The Power-Oriented Graphs Modeling Technique

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Georgia Gilmore
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1 Capitolo 0. INTRODUCTION 3. The Power-Oriented Graph Modeling Technique Complex phical tem can alwa be decompoed in baic phical element which interact with each other b mean of energetic port, and power flow. Example of elementar phical tem:

2 Capitolo 3. DYNAMIC MODELING 3.2 The Power-Oriented Graph (POG): i a graphical modeling technique that ue an energetic approach for modeling phical tem. ue the power and energ variable a baic concept for modeling phical tem. the POG block cheme are ea to ue, ea to undertand and can be directl implemented in Simulink. i baed on the ame energetic concept of the Bond Graph modeling technique. See: Karnopp, Margoli, Roemberg, Stem Dnamic - A unified approach, John Wile & Son. Example. A DC electric motor move an hdraulic pump. The phical tem and the correponding POG block cheme: Accumulator Ia L a R a DC Motor V V a V r E m ω m Q u 8 Q α 9 α p Pump 6 7 Tank Filter V a L a φ φ Motor inductance V r E m Motor reitance 2 R a 3 K m C m K m Energ converion EL-MR 4 ṗ p ω m Motor inertia 5 Motor friction C p 6 K p K p Energ converion MR-ID 7 α p Q α V Q u Hdraulic accumulator Hdraulic leak 8 C 0 V 9

3 Capitolo 3. DYNAMIC MODELING 3.3 The energetic approach i ueful for modeling becaue the phical tem are alwa characterized b the following propertie: ) a phical tem tore and/or diipate energ ; 2) the dnamic model of a phical tem decribe how the energ move within the tem, 3) the energ move from point to point within the tem onl b mean of two power variable. Power ection. The dahed line of the POG cheme repreent the power ection of the tem. The inner product x, = x T of the two power variable x and touched b the dahed line ha the phical meaning of power flowing through the ection. POG block. The POG technique ue two block for modeling phical tem: the Elaboration block and the Connection block. Power ection K m K p L a R a α p C 0 V a K m ω m K p Elaboration block Connection block

4 Capitolo 3. DYNAMIC MODELING 3.4 The Elaboration block i ued for modeling the phical element that tore and/or diipate energ (i.e. pring, mae, damper, capacitie, inductance, reitance, etc.). x x 2 G() Equivalent wa of repreenting the elaboration block:... x x 2 x x 2 G() G() G() G() G() x x 2 x x 2 x x 2 Input path inverion Output path inverion Upide down Input path inverion + upide down Output path inverion + upide down The black pot within the ummation element repreent, when it i preent, a minu ign that multiplie the entering variable. Rule for inverting a path: ) Invert each line of the path; x x 2 x x 2 2) Invert each block of the path; 3) In the ummation block invert the ign of the variable which belong to the path; G() G ()

ummation block are all witched to the oppoite value.

5 Capitolo 3. DYNAMIC MODELING 3.5 Example of path inverion: a) Path to be inverted b) Inverted path c) Equivalent cheme p ω m The POG block cheme doe not change i the ign of a ummation block are all witched to the oppoite value. The Connection block i ued for modeling the phical element that tranform the power without loe (i.e. neutral element uch a gear reduction, tranformer, etc.). x K x 2 K T 2 Equivalent wa of repreenting the connection block: x K x 2 K T 2 K T 2 K T 2 x K x 2 x K x 2 Inverted path Upide down Matrix K can alo be rectangular or time varing. Inverted path + upide down

