Model Predictive Control Toolbox 3
Design and simulate model predictive controllers
Model Predictive Control Toolbox provides MATLAB functions, a graphical user interface (GUI), and Simulink blocks for designing and simulating model predictive controllers in MATLAB and Simulink. These controllers optimize the performance of multi-input/multi-output systems that are subject to input and output constraints. The toolbox lets you define an internal plant model used by the model predictive controller in three ways. You can estimate the model from experimental data (with System Identification Toolbox), obtain it from a linearized Simulink model, or specify it directly as a linear time invariant object, such as a transfer function, or a state space model. The plant model can include delays. You can implement the model predictive controller by generating C code (with Real-Time Workshop).
mo mv MPC ref MPC Controller
Feed rpm Pri gap set point Pri dil flow set point Sec. gap set point Pri Vibration Pri. consistency Sec. vibration Pri. motor load Sec. consistency Sec. motor load

Key features

Graphical user interface and MATLAB commands for designing and simulating model predictive controllers Ability to define an internal linear plant model from experimental data or linearized Simulink model Simulink blocks for designing and simulating model predictive controllers directly in Simulink Control of nonlinear plants using multiple model predictive controllers with bumpless control transfer Ability to handle time-varying constraints and weights, offdiagonal weights, and custom unmeasured disturbance models Ability to generate C code for application deployment (with Real-Time Workshop)
Set points [0 0.8.5 0.3 6]
Nominal fiber water filling factor Fb0 Fbh0 Nominal inlet slurry density
Sec. dilution set point Chip fiber density Chip mixture density
Thermomechanical Pulping Refining Line
Pri. motor load Sec. consistency Sec. motor load
Using one of the two blocks available in Model Predictive Control Toolbox to design and simulate a controller directly in Simulink.
Accelerating the pace of engineering and science
Specifying the inputs and outputs to a controller and showing controller structure in one view in the Control and Estimation Tools Manager.
Plant Model Importer for bringing a model into the toolbox either from the MATLAB workspace or a MAT-file.
Working with Model Predictive Control Toolbox
Model Predictive Control Toolbox uses the Control and Estimation Tools Manager, a GUI that organizes your controller development into projects, enabling you to manage the design and evaluation of multiple controllers. The Control and Estimation Tools Manager simplifies the tasks of importing plant models and previously designed controllers and defining plant inputs and outputs, their units, and their nominal values. It shows your controller structure in one view by indicating the number of set points, manipulated variables, disturbances, and measured and unmeasured outputs. With the Control and Estimation Tools Manager, you can: efine internal plant models used in calcuD lating future control actions Design a model predictive controller imulate the closed-loop behavior of the S controller with linear models

Defining Internal Plant Models
Model predictive controllers base their control actions on an internal plant model of the process. The internal model lets the controller forecast future process behavior and respect output constraints. The ability to model process interactions makes model predictive control easier to maintain and often better performing than multiple proportionalintegral-derivative (PID) control loops, which require individual tuning and other techniques to reduce loop coupling. Model Predictive Control Toolbox uses linear time invariant (LTI) models, enabling you to use transfer function model structures common to all MathWorks control system design products. You can import multiple LTI models into the toolbox from the MATLAB workspace or a MAT-file. The toolbox also lets you directly import multiple models estimated in System Identification Toolbox. Using Simulink Control Design and Simulink, you can extract a linearized form of the Simulink model that is automatically imported as the internal plant model of the controller.

