Dissipativity, stabilization, and regulation of cascade-connected systems

This paper presents a technical framework utilizing the dissipativity mechanism that gives rise to the solution of the global regulation problem of the cascade-connected systems subject to both dynamic uncertainty and static uncertainty without knowing the bound of the static uncertainty. Additionally, this paper shows that the nonlinear robust servomechanism problem for the lower triangular systems can be cast into the regulation problem for the cascade-connected systems, thus leading to a complete solution of the nonlinear robust servomechanism problem for the lower triangular systems for the most general case where both the uncertain parameters and the exogenous signals can be arbitrarily large.     

GLOBAL ROBUST SERVOMECHANISM OF A CLASS OF NONLINEAR SYSTEMS USING OUTPUT FEEDBACK

The robust servomechanism problem (alternatively, output regulation problem) of the class of nonlinear systems in lower triangular form has been extensively studied in recent years. The semi‐global solution was first given by either state feedback or output feedback. The global solution by state feedback was given very recently. However, the global solution by output feedback has long been an open problem. In this paper, we present a set of solvability conditions of the global robust servomechanism problem for this class of nonlinear systems by output feedback.     

Global Robust Stabilization of Cascade-Connected Systems With Dynamic Uncertainties Without Knowing the Control Direction

This note addresses the global robust stabilization problem for cascade-connected nonlinear systems assuming the control direction is unknown and the system is subject to both dynamic and static uncertainties. A possible application of the main result of this note to the global robust servomechanism problem for nonlinear systems in lower triangular form is also discussed.     

Global robust output regulation of lower triangular systems with unknown control direction

This paper presents the solvability conditions for the global robust output regulation problem for lower triangular nonlinear systems assuming the control direction is unknown. The approach used is an integration of the robust stabilization technique and Nussbaum gain technique.     

Global robust output regulation of lower triangular systems with nonlinear exosystems and its application

This paper studies the global robust output regulation problem for lower triangular systems subject to nonlinear exosystems. By employing the internal model approach, this problem can be boiled down to a global robust stabilization problem of a time-varying nonlinear system in the cascade-connected form. Then, a set of sufficient conditions for the solvability of the problem is derived, and thus, leading to the solution to the global robust output regulation problem. An application of the main result of this paper is also proposed.     

A general formulation and solvability of the global robust output regulation problem

While the output regulation problem for linear systems accommodates both bounded and unbounded exogenous signals, the existing formulation of the output regulation problem for nonlinear systems only allows bounded exogenous signals produced by an exosystem with a neutrally stable equilibrium. In particular, when it comes to the robust output regulation problem, the only admissible exogenous signal is a finite combination of step and sinusoidal functions. In this paper, we give a general formulation of the robust output regulation problem that admits unbounded exogenous signals, and contains the previous formulations as special cases when the system is linear or when the exogenous signals are bounded. Then, we give conditions under which the problem can be solved by solving a robust stabilization problem of an augmented system. Finally, we apply our general result to obtain the solvability conditions of the general robust output regulation problem for the class of lower triangular nonlinear systems.     

Global Robust Output Regulation by State Feedback for Strict Feedforward Systems

This note studies the global robust output regulation problem by state feedback for strict feedforward systems. By utilizing the general framework for tackling the output regulation problem , the output regulation problem is converted into a global robust stabilization problem for a class of feedforward systems that is subject to both time-varying static and dynamic uncertainties. Then the stabilization problem is solved by using a small gain based bottom-up recursive design procedure.     

Data-Driven Global Robust Optimal Output Regulation of Uncertain Partially Linear Systems

In this paper, a data-driven control approach is developed by reinforcement learning (RL) to solve the global robust optimal output regulation problem (GROORP) of partially linear systems with both static uncertainties and nonlinear dynamic uncertainties. By developing a proper feedforward controller, the GROORP is converted into a global robust optimal stabilization problem. A robust optimal feedback controller is designed which is able to stabilize the system in the presence of dynamic uncertainties. The closed-loop system is ensured to be input-to-output stable regarding the static uncertainty as the external input. This robust optimal controller is numerically approximated via RL. Nonlinear small-gain theory is applied to show the input-to-output stability for the closed-loop system and thus solves the original GROORP. Simulation results validates the efficacy of the proposed methodology.      

An adaptive regulation problem and its application

This paper studies an adaptive regulation problem for the modified FitzHugh-Nagumo neuron model under external electrical stimulation. We first present the solution of the global robust output regulation problem for output feedback system with an uncertain exosystem which models the external electrical stimulation with unknown frequency and amplitude. Then, we show that the robust control problem for the modified FitzHugh-Nagumo neuron model can be formulated as the global robust output regulation problem and the solvability conditions for the output regulation problem for the modified FitzHugh-Nagumo neuron model are all satisfied. Then, we apply the obtained output regulation result to constructing an output feedback control law for the modified FitzHugh-Nagumo neuron model to achieve global stability of the closed-loop system in the presence of uncertain parameters and external stimulus. An example is given to show that the proposed algorithm can completely reject the external electrical stimulation.     

Robust Tracking Controller Design for a Class of Block Lower‐Triangular Systems

A class of multi‐input multi‐output (MIMO) block lower‐triangular systems are considered with the dynamic and static uncertainties related to the states of the former subsystems. A design procedure of robust tracking controller is presented, which avoids the problem of the “explosion of complexity” and can be implemented very simply. The closed‐loop system possesses robust properties that, 1) all signals involved are semi‐global uniformly ultimately bounded for bounded differentiable reference inputs; 2) for given initial tracking errors and uncertainty boundary, controller parameters can be found to guarantee that the tracking errors can be smaller than a specific positive constant after a limited time. Two numerical instances are given at last.     

