Well-defined Relative Degree Research Articles

Output stabilization for a class of nonlinear systems via high-gain observer with limited gain power

We address the problem of output stabilization for the class of nonlinear minimumphase systems with a well-defined relative degree whose output is affected by measurement noise. In the proposed control structure the derivatives of the output are estimated by means of a new high-gain observer proposed in Astolfi and Marconi [2015], which, with respect to high-gain observers conventionally used in literature, has limited gain power. The advantage in using the proposed observer is in its implementation for systems with high relative degree and in the sensitivity to the sensor noise. Simulation results are presented to show the effectiveness of the proposed control structure.

Left inversion of analytic nonlinear SISO systems via formal power series methods

Given a single-input, single-output (SISO) system, F, and a function y in the range of F, the left inversion problem is to determine an input u such that y=F[u]. The goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where F is an analytic mapping in the sense that it has a convergent Chen–Fliess functional expansion, and y is a real analytic function. In particular, it will be shown that given a certain condition on the generating series c of F, a corresponding unique analytic u can always be determined via operations on formal power series. The condition on c turns out to be equivalent to having a well-defined relative degree when F has an input-affine analytic state space realization with finite dimension. But the method is applicable even when F does not have such a realization. The technique is demonstrated on four examples, including a continuous stirred chemical reactor.

Robust control strategies for a class of nonlinear systems with an ill-defined relative degree

Differential geometry theory has become one of the most promising design tools for nonlinear processes in the past decade. Several differential geometry techniques, such as input–output linearization and input-state linearization, have received considerable attention in the literature. However, one of the main assumptions often needed in the design of nonlinear controller by means of differential geometry is a nonlinear system with a well-defined relative degree. Nonlinear systems with an ill-defined relative degree will have an input–output linearizing law with singularities. This problem restricts the application of the feedback linearization technique in many engineering systems. One approach to this problem is to use approximate linearization techniques but this tends to yield very sluggish responses. In this paper, we investigate the applicability of a composite control scheme consisting of heuristic-based approximate linearization and backstepping design for a nonlinear system with an ill-defined relative degree. Two procedures are used to deal with this problem. In the first step, a function approximates the Lie derivative of the process output along with the input function to avoid the occurrence of singularity. Then, a nonlinear controller with an integral action-based back-stepping design is proposed to ensure that both a stable process and improved performance are obtained. Numerical simulations that illustrate the effectiveness of this approach are given.

Robust adaptive learning control for nonlinear systems with extended matching unstructured uncertainties

SUMMARYThe exponential output tracking problem for a class of single‐input, single‐output uncertain nonlinear systems, including systems with extended matching unstructured uncertainties and without a well‐defined global relative degree, is addressed. Conditions on the uncertain system dynamics are derived, which allow us to design a state‐feedback learning control achieving semi‐global exponential output tracking of sufficiently smooth and periodic reference signals of known period, while guaranteeing ℒ︁2 and ℒ︁∞ transient performances during the learning phase. The application of the proposed learning approach to the position tracking control problem for uncertain permanent magnet step motors with non‐sinusoidal flux distribution and uncertain position‐dependent load torque allows us to provide a solution to a yet unsolved problem. Copyright © 2011 John Wiley & Sons, Ltd.

Path following controller design for a class of mechanical systems

AbstractWe present an approach to designing feedback controllers that solve a path following control problem for a large class of mechanical systems. Our approach involves set stabilization via transverse feedback linearization: a coordinate and feedback transformation that yields a normal form convenient for control design making desired path attractive. Our approach also involves refining the normal form by imposing additional structure on the system, such that the dynamics governing the output motion along the path are linear and controllable. This approach amounts to local input-output feedback linearization for a particular choice of “virtual output”. We present necessary and sufficient checkable conditions for the particular virtual output to yield a well-defined relative degree. When these conditions hold, it is possible to put the mechanical system in the desired normal form, simplifying path following controller design.

Adaptive path following controller of underactuated ships using serret-frenet frame

This paper presents an adaptive path following control law to steer underactuated ships along a predefined path at a constant forward speed with uncertain parameters due to changes of added mass matrices. The proposed controller is based on analytic model predictive control and model reference adaptive control. The Serret-Frenet frame is used to describe the ship dynamics. The analytic model predictive control provides a systematic method rather than try-and-error method to get appropriate control parameters to guarantee the stability of the closed-loop system, and the well-defined relative degree is guaranteed by introducing output-redefinition. An identification algorithm based on model reference adaptive control is used to identify the uncertain parameters. Numerical simulations are provided to demonstrate the validity of the proposed control law.

Path following control of underactuated ships based on unscented Kalman filter

A path following controller is proposed to force an underactuated surface ship which is suffering from disturbance to follow a predefined path. The controller is based on analytic model predictive control and unscented Kalman filter (UKF) techniques. The analytic model predictive control provides a systematic method to get appropriate controller parameters to guarantee the stability of the closed-loop system, and the well-defined relative degree is guaranteed by introducing output-redefinition. The UKF is used to estimate the states and uncertain parameters due to time-varying added mass matrices. With help of the proposed UKF-based controller, the underactuated ship with time-varying parameters can follow a desired straight path. Simulation results are presented to demonstrate the effectiveness of the proposed controller.

