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PI CONTROLLER DESIGN FOR TIME DELAY SYSTEMS
PI CONTROLLER DESIGN FOR TIME DELAY SYSTEMS M. Bakoˇ sov´ a, K. Vanekov´ a, J. Z´ avack´ a Department of Information Engineering and Process Control, Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinsk´eho 9, 812 37 Bratislava, Slovak Republic fax: +421 2 52496469 and e-mail: [email protected], [email protected], [email protected] Abstract: The paper presents an approach to PI controller design for systems with varying time delay, which can be interpreted as systems with interval parametric uncertainty. The transfer function of the controlled system is modified by approximation of the time delay term by its Pade expansion or Taylor expansion in the numerator. PI controllers, that are able to assure robust stability of the feedback closed loop with the modified controlled system are found by the graphical method. Then the pole-placement method is used to specify those controller parameters, which assure certain quality of the control performance. Designed PI controllers are implemented using PLC SIMATIC S7-300 and tested by experiments on a laboratory electronic equipment with varying time delay. Keywords: PI controller, pole-placement method, robust control, time-delay system 1. INTRODUCTION Presence of a time delay in an input-output relation is a common property of many technological processes. Plants with time-delays can often not be controlled using controllers designed without consideration of a time delay. These controllers can destabilize the closed-loop system. There are several approaches for time delay treatment. One of them is based on using time-delay compensators based on Smith predictor, see e.g. Majhi and Atherton (1998). The other approach is based on an approximation of a time delay. After the approximation, a modified transfer function of the controlled process is obtained and this transfer function does not contain the term representing the time delay. Then, classical approaches to controller design can be used, as e.g. polynomial approach (Dost´al et al. (2000)), robust control approach (Zhong (2006)) and others. PI controllers belong to the most frequently used controllers in industry. A method for PI controller design for systems with varying time delay and parametric uncertainty is presented in this paper. The method combines several known approaches. The term representing the time delay in the transfer function of the controlled system is approximated by Taylor expansion in numerator or Pad´e expansion (Dost´ al et al. (2000)) at first. The modified transfer function with interval polynomials in the numerator and the denominator is obtained. Then, the graphical method is used, which designes robust PI controllers for systems with interval parametric uncertainty. The method is based on obtaining robust stability regions for parameters of PI controllers (Tan and Kaya (2003), Z´avack´a et al. (2007)). In the last step, the PI controllers
from robust stability regions are selected using the poleplacement method (Mikleˇs and Fikar (2007)) 2. CONTROLLED SYSTEM Consider a controlled time-delay system with parametric uncertainty described by the transfer function [K − , K + ] −[D− ,D+ ]s e (1) G(s) = − + [T , T ]s + 1 The reason for the controlled system description (1) consists in the fact that many aperodic processes can be identified in this form. One of possibilities for treating the system (1) is to transform the transfer function into the form of rational function only, i.e. the time-delay term is approximated, see e.g. Dost´al et al. (2000). Various approximations of the time-delay term can be used, e.g. the first order or the higher order Taylor expansions in the denominator or in the numerator, the first order or the higher order Pad´e expansions, etc. Then, the mutual dependence between polynomial coefficients in the numerator and the denominator of the modified transfer function can be ignored and the polynomials in the transfer function can be overbounded by interval polynomials. In other words, the time-delay system (1) can be approximated by the system without the time delay described by the transfer function Pm − + i [b , b ]s N (s, b) G(s, b, a) = = Pni=0 i− i+ i m ≤ n (2) D(s, a) i=0 [ai , ai ]s where N (s, b), D(s, a) are polynomials in s with coefficients changing their values between lower and upper + − + bounds b− i , b i , ai , ai .
Fk (ω) = −ω 2 Nko (−ω 2 ) Gk (ω) = Nke (−ω 2 ) Hk (ω) = Nke (−ω 2 ) Ik (ω) = Nko (−ω 2 )
Fig. 1. Control system
2
(9) 2
Jl (ω) = ω Dlo (−ω ) Kl (ω) = −Dle (−ω 2 )
3. ROBUST PI CONTROLLER DESIGN
we can write (7) and (8) as Consider the closed-loop feedback control system depicted in Fig. 1, where G(s, b, a) is the controlled system (2), C(s) is the controller, w is the reference, u is the control input and y is the controlled output. In compliance with the sixteen plant theorem (Barmish (1994)), a first order controller C(s) = kp +
ki s
(3)
robustly stabilizes an interval plant (2) if and only if it stabilizes its 16 Kharitonov plants, which are defined as Gkl (s) =
Nk (s) Dl (s)
(4)
where k ∈ {1, 2, 3, 4}, l ∈ {1, 2, 3, 4} and N1 (s) to N4 (s) and D1 (s) to D4 (s) are the Kharitonov polynomials for the numerator and the denominator of the interval system (2) (Barmish (1994)). The method for finding robust stabilizing PI controllers is based on plotting the stability boundary of the closed loop in the plane of controller parameters for frequency ω ≥ 0 (D-partition method (Neymark (1978)). Substituting s = jω into (4) and decomposing the numerator and the denominator polynomials of (4) into their even and odd parts leads to
Gkl (jω) =
Nke (−ω 2 ) + jωNko (−ω 2 ) Dle (−ω 2 ) + jωDlo (−ω 2 )
(5)
where even parts of polynomials are represented by terms with even powers of ω and odd parts are defined analogically.
