global stabilization and restricted tracking with bounded feedback for multiple oscillator systems this paper is concerned with the problems of global stabilization and restricted tracking for a multiple oscillator system by bounded feedback a nonlinear feedback law consisting of nested saturation functions is proposed to achieve global stabilization to improve the transient performance of the closed loop system the parameters in this feedback law can be tuned to assign the poles of the closed loop system when none of the saturation elements in the control law is in effect the proposed feedback law is also shown to achieve restricted tracking simulation results are given to show the effectiveness of the proposed feedback law

nonlinear control of feedforward systems with bounded signals the stabilization problem for a class of nonlinear feedforward systems is solved using bounded control it is shown that when the lower subsystem of the cascade is input to state stable and the upper subsystem not exponentially unstable global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the upper part of the state only this is made possible by invoking the bounded real lemma and a generalization of the small gain theorem thus stabilization is achieved with typical saturation functions saturations of constant sign or quantized control moreover the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback finally a new class of stabilizing control laws for a chain of integrators with input saturation is proposed some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems

on constructing constrained control lyapunov functions for linear systems this works considers the problem of control of linear systems with input constraints the notion of control lyapunov functions clfs is first generalized and relaxed to define a constrained control lyapunov function cclf and a constructive procedure for cclf is subsequently presented

synchronization of identical linear dynamic systems subject to input saturation abstract this paper investigates the synchronization problem for a group of agents with identical linear dynamics subject to input saturation two classes of linear systems asymptotically null controllable with bounded control ancbc systems and double integrator systems are studied for ancbc systems it is shown that a linear protocol with the control gain obtained via parametric lyapunov equations can semiglobally synchronize the undirected topology provided that its augmented directed topology has a spanning tree for a special case of ancbc the double integrator systems it is established that if the augmented directed topology has a spanning tree then in the presence of input saturation using linear protocols with positive control gains can achieve global synchronization two numerical examples are given to demonstrate the effectiveness of the theoretical results

technical communique semi global stabilization of linear time delay systems with control energy constraint this note is concerned with the problem of semi globally stabilizing a linear system with an input delay and a constraint on the energy of its input under the condition of null controllability with vanishing energy the parametric lyapunov equation based l 2 low gain feedback is adopted to solve the problem the proposed approach is applied to the linearized model of the relative motion in the orbit plane of a spacecraft with respect to another spacecraft in a circular orbit around the earth to validate its effectiveness

nonlinear control of constrained linear systems via predictive reference management a method based on conceptual tools of predictive control is described for solving set point tracking problems wherein pointwise in time input and or state inequality constraints are present it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time a virtual sequence among a family of linearly parameterized command sequences by solving a convex constrained quadratic optimization problem and feeds the primal system according to a receding horizon control philosophy the overall system is proved to fulfill the constraints be asymptotically stable and exhibit an offset free tracking behavior provided that an admissibility condition on the initial state is satisfied though the cg can be tailored for the application at hand by appropriately choosing the available design knobs the required online computational load for the usual case of affine constraints is well tempered by the related relatively simple convex quadratic programming problem

on analytical approximation of the maximal invariant ellipsoids for linear systems with bounded controls analytical approximation of the maximal invariant ellipsoids for linear systems with actuator saturation is established simultaneously a necessary and sufficient condition for such approximation equalling to the exact maximal invariant ellipsoid is presented in addition a geometric property of the maximal invariant ellipsoid in ldquominimal energy control with guaranteed convergence raterdquo problem is proposed all results are given in explicit forms and easy to implement in system analysis and design two numerical examples are used to show the effectiveness of the proposed approach

output regulation of linear systems with bounded continuous feedback this paper studies the classical problem of output regulation for linear systems subject to control constraint the asymptotically regulatable region the set of all initial conditions of the plant and the exosystem for which output regulation is possible is characterized in terms of the null controllable region of the antistable subsystem of the plant continuous output regulation laws of both state feedback type and error feedback type are constructed from a given stabilizing state feedback law it is shown that a stabilizing feedback law that achieves a larger domain of attraction leads to a feedback law that achieves output regulation on a larger subset of the asymptotically regulatable region a feedback law that achieves global stabilization on the asymptotically null controllable region leads to a feedback law that achieves output regulation on the entire asymptotically regulatable region

saturated delayed controls for feedforward nonlinear systems the saturated delayed feedback is presented to globally stabilize a class of feedforward nonlinear systems whose nominal dynamics is the cascade of multiple oscillators and multiple integrators key strategies are stated as follows 1 to compensate the arbitrarily large input delay and to carry out the recursive analysis we transform the concerned system into a normal form 2 in the convergence analysis for multiple oscillators that are relatively difficult to treat we combine the saturated delayed terms define the composite lyapunov functions and use the cauchy formula 3 for ease of the analysis on nonlinear terms we employ saturation levels to restrict them as applications the usual feedforward nonlinear systems with multiple integrators as the nominal dynamics and the null controllable linear systems are dealt with in some new manners and a saturated delayed controller is presented for the well known tora system

composite quadratic lyapunov functions for constrained control systems a lyapunov function based on a set of quadratic functions is introduced in this paper we call this lyapunov function a composite quadratic function some important properties of this lyapunov function are revealed we show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids these results are used to study the set invariance properties of continuous time linear systems with input and state constraints we show that for a system under a given saturated linear feedback the convex hull of a set of invariant ellipsoids is also invariant if each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators then their convex hull can also be made invariant by the same actuators for a set of ellipsoids each invariant under a separate saturated linear feedback we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant

an analysis and design method for discrete time linear systems under nested saturation this paper considers a discrete time linear system under nested saturation nested saturation arises for example in systems with actuators subject to both magnitude and rate saturation a condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant moreover this condition is shown to be equivalent to linear matrix inequalities lmis in the actual and auxiliary feedback gains as a result the estimation of the domain of attraction for a given set of feedback gains is then formulated as an optimization problem with lmi constraints by viewing the feedback gains as extra free parameters the optimization problem can be used for controller design

global predictive regulation of null controllable input saturated linear systems a predictive regulator is described for stabilizing linear plants subject to polyhedral input constraints while the asymptotic stability of general linear plants is ensured subject to a set membership condition on the initial state global asymptotic and semiglobal exponential regulation is achieved for input saturated plants which are asymptotically null controllable with bounded inputs the design knobs of the regulator can be chosen off line so as to compromise between nonconservative regulation performance and computational complexity and an online optimization problem of prequantifiable numerical burden results

linear systems subject to input saturation and time delay global asymptotic stabilization this note deals with the problem of global asymptotic stabilization of linear systems by bounded static feedbacks subject to time delay under standard necessary conditions we provide two constructions of controllers of nested saturation type which extends to the general case partial results of previous papers for arbitrary small bound on the control and large constant delay to validate the approach a third order integrator and oscillator with multiplicity two example is presented

brief predictive pi control of linear plants under positional and incremental input saturations a predictive control strategy is developed for set point tracking of lti plants in the presence of joint positional and incremental rate input saturation constraints the resulting control algorithm is built so as to provide an integral action capable to asymptotically reject arbitrary constant disturbances this task is achieved by possibly modifying on line the set point in the case this becomes incompatible with constraints and disturbances when applied to asymptotically null controllable with bounded inputs ancbi plants viz all stabilizable linear systems with eigenvalues in the closed unit circle the control law is shown to globally provide set point tracking in the presence of arbitrary constant disturbances and initial state for non ancbi plants viz systems with exponentially unstable modes the control law features the mentioned properties locally viz in bounded regions of initial states and disturbances

l_p stabilization of integrator chains subject to input saturation using lyapunov based homogeneous design consider the n th integrator dot x j nx sigma u e n where x in mathbb r n u in mathbb r j n is the n th jordan block and e n 0 cdots 0 1 t in mathbb r n we provide easily implementable state feedback laws u k x which not only render the closed loop system globally asymptotically stable but also are finite gain l p stabilizing with arbitrarily small gain as in a saberi p hou and a stoorvogel ieee trans automat control 45 2000 pp 1042 1052 these l p stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite time stabilization of linear systems we also provide additional l infty stabilization results for the case of both internal and external disturbances of the n th integrator namely for the perturbed system dot x j nx e n sigma k x d d where d in mathbb r and d in mathbb r n

attaining mean square boundedness of a marginally stable stochastic linear system with a bounded control input we construct control policies that ensure bounded variance of a noisy marginally stable linear system in closed loop it is assumed that the noise sequence is a mutually independent sequence of random vectors enters the dynamics affinely and has bounded fourth moment the magnitude of the control is required to be of the order of the first moment of the noise and the policies we obtain are simple and computable

on global leader following consensus of identical linear dynamic systems subject to actuator saturation abstract this paper studies the leader following consensus problem for a group of agents with identical linear systems subject to control input saturation we focus on two classes of linear systems neutrally stable systems and double integrator systems for neurally stable systems we establish that global consensus can be achieved by linear local feedback laws over a fixed communication topology and with proper choices of relative potential functions global consensus can also be achieved over a switching communication topology for double integrator systems we establish that global consensus can be achieved by linear local feedback laws over a fixed communication topology and with the help of a simple saturation function in the local feedback laws global consensus can also be achieved over a switching communication topology simulation results illustrate the theoretical results

