Introduction to LabVIEW Control Design Toolkit by Finn Haugen ...

Introduction to
LabVIEW Control Design Toolkit 1.0
by
Finn Haugen
January 30, 2005
Freeware!
Contents:
1 Preface
2 The contents of the Control Design Palette
3 Creating models
3.1 Creating continuous-time (s-)transfer functions
3.2 Creating discrete-time (z-)transfer functions
3.3 Creating continuous-time state-space models
3.4 Creating discrete-time state-space models
3.5 Standard transfer functions
3.6 PID controllers
3.7 Writing models to file. Reading models from file
3.8 Getting information about a model
3.9 Converting Control Design models to/from Simulation Module models
4 Connecting models
4.1 Series connection
4.2 Feedback connection
5 Calculating transfer functions from state-space models
6 Discretizing continuous-time models
7 Simulation (time responses)
8 Frequency response
9 An application: Control system analysis and simulation
1 Preface
This document gives an introduction to the Control Design Toolkit version 2.0 for LabVIEW
7.1. (LabVIEW is produced by National Instruments.) It is assumed that you have basic
knowledge about LabVIEW programming.
The introduction is based on simple examples - all downloadable via hyperlinks. Only the basic
functions are demonstrated. You can search for a function via the Help menu in LabVIEW or
just browse for it on the Control Design palette in the Functions palette in LabVIEW. Chapter 2
of this document list all functions available in the Control Design Toolkit.
Each function has several input parameters or arguments. You should always use Help (via
right-clicking on the function block) to get information about these parameters before you use
the function in your program.

The Control Design Toolkit was initially launched in Spring 2004. It expands LabVIEW's
capabilities for control system and dynamic system analysis and design considerably. The set of
functions available is comparable with the Control System Toolbox in Matlab and the similar
control system function category in Octave.
Included in version 2.0 is the Control Design Assistant, which is an interactive tool which can
be used independent of LabVIEW, and without LabVIEW programming (you can however
create LabVIEW code from your Control Design Assistant project). The Control Design
Assistant is available from the Start / Programs / National Instruments meny on your PC and
from the Tools / Control Design Toolkit in LabVIEW.
The VIs in the examples does not contain any while loops. Consequently, the VIs run just once.
If you want a VI to run continuously with a well-defined time step between each while loop
execution, possibly while you are adjusting some parameters, you can place the block diagram
code in while loop.
If you have comments or suggestions for this document please send them via e-mail to finn@
techteach.no.
In the text, CDT will be used as an abbreviation for Control Design Toolkit.
The date shown in the beginning of the document indicates the version of the document. The
document may be updated any time. Changes from previous versions will be described in the
Preface.
2 The contents of the Control Design Palette
Once the Control Design Toolkit is installed, the Control Design palette is available from the
Functions palette. The Control Design palette is shown in the figure below.
The Control Design palette
Below is a list of functions (and possible subpalettes) on the Control Design palette. (It may be
wise to just browse the list to get a quick impression of the possibilities.)
The Model Construction palette, with the following functions and/or subpalettes:
Construct State-Space Model
Construct Transfer Function Model
Construct Zero-Pole-Gain Model
Construct Random Model
Construct Special Model:
First order with (or without) time delay
Second order with (or without) time delay
Delay Pade Approximation

PID Parallel
PID Academic (parallel form)
PID Serial
Draw Transfer Function Equation (for displaying the transfer function nicely on the
screen, as writing on paper)
Draw Zero-Pole-Gain Equation
Read Model From File
Write Model From File
Model Information palette (containing functions for setting and getting model
information or properties)
The Model Conversion palette, with the following functions:
Convert to State-Space Model
Convert to Transfer Function Model
Convert to Zero-Pole-Gain Model
Convert Delay with Pade Approximation
Convert Delay to Poles at Origin
Convert Continuous to Discrete (with various methods, e.g. Euler, Tustin, zero
order hold)
Convert Discrete to Discrete (changing the sampling interval)
Convert Discrete to Continuous
Convert Control Design to Simulation (converting models used in Control Design
Tookit for use in Simulation Module)
Convert Simulation to Control Design (converting models used in Simulation
Module for use in Control Design Tookit)
The Model Interconnection palette, with the following functions and/or subpalettes:
Serial
Parallell
Feedback
Append
Rational Polynomial palette with functions for combining polynomials
The Model Reduction palette, with the following functions:
Minimal Realization
Model Order Reduction
Minimal State Realization
Remove IO (input or output) from Model
Select IO (input or output) from Model
The Time Response palette, with the following functions and/or subpalettes:
Step Response (step input)
Impulse Response (impulse input)
Initial Response (response from initial state, with zero input)
Linear Simulation (with user-defined input signal)
Get Time Response Data
The Frequency Response palette, with the following functions:
Bode (calculating frequency response data and plotting the data in a Bode diagram)
Nyquist
Nichols
Singular Values
All Margins
Gain and Phase Margin
Evaluate at Frequency
Bandwidth
Get Frequency Response Data
The Dynamic Characteristics palette, with the following functions:
Root Locus
Pole-Zero Map
Damping Ratio and Natural Frequency
DC Gain
Stability

