fundamental limits in sensitivity minimization multiple input multiple output mimo plants this paper provides a bridge between the h sub spl infin literature and that of fundamental limits by applying operator theoretic tools developed for the standard frequency domain model matching approach to sensitivity minimization explicit closed form expressions are given for tight bounds on the h sub spl infin norm of weighted sensitivity functions for output feedback control of linear multivariable systems

on the structure of h sup infinity control systems and related extensions an existence condition of a h sup infinity controller which achieves a prescribed norm bound of the closed loop transfer function is derived in the frequency domain based on a generalization of the notion of j lossless systems this condition is regarded as a frequency domain representation of the well known existence condition in the state space represented in terms of the two algebraic riccati equations a notion of j orthogonal complement which is introduced as a generalization of the usual orthogonal complement play an important role in clarifying the fundamental frequency domain structure of the model matching problem and simplifying the computation of controllers the results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank it is shown that the proper controller achieving the prescribed norm bound exists even in this case in nonstandard cases the controller order can be smaller than the plant order

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

one step extension approach to optimal hankel norm approximation and h sup infinity optimization problems a methodology is presented for hankel approximation and h sup infinity optimization problems that is based on a new formulation of a one step extension problem which is solved by the sarason interpolation theorem the parameterization of all optimal hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an hermitian matrix composed directly from the coefficients of a given transfer function matrix phi rather than starting with the state space realization of phi the authors use polynomial coefficients of phi as input data in terms of these data a natural basis is given for the finite dimensional sarason model space and all computations involve only manipulations with finite matrices

an image multiresolution representation for lossless and lossy compression we propose a new image multiresolution transform that is suited for both lossless reversible and lossy compression the new transformation is similar to the subband decomposition but can be computed with only integer addition and bit shift operations during its calculation the number of bits required to represent the transformed image is kept small through careful scaling and truncations numerical results show that the entropy obtained with the new transform is smaller than that obtained with predictive coding of similar complexity in addition we propose entropy coding methods that exploit the multiresolution structure and can efficiently compress the transformed image for progressive transmission up to exact recovery the lossless compression ratios are among the best in the literature and simultaneously the rate versus distortion performance is comparable to those of the most efficient lossy compression methods

minimization of a regulated response to a fixed input the problem of minimizing a regulated output due to a specific bounded input is solved in the siso single input single output discrete time case it is shown that in most situations the optimal solution will not exist but given any epsilon 0 it is possible to find a suboptimal solution with performance within epsilon of the optimal performance the proposed approach is extended to the problem of shaping the error transient response it is shown how time domain templates can be used in this problem in the same way that frequency weighting functions are used in the h sup infinity and h sup 2 problems

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

disturbance rejection of left invertible systems abstract a new approach is presented for the study of the disturbance rejection of left invertible systems via proportional state feedback reducing the problem to that of solving a linear algebraic system of equations on the basis of this system of equations the necessary and sufficient conditions for the existence of a solution to the disturbance rejection problem are established in a form of simple algebraic criteria thus reducing the computational effort over known results and the general analytical expressions of all solutions of the controller matrices are derived these solutions are parameterized only by an arbitrary matrix

optimal h sup infinity norm computation for multivariable systems with multiple zeros b c chang and j b peason s see ibid vol ac 29 p 880 7 oct 1984 computation of the optimal h sup infinity norm is generalized to the case with multiple right half plane zeros d sarason s 1967 interpolation theory is used to reduce the problem to a simple eigenvalue or singular value computation

an efficient algorithm on optimal hankel norm approximation for multivariable systems this note provides a novel methodology for hankel approximation and h sup infinity optimization problems based on a new formulation of the one step extension problem which is solved by the sarason interpolation theorem the present method does not require an initial balanced realization and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a hermitian matrix composed directly from the coefficients of the given transfer function matrix

an explicit expression for the minimum phase image of transfer function matrices a simple expression is obtained for the transfer function matrix of the minimum phase image of a left invertible continuous time invariant system with zeros in the right half plane that may be of multiplicities greater than one this expression is obtained by multiplying the system transfer function matrix from the right by a special inner matrix and it is explicitly given in terms of the input zero directions that correspond to the zeros of the system in the right half plane the result provides a simple expression for the inner outer factorization of transfer function matrices and it can thus be used in h sub infinity optimal control

