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 c) Zero initial condition. or linear ytem it i alway poible to witch the poition of two cacade connected linear block without modifying the input-output dynamic behavior of the conidered ytem. f the three block cheme were implemented in Simulink, their initial condition would be φ = φ, = and =, repectively. Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.2 onnecting OG block cheme et u conider the following two electric circuit and OG cheme: A B b R A B φ R A A B B b φ R b Two OG block cheme can be directly connected only if: ) the two power ection are oriented in the ame way ; 2) the two power ection hare the ame poitive power flow. Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.3 The following two OG cheme ANNOT be directly connected becaue they are NOT oriented in the ame way : φ R A A B B b The following two OG cheme ANNOT be directly connected becaue they do NOT hare the ame poitive power flow : a b (a) φ R A A (A) B B (B) The two above OG cheme AN be directly connected along ection (a)-(b) becaue they hare the ame orientation and poitive power flow : (b) b a = b B B (B) (a)=(b) φ R A A (A) Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.4 Algebraic loop et u conider the following electric circuit: b r R r2 n the correponding OG block cheme i preent an algebraic loop : φ r R R r2 The correponding OG tate pace model can till be determined: [ ] [ ] R R [ ] [ ][ ] R = 2 a + ( ) }{{} Ã where i the determinant of the OG cheme without the dynamic block: = + R. ThecoefficientA ij ofmatricaithe tranfergain (computeduingthe Maon formula ) of the path that link the j-th tate variable x j with the input of the i-th integrator. Zanai Roberto - Sytem Theory. A.A. 25/26 b b
apitolo 3. DYNAM MODENG 3.5 The algebraic loop can alo be eliminated graphically a follow: φ r R r2 b r φ R R R r2 b The ame reult can alo be obtained uing the following mathematical approach. Add a puriou phyical element to the original ytem b r R uch to eliminate the algebraic loop in the correponding OG cheme: φ R r r2 Zanai Roberto - Sytem Theory. A.A. 25/26 b
apitolo 3. DYNAM MODENG 3.6 The correponding OG tate pace model can now be written eaily: ] = R + [ a b When = one obtain the following contraint: ) + ( R R2 + = which can be rewritten a follow: = R R + + R R + et u conider the following congruent tate pace tranformation: R x = Tx where T = R R + R + The tranformed and reduced matrice have the following tructure: [ ] [ ] x=tx=, =T T T= [ ] R A = T T R AT = + R [ ] R +, B=T T B=. R R + R + One can eaily verify that the obtained reduced ytem i equal to the ytem ( ) obtained applying the Maon formula to olve the algebraic loop. One can eaily verify that the two matrice A and à are equal: [ ] R R A= + R R + = R R [ + R R2 (+ R R R2 ) = ] R =à R R + R + R (+ R R2 ) (+ R R2 ) R Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.7 Example. Electric circuit. et u conider the following electric circuit: K K2 R b K3 2 2 3 4 R a K K2 K3 3 4 b The correponding.o.g. block cheme i: K φ R a K K2 φ 2 2 2 3 3 Q 3 K2 K3 R b 4 4 Q 4 The OG tate pace dynamic model ha the following tructure: R a R a 2 R 2 a R a [ ] 2 = a 3 3 + R b R b 3 b }{{} 4 4 }{{}}{{} R b R 4 u b }{{}}{{} ẋ }{{} x B A The connection coefficient appear within matrix A only when one type of energy (i.e. magnetic energy in ) convert to another different type of energy (i.e. electrotatic energy in ). A diipative parameter (i.e. R a and R b ) appear 4 time in a ymmetric way within matrix A when the correponding phyical element connect two dynamic element of the ame type. Zanai Roberto - Sytem Theory. A.A. 25/26 K3 b
apitolo 3. DYNAM MODENG 3.8 Example. A chain of carriage and nubber: The ymbolic cheme: b b 2 b 3 K K 3 K 3 M M 2 M 3 ẋ ẋ 2 ẋ 3 The correponding OG block cheme: R 2 R 3 3 M K b M 2 K 2 b 2 M 3 K 3 b 3 ẋ ẋ 2 The OG tate pace dynamic model i: M K M2 K 2M3 K 3 ẍ Ṙ ẍ 2 Ṙ2 ẍ 3 Ṙ3 = ẋ 3 b b b b b 2 b 2 b 2 b 2 b 3 ẋ R ẋ 2 ẋ 3 R 3 + Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.9 The acro dynamic element D e connected in parallel can be graphically repreented only by the following particular OG cheme: Element De Electrical Mech. Tra. Mech. Rot. Hydraulic apacitor Ma nertia Hyd. apacitor 2 2 τ τ 2 M J ẋ ẋ ω ω Q Q 2 i OG cheme Q 2 ẋ p M 2 ẋ τ ω p r J τ 2 ω Q i Q 2 The through dynamic element D f connected in erie can be graphically repreented by only one particular OG cheme: Element Df Electrical Mech. Tra. Mech. Rot. Hydraulic nductor Spring Rot. Spring Hyd. nductor E τ τ Q ẋ ẋ 2 ω ω 2 2 E r i Q OG cheme φ ẋ x E ẋ 2 ω τ θ E r ω 2 τ Q φ i i 2 Q Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3. The acro dynamic element D e connected in erie can be graphically repreented by the following two different OG cheme: Element De Electrical Mech. Tra. Mech. Rot. Hydraulic apacitor Ma nertia Hyd. apacitor M τ J τ Q ẋ ẋ 2 ω ω 2 2 i Q OG cheme Q ẋ M p ẋ 2 ω τ J p r ω 2 τ Q i 2 Q OG cheme 2 Q ẋ M p ẋ 2 ω τ J p r ω 2 τ Q i 2 Q The two OG cheme have oppoite input/output power variable. The choice between the two OG block cheme depend on Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3. The through dynamic element D f connected in parallel can be graphically repreented by the following two different OG cheme: Element Df Electrical Mech. Tra. Mech. Rot. Hydraulic nductor Spring Rot. Spring Hyd. nductor 2 2 τ τ 2 Q Q 2 ẋ E ẋ ω E r ω i OG cheme Q 2 ẋ E p ẋ τ ω E r p r τ 2 ω Q i Q 2 OG cheme 2 Q 2 ẋ E p 2 ẋ τ ω E r p r τ 2 ω Q i Q 2 The two OG cheme have oppoite input/output power variable. The choice between the two OG block cheme depend on Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.2 Acro-variable generator: Eletttrico Mecc. Tra. Mecc. Rot. draulico τ Q Generator ẋ e ẋ e ω ω ẋ e ω OG τ Q Through-variable generator: Eletttrico Mecc. Tra. Mecc. Rot. draulico τ Q τ Q Generator ẋ e ω ẋ e ω OG τ Q Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.3 Example. et u conider the following electric circuit: 2 3 2 3 r R r2 r3 R 3 b The correponding OG block cheme: r R 2 r2 3 r3 R 3 }{{} 2 3 b The OG tate pace dynamic model i: R R ] 2 2 = R R 2 + [ a 3 3 R 3 3 R b 3 [ ] [ ] [ ][ ] a = x+ r3 R 3 R 3 b Note that, in thi cae, the power matrix A i ymmetric becaue the ytem i characterized only by one type of tored energy, in thi cae the magnetic energy tored in the three inductance, 2 and 3. Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.4 Example. et u conider the following electric circuit: r R r2 r3 R 3 r r2 2 r3 3 2 3 3 b The correponding OG block cheme: a r 2 r2 3 r3 R 2 R 3 3 r }{{} r2 2 3 r3 b The OG tate pace dynamic model i: R R R R 2 2 = R R R 2 + R 3 3 R R R R 3 R R 3 3 [ ] [ ] [ ][ ] r a = x+ 3 R R R R 3 R R 3 b The power matrix A i ymmetric becaue the ytem i characterized only by one type of tored energy. [ a b ] Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.5 Example. et u conider the following electric circuit: 3 3 l c c 2 l3 b c l2 2 d b The correponding OG block cheme: c l c 2 l2 r2 d l3 3 b 2 r2 3 b The ytem ANNOT be modeled uing the integral cauality : the inductance 3 i graphically repreented uing derivative cauality. rom the phyical cheme it i evident that the tate variable, 2 and 3 are linked by the following contraint: = 2 + 3 Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.6 The ytem can be modeled adding a puriou dynamic element 2 in parallel connection between element and 3: r2 3 3 l c c 2 r2 c2 c2 2 l3 b c l2 2 b The new OG block cheme i: c l c 2 2 l2 r2 r2 c2 2 The correponding OG tate pace dynamic model i: 2 2 3 c 2 = c2 3 [ ] [ ] = x b c 2 c2 3 + When 2 = fromtheforthequationoneobtainthefollowingcontraint: 2 3 = 3 = 2 et u conider the following tate pace congruent tranformation: x = Tx = c 2 Zanai Roberto - Sytem Theory. A.A. 25/26 c 2 c2 3 l3 3 3 [ a b ] b b
apitolo 3. DYNAM MODENG 3.7 Uing the congruent tranformation x = Tx one obtain the following tranformed and reduced OG dynamic ytem: + 3 3 3 2 + 3 c 2 [ b = ] [ ] = x c 2 + [ a b The tate variable c2 and 3 are no more preent becaue 2 = and the contraint 3 = 2. ThetranformedytemiNOmoredecoupledbecauematrix = T T T i not diagonal. The reduced ytem ha till the tructure of a OG dynamic model. Matrix, for example, i till ymmetric and definite poitive. alculation uing Matlab: -- Matlab command -------------------------------------------------------- ym 2 3 2 R2 T = 2=; M=diag([ 2 2 3]); [ + 3,, -3] AM=[-R2 - R2 - ; [,, ] ; [ -3,, 2 + 3] R2 -R2 ; - -; AT = ]; BM=[ ; ; ; ; -]; [ -R2, -, R2] M=[ ; [,, ] ]; [ R2,, -R2] T=ym([ ; ; BT = ; ; [, -] -]); [, ] T=T. *M*T [, ] %% AT=implify(T. *AM*T) T = BT=T. *BM T=M*T [,, ] [,, -] --------------------------------------------------------------------------- ] Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.8 Example. Tank with variable ection. Q input volume flow rate input preure volume of liquid in the tank z height of the liquid a(z) area of the liquid at height z A area of the bae ection α lope of the liquid area preure per unit of height p Two equivalent OG block cheme can be ued: Φ () Q p A 2 +2α A α Q Q p 2 Ap +α z A α a(z) [Ψ()] The area of the liquid a(z) at height z and the volume of liquid re: a(z) = A+αz, = z a(z)dz = Az + αz2 2 Since z = p, function =Φ() and =Φ () are defined a follow: = Φ() = A p + α2 2p 2, = Φ A2 +2α A () = p. α unction Ψ() and [Ψ()] can be obtained a follow: Ψ() = Φ() = A + α p p 2, [Ψ()] = p 2 Ap +α The energy E tored in the ytem i: E = Φ ()d = p 3α 2 [ (A 2 +2α) 3 2 3αA A 3 ]. Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.9 Example. A tranlating ma. x poition of the ma ẋ velocity of the ma m value of the ma m value of the ma at ret c velocity of the light p momentum of the ma x(t) m The ytem can be decribed uing one of the following OG cheme: Φ (p) ẋ p p m 2 +p2 c 2 ẋ ( )3 2 ẋ2 c 2 m ẍ [Ψ(ẋ)] Taking into account the relativitic effect, ma m and p = Φ(ẋ) are: m m m = ẋ, p = Φ(ẋ) = mẋ = ẋ2 c 2 ẋ2 c 2 The invere function Φ (p) can be expreed a follow: ẋ = Φ p (p) = m 2 + p2 c 2 unction Ψ(ẋ) and [Ψ(ẋ)] can be computed a follow: Ψ(ẋ) = Φ(ẋ) ẋ = m (, )3 v2 2 c 2 [Ψ(ẋ)] = ( v2 c 2 )3 2 The energy E tored in the ytem can be expreed a follow: p [ ] p E = Φ p (p)dp = dp = m c 2 + p2 m 2 + p2 m 2 c 2 c2 m Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.2 Nonlinear hydraulic capacitor. Q input volume flow rate Q 2 output volume flow rate preure of the capacitor volume of liquid within the capacitor G volume of ga within the capacitor preure of the empty capacitor volume of the empty capacitor Q Q 2 Ga The ytem can be decribed uing the following OG cheme: Φ () Q Q 2 Q 2 ẍ Q 2 [Ψ()] The ga preure within the capacitor atifie the following equation: G =, G = = Φ () = The contitutive equation of the nonlinear hydraulic capacitor i: = Φ() = unction Ψ() and [Ψ()] can be computed a follow: Ψ() = Φ() = 2, [Ψ()] = 2 The energy E tored in the ytem can be expreed a follow: E = Φ (p)dp = d = ln ( ) Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.2 General nonlinear vectorial cae n the general nonlinear vectorial cae following OG cheme can be ued: u B(x) y B T (x) q Φ (q) x A(x) u B(x) y B T (x) [Ψ(x)] ẋ x A(x) The correponding OG tate pace equation are the following: dφ(x) { = A(x)x+B(x)u Ψ(x)ẋ = A(x)x+B(x)u dt y = B T (x)x y = B T (x)x t can be proved that vector Φ (q) can be computed a follow: Φ (q) = E(q) q T where E(q) i the energy tored within the ytem expreed a a function of the energy variable q. t can be proved that matrix Ψ(x) can be computed a follow: Ψ(x) = Φ(x) = 2 E(q) x T q T q q=φ(x) where q = Φ(x) i the contitutive relation of the nonlinear vectorial dynamic element. The energy E tored in the ytem can be alway computed a follow: E = q Φ (q)dq Zanai Roberto - Sytem Theory. A.A. 25/26
apitolo 3. DYNAM MODENG 3.22 Relative degree of a tranfer function G() et u conider a generic block cheme: a a a φ φ R a E m K m m K m ṗ p J m ω m b m p K p K p α p Q α Q u Q or each tranfer function G() = Y() U() which link an input u(t) to the output y(t), the following propertie hold: ) the order of function G() i equal to the number n of independent dynamic element which tore energy within the ytem; 2) the pole of function G() are equal to the olution of equation () = where () i the determinant of the block cheme; 3) the relative degree of function G() i equal to the minimum number r of integrator which i preent in the et of all the path that link the input u(t) to the output y(t); 4) if there i only one path that link the input u(t) to the output y(t), then the zero of function G() are equal to the olution of equation () = where () i the determinant of the reduced block cheme obtained from the original one eliminating all the block touched by the path ; G() = ha 3 pole and zero becaue the relative degree i r = 3; G() = a ha 3 pole and 2 zero becaue the relative degree i r = ; Note: the higher i the relative degree the more difficult i the control. Zanai Roberto - Sytem Theory. A.A. 25/26