Nontriangular Systems Research Articles

Simple But Useful Contributions to High-Gain Observer for Non-Triangular Systems

Simple But Useful Contributions to High-Gain Observer for Non-Triangular Systems

Predictor-Based Neural Dynamic Surface Control of a Nontriangular System With Unknown Disturbances

For a class of nontriangular nonlinear systems in presence of unknown disturbances, we propose a predictor-based neural dynamic surface control (PNDSC) strategy in this paper. This nontriangular system is transformed via the mean value theorem, and a predictor is then constructed. To avoid an algebraic loop problem, partial state vectors are employed as input signals of neural networks (NNs) for approximating unknown dynamics, and compensation items are designed to compensate for approximation errors from NNs. Different from the traditional NDSC, the PNDSC in this paper utilizes prediction errors to update learning parameters for improving NNs’ learning behaviors with overlarge adaptive gains. On the basis of improved NNs’ approximation behaviors, a predictor-based NNs disturbance observer (PNNDO) is constructed for compensation for external disturbances and approximation errors from NNs. Furthermore, with predictors, a normalization method of weights is developed to reduce the number of online learning parameters. On the basis of the aforementioned result, measurement noises are taken into account in our predictor-based neural control strategy. We employ predictor states, rather than measurement information paralyzed by noises, in design of our control strategy. This reduces high-frequency oscillations in control input. A Lyapunov-based stability analysis shows that all signals are ultimately bounded in the closed-loop system. Finally, the effectiveness of the proposed control strategy is verified by a numerical example and a permanent magnet brushless DC motor system.

Dynamical Adaptive Integral Sliding Backstepping Control of Nonlinear Nontriangular Uncertain Systems

We present a control strategy for nonlinear nontriangular uncertain systems. The proposed control method is a synergy between the dynamic adaptive backstepping (DAB) and integral sliding mode (ISM) and is referred to as DAB-ISMC. Our main objective is to find a recursive procedure to transform a nontriangular system into an implementable form that enables designing a control law which almost eliminates the reaching-phase. The proposed method further facilitates minimization of chattering which is believed to be a shortcoming of the sliding mode control. In this methodology, the ISM, as an integrated subsystem of DAB, is introduced at the final stage of backstepping. This strategy works very well to obtain a system that is robust against model imperfections, matching and unmatching uncertainties. The DAB-ISMC method is applied on a continuous stirred tank reactor (CSTR) and simulation results obtained on Matlab are found to be very promising.

A Dynamic Scaling Based Control Redesign Procedure for Uncertain Nonlinear Systems with Input Unmodeled Dynamics

The global asymptotic stabilization problem for a general class of uncertain nonlinear systems with dynamic input uncertainties (input unmodeled dynamics) is considered. The system structure addressed in this paper is comprised of a nominal system (which can have a general nonlinear dynamics structure and is assumed to have a nominal control law to stabilize the nominal system) and an appended input unmodeled dynamics. The proposed approach is based on a control redesign utilizing the nominal control law and incorporating a singular-perturbation-like dynamic system extension along with a dynamic scaling to address the dynamic input uncertainties. The control input uncertainty structure is modeled through an uncertain non-affine function of the control input and the unmeasured state of an uncertain input unmodeled dynamics subsystem. The proposed control redesign approach is applicable to a wide class of nonlinear systems including triangular as well as non-triangular system structures as long as a set of structural and inequality conditions on the system dynamics are satisfied and provides a global robust output-feedback stabilizing controller.

A high gain observer with updated gain for a class of MIMO non-triangular systems

A high gain like observer with updated gain is proposed for a class of cascade nonlinear and non triangular systems that are observable for any input. The objective of the gain adaptation is to perform an admissible tradeoff between state reconstruction dynamics on the one hand versus noise amplification on the other hand. To this end, the gain of the proposed observer is tuned through a single scalar time-varying parameter governed by an adequate differential Riccati equation. The involved adaptation process is mainly driven by the power of the output observation error norm computed on a moving horizon window. Simulation results are given to show the effectiveness of the proposed observer, namely its exponential convergence and its insensitivity with respect to noise measurements.

High gain observer for a class of non-triangular systems

This paper presents a high gain observer for a class of MIMO nonlinear systems involving some uncertainties. The latter is particularly composed of cascade subsystems where each subsystem is associated with a subset of the output variables, and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems. The main contribution consists in extending the available results to allow more interconnections between the subsystems. Of fundamental interest, it is shown that the underlying observation error exponentially converges to zero in the absence of uncertainties. Moreover, the observation error can be made as small as desired by properly specifying the observer design parameter in the case where uncertainties are considered.

