h reinforcement learning control of robot manipulators using fuzzy wavelet networks in this paper an h reinforcement learning controller based on a fuzzy wavelet network fwn is proposed to perform a position tracking task for a robot manipulator the proposed controller adopts the actor critic reinforcement learning control scheme the primary reinforcement is generated by a performance measurement unit the learning unit of the controller consists of an associative search network asn and an adaptive critic network acn the asn is employed to approximate unknown nonlinear functions in the robot dynamics and the acn is utilized to construct a more informative signal than the primary reinforcement alone to tune the asn since the fwn can provide accurate function approximation both the asn and acn are implemented by the fwn in addition the proposed controller requires no prior knowledge about the dynamics of the robot manipulators and no off line learning phase moreover by employing the h control theory it is possible to attenuate the effects of the approximation errors of the fwns and external disturbances to a prescribed level in contrast to the general h problem only simple equations rather than riccati equations should be solved computer simulations on a scara robot with 3 degrees of freedom confirm the effectiveness of the fwn based controller with h stabilization

supervisory control of families of linear set point controllers 2 robustness a simply structured high level controller called a supervisor has recently been proposed in part i of this article ibid vol 41 1996 for the purpose of orchestrating the switching of a sequence of candidate set point controllers into feedback with an imprecisely modeled siso process so as to cause the output of the process to approach and track a constant reference input the process is assumed to be modeled by an siso linear system whose transfer function is in the union of a number of subclasses each subclass being small enough so that one of the candidate controllers would solve the set point tracking problem if the process transfer function was to be one of the subclass members this paper proves that without any further modification the same supervisor described in part i can also perform this function in the face of norm bounded unmodeled dynamics and moreover that none of the signals within the overall system can grow without bound in response to bounded disturbance and noise inputs whether they are constant or not

controllability and observability for a class of controlled switching impulsive systems in this note we study the controllability and observability problem for a class of controlled switching impulsive systems by proposing several formula of variation of parameters for this type of time varying systems and employing the characteristic polynomial theory of matrix we establish necessary and sufficient conditions for controllability and controlled observability with respect to a given switching time sequence specializing the obtained results to the case of time invariant linear switching impulsive systems we derive some simple algebraic criteria which include the results reported in the literature for time invariant linear switching systems linear impulsive systems and classical linear systems one example is worked out for illustration

reachability realization and stabilizability of switched linear discrete time systems abstract in this paper the reachability realization of a switched linear discrete time system which is a collection of linear time invariant discrete time systems along with some maps for switching among them is addressed the main contribution of this paper is to prove that for a switched linear discrete time system there exists a basic switching sequence such that the reachable controllable state set of this basic switching sequence is equal to the reachable controllable state set of the system hence the reachability controllability can be realized by using only one switching sequence we also discuss the stabilizability of switched systems and obtain a sufficient condition for stabilizability two numeric examples are given to illustrate the results

on reachability and stabilization of switched linear systems reachability and stabilization issues for switched linear control systems are addressed a necessary and a sufficient condition are presented for reachability under mild assumptions we prove that the switched linear control systems are stabilizable in addition we show that an element driven switching stabilization strategy can be explicitly constructed

tight bounds on computing error correcting codes by bounded depth circuits with arbitrary gates we bound the minimum number w of wires needed to compute any asymptotically good error correcting code c 0 1 n 0 1 n with minimum distance n using unbounded fan in circuits of depth d with arbitrary gates our main results are 1 if d 2 then w n lgn lglgn 2 2 if d 3 then w nlglgn 3 if d 2k or d 2k 1 for some integer k 2 then w n k n where 1 n lgn i 1 n i n and the operation gives how many times one has to iterate the function i to reach a value at most 1 from the argument n and 4 if d lg n then w o n for depth d 2 our n lgn lglgn 2 lower bound gives the largest known lower bound for computing any linear map the upper bounds imply that a necessarily dense generator matrix for our code can be written as the product of two sparse matrices using known techniques we also obtain similar but not tight bounds for computing pairwise independent hash functions our lower bounds are based on a superconcentrator like condition that the graphs of circuits computing good codes must satisfy this condition is provably intermediate between superconcentrators and their weakenings considered before

on controllability of switched linear systems in this paper the controllability of a switched linear system which is a collection of linear time invariant systems along with some maps for switching among them is addressed first a controllable state set is defined as a basic tool to discuss the controllability condition next it is proved that the controllability of a multi input system is equivalent to that of a single input system finally a sufficient and necessary condition for controllability of third order systems is presented according to the proof of this criterion we state a conjecture on the controllability condition of n dimensional systems a numeric example is given to illustrate the result

