REGULATION AND INPUT DISTURBANE SUPPRESSION FOR PORTONTROLLED HAMILTONIAN SYSTEMS Lua Gentili Arjan van der Shaft ASYDEIS University of Bologna viale Risorgimento 4136 Bologna Italy email: lgentili@deisuniboit Faulty of Mathematial Sienes University of Twente POBox 17 75 AE Enshede The Netherls email: ajvershaft@mathutwentenl Abstrat: In this paper the output feedbak regulation problem for portontrolled Hamiltonian systems (PHS) is addressed Following the nonlinear output regulation theory the regulator whih solves the problem is given by a parallel onnetion of two subontrollers: an internal model unit a regulator to stabilize the extended system omposed by the plant the internal model unit The main idea is to use the PHS theory in order to design that stabilizer ontroller: as in many ases the plant to be addressed is indeed a mehanial/eletri system it is very easy to think about it as a PHS the paper shows the onditions to fulfill in order to design the internal model unit as a PHS allowing to use the powerful energyshaping theory in order to stabilize the extended system Moreover the same tehniques are used to design an internal model based ontroller able to globally solve a problem of input disturbane suppression Keywords: Hamiltonian systems nonlinear output regulation internal model dumping injetion input disturbane suppression 1 INTRODUTION The problem of ontrolling the output of a system in order to ahieve asymptoti rejetion of some undesired disturbanes is a entral problem in ontrol theory The lassial solution developed for the output feedbak regulation of general nonlinear systems in (Byrnes et al 1997b) (Byrnes et al 1997a) shows that the regulator whih solves the problem is given by a parallel onnetion of two subontrollers: an internal model unit a stabilizer ontroller to stabilize the extended system omposed by the plant the internal model unit This paper is devoted to investigate some elegant tehnique to design this stabilizer ontroller onsidering general portontrolled Hamiltonian systems This work was supported through a European ommunity Marie urie Fellowship in the framework of the TS ( dissipation) (PHS) taking advantage of their peuliar properties In (Mashke van der Shaft 199) PHS were introdued as a generalization of Hamiltonian systems desribed by Hamiltonian s anonial equations whih may represent general physial systems (ie mehanial eletri eletromehanial systems nonholomi systems their ombinations) As in many ases the plant to address is indeed a mehanial/eletri system it is very easy to think about it as a PHS the main idea is to use the PHS peuliar properties the lassial passivitybased stabilization theory in order to design the stabilizer ontroller to use the internal model unit in the output regulation framework This paper shows the onditions to fulfill in order to design an internal model unit as a PHS allowing to use the powerful energyshaping theory in order to stabilize the extended system that is still a PHS as
? I _ s ^ interonnetion of PHS systems The paper is organized as follows In setion we briefly reall the lassial theory regarding the nonlinear output feedbak regulation introduing some results that will be useful in the following In setion 3 a loal solution for a regulation problem of PHS is presented; moreover it is shown that a loal solution for a traking problem is also available In setion 4 previous results are used to design a regulator to solve globally the same output feedbak regulation problem for PHS Moreover in setion 5 an input disturbane suppression problem is onsidered In the last setion 6 some simulation results regarding a permanent magnet synhronous motor affeted by some voltage disturbane are shown to onfirm the effetiveness of the design INTRODUTION TO OUTPUT FEEDBAK NONLINEAR REGULATION In order to introdue the main ontribution of this paper it is neessary to briefly reall the main theory regarding the nonlinear output regulation (see (Byrnes et al 1997b)); to that aim onsider a nonlinear system desribed by differential equations of the form (1)!#" state IR$ % ontrol input IR In a regulation problem the output ' ( IR is the output of the plant affeted by some exogenous )+*" disturbanes IR ; we assume to be / funtions 34545 6 (for some large 34578 1 ) of their arguments 378 also The seond equation in (1) desribes an autonomous * system (exosystem) defined in a neighborhood of the origin of IR Moreover the exosystem have to satisfy a basi assumption: 9: Hypothesis H1 is a stable equilibrium for *)" the exosystem there exists a neighborhood * ; of the origin the property that eah initial * is Poisson stable <3= ondition ; Remark This hypothesis implies that the matrix > whih haraterizes the ' linear approximation of the system a the equilibrium has all its eigenvalues on the imaginary axis Remark In order to introdue the main result in (Byrnes et al 1997b) let also reall the definition of immersion of a system into another: onsider two autonomous systems desribed by: @ A? @ A B A B D! defined in different state spae! but the same output spae E IR 3FG Assuming 3H 3H 3H7 let indiate the two systems as I!JKJDL I! DL respetively System I!JKJDL is said to be immersed in system! DL if there exists a M/ map NPO!Q! 1SR'T N 3U7 AV 3WXHY suh that N AZV N 3WX [ FF! S N A N A A< N A The general problem to deal is to find a dynami output feedbak ontrol law to obtain a loally (globally) asymptotially stable losedloop system in whih the response of the regulated output asymptotially onverges to as time tends to ] To state the main proposition about nonlinear error feedbak regulation for onveniene set: ^ `_ UaAb ededjfg h_ UaAb ededjf `_ HaAb edjf () Proposition 1: Assume Hypothesis H1 The error feedbak regulation problem for system (1) is solvable iff there exists M/ 1BRji k lh mappings lh3=m nmop op3=m both defined *(q "`* in a neighborhood of the origin satisfying the onditions (regulator equations(see (Byrnes et al 1997b))) l U'lHr5opr lhr=7 the autonomous system In; *jjr5opjl is immersed into a system ut s vwyxz s t3f` xz3` by an immersion / map N Moreover alling { } e 5ƒ 5ƒ u peˆ e 5ƒ 5ƒ the pair ^ u{ a Š_ g a has to be stabilizable for some matrix the pair Œ Š_ g { a has to be detetable A ontroller that solves the problem of error feedbak regulation ould be seen as a parallel onnetion of two subontrollers (see fig1): the role of the internal model subsystem is to render invariant the manifold identified by I PhlHr s N rml : regulator equations assure that it s a zero error manifold xz as on s that manifold the ontrol input is exatly xz N Žyop ; the role of the stabilizer ontroller is to stabilize in first approximation the system omposed by the plant the internal model unit; in other words the role of that stabilizer unit is to make loally (globally) exponentially attrative the same (zero error) manifold The main idea is to use the PHS theory in order to design suh stabilizer ontroller: the prinipal issue to deal is to desribe the internal model unit in the PHS framework
? 9 <<<<<; = A E Stabilizer ontroller ξ = K ξ + L e u = M ξ Internal Model ξ1 = φ ( ξ1 ) + N e u1 = γ ( ξ1 ) Fig 1 ontroller sheme u + + u Exosystem w = s ( w ) w x = f (x u w) e = h(x w) = h(x) + q(w) Plant 3 LOAL SOLUTION: REGULATION AND TRAKING WITH HAMILTONIAN INTERNAL MODEL UNIT To illustrate the onditions to fulfill in order to design the internal model unit as a portontrolled Hamiltonian system let now onsider a loal output regulation problem; this simple ase is interesting as it shows how is possible to solve loally also a nonlinear output traking problem when the output of the plant is required to trak a notknown exogenous signal To onsider both regulation traking problem let assume a plant desribed by @ D U A (3) Z XZ IR$ is the plant state IR is the ontrol input IR the error signal representing the regulated < output of the plant or the traking error IR the exogenous signal representing the disturbane < or the referene input Let assume that is generated by an autonomous exosystem (satisfying Hypothesis 1) desribed by wh > < <3U (4) the (perfetly known) matrix > is defined by L (5) > diagi > > e > / > 7 > `_ a z T 1 (6) Proposition : It is possible to design for system (3) an internal model unit desribed as a PHS moreover the output regulation (or traking) problem ^ for system (3) is loally solvable if (defining as in ()) the pair (^ g ) is stabilizable the pair (^ ) is detetable there exist a mappings lh op lh3h op3h satisfying the onditions l > lhr5opr n lhr (7) op polynomial of the form opu g e (8) m( i1 T for T i 4 rj P!!!! " e! (9) Proof: Main proposition in (Huang 1) states that onditions (8) (9) are equivalent to require the existene of exists some set of # real numbers $ $ e $%! suh that ' %(*) opu+$ op $ ' (*) op $%! ' %! (*) op (1) Moreover (see (Huang 1)) onditions (8) (9) assure the existene of ; j ; e ; 1 $/ # i3 / T j I4 5/ / / R e 476 / T 76 i L suh that 8 $/ 8 ; H 8 % $ $ 8 $%! 8 %! (11) From (11) we immediately found out that $ for z i3: 74 T i ondition (1) implies that I * > 5opJL is immersed by a map N into the linear observable system (see (Byrnes et al 1997a)) defined by? s { s s (1) { diag { e { A diag e T?@ T { @@@@ B T F /D >$ >$" $% T! It is easy to realize that system (1) is equivalent therefore immersed by means of a simple linear transformation to the linear system W<5EWŽz +F W E diag E E e F $/ diag F e F F B T T T *D E E HG ; z ; JILK T i W NM / s The linear transformation is defined by M! diag M e M MPO FRQSETQUFRQ ETQ %! FRQV To design a suitable internal model unit we E< have to hoose a matrix suh that the pair ( ) is stabilizable for instane diag e B T T T D Q WF Q Now we an onlude that the regulation problem an be solved (see (Byrnes et al 1997a)) moreover the internal model unit an be written as a PHS:
W W W Q W E is a skewsymmetri matrix the Hamiltonian funtion is defined as W Q W WF Q An immediate onsequene of the previous statement is that if our system satisfies all onditions of Proposition moreover the unfored plant an be desribed as a PHS ie 5z( HA A A 5H AU Q A (13) ''!%" IR$ ' h" IR E being the dual spae H of O! Q IR is the energy funtion w Q w Q then to omplete the design of a ontroller that solves loally the output regulation traking problem we have only to study a stabilizer ontroller for the PHS desribed by _ W a ( _ HA A Q A Q a _ AM a _ A _ a `_ W a _ A a _ a A Q Q a _ A W a (14) as new ontrol input Remark Note that as we need to design a stabilizer ontroller for the onnetion between the unfored plant the internal model unit in order to obtain a loal solution (ie to stabilize the unfored plant in first approximation) it is possible to onsider both regulation traking problems; in fat in a traking problem ASu we only need that the real output of the plant is dual to the input signal ating on the system that is always fulfilled onsidering Hamiltonian systems 4 GLOBAL SOLUTION: REGULATION WITH AN HAMILTONIAN INTERNAL MODEL UNIT Thanks to result stated in Proposition it is now possible to extend our onsiderations propose how to solve the global problem of nonlinear output feedbak regulation onsidering a portontrolled Hamiltonian system subjet to some exogenous disturbane: ydu( H 'U Q > (15) the exosystem is defined ^ as in (4) (5) (6) Proposition 3: Defined g as in () assume that the pair (^ g ) is stabilizable the B(lH pair (^ ) is detetable there exist a mappings op lh3s op3v satisfying : 545 is op the onditions (7) polynomial of the form (8) (9) If the unfored system stable then it s possible to design an output feedbak ontroller able to assure the output going globally asymptotially to zero Proof: As Proposition holds we are able to design an Hamiltonian internal model unit to write the whole system (plant+internal model unit+exosystem) as a PHS of the form: H Q ( Q > MF F F 3WX ) W Ž Q F Œ _ 3WX ) W a (16) ) W Q W Q R total Hamiltonian defined by From Proposition 1 we know that the internal model unit assures that there exists an invariant zerooutput lh manifold for system (16) namely N n N WM N; N ; is the immersion map to define system (1) Showing that it is possible to make this manifold globally attrative by a simple dumping injetion ( ) will end the proof Let onsider the time derivative of the total Hamiltonian of (16): Q (17) LaSalle invariant priniple guarantees that system (16) tends to the largest invariant manifold ompatible We have simply D9 to show that there s only one manifold assuring at the same time ` that this manifold is just lh the one above defined (namely N z lh op Indeed are mappings providing zero output for system D (15) then it is the unique lh manifold providing ; moreover as is a ontrolled invariant zero output manifold for system (15) then op on that manifold applying the right ontrol input we have! b ) f op QD Q lhr $ b ) f As the trajetories of the system on that manifold are haraterized by quasiperiodial energy values (there exists a time 1 T M M suh that 1 M U for 1 % ) system must satisfies then
W M lhr b ) f 7 It s easy to realize from equation (17) that proposition 3 is proved by LaSalle invariant priniple as lh N is the only manifold providing DH on that manifold b ) f 7 lhr It is worth to note that (15) unfortunately does not allow to study traking problems as output map (dual to the input ) is the real output of the plant not the traking error For traking problem the system to deal should be @ DH( HA ) A > ' > Q A A A A Q ) (18) It easy to realize that as input ( ) the output we want to ontrol to zero ( ) are not dual system (18) doesn t fit in the PHS framework Then while a loal study pointed out the possibility of traking loally unknown trajetories the traking problem is still a big issue for a global haraterization of the problem 5 INPUT DISTURBANES SUPPRESSION To omplete our disussion let disuss about another important issue in the output regulation framework; onsider the ase of an unknown exogenous disturbane ating on the ontrol input hannel: we want to globally regulate the output of the plant in despite of the presene of that input disturbane In order to present the main result let onsider a portontrolled Hamiltonian system of the form: @ UP HA A A A AU QUA A > A (19) the