The system-specific perspective

System-specific PI control theory is based on the fundamental algorithm for PID control systems, it was developed from a 'back-to-basics' perspective considering fluid systems and motion systems independently. The resulting PID control tuning methods are not based on yet another interpretation of conventional theory. A ‘back-to-basics' perspective was deemed desirable to study PID control tuning because PID control tuning methods in textbooks, and the literature, appear to be impractical as evidenced by the numerous Web sites on the topic of PID control tuning. Conventional PID feedback control theory, and conventional PID control tuning methods - originated in the 1940's - were developed for analog, not digital, control systems. System-specific PI control theory is based on digital control systems.

PID feedback control may be used to maintain an office temperature at a desired set point or to maintain the speed of a car constant by cruise control. For fluid system control, the control signal is generally updated at intervals of the order of seconds, not milliseconds - the signal update interval requires specification. For motion system control, the control signal is generally updated at intervals determined by the sampling rate - the magnitude of the sampling rate generally requires experimental determination. Fluid control systems and motion control systems are very different. A generalized PID control theory applicable to all types of systems is inherently complex. A generalized PID control theory may not provide insight to a specific system. System-specific PI control theory does not camouflage the physical significance of system parameters effecting PI coefficients and system response characteristics. System-specific PI control theory is, therefore, relatively simple and yields relatively simple analytical tuning methods.

System-specific PI control theory facilitates the understanding of the physical significance of the system parameters involved in the determination of the PI coefficients. An understanding of the physical significance of the system parameters involved facilitates the generalization of the theory (as detailed in Chapter 14: GENERALIZED METHODOLOGY) and subsequently its application to systems not covered in this book.

Second edition

The second edition contains an extension to the theory that simplifies the use of variable PI coefficients and also simplifies implementation. Variable PI coefficient may yield superior control (relative to constant PI coefficients) of non-linear systems (modulated capacity versus control signal). Non-linear systems are common.

Terminology, units

As previously noted, system-specific theory is not another interpretation of conventional control theory. System-specific PI control theory uses terminology best suited to convey a concept. System-specific PI control theory is not adapted to conventional terminology. Conventional terminology is not considered sacred. Equations for PI coefficients derived with system-specific theory are not based on a time constant. Whereas the validity of any solution to an equation is independent of the method of solution, equations are solved by the simplest method available. Transfer functions are not used in the solution of equations in system-specific theory.

Error and PI coefficients are dimensional. In system-specific PI control theory, these parameters are not non-dimensionalized as in conventional PID control theory. Non-dimensionalizing the parameters may be desirable for analog control systems but is not required for digital control systems. Non-dimensionalizing camouflages the physical signaificance of these, and other, system parameters. Control signal is expressed as a percentage because several conventional control signal (e.g., 4-20 ma dc, 0-5 v dc, 0-10 v dc, etc.) are commonly used. Thus, error has the units of the controlled property, and the PI coefficients have the units of percent per unit controlled property.

Tuning PI control systems

Tuning a PI control system requires

• identifying the type of system,
• selecting the appropriate equations for the type of system,
• specifying the desired system response characteristic,
• evaluating the parameters in the selected equations using physical (not abstract) system properties, and
• calculating the PI coefficients.

Tuning methods based on system-specific PI control theory enable the calculation of the coefficients required to implement a control system from physical characteristics of a system. Calculations may be performed with a simple calculator, software is not required.

There are usually several methods of solving any control problem. However, not all solutions, i.e., system response characteristics, meet the same criteria. System-specific PI control theory is designed to yield response characteristics that are robust, attenuate errors rapidly with minimal overshoot, and have a zero steady-state off-set.

System classifications

System-specific PI control theory classifies systems as

• I systems - I systems require only an integral coefficient. (The proportional coefficient may be set equal to 0 or the control algorithm may be simplified.)

• PI systems - PI require both an integral coefficient and a proportional coefficient.

• P systems - P systems require only a proportional coefficient. (The integral coefficient may be set equal to 0 or the control algorithm may be simplified.)

I systems and P systems are first-order systems. PI systems are second-order systems. A first-order system - represented by a first-order differential equation - has a response (error versus time) characteristic represented by an exponential function. Only one coeffcient is required to determine the shape of the exponential function. A second-order system - represented by either a second-order differential equation - has a response characteristic represented by a function consisting of an exponential term multipled by a linear algebraic term (critically damped solution) or an attenuated sinusoidal function (underdamped solution). Two coeffcient are required to determine the shape of either function.

First-order and second-order systems therefore require 1 or 2 coefficients, respectively. Additional coefficients for first-order and second-order systems are not necessary and may have a detrimental effect on system performance.

Higher order systems may be considered to consist of series control loops. (Parallel control loop systems are also possible.)

System-specific PI control theory and applications

System-specific PI control theory is presented in a reference book and an application - ControlProblems - containing problems. The reference book is formatted so that it does not have to be read cover to cover. A reader may focus only on a specific system of interest. Problems are presented for illustration purposes. Only 14 basic problems are presented. Numerical models of the 14 problems are presented in the application. However, the problems have user-specified options. User-specified PI coefficients or default PI coefficient may be selected.

The procedure for applying system-specific PI control theory to additional systems not presented in the reference book or the ControlProblems application is detailed in Chapter 14: GENERALIZED METHODOLOGY.

It should be noted that the book does not contain a references or bibliography section. Although a few papers based on the precursor to system-specific theory have been published, these papers are now obsolete.

System-specific PI Control Theory for Fluid and Motion Systems