Linear Feedback Law Research Articles

Synchronizing linear systems via partial-state coupling

A basic result in the synchronization of linear systems via output coupling is presented. For identical discrete-time linear systems that are detectable from their outputs and neutrally stable, it is shown that a linear output feedback law exists under which the coupled systems globally asymptotically synchronize for all fixed connected (asymmetrical) network topologies. An algorithm is provided to compute such a feedback law based on individual system parameters.

Interpolated robust time‐optimal controllers for linear systems

Interpolation techniques are well known in reducing the online computational effort of optimal control strategies, such as model predictive control. A general interpolation-based time-optimal control (TOC) for a constrained linear system with bounded disturbances is presented. The terminal controller is nonlinear, which interpolates among multiple predetermined linear feedback laws, and the corresponding terminal set is equal to or sometimes even larger than the convex hull of all constituent terminal sets. Starting from this relatively large terminal set, a large domain of attraction can be obtained by using a short horizon, consequently leading to a low online computational effort. By using multi-parametric programming, the proposed controller can be determined offline, thus further relaxing the online computational effort. Compared with the standard TOC implementations, the proposed approach provides an additional advantage combining ‘a larger domain of attraction’, ‘a better asymptotic behaviour’, and ‘a lower online computational effort’. The performance of the proposed approach is assessed via two numerical examples.

On IQC approach to the analysis and design of linear systems subject to actuator saturation

This paper establishes IQC-based (Integral Quadratic Constraints) conditions under which an ellipsoid is contractively invariant for a single input linear system under a saturated linear feedback law. Based on these set invariance conditions, the determination of the largest such ellipsoid, for use as an estimate of the domain of attraction, can be formulated and solved as an LMI optimization problem. Such an LMI problem can also be readily adapted for the design of the feedback gain that achieves the largest contractively invariant ellipsoid. While the advantages of the proposed IQC approach remain to be explored, it is shown in this paper that the largest contractively invariant ellipsoid determined by this approach is the same as the one determined by the existing approach based on expressing the saturated linear feedback as a linear differential inclusion (LDI), which is known to lead to non-conservative result in determining the largest contractively invariant ellipsoid for single input systems.

Identifiability of linear superstructures under feedback-Taking base-isolated structures as example

The identifiability condition for substructural identification in free-vibration systems is investigated by spectral analysis and parametric methods in the framework of closed-loop systems. Substructures governed by linear and nonlinear feedback laws are both considered. The feedback law mechanism is shown to have greater influence on identifiability than does the model structure or the identification method. Copyright © 2008 John Wiley & Sons, Ltd.

Robust Explicit Time Optimal Controllers for Linear Systems via Decomposition Principle

One of the key problems in time optimal control (TOC) is the inherent computational complexity, which restricts its application to low dimensional systems. Considering a constrained linear system with bounded disturbances, this paper proposes a novel approach to reduce the computational complexity of TOC, where the terminal controller is nonlinear. It comprises several predetermined local linear feedback laws, resulting in a large terminal set. Starting from this relatively large terminal set, a large domain of attraction of the proposed TOC controller can be obtained by using a short horizon, and consequently leads to a low on-line computational effort. Furthermore, by formulating a suitable cost function, as time evolves, the TOC controller reaches the desired controller to obtain a good asymptotical behavior. The performance of the proposed approach is assessed via a numerical example.

Analysis and Design of Switched Linear Systems in the Presence of Actuator Saturation and L2 Disturbances

This paper considers the problem of disturbance tolerance/rejection for a family of linear systems subject to actuator saturation and ℒ2 disturbances. For a given set of linear feedback gains, a given switching scheme and a given bound on the energy of the disturbances, conditions are established in terms of linear or bilinear matrix inequalities under which the resulting switched system is bounded state stable, that is, trajectories starting from a bounded set will remain inside the set or a larger bounded set. With these conditions, both the problem of assessing the disturbance tolerance/rejection capability of the closed-loop system and the design of feedback gain and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories from a given set remain bounded. Disturbance rejection is measured by the restricted ℒ2 gain over the set of tolerable disturbances. In the event that all systems in the family are identical, the switched system reduces to a single system under a switching feedback law. It will be shown that such a single system under a switching feedback law has stronger disturbance tolerance/rejection capability than a single linear feedback law can achieve.

