CDM（係数図法）

Introduction
Coefficient Diagram Method (CDM) is an effective control design theory, and very good control systems can be designed without much design experiences. The field adjustments are also easy. CDM is classified as "Algebraic Design Approach", and some what between "Classical Control" and "Modern Control".

There are five features in CDM.
          (1) Polynomials or polynomial matrices are used in design.
          (2) Characteristic polynomial and controller are simultaneously designed.
          (3) "Coefficient Diagram" is effectively used.
          (4) "Sufficient condition of stability by Lipatov" is the theoretical basis.
          (5)  "Improved version of Kessler standard form" is adopted.

Polynomial Expression
As to the mathematical expression, the transfer function (classical control) and the state space (modern control) are commonly used.  The transfer function expression (the first method) is easy to handle, but it becomes inaccurate when pole-zero cancellation occurs.  The state space expression (the second method) is accurate and well-suited in machine computation, but manual handling is not easy.

The third method is the polynomial expression, where the denominator and the numerator of the transfer function are to be handled as the independent entity representing some concrete physical entity.  This expression enjoys the easiness of handling of the transfer function together with the rigor of the state space, because it is equivalent to the state space expression in control or observer canonical form.

Simultaneous design
All the control system design for linear time-invariant dynamic systems boils down to proper selection of the characteristic polynomials (denominator of transfer functions) and proper selection of the numerator polynomials. Especially the proper selection of the characteristic polynomial is essential in designing a good control system with proper balance of stability, response, and robustness.

In any control system, the controller has practical limitation, such as
         Low order,
         Minimum phase if possible,
         Stable, unless unstable controller is absolutely necessary,
         Bandwidth limitation,
         Power limitation.
Controller limitations impose strict limitation on the choice of characteristic polynomial. If characteristic polynomial is chosen without this consideration, robustness will be lost, while stability and response are satisfied.   Simultaneous design of characteristic polynomial and controller is considered essential for the good controller design

Coefficient diagram
CDM gives the way to directly design the characteristic polynomial under controller limitation, where “Coefficient Diagram” is  effectively utilized.  The coefficient diagram is a semi-log diagram where the coefficients of characteristic polynomial are shown in the ordinate in logarithmic scale and the numbers of powers corresponding to the coefficients are shown in the abscissa in linear scale.

Coefficient diagram gives enough information on stability, response, and robustness in a much clearer way compared with Bode/Nyquist diagram  The convexity of the curve is a measure of stability. The general inclination of the curve is a measure of response speed. The variation of the shape of the curve due to plant/controller parameter variation is a measure of robustness.


Sufficient condition of stability by Lipatov
Routh stability condition is commonly used.  Lipatov proposed sufficient condition for stability and instability in 1978.  Because it is only sufficient conditions, not necessary and sufficient, the condition is much simpler compared with Routh.  When integrated with CDM, it gives information even  about "the degree of stability", important information in practical design.

Standard form
Kessler proposed a standard form in 1960.  Improvement has been made such that undesirable overshoot is taken out, and design flexibility is much enhanced.

Further topics
1.  References are provided in a separate page.  Some important papers can be down-loaded as pdf file.
2.  To expedite CDM design, MATLAB based CAD system is developed.  Its explanation is           in a separate page, and can be down-loaded as zip file.
3.  Preparation of the text book of CDM is under process.  Meanwhile the completed part can be down-loaded as pdf file.

Further explanation of CDM
Further explanation of CDM will be given in the following items.
         Simple example, position control
         Mathematical model
         Mathematical relations
         Coefficient diagram
         Stability condition
         Canonical transfer function
         Standard form
         Design example 1, third order system
         Design example 2,  PID control

Conclusions
CDM uses "Stability index" as the stability measure, and "Equivalent time constant" as response measure.  For the specified "Settling time", CDM gives a controller
          (1)  Without undesirable overshoot,
          (2)  Lowest order,
          (3)  Narrowest bandwidth.
These features guarantees good robustness, good noise attenuation, low cost, and easy adjustment.

CDM can be considered as "Generalized PID", because controller parameter selection rules are given like PID, but controller order can be higher.　CDM can be considered as improved LQG, because weights selection rules are given, and controller order is much lower.