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2 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 6 Tomislav B. Šekara and Miroslav R. Matašek University of Belgrade/Faclty of Electrical Engineering, Serbia 1. Introdction Classification of processes and tning of the PID controllers is initiated by Ziegler and Nichols (194). This methodology, proposed seventy years ago, is still actal and inspirational. Process dynamics characterization is defined in both the time and freqency domains by the two parameters. In the time domain, these parameters are the velocity gain K v and dead-time L of an Integrator Pls Dead-Time (IPDT) model G ZN (s)=k v exp(-ls)/s, defined by the reaction crve obtained from the open-loop step response of a process. In the freqency domain these parameters are the ltimate gain k and ltimate freqency ω, obtained from oscillations of the process in the loop with the proportional controller k=k. The relationship between parameters in the time and freqency domains is determined by Ziegler and Nichols as L 4, Kv, ZN k. (1) However, for the process G p (s)=g ZN (s) in the loop with the proportional controller k, one obtains from the Nyqist stability criterion the same relationship (1) with =1. As a conseqence, from (1) and the Ziegler-Nichols freqency response PID controller tning, where the proportional gain is k=0.6k, one obtains the step response tning k=0.3/(k v L). Ths, for =ε ZN one obtains k=1./(k v L), as in (Ziegler & Nichols, 194), while for =1 one obtains k=0.945/(k v L), as stated in (Aström & Hägglnd, 1995a). According to (1), the same vales of the integral time T i =/ω and derivative time T d =0.5/ω are obtained in both freqency and time domains, in (Ziegler & Nichols, 194) and from the Nyqist analysis. This will be discssed in more detail in Section. Tning formlae proposed by Ziegler and Nichols, were improved in (Hang et al., 1991; Aström & Hägglnd, 1995a; 1995b; 004). Besides the ltimate gain k and ltimate freqency ω of process G p (s), the static gain K p =G(0), for stable processes, and velocity gain K lim sg ( s), for integrating processes, are sed to obtain better process dynamics v s0 p characterization and broader classification (Aström et al.,199). Stable processes are approximated by the First-Order Pls Dead-Time (FOPDT) model G FO (s)=k p exp(-ls)/(ts+1) and classified into for categories, by the normalized gain 1 =K p k and normalized dead-

3 118 Frontiers in Advanced Control Systems time 1 =L/T. Integrating processes are approximated by the Integrating First-Order Pls Dead-Time (IFOPDT) model G IF (s)=k v exp(-ls)/(s(t v s+1)) and classified into two categories, by the normalized gain =K v k /ω and normalized dead-time =L/T v. The idea of classification proposed in (Aström et al., 199) was to predict the achievable closed-loop performance and to make possible performance evalation of feedback loops nder closedloop operating conditions. In the present chapter a more ambitios idea is presented: define in advance the PID controller parameters in a classification plane for the prpose of obtaining a PID controller garanteeing the desired performance/robstness tradeoff for the process classified into the desired region of the classification plane. It is based on the recent investigations related to: I) the process modeling of a large class of stable processes, processes having oscillatory dynamics, integrating and nstable processes, with the ltimate gain k (Šekara & Matašek, 010a; Matašek & Šekara, 011), and optimizations of the PID controller nder constraints on the sensitivity to measrement noise, robstness, and closed-loop system damping ratio (Šekara & Matašek, 009,010a; Matašek & Šekara, 011), II) the closed-loop estimation of model parameters (Matašek & Šekara, 011; Šekara & Matašek, 011b, 011c), and III) the process classification and design of a new Gain Schedling Control (GSC) in the parameter plane (Šekara & Matašek, 011a). The motive for this research was the fact that the thermodynamic, hydrodynamic, chemical, nclear, mechanical and electrical processes, in a large nmber of plants with a large nmber of operating regimes, constittes practically an infinite batch of transfer fnctions G p (s), applicable for the process dynamics characterization and PID controller tning. Since all these processes are nonlinear, some GSC mst be applied in order to obtain a high closed-loop performance/robstness tradeoff in a large domain of operating regimes. A direct soltion, mostly applied in indstry, is to perform experiments on the plant in order to define GSC as the look-p tables relating the controller parameters to the chosen operating regimes. The other soltion, more elegant and extremely time-consming, is to define nonlinear models sed for predicting accrately dynamic characteristics of the process in a large domain of operating regimes and to design a continos GSC (Matašek et al., 1996). However, both soltions are dedicated to some plant and to some region of operating regimes in the plant. The same applies for the soltion defined by a nonlinear controller, for example the one based on the neral networks (Matašek et al., 1999). A real PID controller is defined by Fig. 1, with C(s) and C ff (s) given by ks d kski Cs () FC () s sts ( 1) f k s k C () s FC () s, kff bk, FC ( s) 1, 0 b. () s ff i, ff r C () s ff d Gp( s) n y Cs () Fig. 1. Process G p (s) with a two-degree-of-freedom controller.