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

optimization of disturbance rejection in systems with saturating actuators an extension of the lqr lqg methodology to systems with saturating actuators referred to as slqr slqg is obtained the development is based on the method of stochastic linearization using this method and the lagrange multiplier technique a solution to the slqr slqg problem is derived this solution is given by the standard riccati equations coupled with two transcendental equations which define the variance of the signal at the input of the saturation and the lagrange multiplier associated with the quadratic performance index it is shown that under the standard stabilizability and detectability conditions these equations have a unique solution which can be found by a simple bisection algorithm when the saturation is removed these equations reduce to the standard lqr lqg solution

a partial parameterization of nonlinear output feedback controllers for saturated linear systems this paper addresses the stabilization problem of linear systems subject to input saturation the major purpose is to introduce nonlinearity into control laws so as to expand the design freedom for performance enhancement to this end a new approach is developed which involves a partial differential matrix inequality pdmi a class of stabilizing feedback laws is explicitly obtained by solving this pdmi analytically and the feedback laws are parameterized by a nonlinear function further it is revealed that any linear observer can be used to realize the output feedback stabilization numerical examples including the seek control of a hard disk drive show that the introduced nonlinearity does contribute to the improvement of system performance the application to integral control is also discussed

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

null controllability of linear systems with constrained controls the paper considers the problem of steering the state of a linear time varying system to the origin when the control is subject to magnitude constraints necessary and sufficient conditions are given for global constrained controllability as well as a necessary and sufficient condition for the existence of a control satisfying the constraints which steers the system to the origin from a specified initial epoch x 0 t 0 the global result does not require zero to be an interior point of the control set omega and the theorem for constrained controllability at x 0 t 0 only requires that omega be compact not that it contain zero the results are compared to those available in the literature furthermore numerical aspects of the problem are discussed as is a technique for determining a steering control

stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions in this paper we develop a systematic lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration the only assumptions we make about the system are well posedness of the algebraic loop and local stability problems to be considered include the estimation of the domain of attraction the reachable set under a class of bounded energy disturbances and the nonlinear l2 gain the regional analysis is established through an effective treatment of the algebraic loop and the saturation deadzone function this treatment yields two forms of differential inclusions a polytopic differential inclusion pdi and a norm bounded differential inclusion ndi that contain the original system adjustable parameters are incorporated into the differential inclusions to reflect the regional property the main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain lyapunov function where the pdi or the ndi is valid with quadratic lyapunov functions conditions for stability and performances are derived as linear matrix inequalities lmis to obtain less conservative conditions we use a pair of conjugate non quadratic lyapunov functions the convex hull quadratic function and the max quadratic function these functions yield bilinear matrix inequalities bmis as conditions for stability and guaranteed performance level the bmi conditions cover the corresponding lmi conditions as special cases hence the bmi results are guaranteed to be as good as the lmi results in most examples the bmi results are significantly better than the lmi results

l and l 2 semi global stabilisation of continuous time periodic linear systems with bounded controls this article is concerned with l and l 2 semi global stabilisation of continuous time periodic linear systems with bounded controls two problems namely l semi global stabilisation with controls having bounded magnitude and l 2 semi global stabilisation with controls having bounded energy are solved based on solutions to a class of periodic lyapunov differential equations pldes resulting from the problem of minimal energy control with guaranteed convergence rate under the assumption that the open loop system is asymptotically null controllable with constrained controls periodic feedback are established to solve the concerned problems the proposed plde based approaches possess the advantage that the resulting controllers are easy to implement since the designers need only to solve a linear differential equation a numerical example is worked out to illustrate the effectiveness of the proposed approach

semi global stabilization of linear systems with position and rate limited actuators abstract it is shown that if a linear system is asymptotically null controllable with bounded controls then when subject to both actuator position and rate saturation it is semi globally stabilizable by linear state feedback if in addition the system is also detectable then it is semi global stabilizable via linear output feedback

a circle criterion for strong integral input to state stability abstract we present sufficient conditions for integral input to state stability iiss and strong iiss of the zero equilibrium pair of continuous time forced lur e systems where by strong iiss we mean the conjunction of iiss and small signal iss our main results are reminiscent of the complex aizerman conjecture and the well known circle criterion we derive a number of corollaries including a result on stabilisation by static feedback in the presence of input saturation in particular we identify classes of forced lur e systems with saturating nonlinearities which are strongly iiss but not iss

an algebraic approach to bounded controllability of linear systems in this paper we present an algebraic approach to the proof that a linear system with matrices a b is null controllable using bounded inputs if and only if it is null controllable with unbounde

semi global stabilizability of linear null controllable systems with input nonlinearities an h sub spl infin based lyapunov proof is provided for a result established by lin and saberi 1993 if a linear system is asymptotically null controllable with bounded controls then when subject to input saturation it is semi globally stabilizable by linear state feedback a new result is that if the system is also detectable then it is semi global stabilizable by completely linear output feedback further an extension which relaxes the requirements on the input characteristic is obtained

