National Instruments ICDM - page 90

Chapter 11 Introduction to MIMO Design

Xmath Interactive Control Design Module 11-4 ni.com

Notice that in the SISO case, these “complementary pairs” of transfer

functions (obtained by swapping P and C) are the same. It is important to

remember that in the MIMO case they can be different; they even have

different dimensions if ny≠nu.

In addition to these transfer functions you encounter two (complementary)

open-loop transfer functions:

• The loop transfer function L is defined as L=PC. This is the transfer

function of the loop cut at the sensor (or the error e).

• The actuator loop transfer function (or complementary loop transfer

function) Lact is defined as Lact=CP. This is the transfer function of the

loop cut at the actuator.

Integral Action

Integral action can be quite complicated in the MIMO setting. The simplest

case occurs when the plant is “square”, that is, ny=nu, the plant has no

poles at s= 0, and no zeros at s=0—which in this case means debt

P(0) ≠0.

Suppose the controller has the form

where has no poles at s=0 and the (constant) matrix is non singular

(forexample, R0=I). Then you have T(0) = I, so that you have perfect

asymptotic decoupling and tracking of constant reference inputs. You also

have perfect asymptotic rejection of constant actuator disturbances. The

condition on C means the controller has an integrator in each of its

channels. In this case the integrators can be thought of as either acting on

the sensor signals or acting on the actuator signals.

You often have integrators associated with some of the sensors, or some of

the actuators. Then the matrix can be less than full rank, and you generally

get the benefits of integral action in only some of the I/O channels; that is,

only some of the entries of T(0) are 1 (or 0).

When things get more complicated. The rank of R0 is at most

r=max{nynu}. If you ask ICDM to insert more than r integrators (say,

using an integrator in each sensor channel for a plant with three actuators

and five sensors), you will have a closed-loop system that cannot be

stabilized—it will have an uncontrollable or unobservable mode at s=0.

If this happens, ICDM will warn you. Even if you do not have excess

integrators, you should realize that you will get perfect asymptotic tracking

Cs() 1s⁄()R0C

˜s()+=

C

˜

nynu

≠

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