Designing Controllers

The toolbox lets you design controllers in MATLAB or in Simulink.
Designing Controllers in MATLAB
You can design multiple controllers and use simulation to determine the optimal design. For each controller design, you can select a plant model and specify the following controller parameters: Prediction and control horizons onstraints on the manipulated and C output variables eighting factors on input and output W variables odels for measurement noise and for M unmeasured input and output disturbances The toolbox supports time-varying constraints and weights, off-diagonal weights, and custom unmeasured disturbance models.
Designing Controllers in Simulink
Model Predictive Control Toolbox, when used with Simulink Control Design, can generate a controller directly in a Simulink model. Using an MPC block and the appro-
Setting constraints on manipulated and output variables with the Control and Estimation Tools Manager.
priately connected block inputs and outputs, Simulink Control Design can extract a linearized plant model and generate a controller. Model Predictive Control Toolbox uses the same GUI to specify the controller parameters in Simulink as to design a controller in MATLAB. You can use the Multiple MPC Controllers block for controlling a nonlinear Simulink model over a wide range of operating conditions. With this block you can design a model predictive controller for each operating point and switch between model predictive controllers at run time. The Multiple MPC Controllers block ensures bumpless control transfer from one model predictive controller to another. You can create linear plant models for controller design at each operating point either by linearizing a Simulink model with Simulink Control Design or by specifying the plant model directly.

10 Feed Concentration 298.15 Feed Temperature
CAi Ti Coolant temp T and C
product temperature product concentration Measurements 1

Chemical Reactor

switch MV mv Multiple mo MPC ref Multiple MPC Controllers
CSTR_Setpoints Set points
Simulating Closed-Loop Behavior
You can simulate your controller in MATLAB or Simulink to evaluate its performance.
Multiple MPC Controllers block (red) for controlling nonlinear models over a wide operating range using multiple model predictive controllers with bumpless control transfer. With this block you can design a model predictive controller for each operating point and switch between model predictive controllers at run time.

Simulating in MATLAB

You can use MATLAB functions or the Control and Estimation Tools Manager to run closed-loop simulations of your model predictive controller against linear plant
w w w. m a t h w o r k s. c o m
Configuring and running a simulation to test a controller using the Control and Estimation Tools Manager.
models. The Control and Estimation Tools Manager lets you set up multiple simulation scenarios. For each scenario you can input controller set points and unmeasured disturbances from the following signal profiles: Constant Step Pulse Ramp Sine Gaussian You can compare controller and plant model configurations to judge the effects of model mismatch and different weighting factors on constraints and variables. Constraints can be disabled to evaluate the characteristics of the closed-loop dynamics, such as stability and damping.
or rapid prototyping. (For a list of supported targets, see the section Using Model Predictive Control Toolbox with Real-Time Workshop in the product documentation.) Using OPC Toolbox, you can connect a controller operating in MATLAB directly to an OPC-compliant system.

Required Products

MATLAB Simulink (for using toolbox blocks) Control System Toolbox
Rapid prototyping of a model predictive controller on PC-compatible hardware using Real-Time Workshop and xPC Target.

Related Products

OPC Toolbox Read, write, and log data from OPC servers Real-Time Workshop Generate C code from Simulink models and MATLAB code Simulink Control Design Design and analyze control systems in Simulink System Identification Toolbox Create linear and nonlinear dynamic models from measured input-output data

Learn More

www.mathworks.com/products/mpc

Resources

visit www.mathworks.com Technical Support www.mathworks.com/support Online User Community www.mathworks.com/matlabcentral Demos www.mathworks.com/demos Training Services www.mathworks.com/training Third-Party Products and Services www.mathworks.com/connections Worldwide CONTACTS www.mathworks.com/contact e-mail info@mathworks.com

Simulating in Simulink

You can use Simulink blocks provided with Model Predictive Control Toolbox to run closed-loop simulation of your model predictive controller against a nonlinear Simulink model.
Deploying Model Predictive Controllers
The toolbox provides two ways to deploy a controller in an application. You can use Real-Time Workshop to generate C code from Simulink blocks provided with Model Predictive Control Toolbox and deploy the code to a target system for implementation
Platform and System Requirements
For platform and system requirements, visit www.mathworks.com/products/mpc
2008 The MathWorks, Inc. MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders.