Norm
Covariance Response
Total Delay
Distribute Delay
Parametric Time Response
The State Space Model Analysis palette, with the following functions:
Controllability Matrix
Observability Matrix
Grammians
Canonical State-Space Realization
Balance State-Space Model (Diagonal)
Balance State-Space Model (Grammians)
Controllability Staircase
Observability Staircase
State Similarity Transform
The State Feedback Design palette, with the following functions:
Ackermann
Pole Placement
Linear Quadratic Regulator
Kalman Gain
State Estimator
State-Space Controller
Augment Output with States
3 Creating models
3.1 Creating and displaying continuous-time (s-)transfer functions
The Model Construction palette contains several functions for creating models. The resulting
model is represented as a cluster. This cluster can be used as input argument to other functions,
e.g. for simulation, frequency response analysis, etc.
On the Model Construction palette there are also functions for displaying the transfer function
nicely on the front panel.
Example 3.1.1: Creating and displaying a continuous-time (s-)transfer function
The VI shown below creates the following transfer function using the CD Construct Transfer
Function Model function (CD means Control Design):
H(s) = e -4s 3/(1+2s) = e -4s 3s 0 /(1s 0 +2s 1 )
(a first order transfer function with gain 3, time constant 2, and time delay 4s). In the VI the CD
Draw Transfer Function function displays the transfer function nicely in on the front panel
(using a picture indicator which can be created by right-clicking on the Equation output of the
function).

Front panel and block diagram of create_tf_cont.vi.
End of Example
Note: If the time delay is zero, the Delay input argument of the CD Construct Transfer
Function Model function can be unwired since the default value of the time delay is zero.
Also note: The CD Construct Transfer Function Model function has an input parameter
called Sampling Time. When creating continuous-time models this input must be either unwired
(as in Example 3.1) or wired with value zero. If a non-zero sampling time is connected, a
discrete-time transfer function will be created (with the numerator and denominator coefficients
as defined in the Numerator and Denominator arrays). Cf. Section 3.2.
3.2 Creating discrete-time (z-)transfer functions
Example 3.2.1: Creating a discrete-time (z-)transfer function
The VI shown below creates the following transfer function:
H(z) = z -5 0.4/(-0.6+z) = z -5 0.4z 0 /(-0.6z 0 +1z 1 )
with sampling time 0.1s. The factor z -5 represents a time delay of integer 5 samples (or time
steps), not 5 seconds. (For the present transfer function, the time delay in seconds is 0.1*5 =
0.5s.)
Front panel and block diagram of create_tf_discrete.vi.
End of Example
3.3 Creating continuous-time state-space models
Example 3.3.1: Creating a continuous-time state-space model

The VI shown below creates the following continuous-time state-space model using the CD
Construct State-Space Model function:
In the VI the matrices are represented by arrays. For all models (no matter the order or
dimension of the system) these arrays are 2-dimensional arrays.
End of Example
Front panel and block diagram of create_cont_ss_model.vi.
Note: The CD Construct State-Space Model function has an input parameter called Sampling
Time. When creating continuous-time models this input must be either unwired (as in Example
3.3) or wired with value zero. If a non-zero sampling time is connected, a discrete-time statespace
model will be created (with the system matrices as defined by the 2-dimensional arrays A,
B, C, and D).
3.4 Creating discrete-time state-space models
Example 3.4.1: Creating a discrete-time state-space model

