circular planar graphs and resistor networks abstract we consider circular planar graphs and circular planar resistor networks associated with each circular planar graph there is a set p q of pairs of sequences of boundary nodes which are connected through a graph is called critical if removing any edge breaks at least one of the connections p q in we prove that two critical circular planar graphs are y equivalent if and only if they have the same connections if a conductivity is assigned to each edge in there is a linear from boundary voltages to boundary currents called the network response this linear map is represented by a matrix we show that if is any circular planar resistor network whose underlying graph is critical then the values of all the conductors in may be calculated from finally we give an algebraic description of the set of network response matrices that can occur for circular planar resistor networks

on some minimal supervarieties of exponential growth abstract in the present paper we investigate minimal supervarieties of given superexponent over fields of characteristic zero we show that any minimal supervariety of finite basic rank is generated by one of the minimal superalgebras introduced by giambruno and zaicev in 2003 furthermore it is proved that any minimal superalgebra whose graded simple components of the semisimple part are simple generates a minimal supervariety finally we state that the same conclusion holds when the semisimple part of a minimal superalgebra has exactly two arbitrary graded simple components

exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

minimal superalgebras generating minimal supervarieties it has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by giambruno and zaicev trans am math soc 355 5091 5117 2003 in the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided as a consequence we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety

codimension growth and minimal superalgebras a celebrated theorem of kemer 1978 states that any algebra satisfying a polynomial identity over a field of characteristic zero is pi equivalent to the grassmann envelope g a of a finite dimensional superalgebra a in this paper by exploiting the basic properties of the exponent of a pi algebra proved by giambruno and zaicev 1999 we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero in particular we prove that these algebras can be realized as block triangular matrix algebras over the base field the importance of such algebras is readily proved a is a minimal superalgebra if and only if the ideal of identities of g a is a product of verbally prime t ideals also such superalgebras allow us to classify all minimal varieties of a given exponent i e varieties v such that exp v d 2 and exp u d for all proper subvarieties u of v this proves in the positive a conjecture of drensky 1988 as a corollary we obtain that there is only a finite number of minimal varieties for any given exponent a classification of minimal varieties of finite basic rank was proved by the authors 2003 as an application we give an effective way for computing the exponent of a t ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras

semiglobal stabilization of linear discrete time systems subject to input saturation via linear feedback an are based approach we revisit the problem of semiglobal stabilization of linear discrete time systems subject to input saturation and give an algebraic riccati equation are based approach to the proof of a fact established earlier lin and saberi 1995 i e a linear discrete time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open loop poles are located inside or on the unit circle moreover we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work

electroid varieties and a compactification of the space of electrical networks abstract we construct a compactification of the space of circular planar electrical networks studied by curtis ingerman morrow 4 and colin de verdiere gitler vertigan 3 using cactus networks we embed this compactification as a linear slice of the totally nonnegative grassmannian and relate kenyon and wilson s grove measurements to postnikov s boundary measurements intersections of the slice with the positroid stratification leads to a class of electroid varieties indexed by matchings the uncrossing partial order on matchings arising from electrical networks is shown to be dual to a subposet of affine bruhat order the analogues of matroids in this setting are certain distinguished collections of non crossing partitions

exponential growth for codimensions of some p i algebras abstract by the giambruno zaicev theorem for associative p i algebras the exponential rate of growth of the codimensions of such a p i algebra is always a positive integer here we calculate that integer for various generic p i algebras which are given by a single identity these include capelli type identities and the various powers of the standard polynomials

proper identities lie identities and exponential codimension growth abstract the exponent exp a of a pi algebra a in characteristic zero is an integer and measures the exponential rate of growth of the sequence of codimensions of a a giambruno m zaicev on codimension growth of finitely generated associative algebras adv math 140 1998 145 155 a giambruno m zaicev exponential codimension growth of p i algebras an exact estimate adv math 142 1999 221 243 in this paper we study the exponential rate of growth of the sequences of proper codimensions and lie codimensions of an associative pi algebra we prove that the corresponding proper exponent exists for all pi algebras except for some algebras of exponent two strictly related to the grassmann algebra we also prove that the lie exponent exists for any finitely generated pi algebra the value of both exponents is always equal to exp a or exp a 1

