Karakteristik oran atama yöntemi ile kaskad kontrolör tasarımı
Cascade controller design using characteristic ratio assignment method
Yazar: AHMET CAN ERDEM
Danışman: PROF. DR. MÜJDE GÜZELKAYA
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Kontrol ve Otomasyon Mühendisliği Ana Bilim Dalı / Kontrol ve Otomasyon Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2021
Konu: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol = Computer Engineering and Computer Science and Control

Özet:
Teknolojide yaşanan gelişmelerle birlikte tasarlanan kontrol sistemleri için otomasyon kaçınılmaz bir ihtiyaç olmuştur. Otomasyonun en çok tercih edildiği kontrol dallarından biri de süreç kontrolüdür. Süreç kontrolündeki ana amaç, süreç hakkında bilgi veren değişkenlerin gözlenerek bu değişkenlere etki etmekte olan ayar değişkenlerinin en etkin bir biçimde kontrolünü sağlamaktır.  Modern mühendislikle süreç kontrolü uygulamalarına örnek olarak kimyasal süreçler (sıcaklık kontrolü, derişim kontrolü vb.), otomotiv (motor kontrolü, şanzıman kontrolü vb.), uçuş kontrolü (insansız hava ve kara araçları vb.), robot ve haberleşme (kablosuz haberleşme için güç kontrolü) verilebilir. 

Belirli bir amacı gerçekleştirmeye çalışan bileşenlerin meydana getirmiş olduğu kontrol sistemleri literatürde iki guruba ayrılmaktadır. Bunlar sırasıyla açık çevrim ve kapalı çevrim sistemlerdir. Açık çevrim sistemlerde çıkış işaretinden geri besleme alınmazken, kapalı çevrim sistemlerde giriş ve çıkış arasındaki hatayı azaltmak adına çıkış işaretinden geri besleme alınmaktadır. Sistemlerin bozucular gibi çevresel değişikliklere karşı dayanıklılığını arttırmak adına kontrolör tasarımında geri besleme önemli yer arz etmektedir. Birçok sistemde tek çevrimli kapalı çevrim istenilen başarımı elde edilmesi mümkün olsa da hareket kontrolü gibi sistemlerde istenilen başarım kriterinin elde edilmesi mümkün olmayabilir. İstenilen başarım kriterleri sağlanamadığı durumlarda sistemin başarımını iyileştirmek adına kaskad kontrolör yapısı kullanılır.

Kaskad kontrol yapıları seri kaskad ve paralel kaskad olmak üzere ikiye ayrılabilir. Seri kaskad kontrol yapısı iç ve dış olarak adlandırılan iç içe geçmiş kontrol çevrimlerinden oluşmaktadır. Paralel kaskad yapısında seri kaskad kontrol yapısından farklı olarak seri kaskad kontrolde iç çevrimdeki kontrolör sadece iç çevrimdeki sistemi kontrol ederken, paralel kaskad yapısında iç çevrimdeki kontrolör hem iç hem dış çevrimdeki sistemleri beraber kontrol etmektedir.

Seri kaskad kontrol yapısında kontrol edilen sistemler için kullanılan sistem modelleri yüksek mertebeden olmayan sistemler ya da yüksek mertebeden sistemler olabilir. Yüksek mertebeden olmayan birinci ya ikinci mertebeden kaskad sistemler için geleneksel kontrolör tasarım yöntemleri iyi sonuçlar vermesine rağmen, sistemin mertebesinin artmasıyla kaskad sistem daha karmaşık hale gelmekte ve istenilen sonuçların elde edilmesi zorlaşmaktadır. Yüksek mertebeden kaskad sistemler için bu sistemleri yaklaşık olarak ifade eden birinci mertebeden ölü zamanlı modeller kullanılarak kontrolör tasarımı yapılabilir veya bu sistemler için yüksek mertebeli kontrolörler tasarlanabilir. 

Bu çalışmada öncelikle yüksek mertebeden sistemler için "Karakteristik Oran Atama" yöntemiyle nasıl kontrolör tasarlanacağı anlatılmış ve farklı sistemler ele alınarak kontrolör tasarımları yapılmıştır. Bu yöntem daha sonra, iç çevrimde yükselme zamanını azaltmak ve dış çevrimde ise aşımsız bir sistem cevabı etmek üzere kaskad kontrolör tasarımında kullanılmıştır. 

Karakteristik Oran Atama ile yapılan kaskad kontrolör tasarımı, literatürde yer alan "Model Ayrıştırma Temelli Kaskad Kontrolör" yöntemi ile karşılaştırılmıştır. Tasarlanan kontrolörler için karşılaştırma kriterleri olarak bozucu bastırma süreleri ve referans takip başarımı seçilmiştir. Referans takip başarımı ITSE (Integral Time Square Error) kriterine göre belirlenmiştir.

Abstract:
The recent technological developments resulted in a necessity of automation for control systems. One of the most preferred branches of control in terms of automation is process control. The main purpose of process control is to observe the variables that provide information about the process and to control the manipulated variables that affect these variables most effectively. Chemical processes (such as temperature control and concentration control), automotive (such as motor control and transmission control), flight control (such as unmanned aerial and land vehicles), robotics (voice recognition, digital dictation), and communication (such as power control for wireless communication) can be given as examples for process control applications in modern engineering.

In the literature, the systems that consist of components with a specific purpose are divided into two different categories: open-loop and closed-loop systems. Open-loop systems do not have any feedback from the output signal, while closed-loop systems use feedback to minimize the error between the input and output signals. The feedback has a significant role in increasing the endurance of the system in case of environmental changes such as disturbances. Even if a single closed-loop control can achieve the desired performance in many systems, it may not be enough to meet the performance criteria in systems such as motion control. In case of performance criteria cannot be met, the cascade controller structure is used to improve the system performance. The dynamic performance of the system which consists of nested loops is increased by rejection of any disturbing effects that occur in the inner loop used in cascade control without affecting the system output. These nested loops are called secondary (inner) and primary (outer) loops, or in other words slave and master loops. Although it is possible to create an infinite number of loops, generally two loops are used as inner (secondary) and outer (primary) loops.


Cascade control structures can be divided into two categories. The first cascade control structure is named as serial cascade and the second one is named as parallel cascade, respectively. The serial cascade control structure consists of an inner and an outer nested control loop. For the serial cascade control structure, the inner loop controller controls only the inner loop system. On the other hand, the inner loop controller controls both the inner loop system and the outer loop system at the same time for the parallel cascade control structure which is found as an alternative to the serial cascade control structure.

The model of the system to be controlled can be both high order systems or non-high order systems for serial cascade control structure. Although the conventional controller design methods show good results for the cascade systems with the first or second order, it is hard to have good results in higher-order systems as the cascade system becomes more complex. First-order dead-time model approximations can be used for controller design or high-order controllers can be designed for the high order cascade systems. In systems where cascade control is preferred, such as chemical processes, overshoot that may occur in the system response may cause irreversible effects on the system, the overshoot in the system response is an undesirable time-domain characteristic. On the other hand, the presence of zeros of the controller designed in the inner loop in cascade control can lead to overshoot in the inner loop step response. Furthermore, the purpose of the controller in the outer loop is to tolerate the inner loop overshoot by ensuring that the closed-loop response is a non-overshooting response. 

Overshoot in the closed-loop system response is not a desired time-domain characteristic in cascade control systems, it is aimed to increase the disturbance rejection performance of the inner loop compared to the outer loop as a rule of thumb. The PI-PD controller structure can be provided as a good example for the aforementioned cascade control structure. The PD controller, which is placed in the feedback in the inner loop, is selected so that the inner loop step response has fast dynamics even though there may be an overshoot. In the outer loop, the PI controller is chosen to obtain a non-overshooting closed-loop response regardless of the speed of the outer loop dynamics. In this study, the controller design criteria is to increase the speed of the inner loop system response and to prevent overshoot of the system response in the outer loop, respectively. 

In this study, instead of using low order system models and low order controllers for serial cascade control of high order systems, the high order system models are directly used with a high-order controller design method based on "Characteristic Ratio Assignment (CRA)". This controller design method is based on the relationship between the "coefficients of the characteristic polynomial " of the systems which is proposed by Naslin \cite{haeri2005determine}. In this method, time-domain characteristics, namely, the rise time, the overshoot, or the undershoot of the system can be adjusted using the characteristic ratio assignment by the selection of the time constant "$\tau$" and principal characteristic ratio "${\alpha }_1$". 

In order to explain the effect of characteristic ratio values on system response, firstly, the overshoot values are examined for different ${\alpha }_1$values for the all-pole systems from second to eighth order in case of $\tau$=1 for unit step inputs. Then, appropriate ${\alpha }_1$ values to obtain non-overshooting response are determined for different orders of closed-loop characteristic polynomials. If the order of the closed-loop characteristic polynomial is fourth-order or higher, it is possible by selecting an appropriate ${\alpha }_1$ value. If the closed-loop characteristic polynomial is second-order or third-order, non-overshooting system response can be obtained for the all-pole systems with the selection of ${\alpha }_1$ as 4 and 3, respectively. Since overshoot can be observed in systems that have system zeros, the all-pole systems, the ${\alpha }_1$ value of which are determined for non-overshoot, are examined by adding zero to their structure. It is observed that a non-overshooting response can be obtained by choosing ${\alpha }_1$ or $\tau$ values slightly higher than their values determined for all pole cases. The rise time is found to be generally in the range of 
\newline
[$\tau$-1 $\tau$+1] seconds with respect to the chosen $\tau$ time constant. Although it is theoretically possible to obtain very fast system responses with the Characteristic Ratio Assignment method, via choosing the value of $\tau$ smaller, the design will not be realistic that much since the amplitude of the control signals needs to be very high to provide these high-speed system responses and it may also not be possible to apply such high control signals due to nonlinearities or constraints which are not accounted for.

In order to apply Characteristic Ratio Assignment based high order controller design methods to serial cascade control systems, various inner and outer high order system models with different features are considered. The first setup in this work is chosen as high order all-pole systems for the inner and outer loop in the control structure. Secondly, the inner loop system is taken as an all-pole system and the outer loop system is taken as a system that has a zero in the left-half plane. In the third case, an all-pole system and a non-minimum phase system are considered in the inner loop and the outer loop, respectively. Finally, the inner loop system is selected as a non-minimum phase system and the outer loop system is selected as a system with the free integrator. 

The proposed cascade control method is compared with the "Model Separation-based Cascade Controller (MSCC)" method for all cases. The comparison criterion is selected as ITSE (Integral Time Square Error) and the disturbance rejection and reference tracking performances of both methods are examined in MATLAB (MATrix LABoratory) Simulink environment, which is the software of Mathworks commonly used in academia and industry. Simulation results also show that Characteristic Ratio Assignment based high order serial cascade control method performs better than Model Separation-based Cascade Controller in disturbance rejection. In the Model Separation-based Cascade Controller method first-order dead-time or second-order dead-time model approximations of high order system models are used. Because of this approach, an extra overshoot may be observed when Model Separation-based Cascade Controller is applied to high order system models that fit real-time systems better. 

In the light of the simulation results, good results are obtained in cascade controller design with the Characteristic Ratio Assignment method, especially in terms of disturbance rejection performance. Characteristic Ratio Assignment based method needs a fewer number of controller design parameters. Compared to Model Separation-based Cascade Controller controller design methods, the fewer number of controller design parameters with the Characteristic Ratio Assignment controller simplifies the controller designer's job. While no overshoot is observed in the simulation made over the dead-time system using Model Separation-based Cascade Controller, overshoot may be observed in the closed-loop unit-step response on the real-time system. This difference may be caused by the modeling error since the controllers are designed over the dead-time system models of high-order systems. Moreover, when the coefficients of the higher-order terms are too small, which is a common case for the Characteristic Ratio Assignment based design method, the design can be carried out by omitting the terms with relatively small coefficients. Not only for higher-order controllers but also for a second-order controller design (such as the design of PID (Proportional - Integral - Derivative) controller), the controller designer can freely synthesize the controller by using the characteristic ratios proposed by Naslin, to choose the coefficients of the closed-loop characteristic polynomial.