perfect and subperfect regulation in linear multivariable control systems this paper is concerned with the synthesis of a high gain feedback control called perfect regulation p r for linear multivariable systems with external signals the existence condition for p r is derived in which the minimum phase property plays an essential role the use of an observer is discussed for achieving p r under restricted state observation some asymptotically ideal feedback properties are demonstrated such as the complete desensitization the complete servo performance with decoupling and the complete disturbance rejection which formulate the loop tightness for multivariable systems a design method for applying p r to non minimum phase systems called subperfect regulation s p r is proposed based on the factorization of the plant transfer function matrix into the minimum phase part and the totally non minimum phase part it is a multivariable extension of a well known design technique for scalar systems to overcome the difficulty of phase non minimality computational procedures for p r and s p r are discussed an illustrative example is shown

disturbance decoupled estimation for linear differential algebraic systems we study the disturbance decoupled estimation problem for linear differential algebraic systems which are not necessarily regular we introduce the notion of partial state observers following a rec

high gain observers in the state and estimation of robots having elastic joints the paper deals with the state and parameter estimation of robots having elastic joints the estimation of the unknown parameters of the robot is reduced to the estimation of its state variables by a state space immersion the luenberger observer is used in the state estimation of the extended robot model the use of high gains is studied in the cancellation of nonlinearities in order to simplify the observer design the high gain induces a time scale separation between the robot model and the observer and therefore the singular perturbation theory can be used in the stability analysis of the estimate error dynamics in particular it is shown that the error dynamics reaches the stable equilibrium in a very fast transient ensuring that the slow dynamics of the observer is just that of the given robot

asymptotic behaviour of the solution of the projection riccati differential equation the solution of the riccati differential equation rde is shown to be asymptotically close to the solution of the projection riccati differential equation prde the asymptotic behaviour of the latter is analyzed in an explicit formula the almost periodic asymptote of the solution of the prde is computed by an algorithm based upon the concepts of an aperiodic almost periodic generator apg decomposition of a linear map and unit row staircase form of a polynomial matrix the analysis ultimately provides a convergence criterion in particular it is shown that the solution of the prde always converges in the aperiodic case

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

on observability subspaces the paper presents an in depth study of topics in geometric control pertaining to observer theory from a functional point of view we give characterizations of several classes of subspaces including observability almost observability and reconstructibility subspaces we solve completely the problem of spectral assignability for observer dynamics by generalizing rosenbrock s pole placement theorem these results are then applied to observer theory

a geometric approach to the synthesis of failure detection filters a geometric formulation of beard s failure detection filter problem is stated using the concepts of c a invariant and unobservability subspaces the notions of output separable and mutually detectable families of subspaces introduced by beard are also clarified it is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum moreover it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions finally the complete duality between a generalization of beard s detection filter problem and the restricted control decoupling problem is illustrated

nonlinear h sub spl infin control and its application to rigid spacecraft the problem of h sub spl infin control of a rigid spacecraft with three control torques and disturbances is addressed the hamilton jacobi isaacs inequality for h sub spl infin feedback design is solved an h sub spl infin suboptimal feedback is given and the stability properties of the closed loop system are studied

ale behavior for two sinusoidal signal models the adaptive line enhancer ale weight covariance matrix is determined for a complex monochromatic sine wave signal in an independent circular gaussian noise background the weight covariance matrix is shown to differ significantly from that obtained when the signal is a narrow band sine wave process in spite of this result the misadjustment factors which characterize the mean norm squared error behavior of the ale are essentially indistinguishable for small values of the adaptation parameter noise power product 2

feedback block decoupling and infinite structure of linear systems abstract using factorizations at infinity of the transfer matrix two interesting necessary and sufficient conditions arc obtained for the block decoupling problem of linear systems by state feedback and a constant non singular input transformation these conditions are expressed in a very simple way in terms of integers which are feedback invariants for the system under consideration these feedback invariants are related either to the system infinite structure or to the reachability indices of the feedback reduced system

pole assignment in the regular row by row decoupling problem by applying the canonical decomposition of the right invertible system c a b obtained in wei cheng and wang 2010 in this paper we derive a general formula of all solutions to the regular row by row decoupling problem based on this formula we characterize all attainable transfer function matrices for the decoupling and pole assignment problem in general cases the results obtained in this paper extend those in the literature

semiglobal l sub 2 performance bounds for disturbance attenuation in nonlinear systems this paper addresses the problem of disturbance attenuation in the sense of l sub 2 gain with internal stability and no cost on the control input for nonlinear systems when the problem of almost disturbance decoupling is not solvable i e when the least achievable attenuation level between disturbance and regulated output is nonzero two different approaches are provided which facilitate the estimation of an upper bound for such a value

almost invariant subspaces transmission and infinite zeros a lattice interpretation in the first part of the paper it is shown that the most important almost controlled invariant and almost conditionally invariant subspaces introduced by j c willems 4 5 are elements of the free distributive lattice generated by the well known subspaces download download full size image and download download full size image in the second part it is shown how the transmission zeros and the infinite zero structure may be obtained by duality within a state feedback context or an output injection one

tracking model of an adaptive lattice filter for a linear chirp fm signal in noise the paper studies the behavior of the partial correlation parcor coefficients and the output misadjustment of the stochastic gradient adaptive lattice filter in response to a complex linear chirp fm signal in white gaussian noise analytic expressions for the optimal parcor coefficients of the filter are derived analytic as well as iterative models for a three stage filter are also derived the analytic expressions show that the tracking and convergence properties of the filter are separate phenomena simulation results also show that the spectral contents of the parcor coefficients for the stochastic gradient update algorithm consist of a stationary and a linearly swept component a single stage model is developed to explain this behavior finally output misadjustment plots for the filter show that an optimum value for the forgetting factor can be obtained to minimize the misadjustment but the value required to achieve local minimum misadjustment varies with each stage of the filter it is shown that in applications where the input has a high signal to noise ratio snr the misadjustment decreases rapidly at each successive stage thus implying that relatively short filter lengths are sufficient to provide effective tracking

adaptive recovery of a chirped sinusoid in noise ii performance of the lms algorithm the authors study the ability of the exponentially weighted recursive least square rls algorithm to track a complex chirped exponential signal buried in additive white gaussian noise power p sub n the signal is a sinusoid whose frequency is drifting at a constant rate psi lt is recovered using an m tap adaptive predictor five principal aspects of the study are presented the methodology of the analysis proof of the quasi deterministic nature of the data covariance estimate r k a new analysis of rls for an inverse system modeling problem a new analysis of rls for a deterministic time varying model for the optimum filter and an evaluation of the residual output mean square error mse resulting from the nonoptimality of the adaptive predictor the misadjustment in terms of the forgetting rate beta of the rls algorithm it is shown that the misadjustment is dominated by a lag term of order beta sup 2 and a noise term of order beta thus a value beta sub opt exists which yields a minimum misadjustment it is proved that beta sub opt m 1 rho psi sup 2 sup 1 3 and the minimum misadjustment is equal to 3 4 p sub n m 1 beta sub opt where rho is the input signal to noise ratio snr

the state feedback h problem at optimality abstract we discuss the infinite horizon h control problem by state feedback for a linear time invariant plant which is not restricted with respect to its finite or infinite zero structure we provide an algebraic characterization for the optimal value to be attained in terms of the solvability of a nonstrict riccati inequality and propose possibilities for verifying these criteria if the optimum is not achieved we give a complete test for the need of high gain feedback in order to approach the optimal value

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

algebraic characterizations of almost invariance this paper stresses the algebraic as opposed to analytic nature of the concept of almost invarianoe which has been introduced by willems 1980 as a means of studying asymptotic aspects of linear systems based on an interpretation in terms of difference equations it is shown that the basic properties and a number of characterizations of tho four fundamental classes of subspaces can be derived in a relatively simple and purely algebraic fashion among other things this paper also extends the results of hautus on the frequency domain interpretation of ordinary a b invariant subspaces re derives the pencil characterization of jaffe and karcanias 1981 and obtains a new hybrid type of characterization the last result also loads to a rank test for almost invariance

