STABILIZATION OF NONNECESSARILY INVERSELY STABLE FIRST-ORDER ADAPTIVE SYSTEMS UNDER SATURATED INPUT

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1 SELECED OPICS in SYSEM SCIENCE and SIMULAION in ENGINEERING SABILIZAION OF NONNECESSARILY INVERSELY SABLE FIRS-ORDER ADAPIVE SYSEMS UNDER SAURAED INPU M. De la Sen and O. Barambones Abstract- is paper presents an indirect adaptive stabilization sceme or irst-order continuous-time systems under saturated input wic is described by a sigmoidal unction. e singularities are avoided troug a modiication sceme or te estimated plant parameter vector so tat its associated Sylvester matrix is guaranteed to be non-singular and ten te estimated plant model is controllable. e modiication mecanism involves te use o a ysteresis switcing unction. An alternative ybrid sceme, wose estimated parameters are updated at sampling instants is also given to solve a similar adaptive stabilization problem. Suc a sceme also uses ysteresis switcing or modiication o te parameter estimates so as to ensure te controllability o te estimated plant model. Keywords: ybrid dynamic systems, discrete systems, saturated input, control, stabilization I. INRODUCION e inputs to pysical systems usually present saturation penomena wic limit te amplitudes wic excite te linear dynamics, [-]. Also, te adaptive stabilization and control o linear continuous and discrete systems as been successully investigated in te last years. Classically, te plant is assumed to be inversely stable and its relative degree and its igrequency gain sign are assumed to be known togeter wit an absolute upper-bound or tat gain in te discrete case. Attempts o relaxing suc assumptions ave been made or continuous systems, [5-7]. e assumption on te knowledge o te order can be relaxed by assuming a known nominal order and considering te exceeding modes and unmodelled dynamics, [ 3-6], [9]. e assumption on te knowledge o te ig requency gain as been removed in [ 6] and [ 7] and te assumption o te plant being inversely stable as been successully removed in te discrete case and more recently in te continuous one, [ -6]. e problem as been solved by using eiter excitation o te plant signals or by exploiting te properties o te standard least-squares covariance matrix combined wit an estimation modiication rule based upon te use o a ysteresis switcing unction, [ -6], [ 8]. M. De la Sen is wit te Institute o Researc and Development o Processes, Campus o Leioa, 488 SPAIN ( wepdepam@lg.eu.es. O. Barambones is wit Department o Systems Engineering and Automatic Control, Nieves Cano, 6- Vitoria, University o Basque Country, SPAIN ( oscar.barambones@eu.es. Suc an estimates modiication tecnique guarantees tat te modiied estimated plant model is controllable at all time provided tat te plant is controllable. is paper presents an adaptive stabilization algoritm or irst - order continuous - time systems wit a zero wic can be eiter stable or unstable under saturated input. e saturating device is modelled by a sigmoidal unction. Suc an approac is a very good approximation to te common saturations usually modelled as piecewisecontinuous unctions. Also, it is an exact model or saturations inerent to practical MOS-type ampliiers. e adaptive sceme uses a parameter modiication rule wic guarantees tat te absolute value o te determinant o te Sylvester matrix associated wit te modiied parameter estimates is bounded rom below by a positive tresold and, tus, te estimated model is guaranteed to be controllable. at eature is te main contribution o tis manuscript.e results are ten extended to te case wen an adaptive stabilizer, wic re-updates at sampling instants te plant estimates, modiied estimates and controller parameters, is used or te above continuous - time plant. is strategy results in a ybrid closed-loop system because o te discrete nature o te updating procedure o te parametrical estimation / modiication. II. ADAPIVE SABILIZAION A. Plant, Estimation / Modiication Sceme and Adaptive Stabilization Law Consider te ollowing continuous-time irst-order controllable system under saturated input: + a y b u & + bu (.a > ν u e u sat (u tan( ν u ν (.b ν u + e were te saturated input u to te plant (.a is modelled by a sigmoidal unction (.b, []. o simpliy te writing, te argument (t is omitted and all te constants are denoted by superscripts by. Eqn..a can be rewritten as a y + b u& + bu+ b (u& u & + b(u u ( ISSN: 79-57X 5 ISBN:

2 SELECED OPICS in SYSEM SCIENCE and SIMULAION in ENGINEERING Note tat te equivalence between (.a and ( is an identity were positive and negative terms concerned wit te unsaturated input and its time-derivative are cancelled in te rigt- and-side o (. Deine iltered signals + u& d u u ; u & d u + u ; d y + y (3 or some scalar d > so tat one gets rom ( or iltered signals d t θ ϕ a y + b u& + bu +ε e (4.a d t e (4.b a y + b u& + b u + b (u& u& + b (u u +ε were θ [ b,b,a,b,b, ε ] (5.a d t ϕ [u&,u, y,u& u&,u u,e ] (5.b were ε y ( u ( as been included in θ to obtain (4 witout neglecting te exponentially decaying term due to initial conditions o te ilters / ( s + d used in (4 as proposed in [3], [5] and[6]. Also, te overparametrization o (5.a-(5.b, in te sense tat te coeicients o te numerator polynomial are estimated twice wit dierent regressors, allows describing (4.a as driven by u and u - u. is idea will be ten exploited or te stability analysis o te adaptive stabilizer. e parameter vector θ can now be estimated by using te least-squares algoritm e θ ϕ (6 θ & Pϕe (7 P & Pϕϕ P;P( P ( > (8 were e is te prediction error, θ ( θ, θ, θ 3, θ 4, θ 5, θ 6 is te estimate o θ, deined in (5.a, and P is te covariance matrix. e use o (4.b into (6 yields dt θu& +θu θ3 y +θ4(u& u& +θ5(u u +θ6e + e e ollowing modiication rule o te parameter estimates is used to guarantee te controllability o te estimated plant model θθ+ P β ( (9 wit β being a vector wic can be cosen to be equal to one o te ollowing vectors 6{ [,, L L, ] β ; β v ; β 3 - β (.a β 4 p + p 3 ; β 5 - β 4 ; β 6 p - p 3 (.b β 7 - (p + p 3 ; v (θ - θ 4 p 3 + θ 3 ( p (p - p 5 (.c and wose current value is selected rom a ysteresis switcing unction wic is deined by te ollowing rule. Deine c ( β (θ θ 4 θ 3 (θ θ 5 Det θ 3 θ θ 4 θ 3 θ θ 5 wic is te absolute value o te Sylvester matrix o te modiied parameter estimates associated wit te estimation o te plant numerator and denominator polynomials obtained rom (8-(9 and (-(. Assume tat β (t β i (t and c(β j (t + c(β m (t + or some j,,..., 7 wit j i and all m,,..., 7. us, or some preixed design scalar α (, ] : β(t + β (t + j i c(β j (t + (+α c(β i (t + β i (t + oterwise ( were p i denotes te i-t column o P. is modiication strategy, irst proposed in [3] or te linear continuous-time case and ten extended in [5-6] to linear ybrid systems, guarantees tat te parametrical error lies in te image o te o P (see [3], wile allowing tat te diopantine equation, wic will be ten used or te syntesis o te adaptive stabilizer, will ave no cancellations at any time. It will be ten sown tat te two ollowing conditions are satisied: C β converges C c ( β δ >. wic will be ten required in te proos o convergence and stability. Eqn. 9 can be rewritten as dependent o te modiied estimates (-( as ollows : dt θu& +θ u θ3 y +θ 4(u& u& +θ5(u u +θ 6e + e β Pϕ dt +θ 5 (u u +θ 6e + e β Pϕ (3 ISSN: 79-57X 6 ISBN:

3 SELECED OPICS in SYSEM SCIENCE and SIMULAION in ENGINEERING e iltered control input u to te saturating device and its uniltered version u are generated as ollows: u& su r y ; u d u + u (d s & u r y (4 wit te parameters r and s o te adaptive stabilizer being calculated or all time rom te diopantine polynomial equation (D+θ 3 (D+s +[(θ θ 4 D+(θ θ 5 ] r C (D D + c D+ c (5 de wit D d / dt in (5.a and C ( D being a strictly Hurwitz polynomial tat deines te suited nominal closed-loop dynamics. B. Stability and Convergence Results ey are summarized in te ollowing main result: eorem. Consider te plant ( subject to te estimation sceme (6 -(8, te modiication sceme (-( and te control law (4-(5. Assume tat eiter a ( i. e., te open- loop plant is stable or y( b a b a i a < ( i. e., te initial condition is suiciently small i te plant is unstable. us, te resulting closed-loop sceme as te ollowing properties: (i e modiied estimated plant model is controllable or all time or te cosen β in suc a way tat c ( β δ >. (ii θ θ θ L and e and P ϕ are in L L. (iii θ, P, β, θ, s and r are uniormly bounded and converge asymptotically to inite limits. Also, te number o switces in β is inite. Also, θ & L L. (iv e signals u, u and y and teir corresponding iltered signals are in L L. e signals u, u, u, u, y and y converge to zero and teir time-derivatives are in L L so tat tey converge to zero asymptotically. An outline proo o eorem is given in Appendix A. Note tat te requirement o te initial conditions being suiciently small wen te plant is unstable is a usual requirement or stabilization in te presence o input saturation since it is impossible to globally stabilize an openloop unstable system wit saturated input. is avoids te closed- loop system trajectory to explode. Suc a penomenon occurs wen te initial time- derivative o te state vector is positive and continues to be positive or all time because its sign cannot be modiied or any input value witin te allowable input range. Note also tat eorem (i -(iii imply tat Conditions C-C or te β (. - unctions o te modiication sceme are ulilled. Finally, note tat te controllability o te modiied estimation sceme allows to keep coprime te modiied estimates o te polynomials or zeros and poles. us, te diopantine equation (5 associated wit te controller syntesis is solvable or all time witout any singularities. e mecanism wic is used to ensure local stability or unstable plants and global one or stable ones is to guarantee te boundedness o all te unsaturated iltered and uniltered signals rom te regressor bondedness wile te saturated ones are bounded by construction. is also ensures te identiication (or adaptation error to be bounded or all sampling time since te unmodiied and modiied plant parameter estimates as well as tose o te adaptive controller are all bounded. e act tat te control signal is bounded is ensured since it is saturated. In te unsaturated control case, te control boundedness as to be proven explicitly (see, or instance, [-4] irrespective o te particular teoretical design or application. On te oter and, it turns out te main uture interest o appliying saturating controls to oterwise positive systems in te presence o delays or under ybrid controls (see, [5-7]. Related researc would be an interesting uture investigation ield. III. ADAPIVE ESIMAES AND CONROL Now, te continuous-time plant ( is subject to te control law (4-(5 under te saturating sigmoidal unction (.b but te estimation algoritm (6-(8 only updates parameters at te sampling instants t k + t k + ( k + o te sampling period wile te regressor is evaluated at all time or re-updating te various estimates at sampling instants only. e estimation modiication and calculation o te controller parameters is also updated at sampling instants. e discrete-time parameter estimation and inverse o te covariance matrix adaptation laws are: θ k θ k + Δ θ k θ k ϕ [(k dτ P k + c k ( ϕ [(k dτ θ k (6.a Pk+ Pk +ΔPk Pk ϕ [(k dτ + + c k ( ϕ [(k dτ (6.b c k c k λ max de 4 d τ (P k + 4 d τ (6.c ISSN: 79-57X 7 ISBN:

4 SELECED OPICS in SYSEM SCIENCE and SIMULAION in ENGINEERING wit P (P ( > and θ k θ k θ or all integer k.e main result o tis section is announced as ollows: eorem. Consider te plant ( subject to te estimation sceme (6 and (6, i.e., te parameter estimates are only updated at sampling instants, te modiication sceme (- (, wit ( being updated only at t k, and te stabilizing control law (4-(5. us, te resulting closedloop sceme ulils te same properties o eorem under te same assumptions. e proo o eorem is outlined in Appendix B. IV. CONCLUSIONS is paper as developed a continuous-time adaptive stabilizer or a continuous-time irst-order controllable plants wic can ave an unstable zero and is subject to an input saturation o sigmoidal unction type. e mecanism used to guarantee te sceme s closed-loop stability is a modiication sceme o te parameter estimates wic is based on te use o a ysteresis switcing unction. e switces are built so tat te modiied plant estimated model is controllable and ten it as no pole-zero cancellation. An alternative adaptive stabilizer wic only modiies te parameter estimates at sampling instants, but wic is based on continuous-time input / output measurements, is also addressed or te same kind o simple plant. e resulting closed-loop system is o a ybrid nature because o te discrete updating o te estimation sceme. A similar ysteresis switcing unction, wic operates at sampling instants, is also used in tat case so as to guarantee te controllability o te modiied estimated plant model. APPENDIX A. Outline o proo o eorem Deine te Lyapunov unction candidate V / θ P θ by using te parametrical error θ θ θ and te inverse o te covariance matrix. It ollows tat P θ is constant or all time so tat θ θ + Pβ.us, ( β < δ c were ( + + v + p p p max β 3, θ θ 4 θ 4 + θ 5 θ b a b v p 5 p + p p 4 ( ( θ + p ( θ θ It ollows directly tat c ( β (θ θ 4 θ 3 (θ θ ( v + ( p p β p > + β 4 3 since +, v, p 3 and p p 4 cannot be simultaneously zero since c ( β >. i + so tat c ( β >. I β ± v ten c ( β >. I v ten β equalizes one o te combinations ± ( p p 4 ± p 3 and c ( β >. Property (i as been proven. Property (ii is proven as ollows. First note tat V& e wat implies tat V V ( <. en, e(t is bounded and square-integrable and te parametrical error is also bounded or al time. Finally, d( tr P / dt ϕ P ϕ wat implies tat P ϕ is bounded and square-integrable. Properties (iii-(iv ollow rom te act tat P is non-increasing and positive semideinite rom its updating rule so tat it converges. Also, θ ( θ ( t θ ( τ dτ P( τ ϕ ( τ e( τ dτ t & t t P( τ ϕ( τ + e( τ d τ < ( or all time. It ollows tat te parametrical error converges asymptotically to a inite limit. From tis partly result, te remaining o te proo ollows by calculating a bounded upper-bound o te norm-square integral o te time derivative o te estimate time-.derivative. It ollows tat θ & is bounded and square-integrable. en, using te Diopantine equation or te controller syntesis, it ollows tat te modiied estimated vector θ also converges asymptotically as well as tey converge te various controller parameters. B. Outline o proo o eorem One gets rom (6 tat Δ θ k P k Δ P k θ k wit te one-step incremental error being: Δ θ k θ k θ k and Δ P k P k P + k en, or a Lyapunov sequence candidate V k θ k P k θ k, one gets a one-step increment rom 86 : Δ V k V k V k ( P θ k Δ k I P k Δ P k P k Δ P k θ k i c k c k. en, te candidate is a Lyapunov sequence wit bounded eigenvalues o te covariance matrix implying strictly positive eigenvalues o its inverse, wat leads to te results o eorem. ACKNOWLEDGMENS e autors are grateul to te Spanis Ministry o Education by its partial support o tis work troug Grant ISSN: 79-57X 8 ISBN:

5 SELECED OPICS in SYSEM SCIENCE and SIMULAION in ENGINEERING DPI6-74. ey are also grateul to te Basque Government by its support troug Grants GIC743-I-69-7and SAIOEK S-PE8UN5. REFERENCES [] M. De la Sen, " Design o a discrete robust linear eedback controller wit nonlinear saturating actuator", Int. J. o Systems Sci., Vol., No. 3, pp , 994. [} B. Muller, J. Reinardt and M.. Strickland, Pysics o Neural Networks. An Introduction, Berlin: Springer Verlag, 995. [3] A. Feuer and A.S. Morse, "Adaptive control o single-input single-output linear systems", IEEE rans. Automat. Contr., Vol. AC-3, pp , 