Design of Output Feedback Compensator

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2 Design of Output Feedbak Copensator Vanita Jain, B.K.Lande Professor, Bharati Vidyapeeth s College of Engineering, Pashi Vihar, New Delhi-0063 Prinipal, Shah and Anhor Kuthhi Engineering College, Chebur, Mubai Abstrat This paper disusses the proble of stabilization of physial systes of a very high order and the redued order odel of the higher order systes. The ethodology uses the onept of paraeterization of all opensators that stabilize a given plant. The output feedbak opensator is paraeterized by one paraeter. It stabilizes the higher order syste as well as its redued order odel. The opensator design ethod does not involve higher order syste for its oputation. Keywords: Output feedbak; Copensator; Redued order odel.. Introdution The atheatial odels of any engineering systes are frequently of high diension. These odels are often too large for the purpose of, analysis and ontroller design. The ost and oplexity of the ontroller inreases as the syste order inreases [-3]. This proble an be overoe if a good redued order odel is available for the original higher order syste and if it is possible to design a ontroller using the redued odel, whih will stabilize the original higher order syste when plaed in the losed-loop. Hene for the ost and tie saving in design, and for siplifying ipleentation, redued order odels are highly desirable for engineers in analysis, synthesis, and siulation of opliated high order syste. Model redution onept originated towards the end of the 9th entury when Pade approxiations were introdued in [4]; interest of researhers in this very iportant field was spurred after the work of Rosenbrok [5] on distillation oluns. Today, there exist a variety of onepts and tehniques that have a oon goal of reduing the diension of the atheatial odel of a large-sale syste in order to siplify the design of ontrol and estiation shees. Aong the iportant ethods are Pade approxiation [6], Routh approxiations [7], oent athing tehniques [8] and balaned realization [9,0]. A good review on the available tehniques and oparison an be found in IJERTVIS009 05

3 Mahoud [] and Fortuna []. Linear atrix inequality based odel redution tehniques have been developed in [3] and iproved Routh-Pade approxiants has been given in [4]. Its liitation is that it works for stable systes only. Redued order ontroller has been designed for nonlinear systes also [ Astolfi]. A very less effort has been put to use the redued odels for designing ontroller for the higher order syste. This is priarily due to the fat that if a stabilizing ontroller is designed fro the redued odel and applied to the higher order syste, it does not guarantee the stability of the losed- loop syste [5,6]. In [3], Laba shows that, if a state feedbak ontrol is designed fro the Davison [7] and applied to the higher order syste, it results in a stable losed-loop syste. But the ethod needs the syste states to be fully available for feedbak. Suh a ontroller annot be ipleented for a syste where soe of the state variables are not aessible. Soe other ethods are reported in [8-0]. This paper desribes a tehnique for designing a stabilizing ontroller for a higher order syste using its redued order odel. The ethod uses the paraeterization of all opensators that stabilize a given plant []. It is shown that the output feedbak opensator, whih is obtained fro the redued odel, not only stabilizes the redued odel but also the higher order syste.. Proble Stateent Let a stable single-input single-output (SISO) linear tie-invariant syste of order 'n' is represented as G( s) np( s)/ d p( s) (.) Gs represents a higher order syste whose redued order odel of order '' is represented by M ( s) n( s) / d( s) (.) We assue that there are no pole zero anellations. The proble is to find a ontroller using the redued odel that will stabilize both the higher order syste Gs and its redued order odel Ms and the order of opensator should be less than the order of the higher order syste Gs. The tehnique of paraeterization of all stabilizing ontrollers for a syste [] is used to design the ontroller for higher order syste via its redued odel. Assue that a given syste P( s) ( s) and is expressed as a ratio of rational funtion as: P( s) n( s) / d( s); n( s), d( s) S (.3) Let C ' ( s) ( s) be a stabilizing ontroller of syste Ps expressed as a ratio of rational funtions ' ' ' ' ' C ( s) n ( s)/ d( s); n ( s), d( s) S (.4) where s is the set of real rational funtions in the variable s and S is the subset of s onsisting of all rational funtions that are bounded at infinity and whose all poles have negative real parts. The syste (.3) and the stabilizing ontroller (.4) are expressed as a ratio of two stable rational funtions, whih is alled fatorization approah []. Frational fatorization of IJERTVIS009 06

