CONTROL OF NON-LINEAR SYSTEM USING BACKSTEPPING

Transcription

1 CONTOL OF NON-LINEA SYSTEM USING BACKSTEPPING Anamika as Ojha Ahala Khandelwal Asst Prof Department of Eletronis and Communiation Engineering Aropolis Institute of Tehnolog & esearh Madha Pradesh India Asst Prof Department of Eletrial Engineering Shri aishnav Institute of Tehnolog & Siene Madha Pradesh India Astrat The defining attriute of a servomehanism is that the ontrolled output of a mehanism is automatiall ompared with the ontrolling input. The differene etween the settings or positions of the output and the input is alled the error signal whih ats to ring the output to its desired value.the paper work shows the appliation of akstepping tehnique to stailie a nonlinear dnamial sstem. An nonlinear dnamial sstem an e stailied onl means of Lapunov funtion approah. In this approah akstepping tehnique is invarial adopted. In ase of normal akstepping tehnique stailiing funtion is designed at ever step of dnamis starting from the output towards input. The Bakstepping approah provides a reursive method for stailiing the origin of a sstem in strit-feedak form. The ontroller ontains the terms of sstem state variales referene input & its derivatives and error. The asi aim of the ontroller is to settle down error and its derivatives to ero. The ontroller is designed through Lapunov ased funtion so as to render strong losed loop stailit.in the theor of ontrol sstems Lapunov funtions are salar funtions that ma e used to prove the stailit of equilirium of a ontrol sstem. Kewords: Lapunov funtion akstepping nonlinear sstem stailit derivative *** INTODUCTION Prior moving towards the projeted tehnique we must e aquainted with some ommon terminolog.we must have a lear omprehension of ontrol sstem. A ontrol sstem is a seamless interonnetion of omponents assoiated in suh a manner as to ommand and regulates itself or some other sstem. A ontrol sstem that manages the ehaviour of another sstem is alled a Controller. On the foundation of numer of input-output a sstem an either e eluidated as a MIMO sstem with multiple inputs and outputs therefore needing more than one ontroller. And SISO sstem omprising of a single input and single output hene having onl a single ontroller. Depending on the set-up of the sstem adjusting the sstem's input variale will influene the operating parameter otherwise known as the ontrolled output variale. On reeiving the error signal that marks the disrepan etween the desired output and the atual output the ontroller will then sta to regulate ontrolled output ehavior. The sstem ma e linear or nonlinear that is eing ontrolled a ontroller. In the same wa the methods are different for designing a ontroller for linear and non-linear sstems. A linear sstem is one whih satisfies the properties of superposition and saling or homogeneit. One more aspet of a linear sstem is that its output is diretl proportional to its input. Also these sstems have one equilirium point at the origin. A non-linear sstem is one whih does not oe an of the aove properties. Eah sstem must assure the stailit for losed loop ehaviour. Stailit needs to e preisel defined for nonlinear sstems. For linear sstems this an e otained diretl plaing the poles.. Where Do Nonlinearities ome from? Man of phsial quantities suh as a vehile s veloit or eletrial signals have an upper ound. When that upper ound is reahed linearit is lost. The differential equations governing some sstems suh as some thermal fluidi or iologial sstems are nonlinear in nature. It is therefore advantageous to onsider the nonlinearities diretl while analing and designing ontrollers for suh sstems. Mehanial sstems ma e designed with aklash this is so a ver small signal will produe no output (for eample in gearoes. In addition man mehanial sstems are sujet to nonlinear frition. elas whih are part of man pratial ontrol sstems are inherentl nonlinear. Finall ferromagneti ores in eletrial mahines and transformers are often desried with nonlinear magnetiation urves and equations.. METHODOLOGY. Bakstepping Non-linear sstems use speifi theories and methods to ensure stailit without regard to the inner dnamis of the sstem. Bakstepping and Integrator akstepping ontrol are few of the methods. Bakstepping is a tehnique developed ira 990 Petar. Kokotovi and others for designing stailiing ontrols for a speial lass of nonlinear dnamial sstems. These sstems are uilt from susstems that radiate out from an irreduile susstem that an e stailied using some other method. Beause of this reursive struture the designer an start the design proess at the known-stale sstem and "ak out" new ontrollers that progressivel stailie eah outer susstem. The proess terminates when the final eternal ontrol is reahed. Hene this proess is known as Bakstepping olume: 04 Issue: 05 Ma

