Sliding Mode Compensation, Estimation and Optimization Methods inAutomotive Control Problems

Ibrahim Haskara∗, Cem Hatipoglu †andUmit Ozguner‡

The Ohio State UniversityDepartment of Electrical Engineering

Columbus, OH 43210

Keywords: Sliding modes, automotive control, distur-bance/state estimation, equivalent control, optimization, fric-tion compensation, traction control, pneumatic throttle actu-ator, internal combustion engine.

Abstract

In this paper, we provide a broad overview of a number of re-cent automotive applications in a tutorial fashion where sev-eral analytical design tools of the sliding mode control the-ory were primarily used. The design methods utilized arefirst discussed from a theoretical point of view in three maincategories: online functional optimization, disturbance/stateestimation and friction compensation. The first automotivecontrol example reported in this paper is a traction controldesign which comprises the presented optimization and esti-mation methods as well as several singular perturbation ar-guments. A position tracking control problem of a throttlesystem having inherent coulomb friction and stiff positionfeedback is then discussed. A previous sliding mode posi-tion tracking control of a pneumatic throttle actuator for aninternal combustion engine is also summarized.

1 Introduction

Generally speaking, automotive control problems are highlynonlinear and subject to high amount of disturbances and un-certainties. In most cases, the system to be controlled mayoperate at diverse operating regimes and include significantnonlinear couplings which make the abundant tools of thelinear control system literature not well-suited for a wide

∗ I. Haskara is currently working at Visteon/Ford Motor Company as aPowertrain Control Systems Engineer; Visteon Technical Center, AdvancedPowertrain Control Systems Group, 17000 Rotunda Drive, Room C.395,Dearborn, MI 48121, Voice: (313) 845-8700, Fax: (313) 322-3529, E-mail:ihaskara@visteon.com;†C. Hatipoglu is currently working for Bendix Commercial Vehicle Sys-

tems, a division of Honeywell International as a Senior Control Design En-gineer; 901 Cleveland Street, Elyria, OH 44035; Voice: (440) 329-9721,Fax: (440) 329-9780, E-mail: Cem.Hatipoglu@honeywell.com;‡U. Ozguner is a Professor of Electrical Engineering and the TRC Inc.

Chair for ITS at the Ohio State University; 2015 Neil Avenue, 205 DreeseLaboratory, Columbus OH 43210, Voice: (614) 292-5940, Fax: (614) 292-7596, E-mail: umit@ee.eng.ohio-state.edu.

range robust operation. On the other hand, sliding mode con-trol theory ([22], [24]) has been investigated in detail over thelast there decades and it currently offers numerous systematicdesign methods applicable to several industrial control prob-lems. The use of sliding mode control ideas in automotivecontrol applications has also been reported (see, for instance,[2], [3], [6], [11], [16], [25], [27]).

This paper presents a couple recent automotive applica-tions which blend several sliding mode control design meth-ods. The theory behind the optimization, estimation and fric-tion compensation tools used in these applications is firstdiscussed in Section 2. The optimization method originatesfrom the on-line unimodular functional optimization methodof [24], [1]. The results of a comphrensive investigation ofthe use of the equivalent control idea for state/disturbanceestimation purposes are next summarized from [24], [4], [7],[8]. The friction compensation method is based on a recentdevelopment where the handling of non-smooth nonlineari-ties operating on manifolds in the state/control space is ex-amined in a broader context via sliding motions [14]. Theoptimization and estimation methods are then used in Sec-tion 3 on a traction control problem where the accelerationcharacteristics of a vehicle are to be optimized in an enginecontrol framework via dynamic spark advance. The frictioncompensation method is exemplified on a position trackingcontrol problem of a throttle system with inherent coulombfriction and stiff position feedback in Section 4. Finally, Sec-tion 5 summarizes a previous sliding mode position trackingcontrol design for a pneumatic throttle actuator of an internalcombustion engine.