on the robust stability of linear time invariant plants with unstructured uncertainty we consider general families of linear time invariant plants described by a nominal plant model with unstructured uncertainty we show under a rather weak assumption that when a family of systems is not robustly stabilizable by any linear time invariant feedback controller then no nonlinear time varying controller can robustly stabilize the given family

reversible integer to integer wavelet transforms for image compression performance evaluation and analysis in the context of image coding a number of reversible integer to integer wavelet transforms are compared on the basis of their lossy compression performance lossless compression performance and computational complexity of the transforms considered several were found to perform particularly well with the best choice for a given application depending on the relative importance of the preceding criteria reversible integer to integer versions of numerous transforms are also compared to their conventional i e nonreversible real to real counterparts for lossy compression at low bit rates reversible integer to integer and conventional versions of transforms were found to often yield results of comparable quality factors affecting the compression performance of reversible integer to integer wavelet transforms are also presented supported by both experimental data and theoretical arguments

logarithmic integrals interpolation bounds and performance limitations in mimo feedback systems we study performance limitation issues found in linear multivariable feedback systems our main contributions include bode and poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions these results characterize and quantify explicitly how open loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance the role of time delay is also studied in this context most notably we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros and in particular on how pole and zero directions are aligned the latter is characterized by angles measuring the mutual orientation between zero and pole directions and it is shown to play a crucial role in multivariable system design

optimal disturbance reduction in linear multivariable systems this paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory the approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation the resulting computations are easily implemented using existing methods

h optimal control part 2 solution for controllers abstract the problem is considered of bounding the h norm of the transfer function t y1u1 relating the output y 1 to the input u 1 for the feedback configuration shown in fig 1 subject to the closed loop system being internally stable a new criterion for internal stability is derived in terms of the q parameterization where q k i p22 k 1 by means of such a stability theory it is shown that the h control problem can be decomposed into two one sided model matching problems of the form considered in part 1 hung 1989 of the paper a closed form state space solution with a tight mcmillan degree bound is obtained for h optimal or suboptimal controllers an iterative algorithm for obtaining the h optimal controller is provided computationally the algorithm requires the solution of only two algebraic riccati equations in each iteration for which the upper bound on t y1u1 is adjusted

multivariable stability margin optimization a spectral tangential interpolation approach in this paper we show that many problems of multivariable stability margin optimization can be reduced to a spectral tangential interpolation problem of nevanlinna pick type where the interpolating transfer matrices are bounded not in infinity norm but in spectral radius this equivalence is established via conformal mappings necessary and sufficient conditions for the existence of solutions to the spectral interpolation problem are given in terms of the positivity of an associated nevanlinna pick matrix we conclude by applying this approach to solve a multivariable gain margin problem

sensitivity tradeoffs for multivariable plants this paper gives a characterization of the smallest upper bound on the norm of the sensitivity matrix over a frequency interval with the constraint that the norm remain bounded at all frequencies the matrix nevanlinna pick interpolation theory is applied to give a necessary and sufficient condition for the existence of a sensitivity matrix meeting the upper bound conditions on the j omega axis and the matrix nevnnlinna pick algorithm is used to compute the bounds a scalar example is also presented to demonstrate the tradeoffs between the bounds inside and outside a given operating frequency band

a computational algorithm for the super optimal solution of the model matching problem abstract in this paper we develop a computational algorithm to implement the strengthened minimization technique 1 in the model matching problem which arises in robust controller design and weighted sensitivity minimization a general algorithm is proposed for constructing unitary matrices and the extremal functions for the h optimization problem are diagonalized using the constructed unitary matrices we impose stronger minimization conditions to force the solution of the optimization problem to be unique and explain their physical significance

directional interpolation in the state space abstract this paper presents a new state space characterization of the directional interpolation based on the observation that the generalized pick matrix is regarded as a solution of a lyapunov type equation a simple representation of the set of all solutions is derived in terms of linear fractional transformations the computation of the associated j unitary matrix is simpler than the well known procedure based on the glover s scheme or the j spectral factorization of ball and ran