Generalized State Scaling and Applications to Feedback, Feedforward, and Nontriangular Nonlinear Systems

We propose a global high-gain scaling-based state-feedback controller for a general class of nonlinear systems containing uncertain functions of all the states and the control input as long as polynomial bounds on ratios of some uncertain system terms are available. The design is based on a high gain scaling involving appropriate powers of a high gain scaling parameter which is a dynamic signal driven by the state. The design is applicable to both lower triangular (strict-feedback) and upper triangular (feedforward) systems, and also to nontriangular systems as long as a set of linear inequalities involving powers of the polynomial bounds on the ratios of uncertain system terms and scaling orders is solvable. The stability analysis is based on our recent results on uniform solvability of coupled state-dependent Lyapunov inequalities

Robust disturbance attenuation of nonlinear systems using output feedback and state-dependent scaling

This paper presents an approach to output feedback stabilization with L 2 disturbance attenuation for nonlinear systems in the presence of dynamic uncertainties. A new formulation of state-dependent scaling is introduced into the output feedback design, which unifies treatment of nonlinear and linear gains. The effect of disturbance on the controlled output, which is allowed to be any function of output measurements, can be always attenuated to an arbitrarily small level with global asymptotic stability if the plant belongs to a wide class of triangular systems whose uncertainties do not necessarily have finite linear-gains. The uncertain dynamics is not limited to input-to-state stable systems either. The approach is not only a natural extension of popular approaches in robust linear control, but also advantageous to numerical computation which is applicable to non-triangular systems as well as triangular systems.

Towards the complete classification of homogeneous two-component integrable equations

In this article we suggest an improved method for classifying general two-component integrable evolution equations, homogeneous in a given weighting scheme. This method relies on linear changes of variables and on an appropriate splitting of the solution space. To illustrate the method, we implement the classification of coupled KdV-type equations. We show that there are five nontriangular systems possessing higher order generalized symmetries. One of these systems is previously unknown. This seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme, without any restrictions on the form of the main matrix.

Local stabilization and controllability of a class of non-triangular nonlinear systems

This paper considers the problems of local asymptotic stabilization using continuous static state feedback (LCFS) and small time local controllability (STLC) of single-input nonlinear systems. A specific class of nontriangular systems, called the essentially triangular form, is introduced. For this class of systems sufficient conditions for LCFS and necessary and sufficient conditions for STLC are obtained. In particular, it is shown that STLC implies LCFS. A motivating physical example of the ETF system is presented.

Maple Design of Dynamical Sliding Adaptive Backstepping Controllers

BACK, a Maple symbolic algebra package, is described for the design of dynamical adaptive nonlinear controllers for regulation and tracking tasks of a class of observable minimum phase uncertain nonlinear systems. This package also allows the design of non-adaptive controllers for systems without uncertainty, and adaptive sliding mode controllers (SMC) of first and second order, to provide robustness in the presence of disturbances. The design procedure employs the basic ideas of the adaptive backstepping algorithm with tuning functions via input-output linearization, and is applicable to both triangular and nontriangular systems. Recent algorithms developed for semi-strict nonlinear systems with disturbances, have been included.

Moment stability of linear stochastic non-triangular large-scale systems

The second-moment global exponential stability of large-scale stochastic systems is presented in this paper. The particular class of non-triangular linear parametric stochastic systems is considered. For this class, the second moments are treated as the new state space and the standard procedure of analysis of large-scale deterministic systems is applied. An example is given to illustrate the criteria.

The Causal Interpretation of Non-Triangular Systems of Economic Relations

The Causal Interpretation of Non-Triangular Systems of Economic Relations

On the Causal Interpretation of Non-Triangular Systems of Economic Relations: A Rejoinder

The Strotz-Wold Reply, however, penetrates to the heart of the issue, almost, and it will probably be illuminating to make a rejoinder. To begin with I should like to say that they are justified in taking exception to the unfortunate wording of the headnote. Though, when applied to the sequence of recipes given out by Wold, my remarks do not lack for verisimilitude, they are certainly not true to the letter of the Strotz-Wold article, and I should have taken pains not to convey the impression that I thought they were.2 Accepting their just remonstrance, I turn to the text of my article. The main contention of my paper was that the classical scientific notion of causality (not, however, the doctrine of cause and effect) 3 is adequate to rationalize the construction of non-triangular or interdependent mechanical models in economics; that employment of the classical notion in the case of non-triangular systems neither leads to paradox nor calls for novel categories of causality, e.g., circular causality, bi-causality, vector causality,