VERIFICATION OF ROBUST PROPERTIES OF DIGITAL CONTROL

CLOSED-LOOP SYSTEMS

Vladimír Bobál, Ľuboš Spaček and Peter Hornák Tomas Bata University in Zlín Faculty of Applied Informatics

Nad Stráněmi 4511 760 05 Zlín

Czech Republic E-mail: bobal@fai.utb.cz

KEYWORDS

Digital Control, Polynomial Methods, Robustness, Robustness Margins, LQ Method, Simulation of Control Loop Systems. ABSTRACT

Robustness is specific property of closed-loop systems when the designed controller guarantees control not only for one nominal controlled system but also for all predefined class of systems (perturbed models). The robust theory is mainly exploited for design of the continuous-time systems. This paper deals with an experimental simulation investigation of robust properties of digital control closed-loop systems. Minimization of the Linear Quadratic (LQ) criterion was used for the design of control algorithm. Polynomial approach is based on the structure of the controller with two degrees of freedom (2DOF). Four types of process models (stable, non-minimum phase, unstable and integrating) were used for controller design. The Nyquist plot based characteristics of the open-loop transfer function (gain margin, phase margin and modulus margin) served as robustness indicators. The influence of change of process gain was chosen as a parametric uncertainty. The experimental results demonstrated that a robustness of examined digital control closed-loop systems could be improved by addition of user-defined poles (UDP). INTRODUCTION

One of possible approaches to digital control systems is the polynomial theory. Polynomial methods are design techniques for complex systems (including multivariable), signals and processes encountered in control, communications and computing that are based on manipulations and equations with polynomials, polynomial matrices and similar objects. Systems are described by input-output relations in fractional form and processed using algebraic methodology and tools (Šebek and Hromčík 2007). The design procedure is thus reduced to algebraic polynomial equations. Controller design consists of solving polynomial (Diophantine) equations. The Diophantine equations

can be solved using the uncertain coefficient method – which is based on comparing coefficients of the same power. This is transformed into a system of linear algebraic equations (Kučera 1997). It is obvious that the majority processes met in industrial practice are influenced by uncertainties. The uncertainties suppression can be solved by implementation of either adaptive control or robust control. The robust control and the adaptive control are viewed as two control techniques, which are used for controller design in the presence of process model uncertainty - process model variations (Landau 1999; Landau et al. 2011). The design of a robust controller deals in general with designing the controller in the presence of process uncertainties. This can be simultaneously: parameter variations (affecting low- and medium-frequency ranges) and unstructured model uncertainties (often located in high-frequency range). The aim of this paper is the experimental examination of the robustness of digital controllers based on LQ method. Robustness is the property when the dynamic response of control closed loop (including stability of course) is satisfactory not only for the nominal process transfer function used for the design but also for the entire (perturbed) class of transfer functions that expresses uncertainty of the designer in dynamic environment in which a real controller is expected to operate. The design of a robust digital pole assignment controller is investigated in (Landau and Zito 2006), the robust stability of discrete-time systems with parametric uncertainty is analysed in (Matušů 2014). A more comprehensive discussion of robustness is taken when the design based on frequency methods is considered. One can readily compare the system gain at the desired operating point and the point(s) of onset of instability to determine how much gain change is acceptable. Only this method will be used for investigation of the robustness of digital control stable, unstable, non-minimum phase and integrating processes. The paper is organized in the following way. The fundamental principle of the robustness of digital

Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD)

control-loop in Section 2.presented in robust propetheir results concludes thi

ROBUSTNECONCEPTS

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Figure DESIGN OFCONTROLL

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] [ ]2) ( )uk q u k+

nalization conput to the vrion minimizaectral factorizhe system

( ) ( )1z D z δ− =

so that d0 = 1

troller can befunctions

)11 z−−

)2

21

q zz

−

−− (12)

olving linearFrom first

1( )D z − (13)

ck controllernomials Q, P.olynomial.

signal w isthe controllerng polynomial

1( )z− (14)

e, polynomialynomial whichn. Then it isequation (14)

(15)

) ( )1y k

z− (16)

in this paperra 1991).

y to minimizeization of the

] }2 (17)

nstant, whichvalue of theation (17) willzation for an

( )1D zδ − (18)

.

e

r t

r .