relaxed lyapunov criteria for robust global stabilisation of non linear systems the notion of the restricted robust control lyapunov function rclf is introduced and is exploited for the design of robust feedback stabilisers for non linear systems particularly it is shown for systems with input constraints that restricted rclfs can be easily obtained while rclfs are not available moreover it is shown that the use of restricted rclfs usually results in different feedback designs from the ones obtained by the use of the standard rclf methodology using the restricted rclfs feedback design methodology a simple controller that guarantees robust global stabilisation of a perturbed chemostat model is provided

neural network based near optimal control for a class of discrete time affine nonlinear systems with control constraints in this paper the near optimal control problem for a class of nonlinear discrete time systems with control constraints is solved by iterative adaptive dynamic programming algorithm first a novel nonquadratic performance functional is introduced to overcome the control constraints and then an iterative adaptive dynamic programming algorithm is developed to solve the optimal feedback control problem of the original constrained system with convergence analysis in the present control scheme there are three neural networks used as parametric structures for facilitating the implementation of the iterative algorithm two examples are given to demonstrate the convergence and feasibility of the proposed optimal control scheme

brief piecewise affine lyapunov functions for discrete time linear systems with saturating controls this paper is concerned with piecewise affine pwa functions as lyapunov function candidates for stability analysis of time invariant discrete time linear systems with saturating closed loop control inputs using a pwa model of saturating closed loop system new necessary and sufficient conditions for a pwa function be a lyapunov function are presented based on linear programming formulation of these conditions an effective algorithm is proposed for construction of such lyapunov functions for estimation of the region of local asymptotic stability compared to piecewise linear functions like minkowski functions pwa functions are more adequate to capture the dynamical effects of saturation nonlinearities giving strictly less conservative results the complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension being hence appropriate for problems with large state dimension but with few saturating inputs

robust stabilization via saturated feedback in this paper we deal with the problem of stabilization of uncertain systems in the presence of input constraint first algebraic conditions are derived for input to state stability of linear system with saturated linear feedback of low dimension then a recursive design procedure is derived for robust stabilization of block upper triangular nonlinear systems with feedforward structure

saturated stabilization for an uncertain cascaded system subject to an oscillator abstract this paper investigates saturated stabilization of an uncertain cascaded system that consists of an uncertain double integrator and an uncertain oscillator the saturation reduction analysis is carried out by using the time interval based contradiction approach while the exponential stability analysis of the corresponding reduced system is conducted by using the hurwitz lemma in doing the saturation reduction analysis an intractable problem is how to estimate the states of the oscillator fully using the property of saturation functions as well as the technique of lyapunov inequalities we can first estimate the states of the oscillator and then achieve the time derivative bounds that some saturated terms possess in a finite time this has helped to carry out the saturation reduction analysis as the applications of the suggested algorithm the stabilizing controllers have been provided for two benchmark systems the uncertain two mass spring and the translational oscillator with a rotational actuator tora

non rational dynamic output feedback for time delay systems with saturating inputs this paper addresses the synthesis of stabilizing non rational dynamic output feedback controllers for time delay systems in the presence of control saturation based on lyapunov krasovskii functionals the use of a modified sector condition and a classical change of variables theoretical lmi conditions that allow to simultaneously design the controller and a static anti windup loop are derived two synthesis objectives are considered the maximization of an estimate of the basin of attraction of the system and the guarantee that the trajectories of the system are bounded in the presence of bounded disturbances while ensuring an overbound on the gain from the disturbance to the regulated output

finite difference lattice boltzmann methods for binary fluids we investigate two fluid bhatnagar gross krook bgk kinetic methods for binary fluids the developed theory works for asymmetric as well as symmetric systems for symmetric systems it recovers sirovich s theory and is summarized in models a and b for asymmetric systems it contributes models c d and e which are especially useful when the total masses and or local temperatures of the two components are greatly different the kinetic models are discretized based on an octagonal discrete velocity model the discrete velocity kinetic models and the continuous ones are required to describe the same hydrodynamic equations the combination of a discrete velocity kinetic model and an appropriate finite difference scheme composes a finite difference lattice boltzmann method the validity of the formulated methods is verified by investigating i uniform relaxation processes ii isothermal couette flow and iii diffusion behavior

finite time stability analysis and stabilization by bounded linear time varying feedback abstract this paper studies finite time stability analysis and finite time stabilization of linear systems by bounded linear time varying feedback on the one hand both local and global finite time stability of general nonlinear time varying systems is investigated by the comparison principle and the notion of finite time escaping functions some lyapunov like stability theorems are established on the other hand finite time and prescribed time stabilization of linear systems by bounded linear time varying feedback are revisited based on the proposed finite time stability theorems two classes of new time varying high gain functions are proposed to reduce the regulation time some connections of the proposed results to existing results on lyapunov inequality based finite time stability analysis and nonlinear feedback based finite time stabilization are revealed the effectiveness of the proposed methods is illustrated by a numerical example