6 Capitolo 3. DYNAMIC MODELING 3.6 The main Energetic domain encountered in modeling phical tem are: electrical, mechanical(tranlational and rotational) and hdraulic. Each energetic domain i characterized b two power variable. POG variable Electrical Mech. Tra. Mech. Rot. Hdraulic Acro-var.: V Voltage ẋ Velocit ω Angular vel. P Preure Through-var.: I Current F Force τ Torque Q Volume flow rate In each dahed line of the POG cheme the product P = of two power variable and ha the phical meaning of power P flowing through that particular power ection. K m C m C r C p K p L a R a Energ converion EL-MR Energ converion MR-ID α p C 0 V a V r E m K m ω m K p Q p Q u V a V r E m ω m C m ω m C r ω m C p Q p Q u }{{}}{{}}{{} Electric power flow Mechanical power flow Hdraulic power flow }{{}}{{} E m = ω m C r ω m C p = Q p The connection block convert the power without generating nor diipating energ. The input power flow x T i alwa equal to the output power flow x T 2 2 : x K x 2 x T =< x T, >=< x T,K T 2 > =< (Kx ) T, 2 >=< x T 2, 2 > =x T 2 2 K T 2

7 Capitolo 3. DYNAMIC MODELING 3.7 The power variable can be divided in two group: ) the acro-variable (voltage V, velocit ẋ, angular velocit ω and preure P) which are defined between two point of the pace: V ẋ ω P 2) through-variable (current I, force F, torque τ and volume flow rate Q) which are defined in each point of the pace: I F τ Q Dnamic tructure of the energetic domain. Each domain i characterized b onl 3 different tpe of phical element: 2 dnamic element D e and D f which tore the energ (i.e. capacitor, inductor, mae, pring, etc.); tatic element R which diipate (or generate) the energ (i.e. reitor, friction, etc.); The dnamic of phical tem can be decribed uing 4 variable: 2 energ variable q e and q f which define how much energ i tored within the dnamic element; 2 power variable and whichdecribehowtheenergmove within the tem.

8 Capitolo 3. DYNAMIC MODELING 3.8 Dnamic tructure of the energetic domain: Electrical Mech. Tra. Mech. Rot. Hdraulic D e C Capacitor M Ma J Inertia C I Hd. Capacitor q e Q Charge p Momentum p Ang. Momentum V Volume V Voltage ẋ Velocit ω Ang. Velocit P Preure D f L Inductor E Spring E Spring L I Hd. Inductor q f φ Flux x Diplacement θ Ang. Diplacement φ I Hd. Flux I Current F Force τ Torque Q Volume flow rate R R Reitor b Friction b Ang. Friction R I Hd. Reitor Graphical repreentation of the phical element: Electrical Mech. Tra. Mech. Rot. Hdraulic Capacitor Ma Inertia Hd. Capacitor I F τ Q Element De V C M ω J P C i Inductor Spring Rot. Spring Hd. Inductor I F τ Q Element Df V L E ω E t P L i Reitor Friction Ang. Friction Hd. Reitor I F τ Q Element R V R d ω d t P R i

9 Capitolo 3. DYNAMIC MODELING 3.9 The dnamic element D e i characterized b: ) an internal energ variable q e (t); 2) a through-variable (t) a input variable; 3) an acro-variable (t) a output variable; 4) a contitutive relation q e = Φ e ( ) which link the internal energ variable q e (t) to the output power variable (t); 5) a differential equation q e (t) = (t) (t) which link the internal energ variable Element D e q e (t) to the input power variable (t); 6) the energ E e tored in the dnamic element D e i function onl of the internal energ variable q e : E e = t 0 (t) (t)dt = qe where the following ubtitution have been ued: (t) = Φ e (q e ) Dnamic orientation and tored energ: 0 Φ e (q e ) (t) Φ e (q e )dq e = E e (q e ). dq e = (t)dt q e (t) ) Integral 2) Derivative Stored energ E e Φ e (q e ) q e Φ e ( ) q e E e E e q e Φ e (q e ) TO BE USED DO NOT USE