A global state feedback output regulating control for uncertain systems in strict feedback form

A family of single-input, single-output, nonlinear systems in strict feedback form with uncertain constant parameters which appear linearly and belong to a known compact set Π is considered. A global robust output regulation problem via state feedback is addressed and solved under the assumptions that the regulator equations are solvable in Π , the output equation does not depend on uncertain parameters, no modelled disturbances affect the system and the output reference signal is generated by a known exosystem whose state is bounded and available for feedback. Robust adaptive techniques are used to guarantee closed loop boundedness and to achieve, without requiring persistency of excitation, global asymptotic output regulation for a class of feedback linearizable systems which may have unbounded uncertain tracking dynamics or may not have a well-defined global relative degree.

Incorporating a class of constraints into the dynamics of optimal control problems

AbstractA method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well‐defined relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions. Copyright © 2009 John Wiley & Sons, Ltd.

Approximate Adaptive Output Feedback Stabilization via Passivation of MIMO Uncertain Systems Using Neural Networks

An adaptive output feedback neural network controller is designed, which is capable of rendering affine-in-the-control uncertain multi-input-multi-output nonlinear systems strictly passive with respect to an appropriately defined set. Consequently, a simple output feedback is employed to stabilize the system. The controlled system need not be in normal form or have a well-defined relative degree. Without requiring a zero-state detectability assumption, uniform ultimate boundedness, with respect to an arbitrarily small set, of both the system's state and the output is guaranteed, along with boundedness of all other signals in the closed loop. To effectively avoid possible division by zero, the proposed adaptive controller is of switching type. However, its continuity is guaranteed, thus alleviating drawbacks connected to existence of solutions and chattering phenomena. Simulations illustrate the approach.

On Robust Output Based Finite-Time Control of LTI systems using HOSMs

Abstract Finite-Time stability of Linear Time Invariant systems with matched perturbations using dynamic output feedback is achieved under the assumptions of well-defined relative degree and a known bound of the perturbations. The approach is based on high order sliding modes, using global controllers and differentiator. A separation criteria that allows to detect the convergence of the differentiator and posterior gain adaptation is presented. Analysis of the performance under noise and sampling is presented.

Adding robustness to nominal output-feedback controllers for uncertain nonlinear systems: A nonlinear version of disturbance observer

The linear disturbance observer (DOB) approach has been successfully employed as a tool for robust control and disturbance rejection in practice. In this paper, we present a nonlinear DOB which inherits all the benefits of the classical linear DOB approach. In particular, we prove that the proposed nonlinear DOB recovers not only the steady-state performance but also the transient performance of the nominal closed-loop system under plant uncertainties and input disturbances. One novel feature of the proposed controller is that it is an add-on type inner-loop output-feedback controller; thus any type of outer-loop controller can be combined to enhance the closed-loop system performance. Our result is a semi-global one and the stability proof is given when the nonlinear system has a well-defined relative degree with a stable zero dynamics.

Sliding Mode Control and Feedback Linearization for Non-regular Systems

This paper presents an approach for sliding mode control(SMC) and feedback linearization(FBL) of systems with relative order singularities. Traditionally, SMC and FBL are designed by taking derivatives of the output until the control signal appears(at the rth1 derivative). When a system does not have a well-defined relative degree (r1), the coefficient that multiplies u vanishes for some region of the state-space S1. In this instance, conventional SMC and FBL techniques fail. The presented approach differentiates further the output until the control input appears again (the secondary relative degree, r2) and a differential equation in u is acquired. It may be possible to solve for a dynamic compensator, or in the neighborhood of the singularity, N1, the equations degenerate to a polynomial form. Preliminary results show that at the singularity region, S1, the control-derivative term disappears and the differential equation is degenerated to a center manifold defined by a polynomial (quadratic in general) equation on u. The solution to the quadratic equations is discussed. When this equation has only real roots, the system is well defined at the singularity. A switching controller can be designed to switch from the rth1 controller when system is far away from the singularity to the rth2 controller when the system is in the neighborhood of the singularity. We demonstrate the controller applied to the ball and beam system.

Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators☆

This paper is on the design of robust output feedback integral control for minimum-phase nonlinear systems with a well-defined relative degree. Previous work has shown how to design such controllers to achieve asymptotic regulation by a two-step process. First, robust control is designed to bring the trajectories to a small neighborhood of an equilibrium point. Within this neighborhood, the control then acts as a high-gain feedback that stabilizes the equilibrium point. The asymptotic regulation achieved by integral action happens at the expense of degrading the transient performance. In this paper, we present an approach to improve the transient performance. The control design is a continuous sliding mode control with integral action. However, the integrator is introduced in such a way that it provides integral action only “conditionally”, effectively eliminating the performance degradation . There are two main results in the paper: the first is asymptotic regulation and the second confirms the transient performance improvement by showing that the output feedback continuous sliding-mode control with integral action can be tuned to recover the performance of a state feedback ideal sliding mode control without integral action.

Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators

This paper is on the design of robust output feedback integral control for minimum-phase nonlinear systems with a well-defined relative degree. Previous work has shown how to design such controllers to achieve asymptotic regulation by a two-step process. First, robust control is designed to bring the trajectories to a small neighborhood of an equilibrium point. Within this neighborhood, the control then acts as a high-gain feedback that stabilizes the equilibrium point. The asymptotic regulation achieved by integral action happens at the expense of degrading the transient performance. In this paper, we present an approach to improve the transient performance. The control design is a continuous sliding mode control with integral action. However, the integrator is introduced in such a way that it provides integral action only “conditionally”, effectively eliminating the performance degradation. There are two main results in the paper: the first is asymptotic regulation and the second confirms the transient performance improvement by showing that the output feedback continuous sliding-mode control with integral action can be tuned to recover the performance of a state feedback ideal sliding mode control without integral action.

Identification and control of a nonlinear discrete-time system based on its linearization: a unified framework.

This paper presents a unified theoretical framework for the identification and control of a nonlinear discrete-time dynamical system, in which the nonlinear system is represented explicitly as a sum of its linearized component and the residual nonlinear component referred to as a "higher order function." This representation substantially simplifies the procedure of applying the implicit function theorem to derive local properties of the nonlinear system, and reveals the role played by the linearized system in a more transparent form. Under the assumption that the linearized system is controllable and observable, it is shown that: 1) the nonlinear system is also controllable and observable in a local domain; 2) a feedback law exists to stabilize the nonlinear system locally; and 3) the nonlinear system can exactly track a constant or a periodic sequence locally, if its linearized system can do so. With some additional assumptions, the nonlinear system is shown to have a well-defined relative degree (delay) and zero-dynamics. If the zero-dynamics of the linearized system is asymptotically stable, so is that of the nonlinear one, and in such a case, a control law exists for the nonlinear system to asymptotically track an arbitrary reference signal exactly, in a neighborhood of the equilibrium state. The tracking can be achieved by using the state vector for feedback, or by using only the input and the output, in which case the nonlinear autoregressive moving-average (NARMA) model is established and utilized. These results are important for understanding the use of neural networks as identifiers and controllers for general nonlinear discrete-time dynamical systems.

Initial shift issues on discrete-time iterative learning control with system relative degree

This paper deals with the initial shift problem that arises from discrete-time iterative learning control. A unified learning scheme is considered for a class of nonlinear systems with well-defined relative degree, which adopts the error data with anticipation in time and provides wider freedom for the updating law formation. The sufficient convergence condition is derived to enable the system to possess asymptotic tracking capability and the converged output trajectory can be assessed by the initial condition. The tracking performance is improved further by the introduction of initial rectifying action and the complete tracking is achieved over a specified interval.

Improved Performance of Universal Integral Regulators

The paper is concerned with the design of an integral control for minimum-phase nonlinear systems that have a well-defined relative degree. Earlier work by the author and coworkers has shown how integral control can be combined with robust control techniques to achieve nonlocal and semiglobal regulation results. The results are applicable to output feedback control problems through the use of high-gain observers and saturation nonlinearities. When used with sliding-mode control, the technique produces universal integral regulators that have been shown to coincide with PI and PID controllers, followed by limiters, for relative-degree-one systems and relative-degree-two systems. The use of integral action improves the steady-state response at the expense of degrading the transient response. The present paper summarizes two recent techniques to modify the universal integral regulator so as to improve its transient performance. One technique uses antiwindup integration, while the other uses nonlinear integrators.

Closed‐loop iterative learning control for non‐linear systems with initial shifts

AbstractThis paper is concerned with the problem of the iterative learning control with current cycle feedback for a class of non‐linear systems with well‐defined relative degree. The tracking error caused by a non‐zero initial shift is detected as extended D‐type learning algorithm is applied. The defect is overcome by adding terms including the output error, its derivatives as well as integrals. Asymptotic tracking of the final output to the desired trajectory is guaranteed. As an alternative approach, an initial rectifying action is introduced in the extended D‐type learning algorithm and shown effective to achieve the desired trajectory jointed smoothly with a transitional trajectory from the starting position. Also these algorithms with adjustable tracking interval ensure better robustness performance in the presence of initial shifts. Numerical simulation is conducted to demonstrate the theoretical results. Copyright © 2002 John Wiley & Sons, Ltd.

Analysis of Nonlinear Discrete-Time Systems with Higher-Order Iterative Learning Control

In this paper, an iterative learning control method is proposed for a class of nonlinear discrete-time systems with well-defined relative degree, which uses the output data from several previous operation cycles to enhance tracking performance. A new analysis approach is developed, by which the iterative learning control is shown to guarantee the convergence of the output trajectory to the desired one within bound and the bound is proportional to the bound on resetting errors. It is further proved effective to overcome initial shifts and the resultant output trajectory can be assessed as iteration increases. Numerical simulation is carried out to verify the theoretical results and exhibits that the proposed updating law possesses good transient behavior of learning process so that the convergence speed is improved.