global stabilization with almost disturbance decoupling of a class of uncertain non linear systems in this paper we study the global stabilization of a class of non linear uncertain systems while decoupling the output from external disturbances to any specified degree of accuracy the class of non linear systems considered could be viewed as a generalization of uniform rank multivariable linear systems which have the property that all their infinite zeros are of the same order although the plants considered are non linear the compensator developed here is linear our compensator which achieves global exponential stability and almost disturbance decoupling of the output requires only output feedback rather than full state feedback model uncertainty is dealt with either by restricting the uncertain elements to particular subspaces satisfying the so called matching conditions and or by treating a family of plants for which a single controller simultaneously globally exponentially stabilizes all the plants in while achieving almost disturbance decoupling of the output

brief paper some new results for system decoupling and pole assignment problems in a related article we derived a canonical decomposition of the right invertible system c a b and applied this canonical decomposition to study the smith form of the matrix pencil p s a sibc0 and findout the finite zeros and infinite zeros of p s the range of the ranks of p s for email protected c and the controllability of the right invertible system in this paper we will apply this canonical decomposition of the right invertible system c a b to deduce the triangular decouple upon to row permutation provide some new results of the row by row decoupling and associated pole assignment problems

algebraic characterization of controllability subspaces in the current base and its practical computation from the concepts of input chain and controllability chain introduced by heymann and in view of the generalized eigenvector assignment a new algebraic characterization of controllability subspaces css is developed compared with the previous ones based on matrix pencil theory it presents both the advantages of being stated in the current base and of being derived without resorting to use of canonical forms a practical algebraic algorithm is provided in order to compute the controllable structure of a given pair a b this approach is then applied to characterize css contained in a given subspace and span css via eigenvector assignment

continued fraction decomposition of linear systems in the state space abstract various results pertaining to a continued fraction decomposition of matrix formal power series and in particular of multi input multi output linear constant systems are presented from a state space point of view

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

sampling infinite zeros and decoupling of linear systems abstract in order to understand more fully some of the trade offs involved in using a sampled data representation of a continuous time system the effects of time sampling on the ability to achieve disturbance decoupling and input output decoupling for linear systems are investigated it is shown that disturbance decouplability is lost through sampling whereas row by row dynamic input output decouplability is preserved in a very strong way these results are obtained by analyzing the structure at infinity of a sampled data system

the fixed poles of the disturbance decoupling problem and almost stability subspace mathcal v star_ b g rm ker c this paper is concerned with the fixed poles of the disturbance decoupling problem for linear time invariant systems by state feedback and measurement feedback algebraic characterizations for these fixed poles are given using the reducing subspace technique in numerical linear algebra these algebraic characterizations lead that the fixed poles can be directly computed by numerically reliable algorithms as an additional application of the reducing subspace technique we also develop a numerically stable method for computing the almost stability subspace mathcal v star b g rm ker c which is an extension of the method given by van dooren for computing the invariant subspaces in geometric control theory

decoupling by restricted static state feedback the general case in this paper we tackle the general block decoupling problem for linear constant dynamical systems c a b with m inputs and p outputs with restricted static state feedback in other words with control laws of type u gfx gv we give a necessary and sufficient condition of existence for such laws which generalizes the one previously given in 2 for the simple case k p leq m where k denotes the number of blocks to be decoupled

disturbance localization and pole shifting by dynamic compensation this paper concerns the problem of disturbance localization and pole shifting by dynamic compensation in multivariable linear systems with inaccessible state vector a necessary and sufficient condition for the solvability of the problem is given two synthesis procedures of the compensator are presented one is based on an observer and the other is based on a dual observer the observer designed by the proposed procedure acts as as unknown input observer for a linear function an improved procedure of designing a reduced order unknown input observer is also given the result may be applicable to reducing the compensator order

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

on the structure at infinity of linear block decouplable systems the general case the aim of this note is to show that a linear system c a b is block decouplable by means of static state feedback laws f g with g invertible if and only if the infinite structure of c a b equals the union of the infinite structures of the subsystems c i a b extracted from the given output partition this result generalizes the ones recently obtained for morgan s problem 6 and for the particular cases of right invertible 7 or left invertibie 8 systems