4 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 119 An effective implementation of the control system () is defined by relations ki U() s kbr() s Yf() s R() s Yf() s kdsyf() s FC () s s Ys () Y () s Ts 1, (3) for F C (s) 1 as in (Panagopolos et al., 00; Matašek & Šekara, 011). When the proportional, integral, and derivative gains (k, k i, k d ) and derivative (noise) filter time constant T f are determined, parameter b can be tned as proposed in (Panagopolos et al., 00). The PID controller (), F C (s) 1, can be implemented in the traditional form, when noise filtering affects the derivative term only if some conditions are flfilled (Šekara & Matašek, 009). The derivative filter time constant T f mst be an integral part of the PID optimization and tning procedres (Isaksson & Graebe, 00; Šekara & Matašek, 009). For F C (s) given by a second-order filter, one obtains a new implementation of the Modified Smith Predictor (Matašek & Micić, 1996, 1999). The MSP-PID controller (3) garantees better performance/robstness tradeoff than the one obtained by the recently proposed Dead-Time Compensators (DTC s), optimized nder the same constraints on the sensitivity to measrement noise and robstness (Matašek & Ribić, 01). Robstness is defined here by the maximm sensitivity M s and maximm complementary sensitivity M p. The sensitivity to measrement noise M n, M s, and M p are given by, f f M s max 1 1 L ( i ) M, p max L( i) 1 L ( i ), M n max C n ( i ), (4) where L(s) is the loop transfer fnction and C n (s) is the transfer fnction from the measrement noise to the control signal. In the present chapter, the sensitivity to the high freqency measrement noise is sed M n =M n, where M n = C n (s) s.. Modeling and classification of stable, integrating, and nstable plants A generalization of the Ziegler-Nichols process dynamics characterization, proposed by Šekara and Matašek (010a), is defined by the model G A exp( s) 1 () s s A exp( s) k m,, kg p(0) A, (5) 1 kg(0) p where φ is the angle of the tangent to the Nyqist crve G p (iω) at ω and G p (0) is the gain at the freqency eqal to zero. Ths, for integrating processes G p (0)= and A=ω. Adeqate approximation of G p (s) by the model G m (s) is obtained for ω ω, where arg{g p (i )}=. It is demonstrated in (Šekara & Matašek, 010a; Matašek & Šekara, 011, Šekara & Matašek, 011a) that this extension of the Ziegler-Nichols process dynamics characterization, for a large class of stable processes, processes with oscillatory dynamics, integrating and nstable processes, garantees the desired performance/robstness tradeoff if optimization of the PID controller, for the given maximm sensitivity M s and given sensitivity to measrement noise M n, is performed by applying the freqency response of the model (5) instead of the exact freqency response G p (iω).

5 10 Frontiers in Advanced Control Systems Ziegler and Nichols sed oscillations, defined by the implse response of the system kg p() s Gp() s, (6) 1 kg( s) to determine k and ω, and to define tning formlae for adjsting parameters of the P, PI and PID controllers, based on the relationship between the qarter amplitde damping ratio and the proportional gain k. Oscillations defined by the implse response of the system (6) are sed in (Šekara & Matašek, 010a) to define model (5), obtained from G m (s) G p (s) and the relation m kg m s s p kg () s A exp( s). (7) 1 ( ) Then, by analyzing these oscillations, it is obtained in (Šekara & Matašek, 010a) that amplitde A=ω /(1+), =k G p (0), and dead-time τ is defined by ω and a parameter φ, given by Gp arg ( i ). (8) Other interpretation of amplitde A= A 0, obtained in (Matašek & Šekara, 011), is defined by A 0 Gp( i) k 1. (9) Amplitdes A and A 0 are not eqal, bt they are closely related for stable and nstable processes, as demonstrated in (Matašek & Šekara, 011) and Appendix. Parameter A 0 is not sed for integrating processes, since for these processes A=ω. The qadrplet {k, ω, φ, A} is sed for classification of stable processes, processes with oscillatory dynamics, integrating and nstable processes in the -φ parameter plane, defined by the normalized model (5), given by exp( s ) G s s n n( n,, ), n sn 1exp( sn) s, (10) where =A/ω. From the Nyqist criterion it is obtained that the region of stable processes is defined by 0 / 1,0 1 (Šekara & Matašek, 011a). Integrating processes, since A=ω, are classified as 1, 0 / processes, while nstable processes are otside these regions. It is demonstrated that a large test batch of stable and integrating processes sed in (Aström & Hägglnd, 004) covers a small region in the -φ plane. To demonstrate that besides k and ω, parameters φ and G p (0) mst by sed for the classification of processes, Nyqist crves are presented in Fig. for stable, integrating and

6 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 11 nstable processes having the same vales k =1 and ω =1. For processes having also the same vales of φ, the distinction of the Nyqist crves in the broader region arond the critical point reqires the information abot gain G p (0), as demonstrated in Fig. -a. On the other hand, the reslts presented in Fig. -b to Fig. -d demonstrate that for the same vales of k, ω, and G p (0) the distinction of the Nyqist crves in the region arond the critical point is obtained by applying parameter φ. This fact confirms importance of parameter φ in process modeling for controller tning, taking into accont that optimization of the PID controller nder constraints on the robstness is performed in the region arond ω. a) b) c) d) Fig.. Nyqist crves of processes with the same vales k =1, ω =1: a) φ=/4, stable G p (0)=1 (dashed), integrating G p (0)= (solid), nstable G p (0)= (dashed-dotted); b) stable processes with G p (0)=1, for φ=/4 (dashed), φ=/6 (solid), φ= /3 (dashed-dotted); c) integrating processes with φ=1 (dashed), φ= /4 (solid), φ=1. (dashed-dotted); d) nstable processes with G p (0)=, for φ= /4 (dashed), φ= /6 ( solid), φ= /3 (dashed-dotted). For the lag dominated process G p1 ( s) 1 / cosh s, (11) and the corresponding models, the step and implse responses, with the Nyqist crves arond ω, are presented in Fig. 3. The models are Ziegler-Nichols IPDT model G ZN (s)=k v exp(-ls)/s and model (5), with A=ω k G p (0)/(1+k G p (0)) and A=A 0. The set-point and load distrbance step responses of this process, in the loop with the optimal PID controller (Matašek & Šekara, 011) and PID controller tned as proposed by Ziegler and Nichols (194), are compared in Fig. 4-a. In this case k = , ω = and K v =0.951,