brief paper a full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs this paper studies open loop stabilization problem for bimodal systems with continuous vector field it is based on the earlier work of the authors on the controllability problem for the same class of systems a full characterization of stabilizability is established by presenting algebraic necessary and sufficient conditions it is also shown that controllability implies stabilizability for these systems in a very similar fashion to the linear case

technical communique reachability analysis of constrained switched linear systems in this note we investigate the reachability of switched linear systems with switching input constraints we prove that under a mild assumption of the feasible switching signals the reachability set is the reachable subspace of the unconstrained system we also address the local reachability for switched linear systems with input constraints and present a complete criterion for a general class of switched linear systems

a reinforcement learning adaptive fuzzy controller for robots in this paper a new reinforcement learning scheme is developed for a class of serial link robot arms traditional reinforcement learning is the problem faced by an agent that must learn behavior through trial and error interactions with a dynamic environment in the proposed reinforcement learning scheme an agent is employed to collect signals from a fixed gain controller an adaptive critic element and a fuzzy action generating element the action generating element is a fuzzy approximator with a set of tunable parameters and the performance measurement mechanism sends an error metric to the adaptive critic element for generating and transferring a reinforcement learning signal to the agent moreover a tuning algorithm of the proposed scheme that can guarantee both tracking performance and stability is derived from the lyapunov stability theory therefore a combination of adaptive fuzzy control and reinforcement learning scheme is also concerned with algorithms for eliminating a sequence of decisions from experience simulations of the proposed reinforcement adaptive fuzzy control scheme on the cart pole balancing problem and a two degree of freedom 2dof manipulator scara robot arm verify the effectiveness of our approach

robust stabilizing control laws for a class of second order switched systems abstract for a class of second order switched systems consisting of two linear time invariant lti subsystems we show that the so called conic switching law proposed previously by the present authors is robust not only in the sense that the control law is flexible to be explained further but also in the sense that the lyapunov stability resp lagrange stability properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations resp nonvanishing perturbations the analysis is possible since the conic switching laws always possess certain kinds of quasi periodic switching operations we also propose for a class of nonlinear second order switched systems with time invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system provided that the conic switching law exponentially stabilizes the linearized switched systems consisting of the linearization of each nonlinear subsystem this switched control law is robust in the sense mentioned above

simulation based optimization of markov reward processes this paper proposes a simulation based algorithm for optimizing the average reward in a finite state markov reward process that depends on a set of parameters as a special case the method applies to markov decision processes where optimization takes place within a parametrized set of policies the algorithm relies on the regenerative structure of finite state markov processes involves the simulation of a single sample path and can be implemented online a convergence result with probability 1 is provided

stability analysis of switched stochastic systems in this paper for a class of switched stochastic ss systems the moment stability m s and sample path stability sp s are investigated respectively and there are two main contributions first based on accurate estimations for the powers of solution of a special nonswitched stochastic nss system by employing the concepts of a lyapunov function and describing the switching laws with the average dwell time and the subsystems three sufficiency theorems of p th m s are given for the ss systems then for the sp s of such systems based on the results of p th m s two sufficiency theorems are obtained for p 2 and p 2 respectively

controllability of timed continuous petri nets with uncontrollable transitions this paper is concerned with controllability of timed continuous petri nets under infinite server semantics with uncontrollable transitions which are a class of hybrid systems piecewise linear this class of hybrid systems is suitable for representing biological systems high traffic information networks heavily loaded supply chains etc by adopting a control theory approach the contribution of this paper is the characterisation of controllability over sets of equilibrium markings potential equilibrium points first inside a single marking region or linear mode and later extended to several regions

convergence domain for time varying switched systems abstract in this paper we investigate the convergence domain for time varying switched systems we construct multiple lyapunov functions and each lyapunov function is decreasing outside a ball whose radius is time varying we give a relatively accurate convergence domain for a general time varying switched system finally a numerical example a mass spring damper system is provided to show the effectiveness of the theoretical results

a constructive approach to reachability realization of discrete time switched linear systems this paper studies the reachability of discrete time switched linear systems a new constructive approach to designing switching sequences is developed so that not only the reachable subspace is realized but also the required number of switchings is reduced significantly detailed comparison between the obtained result and that reported in the open literature is also made which verifies the advantage of the proposed approach