8061V04 10/08

4-14 4-14 4-19 4-28 4-32

Examples
Getting Started. Automated Tuning of Simulink PID Controller Block. A-2 A-2
Simulink Control Design lets you design and analyze control systems modeled in Simulink. You can automatically tune the gains of the Simulink PID Controller block to meet performance requirements. With this product you can also non-intrusively find trim points and compute exact linearizations of Simulink models at various operating conditions. Simulink Control Design provides tools for computing simulation-based frequency responses without modifying your model. A graphical user interface lets you design and analyze arbitrary control structures modeled in Simulink, such as cascaded, pre-filter, regulation, and multi-loop architectures. Learn More
What Is a Steady-State Operating Point? on page 2-2 Steady-State Operating Points (Trimming) From Specifications on page 2-3 magball Simulink Model on page 2-9
What Is a Steady-State Operating Point?
A steady-state operating point of the model, also called equilibrium or trim condition, includes state variables that do not change with time. A model might have several steady-state operating points. For example, a hanging pendulum has two steady-state operating points. A stable steady-state operating point occurs when a pendulum hangs straight down. That is, the pendulum position does not change with time. When the pendulum position deviates slightly, the pendulum always returns to equilibrium; small changes in the operating point do not cause the system to leave the region of good approximation around the equilibrium value. An unstable steady-state operating point occurs when a pendulum points upward. As long as the pendulum points exactly upward, it remains in equilibrium. However, when the pendulum deviates slightly from this position, it swings downward and the operating point leaves the region around the equilibrium value. When using optimization search to compute operating points for a nonlinear system, your initial guesses for the states and input levels must be in the neighborhood of the desired operating point to ensure convergence. When linearizing a model with multiple steady-state operating points, it is important to have the right operating point. For example, linearizing a pendulum model around the stable steady-state operating point produces a stable linear model, whereas linearizing around the unstable steady-state operating point produces an unstable linear model. Examples and How To Steady-State Operating Points (Trimming) From Specifications on page 2-3 Steady-State Operating Points From Simulation More About Steady-State Operating Point (Trimming) Operating Point Object Includes a Subset of Simulink Model States
Steady-State Operating Points (Trimming) From Specifications

This example shows how to compute a steady-state operating point, or equilibrium operating point, by specifying known (fixed) equilibrium states and minimum state values. Code Alternative Use findop to find operating point from specifications. For examples and additional information, see the findop reference page.

1 Open Simulink model.

sys = 'magball'; open_system(sys)
2 In the Simulink model window, select Tools > Control Design > Linear
Analysis. This action starts a new project in the Control and Estimation Tools Manager.
3 Select Operating Points and click the Compute Operating Points tab.
By default, all model states are specified to be at equilibrium (Steady State). The Inputs and Outputs tabs are empty because this model does not have root-level input and output ports, respectively.
4 In the States tab, select Known for the height state.
The height of the ball should match the reference signal height. This height value should remain fixed during the optimization.
5 Enter 0 for the minimum bound of the Current state.
6 Click Compute Operating Points.
This action uses numerical optimization to find the operating point that meets your specifications. The Computation Results tab shows that the optimization algorithm terminated successfully. The (Maximum Error) Block area shows the progress of reducing the error of a specific state or output during the optimization.
7 Click Operating Point to evaluate whether the resulting operating point
values meet the specifications. The Actual dx values are near zero (Desired dx), which indicates that the operating point meets the steady state specification. The Actual Value of the states falls within the Desired Value bounds.
8 (Optional) Select File > Generate MATLAB Code to automatically
generate a MATLAB script. This script contains all commands for the most recent Compute Operating Points operation in Control and Estimation Tools Manager. Tip Alternatively, click to generate code.
Related Examples Steady-State Operating Points (Trimming) From Specifications on page 2-3 Steady-State Operating Points From Simulation More About magball Simulink Model on page 2-9 Steady-State Operating Point (Trimming)

magball Simulink Model

The Simulink model magball includes the nonlinear Magnetic Ball Plant in a single-loop feedback system.
The Magnetic Ball Plant subsystem is shown in the following figure.
The Magnetic Ball Plant model represents an iron ball of mass M. This ball moves under the influence of the gravitational force, Mg, and an induced
i2 magnetic force,. The presence of the squared term in the induced h magnetic force results in a nonlinear plant.
The inductor in the electric circuit, shown in the following figure, causes the induced magnetic force. This circuit also includes a voltage source and a resistor.
The following table describes the variables, parameters, differential equations, states, inputs, and outputs of the Magnetic Ball Plant subsystem. Variables h is the height of the ball. i is the current. V is the voltage in the circuit. Parameters M is the mass of the ball. g is the gravitational acceleration. is a constant related to the magnetic force. L is the inductance of the coil. R is the resistance of the circuit. Differential equations The height of the ball, h, is described in the following equation:

