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R2012a Documentation → Control System Toolbox
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Contents

Index

• Getting Started

Key Features

What Is a Plant?

Linear Model Representations

SISO Example: The DC Motor

Building SISO Models

Constructing Discrete Time Systems

Adding Delays to Linear Models

LTI Objects

MIMO Example: State-Space Model of Jet Transport Aircraft

Constructing MIMO Transfer Functions

Accessing I/O Pairs in MIMO Systems

Arithmetic Operations for Interconnecting Models

Feedback Interconnections

Available Commands for Continuous/Discrete Conversion

Available Methods for Continuous/Discrete Conversion

Digitizing the Discrete DC Motor Model

Model Order Reduction Commands

Techniques for Reducing Model Order

Example: Gasifier Model

Plot Types Available in the LTI Viewer

Example: Time and Frequency Responses of the DC Motor

Right-Click Menus

Displaying Response Characteristics on a Plot

Changing Plot Type

Showing Multiple Response Types

Comparing Multiple Models

What is the Linear Simulation Tool?

Opening the Linear Simulation Tool

Working with the Linear Simulation Tool

Importing Input Signals

Example: Loading Inputs from a Microsoft Excel Spreadsheet

Example: Importing Inputs from the Workspace

Designing Input Signals

Specifying Initial Conditions

When to Use Functions for Time and Frequency Response

Time and Frequency Response Functions

Plotting MIMO Model Responses

Data Markers

Plotting and Comparing Multiple Systems

Creating Custom Plots

Choosing a PID Controller Design Tool

Designing PID Controllers with the PID Tuner GUI

PID Controller Design for Fast Reference Tracking

Designing PID Controllers at the Command Line

Tune Compensator Parameters Using the SISO Design Tool

Components of the SISO Tool

Design Options in the SISO Tool

Opening the SISO Design Tool

Using the SISO Design Task Node in the Control and Estimation Tools Manager

Importing Models into the SISO Design Tool

Feedback Structure

Analysis Plots for Loop Responses

Using the Graphical Tuning Window

Exporting the Compensator and Models

Storing and Retrieving Intermediate Designs

What Is Bode Diagram Design?

Bode Diagram Design for DC Motor

Adjusting the Compensator Gain

Adjusting the Bandwidth

Adding an Integrator

Adding a Lead Network

Moving Compensator Poles and Zeros

Adding a Notch Filter

Modifying a Prefilter

What Is Root Locus Design?

Root Locus Design for Electrohydraulic Servomechanism

Changing the Compensator Gain

Adding Poles and Zeros to the Compensator

Editing Compensator Pole and Zero Locations

Viewing Damping Ratios

What Is Nichols Plot Design?

Nichols Plot Design for DC Motor

Opening a Nichols Plot

Adjusting the Compensator Gain

Adding an Integrator

Adding a Lead Network

Supported Automated Tuning Methods

Loading and Displaying the DC Motor Example for Automated Tuning

Applying Automated PID Tuning

When to Use Multi-Loop Compensator Design

Workflow for Multi-Loop Compensator Design

Position Control of a DC Motor

Multiple Models Represent System Variations

Control Design Analysis Using the SISO Design Tool

Specifying a Nominal Model

Frequency Grid for Multimodel Computations

How to Analyze the Controller Design for Multiple Models

When to Use Functions for Compensator Design

Root Locus Design

Pole Placement

Linear-Quadratic-Gaussian (LQG) Design

Design an LQG Regulator

Design an LQG Servo Controller

Design an LQR Servo Controller in Simulink

• User's Guide

• Blocks

• Functions

• GUIs

• Numeric Linear Models

• Generalized Linear Models

• Data Extraction

• Conversions

• System Interconnections

• System Gain and Dynamics

• Time Domain Analysis

• Frequency Domain Analysis

• Model Simplification

• LQR/LQG Design

• Frequency Response Data Models

• Time Delays

• Model Dimensions and Characteristics

• Overloaded and Arithmetic Operators

• Matrix Equation Solvers

• Preferences

• Visualization of Model Dynamics and Responses

• Plot Customization

Examples

• Release Notes

lqr - Linear-Quadratic Regulator (LQR) design

Syntax

[K,S,e] = lqr(SYS,Q,R,N) [K,S,e] = LQR(A,B,Q,R,N)

Description

[K,S,e] = lqr(SYS,Q,R,N) calculates the optimal gain matrix K.

For a continuous time system, the state-feedback law u = –Kx minimizes the quadratic cost function

subject to the system dynamics

In addition to the state-feedback gain K, lqr returns the solution S of the associated Riccati equation

and the closed-loop eigenvalues e = eig(A-B*K). K is derived from S using

For a discrete-time state-space model, u[n] = –Kx[n] minimizes

subject to x[n + 1] = Ax[n] + Bu[n].

[K,S,e] = LQR(A,B,Q,R,N) is an equivalent syntax for continuous-time models with dynamics

In all cases, when you omit the matrix N, N is set to 0.

Tips

lqr supports descriptor models with nonsingular E. The output S of lqr is the solution of the Riccati equation for the equivalent explicit state-space model: