Accelerating the pace of engineering and science

# Documentation

## Using the SISO Design Task in the Controls & Estimation Tools Manager

### Architecture

Use the Architecture tab for:

Architecture Pane on the SISO Design Task Node

#### Block Diagram Structure Modifications

Click Control Architecture to change the feedback structure and label signals and blocks. The following pane appears:

Select an architecture from the list of block configurations. These include compensator in the forward path, compensator in the feedback path, feedforward controller, and various multi-loop configurations. The window automatically updates to show the selected configuration.

Each configuration has associated Signs and Blocks and Signals panes. This figure shows the Signs pane.

The Blocks and Signals pane displays the generic identifier, for example F for the prefilter block, and a default name.

#### Loop Configuration

Click Loop Configuration to configure loops for multi-loop design by opening signals to remove the effects of other feedback loops.

To specify openings for a given open loop, select the loop in the combo box. Click Highlight Feedback Loop to see the effects of the selected openings.

For an example of how to use this window in design, see Multi-Loop Compensator Design.

#### Model Import

Click System Data on the Architecture pane to import models into your system. This opens the System Data dialog box, shown below.

You can import models for the plant (G), compensator (C), prefilter (F), and/or sensor (H). G or H or both are LTI models or row or column arrays of LTI models. If both G and H are arrays, their sizes must match.

To import a model:

1. Select a system in the System column and click Browse. The Model Import dialog box opens, as shown in the next figure.

2. Select a model from the Available Models list. You can import models from:

• The MATLAB® workspace

• A MAT-file

3. Click Import, then click Close. You can now see the model loaded into the system selected in the System Data dialog.

4. Click OK. The Graphical Tuning window is updated with the model you loaded.

Alternatively, you can import by entering a valid expression or variable (double, LTI object or row or column array of LTI objects) in the Data column in the System Data window.

#### Sample Times for Continuous/Discrete Conversions

Click Sample Time Conversion to convert the sample time of the system or switch between different sample times to design different compensators.

Choose from Zero-Order Hold, First-Order Hold, Impulse Variant, Tustin, Tustin w/Prewarping, and Matched Pole-Zero.

For a full description, see Continuous/Discrete Conversions Using the Sample Time Conversion Dialog Box.

#### Multimodel Configuration

The Multimodel Configuration button is enabled only when you import or open the SISO Design Tool GUI with a row or column arrays of LTI models for the plant G or sensor H or both. The LTI arrays model system variations in the plant and sensor. If both G and H are arrays, their sizes must match.

Click Multimodel Configuration to specify the nominal model and frequency grid for multimodel computations. This action opens the Multimodel Configuration Dialog window, as shown in the next figure.

### Compensator Editor

Use the Compensator Editor for adding or editing gains, poles, and zeros.

Compensator Editor Pane on the SISO Design Task Node

1. Enter the compensator gain in the text box in the top part of the pane.

2. Add or remove compensator poles and zeros by right-clicking in the Dynamics table.

3. Adjust pole and zero settings by entering values directly in the Edit Selected Dynamics group box.

### Graphical Tuning

Use the Graphical Tuning pane for

#### Configuring Design Plots for the Graphical Tuning Window

Click the Graphical Tuning tab to configure design plots displayed in the Graphical Tuning Window.

In the Graphical Tuning window, use design plots to graphically manipulate system response. These design plots are dynamically linked to the SISO Design Task. When you change the dynamics of your compensator in either the SISO Design Task or the Graphical Tuning window, the design updates in both places.

For open-loop responses, the available plot types are:

• Root locus

• Nichols

• Bode

For closed-loop responses, the available plot type is Bode.

For row or column arrays of LTI models, the design plots show the individual response of all models in the array by default. For more information, see Using the Graphical Tuning Window in the Getting Started Guide.

#### Selecting New Loops to Tune

Click Select New Open/Closed Loops to Tune to open a window for specifying new loops to tune.

Use the pull down menus to select the desired closed loop to tune by specifying the input, output, and blocks for tuning. Using the dialog box, you can select additional closed loops to tune.

Any loop you specify is displayed in the Summary of Available Loops to Tune in the Graphical Tuning pane. The list is also available in the Design plots configuration table of the same pane. You can use the latter for configuring design plots.