dynamics of flexible sliding beams non linear analysis part ii transient response in this paper the axially rigid sliding beam problem undergoing small deformations is examined first its governing equation and boundary conditions transformed to the fixed domain and the well known galerkin s approach used to study the transient response of this problem the results obtained are then compared with those in the literature subsequently the authors approach is extended to the non linear axially inextensible sliding beams undergoing large amplitude vibrations and solve several examples to show the differences between the solutions obtained via linear and non linear solvers

brief paper a predictive controller with artificial lyapunov function for linear systems with input state constraints this paper copes with the problem of satisfying input and or state hard constraints in set point tracking problems stability is guaranteed by synthesizing a lyapunov quadratic function for the system and by imposing that the terminal state lies within a level set of the function procedures to maximize the volume of such an ellipsoidal set are provided and interiorpoint methods to solve on line optimization are considered

saturations based nonlinear controllers with integral term validation in real time popular saturations based nonlinear controller usually include proportional and derivative components of the state or output the fact that in many applications these components do not suffice to insure the convergence to the desired output values motivate the addition of an integral term in this paper three configurations of nonlinear controllers based on saturation functions are improved with an integral component the stability of the three algorithms is analysed using the lyapunov theory simulation results validate the proposed control laws when they are applied to nonlinear systems with constant and unknown perturbations real time experiments realised with a quad rotor aerial vehicle and a hovercraft vehicle show that the proposed scheme can follow autonomously some trajectories and that it could be robust with respect to delays

global consensus for discrete time multi agent systems with input saturation constraints in this paper we consider the global consensus problem for discrete time multi agent systems with input saturation constraints under fixed undirected topologies we first give necessary conditions for achieving global consensus via a distributed protocol based on relative state measurements of the agent itself and its neighboring agents we then focus on two special cases where the agent model is either neutrally stable or a double integrator for the neutrally stable case any linear protocol of a particular form which solves the consensus problem for the case without input saturation constraints also solves the global consensus problem for the case with input saturation constraints for the double integrator case we show that a subset of linear protocols which solve the consensus problem for the case without saturation constraints also solve the global consensus problem for the case with input saturation constraints the results are illustrated by numerical simulations

on stability and stabilization of the linearized spacecraft attitude control system with bounded inputs lyapunov stability of the roll yaw and pitch subsystems of a linearized spacecraft attitude control system are re examined along the lines of a previously proposed jordan form based approach it is found that the roll yaw subsystem is lyapunov stable for inertially symmetric spacecraft with repeated eigenvalues on the imaginary axis which cannot be asserted by the existing characteristic equation based approach globally stabilizing controllers with bounded linear feedback are then designed for such spacecraft and the optimal feedback gain is designed such that the convergence rate of the linearized closed loop system is maximized the proposed method is also applicable to inertially symmetric spacecraft with magnetic actuators

construction of interval observers for continuous time systems with discrete measurements we consider continuous time systems with input output and additive disturbances in the particular case where the measurements are only available at discrete instants and have disturbances to solve a state estimation problem we construct continuous discrete interval observers that are asymptotically stable in the absence of disturbances these interval observers are composed of two copies of the studied system and of a framer accompanied with appropriate outputs which give componentwise upper and lower bounds for the solutions of the studied system

global analysis of piecewise linear systems using impact maps and surface lyapunov functions this paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems pls this methodology infers global properties of pls solely by studying the behavior at switching surfaces associated with pls the main idea is to analyze impact maps i e maps from one switching surface to the next switching surface such maps are known to be unfriendly maps in the sense that they are highly nonlinear multivalued and not continuous we found however that an impact map induced by an linear time invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state this representation of impact maps allows the search for surface lyapunov functions sulf to be done by simply solving a semidefinite program allowing global asymptotic stability robustness and performance of limit cycles and equilibrium points of pls to be efficiently checked this new analysis methodology has been applied to relay feedback on off and saturation systems where it has shown to be very successful in globally analyzing a large number of examples in fact it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology examples analyzed include systems of relative degree larger than one and of high dimension for which no other analysis methodology could be applied this success in globally analyzing certain classes of pls has shown the power of this new methodology and suggests its potential toward the analysis of larger and more complex pls

the almost disturbance decoupling problem with internal stability for linear systems subject to input saturation state feedback case abstract we consider the almost disturbance decoupling problem addp and or almost d bounded disturbance decoupling problem addp d with internal stability for linear systems subject to input saturation and input additive disturbance via linear static state feedback we show that the almost d bounded disturbance decoupling problem with local asymptotic stability addp las d is always solvable via linear static state feedback as long as the system in the absence of input saturation is stabilizable no matter where the poles of the open loop system are and the locations of these poles play a role only in the solution of addp gas d addp sgas d or addp gas where semi global or global asymptotic stability is required

