ITERATIVE FEEDBACK TUNING APPROACH TO A CLASS OF

STATE FEEDBACK-CONTROLLED SERVO SYSTEMS

Mircea-Bogdan Rădac, Radu-Emil Precup

Dept. of Automation and Appl. Inf., “Politehnica” University of Timisoara

Bd. V. Parvan 2, 300223 Timisoara, Romania

Emil M. Petriu

School of Information Technology and Eng., University of Ottawa

800 King Edward, Ottawa, ON, K1N 6N5 Canada

Stefan Preitl, Claudia-Adina Dragoş

Dept. of Automation and Appl. Inf., “Politehnica” University of Timisoara

Bd. V. Parvan 2, 300223 Timisoara, Romania

Keywords: Iterative Feedback Tuning, Servo systems, State feedback control systems.

Abstract: An original control structure dedicated to a class of second-order state feedback control systems is presented

in the paper. The controlled processes are accepted to be characterized by second-order servo systems with

integral component. Optimal state feedback control systems are designed for those processes making use of

the Iterative Feedback Tuning (IFT) approach. The state feedback control system structure is extended with

an integral component to ensure the rejection of constant disturbances. A case study concerning the position

control of a DC servo system with backlash is included. Real-time experimental results validate the

theoretical part of the IFT approach.

1 INTRODUCTION

The second-order servo systems with integral

component are applied widely as controlled

processes in real-world applications including

mechatronics, electrical drives, sub-systems in

power plant control systems, positioning systems in

manipulators, mobile robots, machine tools, flight

guidance and control (Škrjanc et al., 2005; Gomes et

al., 2007; Petres et al., 2007; Barut et al., 2008;

Costas-Perez et al., 2008; Denève et al., 2008; De

Santis et al., 2008; Orlowska-Kowalska and Szabat,

2008; Precup et al., 2008b; Vaščák, 2008). Those

controlled processes are acknowledged as particular

cases of benchmark systems (Åström and Hägglund,

2000; Isermann, 2003; Horváth and Rudas, 2004;

Kovács, 2006). Accepting that they are linearized

versions of nonlinear servo systems, the parameters

are variable with respect to the operating points.

Hence the parameter variation makes their control a

challenging task when very good control system

performance indices are required. Their control

problems become even more challenging when low-

cost automation solutions are needed in the design

and implementation of the control system structures.

One control solution to cope with the accepted

class of processes described is represented by state

feedback control systems. Since the main control

aims, high performance indices in reference input

tracking and regulation with respect to several types

of load disturbance inputs, are difficult to be

fulfilled, one typical approach is to design optimal

control systems. The improvement of the control

system performance indices (fore example settling

time and overshoot) is enabled by the minimization

of appropriately defined objective functions

resulting in optimal state feedback control systems.

An alternative to the minimization of the objective

functions is represented by Iterative Feedback

Tuning (IFT) (Hjalmarsson et al., 1994, 1998). IFT

41

R

ˇ

adac M., Precup R., Petriu E., Preitl S. and Drago¸s C. (2009).

ITERATIVE FEEDBACK TUNING APPROACH TO A CLASS OF STATE FEEDBACK-CONTROLLED SERVO SYSTEMS.

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics - Intelligent Control Systems and Optimization,

pages 41-48

DOI: 10.5220/0002204400410048

Copyright

c

SciTePress

algorithms make use of the input-output data

measured from the closed-loop system during its

operation to calculate the estimates of the gradients

and Hessians of the objective functions. Several

experiments are done per iteration and the updated

controller parameters are calculated based on the

input-output data and the estimates.

The application of IFT to one-degree-of-freedom

controllers needs two experiments per iteration. The

first experiment is referred to as the normal one and

it corresponds to the usual operation of the control

system. The second experiment is the gradient one.

The reference input in the gradient experiment is the

control error of the first experiment. An additional

normal experiment is needed in case of two-degree-

of-freedom controllers. Even more experiments are

needed to tune the state feedback controllers and the

Multi Input-Multi Output (MIMO) ones. So it is

natural to strive for the alleviation of the number of

experiments (Hjalmarsson and Birkeland, 1998;

Hjalmarsson, 1999; Jansson and Hjalmarsson,

2004).

The paper aims three main contributions. The

first contribution of the paper is the proposal of an

IFT algorithm resulting in a method to obtain the

partial derivatives needed in the calculation of the

gradient of the objective function in state feedback

control systems. The second contribution concerns

the new experiments to be done in the IFT of the

accepted class of second-order state feedback

control systems dedicated to servo systems. The

third contribution involves the highlighting of the

specific aspects related to the actuator saturation

problem proved by the low-cost implementation and

the real-time experimental results included. The

main advantages of the contributions are the

simplification of the experiments and the smooth

decrease of the objective function. Thus the local

minimum will be reached.

The paper treats the following topics. The

controlled processes and the new IFT algorithm

dedicated to the accepted class of state feedback

control system are presented in Section 2. Next,

Section 3 points out original and attractive aspects

concerning the actuator saturation problem. A case

study concentrated on the state feedback position

control of a DC servo system with backlash is

described in Section 4. The real-time experimental

results validate the IFT algorithm. The conclusions

are drawn in Section 5.

2 CONTROLLED PROCESS AND

IFT ALGORITHM

The controlled process as part of servo systems is

characterized by the following state-space model:

⎥

⎦

⎤

⎢

⎣

⎡

ω

α

=

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

⎥

⎦

⎤

⎢

⎣

⎡

ω

α

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

−

=

⎥

⎦

⎤

⎢

⎣

⎡

ω

α

2

2

1

0

1

0

10

I

y

y

u

T

K

T

s

s

s

,

(1)

where α=x

1

is the first state variable usually

representing the (angular) position, ω=x

2

is the

second state variable usually representing the

(angular) speed, u is the control signal, y

1

and y

2

are

the controlled outputs, and I

2

is the identity matrix.

The two parameters in (1) are K

S

>0 which is the

process gain, and T

S

>0 which stands for the small

time constant or the sum of parasitic time constants.

The two transfer functions from u to ω and u to α

are

)(

,

sP

uω

and

)(

,

sP

uα

, respectively:

)1(

)(,

)1(

)(

,,

s

s

u

s

s

u

sTs

K

sP

sT

K

sP

+

=

+

=

αω

.

(2)

Therefore the integral component can be observed in

(2) when α=x

1

is taken as controlled output. Such

situations correspond to positioning systems.

The state feedback control system structure is

presented in Figure 1. The dotted connection

highlighted is valid only when the experiments

specific to IFT are done. That connection is not

applied during the normal system operation.

Figure 1: IFT-based state feedback control system

structure.

The main variables and blocks illustrated in

Figure 1 represent: IFT – the IFT algorithm, RM –

the reference model,

2

21

][ Rxx

T

∈=x

– the state

vector (T highlights the matrix transposition),

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

42

][

21

KK

c

=k

– the state feedback gain matrix,

)()(

,

sPsP

uα

=

– the transfer function of the

controlled process when the controlled output is

y=x

1

, r – the reference input, e=r–y – the control

error. The other variables will be presented in the

sequel.

If the state feedback gain matrix is regarded as a

controller, then use will be made of its parameters to

minimize the tracking error e

t

between the system

output y and the reference model output y

d

. Let J be

a simple objective function defined over a finite time

horizon N:

∑

=

=

N

t

t

e

N

J

1

2

))((

2

1

)( ρρ

,

(3)

where

m

R∈ρ

is the parameters vector containing at

least the parameters of k

c

and e

t

is the tracking error:

d

t

yye −= )()( ρρ

.

(4)

The IFT results (Hjalmarsson et al., 1994, 1998;

Pfeiffer et al., 2006) are employed to find the

solution ρ

*

to the optimization problem

)(minarg

*

ρρ J

SD∈ρ

=

,

(5)

where several constraints can be imposed regarding

the process and the closed-loop system. One

constraint concerns the stability of the system and

SD represents the stability domain (Precup et al.,

2008).

Solving the optimization problem (5) requires

finding the parameters vectors that make the

gradient equal to zero:

0]...[

1

=

ρ∂

∂

ρ∂

∂

=

∂

∂

T

m

JJJ

ρ

.

