Dynamic Feedback Law Research Articles

Robust dynamic feedback stabilization for linear systems with disturbances

In this paper, we are concerned with the dynamic feedback design for linear systems with input disturbance. We propose a new approach to design dynamic feedback such that the disturbance in the control channel can be compensated effectively. An interesting characteristic about the disturbance robustness of dynamic feedback is found. Thanks to this interesting characteristic, we just need to consider the disturbance-free system to make the dynamic feedback design. This fact is well validated by several dynamic feedback laws proposed in this paper. Our results are available for the regular linear system with harmonic disturbance. Moreover, the stabilization of an unstable heat equation with general periodic disturbance is also investigated. In order to validate the theoretical results visually, some numerical simulations are given to show the effectiveness of the proposed dynamic feedback.

Boundary output-feedback control for a moving rigid body-liquid system in the presence of shallow-water sloshing

We solve the boundary output-feedback tracking problem for a moving rigid body-liquid system described by the linearized Saint-Venant PDEs. The control input is an external force acting on the rigid body. The control law is a boundary state-feedback type augmented with a Luenberger boundary observer, leading to a dynamic feedback law. The main complexity lies in the fact that we can only measure the position and velocity of the rigid body. To get around this limitation, the observer estimates the liquid levels at the cavity walls. We utilize a control Lyapunov functional methodology (with two different functionals for the observer and the dynamic feedback law) and achieve a slosh-free asymptotic convergence to the prespecified equilibrium point (in the sense of an appropriate norm). Finally, through a simulation, we demonstrate the feasibility of the boundary output-feedback controller for the (linear and nonlinear) system.

Influence of Discretization in Dynamically Embedded Model Predictive Control

The paper analyzes the closed-loop stability of Dynamically Embedded Model Predictive Control for input-constrained continuous-time nonlinear systems. Given a stabilizing continuous-time optimal control problem, the proposed method performs a discrete approximation to obtain a finite number of optimization variables. The resulting optimization problem is then embedded into a dynamic feedback law that evolves in parallel to the system. Using Input-to-State Stability, it is shown that the dynamic interconnection between the ideal continuous-time model predictive controller and the dynamically embedded solver is asymptotically stable for a sufficiently small discretization step and sufficiently fast solver dynamics. Numerical results, however, highlight a counter-intuitive behavior: as the discretization step decreases, the stability of the closed-loop system tends to deteriorate. This suggests that, although the discretization should be sufficiently accurate to correctly capture the behavior of the system, oversampling the system dynamics may be just as harmful as undersampling.

Feedback Control of a Motorized Skateboard

This paper develops a feedback control solution for the automatization of a motorized skateboard , also called symmetric snakeboard in the robotics and automatic control literature. Controlling this system is particularly challenging due to the association of kinematic and dynamic nonintegrable constraints to which it is subjected. The transverse function approach, combined with backstepping, is used to derive dynamic feedback laws that ensure practical stabilization of arbitrary reference trajectories in the Cartesian space. Simulation results illustrate the proposed approach.

Stabilization of some composite stochastic control systems with nontrivial solutions

This paper deals with the global stability for some composite stochastic control systems with nontrivial solutions by means of dynamic feedback laws. In particular, we establish feedback law for global asymptotic stabilization of a control subsystem deduced from the composite stochastic system and apply the result to stabilize the original composite stochastic system. Under this framework, firstly we derive necessary and sufficient conditions for the existence of feedback law which renders a ball that contains all nontrivial equilibrium solutions of the stochastic control subsystem globally asymptotically stable in probability. Secondly, for the composite system, we give the sufficient conditions for practical global asymptotic stability of the composite system based upon the feedback law derived for the subsystem. Numerical examples are given to validate our results.

