state dependent scaling design for a unified approach to robust backstepping global robust stabilization of nonlinear systems is characterized as a state dependent scaling problem by choosing scaling functions recursively one can construct robust control laws for an extended class of strict feedback systems

optimal diagonal scaling for infinity norm optimization abstract a solution is presented for the previously unsolved diagonally scaled multivariable infinity norm optimization problem of minimizing d s a s s x s d 1 s over the set of stable minimum phase diagonal d s and stable x s this problem is of central importance in the synthesis of feedback control laws for robust stability and insensitivity in the presence of structured plant uncertainty the result facilitates the design of feedback controllers which optimize the excess stability margin 3 or equivalently the structured singular value 4 of diagonally perturbed feedback systems

the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

a framework for robustness analysis of constrained finite receding horizon control a framework for robustness analysis of input constrained finite receding horizon control is presented under the assumption of quadratic upper bounds on the finite horizon costs we derive sufficient conditions for robust stability of the standard discrete time linear quadratic receding horizon control formulation this is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms lending itself to s procedure tools which are used to convert robustness questions to tractable convex conditions robustness with respect to plant model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities simple examples demonstrate the approach

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

an alternative solution to the h optimal control problems abstract in this paper we present an alternative solution to the problem min x h n n a bxc where a b rmand c are rational matrices in h n n the solution circumvents the need to extract the matrix inner factors of b and c providing a multivariable extension of sarason s h interpolation theory 1 to the case of matrix valued b s and c s the result has application to the diagonally scaled optimization problem int d a bxc d 1 where the infimum is over d x h n n d diagonal

generalized l sub 2 synthesis introduces a general framework for control design which can be used to solve several open problems in robust and optimal control these include systems subject to independently norm bounded disturbances robust stability for systems subject to element by element bounded structured uncertainty certain classes of robust performance problems and systems subject to deterministic white noise disturbances this is achieved by combining concepts such as h infinity optimization linear matrix inequalities and integral quadratic constraints necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

higher integrability of iterated operators on differential forms abstract in this paper we first prove the local higher integrability and higher order imbedding theorems for the iterated operators defined on differential forms then we prove the global higher integrability and higher order imbedding inequalities for these operators finally we demonstrate applications of the main results by examples

robust performance of periodic systems robust performance of sampled data systems to structured periodic and quasi periodic uncertainty is considered and necessary and sufficient conditions are derived the conditions are finite dimensional and explicitly computing them is investigated the results are illustrated with an example this work is readily extended to yield exact and finite dimensional robust performance conditions for structured arbitrary time varying uncertainty

some strong p q type inequalities for the homotopy operator abstract we prove the boundedness of the homotopy operator in the weighted l p spaces as extensions of p p type norm inequalities for the homotopy operator we also establish some strong p q type norm inequalities with a r weights and power type weights for the homotopy operator acting on differential forms finally we obtain some estimates for singular integrals

robust state estimation for uncertain systems with averaged integral quadratic constraints this paper considers a robust state estimation problem for a new class of uncertain systems the new uncertainty class introduced in the paper involves structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint this uncertainty class combines aspects of norm bounded and stochastic uncertainty descriptions the solution to the robust state estimation problem is obtained by solving a parametrized riccati differential equation of the game type

robust stability analysis of time varying systems using time varying quadratic forms abstract we consider the robust stability of time varying linear systems subject to structured time varying uncertainty we provide an alternate proof of the result that robust stability holds if and only if a scaled small gain condition holds asymptotically the new proof employs an extension of the so called s procedure losslessness theorem to time varying quadratic forms which is of independent interest

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

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

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

necessary and sufficient conditions of stability a multiloop generalization of the circle criterion a linear time invariant system with a vector output and a vector input is described this system is closed by an uncertain nonlinear time varying feedback the only information on this feedback is given by several integral quadratic inequalities i e the uncertainties under consideration generalize the so called conic nonlinearities necessary and sufficient frequency domain conditions of stability are obtained an advanced version of the s procedure losslessness theorem and some other tools of the absolute stability theory are used

robustness margin calculation with dynamic and real parametric uncertainty the problem of robust stability in feedback control systems with real uncertain parameters and unmodeled dynamics is considered a robust margin r sub m is defined and an algorithm to calculate r sub m and the closely related multivariable stability margin k sub m is developed how robust performance for real parametric uncertainty can be handled in this framework is also shown the result essentially provides a method to check whether or not the infinity norm of a closed loop transfer matrix is less than a specified level over a bounded set of real parametric variations

