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MATLAB-Toolboxes for Control

Linear Control

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Identification

Identification ToolboxFrequency-Domain ID ToolboxSimulink®

Control Design Process

Modeling Tools

Design and Analysis

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

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)

...( )

, ... numerator coefficients

, ... denominator coefficients

p s p s pH s

q s q s q

Transfer Function

Consider a linear time invariant (LTI) single-input/single-output system

Applying Laplace Transform to both sides with zero initial conditions

'' 6 ' 5 4 ' 3y y y u u

( ) 4 3( )

( ) 6 5

Y s sG s

U s s s

Transfer Function

>> num = [4 3];

>> den = [1 6 5];

>> sys = tf(num,den)Transfer function:

4 s + 3

-----------------

s^2 + 6 s + 5

Transfer Function

>> [num,den] = tfdata(sys,'v') num = 0 4 3 den = 1 6 5

Zero-pole-gain model (ZPK)

( )( ) ... ( )( )

( )( ) ... ( )

, ... the zeros of H(s)

, ... the poles of H(s)

s p s p s pH s K

s q s q s q

'' 6 ' 5 4 ' 3y y y u u

Consider a Linear time invariant (LTI) single-input/single-output system

Applying Laplace Transform to both sides with zero initial conditions

( ) 4 3 4( 0.75)( )

( ) ( 1)( 5)6 5

Y s s sG s

U s s ss s

>> sys1 = zpk(-0.75,[-1 -5],4) Zero/pole/gain:4 (s+0.75)-----------(s+1) (s+5)

>> [ze,po,k] = zpkdata(sys1,'v')

-0.7500

state vector

input and output vectors

, , state-space matrices

x A x B u

y C x D u

u and y

A B C and D

State-Space Model (SS)

Consider a Linear time invariant (LTI) single-input/single-output system

State-space model for this system is'' 6 ' 5 4 '' 3y y y u u

' 0 1 0

' 5 6 1

Control System Toolbox

>> sys = ss([0 1; -5 -6],[0;1],[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

State-Space Models

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.

Transfer function State Space

Zero-pole-gain

tf2zpzp2tf ss2zp

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.

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.

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.

1s5s4s2

2s3)s(G

)t5.0sin()t(u

ProblemProblem Given the LTI systemGiven the LTI system

Plot the following responses for:Plot the following responses for: The impulse response using the The impulse response using the impulseimpulse command. command. The step response using the The step response using the stepstep command. command. The response to the input calculated using The response to the input calculated using

both the both the lsimlsim commands commands

Time Response of Systems

Root locus analysisRoot locus analysis Frequency response plotsFrequency response plots

BodeBode Phase MarginPhase Margin Gain MarginGain Margin

NyquistNyquist

Root Locus The root locus is a plot in the s-plane of all The root locus is a plot in the s-plane of all

possible locations of the poles of a closed-loop possible locations of the poles of a closed-loop system, as one parameter, usually the gain, is system, as one parameter, usually the gain, is varied from 0 to varied from 0 to ..

By examining that plot, the designer can make By examining that plot, the designer can make choices of values of the controller’s parameters, choices of values of the controller’s parameters, and can infer the performance of the controlled and can infer the performance of the controlled closed-loop system. closed-loop system.

Plot the root locus of the following Plot the root locus of the following systemsystem

( 8)( )

( 2)( 8 32)

K sG s

s s s s

Root Locus

Root Locus>> rlocus(tf([1 8], conv(conv([1 0],[1 2]),[1 8 32])))>> rlocus(tf([1 8], conv(conv([1 0],[1 2]),[1 8 32])))

Typically, the analysis and design of a Typically, the analysis and design of a control system requires an examination of control system requires an examination of its frequency response over a range of its frequency response over a range of frequencies of interest.frequencies of interest.

The MATLAB Control System Toolbox The MATLAB Control System Toolbox provides functions to generate two of the provides functions to generate two of the most common frequency response plots: most common frequency response plots: Bode Plot (bode command) and Nyquist Plot Bode Plot (bode command) and Nyquist Plot (nyquist command).(nyquist command).

Frequency Response: Bode and Nyquist Plots

Problem Problem Given the LTI systemGiven the LTI system

Draw the Bode diagram for 100 values of Draw the Bode diagram for 100 values of frequency in the interval .frequency in the interval .

1G(s )

s(s 1)

110 10

Control System ToolboxFrequency Response: Bode Plot

>>bode(tf(1, [1 1 0]), logspace(-1,1,100));>>bode(tf(1, [1 1 0]), logspace(-1,1,100));

Frequency Response: Bode Plot

The loop gain Transfer function The loop gain Transfer function G(s)G(s) The The gain margingain margin is defined as the multiplicative is defined as the multiplicative

amount that the magnitude of amount that the magnitude of G(s) G(s) can be increased can be increased before the closed loop system goes unstablebefore the closed loop system goes unstable

Phase marginPhase margin is defined as the amount of additional is defined as the amount of additional phase lag that can be associated with phase lag that can be associated with G(s)G(s) before the before the closed-loop system goes unstableclosed-loop system goes unstable

Frequency Response: Nyquist Plot

16s24.320s81.1604s2.24s

640s1280)s(G

4 310 10

ProblemProblem Given the LTI systemGiven the LTI system

Draw the bode and nyquist plots for 100 values of frequencies Draw the bode and nyquist plots for 100 values of frequencies in the interval . In addition, find the gain and phase in the interval . In addition, find the gain and phase margins. margins.

Control System ToolboxFrequency Response: Nyquist Plot

w=logspace(-4,3,100);w=logspace(-4,3,100);

sys=tf([1280 640], [1 24.2 1604.81 320.24 16]);sys=tf([1280 640], [1 24.2 1604.81 320.24 16]);

bode(sys,w)bode(sys,w)

[Gm,Pm,Wcg,Wcp]=margin(sys)[Gm,Pm,Wcg,Wcp]=margin(sys)

%Nyquist plot%Nyquist plot

figurefigure

nyquist(sys,w)nyquist(sys,w)

The values of gain and phase margin and corresponding frequencies are The values of gain and phase margin and corresponding frequencies are

Gm = 29.8637 Pm = 72.8960 Wcg = 39.9099 Wcp = 0.9036Gm = 29.8637 Pm = 72.8960 Wcg = 39.9099 Wcp = 0.9036

Frequency Response Plots

bode - 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.

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.

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.

Analysis Tool: ltiview

File->Import to import system from Matlab workspace

Design Tool: sisotool

Design with root locus, Bode, and Nichols plots of the open-loop system.Cannot handle continuous models with time delay.

%Define the transfer function of a plantG=tf([4 3],[1 6 5])

%Get data from the transfer function[n,d]=tfdata(G,'v')

[p,z,k]=zpkdata(G,'v')

[a,b,c,d]=ssdata(G)

%Check the controllability and observability of the system

ro=rank(obsv(a,c))rc=rank(ctrb(a,b))

%find the eigenvalues of the systemdamp(a)

%multiply the transfer function with another transfer function

T=series(G,zpk([-1],[-10 -2j +2j],5))

%plot the poles and zeros of the new systemiopzmap(T)

%find the bandwidth of the new systemwb=bandwidth(T)

%plot the step responsestep(T)

%plot the rootlocusrlocus(T)

%obtain the bode plotsbode(T)margin(T)

%use the LTI viewerltiview({'step';'bode';'nyquist'},T)

%start the SISO toolsisotool(T)

M-File Example

7/2/2015MATLAB1. MATLAB MATLAB and Toolboxes MATLAB and Control Control System Toolbox Simulink...

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