state space models312
Post on 19-Feb-2018
222 Views
Preview:
TRANSCRIPT
-
7/23/2019 State Space Models312
1/19
STATE SPACE MODELS
MATLAB Tutorial
-
7/23/2019 State Space Models312
2/19
Why State Space Models
The state space model represents a physical
system as nfirst order differentialequations. This form is better suited for
computer simulation than an nthorder input-
output differential equation.
-
7/23/2019 State Space Models312
3/19
Basics
Vector matrix format generally is given by
!here yis the output equation" and xis the state vector
-
7/23/2019 State Space Models312
4/19
PARTS OF A STATE SPACE
REPRESENTATION
State Vaia!les a subset of system variables !hich
if #no!n at an initial time t$along !ith subsequent
inputs are determined for all time t%t$&
State E"#atio$s n linearly independent first order
differential equations relating the first derivatives of
the state variables to functions of the state variables
and the inputs.
O#tp#t e"#atio$s algebraic equations relating the
state variables to the system outputs.
-
7/23/2019 State Space Models312
5/19
E%AMPLE
The equation gathered from the free body diagram is
mx"& bx'& kx- f't( ) $
*ubstituting the definitions of the states into the
equation results in
mv'& bv& kx- f't( ) $
*olving forv' gives the state equation
v') '-b/m)v& '-k/m)x & f't(+m
The desired output is for the position"x" so
y)x
-
7/23/2019 State Space Models312
6/19
Co$t&
,o! the derivatives of the state variables are
in terms of the state variables" the inputs"and constants.
x' = v
v'' ()k/m*x + ()b/m* v+ f(t*,m
y)x
-
7/23/2019 State Space Models312
7/19
P-TTIN. INTO VECTOR)MATRI%
FORM
ur state vector consists of t!o variables"x
and vso our vector-matrix !ill be in the form
-
7/23/2019 State Space Models312
8/19
Expla$atio$
The first ro! of Aand the first ro! of Bare
the coefficients of the first state equation forx'. Li#e!ise the second ro! of Aand the
second ro! of Bare the coefficients of the
second state equation for v'. Cand Dare the
coefficients of the output equation for y.
-
7/23/2019 State Space Models312
9/19
E%ACT REPRESENTATION
-
7/23/2019 State Space Models312
10/19
/OW TO INP-T T/E STATE
SPACE MODEL INTO MATLAB
n order to enter a state space model into MATLAB"
enter the coefficient matrices A" B" C" and Dinto
MATLAB. The syntax for defining a state space
model in MATLAB is
statespace ) ss'A" B" /" 0(
!here A" B" C" and Dare from the standard vector-
matrix form of a state space model.
-
7/23/2019 State Space Models312
11/19
Exa0ple
1or the sa#e of example" lets ta#e m ) 2" b ) 3" and # ) 4.
%% m ) 25
%% b ) 35
%% # ) 45
%% A ) 6 $ 7 5 -#+m -b+m 85
%% B ) 6 $ 5 7+m 85
%% / ) 6 7 $ 85 %% 0 ) $5
%% statespace9ss ) ss'A" B" /" 0(
-
7/23/2019 State Space Models312
12/19
O#tp#t
This assigns the state space model under
the name statespace9ss and output thefollo!ing
a =
x1 x2
x1 0 1 x2 -1.5 -2.5
-
7/23/2019 State Space Models312
13/19
Co$t&
b =
u1
x1 0 x2 0.5
c =
x1 x2
y1 1 0
-
7/23/2019 State Space Models312
14/19
Co$t&
d =
u1 y1 0
Continuous-time mode.
-
7/23/2019 State Space Models312
15/19
E%TRACTIN. A1 B1 C1 DMATRICES FROM A STATE SPACEMODEL n order to extract the A" B" /" and 0
matrices from a previously defined statespace model" use MATLAB:s ssdata
command.
6A" B" /" 08 ) ssdata'statespace(
!here statespace is the name of the statespace system.
-
7/23/2019 State Space Models312
16/19
Exa0ple
%% 6A" B" /" 08 ) ssdata'statespace9ss(
The MATLAB output !ill be! =
-2.5000 -0.#50 $.0000 0
-
7/23/2019 State Space Models312
17/19
Co$t&
% =
0.2500 0
C =
0 0.5000
& =
0
-
7/23/2019 State Space Models312
18/19
STEP RESPONSE -SIN. T/E
STATE SPACE MODEL
nce the state space model is entered into MATLAB it is easy
to calculate the response to a step input. To calculate the
response to a unit step input" use step'statespace(
!here statespace is the name of the state space system.
1or steps !ith magnitude other than one" calculate the step
response using
step'u ; statespace(
!here u is the magnitude of the step and statespace is the
name of the state space system.
-
7/23/2019 State Space Models312
19/19
top related