1
Engine Idle Speed Control via Maximal Safe Set Computation in the Crank-Angle Domain A. Balluchi2 , L. Benvenuti3 , L. Berardi1 , E. De Santis1 , M.D. Di Benedetto1 , G. Girasole1 , G. Pola1 1
Dipartimento di Ingegneria Elettrica, Università de L’Aquila, Monteluco di Roio, 67040 L’Aquila, Italy. 2 PARADES, Via di S. Pantaleo, 66, 00186 Roma, Italy. 3 DIS, Università di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy.
[email protected],
[email protected], lberardi,desantis,dibenede,girasole,
[email protected]
Abstract — In this paper, the idle speed control problem is addressed via maximal safe set computation. A hybrid model of the engine in idle mode is used. The solution is based on the transformation of the continuous dynamics from the time-domain to the crank-angle domain and on the linearization and discretization of these dynamics. A ‡exible and portable algorithm in the discrete-time domain is proposed for the determination of the maximal safe set. Simulation results show the e¢ciency of the proposed approach.
keywords: engine control, hybrid systems, safe set computation I. Introduction The idle-speed control problem deals with the task of maintaining, while in the idle mode, the engine speed into a given range, rejecting torque disturbances due to accessory loads (such as the air-conditioning system and the steering wheel servo-mechanism), and preventing the engine from stalling. Due to the unpredictable behaviour of the torque loads and to the engine operating at very low engine speed, the synthesis of a suitable idle control strategy poses serious challenges to the control designer and makes the idle speed problem one of the hardest in the area of engine control. Techniques used in the past make extensive use of meanvalue models (see [6] for a survey and the references contained therein); only recently there has been a shift towards hybrid models for the engine in idle mode. These models are based on the hybrid systems paradigm, say dynamical systems with interacting continuous and discrete event dynamics. The use of hybrid dynamical systems allows to model the engine behaviour in a cycle accurate way; in fact, the behaviour of each cylinder can be described by a …nite automaton with 4 states, each representing respectively the intake, compression, expansion and exhaust phase. However, even if mostly adherent to physical engine behaviour, the overall engine model obtained by properly composing the 4-stroke cylinder models is somewhat unsuitable and too complex for design purposes. This calls for a di¤erent modeling perspective that captures the peculiarity of the engine in the idle mode and simpli…es at most the hybrid dynamics, ruling out
the unnecessary details. A …rst attempt in this direction can be found in [5]: in that paper the control synthesis methodology is based on the computation of the maximal safe set, that is the set of all the initial conditions starting from which the evolution of the system stays inside the desired range. Once the maximal safe set has been computed, following the procedure in [4], it is always possible to derive the maximal controller, i.e. the set of all possible control strategies that solve the given problem. However the resulting control synthesis techniques turned out to be too complex from an industrial point of view, since tuning the controller to di¤erent car models required extensive recomputation of the control strategy by the control engineers. In other words, the corresponding controller synthesis techniques lacked of portability: varying some parameters in the plant model involved the complete controller re-design done in a non-automated way. In [6] and [7] a di¤erent approach has been used to address the idle speed control issue, based on a divide and conquer methodology. First, the overall control system is divided into subparts and each part is simpli…ed by relaxing the hybrid dynamics in the discrete-time domain. Subsequently, a control strategy for each block in isolation is designed, assuming that the remaining parts can be controlled in such a way as to guarantee the desired behaviour for the whole system. Finally, the correctness of the assumptions made on each subsystem is veri…ed, so that the closed-loop system is guaranteed to behave correctly on the whole. This methodology derives directly from the assume-guarantee reasoning, frequently used in the formal veri…cation of reactive and timed systems. In this paper, starting from a slightly simpli…ed version of the model in [6] and [7]1 , we address the idle speed control problem from a di¤erent perspective: instead of synthesizing a particular control law to be subsequently veri…ed, our methodology is based as in [5] on the computation of the maximal safe set, this time carried out using e¢cient and portable algorithms in the discrete time domain in [3]. Unlike the approach adopted in [5], our methodology is general enough to be easily extended 1 We do not consider the throttle valve dynamics, assuming that it is fast enough to be neglected compared to the manifold and powertrain dynamics.
to di¤erent problems in the hybrid domain, and the ‡exibility and e¢ciency of the algorithms involved in the computation of the safe sets make it appealing from an industrial point of view. The paper is organized as follows: in Section 2 we give a description of the hybrid model of the engine in the idle mode; in Section 3 a formal characterization of the idle speed control problem is given; in Section 4 the procedure used to compute the maximal safe set is described; simulation results are illustrated in Section 5 and Section 6 o¤ers some concluding remarks. II. Idle speed engine model In this section, we brie‡y describe the operation of the engine in the idle speed mode (see [1], [5], [6]). In the manifold dynamics modelling, it is assumed that the mass of mix entering a cylinder during the intake phase is proportional to the manifold pressure, at the end of the same run. Manifold pressure p is regulated by the throttle opening angle ®, that changes the intake duct section. Hence, we have: p_ = ap p + bp ® where ap = ¡6:67¼ s¡1 , bp = 580¼ mbar s¡1 . In order to prevent some undesiderable control laws that produce large excursions of the throttle valve, it is assumed that ® is constrained to belong to [0; ®max ]; with ®max = 20± . The powertrain dynamics can be expressed as Jeq n_ =
30 (T ¡ (Bn + Tl )) ¼
(1)
where n is the engine speed expressed in RPM (Revolutions Per Minute), T is the torque produced by the engine, Tl is the torque disturbance from accessory loads, Bn takes into account the viscous friction assumed to be proportional to the engine speed n (by coe¢cient B), and Jeq is the momentum of inertia of the driveline. Hence, (1) can be rewritten as
where Km is a constant parameter. In the following we consider traditional spark ignition engines without Gasoline Direct Injection (GDI) system and assume that there exists a control loop that maintains the air-to-fuel ratio to its stoichiometry value. Therefore we can model the torque T generated by each cylinder by means of a piecewise constant function that is zero everywhere except in the expansion phase. T depends nonlinearly on the spark advance µ s and the mass of air m loaded into each cylinder (at the end of the intake run). We linearize this nonlinearity around the equilibrium point in idle mode, thus obtaining T = c1 m + c2 µs + c3
(2)
where: c1 = 0:110; c2 = 0:613; c3 = 0:895. The fourcylinder in-line engine can be modelled by means of a single discrete state since at each dead-center only one cylinder generates torque. Hence, torque generation occurs only when µc = 180± with a value given by (2). III. Problem formulation In this section, we describe a formal model of the engine in idle speed mode and recall the formulation of the problem of engine idle-speed regulation from [1], [6]. Since during the engine operation the value of Jeq changes according to clutch insertion/release, some discontinuities in the physical variables occur; moreover the torque generation is driven by the discrete event dynamics modeling the cylinders sub-system. We therefore model the overall dynamics as a hybrid system (see [4],[2] for a formal treatment) with two …nite states, Q and QL , each corresponding to a di¤erent value of Jeq .
Q
QL
n_ = an n + bn (T ¡ Tl ) 30B ¡ ¼J , eq
30 ¼Jeq
(3) ¡2
2
bn = where an = and B = 1:603 10 kg m ¡1 s . The value of Jeq changes according to clutch insertion (Jeq = 1Kgm2 ) or clutch release (Jeq = 0:1Kgm2 ). By integrating the engine speed n(:), we have: µ_ c = Kc n where Kc = 6 rad RPM¡1 and µc (t) is the crankshaft angular position at time t. The cylinders sub-system models the torque generation. The torque generation depends on the phase of the cylinder, the position of the piston, the air mass, the fuel entering during the intake phase and the spark ignition timing. We assume that the air mass loaded into each cylinder during the intake phase depends only on the manifold pressure at the beginning of the compression phase, and we set: m = Km p
A continuous-time dynamical system with continuous control inputs is associated with each …nite state (node): 8 p_ = ap p + bp ® > > > > < n_ = an n + bn (T ¡ Tl ) µ_ c = Kc n Q: > > > T_ = 0 > : m _ =0 8 p_ = ap p + bp ® > > > n_ = aL n + bL (T ¡ T ) > < l n n µ_ c = Kc n QL : > > > T_ = 0 > : m _ =0
(4a)
(4b)
Transitions from Q to QL and viceversa, are forced by clutch insertion/release, i.e. by external discrete disturbances. Self-loops on the single states are invariance
transitions, in the sense that they occur whenever the continuous state goes out of a certain region of the state space. In particular, an invariance transition occurs at each dead center, i.e. when µc = 180± and the torque is generated by the engine. When an invariance transition £occurs, from node i to¤0node j, the continuous state x = p n T m µc 2 X is reset to a new value, according to a reset function Rij , i; j 2 f1; 2g, Rij : X ! X , described by the following table: Starting node i ¡! Destination node j # Q QL
Q
QL (5)
R11 Identity
Identity R22
where R11 = R22 is de…ned as 8 p := p > > > > < n := n µc := 0 > > m := Km p > > : T := c1 m + c2 µs + c3
Due to physical constraints, we assume that both the control variables ®, µs have lower and upper bounds, i.e.: 9 8 · ¸ ¸ · ® = < ® : 0 · ® · ®max ; µs 2U = (6) µs ; : µs min · µs · µ s max
When the engine is in the idle mode, the control objective consists in maintaining the engine speed in a given range (see [1], [6] [5]). In order to …nd a suitable control law for this purpose, we …rst need to …nd the maximal safe set, that is the set of all initial states such that there exists a feasible control law such that engine speed is in the given range. Hence the problem we address is the following: Problem 1: Given the hybrid system (3), (4), (5), …nd the maximal safe set I, under the constraints on the state variable n: jn ¡ n0 j · ¢ where n0 = 800 RP M is the nominal value for the engine speed and ¢ = 50 RP M , and under the constraints (6) on the control variables ®, µs . IV. Maximal safe set computation In this section, we describe the solution technique used to solve Problem 1. The maximal safe set computation is carried out on a simpli…ed discrete structure consisting of only one state, the one corresponding to the open transmission chain Q. We then show that the maximal safe set computed for such a node is also safe for the other con…guration (closed transmission chain), and is therefore
the maximal safe set for the two nodes con…guration.
Q (7) Consider the system (4a) ; (7) with reset function R = R11 . The solution method used to solve Problem 1 is based on the following steps: ² Transformation of the continuous dynamics from the time-domanin to the crank-angle domain; ² Linearization of the continuous dynamics; ² Discretization of the crank-angle domain linear dynamics; ² Application of the algorithm described in [2] for the computation of the safe set. The algorithm in [2] for a single node can be written as follows: Algorithm 2: 9 8 > > > > 2 3 > > > > p > > > > = < 6 n 7 6 7 Initialization: § := 6 µc 7 : jn ¡ n0 j < ¢ > > > 6 7 > > > > > 4 T 5 > > > > ; : m Repeat W := §¡ ¢ ¹ \ R¡1 (W ) § := Z ¡; ¡ Until § = W End 82 9 3 p > > > > > > > 7 > = > > > 4 5 T > > > > ; : m © ª ¹ = [ p n µc T m ]0 : µc = 180± , R¡1 (W ) is ¡ ¡ ¢ ¹ \ R¡1 (W ) is the the inverse reset image of W , Z ¡; ¡ ¹ \ R¡1 (W ). set of states of ¡ that can be controlled to ¡ The discretization step ± of the crank-angle domain linear dynamics is chosen as: ±=
180 N
(8)
for a …xed N 2 N. As a consequence of (8), the dead-center is reached exactly every N sampling times. We observe that N is also the number of back-integration ¹ \ steps needed for the computation of the set Z(¡; ¡ R¡1 (W )) in a single iteration of the algorithm above. When computing the controllability sets Z(¡; W ), the linear discrete-time system that has to be considered is the discretization of the crank-angle domain linear system associated with node Q stripped of the state-variable µc . The continuous state for each node becomes X (h) = [ p(h) n(h) T (h) m(h) ]0 and the dynamics associated with Q is
X (h + 1) = A X (h) + B ® (h) + F Tl (h)
(9)
where, for N = 4, 2
0.8217 6 0 A = 6 4 0 0 2 15.5022 6 0 B = 6 4 0 0
3 0 0 0 0.9858 0.8889 0 7 7, 0 1 0 5 0 0 1 3 2 3 0 7 6 7 7 , F = 6 -0.8889 7 . 5 4 5 0 0
~ k = Mk ¡ Wk FR M
In order to compute the inverse reset image R¡1 (W ), we can apply the same procedure used for the computation of the one-step controllability sets. Indeed, the reset can be seen as a single step of the evolution of the …ctitious system: X(h + 1) = AR X(h) + BR µs (h) + FR
End In order to get a complete description of the operations involved in the algorithm above, we have to show how to 1 (¤k ) and Z 1 (¤k ). compute the sets ZR 1 ² Computation of the set ZR (¤k ): ¤k = fX : Wk X · Mk g, where X 0 = Assume that: £ ¤0 p n T m . First of all, we have to update the value of Mk as to include the a¢ne term FR in it:
(10)
where AR , BR , FR are respectively the matrices of the a¢ne transformation given by the reset: 3 2 1 0 0 0 6 0 1 0 0 7 7 X(h + 1) = 6 4 0 0 0 c1 5 X(h) + 1 0 0 0 3 3 2 2 0 0 6 0 7 6 0 7 7 7 6 +6 4 c2 5 µs (h) + 4 c3 5 . 0 0
Before illustrating how Algorithm 2 specializes to the case of discrete-time linear systems, we introduce the 1 (¤) is the set of states that can following notations: ZR be mapped to ¤ by means of the reset operator; Z 1 (¤) is the one-step controllability set to ¤ contained in ¤. As a consequence of all the remarks above, Algorithm 2 becomes: Algorithm 3: 9 82 3 p > > > > = > > > ; : m Repeat k := k + 1 (Computation of the inverse reset image) 1 ¤k := ZR (¤k¡1 ) (Computation of the N-steps ”safe” controllability set to ¤k¡1 ) For i := 1 to N k := k + 1 ¤k := Z 1 (¤k¡1 ) If ¤k = ¤k¡1 then Break End For Until ¤k = ¤k¡N¡1
Second, we compute the one-step safe controllability set (considering in this case the dynamics in (10) ) to ¤k , i.e. the safe inverse image of the set ¤k using the reset operator R. 1 ZR (¤k ) can be obtained as the projection onto the statespace of the following set of (state-input) inequalities: 8 ~k < Wk (AR X + BR µs ) · M µs min · µs · µ s max : Wk X · Mk
Computation of the set Z 1 (¤k ): ¤k = fX : Wk X · Mk g, Here again, assume that: £ ¤0 0 p n T m . where X = First of all, we have to update the value of Mk as to include the disturbance term in it: ²
~ k = Mk ¡ max Wk F Tl M Tl
Second, we compute the one-step safe controllability set (considering in this case the dynamics in (9) ) to ¤, i.e. Z 1 (¤k ). Z 1 (¤k ) can be obtained as the projection onto the statespace of the following set of (state-input) inequalities: 8 ~k < Wk (AX + B ®) · M 0 · ® · ®max : Wk X · Mk
V. Simulation results
Algorithm 3, with the given choice of parameters and constants, converges in 5 steps. The maximal safe set I, solution to Problem 1, is a bounded polyhedral described by linear inequalities of the form: WX · M where W 2 R44£4 and M 2 R44£1 . projections of I onto the hyperplane £ Figure 1 shows ¤ p n T . It is interesting to note that safe torque values increase as n decreases; it means that for low engine speed, the torque should be high enough to meet the safety speci…cation, robustly with respect to torque disturbances. By initializing Algorithm 3 with the maximal safe set obtained for (4a),(7) and applying it to the hybrid system
VI. Conclusions In this paper, a computational procedure for the determination of the maximal safe set for the engine in idle mode is given. The in‡uence of the throttle valve dynamics and of the nonlinearities in the controlled system model is under investigation. References [1] Balluchi A., Benvenuti L., Di Benedetto M.D., Pinello, C., Sangiovanni-Vincentelli, A., ”Automotive Engine Control and Hybrid Systems: Challenges and Opportunities”, Proceedings of the IEEE, 88, Special Issue on Hybrid Systems, Invited Paper, pp. 888-912, 2000. [2] Berardi, L., De Santis E., Di Benedetto M.D., ”Invariant sets and control synthesis for switching systems with safety speci…cations”. In N.Lynch & B.H.Krogh, Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 1790 pp. 59–72. Springer–Verlag, 2000. [3] Berardi, L., De Santis, E., Di Benedetto, M.D., Pola, G., ”Approximations of Maximal Controlled Safe Sets for Hybrid Systems”. Workshop on ”Hybrid Control and Automotive Applications ”, Berlin, June 7-8, 2001; to be published as a Springer-Verlag book of the Lecture Notes in Computer Science series. [4] Lygeros, J., Tomlin, C., Sastry, S. ”Controllers for reachability speci…cations for hybrid systems”, Automatica, Special Issue on Hybrid Systems, vol. 35, 1999. [5] Balluchi, A., Benvenuti, L., Di Benedetto, M.D., Miconi, G., Pozzi, U., Villa, T., Wong-Toi„ H., Sangiovanni-Vincentelli, A., ”Maximal Safe set Computaion for Idle Speed Control of an Automative Engine”. In Nancy Lynch and Bruce H. Krogh, editors, Hybrid Systems: Computation and Control, volume 1790 of Lecture Notes in Computer Science, pages 32–44. SpringerVerlag, New York, U.S.A., 2000. [6] Balluchi, A., Benvenuti, L., Di Benedetto, M.D., Girasole, G., Sangiovanni-Vincentelli, A., ”Idle Speed Control Design and Veri…cation for an Automotive Engine”. In Proc. International Workshop on “Modeling, Emissions and Control in Automotive Engines”, MECA’01, Salerno, Italy, September 2001. [7] Balluchi, A., Benvenuti, L., Di Benedetto, M. D. and SangiovanniVincentelli, A., “Idle Speed Controller Synthesis Using an Assume–Guarantee Approach.” In Nonlinear and Hybrid Control in Automotive Applications, Springer-Verlag, 2002.
Fig. 1. Projections of the maximal safe set I
Fig. 2. Projections of the maximal safe sets I and IL
(4b) ; (11), the procedure converges in one step. This means that I is a controlled invariant set.
QL (11) Since the reset function Rij , with i 6= j, is the identity, this shows that the set I is the maximal safe set for the overall hybrid system (3), (4), (5). Let IL be the maximal safe set for (4b) ; (11). Figure 2 shows the projections of I and IL on the plane (n; T ).