7 1 Frontiers in Advanced Control Systems L= The PID controller tned as proposed by Ziegler and Nichols is implemented in the form ki ks d Us () kbrs () Ys () Rs () Ys () Ys () s T s 1 f k T, (1), d b 0, ki, kd ktd, Tf Ti Nd where k=0.6k, T i =/ω, T d = /(4ω ), for the freqency domain ZN tning (ZN PID1). For the time domain ZN tning (ZN PID) the parameters are k=1./(k v L), T i =L, T d =L/, or, as sggested by the earlier mentioned Nyqist analysis, proportional gain k is adjsted to k=0.943/(k v L), denoted as the modified time domain ZN tning (ZN ModifPID). In M ( N 1) k parameter N d is adjsted to obtain the same vale of M n =76.37 sed in the n d constrained optimization of the PID in (3), F C (s) 1, where M n = k d /T f. Parameters of the PID controllers and performance/robstness tradeoff are compared in Table 1. It is impressive that Ziegler and Nichols scceeded in defining seventy years ago an excellent experimental tning for the process G p1 (s), which is an infinite-order system that can be represented in simlation by the following high-order system p1 0 k1 G () s exp( Ls)/ ( T s 1), L= (Matašek & Ribić, 009). Also, it shold be k noted here, that Ziegler and Nichols scceeded seventy years ago in obtaining an excellent tning with the IPDT model defined by K v =0.951, L=0.1534, which is an extremely crde approximation of the real implse response of the process G p1 (s), as in Fig. 3-b. Tning method k k i k d T f N d IAE M n M s M p optpid ZN PID ZN PID ZN ModifPID Table 1. Process G p1 (s): comparison of the optimization (optpid) and the Ziegler-Nichols tning in the freqency domain (ZN PID1) and time domain (ZN PID, ZN ModifPID). The Nyqist crves of G p1 (s), G m1 (s), and G m (s) are almost the same arond ω. This is important since the PID controller optimization, based on the experimentally determined freqency response of the process, nder constraints on M s or on M s and M p, is performed arond the ltimate freqency ω. Amplitdes A and A 0 are closely related for the stable and nstable processes, as demonstrated in (Matašek & Šekara, 011) and Appendix. For integrating processes A=ω. This means, that the Ziegler Nichols parameters k and ω, and the Šekara-Matašek parameters φ and A=A 0, for the stable and nstable processes, and A=ω, for integrating processes, constitte the minimal set of parameters, measrable in the freqency domain, necessary for obtaining PID controller tning for the desired performance/robstness tradeoff. This will be demonstrated in the sbseqent sections. 3. Optimization of PI/PID controllers nder constraints on the sensitivity to measrement noise, robstness, and closed-loop system damping ratio PID controllers are still mostly sed control systems in the majority of indstrial applications (Desborogh & Miller, 00) and it is reasonable to predict that PID control

8 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane Gm ZN Gm1,Gp Gp a) b) Time [sec] c) d) Fig. 3. Process G p1 (s), denoted as (G p ), and models G mj (s), j=1,, k = , ω =9.8696, τ= for A= (G m1 ) and A=A 0 = (G m ), and G ZN (s)= K v exp(-ls)/s, K v =0.951, L= (ZN): a) step responses, b) implse responses, c) Nyqist crves of G p1 (s) and G ZN (s), d) Nyqist crves of G p1 (s), G m1 (s) and G m (s) are almost the same arond ω a) 0. optpid ZN PID1 ZN PID Time [sec] b) 0. ZN PID1 ZN ModifPID Time [sec] Fig. 4. Comparison of the optimization and the Ziegler-Nichols (ZN) tning. Process G p1 (s) in the loop with the optpid or ZN PID, tned by sing the rles: freqency domain (ZN PID1), time domain (ZN PID), and time domain with the modified proportional gain k=0.943/(k v L) (ZN ModifPID). In all controllers b=0 and D(s)=-5exp(-.5s)/s. will contine to be sed in the ftre (Aström & Hägglnd, 001). They operate mostly as reglators (Aström & Hägglnd, 001) and rejection of the load step distrbance is of

9 14 Frontiers in Advanced Control Systems primary importance to evalate PID controller performance nder constraints on the robstness (Shinskey, 1990), measred by the Integrated Absolte Error (IAE). Inadeqate tning and sensitivity to measrement noise are the reasons why derivative action is often exclded in the indstrial process control. This is the main reason why PI controllers predominate (Yamamoto & Hashimoto, 1991). However, for lag-dominated processes, processes with oscillatory dynamics and integrating/nstable processes PID controller garantees considerably better performance than PI controller, if adeqate tning of the PID controller is performed (Matašek & Šekara, 011). Moreover, PID controller is a prereqisite for sccessfl advanced controller implementation (Seki & Shigemasa, 010). Besides PI/PID controllers, in single or mltiple loops (Jevtović & Matašek, 010), only Dead-Time Compensators (DTC) are sed in the process indstry with an acceptable percentage (Yamamoto & Hashimoto, 1991). They are based on the Smith predictor (Smith, 1957; Matašek & Kvaščev, 003) or its modifications. However, the area of application of PID controllers overlaps deeply with the application of DTC s, as confirmed by the Modified Smith Predictor, which is a PID controller in series with a second-order filter, applicable to a large class of stable, integrating and nstable processes (Matašek & Ribić, 01). Optimization of the performance may by carried ot nder constraints on the maximm sensitivity to measrement noise M n, the maximm sensitivity M s and maximm complementary sensitivity M p, as done in (Matašek & Ribić, 01). In this case it is recommended to se some algorithm for global optimization, sch as Particle Swarm Optimization algorithm (Rapaić, 008), reqiring good estimates of the range of nknown parameters. Other alternatives, presented here, are recently developed in (Šekara & Matašek, 009, 010a; Matašek & Šekara, 011). For the PID controller (3), for F C (s) 1 defined by for parameters k, k i, k d and T f, optimization nder constraints on M n and M s is redced in (Šekara & Matašek, 009) to the soltion of a system of three algebraic eqations with adeqate initial vales of the nknown parameters. The adopted vales of M n and M s are satisfied exactly for different vales of z. Ths, by repeating calclations for a few vales of the damping ratio of the controller zeros z in the range 0.5 z, the vale of z corresponding to the minimm of IAE is obtained. Optimization methods from (Šekara & Matašek, 009) are denoted as max(k) and max(ki) methods. The improvement of the max(k) method is proposed in (Šekara & Matašek, 010a). It consists of avoiding repetition of calclations for different vales of z in order to obtain the minimal vale of the IAE for a desired vale of M s. In this method, denoted here as method optpid, the constrained optimization is based on the freqency response of model (5). For the PI optimization, an improvement of the performance/robstness tradeoff is obtained by applying the combined performance criterion J c =k i +(1-) (Šekara & Matašek, 008). Ths, one obtains max J c, (13) k i, F(, k, k i ) 0, F(, k, ki )/ 0, (14) where 0ω< and is a free parameter in the range 0<1. The calclations are repeated for a few vales of, in order to find corresponding to the minimm of IAE. The optimization

