Copyright \0) IFAC Advances in Automotive Control, Karlsruhe, Gennany, 2001
APPROXIMATE I/O LINEARIZATION OF A CONTINUOUSLY VARIABLE TRANSMISSION Soren Hohmann, Alexander Schonbohm, Volker Krebs
Institut fur Regelungs- und Steuerungssysteme, Universitiit Karlsruhe (TH), Kaiserstrafle 12, D-76131 Karlsruhe, Germany, Tel. +49 (0)721/608-2462, e-mail: {hohmann . krebs}@irs.etee.uni-karlsruhe.de
Abstract: A substantial portion of the exhaust gas passenger cars produce is emitted during the transient operation phases of a spark ignition engine. Therefore, Continuously Variable Transmissions (CVT) are used in order to operate the engine in its minimal fuel consumption and minimal emission point . In this paper, a new controlJer design procedure is proposed for this task: Approximate input output linearization. It allows to design controlJers for discrete-time nonlinear systems yielding a prescribed input output transient behavior. Although the CVT is not minimum phase, asymptotic stability in the sense of Lyapunov , is achieved. Copyright @20011FAC
Keywords: Transmission ; discrete-time systems; control system design ; non-minimum phase systems; feedback linearization ; nonlinear systems
trol strategies are used to tackle this problem. Classical control structures like for example gain scheduled PID controlJers are used as well as more advanced nonlinear control schemes (see for example (Sackmann and Krebs , 1998; Sackmann and Krebs , 1999; Guzzella and Schmid, 1995; Kolmanovsky et al., 1999)) . In the latter , feedback linearization techniques are appJied. In this approach , a so called virtual output function is calculated to obtain a system which has no zero dynamics (Isidori. 1995). Although this function may be mathematically interpreted as the output of the system. it is no output in a physical sense. This fact complicates the design of controllers , which should achieve a desired transient behayior. The controlled CVT should haw a prescribed transient beha\'ior for comfort reasons. Hence. using the standard feedback linearization technique the yirtual output function has to be chosen properly, which indeed is not easy task .
1. INTRODUCTIO:\"
Passenger cars have a major environmental impact in particular in terms of the exhaust gases they produce. Therefore, much effort has been spent on reducing the emissions in steady state operation of spark ignition engines. However, an important portion of nitrogen oxides pollutants escapes into the tail pipe when the motor accelerates or slows down. A modern way to handle this problem is to use Continuously Variable Transmissions (CYT) , which in contrast to conventional gear boxes. have a continuum of gear ratios. In that way the spark ignition engine can be held on the steady state optimal curve. In particular , a C\-T is necessary if an engine is used which is at low and medium speed in a lean operation mode. If the engine is switched to a rich air-to-fuel-ratio at higher speed leyels , significant torque variations haw to be suppressed by the C\-T .
Therefore. in this contribution we focus on the nonlinear approximate I/O linearization for discretetime non minimum phase systems which has recently been proposed (Hohmann et al.. 2000b: Hohmann and Krebs , 2000). This method achieyes
It is shown by Guzzella that the C\-T has a nonminimum phase behayior (GuzzelJa and Schmid, 1995) . This in fact means that the attainable ma..ximum control performance is limited . Different con29
where the engine inertia is denoted by Be and the car inertia by Bw. 1] is the CVT's efficiency
an approximate linear input output behavior for the physical output of the system, while maintaining asymptotic stability. Standard linear controller design methods can be used to adjust easily the desired closed loop I/O behavior.
A simplified system model for the CVT's hydraulic subsystem is
v=
In section 2 we briefly review a simplified system model of a CVT, which has already been investigated in more detail (see (Guzzella and Schmid, 1995)). In section 3 we outline the main essentials of the controller design method. One part of the control law consists of an optimization procedure and hence has to be executed on-line. Since its realization turns out to be time-consuming, we propose a new dynamic state feedback law avoiding this online optimization. In section 4 it will be shown that a particular coordinate transformation improves the stability properties of the proposed controller. Finally, in section 5 the feedback law is calculated for the CVT and the closed loop behavior is analyzed by simulations.