Global output feedback regulation of uncertain nonlinear systems with unknown time delay

This paper investigates the problem of global output feedback regulation for a class of nonlinear systems with unknown time delay. It is also allowed to contain uncertain functions of all the states and input as long as the uncertainties satisfying certain bounded condition for the considered systems. In this paper, a constructive control technique has been proposed for controlling the systems. By using dynamic high-gain scaling approach and choosing an appropriate Lyapunov-Krasovskii functional, a delay-independent robust adaptive output feedback controller is constructed such that the states of the considered systems achieve global regulation. Two simulation examples are provided to demonstrate the effectiveness of the proposed design scheme.     

Robust H ∞ performance problem for linear systems with nonlinear uncertainties in all system matrices

This paper considers robust performance analysis and H X controller design for a class of systems with time-varying and nonlinear uncertainties. These uncertainties are allowed to exist not only in the state, but also in the control input, measurement output, exogenous input and derivative of state. A new sufficient condition based on LMI is first provided to analyse the robust H X performance problem of the free systems. For the general case, it is shown that a solvability condition for the output feedback control problem can be reduced to that of a set of LMIs with algebraic constraints. Then it is shown that in some cases, the constraints can be eliminated through simplifications and the output feedback controller design methods can be provided in terms of LMIs. In particular, two special cases of the systems with nonlinear uncertainties are discussed thoroughly and the design procedures of output feedback controllers are provided via typical LMIs. Furthermore, for a class of systems with both structured and nonlinear uncertainties, a new solvability condition is presented and the corresponding problem also cast into that of an auxiliary system without uncertainties. A few examples are also given to demonstrate the effectiveness of the proposed approaches.     

Error feedback and a servomechanism problem for uncertain nonlinear systems

The robust servomechanism problem, also known as the problem of robust output regulation, is considered for a family of uncertain triangular systems. In contrast to the previous work where the uncertain vector field of the controlled plant was assumed to be bounded by a linear growth of the unmeasurable states multiplied by a polynomial function of the system output, we show that the polynomial growth condition can be relaxed and robust output regulation by error feedback is still possible, as long as the uncertain system is dominated by a continuous function of the system output multiplied by a linear growth of the unmeasurable states.     

基于强化学习的部分线性离散时间系统的最优输出调节

针对同时具有线性外部干扰与非线性不确定性下的离散时间部分线性系统的最优输出调节问题, 提出了仅利用在线数据的基于强化学习的数据驱动控制方法. 首先, 该问题可拆分为一个受约束的静态优化问题和一个动态规划问题, 第一个问题可以解出调节器方程的解. 第二个问题可以确定出控制器的最优反馈增益. 然后, 运用小增益定理证明了存在非线性不确定性离散时间部分线性系统的最优输出调节问题的稳定性. 针对传统的控制方法需要准确的系统模型参数用来解决这两个优化问题, 提出了一种数据驱动离线策略更新算法, 该算法仅使用在线数据找到动态规划问题的解. 然后, 基于动态规划问题的解, 利用在线数据为静态优化问题提供了最优解. 最后, 仿真结果验证了该方法的有效性.     

Output regulation for a class of uncertain nonlinear systems with nonlinear exosystems and its application

In this paper, we consider the global robust output regulation problem for a class of uncertain nonlinear systems with nonlinear exosystems. By employing the internal model approach, we show that this problem boils down to a global robust stabilization problem of a time-varying nonlinear system in lower triangular form, the solution of which will lead to the solution of the global robust output regulation problem. An example shows the effectiveness of the proposed approach.     

Distributed output regulation for multi-agent systems with norm-bounded uncertainties

In this paper, we consider the distributed robust output regulation problem for multi-agent systems (MASs). It is involved with a group of heterogeneous high-order linear uncertain systems and an linear exosystem. The regulated output is a combination of the output of MAS and the exosystem, which is defined based on controlling demands. Distributed controllers are designed to ensure that the regulated output converges to the origin and meanwhile the closed-loop MAS is stable. The sufficient conditions for the solvability of distributed output regulation problem are given in terms of linear matrix inequalities. And algorithms are proposed to design distributed dynamic controllers with state feedback and output feedback, via the help of internal models. It is shown that, for any time-invariant norm-bounded uncertainties, the given controllers can realise the objective of output regulation.     

Global robust stabilization of feedforward systems with uncertainties

This paper studies the global robust stabilization problem for a class of feedforward systems that is subject to both dynamic and time-varying static uncertainties. A small gain theorem-based bottom-up recursive design is developed for constructing a nested saturation control law. At each recursion, two versions of small gain theorem with restrictions are employed to establish the global attractiveness and local stability of the closed-loop system at the equilibrium point, respectively.     

Global output regulation for output feedback systems with an uncertain exosystem and its application

This paper presents the solvability conditions for the global robust output regulation problem for a class of output feedback systems with an uncertain exosystem by using output feedback control. An adaptive control technique is used to handle the unknown parameter vector in the exosystem. It is shown that this unknown parameter vector can be exactly estimated asymptotically if a controller containing a minimal internal model is employed. The effectiveness of our approach has been illustrated by an asymptotic tracking problem of a generalized fourth‐order Lorenz system. Copyright © 2009 John Wiley & Sons, Ltd.     

Global adaptive tracking for a class of nonlinear time-delay systems

This paper aims to extend the newly developed global adaptive regulation scheme to adaptive tracking for a general class of nonlinear systems with both parameter uncertainties and unknown time delay for all the system states. With a simple yet novel decomposition of some coupling functions derived from the introduction of a reference signal, a backstepping tracking control strategy is proposed which possesses the feature that for each design step, a dynamic gain is introduced to generate an additional term to counteract the system nonlinearities and parameter uncertainties. As a result, a memoryless adaptive state-feedback controller is obtained to achieve the global adaptive tracking.