s r l

l h s )

r

e e

h e l n

Because the experimental process model (10) is second-order with second-degree polynomials( ) ( )1 1,A z B z− − , the second-degree polynomial is also

obtained from the spectral factorization:

( )1 1 21 21D z d z d z− − −= + + (19)

Spectral factorization of polynomials of the first- and the second- degree can be computed by analytical way; the procedure for higher degrees must be performed iteratively (Bobál et al. 2005). The MATLAB Polynomial Toolbox (Šebek 2014) can be used for a computation of spectral factorization of the higher degree polynomials using file spf.m by command

d = spf(a*qu*a' + b*b') (20)

It is known that by using the spectral factorization (18), it is possible to compute only two suitable poles (α, β). It is obvious from equation (13) that in this case a choice of the fourth-degree polynomial D(z) is suitable

( )1 1 2 3 44 1 2 3 41D z d z d z d z d z− − − − −= + + + + (21)

Therefore, the other poles (γ1, γ2) are needed to be user-defined. The application of user-defined poles is significant mainly in nonstandard process models (e.g. non-minimum phase, unstable, integrating or with time-delay). The usage of user-defined poles makes it possible to improve the robustness of these processes. Then the digital 2DOF controller (16) can be expressed in the form

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )0 0 1 2

1 1

1 2

1 1 2

u k r w k q y k q y k q y k

p u k p u k

= − − − − −

+ − − + − (22)

where

1 2 3 40

1 2

1 d d d dr

b b+ + + +

=+

(23)

and parameters 0 1 2 1, , ,q q q p are computed from (13).

SIMULATION VERIFICATION AND RESULTS

A simulation verification of designed algorithms was performed in MATLAB/SIMULINK environment. The robustness of individual control loops was experimentally investigated by changing characteristic polynomial degree by adding user-defined poles. In addition, the influence of a change of the gain of the nominal process model was examined. From the point of view of robust theory, it is possible to consider these experiments as determination of individual robust stability margins that are caused by parametric uncertainty influence. This paper presents robustness properties of the digital second-order control systems that can be described by the following continuous-time transfer function: 1) Stable system:

( )1 2

110 7 1

G ss s

=+ +

(24)

2) Non-minimum phase stable system:

( )2 2

1 210 7 1

sG ss s−

=+ +

(25)

3) Unstable system:

( )3 2

18 2 1

G ss s

=− −

(26)

4) Integrating system:

( ) ( )41

2 1G s

s s=

+ (27)

Let us now discretize (24) - (27) using a sampling period 0 2 sT = . Discrete forms of these transfer functions are (see (10))

( )1 2

11 1 2

0 1281 0 08031 0382 0 24661N

z. z .G.

zz z.

− −−

− −

+=

+− (28)

( )1 2

12 1 2

0 0736 0 28201 0382 0 24661N

z. z .G z.z. z

− −−

− −

+=

+−−

(29)

( )1 2

13 1 2

01 1

0 3104 36563 3248 6487N

z. z .G.

zz . z

− −−

− −

+=

+− (30)

( )1 2

14 1 2

01 0

0 7385 52851 3679 3679N

z. z .G.

zz . z

− −−

− −

+=

+− (31)

Transfer functions (28) – (31) represent nominal models. From the second-order model (10) and two parts (11), (12) of digital 2DOF controller, the open-loop transfer function can be expressed using individual process and controller parameters

( )3 2

1 2 3 44 3 2

1 2 3 4OL

e z e z e z eG z

z f z f z f z f+ + +

=+ + + +

(32)

where

( )( )

1 1 0 2 1 1 2 2 3 1 2 2 1

4 1 0 1 1 1 2 1 2 1 2

3 1 1 1 2 4 2 1

; ; ;; 1; ;

1 ;

e b q e b q b q e b q b qe b q f p a f p a a a

f p a a a f a p

= = + = +

= = + − = − −

= − − + = −

(33)

Experimental process models (28) – (31) were used for simulation experiments. Individual simulation experiments are realized subsequently: • The individual nominal models (28) – (31) NiG for

( )1, 2,3, 4i = were multiplied by the parameter

PiK , then the perturbed models are given as

Pi Pi NiG K G= (34)