All parameters are listed in (Guzzella and Schmid, 1995) . Denoting Xl = Ww and X2 = v the system can be written in state space form : (3)
Since the control strategy will be implemented in an electronic control unit including a microcontroller, a discrete-time control algorithm is required. Thus, our controller design is based on a sampled data system's representation. In a step invariant sampled data system the discretetime model states equal the continuous time states at equidistant time instants, assuming u to be constant during the sampling intervalls T . Now , the discrete model can be derived by a time discretization procedure of the continuous time system outlined in (3) (Kazantis and Kravaris, 1997). In this approach, the system is written in terms of a Taylor-Lie-Series expansion with respect to the vectorfield [);r, u):
2. MODEL DESCRIPTION Conventionally, a powertrain including a CVT consists of the spark ignition engine, often referred to as the thermodynamic subsystem, the CVT realizing a continuum of gear ratios, and the differential and axle subsystem with fixed gear ratio. We focus on the CVT and assume that in particular the engine is kept on the fuel-efficient and exhaustminimal curve. This is indeed possible since the thermodynamic subsystem has - properly controlled - smaller time constants (see for example (Hohmann et al., 2000a)). In this respect, the engine and the CVT are decoupled , i.e. the engine torque Te can be considered as a function of the engine speed We which is approximated by a second order polynomial Te = QW~ + fJw e + 'Y. The external torque T w is approximated by a second order polynomial, as well, including the drag resistance proportional to the square of the wheel speed Ww and the rolling resistance: Tw = aw~ + c. Other disturbances are not considered.
;r(k
= fw (ww, v) + gw (ww, v) u,
(1)
u is the control variable and the functions gw(-) are given by
f w (.) and
fw (ww, v)
=
(1] QV 3 -
+ 1) = l
(;r(k) , u(k))
= ;r(k)
Tvl
+ L (LLf;r~ N
1'=}
;r(k) , u(k)
y(k)=h(;r(k)) ,
(4)
where f is the discrete-time Coo vectorfield, defined in a region D ~ IRn including the origin. y(k) denotes the system's output which in our case is equal to the wheel speed and u(k) E U ~ IR the system's input variable, which influences any future state, i.e. rank {&L~~'U)} = 1 holds for ;r(k) E D . L f ;r denotes the usual multidimensional LieOperator.
A first order continuous time model can be derived (Guzzella and Schmid, 1995) if the Euler equations for the wheel speed and engine speed, the CVT's kinematic relation VWw = W e and the power balance between wheel side and engine side are combined: Ww
(2)
u.
Simulations have shown that this discrete-time model is approximately equal to the continuous one at the sampling instants for N = 1 with T = 100 ms. It should be mentioned, however , that the methods developed in the following section make use of the general discrete-time model ;r(k + 1) = f (;r(k), u(k)). Hence, iflarger sampling intervals are needed, N could be enlarged. 3. CONTROL LAW CONSTRUCTION
a) w~+1]fJv2ww+(1]'Yv - c) 28
The control objectives for the CVT are guaranteed stability in the sense of Lyapunov and a prescribed I/O dynamics should be achieved. For this task usually the well known exact input output linearization
8 w + 1]v - e
30
(Kotta, 1995; Isidori, 1995) is used, provided the system is minimum phase. However, the CVT (4) has a non minimum phase behavior. In the following we propose a controller design procedure, which copes with the non minimum phase behavior, but leaves the output function unchanged such that the closed dynamics can be designed according to the specific needs. In the sequel, some fundamentals of the exact input output linearization are recalled.
It is well known how to tackle this difficulty in the case of linear non minimum phase systems. For example, standard linear LQ controllers follow the desired input output dynamics only approximately but guarantee a stable closed loop (Papageorgiou, 1991) . In a nonlinear context these controllers may yield a stable system's behavior, but in a small neighborhood of the equilibrium point, only.
Therefore, we propose a new nonlinear control law that takes into account not only the nonlinearity Snl ,rd(;'" -rT;.. + Vh + v) in equation (6) , but also the remaining nonlinearities snl ,rd+ 1, . .. , Snl ,n are incorporated in a mean square sense:
This method transforms a given system exactly into a linear and controllable one. The procedure typically consists of two basic steps: (1) a local change of coordinates
(2) a nonlinear state feedback.
Theorem 1. Suppose (5) is locally controllable. The control law u = -rT;.. + Vh + V , with a constant vector rT and
First, for the construction of a map which links the input and output of the system (4), the output time index in (4) is successively replaced by k + 1 and the system's equation (4) is inserted. If the system (4) has a well defined relative degree rd, the input appears after rd time shifts at the shifted output (Kotta, 1995) . We can choose the output zl(k) := y(k), its rd - 1 output time shifts Zi (k) := y (k + i - 1) , i = 2 .. . r d - 1 and (n - r d) additional coordinates Zrd+1 , .. . , Zn to define a local coordinate transformation. That is a differentiable map with differentiable inverse transforming (4) in a Byrnes-Isidori normal form : zi(k + 1) = Zi+l (k) zi (k
+ 1) = if:n,i;" + bi U + + Snl ,i(;", u) y(k) = zJ(k) ,
with §.nl = [0, ... , 0, snl ,rd' .. . , Snl ,n]T, !2 = [0, . . . , 0, brd . . . , bnf locally stabilizes the equilibrium of (5) .
