Easy Tuning of PID Controllers for Specified Performance

Transcription

1 IFAC Coferece o Advaces i PID Cotrol PID' Brescia (Italy, arch 8-, FrPS. Easy Tuig of PID Cotrollers for Specified Performace Š. Bucz*, A. Kozáková* ad V. Veselý* *Istitute of Cotrol ad Idustrial Iformatics, Faculty of Electrical Egieerig ad Iformatio Techology, Slovak Uiversity of Techology i Bratislava, Ilkovičova, SK-8 9 Bratislava, Slovak epublic ( stefa.bucz@stuba.sk, alea.kozakova@stuba.sk, vojtech.vesely@stuba.sk Abstract: The preseted method allows achievig maximum overshoot ad specified settlig time of the closed-loop step respose. It provides a simple way to cotrol liear stable SISO systems eve if the mathematical model is ukow. Tuig rule parameters are based o oe suitably chose poit of the plat frequecy respose obtaied by sie-wave sigal with specified excitatio frequecy, ad the required phase margi. The mai result provided is costructio of empirical charts used to covert time-domai performace specificatios (maximum overshoot ad settlig time ito frequecy domai performace measure (phase margi. The method is applicable for systematic shapig of the closed-loop respose of the plat. The ew approach has bee verified o a set of bechmark examples ad o a real plat as well. Keywords: PID cotroller, performace assessmet, overshoot, settlig time, phase margi, gai crossover. INTODUCTION Tuig methods are typically two-stage procedures cosistig of idetificatio of certai characteristic data of the plat with ukow mathematical model, followed by cotroller desig. Cotroller tuig rules that directly iclude idetified plat data have bee developed experimetally by techological process specialists (Veselý,. The widespread use of their moder versios is due to their simple implemetatio ad possibility to directly itegrate performace specificatios ito cotroller desig algorithms. Although there are about 8 various sources of PID cotroller tuig methods (Åström ad Hägglud,, % of implemeted cotrollers permaetly operate i maual mode, ad 5% of them use factory-tuig without ay up-date with respect to the specific plat. Hece, there is atural eed for effective PID cotroller desig algorithms that eable ot oly modifyig the cotrolled variable but also achievig specified performace (Kozáková et al.,, (Osuský et al.,. ai advatage of the proposed PID tuig method is a fast desig procedure for performace specified i terms of maximum overshoot η max ad settlig time t s, with o eed for exact mathematical model of the plat. Idetificatio of characteristic data of the black-box type plat is carried out usig siusoidal excitatio sigal. The sie-wave egieerig method eables to achieve η max %, 9% ad t s 6,5, 5 for plats with o itegratio behaviour, η max 9.5%, 9% ad t s,5, 5 for itegratig plats (ω c deotes critical frequecy of the plat. The paper is orgaized as follows: sie-wave idetificatio techique is preseted i Sectio, Sectio describes the derived PID cotroller tuig rules based o guarateed phase margi at a suitably chose excitatio frequecy; achieved closed-loop performace is discussed i Sectio. The proposed method has bee verified via simulatio o bechmark examples, ad o a real plat - a DC motor; the results are i Sectios 5 ad 6, respectively.. PLANT IDENTIFICATION BY A SINUSOIDAL EXCITATION INPUT A setup for the proposed sie-wave method is i Fig., where (s is the trasfer fuctio of the plat with ukow mathematical model, ad SW is a switch. Sie-wave ge. w(t e(t u(t y(t PID cotroller (s - SW elay Fig.. ultipurpose loop for the proposed sie-wave method Whe the switch SW is i positio, a siusoidal excitatio sigal with magitude U ad frequecy ω (Fig.a is ijected ito the plat (s, i.e. ( ω t u( t = U si ( The plat output y(t is also siusoidal with the same frequecy ω, magitude Y ad is the phase lag ϕ with respect to the excitatio sigal u(t (Fig. b, i.e. ( ω +ϕ y( t = Y si t ( After readig the values Y a ϕ from the recorded values of u(t ad y(t, a particular poit of the plat frequecy characteristics ( j j arg ( ω ω ( jω e = jϕ( ω [ Y U ] e = ( correspodig to the excitatio frequecy ω ca be plotted i the complex plae (Fig.c.