#### Refocusing on the Graphical Tuning Window

Click Show Design Plot to change the focus to the Graphical Tuning window.

### Analysis Plots

Use the Analysis Plots pane for

#### Customizing Loop Responses

The following sections describe the main components of the Analysis Plots pane.

Analysis Plots.  You can have up to six plots in one LTI Viewer. To add a plot, start by selecting "Plot 1" from the list of plots. Then select a new plot type from the pull down menu. You can choose any of the plots available in the LTI Viewer. Select "None" to remove a plot.

Contents of plots.  Once you have selected a plot type, you can include several open- and closed-loop transfer function responses for display. You can plot open-loop responses for each of the components of your system, including your compensator (C), plant (G), prefilter (F), or sensor (H). In addition, various closed loop and sensitivity response plots are available.

For row or column arrays of LTI models, the analysis plots show the response of the nominal model only by default. For more information, see Analysis Plots for Loop Responses in the Getting Started Guide.

Click Add Responses to open a window with three drop-down menus for selecting open and closed loop responses for various input and output nodes in the control architecture block diagram. This allows you to select additional responses for viewing. The Response table updates automatically to include the selected response.

#### Opening or Changing the Focus to the LTI Viewer

Click Show Analysis Plot to open a new LTI Viewer for SISO Design with the response plots that you selected. All the plots open in one instance of the LTI Viewer.

### Automated Tuning

Use the Automated Tuning pane to select a method for automatic tuning of your compensator design. Automated tuning methods help you design an initial compensator for a SISO loop that satisfies your design specifications.

You can choose among the following design methods:

• Optimization-Based Tuning — Optimize compensator parameters using design requirements implemented in graphical tuning and analysis plots

• PID Tuning — Tune PID controller parameters using the Robust response time tuning algorithm or classic tuning formulas

• Internal Model Control (IMC) Tuning — Obtain a full-order stabilizing feedback controller using the IMC design method

• LQG Synthesis — Design a full-order stabilizing feedback controller as a Linear-Quadratic-Gaussian (LQG) tracker

• Loop Shaping — Find a full-order stabilizing feedback controller with a desired open loop bandwidth or shape

After you select a design method, the pane updates to display the corresponding options.

 Note   If the particular design method you are using does not apply or fails, try selecting different tuning specifications or switch to a different design method.

#### Stability of an Effective Plant for Automated Tuning

Knowing the stability of the effective plant in your model may help you understand which automated tuning methods work for your model. Some of the automated tuning methods only apply to compensators whose open loops ($L=C\stackrel{\wedge }{P}$) have stable effective plants ($\stackrel{\wedge }{P}$).

An effective plant is the system controlled by the compensator you design and contains all elements of the open loop in your model other than this compensator. The following figure shows two examples of effective plants.

#### Generic Work Flow

1. Select an automated tuning algorithm from the Design method drop-down menu.

2. If you select Optimization-Based Tuning, stop here and see Optimization-Based Tuning.

3. Select a compensator from the drop-down menu.

4. Determine how you want the compensator to perform and set the tuning specifications.

5. Click Update Compensator and notice the changes in the associated design and analysis plots.

 Note:   If you encounter a disabled Update Compensator button, try selecting different tuning specifications (Step 4) or switch to a different tuning algorithm (Step 1). The disabled button means that the current method does not work for your model.

#### Optimization-Based Tuning

Optimization-based tuning is available only if you have Simulink® Design Optimization™ software installed. You can use this method to either:

• Directly tune response signals within Simulink models.

• Tune responses of LTI systems using a SISO Design Task.

See "Frequency Domain Response Optimization Example" in the Simulink Design Optimization documentation for more details.

#### PID Tuning

PID (proportional-integral-derivative) control is the most popular control technique used in modern industry.

To use automatic PID tuning, select PID Tuning from the Automated Tuning pane of the Control and Estimation Tools Manager. In most cases, the PID controllers resulting from PID tuning provide acceptable performance. Use the Analysis Plots to verify design results.

Types of PID Controllers.  SISO Design Tool provides automated tuning for the following PID controller types.

