National Instruments ICDM - page 89

Chapter 11 Introduction to MIMO Design

© National Instruments Corporation 11-3 Xmath Interactive Control Design Module

The standard feedback system has two vector input signals, r and dact, and

three vector output signals, e, u, and y. It can therefore be described by the

3×2 block matrix that relates the three output vector signals to the two

input vector signals:

The entries of this block matrix, that is, the transfer functions from r and

dact to e, u, and y, have standard names and interpretations (which agree

with the standard SISO notation):

• The sensitivity transfer function is denoted S and given by

S=(I+PC)–1. The sensitivity transfer function is the transfer function

from reference input r to the error signal e.

• The closed-loop transfer function T is given by T=PC(I+PC)–1. T is

the transfer function from r to y. T can be expressed in several other

ways, for example:

• The actuator effort transfer function C(I+PC)–1 is the transfer

function from r to u, and so is related to the actuator effort required.

For example, its step response matrix shows the closed-loop step

responses from each reference input signal to each actuator signal.

• The transfer function from dact to e, P(I+CP)–1, is denoted Sact and

called the actuator-referred sensitivity transfer function. The

actuator-referred sensitivity transfer function determines the errors

generated by actuator-referred disturbances. It also can be expressed as

(I+PC)–1P. Notice that it is “complementary” to the transfer function

described just above, that is, C(I+PC)–1, in the sense that the two

transfer functions can be obtained from each other by swapping P

and C.

• The transfer function from dact to u, CP(I+CP)–1, is called the

actuator-referred actuator effort transfer function. Notice that it is

related to the closed-loop transfer function by swapping P and C. It can

also be expressed as C(I+PC)–1P.

• The transfer function from dact to y, (–P)(I+CP)–1, is denoted Tact and

called the actuator-referred closed-loop transfer function.

e

u

y

IPC+()

1–PI CP+()

1–

CI PC+()

1–CP I CP+()

1–

PC I PC+()

1–P–ICP+()

1–

=r

dact

TPCICP+()

1–IPC+()

1–PC I S–===

Contents

Main Page Important Information Warranty Copyright Trademarks Patents Conventions Contents Chapter 1 Introduction Chapter 2 Introduction to SISO Design Chapter 3 ICDM Main Window Chapter 4 PID Synthesis Chapter 5 Root Locus Synthesis Chapter 6 Pole Place Synthesis Chapter 7 LQG Synthesis Chapter 8 H-Infinity Synthesis Chapter 9 Chapter 10 Alternate Plant Window Chapter 11 Introduction to MIMO Design Chapter 12 LQG/H-Infinity Synthesis Page Introduction Using This Manual Document Organization Page Commonly-Used Nomenclature Related Publications MATRIXx Help ICDM Overview SISO Versus MIMO Design Starting ICDM Page Introduction to SISO Design SISO Design Overview Basic SISO Terminology Page Overview of ICDM ICDM Windows ICDM Main Window PID Synthesis Window Root Locus Synthesis Window Pole Place Synthesis Window LQG Synthesis Window H-Infinity Synthesis Window Alternate Plant Window Key Transfer Functions and Data Flow in ICDM Summary Origin of the Controller What the ICDM Main Window Plots Show Controller/Synthesis Window Compatibilities Page Using ICDM G G Page Zooming Data-Viewing Plots Interactive Plot Re-ranging Graphically Manipulating Poles and Zeros Editing Poles and Zeros Editing Poles and Zeros Graphically Complex Poles and Zeros Isolated Real Poles and Zeros Nonisolated Real Poles and Zeros and Almost Real Pairs Adding/Deleting Poles and Zeros Adding/Deleting Pole-Zero Pairs Page ICDM Main Window Communicating with Xmath Most Common Usage Default Plants Saving and Restoring an ICDM Session Reading Another Plant into ICDM Reading a Controller from Xmath into ICDM Writing the Plant Back to Xmath Writing the Alternate Plant back to Xmath Writing a Controller on the History List to Xmath ICDM Plots Selecting Plots Ranges of Plots Plot Magnify Windows Page Selecting a Synthesis or History Window Edit Menu PID Synthesis PID Controller Terms Toggling Controller Terms On and Off Page Opening the PID Synthesis Window Manipulating the Controller Parameters Time Versus Frequency Parameters Ranges of Sliders and Plots Controller Term Normalizations Integral Term Normalization Derivative Term Normalization Rolloff Term Normalization Root Locus Synthesis Overview Page Opening the Root Locus Synthesis Window Terminology Plotting Styles Phase Contours Magnitude Contours Slider and Plot Ranges Manipulating the Parameters Design Adding a Pole-Zero Pair Deleting Pole-Zero Pairs Interpreting the Nonstandard Contour Plots Page Pole Place Synthesis Pole Place Modes Normal Mode Integral Action Mode State-Space Interpretation Opening the Pole Place Window Manipulating the Closed-Loop Poles Time and Frequency Scaling Butterworth Configuration Editing the Closed-Loop Poles Slider and Plot Ranges LQG Synthesis LQG Synthesis Window Anatomy Page Synthesis Modes Opening the LQG Synthesis Window Setup and Terminology Standard LQG (All Toggle Buttons Off) Exponential Time Weighting Output Weight Editing State-Space Interpretation Manipulating the Design Parameters Graphically Page H-Infinity Synthesis H-Infinity Synthesis Window Anatomy Page Opening the Synthesis Window Setup and Synthesis Method Central H-Infinity Controller Output Weight Editing Manipulating the Weight Transfer Function Infeasible Parameter Values Page History Window Saving the Current Controller on the History List Opening the History Window History Window Anatomy Selecting the Active Controller Editing the Comments Deleting History List Entries To Continue Designing from a Saved Controller Cycling Through Designs Writing a Saved Design to Xmath Page Alternate Plant Window Role and Use of Plant and Alternate Plant Displaying the Alternate Plant Responses Alternate Plant Window Anatomy Page Normalization Manipulating the Parameters Using the Alternate Plant Window Robustness to Plant Variations Adding Unmodeled Dynamics Ranges of Sliders and Plot Introduction to MIMO Design Basic Terminology for MIMO Systems Feedback System Configuration Transfer Functions Page Page Overview of ICDM for MIMO Design ICDM MIMO Windows Main Window MIMO Plot Window Alternate Plant Window (MIMO Version) Page LQG/H-Infinity Synthesis LQG/H-Infinity Main Window LQG/H-Infinity Weights Window Page Page Decay Rate Window H-Infinity Performance Window Frequency Weights Window Synthesis Modes and Window Usage Opening the LQG/H-Infinity Synthesis Window Setup and Terminology Page Chapter 12 LQG/H-Infinity Synthesis Xmath Interactive Control Design Module 12-10 ni.com Figure 12-6. LQG/H-Infinity Control Design Configuration w How to Select w, u, y, and z H-Infinity Solution Page Main Window Page Multi-Loop Synthesis Multi-Loop Window Anatomy Page Setup and Synthesis Method Multi-Loop Versus Multivariable Design Page Page Page Opening the Multi-Loop Synthesis Window Designing a Multi-Loop Controller Graphical Editor Selecting and Deselecting Loops Editing and Deleting Loops Loop Gain Magnitude and Phase A Using an Xmath GUI Tool Overview Interacting with a GUI Application GUI Functions GUI Objects Page Page Page B Technical Support and Professional Services Index A B C D G H I K L R S T W Z