A Grassmann Matrix Approach for the Computation of Degenerate Solutions for Output Feedback Laws

The paper is concerned with the improvement of the overall sensitivity properties of a method to design feedback laws for multivariable linear systems which can be applied to the whole family of determinantal type frequency assignment problems, expressed by a unified description, the so-called Determinantal Assignment Problem (DAP). By using the exterior algebra/algebraic geometry framework, DAP is reduced to a linear problem (zero assignment of polynomial combinants) and a standard problem of multilinear algebra (decomposability of multivectors) which is characterized by the set of Quadratic Plücker Relations (QPR) that define the Grassmann variety of P. This design method is based on the notion of degenerate compensator, which are the solutions that indicate the boundaries of the control design and they provide the means for linearising asymptotically the nonlinear nature of the problems and hence are used as the starting points to generate linearized feedback laws. A new algorithmic approach is introduced for the computation and the selection of degenerate solutions (decomposable vectors) which allows the computation of static and dynamic feedback laws with reduced sensitivity (and hence more robust solutions). This approach is based on alternative, linear algebra type criterion for decomposability of multivectors to that defined by the QPRs, in terms of the properties of structured matrices, referred to as Grassmann Matrices. The overall problem is transformed to a nonlinear maximization problem where the objective function is expressed via the Grassmann Matrices and the first order conditions for optimality are reduced to a nonlinear eigenvalue-eigenvector problem. Hence, an iterative method similar to the power method for finding the largest modulus eigenvalue and the corresponding eigenvector is proposed as a solution for the above problem.

A New Philosophy in Adaptive Control Design for Robust Attitude Maneuver of Spacecrafts

An adaptive control law has been designed for robust attitude tracking of a spacecraft with uncertain inertia matrix, torque coupling; coefficients and exogenous disturbances. The strict block feedback structure of the attitude dynamics is exploited in designing a back stepping control law, which depends on a compensator variable whose dynamics is designed to adapt to the unknown parameters. Two switching components have been incorporated into the dynamic feedback law to ensure robustness with respect to the disturbances and uncertainties in the torque coupling; coefficients. The spacecraft dynamics is simulated using the dynamic feedback control, first without the switching components and later incorporating the same, in order to demonstrate the effectiveness and robustness of the proposed method. (C) 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Approximate finite-horizon optimal control for input-affine nonlinear systems with input constraints

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimising a cost functional, while satisfying hard constraints on the input. In this framework, the minimum-time optimal control problem and some related problems are of interest for both theory and applications. For linear systems, the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems, the “shape” of the optimal input is in general not known. The control input can be found solving a Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE): it typically consists of a combination of bang-bang controls and singular arcs. In this paper, a methodology to approximate the solution of the HJB PDE is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of two examples: the minimum-time optimal control problem for an industrial wastewater treatment plant and the Goddard problem, i.e. a maximum-range optimal control problem.

Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input

In this paper, we consider the output feedback stabilization problem of a Euler–Bernoulli beam with timedelay in boundary control. A similar question was studied in Shang and Xu (2012, Syst. Control Lett., 61, 1069–1078), wherein the full state of the system was used in the design of the dynamic feedback law. In the present paper, we consider mainly an output-based stabilization problem for the same model. Using the tricks of the Luenberger observer and partial state predictor, we deduce a kind of dynamic feedback control law that exponentially stabilizes the delayed system.

Dynamic robust stabilization of stochastic differential control systems

The stabilization of stochastic differential control systems by means of dynamic feedback laws is provided. We extend the well-known Artstein–Sontag theorem to derive the necessary and sufficient conditions for the dynamic robust stabilization of stochastic differential systems. An explicit formula for feedback law exhibiting dynamic robust stability in probability is established, and a numerical example is also given to illustrate our results.

Approximate Finite-Horizon Optimal Control for Input-Affine Nonlinear Systems with Input Constraints

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimizing a cost functional, while satisfying hard constraints on the input. For linear systems the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems the “shape” of the optimal input is in general not known. The control input can be found solving an Hamilton-Jacobi-Bellman (HJB) partial differential equation (pde): it typically consists of a combination of bang-bang arcs and singular arcs. In the paper a methodology to approximate the solution of the HJB pde arising in the finite-horizon optimal control problem with input constraints is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of an example: the minimum time optimal control problem for an industrial wastewater treatment plant.