4 rational funtions in S is non unique [,]. The stable rational funtions in (.3) and (.4) are seleted suh that they satisfy Bezout identity. n ( s) n( s) d( s) d( s) (.5) ' Assue that P! ( s n ) ( s ) and let C( s) ( s) be a stabilizing ontroller of r! n r! ' ' Ps. () Let n( s) / d( s) and n( s)/ d( s ) be frational fatorizations of Ps () and C() s in S. The set of all stabilizing ontrollers of Ps () is given by n ( s) rd( s) Stab( P( s)) : r Sandd ( s) rn( s) 0 (.6) d ( s) rn( s) To eah r S orresponds a stabilizing ontroller and to eah stabilizing ontroller orresponds a stable rational funtion r S. 3. Copensator Design In this setion, the above onept of ontroller paraeterization is used to derive a ontroller for higher order syste using its redued order odel. As the higher order syste is stable we an hoose rational funtions as: n ( s) G( s), d( s), x ( s) 0 and y ( s), then they satisfy Bezout identity x ( s) n ( s) d( s) y ( s). Siilarly for redued order plant, if we hoose n ( s) M ( s), d( s), x ( s) 0 and y ( s) then they satisfy Bezout identity n ( s) x ( s) d( s) y( s). We an take G/and M /as the frational fatorization of the syste and its redued odel respetively. As G and M are stable, there exists onstants k and k whih stabilize the syste and the odel respetively as shown in Fig.. and Fig... k G k M Input Output Input Output Fig.: Higher order syste with Fig.: Redued order odel with onstant gain k onstant gain k Let the frational fatorization of k and k be k /and k / are trivial fatorization respetively. Fro (.6), the set of stabilizing ontroller for the syste G(s) is IJERTVIS009 07

5 k r Stab( G( s)) : r S rg Siilarly the set of stabilizing ontroller for the redued odel is (.7) k r Stab( M ( s)) : r S (.8) rm Given the plant and redued odel G(s) and M(s), whih are to be stabilized,the paraeterization of two opensators, whih individually stabilize those plants, an be equated to find the oon stabilizing opensator. We get, k r k r rg rm Solving for r, k r kr M k r (.9) r M Gk r G Theore If dd p nd p npdk npd is Hurwitz then there exists an th order opensator ( k ) d Cs (), whih stabilizes the higher order syste and its redued order odel. d n Proof: Fro (.9) we have k r kr M k r r M Gk rg for sipliity we take r S then k km k r M Gk G as M n / d and G n / d, we get p p ( kd d kn kd ) d p r (.0) dd p nd p npdk npd Hene if dd p nd p npdk npd is Hurnitz then r S. Fro (.8). k r Stab( M ( s)) : r S as r and n M rm d so th order opensator ( k ) d Cs () n stabilizes both the syste and the redued order odel of the syste. (.) Reark : Fro (.) we see that, for designing a ontroller for higher order syste, only a redued order odel and heking Hurwitzness of a ertain polynoial is involved in oputation. Hene, the oplexity in obtaining a ontroller for higher order syste is greatly redued and further we get a ontroller of lower order. Reark : The opensator in (.) is paraeterized by one paraeter k. The range of k for whih the feedbak syste shown in Fig.. is stable an be obtained. Siilarly the range of k for whih the polynoial dd p nd p npdk npd is stable an be obtained. Fro these the range of k an be obtained for whih ( k ) d Cs () n IJERTVIS009 08

6 stabilizes the syste and the redued order odel. In this approah k is not required to be oputed. 4. Exaple Consider the ontinuous tie syste having three poles of relatively different order of agnitude desribed by the following transfer funtion [3] Gs () ( s 5)( s )( s 0.) The transfer funtion of redued order odel is 0.835s Ms () s The range of k for whih redued odel is stable is k 0.5. Fro the polynoial dd p nd p knpd npd,the onstraints on k for the stability are: 0.66k , 0.66k k k 5.86 To satisfy all the onstraints k 0.5 and k 0 Thus a opensator ( k )( s0.0969) C( s), k 0.5 (.835 s ) is a stabilizing ontroller for the given higher order syste and its redued order odel. If k k ks ( ) then C( s), k 0.5 (.835 s ) The losed-loop responses of the higher order syste and its redued order odel with this opensator for k.6 is shown in Fig..3. IJERTVIS009 09

