Non-minimum Phase Nonlinear Systems Research Articles

Adaptive neural network output feedback stabilization of nonlinear non‐minimum phase systems

AbstractThis paper presents an adaptive output feedback stabilization method based on neural networks (NNs) for nonlinear non‐minimum phase systems. The proposed controller comprises a linear, a neuro‐adaptive, and an adaptive robustifying parts. The NN is designed to approximate the matched uncertainties of the system. The inputs of the NN are the tapped delays of the system input–output signals. In addition, an appropriate reference signal is proposed to compensate the unmatched uncertainties inherent in the internal system dynamics. The adaptation laws for the NN weights and adaptive gains are obtained using Lyapunov's direct method. These adaptation laws employ a linear observer of system dynamics that is realizable. The ultimate boundedness of the error signals are analytically shown using Lyapunov's method. The effectiveness of the proposed scheme is shown by applying to a translation oscillator rotational actuator model. Copyright © 2009 John Wiley & Sons, Ltd.

Asymptotic rejection of unmatched general periodic disturbances in a class of non-minimum-phase non-linear systems

This article deals with asymptotic rejection of unmatched general periodic disturbances in nonminimum phase non-linear output feedback systems. The steady state responses are defined for unstable systems subject to general periodic disturbances, and the generalised gain and phase are defined for stable systems subject to general periodic disturbances. Based on the new definitions, new results are obtained for the equivalent input disturbance and disturbance estimation. An L p -convergent estimate of the equivalent input disturbance is incorporated in the control design to ensure the asymptotic rejection of unmatched general periodic disturbances while maintaining the stability of the non-linear system.

Global Asymptotic Stabilization for Nonminimum Phase Nonlinear Systems Admitting a Strict Normal Form

In this paper, we address the problem of global asymptotic stabilization by output feedback for nonminimum phase nonlinear systems which admit a strict normal form. We assume the knowledge of an observer and, depending on its properties, we propose various approaches to design the control law. Each of these approaches needs a different stabilizability assumption on the inverse dynamics. In this way, within a unified framework, we recover and extend some already published results and we establish new ones.

Input–output feedback linearization for nonminimum phase nonlinear systems through periodic use of synthetic outputs

Input–output feedback linearization provides a convenient means of extending linear control strategies such as output zeroing or pole placement to the case of nonlinear affine in the input systems, but such extensions cannot be applied in the presence of nonminimum phase characteristics. This paper overcomes this difficulty through the periodic use of a finite number of synthetic outputs which are so constructed as to define embedded dynamics with stable zero dynamics. The efficacy of the method is demonstrated by means of a numerical example.

Adaptive global output feedback stabilisation of some non-minimum phase nonlinear uncertain systems

The output feedback stabilisation of some parameter-perturbed non-minimum phase nonlinear systems is considered. For such systems, it is shown that a robust global asymptotic stability can always be achieved via dynamic output feedback, if a global adaptive observer can be constructed and if the system can be rendered minimum phase with respect to a virtual output, which is a linear combination of state variables. This result generalises some existing approaches in the literature. A simulation example illustrates the effectiveness of the proposed method.

Output tracking for nonlinear non-minimum phase systems and application to PVTOL aircraft

In this paper, we consider a class of multi-input multi-output (MIMO) nonlinear non-minimum phase systems. For such nonlinear system, an output tracking control approach based on decomposition is presented and applied to a planar vertical takeoff and landing (PVTOL) aircraft. The external dynamics of the system can be separated into minimum-phase and non-minimum-phase parts. The internal dynamics of the system is separated to linear part and nonlinear part, and the linear part of the internal dynamics is only related with the non-minimum phase part of the external dynamics and the states of itself. Consequently, two low-dimensional controllers are designed separately to stabilize the two subsystems, and make the overall closed-loop system stable in spite of the existence of disturbances. Through this method, the design of the controller parameters is very simple. The theoretical results are confirmed by computer simulations.

Asymptotic Rejection of General Periodic Disturbances in Nonminimum-Phase Nonlinear Output-Feedback Systems

This paper deals with asymptotic rejection of unmatched general periodic disturbances in nonminimum phase nonlinear output feedback systems. The steady state responses are defined for unstable systems subject to general periodic disturbances, and the generalized gain and phase are defined for stable systems subject to general periodic disturbances. Based on the new definitions, new results are obtained for the equivalent input disturbance and disturbance estimation. A Lp-convergent estimate of the equivalent input disturbance is incorporated in the control design to ensure the asymptotic rejection of unmatched general periodic disturbances while maintaining the stability of the nonlinear system.