capacity achieving codes with bounded graphical complexity and maximum likelihood decoding in this paper the existence of capacity achieving codes for memoryless binary input output symmetric mbios channels under maximum likelihood ml decoding with bounded graphical complexity is investigated graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding irregular repeat accumulate ira codes are studied first utilizing the asymptotic average weight distribution aawd of these codes and invoking divsalar s bound on the binary input additive white gaussian noise biawgn channel it is shown that simple nonsystematic ira ensembles outperform systematic ira and regular low density parity check ldpc ensembles with the same graphical complexity and are at most 0 124 db away from the shannon limit however a conclusive result as to whether these nonsystematic ira codes can really achieve capacity cannot be reached motivated by this inconclusive result a new family of codes is proposed called low density parity check and generator matrix ldpc gm codes which are serially concatenated codes with an outer ldpc code and an inner low density generator matrix ldgm code it is shown that these codes can achieve capacity on any mbios channel using ml decoding and also achieve capacity on any bec using belief propagation bp decoding both with bounded graphical complexity moreover it is shown that under certain conditions these capacity achieving codes have linearly increasing minimum distances and achieve the asymptotic gilbert varshamov bound for all rates

systems control with generalized probabilistic fuzzy reinforcement learning reinforcement learning rl is a valuable learning method when the systems require a selection of control actions whose consequences emerge over long periods for which input output data are not available in most combinations of fuzzy systems and rl the environment is considered to be deterministic in many problems however the consequence of an action may be uncertain or stochastic in nature in this paper we propose a novel rl approach to combine the universal function approximation capability of fuzzy systems with consideration of probability distributions over possible consequences of an action the proposed generalized probabilistic fuzzy rl gpfrl method is a modified version of the actor critic ac learning architecture the learning is enhanced by the introduction of a probability measure into the learning structure where an incremental gradient descent weight updating algorithm provides convergence our results show that the proposed approach is robust under probabilistic uncertainty while also having an enhanced learning speed and good overall performance

a geometric approach for reachability and observability of linear switched impulsive systems abstract this paper is concerned with the reachability and observability of linear switched impulsive systems with singular impulse matrices first some new concepts with respect to the reachability and unobservability are introduced especially span reachability is proposed because the reachable sets of switched impulsive systems do not always constitute subspaces then the geometric characterization of the span reachable and unobservable sets is presented moreover the relations between the span reachable set unobservable set and the invariant subspaces of such systems are discussed finally corresponding criteria applied to linear impulsive systems and linear switched systems are also discussed

coding theorems for turbo code ensembles this paper is devoted to a shannon theoretic study of turbo codes we prove that ensembles of parallel and serial turbo codes are good in the following sense for a turbo code ensemble defined by a fixed set of component codes subject only to mild necessary restrictions there exists a positive number spl gamma sub 0 such that for any binary input memoryless channel whose bhattacharyya noise parameter is less than spl gamma sub 0 the average maximum likelihood ml decoder block error probability approaches zero at least as fast as n sup spl beta where spl beta is the interleaver gain exponent defined by benedetto et al in 1996

stabilization of switched linear systems with time delay in detection of switching signal a feedback stabilization problem for switched linear systems with time delay in detection of switching signal is formulated first online state feedback controller design method for asymptotic stability and exponential stability is given then offline state feedback controller design method for asymptotic stability and exponential stability is given as well

reachability and controllability of switched linear discrete time systems this paper investigates the reachability and controllability issues for switched linear discrete time systems geometric characterization of controllability is presented for reversible systems the controllable sets and the reachable sets are identified in wonham s geometric approach and verifiable conditions for reachability and controllability are also presented

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

robust adaptive critic control of nonlinear systems using fuzzy basis function networks an lmi approach abstract this paper proposes an adaptive critic tracking control design for a class of nonlinear systems using fuzzy basis function networks fbfns the key component of the adaptive critic controller is the fbfn which implements an associative learning network aln to approximate unknown nonlinear system functions and an adaptive critic network acn to generate the internal reinforcement learning signal to tune the aln another important component the reinforcement learning signal generator requires the solution of a linear matrix inequality lmi which should also be satisfied to ensure stability furthermore the robust control technique can easily reject the effects of the approximation errors of the fbfn and external disturbances unlike traditional adaptive critic controllers that learn from trial and error interactions the proposed on line tuning algorithm for aln and acn is derived from lyapunov theory thereby significantly shortening the learning time simulation results of a cart pole system demonstrate the effectiveness of the proposed fbfn based adaptive critic controller

output tracking control of nonlinear switched cascade systems using a variable structure control method this article is concerned with the output tracking control problem for a class of nonlinear switched cascade systems under some average dwell time based switching laws firstly stabilisation problem of this class of switched systems is considered then the result is extended to the output tracking control problem based on the variable structure control technique and the characteristic of the system the variable structure controllers and the average dwell time are designed the effectiveness of the proposed design approach is illustrated with simulation results

adaptive fuzzy controller application to the control of the temperature of a dynamic room in real time this paper presents a direct adaptive fuzzy controller for unknown monotonic nonlinear systems thus not requiring the system model but only a little information about it the plant monotonicity and its delay without any off line pre training the algorithm achieves very high control performance through a three stage algorithm 1 output scale factor 2 adaptation of the fuzzy rule consequents and 3 optimization of the position of the membership functions the design is simple in the sense that both the membership functions and the rule base can be initialized from arbitrary values it can be applied to a large class of monotonic dynamic or static plants due the fact that the system is able to modify its behaviour in real time i e during the control process

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

an analysis of temporal difference learning with function approximation we discuss the temporal difference learning algorithm as applied to approximating the cost to go function of an infinite horizon discounted markov chain the algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic markov chain with a finite or infinite state space we present a proof of convergence with probability one a characterization of the limit of convergence and a bound on the resulting approximation error furthermore our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal difference learning in addition to proving new and stronger positive results than those previously available we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators first we prove that divergence may occur when updates are not based on trajectories of the markov chain this fact reconciles positive and negative results that have been discussed in the literature regarding the soundness of temporal difference learning second we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

a time space tradeoff for sorting on a general sequential model of computation in a general sequential model of computation no restrictions are placed on the way in which the computation may proceed except that parallel operations are not allowed we show that in such an unrestricted environment text time cdot text space omega n 2 log n in order to sort n integers each in the range 1 n 2

stability analysis and control synthesis for switched systems a switched lyapunov function approach this paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete time domain the approach followed in this paper looks at the existence of a switched quadratic lyapunov function to check asymptotic stability of the switched system under consideration two different linear matrix inequality based conditions allow to check the existence of such a lyapunov function the first one is classical while the second is new and uses a slack variable which makes it useful for design problems these two conditions are proved to be equivalent for stability analysis investigating the static output feedback control problem we show that the second condition is in this case less conservative the reduction of the conservatism is illustrated by a numerical evaluation

brief paper periodic stabilizability of switched linear control systems stabilizability via direct observer based state feedback control for discrete time switched linear control systems slcss is investigated in this paper for an slcs the control factors are not only the control input but also the switching signal and they need to be designed in order to stabilize the system as a result stabilization design for slcss is more complicated than that for non switched ones differently from the existing approaches a periodic switching signal and piecewise constant linear state feedback control are adopted to achieve periodic stabilizability for such systems it is highlighted that multiple feedback controllers need to be designed for one subsystem for discrete time slcss it is proved that reachability implies periodic stabilizability via state feedback a necessary and sufficient criterion for periodic stabilizability is also established two stabilization design algorithms are presented for real application moreover it is proved that reachability and observability imply periodic stabilizability via observer based feedback for discrete time slcss periodic detectability as the dual concept of periodic stabilizability is discussed and the stabilization design algorithms via observer based feedback are presented as well

approximate dynamic programming with a fuzzy parameterization dynamic programming dp is a powerful paradigm for general nonlinear optimal control computing exact dp solutions is in general only possible when the process states and the control actions take values in a small discrete set in practice it is necessary to approximate the solutions therefore we propose an algorithm for approximate dp that relies on a fuzzy partition of the state space and on a discretization of the action space this fuzzy q iteration algorithm works for deterministic processes under the discounted return criterion we prove that fuzzy q iteration asymptotically converges to a solution that lies within a bound of the optimal solution a bound on the suboptimality of the solution obtained in a finite number of iterations is also derived under continuity assumptions on the dynamics and on the reward function we show that fuzzy q iteration is consistent i e that it asymptotically obtains the optimal solution as the approximation accuracy increases these properties hold both when the parameters of the approximator are updated in a synchronous fashion and when they are updated asynchronously the asynchronous algorithm is proven to converge at least as fast as the synchronous one the performance of fuzzy q iteration is illustrated in a two link manipulator control problem

performance of ldpc codes under faulty iterative decoding departing from traditional communication theory where decoding algorithms are assumed to perform without error a system where noise perturbs both computational devices and communication channels is considered here this paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low density parity check ldpc codes concentration of decoding performance around its average is shown to hold when noise is introduced into message passing and local computation density evolution equations for simple faulty iterative decoders are derived in one model computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases but arbitrarily small probability of error is not achievable probability of error may be driven to zero in another system model the decoding threshold again decreases smoothly with decoder noise as an application of the methods developed an achievability result for reliable memory systems constructed from unreliable components is provided