d2 h dt2

The current in the circuit, i, is described in the following equation:

di = V iR dt

States

h dh/dt i

Inputs Outputs
Examples and How To Steady-State Operating Points (Trimming) From Specifications on page 2-3
Applications of Linearization on page 3-2 Open-Loop Response of Control System for Stability Margin Analysis on page 3-3 Bode Response of Simulink Model on page 3-7 watertank Simulink Model on page 3-12
Applications of Linearization
Linearization is useful in model analysis and control design applications. After you linearize a Simulink model at a specific operating point, you can use your linear model to: Compute the Bode response of the Simulink model. Evaluate loop stability margins by computing open-loop response. Obtain linear state-space, transfer-function, or zero-pole-gain representation of the combined Simulink model that contains only linear blocks. Analyze and compare plant response near different operating points. Design linear controller Classical control system analysis and design methodologies require linear, time-invariant models. Simulink Control Design automatically linearization the plant when you tune your compensator. See PID Control Design Using Robust-Response-Time Tuning Algorithm on page 4-19. Analyze closed-loop stability. Measure the size of resonances in frequency response by computing closed-loop linear model for control system. Generate controllers with reduced sensitivity to parameter variations and modeling errors (requires Robust Control Toolbox). Examples and How To Open-Loop Response of Control System for Stability Margin Analysis on page 3-3 Bode Response of Simulink Model on page 3-7 Steady-State Operating Points (Trimming) From Specifications on page 2-3 More About Linearizing Nonlinear Models

Open-Loop Response of Control System for Stability Margin Analysis
This example shows how to use Control and Estimation Tools Manager to analyze the open-loop response of a control system. You compute a linear model of the combined controller-plant system without the effects of the feedback signal. A Bode plot of the resulting linear model shows the open-loop response.
sys = 'watertank'; open_system(sys)
The Water-Tank System block represents the plant in this control system and contains all of the system nonlinearities.
3 In the Simulink model window, define the portion of the model to linearize:
Right-click the PID Controller block input signal (the output of the Sum block). Select Linearization Points > Input Point. Right-click the Water-Tank System output signal, and select Linearization Points > Output Point. Right-click the Water-Tank System output signal and select Linearization Points > Open Loop.
Note Do not open the loop by manually removing the feedback signal from the model. Removing the signal manually changes the model operating point.
4 In the Plot linear result in a list, select Bode response plot.

5 Click Linearize Model.

The Bode plot of the open-loop response appears in the LTI Viewer window.
6 Right-click the plot and select Characteristics > Minimum Stability

Margins.

The Bode plot displays the phase margin marker. Click the marker to show a data tip that contains the phase margin value.

7 Close Simulink model.

bdclose(sys);
Related Examples Bode Response of Simulink Model on page 3-7 Steady-State Operating Points (Trimming) From Specifications on page 2-3 More About Linearizing Nonlinear Models watertank Simulink Model on page 3-12
Bode Response of Simulink Model
This example shows how to use Control and Estimation Tools Manager to linearize a model at the operating point specified in the model. The model operating point consists of the model initial state values and input signals. Code Alternative Use linearize. For examples and additional information, see the linearize reference page.
The Water-Tank System block represents the plant in this control system and includes all of the system nonlinearities.

Right-click the PID Controller block output signal, which is the input to the plant. Select Linearization Points > Input Point. Right-click the Water-Tank System output signal, and select Linearization Points > Output Point. The linearization I/O markers appear in the model.

Input Point Output Point

4 Right-click the Water-Tank System output signal and select Linearization

Points > Open Loop.

This command removes the effects of the feedback signal on the linearization without changing the model operating point. The loop opening marker appears in the model.

