2- Constructing and Connecting Dynamic System

OUTLINES

1. Constructing Dynamic System Models

• Transfer Function Models• Zero-Pole-Gain Models• State-Space Models

2. Connecting

• Models in Series• Models in Parallel• Placing Models in a Closed-Loop Configuration

Model Types

1. Linear versus Nonlinear Models2. Time-Variant versus Time-Invariant Models3. Continuous versus Discrete Models

The Control Design and Simulation Module supports :

• Both linear and nonlinear models. • Time-invariant models • Continuous and discrete models.

Model Forms

Systems described as:

– Single-input single-output (SISO) systems.– Single-input multiple-output (SIMO) systems.– Multiple-input single-output (MISO) systems.– Multiple-input multiple-output (MIMO)

systems.

Model Forms

• The first step in working with the Control Design toolkit is inputting a model to work with. The Control Design Toolkit can work with

• Transfer Function (TF)• State Space (SS)• Zero Pole Gain (ZPK) models.

Constructing Dynamic System Models

Use the CD Construct Transfer Function Model VI to create continuous SISO, SIMO, MISO, and MIMO system models in transfer function form.

Constructing Transfer Function.vi

Sampling Time (s) defines whether the model represents a continuous-time system or a discrete-time system.

If the model represents a continuous-time system. Sampling Time (s) must equal zero.

If the model represents a discrete-time system, Sampling Time (s) must be greater than zero and equal to the sampling rate, in seconds, of the discrete system.

• Numerator contains the constant coefficients, in ascending order. The coefficients take the following form: b0 + b1s + ... +bmsm.

• Denominator contains the constant coefficients, in ascending order. The coefficients take the following form: a0 + a1s + ... +ansn.

• Delay is the transport time delay that might exist in the system.

Constructing Transfer Function.vi

SISO Transfer Function Model

SISO Transfer Function ModelUse the CD Draw Transfer Function Equation VI to displays the transferfunction equation of the model.

Display format specifies the format in which this VI displays the equation.

Output (row) specifies the index number of the output row from which to draw the transfer function matrix. The index is zero-based. The default is –1, which draws all outputs.

Input (column) specifies the index number of the input column from which to draw the transfer function matrix. The index is zero-based. The default is –1, which draws all inputs.

SISO Symbolic Transfer Function Model

SISO Transfer Function Model

• If you know the coefficients of the discrete transfer function model, you can enter in the appropriate values for Numerator and Denominator and set the Sampling Time (s) to a value greater than zero. Figure shows this process using a sampling

time of 10 µs.

SISO Transfer Function Model• If you do not know the coefficients of the discrete transfer function model, you

must use the CD Convert Continuous to Discrete VI for the conversion. Set the Sampling Time (s) parameter of this VI to a value greater than zero.

Converting from a continuous model to a discrete model results in thefollowing equation:

SISO Transfer Function ModelConstructing Models Textually• MathScript allows models to be created using m-file

syntax.

SIMO, MISO, and MIMO Transfer Function Models

You can define the transfer function of this MIMO system by using the following transfer function matrix H, where each element represents a SISO transfer function.

Suppose the following equations define the SISO transfer functions between each input-output pair.H11 & H21 has one input U1 and two outputs Y1 & Y2

H11 & H12 has two inputs U1 and U2 and one output Y1

SIMO Transfer Function Model

MISO Transfer Function Model

MIMO Transfer Function Model

Constructing Zero-Pole-Gain Models

Constructing Zero-Pole-Gain Models

• Use the CD Construct Zero-Pole-Gain Model VI to create this continuous zero-pole-gain model.

• You create SIMO, MISO, and MIMO zero-pole-gain models the same way you create SIMO, MISO, and MIMO transfer function models.

• You create symbolic zero-pole-gain models the same way you create symbolic transfer function models.

Constructing State-Space Models

Constructing State-Space Models• Using the RLC Circuit Example, the following equations

define a continuous state-space model.

• The following matrices define a state-space model where R = 20 Ω, L = 50 mH,and C = 10 µF.

Constructing State-Space Models

• Use the CD Construct State-Space Model VIto create this continuous state-space model.

• You create SIMO, MISO, and MIMO state-space models the same way you create SIMO, MISO, and MIMO transfer function models.

Model FormsControl Design and Simulation Module represents dynamic system models in the following three forms: transfer function, zero-pole-gain, and state-space.

Connecting Models

1. Connecting Models in Series

2. Appending Models

3. Connecting Models in Parallel

4. Placing Models in a Closed-Loop Configuration

Connecting Models in Series

Connecting Models in SeriesCD Series VI

Output Model 1 specifies the output number of the first model that is connected to an input of the second model as specified by Input Model 2. Connections uses the index number of the input and output to identify the input-output pair. The indexes are zero-based.

Input Model 2 is the input number of the second model that is connected to the output of the first model as specified by Output Model 1. Connections uses the index number of the input and output to identify the input-output pair. The indexes are zero-based.

Connections specifies which outputs of the first model this VI connects to which inputs of the second model.

Connecting Models in Series•Connecting SISO Systems in Series

3

1 2

2

1

3

Connecting Models in Series• Connecting SISO Systems in Series

• Consider a valve that controls the flow rate of water into a tank.

21

3

1

2

3

Connecting Models in Series•Creating SIMO Systems in Series

SELECT OUTPUT 0 FROM Model 1

AND INPUT 0 FROM Model 2

Select row indexcol index

Model 1

Model 2

The Output of CD Series.vi

This means you have one input and two outputs1

1

2

2

3

3

Connecting Models in Series• Creating SIMO Systems in Series• For example, adding another valve and tank to the example in the

Connecting SISO Systems in Series section of this chapter results in a SIMO system that divides the flow rate between two different tanks.

Connecting Models in Series• Steps:

1

2

3

4

5

6

1

23

4

5

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Connecting Models in Series• Creating MISO Systems in Series

This means you have two input and one outputs1 2

2

1

3

3

Connecting Models in Series• Connecting MIMO Systems in Series

MIMO 1

MIMO 2

1

2

MIMO 1 MIMO 21 2

3

3

Appending Models

Append Model 2 to Model 1 in a vertical vector

Connecting Models in Series (MathScript)

• Use MathScript commands to interconnect models

Connecting Models in Parallel

Connecting Models in Parallel• A parallel connection creates a single model from two separate

systems that share common inputs. • You also can use a parallel connection to add or subtract outputs

of two subsystems and represent them as a single output.• Use the CD Parallel VI to connect systems in parallel.

SISO with positive connection in the output

SISO will execute parallel concatenation with the specified output sign

SISO Vertical Concatenation

SISO Horizontal Concatenation

Parallel Connection in SIMO

Parallel Connection in SIMO

Parallel Connection in MISO

Parallel Connection in MISO

Parallel Connection in MIMO

Parallel Connection in MIMO

Placing Models in a Closed-Loop Configuration

Use the CD Feedback VI to place one or two models in a closed-loop configuration.

Closed-Loop Models

Feedback Sign specifies the sign of all feedback connections. If Feedback Sign is positive (TRUE),

Model 1 This model represents the system in the forward loop path.

Model 2 This model represents the system in the feedback path

Unity Feedback Closed-Loop Configuration

SISO Feedback Closed-Loop Configuration

SIMO Feedback Closed-Loop

MISO Feedback Closed-Loop

MIMO Feedback Closed-Loop

MIMO Feedback Closed-Loop

Feedback connection of discrete-time transfer function models including time delay

Control Design Model Interconnection (Graphical)

• Connect models in series and parallel• Create feedback loops