kp Fk (ω) + ki Gk (ω) = Jl (ω) kp Hk (ω) + ki Ik (ω) = Kl (ω)
(10)
From these equations, parameters of the PI controller are expressed in the form Jl (ω)Ik (ω) − Kl (ω)Gk (ω) Fk (ω)Ik (ω) − Gk (ω)Hk (ω) Kl (ω)Fk (ω) − Jl (ω)Hk (ω) ki = Fk (ω)Ik (ω) − Gk (ω)Hk (ω)
kp =
(11) (12)
For the choice ω ≥ 0, we obtain sets of kp and ki from (11) and (12). After plotting the dependence of ki on kp , the stability boundary locus is obtained, which splits the (kp , ki )-plane into several regions. Then a pair (kp , ki ) is chosen in each region and the stability of the closed loop with the chosen PI controller is tested, e.g. by calculation of closed-loop poles. The stable regions are those, where pairs (kp , ki ) give closed-loop poles with negative real parts. The approach enables to find regions of PI contollers that stabilize the plant Gkl (s). To avoid potential problems with proper choice of frequency ω ≥ 0, the frequency axis can be divided into several intervals (S¨oylemez et al. (2003)) by the real values of ω which fulfill Im[G(jω)] = 0 (13) and such intervals are then sufficient for testing. To obtain PI controllers, which robustly stabilize the system (2), the over described approach must be applied onto all Kharitonov plants. The robust stability region is obtained as the intersection of stability regions of all Kharitonov plants (Tan and Kaya (2003), Z´avack´a et al. (2007)).
The closed loop characteristic equation for (5) and (3) is ∆(jω) = [−kp ω 2 Nko (−ω 2 ) + ki Nke (−ω 2 )− −ω 2 Dlo (−ω 2 )] + jω[kp Nke (−ω 2 )+ +ki Nko (−ω 2 ) + Dle (−ω 2 )] = 0
3.1 Robust PI controller synthesis for the system of the first order (6)
Then, equating the real and imaginary parts of (6) to zero, we obtain kp (−ω 2 )Nko (−ω 2 ) + ki Nke (−ω 2 ) = ω 2 Dlo (−ω 2 ) (7) kp Nke (−ω 2 ) + ki Nko (−ω 2 ) = −Dle (−ω 2 ) After denoting
(8)
Suppose that the controlled system is described by the transfer function + − + [b− 1 , b1 ]s + [b0 , b0 ] (14) G(s, b, a) = − − + [a1 , a+ 1 ]s + [a0 , a0 ] and it is necessary to find robust PI controller (3). The total number of Kharitonov plants (4) for the 1st order system (14) is 4 and k ∈ {1, 2} and l ∈ {1, 2}. The closed loop characteristic equation (6) for e.g. the Kharitonov plant G11 is − − 2 2 ∆(jω) = (−kp b− 1 ω + ki b0 − a1 ω )+ − − + jω(kp b0 + ki b− 1 + a0 ) = 0
(15)
Equating the real and imaginary parts of (15) to zero gives following expressions for calculating of kp , ki in the dependence on the frequency ω − − 2 2 b− 1 ω kp − b0 ki = −a1 ω
b− 0 kp
b− 1 ki
(16)
= −a− 0
+ (17) After the calculation of sets of kp and ki for chosen intervals of ω, the plot representing the stability boundary locus can be driven and the stability region of kp and ki can be found. The robust stability region is obtained as the intersection of stability regions obtained for all Kharitonov plants. 3.2 Robust PI controller synthesis for the system of the second order Suppose that the controlled system is described by the transfer function [b− , b+ ]s + [b− , b+ ] G(s) = − + 12 1 − +0 0 − + (18) [a2 , a2 ]s + [a1 , a1 ]s + [a0 , a0 ] and it is necessary to find robust PI controller (3). The total number of Kharitonov plants (4) for the 2nd order system (18) is 6 and k ∈ {1, 2} and l ∈ {1, 2, 3}. The closed loop characteristic equation (6) for e.g. the Kharitonov plant G11 is
closed-loop pole. Specifying suitable values of parameters ξ, ω0 , α in (22)-(24) leads to the controller which can assure the desired quality of the control response. To obtain unique solution, the system of equations for calculation of controller parameters has to be the system with zero degree of freedom. If higher order characteristic equation is considered, then any of parameters ξ, ω0 or α can be added to unknown variables. 4.1 Pole-placement method for the system of the first order Suppose that for the controlled system (25) b1 s + b0 G(s) = (25) a1 s + a0 it is necessary to find PI controller (3). The closed loop characteristic equation can be written as (23). The closed loop characteristic equation for the controlled system (25) and the PI controller (3) has also the form a0 + b0 kp + b1 ki b0 ki s2 + s+ =0 (26) a1 + b1 kp a1 + b1 kp After the choice of ξ and ω0 and comparison of coefficients in (23) and (26), parameters of the PI controller can be computed from (27) and (28) (b0 − 2ξω0 b1 )kp + b1 ki = 2ξω0 a1 − a0
∆(jω) =
+
− 2 + ki b− 0 − a1 ω )+ − − 2 jω(kp b0 + ki b1 − a− 2ω
−b1 ω02 kp
2 (−kp b− 1ω
+
a− 0)
b− 0 kp
b− 1 ki
= a− 2ω
(20)
+ − (21) After the calculation of sets of kp and ki for chosen intervals of ω, the plot representing the stability boundary locus can be driven and the stability region of kp and ki can be found. The robust stability region is obtained as the intersection of stability regions obtained for all 16 Kharitonov plants.
