performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

interpolativity of at least and at most models of monotone fuzzy rule bases with multiple antecedent variables of the many desirable properties of fuzzy inference systems not all are known to co exist for instance a fuzzy inference system should be interpolative i e if a fuzzy set equivalent to one of the antecedent fuzzy sets appears on the input of such a system the inferred output should be equivalent to the respective consequent fuzzy set similarly the defuzzified output obtained from a system based on a monotone fuzzy rule base should be monotone i e if x x are crisp inputs to the system that are ordered x x then the corresponding defuzzified outputs from the system y y should also be ordered accordingly i e y y however the particular setting that ensures monotonicity need not simultaneously ensure interpolativity recently st pnicka and de baets have investigated and demonstrated the co existence of the above two properties in the case of fuzzy relational inference systems and single input single output siso rule bases an extension of these results to the multiple input single output miso case is not straightforward due to the lack of a natural ordering in higher dimensions in this work we study the miso case and show that similar results can be obtained when the monotone rule base is modeled based on at most and at least modifiers

on the monotonicity of hierarchical sum product fuzzy systems abstract motivated by the authors previous work on the control of queueing systems the assessment of sustainable development and the measurement of material recyclability this paper provides sufficient conditions on the parameters of hierarchical fuzzy systems under which the output of the system is monotonic with respect to its inputs this property could be useful in designing multistage fuzzy inference systems and fuzzy controllers

a sensitivity tradeoff for plants with time delay in this paper we show that the sensitivity function of a scalar feedback system must satisfy an integral constraint when the open loop transfer function is strictly proper and contains a time delay the integral constraint is identical to the classical bode sensitivity integral which holds provided the open loop gain has greater than a one pole rolloff the design implications of the two constraints are somewhat different however and this difference is discussed the relation between open loop phase and a conflict between bandwidth and sensitivity properties is explored

dynamical system design from a control perspective finite frequency positive realness approach nonminimum phase zeros are well known to limit the best achievable control performance when the control gain is allowed to be arbitrarily high on the other hand the phase crossover appears to be a limiting factor for performance when high gain controllers are not allowed in particular the positive realness in a finite frequency range seems crucial for achieving good performance in the presence of control constraints this paper will first give multiple reasons to support this conjecture and then develop a systematic method for designing mechanical systems to achieve the finite frequency positive real ffpr property specifically we present a state space characterization of the ffpr property by generalizing the well known kalman yakubovich popov lemma to deal with a class of frequency domain inequalities that are required to hold within a finite frequency interval the result is further extended for uncertain systems to give a sufficient condition for satisfaction of a robust ffpr property the nominal ffpr result is interpreted in the time domain in terms of input output signals finally we show that certain sensor actuator placement problems to achieve the ffpr property can be reduced to finite dimensional convex problems involving linear matrix inequalities the method is applied to the shape design of a swing arm for magnetic storage devices with the objective of maximizing the control bandwidth achievable with a limited actuator power

comments on robust tracking control for rigid robotic manipulators in the original paper by m zhihong and m palaniswami ibid vol 39 p 154 9 1994 a robust tracking control using a nominal feedback controller and a variable structure compensator for a rigid robotic manipulator with uncertain dynamics was presented it is assumed that the model uncertainty is bounded the author claims that this assumption is not necessarily correct and that the results obtained are therefore untenable

loop gain optimization of unstable non minimum phase feedback systems the paper presents a design method for optimizing the loop gain of unstable non minimum phase feedback systems because obvious gain bandwidth limitations exist in such systems the proposed best loop gain optimization goal is maximization of the stability gain and phase margins which are inherently low in such feedback systems the design procedure is based on coprime factorization of the plant and controller and the bezout identity an optimization process varies iteratively the observer based controller parameters until maximization of the gain and phase margins is achieved the same design technique also allows one to solve the mixed sensitivity h standard control problem for linear time invariant feedback systems

