comments on algorithm for simultaneous stabilization of single input systems via dynamic feedback wu et al 1990 presented a sufficient condition for the simultaneous stabilizability of single input systems their condition is shown in this note by a counterexample to be incorrect the flaw leading to this error is pointed out and the remedy for their error is provided

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

observer based h control of discrete time delay systems with random communication packet losses and multiplicative noises this paper investigates the observer based h control problem for a class of discrete time mixed delay systems with random communication packet losses and multiplicative noises where the mixed delays comprise both discrete and distributed time varying delays the random packet losses are described by a bernoulli distributed white sequence that obeys a conditional probability distribution and the multiplicative disturbances are in the form of a scalar gaussian white noise with unit variance in the presence of mixed delays random packet losses and multiplicative noises sufficient conditions for the existence of an observer based feedback controller are derived such that the closed loop control system is asymptotically mean square stable and preserves a guaranteed h performance then a linear matrix inequality lmi approach for designing such an observer based h controller is presented finally a numerical example is provided to illustrate the effectiveness of the developed theoretical results

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

computation of the structured stability radius via matrix sign function robustness of perturbed state space models of the form x a bdc x is considered where b c are given matrices a is an asymptotically stable matrix and d is the unknown perturbation matrix an efficient algorithm to compute the complex structured stability radius which is based on the properties of the matrix sign function is presented a comparison with previous algorithms shows the efficiency of the new algorithm

a necessary and sufficient condition for the stability of nonnegative interval discrete systems the stability of nonnegative discrete systems with interval uncertainty is discussed for this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals an illustrative example is used as a guideline to demonstrate the effectiveness of the result simple computational tests are suggested and their usefulness in this framework is discussed the possible extension of this result to more general classes of systems is addressed

a new fast algorithm for robust stability analysis of linear control systems with linearly dependent parametric uncertainties abstract in this paper a fast algorithm is proposed for the computation of stability margins in parameter space for linear control systems subject to structured uncertainties the case in which plant transfer function coefficients are affine in a set of physical uncertain parameters is considered the paper shows also how the proposed algorithm can be used to solve another interesting problem in robust control analysis i e the determination of the region of pole location of a closed loop control system including an entire family of uncertain plants

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

modeling and control of ito stochastic networked control systems with random packet dropouts subject to time varying sampling in this note we discuss the modeling and control of a class of ito stochastic networked control systems ncss with packet dropouts that are subject to time varying sampling the system under consideration is first modeled as a continuous time system with input delay that is subject to double randomness via the input delay approach then by assuming the packet drop rate and recurring to the celebrated formula of total probability an equivalent model which takes full advantage of probability distribution characteristics of both packet dropouts and sampling periods is established in particular the probability distribution values of the stochastic delay taking values in two given intervals can be explicitly obtained which is of crucial importance to model and analyze the actual problem in ncss based on that robustly exponentially mean square stability of the system with an h infty performance is guaranteed an h infty controller design procedure is then proposed finally a numerical simulation example is exploited to show the effectiveness and applicability of the results derived and some less conservative results are obtained

resonance problems for the p laplacian systems abstract two existence theorems of the solutions are obtained for the p laplacian systems at resonance under a landesman lazer type condition by critical point theory

linear system stability under parameter uncertainties a sufficient condition for linear system stability under parameter variations is presented it yields parameter bounds which are not conservative and the test is easily implemented on a digital computer

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

strictly hurwitz property invariance of quartics under coefficient perturbation in this note it is shown how the maximum interval centered around each of the coefficients of a prescribed fourth degree strictly hurwitz polynomial may be determined so that the strictly hurwitz property remains invariant when each coefficient might be perturbed to any arbitrary value within its permissible interval

network based h control for singular systems with event triggered sampling scheme in this paper the problem of event triggered h control is investigated for networked singular systems to make full use of the characteristics of networked control system a discrete event triggered communication scheme which only needs to supervise the system state in discrete instants is proposed the sampling interval is employed to unify the networked singular system and the event triggered condition into a new time delay system based on this model sufficient criteria are derived to guarantee the networked singular system regular impulse free and stable with an h norm bound the corresponding state feedback controller is designed a practical example is given to show that the proposed event triggered scheme is more effective and useful than some existing works