Stabilization of Nonlinear Systems with Filtered Lyapunov Functions and Feedback Passivation

In this paper a generalized class of filtered Lyapunov functions is introduced, which are Lyapunov functions with time-varying parameters satisfying certain differential equations. Filtered Lyapunov functions have the same stability properties as Lyapunov functions. Tools are given for designing composite filtered Lyapunov functions for cascaded systems. These functions are used to design globally stabilizing dynamic feedback laws for block-feedforward systems with stabilizable linear approximation.

Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback

We consider a stabilization problem for abstract second-order evolution equations with dynamic boundary feedback laws with a delay and distributed structural damping. We prove an exponential stability result under a suitable condition between the internal damping and the boundary laws. The proof of the main result is based on an identity with multipliers that allows to obtain a uniform decay estimate for a suitable energy functional. Some concrete examples are detailed. Some counterexamples suggest that this condition is optimal.

Input–output linearisation of a fourth-order input-affine system describing the evolution of a three-phase/switch/level (Vienna) rectifier

This study presents an analytical approach for proper selection of output functions to be regulated for the Vienna rectifier such that the resultant closed-loop systems are minimum phase. Specifically, two different adaptive control methodologies based on the input-output linearisation are developed and categorised. In the first category, three output functions are introduced and imposed to be zero by using three dynamic feedback laws. On the basis of states to be regulated, four different cases are studied and it is shown that only one of these cases results in a one-dimensional zero dynamics with an asymptotically stable equilibrium point. In the second category, two output functions are defined and output zeroing problem is solved with two control inputs. In addition, the remaining control is used to feedback linearise the resultant two-dimensional zero dynamics. Eighteen different cases are studied to demonstrate that only in one of these cases, the corresponding zero dynamics is feedback linearisable. Stability properties of the adaptive systems are investigated to show that the proposed adaptive controllers are capable of DC output voltage regulation and power factor correction in the presence of parametric and non-parametric uncertainties. Finally, simulation results are presented to confirm the validity of the developed approaches.

Filtered Lyapunov functions and the stabilization of block feedforward systems

AbstractIn this paper we introduce a generalized class of filtered Lyapunov functions, which are Lyapunov functions with time-varying parameters satisfying certain differential equations. Filtered Lyapunov functions have the same stability and robustness properties of Lyapunov functions. On the other hand, filtered Lyapunov functions can be easily handled to construct composite filtered Lyapunov functions for cascaded systems. Tools for the design of composite filtered Lyapunov functions are given and used to construct globally stabilizing dynamic feedback laws for block-feedforward systems with stabilizable linear approximation.