2 IFAC Coferece o Advaces i PID Cotrol PID' Brescia (Italy, arch 8-, FrPS. u(t Fig.. Time resposes a u(t; b y(t; c locatio of (jω i the complex plae The output siusoid amplitude Y is affected by the excitatio siusoid amplitude U geerated by the sie wave geerator; it is recommeded to choose U =( 7%u max. Thus, idetified plat parameters are represeted by a triple {ω,y (jω /U (jω,φ(ω }. With the SW i positio, the idetificatio is performed i ope-loop, hece this approach is applicable for stable plats oly. Excitatio frequecy ω is take from empirically specified iterval, ad adjusted prior to idetificatio (Bucz ad Kozáková,.. SINE-WAVE ETHOD TUNIN ULES Cosider the SW i Fig. i positio ad adjust the PID cotroller i maual mode ( (s is ow a PID cotroller trasfer fuctio. The closed-loop characteristic equatio A(jω=+L(jω=+(jω (jω= ca be easily broke dow ito the magitude ad phase coditios ( jω ( jω = ( arg ( ω arg ( ω + = 8 + φ (5 where φ is required phase margi, L(jω is the ope-loop trasfer fuctio. raphical iterpretatio of (, (5 is i Fig.. Deote ϕ=arg(ω ṅ, Θ=arg (ω ṅ, ad cosider the ideal PID cotroller i the form ( s = K + + Td s (6 Ti s where K is the proportioal gai, ad T i, T d are itegral ad derivative time costats, respectively. I the frequecydomai, compariso of the right-had side of (6 ( jω = K + jk Td ω (7 Tiω with the right-had side of the PID cotroller i polar form jθ [ cos Θ + si Θ] ( jω = ( jω e = ( jω j (8 U T =π/ω t y(t ϕ T yields a complex equality cos Θ si Θ K + jk Td ω = + j (9 Tiω ( jω ( jω The PID cotroller parameters ca be obtaied from (8 ad (9 usig the substitutio (jω =/ (jω resultig from the magitude coditio (. The complex equatio (9 is the solved as a set of two real equatios cos Θ K = (a ( jω Y t Im Y U ϕ e ω si Θ K Td ω = (b βtdω ( jω where (a is a geeral rule for calculatig the cotroller gai K; substitutig (a ad the ratio β=t i /T d ito (b, a quadratic equatio i T d is obtaied after some maipulatios Td ω TdωtgΘ =. ( β Expressio for calculatig T d is the positive solutio of ( tgθ tg Θ Td = + + ( ω ω β Hece, PID cotroller parameters are calculated usig the expressios (a, T i =βt d ad (, where Θ is obtaied from the phase coditio (5 Θ = 8 + φ arg ( ω = 8 + φ ϕ ( Hece, usig the desiged PID cotroller, the idetified poit of the plat frequecy respose (jω with co-ordiates ( is moved ito the ope-loop frequecy respose poit L located o the uit circle. Hece, the idetified poit of the plat frequecy respose (jω determies the gai crossover poit L of the ope-loop L(jω [ L( jω, arg L( ω ] [ φ ] L L( jω = =, ( for which the desiged PID cotroller guaratees the required phase margi φ. Therefore for the excitatio frequecy ω L(jω =. utual situatio of the poits (jω ad L(jω is show i Fig.. Im - L(jω φ L L(jω Θ Fig.. raphical represetatio of the PID cotroller desig i the complex plae It is recommeded to derive the frequecy ω of the siusoid from the plat ultimate frequecy ω c usig the well-kow relay experimet (otach, 98, i.e. by switchig SW i Fig. ito. The excitatio frequecy is adjusted accordig to the empirical relatio (Bucz ad Kozáková, ωc,. ϕ c (jω (jω ω. 95ω (5 How to trasform the required maximum overshoot η max ad the settlig time t s ito the couple of frequecy-domai e

