Asymptotic Output Tracking Problem Research Articles

Stochastic Relative Degree and Path-Wise Control of Nonlinear Stochastic Systems

In this article, we address the path-wise control of systems described by a set of nonlinear stochastic differential equations. For this class of systems, we introduce a notion of stochastic relative degree and a change of coordinates, which transforms the dynamics to a stochastic normal form. The normal form is instrumental for the design of a state-feedback control, which linearizes and makes the dynamics deterministic. We observe that this control is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">idealistic</i> , i.e <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">.,</i> it is not practically implementable because it employs a feedback of the Brownian motion (which is never available) to cancel the noise. Using the idealistic control as a starting point, we introduce a hybrid control architecture, which achieves <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">practical</i> path-wise control. This hybrid controller uses measurements of the state to perform periodic compensations for the noise contribution to the dynamics. We prove that the hybrid controller retrieves the idealistic performances in the limit as the compensating period approaches zero. We address the problem of asymptotic output tracking, solving it in the idealistic and in the practical framework. We finally validate the theory by means of a numerical example.

Output tracking for a class of non-minimum phase nonlinear systems: A two-point boundary value problem formulation with a hybrid regulator

The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates the prescribed periodic reference signal. The design strategy is illustrated in two steps. First, it is shown that the knowledge of a solution to a certain two-point boundary value problem involving the underlying zero-dynamics of the plant is instrumental and sufficient to construct a state feedback regulator that achieves boundedness of the trajectories and (exact) asymptotic tracking, hence completely circumventing the need for solving partial differential equations. Then, since the computation of the latter solution may be affected by numerical errors that are particularly detrimental in the presence of unstable zero-dynamics, the above architecture is robustified by means of an additional hybrid feedback loop whose trajectories converge to a solution of the two-point boundary value problem. Once the latter scheme has been established and discussed, the extension to the case of output feedback is presented, firstly in the specially structured case in which the input vector field depends only on the measured output and then extended to the generic case. The theory is then corroborated by means of a physically motivated numerical example involving an inverted pendulum on a cart. Interestingly, it is also shown that the solution provided by the hybrid scheme above coincides with the limiting solution of a suitably defined cheap optimal control problem.

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

SummaryThis paper addresses the problem of asymptotic output tracking of a class of semilinear parabolic equations with pointwise in‐domain actuation. First, the assessment of the well‐posedness of the considered systems is performed, and then, the stability of boundary controlled systems is analyzed via Chaffee‐Infante equation and Fisher's equation. The application of the zero dynamics inverse design results in a dynamic control scheme that is implemented by using the technique of trajectory planning for flat systems and the Adomian decomposition method. The convergence of the solution of the original systems to that of the corresponding zero dynamics and the convergence of the solution expressed by an Adomian series are also analyzed. Numerical simulations are carried out to illustrate the effectiveness of the developed approach.

Adaptive output tracking of linear uncertain discrete-time networked systems over switching topologies

ABSTRACTIn this paper, we study the asymptotic output tracking problem for a class of minimum phase uncertain linear discrete-time multi-agent systems over jointly connected digraphs. As the systems contain arbitrarily large uncertain parameters, the robust feedback control technique does not work. We develop a distributed adaptive control law composed of a distributed observer and a purely decentralised adaptive control law. We first establish a stability result for a class of linear switched systems, which extends the existing results. This stability result leads to a novel distributed observer for the leader system which estimates the leader's signal using the output of the leader system locally and passes the estimated leader's signal to each follower. We then further show that the distributed adaptive control law solves our problem under some mild conditions. An example is used to illustrate the effectiveness and generality of our approach.

Optimality principles and decomposition of tracking controllers for weakly dual redundant systems

SummaryIn this paper, the problem of optimizing the output regulation of a weakly dual redundant plant is addressed. When the system is underactuated, only a subset of the outputs can be arbitrarily controlled and the remaining ones are constrained. With a specific focus on the asymptotic output tracking problem for single‐input systems, we investigate the connection between the overall optimal input and the individually optimal controllers, which leads to the perfect tracking of each output reference. Such relationship is analyzed both for open‐loop and closed‐loop controllers for different cost functions and types of references, and the optimality principles are established, provided that suitable structural conditions are met by the plant matrices.

NONLINEAR OBSERVERS APPLIED TO THE CONTROL OF AN OVERHEAD CRANE

This paper addresses the asymptotic output tracking problem for the model of an overhead crane. It is shown that the model is linearizable by dynamic state feedback when the full state is available for measurement. In the more realistic situation when the angle that the rope forms with respect to the vertical axis cannot be measured, two different nonlinear observers are used in order to estimate the actual value. These are: a numerical observer and a high gain observer. Numerical experiments serve to compare the performance of the proposed schemes.

Tracking, Model Matching, Disturbance Decoupling for a Class of Nonlinear Delay Systems

The problem of asymptotic output tracking for nonlinear delay systems is considered in this paper, and a solution is proposed for nonlinear minimum phase delay systems. A theorem is proved which assures the boundedness of the state variables for a class of nonlinear delay systems, when the reference output and its derivatives up to a given order are bounded. Then, the problem of disturbance decoupling is considered for nonlinear delay systems, both in the cases of known and unknown disturbances. A solution is proposed for minimum phase delay systems, under suitable assumptions on the delay relative degrees. Conditions are given that assure boundedness of the state variables. Simulation results on nonlinear delay systems unstable in open loop are also reported showing the effectiveness of the proposed methods.

Asymptotic output tracking of nonlinear differential-algebraic control systems

The problem of asymptotic output tracking is considered for control systems described by a class of nonlinear differential-algebraic equations. Assumptions are introduced, and an algorithm is developed. Based on the algorithm, a nonlinear feedback control law is then constructed which ensures, under appropriate conditions, that the closed-loop system has a unique bounded differentiable solution and the tracking error in the closed-loop system approaches zero when time goes to infinity.

Asymptotic Output Tracking for a Class of Nonlinear Systems with Singular Points

The problem of asymptotic output tracking in the presence of singular points is addressed. A new index called the extended degree of singularity is introduced. A change of coordinates which allows us to make a study of the Asymptotic Output Problem for Nonlinear Systems with singular points is proposed. Finally, an example which illustrates the concepts is given.

Asymptotic Output Tracking for a Large Class of Uncertain SISO Nonlinear Systems: Stability

The problem of asymptotic output tracking in the presence of uncertainties is reexamined. A characterization of the class of uncertainties dealt with in this study is given. A study of the asymptotic output problem plus stability for the class of uncertainties allowed is also discussed. Then we proceed to make a generalization to the case when there are singular points. Finally an example that illustrates the concepts proposed is presented.

Nonlinear Adaptive Control of a Chemical Reactor Using Singular Perturbation Techniques

The problem of controlling a class of nonlinear singularly perturbed systems through its decomposition into two lower order reduced systems is addressed. In particular, the problem of asymptotic output tracking is solved. An adaptation law is also utilized in order to compensate model parametric uncertainties. The proposed methodology is shown to be suitable in dealing with a chemical reactor. In particular, regulation of the output concentration is achieved in an open-loop unstable stationary point.