The parameter PiK was increased (decreased) as far as control closed-loops were in the stability

boundary• These exp

the polefactorizat= γ2 = 0. for all exp

• Obtainedcontrol-loby spectr

• The samethe secon

• Individuaare shown

• Frequencmodels G

• Values oMm were

Figure 3: Co

Figure 4: N

Figure 5: C

y - the critical periments we

es α, β wetion and user- The penalizaperiments.

d critical gainsoops when thal factorizatioe experimenta

nd case, only γal control behan in Figs. 3, 5

cy plots of oGNi were depict

f individual rcomputed (se

ontrol of stablKc1

Nyquist plots o

Control of nonand using U

gain ciK waere realized foere compute-defined polesation factor qu

s were used fhe poles α, β on and γ1 = 0.2al conditions γ2 = 0.4. aviours of mo, 7a, 7b and 1

open-loops wited, see (Figs.robustness maee Tabs. 1 – 4)

le model (28) = 3.55

of open loop w

n-minimum phUDP, Kc2 = 1.7

s determined.

or the case whed by spects (UDF) wereu = 10 was us

for simulation were comput

2, γ2 = 0. were used as

odels (28) – (31. ith the nomin. 4, 6, 8 and 12argins - Gm, P).

and using UD

with model (28

hase model (2972

hen tral e γ1 sed

n in ted

in

31)

nal 2). Pm,

DP

8)

9)

F

IncrThcoFiadonad

F

F

FiNpo

Figure 6: Nyq

n the case of throsses the reherefore, twoontrol closed-igs. 7a, 7b).ddition of UDn the contrarddition 1 2,γ γ .

Figure 7a: Co

Figure 7b: Co

igs. 9 (Kc3a =Nyquist plots c

oints.

quist plots of o

he unstable meal axis in to critical gainloops are in tIt is obviou

DP does not stary the contro

ontrol of unstaUDP, Kc3a

ontrol of unstaUDP, Kc3b

0.78) and 10

cross the real

open loop with

model (30), thetwo points (ns exist and the stability bus from Fig.tabilize the cool process is

able model (303a = 0.78

able model (33b = 1.29

0 (Kc3b = 1.29l axis relative

h model (29)

e Nyquist plot(see Fig. 8).two cases of

boundary (see 7a that the

ontrol process,unstable by

0) and using

0) and using

) show whereely to critical

t . f e e , y

e l

Figure 8: N

Figure 9: Cr

located on

Figure 10: Clocated on

Figure 11: C

-1.5-0.5

0

0.5

1

1.5

2

2.5

10

6 dB

4 dB

2 dB

Imag

inar

y Axi

s

-3-0.5

0

0.5

1

1.5

2

2.52 dB

Imag

inar

y A

xis

Nyquist plots o

ross point of Nn the right side

Cross point of n the left side

Control of inteUDP, K

-1 -0.5

0 dB

20 dB0 dB

B

B

Nyq

-2 -1

20 d10 dB

6 dB

4 dB

Nyq

R

of open loop w

Nyquist plot we from critical

Nyquist plot wfrom critical

egrating modeKc2 = 2.37

0 0.5

-20 dB-10 dB

-6 dB

-4

quist Diagram

Real Axis

0

0 dB

-20 dB-10 dB

-6 dB

-4

dBB

quist Diagram

Real Axis

with model (30

with real axis il point (-1, j0)

with real axis point (-1, j0)

el (31) and usi

1 1.5

dB

-2 dB

1 2

B

4 dB

-2 dB

0)

is )

is

ing

F

Tyar1.

2.3.It reReanlo

5

igure 12: Nyq

Table 1:

U1γ =

1 0.γ =

1 0.2γ =

Table 2

U1γ γ=

1 0.γ =

1 0.2γ =

Table 3:

U1γ =

1 0γ =

1 0.2γ =

Table 4:

U1γ =

1 0.γ =

1 0.2γ =

ypical values re recommend Gain margin

Phase marg Modulus ma

is obviousecommendatioecommendationd integratingocated on the l

quist plots of

Robustness M

UDP 2 0γ = 3

2.2; 0γ = 4

22; 0.4γ = 5

: Robustness M

UDP G2 0γ = 1

22; 0γ = 1

22; 0.4γ = 2

Robustness M

UDP 2 0γ= =

2.2; 0γ =

22; 0.4γ =

* Interval

** Interval*** Interval

Robustness M

UDP

2 0γ = 2

2.2; 0γ = 2

22; 0.4γ = 3

for stability mded in (Landaun: 2mG ≥ ,

in: o30 mP≤ ≤argin: 0mM ≥s from Tab

ons are fulfilleon 1 is fulfillg system (31)left side of th

open loop wit

Margins, Mod

Gm Pm 3.55 63.40 4.18 65.96 5.84 69.84

Margins, Mod

Gm Pm 1.72 -34.56 1.90 -50.87 2.37 62.18

Margins, Mod

Gm Pm * 11.42

** 11.37 *** 10.82

(0.73 – 1.29) l (0.74 – 1.32)l (0.78 – 1.37)

Margins, Mod

Gm Pm 2.37 38.26 2.61 39.31 3.15 40.88

margins in a u and Zito 200

[min: o60≤

0.5 , [min: bs. 1 – 4ed subsequentlled for stable) – their Nyqhe complex pla

th model (31)

del (28)

Mm0.64 0.68 0.74

del (29)

Mm0.41 0.47 0.58

del (30)

Mm0.19 0.20 0.19

)

del (31)

Mm0.50 0.52 0.57

robust design06): 1.6]

0.4] that thesely: e system (28)quist plots areane (see Figs.

n

e

) e .

4 and 12) and they cross the real axis relatively near the point (0, 0). Recommendation 2 is almost fulfilled for system (28) and fully for system (31). Recommendation 3 can be fulfilled for all systems except of unstable system (30). Remarks: The negative –Pm in the system (29) is caused by an unstable zero (see the first and second Nyquist plot – Fig. 6). The unstable system (30) does not fulfil any recommended value. The open-loop of this system has two cross points of Nyquist plot with real axis (-0.73, j0) and (-1.29, j0) – see Figs. 8 -10. The perturbed system is at the stability boundary for

3 3p c aK K= and 3 3p c bK K= and is stable between these

points ( )3 3 3c a p c bK K K< < - see Tab.3. CONCLUSION

It is well known that the robust theory is applied mainly for design of continuous-time controllers for control of stable systems. The paper presents an experimental simulation investigation of robustness of algorithms for digital LQ control. Individual control algorithms were verified using 2DOF modification in the MATLAB/Simulink environment. Four types of process models (stable, non-minimum phase, unstable and integrating) were used for controller design. The influence of change of process gain (parametric uncertainty) was used for determination of the stability boundary. The gain, phase and modulus margins of the Nyquist plot of open-loop sampled-data system were used as robustness indicators. Values of these indicators showed that only stable and integrating processes are in recommended intervals. Simulation experiments demonstrated that robustness margins of examined digital control closed-loop systems can be improved by addition of user-defined poles. REFERENCES Bobál, V., Böhm, J., Fessl, J. and J. Macháček. 2005. Digital

Self-tuning Controllers: Algorithms, Implementation and Applications. Springer-Verlag, London.

Doyle, J., Francis, B. and A. Tannenbaum. 1990. Feedback Control Theory. 1990. Macmillan Publishing.

Horowitz, I., M. 1963. Synthesis of Fedback Systems. Academia Press.

Kučera, V. 1991. Analysis and Design of Discrete Linear Control Systems. Prentice-Hall, Englewood Cliffs, NJ.

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AUTHOR BIOGRAPHIES VLADIMÍR BOBÁL graduated in 1966 from the Brno University of Technology, Czech Republic. He received his Ph.D. degree in Technical Cybernetics at Institute of Technical Cybernetics, Slovak Academy of Sciences, Bratislava, Slovak Republic. He is now Professor at the Department of Process Control, Faculty of Applied Informatics of the Tomas Bata University in Zlín. His research interests are adaptive and predictive control, system identification and CAD for automatic control systems. You can contact him on e-mail address bobal@fai.utb.cz. ĽUBOŠ SPAČEK studied at the Tomas Bata University in Zlín, Czech Republic, where he obtained his master’s degree in Automatic Control and Informatics in 2016. He currently attends PhD study at the Department of Process Control. His e-mail address is lspacek@fai.utb.cz. PETER HORNÁK studied at the Tomas Bata University in Zlín, Czech Republic, where he obtained his master’s degree in Automatic Control and Informatics in 2016. He is currently working in private company focused on industrial robotics. His e-mail address is pet.hornak@gmail.com.