Avoiding eedack-linearization singularity using a quotient method The ield-controlled DC motor case S S Willson, Philippe Müllhaupt and Dominique Bonvin Astract Feedack linearization requires a unique eedack law and a unique dieomorphism to ring a system to Brunovský normal orm Unortunately, singularities might arise oth in the eedack law and in the dieomorphism This paper demonstrates the aility o a quotient method to avoid or mitigate the singularities that typically arise with eedack linearization The quotient method does it y relaxing the conditions on dieomorphism, which can e achieved since there is an additional degree o reedom at each step o the iterative procedure This reedom in choosing quotients and the resulting advantage are demonstrated or a ield-controlled DC motor Using a Lyapunov unction, the domain o attraction o the control law otained with the quotient method is proved to e larger than the domain o attraction o a control law otained using eedack linearization I INTRODUCTION Feedack linearization is a widely studied method or designing control laws or nonlinear system [] In eedack linearization, a system is transormed to Brunovský normal orm using a eedack law and a dieomorphism All controllale linear systems can e rought to Brunovský normal orm [] In the nonlinear setting, the input-aine singleinput system ẋ = (x+g(xu ( is eedack linearizale (FBL (Theorem 43 o [3] i and only i the ollowing conditions are satisied: Involutivity o the distriution { = Span g,ad g,,ad n Full rank o the accessiility matrix { } L = g,ad g,,ad n g, } g where ad g represents the Lie racket o (x and g(x Achieving eedack linearization requires a eedack linearizing output h(x o relative degree n such that the - orm ω(x = h(x elongs to the kernel o The output h(x is then used to otain the eedack law or and the dieomorphism v = L n h(x+l g L n h(xu, u = v Ln h(x L g L n h(x, Φ = h(x L h(x L n h(x, where v is the input to the Brunovský normal orm The domain o attraction o the control law depends on the domain o validity o Φ and Φ and the zeros o the unction L g L n h(x Hence the domain o attraction excludes the singularity points where the determinant o Φ = or L g L n h(x = as well as all the points where a singularity is reached ollowing the trajectory o the closed loop system There are algorithms to determine the eedack linearizing outputs or single-input systems ([4], [5] and or multi-input systems ([6], [7] The algorithm proposed in [4] generates quotients to otain h(x Based on this algorithm, a quotient method [8] has een developed to directly otain the control law without the need o achieving Brunovský normal orm Due to the reedom in choosing the quotient oliation in the design process o the quotient method, it was oserved, in simulation, that the control law was ale to overcome the singularity in eedack linearization [8] The present paper demonstrates the application o the quotient method to a ield-controlled DC motor and proves, using a Lyapunov unction, that the domain o attraction o the control law otained through the quotient method is indeed larger than the domain o attraction o the control law otained through eedack linearization The paper is organized in the ollowing manner The next section riely introduces the steps in the quotient method Section III presents the mathematical model o the ieldcontrolled DC motor and discusses eedack linearization o the DC motor model Section IV descries the steps involved in designing the control law Section V proves the domain o attraction o the control law and presents simulation results Finally, section VI provides concluding remarks II QUOTIENT METHOD The quotient method is an iterative design technique to otain a control law or nonlinear input-aine single-input systems [8] The method proceeds in two stages, namely a orward decomposition stage and a ackward control design stage Both stages require several iterative steps A Forward decomposition At every step o the orward decomposition stage, an equivalence relation ( is deined etween two vector ields For example, or the vector ields m (x and m (x, the equivalence relation is: m (x m (x i m (x m (x = κ(xg(x,