disturbance decoupling by measurement feedback for structured transfer matrix systems structured transfer matrix systems are linear systems given by transfer matrices of which the infinite zero order of each nonzero entry is known while the associated infinite gains are unknown and assumed mutually independent in this paper necessary and sufficient conditions are derived for the generic solvability of the well known disturbance decoupling problem by measurement feedback for structured transfer matrix systems generic solvability here means solvability for almost all possible values for the infinite gains of the nonzero transfer matrix entries the conditions will be stated in terms of a bipartite graph that can be easily associated with a structured transfer matrix system the advantage of this is that the conditions then can be verified by means of graph theoretic algorithms

some generalized notions of observability generalized state space systems are a more natural way of describing dynamical behavior than standard state space systems such systems exhibit a much richer observability structure than standard state space systems three notions of observability are defined for generalized state space systems in contrast to previous work regularity is not assumed geometric characterization of each type of observability and matrix rank tests for each type of observability are derived

decoupled decomposition of the riccati equation the general matrix riccati equation is decomposed into product of two factors where the first one is determined independent of the second factor condition for existence of solution of the riccati equation is given via this decomposition and the existence of solution of the self adjoint matrix riccati equation arising in optimal control and kalman filtering is directly established from that condition

on nonlinear observers in this note we make some remarks about the general observer problem for nonlinear systems we show how the existence of a partial observer without stability requirements is locally equivalent to the existence of a conditioned invariant distribution we argue that by considering not only identity observers we can put the stability issue in a broader perspective

nonlinear morgan s problem case of p 1 inputs and p outputs noninvertible static state feedback solutions of the row by row decoupling problem are investigated for the class of nonlinear systems having one more input component than output components the analysis involves a combination of geometric and algebraic tools but notably avoids the use of regular or degenerate controllability distribution subspaces the main tools used to date for the class of linear systems the main idea is finding within the system a chain of integrators that may be attached to one of the inputs in order to appropriately modify the system s structure at infinity

assigning the kronecker invariants of a matrix pencil by row or column completions abstract the challenge consists in describing the relationships between the kronecker invariants of a matrix pencil and one of its subpencils for a given subpencil an algorithm for constructing a matrix pencil with prescribed kronecker invariants should also be proposed

feedforward control pid control laws and almost invariant subspaces we define feedforward control as a control policy in which the exogenous disturbances are known for all time at the moment when the control is applied it is shown that disturbance decoupling by feedforward control is possible iff it is possible by pid control or iff approximate disturbance decoupling by state feedback is possible

solution of the static state feedback decoupling problem for linear systems with two outputs we give for linear systems with two outputs a necessary and sufficient condition for the solvability of the restricted decoupling problem rdp in terms of some structural invariants a geometric characterization is also given which provides the least dynamical extension needed when rdp is not solvable

time scale structure assignment in linear multivariable systems using high gain feedback abstract a systematic design method for assigning a required time scale structure to a given multivariable system is developed the time scale structure assignment is carried out using high gain feedback of either the state or the output the method developed decomposes a given multivariable system into several single input single output subsystems each of which is designed separately a standard method of design for single input single output systems is developed and this forms a building block for the multivariable system design the state feedback design depends only on the stage sequence of the given system while the output feedback design depends only on the set of integers that specifies the order of infinite zeros in this way our design is made robust with respect to certain parameter uncertainties high gain parameter s can be adjusted on line and this provides a tool for the fine tuning of the desired performance characteristics of a control system

parallel implementation of efficient preconditioned linear solver for grid based applications in chemical physics ii linear systems in chemical physics often involve matrices with a certain sparse block structure these can often be solved very effectively using iterative methods sequence of matrix vector products in conjunction with a block jacobi preconditioner numer linear algebra appl 7 2000 715 in a two part series we present an efficient parallel implementation incorporating several additional refinements the present study paper i emphasizes construction of the block jacobi preconditioner matrices this is achieved in a preprocessing step performed prior to the subsequent iterative linear solve step considered in a companion paper paper ii results indicate that the block jacobi routines scale remarkably well on parallel computing platforms and should remain effective over tens of thousands of nodes

direct approach to almost disturbance and almost input output decoupling a direct solution to the almost input output and disturbance decoupling problem is given the approach is based on a new canonical form for linear systems which is obtained in a fully constructive way together with some elementary approximation results the solution is given in the form of an explicit feedback law parameterized by a scalar e by letting e go to zero the accuracy of the resulting approximate solution can be improved to any desired degree

dynamic input output decoupling of nonlinear control systems the problem of dynamic input output decoupling of nonlinear control systems is studied based on an analytic algorithm the authors obtain necessary and sufficient conditions for the solvability of this problem the solution of the problem is constructed by applying a series of simple precompensations and linking maps some interesting connections with other approaches in nonlinear control theory are discussed the authors also give a few simple examples to illustrate the methods used