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

quadratic stability with real and complex perturbations it is shown that the equivalence between real and complex perturbations in the context of quadratic stability to linear fractional unstructured perturbations does not hold when the perturbations are block structured for a limited class of problems quadratic stability in the face of structured complex perturbations is equivalent to a particular class of scaled norms and hence appropriate synthesis techniques coupled with diagonal constant scalings can be used to design quadratically stable systems

robust performance unknown disturbances and known fixed inputs this paper addresses the robust performance problem when the system under consideration is subjected to norm bounded time varying uncertainty different performance objectives are considered and computable bounds are obtained for the worst case performance in various norms in some important cases it is shown that these bounds are tight giving simple exact formulas for computing robust performance measures one of the important features of these formulas is that they enable the direct and exact calculation of a number of worst cast performance measures while avoiding entirely the iterations needed for computing these measures by existing methods that rely on repeated scalings and spectral radius bound evaluations

duality bounds in robustness analysis abstract multipliers are used in stability theory to reduce conservatism and exploit structural information about system components for an important class of stability problems which gives a resulting infinite dimensional multiplier optimization a corresponding dual problem is stated the dual gives valuable information about the original problem in particular error bounds for its finite dimensional approximations 1997 elsevier science ltd

continuation based numerical detection of after depolarization and spike adding thresholds the changes in neuronal firing pattern are signatures of brain function and it is of interest to understand how such changes evolve as a function of neuronal biophysical properties we address this important problem by the analysis and numerical investigation of a class of mechanistic mathematical models we focus on a hippocampal pyramidal neuron model and study the occurrence of bursting related to the after depolarization adp that follows a brief current injection this type of burst is a transient phenomenon that is not amenable to the classical bifurcation analysis done for example for periodic bursting oscillators in this letter we show how to formulate such transient behavior as a two point boundary value problem 2pbvp which can be solved using well known continuation methods the 2pbvp is formulated such that the transient response is represented by a finite orbit segment for which onsets of adp and additional spikes in a burst can be detected as bifurcations during a one parameter continuation this in turn provides us with a direct method to approximate the boundaries of regions in a two parameter plane where certain model behavior of interest occurs more precisely we use two parameter continuation of the detected onset points to identify the boundaries between regions with and without adp and bursts with different numbers of spikes our 2pbvp formulation is a novel approach to parameter sensitivity analysis that can be applied to a wide range of problems

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

robustness analysis and synthesis for nonlinear uncertain systems a state space characterization of robustness analysis and synthesis for nonlinear uncertain systems is proposed the robustness of a class of nonlinear systems subject to l sub 2 bounded structured uncertainties is characterized in terms of a nonlinear matrix inequality nlmi which yields a convex feasibility problem as in the linear case scalings are used to find a lyapunov or storage function that give sufficient conditions for robust stability and performances sufficient conditions for the solvability of robustness synthesis problems are represented in terms of nlmis as well with the proposed nlmi characterizations it is shown that the computation needed for robustness analysis and synthesis is not more difficult than that for checking lyapunov stability the numerical solutions for robustness problems are approximated by the use of finite element methods or finite difference schemes and the computations are reduced to solving linear matrix inequalities unfortunately while the development in this paper parallels the corresponding linear theory the resulting computational consequences are of course not as favourable

duality in multiplier based robustness analysis frequency domain conditions involving multipliers are useful for robustness analysis the resulting analysis problem is generally convex but infinite dimensional and numerical solutions restricted to finite dimensional subspaces need to be considered the resulting finite dimensional problem can be transformed into a linear matrix inequality which can be solved using efficient algorithms the paper presents a format for the dual of the infinite dimensional problem the dual problem can be used to estimate the conservatism of particular finite dimensional subspaces

model validation a connection between robust control and identification the gap between the models used in control synthesis and those obtained from identification experiments is considered by investigating the connection between uncertain models and data the model validation problem addressed is given experimental data and a model with both additive noise and norm bounded perturbations is it possible that the model could produce the observed input output data this problem is studied for the standard h sub infinity mu framework models a necessary condition for such a model to describe an experimental datum is obtained for a large class of models in the robust control framework this condition is computable as the solution of a quadratic optimization problem