where κ(x is any unction This deines the equivalence class as [m r (x] = {m(x m(x m r (x = κ(xg(x κ(x}, where m r (x is the representative o the equivalence class To deine the representative o the equivalence class, we choose the exact -orm ω(x = γ(x, where γ(x is any chosen unction such that ω(xg(x Then, we deine the representative o any vector ield m(x as m r (x = m(x ω(xm(x g(x ( ω(xg(x Note that m r (x represents the entire equivalence class to which m(x elongs Hence, it can e veriied that m r (x remains unchanged when m(x is replaced y m(x + κ(xg(x, or all κ(x Using this equivalence relationship, one can otain the representative r (x o (x + g(xu By deinition, r (x remains unchanged or all control laws u = κ(x This whole process is acilitated y designing the dieomorphism φ (x z = Φ p (x = φ n (x γ(x, where φ (x to φ n (x are scalar unctions ( such that Φp(x L g φ i = or i =,,n and rank = n, so that Φ g(x Φ p g(x = x=φ p (z, (3 α(z where α(z := L g γ(x Φ p (z In these transormed coordinates, otaining Φ r (x reduces to simply eliminating the last line o Φ (x The eliminated coordinate can e regarded as the input o the system descried y the vector ield Φ r (x, and thus results in a single-input system o dimension reduced y one This process can e repeated n times to otain a single-dimensional system The dieomorphisms otained at each step can e comined to put the system in eedack orm: ẏ = (y,y ẏ = (y,y,y 3 ẏ n = n (y,,y n,u where y to y n represents the coordinates o the new system otained using the comined dieomorphism B Backward control design The control design stage starts y designing a control law or the system otained at the end o the orward stage, ie the irst equation o (4 For this susystem y is considered as the input Next, assign the target value y,d (y to the (4 input y y solving k y = (y,y or y Here, k is any positive constant This target value, which depends on y, is then tracked using proportional eedack The error e = y y,d is deined to track the desired target Next consider the irst two equations o (4 and assume y 3 as the input to this susystem The error deined aove is then driven asymptotically to zero y assigning the stailizing dynamics ė = k e, where k is a positive gain or the proportional eedack controller Sustituting or ė and e and solving or y 3 results in y 3,d (y,y, which is a unction o y and y The whole step can e repeated y deining the error e 3 = y 3 y 3,d to otain y 4,d (y,y,y 3 The ackward stage continues this way until a control law is otained or the original system It is easy to show, using the center maniold theory (see appendix B o [3] and corollary o [8], that the resulting control law is locally asymptotically stale Furthermore, a Lyapunov-type analysis will e used to estimate the domain o attraction o the control law III MODEL OF THE FIELD-CONTROLLED DC MOTOR The example o a ield-controlled DC motor is chosen to illustrate how the possiility o choosing γ(x during the orward decomposition helps avoid the singularity that arises due to the particular choice o γ(x required or eedack linearization The ield-controlled DC motor is a FBL system However, the quotient method is not restricted to FBL systems, and application to non-fbl system are illustrated in [9] and [] The ield-controlled DC motor with negligile shat damping descried in [] is considered: v v a J dω dt = R i +L di dt, = c i ω +L a di a dt +R ai a, = c i i a The irst equation represents the ield circuit, with v,i,r, and L eing the voltage, current, resistance and inductance, respectively The variales v a,i a,r a, and L a in the second equation are the corresponding variales or the armature circuit The term c i ω is the ack electromotive orce induced in the armature circuit The third equation is the equation o motion or the shat, with the rotor inertia J and the torquec i i a produced y the interaction o the armature current with the ield circuit lux The voltage v a is held constant and control is achieved y varying v The system is represented y the third-order model ẋ = (x+gu,
