riccati differential equation in optimal filtering of periodic non stabilizable systems this paper discusses the periodic solutions of the matrix riccati differential equation in the optimal filtering of periodic systems special emphasis is given to non stabilizable systems and the question addressed is the existence and uniqueness of a steady state periodic non negative definite solution of the periodic riccati differential equation which gives rise to an asymptotically stable steady state filter the results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution the convergence of the general solution of the periodic riccati differential equation to a periodic equilibrium solution is also investigated the results are extensions of existing time invariant systems results to the case of periodic systems

stability analysis of continuous time periodic systems via the harmonic analysis asymptotic stability of finite dimensional linear continuous time periodic fdlcp systems is studied by harmonic analysis it is first shown that stability can be examined with what we call the harmonic lyapunov equation another necessary and sufficient stability criterion is developed via this generalized lyapunov equation which reduces the stability test into that of an approximate fdlcp model whose transition matrix can be determined explicitly by extending the gerschgorin theorem to linear operators on the linear space l sub 2 yet another disc group criterion is derived which is only sufficient stability of the lossy mathieu equation is analyzed as a numerical example to illustrate the results

zeros and poles of linear continuous time periodic systems definitions and properties the paper deals with definitions of zeros and poles and their features in finite dimensional linear continuous time periodic fdlcp systems under a harmonic framework more precisely system and transfer zeros and poles in the harmonic wave to wave sense are defined on what we call the regularized harmonic system operators and the harmonic transfer operators of fdlcp systems by means of regularized determinants then their composition and properties related to system structures are examined via the floquet theory and controllability observability decompositions of fdlcp systems the study shows that under mild assumptions the harmonic transfer operators of fdlcp systems are analytic and meromorphic on which zeros and poles are well defined basic zero pole relationships are established which are similar to their linear time invariant counterparts and in particular explicate some interesting harmonic wave to wave behaviors of fdlcp systems the results are significant in analysis and synthesis of fdlcp systems when the harmonic approach is adopted

robust stabilization of singularly perturbed systems abstract we discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models in 1 it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the true system containing both fast and slow modes this phenomenon was then expanded to include the case where the simplified system is strictly proper in 2 the objectives of this note are threefold i to show that given any proper system and any stabilizing controller for it that is proper but not strictly proper there exists a singular perturbation of the system that is destabilized by that controller ii to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter and iii to provide a characterization in the same spirit as 3 4 of the set of all strictly proper controllers that stabilize a given proper plant by combining these results it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system

brief paper the extended periodic lyapunov lemma in this paper the so called lyapunov lemma for linear systems with periodically time varying coefficients is dealt with by this lemma the asymptotic stability of a given periodic system is linked with the existence of a periodic positive definite solution of the associated lyapunov equation under the assumption on the system being observable by relaxing the observability assumption to detectability an extended version of the lemma is obtained where the system stability is linked with the existence of a periodic positive semidefinite solution of the lyapunov equation

adaptive robust output regulation of uncertain linear periodic systems this paper considers the robust output regulation problem for parameterized families of periodic systems to extend the solution of the output regulation problem to the periodic or time varying setup a classification of the immersion mappings based on various non equivalent observability properties is derived the connections between different canonical realizations of internal models that fully exploit such properties for robust and adaptive output regulation design in periodic systems are investigated it is shown how non minimal realizations of suitable periodic internal models are instrumental in achieving the possibility of performing adaptive redesign to deal with parameterized families of exosystem models an important feature of the proposed solution is the fact that a persistence of excitation condition for the exogenous signals is not required for asymptotic regulation