2 olume: 04 Issue: 05 Ma Controller Design of Seond Order Nonlinear Sstem An eample to illustrate Bakstepping Control applied on a seond order single input nonlinear sstem. The sstem is given equations u ( ( Here we onsider a onstant. Let the first error e Taking derivative The Lapunov funtion is seleted as Then ( From equation aove we get ( Let the seond error e Taking derivative u Then ( To make 0 hoose suh that where 0 Then Therefore Taking another Lapunov funtion Then ( ( u From equation ( we get ( u After solving we get as Then ] [ u Therefore the final ontrol input lawu is given 5 u The ontroller is simulated on MATLAB Simulink ersion 7.5 (009 software. The simulation is shown different lok diagram whih onsists of simulation of sstem equations as well as ontroller. u Consider 5 3

3 In Fig first lok is the referene lok of simulation of a seond order nonlinear sstem. It onsists of different loks as r and its derivative lok with unit step signal as its input net lok is a ontroller sstem where the aove given ontrol law is designed than lok diagram of seond order nonlinear sstem lok whih is to e stailied and a reording sstem lok whih stores the result when we run the simulated sstem. The output of the ontroller is stailiing funtion for the seond order nonlinear sstem. A step signal of unit magnitude is the referene input for the sstem given to r and derivative lok. The reording sstem is to displa the simulation results. Fig -: Simulation lok diagram of seond order nonlinear sstem Fig -: Controller sstem lok 3. ESULTS Fig.3 shows the graph of state variale with respet to time t to make error ero state vetor must follow referene input r. In Fig. 3 the initial value of is ero ut within fration of a seond it eome unit. The final value of is one show that state is following the referene input r that is unit step signal. Fig. 4 shows graph of state vetor with respet to time t to make error to e ero must follow alpha whih settle down to -. The initial value of is ero ut within fration of a seond it eomes -. The final value of in the graph is - whih the desired value as it is following olume: 04 Issue: 05 Ma

4 Fig -3: State esponse of State etor Fig -6: State esponse of error 4. CONCLUSION The results shown aove are of seond order and sstems. From the results it an e seen that after appling akstepping ontrol tehnique the stailit of the sstem improved and the performane with respet to time is reliale. We an also onlude that for multiple input sstems it is more easier to ontrol the sstem with this tehnique So we get a fleiilit in designing the ontrol input law during simulation. Bakstepping ontroller reursivel uses Lapunov funtions in eah integrator level to anel the nonlinear terms whih ensures asmptoti stailit. Fig -4: State esponse of State etor Fig.4 and Fig.5 shows the graphs of errors and respetivel. All this errors must die out to ero to get desired result. The initial and final values of these errors are shown in figure. The first two errors and orrespond to sstem state vetors and. Final values of errors and are pratiall ero so we an epet sstem to e stale. EFEENCES []. A Freeman and P Kokotović Bakstepping design of roust ontrollers for a lass of nonlinear sstems Proeedings of the IFAC Nonlinear Control Sstems Design Smposium Bordean Frane June 99 pp []. H K Khalil Nonlinear Sstems New York: Mamillan 99. [3]. D M Dawson J J Carroll and M Shneider Integrator akstepping ontrol of a rushed DC motor turning a rooti load IEEE Transations on Control Sstem Tehnolog ol. pp [4]. Z P Jiang and J B Pomet Comining akstepping and time varing tehniques for a new set of adaptive ontrollers Proeedings of the 33 rd IEEE onferene on Deision and Control Lake Buena ista FL Deemer 994 pp [5]. Miroslav Krstić Ioannis Kanellakopoulos and Petar Kokotović Nonlinear and Adaptive Control Design A Wile-Intersiene Puliation 995 John Wile & sons In. [6]. A Freeman and P Kokotović oust Nonlinear Control Design. Boston M A Birkhauser 996. Fig -5: State esponse of error olume: 04 Issue: 05 Ma

5 BIOGAPHIES Mrs Anamika as Ojha Asst Prof Aropolis Institute of Tehnolog and esearh Indore Madha Pradesh Ms Ahala Khandelwal Asst Prof Shri aishnav Institute of Tehnolog and Siene Indore Madha Pradesh olume: 04 Issue: 05 Ma