Loop opening

The Analysis I/Os tab of the Control and Estimation Tools Manager updates to show linearization input and output points.
Note Do not open the loop by manually removing the feedback signal from the model. Removing the signal manually changes the operating point of the model.
5 Click the Operating Points tab.
The model operating point is already selected.
6 In the Plot linear result in a list, select Bode response plot.

7 Click Linearize Model.

The Bode plot of the linearized system appears in the LTI Viewer window. This Bode plot looks like a stable first-order response, as expected.

8 Close Simulink model.

Related Examples Open-Loop Response of Control System for Stability Margin Analysis on page 3-3 Steady-State Operating Points (Trimming) From Specifications on page 2-3 More About Linearizing Nonlinear Models watertank Simulink Model on page 3-12

watertank Simulink Model

The Simulink model watertank model includes the nonlinear Water-Tank System plant and a PI controller in a single-loop feedback system.
The Water-Tank System is shown in the following figure.
Water enters the tank from the top at a rate proportional to the voltage, V, applied to the pump. The water leaves through an opening in the tank base at a rate that is proportional to the square root of the water height, H, in the tank. The presence of the square root in the water flow rate results in a nonlinear plant.
The following table describes the variables, parameters, differential equations, states, inputs, and outputs of the Water-Tank System. Variables H is the height of water in the tank. Vol is the volume of water in the tank. V is the voltage applied to the pump. Parameters A is the cross-sectional area of the tank. b is a constant related to the flow rate into the tank. a is a constant related to the flow rate out of the tank.

Differential equation

d dH Vol = A = bV a H dt dt

States Inputs Outputs

Automated Tuning of Simulink PID Controller Block on page 4-2 Designing a PID Controller Using Robust-Response-Time Tuning Algorithm and Bode Graphical Design on page 4-14
Automated Tuning of Simulink PID Controller Block
In this section. Introduction on page 4-2 Introduction of the PID Tuner on page 4-2 Opening the Model on page 4-3 Design Overview on page 4-3 Opening the PID Tuner on page 4-3 Initial PID Design on page 4-4 Displaying PID Parameters on page 4-5 Adjusting PID Design in the PID Tuner on page 4-7 Completing the Design in the Extended Design Mode on page 4-8 Writing the Tuned Parameters to PID Controller Block on page 4-12 Completed Design on page 4-12
Introduction Introduction of the PID Tuner
PID Tuner provides a fast and widely applicable single-loop PID tuning method for the Simulink PID Controller blocks. With this method, you can tune PID parameters to achieve a robust design with the desired response time. A typical design workflow with the PID Tuner involves the following tasks: (1) Launch the PID Tuner. When launching, the software automatically computes a linear plant model from the Simulink model and designs an initial controller. (2) Tune the controller in the PID Tuner by manually adjusting design criteria in two design modes. The tuner computes PID parameters that robustly stabilize the system.
(3) Export the parameters of the designed controller back to the PID Controller block and verify controller performance in Simulink.

Opening the Model

Take a few moments to explore the model. Open the engine speed control model with PID Controller block
open_system('scdspeedctrlpidblock');

Design Overview

In this demo, you design a PI controller in an engine speed control loop. The goal of the design is to track the reference signal from a Simulink step block scdspeedctrlpidblock/Speed Reference. The design requirement are: Settling time under 5 seconds Zero steady-state error to the step reference input. In this example, you stabilize the feedback loop and achieve good reference tracking performance by designing the PI controller scdspeedctrl/PID Controller in the PID Tuner.

Opening the PID Tuner

To launch the PID Tuner, double-click the PID Controller block to open its block dialog. In the Main tab, click Tune.

