Optimal control of inverted pendulum system using PID controller, LQR and MPC

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1 IOP Conferene Series: aterials Siene and Engineering PAPER OPEN ACCESS Otimal ontrol of inverted endulum system using PID ontroller, LQR and PC To ite this artile: Elisa Sara Varghese et al 7 IOP Conf. Ser.: ater. Si. Eng Related ontent - Fuzzy modeling and ontrol of rotary inverted endulum system using LQR tehnique A Fairus, Z ohamed and N Ahmad - Pendulums are magnetially ouled Yaakov Kraftmakher - Comarison of PID and Fuzzy Controller for Position Control of AR.Drone A Prayitno, V Indrawati and I I Trusulaw View the artile online for udates and enhanements. This ontent was downloaded from IP address on 8//8 at 6:39

2 . 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 Otimal ontrol of inverted endulum system using PID ontroller, LQR and PC Elisa Sara Varghese, Anju K Vinent and Bagyaveereswaran V Shool of Eletrial Engineering, VIT University, Vellore - 634, Tamil Nadu, India. vbagyaveereswaran@vit.a.in Abstrat. Inverted endulum is a highly nonlinear system. Here we roose an otimal ontrol tehnique for the ontrol of an inverted Pendulum - art system. The system is modeled, linearized and ontrolled. Here, the ontrol objetive is to ontrol the system suh that when the art reahes a desired osition the inverted endulum stabilizes in the uright osition. Initially PID ontroller is used to ontrol the system. Later, Linear Quadrati Regulator (LQR) a well-known otimal ontrol tehnique whih makes use of the states of the dynamial system and ontrol inut to frame the otimal ontrol deision is used. Various ombinations of both PID and LQR ontrollers are imlemented. To validate the robustness of the ontroller, the system is simulated with and without disturbane. Finally the system is also ontrolled using odel Preditive ontroller (PC). PC has well reditive ability to alulate future events and imlement neessary ontrol ations. The erformane of the system is omared and analyzed..aintrodution The inverted endulum is a highly unstable system whih is very diffiult to ontrol, thus making it a signifiant lassial roblem for researh in ontrol system domain []. Being utmost imortant it is used for analyzing and designing of ontrol laws. The system omrises of a art on whih a ole mounted and it moves horizontally. It osses a stable and unstable equilibrium state. The endulum orients downwards in the stable equilibrium state and in vertially uward diretion in the latter state. Hene in order to align the endulum in the latter osition a fore is required []. To frame a ontrol law for a system first we have to obtain a dynamial model of that system. The ontrol law is designed based on this dynami model so that required erformane is ahieved. ost of the systems that have dynamial model are non-linear []. Being a non-linear system designing of ontrol laws for inverted endulum is very hallenging. PID ontrollers are broadly utilized as a art of roess ontrol framework. To obtain best behavior from the system we use otimal ontrol tehniques with PID ontroller [3]. All ontrol roblems has a erformane inde. The Otimal ontrol tehnique either maimizes or minimizes this erformane inde. any otimal methods and algorithms are used to obtain best erformane. In this aer we use LQR method to obtain otimal ontrol of inverted endulum, whih is highly non-linear system[4]. In order to omare the erformane, the system is ontrolled using model reditive ontroller (PC). PC otimizes the urrent arameters and it an redit the future values and thus it takes ontrol ation aordingly [9]. Initially system is ontrolled using PID ontroller and then ontrolled using PC. Later LQR, the otimal ontrol tehnique is used to ontrol the system. In all Content from this work may be used under the terms of the Creative Commons Attribution 3. liene. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal itation and DOI. Published under liene by Ltd

3 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 the above ases the system is analyzed with and without disturbanes. The system erformanes are omared. This aer has five setions. Setion one gives a brief idea about the aer. Setion two deals with the modeling and linearization of the system. Setion three desribes the designing of various ontrol strategies. Setion four deits the simulation results and observations. The aer onludes with the results obtained in setion five.. odelling of Pendulum -Cart System Here we onsider a endulum art system.figure. reresents the free body diagram of the system. Here we assume that the rod of the endulum is mass-less and the hinge to whih the endulum is fied is fritionless. The mass. of the. endulum is onentrated at the enter. of gravity. of the endulum whih. is loated at endulum ball s enter. The mass of the art is reresented as and the mass of endulum is reresented as m. The ontrol fore u(t) ats along the diretion of the art. The rod s length is reresented as l.the. angle by whih the. endulum is. tilted reresented as θ [5]. The fore equation along the diretion is [5] Figure. Pendulum-art system C +m (), () We have (3) The torque balane of the system is reresented in Fig.

4 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 Figure.Vetor diagram of torque balane The torque balane equation is: (4) Where is the fore along diretion and is the fore along y diretion. After simlifying we get (5) Thus we have (6) (7) The equation (6) and (7) reresents the endulum - art system s Dynamial model... State sae model of system The equations (6) and (7) are to be reresented in state sae []. (8) The state variables are,,, (9) The state sae equations are d dt Where, d dt f d f 3 dt f. 4 f ()

5 th ICSET-7 IOP Conf. Series: aterials Siene and Engineering 63 (7) 57 doi:.88/ x/63/5/57 () ( ) () (3) (4) The outut equation is reresented C (5).. Linearisation of the inverted endulum The objetive is to lae the endulum in eret osition that is θ =.The system is linearized about this equilibrium oint by jaobian method [6]. The system is linearized with the ondition =.The linearized system is in the form (6) After linearization the terms of the jaobian matri are found systematially. We get ) ( ), ( C g m l g m u J (7) After differentiating with reset to u we get, l u l m m l m l m u f u f u f u f u J os ) ( os os 4 ), ( 3 (8) After simlifying we get

6 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 d dt ( m ) g l m g l u (9) as This is the linearized form of the system in erturbation form. Equation 9 an be generalized () 3. Control methods Different methods of ontrol. are used to bring the system. to stable osition. 3.. PIDkontrol First we use PID ontroller to bring the inverted endulum at equilibrium oint when the art reahes at desired osition. Here we use PID ontrollers. One PID ontroller to regulate the.art osition and other one to ontrol endulum. The PID ontroller equation is () () Here. reresent the osition error, reresent error angle, reresent art ontrol signal and reresents the endulum angle ontrol signal []. The referene osition is.m and the referene angle is.the ontroller s tuning has large effet on the ontrol ation. The. arameters of the ontrollers are found through trial and error method. The system using PID ontroller is simulated in ATLAB[ ]. 3.. OtimaloontrolousingyLQR Otimal ontrol tehniques are used to get best erformane from the system with reset to the seified riteria [8]. The main objetive of otimal tehnique to find the best ontrol signals that results the best erformane of the..system by satisfying the seified hysial.onstraints and by maimizing or minimizing the erformane inde []. Linear. quadrati regulator. (LQR) is a ontrol tehnique that. is otimal. Linear quadrati regulator (LQR),is a well known otimal ontrol tehnique, that makes use of dynamial. system s states and to frame the otimal ontrol law. They denote a robust ontrol tehnique [4]. 3.. Framing of otimal law By taking the initial onditions =[ ]. T. The state sae. equation is Where (3) 5

7 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 The feedbak signal is u = -K. We get. (4) The ost funtion is (5) Where Q is a.ositive semi.definite matri and.r is the ositive definite symmetri onstant.matri. K is obtained in LQR gain vetor from K=R - B T P (6) P is a ositive. definite onstant matri. and it is found from the algebrai. Riati equation. (ARE) A\. T P+ P.A PBR - B T P +.Q = (7) For LQR we give four states as feedbak. They are art osition (), art veloity (, endulum angle (θ) and angular veloity ( ).These are onsidered as available measurements. The LQR outut is added negatively to PID ontrol to obtain otimal ontrol. The system is simulated using different ombinations of PID and LQR tehnique odel Preditive Control The funtion of PC is to omute the value to be used in the inut in real time for roblem otimization. PC makes use of the inut and outut values from the earlier roess to redit the following outut in a definite size of horizon of time. The otimisation starts after the alulation of time at that desired horizon []. PC alulates the otimal signal orresonding to eah inut by onsidering the onstraints. It redits the future by making use of the inut outut values of the reeding state. After that the time is shifted and the alulation of inut is done again to obtain an otimal solution whih an be used for the further alulations []. The roess reeats in suh a manner so as to always rodue an otimal ontrol. 4. Simulation Results The non-linear inverted endulum is modelled, analysed and ontrolled using the atlab-simulink models for ases. of with and without disturbane. inut []. The system arameters are. art s ass ( C ) =.4 kg; endulum s mass ( )=.3 kg; Cart trak s length (L C =±.5m); the oeffiient of frition between art and.ole rotation is taken as.negligible. The values of inut. disturbane arameters used for the.simulation are seed = 334, magnitude of band limited white noise ower as. and the samling values time is taken as. [5]. Net ste is to design LQR for whih the system matries are required and are obtained as: A B.467 6

8 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 C D By hoosing R = and Q 5 5 The gain vetor of the LQR is attained as K The otimal ontrol of the inverted endulum is imlemented using three ontrol ombinations []:. Control using two PID ontrollers i) Cart PID ii) Angle PID. Control using two PID ontrollers (ombination of both art and angle PID) with LQR 3. Control using one PID with LQR The values of PID arameters whih are tuned for these ontrol shemes for the ases of with and without disturbane is tabulated in tables and.. Table. Parameters of PID ontroller without disturbane Control Shemes Angle PID ontrol Cart PID ontrol K a K ia K da K a K ia K da PID LQR + PID LQR + PID Table. Parameters of PID ontroller with disturbane Control Shemes Angle PID ontrol Cart PID ontrol K a K ia K da K a K ia K da PID LQR + PID LQR + PID Where K a, K ia and K da are the rootional, integral and differential gains of the endulum resetively, and K a, K ia and K da are the rootional, integral and differential gains of the art resetively The first ontrol method is using only PID ontrollers is simulated in ATLAB Simulink and both onditions of without and with disturbane are deited in Fig 3 and Fig 5 resetively. The disturbane inut added to the system is band limited white noise. The arameters onsidered for measurement are the osition of the art and.endulum s angle. 7

9 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 The referenes for art osition and endulum angle are.m and rad resetively. The results for simulation are shown in figures 4 and 6 resetively. Figure 3. Control of system with two PID ontrollers without disturbane.. Control Angle and Angular veloity.. thetta thetta dot Position and Veloity..5 dot Figure 4. Resonses of ontrol fore, endulum angle and art osition without disturbane. 8

10 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 Figure 5. Control of system with two PID ontrollers with disturbane Control - Angle and Angular veloity thetta. thett dota.5 Position and Veloity dot Figure 6. Resonses of ontrol fore, endulum angle and art osition with disturbane. Figure 4 and figure 6 shows that for the ase of with disturbane the endulum after two small overshoots stabilises in the vertially uward osition and in the ase of without disturbane stabilises with minor osillations. The Simulink models of the nonlinear endulum art system for the otimal ontrol utilizing two PID ontrollers (angle PID and art PID) with LQR ontrol strategy for both instanes of with and without disturbanes inut are shown in figs. 7 and 9.resetively []. The LQR is fed with the states of the system i.e,,, whih is designed using system s linear sae.model. The variables of ontrol are hosen as angle and art osition [4]. The inut disturbane added to the system 9

11 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 is band limited white noise. The referenes taken are as before. Figs. 8 and shows the simulation results. The effiieny and strength of the system is deited by the simulation results. * Figure 7. Control using two PID ontrollers with LQR without disturbane Control(u) Angle And Angular dislaement Cart Position And Veloity thetta thetta dot dot Figure 8. Resonses of ontrol signal, endulum angle, angular dislaement, Cart osition and art veloity without disturbane

12 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 Figure 9. Control using two PID and LQR with disturbane Control (u) Angle And Angular Position Cart Position & Veloity Thetta Thetta dot Time (s). dot Figure. Resonses of ontrol signal, endulum angle, angular dislaement, Cart osition and art veloity with disturbane The net ontrol strategy using one PID ontroller and LQR without and with disturbane is modelled in Simulink atlab as shown in figs. and 3 PID ontroller is used here for art osition ontrol only and not for angle ontrol []. This method resembles the seond one in all asets aart from the number of ontrollers used. Cart PID ontroller is only and it is set with a referene of.m.

13 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 Figure. Control of the.system. using art PID. and LQR without disturbane. Control(u) Angle and angular veloity thetta thetta dot Cart osition and veloity..5 dot Figure. Resonses of ontrol signal, endulum angle, angular dislaement, Cart osition and art veloity with.disturbane Figure 3. Control of inverted endulum using art PID and LQR with disturbane

14 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 Control(u) - Angle and Angular veloity thetta thetta dot Cart osition and veloity..5 dot Figure 4. Resonses of ontrol signal, endulum angle, angular dislaement, Cart osition and art veloity with disturbane As the LQR is.designed using the.state sae feedbak of the. system with.the endulum in the vertial osition the angle ontrol is automatially taken are of [8]. The resonses.of angle,. angular veloity,.art osition,. art veloity and. ontrol signal u.are lotted as shown in figs and 4. The results of simulation justify the effetive robustness of the system. The system is now ontrolled using odel reditive Controller as shown in Figure 5. The resonse obtained is lotted in Figure 6. Figure 5. PC Control of Inverted Pendulum 3

15 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57.5 Contol(u) Angle Position Figure 6. Resonses Of Control signal, Pendulum angle and art osition. The maimum. absolute values of different states of the system and ontrol signal for ases of with and without disturbanes are tabulated in tables 3 and 4 resetively. Table 3. System states and ontrol signal s maimum absolute magnitudes in the absene of disturbane Control Shemes P.ID PID +LQR PID + LQR PC Table 4. System states and ontrol signal s maimum absolute magnitudes in the resene of disturbane Control Shemes PID PID + LQR PID + LQR By relating the resonses of the above ontrol shemes it is found that the ones using LQR ontrol tehnique stands suerior to the one using PID ontrol only. Although the sheme using two PID and LQR and single PID and LQR are similar but it is found that the former is a little imroved 4

16 4th ICSET-7 IOP Conf. Series: aterials Siene and Engineering (7) 57 doi:.88/ x/63/5/57 omared to the latter as it onsist of angle ontroller whih in turn ontributes to additional degree of freedom. On the other hand the ombination of one PID and LQR is simle in struture. From all these results we observe that these ontrol strategies are rofiient and robust. Among the strategies the PID +LQR is seleted as the suerior one for the otimal ontrol of the system due to its simle struture, robustness and effiieny. 5. Conlusions To ontrol the non-linear inverted endulum, PID ontrol and LQR, a ontrol strategy for otimal ontrol has been imlemented. It is analysed for ases of using disturbane and without disturbane is eeuted. The LQR is diretly fed with all the states of the system whih are obtainable for measurement. The design of LQR is done using the model s state sae. LQR s otimal value is added to the PID ontroller to ahieve the resulting otimal ontrol. atlab is used to develo the models and to analyse the resonse oututs. The tuning of the ontrollers to ahieve the most otimal ontrol is attained by trial and error method. The results of the simulation justify the relative effetiveness of using LQR ontrol method. The endulum stabilises in the vertial osition and the art reahes the desired osition even in the resene of disturbane inut. The study of the ontrol resonses deit that the erformane of the PID +LQR ontrols method is suerior to the one using PID only. This onludes with the statement that, of all the ontrol shemes erformed, LQR + PID is effetive, robust and simle.. Referenes [] Prasad L B, Tyagi B and Guta H O Otimal ontrol of nonlinear inverted endulum dynamial system with disturbane inut using PID ontroller & LQR IEEE Int. Conf. on Control System Comuting and Engineering (ICCSCE) 54-5 [] Ogata K 5 odern Control Engineering 4th edition (New Delhi:Pearson Eduation Pvt Ltd.) [3] Sharif B A and Uar A 3 State feedbak and LQR ontrollers for an inverted endulum system Int. Conf. on Tehnologial Advanes in Eletrial Eletronis and Comuter Engineering (TAEECE) [4] Grasser F, D'arrigo A, Colombi S and Rufer A C A mobile inverted endulum IEEE Transations on industrial eletronis 49() 7-4 [5] Ibanez C A, Frias O G and Castanon S 5 Lyaunov-based ontroller for the inverted endulum art system Nonlinear Dynamis 4(4) [6] Ang K H, Chong G and Li Y 5 PID ontrol system analysis, design, and tehnology IEEE transations on ontrol systems tehnology 3(4) [7] Pathak K, Franh J and Agrawal S K 5 Veloity and osition ontrol of a wheeled inverted endulum by artial feedbak linearization IEEE Transations on robotis (3) 55-3 [8] Kumar E V and Jerome J 3 Robust LQR ontroller design for stabilizing and trajetory traking of inverted endulum Proedia Engineering [9] agni L, De Niolao G, Sattolini R and Allgöwer F 3 Robust model reditive ontrol for nonlinear disrete time systems International Journal of Robust and Nonlinear Control 3 (3 4) 9-46 [] Askari, ohamed H A, oghavvemi and Yang S S 9 odel reditive ontrol of an inverted endulum Int. Conf. for Tehnial Postgraduates (TECHPOS) - 4 [] Ray G, Das S K and Tyagi B 7 Stabilization of inverted endulum via fuzzy ontrol Journal Institution Of Engineers India Part el Eletrial Engineering Division