Constructive approximation of non‐linear discrete‐time systems

It is known that large classes of approximately‐finite‐memory maps can be uniformly approximated arbitrarily well by the maps of certain non‐linear structures. As an application, it was proved that time‐delay networks can be used to uniformly approximate arbitrarily well the members of a large class of causal nonlinear dynamic discrete‐time input–output maps. However, the proof is non‐constructive and provides no information concerning the determination of a structure that corresponds to a prescribed bound on the approximation error. Here we give some general results concerning the problem of finding the structure. Our setting is as follows. There is a large family 𝒢 of causal time‐invariant approximately‐finite‐memory input‐output maps G from a set S of real d‐vector‐valued discrete‐time inputs (with d⩾1) to the set of ℝ‐valued discrete‐time outputs, with both the inputs and outputs defined on the non‐negative integers 𝒵₊. We show that for each ϵ>0, any Gϵ𝒢 can be uniformly approximated by a structure map H(G, ·) to within tolerance ϵ, and we give analytical results and an example to illustrate how such a H(G, ·) can be determined in principle. Copyright © 2000 John Wiley & Sons, Ltd.     

Z+ fading memory and extensions of input–output maps

An Erratum for this article has been published in the International Journal of Circuit Theory and Applications 30(4) 2002, 179. Much is known about time‐invariant non‐linear systems with inputs and outputs defined on Z₊ that possess approximately‐finite memory. For example, under mild additional conditions, they can be approximated arbitrarily well by the maps of certain interesting simple structures. An important fact that gives meaning to results concerning such systems is that the approximately‐finite‐memory condition is known to be often met. Here we consider the known proposition that if a causal time‐invariant discrete‐time input–output map H has fading memory on a set of bounded functions defined on all of the integers Z, then H can be approximated arbitrarily well by a finite Volterra series operator. We show that in a certain sense, involving the existence of extensions of system maps, this result too has wide applicability. Copyright © 2001 John Wiley & Sons, Ltd.     

A note on representation theorems for linear discrete‐space systems with stochastic inputs

The cornerstone of the theory of discrete‐space single‐input–single‐output linear systems is the idea that every such system has an input–output map that can be represented by a convolution or the familiar generalization of a convolution. This thinking involves an oversight which, for the case of bounded inputs mapped continuously into bounded outputs, was recently corrected by adding an additional term to the representation. Here, we give a corresponding corrected result for discrete‐space systems with stochastic inputs. Copyright © 2001 John Wiley & Sons, Ltd.     

A‐map representations and asymptotically almost periodic responses

It was recently shown that for each member G of a large class of causal time‐invariant non‐linear input–output maps, with inputs and outputs defined on the non‐negative integers, there is a functional A on the input set such that (Gs) (k) has the representation A(F_ks) for all k and each input s, in which F_k is a simple linear map that does not depend on G. In this paper, we consider non‐linear maps G that have such ‘A‐map representations’. We observe that these Gs have extensions to a domain of inputs defined on the set of all integers. We show that these extensions possess some interesting properties including a certain important uniqueness property. As an application, we show that under the (very often satisfied) conditions of time invariance, causality, and approximately finite memory, and under typically mild boundedness and continuity conditions, the response of G to a discrete‐time asymptotically almost periodic input is an output that is always an asymptotically almost periodic function, and that the almost periodic part of the output is independent of the transient part of the input. We also give corresponding results for a continuous‐time case. Copyright © 2001 John Wiley & Sons, Ltd.     

On the representation of linear system half‐line maps

An expression is given for the most general input–output map associated with the members of a certain important large family of causal linear shift‐invariant systems with bounded inputs. This expression is a function‐space limit of a convolution. Also given is a necessary and sufficient condition under which the limit can be written as a convolution with an integrable impulse‐response function. Related material concerning engineering education, as well as additional results for other sets of inputs, is also given. Copyright © 2005 John Wiley & Sons, Ltd.     

Identification of dual‐rate systems based on finite impulse response models

Two identification algorithms, a least squares and a correlation analysis based, are developed for dual‐rate stochastic systems in which the output sampling period is an integer multiple of the input updating period. The basic idea is to use auxiliary FIR models to predict unmeasurable noise‐free (true) outputs, and then use these and system inputs to identify parameters of underlying fast single‐rate models. The simulation results indicate that the proposed algorithms are effective. Copyright © 2004 John Wiley & Sons, Ltd.     

On the representation of linear system time‐varying half‐line maps

An expression is given for the most general continuous causal linear input–output map that takes the space of (Lebesgue) integrable functions into itself. This expression is a function‐space limit of an integral. As an application, a representation result is given for an important family of linear maps that take the space of bounded measurable functions into itself and satisfy a certain continuity condition. Some related material concerning engineering education is also given. Copyright © 2006 John Wiley & Sons, Ltd.     

A measurement‐based approach with accuracy evaluation and its applications to circuit analysis and synthesis

In this paper, we evaluate a useful general model for linear direct current (DC) networks, containing design parameters, multiple inputs and outputs. The outputs are rational multivariate functions of the parameters and linear functions of the inputs. For an unknown linear system, the unknown coefficients of these functions can be determined by taking a suitable number of measurements and solving a set of linear equations constructed from selected measurement data. The rational functions are useful to support the design such that outputs can be constrained to a desired interval. An interesting result was the particular case where the parameter varied is located at the output terminal. This is an alternative way to estimate the Thévenin equivalent of a circuit. Real data are in general noisy with accuracy dependent on the tolerance of components and on the measurement device used. We present a detailed analysis of the application of the measurement‐based approach under real experiments. Initially, a resistive DC circuit case is presented, and the results are verified by simulation and experimental laboratory data. Finally, practical circuits, such as a conditioning signal stage with an operational amplifier, and a boost DC–DC converter are considered. Copyright © 2017 John Wiley & Sons, Ltd.     

Adaptive stabilization of uncertain nonlinear time‐delay systems with quantized input and output

This paper investigates the problem of global output feedback stabilization for a class of uncertain nonlinear time‐delay systems with unknown time delay in the states and the input in which both the input and the output are logarithmically quantized. The nonlinear functions of such systems are not completely known and satisfying certain bounded condition depending on the unmeasured states and the input. We construct a new dynamic high‐gain observer where only an output quantization instead of the output is available for measurement to dominate the unknown nonlinear functions view as external disturbances. A scaled change of coordinates and an appropriate Lyapunov‐Krasovskii functional are derived to achieve the global stabilization in the sense that all the states of such systems are defined, bounded in the maximal interval [0, +∞), and converge to the zero equilibrium. A numerical example is provided to illustrate the result.     

Interval observers for linear functions of state vectors of linear fractional‐order systems with delayed input and delayed output

This paper addresses the problem of interval observer design for linear functions of state vectors of linear fractional‐order systems, which are subjected to time delays in the measured output as well as the control input. By using the information of both the delayed output and input, we design two linear functional state observers to compute two estimates, an upper one and a lower one, which bound the unmeasured linear functions of state vectors. As a particular case with output delay only, we design a linear functional state observer to estimate (asymptotically) the unmeasured linear functions of state vectors. Existence conditions of such observers are provided, and some of them are translated into a linear programming problem, in which the observers' matrices can be effectively computed. Constructive design algorithms are introduced. Numerical examples are provided to illustrate the design procedure, practicality, and effectiveness of the proposed design method.     

Coordinated semi‐adaptive closed‐loop control for infusion of two interacting medications

This paper presents a coordinated and semi‐adaptive closed‐loop control approach to the infusion of 2 interacting medications. The proposed approach consists of an upper‐level coordination controller and a lower‐level semi‐adaptive controller. The coordination controller recursively adjusts the reference targets based on the estimated dose‐response relationship of a patient to ensure that they can be achieved by the patient. The semi‐adaptive controller drives the patient outputs to the reference targets while estimating the patient's dose‐response relationship online. In this way, the controller is resilient to unachievable caregiver‐specified reference targets and responsive to the medication needs of individual patients. To establish the proposed approach, we developed the following: (1) a linear two‐input–two‐output dose‐response model; (2) a two‐input–two‐output semi‐adaptive controller to regulate the patient outputs while adapting high‐sensitivity parameters in the patient model; and (3) a coordination controller to adjust the reference targets that reconcile caregiver inputs and medication use. The proposed approach was applied to an example scenario in which cardiac output and respiratory rate are regulated via infusion of propofol and remifentanil in an in silico simulation setting. The results show that the coordinated semi‐adaptive control could (1) track achievable reference targets with consistent transient and steady‐state performance and (2) resiliently adjust the unachievable reference targets to achievable ones.     

Self‐tuning control for polynomial systems using a receding horizon observer and control strategy

A receding horizon observer and control scheme is introduced for non‐linear systems described by polynomial maps. This control scheme has a natural interpretation as a two‐stage adaptive or self‐tuning control algorithm. The non‐linear feedback that results is defined only on the basis of past input and output measurements. The computational complexity aspects of this approach to adaptive or self‐tuning control are briefly discussed. A linear system and a Hénon map example are used to illustrate the ideas. Copyright © 2004 John Wiley & Sons, Ltd.     

Current‐mode dual‐output ICCII‐based tunable universal biquadratic filter with low‐input and high‐output impedances

A new tunable current‐mode (CM) biquadratic filter with three inputs and three outputs using three dual‐output inverting second‐generation current conveyors, three grounded resistors and two grounded capacitors is proposed. The proposed circuit exhibits low‐input impedance and high‐output impedance which is important for easy cascading in the CM operations. It can realize lowpass, bandpass, highpass, bandreject and allpass biquadratic filtering responses from the same topology. The circuit permits orthogonal controllability of the quality factor Q and resonance angular frequency ω_o, and no component matching conditions or inverting‐type input current signals are imposed. All the passive and active sensitivities are low. Hspice simulation results are based on using TSMC 0.18 µm 1P6M process complementary metal oxide semiconductor technology and supply voltages ±0.9 V to verify the theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Time‐varying linear systems and input–output stability

One useful strategy for establishing input–output stability is to seek conditions on the system map so that a certain lower‐bound condition is met. We show here that, for a large class of time‐varying linear systems, the lower‐bound condition for stability is in fact necessary. Copyright © 2000 John Wiley & Sons, Ltd.     

Adaptive output tracking for a class of non‐linear time‐delay systems

In this paper the output tracking control problem for a class of non‐linear time delay systems with some unknown constant parameters is addressed. Such a problem is solved in the case that the non‐linear time‐delay system has full delay relative degree and stable internal dynamics. It is supposed moreover that the output and its time derivatives until n?1, where n is the length of the state vector (euclidean part), do not depend explicitly on the unknown parameters. This work is the first step towards the application of the methodologies of adaptive control for non‐linear delayless systems, based on tools of differential geometry, to non‐linear time‐delay systems too. Copyright © 2004 John Wiley & Sons, Ltd.     

Convergence theory for multi‐input discrete‐time iterative learning control with Coulomb friction,continuous outputs,and input bounds

In this paper we consider the problem of discrete‐time iterative learning control (ILC) for position trajectory tracking of multiple‐input, multiple‐output systems with Coulomb friction, bounds on the inputs, and equal static and sliding coefficients of friction. We present an ILC controller and a proof of convergence to zero tracking error, provided the associated learning gain matrices are scalar‐scaled with a sufficiently small positive scalar. We also show that non‐diagonal learning gain matrices satisfying the same prescribed conditions do not lead to the same convergence property. To the best of our knowledge, for problems with Coulomb friction, this paper represents a first convergence theory for the discrete‐time ILC problem with multiple‐bounded‐inputs and multiple‐outputs; previous work presented theory only for the single‐input, single‐output problem. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimal expansions of multivariable ARX processes on Laguerre bases via the Newton–Raphson method

In this article, an optimal linear MIMO system approximation by using discrete‐time MIMO autoregressive with exogenous input (ARX) model is proposed. Each polynomial function of the MIMO ARX model associated with the inputs and with the outputs is expanded on independent Laguerre orthonormal basis. The resulting model is entitled MIMO ARX–Laguerre model. The optimal approximation of which is ensured once the poles characterizing each Laguerre orthonormal basis are set to their optimal values. In this paper, a new method to estimate, from input/output measurements, the optimal Laguerre poles of the MIMO ARX–Laguerre model is proposed. The method consists in applying the Newton–Raphson's iterative technique in which the gradient and the Hessian are expressed analytically. The proposed algorithm is tested on a numerical example and on a benchmark system. Simulation results show the effectiveness of the proposed optimal modeling method. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust output tracking of uncertain nonlinear time‐delay systems with unknown dead‐zone inputs via control schemes with a simple structure

The adaptive robust output tracking control problem is considered for a class of uncertain nonlinear time‐delay systems with completely unknown dead‐zone inputs. A new design method is proposed so that some adaptive robust output tracking control schemes with a rather simple structure can be constructed. It is not necessary to know the nonlinear upper bound functions of uncertain nonlinearities. In fact, the constructed output tracking control schemes are structurally linear in the state and have a self‐tuning control gain function that is updated by an adaptation law. In this paper, the dead‐zone input is nonsymmetric, and its information is assumed to be completely unknown. In addition, a numerical example is given to describe the design procedure of the presented method, and the simulations of this numerical example are implemented to demonstrate the validity of the theoretical results.     

Digital architectures realizing piecewise‐linear multivariate functions: Two FPGA implementations

Digital architectures for the circuit realization of multivariate piecewise‐linear (PWL) functions are reviewed and compared. The output of the circuits is a digital word representing the value of the PWL function at the n‐dimensional input. In particular, we propose two architectures with different levels of parallelism/complexity. PWL functions with n = 3 inputs are implemented on an FPGA and experimental results are shown. The accuracy in the representation of PWL functions is tested through three benchmark examples, two concerning three‐variate static functions and one concerning a dynamical control system defined by a bi‐variate PWL function. Copyright © 2009 John Wiley & Sons, Ltd.     

Direct adaptive control for non‐linear uncertain systems with exogenous disturbances

A direct adaptive non‐linear control framework for multivariable non‐linear uncertain systems with exogenous bounded disturbances is developed. The adaptive non‐linear controller addresses adaptive stabilization, disturbance rejection and adaptive tracking. The proposed framework is Lyapunov‐based and guarantees partial asymptotic stability of the closed‐loop system; that is, asymptotic stability with respect to part of the closed‐loop system states associated with the plant. In the case of bounded energy L₂ disturbances the proposed approach guarantees a non‐expansivity constraint on the closed‐loop input–output map. Finally, several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.