10 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 15 in this method, denoted here as opt, is performed for the desired vale of M s. For =1 one obtains the same vales of parameters k and k i as obtained by the method proposed in (Aström et al., 1998), denoted here as opt1. The most general is the new tning and optimization procedre proposed in (Matašek & Šekara, 011). Besides the tning formlae, the optimization procedre is derived. For the PID and PI controllers it reqires only obtaining the soltion of two nonlinear algebraic eqations with adeqate initial vales of the nknown parameters. PID optimization is performed for the desired closed-loop system of damping ratio and nder constraints on M n and M s. Ths, for =1 the critically damped closed-loop system response is obtained. PI optimization is performed nder constraint on M s for the desired vale of. The procedre proposed in (Matašek & Šekara, 011) will be discssed here in more details, since it is entirely based on the concept of sing oscillators (6)-(7) for dynamics characterization of the stable processes, processes having oscillatory dynamics, integrating and nstable processes. The method is derived by defining a complex controller C(s)=k (1+C * (s)), where the controller C * (s), given by s Es ()/ Λ() s C () s, Ε() s s 1s1, Λ() s s s1, (15) A Λ()1 s Es ( )exp( s)/ Λ( s) is obtained by spposing that in Fig. 1 process G p (s) is defined by oscillator G p () s in (6), approximated by (7). Complex controller C(s)=k (1+C * (s)) is defined by the parameters k,,, A and by the two tning parameters and, with the clear physical interpretation. Parameter is proportional to the desired closed-loop system time constant. Parameter is the desired closed-loop system damping ratio. Then, by applying Maclarin series expansion, the possible internal instability of the complex controller C(s) is avoided and parameters of PID controller C(s) in Fig. 1 are obtained, defined by: T f ( 1 ) 3 1/ (1 M / k )/ 1 n 1, (16) 1 1 f k k ( T ) 1, ki k1, (17) d 1 f 1 3 f k k ( T )( ) 1/ k T. (18) Parameters 1,, 1, and 3, from (Matašek & Šekara, 011), depends on, and k, ω, τ, A. They are given in Appendix. Generalization of this approach is presented in (Šekara & Trifnović, 010; Šekara et al., 011). For the desired closed-loop damping ratio =1, =1/ω, and for T f 1/( N ), (19) one obtains (Matašek & Šekara, 011) the PID tning that garantees set-point and load distrbance step responses with negligible overshoot for a large class of stable processes, processes with oscillatory dynamics, integrating and nstable processes. Tning formlae

11 16 Frontiers in Advanced Control Systems defined by (17)-(19) are denoted here as method tnλ. Absolte vale of the Integrated Error (IE), approximating almost exactly the obtained IAE, is given by IE =1/( k 1 ). Here the vale T f =1/(10ω ) is sed, as in (Matašek & Šekara, 011). To demonstrate the relationship between PID controller, tned by sing the method tnλ, and complex controller C(s)=k (1+C * (s)), obtained for =1/ω and =1, the freqency responses of these controllers, tned for the process p 4 G () s 1/( s 1), (0) are presented in Fig. 5-a. For this process, parameters k,,, A, and are given in Appendix. The load distrbance nite step responses, obtained for G p (s) in the loop with the PID controller and complex controller C(s), are presented in Fig. 5-b. Frther details abot the relationship between these controllers are presented in (Matašek & Šekara, 011; Trifnović & Šekara, 011; Šekara et al., 011). a) b) Fig. 5. Comparison of the complex controller C(s)=k (1+C * (s)) with PID controller, both tned for G p (s): a) Bode plots of the controllers and b) the load nite step distrbance responses of G p (s) in the loop with these controllers. By applying tning formlae (17)-(19), the desired closed-loop damping ratio =1 is obtained with the acceptable vales of maximm sensitivity M s and maximm sensitivity to measrement noise M n. However, when a smaller vale of M n is reqired for a desired vale of M s and the desired closed-loop damping ratio, the other possibility is to determine the closed-loop time constant and the corresponding ω 0, by sing (16)-(18) and by solving two algebraic eqations: m s 1 C( i) G ( i) 1/ M 0, (1) (1 C( i) G ( i) )/ 0. () m In this case, the PID controller in (3), F C (s) 1, is obtained for the desired critical damping ratio =1 of the closed-loop system and the desired vales of M n and M s. This is the niqe possibility of the procedre (16)-(18) and (1)-() proposed in (Matašek & Šekara, 011). Moreover, by repeating the calclations for a few vales of, the vale of is obtained

