INC 69, 48 Dynamics Systm and Modlling: Linar Graph Modling II Dr.-Ing. Sudchai Boonto Assistant Profssor Dpartmnt of Control Systm and Instrumntation Enginring King Mongkut s Unnivrsity of Tchnology Thonuri Thailand
Stat-Spac Systm Rprsntation Th modling systm is writtn in th form: ẋ = Ax + Bu () y = Cx + Du () Th matrics A R n n and B R n n u ar proprtis of th systm. Th output quation matrics C R ny n and D R n y n u ar dtrmind y th particular choic of output varials. ˆ Systm ordr n and slction of a st of stat varials from th linar graph. ˆ Gnration of a st of stat quations and th systm A and B matrics. ˆ Dtrmination of a suital st of output quation and drivation of th appropriat C and D. INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Simpl RLC xampl R R V s (t) + C L V s (t) C L ˆ Thr ar 6 possil varials : i c, v c, i L, v L, i R, v R and i s ˆ Thr ar constitut rlations: dv C = i C i C, i L = L v L, i R = R v R Using a continuity quation and two compatiility quations i C = i R i L, v L = v C, v R = V s v C INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Simpl RLC xampl W hav dv C di L = L v C A = = RC v C + C i L + RC V s [ /RC /C /L ], B = [ ] /RC If th varials i R, v R, v L and i C ar of intrst as output varials: i R = R v C + R V s, v R = v C + V s v L = v C, i C = R v C i L + R V s INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 4/8
Simpl RLC xampl So if th output vctor is dfind to y = [ i R v R v L i C ] T Th C and D matrics ar /R C = /R, D = /R /R INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 5/8
cutst + Vs(t) L R R C C a d c f cutst is a st of ranchs in a graph, which whn cut off, will divid th graph into two disconnctd pics. a d c f a d c f a d c f INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 6/8
asic cutst f f f d d d a c a c a c Figur: thr possil asic cutsts ˆ asic cutst is th cutst that contains only on tr ranch and svral co-tr links. ˆ th continuity quations ar corrsponding to th asic cutsts ar indpndnt. INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 7/8
asic loopst f f f d d d a c a c a c Figur: thr possil asic loopsts ˆ asic loopst is a loop that contains only on co-tr link and svral tr ranchs. ˆ th compatiility quations ar corrsponding to th asic loopsts ar indpndnt. INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 8/8
Idntify th particular tr A normal tr for a connctd systm graph is formd y th following stps:. Draw th systm graph nods.. Th tr should includ all ffort sourcs as tr ranchs.. Th tr should includ a maximum numr of capacitors lmnts. 4. Th tr should includ a maximum possil numr of rsistor lmnts. 5. Th tr may thn includ th ncssary numr of inductor lmnts to complt th tr. INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 9/8
Drivation of diffrntial quations Or ˆ if th varial is a capacitor lmnt voltag, idntify th asic cutst containing that capacitor lmnt voltag. Th diffrntial quation is givn y th continuity quation for that asic cutst. ˆ if th varial is an inductor lmnt currnt, idntify th asic loop containing that inductor lmnt currnt. Th compatiility quation for that asic loop will yild th dsird diffrntial quation. ˆ Slct th stat varials as ffort varials on capacitor nrgy storag lmnts in th normal tr ranchs, and flow varials on inductor nrgy storag lmnts in th links. INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Exampl C f L R d + V s(t) C R a c a f c d Th continuity quation i i c i d = dv C = C (i i d ) INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Exampl First i = i L, which is a stat varial. And i d can xprssd as follows: i d = v d /R Th asic loop that involvs R (loop cafd) Thn v d = v c v a + v f = v C E + v C dv C = C (i L v d R ) = C i L R C v C R C v C + R C E INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Exampl f d Th continuity quation givs a c i f + i d + i = which in turn givs, dv C = ( i d i ) C = ( v d v ) [ = C C R R = v C ( + R C C R R v C E + v C v ] C E R R ) ( ) v C + C + R R E INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Exampl f d Th compatiility quation givs a c v + v c v a = which in turn givs, di L = L ( v c + v a ) = L v C + L E INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 4/8
Mchanical Systm k F s(t) k m v k (a) linar graph F (t) k m k F s(t) k m () normal tr INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 5/8
Mchanical Systm th stat varials ar v m (capacitor lmnt of th ranch) and F k (inductor lmnt of th link) and dv m = m F m, v = F, df k = k v k, F = v v k = k df k Thr ar two compatiility quations: v k = v k v, v = v m and thr continuity quations: F k = F s (t) F k, F = F k, F m = F s (t) F Th rsult is [ ] [ vm m F = k k k (k +k ) ] [ vm F k INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II ] [ ] [ ] + m F s(t) + k F s(t) k +k 6/8
Mchanical Systm Th rsult showing th dpndnc on th drivativ of th input F s (t). Th output quation for v is: v = F k or in matrix form [ v = ] [ ] v m + [ ] F F s (t) k Th stat varials may transformd as x = A x + (AE + B)u or: [ ] [ ] x m ] [ x = x k k + x (k +k ) th corrsponding output v = C x + (CE + D)u + F u m k k (k +k ) F s(t) v = x + k (k + k ) F s(t) INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 7/8
Mchanical Systm (In class work) F s (t) v k m m F m s(t) k k v (c) linar graph m k k m F m s(t) k (d) normal tr INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 8/8
Mchanical Systm (In class work) At a asic cutst : cut st cut st k i m + i + i k + i k = dv m = m v m m F k m F m s (t) k At a asic cutst : i m i Fs i Fk = m F k dv m = m F k + m F s INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 9/8
Mchanical Systm (In class work) At a asic loop st : k v k = v m df k At a asic loop st : = k v m m F m s (t) k v m + v k + v + v m = v m + k df k + F + v m = F = F k df k = k F k k v m + k v m INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Mchanical Systm (In class work) Th stat-spac systm is v m m m m v m m = F k k F k k k k v m v m F k F k m + F s INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Mchanical Systm (In class work) ω τ s (t) l θ m τ s(t) k J τ s(t) k J mg (a) Nonlinar pndulum () Linar graph (c) Normal graph Th stat-spac varials ar τ k, ω J. By continuity quation, w hav i k + i + i J = i s τ k + τ + τ J = T s τ k + ω + J dω J = T s dω J = J ω J J τ k + J T s Not: ω J = ω INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Mchanical Systm (In class work) Th compatiility quation in th asic loop containing τ k w hav v k = v J ω k = ω J Sinc τ k = mgl sin θ thn dτ k = mgl cos θ dθ k = (mgl cos θ)ω J Th nonlinar stat-spac quation is ω J = J ω J J τ k + J T s τ k = (mgl cos θ)ω J INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II /8
Fluid Systm (In class work) Q s(t) l h r = r + k Ch orific R r long pip: l orific R Q s(t) R C R (a) Fluid distriution systm l () Linar graph Q s(t) R C R (c) Normal graph INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 4/8
Fluid Systm (In class work) Th fluid flow through an orific is Q = C P sgn( P ). Th tank is shapd as th frustum of a con, thrfor has a volum which is a nonlinar functin of th hight of th fluid in th tank. h h h V = πr dh = π(r + k C h) dh = π ( r + r k C h + kc h) dh h = π ( ( r + r k C h + kc h) dh = π rh + r k C h + k h ) C For an opn tank th prssur at th as is P C = ρgh, thn whr V = πr ρg P C + πr k C (ρg) P C + πk C (ρg) P C = K t P C + Kt P C + Kt P C K t = πr ρg, K t = πr k C (ρg), and πk C (ρg) INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 5/8
Fluid Systm (In class work) Thr ar two stat-spac varials, Q l and P c. Th continuity quation around nod is Sinc P R = P C and i Qs i R i C i l = Q s Q R Q C Q l = dv = Q C = C dp C = [ K t + K t P C + K t P ] dp C C Q R = K P R sgn(p R ), P R = P C [ ] dp C = K t + K t P C + K t PC Q C [ ] [ ] = K t + K t P C + K t PC Q s K PC sgn(p C ) Q l INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 6/8
Fluid Systm (In class work) Th compatiility is P l = P C P R I l dq l dq l = P C P R, P R = = [ P C ] I l K Q l Q l and P l = I l dq l K Q R Q R, thn Q R = Q l Th nonlinar stat-spac quation is dq l = [ P C ] I l K Q l Q l [ ] dp C [ ] = K t + K t P C + K t PC Q s K PC sgn(p C ) Q l INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 7/8
Rfrnc. Wllstad, P. E. Introduction to Physical Systm Modlling, Elctronically pulishd y: www.control-systms-principls.co.uk,. Banrj, S., Dynamics for Enginrs, John Wily & Sons, Ltd., 5. Rojas, C., Modling of Dynamical Systms, Automatic Control, School of Elctrical Enginring, KTH Royal Institut of Tchnology, Swdn 4. Fain, B., Analytical Systm Dynamics: Modling and Simulation Springr, 9 INC 69, 48 Dynamics Systm and Modlling:, Linar Graph Modling II 8/8