A Convex Optimization Approach to Control Design for ... - WSEAS

WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad KarimiA Convex Optimization Approach to Control Design for SwitchingPower Converters with Time DelaysMohammad KarimiIslamic Azad University, South Tehran Branch, Tehran, Iran(E-mail: mkarimi71@gmail.com)Abstract- In this paper a convex optimization approach to design a robust state feedback control is proposed forDC-DC converter systems with mixed delays. Based on the Lyapunov theory, some required sufficientconditions are established in terms of delay-dependent linear matrix inequalities for the stochastic stability andstabilization of the considered system using some free matrices. The desired control is derived based on aconvex optimization method.Key-Words: Robust control, DC-DC converter, Lyapunov function, delay, LMI1. IntroductionThe main task of DC-DC converters is theadaptation of the voltage and current levels betweensources and loads while maintaining a low powerloss in the conversion [1]-[2]. With the extensiveuse of DC-DC converters in different industryapplications (e.g. power supplies for personalcomputers, DC-motor drive, telecommunicationsequipment, etc.), improving their performancesbecomes an interesting problem in recent years [1]-[7]. Recently, different converter circuits (buckconverter, boost converter, buck-boost converter,Cuk converter, etc.) are known. According to eachapplication purpose (increase or decrease themagnitude of the DC voltage and/or invert itspolarity), the converter circuit was chosen. Amongthem, we consider here, the control of the basicPulse-Width-Modulation (PWM) buck converters,but it could be easily adapted for other converters.On the other hand, in recent years more attentionhas been devoted to the study of stochastic hybridsystems, where the so-called Markov jump systems.These systems represent an important class ofstochastic systems that is popular in modelingpractical systems like manufacturing systems, powersystems, aerospace systems and networked controlsystems that may experience random abrupt changesin their structures and parameters [8]-[13]. Randomparameter changes may result from randomcomponent failures, repairs or shut down, or abruptchanges of the operating point. Many such eventscan be modeled using a continuous time finite-stateMarkov chain, which leads to the hybrid descriptionof system dynamics known as a Markov jumpparameter system [14]; such a description will beutilized in the paper. The state of a Markov jumpparameter system is described by continuous rangevariables and also a random discrete event variablerepresenting the regime of system operation. A greatnumber of results on robust stability, stabilization,control and filtering problems related to suchsystems have been reported in the literatures ([8],[9], [15]).On another research front line, time delays for manydynamic systems have been much investigated; seefor example [16]. Time-delayed systems represent aclass of infinite-dimensional systems largely used todescribe propagation and transport phenomena orpopulation dynamics. Delay differential systems areassuming an increasingly important role in manydisciplines like economic, mathematics, science, andengineering. For instance, in economic systems,delays appear in a natural way since decisions andeffects are separated by some time interval. Thedelay effects on the stability of systems includingdelays in the state and/or input is a problem ofrecurring interest since the delay presence mayinduce complex behaviors for the schemes [16]-[17].On the other hand, stability of neutral delay systemsproves to be a more complex issue because thesystem involves the derivative of the delayed state.Especially, in the past few decades increasedattention has been devoted to the problem of robustdelay-independent stability or delay-dependentstability and stabilization via different approachesfor linear neutral systems with delayed state and/orinput and parameter uncertainties (see [18]-[19]).Among the past results on neutral delay systems, theLMI approach is an efficient method to solve manycontrol problems such as stability analysis andE-ISSN: 2224-2856 56 Issue 2, Volume 8, April 2013

WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad Karimistabilization [20] and control problems [21]-[24]. It is also worth citing that some appreciableworks have been performed to design a guaranteedcost(observer-based) control for the neutral systemperformance representation [25]-[27].In this paper, we are concerned to develop a robustcontrol problem for Markovian switchingsystems with mixed discrete, neutral and distributeddelays. The main merit of the proposed method isthe fact that it provides a convex problem such thedelay-dependent control gains can be found fromthe LMI formulations. Some required sufficientconditions are established in terms of LMIscombined with the Lyapunov-Krasovskii method forthe existence of the desired control. Numericalexamples are given to illustrate the use of our resultsfor a DC-DC converter model.Notation: The notations used throughout the paperare fairly standard. and represent identity matrixand zero matrix; the superscript stands for matrixtransposition. ‖ ‖ refers to the Euclidean vectornorm or the induced matrix 2-norm.represents a block diagonal matrix and the operator( ) represents . Let ) anddenotes the expectation operator with respect tosome probability measure . If ( ) is a continuous-valued stochastic process on), we let( ) for which isregarded as a ( ] )-valued stochasticprocess. The notation means that is realsymmetric and positive definite; the symbol denotes the elements below the main diagonal of asymmetric block matrix.2. Problem DescriptionConsider a class of uncertain time-delay systemswith Markovian switching parameters and mixedneutral, discrete and distributed delays and normboundedtime-varying uncertainties represented byx(t) A (r(t)) x(t d) (A (r(t)) A (t,(r(t)))) x(t) (A (r(t)) A (t,(r(t)))) x(t h) (A (r(t)) A3(t,(r(t)))) x(s) ds (A5(r(t)) A4(t,(r(t)))) x(t d) (B (r(t)) B (t,(r(t)))) u(t) (B (r(t)) B (t,(r(t)))) w(t),12412tt1x(t) (t),t ,0r(t) r0 , t ,0z(t) C(r(t)) x(t) C (r(t))x(t d) C (r(t))x(t h)t C(r(t)) x(s) dsD(r(t))u(t),td(1a)(1b)(1c)(1d)nmwhere x(t), u(t)s, andz(t)z2are state, input, disturbance and controlled1 w(t) L [0, )3h22 output,Crespectively.id (r(t)),Ch(r(t)),C(r(t))))and D(r(t are matrixfunctions of the random jumping process {r(t)} .{ r(t), t 0} is a right-continuous Markov process onthe probability space which takes values in a finitespace S {1,2, ,s}with generator [ ij ] (i, jS)given by ij o( ),if i jP{r(t ) jr(t) i} 1 ii o( ),if i j(2)where 0 , lim 0 o( )/ 0 and , for ,is the transition rate from mode i at time t to modejat time t and .The time-varyingfunction (t) is continuous vector valued initialfunction and h,dand are constant time delayswith . Moreover, the normboundeduncertainties are defined as follows:(3a)(3b)where (t,r(t)) is the uncertain time-varying matrixfunction of the random jumping process, whichsatisfies T (t,r(t)) (t,r(t)) I for t 0; r(t) iSandE i (r(t)) and H 1 (r(t))are known real constantmatrices of the random jumping process withappropriate dimensions.Definition 1. Uncertain time-delay system (1) withMarkovian switching parameter in (2) is said to bestochastically mean square stable if, when ( ) ,for any finite ( ) defined on ], andthe following condition is satisfied‖ ( )‖A (r(t)),B (r(t)),C(r(t)),ij 0j sijj1,jiA(t,r(t)) H (r(t)) (t,r(t))E(r(t)),i1B(t,r(t)) H (r(t)) (t,r(t))Ej1‖ ( )‖where ( ) is the trajectory of the system state frominitial system state ( ) and initial mode , andis a positive constant.Definition 2. The performance measure of thesystem (1) is defined as (∫ ( ) ( )( ) ( )] ), where the positive scalar isgiven.Assumption 1. The full state variableavailable for measurement.iii4j(r(t)),ii 1,2,,4j 1,2( ) isE-ISSN: 2224-2856 57 Issue 2, Volume 8, April 2013

WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad KarimiIn this paper, the author’s attention will be focusedon the design of the following robust modedependentdelayed state feedback control law,( ) ( ) ( )∫ ( )(4)where the matrices of theappropriate dimension is to be determined such thatfor any ( ) the resulting closed-loopsystem is stochastically stable and satisfies annorm bound , i.e. .3. Main ResultsIn this section, we first investigate both thestochastic stability and performance of thesystem (1) with norm-bounded uncertaintyparameters. A new delay-dependent stochasticstability condition by a discretization technique isproposed in Theorem 1. Then, we will show theprocedure to design the controller gains, which guarantee the resulting closedloopsystem is stochastically stable and satisfies annorm bound .We choose a stochastic Lyapunov-Krasovskiifunctional candidate ( )aswhereV(x,x 4 t ,t,i) V i (x,xt,t,i)i10TT1xt, t,i) x(t) P1ix(t) 2 x(t) Q(i) x(t )dhV (x,V (x, x , t,i) 3tV2(x, xt, t,i) x(s)0 0 x(t s)4ddV (x, xtt0 0 x(t s) x(s) td,t, i)hhTTthSx(s) ds R x(t )ds d, ,(5)Theorem 1. The time-delay system (1) withMarkovian switching parameters in (2) and withoutthe norm-bounded uncertainties in (3) isstochastically mean square stable with anperformance level , if there exist somematrices , , andTTU x(s) 1ds x(t )T x(t )ds d, [ x( )t ( t s) x( )0 tstttsT0dd] UT2UT[ x( )d]2tsx( )H x(t )dddsdspositive definite matrices () satisfying the following LMIsP1iQi 0 , (6a) R Sei 1i(Pwhere [ ] andProof. See [28].TTT P2iA2i Qi C C i hi diag{CiCi,0} T P3iA2i T S C CT2i PTP3iA0TA P A4i U)AP( [ ]) {∑( ) }Remark 1. Note that the matrix1i1i4i14i4i[ ] UTPB 2i 2i T P3iB2i0 0 00 0 2 I(6b)(or, equivalently, the matrix ) is non-singular dueto the fact that the only matrix which can benegative definite in the first block on the diagonal ofLMI (6b) is .Remark 2. If the switching modes are notconsidered, i.e. S {1} , the jump linear system issimplified into a general linear system withnonlinearities and time delays. Then it is easy toconclude a criterion from Theorem 1, which can beused to determine the stability of such a system.In the following, we present a condition for thestability of the time-delay system (1) withMarkovian switching parameters in (2) and normboundeduncertainties in (3).T2iCATP3iA0Ti3iC C00ihi3iTihiC2sT T 2i P 2iA5i CiCdi ( ijP1j Qi0)A4ij1isTT3i CdiCdi H A4iijP1jA4ij1 2i TP3iA0,,3i5iE-ISSN: 2224-2856 58 Issue 2, Volume 8, April 2013

WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad KarimiTheorem 2. Under Assumption 1, a state feedbackcontroller given in the form (4) exists such that thetime-delay system (1) with Markovian switchingparameters in (2) is stochastically stable with anperformance level , if there exist some scalars, matrices ̅ ̃ ̃ , ̃ ̃ ̃̃ ,, and positive definite matrices̃ ̃ ̃ ̃ (following LMIs) satisfying the~P Qˆ 1ii 0 , Rˆ Ŝ(7a)ˆˆ ˆeidi ei iI 0 0iI (7b)where ̂ ] ̂̂̃[where([] and̃̃̃( )̃]])be designed as ̅ ̅̅̅Proof. See [28].Remark 3. By setting and minimizingsubject to LMIs (26), we can obtain the optimalperformance level (by √ ) and thecorresponding control gains as well.4. DC-DC convertersThere are three kinds of switching mode DC-DCconverters, buck, boost and buck-boost. The buckmode is used to reduce output voltage, whilst theboost mode can increase the output voltage. In thebuck-boost mode, the output voltage can bemaintained either higher or lower than the sourcebut in the opposite polarity. The simplest forms ofthese converters are schematically represented inFigure 1. These converters consist of the samecomponents, an inductor, L, a capacitor, C and aswitch, which has two states u = 1 and u = 0. Allconverters connect to a DC power source with avoltage (unregulated), and provide a regulatedvoltage, to the load resistor, R by controlling thestate of the switch. In some situations, the load alsocould be inductive, for example a DC motor, orapproximately, a current load, for example in acascade configuration. For simplicity, here, onlycurrent and resistive loads are to be considered.∑ ̃ ( ̃ )[̃ ̃ ̃ ̃̃],( )(a) Boost[[(∑ ̃ ̃ )],(∑ ̃ ̃ ))̃],(b) Buck[], [ ],( )[ ]and ̃ ( ) ∑ ̃ ,, ,( ). Moreover, the controller gains in (4) can(c) Buck-boostFigure 1. Switching-mode DC-DC converters: (a)buck, (b) boost and (c) buck-boost.E-ISSN: 2224-2856 59 Issue 2, Volume 8, April 2013