This solution yields a closed loop, which is approximately linear in a large region and stable.
Proof. We assume that {fR = Q and UR = 0 is an equilibrium point of (4) and furthermore for simplicity v = O. If the system is locally controllable, it is always possible to find a constant vector r , which shifts all eigenvalues of the systems' linear part inside the unit circle. Consequently, there exist positive definite matrices Q and P with
i=1 , . . . ,rd- 1
i=rd , ... , n
(5)
-=
(dun - !2r.T) T P (dlin - !2r.T) - E.. -Q. If we choose a quadratic form V(;..(k)) = ;..T p;.. as a Lyapunov function candidate for the nonlinear controlled system it can be shown along the same lines reported in (Hohmann et al., 2000b; Hohmann and Krebs, 2000) that 6.V = V (;..(k + 1)) - V(;..(k)) < O.
where the right hand side is already decomposed into its first order approximate part (linear part) 4n,i;" + bi u and the remaining nonlinear terms Snl ,i(;", u). In the following, we suppose that the coordinates Zrd +1, . .. , Zn have already been chosen, even though they are not yet uniquely defined. However, in section 4, a procedure is outlined, how to choose these coordinates in an optimal way.
Remark 1. It is obvious, if II§.nl (;.., -rT;.. + Vh + v) +!2 Vh l\z = 0 globally, we obtain a globally asymptotic closed loop.
Second, a static state feedback u = -rT;.. + Vh + v is used to obtain a linear closed loop behavior for the shifted output Zl = y(k) , . .. , Zrd = y(k + rd -1). In this control law rT adjusts the desired linear I/O dynamics, v is the new input variable and Vh is determined by
In general , one cannot expect, that (7) can be solved in closed form . On the other hand, an online solution by numerical minimization procedures may be too time-consuming. Therefore, we propose the following alternative, which proved to be a good approximation to the optimal solution of (7) in many practical situations. If (7) is developed in a Taylor series with respect to V h at the last control value vdk - 1), a linear equation in the variable vh(k) is obtained:
Considering the CVT, using the above control law, a closed loop system is obtained, which has an unstable unobservable part (Monaco and NormandCyrot, 1988; Kotta, 1995). 31
~
112nl (~,u) + Qvh(k - 1)+ +
82n~~~' u) I h
(vh(k) - vh(k -
1))11
v h(k-l)
If we suppose that the system (4) is already in the normal form (5) , which is denoted by 1: 1, one has to determine a transformation ~ = t (!l) with
(8) 2
with u = -r.T ~ + vh(k -1) + v. This is a linear least squares problem with respect to Vh which is usually solved in terms of the Moore-Penrose pseudoinverse denoted by [.]+:
vh(k)
= vh(k -
1) -
[82n~~~' u) I h
. {2nl (~, u)
vh(k-l)
-t () !l
[
l 1 -t- ( ) -~
Ird!l(~)
~-~
'
~
Ird~(z) ] ,
[ Cl
--~
(10)
-
where Ird denotes the rd dimensional identity matrix. The functions t n - rd (!l) need to be derived according to
1+
+ QVh(k - In
=
trd+~~.~tn (~) (~l£ (~~n e) ) ,
(ll)
(9) where e is defined as before, but written in T}coordinates. This off-line optimization problem is by no means trivial to solve.
Hence, (9) yields an approximate solution of the time consuming on-line optimization problem, which therefore can be avoided.
Therefore, we propose an indirect way to obtain the appropriate change of coordinates ~ = t (!l) ' The idea is presented in three steps:
In Figure (1) the structure of the controlled system is depicted. Obviously, the control law consists of a change of coordinates, a linear controller and the described optimization procedure (nonlinear controller) .
(1) A system's structure (1: 2 ) is presented, which leads to a vanishing error ei. (2) It is outlined that 1:2 is feedback linearizable using a linear transformation T} = V ~. The resulting system is denoted by t"3 . (3) Conditions for a feedback linearizing transformation ~ = p. (~) are given, such that ~ = t (!l) can be computed.
L
Original system Oin x-coordinates
l
Y:
8yrnes-Isidori
L10(?::~:r~:ates
£=!(~)
£=r6)
Transformation in BymesIsidori normal form
~=('(£)
Fig. 1. Control strategy applied to the CVT
L
~
of
2 Desired system in l1-coordinates
The CVT has a relative degree 1, i.e. the input appears after 1 time shift at the output. Furthermore, the system (4) is already in the normal form (5). But, the second part of the change of coordinates Z2 = Z2(V,Ww) has not yet been specified, which is the objective of the following section.
~=!::~ -
~=fl (~)
=!::-1 11 -
~L
3 Nonlinear controllability normal form in ~-coordinates
Fig. 2. Used coordinate transformations Figure 2 depicts the interdependence between these three steps. The mean square error (7) is zero, if the set of nonlinear equations Snl , i(~ ' u) + bi Vh = 0, i = rd , .. . , n can be solved uniquely with respect to Vh for every v . The solution Vh does not change, if Snl , i(~' u) + b; Vh + biv = 0 is multiplied with a constant 9i . Hence, we search a change of coordinates t (!l) mapping 1:1 into the system 1: 2:
4. REQUIREMENTS FOR A MINIMUM NORM ERROR The nonlinearities in 2nl (~, u) depend on the coordinates Zrd+1 , . . . , Zn which have not been defined so far. Since these nonlinearities influence the compensation error e = IIQVh + 2nl(~, u) 1 12' the objective of this section is to minimize this norm with respect to the coordinate functions Zrd+1, . . . ,Zn' Naturally, we have to minimize the maximum of min e in D ,
T}i(k T}i(k
i.e. the worst case has to be considered.
+ 1) = T}i+l (k) + 1) = ----
32
,i=I, ... ,rd
(12) where
grd' . .. ,gn
First, the first order approximation of (4) at the nominal point WwO = 125.622 and VwO = 2.6 yields:
are constant.
A . = [0.9970.115] 0 1 .
Obviously, (12) is feedback linearizable. If u + dn /(7], u) is substituted by a new input variable, then a similarity transformation 7] = 11 ~ transforms ~2 into linear controllability normal fo;:m ~3, provided the linear part of (12) is controllable. With resubstituting the original input the system appears in nonlinear controllability normal form , the transformation to which is denoted by ~ = cp (~) .
(15)
£24 m
The linear controller r is calculated using Ackermann's formula. The eigenvalues are placed in an iterative design procedure at )'1 = 0.93 and '\2 = 0.92 such that a fast closed loop transition behavior combined with a low undershoot is obtained. Second, basing on the procedure outlined in section
This fact facilitates the situation significantly: The well known feedback linearization transformation can be used for the calculation of (!l) if it fulfills the following structural properties: Substituting 7]= V~ yields -
4 and (Hohmann et al., 2000b), the following change
1.
of coordinates holds for the CVT:
J-1(z)= [8.93 -8.93] . [Zl] 8.90 -7.90 Z2
t ({)= [1.n_~:(t·~) ]= [~d ]~+[tn-~ W] .(13) ~
!:i
The matrix N denotes the linear transformation part, M is a constant matrix and n - rd (~) is the nonlinear transformation part. N renders the linear part of (5) into linear controllability normal form . In a similar way, the inverse ~ = cp-l (~) is separated into its linear and nonlinear-part:
+ [-8.93 0
0 ] -7.90
. [(0.019 ZI Z2 - 0.002 z~ (0.019 ZI Z2 - 0.002 ZI
+ 0.015 z~) ](16) + 0.015 z2)
1] . [Zl] V= [ -0.89 -0.99 1 Z2
(17)
Using
t
yields
~ = t- (~) = V-I [I;;] ~+V-I [tn~:~(~) l'(14) l
-7]
'----'"'" !:i-I
+
where the constant matrix 111 and the nonlinear functions have to be determined. Substituting (14) in 7] = V~ , the transformation Cl (~) is obtained, which we-have looked for.
tion using a nonlinear map € =
(~)
=
I
[ZI]
Z2
+
0 ] [0.0195 Zl Z2 - 0.002 zl + 0.015 Z2 2
2
• (18)
Third, we have to minimize the nonlinearities of the transformed system: Cl (1.. (i (!l) ,u)) - Aun!l' The resulting control law is expressed in Z coordinates A-I by resubstituting
In practice, € = cp- l (~) is obtained by an approximation of a -given feedback linearizing transformaA-I
= 1.A-I (~)
1. :
meeting the
u = -rT ~ + Vh
Remark 2. A practical procedure to derive the transformation Cl (~) without calculating feedback linearization transformation explicitly, is recently reported (see (Hohmann et al., 2000b)).
Vh
= argmin {Cl Vh
(19)
(J- (~, -r ~ + v + Vh)) T
-ARlinc (~) } l
(20)
where A R1in denotes the controlled linear system. 5. CONTROL LAW CALCULATION AND SIMULATIONS OF THE CVT
This controller was implemented using (9). Figure 3 shows the closed loop behavior of the wheel speed. The gear ratio is depicted in figure 4, respectively. The initial state is set to [112 .57,2.30).
The control law design may be summarized in three steps: (1) the transformed system is controlled using a linear controller r (2) the system is transformed in 7]-coordinates with
Remark 3. If the difference between the desired wheel speed and the actual wheel speed is too large, this controller may violate an upper engine speed limit. In this case, the controller switching strategy used by Guzzella can prevent that the gear ratio exceeds its upper limits (Guzzella and Schmid, 1995) .
A-I
1.
(3) the nonlinearities of the controlled system are minimized and the control law is expressed in z-coordinates. 33
input output behavior, while maintaining asymptotic stability of the whole system. Simulations show that the controller applied to the CVT meets the control objectives: A fast non overshooting transient behavior with a prescribed linear input output dynamics.
I2 6 r - - - - - - - = = = = = = = = = = = !
x ,(t)
[rad I s]
124 122 120
7. REFERENCES
2
4
6
8
10 12 t I[s]
14
16
18
Guzzella, L. and A. M. Schmid (1995). Feedback linearization of spark-ignition engines with continuously variable transmissions. IEEE Transactions on control systems technology 3(1), 5460. Hohmann, S. and V. Krebs (2000) . Exakte und niiherungsweise Zustandsriickfiihrlinearisierung. A utomatisierungstechnik (at) . Hohmann, S., M. Sackmann and V. Krebs (2000a) . Nonlinear torque control of a spark ignition engine. In: Proceedings of the 9th IFAC Symposium on Control in Transportation System 2000 (E. Schnieder, Ed.). pp. 617- 622. Hohmann, S., M. Schlotterer and V. Krebs (2000b). Approximate I/O Linearization for Nonlinear Discrete-Time Systems, to appear in the proceedings of CONTROLO 2000 Portugal. Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag, Berlin. Kazantis, N. and K. Kravaris (1997). Systemtheoretic properties of sampled-data representations of nonlinear systems obtained via Taylor-Lie series. International Journal of Control67,997- 1020. Khalil, H. (1996) . Nonlinear Systems. Prentice Hall, Upper Saddle River, New Jersey. Kolmanovsky, Ilya, Jing Sun and Leyi Wang (1999). Coordinated control of lean burn gasoline engines with continuously variable transmissions. In: Proceedings of the American Control Conference , San Diego. pp. 2673- 2677. Kotta, U. (1995). Inversion Method in the Discretetime Nonlinear Control Systems Synthesis Problems. Vo!. 205. Springer-Verlag, Berlin. Monaco, S. and D. Normand-Cyrot (1988) . Zero dynamics of sampled data systems. Systems fj Control Letters 11, 229- 234. Papageorgiou, Markos (1991). Optimierung. Oldenbourg Verlag, Miinchen . Sackmann, Martin and Volker Krebs (1998). Nonlinear control of a continuously variable transmission using hyperstability theory. In: Proceedings of the IAR-Annual Meeting, Mulhouse, France. pp. 38-44. Sackmann, Martin and Volker Krebs (1999). Nonlinear control of a continuously variable transmission using hyperstability theory. In: Proceedings: European Control Conference, Karlsruh e, Germany.
20
Fig. 3. Closed loop behavior of the wheel speed (-) compared to the desired linear system (. - ._) x2 (t) 6 5.5 5 4.5 4 3.5
2
4
6
8
10
12
14
16
18
20
t /[s]
Fig. 4. Closed loop behavior of the gear ratio (-) compared to the desired linear behavior (. _ ._) The dashed line denotes the desired linear dynamics, the solid line represents the controlled nonlinear system. Obviously, the closed loop behavior is approximately equal to the linear dynamics. The nonlinear controller yields a good performance of the closed loop and a fast transition to the steady state. In contrast to existing control strategies, here, the input output behavior simply depends on the eigenvalues of the linear closed loop. Hence, it is relatively easy to obtain the desired input output dynamics, although the system is nonlinear. 6. CONCLUSION In this paper a recently proposed approximate I/O linearization technique is applied to control a CVT, which is not minimum phase. We first reviewed the approximate I/O linearization technique , consisting of a linear and nonlinear control law . It is shown that a change of coordinates, transforming the system into a particular structure, improves the stability properties of the closed loop. Furthermore, standard linear design methods can be used to obtain a linear 34