A relation between the output regulation and the observer design for nonlinear systems

Formation control for nonlinear multi-agent systems by robust output regulation

A two-cart with inverted-pendulum system is a nonlinear nonminimum-phase system with nonhyperbolic zero dynamics. This system was introduced to study the output regulation problem for nonlinear systems with nonhyperbolic zero dynamics. In this paper, we will first note that nonhyperbolicity is not necessary to the applicability of the output regulation theory. In particular, for the two-cart with inverted-pendulum system, the problem of output regulation for a class of sinusoidal reference input is actually solvable notwithstanding the presence of the nonhyperbolicity. A control law based on the standard output regulation theory is then synthesized and its performance is illustrated through computer simulation.

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.

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 two-cart with an inverted-pendulum system is a nonlinear, nonminimum phase system with nonhyperbolic zero dynamics. Devasia introduced this system to study the asymptotic tracking problem for nonlinear systems with nonhyperbolic zero dynamics and pointed out that the nonhyperbolicity may be challenging to the application of the standard inversion-based tracking technique. We first show that nonhyperbolicity is not necessary for the applicability of the output regulation theory. In particular, the problem of asymptotic tracking of the two-cart with an inverted-pendulum system to a class of sinusoidal reference inputs is actually solvable by the standard output regulation theory. Moreover, an approximation method for calculating the center manifold equation associated with the output regulation problem for general nonlinear systems is given. This approach does not rely on the hyperbolicity condition and, hence, applies to a large class of nonlinear systems.

In this work, we investigate the cooperative output regulation problem of multi-agent uncertain nonlinear systems in lower triangular form with unknown leader. The cooperative output regulator construction is done based on a nonlinear internal model approach. Specifically, nonlinear internal models are first constructed, converting the cooperative output regulation problem into a cooperative stabilization problem of the augmented system, and then an integral input-to-state stable (iISS) based approach is proposed to solve the latter cooperative stabilization. It finally gives rise to a constructive state feedback design for cooperative nonlinear output regulation problem.

Supervisory fault-tolerant regulation for nonlinear systems

In this paper, the global output regulation problem (ORP) for nonlinear output feedback affine systems with unknown functions (UFs) and unmeasured states is studied. Firstly, a Lemma is proposed to solve not only the global ORP of nonlinear systems, but also the ‘explosion of terms’ problem of backstepping. Secondly, the control scheme is designed based on backstepping method, which avoids an Assumption to solve the ORP. Therefore, the algorithm reduces the conservatism and increases the applicability. Finally, the algorithm is applied to Duffing system to verify the effectiveness of the algorithm.

This paper considers the observer design problem for one-sided Lipschitz nonlinear systems with unknown inputs. The systems under consideration are a larger class of nonlinearities than the well-studied Lipschitz systems and have inherent advantages with respect to conservativeness. For such systems, we first propose a full-order nonlinear unknown input observer (UIO) by using the linear matrix inequality (LMI) approach. Following a similar design procedure and using state transformation, the reduced-order nonlinear UIO is also constructed. Sufficient conditions to guarantee existence of full-order and reduced-order UIOs are established by carefully considering the one-sided Lipschitz condition together with the quadratic inner-bounded condition. Based on the matrix generalized inverse technique, the UIO conditions are formulated in terms of LMIs. Moreover, the proposed observers are applied to a single-link flexible joint robotic system with unknown inputs. Simulation results are finally given to illustrate the effectiveness of the proposed design.

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time-delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H ∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.

This paper considers the reduced-order observer design problem for one-sided Lipschitz nonlinear systems with unknown inputs. The systems under consideration are a larger class of nonlinear systems than the well-studied Lipschitz systems and have inherent advantages with respect to conservativeness. For such systems, we propose a reduced-order unknown input observer (UIO) by using the linear matrix inequality (LMI) technique. Sufficient conditions that guarantee the existence of the reduced-order UIO are established and transformed into the format of LMIs via using the matrix generalized inverse technique. Simulation results are finally given to illustrate the effectiveness of the proposed design.

This paper investigates the formation control problem of heterogeneous discrete-time nonlinear multi-agent systems with structural uncertainties. The control objective is to make the outputs of agents reach a desired formation while tracking a leader. The formation control problem of heterogeneous discrete-time multi-agent systems is first converted into a robust output regulation problem, and then, a distributed formation control scheme based on output regulation framework is proposed. A $p$ -copy internal model is synthesized into the distributed formation control law such that the overall closed-loop system can be robust to structural uncertainties. It is shown that the heterogeneous discrete-time nonlinear multi-agent system can reach the desired formation regardless of small parameter permutations of agents under the proposed control law. Finally, simulation and experiment studies are provided to demonstrate the effectiveness of the proposed control law.

In this study, differential stability is introduced for non-linear systems, and this concept is further exploited in reduced-order observer design for non-linear systems with non-linearities of unmeasured state variables, a more general class of non-linear systems than the systems with linear observer errors. It has been shown that if the dynamics of unmeasured state variables under a state transformation is differentially stable, a reduced-order observer can be designed to produce asymptotically convergent estimates of the unmeasured state variables. A systematic design method is then introduced for a class of multi-output non-linear systems. For such a system, a non-linear term of the unmeasured state variables enter the system through a coupling matrix. It is found that a reduced-order observer can be designed if the linear part with the coupling matrix as the input matrix has no unstable invariant zeros. A further exploitation is presented for a class of single-output non-linear systems with non-linearity of unmeasured state variables. In this case, the coupling vector is allowed to be a vector field which depends on the system output.

Reduced-order observer-based control design for nonlinear stochastic systems

This paper deals with the controller design problem of nonlinear quadratic Markov jumping systems with input saturation. Both mode-dependent and mode-independent state feedback controllers are designed. By using the concept of domain of attraction in mean square sense, sufficient conditions for stochastic stabilisation for nonlinear quadratic systems are derived in terms of linear matrix inequalities. Certain existing results are improved. Simulation examples are presented to illustrate the effectiveness of the proposed technique.