new results on the synthesis of pid controllers this paper considers the problem of stabilizing a first order plant with dead time using a proportional integral derivative pid controller using a version of the hermite biehler theorem that is applicable to quasi polynomials the complete set of stabilizing pid parameters is determined for both open loop stable and unstable plants the range of admissible proportional gains is first determined in closed form for each proportional gain in this range the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid a triangle or a quadrilateral for the case of an open loop unstable plant a necessary and sufficient condition on the time delay is determined for the existence of stabilizing pid controllers

connections between discriminants and the root distribution of polynomials with rational generating function abstract let h m z be a sequence of polynomials whose generating function m 0 h m z t m is the reciprocal of a bivariate polynomial d t z we show that in the three cases d t z 1 b z t a z t 2 d t z 1 b z t a z t 3 and d t z 1 b z t a z t 4 where a z and b z are any polynomials in z with complex coefficients the roots of h m z lie on a portion of a real algebraic curve whose equation is explicitly given the proofs involve the q analogue of the discriminant a concept introduced by mourad ismail

stabilizing a class of time delay systems using the hermite biehler theorem abstract in this paper we use the hermite biehler theorem to establish results for the design of fixed order controllers for a class of time delay systems we extend results of the polynomial case to quasipolynomials using the property of interlacing in high frequencies of the class of time delay systems considered

new result on pid controller design of lti systems via dominant eigenvalue assignment this note considers the problem of assigning the dominant eigenvalues of a linear time invariant lti system to the desired positions by using proportional integral derivative pid controllers the procedure is based on first setting some rightmost eigenvalues of the system at the desired positions and then guaranteeing the dominance of those rightmost eigenvalues by using the generalization of the hermite biehler theorem it is worth pointing out that this work aims to ascertain the gains of pid controllers in a straightforwardly computational way which plays an important role in practical applications

brief paper fixed structure h controller synthesis a meta heuristic approach using simple constrained particle swarm optimization this paper provides a design method of fixed structure robust controllers satisfying multiple h norm specifications by using a sort of randomized algorithms first a new tool to perform general constrained optimization is developed which does not need any gradient or derivative of the objective function this tool is based on a particle swarm optimization pso which has attracted a lot of attention recently in the evolutionary computation area due to the empirical evidence of its superiority in solving various non convex problems second it is shown how to design a fixed structure controller satisfying given multiple h specifications by using the developed optimization tool third its effectiveness is evaluated through various numerical examples because it is difficult to guarantee the performance of the proposed method theoretically due to the probabilistic nature of pso the simulation results demonstrate its effectiveness clearly

robust pid controller tuning based on the constrained particle swarm optimization this paper proposes a novel tuning strategy for robust proportional integral derivative pid controllers based on the augmented lagrangian particle swarm optimization alpso first the problem of pid controller tuning satisfying multiple h performance criteria is considered which is known to suffer from computational intractability and conservatism when any existing method is adopted in order to give some remedy to such a design problem without using any complicated manipulations the alpso based robust gain tuning scheme for pid controllers is introduced it does not need any conservative assumption unlike the conventional methods and often enables us to find the desired pid gains just by solving the constrained optimization problem in a straightforward way however it is difficult to guarantee its effectiveness in a theoretical way because pso is essentially a stochastic approach therefore it is evaluated by several simulation examples which demonstrate that the proposed approach works well to obtain pid controller parameters satisfying the multiple h performance criteria

low order stabilization of lti systems with time delay this paper considers the problem of stabilizing a single input single output siso linear time invariant lti plant with known time delay using a low order controller such as a proportional p a proportional integral pi or a proportional integral derivative pid controller for the siso lti system with time delay the closed loop characteristic function is a quasipolynomial that possesses the following features all its infinite roots are located on the left of certain vertical line of the complex plane and the number of its unstable roots is finite necessary and sufficient conditions for the stability of lti systems with time delay are first presented by employing an extended hermite biehler theorem applicable to quasi polynomials based on the conditions analytical algorithms are then proposed to compute the stabilizing sets of p pi and pid controllers the resulting characterizations of the stabilizing sets for p pi and pid controllers are analogous to the youla parameterization of all stabilizing controllers for plants without time delay numerical examples are provided to illustrate the proposed algorithm

distributed control of a class of flexible mechanical systems with global constraint in this paper the constrained problem is investigated for both flexible string model and euler bernoulli beam model with the tip payload based on an infinite dimensional generalisation of a distributed control method the control objectives are to develop the control law so that the motion of flexible mechanical systems can track a desired reference signal and ensure that the string or beam remain in a constrained space we prove that with the proposed control the tracking error is exponentially stable without violation of the constraint the proof of convergence is based on an integral barrier lyapunov function iblf and extensive simulations are provided to illustrate the performance of the control system

algebraic multidimensional phase unwrapping and zero distribution of complex polynomials characterization of multivariate stable polynomials we define the multidimensional unwrapped phase for any finite extent multidimensional signal that may have its zero on the distinguished boundary of the unit polydisc by using this definition we deduce that multivariate stable polynomials can be simply characterized in terms of the proposed unwrapped phase a rigorous symbolic algebraic solution to the exact phase unwrapping problem for multidimensional finite extent signals is also proposed this solution is based on a newly developed general sturm sequence and does not need any numerical root finding or numerical integration technique furthermore it is shown that the proposed algebraic phase unwrapping algorithm can be used to determine the exact zero distribution of any univariate complex polynomial without suffering the so called singular case problem

on the stabilizing pid controllers for integral processes in this note we show how to determine the set of stabilizing parameters of a proportional integral derivative controller for an integrator plus dead time process in particular by exploiting a version of the hermite biehler theorem applicable to quasipolynomals the admissible range of the proportional gain is computed first then for each value of the proportional gain in that range the set of stabilizing values of the integral and derivative gain are found it is shown that the procedure is greatly simplified with respect to the case of a second order integral process with dead time

polynomials with rational generating functions and real zeros abstract this paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial type denominator we show that every member of a two parameter family consisting of such generating functions gives rise to a sequence of polynomials p m z m 0 that is eventually hyperbolic moreover the real zeros of the polynomials p m z form a dense subset of an interval i r whose length depends on the particular values of the parameters in the generating function

computation of all stabilizing pid gains for digital control systems this note determines the set of all proportional integral derivative pid gains that can stabilize a given discrete time plant of arbitrary order utilizing earlier results obtained for the continuous time case and the bilinear transformation it is shown that a complete solution can be obtained using linear programming several numerical examples are presented to illustrate the procedure

brief pi stabilization of first order systems with time delay this paper considers the problem of stabilizing a first order plant with dead time using a pi controller using a version of the hermite biehler theorem applicable to quasipolynomials a complete and constructive characterization of all stabilizing pi gain values is obtained

design of digital pid controllers using the parameter space approach in this paper a parameter space approach is taken for designing digital pid controllers the stability domains of the coefficients of the controllers are computed the existing continuous time results are extended to the case of discrete time systems in this approach the stability region is obtained in the plane of two auxiliary controller coefficients by assuming a fixed value for a third auxiliary controller coefficient the stability region is defined by several line segments or equivalently by several linear equalities and inequalities then through mapping from the auxiliary coefficient space to the original controller coefficient space exact stability domain in the kp ki kd space is obtained the method is also extended for locating the closed loop poles of pid control systems inside the circles with arbitrary radii centred at the origin of the z plane the results can be used in the design of dead beat control systems

stability domains of the delay and pid coefficients for general time delay systems abstracttime delays are encountered in many physical systems and they usually threaten the stability and performance of closed loop systems the problem of determining all stabilising proportional integral derivative pid controllers for systems with perturbed delays is less investigated in the literature in this study the rekasius substitution is employed to transform the system parameters to a new space then the singular frequency sf method is revised for the rekasius transformed system a novel technique is presented to compute the ranges of time delay for which stable pid controller exists this stability range cannot be readily computed from the previous methods finally it is shown that similar to the original sf method finite numbers of singular frequencies are sufficient to compute the stable regions in the space of time delay and controller coefficients

colored stability crossing sets for siso delay systems this paper considers the stability of a single input single output siso system with an input output i o delay called siso delay system hereafter we first derive an algebraic criterion to solve the complete stability w r t the delay tau next the stability analysis is extended to the case with additional free plant and or controller parameters denoted by a vector x by adopting a mathematical tool for polynomial algebra the discrimination system the complete classification of the critical imaginary roots cirs in the whole x tau parameter space may be obtained furthermore with the above results we present the stability crossing sets scss in a new form called the colored scss for each point on the scs the asymptotic behavior information of the corresponding cir can be coded by a certain color red green or yellow hence the stability in the x tau space can be analyzed easily and intuitively moreover the geometry of the scss can be investigated from a new perspective which will help not only to better understand the stability of the siso delay system but also to considerably simplify the procedure for generating the colored scss

stable polyhedra in parameter space a typical uncertainty structure of a characteristic polynomial is p s a s q s b s with a s and b s fixed and q s uncertain in robust controller design q s may be a controller numerator or denominator polynomial an example is the pid controller with q s k i k ps k ds 2 in robustness analysis q s may describe a plant uncertainty for fixed imaginary part of q email protected it is shown that hurwitz stability boundaries in the parameter space of the even part of q email protected are hyperplanes and the stability regions are convex polyhedra a dual result holds for fixed real part of q email protected also s stability with the real parts of all roots of p s smaller than s is treated under the above conditions the roots of p s can cross the imaginary axis only at a finite number of discrete singular frequencies each singular frequency generates a hyperplane as stability boundary an application is robust controller design by simultaneous stabilization of several representatives of a s and b s by a pid controller geometrically the intersection of convex polygons must be calculated and represented tomographically for a grid on k p

static output feedback stabilization necessary conditions for multiple delay controllers this note focuses on the static output feedback stabilization problem for a class of single input single output systems when the control law includes multiple distinct delays we are interested in giving necessary conditions for the existence of such stabilizing controllers illustrative examples second order system chain of integrators or chain of oscillators are presented and discussed

robustness of optimal designs for correlated random variables abstract suppose that y y i is a normal random vector with mean xb and covariance 2 i n where b is a p dimensional vector b j x x ij is an n p matrix with x ij 1 1 this corresponds to a factorial design with 1 1 representing low or high level respectively or corresponds to a weighing design with 1 1 representing an object j with weight b j placed on the left and right of a chemical balance respectively e optimal designs z are chosen that are robust in the sense that they remain e optimal when the covariance of y i y i is 0 for i i within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the a and d criteria

brief fast calculation of stabilizing pid controllers a fast method for the computation of stabilizing pid controllers is proposed it is shown in particular that for a fixed value of the proportional term k p the resulting stabilizing pid compensators form a finite set of disjoint polyhedral sets in the parameter space a complete characterization and a practical way of finding such stabilizing regions are given moreover the bounds on stabilizing k p parameters are derived the method therefore has several advantages over recent results and is very promising for real time and or robust control applications

all fixed order h sub spl infin controllers observer based structure and covariance bounds this note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant which yields for the closed loop transfer matrix a specified h sub spl infin norm bound the algebraic results of covariance control are applied to the h sub spl infin control problem to yield a parameterization in terms of the lyapunov matrix which carries many system properties such as h sub 2 performance covariance bounds system entropy at infinity etc all low order h sub spl infin controllers are shown to have observer based structure for reduced order models of the plant and are characterized by two riccati equations with a coupling condition

technical communique synthesis of pid controllers for a class of time delay systems in this paper we use the hermite biehler theorem to establish results on the design of proportional plus integral plus derivative pid controllers for a class of time delay systems using the property of interlacing at high frequencies of the class of systems considered and linear programming we obtain the set of all stabilizing pid controllers as far as we know previous results on the synthesis of pid controllers rely on the solution of transcendental equations this paper also extends previous results on the synthesis of proportional controllers for a class of delay systems of retarded type to a larger class of delay systems

fixed structure pid controller design for standard h control problem a design method of multivariable pid controllers that satisfy the performance index of the standard h control problem and linear constraints on the control structure is proposed nonconvex convex matrix inequalities obtained from the h norm condition for each frequency are approximated by linear matrix inequalities respectively a sequence of pid gains is obtained by solving a feasibility problem of the linear matrix inequalities starting from a stabilizing pid gain iteratively the pid gain converges to a local optimal solution and the sequence of the h norm bound is monotonically nonincreasing

controller design for delay systems via eigenvalue assignment on a new result in the distribution of quasi polynomial roots this paper considers the eigenvalue distribution of a linear time invariant lti system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment first a new result on the root distribution for a class of quasi polynomials is developed based on the extension of the hermite biehler theorem then such result is applied to proportional integral pi controller parameter design for a first order plant with time delay through pole placement the complete region of pi gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed loop system with delay plant are assigned to desired positions in the complex plane furthermore on the basis of the previous result this paper also extended the pi control to the proportional integral derivative pid control it is worth pointing out that this work aims to improve the performance of the closed loop system on the premise of guaranteeing the stability

generalizations of the hermite biehler theorem abstract the hermite biehler theorem gives necessary and sufficient conditions for the hurwitz stability of a polynomial in terms of certain interlacing conditions in this paper we generalize the hermite biehler theorem to situations where the test polynomial is not necessarily hurwitz the generalization is given in terms of an analytical expression for the difference between the numbers of roots of the polynomial in the open left half and open right half planes the result can be used to solve important stabilization problems in control theory and is therefore of both academic as well as practical interest

generalizations of the hermite biehler theorem the complex case abstract the hermite biehler theorem gives necessary and sufficient conditions for the hurwitz stability of a polynomial in terms of certain interlacing conditions in this paper we extend our earlier generalization of the hermite biehler theorem for real not necessarily hurwitz polynomials to the domain of polynomials with complex coefficients this result which is of interest in its own right can also be used to analytically solve an important stabilization problem in control theory

reliable computation of pid gain space for general second order time delay systems abstractthis paper addresses the problem of determining the stability gain space of a pid controller for general second order time delay systems first a review of existing results and the associated drawbacks is presented subsequently a new algorithm to compute the entire pid stability gain space is developed the new algorithm is based upon existing results on the relationship between the stability of a quasi polynomial and its derivatives an extended version of the hermit biehler theorem and also the nyquist criterion the algorithm entails extraction of an admissible range for the pid parameter kp and then based on this range a stability region in the ki kd plane is computed well known examples are studied to demonstrate the reliability and accuracy of the results

pid controller design for robust performance this note is devoted to the problem of synthesizing proportional integral derivative pid controllers for robust performance for a given single input single output plant in the presence of uncertainty first the problem of robust performance design is converted into simultaneous stabilization of a complex polynomial family an extension of the results on pid stabilization is then used to devise a linear programming design procedure for determining all admissible pid gain settings the most important feature of the proposed approach is that it computationally characterizes the entire set of the admissible pid gain values for an arbitrary plant

new results on eigenvalue distribution and controller design for time delay systems this paper considers the eigenvalue distribution of a linear time invariant lti system with commensurate time delays and its application to proportional integral derivative pid controller design for a delay plant via dominant eigenvalue assignment a new result on the root distribution of a quasi polynomial is first produced by applying part of pontryagin s conclusions this result which gives a necessary and sufficient condition can be directly used to judge the number of the right half plane eigenvalues of the characteristic equation of a time delay system based on the proposed result necessary and sufficient conditions on dominant eigenvalue assignment for pid control of time delay systems are presented and an algorithm is then provided to determine the pid controller gains the proposed approaches can assign some one or two eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones which enables us to design the controller according to the desired performance indexes for a standard first order system or a standard second order system in addition to stability the method is effective for the closed loop characteristic equation being retarded type or neutral type the controller gains to achieve the control objective can be characterized by a straightforward computation further a result on degradation to proportional integral pi control of time delay systems is given

controller synthesis free of analytical models three term controllers the main focus of this paper is on direct data driven synthesis and design of controllers we show that the complete set of stabilizing proportional integral derivative pid and first order controllers for a finite dimensional linear time invariant plant possibly cascaded with a delay can be calculated directly from the frequency response nyquist bode data p jomega for omega epsi 0 infin without the need of producing an identified analytical model it is also shown that complete sets achieving guaranteed levels of performance measures such as gain margin phase margin and hinfin norms can likewise be calculated directly from only nyquist bode data the solutions have important new features for example it is not necessary to know the order of the plant or even the number of left half plane or right half plane poles or zeros the solution also identifies in the case of pid controllers an exact low frequency band over which the plant data must be known with accuracy and beyond which the plant information may be rough or approximate these constitute important new guidelines for identification when the latter is to be used for control design the model free approach to control synthesis and design developed here is an attractive complement to modern and post modern model based design methods which require complete information on the plant and generally produce a single optimal controller a discussion is included with illustrative example of the sharp differences between model free and model based approaches when computing sets of stabilizing controllers for example it is shown that the identified model of a high order system can be non pid stabilizable whereas the original data indicates it is pid stabilizable the results given here are also a significant improvement over classical control loop shaping approaches since we obtain complete sets of controllers achieving the design specifications it can enhance fuzzy and neural approaches which are model free but cannot guarantee stability and performance finally these results open the door to adaptive model free fixed order designs of real world systems

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

technical communique h design with fractional order pdµ controllers this article focuses on h performance design for fractional order systems with time delay using fractional order proportional derivative pd m controllers first the stabilizing parameters region in proportional derivative plane of pd m controller for a fixed derivative order is determined in terms of a graphical stability criterion applicable to fractional order time delay systems then in the stabilizing region the pairs of proportional and derivative gains of pd m controller are calculated for a range of frequencies which satisfy the h norm constraint of complementary sensitivity function and define the h boundary curve finally by changing the derivative order of pd m controller we observe the relationship between the h curve and the derivative order examples are followed to illustrate the design procedure

a new approach to digital pid controller design in this note we present a new approach to the problem of designing a digital proportional integral derivative pid controller for a given but arbitrary linear time invariant plant by using the tchebyshev representation of a discrete time transfer function and some new results on root counting with respect to the unit circle we show how the digital pid stabilizing gains can be determined by solving sets of linear inequalities in two unknowns for a fixed value of the third parameter by sweeping or gridding over this parameter the entire set of stabilizing gains can be recovered the precise admissible range of this parameter can be predetermined this solution is attractive because it answers the question of whether there exists a stabilizing solution or not and in case stabilization is possible the entire set of gains is determined constructively using this characterization of the stabilizing set we present solutions to two design problems 1 maximally deadbeat design where we determine for the given plant the smallest circle within the unit circle wherein the closed loop system characteristic roots may be placed by pid control and 2 maximal delay tolerance where we determine for the given plant the maximal loop delay that can be tolerated under pid control in each case the set of controllers attaining the specifications is calculated illustrative examples are included

uniquely e optimal designs with n 2 mod4 correlated observations abstract in this paper we consider uniquely e optimal and highly e efficient designs described by a linear model with design matrices with elements in 1 0 1 the errors are assumed to be equally positively correlated and to have equal variances these designs correspond to chemical balance weighing designs or to three level factorial designs designs that satisfy certain conditions are proved to be uniquely e optimal designs when the number of observations n 2 mod 4 constructions of such designs are presented given the existence of an s n matrix and a hadamard matrix it is also proved that the constructed uniquely e optimal designs are not in general a or d optimal finally the high e efficiency of the designs of masaro and wong 2008 11 12 and certain other designs is shown these designs can be a good substitute for unknown e optimal designs in some cases

hyperbolic polynomials and linear type generating functions abstract in this paper we consider sequences of polynomials h m z m 0 generated by a relation m 0 h m z t m 1 p t z t r q t where p and q are real polynomials and r n r 2 in the main result of the paper cf theorem 1 we give a necessary conditions on p and q and their zeros to ensure that for all sufficiently large m the zeros of the polynomials h m z are real we also show that the set of all zeros of the h m z s for m 1 is dense in a real ray

h pid controller design for lur e systems and its application to a ball and wheel apparatus in this paper we consider the problem of synthesizing proportional integral derivative pid controllers for a given lur e system in the presence of exogenous energy bounded disturbance based on the circle criterion sufficient conditions for guaranteeing closed loop absolute stability and achieving disturbance attenuation are given in terms of the multi objective h specifications the results from the earlier work are then used to solve the resulting multi objective h pid design problem it is shown that for a fixed proportional gain and by sweeping over a variable the set of admissible integral and derivative gain values can be determined constructively using linear programming techniques the proposed synthesis method is used to design a pid controller for the ball and wheel system and the experimental results are presented to show effectiveness of the proposed method

dynamic similarity approach to control system design delayed pid control loop abstractthe paper deals with a novel approach to dimensional analysis based design of control systems the case of delayed pid control loop is used to show the benefits of the dynamic similarity concept in providing the control with response resulting from a constrained integral absolute error iae optimisation the particular aim is to achieve this response in disturbance rejection ability of the control loop to reach a generalised description in evaluating the control loop dynamics a dimensionless model is applied resulting from the dimensional analysis of the control loop parameters by means of originally introduced similarity numbers of laggardness and swingability explicit relationships between the dominant pole coordinates and the pid controller gains are obtained the proposed optimum setting of control loop constrained in damping is referred to the ultimate oscillations of the plant

robustness of a optimal designs abstract suppose that y y i is a normal random vector with mean xb and covariance 2 i n where b is a p dimensional vector b j x x ij is an n p matrix a optimal designs x are chosen from the traditional set d of a optimal designs for 0 such that x is still a optimal in d when the components y i are dependent i e for i i the covariance of y i y i is with 0 such designs depend on the sign of the general results are applied to x x ij where x ij 1 1 this corresponds to a factorial design with 1 1 representing low level or high level respectively or corresponds to a weighing design with 1 1 representing an object j with weight b j being weighed on the left and right of a chemical balance respectively

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

root counting phase unwrapping stability and stabilization of discrete time systems in this paper we develop a procedure for determining the phase unwrapping of a real polynomial or rational function along the unit circle by representing the unit circle image in terms of tchebyshev polynomials a formula for the unwrapped phase is determined in terms of the zeros and signs of these polynomials the root distribution with respect to the unit circle can thus be determined in terms of the tchebyshev representation this result is applied to the problem of feedback stabilization of a digital control system by constant gain or by a two parameter controller the solution results in a determination of the entire set of stabilizing gains as a solution of sets of linear inequalities this is in sharp contrast to the solution via classical conditions which result in nonlinear inequalities the result also gives a new characterization of schur stability in terms of the tchebyshev representation which may be of independent interest

brief synthesis of h pid controllers a parametric approach this paper considers the problem of synthesizing proportional integral derivative pid controllers for which the closed loop system is internally stable and the h norm of a related transfer function is less than a prescribed level for a given single input single output plant it is shown that the problem to be solved can be translated into simultaneous stabilization of the closed loop characteristic polynomial and a family of complex polynomials it calls for a generalization of the hermite biehler theorem applicable to complex polynomials it is shown that the earlier pid stabilization results are a special case of the results developed here then a linear programming characterization of all admissible h pid controllers for a given plant is obtained this characterization besides being computationally efficient reveals important structural properties of h pid controllers for example it is shown that for a fixed proportional gain the set of admissible integral and derivative gains lie in a union of convex sets

brief analytical pid parameter expressions for higher order systems most approaches to feedback synthesis are numerical design techniques that produce their results in form of numerical values for the controller parameters in contrast analytical design techniques produce the controller parameters as explicit expressions that are a function of open loop system parameters and the desired closed loop advantages include fast commissioning in terms of physical system parameters gain scheduling without extra design effort and analytical sensitivity insights so far it has not been known how to derive analytical pid expressions for higher than second order systems the paper proposes a technique to address this problem analyses its properties demonstrates computational simplicity and shows that its performance is comparable to known numerical techniques