output regulation for linear systems subject to input saturation the purpose of this paper is to examine the problem of controlling a linear time invariant system subject to input saturation in order to have its output track or reject a family of reference or disturbance signals produced by some external generator it is shown that a semiglobal framework for this problem is a natural one within this framework a set of solvability conditions are given and feedback laws that solve the problem are constructed the theory developed in this paper incorporates earlier results established for linear systems without input saturation

antiwindup for stable linear systems with input saturation an lmi based synthesis this paper considers closed loop quadratic stability and l sub 2 performance properties of linear control systems subject to input saturation more specifically these properties are examined within the context of the popular linear antiwindup augmentation paradigm linear antiwindup augmentation refers to designing a linear filter to augment a linear control system subject to a local specification called the unconstrained closed loop behavior building on known results on h sub spl infin and lpv synthesis the fixed order linear antiwindup synthesis feasibility problem is cast as a nonconvex matrix optimization problem which has an attractive system theoretic interpretation the lower bound on the achievable l sub 2 performance is the maximum of the open and unconstrained closed loop l sub 2 gains in the special cases of zero order static and plant order antiwindup compensation the feasibility conditions become convex linear matrix inequalities it is shown that if and only if the plant is asymptotically stable plant order linear antiwindup compensation is always feasible for large enough l sub 2 gain and that static antiwindup compensation is feasible provided a quasi common lyapunov function between the open loop and unconstrained closed loop exists using the solutions to the matrix feasibility problems the synthesis of the antiwindup augmentation achieving the desired level of l sub 2 performance is then accomplished by solving an additional lmi

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

a switching anti windup design based on partitioning of the input space abstract this paper revisits the problem of enlarging the domain of attraction of a linear system with multiple inputs subject to actuator saturation by designing a switching anti windup compensator the closed loop system consisting of the plant the controller and the anti windup compensator is first equivalently formulated as a linear system with input deadzone we then partition the input space into several regions in one of these regions all inputs saturate with the time derivative of the saturated input being zero in each of the remaining regions there is a unique input that does not saturate the time derivative of the deadzone function associated with the unsaturating input is zero by utilizing these special properties of the inputs on an existing piecewise lyapunov function of the augmented state vector containing the deadzone function of inputs we design a separate anti windup gain for each region of the input space the switching from one anti windup gain to another is activated when the input signals leave one region for another which can be implemented online since only the measurement of the input signals is required simulation results indicate that the proposed approach has the ability to obtain a significantly larger estimate of the domain of attraction than the existing approaches

on the output regulation for linear systems in the presence of input saturation some results on the problem of achieving output regulation in the presence of sinusoidal constant exogenous signals for linear systems subject to input saturation are presented in particular the issue of enlarging the set of initial conditions for which the problem has solutions is addressed

simple saturated designs for ancbc systems and extension to feedforward nonlinear systems this article re examines the robust stabilisation of the asymptotically null controllable with bounded controls ancbc systems and extends the established algorithm to a wide class of feedforward nonlinear systems whose nominal dynamics contains both multiple integrators and multiple oscillators based on the notion of converging input bounded state cibs rather than small input small state siss the computation burden in sussmann et al sussmann h j sontag e d and yang y 1994 a general result on the stabilization of linear systems using bounded controls ieee transactions on automatic control 39 2411 2425 is reduced and a class of simple saturated control laws is presented for the cibs stabilisation of ancbc systems then by combining the technique of dealing with higher order terms the algorithm for ancbc systems is extended to feedforward nonlinear systems

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

brief paper an improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation this paper is concerned with the treatment of saturation nonlinearity existing commonly in practical control systems an improved treatment of such nonlinearity is proposed which includes the well known linear differential inclusion method for handling saturation nonlinearity as a special case to show the effectiveness of this new treatment the problem of analysis and design of linear systems subject to nested saturation function is revisited the proposed results not only complement the existing work on this problem but also are less conservative than the existing methods four numerical examples are worked out to illustrate the applicability of the established approaches

a parametric lyapunov equation approach to the design of low gain feedback low gain feedback has found several applications in constrained control systems robust control and nonlinear control low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic riccati equation are the eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter while the are approach requires the solution of an are for each value of the parameter this note proposes an alternative approach to low gain feedback design based on the solution of a parametric lyapunov equation such an approach possesses the advantages of both the eigenstructure assignment approach and the are based approach it also avoids the possible numerical stiffness in solving a parametric are and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach

output feedback stabilization of linear systems with actuator saturation the note presents a method for designing an output feedback law that stabilizes a linear system subject to actuator saturation with a large domain of attraction this method applies to general linear systems including strictly unstable ones a nonlinear output feedback controller is first expressed in the form of a quasi lpv system conditions under which the closed loop system is locally asymptotically stable are then established in terms of the coefficient matrices of the controller the design of the controller coefficient matrices that maximizes an estimate of the domain of attraction is then formulated and solved as an optimization problem with lmi constraints

command governors for constrained nonlinear systems a method is described for set point tracking in nonlinear systems when pointwise in time input and or state inequality constraints are to be enforced it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time the system input via a receding horizon strategy from a virtual sequence amongst all possible command sequences by solving a constrained quadratic optimization problem provided that the initial state is admissible the overall system is proved to fulfil the constraints and have desirable performance stability properties

local stabilization of discrete time linear systems with saturating controls an lmi based approach deals with the problem of local stabilization of linear discrete time systems subject to control saturation a linear matrix inequalities based framework is proposed in order to compute a saturating state feedback that stabilizes the system with respect to a given set of admissible initial states and in addition guarantees some dynamical performances when the system operates in the zone of linear behavior i e when the controls are not saturated

global stabilization of classes of linear control systems with bounds on the feedback and its successive derivatives in this paper we address the problem of globally stabilizing a linear time invariant lti system by means of a static feedback law whose amplitude and successive time derivatives up to a prescribed order p are bounded by arbitrary prescribed values we solve this problem for two classes of lti systems namely integrator chains and controllable skew symmetric systems with single input for the integrator chains the solution we propose is based on the nested saturations introduced by a r teel we show that this construction fails for skew symmetric systems and propose an alternative feedback law we illustrate these findings by the stabilization of the third order integrator and the harmonic oscillator with prescribed bounds on the feedback and its first two derivatives

design of linear systems with saturating linear control and bounded states a theorem is developed based on the circle criterion which gives sufficient conditions for the stability of a saturating linear control for a linear system with bounded states this theorem can be applied to determine a range of gains for the linear control which ensures stability it is shown by example that this approach results in an increased speed of response over that obtained by previous methods which do not account for state bounds

an input output approach to achieve adaptive tracking for saturating stable systems the problem of controlling time invariant linear systems subject to parametric uncertainty and input saturation constraint is dealt with using a pole placement adaptive controller it is shown that the resulting adaptive control system can be described by a non linear feedback scheme then the l 2 stability condition for the control system turns out to be the positive realness of a transfer function involving the plant estimated model and the specified closed loop poles this makes it necessary to adapt on line the desired closed loop poles to the time varying estimated model the proposed adaptive controller keeps bounded all the closed loop sequences whatever the initial conditions furthermore it ensures a quite interesting output reference tracking behaviour more precisely the quality of the tracking depends on the reference sequence rate the slower the reference the better the tracking

an analysis and design method for linear systems under nested saturation this paper considers a linear system under nested saturation nested saturation arises for example when the actuator is subject to magnitude and rate saturation simultaneously a condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant moreover this condition is shown to be equivalent to linear matrix inequalities lmis in the actual and auxiliary feedback gains as a result the estimation of the domain of attraction for a given set of feedback gains can be formulated as an optimization problem with lmi constraints by viewing the feedback gains as extra free parameters the optimization problem can be used for controller design

synthesis of controllers for continuous time delay systems with saturating controls via lmis the stabilization of linear continuous time systems with time delay in the state and subject to saturating controls is addressed sufficient conditions obtained via a linear matrix inequality lmi formulation are stated to guarantee both the local stabilization and the satisfaction of some performance requirements the method of synthesis consists in determining simultaneously a state feedback control law and an associated domain of safe admissible states for which the stability of the closed loop system is guaranteed when control saturations effectively occur

observer based adaptive nn control for a class of uncertain nonlinear systems with nonsymmetric input saturation this paper is concerned with the problem of adaptive tracking control for a class of uncertain nonlinear systems with nonsymmetric input saturation and immeasurable states the radial basis function of neural network nn is employed to approximate unknown functions and an nn state observer is designed to estimate the immeasurable states to analyze the effect of input saturation an auxiliary system is employed by the aid of adaptive backstepping technique an adaptive tracking control approach is developed under the proposed adaptive tracking controller the boundedness of all the signals in the closed loop system is achieved moreover distinct from most of the existing references the tracking error can be bounded by an explicit function of design parameters and saturation input error finally an example is given to show the effectiveness of the proposed method

global stabilization of oscillators with bounded delayed input the problem of globally asymptotically stabilizing by bounded feedback of an oscillator with an arbitrary large delay in the input is solved a first solution follows from a general result on the global stabilization of null controllable linear systems with delay in the input by bounded control laws with a distributed term next it is shown through a lyapunov analysis that the stabilization can be achieved as well when neglecting the distributed terms it turns out that this main result is intimately related to the output feedback stabilization problem

