Design of PID Controller for SISO and TITO Systems A Study with Heuristic Algorithms

Transcription

1 ISSN (Print : ISSN (Online : Design of PID Controller for SISO and TITO Systems A Study with Heuristic Algorithms R. Manibharathi, K. Mano Kumar, K. Kumaran 3, Dr. N. Sri Madhava Raa 4 Deartment of Electronics and Instrumentation Engineering, St. Joseh s College of Engineering, OMR, Chennai 69. Tamilnadu, India. mani.barathi3@gmail.com manoyendo95@gmail.com 3 kumaran495@gmail.com 4 nsrimadhavaraa@stosehs.ac.in Abstract This aer rooses a study on heuristic algorithms based PID controller design for a class of Single Inut Single Outut (SISO and Two Inut and Two Outut (TITO systems. In this work, well known heuristic methods such as Particle Swarm Otimization (PSO, Imroved PSO (IPSO, Bacterial Foraging Otimization (BFO, hybrid algorithm (PSO+BFO and Firefly Algorithm (FA are chosen to find the otimal K, K i, K d values based on the chosen SISO and TITO rocess models. For the SISO systems, the well known PID and set-oint filter based PID controllers are imlemented and for the TITO rocess models, a decentralised PID controller design is roosed. In this work, the otimization rocess is focused to search the best ossible controller arameters (K, K i, K d by minimizing the multi obective erformance index chosen to guide the heuristic search. The effectiveness of the roosed scheme has been confirmed through a comarative study with heuristic rocedures and the classical controller tuning methods roosed in the literature. The results show that, heuristic algorithm assisted aroach rovides enhanced erformance in effective reference tracking with minimal ISE and IAE values. Finally the robustness of the SISO system is validated by oerating the unstable systems in the resence of a measurement noise. The results testify that the PSO-BFO tuned set-oint filter based PID erforms well in tracking the change in reference signal even in the noisy environment. The FA based rocedure on the TITO system also offered better result on the TITO rocess comared with others aroaches existing in the literature. Keyword-PID controller, SISO system, MIMO system, Heuristic aroaches, Performance analysis I. INTRODUCTION In chemical industry, most of the imortant rocessing units are highly non-linear in nature and for economical and/or safety reasons, these rocess loos to be oerated in safe regions with the hel of a PID controller [-3]. Based on its oerating region and its non-linearity, these loos can be modelled as SISO and MIMO systems. During the closed loo oeration, otimized controller arameters for such systems are essential to minimize the waste and to maximize the roduction rate. Fine tuning the controller arameters for non-linear systems are highly comlex than oen loo stable systems. Hence, in recent years, heuristic algorithm assisted aroaches are widely adoted by the researchers. In recent years, many efforts have been attemted to design otimal and robust controllers for chemical systems [4-6]. Normally, most of these classical controller tuning aroaches require an aroximated transferfunction and state sace model of the system around the oerating oint. In real time alications, the aroximated model arameter may be changing or subect to uncertainty. Also the tuning method roosed for a articular model does not rovide a satisfactory result on the other models. Hence, to overcome these issues, it is necessary to emloy intelligent controller tuning rocedures to identify the best ossible controller arameters for the non-linear chemical systems. In recent years, heuristic algorithm based otimization is emerged as a owerful tool for finding the solutions for a variety of control engineering roblems. Recent work utilises the heuristic rocedures, such as Particle Swarm Otimization [6, 7], Imroved PSO [8], Bacterial Foraging Otimization [9, ], Hybrid algorithm [, ], Bat Algorithm [3], Firefly Algorithm [4], Teaching Learning Based Otimization [5], etc. DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 685