on the factorability of polynomial identities of upper block triangular matrix algebras graded by cyclic groups abstract let f be an algebraically closed field of characteristic zero and g be an arbitrary finite cyclic group in this paper given an m tuple a 1 a m of finite dimensional g simple algebras we focus on the study of the factorability of the t g ideals id g u t a 1 a m of the g graded upper block triangular matrix algebras u t a 1 a m endowed with elementary g gradings induced by some maps when g is a cyclic p group we prove that the factorability of the ideal id g u t a 1 a m is equivalent to the g regularity of all except for at most one the g simple components a 1 a m as well to the existence of a unique isomorphism class of admissible elementary g gradings for u t a 1 a m moreover we present some necessary and sufficient conditions to the factorability of id g u t a 1 a 2 even in case g is not a p group with some stronger assumptions on the gradings of the algebras a 1 and a 2

identities and isomorphisms of graded simple algebras abstract let g be an arbitrary abelian group and let a and b be two finite dimensional g graded simple algebras over an algebraically closed field f such that the orders of all finite subgroups of g are invertible in f we prove that a and b are isomorphic if and only if they satisfy the same g graded identities we also describe all isomorphism classes of finite dimensional g graded simple algebras

young derived sequences of sn characters abstract given the sequence n is an s n character n 0 we construct the young derived sequence y y n n 0 y n n j 0 j o n j cocharacters of p i algebras are always young derived v drensky j algebra 91 1984 1 17 we study the relations between and y and between deg n and deg y n when is supported on a strip their asymptotics as n lead to some interesting integration formulas

standard polynomials are characterized by their degree and exponent abstract by the giambruno zaicev theorem giambruno and zaicev 1999 5 the exponent exp a of a p i algebra a exists and is always an integer in berele and regev 2001 2 it was shown that the exponent exp st n of the standard polynomial st n of degree n is not smaller than the exponent of any polynomial of degree n here it is proved that exp st n is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of st n

simultaneous global external and internal stabilization of linear time invariant discrete time systems subject to actuator saturation simultaneous external and internal stabilization in a global framework for linear time invariant discrete time systems subject to actuator saturation is considered internal stabilization is in the sense of lyapunov while external stabilization is in the sense of p stability with different variations e g with or without finite gain with fixed or arbitrary initial conditions with or without bias several simultaneous external and internal stabilization problems all in the global framework are studied in depth moreover we present a design for appropriate adaptive low and high gain feedback controllers that achieve the intended simultaneous external and internal stabilization whenever such problems are solvable

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

truncated predictor feedback stabilization of polynomially unstable linear systems with multiple time varying input delays this technical note is concerned with stabilization of polynomially unstable linear systems with multiple time varying input delays a truncated predictor feedback tpf which is obtained by truncating the distributed terms in the traditional predictor feedback is proposed the proposed tpf is a linear static partial state feedback with a time varying feedback gain that is constructed from the steady solution to a class of parametric lyapunov differential equations under the assumption that the open loop system is only polynomially unstable and is uniformly controllable it is shown that for sufficiently small value of the design parameter the resulting closed loop system is exponentially stable no matter how large the time varying delays are as long as they are bounded simulation demonstrates these theoretical results

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

a parametric periodic lyapunov equation with application in semi global stabilization of discrete time periodic systems subject to actuator saturation this paper is concerned with semi global stabilization of discrete time linear periodic systems subject to actuator saturation provided that the open loop characteristic multipliers are within the closed unit circle a low gain feedback design approach is proposed to solve the problem by state feedback our approach is based on the solution to a parametric discrete time periodic lyapunov equation the proposed approaches not only generalize the corresponding results for time invariant systems to periodic systems but also reveal some important intrinsic properties of this class of periodic matrix equations a numerical example is worked out to illustrate the effectiveness of the proposed approaches