nevanlinna pick interpolation problem for two frequency scale systems the nevanlinna pick interpolation problem for a parametric set of data is considered sufficient conditions are obtained that guarantees a two frequency scale solution decomposition of the problem into two nevanlinna pick interpolation problems with smaller data size is presented an algorithm is developed which computes the solution as the result of combining the solutions for these two smaller problems

a new approach to synthesis problems in linear system theory we consider a rational nonlinear equation which is involved in the general synthesis design problem of linear multivariable systems the complete solution of this equation subject to causality stability and minimality constraints is provided formal power series and in particular the theory of partial realizations prove of central importance for the design problem

on constrained h optimization problem for siso systems this note is concerned with a control problem of designing a cascade compensator which minimizes the effect of the disturbance inputs or the weighted sensitivity h infty norm on the plant output subject to the constraint of a sensor noise characteristic or robust stability of the closed loop system lower and upper bounds are presented

fast iterative computation of optimal two block h sup infinity norm the two block h sup infinity optimization problem is discussed in which the most computationally demanding work is the computation of the optimal h sup infinity norm denoted by gamma sub 0 the problem of computing gamma sub 0 can be considered as that of finding a gamma such that mu gamma is equal to 1 some new properties of mu gamma are revealed and studied based on these properties and those found by c c chu and j doyle 1985 and z z wang and j b pearson 1984 a very fast computational algorithm for finding gamma sub 0 is proposed

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

h optimal observer based controller design for nonlinear multivariable systems abstract we propose a novel algorithm to synthesize an h optimal observer based controller for a nonlinear multivariable system based on the parametrization of the observer based controller a necessary and sufficient condition to achieve robust stability is developed moreover the nevanlinna pick interpolation theory is employed to test the solvability condition of the robust stabilization problem optimal robustness can be achieved via the technique of h optimization the exact solution of the model matching problem induced by the stability criterion is solved by the method of noniterative computation of the h norm

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

a modification of the nevanlinna algorithm on the solution of the h sup infinity sub m n optimization problem a modification of the nevanlinna algorithm is presented for the solution of the h sup infinity sub m m optimization problem this is an essential modification when the nevanlinna algorithm is applied to find the extremal function of the matrix h sup infinity interpolation problem which deals with the case that the interpolation points have multiplicities for the computation of the minimal h sup infinity norm a pure matrices computational algorithm is used where the formulation is solely in terms of the coefficients of the inner function

scaling properties of surface growth on rough substrates generated by standard models abstract in surface growth simulations usually the substrate is initially considered smooth with no roughness but as known in reality many substrates in nature are initially rough in this article we study the sensitivity of dynamic scaling of different models on rough substrates generated by standard surface growth models we implement the three growth procedures which are as follows for the first two procedures two separate substrates are grown with the random deposition rd and restricted solid on solid rsos for different time steps t 1 before growing them both with the ballistic deposition model bd but in the third growth procedure as an inverse of the second procedure particles deposit on the substrates by the bd model and become rough by the rsos model we conclude that the surfaces need a characteristic growth time t x to lose information about the substrate we show that the characteristic time depends on both the time needed for roughening the substrate and the special surface growth models in addition we show that the dependence of the two parameters t x and t 1 is a power law dependence only for adequate short times of t 1 which is also calculated

necessary and sufficient conditions for robust stabilization of perturbed observer based compensating systems abstract the design of a robust observer based compensator for imprecisely known linear multivariable systems is considered necessary and sufficient conditions for robust stability are derived with respect to additive plant perturbation multiplicative plant perturbation and fractional plant perturbation necessary and sufficient conditions are also given for the existence of such a robust observer based compensator