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

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

almost invariance and noninteracting control a frequency domain analysis abstract we solve a number of feedback synthesis problems in the context of noninteracting control or block diagonal decoupling for finite dimensional linear time invariant systems we consider a plant that apart from a control input and a measurement output has a given number of exogenous input vectors and the same number of exogenous output vectors the decoupling problem studied is to find dynamic compensators from the plant measurement output which in this paper will be assumed to be the full plant state to the plant control input in such a way that the following requirements are met 1 the closed loop transfer matrix is block diagonal 2 the remaining diagonal blocks are stable with respect to an a priori given first stability set and 3 the clossed loop system is internaly stable with respect to an a priori given second in general larger stability set in addition we study the almost version of the above problem in the latter the requirement of exact decoupling is replaced by a requirement of approximate decoupling in the sense that the compensators to be designed should yield off diagonal blocks in the closed loop transfer matrix that are arbitrarily small in h norm necessary and sufficient conditions for the existence of such dynamic compensators are formulated in terms of controlled invariant and almost controlled invariant subspaces

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

smith mcmillan factorizations at infinity of rational matrix functions and their control interpretation in this paper the smith mcmillan factorizations at infinity of a transfer function are studied the non uniqueness of these factorizations is characterized via a multiplicative group of bicausal isomorphisms control interpretations are given in terms of system equivalence under some groups of transformations a characterization of the stabilizer of morse group at a b c is given for irreducible systems

parallel implementation of efficient preconditioned linear solver for grid based applications in chemical physics i block jacobi diagonalization abstract linear systems in chemical physics often involve matrices with a certain sparse block structure these can often be solved very effectively using iterative methods sequence of matrix vector products in conjunction with a block jacobi preconditioner numer linear algebra appl 7 2000 715 in a two part series we present an efficient parallel implementation incorporating several additional refinements the present study paper i emphasizes construction of the block jacobi preconditioner matrices this is achieved in a preprocessing step performed prior to the subsequent iterative linear solve step considered in a companion paper paper ii results indicate that the block jacobi routines scale remarkably well on parallel computing platforms and should remain effective over tens of thousands of nodes

some geometric considerations about the kronecker normal form abstract in the first part of this paper we develop general algorithms from the kronecker normal form of any matrix pencil and give some of their properties in the second part these results are applied to the so called restricted matrix pencil associated with a given c a b system and introduced by jaffe and karcanias 1981 this allows us to make a link with well known algorithms of the geometric approach particularly the a b invariant subspace algorithm and the almost a b controllability subspace algorithm finally relationships between morse s invariants morse 1973 and the invariants of the restricted matrix pencil are exhibited

error probabilities for ds spread spectrum systems using an ale for narrow band interference rejection the effects of the least mean squares adaptive line enhancer ale weight misadjustment errors on the bit error rate are investigated for a direct sequence spread spectrum binary communication system in the presence of strong narrowband interference the converged ale weights are modeled as the parallel connection of a deterministic fir finite impulse response filter and a random fir filter the statistics of the random filter are derived assuming the output of the random filter to be primarily due to the jammer convolved with random filter weights this output is shown to be nongaussian and to cause significant error rate degradation in comparison to a gaussian model error probability expressions are derived for the bit error rate evaluated numerically and compared to the corresponding error probabilities for a gaussian model for the random filter output for some typical system parameter values and error probabilities it is shown that the gaussian model yields performance results that are too optimistic by several decibels

on the existence of a common solution x to the matrix equations aixbj cij i j γ in this paper conditions are derived for the existence of a common solution x to the matrix equations aixbj cij i j where the matrices ai bj cij and x have suitable dimensions and the i j s are index pairs in some set the purpose of this paper is to present for certain specific sets of index pairs verifiable necessary and sufficient solvability conditions that are stated directly in terms of the matrices and that do not use kronecker products