10 Capitolo 3. DYNAMIC MODELING 3.0 The dnamic element D f ha a tructure which i dual repect to the tructure of dnamic element D e. ) an internal energ variable q f (t); 2) an acro-variable (t) a input variable; 3) athrough-variable (t)aoutputvariable; 4) a contitutive relation q f = Φ f ( ) which link the internal energ variable q f (t) to the output power variable (t); 5) a differential equation q f (t) = (t) which link the internal energ variable q f (t) to the input power variable (t); (t) Φ f (q f) (t) q f (t) Element D f The dual tructure can be eail obtained performing the following ubtitution: q e (t) q f (t), (t) (t) and Φ e ( ) Φ f ( ). Dnamic orientation and tored energ: ) Integral 2) Derivative Stored energ E e q f q f E e Φ f (q f) Φ f (q f) Φ f ( ) E e q f TO BE USED DO NOT USE Note: the energ variable q e and q f are the integral of the input power variable (t) and (t): q e = t 0 (t)dt, q f = t 0 (t)dt.

11 Capitolo 3. DYNAMIC MODELING 3. Thetatic element Ricompletelcharacterizedbataticfunction = Φ R ( ) which link the input variable to the output variable. ) Reitance 2) Conductance Φ R ( ) Φ R () Diipated power P d of the tatic element R: P d Φ R ( ) P d The differential equation of a phical element can be obtained impoing the time-derivative of the energ variable equal to the input power variable: ) For D f element: q f (t) = (t) d q f(t) dt 2) For D e element: q e (t) = (t) d q e(t) dt = (t) = (t)

12 Capitolo 3. DYNAMIC MODELING 3.2 Electromagnetic domain: Name Contitutive Rel. Linear cae Differentioal Eq. D e C Capacitor q e Q Charge V Voltage Q = Φ C (V) Q = CV dq dt = I D f L Inductor q f φ Flux φ = Φ L (I) φ = LI I Current dφ dt = V R R Reitance V = Φ R (I) V = RI Mechanic Tranlational domain: Name Contitutive Rel. Linear cae Differentioal Eq. D e M Ma q e P Momentum ẋ Velocit P = Φ M (ẋ) P = M ẋ dp dt = F D f E String q f x Diplacement x = Φ E (F) x = EF F Force dx dt = ẋ R b Friction F = Φ b (ẋ) F = bẋ

13 Capitolo 3. DYNAMIC MODELING 3.3 Mechanic Rotational domain: Name D e J Inertia q e P Ang. Momentum ω Ang. Velocit Contitutive Rel. Linear cae Differentioal Eq. P = Φ J (ω) P = J ω dp dt = τ D f E Rot. Spring q f θ Ang. Diplacement θ = Φ E (τ) τ Torque θ = Eτ dθ dt = ω R b Rot. Friction τ = Φ b (ω) τ = bω Hdraulic domain: Name D e C I Hd. Capacitor q e V Volume P Preure Contitutive Rel. Linear cae Differentioal Eq. V = Φ C (P) V = C I P dv dt = Q D f L I Hd. Inductor q f φ I Hd. Flux φ I = Φ L (Q) φ I = L I Q dφ I dt Q Volume flow rate = P R R Hd. Reitor P = Φ R (Q) P = R I Q

14 Capitolo 3. DYNAMIC MODELING 3.4 Connection of phical element Phical Element. The phical tem are compoed b phical element (PE) (i.e. dnamic element D e and D f or tatic element R) which interact with the external world b mean of two terminal: P a) b) PE PE Each terminal, ee cae a), i characterized b two power variable (, ) and (2, 2 ). Chooing = 2 and = = 2 a new power variable, the power interaction of the PE with the external world can be decribed uing the power ection P in cae b). The value of the power P flowing through the ection i the product of the two power variable (t) and (t): P(t) = (t) (t) The ign and the direction of power P(t) depend on the ign and the reference poitive direction choen for the variable (t) and (t). The ign of the power P flowing through a phical ection A-B are: a) Power P flow from A to B b) Power P flow from B to A Power flow P PE P PE P PE P PE A-B A-B A-B A-B