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

h sub infinity minimum error state estimation of linear stationary processes a state estimator is derived which minimizes the h sub infinity norm of the estimation error power spectrum matrix two approaches are presented the first achieves the optimal estimator in the frequency domain by finding the filter transfer function matrix that leads to an equalizing solution the second approach establishes a duality between the problem of h sub infinity filtering and the problem of unconstrained input h sub infinity optimal regulation using this duality previously published results for the latter regulation problem are applied which lead to an optimal filter that possess the structure of the corresponding kalman filter the two approaches usually lead to different results they are compared by a simple example which also demonstrates a clear advantage of the h sub infinity estimate over the conventional l sub 2 estimate

a new algorithm for the minimal balanced realization of a transfer function matrix abstract a new algorithm is proposed here to obtain a minimal balanced realization directly from the transfer function matrix tfm this method which employs the singular value decomposition svd of an infinite block hankel matrix requires neither an initial minimal realization nor the solution of a lyapunov matrix equation the formulation is solely in terms of the coefficients of the transfer function matrix

an algorithm for super optimal h design the two block case abstract in this paper we consider the strengthened h optimization problem in the two block case an algorithm for the super optimal solution is proposed it is based on results by the authors ieee trans aut control ac 33 833 1988 for the less practical one block case it uses state space models and is numerically implementable a feature of the procedure is an optimality criterion which is a natural generalization of that used in the iterative scheme of doyle lecture notes in advances in multivariable control onr honeywell workshop minneapolis u s a 1984 for the unstrengthened problem

h robust stabilizability abstract in this paper we improve on the results of robust stabilizability obtained by kimura we do this in a constructive way by using an h approach and exhibit an upper bound for the order of robust controllers

an elementary state space approach to rh optimal control this paper presents a new elementary approach to the h infinity optimal control problem no knowledge about function spaces is needed to derive the solutions only a few results in linear system theory some matrix manipulations and the classical cauchy integral theorem are used

state space algorithm for the computation of superoptimal matrix interpolating functions a state space algorithm is studied which generates the unique superoptimal nehari extension of a general rational matrix the procedure is to use a set of all pass transformations to sequentially minimize each frequency dependent singular value of the interpolating function in a dimension peeling algorithm these all pass transformations are determined by the maximal schmidt pairs of a sequence of hankel operators the process terminates when the original problem is reduced to one of rank one at this stage all the available degrees of freedom have been exhausted the work is an extension of that by young 1986 and gives a concrete state space implementation of his operator theoretic arguments in addition bounds are given on the minimum achievable values for s1 e sup ersi e j i 1 2 rank g0 and also the mcmillan degree of the final superoptimal extension here si denotes the ith singular value of a frequency dependent matrix and the numbering is taken to be in decrea

parametrization of stabilizing controllers for multivariable servo systems with two degrees of freedom a necessary and sufficient condition for the existence of a servo controller is derived using the fractional representation approach for the plant whose measured variables are not necessarily coincident with its controlled outputs an independent parametrization of the servo controllers and the attainable transfer characteristics is proposed the possible classes of all controllers with integrity in the error channel or measured output channel is also clarified using the derived parametrization

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

stabilization of linear systems with fixed order controllers abstract this paper deals with the problem of stabilizing a linear time invariant plant with a feedback compensator of fixed dynamic order this is an important unsolved problem in control theory because existing solutions to the regulator problem can only generate controllers that are of high enough order that arbitrary pole placement becomes possible in general the dynamic order of controllers can be much lower if only stabilization is desired in the transfer function domain a new necessary condition for stabilizability by a fixed order controller is derived this result which can be used sequentially to derive a lower bound on the order of a stabilizing controller is based on a classical result 1873 of linear programming known as gordan s theorem of the alternative the stabilization algorithm successively selects hurwitz vectors to make the largest stability hypersphere intersect a linear subspace determined by the plant parameters and the order of the controller occurrence of such an intersection is a sufficient condition for stabilizability and the intersection also determines a stabilizing compensator the algorithm is illustrated by practical examples which demonstrate that a large reduction in controller order is achievable by this method

