improved stability robustness bounds using state transformation for linear discrete systems abstract sufficient condition bounds on the linear time varying perturbations for robust stability of an asymptotically stable linear time invariant discrete time system have been reported recently the development involves both schwartz and triangular inequalities and tends to give conservative results this paper uses state transformation to obtain improvement in these bounds both unstructured and structured perturbation bounds are shown to improve using transformations the proposed analysis is applied to a fourth order macroeconomic system model

robustness analysis a case study the stability test of polynomials whose coefficients depend multilinearly on interval parameters is considered the authors describe and compare four brute force solution approaches these are eigenvalue calculation zero exclusion from a specified value set algebraic tests of real and complex hurwitz roots and the parameter space method they are applied to a simple example with two parameters and third order polynomial an interesting feature of the example is that it can have an isolated unstable point the example may be useful as a benchmark for future approaches to the multilinear problem all four methods are shown to be feasible for the simple example but they require effort

optimal stochastic control for performance and stability robustness a design method for discrete time control systems that maintain stability with minimum performance degradation in the presence of a variety of deterministic and stochastic perturbation disturbances is presented deterministic perturbation and modeling errors are represented by maximum deviations from a nominal linear discrete time system and stochastic perturbations are modeled by generalization of the sector bound nonlinearity concept to the stochastic case based on the given bounds on deterministic and stochastic perturbations an upper bound on the quadratic performance index which is useful in assessing the performance deterioration according to the feedback gain used is found then the optimal feedback gain which minimizes this bound on the performance index is derived in the time invariant case the existence of a stabilizing property of the bound optimal controller which is obtained by minimizing an average cost per stage is shown

intertwiners of uq sl 2 representations and the vector valued big q jacobi transform linear operators r are introduced on tensor products of evaluation modules of uq sl 2 obtained from the complementary and strange series representations the operators r satisfy the intertwining condition on finite linear combinations of the canonical basis elements of the tensor products infinite sums associated with the action of r on six pairs of tensor products are evaluated for two pairs the sums are related to the vector valued big q jacobi transform of the matrix elements defining the operator r in one case the sums specify the action of r on the irreducible representations present in the decomposition of the underlying indivisible sum of uqsl 2 tensor products in both cases bilinear summation formulae for the matrix elements of r provide a generalization of the unitarity property corresponding results are given for the remaining pairs

asymptotic behavior of a differential equation with distributed delays abstract in this paper sufficient conditions are established for the asymptotical behavior of solutions of the delay differential equation x t f t x t g t x t with distributed form delays some applications are also given for the neutral differential equation with distributed delays

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

stability and performance robustness for multivariable linear systems abstract this paper deals with both aspects of stability and performance robustness of linear systems in the time domain a linear time invariant feedback control law which yields the largest set of parameter variations is first determined it is then used as an initial guess for designing a robust control law which produces satisfactory performance for each of several operating points of the system given a priori the stability robustness properties of the resulting control law is also studied the proposed approach is applied to an example yielding larger stability bounds including satisfactory performance in comparison with recently reported results

sensitivity properties of a class of discrete time lqg controllers with computation delays sensitivity properties of the discrete time lqg controller which utilizes the optimal state predictor to compensate l step computation delays are discussed we obtain multiplicative decompositions of the sensitivity matrices at the input and output of a plant which clarify the effect of computation delays and imperfect observation furthermore these decompositions are used to show that duality between the sensitivity properties at the input and those at the output holds for l 1 and to identify the asymptotic sensitivity matrices obtained by loop transfer recovery procedures

models of q algebra representations matrix elements of the q oscillator algebra this article continues a study of function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions here a q analog of the oscillator algebra not a quantum algebra is considered it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed this local approach applies to more general families of special functions e g with complex arguments and parameters than does the quantum group approach it is shown that the matrix elements themselves transform irreducibly under the action of the algebra q analogs of a formula are found for the product of two hypergeometric functions 1f1 and the product of a 1f1 and a bessel function they are interpreted here as expansions of the matrix elements of a group operator

a note on tensor products of q algebra representations and orthogonal polynomials abstract we work out examples of tensor products of distinct generalized slq 2 algebras with a factor from the positive discrete series of representations of one algebra and a factor from the negative discrete series of the other we show that the equation for the common eigenfunctions of the casimir operator and the cartan subalgebra generator is just the three term recurrence relation corresponding to orthogonality for special cases of the askey wilson polynomials and this connection yields an almost immediate resolution of the tensor product representation into a direct integral of irreducible representation an identity for the matrix elements of the group representation operators with respect to the tensor product and the reduced bases follows easily cases where the measures for the orthogonal polynomials are not unique correspond to cases where the tensor products and their resolutions are also nonunique

tensor products of q superalgebra representations and q series identities we work out examples of tensor products for distinct q generalizations of euclidean oscillator and type superalgebras in cases where the method of highest weight vectors will not apply in particular we use the three term recurrence relations for askey wilson polynomials to decompose the tensor product of representations from the positive discrete series and representations from the negative discrete series we show that various q analogues of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and we compute the corresponding matrix elements of the group operators on these representation spaces we show that the matrix elements themselves transform irreducibly under the action of the quantum superalgebra the most important q series identities derived here are interpreted as the expansion of the matrix elements of a group operator via the exponential mapping in a tensor product basis in terms of the matrix elements in a reduced basis they involve q hypergeometric series with base

stability and performance robustness analysis with highly structured uncertainties necessary and sufficient test the author presents an extension of a technique of analyzing the stability of linear time invariant interconnected systems with uncertainties each element of the frequency response matrix is assumed to lie inside a polygon at each frequency it is shown that the exterior boundary of the image of the cartesian product of these polygons under the map det i m delta is contained in the boundary of the image of the cartesian product of the boundaries of these polygons under the same map thus the image can be constructed from the images of the edges the computational technique of r r e de gaston and m g safonov 1988 can then be applied to perform a nonconservative robustness analysis of the feedback system

on the loop transfer recovery one method of model based compensator design for linear multivariate systems consists of state feedback design and observer design athans 1971 a key step in recent work in multivariate synthesis involves selecting an observer gain so the final loop transfer function is the same as the state feedback loop transfer function doyle and stein 1979 goodman 1984 this is called loop transfer recovery ltr this paper shows how identification of the eigenstructure of the compensators that achieve ltr makes possible a design procedure for observer gain luenberger 1971 this procedure is based on the eigenstructure assignment of the observers the sufficient condition for ltr and the stability of the closed loop system is that the plant be minimum phase the limitation of this method might arise when the plant has multiple transmission zeros

robustness of discrete time systems for unstructured stochastic perturbations several stochastic stability robustness measures are presented for nominally exponentially stable linear discrete time systems with unstructured perturbations having second moment bounds dependence of these measures on the stability degree of the nominal system and other parameters used in the procedure is illustrated by using the time evolution of the second moment of the system state and stochastic lyapunov stability results positive super martingale convergence theorems the ability of nominally exponentially stable systems to maintain stability in the presence of unstructured stochastic linear and nonlinear perturbations is demonstrated quantitative results are given to determine the maximum modeling uncertainty which can be tolerated in design upper bounds on the second moments of stochastic perturbations to maintain the mean square and almost sure stability of these systems in the presence of unstructured perturbations are obtained