2 ISSN (Print : ISSN (Online : The work by Rainikanth and Latha [6] reorted that heuristic algorithm based PID controller tuning can be imlemented for the unstable SISO system when the Delay/ constant ratio is below.. PID based tuning results large overshoot which tends to increase the error and overshoot value, when this ratio is greater than.. This henomenon disruts the convergence of soft comuting based search. In this work, the PID controller arameter tuning is roosed for a class of SISO and TITO chemical systems existing in the literature. A comarative study also carried out with the roosed heuristic aroaches and the classical controller tuning methods roosed in the literature. The remaining art of the aer is organized as follows: rincile of various evolutionary algorithms and imlementation of these algorithms to obtain otimized controller value is discussed in Section and 3 resectively. In Section 4, results and discussions on a class of chemical system models are resented. Finally, the conclusion of the resent research work is given in Section 5. II. HEURISTIC ALGORITHMS ADOPTED IN THIS PAPER This section resents the overview of the heuristic algorithms chosen for the study. A. Particle Swarm Otimization PSO is a oulation based stochastic otimization technique insired by social behaviour of bird flocking or fish schooling, and it is widely used in engineering alications due to its high comutational efficiency [6]. PSO algorithm is easy to imlement and there are few arameters to adust comared to other heuristic methods. It is a oulation based evolutionary comutation technique, attemts to mimic the natural rocess of grou communication of individual knowledge, to achieve some otimum roerty. More details regarding the PSO can be found in [,3]. The basic mathematical exression of the PSO is deicted below; t+ t t t t t t Vi = W.Vi + C.R.(Pi Si + C.R.(Gi Si ( t+ t t+ S i = Si + Vi ( t (Wmax - Wmin W = (Wmax Iter x Itermax (3 Where, C, C are ositive constants. C is the cognitive learning rate and C is the global learning rate. R, R are random numbers in the range -. The arameter W is inertia weight that increases the overall erformance of PSO. The larger value of W can favour the global wide-range search and lower value of W imlies a higher ability for local nearby search. B. Imroved PSO Algorithm Chang and Shih [8] have develoed an IPSO algorithm to tune the PID controller for a non linear inverted endulum system. In this, an imroved velocity udating equation is roosed to imrove the algorithm convergence and it is given in Eqn.4. V i (t+ = W t.v i t + C. R. (best - S i t + C. R.(gbest -S i t + C 3. R 3. (ibest - S i t (4 Where, ibest reresents the best article s osition among all articles in the sub-oulation that the i th article belongs to. C 3 and R 3 are ositive constant and random number resectively. C. Bacterial Foraging Otimization BFO algorithm is a biologically insired stochastic search technique based on mimicking the foraging behavior of E.coli bacteria [7]. During foraging, a bacterium can exhibit two different actions: Tumbling or swimming. The tumble action modifies the orientation of the bacterium. During swimming (chemotactic ste the bacterium will move in its current direction. Chemotactic movement is continued until a bacterium goes in the direction of ositive nutrient gradient. After a certain number of comlete swims, the best half of the oulation undergoes reroduction, eliminating the rest of the oulation. In order to escae local otima, an elimination-disersion event is carried out where, some bacteria are liquidated at random with a very small robability and the new relacements are initialized at random locations of the search sace. DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 686

3 ISSN (Print : ISSN (Online : In this work, enhanced BFO discussed in [,4] is adoted; Number of E.Coli bacteria = N N N N N N N c = ; Ns =N re ; Ned ;Nr = ; Ped = 3 4 ed ; d att = W att = N + N r N s N ; and hre = W re = c (5 N N D. Hybrid Otimization Algorithm This algorithm was roosed by Korani et al. [] to imrove the erformance of BFO. In this method, the cost function (ISE is alied for both the PSO and BFO algorithm. The PSO algorithm monitors the BFO to achieve a minimum convergence time with otimized arameters. In hybrid algorithm, after undergoing a chemotactic ste, each bacterium gets mutated by a PSO oerator. The PSO oerator considers only the social comonent and eliminates the cognitive comonent. In this algorithm, due to the information sharing between the PSO and BFO, the hybrid algorithm can rovide the otimized solutions with minimal convergence time comared to a conventional BFO algorithm. E. Firefly algorithm The classical Firefly Algorithm (FA was initially roosed by Yang [8]. FA is a nature