2 Sliding Mode Design Methods

2.1 On-line Functional Optimization

To a certain extent, several automotive control objectivescan be formulated as an optimization problem. For exam-ple, ABS/traction control can be designed to robustly oper-ate around the minimum/maximum point of the tire force-relative slip curve, engine should deliver the desired torquewith the least possible fuel consumption, EGR input needsto be determined so as to minimize the emission forma-tion and so on and so forth. The use of sliding modes for

on-line optimization of an analytically unknown unimodularfunctional has been reported in [24]. The basic idea is tomake the optimization variable (the signal which is desiredto be optimized) follow an increasing/decreasing time func-tion via sliding mode motions. The main difficulty with sucha setup is that the unknown gradient term multiplies the con-trol at the differential equation of the optimization variableso that the system itself possesses a variable structure behav-ior. This idea has been extended in [1] with the introductionof the notion of periodic switching function and applied toABS/traction control problems in [2], [11] and [25]. Next,we briefly discuss the basics of this optimization method.

Consider a unimodular functionaly = f(x) which has aunique extremum at the point(x∗, y∗). The mathematical ex-pression off(x) is unknown. For definiteness, the extremumis selected as the maximum which turns the optimization ob-jective into a maximization one.x is assumed to be the out-put of an integrator which takesu as its input. The controlobjective is to keepx at the vicinity of the unknown optimalx∗ by modulatingx by u using the on-line values ofy. Theperformance output (optimization variable),y, is forced totrack an increasing time function irrespective of the unknowngradient information via sliding modes. Pick any increasingfunction g(t) and try to keepf(x) − g(t) at a constant bya properu. If so, f(x) increases at the same rate withg(t)independent of whetherx < x∗ or x > x∗. To this end, let

s = f(x)− g(t) (1)

so that

s = (∂f/∂x) u− g(t) (2)

With the control law of

u = Msgn sin(2πs/α) (3)

as in [1] with α being a small positive constant, a slidingmotion occurs forM |∂f/∂x| > |g(t)| andx is steered to-wards x∗ while y tracking g(t). The region, defined by|∂f/∂x| < |g(t)|/M , quantifies the region in whichx willbe confined with this control. The idea can be extendedto more general dynamics by adding the derivatives of theperformance variable as well as those ofg(t) to the slidingmanifold expression so as to compensate the relative degreedeficit. In [12], this optimization idea has further been devel-oped for on-line operating point and set point optimizationpurposes by ending up with a two-time scale sliding modeoptimization design. The resulting method allows the op-timization of the closed loop operation of a system by ex-ploiting the extra degree of freedom in the available controlauthority possibly in a different time scale.

2.2 Disturbance Estimation and Compensation

One way of enhancing the robustness of a control system isto estimate the discrepancies between the model used for thecontrol derivation purposes and the actual system by a per-turbation estimator and to incorporate this information into

the control law in a proper way. To this end, we next presenttwo ways of disturbance estimation. The first one is basedon the equivalent control methodology as in [24] and it is inthe continuous-time domain. A discrete-time sliding modedisturbance estimation method is also discussed.

First, consider a SISO nonlinear system

x = f(x) + g(x)u + δ(t, x, u) (4)

wherex ∈ R is the state,u ∈ R is the control,f(x), g(x)are smooth, known functions andδ(t, x, u) is the disturbancefunction which lumps all the disturbances and the uncertain-ties of the system. It is assumed that

|δ(t, x, u)| ≤ ρ(t) (5)

whereρ(t) is a known bounding function. The objective isto estimateδ(t, x, u) from x.

The disturbance estimator is given by

˙x = f(x) + g(x)u + (ρ(t) + η) sgn (x− x) (6)

which basically repeats what is known about the system withan additional discontinuous injection. The error dynamicsfollow from the subtraction of Eq. (6) from Eq. (4) as fol-lows:

ex = δ(t, x, u)− (ρ(t) + η) sgn ex (7)

whereex = x − x. Ideally ex = 0 ∀t ≥ t0 with η > 0,x(t0) = x(t0) so that

[(ρ(t) + η) sgn ex]eq = δ(t, x, u) (8)

The operator[•]eq outputs the equivalent value of its dis-continuous argument which is defined as the continuous in-jection which would satisfy the invariance conditions of thesliding motion (ex = 0, ex = 0) that this discontinuous inputinduces. The equivalent value operator,[•]eq, can be approx-imately realized by an high bandwidth low-pass filter accord-ing to the equivalent control methodology ([24]); i.e,

τ v + v = (ρ(t) + η) sgn ex

v = δ(t, x, u) +O(τ, ε/τ) (9)

where |ex| ≤ ε ∀t ≥ t0 with ε being an arbitrarily smallpositive number.