Nonlinear Optimal Control Techniques for Vibration Attenuation Using Magnetostrictive Actuators

This article addresses the development of a nonlinear control design for attenuating structural vibrations using magnetostrictive transducers operating in nonlinear and highly hysteretic operating regimes. We consider as a prototype a thin plate subjected to exogenous pressure waves and controlled via Terfenol-D transducers at the plate edges; however, the methodology is sufficiently general to encompass a wide range of structures and magnetic transducer designs. Hysteresis inherent to the transducer materials is quantified using a homogenized energy framework and the resulting nonlinear constitutive relations are used to construct a PDE representation and corresponding finite dimensional model of the structural system. We employ optimal control theory to construct nonlinear open loop control inputs which accommodate the hysteresis inherent to the transducers but are not robust with regard to unmodeled dynamics or disturbances. Robustness is incorporated by employing perturbation techniques to provide linear feedback laws acting on measured disturbances. As illustrated via numerical examples, the resulting hybrid control design provides excellent control authority and robustness for transducers operating in hysteretic and nonlinear regimes.

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part I: equilibrium bifurcations

AbstractUnder certain non‐degeneracy conditions, necessary and sufficient conditions for stabilizability are obtained for multi‐input nonlinear systems possessing a simple equilibrium bifurcation with the critical mode being linearly uncontrollable. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the bifurcation of the closed‐loop system is a supercritical pitchfork bifurcation, which is equivalent to local asymptotic stability of the system at the bifurcation point. Stabilizability is reduced to the existence of solutions to a third order algebraic inequality, coupled with a quadratic algebraic equation, with the unknowns being the feedback gains. Explicit conditions for the existence of solutions to the algebraic equation and the inequality are derived. Part of the sufficient conditions are equivalent to the rank conditions of augmented matrices. Conditions under which there exists a stabilizing linear feedback law are also derived. The theoretical results are applied to active control of rotating stall in axial compressors using bleed valve as actuator. Copyright © 2006 John Wiley & Sons, Ltd.

Global Practical Stabilization of Planar Linear Systems in the Presence of Actuator Saturation and Input Additive Disturbance

In this note, we revisit the problem of global practical stabilization for planar linear systems subject to actuator saturation and input additive disturbances. A parameterized linear state feedback law is designed such that, by tuning the value of the parameter, all trajectories of the closed-loop system converge to an arbitrarily small neighborhood of the origin in a finite time and remain in there.

Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

We study the exponential stabilization of the linearized Navier--Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a linear-quadratic control problem. We do not assume that the normal component of the control is equal to zero. In the nonzero case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation, satisfied by Py, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation, satisfied by (I - P)y. Using this decomposition, we show that the feedback law can be expressed as a function only of Py. In the two-dimensional case we show that the linear feedback law provides a local exponential stabilization of the Navier--Stokes equations.

A hybrid feedback linearizing-Kalman filtering control algorithm for a distillation column

This paper studies the design of a discrete-time multivariable feedback linearizing control (FLC) structure. The control scheme included (i) a transformer [also called the input/output (I/O) linearizing state feedback law] that transformed the nonlinear u-y to a linearized v-y system, (ii) a closed-loop observer [extended Kalman filter (EKF)], which estimated the unmeasured states, and (iii) a conventional proportional integral (PI) controller that was employed around the v-y system as an external controller. To avoid the estimator design complexity, the design of EKF for a binary distillation column has been performed based on a reduced-order compartmental distillation model. Consequently, there is a significant process/predictor mismatch, and despite this discrepancy, the EKF estimated the required states of the simulated distillation column precisely. The FLC in conjunction with EKF (FLC-EKF) and that coupled with a measured composition-based reduced-order open-loop observer (FLC-MCROOLO) have been synthesized. The FLC structures showed better performance than the traditional proportional integral derivative controller. In practice, the presence of uncertainties and unknown disturbances are common, and in such situations, the proposed FLC-EKF control scheme ensured the superiority over the FLC-MCROOLO law.

Linear Systems Subject to Input Saturation and Time Delay: Finite‐Gain $L^{\lowercase{p}}$‐Stabilization

This paper deals with finite‐gain stabilization of linear systems by static feedbacks which are bounded and time‐delayed. For neutrally stable continuous‐time linear systems, we provide a saturated linear feedback law ensuring finite‐gain stability with respect to every $L^p$‐norm, $p \in [1, \infty]$, with an arbitrary small bound on the control and large (constant) delay. Moreover, we provide upper bounds for the corresponding $L^p$‐gains which are delay‐independent.

Adaptive synchronization between two different chaotic dynamical systems

This work presents the synchronization between two different chaotic systems by using an adaptive feedback control scheme. The adaptive synchronization problem between an electrostatic system and electromechanical transducer has been investigated. An adaptive linear feedback law with two controllers is proposed to ensure the global chaos synchronization of the nonlinear electrostatic and electromechanical systems. Numerical simulations results are presented to demonstrate the effectiveness of the proposed method.

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.

On improvement of transient performance in tracking control for a class of nonlinear systems with input saturation

This paper studies the technique of the composite nonlinear feedback (CNF) control for a class of cascade nonlinear systems with input saturation. The objective of this paper is to improve the transient performance of the closed-loop system by designing a CNF control law such that the output of the system tracks a step input rapidly with small overshoot and at the same time maintains the stability of the whole cascade system. The CNF control law consists of a linear feedback control law and a nonlinear feedback control law. The linear feedback law is designed to yield a closed-loop system with a small damping ratio for a quick response, while the nonlinear feedback law is used to increase the damping ratio of the closed-loop system when the system output approaches the target reference to reduce the overshoot. The result has been successfully demonstrated by numerical and application examples including a flight control system for a fighter aircraft.

Robust input-output energy decoupling for uncertain singular systems

This paper addresses the robust input-output energy decoupling problem for uncertain singular systems in which all parameter matrices except E exist as time-varying uncertainties. By means of linear matrix inequalities (LMIs), sufficient conditions are derived for the existence of linear state feedback and input transformation control laws, such that the resulting closed-loop uncertain singular system is generalized quadratically stable and the energy of every input controls mainly the energy of a corresponding output, and influences the energy of other outputs as weakly as possible.

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 MPC law with prediction dynamics belonging to a polytopic uncertainty set.

A LYAPUNOV-BASED APPROACH FOR THE CONTROL OF BIOMIMETIC ROBOTIC SYSTEMS WITH PERIODIC FORCING INPUTS

Bio-mimetic Robotics often deploys locomotion mechanisms (swimming, crawling, flying etc…) which rely on repetitive patterns for the actuation schemes. This directly translates into periodic forcing inputs for the dynamics of the mechanical system. Closed loop control is achieved by modulating shape-parameters (e.g. duty cycle) which directly affect the mean values of the forcing inputs. In this work, guided by an intuition inspired by linear systems theory, first a linear feedback law is derived that stabilizes a linearization of the average system, i.e. the system subject only to the average values of the forcing inputs, and then it is shown how this very feedback law can also guarantee boundedness of solutions of the original system. Boundedness is proved my means of a Lyapunov energy function easily derived in the linearized case. Unlike classical results found in literature in the areas of averaging and perturbation theory, this work instead of focussing on the existence of periodic limit cycles, simply restricts its attention on the boundedness of solutions, which directly translates into the possibility of deploying input functions which are continuous but not continuously differentiable.

Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation

In this paper, we present a design procedure of composite nonlinear feedback control for general multivariable systems with actuator saturation. We consider both the state feedback case and the measurement feedback case without imposing any restrictive assumption on the given systems. The composite nonlinear feedback control consists of a linear feedback law and a nonlinear feedback law without any switching element. The linear feedback part is designed to yield a closed-loop system with faster rise time, while at the same time not exceeding the actuator limits for the desired command input levels. The nonlinear feedback law is used to reduce overshoot and undershoot caused by the linear part. As such, a highly desired tracking performance with faster settling time and smaller overshoot can be obtained. The result is illustrated by a numerical example, which shows that the proposed design method yields a very satisfactory performance.

Input–output linearization of nonlinear systems using multivariable Legendre polynomials

This contribution presents a new approach for the numeric computation of the input–output linearizing feedback law, which is obtained exactly in an analytical form. By using a state space embedding technique the nonlinear system to be controlled is described by a higher order system with solely polynomial nonlinearities. Consequently, the nonlinearities of this system can be represented by multivariable Legendre polynomials, so that the derivation of the input–output linearizing feedback controller can be accomplished using the operational matrices of multiplication and of differentiation for Legendre polynomials.