MATLAB and its Control Toolbox

1
MATLAB and its Control Toolbox
2
Outline

• MATLAB
• MATLAB and Toolboxes
• MATLAB and Control
• Control System Toolbox
• Simulink

3
MATLAB

• Aerospace and Defense
• Automotive
• Biotech, Medical, and Pharmaceutical
• Chemical and Petroleum
• Communications
• Computers and Office Equipment
• Education
• Electronics and Semiconductor
• Financial Services
• Industrial Equipment and Machinery
• Instrumentation
• Utilities and Energy

4
The MathWorks Product Family
5
MATLAB Toolboxes
MATLAB
Math and optimization
Signal Processing and communications
SimulinkProduct Family
..
Control System Design and Analysis
Toolboxes Optimization Symbolic Math Partial
Diff. Eq.
Toolboxes Signal Processing Communications
Filter Design Filter Design HDL Coder
Simulink Simulink Accelerator Simulink Report
Generator
Toolboxes Simulink Control Design Simulink
Response Simulink Parameter

6
MATLAB and Control
MATLAB-Toolboxes for Control
Identification
Linear Control
Nonlinear Control
Identification Toolbox Frequency-Domain ID
Toolbox Simulink
Control System Toolbox Simulink Mu Toolbox
Nonlinear Control Toolbox Fuzzy Toolbox Simulink

7
MATLAB and Control

• Control Design Process

8
MATLAB and Control

• Modeling Tools

9
MATLAB and Control

• Design and Analysis

10
Control System Toolbox

• Core Features
• Tools to manipulate LTI models
• Classical analysis and design
• Bode, Nyquist, Nichols diagrams
• Step and impulse response
• Gain/phase margins
• Root locus design
• Modern state-space techniques
• Pole placement
• LQG regulation

11
Control System Toolbox

• LTI Objects (Linear Time Invariant)
• 4 basic types of LTI models
• Transfer Function (TF)
• Zero-pole-gain model (ZPK)
• State-Space models (SS)
• Frequency response data model (FRD)
• Conversion between models
• Model properties (dynamics)

12
Control System Toolbox
Transfer Function
13
Control System Toolbox
Transfer Function

• Consider a linear time invariant (LTI)
  single-input/single-output system
• Applying Laplace Transform to both sides with
  zero initial conditions

14
Control System Toolbox
Transfer Function
gtgt num 4 3 gtgt den 1 6 5 gtgt sys
tf(num,den) Transfer function 4 s
3 ----------------- s2 6 s 5

• gtgt num,den
• tfdata(sys,'v')
• num
• 0 4 3
• den
• 1 6 5

15
Control System Toolbox
Zero-pole-gain model (ZPK)
16
Control System Toolbox
Zero-pole-gain model (ZPK)

• Consider a Linear time invariant (LTI)
  single-input/single-output system
• Applying Laplace Transform to both sides with
  zero initial conditions

17
Control System Toolbox
Zero-pole-gain model (ZPK)
gtgt ze,po,k zpkdata(sys1,'v') ze
-0.7500 po -1 -5 k 4
gtgt sys1 zpk(-0.75,-1 -5,4)
Zero/pole/gain 4 (s0.75) ----------- (s1)
(s5)
18
Control System Toolbox
State-Space Model (SS)
19
State-Space Models
Control System Toolbox

• Consider a Linear time invariant (LTI)
  single-input/single-output system
• State-space model for this system is

20
Control System Toolbox
State-Space Models

• gtgt sys ss(0 1 -5 -6,01,3,4,0)
• a
• x1 x2
• x1 0 1
• x2 -5 -6
• b
• u1
• x1 0
• x2 1

c x1 x2 y1 3 4 d
u1 y1 0
21
Control System Toolbox

• State Space Models
• rss, drss - Random stable state-space models.
• ss2ss - State coordinate transformation.
• canon - State-space canonical forms.
• ctrb - Controllability matrix.
• obsv - Observability matrix.
• gram - Controllability and observability
  gramians.
• ssbal - Diagonal balancing of state-space
  realizations.
• balreal - Gramian-based input/output balancing.
• modred - Model state reduction.
• minreal - Minimal realization and pole/zero
  cancellation.
• sminreal - Structurally minimal realization.

22

Conversion between different models
23
Model Dynamics

• pzmap Pole-zero map of LTI models.
• pole, eig - System poles
• zero - System (transmission) zeros.
• dcgain DC gain of LTI models.
• bandwidth - System bandwidth.
• iopzmap - Input/Output Pole-zero map.
• damp - Natural frequency and damping of system
• esort - Sort continuous poles by real part.
• dsort - Sort discrete poles by magnitude.
• covar - Covariance of response to white noise.

24
Time Response of Systems
Control System Toolbox

• Impulse Response (impulse)
• Step Response (step)
• General Time Response (lsim)
• Polynomial multiplication (conv)
• Polynomial division (deconv)
• Partial Fraction Expansion (residue)
• gensig - Generate input signal for lsim.

25
Time Response of Systems
Control System Toolbox

• The impulse response of a system is its output
  when the input is a unit impulse.
• The step response of a system is its output when
  the input is a unit step.
• The general response of a system to any input can
  be computed using the lsim command.