Assume that the disturbance is also differentiable; i.e,

δ(t, x, u) = ∆(t) (10)

|∆(t)| < ρ(t) (11)

whereρ(t) is a known bound. The derivative of the distur-bance function can be obtained by

˙δ(t) = K sgn (δ − δ)˙δ(t) = K sgn ([(ρ(t) + η) sgn ex]eq − δ)

whereK = ρ(t) + κ, κ > 0. Therefore, in sliding mode,

[K sgn ([ (ρ(t) + η) sgn ex]eq − δ)]eq = ∆(t) (12)

The design is recursive. Equivalent control operators per-form information transfer between two consecutive steps andthe design logic can be repeated to estimate higher orderderivatives of the disturbance function as long as it is contin-uously differentiable to a certain order. However, the overalldesign requires the implementation of the sequential equiv-alent value operators. The approximability of the equivalentcontrol by low-pass filtering was proven in [24]. The relationbetween the estimation accuracy and the filter time constantsin the implementation of the sequential equivalent value op-erators by low-pass filters was examined in [8]. In that paper,an ultimate boundedness analysis was carried out for the es-timation errors and a theoretical rule of thumb was proposedfor the selection of the filter time constants.

Suppose that a baseline control law,un(t, x), has alreadybeen specified such that it would achieve the control objec-tive for the nominal system. The control can then be comple-mented with the estimated disturbance as follows:

u(t, x) = un(t, x)− g−1(t, x)v (13)

wherev is obtained from Eq. (9). With this new control, theclosed loop dynamics are only affected by the residual es-timation error which is naturally easier to deal with a lessconservative control action. Generally, if the disturbanceis matched with respect to the control, a direct cancellationterm can be added to the nominal control law to preserve therobustness whereas for mismatched disturbances the nomi-nal control needs to be devised so as to allow freedom for theuse of the disturbance estimates. A recent study where theseideas were elaborated in detail can be found in [10].

Most of the today’s control algorithms are implemented indiscrete-time. However, the discrete-time implementation ofa continuous sliding mode control law may cause the well-known chattering problem if no chattering reduction methodis employed. As an alternative estimator design where thesampling issues are taken into account at the first place, thecontinuous-time system of Eq. (14)

x = u(t) + δ(t) (14)

is first discretized so as to obtain

xk+1 = xk + Tuk + T δk (15)

wherexk = x(kT ), uk = u(t) for kT ≤ t < (k + 1)T , T isthe sampling time,

δk =1T

∫ (k+1)T

kTδ(t)dt (16)

and it is assumed that the control is applied through a zero-order-hold. Note that,δk cannot be computed unless thefuture values of the external disturbance functionδ(t) areknown. However, ifδ(t) is smoothδk can be predicted byδk−1 which can be computed from

δk =1T

[xk − xk−1]− uk−1 (17)

with anO(T ) accuracy according to

δk − δk−1 =1T

∫ (k+1)T

kTδ(t)dt− 1

T

∫ kT

(k−1)Tδ(t)dt

|δk − δk−1| = 2Tδmax = O(T ) (18)

where|dδ/dt| < δmax.Suppose that the control objective is to regulatex to zero.

Incorporating the estimated disturbance to the control law

uk = − 1T

xk − δk−1 (19)

one getsxk+1 = T [δk − δk−1] = O(T 2) (20)

so that at each sampling instantx is actually forced to anO(T 2) vicinity of zero. This is an increase in performancecompared to the direct discretization of a discontinuous slid-ing mode control law which would achieve only anO(T )accuracy. A detailed study of this idea can be found in [18],[19], [20] and [21] where it has also been shown that thiscontrol leads to anO(T 2) accuracy in sliding motion alsoduring the inter-sampling behavior for sampled-data systemswith control being applied through a zero-order hold.

