Dizertacn

1
LEARNING ALGORITHMS for SERVO- MECHANISM TIME
SUBOPTIMAL CONTROL
M. Alexík, University of Žilina, Slovak Republic
1 - Time Optimal Control - Switching Function
(SwF) 2 - Sliding Mode Control (SMC), Adaptive
Sliding Mode Control 3 - Learning Control (LC)
based on SMC approximation of SwF 4 LC based
on Neural Nets quasi real time computation 5
LC based on Identification real time
computation of SwF 6 - Real Time Simulation
2
Laboratory Model of Servomechanism
(61)x 0.6 kg Cart with variable load
Spring
Hand Control
Time and position Display
Load
Load
DC Drive with gear
Communication with PC RS 232
µP Atmel
GOAL Derivation of Time Optimal Control
algorithm for Servomechanism with variable load.
Time Optimal (feedback) Control -
Sliding Mode Control estimation
of switching function (switching curved line, or
approximation- only line, polynomial).
For variable unknown load of servomechanism and
time suboptimal control is necessary
to apply learning algorithm for looking for
switching function (curved line, line). Problem
Nonlinearities variable fiction, two springs
non sensitivity in output variable
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
3
Physical Model of Servomechanism real time
simulation
Km
S(s)
Km 1/b, Tem m/b
s(Tem s 1)
m Weights(changeable), b coef. of friction
(changeable) then Km, Tem
are also changeable


)
(
)
(
)
(
b
Ax
x
t
u
t
t

-
ù
é
ù
é
é
)
(
)
(
)
(
t
y
w
t
e
1



(t)
x
ú
ê
ú
ê
ê
-
)
(
)
(
)
(
t
y
t
e
t
x


û
ë
û
ë
ë
2
D/A converter ? pulse modulation of
Action variable u(k) u(k)max 5 V, u(k)min
-5 V,
Umax 5 V
Controller output 20 times reduced scale
Umin - 5 V
4
Time Optimal Responses digital simulation
hysteresis (non sensitivity - dead zone) on
controller output
Hysteresis in this simulation examples deS
(-0.05 ? 0.05)
From hysteresis on controller output
L m
Position measurenment 1 m 2600 impulses 1
impulse 0.384 mm
Analog model Real time Hardware in Loop
Simulation
Speed measurenment 0.1 ms-1 260 imp/s 1.3
imp/5 ms
Controller output 20 times reduced scale
t s
Sampling interval 5, 10, 20 ms. Problem with
Interrupts DOS, Linux, W98. XP

From hysteresis on controller output

Why we need hysteresis in the controller output?
Controller output have to be without oscillation
(zero ) in steady state. But then there is small
control error in steady state, which depends from
controller output, sampling interval and plant
dynamics. If good condition also transient state
is without oscillation.
5
Time Optimal Responses real time simulation
speed measurement problem
Sampling interval 5 ms, no filter, no noise
Sampling interval 20 ms
Add special noise signal to the measured position
for elimination of speed quantization error, and
after this filtration. Or state observer for
position and speed as signal from state
reconstruction (see later).
6
Optimal responses and trajectories
Lm
3
1-nominal Jm -T1,K1 2- J 5Jm T2,K2 3- J
10Jm - T3,K3
2
1
y
ts
x1,3(Cp)
x1(t) e(t) rad
positionrad
a3,p
x2e(t)
Cp optimal slope of switching line
Cpe(t) / e(t), e(t)d/dte(t)
Cp3 x1,3 (Cp) / x2,3(C p) tg(a3,p)
rad/s
x2,3(Cp)
Cp3
Cp2
Cp1
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
7
Optimal trajectories and switching curved line
Switching line for w 300 rad/s - Cp1
Switching line for w 100 rad/s Cp3
Cp1 lt Cp3
One Switching curved line (switching function)
but More switching line (depends on set point)
Switching function
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
8
Switching curved line function It can be
computed only for known Km, Tem.
Switching function
deS hysteresis of state variable
measurement
Switching line
9
Sliding Mode Control - SMC
x
x2 m/s
Condition of SMC
yA
yB
Lyapunov function
ugt0
x1m
sA
Sliding mode trajectory slide
along sliding line
Relay control
ult0
Cx instantaneous slope of trajectory
point
sB
10
Adaptive - SMC
x1 , y
1.
2.
3.
1. t-suboptimal control with SL
(Switching Line) 2. t-suboptimal adaptive
control 3. t-optimal control
1.
x1 , t
C
2.
ugt0
?C
ult0
d
3.
Ci - initial slope of switching line
SL for t-optimal control.
Adaptive adjusting of switching line
slope
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
11
Adaptive Algorithm based on Sliding Mode
300
X1
1
rad
3
2
1 - time optimal response
2 - adaptive sliding mode response
3 - conventional sliding mode response
4 - actuating variable for response 2 (times 10)
100
4
1
2
3
Time s
position error - Xe1 rad
0
300
100
1 - time optimal trajectory
-100
2 - adaptive trajectory
Xe2
3 - sliding mode trajectory
rad/s
3
2
1
-300
12
Adaptive adjustment of the switching line slope
100
300
position error - Xe1 rad
0
Xe1
Ct
1 - time optimal trajectory

