D'Azzo - Linear Control Systems

These are various useful notes from chapters of the book.

1.2 Introduction to Control Systems

Open Loop examples:

a) automatic toaster: timer = reference selector

b) electric motor: applied armature voltage = input quantity

------------ ---------

----------> | Ref Selector | ---------> | Dyn Unit | ---------->

Cmd input ------------ Ref input ---------

Closed Loop Single Input Single Output examples:

a) home heating: temp control = ref selector, bimetallic strip = comparator and relay controller

furnace & hot vents = plant

1.4 Historical Background

An early military application of feedback control system was the anti-aircraft radar tracking control system. Its components are:

- a radar antenna detecting position and velocity of the target

- computer to determine gun firing angle to intercept target (lead angle so that shell reaches projected target position)

- output signal of computer as a function of firing angle

- amplifier providing power for drive motors

- 2 positioning motors for vertical and horizontal angle

- feedback signal proportional to gun position

[image from P17]

Early studies of control systems were based on the solution of differential equations by classical methods, but this became impractical. Nyquist published a paper in 1932 dealing with the application of steady-state frequency response techniques to feedback amplifier design. Bode and Black extended this work. In 1948 Evans presented his root-locus theory.

The Laplace transform and the principles of linear algebra are used in the application of modern control theory to system analysis and design. The n-th order differential equation describing the system can be converted into a set of n first-order differential equations expressed in terms o the state variables. These equations can be written in matrix notation which lend themselves well to computer calculation.

Anti-windup:

Integral (integrator) windup refers to the situation in a PID feedback controller where a large change in setpoint occurs (say a positive change) and the integral term accumulates a significant error during the rise (windup), thus overshooting and continuing to increase as this accumulated error is unwound (offset by errors in the other direction). The specific problem is the excess overshooting.

The problem can be addressed by

- Initialising the controller integral to a desired value, for instance to the value before the problem

- Increasing the setpoint in a suitable ramp

- Disabling the integral function until the to-be-controlled process variable (PV) has entered the controllable region

- Preventing the integral term from accumulating above or below pre-determined bounds

- Back-calculating the integral term to constrain the process output within feasible bounds

Integral windup particularly occurs as a limitation of physical systems, compared with ideal systems, due to the ideal output being physically impossible (process saturation: the output of the process being limited at the top or bottom of its scale, making the error constant). For example, the position of a valve cannot be any more open than fully open and also cannot be closed any more than fully closed.

Integral windup was more of a problem in analog controllers. Within modern Distributed Control Systems and Programmable Logic Controllers, it is much easier to prevent integral windup by either limiting the controller output, or by using external reset feedback, which is a means of feeding back the selected output to the integral circuit of all controllers in the selection scheme so that a closed loop is maintained.