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

a set of sums for continuous dual q 2 hahn polynomials an infinite set l y r z r l n0 of linear sums of continuous dual q 2 hahn polynomials with prefactors depending on a complex parameter z is studied the sums l y r z have an interpretation in context with tensor product representations of the quantum affine algebra uq sl 2 involving both a positive and a negative discrete series representation for each l 0 the sum l y r z can be expressed in terms of the sum 0 y r z continuous dual q2 hahn polynomials and their associated polynomials the sum 0 y r z is obtained as a combination of eight basic hypergeometric series moreover an integral representation is provided for the sums l y r z with the complex parameter restricted by zq 1 this gives rise to a bilinear summation formula for the continuous dual q 2 hahn polynomials

design of robust state feedback laws existence criteria and computational methods are presented for finding stabilizing state feedback laws for systems with uncertain state or control matrices the uncertainties are modelled as sector bounded non linearities and feedback laws ensuring global asymptotic closed loop stability are obtained using lyapunov e direct method the algorithms essentially amount to repeatedly solving a parameter dependent riccati equation until the maximal solution becomes positive definite

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

some new properties of the structured singular value j c doyle et al 1982 have shown that a necessary and sufficient condition for robust stability or robust performance in the h sup infinity frame work may be formulated as a bound on the structured singular value mu of a specific matrix m which includes information on the system model the controller the model uncertainty and the performance specifications often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices t which are of direct interest to the engineer e g sensitivity or complementary sensitivity the present paper shows how to derive bounds on sigma t from bounds on mu m

discrete time lqg ltr technique for systems with time delays loop transfer recovery ltr is applied for lqg controllers with prediction estimators for a class of square minimum phase discrete time linear systems with time delays we consider the class of plants described by the triple a b c whose time delay structure is characterized by the following hypothesis on the plant impulse response cb cab cad 2b 0 det cad 1b 0 when the estimator gain is fixed by recovery technique it is shown that the loop transfer matrix at the plant input is equal to the transfer matrix obtained with the optimal digital regulator having d sample controller delays described in 6 the dual result corresponding to the situation where the feedback gain is fixed by recovery technique is also analysed

tensor products of uq sl 2 modules and the big q2 jacobi function transform four tensor products of evaluation modules of the quantum affine algebra uq sl 2 obtained from the negative and positive series the complementary and the strange series representations are investigated linear operators r z satisfying the intertwining property on finite linear combinations of the canonical basis elements of the tensor products are described in terms of two sets of infinite sums r t r t z 0 and r t r t z 0 involving big q2 jacobi functions or related nonterminating basic hypergeometric series inhomogeneous recurrence relations can be derived for both sets evaluations of the simplest sums provide the corresponding initial conditions for the first set of sums the relations entail a big q2 jacobi function transform pair an integral decomposition is obtained for the sum r t a partial description of the relation between the decompositions of the tensor products with respect to uqsl 2 or with respect to its complement in uq sl 2 can be formulated in terms of askey wilson

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

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

on the stability bounds of singularly perturbed systems the stability bound problem of linear time invariant singularly perturbed systems is considered a set of new stability conditions based on the frequency domain representation is derived the stability criteria can be easily verified by computing certain singular values within finite frequency intervals illustrative examples show that the proposed criteria actually induce a less conservative epsilon bound than the existing criteria and for certain cases an infinite epsilon bound may be obtained

lqg controller synthesis with plant and noise spectral uncertainties in this paper we consider the problem of robust controller synthesis with plant and noise spectral uncertainties in lqg optimal systems we frame an algorithm for synthesizing the robust controller to minimize the least favourable cost functional j via specifying the weighting matrices q and r the wiener hopf technique in the lqg optimal system the saddle point theory and a property of the contractive operator are employed to solve this problem

improved robust stability bounds for discrete time linear regulators with computational delays abstract recently obtained stability robustness bounds on linear time varying perturbations of an asymptotically stable linear time invariant discrete time system are applied to linear regulators with computational delays the bounds were developed using lyapunov theory and singular value decomposition for unstructured and structured time varying perturbations the new structured perturbation bound may provide improved results for the elemental perturbations over a bound based on the unstructured perturbation bound of ishihara 1988 automatica 24 696 700 as indicated both by an example and by sufficient conditions

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

guaranteed stability margins for the discrete time linear quadratic optimal regulator the stability properties of the linear quadratic deterministic optimal regulator are investigated in the discrete time case guaranteed phase and gain margins are obtained for the regulator in terms of the singular values of the matrices of the regulator state space description these margins are compared to their continuous time counterparts and their implication on the regulator robustness is discussed

robustness with simultaneous pole and zero movement across the jω axis in this paper we present some new tests for assessing the robustness of the stability property of a feedback design the tests offer a wider scope of application than those previously described in the literature the main result is a robustness test which can be used to assess closed loop stability when the nominal plant model and perturbed plant model representing uncertainty have different numbers of poles and zeros in the right half plane this test also indicates over which frequencies good sensitivity and robust stability are compatible objectives and over which frequencies a compromise has to be made contrary to popular belief if there are no uncertain zeros which can move across the imaginary axis then good sensitivity and robust stability are compatible objectives over the same frequencies

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

robust stability bounds for a class of discrete time regulators with computation delays abstract robust stability of discrete time regulators which utilize state predictors to compensate computation delays is considered novel expressions for the return difference matrices and the complementary sensitivity matrices at the input and the output of the regulator are found to obtain simple bounds for unstructured perturbations robust stability for perturbations of the system matrix and or the gain matrix is also considered under certain restrictions on the nominal system simple bounds for the perturbations are obtained directly from the characteristic equation it is shown that as far as the effect of the computation delays is concerned these bounds are explicitly related to those for the unstructured perturbations

an m matrix and majorant approach to robust stability and performance analysis for systems with structured uncertainty uncertain multiinput multioutput systems described in the frequency domain are discussed the theory of nonnegative matrices and m matrices is used along with majorant bounding techniques to develop robust stability and performance results for two types of uncertainty the first type is uncertainty with norm bounded subblocks the second type is uncertainty that has subblocks with known patterns but unknown gains this latter uncertainty representation allows the elements of a given uncertainty block to be correlated for uncertainties of this type a recursive analysis methodology is developed which yields increasingly nonconservative results throughout this paper the performance analysis determines norm bounds on the deviations of the outputs from their nominal values the results are illustrated by a numerical example

the majorant lyapunov equation a nonnegative matrix equation for robust stability and performance of large scale systems a new robust stability and performance analysis technique is developed the approach involves replacing the state covariance by its block norm matrix i e the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics a bound i e majorant for the block norm matrix is given by the majorant lyapunov equation a lyapunov type nonnegative matrix equation existence uniqueness and computational tractability of solutions to the majorant lyapunov equation are shown to be completely characterized in terms of m matrices two examples are considered for a damped simple harmonic oscillator with uncertain but constant natural frequency the majorant lyapunov equation predicts unconditional stability and for a pair of nominally uncoupled oscillators with uncertain coupling the majorant lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators a result not predictable from quadratic or vector lyapunov functions

reduced order dynamic compensator design for stability robustness of linear discrete time systems a reduced order dynamic compensator design is presented with stability robustness for linear discrete systems by including a stability robustness component in addition to the standard quadratic state and control terms in the performance criterion the robustness component is based on an unstructured perturbation stability bound for time varying perturbations the controller design is developed by the parameter optimization technique and involves the solution of five algebraic matrix equations four of which are discrete time lyapunov matrix equations

models of q algebra representations tensor products of special unitary and oscillator algebras this paper begins a study of one and two variable function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions the algebras considered are the quantum algebra uq su2 and a q analog of the oscillator algebra not a quantum algebra in each case a simple one variable model of the positive discrete series of finite and infinite dimensional irreducible representations is used to compute the clebsch gordan coefficients it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed it is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations it is also demonstrated that general q hypergeometric functions can occur as

on the stability region of scalar delay differential equations on considere l equation differentielle scalaire a retard x t x t x t q ou r et q 0 on definit les regions de stabilite des solutions

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

extensions of the small mu test for robust stability the notion of a topologically arcwise connected set is used to develop a general graph topology based robustness test the test provides a switching criterion among different inequality tests and possibly among different nominal models when it is applied to extend the small mu test the result reduces conservatism and allows a more natural application of the test