Lectue 3 Eaple 6.3 Conside an LI syste A Bu with a Huwitz ati A and a unifoly ounded in tie input ut. hese two facts iply that the state t is ounded. hus, the state deivative t is ounded. Let y C epesent the syste output. hen y C is ounded and, consequently the syste output y t is a unifoly continuous function of tie. Moeove, if the input u is constant then the output u t y t tends to a liit, as t. he latte coined with the fact that y is unifoly continuous iplies, (y Baalat s Lea), that the output tie deivative y asyptotically appoaches zeo. o facilitate applications of Baalat s lea to the analysis of nonautonoous dynaic systes, we state the following iediate coollay. Coollay 6. (Lyapunov-like Lea) V V t, can e found such that If a scala function V t, is lowe ounded, V t, f t, V t, then V t is unifoly continuous in tie,,, as t., Notice that the fist two assuptions iply that V t, tends to a liit. he latte coupled with the 3 d assuption poves (using Baalat s lea) the coollay. Eaple 6.4 Conside again the closed-loop eo dynaics of an adaptive contol syste fo Eaple 6.1. Choosing V e, e, it was shown that along the syste tajectoies: V e, e. he nd tie deivative of V is V e, 4ee 4ee wt Since wt is ounded y hypothesis, and etand t wee shown to e ounded, it is clea that V is ounded.. Hence, V is unifoly continuous and y the Baalat s lea (o the Lyapunov-like lea), V which in tun indicates that the tacking eo et tends to zeo, as t. 6. Basic Concepts and Intoduction to Adaptive Contol 3
Intoduction Since the 195 s adaptive contol has fily eained in the ainstea of contols and dynaics eseach, and it has gown to ecoe a well-foed scientific discipline. One of the easons fo the continuing populaity and apid gowth of adaptative contol is its clealy defined goal to contol dynaical systes with unknown paaetes. Reseach in adaptive contol stated in connection with the design of autopilots fo high-pefoance aicaft. But inteest in the suject has soon diinished due to the lack of insights and the cash of a test flight, (NASA X-15 poga). he last decade has witnessed the developent of a coheent theoy fo adaptive contol, which has lead to any pactical applications in the aeas such as aeospace, ootics, cheical pocess contol, ship steeing, ioengineeing, and any othes. he asic idea in adaptive contol is to estiate the uncetain plant and / o contolle paaetes on-line ased on the easued syste signals and use the estiated paaetes in contol input coputation. An adaptive contolle can thus e egaded as an inheently nonlinea dynaic syste with on-line paaete estiation. Geneally speaking, the asic ojective of adaptive contol is to aintain consistent pefoance of a syste in the pesence of uncetainty o unknown vaiation in plant paaetes. hee ae two ain appoaches fo constucting adaptive contolles: Model efeence adaptive contol (MRAC) ethod. Self-tuning contol (SC) ethod. Scheatic epesentation of an MRAC syste is given in Figue 7.1. Figue 7.1: Model Refeence Adaptive Contol Syste he MRAC syste is coposed of fou pats: Plant of a known stuctue ut with unknown paaetes Refeence odel fo specification of the desied syste output Feedack / feedfowad contol law with adjustale gains, (contolle) Paaete / gain adaptation law Scheatic epesentation of an SC syste is given in Figue 7.. 4
Figue 7.: Self-uning Contol Syste An SC syste coines a contolle with an on-line (ecusive) plant paaete estiato. A efeence odel can e added to the achitectue. Basically, SC syste pefos siultaneous paaete identification and contol. he contolle paaetes ae coputed fo the estiates of the unknown plant paaetes, as they wee the tue ones. his idea is often efeed to as the Cetainty Equivalence Pinciple. By coupling diffeent contol and estiation schees, one can otain a vaiety of self-tuning egulatos. When the tue plane paaetes ae unknown, the contolle paaetes ae eithe estiated diectly (diect schees) o coputed y solving the sae design equations using plant paaete estiates (indiect schees). MRAC and SC systes can e designed using oth diect and indiect appoaches. Ou focus will e on the design, analysis and evaluation of the diect MRAC systes fo continuous plants with uncetain dynaics. Definition: acking Contol Pole In paticula, we conside tacking poles fo continuous plants opeating in the pesence of odeling uncetainties, envionental distuances, and contol failues. State feedack / feedfowad tacking contol will e designed fo uncetain dynaical systes in the fo f t,, u, (6.1) y h, whee is the state, u is the contol, is a vecto of unknown constant paaetes, y is the contolled output. It is assued that the syste state vecto is availale (easued on-line). he tacking pole is to design the contol input u so that the contolled output y t tacks a given efeence signal t in the pesence of the syste uncetainties, that is the output tacking eo ey t yt t (6.) ecoes sufficiently sall, as t. Moeove, it is equied that duing tacking, all the signals in the coesponding closed-loop syste eain ounded. 5