• P — Proportional-only control

• I — Integral-only control (available only for the Robust response time tuning method)

• PI — Proportional-integral control

• PD — Proportional-derivative control (available only for the Robust response time tuning method)

• PDF — Proportional-derivative control with a low-pass filter on the derivative term (available only for the Robust response time tuning method)

• PID — Proportional-integral-derivative control

• PIDF — Proportional-integral-derivative control with a low-pass filter on the derivative term

PID Tuning Methods.  SISO Design Tool provides a Robust response time algorithm for interactive tuning, as well as six well-known classical tuning methods.

Robust response time. This method computes PID parameters to robustly stabilize your system based on the bandwidth and phase margin that you specify. Using the robust response time method you can:

• Tune interactively, adjusting bandwidth and phase margin to achieve your desired balance between performance and robustness

• Tune any type of PID controller (P, I, PI, PD, PDF, PID, or PIDF)

• Tune all PID parameters, including the derivative filter

• Design for plants that are stable, unstable, or integrating

Classical design formulas. SISO Design Tool includes the following well-known PID design formulas:

• Approximate MIGO frequency response — Closed-loop frequency-domain approximate M-constrained integral gain approximation (see [1], Section 7.5).

• Approximate MIGO step response — Open-loop time-domain approximate M-constrained integral gain approximation (see [1], Sections 7.3–7.4).

• Chien-Hrones-Reswick — Approximates the plant as a first-order model with a time delay and computes PID parameters using a Chien-Hrones-Reswick look-up table for zero overshoot and disturbance rejection (see [1], Section 6.2).

• Skogestad IMC — Approximates the plant as a first-order model with a time delay and computes PID parameters using Skogestad design rules (see [2]).

(This method is different from selecting Internal Model Control (IMC) Tuning as the full-order compensator tuning method).

• Ziegler-Nichols frequency response — Computes controller parameters from a Ziegler-Nichols lookup table, based on the ultimate gain and frequency of the system (see [1], Section 6.2).

• Ziegler-Nichols step response — Approximates the plant as a first-order model with a time delay that and computes PID parameters using the Ziegler-Nichols design method (see [1], Section 6.2).

The classical design formulas:

• Require a stable or integrating plant.

• Can design for P, PI, PID, or PID with derivative filter.

• Cannot tune the derivative filter. If you select PID with derivative filter, classical design formulas set the filter time constant to Td/10, where Td is the tuned derivative time.

Automated Tuning using the Robust Response Time Method.  To use the robust response time tuning method:

1. Select Robust response time from the Tuning method menu.

2. Choose a controller type by clicking the corresponding radio button. To include a first-order filter on the derivative action for PD or PID controller type, check the Design with first order derivative filter checkbox.

 Note:   If you are tuning a PID Controller block in a Simulink model and your block is type P, I, or PI, select the same controller type as the block. If your block is type PD or PID, select the corresponding button and check the Design with first order derivative filter checkbox.
3. Click Update Compensator to design a controller of the selected type.

By default, the SISO Design Tool automatically computes controller parameters for balanced performance and robustness.

4. Analyze the response using the analysis plots you select on the Analysis Plots pane.

5. To design a controller interactively, select Interactive (adjustable performance and robustness) from the Design mode menu. This selection activates the Bandwidth and Phase Margin sliders.

Adjust the bandwidth and phase margin to achieve your desired controller performance. For example:

• Increase the bandwidth for a more aggressive controller.

• Increase the phase margin to reduce overshoot.

You can adjust the bandwidth and phase margin values by:

• Moving the sliders

• Entering values in the text field

• Incrementally adjusting the values in the text field using the up and down arrows.

To increase or decrease the bandwidth by a factor of 10, click the right or left double arrows, respectively.

 Note:   Click the Reset bandwidth and phase margin button at any time to return the sliders to the bandwidth and phase margin of the default compensator design for your plant.
6. Click Update Compensator again. SISO Design Tool computes new controller parameters for the specified target bandwidth and phase margin. Repeat steps 4-6 as necessary to achieve your desired performance.

 Note:   SISO Design Tool displays the tuned compensator in zpk form in the Compensator section of the Automated Tuning Pane. To convert the compensator to parallel or standard PID form, export the compensator to the MATLAB workspace, as described in SISO Design Task Node Menu Bar. Then use the pid or pidstd commands to convert the exported compensator to parallel or standard PID parameters, respectively.