a− 0
The pole-placement control design belongs to the class of well-known analytical methods where transfer function of the controlled process is known (Mikleˇs and Fikar (2007)). In this method, only the closed-loop denominator that assures stability is specified. The advantage of this approach is its usability for a broad range of systems. If the controller is of PID structure then the characteristic equation can be one of the following
(s +
(28)
4.2 Pole-placement method for the system of the second order Suppose that for the controlled system (29) b1 s + b0 (29) G(s) = 2 a2 s + a1 s + a0 it is necessary to find PI controller (3). The closed loop characteristic equation can be written as (24). The closed loop characteristic equation for the controlled system (29) and PI controller (3) has also the form a1 + b1 kp 2 a0 + b1 ki + b0 kp b0 ki s3 + s + s+ = 0 (30) a2 a2 a2 After comparison of coefficients in (24) and (30), parameters of the PI controller can be computed from (31)-(33)
4. POLE-PLACEMENT METHOD
s + ω0 = 0
(22)
s + 2ξω0 s + ω02 = 0 α)(s2 + 2ξω0 s + ω02 ) = 0
(23)
2
(27)
= 0 (19)
Equating the real and imaginary parts of (19) to zero gives following expressions for calculating of kp , ki in the dependence on the frequency ω − − 2 2 b− 1 ω kp − b0 ki = −a1 ω
+
b0 ki = a1 ω02
(24)
or the combination of (23) and (24), where ξ is relative damping, ω0 natural undamped frequency, and α is a
b1 kp = a2 α + 2ξω0 a2 − a1 b0 kp + b1 ki = 2ξω0 a2 α +
ω02 a2
− a0
b0 ki = ω02 a2 α
(31) (32) (33)
It is clear that unique solution kp ki is obtained if only two of parameters ξ, ω0 or α are chosen. 5. CONTROLLED LABORATORY PROCESS Controlled laboratory process (Fig. 2) is an electronic model of a linear system with a varying time delay (Babir´ad (2006)). The process was identified from step responses measured in various working areas by Strejc
0.12
0.1
ki
0.08
0.06
0.04
0.02
0
−1
−0.5
0 kp
0.5
1
Fig. 2. Controlled laboratory process method (Mikleˇs and Fikar (2007)) in the form of a transfer function of the first order with time delay − + K G(s) = − + e−[D ,D ]s (34) [T , T ]s + 1 where K is the gain of the process, which is constant, and T − , T + , D− , D+ are the minimum and maximum values of the time constant and the time delay of the process with following values K = 0.97 T
−
(35)
= 16s = 0.27min,
D = 16s = 0.27min, −
T
+
= 19s = 0.31min
(36)
D = 39s = 0.65min
(37)
+
Using the first order Taylor expansion in numerator, the modified transfer function of the controlled system is −K[D− , D+ ]s + K G(s) = (38) [T − , T + ]s + 1 and after overbounding polynomials in (38) with interval polynomials, the transfer function of the first order is obtained in the following form G(s) =
[−37.83, −15.52]s + 0.97 [16, 19]s + 1
(39)
Similarly, using the first order Pad´e expansion, the modified transfer function of the controlled system is G(s) =
−K [D [T − , T + ] [D
− ,D + ]
s2 +
,D+ ] s+K 2 − + − ([T , T + ] + [D 2,D ] )s
−
+1 (40) and after overbounding polynomials in (40) with interval polynomials, the transfer function of the second order is obtained in the following form [−18.91, −7.76]s + 0.97 G(s) = . (41) [128, 370.5]s2 + [24, 38.5]s + 1 2
5.1 PI controller design for the controlled process Methods described in Section 3, Section 4, and in Tan and Kaya (2003), Z´ avack´ a et al. (2007) are used for PI controller design. Consider the controlled system (39). The robust stability region of PI controller parameters is found at first. The first Kharitonov plant is obtained for the maximum time delay and has the form
Fig. 3. Choices of PI controllers for the 1st order system −37.83s + 0.97 (42) 16s + 1 The expressions for calculation of kp and ki in the dependence on ω are G11 (s) =
605.28ω 2 − 0.97 (43) 1431.1ω 2 + 0.94 53.35ω 2 ki = (44) 1431.1ω 2 + 0.94 After a suitable choice of interval for ω, the stability boundary locus as the dependence of ki on kp is plotted and the stability region is found by the choice of testing points in two obtained regions. Analogical procedure must be repeated for all Kharitonov plants that can be created for (39). The stability boundaries for Kharitonov plants are depicted in Fig. 3 by blue lines. The stability regions lie in the intersection of regions below the boundaries. kp =
In the second step the pole-placement method is used for the reasonable choice of the controller from the robust stability region. The quality of the control response can be prescribed by the choice of the relative damping ξ and the natural undamped frequency ω0 in (23). The more important parameter from our point of view is the relative damping ξ and PI controllers were designed for ξ = 1 for the aperiodic and ξ = 0.5 for the periodic control response. The natural undamped frequncy ω0 was chosen in the same interval as the frequency ω. Expressions for obtaining PI controller parameters in dependence on prescribed ξ and ω0 are 31.04ξω0 + 605.28ω02 − 0.97 (45) 73.39ξω0 + 1431.1ω02 + 0.94 53.35ω02 ki = (46) 73.39ξω0 + 1431.1ω02 + 0.94 After calculation of kp and ki for chosen ξ and ω0 , the curves representing the loci of PI controllers were plotted (Fig. 3). The curve for ξ = 0.5 is represented by magenta line and for ξ = 1 by green line. Then two PI controllers where choosen as it is shown in Tab. 1 and Fig. 3 (red stars). kp =