A new perspective on stable inversion of non-minimum phase nonlinear systems

Abstract We present a new perspective on the problem of stable inversion of nonlinear non-minimum phase systems. It is based on the notion of convergent systems. The machinery of convergent systems allows us to obtain novel qualitative and quantitative conditions for solving this problem. These conditions provide insight into the dynamics behind the stable inversion problem and make it possible to treat this problem in a non-local way. Qualitatively, they cover the conditions for the stable inversion of non-minimum phase nonlinear systems previously reported in literature and allow us to solve this problem for a broader class of systems. The proposed approach is supported with a novel computational method.

On-line references reshaping and control reconfiguration for non-minimum phase nonlinear fault tolerant control

In this paper we consider the problem of graceful performance degradation, for affine non-minimum phase nonlinear systems. The method is an optimization based scheme, that gives a constructive way to re-shape on-line the output reference for the post-fault system, and explicitly take into account the actuators and states saturations. The on-line output reference reshaping is associated with an on-line, MPC-based, controller reconfiguration, that forces the post-fault system to track the new output reference. The effect of FDD uncertainties on the online controller reconfiguration stability are studied, to ensure at least boundeness of the closed-loop system's states. The reshaping and reconfiguration schemes are applied to the Caltech ducted fan numerical example, which is described by a non-minimum phase nonlinear model.

Explicit construction of [formula omitted] control law for a class of nonminimum phase nonlinear systems

This paper addresses a nonlinear H ∞ control problem for a class of nonminimum phase nonlinear systems. The given system is first transformed into a special coordinate basis, in which the system zero dynamics is divided into a stable part and an unstable part. A sufficient solvability condition is then established for solving the nonlinear H ∞ control problem. Moreover, based on the sufficient solvability condition, an upper bound of the best achievable L 2 gain from the system disturbance to the system controlled output is estimated for the nonlinear H ∞ control problem. The proofs of our results yield explicit algorithms for constructing required control laws for solving the nonlinear H ∞ control problem. In particular, the solution to the nonlinear H ∞ control problem does not require solving any Hamilton–Jacobi equations. Finally, the obtained results are utilized to solve a benchmark problem on a rotational/translational actuator (RTAC) system.

Explicit constructions of global stabilization and nonlinear H∞ control laws for a class of nonminimum phase nonlinear multivariable systems

AbstractThis paper investigates a global stabilization problem and a nonlinear H∞ control problem for a class of nonminimum phase nonlinear multivariable systems. To avoid the complicated recursive design procedure, an asymptotic time‐scale and eigenstructure assignment method is adopted to construct the control laws for the stabilization problem and the nonlinear H∞ control problem. A sufficient solvability condition is established onthe unstable zero dynamics of the system for global stabilization problem and nonlinear H∞ control problem, respectively. Moreover, based on the sufficient solvability condition, an upper bound of the achievable L2‐gain is estimated for the nonlinear H∞ control problem. Copyright © 2007 John Wiley & Sons, Ltd.

A NEW PERSPECTIVE ON STABLE INVERSION OF NON-MINIMUM PHASE NONLINEAR SYSTEMS

We present a new perspective on the problem of stable inversion of nonlinear non-minimum phase systems. It is based on the notion of convergent systems. The machinery of convergent systems allows us to obtain novel qualitative and quantitative conditions for solving this problem. These conditions provide insight into the dynamics behind the stable inversion problem and make it possible to treat this problem in a non-local way. Qualitatively, they cover the conditions for the stable inversion of non-minimum phase nonlinear systems previously reported in literature and allow us to solve this problem for a broader class of systems. The proposed approach is supported with a novel computational method.

ADAPTIVE STABILIZATION OF NON-MINIMUM PHASE NONLINEAR SYSTEMS USING NEURAL NETWORKS

This paper, presents a direct adaptive control design method for uncertain nonlinear non-minimum phase systems. First, an appropriate reference signal is designed such that the internal dynamic subsystem is input-to-state practical stable. Then an output feedback control, which does not rely on the state estimation, is designed such that the output of system asymptotically tracks this reference signal. This controller is comprised of a dynamic linear controller an adaptive neural network and a discontinuous robustifying term. Stability of the overall system is guaranteed using the small gain theorem. The effectiveness of the proposed scheme is shown in simulations using a non-minimum phase non-affine system.

Precision preview-based stable-inversion for nonlinear nonminimum-phase systems: The VTOL example

This article quantifies the importance of the future desired trajectory in determining the exact-output-tracking input for nonlinear, nonminimum-phase systems by using system inversion techniques. It is intuitive that the effect of the desired output's distant-future values, on the output-tracking input at the current time instant, should be small. Therefore, at a current time instant ( t c ) , preview information of the desired output in a finite-time window [ t c , t c + T p ] should be sufficient to compute the output-tracking input with an arbitrarily small prescribed error, if the preview time T p is sufficiently large. The contribution of this article is the quantification of the needed preview time T p by using the benchmark VTOL aircraft model as an example. Additionally, simulation results are presented to evaluate the efficacy of the finite-preview-based stable-inversion approach.

Iterative learning control with advanced output data for nonlinear non-minimum phase systems

This paper investigates iterative learning control of nonlinear discrete time non-minimum phase systems in tracking problems. The main objective of this paper is to find an input-to-output mapping in order to stabilize the non-minimum phase systems and to obtain an input update law for handling uncertain systems. In conventional approaches on the tracking of non-minimum phase systems, zero dynamics is stabilized from the system equations and the input is calculated from the state information. For the learning of uncertain systems, conventional approaches depend on the output-to-state and state-to-input mappings. In the proposed method, the inverse system is stabilized using the input-to-output mapping for nonlinear non-minimum phase systems. A new input update law is proposed based on the relative degree and the number of non-minimum phase zeros. This makes the overall proposed learning system have a simple structure as in the classical ILC.

Adaptive estimation and rejection of unknown sinusoidal disturbances in a class of non-minimum-phase nonlinear systems

The author deals with adaptive estimation of unknown disturbances in a class of non-minimum phase nonlinear systems, and the stabilisation and disturbance rejection based on the estimated disturbances. The unknown disturbances are the combination of sinusoidal disturbances with unknown frequencies, unknown phases and amplitudes. The only information of the unknown disturbances is the number of distinctive frequencies inside. The class of nonlinear systems considered by the author consists of nonlinear systems in the output feedback form and the systems may be non-minimum phase, that is with unstable zero dynamics. An adaptive estimation algorithm is developed to give exponentially convergent estimates of the unknown disturbance and the system states. The asymptotic convergent estimates of unknown frequencies are also obtained. The proposed estimation algorithm works for both minimum phase and non-minimum phase nonlinear systems in output feedback form. Disturbance rejection with stabilisation is achieved by combining the control designed for the stabilisation of the disturbance-free system and the exponentially convergent estimate of the disturbance. Under the conditions specified for the control design of the system with no disturbance, the overall stability and complete rejection of the unknown disturbance are guaranteed by the proposed control algorithm.

A Constructive Approach to Local Stabilization of Nonlinear Systems by Dynamic Output Feedback

The note considers the problem of local stabilization of nonlinear systems by dynamic output feedback. A new concept, namely, local uniform observability of feedback control law, is introduced. The main result is that if a nonlinear system is Nth-order approximately stabilizable by a locally uniformly observable state feedback, then it is stabilizable by dynamic output feedback. Based on the approximate stability, a constructive method for designing dynamic compensators is presented. The design of the dynamic compensators is beyond the separation principle and can handle systems whose linearization might be uncontrollable and/or unobservable. An example of nonminimum phase nonlinear systems is presented to illustrate the utility of the results.

Adaptive output feedback control methodology applicable to non-minimum phase nonlinear systems

An adaptive output feedback control methodology is developed for a class of uncertain multi-input multi-output nonlinear systems using linearly parameterized neural networks. The methodology can be applied to non-minimum phase systems if the non-minimum phase zeros are modeled to a sufficient accuracy. The control architecture is comprised of a linear controller and a neural network. The neural network operates over a tapped delay line of memory units, comprised of the system's input/output signals. The adaptive laws for the neural-network weights employ a linear observer of the nominal system's error dynamics. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. Simulations of an inverted pendulum on a cart illustrate the theoretical results.

Global compensation of unknown sinusoidal disturbances for a class of nonlinear nonminimum phase systems

A class of output feedback stabilizable nonlinear systems with known output dependent nonlinearities and affected by unknown sinusoidal disturbances is considered: Nonminimum phase systems are also allowed. The problem of designing a global output feedback compensator which drives the state of the system exponentially to zero is solved when the disturbance consists of a known number of biased sinusoids with any unknown bias, magnitudes, phases, and frequencies.

Semi-global stabilisation of a class of non-minimum phase non-linear output-feedback systems

The stabilisation of non-linear systems with unstable zero-dynamics is considered. The non-linear systems are assumed to be transformable to the non-linear output feedback form. The control design starts from the state feedback, and the state variables associated with the zero-dynamics are stabilised first. The state feedback control design is then carried out using backstepping. An observer with linear error dynamics is constructed with a design parameter which can be set at any positive value. The dynamic output feedback control is then designed, by substituting the state in state feedback control with the observed one and imposing a saturation on the control input. The saturation level depends on the domain of the initial state. The proposed control design asymptotically stabilises the system for any prescribed domain of the initial state.