v c(t)WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad Karimi5. Averaged model of Basic PWMbuck converterFigure 1 shows the basic circuit of the nonlinearPWM buck converter proposed in [3]-[4] with anexternal disturbance ( ) as proposed in [1]-[2].is the on-state resistance of the MOSFETtransistor, is the winding resistance of inductor,is the threshold voltage of the diode and isthe equivalent series resistance of the filtercapacitor. By applying the Kirchhoff’s voltage law(KVL) and Kirchhoff’s current law (KCL) in onstateof the MOSFET transistor case, we obtain[29]:1 (Rl R Rc) i(t) lil(t d) L E v(t) v (t d) R Rccc CRc 1(V Lin R0 R Rc il(t)) L R R CRc ic10 RRcwhere E and R Rc . 0 c2R Rcm R RLRc R RCRRFigure 2. Schematic of a basic PWM buckconverter.(8)Now, in off-state of the MOSFET transistor caseand by applying of KVL and KCL, we get:1 R Rc (Rl R Rc) i(t) i (t d) LLR il(t)ll c E v (t) v (t d) R Rc R Rc vc(t)cc CRcCRRc R Rc Vd LL iload(t) R R0 c CR c (9)Using Averaging Method of on One Time ScaleDiscontinuous system (AM-OTS-Ds) [30], thecload(t)ccc il(t) vc(t) global dynamical behavior of the DC-DC converteris modeled as:1 R Rc (Rl R Rc) i(t) i (t d) LLR il(t)ll c E v (t) v (t d) R Rc R Rc vc(t)cc CRcCRRc R Rc Vd Vin Vd Rmil(t) d(t)LL L iload(t)R Rc0 0 CR c (10)where ( ) is the duty cycle.6. Simulation resultsIn this section, with the aid of MATLAB LMIToolbox [31], we use the averaged model of BasicPWM buck converter to illustrate the effectivenessand advantage of our design method. In thissimulation, the PWM frequency equals to 1 kHz.The simulation parameters used in this work are asfollows [3]:with the input current i load 0.25sin(1000t) . The DC-DC converter parameters are switching between twomodes, i.e., 1.0 and 1.2 nominal values, under (8a) thefollowing Markovian transition matrix[ ].The optimization problem described in (7) is solvedfor initial condition x 0 0 , h=15 ms and c=0.3.Responses of the output voltage of PWM buckconverter and the inductance current of PWM buckconverter are, respectively, plotted in Fig. 3 and Fig.4.3020100-100 0.05 0.1 0.15 0.2t[s]Figure 3. Response of the output voltage of PWMbuck converterE-ISSN: 2224-2856 60 Issue 2, Volume 8, April 2013

i l(t)WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad Karimi1510500 0.05 0.1 0.15 0.2t[s]Figure 4. Response of the inductance current ofPWM buck converter.7. ConclusionThe problem of robust state feedback controlwas proposed for a class of Markovian swtichingsystems with mixed discrete, neutral anddistributed delays. Some required sufficientconditions were derived in terms of delaydependentLMIs using some free matrices and theLyapunov-Krasovskii functional theory. Thedesired control is derived based on a convexoptimization method. Numerical examples weregiven to illustrate the use of our results for a DC-DC converter model.References[1] C.Olalla, R. Leyva, A. El Aroudi, I. Queinnec, S.Tarbouriech, Control of DC-DC ConvertersConverters with Saturated Inputs, IEEE Conf.on Industrial Electronics, IECON'09, Porto,2009.[2] C. Olalla , I. Queinnec, R. Leyva, A. ElAroudi,Robust optimal control of bilinear DC–DCconverters, Control Eng. Practice, In Press,2011.[3] Y. Li, Z. Ji, T-S Modeling, Simulation andControl of the Buck onverter, 5 th Inter.Conference on Fuzzy Systems and KnowledgeDiscovery, 18-20 Oct. 2008, Schandong, China,pp. 663-667.[4] K. Y. Lian, J.J. Liou, LMI-based integral fuzzycontrol of DC-DC converters, IEEE transactionon Fuzzy System, 14(2006), 71-80.[5] D. Cortés, Jq. Alvarez, Ja. Alvarez-Gallegos,Feedforward and feedback robust control of thebuck converter, 15 th World IFAC Congress,Barcelona, Spain, (2002).[6] I. Gadoura, T. Suntio, K. Zenger, Roubustcontrol design for paralleled DC/Dc converterswith current sharing, 15 th World IFACCongress, Spain (2002).[7] D. Alejo, P. Maussion, J. Fauche1, Multiplemodel control of a Buck DC/DC converter,Mathematics and Computers in Simulation 63(2003) 249–260.[8] Boukas, E.K., and Benzaouia, A.. Stability ofdiscrete-time linear systems with Markovianjumping parameters and constrained control.IEEE Transactions on Automatic Control, 47(3),516-521, 2002.