If ey t then we say that an asyptotic output tacking is achieved. In geneal, it ight not e feasile to achieve asyptotic tacking. In that case, the goal will e to achieve ultiate oundedness of the tacking eo within a pescied toleance, that is ey t, t (6.3) whee is the pescied sall positive nue. MRAC Design of 1 st Ode Systes Suppose that a plant contains unknown constant paaetes, without any infoation aout thei ounds. he plant dynaics ae au f (6.4) whee is the state, u is the contol input, a and ae unknown constants. It is assued that the sign of is known, while the unknown and possily nonlinea function f is linealy paaeteized in tes of N unknown constant paaetes i and known asis functions i. N i i (6.5) i1 f In (6.5), N i i R assued that the egesso coponents denotes the known egesso vecto. It is i ae Lipschitz-continuous functions of the syste state. A efeence odel is descied y the 1 st ode diffeential equation a t (6.6) whee a and ae the desied constants and t is the efeence input. he tacking task is to design a contol law ut such that all signals in the syste eain ounded, while the tacking eo et t t tends to zeo asyptotically, as t. Notice, that the tacking task ust e accoplished in the pesence of N unknown constant paaetes: a,,,, N. Fist, we define an ideal contol solution, as if the unknown paaetes wee known. he ideal contol is foed using feedack / feedfowad achitectue uideal k k (6.7) Sustituting (6.7) into (6.4), the closed-loop dynaics can e witten. ak k t (6.8) Copaing (6.8) with the desied efeence odel dynaics (6.6), it iediately follows that ideal gains k and k ust satisfy the following atching conditions a k a (6.9) k It is clea that the ideal gains in (6.9) (whichae not known!) always eist. 1 6
Based on (6.7), the following tacking contol solution is foed, u kˆ kˆ ˆ (6.1) whee the feedack gain k ˆ, the feedfowad gain k ˆ, and the estiated vecto of paaetes ˆ will e found to achieve asyptotic tacking of the efeence odel tajectoies. We sustitute (6.1) into the syste dynaics (6.4). hen the closed-loop syste ecoes: akˆ kˆ ˆ (6.11) Using the atching conditions (6.9), yields: a k kˆ k kˆ k ˆ (6.1) Define paaete estiation eos: k ˆ k k k ˆ k k (6.13) ˆ hen the closed-loop dynaics of the tacking eo signal et t t can e otained y sutacting (6.6) fo (6.1). et t t aek k (6.14) Conside the Lyapunov function candidate. 1 1 1 V e, k, k, e k k (6.15) whee,, and ae the so-called ates of adaptation. aking the tie deivative of V, along the tajectoies of (6.14), gives: 1 ˆ 1 ˆ 1 V e, k, k, ee k k k k ˆ 1 ˆ 1 ˆ 1 e a ˆ e k k k k k k 1 a sgn ˆ e k e k 1 1 ˆ k ˆ e k e sgn sgn Using (6.16), the adaptive laws ae chosen to enfoce closed-loop staility. ˆ k esgn ˆ k esgn ˆ esgn In fact, due to (6.17), the tie deivative ecoes negative sei-definite, that is, (6.16) (6.17) 7
V e, k, k, a e t (6.18) which iediately iplies that the signals e, k, k, ae unifoly ounded. he latte coupled with the fact that t ae ounded and is a constant vecto, t, t and the estiated vecto of paaetes iplies that the syste state ˆ t ae unifoly ounded. It was assued that the vecto of the coponents i of the egesso vecto wee Lipschitz-continuous functions of. heefoe, they ae unifoly ounded. Hence, the contol ut in (6.1) is unifoly ounded. Consequently, oth t and t ae unifoly ounded. Futheoe, diffeentiating (6.18) yields: V e, k, k, 4aete t (6.19) heefoe, V is ounded and, consequently, V is unifoly continuous function of tie. Since V is lowe ounded, V is negative sei-definite and unifoly continuous, then all the thee conditions of the Lyapunov-like lea (Coollay 6.) ae satisfied, and hence: liv t (6.) t Due to (6.18), we can finally conclude that the tacking eo goes to zeo asyptotically, as t. Moeove, since the Lyapunov function is adially unounded, the contol solution is gloal, that is the closed-loop tacking eo dynaics ae gloally asyptotically stale. he tacking pole is solved. We suaize ou esults in the theoe elow. heoe 6.1 Fo the uncetain dynaical syste in (6.4), with the contolle in (6.1), and the adaptive laws in (6.17), the closed-loop state t asyptotically tacks the state t of the efeence odel in (6.6), while all the signals in the closed-loop syste eain ounded. Moeove, the closed-loop tacking eo dynaics in (6.14) ae gloally asyptotically stale. 7. Dynaic Invesion ased MRAC Design fo 1 st Ode Systes Using siila (as in pevious section) design appoach, a dynaic invesion (DI) ased adaptive contol laws can e deived. Conside the uncetain scala dynaical syste au f (7.1) Let the constants a and e unknown. Assue that, whee is the known lowe ound of. Also assue that the unknown possile nonlinea function f is 8
linealy paaeteized in tes of the unknown constants i and known ounded continuous asis functions : i f N Let the efeence odel dynaics e specified as: a t i i (7.) i1 a, t Rewite the syste dynaics in the fo: a ˆ u ˆ fˆaˆa ˆ u fˆ f whee â, ˆ, and ˆ f, and f N ˆ ˆ ˆ f a f (7.3) (7.4) epesent the on-line estiated quantities, while a, ae the coesponding appoiation eos. Using i i (7.5) i1 the function appoiation eo can e witten as: N ˆ ˆ ˆ f f f i ii (7.6) i1 Conside the following dynaic invesion ased adaptive contolle 1 ˆ u a ˆ a ˆ (7.7) Sustituting (7.7) into the nd te of (7.4), yields a au (7.8) Let e e the tacking eo signal. Its dynaics can e otained y sutacting (7.3) fo (7.8). e a eau (7.9) Conside the following Lyapunov function candidate: 1 1 1 V e, a,, e a a (7.1) whee a,, will eventually ecoe the adaptation ates. he tied deivative of V along the tajectoies of the eo dynaics (7.9) can e coputed: 1 1 ˆ 1 V e, a,, ee aaˆ ˆ a 1 1 ˆ 1 e a ˆ ˆ e a u a aa 1 1 1 ˆ ˆ a ˆ e a a a e ue e Based on (7.11) and in ode to ake V, the adaptive laws ae chosen as: i (7.11) 9
aˆ a e ˆ ue (7.1) ˆ e In fact, this leads to V e, a,, a e (7.13) heefoe, the signals e, a,, ae ounded. Since t is ounded, then the efeence odel state is ounded. Hence,, aˆ, ˆ, ˆ ae ounded. Due to the division y ˆ in (7.7) and in ode to keep the contol signal u ounded, we need to odify adaptive laws (7.1). Conside the following odification of the nd equation in (7.1): ˆ ˆ, if ˆ ue ue (7.14), if ˆ ue Basically, the intent is to stop the adaptation if the ˆ eaches its lowe liit and its tie deivative is negative. One needs to veify that this odification does not advesely effects the closed-loop staility. Foally, we need to show that 1 ˆ ue (7.15) When ˆ, the adaptive law (7.14) is the sae as the coesponding law in (7.1) and, theefoe V a e. Suppose that thee eists such that ˆ. Since ˆ. If u e then again V a e, while then ˆ ue iplying that ˆ t inceases locally fo t 1 u e then accoding to (7.14), ue ue. On othe hand, if ˆ. As a esult, V a e. hus, odification (7.14) always contiutes to aking the tie deivative of V to e negative-seidefinite. he adaptive laws can now e witten eplicitly. aˆ a, if ˆ ˆ u u ˆ (7.16), if ˆ ue ˆ Net, a foal poof is given to show that the DI ased adaptive contol in (7.7) povides asyptotic tacking of the efeence odel state. Since V then 3
e, a,, ae ounded. he latte iplies that, aˆ, ˆ, ˆ ae ounded. Due to odification (7.14), ˆ and consequently u is ounded. his in tun iplies that is ounded. Moeove, since is ounded, then is ounded and theefoe e is ounded. Because of (7.14) V e, a,, a e (7.17) fo all t. Since V is ounded fo elow y zeo and its deivative is sei-negative, V conveges to a liit, as a function of tie. Integating oth sides of (7.17) yields: t V t V a e d (7.18) o, equivalently: t 1 e d V V t (7.19) a t Let W t e d. Fo (7.19) it follows that W t tends to a finite liit, as t. At the sae tie its tie deivative is W t e t. Since W t ee t then W t is unifoly continuous. Using Baalat s Lea, iplies that liw t. hus, li e t and the tacking pole is solved. t t Reak 7.1 Modification (7.14) is a special case of the well-known Pojection Opeato. Since the ight hand side of (7.14) is not Lipschitz the closed-loop syste does not satisfy the sufficient conditions to have a unique tajectoy, given an initial state. Coesponding solutions can e defined siila to the case of vaiale stuctue systes such as systes with sliding odes. Nevetheless, a continuous vesion of the Pojection Opeato eists and will e coveed late in the couse. 31