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

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

paper uniformly optimal control of linear feedback systems the problem of uniformly optimal control is posed in the context of linear time varying discrete time systems the controller is optimal uniformly over a pre specified set of exogenous signals existence of an optimal controller is proved and a formula for the minimum cost is derived the time invariant case is treated in the frequency domain it is shown that for time invariant systems an optimal time varying controller is no better than an optimal time invariant controller

almost disturbance decoupling with internal stability the almost disturbance decoupling problem with internal stability is solved in terms of the geometric concepts of linear systems theory the solution gives necessary and sufficient conditions for the existence of a dynamic output feedback controller such that in the closed loop system the disturbances are quenched say in the h sup infinity sense up to any degree of accuracy while maintaining a stable system matrix

matrix valued nevanlinna pick interpolation with complexity constraint an optimization approach over the last several years a new theory of nevanlinna pick interpolation with complexity constraint has been developed for scalar interpolants in this paper we generalize this theory to the matrix valued case also allowing for multiple interpolation points we parameterize a class of interpolants consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros this is a complete parameterization and for each choice of interpolant we provide a convex optimization problem for determining it this is derived in the context of duality theory of mathematical programming to solve the convex optimization problem we employ a homotopy continuation technique previously developed for the scalar case these results can be applied to many classes of engineering problems and to illustrate this we provide some examples in particular we apply our method to a benchmark problem in multivariate robust control by constructing a controller satisfying all design specifications but having only half the mcmillan degree of conventional h sup spl infin controllers we demonstrate the advantage of the proposed method

sensitivity reduction by strongly stabilizing controllers for mimo distributed parameter systems this note investigates a sensitivity reduction problem by stable stabilizing controllers for a linear time invariant multi input multioutput distributed parameter system the plant we consider has finitely many unstable zeros which are simple and blocking but can possess infinitely many unstable poles we obtain a necessary condition and a sufficient condition for the solvability of the problem using the matrix nevanlinna pick interpolation with boundary conditions we also develop a necessary and sufficient condition for the solvability of the interpolation problem and show an algorithm to obtain the solutions our method to solve the interpolation problem is based on the schur nevanlinna algorithm

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

directional interpolation via all pass transfer function matrices and its application in hankel norm approximations a new approach to the problem of multivariable interpolation via all pass transfer function matrices that are not necessarily stable is presented the approach applies to both state space and classical function theoretic arguments and obtains a very simple expression for the all pass matrix that satisfies the interpolation requirement unlike the solution that is obtained by the generalized nevanlinna pick algorithm this expression is derived in closed form explicitly in terms of the interpolation parameters it allows a detailed investigation of the structure of the all pass solution and it is readily used in hankel norm approximations of linear multivariable systems

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

polynomial and rational matrix interpolation theory and control applications a generalization of polynomial interpolation to the matrix case is introduced and applied to problems in systems and control it is shown that this generalization is most general and it includes all other such interpolation schemes that have appeared in the literature the polynomial matrix interpolation theory is developed and then applied to solving matrix equations solutions to the diophantine equation are also derived the relation between a polynomial matrix and its characteristic values and vectors is established and it is used in pole assignment and other control problems rational matrix interpolation is discussed it can be seen as a special case of polynomial matrix interpolation it is then used to solve rational matrix equations including the model matching problem

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

a stable state space realization in the formulation of h norm computation in the two block hinftyoptimization problem usually we are given the state space realizations of the proper rational matrices r 1 s and r 2 s whose poles are all the open right half plane two problems are studied in the note the first is the evaluation of phi s r 1 s at s s k k 1 2 n where phi s is an inner function whose zeros s k k 1 2 n are the poles of r 1 s this evaluation is essential if chang and pearson s method is used for computing the optimal hinftynorm the problem is solved in state space via the solutions of lyapunov equations neither polynomial matrix manipulations nor numerical pole zero cancellations are involved in the evaluation the second problem is to find a stable state space realization of s s u s r 2 s where u s is an inner matrix this problem arises in the spectral factorization of gamma 2 r 2 ast r 2 doyle and chu had a method for constructing stable s s based on a minimal realization of r 2 s an alternate method is proposed the alternate method does not require a minimal realization of r 2 s and only a lyapunov equation is involved

two properties of l sub 1 optimal controllers the solution of the l sub 1 sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better known h infinity sensitivity minimization problem or the lqg linear quadratic gaussian problem is given of a one parameter family of first order plants where the order of the l sub 1 optimal compensator can be arbitrarily large and thus it is impossible to bound the order of an l sub 1 optimal compensator in terms of the order of the plant a plant is considered which has a continuous one parameter family of l sub 1 optimal compensators and thus l sub 1 optimal compensators need not be unique the author s two examples are considered to answer two questions left open by m a daleh and j b pearson ibid vol 32 p 314 23 1987

