Plane geometry and convexity of polynomial stability regions

The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible

Адаптивное двухканальное корректирующее устройство для систем автоматического регулирования

Предложено адаптивное псевдолинейное двухканальное корректирующее устройство динамических свойств систем автоматического регулирования. Проведено исследование свойств систем автоматического регулирования с адаптивным псевдолинейным двухканальным корректирующим устройством. Показана эффективность предложенного корректора в системах автоматического регулирования с нестационарными параметрами

On stable cones of polynomials via reduced Routh parameters

summary:A problem of inner convex approximation of a stability domain for continuous-time linear systems is addressed in the paper. A constructive procedure for generating stable cones in the polynomial coefficient space is explained. The main idea is based on a construction of so-called Routh stable line segments (half-lines) starting from a given stable point. These lines (Routh rays) represent edges of the corresponding Routh subcones that form (possibly after truncation) a polyhedral (truncated) Routh cone. An algorithm for approximating a stability domain by the Routh cone is presented

A geometric description of the set of stabilizing PID controllers

This article developed a new method to described the set of stabilizing PID control. The method is based on D-parameterization with natural description of the set. It was found that the stability crossing surface is a ruled surface that is completely determined by a curve known as discriminant. The discriminant is divided into sectors at the cusps. Corresponding to the sectors, the stability crossing surface is divided into positive and negative patches. A systematic study is conducted to identify the regions with a fixed number of right half-plane characteristic roots. The crossing directions of characteristic roots for positive patches and negative patches are also studied. As a result, a systematic method is developed to identify the regions of PID parameter such that the system is stabilized

Continuity argument revisited: geometry of root clustering via symmetric products

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as a symmetric product morphism. We describe the topology of strata up to homotopy equivalence and, for many important cases, up to homeomorphism. Adjacencies between strata are also described. Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.Comment: 45 pages, 4 figure

Stabilizing constant diagonal controllers for TITO systems

Proses kontrolün ilk ve en önemli amacı sistemi kararlı kılmaktır, performans üzerine yapılacak her eylem bu temel koşul sağlandığı müddetçe bir anlam taşır. Kontrolör tasarımında kullanılan yöntemlerden biri sistemi kararlı kılacak tüm kontrolörleri bulmak ve ardından, tasarımdaki diğer beklentileri sağlamak üzere, bu sınıf içinden uygun bir kontrolörde karar kılmaktır. Bu hedef doğrultusunda yapılanlar şu özellikleri de barındırmalıdır ki, iyi bir kontrolör tasarım sürecinden söz edilebilsin. Kontrolör kırılgan olmamalıdır, yani kontrolör de parametre değişimlerine karşı sistemin kararlılığını ve mümkünse performansını zedelememelidir. Kontrolörün düşük mertebeden olması, parametre ayarlama sürecinde büyük kolaylıklar sağlar, zira ne kadar az parametre o kadar basit bir tasarım süreci demektir. Öte yandan pratik anlamı dolayısıyla da az parametre, ya da düşük mertebe endüstride görev alan uygulamacılar için problemlerin daha hızlı çözülmesi anlamına gelir. Bu çalışmada çok değişkenli kontrol yapıları arasında önemli bir yer tutan karakteristik değerler ve karakteristik değer eğrileri ele alınmış, rasyonel polinomlar cisminde indirgenebilir olan karakteristik denklemler için karakteristik değerlerin reel ekseni kesim noktaları ve bu noktalar civarındaki davranışları üzerinde durulmuştur. Zira bu değerler çok girişli çok çıkışlı sistemleri kararlı kılan kontrolörlerin bulunmasında ya da verilen bir kazanç değeri için kararlılığın analiz edilmesinde hızlı hesaplanabilirlikleri yönünden önem arz etmektedirler. Bu çalışma kapsamında iki girişli iki çıkışlı (TITO) sistemleri kararlı kılan sabit köşegen kontrolörler üzerinde durulmuş, karakteristik değer eğrilerinin reel ekseni kestikleri yerler ve eğrilerin bu noktalardaki geçiş yönlerinden hareket ederek hızlı bir algoritma geliştirilmiş ve bu tür sistemleri kararlı kılan kontrolörler örnek sistem üzerinde sınanarak hesaplanmıştır. Anahtar Kelimeler: Bilgisayar cebri, blok diyagram indirgeme, PID kontrol, parametre uzayı yaklaşımı, karakteristik değer eğrileri.A control system is an interconnection of compo-nents to perform certain tasks and to generate desired output signal, when it is driven by the input signal. In contrast to an open-loop system, a closed-loop control system uses sensors to measure the actual output to adjust the input in order to achieve desired output. Most industrial control systems are no longer single-input and single-output (SISO) but multi-input and multi-output (MIMO) systems with a high coupling between the channels. In the design of all dynamical systems stability is the most important property. When a dynamic system is described by its input-output relationship such as a transfer function, the system is stable if it generates bounded outputs for any bounded inputs; bounded-input bounded-output (BIBO) stability. For a linear, time-invariant system modeled by a transfer function matrix, the BIBO stability is guaranteed if and only if all the poles of the transfer function matrix are in the open-left-half complex plane. A reasonable approach to controller design is to find the set of all stabilizing compensators and then using a member of this set to satisfy further design criteria. A complete parameterization of all stabilizing controllers for a given system was suggested by Youla. An important disadvantage of this parameterization is that the order of the controller cannot be fixed. As a result, the order of the controller tends to be quite high most of the time. Therefore, in the last few years computation of all stabilizing controllers of a given order is examined by several researchers. It is a common fact that it is more difficult to design controllers for MIMO systems because there are usually interactions between different control loops. To overcome this difficulty decentralized controllers are considered which have fewer tuning parameters compared to general multivariable controllers. For example, decentralized PID (P: proportional, I: integral, D: derivative)  controllers are widely used in process control due their simplicity and facility in working in case of actuator and/or sensor failure because it is relatively easy to tune manually as only one loop is directly affected by the failure. If a MIMO system described by a  transfer-function matrix G(s) is diagonal dominant over the bandwidth of interest, or there exists an input compensator matrix C(s) to achieve diagonal dominance, then the stability and time domain behavior of the system can be inferred from the diagonal elements of G(s)C(s). The relative gain array, the (inverse) Nyquist array approach, the block Nyquist array method, the Perron-Frobenius scaling procedure and the characteristic locus method are among the analysis and design methods to reduce the interaction in a multivariable system. However, these approaches do not provide the set of all stabilizing controllers. Generalizing the Nyquist stability criterion for MIMO case is particularly important because plotting the characteristic values of the open-loop transfer function enables us to check the stability of the closed-loop system for a gain parameter. A stable characteristic polynomial, becomes unstable if and only if at least one root crosses the imaginary axis. The parameter values of the root crossing form the stability boundaries in the parameter space, which can be classified into three cases: the real root boundary, where a root crosses the imaginary axis at the origin, the infinite root boundary, where a root leaves the left half plane at infinity and the complex root boundary, where a pair of conjugate complex roots crosses the imaginary axes. These stability boundaries separate regions in which the number of closed loop system unstable poles does not change in parameter space.The main objective of this work is to develop an efficient and fast algorithm that can be used in finding all stabilizing constant controllers of diag(k,k)-type for TITO processes. Recall that a TITO system has two characteristic values that are in the field of rational functions, if the characteristic equation is reducible in this field. Hence Nyquist stability criterion can be applied to both of the characteristic values in order to determine the conditions for stability.  Recall that the generalized Nyquist theorem requires that the net sum of encirclements of the point -1 by the characteristic values equal to the number of open-loop unstable poles of the system. Hence, it is of special importance to determine where the characteristic locus intersects with the real axis, i.e. where the imaginary part of, is zero. The direction of these crossings is also important, because the net count of crossings at an intersection point will indicate whether there are closed-loop poles to cross the imaginary axis.  Keywords: Computer Algebra, block diagram reduction, PID control, parameter space approach, characteristic value plots

Advanced modeling of complex robotic systems and mechanisms and applications of modern control theory

Предмет истраживања ове докторске дисертације јесте моделовање и управљање механичких система, са посебним акцентом на роботске системе типа отвореног кинематичког ланца. При формирању математичког модела робота коришћен је тзв. Родригов приступ, који је посебно погодан за примену на рачунару и аутоматско формирање диференцијалних једначина кретања система. За описивање дисипативних сила које се јављају у зглобовима роботских сегмената при кретању система, уводи се модел трења фракционог типа који представља генерализацију стандардног модела...The main research topics of this doctoral dissertation are modeling and control of mechanical systems, with special emphasis on robotic systems with open-chain structure. The so-called Rodriquez’s approach, which is well suited for usage on a computer and for automatic derivation of differential equations of motion, has been used for obtaining mathematical model of robot. In order to define more accurately dissipative forces occurring in robot joints when the system is in motion, a fractional order friction model, which represents a generalization of the standard model, has been introduced..

Advanced modeling of complex robotic systems and mechanisms and applications of modern control theory

Predmet istraživanja ove doktorske disertacije jeste modelovanje i upravljanje mehaničkih sistema, sa posebnim akcentom na robotske sisteme tipa otvorenog kinematičkog lanca. Pri formiranju matematičkog modela robota korišćen je tzv. Rodrigov pristup, koji je posebno pogodan za primenu na računaru i automatsko formiranje diferencijalnih jednačina kretanja sistema. Za opisivanje disipativnih sila koje se javljaju u zglobovima robotskih segmenata pri kretanju sistema, uvodi se model trenja frakcionog tipa koji predstavlja generalizaciju standardnog modela...The main research topics of this doctoral dissertation are modeling and control of mechanical systems, with special emphasis on robotic systems with open-chain structure. The so-called Rodriquez’s approach, which is well suited for usage on a computer and for automatic derivation of differential equations of motion, has been used for obtaining mathematical model of robot. In order to define more accurately dissipative forces occurring in robot joints when the system is in motion, a fractional order friction model, which represents a generalization of the standard model, has been introduced..