Coefficient diagram method: Difference between revisions

Revision as of 15:57, 7 June 2006

Latest revision as of 12:10, 30 June 2025

(44 intermediate revisions by 24 users not shown)

Line 1:

In [[control theory]], the '''coefficient diagram method''' (CDM) is an [[algebra]]ic approach applied to a [[polynomial]] loop in the [[parameter space]]. A special diagram called a "''coefficient diagram''" is used as the vehicle to carry the necessary information and as the criterion of good design.<ref>S. Manabe (1998), "''Coefficient Diagram Method''", 14th IFAC Symp. on Automatic Control in Aerospace, Seoul.</ref> The performance of the closed-loop system is monitored by the coefficient diagram.

'''COEFFICIENT DIAGRAM METHOD'''

<nowiki>

The Coefficient Diagram Method (CDM), recently developed and introduced by [http://www.cityfujisawa.ne.jp/~manabes/ Prof. Shunji Manabe] in 1991.

CDM is an algebraic approach applied to polynomial loop in the parameter space, where a special diagram

called coefficient diagram is used as the vehicle to carry the necessary information, and as the criteria

of good design (Manabe,1998). The performance of the closed loop system is monitored on coefficient diagram.

The simplicity of the controller structure made it very powerful for systems with uncertainties such as

robotic manipulator (Ucar and Hamamci, 2000).

The most considerable advantages of CDM can be listed as follows:<ref>S.E. Hamamci, "''A robust polynomial-based control for stable processes with time delay''", Electrical Engineering, vol: 87, pp.163–172, 2005.</ref>

Generally, the problem of a control system design consists of choosing a proper controller considering

the system dynamics, which is to be controlled, and desired performance specifications. There are three

main theory for a design procedure: Conventional Control Theory, Modern Control Theory and Algebraic Approach.

The main difference among these theories is the design approach used to obtain the controller and the

mathematical expressions used to represent the system . Classical Control Methods, such as Frequency Response

Method and Root-Locus Method, use the transfer function for the system representation. However, this representation

can lead to undesired results because of pole-zero cancellations due to uncontrollable or unobservable

situations. Modern Control Methods, like Pole-Placement, Optimal Control (LQR) and H¥ , use state-space

representation. This representation, especially as the plant degree gets larger, involves complex calculations

which require the use of a computer. Algebraic methods like Pole-placement direct method and CDM use polynomial

expressions. In this representation, since the numerator and denominator of the transfer function are

considered independently from each other, better results can be achieved against pole-zero cancellations.

In this approach, the type and degree of the controller polynomials and characteristic polynomial of the

closed-loop system are defined at the beginning. Considering the design specifications, coefficients of the

polynomials are found later in the design procedure. In algebraic methods, CDM is the one which gives

the most proper results with the the easiest procedure. In CDM, design specifications are equivalent time

constant (t ), stability indices (yi) and stability limits (yi*). These parameters have certain relations

which will be explained later with the controller polynomials (Manabe, 1994).

#The design procedure is easily understandable, systematic and useful. Therefore, the coefficients of the CDM controller polynomials can be determined more easily than those of the [[PID controller|PID]] or other types of controller. This creates the possibility of an easy realisation for a new designer to control any kind of system.

</nowiki>

# There are explicit relations between the performance parameters specified before the design and the coefficients of the controller polynomials as described in.<ref>S. Manabe (1998), "''Coefficient Diagram Method''", 14th IFAC Symp. on Automatic Control in Aerospace, Seoul.</ref> For this reason, the designer can easily realize many [[control system]]s having different performance properties for a given control problem in a wide range of freedom.

# The development of different tuning methods is required for time delay processes of different properties in PID control. But it is sufficient to use the single design procedure in the CDM technique. This is an outstanding advantage.<ref>S.E. Hamamci, I. Kaya and D.P. Atherton, "''Smith predictor design by CDM''", Proceedings of the ECC’01 European Control Conference, Semina´rio de Vilar, Porto, Portugal, 2001.</ref>

# It is particularly hard to design robust controllers realizing the desired performance properties for unstable, integrating and oscillatory processes having poles near the imaginary axis. It has been reported that successful designs can be achieved even in these cases by using CDM.<ref>S. Manabe, "''A low cost inverted pendulum system for control system education''", The 3rd IFAC Symposium on advances in Control Education, Tokyo, 1994.</ref>

# It is theoretically proven that CDM design is equivalent to LQ design with proper state augmentation. Thus, CDM can be considered an ‘‘improved LQG’’, because the order of the controller is smaller and weight selection rules are also given.<ref>S. Manabe, "''Analytical weight selection for LQ design''", Proceedings of the 8th Workshop on Astrodynamics and Flight Mechanics, Sagamihara, ISAS, 1998.</ref>

It is usually required that the controller for a given plant should be designed under some practical limitations.

REFERENCES

The controller is desired to be of minimum degree, [[minimum phase]] (if possible) and stable. It must have enough bandwidth and power rating limitations. If the controller is designed without considering these limitations, the robustness property will be very poor, even though the stability and [[time response]] requirements are met. CDM controllers designed while considering all these problems is of the lowest degree, has a convenient bandwidth and results with a unit step time response without an overshoot. These properties guarantee the robustness, the sufficient [[damping]] of the disturbance effects and the low economic property.<ref>S. Manabe and Y.C. Kim, "''Recent development of coefficient diagram method''", Proceedings of the ASSC’2000 3rd Asian Control Conference, Shanghai, 2000.</ref>

Although the main principles of CDM have been known since the 1950s,<ref>D. Graham and R.C. Lathrop, "''The synthesis of optimum transient response: criteria and standard forms''", AIEE Trans., vol:72, pp.273–288, 1953.</ref><ref>P. Naslin, ''Essentials of optimal control'', Boston Technical Publishers, Cambridge, MA, 1969.</ref><ref>A.V. Lipatov and N. Sokolov, "''Some sufficient conditions for stability and instability of continuous linear stationary systems''", Automat. Remote Control, vol:39, pp.1285–1291, 1979.</ref> the first systematic method was proposed by [[Shunji Manabe]].<ref>Y.C. Kim and S. Manabe, "''Introduction to coefficient diagram method''" Proceedings of the SSSC’01, Prague, 2001.</ref> He developed a new method that easily builds a target [[characteristic polynomial]] to meet the desired time response. CDM is an algebraic approach combining classical and modern control theories and uses polynomial representation in the mathematical expression. The advantages of the classical and modern control techniques are integrated with the basic principles of this method, which is derived by making use of the previous experience and knowledge of the controller design. Thus, an efficient and fertile control method has appeared as a tool with which control systems can be designed without needing much experience and without confronting many problems.

Many control systems have been designed successfully using CDM.<ref>S. Manabe, "''A low cost inverted pendulum system for control system education''", The 3rd IFAC Symposium on advances in Control Education, Tokyo, 1994.</ref><ref>S.E. Hamamci, M. Koksal and S. Manabe, "''On the control of some nonlinear systems with the coefficient diagram method''", Proceedings of the 4th Asian Control Conference, Singapore, 2002.</ref> It is very easy to design a controller under the conditions of stability, [[time ___domain]] performance and robustness. The close relations between these conditions and coefficients of the characteristic polynomial can be simply determined. This means that CDM is effective not only for control system design but also for controller parameters tuning.

1. S. Manabe, "Coefficient Diagram Method", 14th IFAC Symp. on Automatic Control in Aerospace, Seoul,1998.

2. A. Ucar and S.E. Hamamci, "A Controller Based on Coefficient Diagram Method for the Robotic Manipulators", ICECS2K The 7th IEEE Int.Conf. on Electronics, Circuits&Systems,, Kaslik, Lebanon, 2000.

3. S. Manabe, "A low cost inverted pendulum system for control system education", The 3rd IFAC Symposium on advances in Control Education, Tokyo, 1994.

==See also==

[http://http://web.inonu.edu.tr/~shamamci/mra_cdm.htm Coefficient Diagram Method]

*[[Polynomials]]

==References==

<references/>

==External links==

*[httphttps://archive.today/20051210021648/http://web.inonu.edu.tr/~shamamci/mra_cdm.htm Coefficient Diagram Method]

.

[[Category:Polynomials]]

[[Category:Control theory]]