(6)

Making use of (3) and (4) the equation (6) will be

transformed into

0])([

1

1

=−

∂

∂

∑

=

N

t

d

T

yy

y

N

ρ

ρ

.

(7)

The partial derivatives

i

y

ρ∂

∂

should be calculated

to obtain the components of the gradient,

i

J

ρ∂

∂

,

mi ,1=

. The new IFT approach to be described as

follows will employ specific experiments to obtain

those components. Use will be made of the

following notation:

i

ρ∂

α

∂

=α'

(8)

to highlight the partial derivative of the variable α

taken with respect to ρ

i

and obtain the simplicity of

the presentation.

The state-space model (1) can be reconsidered by

including one additional state variable to the state

variable. That variable is x

3

=x

R

and it corresponds to

the integrator inserted into the control system

structure. Thus its gain K

R

will be subject to IFT as it

is shown in Figure 1. The extended state-space

model of the process is

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

+

+

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−

−−−=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

R

r

s

s

R

Rss

s

s

R

x

I

y

y

y

rK

T

K

x

KKKK

T

KK

x

ω

α

ω

α

ω

α

3

3

2

1

21

1

0

001

1

010

,

(9)

where the parameter K

r

is not included in the tuning

scheme. Its value is set prior to the application of

IFT. One way to choose K

r

is to keep a connection

between the steady-state value of r and the steady-

state value of r

x

for which the desired r can be

tracked by the steady-state value of y. That value of

r

x

can be subject to the experimental identification of

the state feedback control system.

The preparation of the experimental scheme

needed in the calculation of the gradient starts with

the reconsideration of the input-output relations

specific to the control system structure presented in

Figure 1. Observing that generally

xIyykxk

P

y

3

for =−=−=

=

cxcx

rru

u

,

(10)

the following relationships hold:

T

R

T

ERc

Ecr

RRrRRrx

xxKKK

rKxKxK

xKrKuxKrKr

][],[

,

,

21

2211

=−−=

+=−−

−+

=

+

=

xK

xK

.

(11)

Next the gradient of y with respect to each

parameter can be calculated, where the parameters

are the m=3 components of the parameters vector

T

R

KKK ][

21

=ρ

.

(12)

ITERATIVE FEEDBACK TUNING APPROACH TO A CLASS OF STATE FEEDBACK-CONTROLLED SERVO

SYSTEMS

43

Since y and u are functions of ρ it is justified to

apply

'' uPy =

,

(13)

leading to

'''

EcEc

u xKxK +=

.

(14)

In addition, accepting the MIMO formalism

suggested in (10), the following relationship can be

expressed:

''' yKyK

cc

u +=

.

(15)

Equation (15) is of great importance for the new

approach. The first term in the right-hand side of

(15),

yK '

c

, needs to be added to the control signal

to obtain the desired experimental scheme. That

term contains the unmodified output vector (in the

MIMO framework) so the idea is to obtain it from

one first initial experiment (Hjalmarsson et. al.,

1998). The second term in the right-hand side,

'yK

c

, is measured from the control system

structure. Therefore the experimental scheme to

calculate the gradients results in terms of Figure 2

(without the blocks RM and IFT for the sake of

simplicity).

Figure 2: Experimental scheme to calculate the gradients

in the IFT-based state feedback control system structure.

The block

'

c

K

in Figure 2 plays the role of filter.

It differs from one experiment to another one

depending on the actual parameter with respect to

which the gradient is computed.

Since the calculation of the gradients has been

derived in the MIMO framework, m+1=4

experiments are done with it. The first experiment,

referred to also as the normal one, is done with the

control system structure presented in Figure 1 in

order to measure the controlled output y. The next

m=3 experiments, called the gradient experiments,

are done with the experimental scheme presented in

Figure 2. These experiments are done separately for

each parameter in K

c

(defined in (11)) considering

the zero values of the other m–1=2 parameters

(because their derivatives with respect to the current

parameter are zero).