a characterization of minimal varieties of zp graded pi algebras abstract let f be a field of characteristic zero and p a prime in the present paper it is proved that a variety of z p graded associative pi f algebras of finite basic rank is minimal of fixed z p exponent d if and only if it is generated by an upper block triangular matrix algebra u t z p a 1 a m equipped with a suitable elementary z p grading whose diagonal blocks are isomorphic to z p graded simple algebras a 1 a m satisfying dim f a 1 a m d

pi algebras and their cocharacters abstract all pi algebras r satisfy identities of the form f x y x 1 y 1 x 2 y 2 y n 1 x n theorem a the existence of these identities imply also that cocharacters x n r lie in a hook shaped strip of width depending on the degree of the minimal identity of r theorem c this extends a characterization of rings satisfying a capelli identity theorem b

logical composition of lyapunov functions this article introduces the use of r functions to compose single lyapunov functions lfs via classic boolean operators with the aim to obtain a rich family of non conventional generally non convex functions the main benefit of the proposed composition is the nice geometric interpretation since it corresponds to intersection and union operations in the phase space region the composition of lfs is parameterised through a variable and classic compositions of lfs through min and max operations are recovered as a special case for a particular value of the proposed logical composition is applied to region of asymptotic stability ras estimation problems where the union of several lfs corresponds to the union of the ras estimates obtained from the separate use of each lf likewise the intersection of several lfs defined on independent subsets of the state space variables provides a single lf for the overall dynamical system sufficient conditions for the composition function to be an lf are provide

simultaneous semi global lp stabilization and asymptotical stabilization for singular systems subject to input saturation abstract the simultaneous semi global l p stabilization and asymptotic stabilization in both semi global and global cases are considered for continuous time linear singular systems subject to input saturation based on the asymptotical property of the parametric generalized algebraic riccati equations a low and high gain state feedback control law and a scheduling technique are proposed when applied to the singular system subject to input saturation and external disturbance the semi global finite gain l p stabilization and semi global global asymptotic stabilization are obtained simultaneously the obtained results reduce to the existing ones for non singular linear systems when the linear singular system reduces to a non singular system furthermore our results also improve the semi global stabilization results for singular systems with saturation in the absence of the external disturbance

on the minimality of algebras with involution abstract in di vincenzo and la scala 2007 1 given a n tuple a 1 a n of finite dimensional simple algebras over a field of characteristic zero a block triangular matrix algebra with involution denoted by r ut a 1 a n was introduced and it was proved that any finite dimensional algebra with involution which is minimal with respect to its exponent is pi equivalent to r for a suitable choice of the algebras a i motivated by a conjecture stated in the same paper here we show that r is minimal when either it is symmetric or n 2

on asymptotic stabilizability of linear systems with delayed input this paper examines the asymptotic stabilizability of linear systems with delayed input by explicit construction of stabilizing feedback laws it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable i e all the open loop poles are on the closed left half plane a simple example shows that such results would not be true if the open loop system is exponentially unstable it is further shown that such systems when subject to actuator saturation are semiglobally asymptotically stabilizable by linear state or output feedback

minimal varieties of algebras of exponential growth abstract the exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer through this scale we can now classify all minimal varieties of given exponent and of finite basic rank as a consequence we describe the corresponding t ideals of the free algebra and we compute the asymptotics of the related codimension sequences verifying in this setting some known conjectures we also show that the number of these minimal varieties is finite for any given exponent we finally point out some relations between the exponent of a variety and the gelfand kirillov dimension of the corresponding relatively free algebras of finite rank

distributed and truncated reduced order observer based output feedback consensus of multi agent systems this technical note is concerned with output feedback consensus of both continuous time and discrete time multi agent systems mass characterized by high order linear systems with directed communication topologies distributed reduced order observer based protocols are established by only using the relative outputs and inputs of neighboring agents it is shown that consensusability by the proposed output feedback protocols is equivalent to the consensusability by state feedback protocol under the condition that the open loop dynamics of the mass is not exponentially unstable a truncated reduced order observer based protocol utilizing only the relative outputs of neighboring agents is also established a numerical example is given to illustrate the effectiveness of the proposed approaches

graded polynomial identities on upper block triangular matrix algebras let g be an arbitrary abelian group in the present paper we study the question of whether g graded upper block triangular matrix algebras over a field of characteristic zero are determined up to g graded isomorphism by their g graded polynomial identities we obtain some results for an elementary grading and as a consequence for any grading over an algebraically closed field