Initial PID Design

When the PID Tuner launches, the software computes a linearized plant model seen by the controller. The software automatically identifies the plant input and output, and uses the current operating point for the linearization. The plant can have any order and can have time delays. The PID Tuner computes an initial PI controller to achieve a reasonable tradeoff between performance and robustness. By default, step reference tracking performance displays in the plot. The following figure shows the PID Tuner dialog with the initial design:
Displaying PID Parameters
Click the Show parameters arrow to view controller parameters P and I, and a set of performance and robustness measurements. In this example, the
initial PI controller design gives a settling time of 2 seconds, which meets the requirement. The following figure shows the parameter and performance tables:
Adjusting PID Design in the PID Tuner
The overshoot of the reference tracking response is about 8 percent. Because the response performance is limited in many systems with time delays, you need to slow down response speed to reduce overshoot. Move the response time slider to the left to increase the closed loop response time. Notice that when you adjust response time, the response plot and the controller parameters and performance measurements update. The following figure shows an adjusted PID design with an overshoot of zero and a settling time of 4 seconds. The designed controller effectively becomes an integral-only controller.
Completing the Design in the Extended Design Mode
To reduce the overshoot while maintaining the settling time of 2 seconds, you must tradeoff between controller performance (measured by settling time)
and robustness (measured by overshoot). You can perform such a trade-off in the Extended design mode of the PID Tuner. To switch to the Extended design mode, select Extended in the Design Mode dropdown menu in the toolbar. The following figure shows the PID Tuner in the Extended design mode with the integral only controller designed in the previous section:
There are two sliders in the Extended design mode. You can adjust performance with the Bandwidth slider. Large bandwidth results in fast response. You can also adjust robustness with the Phase margin slider. Large phase margin results in small overshoot. Move around both sliders to achieve the settling time of 2 seconds and zero overshoot. One way to achieve this is

Bandwidth of 1.23 rad/sec Phase margin of 72 degree The following figure shows the PID Tuner with these settings:
Writing the Tuned Parameters to PID Controller Block
After you are happy with the controller performance on the linear plant model, you can test the design on the nonlinear model. To do this, click Apply in the PID Tuner. This action writes the parameters back to the PID Controller block in the Simulink model. The following figure shows the updated PID Controller block dialog:

Completed Design

The following figure shows the response of the closed-loop system:
The response shows that the new controller meets all the design requirements.
You can also use the SISO Compensator Design Tool to design the PID Controller block. When the PID Controller block belongs to a multi-loop design task. See the demo "Single Loop Feedback/Prefilter Compensator Design".
bdclose('scdspeedctrlpidblock')
Designing a PID Controller Using Robust-Response-Time Tuning Algorithm and Bode Graphical Design
In this section. About This Tutorial on page 4-14 PID Control Design Using Robust-Response-Time Tuning Algorithm on page 4-19 PID Control Design Using Bode Graphical Tuning on page 4-28 Closed-Loop Simulation of Simulink Model on page 4-32

About This Tutorial

Objectives on page 4-14 About the Model on page 4-14 Requirements for the Compensator Design on page 4-18 Overview of the Compensator Design Process on page 4-18

Objectives

In this tutorial, you learn how to use the Simulink Control Design GUI to design a PID controller for a single-loop feedback system that is operating at the operating conditions specified in the Simulink model. You accomplish the following tasks: Configure the model and GUI for compensator design. Design a PID compensator using the robust-response-time tuning algorithm and Bode graphical design. Simulate the closed-loop nonlinear model.

About the Model

watertank_comp_design Simulink Model on page 4-15 Water-Tank Subsystem on page 4-15

Controller Subsystem on page 4-18 watertank_comp_design Simulink Model. The watertank_comp_design model, shown in the following figure, contains the Water-Tank System plant and a simple proportional-integral-derivative (PID) controller, called Controller, in a single-loop feedback system.
To view the Water-Tank System and the Controller, double-click the corresponding subsystem in the watertank_comp_design model. For descriptions of these subsystems, see the following topics: Water-Tank Subsystem on page 4-15 Controller Subsystem on page 4-18 Water-Tank Subsystem. The Water-Tank subsystem of the watertank_comp_design model appears in the following figure.
This model represents the water-tank system depicted in the following figure.
The following table describes the variables, parameters, differential equations, states, inputs, and outputs of the water-tank system. Variables H is the height of water in the tank. Vol is the volume of water in the tank. V is the voltage applied to the pump. Parameters A is the cross-sectional area of the tank. b is a constant related to the flow rate into the tank. a is a constant related to the flow rate out of the tank. Differential equation