robust synthesis in sub 1 a globally optimal solution in this paper a method to synthesize controllers that achieve globally optimal robust performance within any prespecified tolerance against structured norm bounded time varying and or nonlinear uncertainty is developed the performance measure considered is the infinity to infinity induced norm of a system s transfer function the method developed utilizes linear relaxation techniques to solve the infinite dimensional nonconvex problem via finite dimensional linear programming problems

analysis of robust h 2 performance using multiplier theory in this paper the problem of determining the worst case h2 performance of a control system subject to linear time invariant uncertainties is considered a set of upper bounds on the performance is derived based on the theory of stability multipliers and the solution of an original optimal control problem the numerical issues raised by the resulting computational problems are discussed in particular newly developed interior point convex optimization methods combined with linear matrix inequalities apply very well to the fast and accurate solution of these problems the new results compare favorably with prior ones the method can be extended to other types of perturbations

the slow passage through a hopf bifurcation delay memory effects and resonance this paper explores analytically and numerically in the context of the fitzhugh nagumo model of nerve membrane excitability an interesting phenomenon that has been described as a delay or memory effect it can occur when a parameter passes slowly through a hopf bifurcation point and the system s response changes from a slowly varying steady state to slowly varying oscillations on quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation this delay and its dependence on the speed of the parameter variation are described the model involves several parameters and particular singular limits are investigated one in particular is the slow passage through a low frequency hopf bifurcation where the system s response changes from a slowly varying steady state to slowly varying relaxation oscillations we find in this case the onset o

on robust stability with structured time variant perturbations abstract we consider the problem of robust stability for uncertain systems described in terms of multiple scalar structured time invariant perturbations which are norm bounded the issue we address is the role that the type of perturbation norm plays in the robust stability conditions to this end the l p induced norm for 1 p on the perturbations is considered and it is shown that the structured singular value is the necessary and sufficient condition for robust stability for all these norms we conclude that for time invariant perturbations in contrast to the time varying case robust stability conditions are independent of the type of perturbation norms

a parametrization of all the unfalsified plant models for mimo systems the results of the parametrization of all the unfalsified models for single input single output systems are extended to multi input multi output mimo systems in this parametrization the assumed a priori plant and noise information are upper bounds of the h sub spl infin norm of the plant matrix valued transfer function and the noise magnitude the parametrization is on the basis of several series of plant time domain identification experiment data it is shown that for mimo systems all the unfalsified plant matrix valued transfer functions can still be expressed by the linear fractional transformation of a fixed matrix valued transfer function and an h sub spl infin norm bounded structure fixed but uncertain matrix valued transfer function however the high dimensions of the matrix valued transfer functions involved in the parametrization hamper the direct application of these results to robust controller design to overcome these difficulties further efforts are needed

a numerical method for computing the structured singular value abstract this article suggests a new approach to computing doyle s structured singular value ssv of a matrix the ssv is a notion important in robust control and several iteration schemes exist for approximation a solution 1 2 4 8 10 11 our idea is to pick a special case of the general problem which we believe to be natural and give an algorithm for studying it based on off the shelf packages once we are committed to this special case we discuss a very general plant uncertainty problem it embraces real as well as complex plant perturbations of many kinds the idea is simple and we believe very natural to the problem

stabilization of uncertain systems with norm bounded uncertainty a control a robust stabilization problem in a state space setting is treated it is assumed that the states are available for feedback using a fixed lyapunov function approach quadratic stability it is shown that an open loop stabilizability condition is equivalent to the existence of a stabilizing memoryless linear state feedback controller as a consequence it is shown that the existence of a quadratically stabilizing nonlinear time varying dynamic state feedback controller implies the existence of a quadratically stabilizing memoryless linear time invariant state feedback compensator

noncommuting multidimensional realization theory minimality reachability and observability a generalization of the notions of minimality reachability and observability for parameter dependent and multidimensional systems modeled by linear fractional transformations over noncommuting operators is presented the multidimensional system results developed herein are formally similar to and directly specialize to the standard results when one dimensional systems are considered