Consider now the controlled system (41). The robust stability region of PI controllers, that stabilize (41), is shown in Fig. 4, and it is found as the intersection of stability regions of all Kharitonov plants of (41). Then, the pole-placement method is used and the loci of PI
Table 1. Controller parameters for the 1st order system C1 C2
ξ 0.5 1
kp 0.370 0.371
16 14
ki 0.0224 0.0232
12
w,y [%]
10 0.25
8 6
0.2
4 w y − Dmin y − Dmax y − Dnom
0.15 k
i
2 0 0
0.1
2
4
time[min]
6
8
10
0.05
Fig. 5. Control responses for controller C1 0 −1
0
1
2 kp
3
4
5
Fig. 4. Choices of PI controllers for the 2nd order system Table 2. Controller parameters for the 2nd order system C3 C4
ξ 0.5 1
kp 0.543 0.235
ki 0.0234 0.0111
controllers are found so, that the relative damping ξ and the natural undamped frequency ω0 in (24) are chosen. PI controllers are designed for ξ = 1 for the aperiodic control response and ξ = 0.5 for the periodic control response. The natural undamped frequency ω0 is chosen in the same interval as ω. The real pole α is added to unknown variables, and so the set of equations (31)-(33) has unique solution. The calculated real pole α has to be checked during computation, because it has to lie in the left half of the complex plane. After calculation of kp and ki for chosen ξ and ω0 , the curves representing the loci of PI controllers are plotted (Fig. 4). The curve for ξ = 0.5 is represented by magenta line and for ξ = 1 by green line Then two PI controllers are chosen as it is shown in Tab. 2 and Fig. 4 (red stars). It is necessary to note that designed PI controllers robustly stabilize only interval processes (39) and (41). Similarly, the control responses with prescribed properties will be obtained only for the process used in the pole-placement design. The possibility to use designed PI controllers for control of time-delay processses was verified by experiments. 6. CONTROL OF A LABORATORY PROCESS 6.1 SIMATIC S7-300 Industrial control system SIMATIC S7-300 (Koˇzka and Kvasnica (2001)), (Siemens (1996)) was used for implementation of designed PI controllers. SIMATIC includes programming (STEP7) and visualization (WinCC) software, which are used for programming of programmable logic controller (PLC). The organization
block OB35 includes a function block of a PID controller (FB41). 6.2 Experimental results The laboratory process (Fig. 2) was controlled by the designed PI controllers (Tab. 1) and (Tab. 2). Twelve various control experiments are presented. Control responses of the process obtained using PI controllers C1 - C4 for three different values of time delay are shown in Figs. 5, 6, 7 and 8. Here, w is the setpoint, y the controlled output and Dmin , Dmax , Dnom are the minimum, maximum and nominal time delays of the controlled system, respectively. More ”aggressive” PI controllers are controllers C1 , C2 , design of which was based on the Taylor aproximation of the time delay. The controller C1 (ξ = 0.5) lead to periodic control response for maximum and nominal time delay, but the response for the minimum time delay was aperiodic. The controller C2 (ξ = 1) was able to assure the aperiodic control response for minimum and nominal time delays. The control response for maximum time delay was periodic. Less ”aggressive” PI controllers are C3 , C4 , design of which was based on Pad´e approximation of the time delay. The controller C4 (ξ = 1) was able to assure the aperiodic control response for all time delays. The controller C3 (ξ = 0.5) lead to periodic control response for nominal and maximum time delay, but the response for the minimum time delay was aperiodic. The reason is, that the pole-placement method was applied onto the model with maximum time delay. According to the presented results, it can be stated that all four tested PI controllers are able to control the laboratory time-delay process with acceptable behavior. 7. CONCLUSION A laboratory electronic equipment was identified as a system with time delay and parametric uncertainty. For this process, PI controllers were designed via combination of two methods, the graphic method for robust PI controller design for interval plants and the pole-placement method.
ACKNOWLEDGEMENTS The authors are pleased to acknowledge the financial support of the Scientific Grant Agency of the Slovak Republic under grants No. 1/0537/10 and 1/0071/09 and the Slovak Research and Development Agency under the contract No. APVV-0029-07.
12
10
w,y [%]
8
6
REFERENCES
4 w y − Dmin y − Dmax y − Dnom
2
0 0
1
2
3 4 time[min]
5
6
7
Fig. 6. Control responses for controller C2
12
10
w,y [%]
8
6
4 w y − Dmin y − Dmax y − Dnom
2
0 0
1
2
3 time[min]
4
5
6
Fig. 7. Control responses for controller C3 12
10
w,y [%]
8
6
4 w y − Dmin y − Dmax y − Dnom
2
0 0
1
2
3
4 time[min]
5
6
7
8
Fig. 8. Control responses for controller C4 The pole-placement method was used for the choice of PI controllers from the robust stability region. Designed PI controllers were implemented using the industrial control system SIMATIC and tested by experiments on the laboratory equipment. Although designed PI controller do not assure robust stabilizability of the original timedelay process, obtained experimental show that proposed approach leads to the design of PI controllers that are suitable for control of real processes with varying timedelay and parametric uncertainty.
