Single-input Single-output Nonlinear Systems Research Articles

Extremum seeking with sliding mode gradient estimation and asymptotic regulation for a class of nonlinear systems

This paper presents a model-based extremum seeking approach for a class of single-input–single-output nonlinear systems, with the analytic form of the performance function unknown a priori. We focus on a practically implementable design with robustness to model uncertainties and disturbances. A discrete-time sliding mode gradient estimator is developed for estimating the gradient of the performance profile. Based on the estimate, a variable structure output feedback regulator is proposed to enforce the system states toward the optimal trajectory. We analyze convergence conditions of the switching system toward a neighborhood of the optimal trajectory, and establish an ultimate bound on the size of the neighborhood. The robustness of the proposed controller is discussed with respect to measurement noise.

Indirect Adaptive Fuzzy Control for a Class of Uncertain Nonlinear Systems with Unknown Control Direction

In this paper, the authors propose two indirect adaptive fuzzy control schemes for a class of uncertain continuous-time single-input single-output (SISO) nonlinear dynamic systems with known and unknown control direction. Within these schemes, fuzzy systems are used to approximate unknown nonlinear functions and the Nussbaum gain technique is used to deal with the unknown control direction. This paper first presents a singularity-free indirect adaptive control algorithm for nonlinear systems with known control direction, and then this control algorithm is generalized for the case of unknown control direction. The proposed adaptive controllers are free from singularity, allow initialization to zero of all adjustable parameters of the used fuzzy systems, and guarantee asymptotic convergence of the tracking error to zero. Simulations performed on a nonlinear system are given to show the feasibility of the proposed adaptive control schemes.

Direct adaptive neural network controller for a class of nonlinear systems based on fuzzy estimator of the control error

A new approach of direct adaptive control of single input single output nonlinear systems in affine form using single-hidden layer neural network (NN) is introduced. In contrast to the algorithms in the literature, the weights adaptation laws are based on the control error and not on the tracking error or its filtered version. Since the control error is being expressed in terms of the NN controller, hence its weights updating laws are obtained via back-propagation concept. A fuzzy inference system (FIS) with heuristically defined rules is introduced to provide an estimate of this error based on the past history of the system behaviour. The stability of the closed loop is studied using Lyapunov theory. A fixed structure is then proposed for the FIS and the design parameters reduce to the parameters of the NN. The method is reproducible and does not require any pre-training of the network weights.

Adaptive Fuzzy Sliding Mode Control of Uncertain Nonlinear SISO Systems

In order to improve the performance of single-input single-output (SISO) nonlinear systems with uncertainties, an adaptive fuzzy sliding mode controller (AFSMC) that combines linearization feedback is presented in this paper. The fuzzy logic system is used to approximate the unknown system function and the AFSMC algorithm is designed by used of sliding mode control techniques. Based on the Lyapunov theory, the continuous function is designed to eliminate the chatting action of the control signal. The simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the global asymptotic stability if all the signals involved are uniformly bounded. Simulation results have shown that the proposed controller shows superior tracking performance. AFSMC can effectively achieve desired performance and have much more advantages over conventional SMC.

Design of self tuning fuzzy controllers for nonlinear systems

Nonlinearities present in the systems make their controller design a non-trivial task. The difficulty further increases in case of multi-input–multi-output (MIMO) systems with increased number of variables and interactions between them. In this paper, fuzzy based intelligent control schemes are designed for control of nonlinear single-input–single-output (SISO) and MIMO systems. The comparative study of the designed self tuning fuzzy controller with a standard Takagi–Sugeno (TS) fuzzy controller is discussed with application to a shell and tube heat exchanger (nonlinear SISO system) and a coupled two tank system (nonlinear MIMO system). Online tuning of the membership functions and control rules of fuzzy controller is carried out using simulated annealing (SA) to obtain improved performance by minimizing the error function. Experimental results demonstrate the effectiveness of the control scheme.

New approaching condition for sliding mode control design with Lipschitz switching surface

In this paper, we concern the approaching condition of sliding mode control (SMC) with a Lipschitz switching surface that may be nonsmooth. New criteria on the relation between phase trajectories and an arbitrary Lipschitz continuous surface are examined firstly. Filippov’s differential inclusion is adopted to describe the dynamics of trajectories of the closed-loop system with SMC. Compared with Filippov’s criteria for only smooth surface, new criteria are proposed by utilizing the cone conditions that allow the surface to be nonsmooth. This result also yields a new approaching condition of SMC design. Based on the new approaching condition, we develop the sliding mode controller for a class of nonlinear single-input single-output (SISO) systems, of which the switching surface is designed Lipschitz continuous for the nonsmooth sliding motion. Finally, we provide a numerical example to verify the new design method.

Observability analysis by Poincaré normal forms

This paper deals with quadratic equivalence, normal forms of observability, characteristic matrices and normal quadratic numbers for nonlinear Single-Input Single-Output systems. We investigated both cases: nonlinear systems linearly observable and nonlinear systems with one linear unobservable mode. Particularly, the effect of the normal quadratic numbers on the observer design is pointed out. Finally, a faster observability analysis is proposed using characteristic matrices and normal quadratic numbers. Throughout the paper, academic examples as well as bio-reactor example highlight our purpose.

An Iterative Learning Control for a Class of Partially Feedback Linearizable Systems

A class of single-input single-output nonlinear systems which are partially linearizable by state feedback is considered: feedback linearizable systems are included in such a class; no parametrization is required for the uncertainties which are required to satisfy the matching condition. Periodic reference signals with known period T are to be tracked by the output. Provided that known bounding functions on the uncertainties are available, a state feedback iterative learning control is designed which achieves asymptotic output tracking and guarantees bounded closed loop signals from any intial condition. The novel control tecnhique is illustrated for a single-link robot arm.

Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems

This work presents an adaptive fuzzy sliding mode controller (AFSMC) that combines a robust proportional integral control law for use in designing single-input single-output (SISO) nonlinear systems with uncertainties and external disturbances. The fuzzy logic system is used to approximate the unknown system function and the AFSMC algorithm is designed by used of sliding mode control techniques. Based on the Lyapunov theory, the proportional integral control law is designed to eliminate the chattering action of the control signal. The simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the global asymptotic stability in the Lyapunov sense if all the signals involved are uniformly bounded. Simulation studies have shown that the proposed controller shows superior tracking performance.

When does QP yield the exact solution to constrained NMPC?

It is well known that the optimal control sequence for a linear system with a quadratic cost and linear inequality constraints over a finite optimisation horizon can be computed by means of a quadratic programme (QP). The aim of this article is to investigate when the optimal control sequence for a non-linear single-input single-output system also can be computed via QP. Our main contribution is to show that the optimal control sequence for non-linear systems, with a quadratic cost and linear inequality constraints can be computed in exact form via QP provided the optimisation horizon is no larger than a critical quantity that we name the ‘input–output linear horizon’. The results do not require any linearisation technique and are applicable to general non-linear systems, provided their input–output linear horizon is larger than their relative degree.

Barrier Lyapunov Functions for the control of output-constrained nonlinear systems

In this paper, we present control designs for single-input single-output (SISO) nonlinear systems in strict feedback form with an output constraint. To prevent constraint violation, we employ a Barrier Lyapunov Function, which grows to infinity when its arguments approach some limits. By ensuring boundedness of the Barrier Lyapunov Function in the closed loop, we ensure that those limits are not transgressed. Besides the nominal case where full knowledge of the plant is available, we also tackle scenarios wherein parametric uncertainties are present. Asymptotic tracking is achieved without violation of the constraint, and all closed loop signals remain bounded, under a mild condition on the initial output. Furthermore, we explore the use of an Asymmetric Barrier Lyapunov Function as a generalized approach that relaxes the requirements on the initial conditions. We also compare our control with one that is based on a Quadratic Lyapunov Function, and we show that our control requires less restrictive initial conditions. A numerical example is provided to illustrate the performance of the proposed control.

Transformation the Nonlinear System into the Observer Form: Simplification and Extension

The paper addresses the problem of transforming the discrete-time single-input single-output nonlinear control system into the observer form using the state and output transformations. The necessary and sufficient solvability conditions are formulated in terms of differential one-forms, associated with the input-output equation of the control system. These conditions simplify the existing conditions [10] and extend them for the systems with inputs. The procedures to find the state and output transformations are given.