robust stability of lure systems with multiple nonlinearities the stability of a perturbed lure system is studied through the popov lyapunov approach the multivariable popov criterion is used to guarantee the stability of the nominal system and then the stability of the perturbed system is investigated by using the lyapunov function developed for the nominal system a robustness measure which gives some bounds on allowable perturbations for the system is derived evading the use of norm forms a feature of the proposed measure is that the observation of the range for individual parameter deviation is feasible

a note on robust pole placement pole placement algorithm in single input single output siso systems is discussed with respect to the corresponding real stability radius of the resulting closed loop polynomial

simple robust stability tests for polynomials with real parameter uncertainty in this paper we consider several problems related to the robust stability of polynomials whose coefficients depend on real parameters we first show that if these parameters appear linearly in the coefficients the computation of the largest stability hypercube with a fixed centre does not require a frequency sweep its size can be determined by considering a finite set of frequencies we also demonstrate that for some polynomials where the coefficients themselves are polynomials in the parameters the determination of stability can be made in a manner that involves a single frequency sweep and where at each frequency a finite number of simple tests are carried out the methods are based on the nyquist theorem and value set ideas and the approach provides analytical solutions

fast computation of the multivariable stability margin for real interrelated uncertain parameters an alternative implementation of an algorithm for the computation of the multivariable stability margin to check the robust stability of feedback systems with real parametric uncertainty is proposed this method eliminates the need for the frequency search by reducing it to the testing of a finite number of conditions these conditions have a special structure that allows a significant improvement in the speed of computation

computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices a method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known stable nominal system a globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented using the polynomial structure of the problem the algorithm generates a convergent sequence of interval estimates of the global extremum these intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure some numerical examples are reported showing attractive features of the algorithm from the point of view of computational burden and convergence behavior

robustness measures for linear systems with application to stability radii of hurwitz and schur polynomials in this paper we consider robustness measures stability radii for system matrices which are subjected to structured real and complex perturbations of the form a a bdc where b c are given matrices our object is twofold a to present a number of new results mainly concerning the real stability radius and its differences from the complex one b to give an overview of our approach to the robustness analysis of linear state space systems including basic properties and characterizations of the complex stability radius applying the results to the special case where a is in companion form and b 0 0 0 1 t we are able to determine stability radii for hurwitz and schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector computable formulae are obtained and illustrated by several examples

maximal polytopic stability domains in parameter space for uncertain systems abstract some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented the case in which the coefficients are affine in a set of system physical parameters is considered assuming that the parameters are subject to polytopic perturbations of a given structure for these cases which often arise in system stability robustness analysis as for example in the robustification of controllers for multivariable closed loop systems a procedure is given to estimate maximal stability polytopic regions in the space of parameters this result is also used to derive simple necessary and sufficient conditions for the discrete time stability of convex combinations of two polynomials

multiplicity of solutions for resonant elliptic systems we establish the existence and multiplicity of solutions for some resonant elliptic systems the results are proved by applying minimax arguments and morse theory

improved lmi representations for the analysis and the design of continuous time systems with polytopic type uncertainty simple modifications to the representation of the bounded real lemma brl in the linear continuous time invariant case are introduced these modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic type uncertainties a different lyapunov function accompanies each of the vertices of the uncertainty polytope thus eliminating the need for quadratic stability or stabilizability the advantages of these new representations are demonstrated by way of two examples

admissible additive perturbations for the general feedback system with two input two output plant and compensator the class of all admissible additive perturbations of the plant which do not destabilize the closed loop is established

allowable coefficient perturbations with preserved stability of a hurwitz polynomial given a polynomial p s ansn an 11sn 1 a1s a0 which satisfies the routh hurwitz criterion a procedure is given for finding non conservative upper bounds on the allowable variations of the coefficients such that the perturbed polynomial maintains the stability property

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

robust stability of linear systems described by higher order dynamic equations the stability radius of higher order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied different perturbation norms are considered the aim is to derive robustness criteria that are expressed directly in terms of the original data previous results on robust stability of hurwitz and schur polynomials are extended to monic matrix polynomials for disturbances acting via a uniform input matrix computable formulas are obtained whereas for perturbations with multiple input matrices structured singular values are involved

existence and multiplicity of nontrivial solutions for quasilinear elliptic systems abstract the existence and multiplicity of nontrivial solutions are obtained for the quasilinear elliptic systems by the linking argument the cohomological index theory and the pseudo index theory