3 IFAC Coferece o Advaces i PID Cotrol PID' Brescia (Italy, arch 8-, FrPS. parameters (ω,φ eeded for idetificatio ad PID cotroller coefficiets tuig is described i the followig subsectio.. CLOSED-LOOP PEFOANCE UNDE THE SINE-WAVE TYPE PID CONTOLLE Lookig for appropriate trasformatio :(η max,t s (ω,φ, cosider typical phase margis φ give by the set (j=...8 {,,,5,6, 7,8, } φ = 9 (6 j split ito 5 equidistat sectios ω =,5ω c, ad geerate the set of excitatio frequecies (k=,,6 {(.,.5,.5,.65,.8,. ω } { σ ω } ω = = (7 k 95 Each elemet i (7 represets a differet idetificatio level ω k. Fig.6 ad Fig.7 show closed-loop step resposes for plats ( s =, ( s + c,5s e ( s = s + k c (8 uder PID cotrollers desiged for three values of phase margi φ =,6,8 o four differet excitatio levels σ =ω =,; σ =ω =,5; σ =ω =,5; ad σ 5 =ω 5 =,8, demostratig qualitative effect of ω k ad φ j o closed-loop step respose. Achievig t s ad η max was tested by desigig PID cotroller for a vast set of bechmark examples (Åström ad Hägglud, for excitatio frequecies ad phase margis expressed by Cartesia product φ j ω k of the sets (6 ad (7 for j=,...,8, k=,...,6. Obtaied depedeces η max =f(φ,ω ad t s =(φ,ω are plotted i Fig. (for o-itegratig plats ad i Fig.5 (for itegratig plats, where the relative settlig time τ s is t s weighted by the plat ultimate frequecy ω c. aximum overshoot η max [%] Depedeces ηmax=f(φ,ω for differet idetificatio levels ω ω =.ω c ω =.5ω c ω =.5ω c ω =.65ω c ω =.8ω c ω =.95ω c equired phase margi φ equired phase margi φ Fig.. Depedeces: a η max =f(φ,ω, b τ s =ω c t s =f(φ,ω for cotrolled plats without itegral behaviour, β=t i /T d = aximum overshoot η max [%] Depedeces ηmax=f(φ,ω for differet idetificatio levels ω ω =.ω c ω =.5ω c ω =.5ω c ω =.65ω c ω =.8ω c ω =.95ω c Požadovaá equired phase fázová margi bezpecost φ φ Fig.5. Depedeces: a η max =f(φ,ω, b τ s =ω c t s =f(φ,ω for cotrolled plats with itegral behaviour, β=t i /T d = elative settlig time τs=ωcts=f(φ,ω elative settlig time τs=ωcts=f(φ,ω Depedeces τs=f(φ,ω g g for differet idetificatio levels ω ω =.ω c ω =.5ω c ω =.5ω c ω =.65ω c ω =.8ω c ω =.95ω c Depedeces gτs=f(φ,ω for differet idetificatio levels ω ω =.ω c ω =.5ω c ω =.5ω c ω =.65ω c ω =.8ω c ω =.95ω c Požadovaá equired phase fázová margi bezpecost φ φ Cotrolled variable y(t Cotrolled variable y(t Closed-loop time resposes, ω.6 =.ω c...8 =.6 =6. = Closed-loop time resposes, ω =.5ω c...8 =.6 =6. = Fig.6. Closed-loop step resposes of the plat (s uder PID cotrollers desiged for various φ ad ω Cotrolled variable y(t Cotrolled variable y(t.6 Closed-loop time resposes, ω =.ω c...8 =.6 =6. =8. Closed-loop time resposes, ω.6 =.5ω c...8 =.6 =6. =8. Fig.7. Closed-loop step resposes of the plat (s uder PID cotrollers desiged for various φ ad ω It is a well-kow fact that the maximum overshoot η max ca be estimated from the desired phase margi φ, ad similarly the settlig time t s ca be estimated from the ope-loop gai crossover frequecy ω a *. Accordig to eiisch, aalytical depedeces η max =f(φ ad t s =f(ω a * derived for secod order closed-loop trasfer fuctios are (Bucz ad Kozáková, η.9φ 6.55 for φ 8, 7 (9 max = + η.5φ 88.6 for φ, 8 ( max = + * a.6 Closed-loop time resposes, ω =.5ω c...8 =.6 =6. = Closed-loop time resposes, ω.6 =.8ω c...8 =.6 =6. = * a π π < t s < ( ω ω The above eiisch formulae are useful to express desired closed-loop dyamics i classical aalytical desig procedures. Cotrolled variable y(t Cotrolled variable y(t Cotrolled variable y(t Cotrolled variable y(t.6 Closed-loop time resposes, ω =.5ω c...8 =.6 =6. = Closed-loop time resposes, ω.6 =.8ω c...8 =.6 =6. =8.