exosystem is still defined as in (4) (5) (6) ^ Proposition 4: Defining g as in () assume that the pair (^ g ) is stabilizable the pair (^ ) is detetable '}opv there exist a mappings j lh lh3} op3 op' satisfying the onditions (7) polynomial of the form (8) (9) If the unfored system 545 is stable then it s possible to design an output feedbak ontroller able to assure the output going globally asymptotially to zero Proof: As Proposition holds we are able to design an Hamiltonian internal model unit to write the whole system (plant+internal model unit+exosystem) as: HA A Q P AMF A Q F > F _ A 3WX ) W a AMF ΠA MF ΠA Q Π_ A 3WX ) W a () W Q W ) Q F Q E as defined in setion 3 As system I > 5opJL is immersed into the linear observable system WF EWn knf W by a nonlinear map defined by N ' N we an state the following: N < N > m E N op< U+F N W N time Defining a new oordinate as deriving we obtain: EW E N F Q Q A E F Q Q A 5E Q Q A Taking in aount that F F W F N + Q we ould rewrite system () as a PHS: HA A Q ( AMF A Q F > F _ A ) A a A Q F Π_ A 3WX ) a (1) As system (1) is similar to (16) ) Q Q lh an invariant zero output manifold now defined by the proof an be ompleted following the one stated in setion 4 6 PERMANENT MAGNET SYNHRONOUS MOTOR EXAMPLE In this setion in order to point out the physial effetiveness of the input disturbane suppression result we show some simple simulation results regarding a well known eletromehanial problem: we want to stabilize a permanent magnet synhronous motor around its equilibrium point robustly in despite of some voltage disturbanes ourring to the ontrol inputs A permanent magnet synhronous motor (in a rotating referene ie the dq frame) an be written as a portontrolled Hamiltonian system dissipation (see
' Q Q 5 1 15 5 5 5 1 5 1 15 5 1 5 1 15 5 1 1 5 1 15 5 input voltage disturbanes Fig From upper plot to lower: traking error (van der Shaft 1999) (Ortega et al 1999)) for the state vetor ' y ' ' are the urrents the angular veloity ' the stator indutanes the inertia momentum the number of pole pairs The Hamiltonian is defined by A< Q! HA while A A are determined as ' " HA< ' " { { h A< AU h T T T the stator winding resistane { a onstant term due to interation of the permanent magnet the magneti material in the stator ' ' Inputs are the stator voltages Q the load torque onsidering a onstant load torque N it is easy to realize that there exists an equilibrium point desribed by " ; the whole system an be rewritten in the new (error) oordinates + 7 z " " " Q as H D D A B Q A H A is defined by ' ' " ' " " H AU ' " " ' ' " ' { { () L ' L" are two sinusoidal disturbanes ating on the voltage inputs It is immediate to hek that system () satisfies all onditions imposed in Proposition 4; we simulate the behavior of system () onsidering # $ o Lu T# $ o L"u i# $ o onneted an internal model unit a dumping injetion designed following the proedure introdued in setion 5 Fig shows the traking errors the input disturbanes 7 ONLUSIONS In this paper the output feedbak regulation problem for portontrolled Hamiltonian systems (PHS) is disussed Main results stated in setion 3 for a loal solution of the problem in setion 4 for a global solution show the onditions to fulfill in order to design following the lassial nonlinear output regulation theory a regulator whih solves the problem as a parallel onnetion of two subontrollers both onserving the PHS struture: an internal model unit a stabilizer ontroller to stabilize the extended system omposed by the plant the internal model unit that is still a PHS Moreover in setion 5 the same tehniques are used to design an internal model based ontroller able to globally solve a problem of input disturbane suppression ie to globally stabilize a system affeted by an unknown exogenous input through the input hannel In setion 6 some simulation results regarding a permanent magnet synhronous motor affeted by some voltage disturbane are shown to onfirm the effetiveness of the design 8 REFERENES Byrnes I F Delli Prisoli A Isidori W Kang (1997a) Struturally stable output regulation of nonlinear systems Automatia 33() 369 385 Byrnes I F Delli Prisoli A Isidori (1997b) Output regulation of unertain nonlinear systems Birkhäuser Boston Huang J (1) Remarks on the robust output regulation problem for nonlinear systems IEEE Transation on Automati ontrol Mashke BM AJ van der Shaft (199) Portontrolled hamiltonian system: modelling origins system theoreti approah Pro nd IFA NOLOS Bordeaux pp 888 Ortega R AJ van der Shaft BM Mashke G Esobar (1999) Stabilization of portontrolled hamiltonian systems: Passivation energy balaning submitted for publiation ' van der Shaft AJ (1999) gain Passivity Tehniques in Nonlinear ontrol Springer Verlag London UK