control of linear systems subject to time domain constraints with polynomial pole placement and lmis this note focuses on the control of continuous time linear systems subject to time domain constraints input amplitude limitation output overshoot on closed loop signals using recent results on positive polynomials it is shown that finding a youla kuc spl breve era polynomial parameter of fixed degree hence a controller of fixed order such that time domain constraints are satisfied amounts to solving a convex linear matrix inequality lmi optimization problem as soon as distinct strictly negative closed loop poles are assigned by pole placement proceeding this way time domain constraints are handled by an appropriate choice of the closed loop zeros

discrete time l and l2 norm vanishment and low gain feedback with their applications in constrained control the existing low gain feedback which is a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero has been designed in very specific ways in this paper by recognizing the l and l 2 slow peaking phenomenon that exists in discrete time systems under low gain feedback more general notions of l and l 2 norm vanishment are considered so as to provide a full characterization of the nonexistence of slow peaking phenomenon in some measured signals low gain feedback that does not lead to l and l 2 slow peaking in the control input are respectively referred to as l and l 2 low gain feedback based on the notions of l and l 2 vanishment not only can the existing low gain feedback be recognized as an l low gain feedback but also a new design approach referred to as the l 2 low gain feedback approach is developed for discrete time linear systems parallel to the effectiveness of l low gain feedback in magnitude constrained control the l 2 low gain feedback is instrumental in the control of discrete time systems with control energy constraints the notions of l and l 2 vanishment also result in a systematic approach to the design of l and l 2 low gain feedback by providing a family of solutions including those resulting from the existing design methods

design of saturation based switching anti windup gains for the enlargement of the domain of attraction this technical note proposes a saturation based switching anti windup design for the enlargement of the domain of attraction of the closed loop system a widely adopted method of dealing with an m dimensional saturated linear feedback is to express it as a linear combination of a set of 2m auxiliary linear feedbacks for each value of the state these auxiliary linear feedbacks form a convex polyhedron of 2m vertices in the input space we propose to divide this convex polyhedron into several convex sub polyhedrons each of which is defined as the convex hull of a subset of the auxiliary feedbacks whenever the value of the saturated linear feedback falls into a sub polyhedron it can be expressed as a linear combination of a subset of the 2m auxiliary linear feedbacks that define the sub polyhedron and thus a less conservative result can be expected a separate anti windup gain is designed for each sub polyhedron by using a common quadratic lyapunov function and implemented when the value of the saturated linear feedback falls into this sub polyhedron simulation results indicate that such a saturation based switching anti windup design has the ability to significantly enlarge the domain of attraction of the closed loop system

lyapunov functions for stable cascades and applications to global stabilization generalizes lyapunov constructions for a cascade of two nonlinear systems one of which is stable rather than asymptotically stable a new cross term construction in the lyapunov function allows us to replace earlier growth conditions by a necessary boundedness condition this method is instrumental in the global stabilization of feedforward systems and new stabilization results are derived from the generalized construction

duality and uniqueness of convex solutions to stationary hamilton jacobi equations value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality hamilton jacobi characterizations and the conjugacy of primal and dual value functions are of main interest close ties between the uniqueness of convex solutions to a hamilton jacobi equation the uniqueness of such solutions to a dual hamilton jacobi equation and the conjugacy of primal and dual value functions are displayed simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon for which the value functions are conjugate leads to sufficient conditions on the conjugacy of the infinite time horizon value functions consequently uniqueness results for the hamilton jacobi equation are established little regularity is assumed on the cost functions in the control problems correspondingly the hamiltonians need not display any strict convexity and may have several saddle points

global asymptotic controllability implies input to state stabilization the main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable gac control affine systems that render the closed loop systems input to state stable iss with respect to actuator errors extensions for fully nonlinear gac systems with actuator errors are also discussed our controllers have the property that they tolerate small observation noise as well

saturation based switching anti windup design for linear systems with nested input saturation this paper proposes a saturation based switching anti windup design for the enlargement of the domain of attraction of a linear system subject to nested saturation a nestedly saturated linear feedback is expressed as a linear combination of a set of auxiliary linear feedbacks which form a convex hull where the nestedly saturated linear feedback resides this set of auxiliary linear feedbacks is then partitioned into several subsets the auxiliary linear feedbacks in each of these subsets form a convex sub hull of the original convex hull when the value of the nestedly saturated linear feedback falls into a convex sub hull it can be expressed as a linear combination of the subset of all the auxiliary feedbacks that form the convex sub hull a separate anti windup gain is designed for each convex sub hull by using a common quadratic lyapunov function and is implemented when the value of the nestedly saturated linear feedback falls into this convex sub hull simulation results indicate that such a saturation based switching anti windup design has the ability to significantly enlarge the domain of attraction of the closed loop system