978. [4] A.S. Morse, " Global stability o parameter-adaptive control systems", IEEE rans. Automat. Contr., Vol. AC-5, No. 3, pp , 98. [5] K.S. Narendra, Y. Lin and L.S. Valavani, " Stable adaptive controller design. Part II: Proo o stability", IEEE rans. Automat. Contr, Vol. AC-5, No. 3, pp , 98. [6] R.D. Nussbaum, "A state approac to te problem o adaptive control assignment", MCSS, Vol., pp , 989. [7] R.D. Nussbaum, " Some remarks on a conjecture in parameter adaptive control ", Systems and Control Letters, Vol., pp , 989. [8] R. Mudgett and A.S. Morse, "Adaptive stabilization o linear systems wit unknown ig requency gains", IEEE rans. Automat. Contr., Vol. AC-3, pp , 985. [9] G. Kreisselmeier and M.C. Smit, " Stable adaptive regulation o arbitrary n-t order plants", IEEE rans. Automat. Contr., Vol. AC- 3, pp , 986. [] V. Etxebarria and M. De la Sen, " Adaptive control based on special compensation metods or time-varying systems subject to bounded disturbances", Int. J. o Control, Vol. 6, No. 3, pp , 995. [] M. De la Sen and A. Pena, " Syntesis o controllers or arbitrary pole-placement in discrete plants including unstable zeros wit extensions to adaptive control", J.o te Franklin Institute, Vol. 335.B, No. 3, pp. 47-5, 988. [] P. De Larminat, "On te stabilization condition in indirect adaptive control", Automatica, Vol., pp , 984. [3] R. Lozano, A. Osorio and J. orres, "Adaptive stabilization o nonminimum pase irst-order continuous-time systems ", IEEE rans. Automat. Contr., Vol. AC-39, No. 8, pp , 99. [4] R. Lozano and X. Zao, "Adaptive pole placement witout excitation probing signals ", IEEE rans. Automat. Contr., Vol. AC- 39, No., pp , 994. [5] M. De la Sen, "On te robust adaptive stabilization o a class o nominally irst-order ybrid systems", IEEE rans. Automat. Contr., Vol. 44, No. 3, pp , 999. [6] M. De la Sen, " Robust adaptive stabilization o time- varying irst- order ybrid systems wit covariance resetting ", Int. J. o Nonlinear Mecanics, Vol. 33, No., pp , 997. [7]. H. Lee and K. S. Narendra, "Stable discrete adaptive control wit unknown ig- requency gain ", IEEE rans. Automat. Contr., Vol. AC-3, No. 5, pp , 986. [8] M. De la Sen, "Multirate ybrid adaptive control", IEEE rans. Automat. Contr., Vol. AC-3, No. 5, pp , 986. [9] R. Middleton, G.C. Goodwin, D.J. Hill and D.Q. Mayne, "Design issues in adaptive control ", IEEE rans. Automat. Contr., Vol. AC-33 No., pp. 5-58, 988. [] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input - Output Properties, New York : Academic Press, 975. [] N. Luo and M. De la Sen, State-eedback sliding mode control o a class o uncertain time-delay systems, IEE Proceedings-D Control eory and Applications, Vol. 4, No. 4, pp. 6-74, 993. [] M. De la Sen and S. Alonso Quesada, Robust adaptive regulation o potentially inversely unstable irst-order systems, Journal o te Franklin Institute- Engineering and Applied Matematics, Vol. 336, No. 4, pp , 999. [3] N. Luo, J. Rodellar, J. Vei and M. De la Sen, Composite semiactive control o a class o seismically excited structures, Journal o te Franklin Institute- Engineering and Applied Matematics, Vol. 338, No. -3, pp. 5-4,. [4] N. Luo, J. Rodellar, J. Vei and M. De la Sen, Output eedback sliding mode control o base isolated structures, Journal o te Franklin Institute- Engineering and Applied Matematics, Vol. 337, No. 5, pp ,. [5] M. De la Sen, About te positivity o a class o ybrid dynamic linear systems, Applied Matematics and Computation, Vol. 89, No., pp , 7. [6] M. De la Sen, Quadratic stability and stabilization o switced dynamic systems wit uncommensurate internal point delays, Applied Matematics and Computation, Vol. 85, No., pp , 7. [7] M. De la Sen, On positivity o singular regular linear timedelay time-invariant systems subject to multiple internal and external incommensurate point delays, Applied Matematics and Computation, Vol. 9, No., pp. 38-4, 7. ISSN: 79-57X 9 ISBN: