Architectural Issues in SISO Control

IntroductionInternal Model PrincipleFeed Forward ControlCascade ControlSummaryReferences

To adjust the control system architecture To achieve given performance objectives

Prime examples of architectural changes Feed forwardCascade control

Signals of interest Output of a linear dynamic system Having zero input certain specific initial conditions. Example of such a signal is a constant

The generalization of differential equation

Laplace transform of the disturbance:

where d(s) is the disturbance generating polynomial defined by

Example :-A disturbance takes the following form

Then the generating polynomial is given by

K1, K2 and K3 are related to the initial state xd(0) in the state space model

Disturbance Entry Points

For output disturbance -Go(s) =Go1(s)For input disturbance - Go(s) =Go2(s)R(S)U(S)Y(S)Dg(s)

Effect of disturbance vanishes whenDg(t) -> 0 as t -> d(s) is a factor of the numerator of S0(s)Go2(s)

A sufficient condition for steady state disturbance compensation is that the generating polynomial be included as part of the controller denominator. This is known as the Internal Model Principle (IMP).

H(s)C(s)Go(s)R(s)E(s)Y(s)

To use IMP for Reference Tracking

Set H(s) = 1 (Reference feed forward transfer function)Include reference generating polynomial in the denominator of C(s)G0(s)So(i) = 0 where i = 1, . . .n are the poles of the reference generating polynomialTo(s) = 1E(s) = 0 indicates as t e(t) = 0

To achieve Robust Tracking-The reference generating polynomial must be in the denominator of C(s)G0(s) i.e. IMP has to be satisfied for the referenceWhen the reference and the disturbance generating polynomial share some roots, then these common roots need only be included once in the denominator of C(s)IMP application- PWM invertersActive filtersUPS system

Use of the IMPProvides complete disturbance compensation and reference tracking in steady state for certain classes of signals

Leaves unanswered the issue of transient performance (how the system responds during the initial phase following a change in the disturbance or reference signal)

Essential idea of reference feed forward -

To use H(s) to invert T0(s) at certain key frequencies i.e. so that H(s)T0(s) = 1 at the poles of the reference model

E(s) = R(s) Y (s) = [1 H(s)To(s)]R(s)

Yd(s) = So(s)Go2(s)[1 + Go1Gf (s)]Dg(s) Ud(s) = Suo(s)[Go2(s) + Gf (s)]Dg(s)FeaturesFeed forward block transfer function Gf(s) must be stable and proper, since it acts in open loop.Gf(s)= -[1/Go1(s)]Usually G01(s) will have a low pass characteristic, Gf(s) - high pass characteristic.

Core Idea -To feedback intermediate variables that lie between the disturbance injection point and the output. This gives rise to cascade control.

Main features of cascade control areCascade control is a feedback strategy.A second measurement of a process variable is required. Measurement noise in the secondary loop must be considered in the design, since it may limit the achievable bandwidth in this loop.Well proven technology when two or more system feed sequentially into each other.

Internal model principle :-Compensation for classes of references and disturbancesFeed forward :- Effective technique for improving responses to set point changesCascade control :-Well proven technique applicable when two or more systems feed sequentially into each other

CONTROL SYSTEM DESIGN Graham C. Goodwin, Stefan F. Graebe, Mario E. Salgado

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