4 IFAC Coferece o Advaces i PID Cotrol PID' Brescia (Italy, arch 8-, FrPS. However, with icreased order of the closed-loop trasfer fuctio they fail to be valid, ad are ot applicable i tuig methods if the mathematical model of the plat is ukow. Let us express the closed-loop settlig time t s similarly as i ( γπ t s = ( ω where γ represets the shape factor of the closed-loop step respose. I the eiisch relatio for a d order closed-loop system its value usually rages from to, depedig o dampig coefficiet specificatios (Bucz ad Kozáková,. I the proposed sie-wave method, γ chages more cosiderably withi the iterval (.5;6 strogly depedig o the phase margi φ at the give excitatio frequecy ω. To explore settlig times of closed-loops with differet dyamics it is useful to defie a ew performace measure, the so-called relative settlig time t ω = πγ ( φ ( s Substitutig for ω =σω c ito ( we ca defie the relative settlig time τ s =t s ω c as follows π t s ω c = γ ( φ ( σ The relative settlig time ( relates the settlig time t s with the plat ultimate frequecy ω c, whereby the left-had side of ( is idepedet from the excitatio frequecy ω. This empirical depedece is plotted i Fig.b (for o-itegratig plats ad Fig.5b (for itegratig plats for differet idetificatio levels ω k, showig that with icreasig the desired phase margi φ, the relative settlig time first drops ad after achievig its optimal value τ s_opt grows agai quadratically. Empirical depedeces i Fig. ad Fig.5 have bee approximated by quadratic regressio curves ad are called B-parabolas (Bucz ad Kozáková,. B-parabolas are a useful tool to carry out the trasformatio :(η max,t s (ω,φ that eables to choose appropriate values of phase margi ad excitatio frequecy φ ad ω, respectively, to guaratee the performace specified i terms of maximum overshoot η max ad settlig time t s. Note that pairs of B-parabolas at the same level are always to be used. The sie-wave type PID cotroller desig procedure. Set the PID cotrol ito maual mode. Fid the critical frequecy ω c of the plat usig the multipurpose loop i Fig. (SW i positio.. From the required settlig time t s calculate the relative settlig time τ s =ω c t s.. O the vertical axis of the plot i Fig.b or Fig.5b fid the value τ s calculated i Step.. Choose the excitatio level σ (e.g. σ 5 =ω 5 =,8. 5. For τ s, fid the correspodig phase margi φ o the parabola τ s =f(φ,ω at excitatio level foud i Step. 6. ead φ from Step 5 o the horizotal axis of the plot i Fig.a or Fig.5a, ad fid the correspodig maximum overshoot η max o the parabola η max =f(φ,ω at the excitatio level foud i Step. 7. If the foud η max is iappropriate, repeat steps to 6 for parabolas τ s =f(φ,ω, ad η max =f(φ,ω correspodig to other levels σ k =ω k (related with the choice σ 5 =ω 5 =,8 for σ k ={,;,5;,5;,65;,95}, k=...,6. epeat util both required performace measures η max ad t s are satisfied. 8. Usig the critical frequecy ω c (from Step ad the chose excitatio level σ (from Step, calculate the excitatio frequecy ω accordig to ω =σω c. 9. Idetify the plat usig siusoidal excitatio sigal with frequecy ω specified i Step 8 (SW is i positio.. Specify ϕ=arg(ω, ad (ω. Calculate the cotroller argumet Θ by substitutig ϕ ad φ ito (.. Calculate the PID cotroller parameters by substitutig the idetified values ϕ=arg(ω, (ω ad specified φ ito tuig rules (a, T i =βt d ad (. The above PID cotroller desig procedure has bee itegrated ito the auto-tuig algorithm of the preseted sie-wave method. To estimate computatio time t h of PID coefficiets, followig approximate relatio ca be used π (... + t h = t 7 + (5 ωc ω where t -7 is covergece time of the iteratio fragmet (steps 7 of the above procedure. It is evidet, that computatio time of PID coefficiets depeds o the plat dyamics. 5. VEIFICATION OF THE SINE-WAVE ETHOD ON BENCHAK EXAPLES Usig the sie-wave method let us desig ideal PID cotrollers (6 for the followig plats A ( s =, (,s + B ( s, s(7,5s +,s = e (6 The cotrol objective is to secure two differet performaces: (η max,τ s =(%,; (η max,τ s =(5%, for the plat A, (η max,τ s =(%,; (η max,τ s =(%, for the plat B. PID cotroller desig for the plat A (s Critical frequecy of the plat idetified by the otach test is ω c =7,6[rad/s]. The prescribed closed-loop settlig time is t s =τ s =/7,6[s]=69,[ms]. For the first expected performace (η max ;τ s =(%; a satisfactory choice is (φ ;ω =(5 ;,5ω c resultig from the B,5 parabola i Fig.. The secod performace i terms of (η max ;τ s =(5%; ca be achieved by choosig (φ ;ω = =(7 ;,8ω c resultig from the B,8 parabola i Fig.. Siewave type idetificatio of A (s at excitatio frequecies ω =,5ω c ad ω =,8ω c is depicted i Fig.8.