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

a necessary and sufficient minimality condition for uncertain systems a necessary and sufficient condition is given for the exact reduction of systems modeled by linear fractional transformations lfts on structured operator sets this condition is based on the existence of a rank deficient solution to either of a pair of linear matrix inequalities which generalize lyapunov equations the notion of gramians is thus also generalized to uncertain systems as well as kalman like decomposition structures a related minimality condition the converse of the reducibility condition may then be inferred from these results and the equivalence class of all minimal lft realizations defined these results comprise the first stage of a complete generalization of realization theory concepts to uncertain systems subsequent results such as the definition of and rank tests on structured controllability and observability matrices are also given the minimality results described are applicable to multidimensional system realizations as well as to uncertain systems connections to formal powers series representations also exist

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

on l robust stability and performance with l 2 and l perturbations ubscrpt perturbations in this paper we will consider systems with linear time invariant perturbations we will analyze robust performance in the l2 and l settings the l2 setting gives rise to the familiar case of structured singular values and a stability criterion is given by the small theorem we show that although the necessary and sufficient criterion of robust stability for the l case l stability with structured l gain bounded perturbations is the same small criterion a system with l2 gain bounded perturbations is never l stable

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

a maximum modulus theorem for linear fractional transformations abstract a main result in the structured singular value theory is a maximum modulus like theorem for a special case of linear fractional transformations we extend that result to general linear fractional transformations

robust performance against time varying structured perturbations in this paper we consider the problem of robust performance analysis for some nominal system m z against bounded linear time varying structured feedback perturbations we introduce a natural input output notion of rate of variation for a linear time varying operator we then exhibit upper and lower bounds on the maximum rate of variation of these perturbations against which robust performance is achievable using these bounds we show that the existence of frequency dependent d scales that render the norm of m z less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one this result suggests that it is natural and well justified to deal with the frequency dependent upper bound for the spl mu norm rather than spl mu itself this is reassuring given that the upper bound is easily and reliably computable while computation of complex spl mu appears difficult

controller design for plants with structured uncertainty this paper addresses the problem of designing feedback controllers to achieve good performance in the presence of structured plant uncertainty and bounded but unknown disturbances a general formulation for the performance robustness problem is presented and exact computable conditions are furnished these conditions are then utilized for synthesizing robust controllers which involves solving l1 optimization problems these solutions are computed using the duality theory of lagrange multipliers approximations and computational issues are discussed

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

an optimization algorithm for checking feasibility of robust h infinity control problem for linear time varying uncertain systems an algorithm for checking feasibility of the robust h control problem for systems with time varying norm bounded uncertainty is suggested this algorithm is an iterative procedure on each step of

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

complex analytic theory of the μ function in this paper we consider the determinant of the multivariable return difference nyquist map crucial in defining the complex m function as a holomorphic function defined on a polydisk of uncertainty the key property of holomorphic functions of several complex variables that is crucial in our argument is that it is an open mapping from this single result only we show that in the diagonal perturbation case all preimage points of the boundary of the horowitz template are included in the distinguished boundary of the polydisk in the block diagonal perturbation case where each block is norm bounded by one a preimage of the boundary is shown to be a unitary matrix in each block finally some algebraic geometry together with the weierstrass preparation theorem allows us to show that the deformation of the crossover under holomorphic variations of certain parameters is continuous q 1999 academic press

an implicit small gain condition and an upper bound for the real structured singular value in this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem the implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and unlike prior bounds does not require frequency dependent scales or multipliers numerical results show that the implicit small gain bound compares favorably with real bounds that involve frequency dependent scales and multipliers

systems with structured uncertainty relations between quadratic and robust stability the relation between the notions of robust stability and quadratic stability for uncertain systems with structured uncertainty due to both real and complex parameter variations is discussed examples are presented to demonstrate that for systems containing at least two uncertain blocks the notions of robust stability for complex parameter variations and quadratic stability for real parameter variations are not equivalent a byproduct of these examples is that for this class of systems quadratic stability for real perturbations need not imply quadratic stability for complex perturbations this is in stark contrast with the situation in the case of unstructured uncertainty for which it is known that quadratic stability for either real or complex perturbations is equivalent to robust stability for complex perturbations and thus equivalent to a small gain condition on the transfer matrix that the perturbation experiences