States Inputs

Outputs
Controller Subsystem. The Controller subsystem appears in the following figure.
This model contains a PID Controller block that controls the height of the water in the Water-Tank System.
Requirements for the Compensator Design
The PID controller you design in this tutorial must control the Water-Tank System response such that the: Overshoot is less than 5%. Rise time is less than 5 seconds.
Overview of the Compensator Design Process
The process for designing a compensator for the Water-Tank System in this tutorial includes the following tasks: Configuring the model and GUI for the design. Designing a PID compensator using the robust response time tuning algorithm. Tuning the compensator using the Bode design technique.
Simulating the closed-loop Simulink model with the compensator design to analyze the system dynamics. Simulink Control Design tools work only with linear plant models. Because the Water-Tank System is nonlinear, Simulink Control Design automatically linearizes the model about the model operating point, by default. The linearization provides a valid approximation of the nonlinear model in a region around the operating point. For more information about linearization and how the operating point impacts linearization results, see Chapter 3, Linearization.
PID Control Design Using Robust-Response-Time Tuning Algorithm
In this portion of the tutorial, you design a compensator using the automated PID robust-response-time tuning algorithm. This tuning method tunes the PID gains to maximize bandwidth and optimize phase margin.

1 Open the watertank_comp_design model by typing the model name in
the MATLAB Command Window:

watertank_comp_design

The command opens the watertank_comp_design model in Simulink, as shown in the following figure.
2 In the watertank_comp_design model window, select Tools > Control
Design > Compensator Design. This action opens the Control and Estimation Tools Manager with the Simulink Compensator Design Task node selected.
3 Select the PID Controller block as the block to tune. a In the Tunable Blocks tab, click Select Blocks.
This action opens the Select Blocks to Tune window.
b In the watertank_comp_design tree, select the Controller subsystem. c Select the Tune? check box for PID Controller.
d Click OK. 4 Define the closed-loop systems for which you want to analyze the response.
The input and output points of the closed-loop path are already defined in the watertank_comp_design model. If you needed to add or define them, you would use the following steps:
a In the watertank_comp_design model, right-click the output of the
Desired Water Level block, and select Linearization Points > Input Point. This action displays the symbol on the signal line. This symbol indicates the input of the closed-loop path.
b Right-click the output signal from the Water-Tank System, and select
Linearization Points > Output Point. This action displays the symbol on the signal line. This symbol indicates the output of the closed-loop path.
The Simulink model now resembles the following figure.
5 In the Control and Estimation Tools Manager, click Tune Blocks to open
the Design Configuration Wizard. Click Next.
6 Step 1 of the Design Configuration Wizard prompts you to select the design
plots you will use to tune the controller. Accept the default settings and click Next.
7 In Step 2 of the Design Configuration Wizard, specify the type of plot for

analyzing the response.

a In the Analysis Plots area, select Step for the Plot Type corresponding

to Plot 1.

b In the Plots section of the Contents in Plots pane, select 1 for Closed
Loop from Desired Water Level to Water-Tank System.

8 Click Finish.

The software performs the following actions: Linearizes the Simulink model about the operating point specified in the model. Creates a SISO Design Task node under the Simulink Compensator Design Task node. Opens the following plot windows: LTI Viewer for SISO Design Task window, which shows the closed-loop Step Response plot of the linearized model SISO Design for SISO Design Task window, which is empty You do not use in this window in this section of the tutorial. Keep this window open for the next section of the tutorial. The Control and Estimation Tools manager resembles the following figure.