vector valued versus scalar valued figures of merit in h feedback system design abstract in this paper we adopt the point of view that the real figure of merit in h feedback systems design is the vector valued performance measure w1s w2t where w1s is the frequency weighted sensitivity function and w2t w2 i s the weighted complementary sensitivity function a compensator c0 is optimal if its induced performance w1s c0 w2t c0 is a minimal element of the set of achievable performances in the w1s w2t plane this set is shown to be convex and the fundamental limitations on achievable feedback performance take the geometric interpretation of a polygon bounding from below this convex set the h theory deals with feedback system performance tradeoffs by lumping the two conflicting objective functions s and t into a scalar valued criterion of the form p p w 1 s p p p w 2 t p 1 p where p p are scalar weighting factors and w1 w2 are frequency dependent weighting functions in this paper we develop strategies for weighting functions and weighting factors selections so as to direct the resulting scalar valued criterion design to a minimal element of the set of achievable performances in the w1s w2t plane if this is possible it appears that a scalar valued criterion is most likely to direct the design toward a performance acceptable from the vector valued criterion point of view if p 2 and w1 w2 are nonoverlapping

the explicit structure of inner matrices and its application in h sup infinity optimization an expression for inner matrices is obtained which allows a detailed investigation of their structure the expression is used to solve the h sup infinity interpolation problem in a simple way a solution to this problem is obtained explicitly in terms of interpolation parameters without solving any equation it is found that the resulting solution encounters a reduction in its order at the critical point and the characteristics of the resulting reduced order solution are investigated

conjugation and h control of discrete time systems we establish conjugation notion in discrete time systems first introduced into the h control theory of continuous time systems by kimura 1989 in discrete time systems conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle with the aid of j lossless conjugation conjugation by a j lossless system it is shown that the parametrization of sub optimal solutions of h model matching problems is reduced to a lyapunov type equation the parametrization of all solutions is given in an extremely simple way it is further proved that the j lossless conjugation of the h model matching problem is a natural state space representation of classical interpolation in discrete time systems

the structure of inner matrices that satisfy multiple directional interpolation requirements the structure of an inner matrix that satisfies directional interpolation requirements for nondistinct values at the right half plane is derived explicitly in closed form in terms of the interpolation directions the simple expression that is obtained for the required inner transfer function matrix has the same structure as the one that has already been found for distinct interpolation requirements as a result this expression provides a comprehensive treatment in the frequency domain of interpolation problems that arise in many areas of linear system optimization

a transfer function approach to the problems of discrete time systems h sub infinity optimal linear control and filtering a solution is derived to the problems of h sub infinity optimal linear state regulation and filtering the solution method for both problems is based on a transfer function approach which applies standard spectral factorization return difference relations are given which are extensions of the relations associated with the linear quadratic problem

an h 3wk0e minimax approach to the design of robust control systems part ii all solutions all pass form solutions and 10 best solution abstract we characterize all solutions to a robustness optimization problem as the solutions of a two parameter interpolation problem from this characterization it is easy to show that an all pass form solution always exists as long as a solution exists we also study the possibility of using non all pass form solutions and by introducing other optimization objectives motivated by improvements in disturbance rejection and robust stability we search for the best solution

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

on the design of linear multivariable feedback systems via constrained nondifferentiable optimization in h sup infinity spaces the design discussed is of linear lumped time invariant multivariable feedback systems subject to various frequency and time domain performance specifications the approach is based on the use of stabilizing controller parametrizations which result in the formulation of feedback system design problems as convex nondifferentiable optimization problems these problems are solvable by recently developed nondifferentiable optimization algorithms for the constrained minimization of regular uniformly locally lipschitz continuous functions in r sup n