nonlinear output feedback tracking with almost disturbance decoupling addresses the design of global output feedback controls for a class of uncertain nonlinear single input single output systems which are globally transformable into an observable minimum phase system whose nonlinearities depend on the output only they may be affected by unknown time varying disturbances or parameter variations entering linearly in the state equations the proposed dynamic controller is of order spl rho 1 spl rho is the relative degree of the given system and ensures that the closed loop system enjoys for any initial condition and for any smooth bounded output reference signal with bounded time derivatives up to order spl rho the following properties input to state stability with respect to disturbance inputs and almost disturbance decoupling i e the influence of disturbances both on the l sub 2 and on the l sub spl infin norm of the output tracking error is arbitrarily attenuated by increasing a positive scalar control parameter when disturbances are zero the reference signal is exponentially tracked by the output and the equilibrium point corresponding to the reference signal is globally asymptotically stable

further results on morgan s problem this paper deals with the row by row decoupling problem known as the morgan problem with static state feedback the solution of this problem is well known when some artificial constraints on the feedback laws are imposed such as regular input transformations or restricted state feedbacks however in its most general setting the problem remains unsolved we provide here a new sufficient condition of its solvability the fact that this condition turns out to be also necessary in the particular case of systems having 2 outputs and 3 inputs enforces its interest

a note on stabilization by dynamic high gain output feedback in this note we study the problem of stabilization by dynamic output feedback for a certain class of linear systems we establish the existence of pairs of subspaces im upsilon a where im is a minimum phase input subspace contained in an almost stabilizability snbspace upsilon a by applying a recent theorem by sehumacber this result leads to the existence of stabilizing high gain compensators for the linear systems under consideration in particular it will be shown that invertible minimum phase systems can always be stabilized by dynamic compensators of dynamic order equal to the excess of poles over zeros minus the number of inputs

inner and outer geometry for singular systems with computation of subspaces abstract output nulling and unknown input subspaces are defined for singular systems with recursions given for their computation inner i e domain subspaces are distinguished from outer i e codomain subspaces a general duality theory is presented and the composite image composite pre image and composite subspaces are introduced it is shown how output nulling and unknown input subspaces may be computed via the singular system structure algorithm and a new square root algorithm the latter algorithm is shown to be related to the solution of a riccati equation for singular systems a simplified riccati solution procedure is introduced that only requires the solution of a non square lyapunov equation and also applies in the state variable case

reduction of observer order by differentiation almost controllability subspace covers and minimal order pid observers this note generalizes the geometric theory around minimal and reduced order observers to the situation in which differentiation of certain components of the observed output is allowed a geometric theory involving the notion of pid observer is introduced using the concept of almost complementary observability subspace a generalization of the notions of dynamic cover and cover index is defined in the context of almost controllability subspaces we prove a result on the existence of minimal dimension covers for one dimensional subspaces these ideas are used to define the concept of minimal order pid observer and to establish the existence of minimal order pid observers for a single linear functional of the state

a transfer matrix approach to feedback invariants of linear systems application to decoupling abstract the transfer matrix approach allows one to exhibit fairly easily fundamental invariants of multivariable systems under the action of transformation groups these transformation groups include the classical state feedback state and input changes of coordinates then the solvability of numerous classical control problems e g decouplability disturbance rejection model following can be expressed directly in terms of these invariants in this expository paper we present in a unified way three lists of feedback invariants related to the infinite behavior of linear systems these invariants turn out to be crucial in studying feedback decoupling problems the main tools for this study are factorization of transfer matrices over the p i d of proper rational functions when dealing with stability we factor transfer matrices over the p i d of proper rational stable functions we illustrate the use of these invariants by studying some decoupling problems

generalized linear systems geometric and structural approaches abstract this paper presents a new way of introducing invariant subspaces for generalized systems this comes from somewhat known geometric algorithms but in most cases with initial conditions different from those usually seen in this context these definitions are shown to be consistent with other ones directly deduced from classical approaches like matrix pencil tools or discrete algebraic formalism the key starting point for their introduction is the extension of structural descriptions for these generalized systems such as have been available for proper systems many results from the proper case can thus be extended for instance the geometric definition for controllability indices and something similar to the famous morse canonical decomposition of the strictly proper case some links with the inversion algorithm are also sketched