Xe2
2 - adaptive trajectory
IF C_1 C3 gt Ct C4 (C5)
3 - sliding mode trajectory
THEN Change Cp
5
Xe1(1)
-100
4
3
Xe2
2
rad/s
1
3
Xe2(1)
2
1
Cp
1,2,3
-300
Cp
Es - angular speed error
C
t
Cp0
Copt
13
Optimal trajectory of all II. Order
Systems slope of switching line on the optimal
trajectory be on the decrease
x1
S1
S2
S3
S4
x2
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
14
Automatic generation of suboptimal responses and
trajectories
15
Generation of suboptimal trajectories
4
1
Point for slope of Suboptimal switching line
Learning looking for Points for suboptimal
switching line look up table (memory) for its
classification (identification)
of Load (parameters of transfer function
parameters of controlled process)
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
16
Learning Controller based on SMC basic problems
1- SL - Switching line 2- LSC - Linear
switching curve 3- NS - Neural network 4- SCL
- Switching curved line.
(Identification of Km, Tem and
computation of SCL)
1- slope of SL and polynomials parameters 2-
LSC points 3- NS veights 4- structure of SC
function
After learning process, recognition of number
of load Km, Tm
Possibilities of Learning (historical
evolution) 1- fractional changing of SL slope and
polynomial interlace 2- adaptation of LSC profile
(online and offline) 3- simulation of finishing
trajectories on neuro model (1,2,3 off
line learning) 4- continuous identification of
process parameters (Km, Tem) (on line
learning)
Classification option 1 -Hopfield net 2 -fuzzy
clustering 3 -ART net (1-3 classific. off
line) 4 Parameters identification (on line)
Classification problems non linearity's in Km,
Tem bring about changing instantaneous values of
this parameters and then also changing of step
response for the same number of load.
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
17
Learning algorithm based on switching line SL
Real Time Simulation Experiments
c_sus Cmin Cmax krok e(0)
c_sys a0 a1 a2
Memory - look up tables
SMC

u
u
x
s(x) -x2-Cx1
plant
u-
Polynomial approximation of switching
function
Memory
c_sys
Learning algorithm
Classification
18
Classification - Hopfield NET
Stochastic asynchronous dynamics
scale adaptation
y(t)
Pattern coding
Transient response
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
19
Classification - Hopfield net (N255)
disadvantages speed, number of pattern limited ,
pattern numbering
Advantage quality
Output
Input
Input
Output
2. S2
n.c.1
1. S1
n.c.2
n.c.3
3. S4
4. S5
s.c.3
5. S1
6. S4
s.c.1
s.c.3
7. S5
8. S2
s.c.3
s.c.2
s.c.2
9. S2
10. S4
s.c.3
11. S3
12. S3
n.c.4
s.c.3
Evolution of nets energy according to number of
iteration
20
Fuzzy classification
Plant Input data (x) Output (y)
S1 x1, x2, ..., x251 1
S2 x1, x2, ..., x252 2
... ... ...
Sn x1, x2, ..., x25n n
Parameters estimation in consequent rules of
fuzzy classifier
Data clustering (counts of rules and membership
functions )
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
21
Fuzzy classification
Disadvantages too lot of parameters, necessity
to keep data patterns
Advantage quality
1. S1
2. S3
4. S3
3. S3
5. S2
6. S3
7. S2
8. S3
10. S3
9. S4
11. S4
12. S2
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
22
Classification ART network

