Nonlinear Output Regulation: A Unified Design Philosophy

Introduction: the basic ingredients of asymptotic output regulation the computation of the steady-state response highlights of output regulation for linear systems. Output regulation of nonlinear systems: the regulator equations the internal model necessary and sufficient conditions for local output regulation the special case of harmonic exogenus inputs approximate regulation. Existence conditions for regulator equations: linear regulator equations and transmission zeros the zero dynamics of a nonlinear system existence of solutions of the nonlinear regulator equations. Robust output regulation: structurally stable local regulation robust local regulation the special case of systems in triangular form a globally defined error-zeroing invariant manifold semiglobal robust regulation.

This paper deals with output feedback stabilisation of a class of nonlinear systems with nonlinearity of the unmeasured state variables. The nonlinearity of the unmeasured state complicates the zero dynamics of the nonlinear system and makes the control design more difficult. In this paper, the zero dynamics analysed for the class of nonlinear systems with respect to the original output and then with respect to a virtual output. It is demonstrated that nonlinear systems with unstable zero dynamics may have stable zero dynamics with the virtual output. Based on the analysis, conditions are then identified for nonlinear systems which can be stabilised by control design with estimated state, including the estimated virtual output, and this removes a restriction of a recent result that requires the transformed system has no zero dynamics. Examples are included to demonstrate that the proposed control design can be used to stabilise nonlinear systems with unstable nonlinear zero dynamics.

In recent years, the tracking problem of spherical inverted pendulum (SIP) system with multi-input, multi-output nature and unstable zero dynamics has been well addressed based on continuous-time nonlinear output regulation (NOR) theory. For the convenience of digital implementation, this paper further investigates the approximate NOR problem of the SIP system in discrete-time framework. The key for solving the discrete-time NOR problem lies in how to solve a set of algebraic functional equations known as discrete regulator equations (DRE). Since the equations are very complicated, the accurate solution of the DRE can not be obtained. In this paper, we first show that the DRE associated with the SIP system are solvable by center manifold theorem and then use neural network approach to tackle with the tracking problem. Finally, we compare our method with polynomial approximation method.

This paper addresses the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> output regulation problem of affine nonlinear systems with unmeasurable states and completely unknown dynamics. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> output regulation problem is formulated into a two-player zero-sum differential game problem with the control input and disturbance being the two adversary players. A model-based global state- feedback optimal solution to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> output regulation problem is presented by solving a Hamilton-Jacobi-Isaacs (HJI) equation, and a state- feedback reinforcement learning (RL) algorithm is proposed to solve the HJI equation without knowledge of system dynamics. A linear data-driven observer is designed to estimate the unmeasurable states and a novel output- feedback RL algorithm is developed based on the state- feedback RL framework by employing a critic-actor-disturbance neural network (NN) with a hidden layer. The salient found of this work is that, under the proposed output- feedback RL algorithm, the closed-loop system can reach globally stable. A numerical example is provided to demonstrate the effectiveness of the proposed solution.

A preliminary analysis of the local regulator problem is presented. It is then demonstrated that it is possible to isolate the main ingredients of a global solution to the output regulation problem. The authors then summarize some recent advances in the design of globally stabilizing feedback laws that were made possible by suitable use of the notion of passivity and, in particular, by the solution to the problem of when a given nonlinear system can be rendered passive via state feedback. Some of the results obtained are applied to the problem of deriving globally/asymptotically stabilizing feedback laws for certain classes of nonlinear systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Harmonic internal models for structurally robust periodic output regulation

This paper studies the semi-global output regulation problem for a class of nonlinear systems in the normal form under disturbances generated from nonlinear exosystems. By using a circle criterion and high gain design technique, a new nonlinear internal model is proposed to deal with the nonlinearities of the exosystems and to generate the desired input term for output regulation. It is shown that the proposed partial-state feedback control method ensures the semi-global convergence of the regulated variable.