Observability and observer‐based control of hybrid systems

Addressing significant application domains is important to gain further understanding of the implications of hybrid modeling on control algorithms and to evaluate whether using hybrid formalism can be of substantial help in solving complex, real-life, control problems. In many application domains, hybrid controller synthesis problems are addressed by assuming full hybrid state information, although in many realistic situations state measurements are not available. Hence, to make hybrid controller synthesis relevant, the design of hybrid state observers is of fundamental importance. Moreover in many cases, for example, in communication systems or in fault detection, algorithms for state estimation are important by themselves. Observability has been extensively studied both in the continuous and in the discrete state domains. More recently, various researchers investigated observability of hybrid systems. The definitions of observability and the criteria for assessing this property varied depending on the class of systems under consideration and on the knowledge that is assumed at the output. We believe that this Special Issue, dedicated to observability for hybrid systems, is a useful contribution for the control community as it examines a topic that is in continuous evolution and that offers great opportunities for novel research. Six papers have been selected to reflect the most current research activities. The first two papers address discrete-time linear systems that may switch in an unknown and unpredictable way among different modes taken from a finite set (switching systems). The paper by M. Baglietto, G. Battistelli and L. Scardovi tackles the problem of mode observability with unknown but bounded noises affecting both the system and measurement equations. The paper by P. Caravani and E. De Santis presents conditions of stabilizability for switching systems in terms of a new definition of control invariance. These conditions are based on the realization of a discrete observer that allows the reconstruction of the discrete state in certain intervals of the time basis. The paper by M.D. Di Benedetto, S. Di Gennaro and A. D'Innocenzo proposes a novel definition of observability motivated by safety critical applications given with respect to a subset of critical discrete states. These states model unsafe or unallowed behaviors. This definition is adapted for the class of discrete event systems, for the class of switching systems, and for the class of hidden Markov models. A safety control problem for discrete time block-triangular order preserving hybrid automata with imperfect continuous state information is solved in the paper by D. Del Vecchio. A dynamic feedback law is constructed to guarantee that the continuous state is always outside a bad set. The proposed algorithms, which have linear complexity in the number of variables, are applied to a collision avoidance problem arising in the context of intelligent transportation. The last two papers deal with stochastic systems. The paper by S. Battilotti presents an observer design for a class of single-output nonlinear systems with Markov jumps. The Markov jump process interferes with a deterministic nonlinear dynamics at random times and retains its state for a certain amount of time (dwell time). The novelty is given by the reset mechanism adopted for the estimation process itself, depending on the reset values of the state process. In the paper by A. Germani, C. Manes and P. Palumbo, the state estimation problem is investigated for a class of stochastic linear switching-output systems. In these systems, the output matrix switches in a finite set of possible values according to a not directly measured discrete Markov sequence. The proposed real-time algorithm is based on the optimal polynomial filtering approach, which achieves simultaneous estimation of both the continuous system state and the switching parameter. We conclude by thanking the authors for the quality of their contributions and for the effort in completing their papers on time with the tight deadlines of this Special Issue. We are grateful to Prof. Mike J. Grimble, the Managing Editor of this Journal, for his help and support and to the reviewers who contributed greatly to make this issue a significant contribution to Observability and Observer Design for Hybrid Systems.

Observer‐based control of block‐triangular discrete time hybrid automata on a partial order

AbstractThe safety control problem for the class of discrete time block‐triangular order preserving hybrid automata with imperfect continuous state information is addressed. A dynamic feedback law is constructed in order to guarantee that the continuous state is always outside a bad set. The order preserving properties of the dynamics are exploited to construct state estimation and control algorithms that have linear complexity in the number of variables. Such algorithms adopt an interval abstraction approach, in which sets of interest are represented and propagated only through suitable upper and lower bounds. The proposed algorithms are applied to a collision avoidance problem arising in the context of intelligent transportation. Copyright © 2008 John Wiley & Sons, Ltd.

Robust H ∞ Control for Uncertain Nonlinear Systems

This paper studies the robust H∞ disturbance attenuation with internal stability for uncertain nonlinear control systems. By adding one power integrator technique, this paper designs a explicit smooth robust dynamic feedback law while rejecting the disturbance to any specified degree of accuracy. Further, the example and simulation results show the effectiveness of the proposed schemes.

Optimizing prediction dynamics for robust MPC

A convex formulation is derived for optimizing dynamic feedback laws for constrained linear systems with polytopic uncertainty. We show that, when it exists, the maximal invariant ellipsoidal set for the plant state under a dynamic feedback law incorporating any chosen static feedback gain is equal to the maximal invariant ellipsoidal set under any linear feedback law. The dynamic controller and its associated invariant set define a computationally efficient robust model predictive control (MPC) law with prediction dynamics belonging to a polytopic uncertainty set.

Output feedback stabilization and approximate and restricted tracking for a class of cascaded systems

In this note, we discuss the problems of output feedback stabilization for a class of cascaded systems and of (approximate and restricted) output regulation for general nonlinear systems. It is shown that (global) output feedback stabilization for a class of systems in feedforward form can be achieved with a dynamic feedback law, yielding bounded control, and relying on the introduction of a reduced-order observer. The above result, together with standard tools borrowed from the output regulator theory, is instrumental to construct dynamic control laws achieving (approximate) disturbance rejection and output tracking in the presence of (small) disturbance/reference signals generated by a known exosystem.