Nonlinear Single-input Single-output Research Articles

Measuring a linear approximation to weakly nonlinear MIMO systems

The paper addresses the problem of preserving the same LTI approximation of a nonlinear MIMO (multiple-input multiple-output) system. It is shown that when a nonlinear MIMO system is modeled by a multidimensional Volterra series, periodic noise and random multisines are equivalent excitations to the classical Gaussian noise, in a sense that they yield in the limit, as the number of the harmonics M → ∞ , the same linear approximation to the nonlinear MIMO system. This result extends previous results derived for nonlinear SISO (single-input single-output) systems. Based upon the analysis of the variability of the measured FRF (frequency response function) due to the presence of the nonlinearities and the randomness of the excitations, a new class of equivalent input signals is proposed, allowing for a lower variance of the nonlinear FRF measurements, while the same linear approximation is retrieved.

A novel multiple-controller incorporating a radial basis function neural network based generalized learning model

A new adaptive multiple-controller is proposed incorporating a radial basis function (RBF) neural network based generalized learning model (GLM). The GLM assumes that the unknown complex plant is represented by an equivalent stochastic model consisting of a linear time-varying sub-model plus a non-linear RBF neural-network learning sub-model. The proposed non-linear multiple-controller methodology provides the designer with a choice, through simple switching, of using: either, a conventional proportional-integral-derivative (PID) controller, a PID structure based pole (only) placement controller, or a newly developed PID structure based (simultaneous) zero and pole placement controller. Closed-loop stability analysis of the multiple-controller framework is discussed and sample simulation results using a realistic non-linear single-input single-output (SISO) plant model are used to demonstrate the effectiveness of the multiple-controller with respect to tracking desired set-point changes and dealing with sudden introduction of disturbances.

New stable adaptive fuzzy approach to control uncertain nonlinear SISO systems

This paper deals with the synthesis of a new robust adaptive fuzzy control for a class of nonlinear and disturbed single-input single-output (SISO) systems. To attenuate the effect of both of the approximation errors and the external disturbances to a prescribed level, two signals are added to the indirect adaptive fuzzy control law: the first deduced from a fuzzy system allows approximation errors and external disturbances to eliminated; the second signal deduced from the Riccati equation attenuates the effect of the residual errors to a prescribed level. To illustrate the efficiency of the proposed approach, a simulation example is presented.

Constrained predictive control of a greenhouse

A scheme for temperature control of a greenhouse is presented. The present work proposes an approach based on a combination of two different control schemes: Feedback Linearization (FL) and standard linear Model Predictive Control (MPC), using their advantages. The treated greenhouse is considered a non-linear Single-Input–Single-Output process and subject to strong external disturbances. Since the methodology used for solving the MPC + FL approaches generally leads to an optimization problem subject to state-dependent non-linear constraints, an alternative for its implementation is discussed. Two control techniques are compared, namely MPC + FL and Non-linear Model Predictive Control (NLMPC).

Equivalence of Different Realization Methods for Higher Order Nonlinear Input–Output Differential Equations

Several state space realizability conditions and realization algorithm for nonlinear single-input single-output higher order input–output differential equation are compared. Three different necessary and sufficient state space realizability conditions, respectively, given in terms of integrability of the subspaces of one-forms, involutivity of the conditionally invariant distributions, and commutativity of iterative Lie brackets of vector fields, are proved to be equivalent. Moreover, the algorithm-based solutions are demonstrated for computing the integrable basis of the subspaces of oneforms. Finally, explicit formulas are provided for calculation of the differentials of the state coordinates which, in case the necessary and sufficient realizability conditions are satisfied, can be integrated to obtain the state coordinates.

REALIZATION OF NONLINEAR DISCRETE-TIME COMPOSITE SYSTEMS: COMPUTATIONAL ASPECTS

The paper is devoted to computational aspects of the problems of realization of discrete-time nonlinear single-input single-output composite systems The main contributions are made in two directions. First, we study the relations between three different methods that for any non-realizable system construct a “compensating system” which will result in a realizable series or parallel connection. Second, we implement these methods and algorithms in the computer algebra system Mathematica. Results are illustrated by examples.

Realization of Nonlinear Discrete-Time Composite Systems

The present contribution is devoted to the problems of realization of discrete-time nonlinear single-input single-output composite systems. Namely, re-alization of series and closed-loop connections are under consideration. It is proved that for any non-realizable system one can construct a “compensating system” which will result in a realizable series or feedback connection. A constructive algorithms are given to obtain such a system for the cases of series and closed-loop connections.

A new approach for neural control of nonlinear discrete dynamic systems

This paper proposes a new approach to control nonlinear discrete dynamic systems, which relies on the identification of a discrete model of the system by a neural network. A locally equivalent optimal linear model is obtained from the neural network model at every operating point the system goes through during the control task. Based on the linear model, a linear state-space control design technique can be used to design a local control action to be applied at that particular operating point. The design procedure is repeated at every operating point of the system during the control task. The proposed approach was applied in three examples, which involved a linear and two nonlinear discrete single-input single-output (SISO) systems. In the first two examples, pole placement was chosen as the linear design technique. This method led to satisfactory tracking of the reference input but a steady-state error was present due to modeling inaccuracies. In the third example, the linear design technique involved an optimal control scheme with an integrator. This method led to satisfactory tracking of the reference input with zero steady-state error.

Generalization of Transfer Equivalence for Discrete-Time Non-Linear Systems: Comparison of Two Definitions

The problem of system reduction is studied for discrete-time non-linear single-input single-output systems described by high-order input–output (i/o) difference equations, that is, given the i/o equation, we can find a minimal representation which is equivalent to the original system with the order being as small as possible. Comparison of two definitions of reducibility is provided. The reducibility concepts addressed are both generalizations of the well-known notion of transfer equivalence to the case of non-linear control systems. Two roles of transfer equivalence are covered by these extensions. The first is that of identity of the outputs for any fixed control sequence under zero initial conditions. The second property is that of pole/zero cancellation that may occur in the transfer function of equivalent systems. The relationship between two reducibility definitions related to two equivalence notions is examined and the reducibility definition which extends pole/zero cancellation property is shown to be stronger. Finally, the computational aspects of system reduction are discussed.

Transition control using a state-shared model approach

Transition control is defined in this work as a type of control method that is employed when the plant transitions from one steady state to another as a result of a set point change. Recent approaches have relied on multiple models and centralized or decentralized controller designs to address this issue. This work introduces and develops a transition control framework that consists of multiple fixed and adaptive models within a state-shared non-minimal realization and an H ∞ controller design. The efficacy of this transition control framework is demonstrated on two nonlinear single-input single-output reactors in the face of modeling errors, parameter uncertainties and disturbances.

Fuzzy model predictive control of non-linear processes using genetic algorithms

This paper introduces a new fuzzy control technique, which belongs to the popular family of control algorithms, called Model Predictive Controllers. The method is based on a dynamic fuzzy model of the process to be controlled, which is used for predicting the future behavior of the output variables. A non-linear optimization problem is then formulated, which minimizes the difference between the model predictions and the desired trajectory over the prediction horizon and the control energy over a shorter control horizon. The problem is solved on line using a specially designed genetic algorithm, which has a number of advantages over conventional non-linear optimization techniques. The method can be used with any type of fuzzy model and is particularly useful when a direct fuzzy controller cannot be designed due to the complexity of the process and the difficulty in developing fuzzy control rules. The method is illustrated via the application to a non-linear single-input single-output reactor, where a Takagi–Sugeno model serves as a predictor of the process future behavior.

Motion synchronization for dual-cylinder electrohydraulic lift systems

This paper presents a nonlinear control algorithm to address the motion synchronization problem for a dual-cylinder electrohydraulic (EH) lift system. A two-step design approach is applied that utilized linear multiple input-multiple output (MIMO) robust control technique to design an outer-loop motion synchronization controller. A nonlinear single input-single output (SISO) perturbation observer-based pressure-force controller is designed for each of the lift cylinders as the inner-loop controller to handle the nonlinearities associated with the EH actuators. Experimental results on a 2-cylinder lift system verified the effectiveness of the proposed approach.

An automated driver based on convergent vector fields

This paper describes a new general framework for the action of an automated driver (or driver model) to provide the control of longitudinal and lateral dynamics of a road vehicle. The context of the problem is assumed to be in high-speed competitive driving, as in motor racing, where the requirement is for maximum possible speed along a track, making use of a reference path (racing line) but with the capacity for obstacle avoidance and recovery from large excursions. While not necessarily representative of a human driver, the analysis provides worthwhile insight into the nature of the driving task and offers a new approach for vehicle lateral and longitudinal control; it also has applications in less demanding applications such as Advanced Cruise Control systems. As is common in the literature, the driving task is broken down into two distinct subtasks: path planning and local feedback control. In the first of these tasks, an essentially geometric approach is taken here, which makes use of a vector field analysis. At each location x the automated driver is to prescribe a vector w for the desired vehicle mass centre velocity; the spatial distribution and global properties of w( x) provide essential information for stability analysis, as well as control reference. The resulting vector field is considered in the context of limited friction and limited mass centre accelerations, leading to constraints on ∇ w. Provided such constraints are satisfied, and using suitable adaptation of w( x) when required, it is shown that feedback control can be applied to guarantee stable asymptotic tracking of a reference path, even under limit handling conditions. A specific implementation of the method is included, using dual non-linear SISO (single-input single-output) controllers.

CONSTRAINED PREDICTIVE CONTROL OF A GREENHOUSE

A scheme for temperature control of a greenhouse is presented. The present work proposes an approach based on a combination of two different control schemes: Feedback Linearization (FL) and standard linear Model Predictive Control (MPC), using their advantages. The treated greenhouse is considered a nonlinear Single-Input Single-Output process and subject to strong external disturbances. Since the methodology used for solving the MPC+FL approaches generally leads to an optimization problem subject to state-dependent nonlinear constraints, an alternative for its implementation is discussed. Two control techniques are compared, namely MPC+FL and Nonlinear Model Predictive Control (NLMPC).

Transfer equivalence and realization of nonlinear higher order input–output difference equations

Two fundamental modelling problems in nonlinear discrete-time control systems are studied using the language of differential forms. The discrete-time nonlinear single-input single-output systems to be studied are described by input–output (i/o) difference equations, i.e. a high order difference equation relating the input, the output and a finite number of their time shifts. A new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case. Our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous-time systems. The second problem to be addressed is the realization problem. The i/o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization. Necessary and sufficient conditions are given for the existence of a (local) state-space realization of the irreducible i/o difference equation. These conditions are formulated in terms of the integrability of certain subspaces of one-forms, classified according to their relative degree. The sufficiency part of the proof gives a constructive procedure (up to finding the integrating factors and integration of the set of one-forms) for obtaining a locally observable and accessible state-space system. If the system is not in the irreducible form, one has first to apply the reduction procedure to transform the system into the irreducible form.

Irreducibility and Reduction of Discrete-Time Nonlinear Control Systems: An Alternative approach

The problem of irreducibility and system reduction is considered for discrete-time nonlinear single-input single-output systems described by high order input-output (i/o) difference equations. The two seemingly different irreducibility criteria, given in the literature, are shown to be the same except a specific subclass of nonlinear systems. It is suggested how to extend one of those to include the abovementioned subclass. Finally, it is demonstrated how to apply the Maple function univpoly to find the reduced system in case of the polynomial i/o equations.

Constrained Predictive Control of a Greenhouse

A scheme for temperature control of a greenhouse is proposed. The present work proposes an approach based on a combination of two different control schemes: Feedback Linearization (FL) and standard linear Model Predictive Control (MPC), using their advantages. The treated greenhouse is considered a nonlinear Single-Input Single-Output process and subjected to strong external disturbances. Since the methodology used for solving the MPC+FL approaches generally leads to an optimization problem subjected to state-dependent nonlinear constraints, an alternative for its implementation is discussed. Two control techniques are compared, namely; MPC+FL and Nonlinear Model Predictive Control (NLMPC).

Non-linear inverse compensation of an SI engine by system identification for robust performance control

Treating non-linear systems as equivalent time-invariant systems with associated uncertainty is often excessively conservative and significantly reduces the achievable robust performance. This paper presents a method for reducing the associated uncertainty of non-linear uncertain single-input single-output systems based on Horowitz's method for non-linear quantitative feedback theory. The new method implements a non-linear compensator based on an approximate causal inversion of an identified nominal model. The inverse system compensator reduces the uncertainty associated with the linear time-invariant equivalent system. The method is applied and experimentally validated on the idle speed dynamics of an automotive SI engine system using an electric dynamometer.

Transfer Equivalence of Nonlinear Higher Order Difference Equations

The problem of equivalence is considered for discrete-time nonlinear single-input single-output systems described by higher order input-output difference equations. A new definition of equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous time systems. Compared to the existing definition of equivalence of discrete-time nonlinear systems, it avoids many undesirable results.

A general canonical form for feedback passivity of nonlinear systems

In this article projection operators are used for obtaining a natural canonical form, for non-linear single-input single-output systems, which clearly exhibits the conservative, the dissipative and the locally destabilizing forces present in the drift vector field of a given system, for which a storage function is known. When the system's natural 'energy management' structure is revealed, a suitable state-dependent input coordinate transformation makes the system passive or strictly output passive. A direct application of the 'energy shaping plus damping injection' controller design method, whch is a well known controller design option for passive systems, may then be attempted. A bioreactor controller design example is presented, for illustrative purposes, which includes computer simulations.