Once the experiments are done the parameters

vector must be updated. Newton’s algorithm is

generally used as one convenient technique which

iteratively approaches a zero of a function without

knowledge of it’s expression. The update law to

calculate the next parameters vector

1+i

ρ

is

)]([

1

1 i

ii

ii

J

est ρ

∂

∂

γ−=

−

+

ρ

Rρρ

,

(16)

where i is the index of the current iteration /

experiment,

i

γ

is the step size,

)]([

i

J

est ρ

∂

∂

ρ

is the

estimate of the gradient, and the regular matrix R

i

can be the estimate of the Hessian matrix (positive

definite) or the identity matrix. The identity matrix is

employed when simple implementations are needed.

Making use of all aspects presented before the

new IFT algorithm consists of the following steps to

be performed per iteration:

Step 1. Do the normal experiment and measure y

based on the control system structure presented in

Figure 1. Next do the three gradient experiments

making use of the experimental scheme presented in

Figure 2 and measure the closed-loop system output

that gives the gradient of the controlled output,

'y

ρ

y

=

∂

∂

.

Step 2. Calculate the output of the reference

model, y

d

, in terms of the control system structure

presented in Figure 3.

Step 3. Calculate the estimate of the gradient of

the objective function:

∑

=

−

∂

∂

=ρ

∂

∂

N

t

d

T

i

N

J

est

1

])([

1

)]([ yρy

ρ

y

ρ

.

(17)

Step 4. Calculate the next set of parameters

1+i

ρ

according to the update law (16).

Three aspects can be highlighted with respect to

the above presented IFT algorithm. First, prior to the

four steps the designer should set the step size, the

reference model and the initial controller parameters

in the vector

0

ρ

. Second, the first task of the state

feedback controller is to ensure an initially stable

control system. The pole placement design can be

used with this regard. Third, the estimate of the

Hessian matrix should be calculated in the step 3 is

it is used as the matrix R

i

in the update law (16) or

an additional experiment can be employed with this

regard.

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

44

3 ACTUATOR SATURATION

PROBLEM

In many cases the actuator is characterized by a

nonlinear input-output map caused by the actuator

saturation. That is a problem because it introduces

usually nonlinear behaviours in the evolution of the

process. Hence it should be avoided. When making

use of the integrator in the controller the actuator

saturation problem becomes important since the

actuator that enters a deep saturation region requires

usually a longer time to re-enter the active region of

normal operation.

Analyzing the structure illustrated in Figure 2

and used in the gradient experiments it is clear that

when the state vector is injected in the control signal

it may cause saturation. Hence the experiment will

be prevented from calculating the correct gradients.

In the following, an actuator with the active input

range varying from −1 to +1 is considered.

One solution to cope with the above mentioned

problem is to design the experiment in such a

manner that the actuator never enters saturation. For

this, the injected quantity must be in the active

region of the actuator’s input-output static map. The

quantity can be scaled to its maximum value from its

evolution. That is obtained by dividing every sample

to the maximum absolute value from the sample

vector. So it is guaranteed that the new quantity to

be injected will be within the accepted domain of the

actuator input.

It can be shown as follows how the gradient

experiments will be influenced. The general case of

MIMO IFT will be considered. First, the scaled,

added value to the control is defined as

|)(|max ,/)()(

,1

tzMMtztz

Nk

s

=

=

=

.

(18)

Next the gradient of the control signal with respect

to the parameters vector,

'u

, can be expressed in

(19) accepting a MIMO control loop with the

controller transfer function C:

'')('' CyzCyyrCu −=−−=

.

(19)

Equation (19) is divided by M resulting in the

following relationship between the scaled values of

the gradients,

Muu

s

/'' =

and

Myy

s

/'' =

:

'/'

ss

CyMzu −=

.

(20)

Concluding, dividing (13) by (18) the result will

be

''

ss

Puy =

.

(21)

Practically a scaled value of the estimate of the

gradient can be obtained making use of the (20) and

(21). After the gradient experiments are done the

measured values

'

s

y

are multiplied by M. Thus they

will give the normal estimate of the gradient to be

used in the iterative minimization of the objective

function J.

4 CASE STUDY AND REAL-TIME

EXPERIMENTS

The validation of the theoretical approaches is done

in terms of a case study consisting of a position

control, y=α, of a DC servo system with backlash.

The experimental setup illustrated in Figure 3 is

built starting with the INTECO DC motor laboratory

equipment. It makes use of an optical encoder for

the angle measurement and a tacho-generator for the

measurement of the angular speed. The tacho-

generator measurements are very noisy. The speed

can also be observed from the angle measurements.

The control system performance indices such as

settling time and overshoot can be assessed easily.

The process (1) is characterized by the

parameters

88.139

=

s

K

and

s 9198.0=

s

T

, obtained

after experimental identification. The initial

parameters vector has been set to

T

]005.00126.00132.0[

0

=ρ

which has been

obtained to stabilize the system.

A constant reference input has been applied,

rad 150

=

r

. This allows, without any loss of

generality, to pre-tune the parameter K

r

at the value

0133.0

=

r

K

and drop it of the variables in the

optimization problem (5). That value of K

r

has been

obtained by steady-state calculation as a gain that

connects r with α through the steady-state gain of

the inner state-feedback loop. The sampling period

has been set to 0.01 s. The following reference

model has been considered:

)15.1/(1)(

2

++= sssG

RM

.

(22)

Figure 3: Experimental setup.

ITERATIVE FEEDBACK TUNING APPROACH TO A CLASS OF STATE FEEDBACK-CONTROLLED SERVO

SYSTEMS

45

Its corresponding pulse transfer function has been

obtained for the accepted sampling period. The

behaviour of the control system before the

application of the IFT algorithm is illustrated in

Figure 4.

Figure 4: Reference model output and controlled output

(position) versus time before IFT.

The IFT algorithm has been applied according to

the steps presented in Section 3. The parameters

have been set to

0001.0=γ

i

and

3

IR

i

=

. The

behaviour of the control system after 12 iterations is

presented in Figure 5. The control system

performance enhancement is highlighted. It is

reflected by smaller overshoot and settling time.

Figure 5: Reference model output and controlled output

(position) versus time after IFT.

The variation of the objective function versus the

iteration number is illustrated in Figure 6. It shows a

good decrease of the objective function and the fact

that the number of iterations can be even smaller.

Figure 6: Objective function versus iteration number.

5 CONCLUSIONS

The paper has been presented a new approach to the

IFT-based design of state feedback control systems

meant for a class of second-order systems with

integral component. The new IFT algorithm can be

applied without any difficulties to the state feedback

control of systems of arbitrary order.

The case study accompanied by real-time

experimental results validates the theoretical

approaches. The control system designed exhibits

better performance indices compared to the situation

prior to the application of the IFT algorithm.

The static and kinetic frictions were neglected.

They can result in the nonlinearity of the input-

output static map

)(uf

=

ω

. The idealization

considered here simplifies the model to be handled

easily because the nonlinearity is not strong.

The first limitation of the proposed IFT approach

concerns the tuning of the initial parameters of the

controller (grouped in the vector

0

ρ

). That problem

is not simple if nonlinear processes are involved.

The second limitation is that the global optimum

cannot be guaranteed. Hence only quasi-optimal

state feedback control systems can be designed.

The presence of the parameter K

r

presented in

Figure 1 and Figure 2 is not mandatory because the

integrator acts in the direction of error alleviation.

So the control system structure can be simplified.

However its presence is important because it can

influence the initial control error with effects on the

convergence of the IFT algorithm.

The future research will be focused on: the

consideration of more complex objective functions

to include the control signal, the state and output

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

46

sensitivity functions as well, the generalization to

nonlinear processes (Cottenceau et al., 2001;

Johanyák and Kovács, 2007; Savaresi et al., 2006;

Andrade-Cetto and Thomas, 2008; Giua and Seatzu,

2008; Precup et al., 2008a; Dolgui et al., 2009)

including MIMO servo systems, and the mapping of

the results from the linear case onto the parameters

of the fuzzy controllers in the framework of state

feedback fuzzy control systems. The convergence

analysis of all IFT algorithms is needed.

ACKNOWLEDGEMENTS

The paper was supported by the CNMP & CNCSIS

of Romania. The first and fifth authors are doctoral

students with the “Politehnica” University of

Timisoara, Romania, and also SOP HRD

stipendiaries co-financed by the European Social

Fund through the project ID 6998.

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