with the states x = i,x = i a,x 3 = ω, and the input u = v L, (x = ax x +ρ cx x 3 θx x, g =, (5 and the positive constants a = R /L, = R a /L a,c = c /L a,θ = c /J,ρ = v a /L a For ū =, the motor is at equilirium at x =, x = ρ/ and any aritrary x 3 value The aim is to design a control law that drives the system rom any initial condition to the desired operating pointx = (, ρ/, ω, whereω is the desired set point or the angular velocity x 3 The dieomorphism Φ FBL = θx +cx 3 θx (ρ x θ(ρ x ( x +ρ cx x 3 rings the system to Brunovský normal orm [] Next, consider the determinant o ΦFBL given y Det( Φ FBL = 8c θ x 3(x ρ, which shows that Φ FBL is singular at oth x 3 = and x = ρ/ It can also e seen in the accessiility matrix, L = (g,[,g],[,[,g]] = a a cx 3 (a+cx 3 θx θρ aθx +θx whose determinant is given y Det(L = cθx 3 (x ρ, Note that the accessiility matrix looses rank or oth x 3 = and x = ρ/ Consequently, the control law designed through eedack linearization has singularities, and the dieomorphism is not valid there Hence, the domain o attraction o the control law developed using eedack linearization [] is: D FBL = { (x,x,x 3 x > ρ and x 3 > } (6 Physically, the two points o singularity arise due to the presence o two cross terms The irst term is the ack em resulting rom the motion o the shat in the magnetic lux generated y the ield coils cx x 3 = cωl i L a The second term is θx x = cl i i a J, which corresponds to the torque generated at the shat due to the current lowing in the armature coil inside the magnetic lux generated y the ield coils These cross terms represent the mechanisms y which the ield coils are ale to inluence the armature current and produce torque at the shat Now, or ω =, the magnetic lux and thus, the ield current looses its inluence over the armature current Similarly, i the armature current i a =, then the ield current ails to produce any torque on the shat However, due to the presence o a ias voltage in the armature circuit v a, the point o singularity is shited to i a = va R a The two cases, ω = and i a = va R a, represent only a momentary loss o the inluence o the input They ecome singularities ecause the eedack linearization attempts to impose a linear aine structure (Brunovský normal orm to the system In the quotient method, y suitaly using the degree o reedom in the algorithm, we can avoid imposing a strict linear orm and thus having singularities For the ieldcontrolled DC motor, this degree o reedom can only e used in one equation; however, we can choose which singularity to avoid In this paper, we chose to avoid the singularity at i a = va R a since we are interested in a cascade to control successively i, i a and ω, which is justiied y the act that the time constants o the electrical circuits are consideraly smaller than the time constant o the mechanical component The other option to control in turn i, ω and i a would avoid the singularity at ω = IV CONTROL DESIGN USING THE QUOTIENT METHOD Since the quotient method allows avoiding at least one singularity, the resulting domain o attraction o the control law is increased The orward decomposition stage includes two steps, so as to achieve a single-dimensional system Step : This step rings the vector ield g(x into the canonical orm (3 and then shits the equilirium point rom (, ρ/, ω to (,, y designing a suitale dieomorphism The irst two unctions o the dieomorphism are [8]: z = x ρ/ (7 z = x 3 ω (8 The third unction can e chosen to maximize the size o the validity domain o the resulting dieomorphism One such choice is z 3 = x, which results in the gloally valid dieomorphism: z z z 3 = Φ (x,x,x 3 = x ρ/ x 3 ω x The model in z-coordinates ecomes: ż ż = z cz 3 (ω +z θ( ρ +z z 3 + u ż 3 az 3 The last row is removed and ż and ż expressed with z 3 as input to give the ollowing two-dimensional system: ( ( ( ż z c(ω +z = + ż θ( ρ +z z 3 (9 Step : This step rings the input vector ield o (9 into the canonical orm (3 y designing a suitale dieomorphism The irst unction o this dieomorphism is: y = θ z + θρ z + c z +cω z This shit to i a = va R a is not ovious rom the equations; however, the asence o a singularity or i a = clearly indicates the eect o the ias voltage v a