Automated Tuning using Classical Design Formulas.  To use one of the classical design formulas:

1. Select Classical design formulas from the Tuning method menu.

2. Select a controller type (P, PI, PID, or PID with derivative filter) by clicking the corresponding radio button.

 Note:   If you are tuning a PID Controller block in a Simulink model and your block is type P or PI, select the same controller type as the block. If your block is type PID, select PID with derivative filter. If your block is type PD or I, use the Robust response time tuning method.
3. Select the classical design formula you want to use from the Formula menu.

4. Click the Update Compensator button to design a controller using the selected formula.

 Note:   SISO Design Tool displays the tuned compensator in zpk form in the Compensator section of the Automated Tuning Pane. To convert the compensator to parallel or standard PID form, export the compensator to the MATLAB workspace, as described in SISO Design Task Node Menu Bar. Then use the pid or pidstd commands to convert the exported compensator to parallel or standard PID parameters, respectively.

References.

[1] Åström, K. J. and Hägglund, T. Advanced PID Control, Research Triangle Park, NC: Instrumentation, Systems, and Automation Society, 2006.

[2] Skogestad, S., "Simple analytic rules for model reduction and PID controller tuning." Journal of Process Control, Vol. 13, No. 4, 2003, pp. 291–309.

#### Internal Model Control (IMC) Tuning

IMC design generates a full-order feedback controller that guarantees closed-loop stability when there is no model error. It also contains an integrator, which guarantees zero steady-state offset for plants without a free differentiator. You can use this tuning method for both stable and unstable plants.

To design an IMC controller:

1. Specify a value in the Dominant closed-loop time constant field. The initial value is set as 5% of the open-loop settling time. In general, increasing this value slows down the closed system and makes it more robust.

2. Specify a value in the Desired controller order field using the slider. After you obtain a full-order feedback controller, you can try to reduce its order. You may lose performance and closed-loop stability if you reduce the order.

3. Click Update Compensator.

#### LQG Synthesis

LQG tracker design generates a full-order feedback controller that guarantees closed-loop stability. It also contains an integrator, which guarantees zero steady-state error for plants without a free differentiator.

To design an LQG controller:

1. Specify your preference for controller response using the Controller response slider.

• Move the slider to the left for aggressive control response.

This means that large overshoot is more heavily penalized so that the controller acts more aggressively. If you believe your model is accurate and that the manipulated variable has a large enough range, an aggressive controller is more desirable.

• Move the slider to the right for robust control response.

2. Specify your estimation of the level of measurement noise using the Measurement noise slider.

• Move the slider to the left for small measurement noise.

This means that you expect low noise from the process output measurement. Because this measurement is used by the Kalman estimator, process disturbances are picked up more accurately by the estimated states. In this case, the controller is freer from robustness considerations.

• Move the slider to the right for large measurement noise. This results in a controller that is more robust to measurement noise.

3. Specify your preference for controller order using the Desired controller order slider.

4. Click Update Compensator.

#### Loop Shaping

Loop shaping generates a stabilizing feedback controller to match as closely as possible to a desired loop shape. You can specify this loop shape as a bandwidth or an open loop frequency response. If you have Robust Control Toolbox™ software installed, you can use loop shaping for SISO systems. For more information see the section on H-Infinity Loop Shaping in the Robust Control Toolbox User's Guide.

To design a controller using loop shaping:

1. Select a tuning preference by clicking one of these option buttons:

• Target bandwidth — Allows you to specify a target loop shape bandwidth (${\omega }_{b}$). This results in a loop shape of your specified bandwidth over an integrator ($\frac{{\omega }_{b}}{s}$).

• Target loop shape — Allows you to specify the target open loop shape in one of the following representations: state-space, zero-pole-gain, or transfer functions.

2. Set the tuning options available for your selected tuning preference as follows:

• If you chose Target bandwidth, specify the desired Target open-loop bandwidth in the editable box.

• If you chose Target loop shape, do the following:

• Enter the desired Target open-loop shape (LTI).

This can be a state-space representation, a zero-pole-gain representation, or a transfer function.

• Enter the desired Frequency range for loop shaping [wmin,wmax].

3. Specify your preference for controller order using the Desired controller order slider.

4. Click Update Compensator.