Noncommutative geometric structures on stabilizable infinite-dimensional linear systems

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.

On stabilizability of multi-agent systems

In this paper, the robust stabilization and synchronization of a novel chaotic system are presented. First, a novel chaotic system is presented in which this system is realized by implementing a sigmoidal function to generate the chaotic behavior of this analyzed system. A bifurcation analysis is provided in which by varying three parameters of this chaotic system, the respective bifurcations plots are generated and evinced to analyze and verify when this system is in the stability region or in a chaotic regimen. Then, a robust controller is designed to drive the system variables from the chaotic regimen to stability so that these variables reach the equilibrium point in finite time. The robust controller is obtained by selecting an appropriate robust control Lyapunov function to obtain the resulting control law. For synchronization purposes, the novel chaotic system designed in this study is used as a drive and response system, considering that the error variable is implemented in a robust control Lyapunov function to drive this error variable to zero in finite time. In the control law design for stabilization and synchronization purposes, an extra state is provided to ensure that the saturated input sector condition must be mathematically tractable. A numerical experiment and simulation results are evinced, along with the respective discussion and conclusion.

On stabilization for systems with two additive time-varying input delays arising from networked control systems

This paper is about the stabilization of a cascade system composed by an\ninfinite-dimensional system, that we suppose to be exponentially stable, and an\nordinary differential equation (ODE), that we suppose to be marginally stable.\nThe system is controlled through the infinite-dimensional system. Such a\nstructure is particularly useful when applying the internal model approach on\ninfinite-dimensional systems. Our strategy relies on the forwarding method,\nwhich uses a Lyapunov functional and a Sylvester equation to build a\nfeedback-law. Under some classical assumptions in the output regulation theory,\nwe prove that the closed-loop system is globally exponentially stable.\n

\(\mathrm{{H}}_{2}\) and \(\mathrm{{H}}_{\infty }\) optimal control are two important breakthroughs in modern robust control. In recent years, the \(\mathrm{{H}}_{2}\) and \(\mathrm{{H}}_{\infty }\) controller design techniques have gained a lot of research attention. Both have strong theoretical basis and are efficient algorithms for synthesizing optimal controllers. The performance of \(\mathrm{{H}}_{2}\) robust controller is useful to handle stochastic aspects such as measurement noise and capture the control cost. In the robust \(\mathrm{{H}}_{2}\) approach, the controller is designed to minimize an upper bound on the worst case \(\mathrm{{H}}_{2}\) norm for a range of admissible plant perturbations. When a model under analysis is applied with \(\mathrm{{H}}_{2}\) robust controller, the closed loop system will have a good dynamic system performance, but it has poor robustness for the external disturbances of the uncertain system model. The performance of \(\mathrm{{H}}_{\infty }\) is convenient to enforce robustness to model uncertainty, but it is based on compromising system performance. Their combination, the mixed \(\mathrm{{H}}_{2}/\mathrm{{H}}_{\infty }\) allows intuitive quadratic performance specifications of the \(\mathrm{{H}}_{2}\) synthesis with robust stability requirements specifications expressed by the \(\mathrm{{H}}_{\infty }\) synthesis. In time domain aspects, satisfactory time response and closed loop damping can often be achieved by enforcing the closed loop poles into a predetermined region of the left half plane. Combining these requirements to form so-called mixed \(\mathrm{{H}}_{2}/\mathrm{{H}}_{\infty }\) design with regional pole placement constraints allows for more flexible and accurate specification of closed-loop behavior. This chapter introduces the work of design of improved LMI-based robust output feedback controller and related simulations.

Learning-based robust output tracking control for unknown discrete-time nonlinear systems with dynamic uncertainty

This paper investigates the vibration control of geometrically nonlinear beam with Macro Fiber Composite (MFC) actuators using two different adaptive control algorithms. A complete mathematical modeling is presented in order to find the dynamic equation of motion. Then, a robust adaptive fuzzy control algorithm for controlling the proposed mechanical structure is introduced. This controller includes a fuzzy scheme and a robust controller. Based on sliding mode controller a fuzzy system is introduced to mimic an ideal controller. The robust controller is designed based on compensation of the difference between the fuzzy controller and the ideal controller. The parameters of the fuzzy system and uncertainty bound of the robust controller are adjusted adaptively. The adaptive laws are designed based on the Lyapunov stability theorem to reach the stability of the closed-loop system. Meanwhile, for comparison purposes the presented controller is compared with self tuning Ziegler-Nichols PID controller for both robustness and vibration suppression performance aspects. Effectiveness of these two control strategies is evaluated by numerical simulations. Detailed analysis for the closed-loop system is carried out to evaluate the vibration controlling performance under different output excitation, robustness of the closed-loop system under sudden loading and the effect of initial condition on vibration characteristic.

In this article, robust control schemes are presented for time-varying trajectory tracking control of Wheeled Mobile Robots (WMRs) in the presence of disturbance. First, the robust model predictive control (RMPC) is implemented for the kinematic subsystem by solving the modified optimization problem with the fixed starting point. Second, the robust control is extended for the dynamic subsystem by the Backstepping technique and additional observer. This paper shows how to ensure the stability of the closed-loop tracking error system. The performance of the proposed robust control is verified through simulation studies using Casadi tool.KeywordsModel predictive control (MPC)Wheeled mobile robotic systems (WMRs)Output feedback controlTerminal controllerFeasibility

The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy nontrivial commutation relations. For a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice. These differential calculi generate a lattice on a space continuum. The whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus. In this framework one just has to express the Lagrangian for the continuum theory in terms of differential forms. This expression then also makes sense for the deformed differential calculus. There is a natural integral associated with the latter. Integration of the Lagrangian over a space continuum then produces the correct lattice action for a large class of theories. This is explicitly shown for the scalar field action and the action for SU(m) gauge theory.

A robust PID controller based on linear quadratic gaussian approach for improving frequency stability of power systems considering renewables

Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties. In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. In early application of Corless-Leitmann [1] approach to robot manipulators [2, 3], it is difficult to compute uncertainty bound precisely. Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector. Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems. In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. One of the advantage of Spong’s approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. Yaz [6] proposed a robust control law based on Spong’s study [4] and global exponential stability of uncertain system is guaranteed. However, disturbance and unmodelled dynamics are not considered in algorithm of [4, 6]. Danesh at al [7] develop Spong’s approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances. Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators. In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system. A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty. Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty. A decomposition-based control design framework for mechanical systems with model uncertainties is proposed by Liu [10]. In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered. For this purpose, some uncertainty bound estimation functions are developed [11-15] based on a Lyapunov function, thus, stability of

Approach to robust multivariable combustion control design

Robust velocity and height control problem is dealt with for hypersonic vehicles subject to highly nonlinearity and uncertainties. The linear robust controller design for nonlinear systems suffers from the conservation problem, because the nonlinear dynamics are considered as uncertainties. In this paper, a feedback linearization controller is firstly applied to achieve the dynamic decoupling of the velocity and height channels. Then, robust controllers are designed for each channel to restrain the effects of the parametric uncertainties, external disturbances, and nonlinear dynamics, which cannot be counteracted accurately by the feedback linearization technique. This method can guarantee the robust stability and robust tracking properties of the closed-loop control system, while the conservation of the robust controller design can be reduced because the nonlinear information of dynamic system is used. Simulation comparisons are given to show the advantages of the proposed robust nonlinear control method.

Design of an intelligent optimal neural network-based tracking controller for nonholonomic mobile robot systems