Dynamic Feedback Linearization applied to Asymptotic Tracking: generalization about the Turbocharged Diesel Engine outputs choice

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1 Dynamic Feedback Linearization applied to Asymptotic Tracking: generalization about the Turbocharged Diesel Engine outputs choice 9 American Control Conerence Hyatt Regency Riverront, St. Louis, MO, USA June -, 9 Marcelin Dabo, Nicolas Langlois and Houcine Chaouk ThC7.3 Abstract In this paper we apply dynamic eedback linearization to the tracking problem or a turbocharged diesel engine TDE equipped with exhaust gas recirculation EGR valve and variable geometry turbocharger VGT. The model used here is the third-order mean-value model, a reduction o the eighth-order mean-value one, see [3, or sake o simplicity. Our goal is to track desired values o suitably chosen outputs. In act, we irst plan to control the input maniold pressure and the compressor mass low rate instead o the air uel ratio AFR and the EGR raction. Unortunately, the ormer lead to a non-minimum phase system while the latter are not accessible or measurements in a vehicle, see [7. We thus replace the problem o tracking o desired values o the original output y input maniold pressure and compressor mass low rate by that o tracking a suitably constructed modiied output or which the values to be tracked are speciically chosen: namely, when the modiied output ỹ approaches them, the original output converges to the desired values. Simulation results are presented. I. INTRODUCTION In order to comply with more constraining European emission regulations, reer to [ Euro norms see [6 and [7, car manuacturers introduce in some diesel engines two actuators: the exhaust gas recirculation EGR valve and the variable geometry turbocharger VGT. The ormer permits recirculation o exhaust gas into the intake maniold reducing by this way the ormation o NO x while the latter permits the improvement o the relatively low power diesel engine density. But some drawbacks have to be underlined: an important reduction o the amount o resh air leads to an increase in particulate emissions and possibly visible smoke whereas a low amount o EGR raction leads to an increase in NO x emissions, see [7. To render these two actuators more eicient during combustion, several control design methods have been proposed: polynomial control in [, dynamic eedback linearization in [ and [3, optimal nonlinear control in [4, constructive Lyapunov control in [7, indirect passivation in [9, passivation in [8, predictive control in [3, [4 and [, etc. The output y o to-be-controlled variables, which consists o the input maniold pressure p and the compressor mass low rate, leads to a non-minimum phase system. Thereore we propose another choice o output whose zero dynamics are trivial such that i that modiied output ỹ tracks a suitably chosen value, then the original output yt converges asymptotically to its desired value. In [3 we solved the tracking problem or the TDE model using M. Dabo, N. Langlois and H. Chaouk are with IRSEEM, Technopôle du Madrillet, 768 Saint Etienne du Rouvray, France {dabo,langlois,chaouk}@esigelec.r nonlinear predictive control. In [3, Plianos et al. apply dynamic eedback linearization based on the property o latness to a reduced-order TDE model. Pursuing their work, this paper presents a more general result on the application o the dynamic eedback linearization control design method to the simpliied TDE model. Indeed, we see, with notions o geometric control that or each suitably chosen vector output, the third-order nonlinear model remains dynamic eedback linearizable with a trivial zero dynamics. To argue this, we chose a dierent vector output rom that o Plianos et al. in [3 or this study. The paper is organized as ollows: in Section II a description o TDE is given and its simpliied model is presented with a brie remind on the eighth-order one. Vector output is chosen. In Section III we examine the behavior o zero dynamics and propose a modiication o the output. A construction o a dynamic extension o the simpliied model and a general result about the TDE outputs choice are given in Section IV. Section V presents the control law derived rom eedback linearization theory while simulation results are presented in Section VI. Conclusion and some uture research directions are briely given in Section VII. II. TURBOCHARGED DIESEL ENGINE: DESCRIPTION, MODEL AND OUPUTS CHOICE A. Description o TDE unctioning A schematic diagram see Fig. o TDE is presented below. At the top o this diagram is the turbocharger composed o the turbine and the compressor. During the unctioning o TDE, the turbine takes its energy rom the exhaust gas. As it is linked to the compressor via a common shat, the compressor starts rotating, bringing consequently resh air into the combustion chambers via the intercooler and the intake maniold. A part o the exhaust gas is recirculated into the combustion chambers to reduce NO x ormation reer to [7. B. Reduced-order model The ollowing presented nonlinear model is a simpliied model rom the eighth-order one briely outlined in [3: T = k + W egr k e p + T p = k k e p W egr W t + W + P c = τ η mp t P c T T p, where the compressor resp. the turbine mass low rate is related to the compressor resp. the turbine power as ollows: k c = P c p µ /9/$5. 9 AACC 3458

2 see [3. For a detailed description o the ull-order model see [7 and [5. The nomenclature o some TDE variables is summarized in TABLE I, see [7 and [3. TABLE I NOMENCLATURE OF SOME DIESEL VARIABLES Fig.. Turbocharged diesel engine TDE resp. Pt = k t p µ Wt. 3 Despite o the act that the real inputs are the EGR valve and VGT openings, the considered inputs, in this study, or sake o simplicity, are u = W egr and u = W t, see [3. In the sequel, Ṫ and Ṫ are assumed to vanish because their corresponding measured signals T and T have very slow variations, see [7. The uel mass low rate W is regarded as an external disturbance and will not be taken into account or the synthesis o the controller. This yields the ollowing system: = k + u k e p = k k e p u u. 4 P c = τ η mp t P c Replacing and P t by their expressions and 3 and denoting K = ηm τ k t yield the system: where g x = ẋ = x + g xu + g xu, 5 x = k k k k c P c p µ k k e p k k e p Pc τ and g x =, 6 k K p µ 7 and x = p, p, P c belong to the set Ω, see [7, deined by Ω = {p, p, P c : < p < p max < p < p max, < P c < Pc max },, with the maximal values p max, p max, Pc max ollow rom physical limits o TDE. All the parameters o the model k, k, k c, k e, k t, τ and η m are identiied rom the eighth-order mean-value nonlinear model at a constant speed o 6 RPM and a ueling rate o 7. kg/h as said in a previous study, see [3. The ull-order model consists o the equations o the change o pressures, masses and ractions o burned gas in the intake and exhaust maniolds. These six equations are completed by two more: the turbocharger speed and the air mass low rate in the pipe connecting the compressor outlet and the intake maniold, 8 Variable EGR AFR N F F m m p p P c P t W e W t W W egr V V T T T c T e T egr ω tc J tc η c η t η m γ R µ C. Vector output choice Description Exhaust gas recirculation Air uel ratio Engine speed Intake maniold burned gas raction Exhaust maniold burned gas raction Mass o gas in the intake maniold Mass o gas in the exhaust maniold Gas pressure in the intake maniold Gas pressure in the exhaust maniold Compressor power Turbine power Total mass low rate into the engine Compressor mass low rate Turbine mass low rate Fuel mass low rate EGR mass low rate Intake maniold volume Exhaust maniold volume Intake maniold temperature Exhaust maniold temperature Compressor temperature Temperature o the exhaust rom the engine EGR temperature Turbocharger speed Turbocharger moment o inertia Compressor isentropic eiciency Turbine isentropic eiciency Turbocharger mechanical eiciency Speciic heat ratio Speciic gas constant γ γ The output o to-be-controlled variables consists o the input maniold pressure p and the compressor mass low rate instead o the AFR and EGR raction because the latter are not accessible or measuraments in a vehicle, see [7. We thus consider the nonlinear system 4 equivalently, 5-7 with the vector output: p y = 9 and the goal is to track desired constant values p d o p and d o. In the sequel, we suppose that all the components p, p, P c o the state x measured or estimated. III. ZERO DYNAMICS AND CHANGE OF THE OUTPUT Consider the square-mimo m m nonlinear system ẋ = x + m j= g jxu j y = h x,..., h m x t, where x R n, u R m and y R m are the vectors state, control, and output, respectively. To simpliy the exposition, the standard geometric notation or Lie derivatives is used in this paper. For a real-valued 3459

3 unction h on R n and a vector ield on R n, the Lie derivative o h along at x R n is given by: n h L hx = x i x. x i Inductively, we deine L k hx = L L k with L hx = hx. A. Vector relative degree i= hx = Lk h xx, x A system o the orm has a vector relative degree ρ,, ρ m i: i or any x R n LgjL k h ix =, or all i m, all j m, and all k < ρ i ; ii the m m matrix decoupling matrix L g L ρ h x L gm L ρ h x Ax =.. L g L ρm h m x L gm L ρm h m x is nonsingular or all x R n see, e.g., [6. For the third-order model 5-7 with the output 9, the vector relative degree exists and ρ, ρ =, or all p, p, P c Ω. The decoupling matrix is: [ k Ax = µkckpcpµ p K p µ k µ. c p µ The sum o the vector relative degree components is equal to which is less than 3, the dimension o the state space o system 5-7. Thereore one-dimensional zero dynamics exist. An examination o their stability is necessary beore deriving the vector control law assuring tracking the desired output value. B. Zero dynamics Since the goal is to track a reerence signal, which consists o desired ixed values p d and d o the respective components o the output y = p, t, deine the error p p e = d, d between the to-be-controlled variables p, and their desired values. The zero dynamics o the error are obtained by applying the control annihilating identically the error et and thus are given by = k d [ p µ d, 3 η mk tk c p µ which are unstable. Indeed, 3 has a single equilibrium p e = pµ d µ η mk tk c =.7 bar and the eigenvalue λ = 5 o the linearization o 3 at p e is positive, see Fig. and Fig. 3. These numerical values are given or the values p d =.6 bar and Wcd =.7 kg/s that are, indeed, natural or applications. It can be noticed, however, that the instability o the zero dynamics 3 depends neither on the choice o p d nor o d. The system is I-O decouplable Fig.. Fig. 3. zero dynamics and lower bound bar zero dynamics solution bar lower bound time s Unstable zero dynamics with initial condition o p =.6 bar zero dynamics and lower bound bar zero dynamics solution bar lower bound times Unstable zero dynamics with initial condition o p =.8 bar via static eedback. So using I-O linearization we can easily ind a control law that steers the error et, given by, asymptotically to zero see, e.g., [6. Unstable zero dynamics will result, however, in an undesired property: the internal variable p will not remain bounded, it will either reach the limit value p = in inite time i p < p e, see Fig. or will go to plus ininity with an asymptotically constant velocity i p > p e, see Fig. 3. To avoid dealing with unstable zero dynamics, we propose another choice o vector output and a dynamic extension. C. Change o the vector output We will overcome the problem o unstable zero dynamics by changing the output 9 such that the modiied system has trivial zero dynamics, that is, consisting o a single equilibrium point. This can be achieved by keeping the irst component as p and replacing the second output component by a unction h x, where x = p, p, P c, such that, indeed, L g h =. Resolving this equation gives h x = P c + K k p µ p µ, and thus we consider the new output ỹt o to-be-controlled variables deined by ỹ = hx p = P c + K k p. 4 µ p µ In the next subsection we will show that the system 5-7 with the output 4 has, indeed, trivial zero dynamics. 346

4 As we speciied, our problem is to track desired constant values p d o p and d o. A natural question is thus how to reormulate the problem in terms o the components o the new output ỹ, given by 4, in order to achieve a solution o the original tracking goal. Notice that, P c, k and p are linked via the relation = P c c and hence p µ the desired tracking values p d and d determine uniquely the desired value P cd o P c as P cd = d p µ d k c. Notice that given any ixed p d and P cd, there exists a unique point x e = p e, p e, P ce, satisying p e = p d and P ce = P cd, and unique control values u e = u e, u e such that the right hand side o 5-7 has an equilibrium at x e when the controls are evaluated at u e. To see this, recall that the equilibrium set o 5 consists o the points at which we can create an equilibrium by a suitable eedback and thus E = {x : x span{g x, g x}}. Observe that the zero dynamics maniold o the error e, given by, is Zd = {p = p d, = d } and is thus transversal to the equilibrium set E. More precisely, the intersection {x e } = E Zd consists o a single equilibrium point x e = p d, p e, P cd, with uniquely deined p e, and there exist unique control values u e = u e, u e t R such that x e + u e g x e + u e g x e =. We will deine the desired tracking value h d o h, the second component o the new output ỹ given by 4, as h d = h x e = P cd + K p e k µ p µ e. Notice that the zero dynamics maniold Z d o the error p p d ẽ = h x h 5 d passes through the equilibrium point x e. Now a crucial observation is that when the new output ỹt tracks asymptotically the constant value p d, h d and the overall system approaches the equilibrium point x e, then the original output yt tracks asymptotically the desired values p d, d. It ollows that in order to solve the tracking problem, it is enough to show that the zero dynamics corresponding to the new error 5 are asymptotically stable or instance, trivial consisting o x e only. The main idea o changing the output is illustrated in Fig. 4. The zero dynamics, evolving on Zd, o the error between the original output yt and its desired value are unstable. We look or a new output ỹ and or its desired value such that the zero dynamics o the new error are asymptotically stable and their maniold Z d intersect Zd at an equilibrium point x e. A solution can be either asymptotically stable dynamics on Z d as illustrated by Fig. 4 or trivial zero dynamics reduced to the equilibrium x e which will be the case o our TDE model. Fig. 4. Equilibrium set E and zero dynamics IV. DYNAMIC EXTENSION OF THE SIMPLIFIED MODEL AND GENERALIZATION ABOUT THE TDE OUTPUTS CHOICE In this section we will ollow notions o geometric nonlinear control see, e.g., [6 and [. The system 5-7 with the output 4 has the decoupling matrix [ k Ãx = µkckpcpµ p µ, 6 which is not invertible, and thus the system has no a vector relative degree. We can, however, construct a suitable dynamic extension with a well deined relative degree. To this end, put z = u = W egr, ż = v and apply the new vector control [v, v t, where v = u = W t. This yields the ollowing extended nonlinear system = k + z k e p = k k e p z v P c = τ η mp t P c ż = v, 7 which we can rewrite, denoting its extended state by x e = p, p, P c, z t Ω [Wegr min; W egr max min where Wegr and Wegr max are the minimum and maximum values o the EGR mass low rate, as where g e xe = ẋ e = e x e + g e x e v + g e x e v, 8 e x e = k k c P c p µ k ep + z k k e p z and ge xe = Pc τ k K p µ, 9. The extended system 8- with the output 4 has the vector relative degree ρ e, ρ e =, and the invertible 346

5 decoupling matrix A e x e = p k k k c K µ p µ K p µk K z k e p p µ + K τ p µ. The extended system is thus I-O decouplable and has trivial zero dynamics since the sum o the components o its vector relative degree is ρ e + ρ e = + = 4, the dimension o the state space o the extended system. Notice that the original system 5-7, with the output 4, has trivial zero dynamics too because the latter does not depend on invertible endogenous eedback. A natural question is whether a dynamic extension is necessary. In other words, is it possible to choose another output, say ȳ = hx, such that the original system 5-7, with the output ȳ, would have the vector relative degree ρ, ρ =,. In this case, the original system would be static eedback I-O decouplable with trivial zero dynamics with respect to the output ȳ and no extension would be needed. The answer is: this is impossible, independently o the choice o the output ȳ = hx. Indeed, i such an R - valued unction h exists, then the original system 5-7 would be static eedback linearizable. This is not the case, however, because the distribution D = span {g, g } is not involutive since the Lie bracket [ g, g = µk K p µ is independent o g and g. Another way o looking at the extension procedure that we propose is to observe that although the system 5-7 is not static eedback linearizable it is, however, lat, reer to [5 and [3 that is, dynamic eedback linearizable since any 3-dimensional system with noninvolutive distribution D = span {g, g } is so and the components p and h, given by 4, or any other suitably chosen outputs, see [3, o the new output ỹ are actually lat outputs linearizing outputs o the system 5-7. Indeed, we can express the state components p, p, P c as well as the controls u and u o the system using p and h and their time derivatives; when calculating u we will have to dierentiate u which conirms that the system is dynamically but not statically linearizable. V. CONTROL LAW In [3 we solved the tracking problem or the simpliied TDE model using nonlinear predictive control. In this section we will linearize the extended system which will allow us to calculate the control law that assures tracking desired values o the modiied output 4 which, in turn, guarantees tracking the desired values o the original output 9 as we explained in Section III. The extension 8- o the TDE model, together with the output 4, is I-O decouplable and has trivial zero dynamics so the linearizing coordinates and the linearizing eedback can be calculated as ollows, see, e.g., [6 and [. Introduce new coordinates ϕ = h x e = p ϕ = L e h x e = k z k e p + kc p µ P c ϕ = h x e ϕ = L e h x e = Pc τ + K k e p z p µ 3 ollowed by the eedback vx = A e x e [ bx e + w 4 where A e x e is the decoupling matrix, and bx e is given by: k k e + µkcpcpµ P k c p µ c p µ + z k ep + kkc P c bx e τ p µ =. Pc τ k k e K p µ P k c c p µ + z k ep µk K p µ k e p z This yields the ollowing decoupled system: ϕ = ϕ ϕ = w ϕ = ϕ ϕ = w with the output ỹ = ϕ ỹ = ϕ. 5 The desired tracking value p d o ϕ is given while desired tracking value h d o ϕ is deduced rom d and rom the requirement that x e e = p d, p e, P cd, z d is an equilibrium point see Section III. This yields z d = u d = k e p d d p e = P µ cd τk d h d = P cd + K k p e µ p µ e. The tracking control law is now calculated as [ K w = ϕ p d + Kϕ K ϕ h d + K ϕ [ K = ϕ + [ K ϕ K K ϕ + p d K ϕ K h d, 6 7 where Kj i R are chosen so that, or i =,, the polynomials λ Kλ i K i are Hurwitz. This control law guarantees that output ỹt tracks asymptotically the constant values p d, h d and thereore the original output yt tracks asymptotically the desired values p d, d. VI. SIMULATION RESULTS We chose as the output yt to be tracked a concatenation o constant desired values p d o p and d o, corresponding to speciic rate o emission o the engine, in the neighborhood o that chosen by Plianos et al., see [3. The eigenvalues o the closed loop system are picked up as -3 and -4 or the irst subsystem and - and - or 346

6 the second. The simulations have been done under Matlab Simulink V 7., with the ollowing characteristics: all step sizes and tolerances are under the position auto and the used solver is Ode 3s sti/mod. Rosenbrock. The time o simulation is 4 s. We show in Figures 5, 6, 7 zoom o Fig. 6 and 8 the behavior o the to-be-controlled output yt = p t, t as well as that o the second component h xt o the modiied output ỹt. Fig. 5. p d p bar Reerence p d Output p time s Extended model 4 th order: Output p and its reerence signal h no unit Reerence H d Output H time s Fig. 6. Extended model 4 th order: Output h denoted by H and its reerence signal h d denoted by H d h no unit Reerence H d Output H times Fig. 7. Extended model 4 th order: Zoom in on output h denoted by H and its reerence signal h d denoted by H d Fig. 8. kg/s Desired Value d time s Extended model 4 th order: Output and its desired value d VII. CONCLUSIONS AND FUTURE WORKS A. Conclusions In this paper, a 3-dimensional nonlinear TDE model is considered. In order to avoid dealing with unstable zero dynamics we propose a solution o the tracking problem based on two basic ingredients: change o the output which yields a system with a trivial zero dynamics and dynamic extension which allows a simple calculation o a tracking controller. B. Future Works In our study we suppose that all states are accessible or measurements which may not always be the case in practice or instance, the gas pressure in the exhaust maniold is not accessible or measurements. Thereore in our uture works we are planning to use a nonlinear observer or that state o the turbocharged diesel engine TDE and to do a comparative study between our controller based dynamic eedback linearization and that o Plianos et al., see [3. We are planning also to study robustness o the control law with respect to the uel mass low rate W considered as an external perturbation in this study. REFERENCES [ J. F. Arnold, N. Langlois and G. Tremoulière, Control o the air system o a diesel engine: a uzzy multivariable approach, IEEE CCA Conerence, München, Germany, October 4-6, 6. [ M. Ayadi, N. Langlois and H. Chaouk, Polynomial control o nonlinear turbocharged diesel engine model, International Conerence on Industrial Technology, 4, pp [3 M. Dabo, N. langlois, W. Respondek and H. Chaouk, NCGPC with dynamic extension applied to a Turbocharged Diesel Engine, Proc. o 7 th IFAC world Cong., July 6-8, Seoul, S. Korea. [4 H. J. Ferreau, P. Ortner, P. Langthaler, L. del Re, and M. Diehl, Predictive control o a real-world Diesel engine using an extended online active set strategy, Annual Review in Control, vol. 3, 7, pp [5 M. Fliess, J.J. Lévine, P. Martin, and P. Rouchon, Flatness and deect o nonlinear systems: Introductory theory and examples, Int. J. o Contr., Vol. 6, pp , 995. [6 A. Isidori, Nonlinear control systems, Springer Verlag, Englewood Clis, 3rd edn., New York, 995. [7 M. Jankovic, M. Jankovic and I. Kolmanovsky, Constructive Lyapunov Control Design or Turbocharged Diesel Engines, IEEE Tran. on Cont. Syst. Tech., vol. 8,, pp [8 M. Larsen and P. Kokotovic, Passivation design or a turbocharged diesel engine model, IEEE CDC, 998, pp [9 M. Larsen, M. Jankovic and P. V. Kokotovic, Indirect passivation design or diesel engine model, Proceedings o the IEEE International Conerence Applications Ancorage, Alaska, USA,. [ H. Nijmeijer and A. J. van der Schat, Nonlinear Dynamical Control Systems Springer-Verlag, New York, 99. [ P. Ortner and L. del Re, Predictive Control o a Diesel Engine Air Path, IEEE Tran. on Cont. Syst. Tech., vol. 5, 7, pp [ A. Plianos and R. Stobart, Dynamic eedback linearization o Diesel engines with intake variable valve actuation, IEEE Int. Con. on Cont. and Appl., 7, pp [3 A. Plianos, A. Achir, R. Stobart, N. Langlois and H. Chaouk, Dynamic eedback linearization based control synthesis o the turbocharged Diesel Engine, Amer. Cont. Con., 7, pp [4 A. Plianos, A. Achir, R. Stobart and H. Chaouk, Optimal nonlinear control o a Diesel engine air-path system using VGT/EGR, Proc.- Europ. Cont. Con., 7, pp [5 A. Steanopoulou, I. Kolmanovsky and J. S. Freudenberg, Control o variable geometry turbocharged diesel engines or reduced emissions, IEEE Tran. on Cont. Syst. Tech., vol. 8,, pp [6 Transport and E. E. Federation, Euro v and euro vi emission standards, March 4, european Union. [7 Umweltbundesamt, Future Diesel: Exhaust gas legislation or passengers cars, light-duty commercial vehicles and heavy-duty vehicle - updating o limit values or diesel vehicle, Federal Environmental German Agency, Tech. Rep., July