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

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

fundamental design tradeoffs in filtering prediction and smoothing this paper examines fundamental design tradeoffs that apply to all linear filtering prediction and smoothing problems we introduce sensitivity functions that quantify filter performance and we show that their frequency response satisfies constraints imposed by unstable poles and nonminimum phase zeros of the system the constraints allow one to determine a priori whether or not a desired filter performance is attainable and how different arrangements of the measurement system influence the achievable performance

performance limitations of nonminimum phase affine nonlinear systems this paper presents an analytical method to determine the minimum norm of the regulation error as well as that of the trajectory tracking error for nonminimum phase affine nonlinear systems the proposed method derives the hamilton jacobi bellman equation associated with the cheap control problem depending on the order of the system s internal dynamics the proposed technique either constructs an algebraic equation with parametric coefficients for systems with first order internal dynamics or gives a differential equation in the case of systems with second order or higher internal dynamics the energy of the solution of the derived equation is then used as the performance limitation index to represent the minimum value of the error norm the proposed index is well suited for analytical as well as numerical calculations for any arbitrary set of initial values for the internal states simulation studies are performed to show how the proposed method can be applied to obtain the minimum norm of the output for a system

the relationship between real poles and real zeros in siso sampled data systems a relationship between real poles and real zeros of siso single input single output sampled data systems is presented which is independent of the sampling period the relation is stated in terms of a parity property involving the number of real zeros between any two real poles this property allows the investigation of conditions for the preservation of stable or unstable zeros under sampling a sufficient condition for stability of a sampled system with a continuous time plant having all real zeros and a sufficient condition for instability of a two sampled system with an unstable plant is developed

the effect of system zeros on the achievable h sub spl infin estimation level lower bounds on the achievable estimation accuracy of stationary linear processes in the presence of energy bounded exogenous signals are derived these bounds are determined by the zeros of the process transfer function matrix a significant difference is observed between the minimum phase and the nonminimum phase cases in the latter case a lower bound is derived also for the norm of the matrix that solves the corresponding riccati equation the difference between the cases of minimum and nonminimum phase is accentuated when the measurement noise intensity tends to zero it is shown that in the minimum phase case the corresponding filter gain is almost in the range of the process input matrix in the nonminimum phase case the columns of the corresponding gain tend to stay in the space spanned by the rows of the system output matrix

robustness of zero shifting via generalized sampled data hold functions in this paper we study robustness and sensitivity properties of a sampled data feedback system with a generalized sampled data hold function gshf we argue that shifting non minimum phase zeros using gshf control can lead to difficulties unless the zero is outside the closed loop bandwidth

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

only smooth rule bases can generate monotone mamdani assilian models under center of gravity defuzzification at first sight it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model however this is not the case in this study we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone input output behavior in the mamdani assilian models considered in this study the linguistic values of the input variables as well as the output variable are described by trapezoidal membership functions that form a fuzzy partition the rule base is monotone and the crisp output is obtained by the center of gravity cog defuzzification method it is verified that for each of the three basic t norms i e the minimum tm the product tp and the lukasiewicz t norm tl a monotone input output behavior is obtained for any monotone rule base or at least for any monotone smooth rule base the outcome of this study is a guideline for designers of monotone linguistic fuzzy models for the t norms tm and tl models with a single input variable show a monotone input output behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length the latter restriction can easily be bypassed by an auxiliary interpolation procedure for the t norm tp models with a single input variable show a monotone input output behavior for any monotone rule base and any fuzzy output partition when designing a monotone model with more than one input variable one should opt for the t norm tp and use a monotone smooth rule base it is shown that monotonicity of models with two input variables that apply tp is guaranteed for any monotone smooth rule base and any fuzzy output partition finally it is proved that a monotone input output behavior is always obtained for models with three input variables that apply tp and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape

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

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

properties of zeros in digital control systems with computational time delay abstract a digital control system with computational time delay is considered in which the controlled plant is a continuous time linear system and the control input is calculated by a digital computer based on the data of the output measured before a certain time not greater than the sampling period some qualitative properties of the zeros such as the characteristics of the limiting zeros the parity relation between real poles and zeros and the behaviour of the zeros with respect to the computational time delay are investigated and the parity interlacing property and its dual are derived a method of stabilization by a stable digital compensator using the inter sample data is also proposed finally the first order hold case is discussed

parameter conditions for monotonic takagi sugeno kang fuzzy system this paper presents sufficient conditions on the parameters of the takagi sugeno kang tsk fuzzy system under which the output of the tsk fuzzy system becomes monotonic with respect to its input the output of the tsk fuzzy system is first differentiated with respect to its input and the parameter conditions are derived so as to make the derivative nonnegative the parameter conditions are developed for both single input and multi input tsk fuzzy systems where the involved fuzzy membership functions are differentiable everywhere or bar some finite points the derived monotonicity conditions consist of two parts the conditions on the consequent parts and the conditions on the fuzzy membership functions they are further characterized for the most important types of fuzzy membership functions