12 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 17 garanteeing, for desired M n and M s, almost the same vale of the IAE as obtained by the constrained PID optimization based on the exact freqency response G p (i). This PID optimization method is denoted here as the method opta, when the qadrplet {k, ω, φ, A} is sed, or opta 0, when the qadrplet {k, ω, φ, A 0 } is sed. It shold be noted here, that for k d =0 and T f =0, by relations (17) and (1)-() a new effective constrained PI controller optimization is obtained, denoted here as opt3. It is sccessflly compared (Matašek & Šekara, 011) with the procedre proposed in (Aström et al., 1998), opt1. Now, tning defined by (17)-(19) with N=10, =1/ω and =1, method tnλ, will be compared with the optimization defined by (16)-(18), (1)-(), method opta. Both procedres garantee desired critical damping =1, however only the second one garantees the desired vales of M n and M s. Ths, for =1 and for the maximm sensitivity M s obtained by applying method tnλ, the smaller vale of sensitivity to measrement noise M n will be sed by applying PID optimization method opta. The reslts of this analysis are presented in Table and Fig. 6. As in Table 1, controller is tned by sing the model G m (s) in (5) and then applied to processes G p3 (s) to obtain IAE, M s and M p, where G 1.507(3.4s1)( s) () s. (3) (577s 1)(18.1s1)(0.73s1)(104.6s 15s1) p3 Lower vale of IAE is obtained, for almost the same robstness, by sing higher vale of the sensitivity to measrement noise. However, for the lower vale of M n the controller and, as a reslt, the actator activity is considerably redced. Ths, the comparison of the IAE, obtained by the PID controllers with the same robstness, is meaningless if the sensitivity to measrement noise M n is not specified, as demonstrated in Fig. 6. This fact is freqently ignored. method k k i k d T f IAE M n M s M p tnλ opta Table. Process G p3 (s) in the loop with the PID controllers. Tning method (17)-(19), tnλ and optimization (16)-(18), (1)-(), opta for =1. Conclding this section, the constrained PI/PID controller optimization methods proposed in (Matašek & Šekara, 011) is compared with the constrained PID controller optimization method proposed in (Šekara & Matašek, 010a), optpid1, and the constrained PI controller optimization method proposed in (Šekara & Matašek, 008), opt. The test batch of stable processes, processes having oscillatory dynamics, integrating and nstable processes sed in this analysis is defined by transfer fnctions G p1 (s), G p (s), G p3 (s) and 5s s e e Gp4() s, G 3 p5() s, ( s 1) 9s 0.4s1 (4) 5s 5s e e Gp6() s, Gp7() s, ss ( 1)(0.5s1)(0.5s1)(0.15s1) (10s1)(s1) (5)

13 18 Frontiers in Advanced Control Systems with parameters k,,, A, A 0,, presented in Appendix. Comparison of the methods for PID controller tning is presented in Table 3. Comparison of the methods for PI controller tning is presented in Table 4 and Fig. 7. a) b) Fig. 6. Set-point, R(s)=1/s, and load distrbance, D(s)=-10exp(-400s)/s, step responses. G p3 (s) and PID controllers tned by: a) tnλ, b=0.5; b) opta, b=0.6. Measrement noise is obtained by passing niform random noise 1 throgh a low-pass filter F(s)=0.5/(10s+1). Process/ k k i k d T f IAE M n M s M p z method G p3 /max(k) G p3 /optpid G p3 /opta G p3 /opta G p3 /opta G p5 /max(ki) G p5 /optpid G p5 /opta G p5 /opta G p5 /opta G p6 /max(k) G p6 /optpid G p6 /opta G p6 /opta G p7 /max(k) G p7 /optpid G p7 /opta G p7 /opta Table 3. PID controllers, obtained by applying model G m (s) and tning methods: max(k), max(ki); (31)-(35) optpid; (16)-(18), (1)-() opta and opta 0. In Table 3 optimization (16)-(18), (1)-() is performed for stable G p3 (s), G p5 (s) and nstable G p7 (s) processes by sing G m (s) with two qadrplets: {k,,, A}, denoted as opta, and

14 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 19 {k,,, A 0 }, denoted opta 0. As mentioned previosly, for integrating processes A=. Almost the same performance/robstness tradeoff is obtained for A and A 0, as spposed in Section. This reslt is important since it confirms that an adeqate approximation of the freqency response of the stable and nstable processes arond can be sed in the optimization (16)-(18) and (1)-(), instead of the model G m (i) in (5). Obviosly, the same applies for integrating processes. The advantage of the constrained PID controller optimization (16)-(18) and (1)-() is that only two nonlinear algebraic eqations have to be solved, with very good initial conditions for the nknown parameters and 0. Moreover, the optimization is performed for the desired vales of M s, M n and for the desired closedloop system damping ratio. Finally, the reslts of the PI controller optimization are demonstrated in Table 4 and in Fig. 7. By repeating calclations for a few vales of, for the same vales of M s and M p, the same (minimal) vale of the IAE is obtained by applying method opt3, defined by (17) and (1)- (), and the method opt, defined by (13)-(14). As mentioned previosly, method opt is an improvement of the method proposed in (Aström et al., 1998), denoted here as method opt opt1 opt opt Time [sec] a) b) c) d) Fig. 7. Set-point and load distrbance step responses: y(t) (left) and (t) (right). PI controllers from Table 4: opt1 b=0, opt b=0.6, opt3 b=0.6. In a) and b) G p1 (s), D(s)=-exp(-4s)/s; in c) and d) G p4 (s), D(s)=-0.5exp(-80s)/s.

15 130 Frontiers in Advanced Control Systems Process/method k k i IAE M s M p G p1 /opt G p1 /opt G p1 /opt G p3 /opt G p3 /opt G p3 /opt G p4 /opt G p4 /opt G p4 /opt Table 4. PI controllers, obtained for M s = by applying model (5) and methods: (Aström et al., 1998) opt1, (13)-(14) opt, and (17), (1)-() opt3. 4. Closed-loop estimation of model parameters Approximation of process dynamics, arond the operating regime, can be defined by some transfer fnction G p (s) obtained from the open-loop or closed-loop process identification. One two step approach (Hjalmarsson, 005) is based on the application of the high-order ARX model identification in the first step. In the second step, to redce the variance of the obtained estimate of freqency response of the process, cased by the measrement noise, this ARX model is redced to a low-order model G p (s). By applying this procedre an adeqate approximation G p (iω) of the nknown Nyqist crve can be obtained in the region arond the ltimate freqency ω. As demonstrated for the Ziegler-Nichols tning, in Fig. 3- c and Fig. 4-b, sch approximation of the nknown Nyqist crve is of essential importance for designing an adeqate PID controller. The same applies for the sccessfl PID optimization nder constraints on the desired vales of M n and M s, as demonstrated in Table 5 for the vale of A defined as in (5) and for A=A 0. The Closed-Loop (CL) system identification can be performed by sing indirect or direct identification methods. In indirect CL system identification methods it is assmed that the controller in operation is linear and a priory known. Direct CL system identification methods are based only on the plant inpt and otpt data (Agüero et al., 011). Finally, the identification can be based on the simple tests, as initiated by Ziegler and Nichols (194), to obtain an IPDT model (1). Later on, this approach is extended to obtain FOPDT model and the Second-Order Pls Dead-Time (SOPDT) model, for integrating processes characterized by the IFOPDT model. The SOPDT model can be obtained from k, ω, φ, A. In this case it is defined by G e () s as bs c, (6) SO where parameters a, b, c and L are fnctions of k, ω, φ and A, obtained from the tangent rle (Šekara & Matašek, 010a). This model (6) is an adeqate SOPDT approximation of the Nyqist crve G p (i) in the region arond the ltimate freqency ω, for a large class of stable processes, processes with oscillatory dynamics, integrating and nstable processes. Ls

