-order statistics for linear systems; double C(q,k) formulas for extracting autoregressive moving average (ARMA) coefficients; bicepstral formulas; multichannel formulas; harmonic processes; estimates of cumulants; and applications to identification of various systems, including the identification of systems from just output measurements, identification of AR systems, identification of moving-average systems, and identification of ARMA systems

The channel estimation techniques for OFDM systems based on pilot arrangement are investigated. The channel estimation based on comb type pilot arrangement is studied through different algorithms for both estimating channel at pilot frequencies and interpolating the channel.

The present bibliography represents a comprehensive list of references on nonlinear system identi"cation and its applications in signal processing, communications, and biomedical engineering. An attempt has been made to make this bibliography complete by listing most of the existing references up to the year 2000 and by providing a detailed classi"cation group.

There is a growing appreciation of the importance of nonlinearities in evoked responses in fMRI, particularly with the advent of event-related fMRI. These nonlinearities are commonly expressed as interactions among stimuli that can lead to the suppression and increased latency of responses to a stimulus that are incurred by a preceding stimulus. We have presented previously a model-free characterization of these effects using generic techniques from nonlinear system identification, namely a Volterra series formulation. At the same time Buxton et al. (1998) described a plausible and compelling dynamical model of hemodynamic signal transduction in fMRI. Subsequent work by Mandeville et al. (1999) provided important theoretical and empirical constraints on the form of the dynamic relationship between blood flow and volume that underpins the evolution of the fMRI signal. In this paper we combine these system identification and model-based approaches and ask whether the Balloon model is sufficient to account for the nonlinear behaviors observed in real time series. We conclude that it can, and furthermore the model parameters that ensue are biologically plausible. This conclusion is based on the observation that the Balloon model can produce Volterra kernels that emulate empirical kernels. To enable this evaluation we had to embed the Balloon model in a hemodynamic input-state-output model that included the dynamics of perfusion changes that are contingent on underlying synaptic activation. This paper presents (i) the full hemodynamic model (ii), how its associated Volterra kernels can be derived, and (iii) addresses the model's validity in relation to empirical nonlinear characterisations of evoked responses in fMRI and other neurophysiological constraints.

Dynamic textures are sequences of images of moving scenes that exhibit certain stationarity properties in time; these include sea-waves, smoke, foliage, whirlwind but also talking faces, traffic scenes etc. We present a novel characterization of dynamic textures that poses the problems of modelling, learning, recognizing and synthesizing dynamic textures on a firm analytical footing. We borrow tools from system identification to capture the "essence" of dynamic textures; we do so by learning (i.e. identifying) models that are optimal in the sense of maximum likelihood or minimum prediction error variance. For the special case of secondorder stationary processes we identify the model in closed form. Once learned, a model has predictive power and can be used for extrapolating synthetic sequences to infinite length with negligible computational cost. We present experimental evidence that, within our framework, even low dimensional models can capture very complex visual phenomena.

A new LMS-type adaptive filter with a variable step size is introduced. The step size increases or decreases as the mean-square error increases or decreases, allowing the adaptive filter to track changes in the system as well as produce a small steady state error. The convergence and steady state behavior of the algorithm are analyzed. These results reduce to well-known ones when specialized to the constant step size case. Simulation results are presented to support the analysis and to compare the performance of the new algorithm with the usual LMS algorithm and another variable step algorithm. They show that the performance of the new algorithm compares favorably with these existing algorithms.

A new spacecraft attitude estimation approach based on the Unscented Filter is derived. For nonlinear systems the Unscented Filter uses a carefully selected set of sample points to more accurately map the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The filter formulation is based on standard attitude-vector measurements using a gyro-based model for attitude propagation. The global attitude parameterization is given by a quaternion, while a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is derived from the local attitude error, which guarantees that quaternion normalization is maintained in the filter. Simulation results indicate that the Unscented Filter is more robust than the Extended Kalman Filter under realistic initial attitude-error conditions.

The use of kriging models for approximation and global optimization has been steadily on the rise in the past decade. The standard approach used in the Design and Analysis of Computer Experiments (DACE) is to use an Ordinary kriging model to approximate a deterministic computer model. Universal and Detrended kriging are two alternative types of kriging models. In this paper, a description on the basics of kriging is given, highlighting the similarities and differences between these three different types of kriging models and the underlying assumptions behind each. A comparative study on the use of three different types of kriging models is then presented using six test problems. The methods of Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) for model parameter estimation are compared for the three kriging model types. A one-dimension problem is first used to visualize the differences between the different models. In order to show applications in higher dimensions, four two-dimension and a 5dimension problem are also given.

In this contribution we study the statistical properties of a number of closed-loop identi cation methods and parameterizations. A focus will be on asymptotic variance expressions for these methods. By studying the asymptotic variance for the parameter vector estimates we show that indirect methods fail to give better accuracy than the direct method. Some new results for the bias distribution of the direct method will be presented and we also show how di erent methods correspond to di erent parameterizations and how direct and indirect identi cation can be linked together via the noise model. In addition, analysis of a new method for closed-loop identi cation shows that it allows tting the model to data with arbitrary frequency weighting under quite general conditions.

When using the analysis of vibration measurements as a tool for health monitoring of bridges, the problem arises of separating abnormal changes from normal changes in the dynamic behaviour. Normal changes are caused by varying environmental conditions such as humidity, wind and most important, temperature. The temperature may have an impact on the boundary conditions and the material properties. Abnormal changes on the other hand are caused by a loss of sti ness somewhere along the bridge. It is clear that the normal changes should not raise an alarm in the monitoring system (i.e. a false positive), whereas the abnormal changes may be critical for the structure's safety. In the frame of the European SIMCES-project, the Z24-Bridge in Switzerland was monitored during almost one year before it was artiÿcially damaged. Black-box models are determined from the healthy-bridge data. These models describe the variations of eigenfrequencies as a function of temperature. New data are compared with the models. If an eigenfrequency exceeds certain conÿdence intervals of the model, there is probably another cause than the temperature that drives the eigenfrequency variations, for instance damage.

In this chapter we consider reduced basis (RB) approximation and a posteriori error estimation for linear functional outputs of affinely parametrized linear parabolic partial differential equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametrically induced manifolddimension reduction; efficient and effective POD-Greedy RB sampling methods for identification of optimal and numerically stable approximations spaces -rapid convergence; rigorous and sharp a posteriori error bounds for the linear-functional outputs of interest -certainty; and Offline-Online computational decomposition strategies -minimum marginal cost. The RB approach is effective in the real-time context in which the expensive Offline stage is deemed unimportant -for example parameter estimation and control; the RB approach is also effective in the many-query context in which the expensive Offline stage is asymptotically negligible -for example design

We consider the problem of distributed Kalman filtering, where a set of nodes are required to collectively estimate the state of a linear dynamic system from their individual measurements. Our focus is on diffusion strategies, where nodes communicate with their direct neighbors only, and the information is diffused across the network. We derive and analyze the mean and mean-square performance of the proposed algorithms and show by simulation that they outperform previous solutions.

It has recently been shown that gradient-descent learning algorithms for recurrent neural networks can perform poorly on tasks that involve long{term dependencies, i.e. those problems for which the desired output depends on inputs presented at times far in the past.

It is shown how genetic algorithms can be applied for system identification of both continuous and discrete time systems. It is shown that they are effective in both domains and are able to directly identify physical parameters or poles and zeros. This can be useful because changing one physical parameter might affect every parameter of a system transfer function. The estimates of poles and zeros are then used to design a discrete time pole placement adaptive controller. Simulations for minimum and nonminimum phase systems and a system with unmodeled dynamics are presented

Representation, identification, and modeling are investigated for nonlinear biomedical systems. We begin by considering the conditions under which a nonlinear system can be represented or accurately approximated by a Volterra series (or functional expansion). Next, we examine system identification through estimating the kernels in a Volterra functional expansion approximation for the system. A recent kernel estimation technique that has proved to be effective in a number of biomedical applications is investigated as to running time and demonstrated on both clean and noisy data records, then it is used to illustrate identification of cascades of alternating dynamic linear and static nonlinear systems, both single-input single-output and multivariable cascades. During the presentation, we critically examine some interesting biological applications of kernel estimation techniques.

In this paper we present an application of the Wold-Cramer representation of non-stationary processes to the identification of non-minimum phase linear time-invariant (LTI) systems and modeling of non-stationary signals. According to the Wold-Cramer representation, a non-stationary signal can be expressed as an infinite sum of sinusoids with time-varying random magnitudes and phases. For the identification of LTI systems with non-stationary input and output, we relate the Wold-Cramer representations of the system's input and output to obtain estimates of the magnitude and phase frequency responses of the system. Our procedure uses estimators from the evolutionary periodogram recently introduced, permits the identification of non-minimum phase systems, and provides also rational estimates. Furthermore, we show that when the system output is corrupted by additive stationary noise it is possible to avoid the effects of the noise in the identification. If the noise is non-stationary and Gaussian, we show that it can be removed by considering the evolutionary bispectrum, recently introduced. The analysis also provides a model for a non-stationary signal, as the output of the cascade of a linear time-varying (LTV) and an LTI systems with white noise as input. To illustrate the performance of our procedure, we present simulations of the identification of non-minimum phase LTI systems with noisy outputs.

In this paper, we investigate pilot-symbol-aided parameter estimation for orthogonal frequency division multiplexing (OFDM) systems. We first derive a minimum mean-square error (MMSE) pilot-symbol-aided parameter estimator. Then, we discuss a robust implementation of the pilot-symbol-aided estimator that is insensitive to channel statistics. From the simulation results, the required signal-to-noise ratios (SNR's) for a 10% word error rate (WER) are 6.8 dB and 7.3 dB for the typical urban (TU) channels with 40 Hz and 200 Hz Doppler frequencies, respectively, and they are 8 dB and 8.3 dB for the hilly-terrain (HT) channels with 40 Hz and 200 Hz Doppler frequencies, respectively. Compared with the decision-directed parameter estimator, the pilot-symbolaided estimator is highly robust to Doppler frequency for dispersive fading channels with noise impairment even though it has some performance degradation for systems with lower Doppler frequencies.

In biometrics, a human being needs to be identified based on some characteristic physiological parameters. Often this recognition is part of some security system. Secure storage of reference data (i.e., user templates) of individuals is a key concern. It is undesirable that a dishonest verifier can misuse parameters that he obtains before or during a recognition process. We propose a method that allows a verifier to check the authenticity of the prover in a way that the verifier does not learn any information about the biometrics of the prover, unless the prover willingly releases these parameters. To this end, we introduce the concept of a delta-contracting and epsilon-revealing function which executes preprocessing in the biometric authentication scheme. It is believed that this concept can become a building block of a public infrastructure for biometric authentication that nonetheless preserves privacy of the participants.

This paper proposes a probabilistic formulation to assess the effectiveness of the fiber reinforced polymer (FRP) retrofit schemes in enhancing the structural performance of reinforced concrete (RC) bridge columns. Two probabilistic models are proposed to predict the deformation capacities of retrofitted columns. One deformation model corresponds to the flexural failure and the other considers the bond failure in the lap-splice region. A Markov Chain Monte Carlo (MCMC) simulation method is used to estimate unknown model parameters in the context of a Bayesian updating approach. The probabilistic capacity models are used to estimate the fragility curves of three example columns. In this paper, fragility is defined as the conditional probability of failure for given deformation demand. The results compare the column fragilities before and after the application of the retrofit measure. The results from the example columns indicate that the use of FRP composites considerably reduced the fragility for the bond failure mode and is also beneficial but with a moderate impact when considering the flexural failure.

