Asymptotic stabilization via control by interconnection of port-Hamiltonian systems

We study the asymptotic properties of control by interconnection, a passivity-based controller design methodology for stabilization of port-Hamiltonian systems. It is well-known that the method, in its basic form, imposes some unnatural controller initialization to yield asymptotic stability of the desired equilibrium. We propose two different ways to overcome this restriction, one based on adaptation ideas, and the other one adding an extra damping injection to the controller. The analysis and design principles are illustrated through an academic example.     

On the addition of integral action to port-controlled Hamiltonian systems

A technique that provides closed loop integral action depending on the passive outputs of port-controlled Hamiltonian systems is already available. This paper addresses a new method that allows us to add integral action also on system variables having relative degree higher than one, while still preserving the Hamiltonian form and, thus, closed loop stability. The new approach is applied to design speed regulation controllers for the permanent magnet synchronous motor. Closed loop stability and asymptotic rejection of unknown piecewise constant load torques are formally proved. This theoretically predicted control system performance is illustrated via simulation experiments, which also show that the properties hold under parameter uncertainties. This is in line with the usual practice of including integral action in a controller with the aim of improving its closed loop robustness. The fact that the method enhances the range of possible integral actions in the controller, enriched with this robustness property, allows us to assess it as a practically important complement to the well-known interconnection and damping assignment techniques developed in the framework of port-controlled Hamiltonian systems.     

Robust adaptive control of nonlinearly parametrized multivariable systems with unmatched disturbances

A new robust adaptive control scheme is developed for nonlinearly parametrized multivariable systems in the presence of parameter uncertainties and unmatched disturbances. The developed control scheme employs a new integrated framework of a functional bounding technique for handling nonlinearly parametrized system dynamics, an adaptive parameter estimation algorithm for dealing with parameter uncertainties, a nonlinear feedback controller structure for stabilization of interconnected system states, and a robust adaptive control design for accommodating unmatched disturbances. It is proved that such a new robust adaptive control scheme is capable of ensuring the global boundedness and mean convergence of all closed‐loop system signals. A complete simulation study on an air vehicle system with nonlinear parametrization in the presence of an unmatched wind disturbance is conducted, and its results verify the effectiveness of the proposed robust adaptive control scheme.     

Nonlinear stabilizability via encoded feedback: The case of integral ISS systems

In this note we observe how, using the same arguments of Liberzon and Hespanha [(2005). Stabilization of nonlinear systems with limited information feedback, IEEE Transactions on Automatic Control, 50, 910-915] , integral input-to-state stabilizability with respect to measurement errors is enough to prove stabilizability of nonlinear systems with limited information. The utility of the remark lies on the fact that, under a requirement on the data rate analogous to the one in the paper by Liberzon and Hespanha, the stabilizability result is proven for a larger class of systems.     

Full-order observer design for a class of port-Hamiltonian systems

We consider a special class of port-Hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full-order observers using a passivity based approach. The essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state-space, on which the state estimation error is zero. We first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state-space and then perform a partial state feedback passivation which leads to the construction of the observer. We then illustrate this observer design procedure on two physical examples, the magnetic levitation system and the inverted pendulum on the cart system.     

Robust integral sliding mode control for uncertain stochastic systems with time-varying delay

This paper is concerned with sliding mode control for uncertain stochastic systems with time-varying delay. Both time-varying parameter uncertainties and an unknown nonlinear function may appear in the controlled system. An integral sliding surface is first constructed. Then, by means of linear matrix inequalities (LMIs), a sufficient condition is derived to guarantee the global stochastic stability of the stochastic dynamics in the specified switching surface for all admissible uncertainties. The synthesized sliding mode controller guarantees the reachability of the specified sliding surface. Finally, a simulation example is presented to illustrate the proposed method.     

Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation

The main contribution of this paper is a procedure for the control by energy shaping via Casimir generation of infinite dimensional port-Hamiltonian systems based on a particular finite element approximation. The proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve Casimir functions in the closed-loop system when going from the distributed to the (approximated) lumped parameter system. Besides the intrinsic difficulties related to the large number of state variables, the finite element model is generally given in terms of a Dirac structure and is completely a-causal, which implies that the plant dynamics is not given in standard input-state-output form, but as a set of DAEs. Consequently, the classical energy Casimir method has to be extended in order to deal with dynamical systems with constraints, usually appearing in the form of Lagrangian multipliers. The general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line.     

Robust integral control for a class of nonlinear systems with unknown control coefficients

The robust integral control problem is studied for a class of nonlinear systems with input-to-state stable (ISS)unmodeled dynamics in this paper.It does not require a priori knowledge of the control coefficients.Combining the Nussbaum-type gain technique and the backstepping design,we propose a state feedback controller,which could achieve the global asymptotic tracking for any constant reference signal,irrespective of the unmeasured dynamic disturbance.It is shown that the proposed methodology further extends the existing robust nonlinear integral control results.Simulation results verify the correctness of the theoretical analysis.     

Explicit simplicial discretization of distributed-parameter port-Hamiltonian systems

Simplicial Dirac structures as finite analogues of the canonical Stokes–Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input–output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds.     

Robust integral sliding mode control for uncertainsystems with multiple time delays

This paper proposes a robust integral sliding mode (RISM) manifold and the corresponding stabilization control law for uncertain systems with multiple time-varying time delays based on the techniques of linear matrix inequalities (LMI). The sufficient condition for the existence of the RISM manifold is given in terms of LMI, and then, the sliding mode control (SMC) law that can keep the system state on the RISM manifold from the initial time moment is developed. The efficiency and feasibility of the results are illustrated by a numerical example.     

