adaptive robust control of uncertain systems with measurement noise abstract a class of dynamical systems with time varying uncertainty is considered the nominal portion i e the portion without uncertainty of the system is stabilizable the uncertainty is either assumed to satisfy the matching condition or is close to matching moreover the uncertainty is to be cone bounded the bound is however unknown the control design is only based on the deterministic properties related to the bound of uncertainty the feedback information is the state with measurement noise

robust stability parameter dependent perturbations conditions for a system to be robustly stable with respect to a set of perturbations parametrized by possibly uncertain parameters are given in terms of some worst case estimates of the perturbations making use of the hadamard product of matrices a new way of formulating perturbations is introduced the advantage of representing the perturbations in this way is that the magnitude structure of the system can be taken into robustness considerations

robust stability of additively perturbed interconnected systems the concept of generalized block diagonal dominance is used to derive robust stability results for additively perturbed interconnected systems these results characterize the robustness of interconnected systems by putting upper bounds on the spectral norms of the various subblocks of admissible perturbations

on the robustness of lq regulators it is shown that in spite of its large gain and phase margins the linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system may lead to fast unstable closed loop modes

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

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

loop transfer recovery for linear systems with delays in the state and the output this paper presents robustness properties of the kalman filter and the associated lqg ltr method for linear time invariant systems having delays in both the state and the output a circle condition relating to the return difference matrix associated with the kalman filter is derived using this circle condition it is shown that the kalman filter guarantees 1 2 gain margin and 60 phase margin which are the same as those for non delay systems however it is shown that even for minimum phase plants the lqg ltr method cannot recover the target loop transfer function instead an upper bound on the recovery error is obtained by using an upper bound of the solution of the kalman filter riccati equations finally some dual properties between output delayed systems and input delayed systems are exploited

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

an estimation algorithm for multirate sampled digital control systems abstract this paper presents an optimization based algorithm for state estimation of multirate digital control systems the algorithm features a predictor estimator structure and it is derived by employing a time invariant modelling approach to multirate systems special care is taken to ensure the causality of the derived estimation algorithm and to reduce the dimensionality inherent in multirate system modelling while maintaining a reasonably simple implementation the proposed algorithm is developed to ensure convergence in state estimation between base sampling instants as well as between model rate sampling instants

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

stability robustness of lq optimal regulators for the performance index with cross product terms suggested are the guaranteed stability margins of lq optimal regulators for the performance index with cross product terms using modified return difference equality sufficient conditions to guarantee the required stability margins are presented as well as the conditions to guarantee the stability margins of standard lq optimal regulators an example for lq optimal control of a dc motor system is given to demonstrate how the weighting matrices can be chosen to guarantee the required stability margins

improved design technique for uncertain multiple input multiple output feedback systems this paper presents a synthesis technique for linear time invariant n n multiple input multiple output mimo feedback systems with constrained plant p and output feedback due to uncertainty p is known only to be a member of a set x1d4ab p it is required for all pep the n2 system transfer functions tuv be members of specified sets of acceptable outputs a uv u v 1 tdot n the problem is rigorously converted into a number of single input single output siso uncertainty problems whose solutions are guaranteed to solve the original mimo problem the technique has several advantages over previous ones 1 fixed point theory is not needed to rigorously justify the theory the justification is very simple 2 there is significantly less overdesign inherent in the method and 3 if arbitrary small sensitivity is desired over arbitrary large bandwidth then the set p must satisfy certain constraints as 8 it is shown that these constraints are inherent in all linear time invariant compensat

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

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

a state space algorithm for the solution of the 2 block superoptimal distance problem a state space algorithm for computing the solution of the 2 block superoptimal distance problem sodp is presented given a rational and antistable matrix function r s r 11 s r 12 s find all stable approximations q s that lexicographically minimize the singular values of the error function e s r 11 s r 12 s q s conditions are given for which the superoptimal approximation is unique in addition an a priori upper bound on the macmillan degree of the approximation is given the algorithm may be stopped after minimizing a given number of singular values this premature termination of the algorithm carries with it an expected saving in the computational effort and a predictable reduction in the macmillan degree of the approximation the algorithm only requires standard linear algebraic computations and is therefore easily implemented

lqg ltr methods for linear systems with delay in state the lqg ltr linear quadratic gaussian loop transfer recovery method is discussed along with related frequency domain properties for linear systems with delay in state based on the frequency domain inequality and a robust stability condition the guaranteed stability margin of lq regulators and the guaranteed nondivergence margin of kalman filters are obtained to improve the robustness of lqg regulators the asymptotic property of the filter gain is investigated and a loop transfer recovery procedure is suggested with an illustration

two delay robust digital control and its applications avoiding the problem on unstable limiting zeros in the design of linear control systems the existence of unstable zeros makes it difficult to construct many control systems when the usual digital control scheme is used unstable zeros appear in the discrete time model due to the existence of limiting zeros even though the continuous time plant is minimal phase to avoid this unstable zero problem two new digital control schemes called two delay output control and two delay input control are proposed these control systems are proved to have no finite zeros asymptotic inverse systems pole assignable unknown input observers and output feedback controllers having the ltr loop transfer recovery property are developed using the two delay output control zero assignable control systems and model matching control systems are developed using the two delay input control

a quadratic weight selection algorithm a new algorithm is presented which determines a positive semi definite state weighting matrix in the linear quadratic optimal control design problem the algorithm chooses the weighting matrix by attempting to place dosed loop eigenvalues and eigenvectors near desired locations using optimal feedback gains a simplified flight control design example is used to illustrate the algorithm s capabilities

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

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

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

optimal pole shifting for continuous multivariable linear systems a method for shifting the real parts of the open loop poles to any desired positions while preserving the imaginary parts is presented the method is based on the mirror image property which has been reported by molinari in other words molinari s results are extended and then a recursive approach is developed in each step of this approach it is required to solve a first order or a second order linear matrix lyapunov equation for shifting one real pole or two complex conjugate poles respectively the presented method yields a solution which is optimal with respect to a quadratic performance index the attractive feature of this method is that it enables solutions to complex problems to be easily found without solving any non linear algebraic riccati equation

gain optimization for distributed plants abstract in this note we give an explicit solution to the gain margin optimization problem for a wide class of distributed linear time invariant plants essentially we show that the methods of tannenbaum 16 18 applied to lumped systems extend almost immediately to this case as well

calculation of highly structured stability margins abstract the problem of assessing the robust stability of a closed loop system subject to plant uncertainty is addressed earlier techniques allow only a limited amount of structure information uncertainty and thus can often lead to conservative results a technique for increasing the bound over the stability margin is presented which may be applied together with the cross condition number to compute a stability margin for highly structured uncertainty regions a sample application is given to show the improvements that can be obtained by using the proposed technique in practical situations

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

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

model independent comparison between factorization algebras and algebraic quantum field theory on lorentzian manifolds this paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic lorentzian manifolds functorial constructions that map between these two types of theories in both directions are developed under certain natural hypotheses including suitable variants of the local constancy and descent axioms the main result is an equivalence theorem between cauchy constant and additive algebraic quantum field theories and cauchy constant additive and time orderable prefactorization algebras a concept of involution for the latter class of prefactorization algebras is introduced via transfer this involves cauchy constancy explicitly and does not extend to generic time orderable prefactorization algebras

uncertain multiple input multiple output systems with internal variable feedback there is given a linear time invariant multiple input multiple output mimo plant p with internal variables xi and outputs yi available for feedback due to uncertainty p is known only to be in a set p there are given tolerances on the overall system response functions tuv system output yu in response to command input ru to be satisfied pep it is shown how this mimo uncertainty problem may be replaced by a number of single input single output siso multiple loop uncertainty problems a suitable fixed point theorem guarantees that the solution of the siso problems solves the original mimo problem plant elements which appear singly in the mimo structure also appear singly in the siso structures a design example demonstrates how inner loop feedback may enormously decrease the effect of sensor noise at the mimo plant inputs extensions to disturbance attenuation under uncertainty and to uncertain time varying and non linear mimo plants are discussed

the robustness optimization of a multivariable feedback system in hankel norm space this paper is concerned with the optimization of robustness in a multi input multi output linear time invariant finite dimensional feedback system the objective is to derive a closed form solution to the problems of the synthesis of optimal controllers that either minimize the weighted sensitivity function or maximize the excess robust ness with respect to hankel norm criterion an all pass solution for the minimum hankel norm is employed to treat our problems finally the infinite norm case is also discussed

robust stability of linear quadratic state feedback regulator under system uncertainty this paper is concerned with the robustness of lq state feedback regulators in the presence of system uncertainty assuming that the perturbations satisfy the matching condition several robust stability results are explicitly derived for linear non linear time varying and dynamical perturbations in terms of bounds of perturbations and the weighting matrices in the performance index some examples are included to show the applicability and the limitation of the present results

simultaneous stabilization with almost disturbance decoupling uniform rank system abstract in this paper we study the problem of stabilization with almost disturbance decoupling for a family of linear time invariant finite dimensional uniform rank plants with a fixed controller it is shown that for a family of minimum phase plants which have the same order of infinite zeros and have high frequency gain matrices satisfying a certain condition such a fixed controller exists the restriction on the high frequency gain matrices is equivalent to requiring that the sign of high frequency gain for siso plants be known

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

strictly proper projective controllers for disturbance attenuation a design methodology is presented which applies projective controls to the disturbance attenuation problem for large flexible structures through this method low order strictly proper controllers are designed which improve the disturbance attenuation properties of the system a new parameterization of the closed loop system in terms of free parameters in the projective controller leads to an auxiliary minimization problem which determines the parameters of the controller a flexible structure example demonstrates the use of this methodology

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

circle condition and stability margin of the optimal regulator for systems with delays robustness properties of the optimal regulator for systems with delays in the state are characterized first a nyquist stability criterion is developed and the circle condition of the optimal regulator is derived on the basis of these results the stability margin of the optimal regulator is evaluated as by lehtomaki el al 1981 and it is shown finally that the stability margin is equal to that in the finite dimensional case

stability robustness of linear discrete time systems in the presence of uncertainty the stability robustness problem is considered for nominally stable linear discrete time systems using time domain analysis methods and lyapunov theory bounds on the norms of the time varying non linear perturbations are given to maintain the asymptotic stability of these systems in the presence of such perturbations

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

loop shaping design related to lqg ltr for siso minimum phase plants one method of model based compensator design for linear systems consists of two stages state feedback design and observer design a key issue in recent work in multivariable synthesis involves selecting the observer state feedback gain so that the final loop transfer function is the same as the state feedback observer loop transfer function this is called loop transfer recovery ltr athans and stein 1987 kazerooni and houpt 1986 kazerooni et al 1985 doyle and stein 1981 this paper shows how identification of the internal mechanism of the ltr provides simple design rules with little algebra for single input single output siso systems in the siso case the lqg ltr reduces to computation of a compensator that shapes the loop transfer function by i cancelling the zeros of the plant with the compensator poles and ii locating a new set of zeros for the compensator to shape the loop transfer function

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

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

robustness measure bounds for optimal model matching control designs the theorems introduced by r v patel et al 1977 for multivariable linear quadratic state feedback design in the presence of perturbations are extended to optimal model matching control system designs a robustness measure bound is introduced for optimal model matching control systems bounds are obtained for allowable nonlinear time varying perturbations such that the resulting closed loop system remain stable bounds on a special but important class of perturbations in which the perturbation is linear are also derived

stability robustness measures utilizing structural information this paper presents techniques of using weighted l1 and norms to incorporate the structural information of the return difference and the perturbation matrices in the measure of stability robustness of multivariable control systems the flexibility offered by these norms along with the use of non negative matrix theory enables one in most cases to reduce the conservatism which has been a typical concern with the use of singular value based stability robustness tests new stability robustness criteria are derived on the basis of weighted l1 and norms examples are given to demonstrate the merits of these methods over singular value tests

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