Babir´ad, J. (2006). Model of a 2nd order system with a transport delay. Technical documentation, DIEPC IIEAM FCFT STU, Bratislava, Slovakia (in Slovak). Barmish, B. (1994). New Tools for Robustness of Linear Systems. Macmillan, New York, USA. Dost´al, P., Prokop, R., and Bob´al, V. (2000). Polynomial design of simple controllers for time delay systems. In 3rd IFAC Symp. ROCOND 2000, Prague, Czech Republic, June 21-31. Koˇzka, S. and Kvasnica, M. (2001). Programming PLC SIMATIC. DIEPC IIEAM FCFT STU, Bratislava, Slovakia (in Slovak). Majhi, S. and Atherton, D. (1998). A new smith predictor and controller for unstable and integrating processes with time delay. In 37th IEEE Conf. on Decision and Control, Tampa. Mikleˇs, J. and Fikar, M. (2007). Process Modelling, Identification, and Control. Springer Verlag, Berlin, Heidelberg. Neymark, J. (1978). Dynamical Systems and Controlled Processes. Nauka, Moscow, Russia (in Russian). Siemens (1996). Standard Software for S7–300 and S7–400 PID Control. Siemens User Manual, Siemens Automation Group. S¨oylemez, M., Munro, N., and Baki, H. (2003). Fast calculation of stabilizing PID controllers. Automatica 39, 121–126. Tan, N. and Kaya, I. (2003). Computation of stabilizing PI controllers for interval systems. In 11th Mediterranean Conference on Control and Automation, Rhodes, Greece. Z´avack´a, J., Bakoˇsov´a, M., and Vanekov´a, K. (2007). Robust PI controller design for control of a system with parametric uncertainties. In 34th Int. Conf. SSCHE, Tatransk`e Matliare, Slovakia, CDROM 038p. Zhong, Q. (2006). Robust control of time delay systems. Springer Verlag, London.
from robust stability regions are selected using the poleplacement method (Mikleˇs and Fikar (2007)) 2. CONTROLLED SYSTEM Consider a controlled time-delay system with parametric uncertainty described by the transfer function [K − , K + ] −[D− ,D+ ]s e (1) G(s) = − + [T , T ]s + 1 The reason for the controlled system description (1) consists in the fact that many aperodic processes can be identified in this form. One of possibilities for treating the system (1) is to transform the transfer function into the form of rational function only, i.e. the time-delay term is approximated, see e.g. Dost´al et al. (2000). Various approximations of the time-delay term can be used, e.g. the first order or the higher order Taylor expansions in the denominator or in the numerator, the first order or the higher order Pad´e expansions, etc. Then, the mutual dependence between polynomial coefficients in the numerator and the denominator of the modified transfer function can be ignored and the polynomials in the transfer function can be overbounded by interval polynomials. In other words, the time-delay system (1) can be approximated by the system without the time delay described by the transfer function Pm − + i [b , b ]s N (s, b) G(s, b, a) = = Pni=0 i− i+ i m ≤ n (2) D(s, a) i=0 [ai , ai ]s where N (s, b), D(s, a) are polynomials in s with coefficients changing their values between lower and upper + − + bounds b− i , b i , ai , ai .
Fk (ω) = −ω 2 Nko (−ω 2 ) Gk (ω) = Nke (−ω 2 ) Hk (ω) = Nke (−ω 2 ) Ik (ω) = Nko (−ω 2 )
Fig. 1. Control system
2
(9) 2
Jl (ω) = ω Dlo (−ω ) Kl (ω) = −Dle (−ω 2 )
3. ROBUST PI CONTROLLER DESIGN
we can write (7) and (8) as Consider the closed-loop feedback control system depicted in Fig. 1, where G(s, b, a) is the controlled system (2), C(s) is the controller, w is the reference, u is the control input and y is the controlled output. In compliance with the sixteen plant theorem (Barmish (1994)), a first order controller C(s) = kp +
ki s
(3)
robustly stabilizes an interval plant (2) if and only if it stabilizes its 16 Kharitonov plants, which are defined as Gkl (s) =
Nk (s) Dl (s)
(4)
where k ∈ {1, 2, 3, 4}, l ∈ {1, 2, 3, 4} and N1 (s) to N4 (s) and D1 (s) to D4 (s) are the Kharitonov polynomials for the numerator and the denominator of the interval system (2) (Barmish (1994)). The method for finding robust stabilizing PI controllers is based on plotting the stability boundary of the closed loop in the plane of controller parameters for frequency ω ≥ 0 (D-partition method (Neymark (1978)). Substituting s = jω into (4) and decomposing the numerator and the denominator polynomials of (4) into their even and odd parts leads to
Gkl (jω) =
Nke (−ω 2 ) + jωNko (−ω 2 ) Dle (−ω 2 ) + jωDlo (−ω 2 )
(5)
where even parts of polynomials are represented by terms with even powers of ω and odd parts are defined analogically.