finite time behavior of inner systems in this paper we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties for the class of linear time invariant dynamical systems these characteristics are determined by transmission zeros of the inner factor of the system transfer function the relation between nonminimum phase zeros and hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance the results are used to solve various tracking problems both on finite as well as on infinite time horizons a typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller

vibrational feedback control of time delay systems this paper applies vibrational feedback control to time lag systems averaging theory for differential delay systems is presented and then applied to aid in the design of periodic controllers both stabilization and transient issues are discussed an illustrative example is given which demonstrates that the proposed controller 1 provides superior robustness properties in comparison to time invariant controllers when applied correctly 2 is not robust with respect to unmodeled delays and 3 does not have zero placement capabilities in the sense defined in the paper

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

monotone mamdani assilian models under mean of maxima defuzzification in the mamdani assilian models considered in this study the linguistic values of the input variables as well as the output variable are described by trapezoidal membership functions forming a fuzzy partition the rule base is monotone and the crisp output is obtained by the mean of maxima mom defuzzification method it is verified for each of the three basic t norms t m t p and t l whether a monotone input output behaviour is obtained for any monotone rule base or at least for any monotone smooth rule base for models applying t l a monotone input output behaviour can only be guaranteed if the model has a single input variable the monotonicity of models with two or more input variables applying t m cannot be guaranteed for any monotone rule base whereas when applying t p monotonicity is only guaranteed for any monotone rule base when using a fuzzy output partition belonging to a restricted class of nine types of fuzzy partitions which contain at most five linguistic values finally it is proved that a monotone input output behaviour is always obtained for models with a monotone smooth rule base applying t m when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degree of equal length a restriction that can easily be bypassed by an auxiliary interpolation procedure and for models with a monotone smooth rule base applying t p for any fuzzy output partition

commutativity as prior knowledge in fuzzy modeling in fuzzy modeling fm the quantity and quality of the training set is crucial to properly grasp the behavior of the system being modeled however the available data are often not large enough or are deficiently distributed along the input space not revealing the system behavior completely in such cases the consideration of any prior knowledge about the system can be decisive for the accuracy achieved by the fuzzy modeling this paper faces with the integration of mathematical properties satisfied by a system as prior knowledge in fm focusing on the commutativity property as a starting point with this aim several measures are developed to evaluate the commutativity in a fuzzy environment dealing with different elements involved in fm then several approaches are proposed to measure the commutativity degrees of a fuzzy rule with respect to the training set and a simple method is presented to integrate these degrees into the fm task the experimental results show the accuracy improvement gained by the proposed method

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

discrete time pole placement with stable controller the classical polynomial approach to pole placement involves the computation of a particular solution of a given diophantine equation the solution corresponds to a proper controller whose order is fixed by the number of nonminimum phase zeros of the plant transfer function by its order and by the desired closed loop characteristic polynomial this controller might be unstable for some desired closed loop characteristic polynomials even though the plant can be stabilized by a stable controller now unstable controllers are known to deteriorate closed loop system performances compared with stable ones hence we propose a procedure to build for a specific class of plants a stable pole placement controller when the classical pole placement controller is unstable the properties of both stable and unstable controllers are compared via simulations it appears that for the considered plants the stable controller improves at least one of the following properties compared with its unstable counterpart step response transient gain margin measurement noise rejection the results obtained with our procedure are also compared with the results obtained via middleton s approach which consists in modifying the desired closed loop poles so as to obtain a stable and minimum phase controller with the classical pole placement method finally our pole placement design with stable controller is applied to solve a particular simultaneous partial pole placement problem

paper sensitivity integrals for multivariable discrete time systems the purpose of this paper is to develop integral formulae for singular values of the sensitivity function to express design constraints in multivariable discrete time systems we present extensions to both the classical poisson integral and bode s sensitivity integral formulae the main utility of these results is that they can be used to quantify design limitations that arise in multivariable discrete time systems due to such system characteristics as open loop unstable poles and nonminimum phase zeros and to such design requirements as stability and bandwidth constraints these formulae are similar to those for multivariable continuous time systems obtained elsewhere and they reveal that feedback design limitations depend on directionality properties of the sensitivity function and on those of unstable poles and nonminimum phase zeros in the open loop transfer function

