Vol.50, No.6, 445/ Explicit MPC A Controller Design for a Diesel Engine Air-path System Based on Explicit MPC Akira Kojima, Kazuaki Sawado, Ts
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1 Vol.50 No.6 445/ Explicit MPC A Controller Design for a Diesel Engine Air-path System Based on Explicit MPC Akira Kojima Kazuaki Sawado Tsugito Maruyama Yuhei Umeda Hirokazu Anai and Keiji Shimotani A design procedure for the transient control of a Diesel engine air-pass system is proposed and the feature of the control system is investigated based on the simulation and the experimental results. The design method enables to obtain a nonlinear feedback law via Explicit MPC (model predictive control) and further derive a series of interpolated control laws which is applicable to the mode transient control. For the controlled system we focused here sufficient conditions on the stability and the positively invariance of state region are clarified and applied to evaluate the feature of the resulting control law. Key Words: diesel engine model predictive control (MPC) explicit MPC 1. CO 2 1) 3) PM NO x 4) Explicit MPC (Model Predictive Control) 5) 6) Fig. 1 VNT Variable Nozzle Turbo EGR Exhaust Gas Graduate School of System Design Tokyo Metropolitan University 6 6 Asahigaoka Hino Faculty of Engineering Tohoku Institute of Technology 35 1 Yagiyama Kasumimachi Taihaku-ku Sendai Fujitsu Laboratories Ltd Kamikodanaka Nakahara-ku Kawasaki Transtron Inc Endou Fujisawa Received February Fig. 1 Air path and exhaust system Recirculation MAF Mass Air Flow MAP Manifold Air Pressure PM NO x 7) PID 8) 9) TR 0006/14/ c 2014 SICE
2 446 T. SICE Vol.50 No.6 June ) 9) Explicit MPC 5) 6) Fig. 1 EGR VNT MAF MAP x(k +1)=f(x(k)u(k)) xmaf (k) uegr(k) x(k) := u(k) := x map(k) u vnt(k) { 0 uegr(k) u vnt(k) u egr u vnt EGR VNT (%) (1) x maf x map MAF MAP MAF 1 g/cyl MAP kpa (1) 10) x(k +1)=A x(k)+bũ(k) 2a x(k) X X := { x xmin x x max} 2b ũ(k) Ũ Ũ := {ũ uref ũ(k) u ref } 1 := 1 1 T 2c x(k) :=x ref + x(k) u(k) :=u ref +ũ(k) x ref x ref maf u ref := u ref egr := x ref map u ref vnt x ũ x ref u ref x ref u ref x ref = f(x ref u ref ) 3 (2b) (2c) (1) ũ(k) (2) Explicit MPC s Plant A 0 1 EGR VNT Plant B 0 1 VNT 1 mm 3 /st Plant A ( 0) 2000 rpm 20 mm 3 /st A = B = x ref := u ref := Plant A ( 1) 2000 rpm 50 mm 3 /st A = B = x ref := u ref := Plant B ( 0) 1200 rpm 10 mm 3 /st A = B = x ref := u ref := Plant B ( 1) 1200 rpm 40 mm 3 /st A = B = x ref := u ref := ) 2) 3. 1 (3) (2) (2b)
3 (2c) 5) 11) J = k=0 { } x T (k)q x(k)+ũ T (k)rũ(k) Q>0R>0 1) x =0ũ =0 X Ũ 4 2) x(0) X (2b) (2c) J< (4) 5) ũ(k) =F ( x(k)) 5 F ( ) 1) MAF MAP 2) PID 3s 3. 2 (5) 0 1 0: x 0(k +1)=A 0 x 0(k)+B 0ũ 0(k) 6a x(k) =x ref 0 + x 0(k) 6b u(k) =u ref 0 +ũ 0(k) 6c ũ 0(k) =F 0( x 0(k)) 6d 1: x 1(k +1)=A 1 x 1(k)+B 1ũ 1(k) 7a x(k) =x ref 1 + x 1(k) 7b u(k) =u ref 1 +ũ 1(k) 7c ũ 1(k) =F 1( x 1(k)) 7d Fig. 2 Mode transition system x(k) =x ref α + x α(k) 9 u(k) =u ref α +ũ α(k) 10 x ref α u ref α := (1 α) x ref 0 + α x ref 1 11 := (1 α) u ref 0 + α u ref α 1 10 step 0 1 α x α(k) ũ α(k) (Fig. 2) α (6a) (7a) x α(k +1)=A α x α(k)+b αũ α(k) A α := (1 α) A 0 + α A 1 B α := (1 α) B 0 + α B 1 0 α 1 13a 13b 13c (6d) (7d) 3.1 x 0(k) x 1(k) X ũ 0(k) Ũ0 ũ1(k) Ũ1 (k =0 1...) ũ α(k) =F α( x α(k)) F α( x) :=(1 α) F 0( x)+α F 1( x) (6) (7) (14) 14a 14b Ũ j := {ũ u ref j ũ u ref j } (j =0 1) (6d) (7d) 0 u(k) α 0 1 α =
4 448 T. SICE Vol.50 No.6 June 2014 (14) (14) (8) 1 1) X 2) 1) (6d) (7d) (14) X 2) x ref α (14) (13) (14) α 0 =1 α α 1 = α x α(k +1)=A α x α(k)+b αf α( x α(k)) (α 0α 1 0 α 0 + α 1 =1) 16 = α 2 0 {A 0 x α(k)+b 0F 0( x α(k))} + α1 2 {A 1 x α(k)+b 1F 1( x α(k))} +2α 0α {(A0 xα(k)+b1f0( xα(k)) +(A 1 x α(k)+b 0F 1( x α(k))} (17a) (17b) (8) G 0( X ) X G 1( X ) X 21 (2b) X (20) μ i 0 2 μi =1(i =0 1 2) 2 μ i G i( x) X x X 22 (13) (14) (18) (20) (19) (13) (14) α(k) 0 1 (k = ) (6d) (7d) ũ 0(k) =K 0 x 0(k) ũ 1(k) =K 1 x 1(k) A 0 + B 0K 0 A 1 + B 1K 1 (14) ũ α(k) =K α x α(k) 25a G 0( x) :=A 0 x + B 0F 0( x) 17a K α := (1 α(k)) K 0 + α(k) K 1 25b G 1( x) :=A 1 x + B 1F 1( x) 17b (k = ) G 2( x) := 1 {(A0 x + B1F0( x)) + (A1 x + B0F1( x))} 2 17c μ 0 = α μ 1 = α μ 2 =2α 0α 1 0 (μ 0 + μ 1 + μ 2 =1) x α(k +1)= 2 μ i G i( x α(k)) x α(k) X 18 μ i 0 2 μ i =1(i =0 1 2) ( 1) 1 α 0 1 (13) (14) (13) (25) x α(k +1)=(A α + B αk α) x α(k) 0 α(k) 1 (k = ) 26 (26) 2 (26) α(k) 0 1 (k = ) 2 x α(k +1)= μ i(k) A ci x α(k) x α(k) X 27 μ i(k) 0 2 μ i(k) =1 1 2 (k = ) x α(k +1)=A α x α(k)+b αf α( x α(k)) X x α(k) X G 2( X ) X A c0 = A 0 + B 0K 0 A c1 = A 1 + B 1K 1 A c2 = 1 {(A0 + B1K0)+(A1 + B0K1)} 2 α(k) 0 1 (k =
5 ) A α + B αk α = α0(k) 2 A c0 + α1(k) 2 A c1 +2α 0(k)α 1(k) A c2 28 α 0(k) =1 α(k) α 1(k) =α(k) μ 0(k) = α0(k) 2 μ 1(k) = α1(k) 2 μ 2(k) = 2α 0(k)α 1(k) (26) (27) (27) (26) 2 (27) 12) Theorem 3 μ i(k) (i =0 1 2) LMI Hi + Hi T S i Hi T A T ci > 0 A cih i S j (i =0 1 2 j=0 1 2) 29 H i S i (i =0 1 2) LMI (29) (26) α(k) 0 1 (k = ) (Plant A B) ( 1) Plant A step (0.48 s) ( 2) Plant B step (0.48 s) Plant A (4) Qmaf 0 Regr 0 Q = R= 30 0 Q map 0 R vnt Q maf = Q map = R egr = R vnt = (2b) x min = x max = Fig. 3 Feedback gain (Plant A Mode 0) Fig. 3 Fig. 4 Fig. 3 (a) 0 Fig. 3 (b) (EGR VNT) MAF x ref maf EGR VNT MAF EGR VNT % 25% (0.48 s) (17c) G 2( X ) Fig. 5 1(20) X (29) ( A) 2 (14) Fig. 6 Fig
6 450 T. SICE Vol.50 No.6 June 2014 Fig. 6 Transient response (Plant A x u) Fig. 4 Feedback gain (Plant A Mode 1) Fig. 7 Transient response (Plant A: tracking error x ũ) Fig. 5 Transition map of G 2 ( X ) (Plant A) Fig. 6 MAF MAP EGR VNT Fig. 7 Fig. 8 Fig. 8 Transient response (Plant A: accurate simulation x u) (4.3 ) 0 1 Fig. 6 Fig Plant B Plant B 0 1 (30)
7 Fig. 9 Feedback gain (Plant B Mode 0) Fig. 10 Feedback gain (Plant B Mode 1) Q maf = Q map = R egr = R vnt = Fig. 9 (a) Fig. 10 (a) 0 1 VNT Plant A (20) Fig (29) B X Fig. 12 VNT MAF MAP 10% EGR 4. 3 ( 1) 1 Realtime Workshop Rapid Prototyping ECU (Engine Control Unit) Plant A Fig. 13 (a) (b) Fig. 11 Transition map of G 2 ( X ) (Plant B) 3 s (Fig. 8) Fig. 13 (a) 8) ( ) Fig. 13 (b)
8 452 T. SICE Vol.50 No.6 June ms ms 1/5 5. Fig. 12 Transient response (Plant B: accurate simulation x u) Explicit MPC MPC ) Fig. 13 Transient response (Plant A: experimental result) EURO V PID Fig. 13 (a) 1 ENGINE TECHNOLOGY REVIEW /47 (2009) 2 ILQ EGR-VNT /359 (2009) 3 (2008) 4 HONDA S & D Technical Review /193 (2009) 5 A. Bemporad M. Morari V. Dua and E.N. Pistikopoulos: The explicit linear quadratic regulator for constrained systems Automatica /20 (2002) 6 M. Kvasnica P. Grieder and M. Baotic: Multi-Parametric Toolbox (MPT) mpt/ (2004) (2010) (2012) 9 T. Maruyama A. Ejiri Y. Ikari and K. Shimotani: Model predictive control considering disturbances in Diesel engine air intake systems IEEE Multi-conference on Systems and Control /408 (2012) /411 (2011) 11 D. Chmielewski and V. Manousiouthakis: On constrained infinite-time linear quadratic optimal control Systems & Control Letters /130 (1996)
9 12 J. Daafouz and J. Bernussou: Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties Systems & Control Letters /359 (2001) H 2 = A. 1: (29) A c0 = A c1 = A c2 = S 0 = S 1 = S 2 = H 0 = H 1 = H 2 = B. 2: (29) A c0 = A c1 = A c2 = S 0 = S 1 = S 2 = H 0 = H 1 = (ETH) Automatica IET J. Control Theory and Applications Asian J. Control Associate Editor. IEEE ( ) IEEE ( ) IEEE 1991 ( ) Universität Passau ( ) CREST / ( ) ( ) ( )
10 454 T. SICE Vol.50 No.6 June ( ) 2007 ( )
: u i = (2) x i Smagorinsky τ ij τ [3] ij u i u j u i u j = 2ν SGS S ij, (3) ν SGS = (C s ) 2 S (4) x i a u i ρ p P T u ν τ ij S c ν SGS S csgs
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15 C11-4 Numerical analysis of flame propagation in a combustor of an aircraft gas turbine, 4-6-1 E-mail: tominaga@icebeer.iis.u-tokyo.ac.jp, 2-11-16 E-mail: ntani@iis.u-tokyo.ac.jp, 4-6-1 E-mail: itoh@icebeer.iis.u-tokyo.ac.jp,
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MPC MPC R p N p Z p p N (m, σ 2 ) m σ 2 floor( ), rem(v 1 v 2 ) v 1 v 2 r p e u[k] x[k] Σ x[k] Σ 2 L 0 Σ x[k + 1] = x[k] + u[k floor(l/h)] d[k]. Σ k x
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