• Initialisation
• Recognition
• Comparison
• Searching
• Adaptation

x1
x2
xn
Advantages quality, speed
t
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
23
Learning switching curve (LSC) definition
Method for LSC points setting
LSCstep1
LSCstep2
x2
linearized SC
0.5
SC for t-optimal control
0
1
x1
0
-1
1
r
x(n)
x(m)
-0.5
-0.5
SC for t-optimal control
24
Settings of LSC profile (1. Learning step )
Off-line according to trajectory profile For
LSC points..
On-line according to adaptation For LSC points.
1. LSC according to adaptation 2. LSC according
to trajectory 3. SC for t-optimal control. 4.
System output
x2, y
x2, y
1. trajectory 2. LSC 3. SC for t-optimal
control. 4. system output
4.
4.
x1, t
x1, t
1.
1.
2.
2.
?C
3.
3.
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
25
Control on LSC for different set points
According to trajectory profile
According to adaptation
x2, y
3
1. LSC in single steps 2. SC for t-optimal
control. 3. System output
x1, t
together
1.
2.
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
26
Learning algorithm based on LSC Real Time
Simulation Experiments
SMC
u
x
u
sLPK(x)
System
u-
c_sus C
Pamät
c_sus
Learning Algorithm
Classification
27
Learning algorithm based on neuro networks- NN
Basic description
Two Neuro Networks NS1 and NS2. First step
From measured values of input (Umax, Umin) and
output y(k) to set up NS1. Then NS1 can
generated t - optimal phase trajectories and to
set up NS2. Second step t optimal control
with NS2 as the switching function. It is
possible to find t-suboptimal control only from
ONE loop response (with switching line). This
t- suboptimal control is compliance for all set
points (but only for one combination of loads).
NS1- 2 layers (6 and 1) neurons with linear
activation function. (Model of
servo system (output) with inverted time). n
transfer function order (2,3) NS2 - 3
layers, model of switching function. Input
layer 6 neurons with tangential sigmoid
activation function. Hidden layer 6 neurons
with linear activation function Output layer -
1 neuron with linear activation function.
For 2 order transfer function it is needed
from simulation approximately 300 points as the
substitution of switching function.
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
28
Learning algorithm based on NNSteps of
computation

x
(t), y(t)

2

Output (2.step)
1. Step Real time response 2. Step a Off line
computation of switching function
5 sDOS, 3 s Windows on line
computation in progress b Real time
suboptimal time response

Output (1.step)
x
(t),t

1
Phase trajectory (1.step)


Phase trajectory (2.step)

Switching function (1.step)


Switching function
(2.step)

M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
29
Learning algorithm based on NN Real Time
Simulation Experiment
NN switching function
SMC
u
x
u
sNS2(x)
System
u-
c_sus WNS2
Memory
c_sus
Learning algorithm
Clasification
NN model in invert. time
Model NN1
Block of simulation according to NN model
30
Learning algorithm based on NNSteps of
Computation
x
(t)
2
y(t)
Output (2.step)



Output (1.step)

x
(t),t
1

Phase trajectory (1.step)

Phase trajectory (2.step)
Switching function (1.step)


Switching function
(2.step)



31
Learning algorithm based on NN, Simulation
Experiments
Load 12
Load 10
Response quality Settling time, tR 2.83 s
, 3.31 IAE 1.53 Vs , 1.63
Response quality tR 3.68 s , 3.99
IAE 1.76 Vs , 1.81
Load 16
Load 14
Response quality tR 4.17 s , 4.74
IAE 1.89 Vs , 1.93
Response quality tR 4.54 s , 4.88
IAE 1.98 Vs , 2.01
32
Optimal Trajectories for 3. Order Controlled
System Computed by Neuro Networks
Model phase trajectory for uUmax
x2
4

)
(
s
S



)
2
)(
1
)(
7
,
0
(
s
s
s
koncový state
Model phase trajec- tories for uUmax
0.5
Initial switching plain
x1
0
-1
1
Model phase trajectories for uUmin
x3
-0.5
Model phase trajectory for uUmin
x3
y(t)
Second control according to neuro nete NN2
Points of phase trajectories from simulation
Switching plain according to NN2
First control according to switching plain
x2
t
x1
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
33
Classification with Identification
3 possibilities

• Step response of transfer function
• h(t) Km t Km Tem exp ((-1/T) t) Km Tem
• Analytical derivation of parameters Km and
  Tm
• Is possible with static optimisation or
  continuous identification
• 1. Static optimization from
• h(t) Km-1 t Tem (exp((-1/Tem) t) - 1)
• 2. Continuous Identification.
• Parameters of discrete transfer function from
  Identification (ai , bi)
• and recalculation to parameters of continuous
  transfer function Km, Tem
• Advantages Direct calculation of parameters of
  switching function
• Disadvantages Real time calculation of RLS
  algorithm.

Kmx2(t/2)2/Umax2x2(t/2)-x2(t) Tem
-t/ln1-(x2(t)/Km)
Tem T0/ln(1/a2) Kmb1/T0Tem (a2 - 1)
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
34
Classification with Identification
speed measurement problem

Speed measurenment 0.1 ms-1 260 imp/s 1.3
imp/5 ms
x2(t) x1(k) - x1(k-1)/ T0 T0 sampling
interval
x! - position mm
600

4 -set point w 0.6 m

4 -set point w 400 mm
x
mm

1
u(k) V
2 u(k)- control output 5 V
2 u(k)- control output 5 V
3 -controlled variable


200

2
0
1

6

2

4


3

Time s

-
100


1 - control trajectory
1 - control trajectory
Settling time 3.75 s


-500


35
Classification with Identification

M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
36
Classification with Identification
state estimator


y(t)

u(k)

w

s

e

d
S

r

e
(k)

h

-
1

c

z
b

?(
k)

F



d
e
(t)
/dt
Kmx2(t/2)2/Umax2x2(t/2)-x2(t) Tem
-t/ln1-(x2(t)/Km)
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
37
Learning algorithm - Identification state
estimator real
time hardware in loop simulation
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
38
Learning algorithm - Identification state
estimator real
time hardware in loop simulation
Load 12
Load 12
Load 14
Load 16
39
Learning algorithm - Identification state
estimator real
time hardware in loop simulation
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008
40
Comparison of Learning algorithm loop
response quality
real time hardware in loop simulation

7
0.5x

1
0.4u(k)
Identification and
y (t)
Set Point


state estimator


4


Switching function
Controlle
r output
Neuro

identification
-

1

2

0
6

1
4

0.5x
, t
s

1
-1
Set

Integral of ab
-

-


Algo
Settling
point

solute value of

State trajectory
rithm
time s
-


m
tthe error ms
estimator
Switching

4.
06

-

0.4


0.98

function

Trajectory
Neuro
-3
0.4

3.86

0.82

Neuro


Switching function


Identif.
3.64
0.4
0.76

learned with NN
estimation
41
Conclusion and outlook
2- Nowadays, paradigm of optimal and adaptive
control theory culminates. It is needed to solve
problems such as MIMO control, multi level and
large-scale dynamic systems with discrete event,
intelligent control. That demands to turn
adaptive control chapter into appearance of
classical theory. Moreover, we need to classify
adaptive systems with one loop among as classic
ones and focus on multi level algorithms and
hierarchical systems. Then we will be able to
formulate new paradigm of large-scale systems
control and intelligent control.
3 Realization t optimal control based on
sliding mode and Neuro Nets (real time
computation of NS1 and NS2) but also real tike
identification with estimator state have to use
parallel computing. So control algorithm than can
be classified as intelligent control.
M. Alexík, KEGA,06- 08, Žilina, Sept. 2008