15 Capitolo 3. DYNAMIC MODELING 3.5 Integral and derivative caualit. The POG dnamic model of a phicalelement(pe),thatianelementd e,d f orr,canbegraphicall decribed b uing two block cheme having different orientation: Model Model P P P PE Φ( ) Φ( ) 2 Phical Element The two poible orientation of the PE dnamic model are: ) a input and a output: model 2) a input and a output: model. The function Φ( ) hown in the figure mbolicall repreent the dnamic or the tatic equation decribing the phical element. If PE i a tatic element R, the two diagram are both uitable for decribing the mathematical model of the phical element. If PE i a dnamic element D e or D f, the two diagram repreent the two poible caualit mode of the phical element: ) the integral caualit (TO BE USED) i phicall realizable, ueful in imulation and i the preferred dnamic model in the POG technique. 2) the derivative caualit (DO NOT USE) i till a correct mathematical model of the PE, but it i not ued in the POG technique becaue it i not phicall realizable and it i not ueful in imulation.

16 Capitolo 3. DYNAMIC MODELING 3.6 Each Phical Element (PE) interact with the external world through the power ection aociated to it terminal. Two baic connection are poible: erie and parallel. Serie: a Phical Element PE i connected in erie if it terminal hare the ame through-variable = = 2 : P P 2 P VKL P 2 2 PE Kirchhoff voltage law VKL 0 2 f e ( ) 2 Phical Element connected in erie ) POG cheme with output The ummation element i a mathematical decription of the Voltage Kirchhoff Law (VKL) applied to a cloed path which involve the acro variable, 2 and Inverting the input and output path of the POG block cheme ) one obtain the following equivalent POG block cheme: P VKL P 2 2 P VKL P 2 2 f e ( ) f e ( ) 2 2 2) POG cheme with output. 3) POG cheme with output 2.

17 Capitolo 3. DYNAMIC MODELING 3.7 Parallel: a Phical Element PE i connected in parallel if it terminal hare the ame acro-variable = = 2 : P P 2 Kirchhoff current law CKL 2 P CKL P 2 2 PE 2 f f ( ) 2 Phical Element connected in parallel 2 ) POG cheme with output. The ummation element i a mathematical decription of the Current Kirchhoff Law (CKL) applied to a node which involve the through variable, 2 and. Inverting the input and output path of the POG block cheme ) one obtain the following equivalent POG block cheme: P CKL P 2 2 P CKL P 2 2 f f () f f () 2 2 2) POG cheme with output. 3) POG cheme with output 2.

18 Capitolo 3. DYNAMIC MODELING 3.8 Connecting phical element Two phical element PE and PE 2 can be connected a follow: Phical connection Baic POG cheme a) Serie - Serie 3 PE PE 2 4 P VKL P 3 3 VKL2 v a P 4 4 VKL 3 VKL2 4 f e ( ) f f ( ) 2 = = 3 b) Serie - Parallel PE v CKL f3 4 P VKL P 3 3 v a P VKL 2 = 3 PE 2 4 f e ( ) f f (v) v 4 3 CKL2 c) Parallel - Serie CKL 3 PE 2 4 P P 3 3 VKL2 v a P 4 4 PE 3 VKL2 2 = 3 4 f f ( ) f e (v a ) v 4 3 CKL d) Parallel - Parallel CKL v CKL2 f3 4 P P 3 3 v a P 4 4 PE 3 PE 2 4 f f ( ) f e (v a ) 2 = 3 v 4 3 CKL CKL2

19 Capitolo 3. DYNAMIC MODELING 3.9 The baic POG cheme aociated to a PE -PE 2 connection can be drawn in four different wa. For the Serie - Serie connection: a) Serie - Serie PE 3 PE VKL VKL2 2 = 3 the following POG cheme can be ued: Baic POG cheme Inverion of the internal loop P P 3 P 4 P P 3 P 4 3 v a 4 3 v a 4 f e ( ) f f ( ) f e ( ) f f (v a ) 4 2 = 3 Inverion of the input path 4 2 = 3 Inverion of the output path P P 3 P 4 P P 3 P 4 3 v a 4 3 v a 4 f e ( ) f f ( ) f e ( ) f f ( ) = 3 2 = 3 Other four poible POG block cheme can be obtained conidering the upide down verion of the above reported POG cheme.