generalization of integral constraints on sensitivity to time delay systems extends results on integral constraints to time delay systems key technical issues in this extension include properties of nonminimum phase zeros the convergence of blaschke products and the poisson integral the results have implications for scalar and multivariable control system design as well as in filtering problems

integral constraints on sensitivity vectors for multivariable linear systems this paper develops integral constraints on sensitivity vectors for multivariable feedback systems due to either unstable poles or non minimum phase zeros of the plant the results presented cover the continuous rational or distributed case and are based on the use of directions corresponding to zeros or poles of the plant the integral constraints are in vector form and give new insights into the inherent trade offs in sensitivity reduction for multivariable systems and the cost of decoupling

on the role of subspace zeros in retrospective cost adaptive control of non square plants we consider adaptive control of non square plants that is plants that have an unequal number of inputs and outputs in particular we focus on retrospective cost adaptive control rcac which is a direct discrete time adaptive control algorithm that is applicable to stabilisation command following disturbance rejection and model reference control problems previous studies on rcac have focused on control of square plants in the square case rcac requires knowledge of the first non zero markov parameter and the non minimum phase nmp transmission zeros of the plant if any no additional information about the plant or the exogenous signals need be known the goal of the present paper is to consider rcac for non square plants unlike the square case we show that the assumption that the non square plant is minimum phase does not guarantee closed loop stability and signal boundedness the main purpose of this paper is to establish the existence of time invariant input and output subspaces correspond

design sensitivity analysis in large deformation elasto plastic and elasto viscoplastic problems implicit time integration algorithm derived by simo for his large deformation elasto plastic constitutive model is generalized for the case of isotropy and associative flow rule towards viscoplastic material behaviour and consistently differentiated with respect to its input parameters combining it with the general formulation of design sensitivity analysis dsa for non linear finite element transient equilibrium problem we come at a numerically efficient closed form finite element formulation of dsa for large deformation elasto plastic and elasto viscoplastic problems with various types of design variables material constants shape parameters the paper handles several specific issues like the use of a non algorithmic coefficient matrix or sensitivity discontinuities at points of instantaneous structural stiffness change computational examples demonstrate abilities of the formulation and quality of results copyright 2005 john wiley sons ltd

graph theoretic analysis of network input output processes the control of dynamical processes in networks is considered in the case where measurement and actuation capabilities are sparse and possibly remote specifically we study control of a canonical network dynamics when only one network component s state can be measured and only one in general different component can be actuated to do so we characterize the finite and infinite zeros of the resulting siso system in terms of the graph topology using these results we establish graph theoretic conditions under which there are zeros in the closed right half plane these conditions depend on the length strength and number of the paths from the component where the input is applied to the component where the measurements are made then we present the implications of these conditions on the controller design task focusing in stabilizations destabilization of network processes under static negative feedback

monotonic smooth takagi sugeno fuzzy systems with fuzzy sets with compact support in this paper sufficient conditions on monotonicity of takagi sugeno t s fuzzy systems with linear consequent parts with smooth membership functions with compact support i e equal to zero outside a compact set are derived the presented conditions considerably reduce conservatism of those known for t s fuzzy systems utilizing membership functions with the support that is not compact besides two illustrative examples will demonstrate that the monotonicity requirement imposed on fuzzy systems may significantly improve their extrapolation capability

tradeoffs in linear time varying systems an analogue of bode s sensitivity integral bode s integral characterizes some of the tradeoffs imposed by the open loop dynamics of a linear time invariant system here we present an analogous time domain characterization applicable to time varying systems

sensitivity shaping with degree constraint by nonlinear least squares optimization this paper presents a new approach to shaping of the frequency response of the sensitivity function in this approach a desired frequency response is assumed to be specified at a finite number of frequency points a sensitivity shaping problem is formulated as an approximation problem to the desired frequency response with a function in a class of sensitivity functions with a degree bound the sensitivity shaping problem is reduced to a finite dimensional constrained nonlinear least squares optimization problem to solve the optimization problem numerically standard algorithms for an unconstrained version of nonlinear least squares problems are modified to incorporate the constraint numerical examples illustrate how these design parameters are tuned in an intuitive manner as well as how the design proceeds in actual control problems

interpolativity of at least and at most models of monotone single input single output fuzzy rule bases interpolativity is one of the most important properties of a fuzzy inference system it is well known that normal antecedent fuzzy sets forming a ruspini partition constitute a practical setting ensuring interpolativity in case of a fuzzy rule base expressing a monotone relationship another desirable property is the monotonicity of the resulting function after defuzzification unfortunately this goal may often only be reached through the application of the at least and or at most modifiers to the antecedent and consequent fuzzy sets however this approach does not seem compatible with the practical setting of a ruspini partition this paper shows that the situation is less conflicting than it seems and that interpolativity can still be guaranteed in the same practical setting and interestingly from two different modeling points of view this paper addresses the case of single input single output fuzzy rules

on the continuity of type 1 and interval type 2 fuzzy logic systems this paper studies the continuity of the input output mappings of fuzzy logic systems flss including both type 1 t1 and interval type 2 it2 flss we show that a t1 fls being an universal approximator is equivalent to saying that a t1 fls has a continuous input output mapping we also derive the condition under which a t1 fls is discontinuous for it2 flss we consider six type reduction and defuzzification methods the karnik mendel method the uncertainty bound method the wu tan method the nie tan method the du ying method and the begian melek mendel method and derive the conditions under which continuous and discontinuous input output mappings can be obtained guidelines for designing continuous it2 flss are also given this paper is to date the most comprehensive study on the continuity of flss our results will be very useful in the selection of the parameters of the membership functions to achieve a desired continuity e g for most traditional modeling and control applications or discontinuity e g for hybrid and switched systems modeling and control

squaring down by static and dynamic compensators given a system which is not necessarily invertible and which has an unequal number of inputs a method of squaring down that is a method of designing pre compensators and postcompensators such that the resulting system has an equal number of inputs and outputs and is invertible is presented the compensators are in general dynamic and have the property that the additional finite zeros induced by them are assignable to the open left half complex plan furthermore the compensators are asymptotically stable and hence do not deteriorate the robustness and performance of an eventual feedback design also the compensator design preserves the stabilizability detectability and minimum phase properties and the infinite zero structure of the original system thus a method of designing nonsquare systems by converting them to square invertible systems is introduced two applications of such a design philosophy 1 diagonal decoupling with state feedback and 2 almost disturbance decoupling with output feedback are pointed out

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

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

performance bounds in cal h _ infty optimal control for stable siso plants with arbitrary relative degree this note deals with performance bounds for the h infin optimal control of discrete time lti plants the case studied corresponds to stable scalar plants with arbitrary relative degree but no finite non minimum phase zero by using nehari s theorem and a reformulation of the standard youla parameterization a closed form expression for the characteristic polynomial of the associated eigenvalue problem is obtained also we derive an analytic expression for the optimal h infin cost as a function of the plant relative degree

a simplified approach to analyze complementary sensitivity tradeoffs in continuous time and discrete time systems continuous time and discrete time complementary sensitivity bode integrals csbis are investigated via a simplified approach in this note for continuous time feedback systems with unbounded frequency domain the csbi weighted by 1 omega 2 is considered where this simplified method reveals a more explicit relationship between the value of csbi and the structure of the open loop transfer function with a minor modification of this method the csbi of discrete time system is derived and illustrative examples are provided compared with the existing results on csbi neither the cauchy integral theorem nor the poisson integral formula is used throughout the analysis and the analytic constraint on the integrand is removed

sensitivity integrals and transformation techniques a new perspective we provide alternative proofs for the classical bode and poisson type sensitivity integrals and their extensions for both continuous time and discrete time systems our derivation uses the well known properties of laplace and z transformations this derivation helps establish a connection between bode and poisson type sensitivity integrals and laplace and z transforms hence providing an alternative perspective in interpreting these fundamental integral results

a fundamental control performance limit for a class of positive nonlinear systems abstract a fundamental performance limit is derived for a class of positive nonlinear systems the performance limit describes the achievable output response in the presence of a positive disturbance and subject to a sign constraint on the allowable input an explicit optimal input is derived which minimises the maximum output response whilst ensuring that the minimum output response does not fall below a pre specified lower bound the result provides a fundamental performance standard against which all control policies including closed loop schemes can be compared implications of the result are examined in the context of blood glucose regulation for type 1 diabetes

multivariable gain phase and sensitivity integral relations and design trade offs we present several extensions of the classical bode integral relations to multivariable systems the main contributions are an extended gain phase integral formula and an extended sensitivity integral relation these results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems in particular they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single input single output systems both results show that design limitations in multivariable systems depend strongly upon certain directionality properties

inherent design limitations for linear sampled data feedback systems there is a well developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller this theory describes limitations on achievable performance present when the plant has non minimum phase zeros unstable poles and or time delays the parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants this paper develops a theory of design limitations for sampled data feedback systems wherein the response of the analogue system output is considered this is done using the fact that the steady state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental this fact allows fundamental sensitivity and complementary sensitivity functions that re