The second unction y can e chosen to maximize the size o the validity domain o the resulting dieomorphism The choice y = z results in the dieomorphism: ( y y = Φ (z,z = ( θ z + θρ z + c z +cω z z, which, however, has a singularity at z = ω, corresponding to x 3 = (see eq 8 Other choices or y are possile, or example: y = z results in a singularity at z = ρ/, corresponding to x = y = z +z results in a singularity at θz cz + θρ cω =, corresponding to θx cx 3 = y = θz θρz results in a singularity at cθρω +cθω z +cθρz +cθz z =, corresponding to (ρ x x 3 =, that is, either x 3 = or x = ρ, which is the same singularity as in case o eedack linearization With the quotient method, the choice o the last unction o the dieomorphism at each step plays a crucial role in determining the singularity o the resulting control law Transorming using Φ results in: = y θ(y +ρ, = y +g (y,y z 3, where c g (y,y = ω + cy cθy cθρy The last row is removed and ẏ is expressed with y as input to give the ollowing one-dimensional system: = y θ(y +ρ ( Remark : The aility to avoid singularity is harnessed rom the act that ( is not aine in y By choosing y satisying the lemma 4 o [8], it is possile to otain an equation aine in y in lieu o equation ( However, this would result in the same singularities as in eedack linearisation The dieomorphism Φ = Φ Φ can e otained y augmenting Φ with y 3 = z 3 : y y = Φ(x,x,x 3 y 3 = θρ cω + θx + cx 3 x ρ/ x, ( with the inverse transormation: x x = Φ (y,y,y 3 x 3 = y 3 ρ +y c(cω +y θρy θy c This dieomorphism leads to the ollowing model in y- coordinates: = y θ(y +ρ, = y +g (y,y y 3, ( 3 = ay 3 +u The irst unction o Φ is always the static eedack linearizing output unction (Propositions 3 and 4 o [4] However, due to the choice o y = z, this dieomorphism has a singularity at x 3 = Hence, any gloally asymptotically stailizing (GAS control law designed or ( through the quotient method will have the ollowing domain o attraction: D QM = { (x,x,x 3 x 3 < or x 3 > } (3 The ackward stage computes a control law or (, which is given y where u = k 3 (y 3 y 3,d + y 3,d y ( y θ(y +ρ + y 3,d y ( y +g (y,y z 3 +az 3, (4 y 3,d = k (y y,d +y + y,d y ( y θ(y +ρ, g (y,y y,d = θρ+ θ ρ +4θk y θ During the control design stage, the errors e,e and e 3 are deined in order to otain y,d, y 3,d and u The dieomorphism Φ e is otained using these deinitions: e y e = Φ e (y,y,y 3 = y y,d, e 3 y 3 y 3,d which transorms the closed-loop system composed o ( and u given y (4 into the ollowing orm ė 4θe k +θ ρ ė ė 3 where = g e (e,e = k e e θ e k e g e (e,e e 3 k 3 e 3 ( cρ 4θe k +θ ρ + ce 4θe k +θ ρ + ce θρ cθρ + ce k c ω ce +ce θ /, (5
I Φ e is gloally valid and (5 is GAS, then the closedloop system is also GAS As a consequence, the domain o attraction or the original system is D QM The ollowing section proves the GAS property o (5 and presents the condition or the gloal validity o Φ e V DOMAIN OF ATTRACTION This section proves the gloal asymptotically staility (GAS o (5 The proo is divided in two parts Firstly, the GAS o a susystem otained or e 3 = is estalished using a Lyapunov unction Based on this Lyapunov unction, a new Lyapunov unction is proposed in order to prove the GAS o the ull system (5 A Susystem Let us irst consider the ollowing su-system resulting y setting e 3 = in (5: ( ( ė k e = e θ e 4θe k +θ ρ, (6 ė k e Note that the constants k and k are tunale positive gains Lemma : System (6 is gloally asymptotically stale Proo: Consider the Lyapunov unction V = e + C e, where C is a positive constant with C > θ ρ k Consider the time derivate o V, V = e ė +C e ė = k e e e θ e e 4θe k +θ ρ C k e k e e e θ + e e 4θe k +θ ρ C k e = k e e e θ + e 4θe k e +θ ρ e C k e Using Young s inequality, a ǫa + ǫ, with a = e, = 4θe k e +θ ρ e and ǫ = k gives: V k e e e ( θ k e + + 4θe k e +θ ρ e C k e = k e e e θ + (k e +e e θ + θ ρ e C k e = (C k θ ρ e 4k (7 Since C k > θ ρ, V is negative semi-deinite Moreover, or e =, one has: V = k e (8 Next, V = impliese = rom relation (7 ande = rom (8, which implies that V is negative deinite Hence, system (6 is gloally asymptotically stale Also note that e = y = θρ cω + θx + cx 3, which implies: e θρ cω From (5, it is clear that, in order to otain a real solution, one must ensure ψ 4θe k +θ ρ > Next, sustituting the lower ound on e yields an upper ound on k to enorce ψ > : θρ k < cω (9 +θρ Note that Φ e involves y,d, which in turn has ψ Hence, gloal validity o Φ e depends on ensuring positive ψ or all y By choosing k such that (9 is satisied, we will ensure gloal validity o Φ e or all y B Full system Next, consider the ull system (5 and the ollowing theorem Theorem : System (5 is gloally asymptotically stale Proo: Consider the Lyapunov unction V = log(+v + C e 3, where V = e + Ce, C and C are positive constants such that and L to L 8 are deined as C > θ ρ + k k C > 8 L i, k 3 i= L = max cρ 4θe k +θ ρ (+e +e 4 cθρ L = in 4 (+e +e = ce 4θe k +θ L 3 = max ρ (+e +e L 4 = max ce k (+e +e = ck 4 L 5 = max ce θρ (+e +e = cθρ 4 c ω L 6 = in (+e +e = L 7 = max ce (+e +e = c ce L 8 = sup θ (+e +e = cθ,
Comparing (5 and (6 and sustituting rom (7 gives: V (C k θ ρ e e g e (e,e e 3 Next, consider the time derivative o V, V = and sustituting V in V gives: (C k θ ρ e +e +e (C k θ ρ e +e +e C k 3 e 3 V +V C k 3 e 3, e g e (e,e e 3 +e +e + e g e (e,e e 3 +e +e C k 3 e 3 Using Young s inequality with a = e, = g e (e,e e 3 and ǫ = allows writing: (C k θ ρ e e +e + + ge(e,e e 3 +e +e +e C k 3 e 3 Finally, sustituting or g e (e,e gives: (C k θ ρ e (+e +e + (cρ 4θe k +θ ρ cθρ e 3 (+e +e + (ce 4θe k +θ ρ +ce k +ce θρ e 3 (+e +e + ( c ω ce +ce θ (+e +e C k 3 e 3 Using the deinition o L to L 8, V ecomes: (C k θ ρ e (+e +e +L e 3 +L e 3 = +L 3 e 3 +L 4e 3 +L 5e 3 +L 6e 3 +L 7e 3 +L 8 e 3 C k 3 e 3 (C k θ ρ (+e +e e ( C k 3 8 L i e 3 i= Since the explicit expressions o L and L 3 are involved, there are not presented here Nevertheless, L and L 3 exist since oth expressions are continuous unctions that are dierent rom zero somewhere and cρ 4θe k +θ lim ρ (e,e (+e =, +e 4 ce 4θe k +θ lim ρ (e,e (+e +e = a c θ ρ ω 3995 35434 45 3769 57588 TABLE I PARAMETER VALUES FOR THE SIMULATED DC MOTOR Hence, L and L 3 must exist somewhere on the (e,e - plane Since C and C satisies C k > θ ρ C k 3 > + 8 L i, i= negative deinitiveness o V is estalished Hence, the system (5 is gloally asymptotically stale Also, k can e chosen according to (9 to ensure that, despite having a square root in the equation, Φ e is gloally valid and the resulting control input is always real GAS property or (5 implies that the closed-loop system is GAS with the equilirium point: y y y 3 =, which implies: θρ cω + θx + cx 3 x ρ/ = x and x =, x = ρ/ x 3 = ω Hence, the closed-loop system will converge to (x, x, x 3 = (, ρ/, ±ω Since x 3 = is a singularity, the convergence to ±ω depends on the initial condition I x 3 t= >, then x 3 converges to +ω, else i x 3 t= <, x 3 converges to ω The domain o attraction o the control law designed using the quotient method is D QM deined in (3 Furthermore, ased on (6, it is clear that D FBL D QM, and thus D QM is larger than D FBL Hence, using the quotient algorithm, we can initialise the system also rom the points x t= ρ in addition to the points in D FBL This act is clearly seen in the simulation results presented in Figure The target in these simulations is to achieve ω = The simulations are carried out using the DC-motor parameters taken rom [3] and given in Tale I Three initial conditions are chosen to simulate the control law otained using the quotient method The irst (, ρ/(, ; FBL singularity and the second (,, ; FBL impossile initial conditions are outside the domain o attraction o any control law designed using eedack linearization [] due to the presence o singularity at x t= = ρ/ Only the last initial condition (,, ; general lies in D FBL The control law designed using eedack linearization works only or x > ρ/, whereas the control law designed using the
x = i (amps 5 5 3 x = i a (amps x 3 = ω (rad/sec (a x 4 8 Time (sec FBL singularity FBL impossile general FBL singularity FBL impossile general 4 8 Time (sec ( x FBL singularity FBL impossile general 4 8 Time (sec (c x 3 Fig DC motor controlled with the quotient method The state ehavior using the quotient method is depicted or three dierent initial conditions The thin line represents a general case The thick line corresponds to an initial condition that is outside D FBL The dashed line corresponds to a case that is exactly the point o singularity or eedack linearization stems rom the act that there is an additional degree o reedom (in particular the choice o the last unction in the deinition o the dieomorphism at every step o the orward decomposition stage This choice plays a crucial role in determining the singularity o the resulting control law Dierent choices result in dierent domains o attraction or the resulting control law For a particular choice, a Lyapunovased proo o the domain o attraction has een provided By removing singularities, the domain o attraction o the control law designed through the quotient method is shown to e larger than the domain o attraction o the control law designed through eedack linearization To sustantiate the results, simulations are provided with initial conditions that are outside the domain o attraction o any controller designed using eedack linearization Acknowledgments: Support y the Swiss National Science Foundation under Grant FN -696/ is grateully acknowledged REFERENCES [] B Jakuczyk and W Respondek On linearization o control systems Bulletin de l Académie Polonaise des Sciences, 8(9-:57 5, 979 [] P Brunovský A classiication o linear controllale systems Kyernetika, 6(3:73 88, 97 [3] A Isidori Nonlinear Control Systems Springer Verlag, Berlin, Heidelerg, New York, nd edition, 989 [4] Ph Mullhaupt Quotient sumaniolds or static eedack linearization Systems & Control Letters, 55:549 557, 6 [5] I A Tall State and eedack linearization o single-input control systems Systems & Control Letters, 59:49 44, [6] R B Gardner and W F Shadwick The GS algorithm or exact linearization to Brunovsky normal orm IEEE Trans on Automatic Control, 37:4 3, 99 [7] B Gra and Ph Müllhaupt Feedack linearizaility and latness in restricted control system Proc o the 8th IFAC Congress, [8] S S Willson, Ph Müllhaupt, and D Bonvin Quotient method or designing nonlinear control law 5th IEEE Con on Decision and Control, [9] S S Willson, Ph Mullhaupt, and D Bonvin Quotient method or controlling the acroot 48th IEEE Con on Decision and Control, 9 [] D Ingram, S S Willson, Ph Mullhaupt, and D Bonvin Stailization o the cart-pendulum system through approximate maniold decomposition 8th IFAC World Congress, [] H K Khalil Nonlinear Systems Prentice-Hall, Englewood Clis, NJ, 3rd edition, [] M Bodson and J Chiasson Dierential-geometric methods or control o electric motors International Journal o Roust and Nonlinear control, 8:93 954, 998 [3] Yung-chii Liang and V J Gosell DC machine models or spice simulation IEEE Trans on Power Electronics, 5(:6, 99 quotient method does not have this restriction Hence, upon using the quotient method, a larger domain o attraction is achieved VI CONCLUSION This paper has illustrated the application o the quotient method to control a ield-controlled DC motor For this system, singularity arises in eedack linearization due to the nature o the dieomorphism used to otain the Brunovský normal orm The quotient method relaxes this condition y not requiring the Brunovský normal orm The advantage