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

lossless compression of color mosaic images lossless compression of color mosaic images poses a unique and interesting problem of spectral decorrelation of spatially interleaved r g b samples we investigate reversible lossless spectral spatial transforms that can remove statistical redundancies in both spectral and spatial domains and discover that a particular wavelet decomposition scheme called mallat wavelet packet transform is ideally suited to the task of decorrelating color mosaic data we also propose a low complexity adaptive context based golomb rice coding technique to compress the coefficients of mallat wavelet packet transform the lossless compression performance of the proposed method on color mosaic images is apparently the best so far among the existing lossless image codecs

optimal disturbance rejection for nonlinear control systems the problem of optimal disturbance rejection for a general class of multiinput multioutput mimo nonlinear systems achieved over a set of stabilizing compensator operators is formulated in a banach space setting conditions on the operators are imposed on the basis of the inputs as well as the outputs an existence theorem for optimal solutions is established under some realistic and verifiable conditions for solving the optimization problem a generalized recursive nonlinear programming algorithm is passed and a simple example of nonlinear optimal feedback design is given to show that the problem can be reduced to standard min max optimization and is hence solvable

the robust optimal control of uncertain systems state space method the robust optimal control of linear uncertain systems with l sub 2 bounded uncertainties is addressed the optimal state feedback control law in the so called worst disturbance case is derived the lql method minimax state estimation maximin state feedback closed loop control scheme lql means linear quadratic l sub 2 bounded uncertainties is proposed in a sense the lql method uses the idea of the linear quadratic gaussian lqg method to treat h sub infinity optimizing design problems

feasibility of h design specifications an interpolation method given a set of h design specifications the issue is to check whether there exists a controller whose order is free which satisfies these specifications the classical solution which is based on youla parametrization and convex closed loop design is not really satisfactory since it should use an infinite dimensional basis of filters which cannot be done in practice let j the minimal value of the design objective over such an infinite dimensional basis of filters a nevanlinna pick interpolation method is proposed here to compute lower and upper bounds of j by solving the design problem on a finite set of frequencies finite time convergence of the algorithm is proved

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

robust stabilizability for linear systems with both parameter variation and unstructured uncertainty the authors consider the problem of finding a single compensator which stabilizes a plant having both parameter variation and high frequency unstructured uncertainty an uncertain system model is proposed to characterize plants containing those two different uncertainties closed loop stability criteria are then proposed for the model robust stabilizability for single input single output siso systems is studied a number of assumptions describing the set of admissable siso plants are imposed the satisfaction of these assumptions guarantees the existence of a proper stable compensator c s achieving robust stabilization the algorithm used for controller design is recursive in nature and allows the designer to select one compensator coefficient at a time also the minimum order of the desired stabilizer c s depends only on the largest relative degree of the nominal plants which often leads to a lower order compensator

a two sided interpolation approach to h sub infinity optimization problems a solution to the two sided interpolation problem which arises in h sub infinity optimization theory is obtained this solution is found in closed form explicitly in terms of the required interpolation directions it is simple to obtain and it does not require the application of the relatively complicated matrix pick nevanlinna theory the solution obtained is of minimum order due to its simplicity the order reduction which occurs at the minimum value of the h sub infinity norm is clearly explained

synthesis of linear multivariable regulators this paper generalizes the results obtained by the authors in 1 and presents new results on the construction of proper controllers for linear multivariable regulators these results may lead to efficient computation methods that do not involve smith forms or the order relations used in 1 to construct proper controllers the methods are based on the recently developed theory of generalized polynomials 2 we discuss several numerical algorithms and how they can be applied to the regulator problem we also point out an important computational problem in need of an efficient numerical algorithm

disturbance attenuation in lti plants with finite horizon optimality of nonlinear controllers abstract it is an established fact that search for an optimum in the problem of disturbance attenuation in linear time invariant lti plants can be restricted to linear designs when the sole criterion is the minimization of the h norm of an appropriate input output operator what is not known however is whether there exist also optimal nonlinear controllers which would lead to uniformly better performance than linear controllers in a neighborhood of the worst case disturbance it is shown in this paper that for the finite horizon version and with the disturbance energy fixed there do exist optimal minimax controllers outside the linear class which can oout perform the best linear controller locally it is also argued that a differential dynamic games set up provides a natural framework for the time domain derivation of optimal minimax controllers for tracking model matching and disturbance rejection problems in both finite and infinite horizons