kp Fk (ω) + ki Gk (ω) = Jl (ω) kp Hk (ω) + ki Ik (ω) = Kl (ω)
(10)
From these equations, parameters of the PI controller are expressed in the form Jl (ω)Ik (ω) − Kl (ω)Gk (ω) Fk (ω)Ik (ω) − Gk (ω)Hk (ω) Kl (ω)Fk (ω) − Jl (ω)Hk (ω) ki = Fk (ω)Ik (ω) − Gk (ω)Hk (ω)
kp =
(11) (12)
For the choice ω ≥ 0, we obtain sets of kp and ki from (11) and (12). After plotting the dependence of ki on kp , the stability boundary locus is obtained, which splits the (kp , ki )-plane into several regions. Then a pair (kp , ki ) is chosen in each region and the stability of the closed loop with the chosen PI controller is tested, e.g. by calculation of closed-loop poles. The stable regions are those, where pairs (kp , ki ) give closed-loop poles with negative real parts. The approach enables to find regions of PI contollers that stabilize the plant Gkl (s). To avoid potential problems with proper choice of frequency ω ≥ 0, the frequency axis can be divided into several intervals (S¨oylemez et al. (2003)) by the real values of ω which fulfill Im[G(jω)] = 0 (13) and such intervals are then sufficient for testing. To obtain PI controllers, which robustly stabilize the system (2), the over described approach must be applied onto all Kharitonov plants. The robust stability region is obtained as the intersection of stability regions of all Kharitonov plants (Tan and Kaya (2003), Z´avack´a et al. (2007)).
The closed loop characteristic equation for (5) and (3) is ∆(jω) = [−kp ω 2 Nko (−ω 2 ) + ki Nke (−ω 2 )− −ω 2 Dlo (−ω 2 )] + jω[kp Nke (−ω 2 )+ +ki Nko (−ω 2 ) + Dle (−ω 2 )] = 0
3.1 Robust PI controller synthesis for the system of the first order (6)
Then, equating the real and imaginary parts of (6) to zero, we obtain kp (−ω 2 )Nko (−ω 2 ) + ki Nke (−ω 2 ) = ω 2 Dlo (−ω 2 ) (7) kp Nke (−ω 2 ) + ki Nko (−ω 2 ) = −Dle (−ω 2 ) After denoting
(8)
Suppose that the controlled system is described by the transfer function + − + [b− 1 , b1 ]s + [b0 , b0 ] (14) G(s, b, a) = − − + [a1 , a+ 1 ]s + [a0 , a0 ] and it is necessary to find robust PI controller (3). The total number of Kharitonov plants (4) for the 1st order system (14) is 4 and k ∈ {1, 2} and l ∈ {1, 2}. The closed loop characteristic equation (6) for e.g. the Kharitonov plant G11 is − − 2 2 ∆(jω) = (−kp b− 1 ω + ki b0 − a1 ω )+ − − + jω(kp b0 + ki b− 1 + a0 ) = 0
(15)
Equating the real and imaginary parts of (15) to zero gives following expressions for calculating of kp , ki in the dependence on the frequency ω − − 2 2 b− 1 ω kp − b0 ki = −a1 ω
b− 0 kp
b− 1 ki
(16)
= −a− 0
+ (17) After the calculation of sets of kp and ki for chosen intervals of ω, the plot representing the stability boundary locus can be driven and the stability region of kp and ki can be found. The robust stability region is obtained as the intersection of stability regions obtained for all Kharitonov plants. 3.2 Robust PI controller synthesis for the system of the second order Suppose that the controlled system is described by the transfer function [b− , b+ ]s + [b− , b+ ] G(s) = − + 12 1 − +0 0 − + (18) [a2 , a2 ]s + [a1 , a1 ]s + [a0 , a0 ] and it is necessary to find robust PI controller (3). The total number of Kharitonov plants (4) for the 2nd order system (18) is 6 and k ∈ {1, 2} and l ∈ {1, 2, 3}. The closed loop characteristic equation (6) for e.g. the Kharitonov plant G11 is
closed-loop pole. Specifying suitable values of parameters ξ, ω0 , α in (22)-(24) leads to the controller which can assure the desired quality of the control response. To obtain unique solution, the system of equations for calculation of controller parameters has to be the system with zero degree of freedom. If higher order characteristic equation is considered, then any of parameters ξ, ω0 or α can be added to unknown variables. 4.1 Pole-placement method for the system of the first order Suppose that for the controlled system (25) b1 s + b0 G(s) = (25) a1 s + a0 it is necessary to find PI controller (3). The closed loop characteristic equation can be written as (23). The closed loop characteristic equation for the controlled system (25) and the PI controller (3) has also the form a0 + b0 kp + b1 ki b0 ki s2 + s+ =0 (26) a1 + b1 kp a1 + b1 kp After the choice of ξ and ω0 and comparison of coefficients in (23) and (26), parameters of the PI controller can be computed from (27) and (28) (b0 − 2ξω0 b1 )kp + b1 ki = 2ξω0 a1 − a0
∆(jω) =
+
− 2 + ki b− 0 − a1 ω )+ − − 2 jω(kp b0 + ki b1 − a− 2ω
−b1 ω02 kp
2 (−kp b− 1ω
+
a− 0)
b− 0 kp
b− 1 ki
= a− 2ω
(20)
+ − (21) After the calculation of sets of kp and ki for chosen intervals of ω, the plot representing the stability boundary locus can be driven and the stability region of kp and ki can be found. The robust stability region is obtained as the intersection of stability regions obtained for all 16 Kharitonov plants.