insired metaheuristic algorithm, in which flashing illumination atterns generated by fireflies are modelled using a suitable mathematical exression. In this work, the recent version of the FA discussed in [9, ] is considered. The association of an attracted firefly towards a mate can be exressed as: t + t γ d i t t X i = X i + βe ( X X i + α (rand - ½ where t X i is early location; t X i + γ d i t t is udated location; βe ( X Xi is attraction among fireflies; β is reliminary attractiveness; γ is absortion coefficient; α is randomization oerator and rand is random number [,]. In this aer, the following values are chosen for FA arameters: α =.5; β =.and γ =. III. IMPLEMENTATION The chosen roblem is to design the PID controller for the rocess models. A generalized close loo control system is deicted in Fig.. The controller G c (s has to rovide closed loo stability, smooth reference tracking and load disturbance reection [, ]. (6 R(s E(s U c (s G c (s - Y m (s G (s D(s Y(s Fig.. Block diagram of a closed loo control system Closed loo resonse of the above system with set-oint R(s and disturbance D(s can be exressed as; G GC Y = R( s + D( s + G GC + G GC (7 The final steady state resonse of the system for the reference tracking and the disturbance reection is resented in Eqn.3 and Eqn.4 corresondingly. G (sg C (s A YR ( = lim s YR (s = lim s x = A t t + G (sg C (s s L YD ( = lims x = t + G (sg C (s s Where: A = amlitude of reference signal ; D = disturbance (8 (9 To achieve a satisfactory Y R ( and Y D (, it is necessary to have otimally tuned values for K, K i and K d. In this study, a non-interacting form of arallel PID controller is considered to achieve the referred resonse. DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 687

4 ISSN (Print : ISSN (Online : The arallel PID structure is given below: T de(t G C(s = K e(t + Ki e(t dt + Kd ( dt G PID (s = K + + Td s ( Ti s Where: K / T i = K i ; K*T d = K d. For the filtered PID, the filter time constant is chosen as the Ti [3]. The multile obective function for controller otimization was chosen. In this work erformance criterion with four functions, such as CF, M, t s, and E ss as resented below was chosen to guide the heuristic search J min (K, K i, K d = (w.e ss + (w.t s + (w 3. M ( Where: w. w, w 3 -weighting arameters (range is from -, E ss the steady state error, t s settling time and M overshoot. The PID tuning rocess is emloyed to find the best ossible values for K, K i and K d and the set-oint filter arameter T f form the three dimensional search sace by minimizing the obective function. During this search, the erformance criterion J min (K, K i, K d guides the heuristic algorithm to get aroriate values for the controller arameters. IV. RESULTS AND DISCUSSIONS This section resents the results obtained with the roosed aroach. All the simulations are imlemented using the MATLAB software. In this work, the roosed controller design rocedure is imlemented on the well known SISO and TITO rocess loos. During the simulation work, the following values are assigned for the algorithm arameters: number of agents (N = 3, dimension of the search (D = 3, stoing criteria = J min and the number of iterations =. a. SISO systems Process : The first order lus delayed time unstable rocess with the following transfer function model is considered [4-7]; s 4e G( s = (3 4s The rocess has a gain (K = 4, rocess time constant (τ = 4 and time delay (d =. For this rocess d/τ is.5. Many studies have roosed different PID settings for the above model and the values are clearly resented in the literature. The classical PID settings are resented in Table. The evolutionary algorithm based controller tuning is roosed for the system as in Fig.. The final convergence of the controller arameters for the hybrid algorithm is shown in Fig.3 and the otimised K, K i, K d values are tabulated in Table. Fig. 4 shows the convergence of the CF for the various evolutionary search algorithms. Fig.5 deicts the servo resonse of the rocess with classical PID settings roosed in the literature. In this The HC [4] method shows a satisfactory result comared to other methods. Fig.6 shows the reference tracking erformance of the evolutionary methods. The observation is that, the hybrid method rovides a good result for reference tracking erformance. Resonse HC SSC Visioli JSH Fig.. Servo resonses for Process with classical PID arameters DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 688