5 IFAC Coferece o Advaces i PID Cotrol PID' Brescia (Italy, arch 8-, FrPS. Sie-wave type idetificatio of A (s, ω =.5 y(t u(t Sie-wave type idetificatio of A (s, ω =.8 y(t u(t.5 Closed-loop time resposes for system A (s η max * =9,7%, t s * =58,[ms].5 φ =5, ω =,5ω c Fig.8. Sie-wave type idetificatio of the plat A (s at a ω =,5 for (η max,τ s =(%,; b ω =,8 for (η max,τ s =(5%, Idetified poits for the first ad secod desigs are A (j,5ω c =,e -j ad A (j,8ω c =,9e -j65, respectively. Accordig to Fig.9, both poits are located i the Quadrat II of the complex plae, o the Nyquist plot A (jω (blue curve i Fig.9 which verifies the idetificatio. Usig the PID cotroller desiged for (φ ;ω =(5 ;,5ω c, the poit A (j,5ω c is moved ito the gai crossover L A (j,5ω c =e -j o the uit circle, which verifies achievig the phase margi φ =8 - =5 (red Nyquist plot i Fig.9. By desigig PID cotroller for (φ ;ω = =(8 ;,8ω c, the poit A (j,8ω c has bee moved ito L A (j,8ω c =e -j yieldig phase margi φ =8 - =7 (gree Nyquist plot i Fig.9. Imagiary Axis Nyquist plots, φ =5, ω =.5ω c ; φ =7, ω =,8ω c L A (jω 7 5 L A (j,5ω c L A (jω L A (j,8ω c A (j,8ω c A (j,5ω c A (jω eal Axis Fig.9. Nyquist plots of A (s, ad ope-loops for required performaces (η max,τ s =(%, ad (η max,τ s =(5%, Performace read from the closed-loop step respose i Fig.b (red plot η max * =9,7%, t s * =58,[ms] was achieved usig PID cotroller coefficiets (K;T i ;T d = =(.8;.9;.9. Performace i terms of η max * =,89%, t s * =6,5[ms] idetified from the closed-loop step respose i Fig.a (gree plot complies with the required performace. I this case the PID cotroller parameters are (K;T i ;T d =(.986;.88; Fig.. Closed-loop step resposes with A (s ad required performace a (η max,τ s =(%,, b (η max,τ s =(5%, PID cotroller desig for the plat B (s Accordig to plat critical frequecy ω c =,7[rad/s], the required settlig time is t s =τ s =/,7[s]=8,9[s]. Time delay of B (s is D B =,[s]. The first performace specificatio (η max ;τ s =(%; ca be provided usig the B,5 parabolas for β= (Fig.5 at ω =,5 ad for parameters (φ ;ω =(5 ;,5ω c, supplyig the augmeted ope-loop phase margi φḿ =φ +ω D B =5 +, =6, ito the cotroller desig algorithm. The secod performace specificatio (η max ;τ s = =(%, is achievable usig the B,5 parabolas i Fig.5 for β= ad ω =,5 ad parameters (φ ;ω =(6 ;,5ω c. To reject the ifluece of D B, istead of φ =6 the augmeted ope-loop phase margi φḿ =φ +ω D B =6 +,5 =76,5 was supplied ito the PID cotroller desig algorithm. Sie-wave type idetificatio of B (s at ω =,5ω c ad ω =,5ω c is depicted i Fig Sie-wave type idetificatio of B (s, ω =.5 y(t u(t φ =7, ω =,8ω c η max * =,89%, t s * =6,5[ms] Fig.. Sie-wave type idetificatio of the plat B (s at a ω =,5 for (η max,τ s =(%,; b ω =,5 for (η max,τ s =(%, Usig PID cotroller, the first idetified poit B (j,5ω c = =,7e -j (Desig No. was moved ito the gai crossover L B (j,5ω c =e -j7 located o the uit circle ; this verifies achievig the phase margi φ =8-7 =5 (red Nyquist plot i Fig.. Achieved performace (η max * =9,6%, t s * =8,7[s] red from the closed-loop step respose i Fig.a (red plot was obtaied usig desiged cotroller coefficiets (K;T i ;T d =(.78;6.99; Sie-wave type idetificatio of B (s, ω =.5 y(t u(t