Many mechanical systems are subjected to periodic disturbances which may adversely influence control performance. When the plants are nonlinear systems with unknown parameters, this disturbance compensation problem has been considered as adaptive output regulation problem. This paper discusses the problem about adaptive output regulation for nonlinear systems with arbitrary relative degree, which can be transformed into the output feedback form. The nonlinear plant is affected by unknown constant parameters and periodic disturbances with unknown frequencies, magnitudes and phases. The periodic disturbances are considered as signals generated from a linear time invariant exosystem. It is not assumed that the disturbances meet the matching condition for the control input. The design method of the adaptive controller and its parameter update law is based on adaptive backstepping manner for the coordinates-changed extended system, which is generated from the nonlinear plant and the exosystem. Then, it can be achieved that the output asymptotically converges to the output reference and that all signals in the closed-loop system are bounded.

This paper investigates the problem of output regulation of the Arneodo-Coullet chaotic system, which is one of the paradigms of the chaotic systems proposed by A. Arneodo, P. Coullet and C. Tresser (1981). Explicitly, state feedback control laws to regulate the output of the Arneodo-Coullet chaotic system have been derived so as to track the constant reference signals as well as to track periodic reference signals. The control laws are derived using the regulator equations of C.I. Byrnes and A. Isidori (1990), who solved the problem of output regulation of nonlinear systems involving neutrally stable exosystem dynamics. The output regulation of the Coullet chaotic system has important applications in Electrical and Communication Engineering. Numerical simulations are shown to verify the results.

In this brief article we consider the formation control problem of a system of M agents with general non-linear dynamics. We show that the problem of the agents moving in a formation along constant or periodic trajectories can be formulated as a non-linear output regulation (servomechanism) problem and solved using known techniques. Then, we consider the example of formation control of agents with point mass dynamics and apply the procedure. Few illustrative numerical simulation examples are also provided.

Adaptive wavelet output regulation for nonlinear systems

In this paper, we deal with an output regulation for nonlinear systems. Especially, when it comes to the output feedback control problem for nonlinear systems, it would be difficult to solve. From a practical point of view, it is an important issue to design a feedback controller using only output information. We introduce a certain kind of optimal inverse system for the output feedback control design. The optimal inverse system is a new concept. With the optimal inverse system, we obtain feedback controllers using only output information. In the proposed method, Genetic Programming (GP) also plays a key role. We utilize the GP's emergent ability for the control design. However, it is known that GP requires a lot of computation for evaluation of the individuals. The high computation load is one of the obstacles in the GP based control design. We adopt the so-called efficient GP, which is suitably modified for control design methods. Finally, some simulation results are given to demonstrate the effectiveness of the proposed control design.

In this paper, we propose a new method for modeling and output regulation of nonlinear systems with disturbance noises. We can get a semi-global error system between a real plant and a simplified model as a LPV system with two-step system identification method including L0-quasi norm minimizing sparse algorithm. LPV error system can produce compensating input for conventional homogeneous output feedback law, which enables us to release the assumption which isn't satisfied by the real plant, and globally regulate the real plant without its detail information. Furthermore, we propose a new nonlinear filtering method for the output feedback. We conventionally use Unscented Kalman Filter (UKF) which can't reduce non-Gaussian noise effect to nonlinear systems effectively. Instead UKF, we propose Robust Unscented Kalman Filter (RUKF). This filter consists of UKF and Robust Kalman Filtering algorithm using L1 optimization. RUKF can handle non-Gaussian, especially outlier of measurement value, which causes homogeneous output feedback to diverge. We confirm the effectiveness of this method with numerical simulations.

The semi-global stabilization problem and output regulation problem of singular linear systems subject to input saturation are addressed. A reduced-order normal system is obtained by a standard coordinate transformation. It is further shown that the controller that solves the stabilization (output regulation) problem of the reduced-order normal systems also solves the stabilization (output regulation) problem of the original singular systems.

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems