Robust Output Regulation Problem Research Articles

Practical output regulation of uncertain Euler–Lagrange systems

This paper deals with the problem of robust output regulation of uncertain Euler–Lagrange systems. An internal model controller is synthesized by a systematic framework that considers polynomial mappings of the steady-state trajectory. The closed-loop stabilization is guaranteed by using a descriptor differential–algebraic representation of the system. This methodology allows the controller design problem to be cast as a convex optimization problem subject to linear matrix inequalities. Simulations considering a two-link robotic manipulator illustrate the method.

Neural operators for robust output regulation of hyperbolic PDEs

The recently introduced neural operator (NO) has been employed as a gain approximator in the backstepping stabilization control of first-order hyperbolic and parabolic partial differential equation (PDE) systems. Due to the global approximation ability of the DeepONet, the NO provides approximate spatial gain function with arbitrary accuracy. The closed-loop system stability can be ensured by the backstepping controller involving the approximate gain with sufficiently small error. In this paper, the NO theory is leveraged to solve the robust output regulation problem for a class of uncertain hyperbolic PDE systems under the design framework of backstepping-based regulator. The NO is trained offline on a dataset containing a sufficient number of system parameters and corresponding prior solutions of the kernel equation, so as to generate feedback gain for the robust regulator. Once the NO is trained, the kernel equation does not need to be solved ever again, for any new system parameters that do not exceed the range of the training set. Based on the internal model principle, the regulator is inherently robust to a degree of parameter uncertainty and error in approximate gain. Therefore, the tracking error can still converge to 0 if the extended regulator equations are solvable and the parameter uncertainty leads to an asymptotically stable origin. We provide a series of theory proofs and a numerical test under the approximate control and observation gains to demonstrate the robust regulation problem. It is shown that the NO is almost three orders of magnitude faster than PDE solver in generating kernel function, and the loss remains on the order of 10−4 in the test. This provides an opportunity to use the NO methodology for accelerated gain scheduling regulation for PDEs with time-varying system parameters.

Robust output regulation of a class of uncertain nonlinear minimum phase systems perturbed by unknown external disturbances

This paper addresses the problem of robust output regulation for uncertain nonlinear minimum phase systems affected by an unknown external disturbance. We present a novel unknown input observer (UIO)-embedded high-gain stabilizer to completely reconstruct the disturbance in a finite time. A new saturation function is introduced to prevent finite time escape of the closed-loop trajectories. This allows us to achieve a semi-globally exponential regulation of the regulated output. The proposed method features the exact cancellation rather than approximate cancellation, for both structured and/or unstructured external disturbance. The presented controller is finite-dimensional and does not require the nominal model of the plant. Finally, a numerical experiment is conducted to demonstrate the effectiveness of the proposed method.

Event‐triggered cooperative robust output regulation of minimum‐phase linear uncertain multi‐agent systems

AbstractIn this article, we investigate the event‐triggered cooperative robust output regulation problem of a class of minimum‐phase linear uncertain multi‐agent systems. In particular, we first propose a dynamic event‐triggered mechanism and an event‐triggered distributed output feedback control law. Then, by the internal model, the event‐triggered cooperative robust output regulation problem is converted into the event‐triggered cooperative robust stabilization problem of an augmented system composed of the distributed internal model and the original multi‐agent system. Next, we show that the event‐triggered cooperative robust stabilization problem can be solved by the proposed event‐triggered distributed output feedback control law together with a dynamic event‐triggered mechanism, and thus the event‐triggered cooperative robust output regulation problem is solved exactly in the sense that the tracking error of each subsystem converges to zero asymptotically while preventing the Zeno behavior from happening. Finally, we provide an example to illustrate the proposed control approach.

Event-triggered robust cooperative output regulation for a class of linear multi-agent systems with an unknown exosystem

This paper investigates the robust cooperative output regulation problem for a class of heterogeneous uncertain linear multi-agent systems with an unknown exosystem via event-triggered control (ETC). By utilizing the internal model approach and the adaptive control technique, a distributed adaptive internal model is constructed for each agent. Then, based on this internal model, a fully distributed ETC strategy composed of a distributed event-triggered adaptive output feedback control law and a distributed dynamic event-triggering mechanism is proposed, in which each agent updates its control input at its own triggering time instants. It is shown that under the proposed ETC strategy, the robust cooperative output regulation problem can be solved without requiring either the global information associated with the communication topology or the bounds of the uncertain or unknown parameters in each agent and the exosystem. A numerical example is provided to illustrate the effectiveness of the proposed control strategy.

Robust Cooperative Output Regulation for Networks of Hyperbolic PIDE–ODE Systems

In this paper the robust cooperative output regulation problem for multi-agent systems (MAS) with general heterodirectional hyperbolic PIDE-ODE agents is considered. This setup also covers networks of ODEs with arbitrarily long input and output delays. The output of the agents can be defined at all boundaries, in-domain and may depend on the ODE state, while disturbances act on the agents in-domain, at the boundaries, the output and the ODE. The communication network is described by a constant digraph and if its Laplacian is reducible, then heterogeneous agents are permitted also in the nominal case. The solution is based on the cooperative internal model principle, which requires to include a diffusively driven internal model in the controller. The corresponding state feedback regulator design starts with a local backstepping stabilization of the coupled hyperbolic PIDE-ODE systems. It is shown that the remaining simultaneous stabilization of the MAS can be traced back to the simultaneous stabilization of the finite-dimensional cooperative internal model. Solvability conditions in terms of the network topology and the agents transfer behavior are presented. The new design method is applied to the formation control of a platoon of uncertain heavy ropes carrying loads to verify its applicability. Simulations confirm the synchronization performance achieved by the resulting networked controller.

Robust switching control design using Lur'e Lyapunov functions

AbstractThis article addresses the problem of robust output regulation for uncertain switched systems. A general result is first proposed for the case of uncertain and nonlinear systems. In order to cope with model uncertainties, the proposed approach implements the internal model principle. Local asymptotic stability of the closed‐loop system as well as zero steady state error on the controlled output are proven using Lur'e Lyapunov functions. Then, this result is specialized to the linear case. Sufficient conditions ensuring global or local stability of the closed loop system are provided. Furthermore, a constructive method based on LMI is presented. When only local stability is achieved, outcomes of this procedure are not only the controller gains, but also an estimation of the robust domain of attraction. Finally, illustrative examples are proposed, showing the efficiency of the proposed methods.

Cooperative robust output regulation for a class of nonlinear multi-agent systems over jointly connected switching networks

In this paper, we consider the cooperative robust output regulation problem for a class of nonlinear multi-agent systems over jointly connected switching networks. First, by attaching each subsystem with an input driven filter, we obtain an extended system. Second, by integrating the distributed observer approach and the distributed internal model approach, we convert the cooperative output regulation problem of the original system into a distributed robust stabilisation problem of an augmented error system which is also a multi-agent system. Finally, we solve the original problem by stabilising the augmented error system via a distributed output feedback control law.

Robust Output Regulation of Linear Uncertain Systems by Dynamic Event-Triggered Output Feedback Control.

In this article, the robust output regulation problem of the linear uncertain system is investigated by the event-triggered control approach. Recently, the same problem is addressed by an event-triggered control law where the Zeno behavior may happen when time tends to infinity. In comparison, a class of event-triggered control laws is developed to achieve output regulation exactly, and meanwhile, explicitly exclude the Zeno behavior for all time. In particular, a dynamic triggering mechanism is first developed by introducing a dynamic changing variable with specific dynamics. Then, by the internal model principle, a class of dynamic output feedback control laws is designed. Later, a rigorous proof is provided to show that the tracking error of the system converges to zero asymptotically while prohibiting the Zeno behavior for all time. Finally, we give an example to illustrate our control approach.

Robust cooperative output regulation for heterogeneous nonlinear multi-agent systems with an unknown exosystem subject to jointly connected switching networks

This paper investigates the robust cooperative output regulation problem for heterogeneous lower triangular nonlinear multi-agent systems with an unknown exosystem over jointly connected switching networks. The problem has been studied for the exactly known exosystem over switching networks. However, the existing result for the unknown exosystem is still limited to the static networks. To ensure that all followers acquire the reference trajectory generated by the unknown exosystem through the jointly connected switching networks, by combining a set of auxiliary filtering variables and fixed-time stability theory, an adaptive distributed observer is designed. On the basis of the adaptive distributed observer and the distributed internal model approach, we propose a distributed controller under several standard assumptions to solve the problem. Compared with the similar work subject to the static networks, the controller in this paper is applicable to the more general communication network while weakening the assumptions of the controlled system.

Cooperative robust output regulation of linear uncertain multi‐agent systems: A dynamic event‐triggered approach

AbstractIn the article, the cooperative robust output regulation problem of multi‐agent systems with general linear uncertain dynamics is investigated by event‐triggered control. Recently, the similar problem is solved practically by a distributed event‐triggered control law where the tracking error of the system tends to some small neighborhood of the origin asymptotically. In comparison, we develop a distributed event‐triggered output feedback control law together with a class of dynamic event‐triggered mechanisms to solve this problem exactly. In particular, we first design a class of dynamic event‐triggered mechanisms. Then, based on the internal model principle, we propose a distributed event‐triggered output feedback control law. By rigorous proofs, we show that under the proposed distributed control law together with the dynamic event‐triggered mechanism, the cooperative robust output regulation problem is solved exactly, which means that the tracking errors of all followers converge to zero asymptotically while prohibiting the Zeno behavior. Finally, we use an example to illustrate our control approach.

Robust Output Regulation Problem with Prescribed Performance for Nonlinear Strict Feedback Systems

Robust Output Regulation Problem with Prescribed Performance for Nonlinear Strict Feedback Systems

Output regulation control for stochastic systems with additive noise

This paper investigates the robust output regulation problem for stochastic systems with additive noises. As is known, for the output regulation control problem, a general method is to regard that the system is disturbed by an autonomous exosystem (which is consisted by external disturbances and reference signals), and for the system disturbed by the white noise, the stochastic differential equations (SDEs) should be utilized in modeling, accordingly, a controller with a feedforward regulator is constructed for the stochastic system with an exosystem, which can not only cancel the external disturbance, but also transform the trajectory tracking problem into the stabilization problem; In consideration of the state variables in stochastic systems cannot be measured completely, we embed an observer to the controller, such that the random interference can be suppressed, and the trajectory tracking can be achieved. Based on the stochastic control theory, the criteria of the exponential practical stability in the mean square is presented for the closed-loop system, finally, through tuning the controller parameters, the mean square of the tracking error can converge to an arbitrarily small neighborhood of the origin.

Robust error feedback output regulation of a 2 × 2 hyperbolic system

Abstract In this paper, we consider the robust error feedback output regulation problem for a linear $ 2 \times 2 $ hyperbolic system with system uncertainties and disturbance signals. Using the backstepping method, the state feedback controller for the system without the system uncertainties and the external disturbance signals (nominal system) is designed. In terms of the measurable regulation error, an observer system is designed to recover the state of the nominal system, and an observer-based error feedback controller is obtained to solve the robust output regulation problem. Moreover, we prove that the controller is robust to the system uncertainties and the disturbance signals, and show that the state of the closed-loop system is bounded. The numerical simulations are presented to illustrate the effectiveness of the theoretical results.

Periodic event-based robust output regulation for a class of uncertain linear systems with inter-event analysis

This paper investigates the robust output regulation problem of uncertain linear systems by using a periodic event-triggered controller. A new type of event triggers based on local sampled signals is proposed to monitor events. An event detector only uses local information, avoiding infinitesimal inter-event times as in some continuous event-triggered schemes. A periodic event-triggered system naturally excludes Zeno behavior as the system operates at discrete-time instants. Convergence of tracking error holds for a class of decreasing triggering positive functions. When exponential functions bound the triggering function, analytical characterization of the relationship among sampling, event triggering, and inter-event behavior is established.

Practical output regulation for nonlinear multiagent systems with event‐triggered control protocol under directed switching networks

SummaryIn this paper, the event‐triggered‐based cooperative global robust practical output regulation problem is studied for nonlinear multiagent systems with directed switching networks. Firstly, the considered problem can be resolved with a prerequisite where the problem is solvable by an input‐to‐state stabilizing controller with measurement feedback. Specifically, the considered problem can be converted into a stabilization one for the so‐called augmented system through the canonical internal model construction. Then by combining the multiple Lyapunov functions (MLF) with the average dwell time (ADT) method, it is demonstrated that this stabilization problem can be solved by output feedback with measurement disturbances, which indicates a solution to the initial problem. At last, a suitable event‐triggering mechanism is proposed to save communication resources and reduce computational burden among all the agents. Finally, a group of FitzHugh‐Nagumo multiagent systems is provided to verify our theoretical results.

Robust Output Regulation for Uncertain Discrete-Time Linear Systems under the Effect of a Sinusoidal Disturbance

This paper studies the robust output regulation problem of uncertain single-input single-output (SISO) discrete-time linear systems. To reject the effect of a sinusoidal disturbance that only the frequency information is known prior, a novel strategy based on adaptive feedforward control (AFC) is developed. Compared with existing regulators for uncertain discrete-time systems, neither the knowledge on the sign of the real part or the imaginary part of the transfer function at the frequency of interest (the so-called strictly positive real (SPR)-like condition), nor persistence of excitation condition is required in this approach. Stability of the closed-loop system is rigorously analyzed using small-gain theorem and Lyapunov-based stability theory. Essentially, the proposed scheme guarantees that all signals of closed-loop system are bounded while the output of system asymptotically converges to zero, which is demonstrated by a numerical example.

Reinforcement learning‐based robust optimal output regulation for constrained nonlinear systems with static and dynamic uncertainties

AbstractThis article investigates the robust optimal output regulation problem for the constrained uncertain nonlinear systems. A two‐step design framework is proposed to overcome the difficulties brought by uncertainties, constraint and performance optimization. First, the feedforward controller is designed by the internal model principle, and the robust output regulation problem is transformed into the robust stabilization problem. Then, the optimal control problem with input saturation is further considered in the robust feedback controller design process. With the help of a non‐quadratic cost functional, actor‐critic algorithm and robust redesign technique are brought together to design the constrained robust optimal feedback controller. Finally, stability analysis based on Lyapunov method shows that all the signals of the closed‐loop system remain bounded, and the tracking error is uniformly ultimately bounded with arbitrarily small ultimate bound. Simulation results illustrate the effectiveness of the proposed methodology.

A Lyapunov Approach to Robust Cooperative Output Regulation of Multiagent Systems Under Infinite Communication Delays

This article investigates the robust cooperative output regulation problem of heterogeneous uncertain linear multiagent systems with switching topologies and infinite distributed communication delays. A novel distributed observer is proposed for each agent to estimate the state of the so-called exosystem by taking consideration of both switching topologies and infinite distributed communication delays. A distributed output feedback controller is then developed based on the internal model principle and the estimated state without using the prior knowledge of communication delays. It is shown via the newly developed Lyapunov-like method that the robust cooperative output regulation problem can be solved under some mild assumptions. Finally, a numerical example is given to demonstrate the effectiveness of the proposed controller.

Robust output regulation of a thermoelastic system

In this paper, we consider the robust output regulation problem for the thermoelastic system where the boundary control input and the output are located at the left end of the wave equation and the disturbance input is located at the right end of the heat equation. We formulate the system as a boundary control system and prove that it is impedance passive and well-posed. We design a finite-dimensional controller based on the internal model principle and show that the controller can achieve robust output tracking and disturbance rejection. The numerical simulations demonstrate that the finite-dimensional dynamic error feedback controller can make the output track the reference signal asymptotically and the state of the closed-loop system is bounded.