The VI shown below creates the following discrete-time state-space model using the CD
Construct State-Space Model function:
x(k+1) = Ax(k) + Bu(k)
y(k) = Cx(k) + Du(k)
where the system matrices A, B, C, and D are as shown in the figure below. In the VI the
matrices are represented by arrays. Note that the matrices are technically 2x2 matrices (arrays),
although there may be only one row and/or column in the matrix.
End of Example
Front panel and block diagram of create_discrete_ss_model.vi.
3.5 Standard transfer functions
Several standard transfer functions are available:
First order with (or without) time delay
Second order with (or without) time delay
Delay Pade Approximation
Example 3.5.1: First order system with time delay
The VI below creates a first order transfer function with gain 2, time constant 3 seconds and
time delay 4 seconds.

Front panel and block diagram of first_order_with_time_delay.vi.
End of Example
3.6 PID controllers
Several verions of PID controls are available as transfer functions:
PID Academic:
PID Parallel:
PID Serial:
Example 3.6.1: PID controller
The VI shown below shows how to create and display an PID Academic controller (which is a
standard parallel PID controller). (The derivative time is set to zero, so the controller is actually
a PI controller.)

Front panel and block diagram of pid_controllers.vi.
End of Example
3.7 Writing models to file. Reading models from file
Models can be written to a file, and later read from that file, using the CD Write Model to File
and CD Read Model from File functions, respectively.
Example 3.7.1: Writing a transfer function model to a file
The VI shown below shows how to write a transfer function model to a file.
Front panel and block diagram of file_write_model.vi.
When the CD Write Model to File function is executed the usual Save File dialog window
appears. (If you have wired a file path to the File Path input of the function, this dialog window
is not opened.) You can give the file any name (the file extension does not matter).
End of Example
A model can be read from a model file using the CD Read Model from File function.
Example 3.7.2: Reading a transfer function model from a file
The VI shown below shows how to read a transfer function model from a file. (The model is the
same as in Example 3.7.1.)
Front panel and block diagram of file_read_model.vi.
When the CD Read Model from File function is executed a File dialog window appears. (If
you have wired a file path to the File Path input of the function, this dialog window is not
opened.)
End of Example

3.8 Getting information about a model
You can get various information about a model by using functions on the Create Model /
Model Information subpalette.
Example 3.8.1: Getting the numerator and denominator coeffiecient arrays of a transfer function
model
The VI shown below shows how to get the numerator and denominator coeffiecient arrays of a
transfer function model using the CD Get Data from Model function.
End of Example
Front panel and block diagram of get_model_data.vi.
3.9 Converting Control Design models to/from Simulation Module models
You can use models created in Control Design Toolkit in a Simulation diagram in the LabVIEW
Simulation Module. However, it is then necessary to first convert the model by using the CD
Convert Control Design to Simulation function.
Example 3.9.1: Converting a Control Design Toolkit model to a Simulation Module model
The VI shown below shows how to convert a transfer function model.
End of Example
Front panel and block diagram of convert_to_simmodule.vi.
4 Connecting models
The Model Interconnection palette contains several functions for connecting models. Series
connection and a feedback connection of transfer functions are described in the following.

4.1 Series connection
Example 4.1.1: Series connection of transfer function models
The VI shown below shows how to get the resulting transfer function of two transfer functions
connected in series using the CD Series function.
End of Example
4.2 Feedback connection
Front panel and block diagram of serial_connection.vi.
In models of feedback control systems, transfer functions are connected in a feedback loop. The
resulting transfer function can be calculated using the CD Feedback function. This functions
works for continuous-time models and for discrete-time models.
Example 4.2.1: Feedback connection of continuous-time transfer function models
The VI shown below shows how to get the resulting transfer function of two continuous-time
transfer functions connected in a feedback loop.

Front panel and block diagram of feedback_connection.vi.
End of Example
Note: For continuous-time models, the CD Feedback function ignores a time delay included in
any of the transfer functions in the feedback loop, that is, the resulting transfer function is
derived assuming the time delays are zero. To actually include the time delay(s), use the CD
Construct Special Model function with the option Delay (Pade Approx.) selected to create a
rational transfer function representing (and approximating) the time delay. Then include this
transfer function in the feedback loop using e.g. the CD Series function. This is demonstrated in
Example 9.1.
The following example shows how to connect discrete-time transfer functions including time
delays in a feedback loop. It is necessary to convert the time delay part of a discrete-time model
to poles at the origin using the CD Convert Delay to Poles at Origin function for the CD
Feedback function to produce the correct transfer function of the combined feedback loop. This
also applies to discrete-time transfer functions which have been derived by discretizing an
original continuous-time transfer function, that is, you have to use the CD Convert Delay to
Poles at Origin function for the CD Feedback function to produce the correct result.
Example 4.2.2: Feedback connection of discrete-time transfer function models including time
delay
In the VI shown below two discrete-time transfer functions are connected in a feedback loop.
One of the transfer functions, H 2
(z), contains a time delay of 2 samples, corresponding to 2
poles at the origin of the z-plane.

End of Example
Front panel and block diagram of feedback_connection_discrete.vi.
5 Calculating transfer functions from state-space models
The CD Convert to Transfer Function Model function converts continuous-time and discretetime
state-space models to transfer function models. The resulting transfer function model is
actually a MIMO (multiple input multiple output) transfer function, i.e. a transfer function
matrix. To get a particular SISO (single input single output) transfer function from this MIMO
transfer function you must apply the CD Get Data from Model function. This is illustrated in
the following example. This example is about a continuous-time model, but the same functions
are used for discrete-time models.
Example 5.1: Calculating transfer function from state-space model
The VI shown below shows how to get the SISO transfer function from input u to output y from
the state-space model

dx/dt = Ax + Bu
y = Cx + Du
where the system matrices are as shown on the VI front panel below.
Front panel and block diagram of convert_ss_to_tf.vi.
Note that the indexing of the rows and the columns start with indices 0, i.e. the first row has
index 0, and the first column has index 0.
End of Example
6 Discretizing continuous-time models
The following example illustrates how to discretize a continuous-time transfer function using
the CD Convert Continuous to Discrete function. The same function can be used to discretize
state-space models. Converting a model the opposite way - from discrete-time to continuoustime
- is done in a similar way using the CD Convert Discrete to Continuous function.
Example 6.1: Discretizing a continuous-time transfer function
The VI shown below shows how to do the discretization using the ZOH method (zero order
hold) with sampling time 0.2s. The original transfer function contains a time delay of 1 second.
This time delay is represented in the discrete-time transfer function by the factor z -5 (since
5*0.2s = 1s).

Front panel and block diagram of convert_cont_to_discrete.vi.
End of Example
7 Simulation (time responses)
The Time Response palette contains several simulation functions for simulating step response,
impulse response, arbitrary input response, and initial state response. The following example
shows how to simulate the step response.
The simulations are run as a "batch" simulation, being completed as fast as the PC allows. If
you want a real-time simulation, i.e. the simulation develops along a real time axis, you can use
LabVIEW Simulation Module. Models created in the Control Design Toolkit can be used in the
Simulation Module by using the models conversion functions demonstrated in Chapter 3.10.
Example 7.1: Simulation of the step response of a continuous-time transfer function
The VI shown below simulates the step response of the following transfer function:
H(s) = 3/(1+2s)

Front panel and block diagram of step_response_tf_model.vi.
CD Step Response simulates with a unity step (amplitude 1) at the model input.
The graph indicator can be created by right-clicking on the Step Response Graph output of the
CD Step Response function.
End of Example
8 Frequency response
The Frequency Response palette contains several functions for generating and plotting
frequency response data - for continuous-time as well as discrete-time models.
Example 8.1: Frequency response of a continuous-time transfer function

Front panel and block diagram of frequency_response.vi.
End of Example
9 An application: Analysis and simulation of control system
Example 9.1: Control system analysis and simulation
The VI shown below shows how to analyze and simulate a feedback control system. The block
diagram code is put inside a while loop with cycle time 100ms to make the program run
continuously. The controller is a PID Academic controller (which has parallel form) with the
following transfer function, H c
(s):

A Pade approximation is used to represent the time delay of the process because the CD
Feedback function works correctly only if there are only rational transfer functions in the
feedback loop.

Front panel and block diagram of controlsys_analysis.vi.
End of Example
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