20 Capitolo 3. DYNAMIC MODELING 3.20 Example. A C-parallel and R-erie connection: Kirchhoff current law I R I 2 I 2 I 3 C V 4 V V 2 Kirchhoff voltage law I V I 3 C R V 4 I 2 V The internal loop and the input path of the POG cheme CANNOT be inverted becaue the capacitor mut be decribed uing it integral caualit model. The output path can be inverted. Example. A R -parallel and R 2 -erie connection: R I 2 I 2 I I 2 V R V 2 R V The following equivalent block cheme can be ued: R 2 3 V 2 Inverion of the internal loop Inverion of the input path Inverion of the output path I V R R 2 I 2 V 2 I V R R 2 I 2 V 2 I V R R 2 I 2 V 2

21 Capitolo 3. DYNAMIC MODELING 3.2 Example. A C -parallel and C 2 -erie connection: I C 2 I c I I 2 V C V 2 V c C C In thi cae there i onl one POG block cheme that can be ued for decribing the given phical tem. V 2 3 V 2 Often, when two phical element are connected, a feedback loop appear in the correponding POG block cheme. The following propert hold. Propert. All the loop of a POG cheme contain an odd number of minu ign(i.e. black pot in the ummation element). V a Example: L a φ φ R a E m K m C m K m ṗ p ω m C p K p K p α p C 0 V Q α V Q u All the five loop of thi block cheme contain one minu ign. Thi rule can be ued to verif the conitenc and the correctne of the conidered POG block cheme.

22 Capitolo 3. DYNAMIC MODELING 3.22 Example of a Parallel - Parallel connection. Baic POG cheme Reduced POG cheme f f 3 F f 5 f 4 G f 2 f F+G f 2 e e 3 e 5 e 4 e 2 e e 3 e 2 Electrical Mech. Tralational I I 2 F F 2 I 3 I 4 F 3 F C M 4 V V 3 V 4 R V 2 v v 3 v 4 d v 2 f = I, F = C, e = V, G = R f = F, F = M, e = v, G = d Mech. Rotational Hdraulic τ τ 2 Q Q 2 τ 3 τ 4 Q J 3 C i Q 4 w w 3 w 4 d t w 2 P P 3 P 4 R i P 2 f = τ, F = J, e = w, G = d t f = Q, F = C i, e = P, G = R i The reduced model i obtained uing the Maon formula: G 0 () = F + G = F+G F

23 Capitolo 3. DYNAMIC MODELING 3.23 Example of a Serie - Serie connection. Baic POG cheme Reduced POG cheme e e 3 F e 5 e 4 R e 2 e F+R e 2 f f 3 f 5 f 4 f 2 f f 3 f 2 Electrical I L R I 2 I 3 I 4 V 3 V 4 V V 2 Mech. Tralational F E d F 2 F 3 F v 4 3 v 4 v v 2 e = V, F = L, f = I, R = R Mech. Rotational e = v, F = E, f = F, R = d Hdraulic K t d t τ τ 2 τ 3 τ 4 w 3 w 4 w w 2 Q L i R i Q 2 Q 3 P Q 4 3 P 4 P P 2 e = w, F = E t, f = τ, R = d t e = P, F = L i, f = Q, R = R i

24 Capitolo 3. DYNAMIC MODELING 3.24 Propert. The direction of the power P flowing through a ection of apogblockchemei poitive ifan even number of minu ign i preent along the path which goe from the input to the output. V a L a φ φ R a E m K m C m K m ṗ p ω m } {{ } P C p K PA p K p B α p Q α C 0 V V Q u }{{} P 2 Let u conider, for example, the power ection A-B which divide the block cheme in two ection: P and P 2. The power P flow from ection P to ection P 2 becaue: the red dahed paththatgoe frombtoawithin ection P 2 contain zero minu ign (i.e. an even number); thebluedahedpaththatgoefromatobwithinectionp contain one minu ign (i.e. an odd number); Uing the previou rule it i poible to compute the poitive direction of the poter flow in each power ection of a POG block cheme: V a L a φ φ R a K m K m ṗ p ω m K p K p α p C 0 V Q α V Q u

25 Capitolo 3. DYNAMIC MODELING 3.25 The ign of the power flow depend on the ign of the power variable. a) Changing the ign of variable V r and Q u one obtain: V a L a φ φ V r R a E m K m C m K m ṗ p ω m C p K p K p Q p α p Q α Q u C 0 V V b) Changing the ign of variable E m, ω m and Q p one obtain: V a L a φ φ V r R a E m K m C m K m ṗ p ω m C p K p K p Q p α p Q α Q u C 0 V V c) Changing the ign of variable C p and one obtain: V a L a φ φ V r R a K m C m ṗ p K m E m ω m C p K p K p α p Q p Q α Q u C 0 V V

26 Capitolo 3. DYNAMIC MODELING 3.26 From POG cheme to State pace model From a POG block cheme: V a L a φ φ R a K m K m ṗ p ω m K p K p α p C 0 V V one can directl obtain the correponding POG tate pace model: L a 0 0 R a K m ω m = K m K p ω m C 0 0 K p α p }{{}}{{}}{{} 0 ẋ A }{{}}{{} B L x [ ] [ ] = x+ u }{{}}{{} C D The following procedure mut be ued: [ Va ] }{{} u ) The component of the tate vector x mut be choen equal to the output power variable of the dnamic element: x = [ ω m V 0 ] T 2) ThecoefficientL a, andc 0 ofthediagonalmatrixl(i.e. theenerg matrix) are the coefficient that link the output power variable to the internal energ variable within the contitutive relation: φ = L a p = ω m L = V = C 0 L a C 0

27 Capitolo 3. DYNAMIC MODELING ) The coefficient A ij of matrix A are the gain of all the path that link the j-th tate variable x j to the i-th input q i of the integrator. V a L a q q R a K m q 2 K m q 2 ω m K p K p α p C 0 q 3 q 3 L a 0 0 R a K m ω m = K m K p ω m C 0 P 0 }{{}}{{} 0 K p α 0 p }{{}}{{} L ẋ }{{} x B [ A ] [ ] = x+ u }{{}}{{} C D [ Va ] }{{} u The element A 23 = K p of matrix A, for example, i the gain of the path that link the third tate variable x 3 = to the input q 2 = ṗ of the econd integrator. 4) The coefficient B ij, C ij and D ij of matrice B, C and D, repectivel, can be determined in a imilar wa. Coefficient B ij are the gain of the path that link the j-th input u j to the i-th input q i of the integrator. The coefficient B 32 =, for example, i the gain of the path that goe from input u 2 = to the input q 3 of the third integrator. Coefficient C ij are the gain of the path that link the j-th tate variable x j to the i-th output i of the tem; Coefficient D ij are the gain of the path that link the j-th input u j to the i-th output i of the tem;

28 Capitolo 3. DYNAMIC MODELING 3.28 Propertie of a linear POG tate pace model ) The energ matrix L i alwa mmetric and poitive definite: L a 0 0 L = L T > 0 L = C 0 2) The energ E tored in the tem can be expreed a follow: E = 2 xt Lx 0 E = 2 L ai 2 a + 2 ω 2 m + 2 C 0P 2 0, Matrix L i characterized b the coefficient of the contitutive relation of the dnamic element of the tem. 3) Thepower P d diipated in the temcanbeexpreedafollow: P d = x T A x P d = R a I 2 a ω 2 m α p P 2 0 (if L i contant) where A i the mmetric part of the power matrix A. R A = (A+AT ) a 0 0 = α p MatrixA icharacterizedballthediipativecoefficientofthetatic element preent in the tem. 4) The power P w reditributed within the tem i zero: P w = x T A w x = 0 where A w i the kew-mmetric part the power matrix A: 0 K A w = (A AT ) m 0 = K m 0 K p 2 0 K p 0 Matrix A w i characterized b all the coefficient of the connecting block preent in the tem.

29 Capitolo 3. DYNAMIC MODELING 3.29 Definition. A Linear Time-invariant Power-Oriented Graph dnamic tem S i characterized b a tate pace differential equation having the following tructure: { Lẋ = Ax+Bu S= = Cx+Du where L i the energ matrix, A i the power matrix, B i the input power matrix, C i the output matrix and D i the inputoutput matrix. Moreover, a POG dnamic tem atifie the following propertie: a) matrix L = L T 0 i a mmetric emidefinite matrix; b) the energ E tored in the tem can be expreed a follow: E = 2 xt Lx 0 c) the power P d diipated in the tem can be expreed a follow: P d = x T Ax. Tranformed POG tem. A POG dnamic tem S can be tranformed uing a congruent tranformation x = T z: { { Lẋ = Ax+Bu x=tz Lż = Az+Bu S= = S= = Cx+Du = Cz+Du where S i the tranformed tem and (if matrix T i contant) L = T T LT, A = T T AT, B = T T B, C = CT The tranformed POG tem S maintain the ame propertie of the original POG tem S. A congruent tranformationx = Tzdoenotrequirethecalculation of the the invere of matrix T. It can be applied alo when T i ingular or rectangular.

30 Capitolo 3. DYNAMIC MODELING 3.30 Model reduction of a POG block cheme When a phical parameter of the tem tend to zero (or to infinit) the POG tem degenerate toward a lower dimenion dnamic tem. The POG dnamic model can be reduced graphicall or analticall Graphical model reduction. Let u conider the following POG cheme: V a L a φ φ R a K m K m 4 ṗ p ω m 5 }{{} ( ) 6 K p K p α p C 0 V V When = 0 the above POG block cheme cannot be ued becaue the term preent in the block cheme i infinite. In thi cae the central part ( ) of the block cheme can be graphicall tranformed a follow: a) Loop top be inverted b) Inverted loop c) Simplified cheme p 4 ω m

31 Capitolo 3. DYNAMIC MODELING 3.3 The implified and tranformed tem ha now the following tructure: K m K p L a φ R a α p C 0 V V a φ K m K p V The correponding POG tate pace model i: [ ] [ ] La 0 Ia 0 C 0 P = R a K2 m bm 0 4 K m K p 6 K m K p α p K2 p [ Ia ] [ ][ Va Analtical model reduction. When = 0, the tate pace dnamic model of the tem can be rewritten a follow: L a 0 0 R a K m 0 0 [ ] ω m = K m K p ω m Va 0 0 C 0 P Q 0 0 K p α p 0 0 The econd equation i an algebraic contraint between the tate variable: K m ω m K p = 0 The angular velocit ω m can be expreed a follow: ] ( ) ω m = K m K p Appling the following congruent tate pace tranformation: 0 [ ] ω m K = m bm K p Ia, x = Tz P 0 0 }{{}}{{}}{{} z x T to the given tem one directl obtain the reduced tem (*).

Equivalent POG block schemes

Equivalent POG block schemes apitolo. NTRODUTON 3. Equivalent OG block cheme et u conider the following inductor connected in erie: 2 Three mathematically equivalent OG block cheme can be ued: a) nitial condition φ φ b) nitial condition