5 ISSN (Print : ISSN (Online : Servo resonse PSO IPSO BFO Hybrid Fig. 3. Servo resonses for Process with otimised PID arameters From Table, it is observed that, the hybrid algorithm based tuning has less number of iteration (67 and it also shows a good erformance measure in reference tracking (such as: ISE, IAE, M and T s comared to PSO, IPSO and BFO algorithms. Process : The second order delayed unstable rocess with the following transfer function is considered. It has one unstable ole and a stable ole [4, 8]. s ex G (s = (4 (s (.5s + Previous studies have roosed different PID settings for the above model (Huang and Chen, 999; Lee, Lee, and Park,. Fig.7. shows the servo resonse of the revious work reorted in literature. In this diagram, the method roosed by LLP rovides the smooth reference tracking erformance comared to PC and HC. Fig.8. shows the reference tracking erformance of the resent study. The resonse roduced by the PSO and IPSO algorithm is more oscillatory comared to BFO and PSO-BFO methods. Resonse HC PC LLP Fig. 4. Servo resonses for Process with conventional PID arameters.4. Resonse PSO IPSO BFO Hybrid Fig. 5. Servo resonses for Process with soft comuting based PID arameters From Fig.5 and Table, it is inferred that, the roosed hybrid method can be used to get an otimal controller arameter with lesser convergence time to rovide a smooth reference tracking erformance than other otimisation algorithms. TABLE I. Controller arameters, filter arameters and the erformance measure of this study Process Method Iteration K K i K d T f ISE IAE M T s Process HC [4] SSC [5] Visioli [6] JSH [7] PSO IPSO BFO DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 689

6 ISSN (Print : ISSN (Online : Process Hybrid HC [4] PC [] LLP [8] PSO IPSO BFO Hybrid Process 3: Jacketed Continuous Stirred Tank Reactor (CSTR studied by Bequette [9] can be reresented by the following equations; dc dt A F ΔE = f(ca,t,t = (CAf CA kex CA (5 V RT dt = f dt (C A,T,T = F V (Tf T + ( ΔH ρ ΔE UA kex CA ( T T C RT VρC (6 dt F f = f3(ca,t,t = ( Tf dt V UA T + V ρ C ( T T (7 Eqn. 5 7 reresents the modeling equations of the case study which discuss the roduction of roylene glycol by the hydrolysis of roylene oxide with sulhuric acid as a catalyst. TABLE II. Modelling arameters of the CSTR Parameter Value Activation factor (E a 3,4 Btu/lbmol Frequency factor (k o 6.96x hr - Heat of reaction (-ΔH 39 Btu/lbmol PO U 75 Btu/hr ft o F ρ C 53.5 Btu/ ft 3 o F R.987 Btu/lbmol o F Residence time (V/F 5 min=.5 hr C Af.3 lbmol/ ft 3 T f 6 o F =59.67R V /V.5 T f o F ρ C 55.6 Btu/ ft 3 o F The rocess transfer function model relating the acket flow rate to the reactor temerature is G (s s x e s s s = (8 3 s The robust stability of the CSTR model is analysed using the Kharitonov s theorem [3]. This method rovides a necessary and sufficient analysis test for the robust stability of olynomials with erturbed coefficients. L U Let the closed loo system with the above characteristic equation are defined by in the interval i i i with a lower and higher bond of arameters of i with i =,,...6. According to Kharitonov s theorem, every characteristic equation in the interval family will be stable if the following four Kharitonov olynomials are Hurwitz: L L U U 3 L 4 L 5 K (s = + s + s + 3 s + 4 s + 5 s +... (9 L U U L 3 L 4 U 5 K (s = + s + s + 3 s + 4 s + 5 s +... ( U L L U 3 U 4 L 5 K (s = + s + s + 3 s + 4 s + 5 s +... ( U U L L 3 U 4 U 5 K (s = + s + s + 3 s + 4 s + 5 s +... ( DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 69

7 ISSN (Print : ISSN (Online : As the rocess arameters (gain, time constant and time delay are erturbed, closed-loo stability changes. The CSTR model for the erturbed rocess arameters around their nominal values with the olynomials (K, K, K, K in s=ω lane were checked for Hurwitz condition. Fig 6. Shows the Kharitonov s rectangles for the CSTR model in S lane, from this lot, it is observed that the oen loo system with the erturbed rocess arameters shows that it is in unstable oerating region. An otimal controller tuning for this rocess is then roosed with the evolutionary methods and the corresonding controller and the filter arameters are resented in Table. Fig 7 shows the servo resonse of the CSTR temerature rofile. The hybrid algorithm based controller erforms better comared to PSO, IPSO and BFO tuned controller. The erformance measure values are clearly resented in Table. 