Broadband macromodeling of large multiport systems by vector fitting can be time consuming and resource demanding when all elements of the system matrix share a common set of poles. This letter presents a robust approach which removes the sparsity of the block-structured least-squares equations by a direct application of the QR decomposition. A 60-port printed circuit board example illustrates that considerable savings in terms of computation time and memory requirements are obtained.

In this paper, a time series algorithm is presented for damage identification and localization. The vibration signals obtained from sensors are modeled as autoregressive moving average (ARMA) time series. A new damage-sensitive feature, DSF, is defined as a function of the first three auto regressive (AR) components. It is found that the mean values of the DSF for the damaged and undamaged signals are different. Thus, a hypothesis test involving the t-test is used to obtain a damage decision. Two damage localization indices LI 1 and LI 2 , are introduced based on the AR coefficients. At the sensor locations where damage is introduced, the values of LI 1 and LI 2 appear to increase from their values obtained at the undamaged baseline state. The damage detection and localization algorithms are valid for stationary signals obtained from linear systems. To test the efficacy of the damage detection and localization methodologies, the algorithm has been tested on the analytical and experimental results of the ASCE benchmark structure. In contrast to prior pattern classification and statistical signal processing algorithms that have been able to identify primarily severe damage and have not been able to localize the damage effectively, the proposed algorithm is able to identify and localize minor to severe damage as defined for the benchmark structure. r

Modern robust control synthesis techniques aim at providing robustness with respect to uncertainty in the form of both additive noise and plant perturbations. On the other hand, most popular system identification methods assume that all uncertainty is in the form of additive noise. This has hampered the application of robust control methods to practical problems. This paper begins to address the gap between the models used in control synthesis and those obtained from identification experiments by considering the connection between uncertain models and data. The model validation problem addressed here is: given experimental data and a model with both additive noise and norm-bounded perturbations, is it possible that the model could produce the observed input-output data? This problem is studied for the standard H ∞ /µ framework models. A necessary condition for such a model to describe an experimental datum is obtained. Furthermore, for a large class of models, in the robust control framework, this condition is computable as the solution of a quadratic optimization problem.

PAPERS Distributed Function Calculation via Linear Iterative Strategies in the Presence of Malicious Agents ...................... ............................................................................................. S.SundaramandC.N.Hadjicostis ... A Kinship Function Approach to Robust and Probabilistic Optimization Under Polynomial Uncertainty .................. ..................................................................... .................... C. Feng, F. Dabbene, and CM Lagoa ... Real-Time Suboptimal Model Predictive Control Using a Combination of Explicit MPC and Online Optimization ...... ............................. ...

Fingerprint friction ridge details are generally described in a hierarchical order at three different levels, namely, Level 1 (pattern), Level 2 (minutia points), and Level 3 (pores and ridge contours). Although latent print examiners frequently take advantage of Level 3 features to assist in identification, Automated Fingerprint Identification Systems (AFIS) currently rely only on Level 1 and Level 2 features. In fact, the Federal Bureau of Investigation's (FBI) standard of fingerprint resolution for AFIS is 500 pixels per inch (ppi), which is inadequate for capturing Level 3 features, such as pores. With the advances in fingerprint sensing technology, many sensors are now equipped with dual resolution (500 ppi/1,000 ppi) scanning capability. However, increasing the scan resolution alone does not necessarily provide any performance improvement in fingerprint matching, unless an extended feature set is utilized. As a result, a systematic study to determine how much performance gain one can achieve by introducing Level 3 features in AFIS is highly desired. We propose a hierarchical matching system that utilizes features at all the three levels extracted from 1,000 ppi fingerprint scans. Level 3 features, including pores and ridge contours, are automatically extracted using Gabor filters and wavelet transform and are locally matched using the Iterative Closest Point (ICP) algorithm. Our experiments show that Level 3 features carry significant discriminatory information. There is a relative reduction of 20 percent in the equal error rate (EER) of the matching system when Level 3 features are employed in combination with Level 1 and 2 features. This significant performance gain is consistently observed across various quality fingerprint images.

The cross-correlation function between two response measurements made on an ambiently excited structure is shown to have the same form as the system's impulse response function. Therefore, standard time domain curve-fitting procedures, which are typically applied to impulse response functions, can now be applied to the cross-correlation functions to estimate the resonant frequencies and modal damping of the structure. This derivation is based on the assumption that the ambient vibration source is a white noise random process. Curve-fitting cross-correlation functions to obtain modal properties offers advantages over standard procedures that identify resonant frequencies from peaks in the power spectrum and damping from the width of the power spectrum. The primary advantage is the ability to identify closely spaced modes and their associated damping. The resonant frequencies of a highway bridge that were identified by curve-fitting the cross-correlation functions, using traffic excitation as the ambient vibration source, are compared to modal properties identified by standard forced vibration testing methods. Results of this comparison showed a maximum discrepancy of 3·63 percent. Similar comparisons for the average modal damping values identified by the two methods showed a 9·82 percent difference. This experimental verification implies that the proposed method of analyzing ambient vibration data can be used to accurately assess the dynamic properties of structures in a non-intrusive manner. 7 1997 Academic Press Limited

A new dynamic time-delay fuzzy wavelet neural network model is presented for nonparametric identification of structures using the nonlinear autoregressive moving average with exogenous inputs approach. The model is based on the integration of four different computing concepts: dynamic time delay neural network, wavelet, fuzzy logic, and the reconstructed state space concept from the chaos theory. Noise in the signals is removed using the discrete wavelet packet transform method. In order to preserve the dynamics of time series, the reconstructed state space concept from the chaos theory is employed to construct the input vector. In addition to denoising, wavelets are employed in combination with two soft computing techniques, neural networks and fuzzy logic, to create a new pattern recognition model to capture the characteristics of the time series sensor data accurately and efficiently. The model balances the global and local influences of the training data and incorporates the imprecision existing in the sensor data effectively. Experimental results on a five-story steel frame are employed to validate the computational model and demonstrate its accuracy and efficiency.

We consider the problem of online prediction when it is uncertain what the best prediction model to use is. We develop a method called Dynamic Model Averaging (DMA) in which a state space model for the parameters of each model is combined with a Markov chain model for the correct model. This allows the "correct" model to vary over time. The state space and Markov chain models are both specified in terms of forgetting, leading to a highly parsimonious representation. As a special case, when the model and parameters do not change, DMA is a recursive implementation of standard Bayesian model averaging, which we call recursive model averaging. The method is applied to the problem of predicting the output strip thickness for a cold rolling mill, where the output is measured with a time delay. We found that when only a small number of physically motivated models were considered and one was clearly best, the method quickly converged to the best model, and the cost of model uncertainty was small; indeed DMA performed slightly better than the best physical model. When model uncertainty and the number of models considered were large, our method ensured that the penalty for model uncertainty was small. At the beginning of the process, when control is most difficult, we found that DMA over a large model space led to better predictions than the single best performing physically motivated model. We also applied the method to several simulated examples, and found that it recovered both constant and time-varying regression parameters and model specifications quite well.

A blind deconvolution methodology for system identification is presented. The methodology requires two or more recorded time histories, that resulted from the convolution of a common ("input") motion with the impulse response functions of filters ("channels") with different properties. No prior information on the input or the channels is necessary. The methodology is based on the reconstruction of signals from information on their phase only. By eliminating the common component, the approach retains the channel characteristics and identifies them in both time and frequency domains. Through deconvolution of each recorded time history with the corresponding channel impulse response function, the input motion can also be recovered. An example application of the approach to synthetic seismic ground motion data is also presented.

Particle methods are a set of powerful and versatile simulation-based methods to perform optimal state estimation in nonlinear non-Gaussian state-space models. The ability to compute the optimal filter is central to solving important problems in areas such as change detection, parameter estimation, and control. Much recent work has been done in these areas. The objective of this paper is to provide a detailed overview of them.