a− 0
The pole-placement control design belongs to the class of well-known analytical methods where transfer function of the controlled process is known (Mikleˇs and Fikar (2007)). In this method, only the closed-loop denominator that assures stability is specified. The advantage of this approach is its usability for a broad range of systems. If the controller is of PID structure then the characteristic equation can be one of the following
(s +
(28)
4.2 Pole-placement method for the system of the second order Suppose that for the controlled system (29) b1 s + b0 (29) G(s) = 2 a2 s + a1 s + a0 it is necessary to find PI controller (3). The closed loop characteristic equation can be written as (24). The closed loop characteristic equation for the controlled system (29) and PI controller (3) has also the form a1 + b1 kp 2 a0 + b1 ki + b0 kp b0 ki s3 + s + s+ = 0 (30) a2 a2 a2 After comparison of coefficients in (24) and (30), parameters of the PI controller can be computed from (31)-(33)
4. POLE-PLACEMENT METHOD
s + ω0 = 0
(22)
s + 2ξω0 s + ω02 = 0 α)(s2 + 2ξω0 s + ω02 ) = 0
(23)
2
(27)
= 0 (19)
Equating the real and imaginary parts of (19) to zero gives following expressions for calculating of kp , ki in the dependence on the frequency ω − − 2 2 b− 1 ω kp − b0 ki = −a1 ω
+
b0 ki = a1 ω02
(24)
or the combination of (23) and (24), where ξ is relative damping, ω0 natural undamped frequency, and α is a
b1 kp = a2 α + 2ξω0 a2 − a1 b0 kp + b1 ki = 2ξω0 a2 α +
ω02 a2
− a0
b0 ki = ω02 a2 α
(31) (32) (33)
It is clear that unique solution kp ki is obtained if only two of parameters ξ, ω0 or α are chosen. 5. CONTROLLED LABORATORY PROCESS Controlled laboratory process (Fig. 2) is an electronic model of a linear system with a varying time delay (Babir´ad (2006)). The process was identified from step responses measured in various working areas by Strejc
0.12
0.1
ki
0.08
0.06
0.04
0.02
0
−1
−0.5
0 kp
0.5
1
Fig. 2. Controlled laboratory process method (Mikleˇs and Fikar (2007)) in the form of a transfer function of the first order with time delay − + K G(s) = − + e−[D ,D ]s (34) [T , T ]s + 1 where K is the gain of the process, which is constant, and T − , T + , D− , D+ are the minimum and maximum values of the time constant and the time delay of the process with following values K = 0.97 T
−
(35)
= 16s = 0.27min,
D = 16s = 0.27min, −
T
+
= 19s = 0.31min
(36)
D = 39s = 0.65min
(37)
+
Using the first order Taylor expansion in numerator, the modified transfer function of the controlled system is −K[D− , D+ ]s + K G(s) = (38) [T − , T + ]s + 1 and after overbounding polynomials in (38) with interval polynomials, the transfer function of the first order is obtained in the following form G(s) =
[−37.83, −15.52]s + 0.97 [16, 19]s + 1
(39)
Similarly, using the first order Pad´e expansion, the modified transfer function of the controlled system is G(s) =
−K [D [T − , T + ] [D
− ,D + ]
s2 +
,D+ ] s+K 2 − + − ([T , T + ] + [D 2,D ] )s
−
+1 (40) and after overbounding polynomials in (40) with interval polynomials, the transfer function of the second order is obtained in the following form [−18.91, −7.76]s + 0.97 G(s) = . (41) [128, 370.5]s2 + [24, 38.5]s + 1 2
5.1 PI controller design for the controlled process Methods described in Section 3, Section 4, and in Tan and Kaya (2003), Z´ avack´ a et al. (2007) are used for PI controller design. Consider the controlled system (39). The robust stability region of PI controller parameters is found at first. The first Kharitonov plant is obtained for the maximum time delay and has the form
Fig. 3. Choices of PI controllers for the 1st order system −37.83s + 0.97 (42) 16s + 1 The expressions for calculation of kp and ki in the dependence on ω are G11 (s) =
605.28ω 2 − 0.97 (43) 1431.1ω 2 + 0.94 53.35ω 2 ki = (44) 1431.1ω 2 + 0.94 After a suitable choice of interval for ω, the stability boundary locus as the dependence of ki on kp is plotted and the stability region is found by the choice of testing points in two obtained regions. Analogical procedure must be repeated for all Kharitonov plants that can be created for (39). The stability boundaries for Kharitonov plants are depicted in Fig. 3 by blue lines. The stability regions lie in the intersection of regions below the boundaries. kp =
In the second step the pole-placement method is used for the reasonable choice of the controller from the robust stability region. The quality of the control response can be prescribed by the choice of the relative damping ξ and the natural undamped frequency ω0 in (23). The more important parameter from our point of view is the relative damping ξ and PI controllers were designed for ξ = 1 for the aperiodic and ξ = 0.5 for the periodic control response. The natural undamped frequncy ω0 was chosen in the same interval as the frequency ω. Expressions for obtaining PI controller parameters in dependence on prescribed ξ and ω0 are 31.04ξω0 + 605.28ω02 − 0.97 (45) 73.39ξω0 + 1431.1ω02 + 0.94 53.35ω02 ki = (46) 73.39ξω0 + 1431.1ω02 + 0.94 After calculation of kp and ki for chosen ξ and ω0 , the curves representing the loci of PI controllers were plotted (Fig. 3). The curve for ξ = 0.5 is represented by magenta line and for ξ = 1 by green line. Then two PI controllers where choosen as it is shown in Tab. 1 and Fig. 3 (red stars). kp =
Consider now the controlled system (41). The robust stability region of PI controllers, that stabilize (41), is shown in Fig. 4, and it is found as the intersection of stability regions of all Kharitonov plants of (41). Then, the pole-placement method is used and the loci of PI
Table 1. Controller parameters for the 1st order system C1 C2
ξ 0.5 1
kp 0.370 0.371
16 14
ki 0.0224 0.0232
12
w,y [%]
10 0.25
8 6
0.2
4 w y − Dmin y − Dmax y − Dnom
0.15 k
i
2 0 0
0.1
2
4
time[min]
6
8
10
0.05
Fig. 5. Control responses for controller C1 0 −1
0
1
2 kp
3
4
5
Fig. 4. Choices of PI controllers for the 2nd order system Table 2. Controller parameters for the 2nd order system C3 C4
ξ 0.5 1
kp 0.543 0.235
ki 0.0234 0.0111
controllers are found so, that the relative damping ξ and the natural undamped frequency ω0 in (24) are chosen. PI controllers are designed for ξ = 1 for the aperiodic control response and ξ = 0.5 for the periodic control response. The natural undamped frequency ω0 is chosen in the same interval as ω. The real pole α is added to unknown variables, and so the set of equations (31)-(33) has unique solution. The calculated real pole α has to be checked during computation, because it has to lie in the left half of the complex plane. After calculation of kp and ki for chosen ξ and ω0 , the curves representing the loci of PI controllers are plotted (Fig. 4). The curve for ξ = 0.5 is represented by magenta line and for ξ = 1 by green line Then two PI controllers are chosen as it is shown in Tab. 2 and Fig. 4 (red stars). It is necessary to note that designed PI controllers robustly stabilize only interval processes (39) and (41). Similarly, the control responses with prescribed properties will be obtained only for the process used in the pole-placement design. The possibility to use designed PI controllers for control of time-delay processses was verified by experiments. 6. CONTROL OF A LABORATORY PROCESS 6.1 SIMATIC S7-300 Industrial control system SIMATIC S7-300 (Koˇzka and Kvasnica (2001)), (Siemens (1996)) was used for implementation of designed PI controllers. SIMATIC includes programming (STEP7) and visualization (WinCC) software, which are used for programming of programmable logic controller (PLC). The organization
block OB35 includes a function block of a PID controller (FB41). 6.2 Experimental results The laboratory process (Fig. 2) was controlled by the designed PI controllers (Tab. 1) and (Tab. 2). Twelve various control experiments are presented. Control responses of the process obtained using PI controllers C1 - C4 for three different values of time delay are shown in Figs. 5, 6, 7 and 8. Here, w is the setpoint, y the controlled output and Dmin , Dmax , Dnom are the minimum, maximum and nominal time delays of the controlled system, respectively. More ”aggressive” PI controllers are controllers C1 , C2 , design of which was based on the Taylor aproximation of the time delay. The controller C1 (ξ = 0.5) lead to periodic control response for maximum and nominal time delay, but the response for the minimum time delay was aperiodic. The controller C2 (ξ = 1) was able to assure the aperiodic control response for minimum and nominal time delays. The control response for maximum time delay was periodic. Less ”aggressive” PI controllers are C3 , C4 , design of which was based on Pad´e approximation of the time delay. The controller C4 (ξ = 1) was able to assure the aperiodic control response for all time delays. The controller C3 (ξ = 0.5) lead to periodic control response for nominal and maximum time delay, but the response for the minimum time delay was aperiodic. The reason is, that the pole-placement method was applied onto the model with maximum time delay. According to the presented results, it can be stated that all four tested PI controllers are able to control the laboratory time-delay process with acceptable behavior. 7. CONCLUSION A laboratory electronic equipment was identified as a system with time delay and parametric uncertainty. For this process, PI controllers were designed via combination of two methods, the graphic method for robust PI controller design for interval plants and the pole-placement method.
ACKNOWLEDGEMENTS The authors are pleased to acknowledge the financial support of the Scientific Grant Agency of the Slovak Republic under grants No. 1/0537/10 and 1/0071/09 and the Slovak Research and Development Agency under the contract No. APVV-0029-07.
12
10
w,y [%]
8
6
REFERENCES
4 w y − Dmin y − Dmax y − Dnom
2
0 0
1
2
3 4 time[min]
5
6
7
Fig. 6. Control responses for controller C2
12
10
w,y [%]
8
6
4 w y − Dmin y − Dmax y − Dnom
2
0 0
1
2
3 time[min]
4
5
6
Fig. 7. Control responses for controller C3 12
10
w,y [%]
8
6
4 w y − Dmin y − Dmax y − Dnom
2
0 0
1
2
3
4 time[min]
5
6
7
8
Fig. 8. Control responses for controller C4 The pole-placement method was used for the choice of PI controllers from the robust stability region. Designed PI controllers were implemented using the industrial control system SIMATIC and tested by experiments on the laboratory equipment. Although designed PI controller do not assure robust stabilizability of the original timedelay process, obtained experimental show that proposed approach leads to the design of PI controllers that are suitable for control of real processes with varying timedelay and parametric uncertainty.
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