8 x 3 Maximum Frequency of Evaluation (r/s = Imag Axis Real Axis x Fig. 6. Kharitonov rectangles for the CSTR model in S lane.4. Resonse PSO IPSO BFO Hybrid Fig.7. Servo resonse of Jacketed CSTR.4. Servo Resonse Original Data with noise Change in delay Change in gain. Fig. 8. Servo resonse of CSTR with erturbed rocess arameters The hybrid algorithm tuned set-oint filter with the PID structure is then tested for the robustness by alying a % increase in the delay, 5% increase in the gain and a measurement noise (band limited white noise with a noise ower of.. From Fig 8. it is observed that, the controller rovides a smooth resonse for the CSTR model with the erturbed rocess arameters. b. TITO system Process 4: The roosed controller design rocedure is then tested on the WB model [4]. s 3s.8e 8.9e y ( s = 6.7s + s + u( s y ( s 7s 3s 6.6e 9.4e u( s.9s + 4.4s + Eqn. 3 shows the model of WB bench mark system. Initially, the IPSO based search is attemted to identify the otimal controller values, later BFO, hybrid algorithm and FA based search are considered. During the controller design rocess, the simulation time T is assigned as 5 sec. A unit ste signal is alied as the reference signal during PID design and the inut signal is considered as for PID tuning. (3 DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 69

8 ISSN (Print : ISSN (Online : TABLE III. Otimized PID values for WB model Process WB Method To Product Bottom roduct K K i K d K K i K d IPSO BFO Hybrid FA Resonse.5 Reference IPSO BFO Hybrid FA (sec Fig. 9. Resonse of to roduct.5 Resonse.5 Reference IPSO BFO Hybrid FA (sec Fig.. Resonse of bottom roduct TABLE IV. Error values for WB model Process WB Method To Product Bottom roduct ISE IAE ISE IAE IPSO BFO Hybrid FA Table 3 resents the otimised controller arameters and Table 4 resents the erformance measure values obtained with heuristic algorithm tuned controllers. In order to confirm the smooth reference tracking erformance, a unity ste signal is assigned as the reference for the bottom roduct at T=5 and the rocess resonse values are recorded as shown in Fig. 9 and. From this result, it is confirmed that, roosed aroach is efficient in offering the better PID arameters for the TITO system. V. CONCLUSION In this work, heuristic aroach based controlling tuning roblem is addressed for a class of SISO and a TITO system. A detailed comarative study between the PSO, IPSO, BFO, hybrid algorithm and FA are resented. In this work, a well known unstable SISO rocess models are considered and the PID controller values are designed. The erformances of the heuristic algorithms are assessed based on the time domain erformance measures. The roosed aroach is also validated on the classical tuning rocedures existing in the literature. Finally, the robustness of the designed controller is confirmed using the Kharitonov s aroach. The simulation result confirms that, the roosed aroach offers better result on the SISO rocess models. Finally, the DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 69

9 ISSN (Print : ISSN (Online : advantage of the heuristic aroach is then confirmed using the WB distillation rocess existing in the literature. From this aer, it is confirmed that, heuristic algorithm based aroaches are very efficient in designing the PID controller for the SISO and TITO rocess models. REFERENCES [] V. Rainikanth, and K. Latha, Controller Parameter Otimization for Nonlinear Systems Using Enhanced Bacteria Foraging Algorithm, Alied Comutational Intelligence and Soft Comuting, vol., Article ID 464, ages,. [] K. Latha, V. Rainikanth, and P. M. Surekha, PSO-Based PID Controller Design for a Class of Stable and Unstable Systems, ISRN Artificial Intelligence, vol. 3, Article ID 54367, ages, 3. [3] V. Rainikanth, and K. Latha, Setoint weighted PID controller tuning for unstable system using heuristic algorithm, Archives of Control Sciences, vol., no.4, ,. [4] V. Rainikanth, and K. Latha, Bacterial foraging otimization algorithm based PID controller tuning for time delayed unstable system, The Mediterranean Journal of Measurement and Control, vol.7, no.,. 97-3,. [5] K. Suresh Manic, S. Devakumar, V. Viayan, and V. Rainikanth, Design of Centralized PI Controller for Interacting Conical Tank System, Indian Journal of Science and Technology, vol.9, no.,. 899, 6. [6] V. Rainikanth, and K. Latha, Otimization of PID controller arameters for unstable chemical systems using soft comuting technique, International Review of Chemical Engineering, vol.3, no.3, ,. [7] R. Kotteeswaran, and L. Sivakumar, Otimal Tuning of Decentralized PI Controller of Nonlinear Multivariable Process Using Archival Based Multiobective Particle Swarm Otimization, Modelling and Simulation in Engineering, vol. 4,. 6, 4. [8] W.D. Chang, and S.P. Shin, PID controller design of nonlinear systems using an imroved article swarm otimization aroach, Communications in Nonlinear Science and Numerical Simulation, Elsevier, vol. 5, no., ,. [9] V. Rainikanth, and K. Latha, Tuning and Retuning of PID Controller for Unstable Systems Using Evolutionary Algorithm, ISRN Chemical Engineering, vol., Article ID , ages,. [] V. Rainikanth, and K. Latha, PID controller tuning for magnetic susension system using evolutionary algorithm, International Review of Mechanical Engineering, vol.6, no.5, [] Wael. M. Korani, Hassen Taher Dorrah, and Hassan M. Emara, Bacterial foraging oriented by article swarm otimization strategy for PID tuning, Proceedings of the 8 th IEEE international conference on Comutational intelligence in robotics and automation, , 8. [] A. Raasekhar, A. Abraham, and V. Snasel, A Hybrid Bacterial Foraging - PSO Algorithm Based Tuning of Otimal FOPI Seed Controller, Acta Montanistica Slovaca, vol.6,no.,.55-65,. [3] K. Suresh Manic, V. Rainikanth, Sarath Ananthasivam, Uma Suresh, Design of Controller in Double Feedback Control Loo An Analysis with Heuristic Algorithms, Chemical Product and Process Modeling, vol., no.4,. 53-6, 5. [4] R. Sivakumar, V. Rainikanth, and D. Sankaran, Multi-loo PI controller design for TITO system: an analysis with BA, FA, PSO and BFO, Australian Journal of Basic Alications Sciences, vol.9, no.6,.49-54, 5. [5] V. Rainikanth, and Suresh Chandra Sataathy, Design of controller for automatic voltage regulator using teaching learning based otimization, Procedia Technology, vol.,. 95-3, 5. [6] J. Kennedy, and R. Eberhart, Particle Swarm Otimization, in roceedings of IEEE International Conference on Neural, , 995. [7] K. M. Passino, Biomimicry of bacterial foraging for distributed otimization and control, IEEE Control Systems Magazine,.5-67,. [8] X.-S. Yang, Firefly algorithms formultimodal otimization, in Stochastic Algorithms: Foundations and Alications, Lecture Notes in Comuter Science, vol. 579, , 9. [9] N. S.M. Raa, V. Rainikanth, and K. Latha, Otsu Based Otimal Multilevel Image Thresholding Using Firefly Algorithm, Modelling and Simulation in Engineering, vol. 4, Article ID , 7 ages, 4. [] V. Rainikanth, and M.S. Couceiro, RGB histogram based color image segmentation using firefly algorithm, Procedia Comuter Science, vol.46, , 5. [] R. Padma Sree, and M. Chidambaram, Control of Unstable Systems, Narosa Publishing House, India, 6. [] V. Rainikanth, and K. Latha, I-PD controller tuning for unstable system using bacterial foraging algorithm: a study based on various error criterion, Alied Comutational Intelligence and Soft Comuting, vol., Article ID 39389, ages,. [3] V. Rainikanth, and K. Latha, Internal model control-roortional integral derivative controller tuning for first order lus time delayed unstable systems using bacterial foraging algorithm, Scientific Research and Essays, vol.7, no.4, ,. [4] H. P. Huang, and C.C. Chen, Auto-tuning of PID controllers for second order unstable rocess having dead time,.j. Chem. Eng. Jn., vol.3, no.4, , 999. [5] R. Padma Sree, M.N. Srinivas, and M. Chidambaram, A simle method of tuning PID controllers for stable and unstable FOPDT systems, Comut. Chem. Eng., vol.8,. 8, 4. [6] A. Visioli, Otimal tuning of PID controllers for integral and unstable rocesses, IEE Proc. Control Theory Al., vol.48, no.,.8 84,. [7] C.S. Jung, H.K. Song, and J.C. Hyun, A direct synthesis tuning method of unstable first-order-lus-time-delay rocesses, J. Process Control, vol.9, , 999. [8] Y. Lee, J. Lee, and S. Park, PID controller tuning for integrating and unstable rocesses with time delay, Chem. Eng. Sci., vol.55, ,. [9] B. Wayne Bequette, Process Control Modeling, Design and Simulation, Prentice Hall of India Pvt Ltd, 3. [3] V.L. Kharitonov, Asymtotic stability of an equilibrium osition of a family of systems of differential equations, Differentsialnye uravneniya, vol. 4, , 978. DOI:.87/iet/7/v9i3/7937 Vol 9 No 3 Jun-Jul 7 693