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

on the relationship between the spectral radius and stability radius for discrete systems the relationship between the spectral radius and the stability radius is established via elementary calculation the result gives a discrete counterpart of the existing result for continuous systems and provides an upper bound for the stability radius the main result of the calculation is summarized in the form of a theorem

a pole placement design approach for systems with multiple operating conditions the author proposes design procedures based on state space pole placement techniques for systems with multiple operating conditions this is the so called simultaneous pole placement problem first the full state feedback problem is studied in which a nonlinear local pole placement solution is proposed the design condition is formulated in terms of the rank condition of a multimode controllability matrix then the output feedback problem is approached using a multimodel controller design which is an extension of the observer design to multimode systems the design is decomposed into separated global pole placement subproblems and a local pole placement subproblem for a system with some operating conditions having modes on the j omega axis but no modes at the origin in the open right half of the complex plane stabilizability and detectability conditions for the design of an asymptotically stabilizing control are established without any restriction on the number of inputs or outputs relations of this approach to other simultaneous control design approaches are pointed out

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

sensitivities of stability constraints and their applications let the real polynomial a s a 0 a 1 s a n s n with the coefficients being known differentiable functions a k x be given and let the constraints g i x 0 determine the strictly hurwitz property of the polynomial a s a simple and efficient method to calculate the derivatives partial g i x partial x j is proposed then the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed also a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

complete characterization of the spherical spectral set the problem of computing the spectral set of any given spherical family of polynomials is considered an algorithm based on the decomposition of the spectral set in three disjoint subsets with relate points having the same dimension of the back image in the parameter space is devised moreover the structure of the polynomial function which characterizes the main subset is analyzed such an analysis leads to significant simplifications for boundary generation and bounds computation of the spectral set

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

the robust root locus abstract in this paper a simple technique is described for generating the root loci of a feedback system which includes perturbations q rl entering affine linearly into the coefficients of the plant denoting the perturbed plant by p s q and the compensator by c s we address the following problem given a bounding set q rl for q find the locus of points z in the complex plane such that 1 kp z q c z 0 for some k 0 and some q q such points z are said to lie on the robust root locus one of the strengths of the technique presented in this paper is that it avoids the combinatoric explosion synonymous with gridding the l dimensional set q and plotting a large number of ordinary root loci associated with gridding the points instead the technique given here exploits only a 2 dimensional gridding of a bounded subset of the complex plane for the more general case when perturbations enter multilinearly into the coefficients a certain convex hull robust root locus is constructed finally the robust root locus technique is illustrated using a model of a supersonic transport plane

calculation of the largest generalized stability hypersphere in the robust stability problem for the maximum setting time and minimum damping ratio cases the results of a procedure which employs a geometric approach to derive the maximal ellipsoidal domains of stability in the physical parameter space for the case in which the characteristic polynomial coefficients depend linearly on the physical parameters of the plant obtained by biernacki et al see ibid vol ac 32 no 6 p 495 505 1987 are extended to a more generalized stability region such as the maximum setting time and the minimum damping ratio cases the method can be used to estimate the largest generalized stable hypersphere in the parameter space for a given controller

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

on sufficient conditions for the stability of interval matrices using gershgorin s diagonal dominance theorem new sufficient conditions for the stability of interval matrices are obtained the new conditions remove a restriction in heinen s 1983 analysis and are applicable to a wider class of problems than previously possible

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

pole assignment with robust stability this paper uses convex analysis for the pole assignment design of discrete time siso systems incorporating robust stability against norm bounded parametric perturbations in the plant transfer function the method involves designing an overparameterized pole assignment controller for the nominal plant with the overparameterization chosen to reduce the size of changes in the closed loop characteristic equation which result from plant perturbations sufficiency bounds on the l sub p norm of the perturbation guaranteeing stability are obtained

a routh like test for the stability of families of polynomials with linear uncertainty abstract the stability of a polynomial can be deduced from a routh table in this paper we show that such tabular tests can also be developed for the robust stability of polynomials whose coefficients depend linearly on real parameters

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions

robust stability and performance via fixed order dynamic compensation the discrete time case a discrete time feedback control design problem involving parametric uncertainty is considered a quadratic bound suggested by recent work on discrete time state space h sub infinity theory is utilized in conjunction with the guaranteed cost approach to guarantee robust stability with a robust performance bound the principal result involves sufficient conditions for characterizing robust full and reduced order controllers with a worst case h sub 1 performance bound