analysis and control of switched linear systems via dwell time min switching abstract this paper addresses analysis and switching control problems of switched linear systems to this end a mixed time driven and state dependent switching strategy is proposed it guarantees an average dwell time even when all subsystems are unstable the switching rule and its associated switching output feedback controllers are designed to stabilize the switched system and satisfy a weighted l 2 gain performance the proposed analysis and switching control approach could refrain frequent switches commonly observed in min switching based designs the effectiveness of the proposed approach has been illustrated through a spherical inverted pendulum example

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

distribution of lattice points visible from the origin let be a region in the plane which contains the origin is star shaped with respect to the origin and has a piecewise c 1 boundary for each integer q 1 we consider the integer lattice points from which are visible from the origin and prove that the 1 st consecutive spacing distribution of the angles formed with the origin exists this is a probability measure supported on an interval m with m 0 its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel

the distribution of the free path lengths in the periodic two dimensional lorentz gas in the small scatterer limit we study the free path length and the geometric free path length in the model of the periodic two dimensional lorentz gas sinai billiard we give a complete and rigorous proof for the existence of their distributions in the small scatterer limit and explicitly compute them as a corollary one gets a complete proof for the existence of the constant term c 2 3 ln 2 frac 27 zeta 3 2 pi 2 in the asymptotic formula h t 2 ln epsilon c o 1 of the ks entropy of the billiard map in this model as conjectured by p dahlqvist

on stability l2 gain and h control for switched systems this paper addresses the issues of stability l 2 gain analysis and h control for switched systems via multiple lyapunov function methods a concept of general lyapunov like functions is presented a necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized lyapunov like functions which enables derivation of improved stability tests an l 2 gain characterization and a design method for stabilizing switching laws a solution to the h control problem for switched systems is also provided

infinite horizon switched lqr problems in discrete time a suboptimal algorithm with performance analysis this paper studies the quadratic regulation problem for discrete time switched linear systems dslqr problem on an infinite time horizon a general relaxation framework is developed to simplify the computation of the value iterations based on this framework an efficient algorithm is developed to solve the infinite horizon dslqr problem with guaranteed closed loop stability and suboptimal performance due to these guarantees the proposed algorithm can be used as a general controller synthesis tool for switched linear systems

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

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

disturbance attenuation for systems with input saturation an lmi approach a new design technique is proposed to improve disturbance attenuation for systems with input saturation the main results rely on a multiobjective approach which combines the quadratic stabilization technique with constraints on inputs for the full state feedback problem the sufficient conditions for the feasibility of a high gain controller are expressed in terms of linear matrix inequalities lmi for output feedback design the controllers are assumed to have an observer based structure the sufficient conditions for the output feedback controller to recover the full state properties are also given in terms of lmi

stability and stabilization of continuous time switched linear systems this paper addresses two strategies for the stabilization of continuous time switched linear systems the first one is of open loop nature trajectory independent and is based on the determination of a minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function to be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition including chattering is expressed being nonconvex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

dynamic output feedback control of switched linear systems this paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete time domains a particular class of matrix inequalities the so called lyapunov metzler inequalities provides conditions for open loop stability analysis and closed loop switching control using state and output feedback switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the hamilton jacobi bellman inequality an upper bound on the optimal cost associated with a closed loop stabilizing switching strategy is provided as well the solution of the output feedback problem is based on the construction of a full order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule throughout the theoretical results are illustrated by means of academic examples a realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported

extensions of factorial states of c algebras let a be a c algebra b be a c subalgebra of a and be a factorial state of b sometimes may be extended to a factorial state of a by a tensor product method of sakai c algebras and w algebras springer verlag berlin heidelberg new york 1971 sometimes there is a weak expectation of a into b and then factorial extensions may be found by a method of sakai and tsui yokohama math j 29 1981 157 160 these two methods are shown to have the same effect and the factorial extensions produced by them are analysed 1985

gd aided iol input output linearisation controller for handling affine form nonlinear system with loose condition on relative degree abstractinput output linearisation iol may encounter a singularity problem when applied to the tracking control of affine form nonlinear system afns which may not have a well defined relative degree the singularity problem has occurred in the area of control for decades in this paper we incorporate the gradient dynamics gd into iol which leads to the gd aided iol method to solve the singularity problem with the proposition of the loose condition on relative degree moreover detailed theoretical analyses on tracking error bound and convergence performance of the corresponding gd aided iol controller are presented simulations and comparisons substantiate that the proposed gd aided iol method is capable of completing the tracking control task and conquering the singularity encountered in the afns

output feedback control for switched linear systems subject to actuator saturation this article is devoted to the output feedback control problem for switched linear systems subject to actuator saturation we consider both continuous and discrete time switched systems using the minimal switching rule nonlinear output feedbacks expressed in the form of quasi linear parameter varying system are designed to satisfy a pre specified disturbance attenuation level defined by the regional 2 l2 gains over a class of energy bounded disturbances the conditions are expressed in bilinear matrix inequalities and can be solved by line search coupled with linear matrix inequalities optimisation a spherical inverted pendulum example is used to illustrate the effectiveness of the proposed approach

the statistics of the trajectory of a certain billiard in a flat two torus we consider a billiard in the punctured torus obtained by removing a small disk of radius 0 from the flat torus 2 with trajectory starting from the center of the puncture in this case the phase space is given by the range of the velocity only let and respectively r denote the first exit time length of the trajectory and respectively the number of collisions with the side cushions when 2 is being identified with 0 1 2 we prove that the probability measures on 0 associated with the random variables and r are weakly convergent as varepsilon rightarrow 0 and explicitly compute the densities of the limits

brief hysteresis based switching algorithms for supervisory control of uncertain systems we address the problem of controlling a linear system with unknown parameters ranging over a continuum by means of switching among a finite family of candidate controllers we present a new hysteresis based switching logic designed specifically for this purpose and derive a bound on the number of switches produced by this logic on an arbitrary time interval the resulting switching control algorithm is shown to provide stability and robustness to arbitrary bounded noise and disturbances and sufficiently small unmodeled dynamics

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

anti windup strategies for discrete time switched systems subject to input saturation this paper deals with the design of anti windup compensator for discrete time switched systems subject to input saturation the cases of static and dynamic anti windup controllers are addressed aiming at maximising the estimate of the basin of attraction of the origin for the closed loop system two aspects of the switching law are taken into account during the design either it is arbitrary or it is a part of the complete control law theoretical conditions allowing to synthesise the anti windup compensator are mainly described through linear matrix inequalities computational oriented conditions are then provided to solve convex optimisation problems that are able to give a constructive solution

brief paper output feedback design for saturated linear plants using deadzone loops in this paper we present an lmi based synthesis approach on output feedback design for input saturated linear systems by using deadzone loops algorithms are developed for minimizing the upper bound on the regional l 2 gain for exogenous inputs with l 2 norm bounded by a given value and for minimizing this upper bound with a guaranteed reachable set or domain of attraction the proposed synthesis approach will always lead to regionally stabilizing controllers if the plant is exponentially unstable to semi global results if the plant is non exponentially unstable and to global results if the plant is already exponentially stable where the only requirement on the linear plant is detectability and stabilizability the effectiveness of the proposed techniques is illustrated with one example