26
Control System Toolbox
Time Response of Systems

• Problem Given the LTI system
• Plot the following responses for
• The impulse response using the impulse command.
• The step response using the step command.
• The response to the input
  calculated using both the lsim commands

27
Control System Toolbox
Time Response of Systems
28
Frequency Domain Analysis and Design

• Root locus analysis
• Frequency response plots
• Bode
• Phase Margin
• Gain Margin
• Nyquist

29
Frequency Domain Analysis and Design
Root Locus

• The root locus is a plot in the s-plane of all
  possible locations of the poles of a closed-loop
  system, as one parameter, usually the gain, is
  varied from 0 to ?.
• By examining that plot, the designer can make
  choices of values of the controllers parameters,
  and can infer the performance of the controlled
  closed-loop system.

30
Frequency Domain Analysis and Design
Root Locus

• Plot the root locus of the following system

31
Frequency Domain Analysis and Design
Root Locus
gtgt rlocus(tf(1 8, conv(conv(1 0,1 2),1 8
32)))
32
Frequency Response Bode and Nyquist Plots

• Typically, the analysis and design of a control
  system requires an examination of its frequency
  response over a range of frequencies of interest.
• The MATLAB Control System Toolbox provides
  functions to generate two of the most common
  frequency response plots Bode Plot (bode
  command) and Nyquist Plot (nyquist command).

33
Control System Toolbox
Frequency Response Bode Plot

• Problem
• Given the LTI system
• Draw the Bode diagram for 100 values of
  frequency in the interval .

34
Control System Toolbox
Frequency Response Bode Plot

• gtgtbode(tf(1, 1 1 0), logspace(-1,1,100))

35
Control System Toolbox
Frequency Response Nyquist Plot

• The loop gain Transfer function G(s)
• The gain margin is defined as the multiplicative
  amount that the magnitude of G(s) can be
  increased before the closed loop system goes
  unstable
• Phase margin is defined as the amount of
  additional phase lag that can be associated with
  G(s) before the closed-loop system goes unstable

36

Control System Toolbox
Frequency Response Nyquist Plot
Problem Given the LTI system Draw the bode
and nyquist plots for 100 values of frequencies
in the interval . In addition,
find the gain and phase margins.
37
Control System Toolbox
Frequency Response Nyquist Plot

• wlogspace(-4,3,100)
• systf(1280 640, 1 24.2 1604.81 320.24 16)
• bode(sys,w)
• Gm,Pm,Wcg,Wcpmargin(sys)
• Nyquist plot
• figure
• nyquist(sys,w)

38
Control System Toolbox
Frequency Response Nyquist Plot
The values of gain and phase margin and
corresponding frequencies are Gm 29.8637 Pm
72.8960 Wcg 39.9099 Wcp 0.9036
39
Control System Toolbox
Frequency Response Plotsbode - Bode diagrams of
the frequency response.bodemag - Bode magnitude
diagram only.sigma - Singular value frequency
plot.Nyquist - Nyquist plot.nichols - Nichols
plot.margin - Gain and phase margins.allmargin
- All crossover frequencies and related
gain/phase margins.freqresp - Frequency response
over a frequency grid.evalfr - Evaluate
frequency response at given frequency.interp -
Interpolates frequency response data.
40
Control System Toolbox

• Design Pole Placement
• place - MIMO pole placement.
• acker - SISO pole placement.
• estim - Form estimator given estimator gain.
• reg - Form regulator given state-feedback and
  estimator gains.

41
Control System Toolbox

• Design LQR/LQG design
• lqr, dlqr - Linear-quadratic (LQ) state-feedback
  regulator.
• lqry - LQ regulator with output weighting.
• lqrd - Discrete LQ regulator for continuous
  plant.
• kalman - Kalman estimator.
• kalmd - Discrete Kalman estimator for continuous
  plant.
• lqgreg - Form LQG regulator given LQ gain and
• Kalman estimator.
• augstate - Augment output by appending states.

42
Control System Toolbox
Analysis Tool ltiview
File-gtImport to import system from Matlab
workspace
43
Control System Toolbox
Design Tool sisotool
Design with root locus, Bode, and Nichols plots
of the open-loop system. Cannot handle
continuous models with time delay.
44
M-File Example

• Define the transfer function of a plant
• Gtf(4 3,1 6 5)
• Get data from the transfer function
• n,dtfdata(G,'v')
• p,z,kzpkdata(G,'v')
• a,b,c,dssdata(G)
• Check the controllability and observability of
  the system
• rorank(obsv(a,c))
• rcrank(ctrb(a,b))
• find the eigenvalues of the system
• damp(a)
• multiply the transfer function with another
  transfer function
• Tseries(G,zpk(-1,-10 -2j 2j,5))

• find the bandwidth of the new system
• wbbandwidth(T)
• plot the step response
• step(T)
• plot the rootlocus
• rlocus(T)
• obtain the bode plots
• bode(T)
• margin(T)
• use the LTI viewer
• ltiview('step''bode''nyquist',T)
• start the SISO tool
• sisotool(T)