gain margin improvement using generalized sampled data hold function based multirate output compensator abstract for a siso strictly proper nonminimum phase continuous time linear time invariant lti plant it was shown in yan w b d o anderson and r r bitmead 1993 ieee trans automat contr that the closed loop gain margin obtained via a generalized sampled data hold function gshf based dynamic compensator is significantly improved over that achieved via a conventional periodic controller used in francis b a and t t georgiou 1988 ieee trans automat contr ac 33 820 832 nevertheless the compensator so designed is not necessarily strictly proper it is well known that it is practically difficult and sometimes impossible to implement a nonstrictly proper compensator due to the requirement of zero computation time further as has been shown in vidyasagar m 1985 systems and control letters 5 413 418 stabilization by a nonstrictly proper controller is never robust against singular perturbations in this paper we propose a new type of gshf based compensator which employs multirate sampling of the plant output using the proposed compensator not only the same level of gain margin as in yan w b d o anderson and r r bitmead 1993 ieee trans automat contr can be achieved but also more importantly the compensator is strictly proper

a harmonic framework for controllability in linear continuous time periodic systems controllability of linear continuous time periodic systems is dealt with via a harmonic analysis approach for the first time in this paper this approach reveals that controllability of continuous time periodic systems can be connected to necessary and sufficient conditions expressed explicitly with fourier coefficients of the system matrices which can be interpreted in a way similar to what we have seen in linear time invariant cases these controllability conditions shed new light upon structural characteristics of continuous time periodic systems that are hard to know by means of existing time domain controllability criteria in the literature controllability canonical decomposition of linear continuous time periodic systems is revisited through state coordinate transforms of strong analytic features the results are heuristic and significant for examining structural characteristics of continuous time periodic systems and extending controllability related techniques that are widely employed in linear time invariant systems to linear continuous time periodic systems

an lmi approach to periodic discrete time unbiased filtering the problem of unbiased filtering for a discrete time linear periodic system is faced by means of linear matrix inequality techniques as leading case we derive the synthesis conditions to obtain an unbiased filter and an unbiased fixed lag smoother enforcing a bound on the h performance on the error dynamics

differential periodic riccati equations a note on the existence of an infinite number of periodic strong solutions abstract the differential matrix riccati equation with periodic coefficients is considered it is shown by means of a counter example that a sufficient condition for the existence of an infinite number of periodic non negative definite solutions given in a previous paper is not appropriately stated a suitable adjustment of the assumptions allows one to work out rigorous sufficient conditions thanks to these results a necessary and sufficient condition for the existence of the periodic strong solution which relied on the incorrect condition can be shown to be still valid

on the gain margin improvement using dynamic compensation based on generalized sampled data hold functions this paper shows that the use of dynamic compensation based on generalized sampled data hold functions gshf can arbitrarily improve the gain margin for continuous time nonminimum phase linear systems the gshf compensator is a particular periodic digital controller the effect of sampling period on the gain margin is analyzed furthermore it is proved that under a mild condition the gain margin improvement can be achieved without forcing the sampling period small an important advantage of periodic compensation over lti compensation is found to be the capability of reducing conflict between gain margin and sensitivity

convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems this paper studies the convergence and properties of the solutions of the riccati difference equation special emphasis is given to systems which are not necessarily stabilizable in the filtering sense particularly those having uncontrollable roots on the unit circle besides generalizing and unifying previous work the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

lp functional and j holomorphic curves in kahler manifolds abstract in this paper we give another variational characterization of j m holomorphic curves in kahler manifolds generalizing our results in 1 we prove that if the functional l p has a critical point or a stable point in the fixed kahler class then the immersion is j m holomorphic or lagrangian

harmonic balance galerkin method for non smooth dynamic systems models of non linear systems frequently introduce forces with bounded continuity resulting in non smooth even discontinuous flow examples include systems with clearances backlash friction and impulses asymptotic methods require smooth differentiable flow and are therefore ill suited for analyzing non smooth systems in these cases the traditional harmonic balance method may be used to obtain approximate periodic solutions but the method suffers from extremely slow convergence in general generalizations of the traditional harmonic balance method are introduced in this paper that result in superior convergence rates and superior modes of convergence these improvements derive from the introduction of one or more expansion functions that possesses the same degree of continuity as the exact solution in particular forming an infinite series of such functions results in an expansion in the same function space of the exact solution this expansion converges pointwise to the exact solution and to all derivatives thereof these improvements are illustrated by example upon re evaluating a classical single degree of freedom model for friction induced vibration

regulators for optimal periodic processes a feedback controller called a periodic regulator is developed for implementing an optimal periodic process to a physical system under state deviations and constant parameter variation the objective of the periodic regulator is to bring the perturbed plant back to a neighboring optimal periodic path in such a manner that the infinite time second variation of the cost is minimized the transition matrix associated with the variational problem evaluated over one period is symplectic and generically possesses two unity eigenvalues coupled in the same jordan box the primary eigenvector associated with the unity eigenvalue corresponds to the orbital velocity direction it is shown that there exists a locally stable subspace in the neighborhood of the periodic orbit moreover a simple condition is given for defining the state variation which guarantees that it is contained in this stable subspace a controller including feedback of the parameter variation is derived which provides first order convergence to a new optimal periodic trajectory finally the periodic regulator developed is applied to a simple two state periodic optimal control problem to demonstrate its convergence properties

existence conditions and properties of the frequency response operators of continuous time periodic systems the definition of the frequency response operator via the steady state analysis in finite dimensional linear continuous time periodic fdlcp systems is revisited it is shown that the frequency response operator is guaranteed to be well defined only densely on the linear space l2 which is different from the usual understanding fortunately however it turns out that this frequency response operator can have an extension onto l2 so that the equivalence between the time domain h2 norm respectively the l2 induced norm and the frequency domain h2 norm respectively the h infty norm of the frequency response operator is recovered under some stronger assumptions it is also shown that the frequency response operator can be viewed as a bounded operator from l1 to l1 which can also be established via the steady state analysis

technical communique zeros of continuous time linear periodic systems zeros of continuous time linear periodic systems are defined and their properties investigated under the assumption that the system has uniform relative degree the zero dynamics of the system is characterized and a closed form expression of the blocking inputs is derived this leads to the definition of zeros as unobservable characteristic exponents of a suitably defined periodic pair the zeros of periodic linear systems satisfy blocking properties that generalize the well known time invariant case finally an efficient computational scheme is provided that essentially amounts to solving an eigenvalue problem

existence conditions and properties for the maximal periodic solution of periodic riccati difference equations the existence and properties of the maximal symmetric periodic solution of the periodic riccati difference equation is analysed for the optimal filtering problem of linear periodic discrete time systems special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption necessary and sufficient conditions for the existence and uniqueness of periodic non negative definite solutions of the periodic riccati difference equation which gives rise to a stable filter are also established furthermore the convergence of non negative definite solutions of the riccati equation is investigated

on the convergence to the strong solution of periodic riccati equations the convergence of the solution of the differential and difference periodic riccati equation with non negative initial condition is studied in particular with reference to the riccati equation for optimal filtering we consider detectable but not necessarily stabilizable systems and no restrictive assumption is imposed on the modulus of the unreachable characteristic multipliers then a new condition for the convergence to the periodic strong solution is provided as a corollary it follows that for any positive definite initial condition the solution of the periodic riccati equation converges to the periodic strong solution the new condition encompasses as particular cases the convergence results already available in the literature

new doubling algorithm for the discrete periodic riccati equation the periodic riccati equation that results from periodic state space models plays an important role in many fields of mathematics science and engineering in most applications it is essential that the solution to the riccati equation be obtained in the shortest possible time such a computationally effective doubling algorithm that solves the discrete periodic riccati equation is proposed in this paper moreover the memory requirements and the calculation burden needed for the sequential implementation of the proposed algorithm are established and compared to the memory requirements and the calculation burden needed for the sequential implementation of classical algorithms the basic conclusion of the above comparison is that the calculation time required to solve the periodic riccati equation using the classical algorithms is in general much greater than the calculation time required to solve the periodic riccati equation by using the proposed algorithm finally the numerical behavior of the proposed algorithm is tested through simulation examples it is established that the proposed algorithm is fast computationally efficient and numerically stable and possesses very good parallelism efficiency

gibbs phenomenon in structural mechanics the concepts of complementary boundary condition and interior condition are discussed guidelines are developed to eliminate the gibbs effect when selecting trial functions for the series expansion

inertia theorems for the periodic liapunov equation and periodic riccati equation the inertia theorems for the time invariant liapunov equation and time invariant riccati equation are generalized to the case were the coefficient matrices are periodically time varying

spectral characteristics and eigenvalues computation of the harmonic state operators in continuous time periodic systems abstract spectral characteristics of what we call the harmonic state operators in finite dimensional linear continuous time periodic fdlcp systems are examined by means of the fredholm theory for the first time it is shown that the harmonic state operator is a closed densely defined fredholm operator on the hilbert space l 2 and its spectrum contains only eigenvalues of finite type in any bounded neighborhood of the origin of the complex plane these spectral characteristics lead us to an asymptotic computation algorithm for the eigenvalues of the harmonic state operator via truncations on its fredholm regularization furthermore the truncation approach also gives us a necessary and sufficient stability criterion in the fdlcp setting which only involves the fourier coefficients of the state matrix asymptotic stability of the lossy mathieu differential equation is investigated to illustrate the results

h filtering for linear periodic systems with parameter uncertainty abstract this paper focuses on h filtering for a class of linear periodic systems with a certain type of norm bounded time varying parameter uncertainty which appears in both the state and output matrices the problem addressed is the design of a linear periodic estimator that guarantees both the quadratic stability and and prescribed h performance on infinite horizon for the estimation error for all admissible parameter uncertainties a solution to this problem is obtained via a riccati equation approach

zeros of discrete time linear periodic systems in this note the concept of zero for linear discrete time siso periodic systems is discussed the definition is based on a time invariant mimo equivalent representation an input output characterization of the zeros is provided which extends a well known property for time invariant systems

component mode iteration for frequency calculations this paper presents component mode iteration a new method for carrying out a form of subspace iteration on the substructure level for structures composed of substructures in component mode iteration substructure displacement is in terms of component modes with the interface displacements as coefficients component modes are chosen such that exact compatibility is assured throughout the structure the reduced structure eigenvalue problem is assembled and solved for lower mode approximations the resulting estimates of lower mode interface motion are used to refine the component modes by approximating the dynamic response of substructures to interface motion with a vector iteration process for this step the substructures are independent and can be processed in parallel reassembly and solution of the reduced structure problem yields improved lower mode estimates further iterations of refinement and reassembly are performed until convergence takes place the initial component modes are generated automatically and at negligible expense and numerical examples show that convergence with these is excellent yielding very accurate results after only one iteration indeed convergence is shown to be significantly faster than in the standard subspace iteration method

elimination of gibbs phenomena from error analysis of finite element results a change of variables is presented that improves the accuracy of fourier half range sine series used in numerical harmonic analysis the change in variables results in an interpolation formula that passes through the endpoints of tabulated data

a note on the maximal solution of the periodic riccati equation the periodic symmetric solutions of the periodic riccati differential equation associated with the filtering problem are considered by the authors it is proven that under the sole assumption of detectability there exists a maximal solution moreover such a solution turns out to be strong i e the characteristic multipliers of the associated closed loop system belong to the closed unit disk the proof relies on an iterative linearization technique which calls for a sequence of periodic lyapunov equations similar results are given for the minimal solution

gibbs phenomenon free fourier series for vibration and stability of complex beams the gibbs phenomenon in fourier series has long been recognized as a drawback in its applications in particular when it is used to represent a function having discontinuities one category namely intrinsic discontinuities could be derived from the nature of the physical problems another namely inherent discontinuities is created undesirably through the employment of fourier series to alleviate these drawbacks two techniques are developed the first aims at eliminating the inherent discontinuities the size of the original domain 0 l is first doubled with a virtual function 0 defined over the neighboring domain l 0 the augmented domain l l is then further extended periodically the virtual function 0 y is chosen so that continuities are achieved at y 0 l 2l consequently no gibbs phenomena will occur at the boundaries the second aims at reproducing the intrinsic discontinuities in the representation function by incorporating piecewise cubic polynomials into the fourier base function the enlarged basis function namely the gibbs phenomenon free fourier series function is able to represent accurately any specific boundary conditions and interior intrinsic discontinuous conditions examples of vibration and buckling analysis of complex beams show that the present method is versatile accurate and efficient

cyclomonotonicity riccati equations and periodic receding horizon control abstract in this paper periodic receding horizon control of periodic and time invariant systems is considered this type of receding horizon control is based on the periodic extension of the control law that solves a finite horizon optimal control problem it is shown that cyclomonotonicity of the solution of the differential riccati equation entails stability of the closed loop system stability analysis hinges on the so called fake two point boundary riccati equation a differential riccati equation whose solution is constrained to satisfy a two point boundary condition in the case of time invariant systems cyclomonotonicity is always achievable by a proper scaling of the terminal condition of the riccati equation it is also proven that cyclomonotonic solutions of the riccati equation not only yield stabilizing controllers but also constant and stabilizing control laws

maximal solution of a certain class of periodic riccati differential equations this paper deals with a generalized version of the well known periodic riccati differential equation which arises in certain optimal filtering problems for linear periodic systems a monotonicity property for symmetric periodic equilibrium solutions is established and it is proved that there exists a maximal symmetric periodic equilibrium solution

frequency domain analysis of linear time periodic systems in this paper we study convergence of truncated representations of the frequency response operator of a linear time periodic system the frequency response operator is frequently called the harmonic transfer function we introduce the concepts of input output and skew roll off these concepts are related to the decay rates of elements in the harmonic transfer function a system with high input and output roll off may be well approximated by a low dimensional matrix function a system with high skew roll off may be represented by an operator with only few diagonals furthermore the roll off rates are shown to be determined by certain properties of taylor and fourier expansions of the periodic systems finally we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

decentralized control of multi channel systems with direct control feedthrough abstract linear time invariant multi channel general proper systems with direct control feedthrough are considered a polynomial matrix method is presented to derive explicit conditions characterizing the systems which can be made controllable or stabilizable through a single channel by applying non dynamic decentralized feedback to all other channels these conditions are then applied to the decen tralized pole assignment and stabilization problems for the systems yielding a complete criterion for determining when the closed loop spectrum can be freely assigned or stabilized with decentralized control it can be expected that the results obtained will have direct application to periodic discrete time control systems

linear periodic systems and multirate filter design we present a systematic procedure for the design of filters intended for multirate systems this procedure is motivated by viewing the equiripple design of filters in linear time invariant systems as a process of obtaining optimum minimax filters for a class of bounded energy input signals the philosophy of designing optimum minimax filters for classes of input signals is extended to multirate systems which are not time invariant we develop a generalized fourier analysis appropriate for linear periodic systems and use it to derive new error criteria for multirate filter design using such criteria yields optimum minimax multirate filters for the input signal class the utility of our method is demonstrated by using it to analyze several multirate systems we give numerical results on the design of a multirate implementation of a narrowband filter and compare our work to previous work on multirate filter design our numerical analysis is based upon a new formulation of the design as a semi infinite linear programming problem

stabilizability and detectability of linear periodic systems abstract in this paper four characterizations of stabilizability and detectability of linear periodic systems are considered two of them look as natural extensions of the classical definitions given for time invariant systems the remaining two are modal characterizations which turn out to be useful in the analysis of the periodic lyapunov and riccati equations it is shown that all these notions of stabilizability and detectability are in fact equivalent to each other one of the various definitions calls for the existence of the kalman canonical decomposition of periodic systems this issue is addressed in the appendix

vibrational feedback control zeros placement capabilities it is shown that in addition to closed loop pole placement vibrational feedback controllers lead in the sense specified below to a possibility of open loop zeros assignability on this basis the superior performance characteristics of continuous time periodic controllers discovered in 3 6 are explained

an efficient general approach to modal analysis of frame resonators with applications to support loss in microelectromechanical systems abstract the design of high q resonators such as xylophone bar resonators xbrs capable of being fabricated using micro electro mechanical systems mems processes is of considerable interest in light of the widespread and rapidly growing use of systems dependent on their availability and performance this paper is concerned with vibration analysis and q optimisation of an xbr with the method extending directly to other planar frames and straightforwardly to more complex structures the rayleigh ritz method is discussed in some detail first treating the discrete case followed by developing and applying a kinematical procedure to an l frame structure attention is given to geometric interpretation of the rayleigh ritz procedure and to developing an intuitive understanding the method before turning to the xbr case having developed an approximation for system dynamics the results are used in conjunction with an analytical model of elastic wave propagation in the substrate to obtain an estimate for the support q factor natural frequencies mode shapes and support q values are presented and compared to finite element models of the same problem with excellent agreement observed at substantially lower computational cost for the first time in the literature the geometric impedance tuning principle underlying the xbr design is validated and quantified including sensitivity to manufacturing error

periodic strong solution for the optimal filtering problem of linear discrete time periodic systems the periodic riccati difference equation prde for the optimal filtering problem of linear periodic discrete time systems is addressed specifically the author provides a number of results on the existence uniqueness and stability properties of symmetric periodic nonnegative definite solutions of the periodic riccati difference equation in the case of nonreversible and nonstabilizable periodic systems the convergence of symmetric periodic nonnegative definite solutions of the periodic riccati difference equation is also analyzed the results have been established under weaker assumptions and include both necessary and sufficient conditions the existence and properties of symmetric periodic nonnegative definite solutions of the prde are established directly from the prde

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

another look at admissible functions abstract problems associated with using global admissible functions in boundary value problems are discussed it is shown that if care is not taken it is possible to construct sets of trial functions that yield inaccurate results which take too long to converge the concept of complementary boundary conditions is introduced and their existence is established complementary boundary conditions indicate that the solution to the boundary value problem cannot take certain values at the boundaries it is shown that trial functions that yield incorrect or very slowly converging results are those that violate the complementary boundary conditions a revision for the definition of admissible and comparison functions is proposed that takes the complementary boundary conditions into consideration

convergence of the classical rayleigh ritz method and the finite element method it is demonstrated in this paper that convergence of the classical rayleigh ritz method can be vastly improved by introducing a new class of admissible functions called quasi comparison functions

estimates of asymptotic trajectory bounds in digital implementations of linear feedback control systems estimates are established for trajectory bounds which are easy to calculate applicable to general controller topologies and dependent on parameters i e eigenvalues and eigenvectors of the idealized linear system i e the control system model neglecting quantizers the approach utilizes the comparison principle in the lyapunov theory and the properties of minimal matrix norms the applicability of the results are demonstrated by means of a specific example

h sub infinity state estimation for linear periodic systems this note deals with the h sub infinity state estimation problem for linear periodic systems the question addressed is the design of an unbiased linear periodic and asymptotically stable estimator that achieves a prescribed h sub infinity performance on an infinite horizon necessary and sufficient conditions for the existence of a periodic estimator have been derived asymptotic properties of the finite horizon estimation problem when the time horizon tends to infinity are also investigated

the simultaneous optimization problem for sensitivity and gain margin the combined sensitivity and gain margin problem for single input single output linear systems is formulated and solved using a complex function interpolation technique it is proved that this problem always has a real rational solution provided it is solvable in the complex irrational sense the sensitivity minimization problem subject to a gain margin constraint and its dual problem are also considered the range of the gain margin constraint is given subject to which the optimal constrained sensitivity is identical with the optimal unconstrained sensitivity it is shown that not unexpectedly the gain margin maximization conflicts with the sensitivity minimization for a nonminimum phase plant