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

stabilization of uncertain linear systems an lft approach this paper develops machinery for control of uncertain linear systems described in terms of linear fractional transformations lfts on transform variables and uncertainty blocks with primary focus on stabilization and controller parameterization this machinery directly generalizes familiar state space techniques the notation of q stability is defined as a natural type of robust stability and output feedback stabilizability is characterized in terms of q stabilizability and q detectability which in turn are related to full information and full control problems computation is in terms of convex linear matrix inequalities lmis the controllers have a separation structure and the parameterization of all stabilizing controllers is characterized as an lft on a stable free parameter

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

optimal constant i o similarity scaling for full information and state feedback control problems abstract in robust control problems capturing all robustness and performance objectives in a single h norm cost function is impossible an alternative approach still untilizing the h norm involves diagonal similarity scaling of certain closed loop transfer functions the set of allowable diagonal scalings is problem dependent and reflects assumptions about the uncertainty and desired performance objectives the scaling set considered here is a prescribed convex set of positive definite matrices we consider the optimal constant scaling problem for the full information h control problem the solution is obtained by transforming the original problem into a convex feasibility problem specifically a structured linear matrix inequality in special cases solvability of the full information problem is equivalent to solvability of the state feedback problem

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

on convexified robust control synthesis in this paper we study a convexified robust control problem and its relation to gain scheduling control it reveals that the robust control synthesis condition becomes convex under a special plant structure moreover for this class of robust control problems the gain scheduling control approach from scaled small gain theorem will not provide any performance improvement over convexified robust controllers based on this observation a convexified robust control synthesis framework with stringent performance and computational efficacy is proposed

robust e p stability and performance in this paper we study robust stability in the lp sense against structured perturbations of bounded lp norm the perturbations may be either linear time varying nonlinear time varying or nonlinear time invariant we develop exact tests for robust stability for each case in terms of convex optimization problems these results are extended to solve the corresponding robust performance problems with performance measured via the induced lp norm

global integrability theorems for a harmonic tensors abstract we extend global integrability theorems for the gradients of a harmonic functions to the exterior derivative of differential forms satisfying rather general nonhomogeneous elliptic equations these include the usual a harmonic equations geometric conditions on the boundary of the domains of integration imply a corresponding exponent of integrability in the process we generalize the weak reverse holder inequality to such differential forms

dimension reduction for semidefinite programs via jordan algebras we propose a new method for simplifying semidefinite programs sdp inspired by symmetry reduction specifically we show if an orthogonal projection map satisfies certain invariance conditions restricting to its range yields an equivalent primal dual pair over a lower dimensional symmetric cone namely the cone of squares of a jordan subalgebra of symmetric matrices we present a simple algorithm for minimizing the rank of this projection and hence the dimension of this subalgebra we also show that minimizing rank optimizes the direct sum decomposition of the algebra into simple ideals yielding an optimal block diagonalization of the sdp finally we give combinatorial versions of our algorithm that execute at reduced computational cost and illustrate effectiveness of an implementation on examples through the theory of jordan algebras the proposed method easily extends to linear and second order cone programming and more generally symmetric cone optimization

structured dissipativity and absolute stability of nonlinear systems the paper considers a problem of absolute stability for a nonlinear uncertain system with structured uncertainty an s procedure for nonlinear systems is established and using this the paper shows that the absolute stability problem is equivalent to an absolute stability problem for a nonlinear uncertain system with unstructured uncertainty the paper also introduces a new notion of structured dissipativeness for nonlinear systems this notion extends the standard notion of dissipativeness in that it allows for multiple supply functions the paper shows that a nonlinear uncertain system containing structured uncertainties will be absolutely stable if and only if a corresponding structured dissipativeness condition is satisfied it is also shown that structured dissipativeness is equivalent to the standard notion of dissipativeness with a parameter dependent supply function

an input output approach to structured stochastic uncertainty we consider linear time invariant systems with exogenous stochastic disturbances and in feedback with structured stochastic uncertainties this setting encompasses linear systems with both additive and multiplicative noise our concern is to characterize second order properties such as mean square stability mss and performance a purely input output treatment of these systems is given without recourse to state space models and thus the results are applicable to certain classes of distributed systems we derive necessary and sufficient conditions for mss in terms of the spectral radius of a linear matrix operator whose dimension is that of the number of uncertainties rather than the dimension of any underlying state space models our condition is applicable to the case of correlated uncertainties and reproduces earlier results for uncorrelated uncertainties for cases where state space realizations are given linear matrix inequality equivalents of the input output conditions are given