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

integral constraints and performance limits on complementary sensitivity discrete time systems abstract we study performance limitation issues for multivariable discrete time feedback systems the complementary sensitivity function is employed as a performance measure and bode and poisson type integral inequalities and h type performance limits are derived the results exhibit frequency dependent constraints as well as best achievable limits on the complementary sensitivity function which are shown to be determined by nonminimum phase zeros unstable poles and time delays in particular the directions of such zeros and poles are seen to play a central role to this effect

on h optimal sensitivity theory for siso feedback systems this paper deals with the design of feedback controllers which minimize the h infty norm of the sensitivity function suitably weighted this approach to the theory of feedback design was introduced by zames 1 and developed by zames and francis 2 in this paper the theory of sarason 3 is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm the problem of achieving small sensitivity over a specified frequency band is studied and the effect of nonminimum phase is elucidated finally a method is introduced for handling plant poles and zeros on the imaginary axis

the lqg ltr procedure for multivariable feedback control design this paper provides a tutorial overview of the lqg ltr design procedure for linear multivariable feedback systems lqg ltr is interpreted as the solution of a specific weighted h2 tradeoff between transfer functions in the frequency domain properties of this solution are examined for both minimum phase and nonminimum phase systems this leads to a formal weight augmentation procedure for the minimum phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights while such arbitrary specifications are not possible for nonminimum phase problems a direct relationship between weights and sensitivities is developed for nonminimum phase siso and certain nonminimum phase mimo cases which guides the weight selection process

implication based models of monotone fuzzy rule bases abstract in many modelling problems there is some inherent monotone relationship between one or more of the input variables and the output variable we consider the prototypical case of an increasing relationship between each of the input variables and the output variable when using fuzzy rule based models this desired monotonicity is reflected in the rule base given an appropriate ordering on the fuzzy sets involved in the respective input and output domains more specifically the larger the antecedent fuzzy sets the larger the consequent fuzzy set however fuzzy rule based modelling involves a final defuzzification step possibly resulting in a function that is no longer monotone in the context of mamdani assilian conjunctive fuzzy models ample attention has been paid to this problem both for the centre of gravity defuzzification and mean of maxima defuzzification methods in this paper we show that for implicative fuzzy models the non monotonicity problem can be circumvented by making explicit the semantics of the fuzzy rules by subjecting the antecedent and consequent fuzzy sets to the at least and or at most modifiers

logarithmic integrals and system dynamics an analogue of bode s sensitivity integral for continuous time time varying systems a new time domain interpretation of bode s integral is presented this allows for a generalization to the class of time varying systems which possess an exponential dichotomy it is shown that the sensitivity function is constrained on average by the spectral values in the dichotomy spectrum of the antistable component of the open loop dynamics

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

limits on achievable robustness against coprime factor uncertainty abstract we consider the problem of robustness optimization against normalized coprime factor uncertainty in single input single output systems we show that loop shapes known from classical analysis to be inconsistent with closed loop robust stability will tend to have poor optimal robustness such loop shapes include those with a high crossover frequency relative to a nonminimum phase zero a low crossover frequency relative to an unstable pole or a rapid rolloff rate near gain crossover our results consist of a set of lower bounds on the optimal cost of the robustness optimization problem each lower bound being appropriate to one of these three problematic loop shapes the lower bounds are derived using the poisson integral and display the qualitative relationship between the loop shape and the level of optimal robustness

a shaping limitation of rational sensitivity functions with a degree constraint this note concerns a certain shaping limitation of sensitivity functions the focus is on a frequency wise infimum of gains of rational sensitivity functions with a degree constraint an explicit infimum is derived for a special case the result is useful for determining the inability of sensitivity functions of low degrees to achieve a specification in the frequency domain and thus for motivating the use of higher degree sensitivity functions to fulfill the specification