Robust H-infinity integral sliding mode control for a class of uncertain switched nonlinear systems

This paper develops a new method to deal with the robust H-infinity control problem for a class of uncertain switched nonlinear systems by using integral sliding mode control. A robust H-infinity integral sliding surface is constructed such that the sliding mode is robust stable with a prescribed disturbance attenuation level γ for a class of switching signals with average dwell time. Furthermore, variable structure controllers are designed to maintain the state of switched system on the sliding surface from the initial time. A numerical example is given to illustrate the effectiveness of the proposed method.     

Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators

We consider infinite-dimensional port-Hamiltonian systems described on jet bundles. Based on a power balance relation we introduce the port-Hamiltonian system representation using differential operators regarding the structural mapping, the dissipation mapping and the input mapping. In contrast to the well-known representation on the basis of the underlying Stokes–Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations. The presented constructions will be specialized to mechanical systems to which class also the presented examples belong.     

Robust adaptive control of nonlinear non-minimum phase systems with uncertainties

This paper, presents a robust adaptive control method for a class of nonlinear non-minimum phase systems with uncertainties. The development of the control method comprises two steps. First, stabilization of the system is considered based on the availability of the output and internal dynamics of the system. The reference signal is designed to stabilize the internal dynamics with respect to the output tracking error. Moreover, a combined neuro-adaptive controller is proposed to guarantee asymptotic stability of the tracking error. Then, the overall stability is achieved using the small gain theorem. Next, the availability of internal dynamics is relaxed by using a linear error observer. The unmatched uncertainty is compensated using a suitable reference signal. The ultimate boundedness of the reconstruction error signals is analytically shown using an extension of the Lyapunov theory. The theoretical results are applied to a translational oscillator/rotational actuator model to illustrate the effectiveness of the proposed scheme.     

Quasi-continuous HOSM control for systems with unmatched perturbations

The control of nonlinear systems subject to unmatched perturbations is studied. A new design algorithm is proposed based on the block control and quasi-continuous higher order sliding modes techniques. The proposed method provides for the finite time exact tracking of a smooth desired signal in spite of unmatched perturbations.     

An algorithm for approximate calculation of the attainable sets of the nonlinear control systems with integral constraint on controls

In this paper, an algorithm for approximate calculation of the attainable sets of nonlinear control systems with integral constraint on controls is presented. A numerical example of the computer application of this algorithm is also given.     

Optimal tracking control for nonlinear large-scale systems with persistent disturbances

This paper studies the optimal control with zero steady-state error problem for nonlinear large-scale systems affected by external persistent disturbances. The nonlinear large-scale system is transformed into N nonlinear subsystems with interconnect terms. Based on the internal model principle, a disturbance compensator is constructed such that the ith subsystem with external persistent disturbances is transformed into an augmented subsystem without disturbances. According to the sensitivity approach, the optimal tracking control law for the ith nonlinear subsystem can be obtained. The optimal tracking control law for the nonlinear large-scale systems can be obtained. A numerical simulation shows that the method is effective.     

A port-Hamiltonian formulation of physical switching systems with varying constraints

This paper extends a generic method to design a port-Hamiltonian formulation modeling all geometric interconnection structures of a physical switching system with varying constraints. A non-minimal kernel representation of this family of structures (named Dirac structures) is presented. It is derived from the parameterized incidence matrices which are a mathematical representation of the primal and dual dynamic network graphs associated with the system. This representation has the advantage of making it possible to model complex physical switching systems with varying constraints and to fall within the framework of passivity-based control.     

Robust output-feedback control for stochastic nonlinear systems with modeling errors

In this paper,the stabilization problem of a stochastic nonlinear system with modeling errors is considered. An augmented observer is first presented to counteract the unmeasurable states as well as modeling errors.An adaptive output feedback controller is designed such that all signals in the closed-loop system are bounded in probability and the output is regulated to the origin almost surely.     

Power-shaping control of reaction systems: The CSTR case

Power-shaping control is a recent approach for the control of nonlinear systems based on the physics of the dynamical system. It rests on the formulation of the dynamics in the Brayton-Moser form. One of the main obstacles for using the power-shaping approach is to write the dynamics in the required form, since a partial differential equation system submitted to sign constraints has to be solved. This work comes within the framework of control design approaches that could possibly generate a closer link between the notions of energy that are specific to reaction systems as derived from thermodynamics concepts, and the dynamic system stability theory. The objective of this paper is to address the design of power-shaping control to reaction systems, and more particularly the step of solving the partial differential equation system. In order to illustrate the approach, we have selected the classical yet complex continuous stirred tank reactor (CSTR) as a case study. We show how using the power-shaping approach leads to a global Lyapunov function for the unforced exothermic CSTR. This Lyapunov function is then reshaped by means of a controller in order to stabilize the process at a desired temperature.     

具有非线性扰动的广义系统的鲁棒H∞控制

研究具有非线性结构扰动广义系统的鲁棒H∞控制和鲁棒H∞保性能控制问题,该不确定性为时间和状态的函数.且满足Lipschitz条件.目的是分别设计系统的鲁棒H∞控制器和鲁棒H∞保性能控制器.应用线性矩阵不等式方法,分别给出了系统的鲁棒H∞控制器和鲁棒H∞保性能控制器存在的充分条件.当这些条件可解时,分别给出了鲁棒H∞控制器和鲁棒H∞保性能控制器的表达式.最后通过一个仿真算例说明了所给出方法的应用.