[9] Shi, P., and Boukas, E.K.. control forMarkovian jumping linear systems withparametric uncertainty. Journal of OptimizationTheory and Applications, 95(1), 75-99, 1997.[10] Shi, P., Boukas, E. K., and Agarwal, K..Kalman filtering for continuous-time uncertainsystems with Markovian jumping parameters.IEEE Transactions on Automatic Control, 44(8),1592-1597, 1999.[11] Shi, P., Boukas, E. K., and Agarwal, K..Control of Markovian jump discrete-timesystems with norm bounded uncertainty andunknown delay. IEEE Trans. on AutomaticControl, 44(11), 2139-2144, 1999.[12] Shi, P., Xia, Y., Liu, G. P., and Rees, D.. Ondesigning of sliding-mode control for stochasticjump systems. IEEE Transactions on AutomaticControl, 51(1), 97_103, 2006.[13] Wang Z., Lam J., Liu X., ‘Exponential filteringfor uncertain Markovian jump time-delaysystems with nonlinear disturbances’ IEEETrans. Circuits and Systems, II: Express Briefs,51(5), 262-268, 2004.[14] Mariton, M.. Jump linear systems in automaticcontrol. New York: Marcel Dekker Inc, 1990.[15] Saif A.W., Mahmoud M.S. and Shi Y., ‘Aparameterized delay-dependent control ofswitched discrete-time systems with time-delay,’Int. J. Innov. Comput., Information and Control,5(9), 2893-2906, 2009.[16] Niculescu S. I., Delay Effects on Stability: ARobust Control Approach (Berlin: Springer),2001.[17] Gu K., Kharitonov V.L., and Chen J., ‘Stabilityof time-delay systems’ Birkhauser, Boston, 2003,ISBN 0-8176-4212-9.[18] Ivansecu, D., Niculescu S.I., Dugard L., DionM. and Verriest E.I., ‘On delay-dependentstability of linear neutral systems’, Automatica,39, 255-261, 2003.[19] Lam, J., Gao, H., and Wang, C., ‘ modelreduction of linear systems with distributeddelay.’ IEE Proc. Control Theory Appl., 152(6),662-674, 2005.[20] Fridman, E., ‘New Lyapunov-Krasovskiifunctionals for stability of linear retarded andneutral type systems.’ Systems and ControlLetters, 43, 309-319, 2001.[21] Fridman, E., and Shaked, U., ‘Delay-dependentstability and control: constant and time-E-ISSN: 2224-2856 61 Issue 2, Volume 8, April 2013

WSEAS TRANSACTIONS on SYSTEMS and CONTROLMohammad Karimivarying delays.’ International Journal ofControl, 76, 48-60, 2003.[22] Gao, H., and Wang, C., ‘Comments and furtherresults on ‘A descriptor system approach tocontrol of linear time-delay systems.’ IEEETransactions on Automatic control, 48, 520-525,2003.[23] Zhao Y., Gao H., Lam J. and Du B., Stabilityand stabilization of delayed T-S fuzzy systems:A delay partitioning approach. IEEE Trans. onFuzzy Systems, 17(4), 750-762, 2009.[24] Karimi H.R. and Gao H., ‘New delaydependentexponential synchronization foruncertain neural networks with mixed timedelays’IEEE Trans. on Systems, Man andCybernetics-B, 40(1), 173-185, 2010.[25] Chen, B., Lam, J., and Xu, S., ‘Memory statefeedback guaranteed cost control for neutralsystems’ Int. J. Innovative Computing,Information and Control, 2(2), 293-303, 2006.[26] Park J.H., ‘Guaranteed cost stabilization ofneutral differential systems with parametricuncertainty’ J. Computational and Appliedmathematics, 151, 371-382, 2003.[27] Lien, C.H., ‘Guaranteed cost observer-basedcontrols for a class of uncertain neutral timedelaysystems.’ J. Optimization Theory andApplications, 126(1), 137-156, 2005.[28] H.R. Karimi, Robust Delay-DependentControl of Uncertain Markovian Jump Systemswith Mixed Neutral, Discrete and DistributedTime-Delays, IEEE Trans. Circuits and SystemsI, 58(8), 1910-1923, 2011.[29] J. Sun, H. Grotstollen, Averaged modelling ofswitching power converters: Reformulation andtheoretical basis, Power electronics specialistsconf., 29-03 jul 1992, Taledo, Spain, pp. 1165-1172.[30] Bellen A., Guglielmi N., and Ruehli A.E.,‘Methods for linear systems of circuit delaydifferential equations of neutral type’ IEEETrans. on Circuits and Systems-I, 46(1), 212-216, 1999.[31] P. Gahinet, A. Nemirovsky, A.J. Laub, and M.Chilali, ‘LMI control Toolbox: For use withMatlab.’ Natik, MA: The MATH Works, Inc.,1995.E-ISSN: 2224-2856 62 Issue 2, Volume 8, April 2013