7 5. Conlusions The effetiveness of a redued order odel is established, as a ontroller designed fro the redued odel works satisfatory when applied to the higher order syste. The ontroller design ethod does not involve higher order syste for its oputation and it ensures the stability of the losed-loop syste. Another feature of this ethod is that the order of the opensator is equal to the order of the redued odel. The efforts required to design the opensator through the redued odel are very less as opared to the one, whih will involve higher order syste for its oputation. Referenes. Y.Shaash, Stable Redued Order Models Using Pade Type Approxiation, IEEE Trans. Autoati Control, 9(5), 65 (974).. A.Bultheel and M.Van Barel, Pade Tehniques for Model Redution in Linear Syste Theory A Survey, Journal Cop. Appl. Math., 4, 40(986). 3. S.S. Laba and S.V Rao, On Sub-Optial Control via Siplified Model of Davison, IEEE Trans. Autoati Control, 9(4),448 (974). 4. H. Kiura, A Further Result on the Proble of Pole Assignent by Output Feedbak, IEEE Trans. Autoati Control, (3), 458 (977). 5. J.L.Willes, Tie-Varying Feedbak for the Stabilization of Fixed Modes in Deentralized Control Systes, Autoatia, 5, 7 (989). 6. R. A Luke, Siultaneous Guaranteed-Cost Vetor-Optial Perforane Design for Colletions of Systes, J. of Guidane, Control, and Dynais, (), 96 (999). 7. M.F. Hutton and B.Friedland, Routh Approxiation for Reduing Order of Linear Tie Invariant Systes, IEEE Trans. Autoati Control, 0(3), 39 (975). 8. N.K.Sinhna and B. Kuszta, Modeling and Identifiation of Dynai Systes (Van Nostrand Reinhold, New York, 983). 9. B.C. Moore, Priniple Coponent Analysis in Linear Systes: Controllability, Observability and Model Redution, IEEE Trans. Autoati Control, 6(), 7 (98). 0. L.Pernebo and L.M. Silveran, Model Redution via Balaned State Spae Representation, IEEE Trans. Autoati Control, 7(), 38 (98).. M.S. Mahoud and M.G. Singh, Large-Sale Systes Modeling (Pergaon Press, Oxford, 98).. L. Fortuna, G. Nunnari.and A. Gallo, Model Order Redution Tehniques with Appliations in Eletrial Engineering (Springer-Verlag, London, 99). 3. K.M.Grigoriadis, Optial H Model Redution via Linear Matrix Inequalities: Continuous and Disrete Tie Cases, Syst. and Control Letters, 6, 3 (995). 4. V.Singh, D.Chandra and H. Kar, Iproved Routh-Pade/Spl Atue /Approxiants: A Coputer Aided Approah, IEEE Trans. Autoati Control, 49(), 9 (004). 5. M.R. Chidabara and R.B. Shainker, Lower Order Generalized Aggregated Model and Sub- Optial Control, IEEE Trans. Autoati Control, 6(),75 (97),. 6. Mishra, R.N., Design of Low Order Control Shees Using Redution Tehniques,International J. Control, 3(5) (980), E.J Davison, A Method for Siplifying Linear Dynai Systes, IEEE Trans. Autoati Control, (),93(966). 8. R. Prasad, J. Pal and A.K. Pant, Multivariable Controller Design via Low Order Models, Journal of Inst. of Engineers (India), 76(3),7 (996). IJERTVIS009 0

8 9. S.K Nagar.and S.K. Singh, An Algorithi Approah for Syste Deoposition & Balaned Realized Model Redution, Journal of the Franklin Institute, 34, 65 (004). 0. P.D.Olivier, On the Relationship between Model Order Redution Proble and the Siultaneous Stabilization Proble, IEEE Trans. Autoati Control, 3(),54 (987).. V. Blondel, Siultaneous Stabilization of Linear Systes, (Leture Notes in Control and Inforation Sienes 9, Spriger-Verlag, Berlin, 994).. M. Vidyasagar, Control Syste Synthesis: A Fatorization Approah, (MIT press, 985). 3. P.Jean Corriou, Proess Control Theory and Appliations, (Springer-Verlag, London, 004). 4. D.Carnevale,D.Karagiannis and A.Astolfi, Redued order observer design for nonlinear systes Pro.European Contr.Conf.,Greee, 559 (007). IJERTVIS009