A fuzzy sliding mode control approach for nonlinear chemical processes

Fuzzy sliding mode control (FSMC) as a robust and intelligent nonlinear control technique is proposed to control processes with severe nonlinearity and unknown models. The performance of the proposed method has been evaluated for both single input single output (SISO) and MIMO nonlinear systems through its application in three severely nonlinear processes that are frequently used as benchmarks of nonlinear process control strategies. The evaluation shows that, despite its lack of dependence on the process model, the proposed method performs almost the same as conventional sliding mode control alternatives that utilize all the information that exists in the mathematical model of the process.

Indirect adaptive control of non-linear systems using multiple identification models and switching

Transient performance improvement of adaptive control of a class of single-input single-output (SISO) non-linear systems is considered in this study. The system under study is assumed to be minimum-phase and input-output linearisable. The non-linear dynamics is also assumed to be linearly parameterised in terms of the unknown parameters. The improvement in the transient performance under large parametric uncertainties is obtained with the use of multiple identification models and switching, and the closed loop system is shown to be stable with the switching mechanism. A new methodology is proposed for the quantitative evaluation of the transient performance. The study is verified by simulation of a non-linear physical system.

Robust limit cycle gradients

For nonlinear single-input single-output control systems, the limit cycle amplitude gradients and frequency gradients are derived. They are utilized for adjustments in size in order to satisfy the operating conditions in an industrial application. The linear plant is considered deteriorated by spherical uncertainties.

Design of a nonlinear controller based on a piecewise-linear Hammerstein model

This paper presents a nonlinear controller, which can be used to control single-input single-output nonlinear systems that can be approximated in terms of Hammerstein models. The proposed nonlinear controller is based on the principles of a classic linear pole placement controller and the piecewise-linear Hammerstein model. The controller can be used to control processes with highly nonlinear or even discontinuous static functions, while keeping simple controller structure and a very low computational burden.

Direct adaptive fuzzy control with a self-structuring algorithm

This paper presents a direct self-structuring adaptive fuzzy control (DSAFC) scheme for affine nonlinear single-input–single-output systems. We show that the only restriction on the control gain is that it be positive. No upper bound on this gain nor its derivative needs to be known. From an initial fuzzy system with a small number of rules, the self-structuring algorithm adds membership functions and rules when needed. To limit the size of the fuzzy system from growing indefinitely, the self-structuring algorithm replaces old membership functions by new ones instead of adding more membership functions so that the number of rules never exceeds a predefined upper bound. The stability of the closed loop system is guaranteed using the Lyapunov synthesis approach. The proposed control scheme is demonstrated by application to an inverted pendulum and a magnetic levitation system.

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.

STATE-SPACE REALIZATION OF NONLINEAR INPUT-OUTPUT DELTA DIFFERENTIAL EQUATIONS ON HOMOGENEOUS TIME SCALES

The paper studies the nonlinear control systems on time scales that serve as models of time and unify the continuous and discrete time. The realization problem for single-input single-output nonlinear delta differential systems is studied through the language of differential forms. Necessary and sufficient conditions are given for the existence of a state-space realization of the nonlinear input-output delta differential equation, i.e. a higher order delta differential equation relating the input, the output and a finite number of their delta-derivatives. The sufficiency part of the proof gives a constructive procedure (up to the integration of one-forms) for obtaining an observable state-space description.

DIRECT ADAPTIVE FUZZY CONTROL OF A CLASS OF NONLINEAR SYSTEMS WITH INPUT SATURATION

This paper presents a direct adaptive fuzzy control scheme for a class of uncertain single-input single-output nonlinear systems with input saturation. The fuzzy controller is designed to mimic an ideal controller which is assumed guarantying the control objective. The free parameters of the fuzzy controller are adjusted by an adaptive law which prevents the presence of input saturation from destroying the system stability. The stability properties of the proposed adaptive controller are obtained via Lyapunov analysis. A numerical simulation example is given to demonstrate the effectiveness of the proposed adaptive control approach.