limitations of feedback properties imposed by open loop right half plane poles an integral constraint on the closed loop transfer function of unstable open loop systems is used to quantify a tradeoff between feedback properties of the closed loop system this results in a lower bound on the peak of the complementary sensitivity function for unstable plants the lower bound illustrates the nature of the tradeoff that is imposed and provides insight into the difficulties imposed on the control of unstable systems it can be used to quantitatively evaluate tradeoffs between the frequency of the open loop unstable pole the peak of the complementary sensitivity functions and the bandwidth of the closed loop system such a relationship has long been recognized and is embodied in the classical design rule of thumb which states that the closed loop bandwidth must be at least twice as large as the frequency of the unstable pole results which give insight into this rule and a theoretical basis for it are provided

monotonicity of siso fuzzy relational inference with an implicative rule base a fuzzy relational inference fri mechanism is appraised based on the different desirable properties it possesses among these properties monotonicity of an fri has not received much attention in this work we investigate the monotonicity of a single input single output fri with an implicative form of the rule base in all the previous works that deal with monotonicity of an fri with the implicative form of rule base the employed fuzzy implications come from a residuated lattice it can be noticed that this rich underlying structure plays a major role in proving the results further we also modify the given monotone rule base this work differs from the previous works in that 1 the fuzzy implications employed in it do not come from any known residuated structure on text 0 1 and 2 the original rule base is employed without any alteration we determine conditions under which monotonicity of an fri where the rule base is modeled by a strict fuzzy implication can be ensured without transforming the original rule base thus the results in this work further augment the case for considering fuzzy implications other than those from the residuated setting to be used in applications

on monotonicity of takagi sugeno fuzzy systems with ellipsoidal regions this paper presents sufficient conditions on monotonicity of a takagi sugeno t s fuzzy system with linear submodels in the consequents of the rules where the input space is considered to be partitioned to ellipsoidal regions such regions commonly arise in practice if clustering algorithms are used to identify a fuzzy model from measured data by the monotonicity it is meant that the partial derivatives of the output of the mapping represented by the fuzzy model with respect to all inputs are nonnegative on a given universe of discourse the conditions are given in the form of linear matrix inequalities with respect to the parameters of the submodels that may be useful in solving associated optimization problems via efficient semidefinite programming techniques the proposed conditions reduce conservatism of the existing ones from two reasons first the conservatism is introduced only once since the conditions are not separated between antecedent and consequent parts of fuzzy rules second the domain of interest where monotonicity is enforced may be restricted the proposed algorithm is illustrated on least squares approximation of a multivariate function by a monotonic t s fuzzy system

tracking sensitivity and achievable h performance in preview control preview control has been explored in the context of linear and nonlinear inverse tracking controllers preview of the reference trajectory has been shown to give significant performance advantages particularly for plants that do not possess a stable inverse without preview it has been known that lack of a stable inverse imposes inherent limitations on the achievable tracking performance of a control loop this has been explored in the context of time and frequency domain integral constraints in this paper we explore the extension of these performance limitations to the preview control case in particular we consider the infimal achievable h performance and poisson sensitivity integral results we also briefly discuss tracking sensitivity properties of preview control

toward necessity of parametric conditions for monotonic fuzzy systems input output monotonicity is an important constraint found in many application domains a monotonic fuzzy system mfs is defined as a takagi sugeno kang tsk system whose output is monotonically increasing or decreasing with respect to one or more inputs this paper reviews the authors previous work which derived the parametric conditions for the mfs and discusses the rationale lying behind the conditions an mfs is developed by creating a monotonic rule base while preserving the relative monotonicity among the membership functions corresponding to the fuzzy rules this paper also proves that the parametric conditions are necessary and sufficient to build a single input zeroth order mfs with two rules only the sufficiency of the conditions holds for a multi input first order or higher order tsk fuzzy system with three or more rules

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

technical communique nyquist stability criterion of intrinsic zeros for continuous time pure imaginary zeros this note treats a stability condition of the zeros of sampled systems corresponding to continuous time pure imaginary zeros with multiplicity one for systems with the relative degree one it is shown that the nyquist plot can be used to check the stability condition

linear deterministic adaptive control fundamental limitations this paper is concerned with the achievable performance of adaptive control algorithms we show that when the only uncertainty is in the form of fixed parameter errors then there exists an adaptive feedback law whose performance can be made arbitrarily close to that achievable when the system is a priori known the result is not intended as a practical strategy instead we use it to make the perhaps obvious point that meaningful results on performance of adaptive control algorithms must account for non ideal factors including at a minimum noise parameter time variations and unstructured uncertainty

controller design for maximizing the size of a step disturbance in non minimum phase unstable uncertain systems considered in this paper is the problem of maximization of tolerable disturbances in the presence of hard time domain constraints on system states and the input for a class of linear time invariant single input single output uncertain systems in particular a methodology proposed by the authors is extended to include non minimum phase and or unstable systems an inner outer factorization separates a nominal minimum phase transfer function from the uncertain plant set thus facilitating the generation of the required frequency domain design region in dealing with unstable systems an intermediate prestabilization step may become necessary

limiting performance of optimal linear filters we study the lowest achievable mean square estimation error in two limiting optimal linear filtering problems first when the intensity of the process noise tends to zero the lowest achievable mean square estimation error is a function of the unstable poles of the system second when the intensity of the measurement noise tends to zero the lowest achievable mean square estimation error is a function of the nonminimum phase zeros of the system we link these results with bode integral characterisations of performance limitations in linear filtering

a dissipation inequality for the minimum phase property the minimum phase property is an important notion in systems and control theory in this paper a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived it is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of byrnes and isidori if the control system is affine in the input and the so called input output normal form exists

on the monotonicity of fuzzy inference methods related to t s inference method yubazaki et al have proposed a single input rule modules connected type fuzzy inference method sirms method whose final output is obtained by combining the products of the importance degrees and the inference results from single input fuzzy rule modules moreover seki et al have proposed a functional type sirms method whose consequent parts are generalized to functions from real numbers it is expected that inference results from the functional type sirms method are monotone if the antecedent parts and the consequent parts of fuzzy rules in the functional type sirms rule modules are monotone however this paper points out that even if consequent parts in the functional type sirms rule modules are monotone the inference results are not necessarily monotone when the antecedent parts are noncomparable fuzzy sets and it clarifies the conditions for the monotonicity of inference results from the functional type sirms method moreover for the takagi sugeno t s inference method the monotonicity condition is clarified in the case of two inputs by using the equivalence relation of fuzzy inference

sensitivity improvement by a stable controller in siso digital control systems abstract lower bounds which are simple functions of the sampling period are developed for h norm sensitivity functions in digital control systems with and without computational time delay it is shown that the sensitivity property can be improved by a stable controller using a new simple scheme called 2 delay digital control in which we use the inter sample information of the output as well as the data at each sampling instant

path following for linear systems with unstable zero dynamics a constructive solution to the path following problem for mimo linear systems with unstable zero dynamics is developed while the original control variable steers the system output along the path the path parameter q is used as an additional control to stabilize zero dynamics with a feedback law which is nonlinear due to the path constraint a sufficient condition for solvability of the path following problem is given in terms of the geometric properties of the path when this condition is satisfied an arbitrary small l 2 norm of path following error can be achieved thus avoiding performance limitations of the standard reference tracking problem imposed by unstable zero dynamics

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

a fundamental limit on the heat flux in the control of incompressible channel flow this paper proves that there are no zero net wall transpiration control strategies that can sustain net heat flux below the laminar level in an incompressible channel flow with constant temperature walls the result represents a fundamental limit on the performance of a controlled nonlinear system as measured by a linear cost function over a broad class of admissible initial conditions and control inputs not a zero sum tradeoff in the frequency domain or time domain both buoyancy effects via the boussinesq approximation and viscous heating effects are accounted for and phenomenological justification for the result is also given the boundedness of solutions of the two way coupled navier stokes energy equations when both buoyancy and viscous heating are accounted for is also discussed and a new proof of existence under an appropriate small data assumption is provided

on overshoot and nonminimum phase zeros it is widely known that real nonminimum phase zeros lead to step response undershoot and that the size of the undershoot necessarily tends to infinity as the settling time tends to zero in this note we show that the presence of two or more real nonminimum phase zeros can lead to step response overshoot in addition to undershoot a lower bound on the overshoot is derived and it is shown that the overshoot like the undershoot necessarily tends to infinity as the settling time tends to zero the results are derived for single input single output linear time invariant continuous time systems and apply to both open loop control and general two degree of freedom closed loop control