16 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 131 The recently proposed new Phase-Locked Loop (PLL) estimator (Matašek & Šekara, 011), its improvement (Šekara & Matašek, 011c), and new relay SheMa estimator (Šekara & Matašek 011b) make possible determination of parameters k, ω, φ and A 0 of the model G m (s) in the closed-loop experiments, withot breaking the control loop in operation. This property of the proposed PLL and SheMa estimators is important for practice, since breaking of control loops in operation is mainly ignored by plant operators, especially in the case of controlling processes with oscillatory dynamics, integrating or nstable processes. The PLL estimator can be applied in the case when the controller in operation is an nknown linear controller, while the SheMa estimator can be applied when the controller in operation is nknown and nonlinear. In that sense, the SheMa estimator belongs to the direct CL system identification methods, based only on the plant inpt and otpt data, as in (Agüero et al., 011). Both procedres, SheMa and PLL, are based on the parameterization presented in (Šekara & Matašek, 010a; Matašek & Šekara, 011). Estimates of parameters k, k, k and,,, obtained for arg G p ( i ) -, 0 and / 36, are sed for determining φ and A 0, as defined in (Matašek & Šekara, 011). In this section, an improvement of the new PLL estimator from (Matašek & Šekara, 011) is presented in Fig. 8. The improvement, proposed by Šekara and Matašek (011c), consists of adding two integrators at the inpt to the PLL estimator from (Matašek & Šekara, 011). Inpts to these integrators are defined by otpts of the band-pass filters AF 1, sed to eliminate the load distrbance. Otpts of these integrators are passed throgh a cascade of the band-pass filters AF m, m=,3,4. All filters AF m, m=1,,3,4, are tned to the ltimate freqency. Sch implementation of the PLL estimator eliminates the effects of the high measrement noise and load distrbance. Blocks AF m, j=1,,3,4, are implemented as presented in (Matašek & Šekara, 011), while implementation of blocks for determining arg{g p (i)} and G p (i) are presented in (Šekara & Matašek, 011c). PLL estimator from Fig. 8 is applied to processes G p8 (s)=exp(-s)/(s+1) and G p9 (s)=4exp(-s)/(4s-1) in the loop with the known PID controller. Estimation of parameters k, k, k and and,,,, is presented in Fig. 9. Highly accrate estimates of k, k, k are obtained in the presence of the high measrement noise and load distrbance. Since these parameters are sed to determine φ and A 0, this experiment demonstrates that highly accrate estimate of the qadrplet {k, ω, φ, A 0 } can be obtained, in the presence of the high measrement noise and load distrbance, by the PLL estimator from (Šekara & Matašek 011c). In Fig. 10, estimation of the nknown Nyqist crve of the nstable process in the loop with the PID controller is demonstrated. The PLL estimator from (Matašek & Šekara, 011; Šekara & Matašek 011c) is a frther development of the idea firstly proposed in (Crowe & Johnson, 000) and sed in (Clarke & Park, 003). The SheMa estimator is a frther development of the estimator proposed by Aström and Hägglnd (1984) as an improvement of the Ziegler-Nichols experiment. The Ziegler and Nichols (194) experiment, sed to determine k and of a process is performed by setting the integral and derivative gains to zero in the PID controller C(s) in

17 13 Frontiers in Advanced Control Systems operation. However, in this approach the amplitde of oscillations is not nder control. This drawback is eliminated by Aström and Hägglnd (1984). The factors inflencing the critical point estimation accracy in this conventional relay setp are: the se of describing fnction method is faced with the fact that higher harmonics are not efficiently filtered ot by the process, presence of the load distrbance d, and presence of the measrement noise n. The first drawback of the conventional relay experiment is eliminated by the modified relay setp (Lee et al., 1995). y AF AF s 1 s arg{ G (i )} p r Cff( s) d Gp( s) n y AF,3,4 AF,3,4 G (i ) p Cs () y F F U / U F F OS arg{ U / U } Y / U F F OS OS arg{ Y / U } OS ref k PLL s 1 1 s Oscillator 1 s cos sin OS Fig. 8. Improved PLL estimator. AF,3,4 is the cascade of band-pass filters AF m, m=1,,3,4. k k k Fig. 9. PLL estimates of k, k, k and,,, in the presence of the high measrement noise and step load distrbance at t=700 s. Process G p8 (s)=exp(-s)/(s+1), for: /36 for 0 t 300 s, 0 for 300<t 500 s and / 36 for 500<t 1000 s.

