Hermite Biehler Research Articles

A New Tuning Method for Stabilization Time Delay Systems Using PIλDμ Controllers

AbstractThis paper presents a new design procedure to tune the fractional order PIλDμ controller that stabilizes a first order plant with time delay. The procedure is based on a suitable version of the Hermite–Biehler Theorem and the Pontryagin Theorem. A Theorem and a Lemma are developed to compute the global stability region of the PIλDμ controller in the (kp,ki,kd) space. Hence, this Theorem and Lemma allow us to develop an algorithm for solving the PIλDμ stabilization problem of the closed loop plant. The proposed approach has been verified by numerical simulation that confirms the effectiveness of the procedure.

A Set of Stabilizing PD Controllers: An Application to Robot Manipulator

This paper considers the problem of designing a class of stabilizing proportional derivative (PD) controllers for multi-input multi-output (MIMO) nonlinear systems. For a MIMO nonlinear system a set of stabilizing PD controllers were designed for each subsystem. The design approach is based on PD stabilization theorem derived from generalized result of the classical Hermite Biehler theorem. By considering the nonlinear terms as interconnections, a linear matrix inequalities optimization problem is formulated to ensure the stability of the composite nonlinear system with the designed decentralized controller parameters. A genetic algorithm based search technique is adopted to select an optimal PD controller gain from a search space of stabilizing controllers in order to have an optimum value of performance index. A two-link robot manipulator system is considered to show the effectiveness of the design procedure.

All stabilizing sets for proportional–integral controller of high-order delay processes

This article considers the problem of determining the complete stabilizing set of proportional–integral controllers, which is applied in a class of high-order processes with time delay in complex-frequency domain. First, we give a necessary condition for a proportional–integral controller to stabilize the process with certain constant delay using Descartes’ rule of signs. Then by employing the generalization of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete set for all stabilizing proportional–integral parameters is derived to stabilize the open-loop stable and unstable systems. In the case of uncertain time delay, we provide a design approach to stabilize the related plant with a robust proportional–integral controller, where the unknown but constant time delay lies inside a known interval. Three examples illustrate the effectiveness of the proposed results.

Derivation of the Real-Rootedness of Coordinator Polynomials from the Hermite–Biehler Theorem

By using the Hermite–Biehler theorem, we give a new proof of the real-rootedness of the coordinator polynomials of type D, which was recently established by Wang and Zhao. As a consequence, we also obtain the compatibility between the coordinator polynomials of type D and those of type C.

Proportional-derivative controllers for stabilisation of first-order processes with time delay

This paper considers the problems of determining the complete stabilising set of proportional-derivative controllers for a first-order process with time delay. First, by employing a version of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete set of all stabilising proportional-derivative parameters for first-order processes with constant time delay are obtained. Next, we provide an approach to design a robust PD controller to stabilise a first-order process with uncertain time delay, which lies inside a known interval.

Modelling and control of flexible joint robot based on Takagi–Sugeno fuzzy approach and its stability analysis via sum of squares

This article proposes a Takagi–Sugeno (T-S) fuzzy model of single-link rotary flexible joint robot. The proposed control method is based on parallel distributed control. The parameters of T-S controller are improved by distributed population genetic algorithm (GA) with chaos GA. Using Hermite–Biehler theorem in distributed population, GA is made to have a fast convergence. Dividing search space into several sub-spaces causes a better response, and chaos disturbance helps the whole algorithm to reach a best answer. The stability of the controller is analysed via the sum of squares programming, and finally, it is implemented on the plant.

Synthesis of a fractional PI controller for a first-order time delay system

This paper considers the problem of stabilizing a first-order plants with known time delay using a fractional-order proportional–integral controller [Formula: see text]. Using a generalization of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete analytical characterization of all stabilizing gain values ([Formula: see text]) is provided. The widespread industrial use of fractional PI controllers justifies a timely interest in [Formula: see text] tuning techniques.

A numerical approach for computing stability region of FO-PID controller

The paper proposes a new procedure which allows to define the parameters of a Fractional Order Proportional Integrative Derivative (FO-PID) controller that stabilizes a first-order plant with time-delay. The stabilizing FO-PID parameters complete set has been determined by the application of the Hermite–Biehler theorem to fractional order systems. The proposed procedure has been verified by computer simulations that confirm the effectiveness of the approach. The diffused PID controller industrial use and the verified capability of their non-integer order form justifies the timely FO-PID controller interest.

A PID controller set of guaranteeing stability and gain and phase margins for time-delay systems

The stabilizing parameter sets and the guaranteed gain-margin (GM) and phase-margin (PM) regions of proportional-integral-derivative (PID) controllers for a class of processes with time-delay are discussed in the paper. The admissible range of stabilizing proportional-gain is first derived by a version of Hermite–Biehler Theorem and the evaluation of some properties of the functions involved in the closed-loop characteristic equation. Then, the stabilizing region in integral-derivative plane, for a fixed proportional-gain, is drawn and identified directly in terms of a graphical stability criterion applicable to time-delay systems. Further, in the stabilizing region, the gain-margin and phase-margin specifications are considered using the same strategy as drawing the stability boundary lines, based on the technique of gain-phase margin tester (GPMT). Illustrating examples are followed in each design step to show the effectiveness of the method.

ROBUST CONTROL FOR UNCERTAIN DELAY SYSTEM

In this paper, the Generalized Kharitonov Theorem for quasi-polynomials is exploited for the purpose of synthesizing a robust controller. The aim here is to develop a controller to simultaneously stabilize a given interval plant family with unknowing and bounded time delay. Using a constructive procedure based on Hermite–Biehler theorem, we obtain all PI gains that stabilize an uncertain first-order delay system. An application example is presented for the temperature control of a heated air stream, process trainer PT326.

Stability of PID-controlled second-order time-delay feedback systems

In the previous papers, the stability of PID-controlled first-order time-delay systems has been investigated by means of several methods, of which the Nyquist criterion, a generalization of the Hermite–Biehler Theorem, and the root location method are well known. Explicit expressions of the boundaries of the stability region, which is the set of controller parameters that give stable closed-loop systems, have been determined. From these studies, one can verify that not all plants can be made stable and then obtain the set of process parameters that allow stable closed-loop systems. With this set, one can implement the stability region of the process parameters. In a recent paper the stability conditions based on Pontryagin’s studies and valid for arbitrary-order plants have been presented. The procedure deduced for the controller parameters is exhaustive, but that deduced for the process parameters requires further mathematical evaluations, whose complexity is proportional to the number of process time constants. In the aforementioned recent paper these evaluations have been performed for a second-order time-delay plant whose transfer function has no zero. The aim of this paper is to execute these calculations for a second-order plant whose transfer function has one zero and to provide the related stability region.

Synthesis of PID controllers for integral processes with time delay

AbstractThis article deals with the problem of determination of the stabilizing parameter sets of Proportional‐Integral‐Derivative (PID) controllers for first‐order and second‐order integral processes with time‐delay. First, the admissible stabilizing range of proportional‐gain is determined analytically in terms of a version of the Hermite–Biehler Theorem applicable to quasi‐polynomials. Then, based on a graphical stability condition developed in parameter space, the complete stabilizing regions in an integral‐derivative plane are drawn and identified graphically, not calculated mathematically, by sweeping over the admissible range of proportional‐gain. An actual algorithm for finding the stabilizing parameter sets of PID controllers is also proposed. Simulations show that the stabilizing regions in integral‐derivative space are either triangles or quadrilaterals. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

New results on the synthesis of FO-PID controllers

In this paper a new procedure that allows to define the parameters of fractional order PIλDμ controller, designed to stabilize a first-order plant with delay-time, is proposed. Using a version of the Hermite–Biehler theorem applicable to quasipolynomials, the complete set of stabilizing PIλDμ parameters is determined. The widespread industrial use of PID controllers and the potentiality of their non-integer order representation justifies a timely interest to PIλDμ tuning techniques.

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.

Sets of stabilising PID controllers for second-order integrating processes with time delay

The procedures for determining the complete sets of stabilising proportional integral derivative (PID) controllers are presented for three classes of second-order integrating processes with time delay. The necessary condition for the PID controllers to stabilise second-order integrating processes with time delay is first derived in the case where frequency, ω, equals zero or infinity. On the basis of the extended Hermite–Biehler theorem, analytical formulas are provided to determine the admissible ranges of the proportional gain (kp), integral gain (ki) and the derivative gain (kd), respectively, and the complete set of stabilising PID controllers is found by sweeping over the entire range of allowable kp values. The stability analysis on the basis of the extended Hermite–Biehler theorem shows that the stability boundary in the controller parameter space depends only on the frequencies in the defined small range. In view of this, the sets of stabilising PID controllers are also determined by sweeping over the admissible ranges of kd and ki, respectively. As a result, for a given second-order integrating process with time delay, the complete set of stabilising PID controllers can be determined by sweeping any one of the PID parameters over the corresponding admissible range.

A new method for the computation of all stabilizing controllers of a given order

A new method is given for computing the set of all stabilizing controllers of a given order for linear, time invariant, scalar plants. The method is based on a generalized Hermite–Biehler theorem and the successive application of a modified constant gain stabilizing algorithm to subsidiary plants. It is applicable to both continuous and discrete time systems.

Stabilizing PID Controllers for a Single-Link Biomechanical Model with Position, Velocity, and Force Feedback

In this paper we address the problem of PID stabilization of a single-link inverted pendulum-based biomechanical model with force feedback, two levels of position and velocity feedback, and with delays in all the feedback loops. The novelty of the proposed model lies in its physiological relevance, whereby both small and medium latency sensory feedbacks from muscle spindle (MS), and force feedback from Golgi tendon organ (GTO) are included in the formulation. The biomechanical model also includes active and passive viscoelastic feedback from Hill-type muscle model and a second-order low-pass function for muscle activation. The central nervous system (CNS) regulation of postural movement is represented by a proportional-integral-derivative (PID) controller. Padé approximation of delay terms is employed to arrive at an overall rational transfer function of the biomechanical model. The Hermite-Biehler theorem is then used to derive stability results, leading to the existence of stabilizing PID controllers. An algorithm for selection of stabilizing feedback gains is developed using the linear matrix inequality (LMI) approach.

Robust phase margin, robust gain margin and Nyquist envelope of an interval plant family

In this paper, robust gain margin, robust phase margin and Nyquist envelope of interval systems are studied. It is first presented that the robust gain and phase margins of a control system with an interval plant family are achieved at one of the Kharitonov plant by using the interlacing theorem and virtual compensator concept. Then, using the generalized Hermite–Biehler theorem, it is shown that the outer boundary of the Nyquist envelope of a stable interval plant is covered by the Nyquist plots of the 16 Kharitonov plants family. Examples are given to illustrate the method presented.

Hermite–Biehler, Routh–Hurwitz, and total positivity

Simple proofs of the Hermite–Biehler and Routh–Hurwitz theorems are presented. The total nonnegativity of the Hurwitz matrix of a stable real polynomial follows as an immediate corollary.

Computation of stabilizing Lag/Lead controller parameters

One of the central problems in control theory relates to the design of controllers for stabilization of systems. The paper deals with the problem of computing all stabilizing values of the parameters of Lag/Lead controllers for linear time-invariant plant stabilization. It is well known that linear controllers of Lag/Lead type are still widely used in many industrial applications. In this paper, an extension of a new approach to feedback stabilization based on the Hermite–Biehler theorem to the Lag/Lead controller structure is given. In addition, the problem of stabilization of uncertain systems defined by an interval plant is studied using the Kharitonov and the Hermite–Biehler theorems. The proposed method is analytical and it can be applied successfully using today’s advanced computer technology. Examples are included to illustrate the method presented.