zhang neural network for online solution of time varying linear matrix inequality aided with an equality conversion in this paper for online solution of time varying linear matrix inequality lmi such an lmi is first converted to a time varying matrix equation by introducing a time varying matrix of which each element is greater than or equal to zero then by employing zhang s neural dynamic method a special recurrent neural network termed zhang neural network znn is proposed and investigated for solving online the converted time varying matrix equation as well as the time varying lmi such a znn model showed in an explicit dynamics exploits the time derivative information of time varying coefficients in addition theoretical analysis and results of the proposed znn model are discussed and presented to show its excellent performance on solving the time varying lmi computer simulation results further demonstrate the efficacy of the proposed znn model for online solution of the time varying lmi and the converted time varying matrix equation

when is d g scaling both necessary and sufficient it is shown that the well known d g scaling upper bound of the structured singular value is a nonconservative test for robust stability with respect to certain linear time varying uncertainties

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

brief the analysis of optimization based controllers many control techniques employ on line optimization in the determination of a control policy we develop a framework which provides sufficient convex conditions in the form of linear matrix inequalities for the analysis of constrained quadratic based optimization schemes these results encompass standard robustness analysis problems for a wide variety of receding horizon control schemes including those with polytopic and structured uncertainty a simple example illustrates the methodology

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

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

extended matrix cube theorems with applications to µ theory in control in this paper we study semi infinite systems of linear matrix inequalities which are generically np hard for these systems we introduce computationally tractable approximations and derive quantitative guarantees of their quality as applications we discuss the problem of maximizing a hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value with the help of our complex matrix cube theorem we demonstrate that the standard scaling upper bound on m is a tight upper bound on the largest level of structured perturbations of the matrixm for which all perturbed matrices share a common lyapunov certificate for the discrete time stability

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

affine characterizations of minimal and mode dependent dwell times for uncertain linear switched systems an alternative approach for minimum and mode dependent dwell time characterization for switched systems is derived while minimum dwell time results require the subsystems to be asymptotically stable mode dependent dwell time results can consider unstable subsystems and dwell times within a possibly unbounded range of values the proposed approach is related to lyapunov looped functionals a new type of functionals leading to stability conditions affine in the system matrices unlike standard results for minimum dwell time these conditions are expressed as infinite dimensional lmis which can be solved using recent polynomial optimization techniques such as sum of squares the specific structure of the conditions is finally utilized in order to derive dwell time stability results for uncertain switched systems several examples illustrate the efficiency of the approach

computational complexity reduction in scaled h synthesis abstract the h control problem with constant scaling is considered this problem is known to reduce to a search for matrices satisfying linear matrix inequalities lmis and an additional nonconvex constraint recently globally convergent algorithms to solve this nonconvex feasibility problem have been proposed a major factor that determines the computational complexity is the number of nonconvex constraints this paper gives verifiable conditions on the plant under which the reduction of nonconvex constraints is possible the conditions appear to be a strong requirement but in fact they are nautrally satisfied in certain problems 1997 elsevier science ltd

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

a study of the gap between the structured singular value and its convex upper bound for low rank matrices the size of the smallest structured destabilizing perturbation for a linear time invariant system can be calculated via the structured singular value spl mu the function spl mu can be bounded above by the solution of a convex optimization problem and in general there is a gap between spl mu and the convex bound this paper gives an alternative characterization of spl mu which is used to study this gap for low rank matrices the low rank characterization provides an easily computed bound which can potentially be significantly better than the standard convex bound this is used to find new examples with larger gaps than previously known

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

global optimization for h sub spl infin control with constant diagonal scaling this paper considers the h sub spl infin control problem with constant diagonal scaling related to the robust control synthesis for systems with structured time varying uncertainties the problem is not convex in general and hence it is difficult to find a global solution the purpose of this paper is to provide an algorithm to find a suboptimal solution with any specified small tolerance from the globally optimal solution for the optimization problem the algorithm based on the triangle covering method is proposed the computational complexity analysis shows that its worst case order is polynomial in the inverse of the tolerance and the size of an a priori given interval of scaling with a fixed number of uncertainty blocks