18 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 133 ref G (i ( )) p9 ref Fig. 10. Estimates (circles) of the Nyqist crve (solid) obtained by the PLL estimator for the desired vales ref arg{g p9 (i)}. Process G p9 (s)=4exp(-s)/(4s-1), the noise-free case. De to its simplicity, the relay-based setp proposed by Aström and Hägglnd (1984) is still a basic part of different methods developed in the area of process dynamics characterization. For example, it is sed to generate signals to be applied for determining FOPDT and SOPDT models, sing a biased relay (Hang et al., 00). However, from the viewpoint of the process control system in operation, the estimation based on this setp, and its modifications, is performed in an open-loop configration: the loop with the controller C(s) in operation is opened and the process otpt is connected in feedback with a relay. In the paper (Šekara & Matašek, 011b) a new relay-based setp is developed, with the controller C(s) in operation. It consists of a cascade of variable band-pass filters AF m, from (Clarke & Park, 003), a new variable band-pass filter F mod proposed by Šekara and Matašek (011b) and a notch filter F NF =1-F mod. Center freqencies of variable band-pass filters AF m and F mod are at. Highly accrate estimates of and k are obtained in the presence of the measrement noise and load distrbance. Also, highly accrate estimates of the Nyqist crve G p (i) at the desired vales of arg{g p (i)} are obtained by inclding into the SheMa the modified relay instead of the ordinary relay. The amplitde of both relays is eqal to =k,0 y ref 0 /4, where k,0 is the ltimate gain obtained in the previos activation of the SheMa, y ref is the amplitde of the set-point r and 0 is a small percent of y ref, for example 0 =0.1% in the examples presented in (Šekara & Matašek, 011b). The proposed closed-loop procedre can be activated or deactivated with small impact on the controlled process otpt. Frther details of the SheMa estimator, inclding the stability and robstness analyses, and implementation details, are presented in (Šekara & Matašek 011b). 5. Gain schedling control of stable, integrating, and nstable processes, based on the controller optimization in the classification parameter plane For a chosen region in the -φ classification plane, presented in Fig. 11, the normalized parameters k n (,φ), k in (,φ), k dn (,φ) and T fn = k dn (,φ) /m n of a virtal PID n controller are calclated in advance by sing the process-independent model G n (iω n,, φ) in (10).

19 134 Frontiers in Advanced Control Systems Then, parameters k, k i, k d and T f of the PID controller (3), F C (s) 1, are obtained, for the process classified in the chosen region of the -φ plane, by sing the estimated k, ω, φ, A and the following relations k k k, k k k, k k k /, T T /. (7) n i in d dn f fn Depending on the method applied to obtain parameters k n, k in, k dn and T fn = k dn /m n of a PID n controller, parameters k, k i, k d and T f of the PID controller (3), F C (s) 1, garantee the desired M s and the sensitivity to measrement noise eqal to M n = k m n, or garantee the Fig. 11. Classification -φ parameter plane, with processes G pj (s), j=1,,...,9. Stable processes are classified in the region 0 1,0 / 1, integrating processes are classified as 1, 0 / processes. Unstable processes are classified otside this region. desired M s, and M n = k m n. Since parameters k n, k in, k dn and T fn = k dn /m n are determined in advance, they can be memorized as look-p tables in the -φ plane. Besides, this can be done for different vales of M s, m n and. These look-p tables define a new Gain Schedling Control (GSC) concept. Important featre of this GSC is that these look-p tables, obtained for some vales of M s, m n and from the model G n (iω n,, φ), are processindependent. Enormos resorces are avoided, reqired for performing experiments on the plant in order to define the standard GSC as the look-p tables of PID controller parameters for this plant and the desired region of operating regimes. Ths, the important and exclsive featre of the new GSC is that a desired performance/robstness tradeoff can be obtained for a large region of dynamic characteristics of processes in different plants and different operating regimes, covered by the look-p tables of parameters k n, k in, k dn in the -φ classification plane. Now, this GSC PID controller tning, performed by sing (7), will be demonstrated by the two different procedres applied for obtaining parameters k n, k in, k dn and T fn = k dn /m n of the PID n controller for integrating and stable processes. Stable processes having a weakly damped implse response are denoted as processes having oscillatory dynamics, while processes with damped implse response are denoted as stable processes. For integrating processes, parameters k n, k in, k dn and T fn = k dn /m n of the PID n controller depend only on angle φ, since =1. In this case, for desired vales of M s and m n, PID controller parameters (7) are obtained from tning formlae for k n (φ), k in (φ) and k dn (φ)

20 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 135 (Šekara & Matašek, 011a). Ths, for integrating process G p6 (s) parameters of the PID n controller are obtained by applying angle φ= in the tning formlae defined in (Šekara & Matašek, 011a) for M s = and m n =, given in Appendix as tn1. The reslts are presented in Table 5, G p6 -tn1. For processes having the oscillatory dynamics look-p tables and tning formlae are derived in (Šekara & Matašek, 011a) for M s = and m n =40, in the region , φ of the -φ classification plane of Fig. 11. These tning formlae, in Appendix denoted as tn, are applied to determine parameters k, k i, k d and T f for the process having the oscillatory dynamics G p5 (s), classified as process =0.1971, φ= (Table 5, G p5 -tn). To illstrate the direct application of the look-p tables from (Šekara & Matašek, 011a, Table A4) and interpolation procedre defined in Appendix, Fig. 17, since this process is classified as =0.1971, φ= (0.3679), the following points are determined from (Šekara & Matašek, 011a, Table A4) and Appendix, Fig. 17: 1,1 =0.15, φ 1,1 =0, 1, =0., φ 1, =0 and, =0., φ, =30. Parameters (k n, k in, k dn ) are defined by: (-.41, , ) for 1,1, φ 1,1, (-1.70, 0.415,.8783) for 1,,φ 1, and (-1.666, ,.3017) for,,φ,. Then, by sing three point interpolation from Appendix, pper triangle ( r =0.0578, r =0.1971), one obtains parameters in Table 5, G p5 -GSC: k=-0.40, k i =0.0384, k d =1.9116, T f = For stable processes, in a large region of the -φ plane, look-p tables of parameters k n, k in and k dn are defined for M s = and m n = (Šekara & Matašek, 011a, Tables A1-A3). These look-p tables are applied in the present paper to determine parameters k, k i, k d and T f for the stable process G p3 (s). This process is classified as process =0.9808, φ= ( ). Ths, for G p3 (s) parameters (k n, k in, k dn ) can be obtained from the three points in the -φ classification plane (Appendix, Fig. 17): 1,1 =0.95, φ 1,1 =30 ;,1 =0.95, φ,1 =40 and, =1, φ, =40 (0.6981). Two points are sed for stable processes (0.5086, , ) for 1,1,φ 1,1 and (0.5013, 0.161, 0.533) for,1,φ,1 from the look-p tables (Šekara & Matašek, 011a, Tables A1-A3), while data (0.5036, , 0.533) for,, φ, are obtained from tning formlae derived for integrating processes in (Šekara & Matašek, 011a), given in Appendix as tn1. Then, by sing three point interpolation from Appendix, Fig. 17 lower triangle ( ll =0.6166, ll =0.1136), one obtains parameters presented in Table 5, G p3 -GSC: k= , k i =0.307, k d = and T f = Process-method k k i k d T f IAE M n M s M p G p3 -GSC G p5 -tn G p5 -GSC G p6 -tn Table 5. PID controllers: stable process G p3 (s), method GSC-Appendix; stable process having oscillatory dynamics G p5 (s), method tn and method GSC-Appendix; integrating process G p6 (s), method tn Experimental reslts Experimental reslts, presented in Fig. 1, are obtained by sing the laboratory thermal plant. It consists of a thin plate made of alminm, L a =0.1m long and h=0.03m wide (Matašek & Ribić, 01). Temperatre T(x,t) is distribted along the plate, from x=0 to

21 136 Frontiers in Advanced Control Systems x=l a, and measred by precision sensors LM35 (TO9), at x=0 and x=l a. The plate is heated by a terminal adjstable reglator LM317 (TO 0) at position x=0. The maniplated variable is the dissipated power of the heater at x=0. The inpt to the heater is the control variable (t) (%), defined by the otpt of the PID controller. The controlled variable is y(t)=t(l a,t), measred by the sensor at position x=l a. Temperatre sensor at x=0 is sed in the safety device, to prevent overheating when 70 C T(0,t). The anti-windp implementation of the PID controller (3), F C (s) 1, is given by bks ki ks d ksk i 1 C Taw r y. (8) T aws 1 ( Taws 1)( Tfs 1) Taws1 The satration element is defined by the inpt C (t) and otpt (t): l, l, l l l, l low C low C low C high high C high. (9) Obviosly, in the linear region l low < C (t)< l high of the satration element, for C (t)(t) one obtains (3), F C (s) 1, from (8) PID PI a) b) Time [sec] c) d) Fig. 1. Experimental reslts. Set-point and load step (-0% change of the controller otpt at t=1600 s) responses of the real plant, with the PI and PID controller: a) control variable (t) and b) controlled variable y(t). The real plant, with the anti-windp PID controller nder the distrbance indced by activating/deactivating the fan: c) (t) and d) y(t).

22 PID Controller Tning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane 137 Transfer fnction G p3 (s), sed for determining parameters of the PID controller applied in the real-time experiment, is obtained previosly in (Matašek & Ribić, 01). By applying a Psedo-Random-Binary-Seqence for (t), the open-loop response y(t) of the laboratory thermal plant is obtained. From these (t) and y(t) a 100-th order ARX model is determined and redced then to the 5-th order transfer fnction G p3 (s) in (Matašek & Ribić, 01). This model of the process is sed here to determine the qadrplet {k,,, A} presented in the Appendix. Ths, the laboratory thermal plant is classified as the process =0.9808, φ= Then, PID controller applied to the real thermal plant is determined by sing look-p tables of parameters k n (,φ), k in (,φ), k dn (,φ), for stable processes, and parameters k n (φ), k in (φ), k dn (φ), for integrating processes, previosly determined in (Šekara & Matašek, 011a). This procedre, sed to obtain PID in Table 5, row G p3 -GSC, and reslts obtained by this PID controller, presented in Fig. 1, demonstrate that in advance determined look-p tables of parameters k n, k in and k dn defines a process-independent GSC applicable for obtaining the desired performance/robstness tradeoff for a real plant classified in the -φ parameter plane. For T i =k/k i and T d =k d /k, parameter T aw =15s is obtained from T aw =pt i +(1-p)T d, for p=0., and l low =0, l high =100%, b=0.5. Closed-loop experiment in Fig. 1-a and Fig. 1-b is sed to demonstrate advantages of the designed PID controller, compared with the PI controller, from Table 4, row G p3 /opt3 defined by: k=8.1355, k i =0.0679, and b=0.5. This experiment starts from temperatre T(L a,t)45 C, as presented in Fig. 1-b. Then at t=1000 s the set point is changed to r=45 C+r 0, r 0 =5 C. At t=1600 s a load distrbance is inserted as a step change of the controller otpt eqal to -0%. Improvement of the performance obtained by the PID controller is evident. As expected, this is obtained with the greater variation of the control signal PID (t) than that obtained by PI (t). This is the reason why PID controller from Table, row tnλ, having a greater vale of M n =17.7, is not applied to the real thermal plant. The closed-loop experiment presented in Fig. 1-c and Fig. 1-d starts from the steady state temperatre T(L a,t)50 C by activating a fan at t=400 s. Then, at t=600 s the fan is switched-off. Action of the fan indced a strong distrbance, as seen from the control signal (t) in Fig. 1-c. It shold be observed that anti-windp action is activated two times, arond 410 s and 65 s. Anti-windp action is effective and rejection of the distrbance is fast, as seen from Fig. 18-d. 6. Conclsion The extension of the Ziegler-Nichols process dynamics characterization, developed in (Šekara & Matašek, 010a; Matašek & Šekara, 011), is defined by the model (5). Based on this model, a procedre is derived for classifying a large class of stable, integrating and nstable processes into a two-parameter -φ classification plane (Šekara & Matašek, 011a). As a reslt of this classification, a new CSC concept is developed. In the -φ classification plane, parameters g n (,φ)={k n (,φ), k in (,φ), k dn (,φ)} and T fn (,φ)= k dn (,φ) /m n, of a virtal PID n controller can be calclated in advance, to satisfy robstness defined by M s and sensitivity to measrement noise defined by m n. Also it is possible to satisfy M s, m n and the closed-loop system damping ratio. Calclation of parameters g n (,φ) and T fn (,φ) is process-independent. The calclation is performed by sing model G n (s n,,φ), defined by the vales of and φ for stable processes in the range 0 1,0 / 1, for integrating processes in the range 1, 0 /, for nstable processes by the vales of the and φ otside these regions.