A parallel-cascade system identification method was used to identify intrinsic and reflex contributions to dynamic ankle stiffness over a wide range of tonic voluntary contraction levels and ankle positions in healthy human subjects. Intrinsic stiffness dynamics were described well by a linear pathway having elastic, viscous, and inertial properties. A velocity-sensitive pathway comprising a delay, a static non-linearity, resembling a half-wave rectifier, followed by a low-pass filter, described reflex stiffness dynamics. The absolute magnitude of intrinsic and reflex stiffness parameters varied from subject to subject but the relative changes with contraction level and position were consistent. Intrinsic stiffness increased monotonically with contraction level while reflex stiffness was maximal at low contraction levels and then decreased. Intrinsic and reflex stiffness both increased as the ankle was dorsiflexed. As a result, reflex mechanics made their largest relative contributions near the neutral position at low levels of activity. The size of the maximum reflex contribution varied widely among subjects, in some it was so small (ca 1%) that it would be unlikely to have any functional importance; however, in other subjects, reflex contributions were large enough (as high as 55% in one case) to play a significant role in the control of posture and movement. This variability may have arisen because stretch reflexes were not useful for the torque-matching task in these experiments. It will be of interest to examine other tasks where stretch reflexes would have a direct impact on performance.

We give a general overview of the state-of-the-art in subspace system identi®cation methods. We have restricted ourselves to the most important ideas and developments since the methods appeared in the late eighties. First, the basics of linear subspace identi-®cation are summarized. Dierent algorithms one ®nds in literature (such as N4SID, IV-4SID, MOESP, CVA) are discussed and put into a unifying framework. Further, a comparison between subspace identi®cation and prediction error methods is made on the basis of computational complexity and precision of the methods by applying them on 10 industrial data sets. #

Active sound attenuation systems may be described using a system identification framework in which an adaptive filter is used to model the performance of an unknown acoustical plant. An error signal may be obtained from a location following an acoustical summing junction where the undesired noise is combined with the output of a secondary sound source. For the model output to properly converge to a value that will minimize the error signal, it is frequently necessary to determine the transfer function of the secondary sound source and the path to the error signal measurement. Since these transfer functions are unknown and continuously changing in a real system, it is desirable to perform continuous on-line modeling of the output transducer and error path. In this article, the use of an auxiliary random noise generator for this modeling is described. Based on a Galois sequence, this technique is easy to implement, provides continuous on-line modeling, and has minimal effect on the final value of the error signal.

The time-frequency character of wavelet transforms allows increased flexibility -as both traditional time and frequency domain system identification approaches can be adapted to examine non-linear and non-stationary response data. However, a number of additional processing concerns must be understood to fully exploit the power of the multi-resolution, dual-domain transform, particularly for the popular Morlet wavelet. Unfortunately, in prior applications of wavelet transforms for system identification, the implications of the aforementioned concerns were often negligible, as these studies considered mechanical systems characterized by higher frequency, broader-band signals. It was the subsequent analysis of Civil Engineering structures that highlighted the need to understand more fully these processing concerns in order to insure the successful identification of dynamic system properties. This study identifies a number of these considerations that are a direct consequence of the wavelet's multi-resolution character, including considerations for selection of wavelet central frequencies and strategies to resolve end effects errors. In total, this study serves as an overview of these processing considerations for Civil Engineering structures, helping to lessen the challenges associated with the transition into the time-frequency domain.

We consider the tracking problem of unknown, robustly stabilizable, multi-input multi-output (MIMO), affine in the control, nonlinear systems with guaranteed prescribed performance. By prescribed performance we mean that the tracking error converges to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, exhibiting maximum overshoot as well as undershoot less than some sufficiently small preassigned constants. Utilizing an output error transformation, we obtain a transformed system whose robust stabilization is proven necessary and sufficient to achieve prescribed performance guarantees for the output tracking error of the original system, provided that initially the transformed system is well defined. Consequently, a switching robust control Lyapunov function (RCLF)-based adaptive, state feedback controller is designed, to solve the stated problem. The proposed controller is continuous and successfully overcomes the problem of computing the control law when the approximation model becomes uncontrollable. Simulations illustrate the approach.