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

stabilization of exponentially unstable discrete time linear systems by truncated predictor feedback abstract predictor state feedback solves the problem of stabilizing a discrete time linear system with input delay by predicting the future state with the solution of the state equation and thus rendering the closed loop system free of delay the solution of the state equation contains a term that is the convolution of the past control input with the state transition matrix thus the implementation of the resulting predictor state feedback law involves iterative calculation of the control signal a truncated predictor feedback law results when the convolution term in the state prediction is discarded when the feedback gain is constructed from the solution of a certain parameterized lyapunov equation the truncated predictor feedback law has been shown to achieve asymptotic stabilization of a system that is not exponentially unstable in the presence of an arbitrarily large delay by tuning the value of the parameter small enough in this paper we extend this result to exponentially unstable systems stability analysis leads to a bound on the delay and a range of the values of the parameter for which the closed loop system is asymptotically stable as long as the delay is within the bound the corresponding output feedback result is also derived

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

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

block triangular matrix algebras and factorable ideals of graded polynomial identities abstract we study the graded polynomial identities of block triangular matrix algebras with respect to the grading defined by an abelian group g in particular we describe conditions for the t g ideal of a such algebra to be factorable as a product of t g ideals corresponding to the algebras defining the diagonal blocks moreover for the factorable t 2 ideal of a superalgebra we give a formula for computing its sequence of graded cocharacters once given the sequences of cocharacters of the t 2 ideals that factorize it we finally apply these results to a specific example of block triangular matrix superalgebra

simultaneous external and internal stabilization of linear systems with input saturation and non input additive sustained disturbances in this paper we study simultaneous external and internal stabilization of the linear system under input saturation and non input additive sustained disturbances for systems that are asymptotic null controllable with bounded control it is shown that a nonlinear dynamic feedback controller can be designed so that the closed loop states remain bounded for any initial condition and for two classes of sustained disturbances and that the equilibrium in the absence of disturbances is globally asymptotically stable

on codimension growth of finitely generated associative algebras abstract letabe a pi algebra over a fieldf we study the asymptotic behavior of the sequence of codimensionscn a ofa we show that ifais finitely generated overftheninv a limn c n a always exists and is an integer we also obtain the following characterization of simple algebras ais finite dimensional central simple overfif and only ifinv a dim a

noncommutative invariant theory of symplectic and orthogonal groups abstract we present a method for computing the hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group we apply our method to compute the hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated respectively by the grassmann algebra and the algebra of 2 2 upper triangular matrices these two varieties are remarkable with the property that they are the only minimal varieties of exponent 2

graded identities of block triangular matrices abstract let f be an infinite field and u t d 1 d n be the algebra of upper block triangular matrices over f in this paper we describe a basis for the g graded polynomial identities of u t d 1 d n with an elementary grading induced by an n tuple of elements of a group g such that the neutral component corresponds to the diagonal of u t d 1 d n in particular we prove that the monomial identities of such algebra follow from the ones of degree up to 2 n 1 our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for fields of characteristic zero and for particular g gradings in the characteristic zero case we also generalize results for the algebra u t d 1 d n c with a tensor product grading where c is a color commutative algebra generating the variety of all color commutative algebras

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

adaptive fuzzy control for a class of nonlinear discrete time systems with backlash an adaptive fuzzy controller design is studied for uncertain nonlinear systems in this paper the considered systems are of the discrete time form in a triangular structure and include the backlash and the external disturbance by using the prediction function of future states the systems are transformed into an n step ahead predictor the fuzzy logic systems flss are used to approximate the unknown functions unknown backlash and backlash inversion respectively a discrete time tuning algorithm is developed to estimate the optimal fuzzy parameters compared with the previous works for the discrete time systems with backlash the main contributions of the paper are that 1 the rigorous restriction for the functional estimation error is removed and 2 the external disturbance is bounded but the bound is not required to be known a novel controller and the adaptation laws are constructed by using the discrete taylor series expansion and the difference lyapunov analysis and thus those limitations in the previous works are overcome it is proven that all the signals in the closed loop system are bounded and that the system output can be to follow the reference signal to a bounded compact set a simulation example is provided to illustrate the effectiveness of the proposed approach

minimal algebras with respect to their exponent abstract given an m tuple a 1 a m of finite dimensional simple algebras we introduce a block triangular matrix algebra with involution denoted as ut a 1 a m where each a i can be embedded as algebra we describe the t ideal of r ut a 1 a m in terms of the ideals t a i and prove that any algebra with involution which is minimal with respect to its exponent is pi equivalent to r for a suitable choice of a 1 a m moreover we show that if m 1 or a i f for all i then r itself is a minimal algebra the assumption for the base field f is characteristic zero

brief paper a parametric lyapunov equation approach to low gain feedback design for discrete time systems low gain feedback a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero has found several applications in constrained control systems robust control and nonlinear control in the continuous time setting there are currently three ways of constructing low gain feedback laws the eigenstructure assignment approach the parametric are based approach and the parametric lyapunov equation based approach the eigenstructure assignment approach leads to feedback gains explicitly parameterized in the low gain parameter the parametric are based approach results in a lyapunov function along with the feedback gain but requires the solution of an are for each value of the parameter the parametric lyapunov equation based approach possesses the advantages of the first two approaches and results in both an explicitly parameterized feedback gains and a lyapunov function the first two approaches have been extended to discrete time setting this paper develops the parametric lyapunov equation based approach to low gain feedback design for discrete time systems

inverse problem in cylindrical electrical networks in this paper we study the inverse dirichlet to neumann problem for certain cylindrical electrical networks we define and study a birational transformation acting on cylindrical electrical networks called the electrical r matrix we use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder this conjecture extends work of curtis ingerman and morrow linear algebra appl 283 1998 pp 115 150 and of de verdiere gitler and vertigan comment math helv 71 1996 pp 144 167 for circular planar electrical networks we show that our conjectural solution holds for certain purely cylindrical networks here we apply the grove combinatorics introduced by kenyon and wilson trans amer math soc 363 2011 pp 1325 1364

stabilization of discrete time linear systems with an unknown time varying delay by switched low gain feedback in this note we investigate the stabilization of discrete time linear systems with an unknown time varying delay existing literature on the stabilization of time delay systems require some knowledge such as an upper bound of the delay in this paper by proposing a switched low gain feedback control law we are able to achieve asymptotic stabilization of the system without any knowledge of the delay in addition the proposed switched low gain feedback law also significantly improves the closed loop system performance in terms of overshoot and convergence speed in comparison with the traditional fixed gain design simulation study illustrates the effectiveness of our theoretical results

semi global exponential stabilization of linear discrete time systems subject to input saturation via linear feedbacks abstract in this paper we show that a linear discrete time system subject to input saturation is semi globally exponentially stabilizable via linear state and or output feedback laws as long as the system in the absence of input saturation is stabilizable and detectable and has all its poles located inside or on the unit circle furthermore the semi globally stabilizing feedback laws are explicitly constructed the results presented here are parallel to our earlier results on the continuous time counterpart lin and saberi 1993

approximation and monotonicity of the maximal invariant ellipsoid for discrete time systems by bounded controls an analytic approximation of the maximal invariant ellipsoid for a discrete time linear system with bounded controls is derived the approximation is expressed explicitly in terms of the coefficient matrices of the system and the positive definite matrix that represents the shape of the invariant ellipsoid it is shown that this approximation is very close to the exact maximal invariant ellipsoid obtained by solving either an lmi based optimization problem or a nonlinear algebraic equation furthermore the necessary and sufficient condition for such an approximation to be equal to the exact maximal invariant ellipsoid is established on the other hand the monotonicity of the maximal invariant ellipsoid resulting from the minimal energy control with guaranteed convergence rate problem is established that shows a trade off between increasing the size of the invariant ellipsoid and increasing the convergence rate of the closed loop system under a bounded control two illustrative examples demonstrate of the effectiveness of the results