a subspace approach to linear dynamical systems abstract in this paper we present an approach to linear dynamical systems which combines the positive features of two well known formulations namely standard state space theory see for e g wonham 1978 9 and behavioural systems theory see polderman and willems 1997 4 our development is also geometric in the tradition of wonham and others but instead of using explicit linear maps we work with linear relations implicitly which amounts to working with subspaces one of our primary motivations is computational efficiency all our computations can be performed on the system as it is without elimination of variables and further unlike the behaviourists who manipulate matrices with polynomial entries we work only with real matrices using our formulation we derive the standard vector space results on controlled and conditioned invariant subspaces of linear dynamical systems duality which is a distinctive feature of state space theory but not of the behavioural view point comes out naturally in our approach too through the use of the adjoint we illustrate our ideas for an important class of dynamical systems viz electrical networks the theory proposed in this paper gives a unified description of both the standard linear dynamical systems and the linear singular systems or the linear descriptor systems see for e g f gantmacher 1959 1 and lewis 1986 2 therefore the algorithms described for the invariant spaces in this paper are also applicable to linear singular systems

the complex lms adaptive algorithm transient weight mean and covariance with applications to the ale the transient and steady state mean and covariance of the complex valued lms adaptive element weights are investigated when the inputs are samples from circularly normal processes it is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix this result permits describing the complete transient behavior of the adaptive line enhancer ale weight covariance matrix in closed form for the case of multiple equal power narrow band statistically independent orthogonal rayleigh fading sinusoids in broad band noise based on these transient covariance results it is shown that for any stage of adaptation there exists an optimum ale gain constant that minimizes the excess mean squared prediction error this result is of particular significance when processing time limited data

a geometric approach to the theory of 2 d systems the authors develop a geometric approach to the theory of 2 d two dimensional systems defining in a suitable way the notion of a sub 1 2 b sub 1 2 invariant subspaces and of controlled invariant subspaces such subspaces are shown to have good computational and feedback properties which make them useful in application in particular sufficient conditions and constructive procedures are obtained for the solutions of disturbance decoupling and model matching problems moreover it is shown that certain structural properties of a 2 d system can be described by means of a set of indexes defined in geometric terms and that such structural indexes can be used to reformulate the sufficient condition for model matching

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

tracking characteristics of the constrained iir adaptive notch filter the tracking behavior of a constrained iir notch filter the coefficients of which are estimated using a constant step size gauss newton algorithm is studied using weak convergence theory and the concept of prescaling it is shown that the mean behavior of the adaptive filter coefficients can be studied by using an ordinary differential equation ode approximate and simple closed form results are derived for the tracking behavior of a second order notch filter computer simulations are presented to substantiate the analysis

sufficient algebraic conditions for stability of cones of polynomials in this paper a sufficient condition for a cone of polynomials to be hurwitz is established such condition is a matrix inequality which gives a simple algebraic test for the stability of rays of polynomials as an application to stable open loop systems a cone of gains c such that the function u kctx is a stabilizing control feedback for all k 0 is shown to exist

a complement about the structure of linear systems the aim of this paper is to establish new relationships between the steps of the well known basic algorithms i s a and c i s a it is suggested that this new structural information will provide a better understanding of the structural modifications and more particularly the increase of the infinite zero structure due to the use of non invertible input transformations

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

stability radius optimization a geometric approach abstract the subject of this paper is a geometric approach to the robustness optimization problem for uncertain finite dimensional linear systems for the structurally perturbed stabilizable pair a b we show that optimally robust feedback stabilization is possible via the class of feedbacks parameterizing a b feedback invariant subspaces of codimension rk b a direct consequence of this geometric approach is a reduction in dimension of the optimization problem

matrix spectroscopy computation of interior eigenstates of large matrices using layered iteration the computation of a small group of interior eigenstates of a large matrix h is an important problem in chemical physics this general problem is approached in this study through the application of a two layer iteration scheme the design of the outer loop is based upon the strategy developed by ericsson and ruhe math comput 35 1251 1980 starting with a variable input energy e this loop uses the lanczos algorithm with no reorthogonalization to develop a reduced tridiagonal representation of the total green function g e diagonalization of this small matrix denoted t yields through a simple mapping excellent approximations to the eigenvalues of h that lie near energy e the inner iteration loop of the algorithm computes the matrix vector product g e mathbf q mathit j where mathbf q mathit j is one of the lanczos vectors both direct and indirect methods may be used for this operation the direct method uses a perturbative expansion of the green function while the indirect method is based upon the iterative solution of a linear algebraic system the indirect method yields a parametrized polynomial representation of the green function with the coefficients chosen variationally the method is adaptive in the sense that these coefficients change during the course of the lanczos iteration convergence of these iterative methods is improved significantly by developing the green function in the partial adiabatic representation conversion to this representation involves diagonalization of the diagonal block of h involving those states whose zeroth order energies surround the energy of interest the two layer scheme is applied to a model hamiltonian matrix that has a variable coupling strength within and between a number of dense bands of states with this scheme it is possible to accurately compute a group of eigenstates near energy e in just a few lanczos iteration steps 10

almost perfect tracking for linear systems abstract two reference input tracking problems that involve arbitrarily small tracking error for time varying inputs and asymptotically zero tracking error for constant inputs are formulated for square invertible linear plants conditions for solvability via high gain feedback are given in terms of plant transmission zeros simplifications in the geometry of almost invariant subspaces for plants that satisfy the conditions are exploited to obtain explicit control law constructions

almost disturbance decoupling for a class of high order nonlinear systems the problem of almost disturbance decoupling with internal stability add is formulated in terms of an l sub 2 l sub 2p instead of an l sub 2 gain for a class of high order nonlinear systems which consist of a chain of power integrators perturbed by a lower triangular vector field a significant feature of the systems considered in the paper is that they are neither feedback linearizable nor affine in the control input which have been two basic assumptions made in all the existing add nonlinear control schemes using the technique of adding a power integrator we solve the add problem via static smooth state feedback under a set of growth conditions that can be viewed as a high order version of the feedback linearization conditions we also show how to explicitly construct a smooth state feedback controller that attenuates the disturbance s effect on the output to an arbitrary degree of accuracy with internal stability

weight modulation effects in the adaptive line enhancer the steady state behavior of the adaptive line enhancer is modeled by the sum of two filters the wiener weight filter and a weight misadjustment filter the effects of randomly time varying coefficients in the weight misadjustment filter are modeled so as to display line broadening of single frequency ale inputs

a complement about almost controllability subspaces abstract we show that for any almost controllability subspace there exist a maximal input chain and a family of state feedbacks which lead to the well known feedback chain expression for this subspace

the almost disturbance decoupling problem with internal stability for linear systems subject to input saturation state feedback case abstract we consider the almost disturbance decoupling problem addp and or almost d bounded disturbance decoupling problem addp d with internal stability for linear systems subject to input saturation and input additive disturbance via linear static state feedback we show that the almost d bounded disturbance decoupling problem with local asymptotic stability addp las d is always solvable via linear static state feedback as long as the system in the absence of input saturation is stabilizable no matter where the poles of the open loop system are and the locations of these poles play a role only in the solution of addp gas d addp sgas d or addp gas where semi global or global asymptotic stability is required

all optimal controls for the singular linear quadratic problem without stability a new interpretation of the optimal cost abstract the singular linear quadratic control problem without stability is solved by means of a generalized dual structure algorithm in order to generate all optimal inputs furthermore it is shown that the optimal cost can be interpreted as the smallest nonnegative rank minimizing solution of a certain matrix inequality the so called dissipation inequality

a geometric approach to proportional plus derivative feedback using quotient and partitioned subspaces abstract in this paper a new characterization of invariant subspaces is presented using the notions of partitioned and quotient subspaces this classification is based on a feedback approach that exhibits the importance of these subspaces for the problem of proportional plus derivative feedback it also provides the ability to decompose the closed loop system into two subsystems that completely characterize the closed loop behavior a feedback free formulation is also considered which opens new horizons for the problem of general semistate feedback by utilizing nonsquare pencils for spectrum assignability the importance of this technique arises from the fact that we deal with reduced order pencils therefore the computational methods are more stable in order to accomplish this we use two generalized lyapunov equations and exploit the generalized hessenberg form

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods