P and PI controller design for systems with unstructured uncertainty

Bu çalışmada, yapısal olmayan belirsizliğe sahip tek girişli-tek çıkışlı doğrusal zamanla değişmeyen sistemleri kapalı çevrimde dayanıklı kararlı kılan oransal (P) ve oransal-integral (PI) kontrolörlerin tam bir kümesini bulmak için yeni yöntemler önerilmiştir. Böyle bir sistemi dayanıklı kararlı kılan tüm oransal kontrolörleri hesaplamak için önerilen yöntem, temelde Nyquist teoreminin bir genelleştirilmesine dayanır. Yapısal olmayan belirsizliklere sahip bir sistemin Nyquist eğrisi, tek bir eğri olmayıp reel ekseni bölgeler biçiminde kesen bir eğri ailesi biçimindedir. Bu eğri ailesi, belirsizlik disklerinden oluşan bir bantın içinden geçer. Nominal sistemi kararlı yapan kazançlar ile belirsizlik bantının reel ekseni kesim yerlerinden bulunan belirsizlik kazanç kümelerinden yararlanarak sistemi kapalı çevrimde dayanıklı kararlı kılan kazançların hesabı hızlıca yapılabilir. İki reel polinomun köklerinin hesabını gerektiren bu yöntemin bir parametre üzerinde herhangi bir tarama yapmayı gerektirmemesinden dolayı literatürdeki yöntemler üzerine avantajı da vardır. PI kontrolör parametrelerinin kutupsal koordinatlarda yazılmasıyla bulunan yeni sistem için birim daire taratılarak ve bu yöntemden elde edilen sonuçlardan da yararlanılarak dayanıklı kararlı yapan PI kontrolörlerin tam bölgesi hesaplanabilir. Ayrıca yapısal olmayan belirsizlik içeren sistemi dayanıklı kararlı kılan PI kontrolörleri bulmak için parametre uzayı yaklaşımını kullanan geometrik tabanlı iki yeni yöntem de önerilmiştir. Bu yöntemler, belirsizlik disklerinin orijini içermesi ve içermemesine göre iki aşama içerirler. Birinci yöntem, dayanıklı kararlı yapan PI kontrolörlerin tam bölgesini hesaplayan yavaş bir yöntemken; diğeri ise dayanıklı kararlı yapan PI kontrolörlerin yaklaşık bölgesini veren hızlı bir yöntemdir.  Anahtar Kelimeler: P ve PI kontrol, dayanıklı kontrol, Nyquist teoremi, parametre uzayı yaklaşımı.In controller design, it is essential to achieve stability of the closed-loop system and various performance specifications. Frequency domain criteria such as gain margin, phase margin and   norms of the closed-loop transfer functions as well as time domain criteria such as settling time, rise time and overshoot can be counted among important performance specifications. Most of the controllers used in the practical world are low order controllers such as P, PI and PID controllers. It is possible to see that methods for finding stabilizing low order compensators can be considered in three main categories: methods based on Nyquist theorem, methods based on a generalized version of the Hermite-Biehler theorem, and methods based on parameter space and the concept of singular frequencies.Robust stabilization of continuous time single-input single-output (SISO) linear time invariant (LTI) systems with multiplicative uncertainties is considered in this study. In particular, it has been shown that all P and PI controllers that robustly stabilize a given uncertain SISO LTI system can be found by utilizing a generalization of the Nyquist theorem and the parameter space approach, respectively. The generalization of Nyquist stability criterion suggests to determine the number of the unstable poles for gain intervals obtained by calculating the location and direction of the crossing of the Nyquist plot with the real axis. A stable characteristic polynomial, whose roots are in the left half plane, 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 (substitute   and   in the characteristic polynomial), the infinite root boundary, where a root leaves the left half plane at infinity (for  ) and the complex root boundary, where a pair of conjugate complex roots crosses the imaginary axes (for  ). These stability boundaries seperate regions in which the number of closed loop system unstable poles do not change in the parameter space.Sometimes, it is not possible to represent uncertainties in a system model with parametric uncertainties. Such uncertainties are usually encapsulated in a norm bounded system block that acts on a nominal system in an additive or multiplicative manner. Although it is possible to find robust controllers that can stabilize systems with such uncertainties by the help of  control theory, the resulting compensators are usually of high order (at least as high as the order of the plant) and therefore impractical in many cases. Several attempts exist to put constraints on the order of   controllers in the literature. However, many of these approaches suffer from computational intractability. In many practical cases direct determination of the set of P and PI controllers that provide robust stability of SISO LTI systems with unstructured uncertainty is required. To the best knowledge of authors, there is no such direct methods available in the literature for this purpose. The main aim of this paper is to provide such methods. Nyquist plot of a system with multiplicative uncertainties is a family of curves rather than a single curve and crosses the real axis in segments of the real axis instead of at single points. Actually, the frequency response of a system with unstructured uncertainty at a given frequency is a disk. A new method is proposed to determine all stabilizing P and PI controllers for a given system with multiplicative uncertainty. The method is applicable to systems with unstable or nonminimum phase transfer functions and/or weight functions. Proposed method involves calculation of roots of two real polynomials and does not require any search or gridding over a parameter (for P control), and as a result offers computational advantages over existing methods in literature. Although it is assumed that the nominal system does not have any poles on the imaginary axis in derivations of formulations, it is actually possible to extend the results to cover such cases rather easily. In this study, two new geometric methods are also proposed to find the set of PI controllers that robustly stabilize a given system with unstructured uncertainty. The first method gives exact set of robustly stabilizing PI controllers; but it is slow. An alternative method suggests approximation set of robustly stabilizing PI controllers; but it is faster than the first one. Keywords: P and PI control, robust control, Nyquist theorem, parameter space approach. 

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Abstract Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Yapısal Olmayan Belirsizlik İçeren Sistemlerde Dayanıklı Kararlılığı Sağlayan P ve PI Tipi Kontrolörlerin Tasarlanması İçin Bir Yöntem

Bu çalışmada, yapısal olmayan belirsizliğe sahip tek girişli-tek çıkışlı doğrusal zamanla değişmeyen sistemleri kapalı çevrimde dayanıklı kararlı kılan oransal (P) ve oransal-integral (PI) kontrolörlerin tam bir kümesini bulmak için yeni bir yöntem önerilmiştir. Böyle bir sistemi dayanıklı kararlı kılan tüm oransal kontrolörleri hesaplamak için önerilen yöntem, temelde Nyquist teoreminin bir genelleştirilmesine dayanır. Yapısal olmayan belirsizliklere sahip bir sistemin Nyquist eğrisi, tek bir eğri olmayıp reel ekseni bölgeler biçiminde kesen bir eğri ailesi biçimindedir. Bu eğri ailesi, belirsizlik disklerinden oluşan bir bantın içinden geçer. Nominal sistemi kararlı yapan kazançlar ile belirsizlik bantının reel ekseni kesim yerlerinden bulunan belirsizlik kazanç kümelerinden yararlanarak sistemi kapalı çevrimde dayanıklı kararlı kılan kazançların hesabı hızlıca yapılabilir. Sadece iki reel polinomun köklerinin hesabını gerektiren bu yöntemin bir parametre üzerinde herhangi bir tarama yapmayı gerektirmemesinden dolayı literatürdeki yöntemler üzerine önemli bir avantajı vardır. PI kontrolör parametrelerinin kutupsal koordinatlarda yazılmasıyla bulunan yeni sistem için birim daire taratılarak ve bu yöntemden elde edilen sonuçlardan da yararlanılarak dayanıklı kararlı yapan PI kontrolörlerin tam bölgesi hesaplanabilir

Zaman Gecikmeli Sistemler İçin Oransal Kontrolörler Kullanılarak Erişilebilecek Maksimum Faz Payını ve Kazanç Payını Hesaplayan Yeni Bir Yöntem

Bu çalışmada, zaman gecikmeli tek girişli-tek çıkışlı sistemlerde oransal kontrolörler için maksimum faz payının ve kazanç payının hesabını veren yeni bir yöntem geliştirilmiştir. Burada önerilen yöntem, temelde Nyquist teoreminin bir genelleştirilmesine dayanır. Kazanç ve faz payı kısıtlamalarının istenen her değeri için kapalı çevrimde sistemi kararlı kılmak her durumda mümkün olamayacağından bunların sınırlarını bulmak amacıyla erişilebilir maksimum kazanç payının ve maksimum faz payının hesabı önem taşımaktadır. Zaman gecikmeli sistemlerin maksimum faz payı hesabı için burada önerilen yöntem, üstel biçimde olan zaman gecikmesi terimini içermeyip tam çözümü verir. Ayrıca bu çalışmada zaman gecikmeli sistemler için maksimum faz payını sağlayan en büyük kazancın hesabı da yapılmıştır