The Step Response plot shows an overshoot that does not meet the overshoot design requirement of less than 5%.
9 In the Automated Tuning tab of the SISO Design Task node in the
Control and Estimation Tools Manager, select PID Tuning as the Design method.
10 In the Specifications area, select the following options:
Controller type: PI Tuning algorithm: Robust response time
11 Click Update Compensator.
This action computes the PI values for the compensator using the robust response time tuning algorithm and updates the Step Response plot. Tip You can view the PI values in the Parameter tab of the Compensator Editor tab in the SISO Design Task node.
12 Evaluate whether the compensator design meets the design requirements
by analyzing the overshoot and the rise time, as follows:
a Right-click the Step Response plot and select the following options:
Characteristics > Peak Response Characteristics > Rise Time These actions add a plot marker to the plot for each characteristic, shown as blue dots.
b Left-click each blue dot to open the corresponding data marker.
The data markers show the following response characteristics:
The overshoot is 11.6%. The rise time is 82.2 seconds.
This system response with the PID compensator exceeds the maximum allowed overshoot of 5%. The rise time is much slower than the required rise time of 5 seconds. You decrease the rise time by increasing the gain of the compensator, as described in PID Control Design Using Bode Graphical Tuning on page 4-28. Tip You can also decrease the rise time by adjusting the loop bandwidth. First, select Interactive (adjustable performance and robustness) from the Design Mode menu. Then, move the Bandwidth slider to the right. Finally, click Update Compensator to design a new compensator for the new target bandwidth.
PID Control Design Using Bode Graphical Tuning
In this example, you decrease the rise time of the Water-Tank System response by increasing the compensator gain using Bode graphical tuning. Bode graphical tuning lets you design a compensator by manipulating Bode diagrams of the open-loop response. This process is also called loop shaping. You must have already designed an initial compensator using PID tuning, as described in PID Control Design Using Robust-Response-Time Tuning Algorithm on page 4-19. To design a compensator using Bode graphical tuning:
1 In the Control and Estimation Tools Manager, select the Graphical
Tuning tab of the SISO Design Task node.
2 In the Plot Type cell that corresponds to Plot 1, select Open-Loop Bode.

This action creates an Open-Loop Bode plot in the SISO Design for SISO Design Task window. This plot shows a Bode plot of the linearized model with the compensator designed using automated PID tuning.
3 In the SISO Design window, drag the Bode Magnitude line upward to
increase the gain. As you adjust the gain, view the affects on the closed-loop response in the Step Response plot.
By increasing the gain, you increase the bandwidth and speed up the response. One possible compensator design that meets the tutorial requirements has the following parameters: P = 5.0368 I = 0.11434 D=0
Tip You can view the parameter values corresponding to the gain adjustment you made in the Bode Magnitude plot in the Compensator Editor tab of the SISO Design Task. You can also adjust the parameter values in this tab.
4 Evaluate whether the compensator design meets the design requirements
a Right-click the Step Response plot and select the following options, if you
have not done so already: Characteristics > Peak Response Characteristics > Rise Time These actions add a plot marker to the plot for each characteristic, shown as blue dots.
The data markers show the following response characteristics: The overshoot is 0.437%. The rise time is 1.72 seconds.
This compensator design satisfies the design requirements of less than 5% overshoot and less than 5 second rise time.
Closed-Loop Simulation of Simulink Model
In this example, you simulate the nonlinear closed-loop Simulink model that includes a PID controller to determine how well the design meets the requirements. You must have already designed the compensator, as described in PID Control Design Using Bode Graphical Tuning on page 4-28.
1 In the Control and Estimation Tools Manager SISO Design Task node,
click Update Simulink Block Parameters. This action writes the compensator parameters into the PID Controller block of the Controller subsystem in the Simulink model.
Tip You can view the PID Controller block parameters in the Function Block Parameters Dialog box. To open this dialog box, double-click the PID Controller block.

2 In the Simulink model, double-click the Scope block to open the Scope block

window.

3 In the Simulink model, click
to simulate the model. Then, click

to autoscale the axis.

This action updates the Scope window with the response of the nonlinear model with the compensator design. This simulation shows that the rise time is less than 5 seconds and there is minimal overshoot. Thus, this compensator design meets the requirements of less than 5% overshoot and less than 5 second rise time.
Use this list to find examples in the documentation.

Getting Started

Chapter 2, Steady-State Operating Points Open-Loop Response of Control System for Stability Margin Analysis on page 3-3 Bode Response of Simulink Model on page 3-7 Chapter 4, PID Control Design
Automated Tuning of Simulink PID Controller Block on page 4-2

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