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

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

endcoding complexity versus minimum distance a bound on the minimum distance of a binary error correcting code is established given constraints on the computational time space complexity of its encoder where the encoder is modeled as a branching program the bound obtained asserts that if the encoder uses linear time and sublinear memory in the most general sense then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing that is the minimum relative distance of the code tends to zero in such a setting the setting is general enough to include nonserially concatenated turbo like codes and various generalizations our argument is based on branching program techniques introduced by ajtai the case of constant depth and or circuit encoders with unbounded fanins are also considered

brief paper design of switching sequences for controllability realization of switched linear systems in this paper we study the problem of designing switching sequences for controllability of switched linear systems each controllable state set of designed switching sequences coincides with the controllable subspace both aperiodic and periodic switching sequences are considered for the aperiodic case a new approach is proposed to construct switching sequences and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d d d 1 1 here d is the dimension of the controllable subspace d email protected i 1 m where a i b i are subsystems for the periodic case we show that the controllable subspace can be realized within d switching periods

state feedback control of switched linear systems an lmi approach this paper addresses the problem of state feedback control of continuous time switched linear systems with arbitrary switching rules a quadratic lyapunov function with a common matrix is used to derive a stabilizing switching control strategy that guarantees i the assignment of all the eigenvalues of each linear subsystem inside a chosen circle in the left hand half of the complex plane ii a minimum disturbance attenuation level for the closed loop switched system the proposed design conditions are given in terms of linear matrix inequalities that encompass previous results based on quadratic stability conditions with fixed control gains although the quadratic stability based on a fixed lyapunov matrix has been widely used in robust control design the use of this condition to provide a convex design method for switching feedback gains has not been fully investigated numerical examples show that the switching control strategy can cope with more stringent design specifications than the fixed gain strategy being useful to improve the performance of this class of systems

controllability and stabilizability of switched linear systems in this paper we study the conjecture made in ieee trans automat control 47 8 2002 1401 concerning the existence of basic switching sequence for switched systems and give it an affirmative answer and a construction method this paper proves that for switched linear system there exists a switching sequence such that the controllable state set of this switching sequence is equal to the controllable state set of the system hence the controllability can be realized by using only one switching sequence then the result is extended to the systems with time delay in controls furthermore we define the stabilizability of switched linear systems and based on certain transformation of state coordinate we give a necessary and sufficient condition for the stabilizability two examples are presented to illustrate the obtained results

adaptive control of robot manipulators using fuzzy logic systems under actuator constraints in this paper a stable fuzzy adaptive controller for trajectory tracking is developed for robot manipulators without velocity measurements taking into account the actuator constraints the controller is based on structural knowledge of the dynamics of the robot and measurements of link positions only the gravity torque including system uncertainty like payload variation etc is estimated by a fuzzy logic system fls the adaptive controller represents an amalgamation of a filtering technique to eliminate velocity measurements and the theory of function approximation using fls to estimate the gravity torque the proposed controller ensures the local asymptotic stability and the convergence of the position error to zero the proposed controller is robust not only to structured uncertainty such as payload parameter variation but also to unstructured one such as disturbances the validity of the control scheme is shown by simulations on a two link robot manipulator

controllability and observability of hybrid systems abstract a special class of hybrid systems is considered whose state space is a cross product space of a euclidean space and a finite state space such models may be used to represent systems subject to known abrupt parameter variations such as commu tated networks they may also be used to approximate some types of time varying systems controllability observability and stability of hybrid systems are inves tigated in particular a necessary and sufficient algebraic condition a simple algebraic criterion and a computationally simple algebraic sufficient test for controllability and observability are derived moreover a simple sufficient stability condition is provided

commuting and stable feedback design for switched linear systems abstract in the paper commuting and stable feedback design for switched linear systems is investigated this problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed loop systems are not only asymptotically stable but also commuting each other a new concept common admissible eigenvector set caes is introduced to establish necessary sufficient conditions for commuting and stable feedback controllers for second order systems a necessary and sufficient condition is established moreover a parametrization of the caes is also obtained the motivation comes from stabilization of switched linear systems which consist of a family of lti systems and a switching law specifying the switching between them where if all the subsystems are stable and commuting each other then the total system is stable under arbitrary switching