6 IFAC Coferece o Advaces i PID Cotrol PID' Brescia (Italy, arch 8-, FrPS. The secod idetified poit B (j,5ω c =8,e -j9 (Desig No. was moved ito L B (j,5ω c =e -j8 achievig the phase margi φ =8-8 =6 (gree Nyquist plot i Fig.. Achieved performace (η * max =9,7%, t * s =8,[s] red from the step respose i Fig.b (gree plot meets the required specificatio ad was obtaied by PID cotroller with coefficiets (K;T i ;T d =(.9;.78; Imagiary Axis Nyquist plots, φ =5, ω =.5ω c ; φ =6, ω =.5ω c L B (jω B (jω L B (j,5ω c 6 5 L B (jω L B (j,5ω c eal Axis Fig.. Nyquist plots of B (s, ad ope-loops for required performaces (η max,τ s =(%, ad (η max,τ s =(%, B(j,5ωc B(j,5ωc DC motor speed y(t [V] DC motor speed y(t [V] Time (s 8 6 DC motor speed time resposes for a η max =%, τ s =; b η max =5%, τ s = Time (s Fig.. Closed-loop time resposes of the DC motor speed for a η max =%, τ s =; b η max =5%, τ s = 7. CONCLUSIONS cotrolled speed referece speed cotrolled speed referece speed esultig closed-loop step resposes depicted i Fig., Fig. ad Fig. prove that PID cotrollers were able to guaratee the required performace measure values. The proposed ew sie-wave type desig method allows successful PID cotroller tuig. Aother importat cotributio of the paper is costructio of empirical plots covertig egieerig time-domai requiremets specified by a process techologist (maximum overshoot ad settlig time ito frequecy domai performace specificatio (i terms of phase margi ad gai crossover frequecy..5.5 Closed-loop time resposes for system B (s φ =5, ω =,5ω c η max * =9,6%, t s * =8,7[s] ACKNOWLEDENT This research work has bee supported by the Scietific rat Agecy of the iistry of Educatio of the Slovak epublic, rat No. // φ =6, ω =,5ω c η max * =9,7%, t s * =8,[s] Fig.. Closed-loop step resposes with B (s, ad required performace a (η max,τ s =(%,, b (η max,τ s =(%, 6. VEIFICATION OF THE SINE-WAVE ETHOD ON A EAL PLANT The sie wave method was applied to cotrol a physical model of a DC permaet maget motor; cotrolled variable was the speed, ad plat iput u(t was armature voltage geerated usig the atlab-ealtime Workshop cotrol system. A speedvoltage geerator was used to sese the output variable y(t. The cotrol objective was to guaratee the performace requiremets: η max =%, η max =5% ad τ s =. EFEENCES Åström, K.J. ad Hägglud, T. (. Bechmark Systems for PID Cotrol. IFAC Workshop o Digital Cotrol PID', pp. 8-8, Terrassa, Spai, April 5-7,. Bucz, Š. ad Kozáková, A. (. PID Cotroller Desig for Specified Performace. Itroductio to PID Cotrollers: Theory, Tuig ad applicatio to frotier areas. Departmet of Chemical Egieerig, CLI, Adyar, Cheai, Idia,, ISBN Kozáková, A., Veselý, V. ad Osuský, J. (. Decetralized Digital PID Desig for Performace. I: th IFAC Symposium o Large Scale Systems: Theory ad Applicatios, Lille, Frace,.-.7., Ecole Cetrale de Lille, ISBN Osuský, J.; Veselý, V. ad Kozáková, A. (. obust decetralized cotroller desig with performace specificatio, ICIC Express Letters, Vol., No., pp. 7-76, ISSN 88-8X, Kumamoto, Japa. otach, V. (98. Avtomatizacija astrojki system upravleija. Eergoatomizdat, oskva, (i ussia Veselý, V. (. Easy Tuig of PID Cotroller. Joural of Electrical Egieerig, Vol.5, No.5-6, pp. 6-9, ISSN 5-6, Bratislava, Slovak epublic.