2.3 State Observation

In many industrial applications, the on-line estimation of sev-eral signals by an observer rather than using a sensor maylead to more sophisticated and cost effective control systems.There are several state observer design methods reported inthe sliding mode control literature. The discussion of allthese methods are beyond the scope of this study. Instead, inthis paper, we are specifically interested in a sliding mode ob-server design method which uses the equivalent control ideaas in the continuous time disturbance estimation method ofSection 2.2. Next this method is presented similar to its orig-inal which was reported in [24]:

Consider a linear system

x = Ax + Buy = Cx (21)

wherex ∈ Rn, u ∈ Rp, y ∈ Rm, the pair(A,C) observable,C has full rank. This system can be transformed into

y = A11y + A12x1 + B1ux1 = A21y + A22x1 + B2u

(22)

whereA11, A12, A21, A22, B1 B2 are constant matrices ofappropriate dimensions. The observer equation for the firstpart of Eq. (22) is selected as follows:

˙y = A11y + B1u + L1sgn(y − y) (23)

Error dynamics are given by

ey = A11ey + A12x1 − L1 sgn ey (24)

whereey = y − y. A sliding motion occurs oney = 0 infinite time with a suitableL1 and in sliding mode

[L1 sgn ey]eq = A12x1 (25)

Therefore, an information onx1 is indirectly availablethrough a low-pass filter.

At the second step, consider

x1 = A22x1 + A21y + B2u

y1 = L−11 A12x1

This reduced order system can also be transformed into

y1 = A31y1 + A32x2 + A33y + B3u

x2 = A41y1 + A42x2 + A43y + B4u

Let the second observer equation be

˙y1 = A31y1 + A33y + B3u + L2 sgn(y1 − y1) (26)

Replacingy1 with its equivalent in Eq. (25) for implementa-tion, ideally a sliding motion can be guaranteed ony1− y1 =0 in finite time as well. This design routine can be repeatedso as to result in a full finite time converging observer. Thedetails and the formulatization of this method can be foundin [4], [7], [8] among others. As in the disturbance estima-tion design, the effects of the repeated use of low-pass fil-tering on the overall estimation accuracy were quantified in[8] in terms of a single variable which parameterizes all thefilter time constants. In [4], [7], a discrete-time equivalentcontrol based sliding mode observer design method was alsoproposed. Next, this method is summarized.

Consider a discrete-time linear system

xk+1 = Φxk + Γuk

yk = Cxk(27)

wherex ∈ Rn, u ∈ Rp, y ∈ Rm, the pair(Φ, C) observableandC has full rank.

The discrete time equivalent control definition of [23] isused for a dual discrete-time design. Transform the originalsystem of Eq. (27) into

yk+1 = Φ11yk + Φ12x1,k + Γ1ukx1,k+1 = Φ21yk + Φ22x1,k + Γ2uk

(28)

and let the corresponding discrete time sliding mode observerbe

yk+1 = Φ11yk + Φ12x1,k + Γ1uk − vk

x1,k+1 = Φ21yk + Φ22x1,k + Γ2uk + Lvk(29)

Error dynamics are as follows:

ey,k+1 = Φ11ey,k + Φ12ex1,k + vk

ex1,k+1 = Φ21ey,k + Φ22ex1,k − Lvk(30)

whereey,k = yk−yk andex1,k = x1,k−x1,k. The equivalentvalue ofvk can be calculated by solvingey,k+1 = 0 for vk

([23]) as follows:

vk,eq = −Φ11ey,k − Φ12ex1,k (31)

A sliding motion occurs oney = 0 in finite step ifvk = vk,eq.In sliding mode

ex1,k+1 = (Φ22 + LΦ12)ex1,k (32)

To implement the observer, we define an auxiliary system

zk+1 = (Φ22 + LΦ12)zk + (Φ21 + LΦ11)ey,k−1 − Ley,k (33)

and replacevk by

vk,eq = −(Φ11 − Φ12L)ey,k − Φ12(Φ21 + LΦ11)ey,k−1

−Φ12(Φ22 + LΦ12)zk(34)

By placing the eigenvalues of(Φ22 + LΦ12) at the origin,vk,eq → vk,eq andey → 0 in finite step. Note that thez-dynamics replace the low-pass filtering of the continuous-time equivalent control based observer design.

2.4 Friction Compensation

This section summarizes the theory behind the friction com-pensation method of [13], [14]. To this end, we first discussthe system induced manifolds concept and the generalizedstiction phenomenon.

Consider the following class of systems

x = f (x, t) + µ · sgn (s (x, t)) + h (x, t)u (35)

with right hand-side discontinuities on thek surfaces,

s (x, t) =[

s1 (x, t) , s2 (x, t) , · · · , sk (x, t)]′

= 0

whereµ = [µ1 · · · µk] ∈ Rn×k, h(x, t) ∈ Rn×p, f(x, t) ∈Rn and the entries off andh are smoothly differentiablefunctions (inCn), u ∈ Rp is the control input ands : Rn →Rk. The signum operator is defined to operate on every entryof its argument.

The system as given in Eq. (35) appears in the form of annth-order system with an input that hask + p componentskof which have already been specified in the form of slidingmode control withµ andσ = {x ∈ Rn : s(x, t) = 0}being the gain and the manifold, respectively. Note that, thisis not exactly the case as these manifolds have not been de-signed and the associated gains have not been selected by thedesigner. Instead, they have been induced by the system it-self. However, this sort of analogy allows us to analyze thesystem using the mathematical tools of sliding mode theory.

Consider one of the candidate stiction manifolds, namelysj for j = 1, · · · , k. Under the assumption of the existenceof sliding mode, i.e. when

sj · sj < 0; j = 1, · · · , k (36)

the system starts to slide on the manifold described by,

σj = {[x1 · · · xn−1 xn] ∈ Rn | sj(x, t) = 0} (37)

Note that the condition of Eq. (36) defines an open regionAj

in the state space. This region can be found by analyzing thederivative ofsj for j = 1, · · · , k

sj =ddt

sj (x1, · · · , xn−1, xn) = qj (x1, · · · , xn−1, xn)(38)

whereqj ∈ R when confined to the trajectories describedby Eq. (35), i.e., when the equality in Eq. (35) is used toreplace the derivatives of the states in Eq. (38), becomes afunction of the statesx1, · · · , xn, the control inputu(t) andthe combination of the discontinuities given on the right handside of the system description.

sj = qj (f1(x), · · · , fn(x), sgn(s1(x)), · · · , sgn(sk(x)),u1, · · · , up, x) + gj (x) sgn(sj(x)) (39)

wheregj : Rn → R is the gain multiplying thejth dis-continuous component, andqj : Rn → R. Then,(sj · sj)becomes negative if,

−gj (x, t) > |qj(·)| (40)

∀x ∈ Rn. The open regionAj is then described by,

Aj = {[x1 · · · xn−1 xn] ∈ Rn | − gj (·) > |qj (·)|} (41)

Recall that this analysis is prior to the controller design.When the condition of Eq. (40) is satisfied for somej, thesystem trajectories of Eq. (35) will get stuck atsj = 0 whichis not a designed manifold. This phenomenon is induced dueto the inherent right hand side discontinuities existing in theoriginal system. Hence, an open stiction region can now bedescribed in the state space as follows,

Rs =k

⋃

j=1

(σj ∩ Aj) (42)

Note that,Aj could differ from an empty set to the entirestate space, but in general describes an open region which isa subset ofx ∈ Rn. It is also affected by the magnitude ofthe control input being generated.

So far, the definition of “generalized stiction” has beengiven. Next, a sliding mode controller design approachwhich guarantees the avoidance of generalized stictionRs\(Rs ∩ Rc) ⊂ Rn in tracking problem for a class ofsystems whereRc is the controlled manifold.

Consider now the following class of SISO systems in theircompanion forms with right hand-side discontinuities on thep surfaces,

x(n) = f (x) + µ · sgn (s (x)) + h (x) v (43)

v = u, y = x

whereµ = [µ1 · · · µp] ∈ R1×p, h(·), f(·) : Rn → R aresmoothly differentiable functions and moreoverh(·) 6= 0 forany x = [x, x, · · · , x(n−1)]T (controllability condition overthe entire state space),u, v ∈ R, u is the control input. It is

allowed that there are uncertainties inf(·) and/orµ, and onlysome nominal valuesf(·) and µ are known with boundederrors∆f(·) = |f(·)− f(·)| and∆µ = |µ− µ|. The controlobjective is to generate a control inputu such that the outputy = x tracks the reference signalxr. Define the trackingerror ase = x− xr.

The controller creates multi-layer quasi-sliding manifoldsso as to compensate for the uncertainties. To this end, let

vd =1

h(·)

{[

x(n)r − κ1e(n−1) − · · · − κne

]

(44)

−f(·)− µ · sgnk(s(·)) + w}

wherew is a fictitious input,sgnk(x) = (2/π) arctan(kx)is a smooth approximation for the signum function withΦk(x) = sign(x) − signk(x) denoting the approximationerror.

Selectσ1 = e(n−1) + c1e(n−2) + · · · + cn−2e + cn−1esuch thatσ1 = 0 will exhibit stable dynamics with negativereal poles. Then pick,

w = (κ1−c1)e(n−1)+· · ·+(κn−1−cn−1)e+κne−β·sgnk(σ1)(45)

which will ensure(σ1 · σ1) < 0 for all |σ1| < γ, providedthatk is picked large enough and

β > ∆f(·) + |µΦk(s(·))|+ |∆µ · sgnk(s(·))|+ ε (46)

Defining the sliding manifold byσ2 = v − vd and pickingthe corresponding control inputv as follows:

u = v = −α · sgn(σ2), s.t. α > (|vd|+ ε) (47)

then σ2 · σ2 < 0 is ensured. In sliding mode,x → xd atthe desired rate. A smooth approximation has been used forthe signum functions of the first layer to assure thatvd isalso bounded arounds = 0 (which may be the case if thesignal to be tracked lies on the hyper-surface described bythe inherent discontinuity or requires crossing the mentionedhyper-plane). Note that the described control law will guar-antee that the system trajectories will be directed towards thesubspace described by|σ1| ≤ γ on then-dimensional statespace. The magnitude ofγ can be manipulated by the de-signer, but cannot be explicitly made zero.

3 Sliding Mode Traction Control

The acceleration characteristics of a vehicle can be improvedin an engine control setup where the dynamic spark advanceis used to dynamically modulate the engine torque [6]. Theprimary reason for wheel spin due to sudden changes in theengine air input is closely related to the tire force characteris-tics of the wheel. The tire force/relative slip curve has ideallyonly one extrema for each of the acceleration and the brak-ing regions. A sufficiently large throttle input might causethe relative slip to move into the positive feedback regionwhere the tire force is decreasing with increasing slip. Since

the availability of a relative slip measurement and an accu-rate analytic expression of the tire force/relative slip curveare quite unrealistic in the current setup the sliding mode op-timization method of [1] were utilized in [11], [25] to ro-bustly operate around the peak driving tire force without anya priori information on the tire force/relative slip curve.

In this section, the previous results on this topic are sum-marized from [11], [25]. The first control logic was to devisean optimal law for the spark angle input in the form of an en-gine torque multiplier so as to keep the relative slip around itsoptimal which would produce the maximum traction force.Then, the same idea is used to optimize the performance ofa baseline dynamic output feedback spark advance controller(DOFSAC) ([6]) with no additional sensor inputs.

3.1 The Model

The plant model includes a static engine torque map, a firstorder transmission model, a nonlinear longitudinal tire forcemodel and the vehicle is considered as a point mass with anaerodynamic drag force. The intake manifold dynamics areneglected for simplicity and no driver model is utilized. Themodel equations can be written as follows:

V = a1V 2 + b1Fd(t, σ)

w = a2w + a3Ψ + b2Fd(t, σ)Ψ = a4n + a5w

n = a6n + a7Ψ + b3Te(n,Θ)u (48)

whereV is the longitudinal speed,w is the wheel speed,Ψ is a transmission variable,n is the engine shaft speed,σ = w/V − 1 is the slip,a1 = −(Aρ/Jv), b1 = (re/Jv),a2 = −(Bw/Jw), a3 = (KT Kg/Jw), b2 = −(re/Jw),a4 = 1, a5 = −Kg, a6 = −(Be/Je), a7 = −(KT /Je),b3 = (1/Je) with relevant physical parameters and the effectof spark retard is modeled as a variable engine torque mul-tiplier denoted byu. The value ofu are then translated intospark timing information using an approximate static map.

The engine-wheel coupling through transmission resultsin a two-time scale behavior. For controller design purposes,this characteristic was utilized to further reduce the order ofthe actual model based on the singular perturbation theory[15]. The slow system dynamics are summarized as follows:

Vs = a1V 2s + b1Fd(t, σs)

ws = a2ws + b2Fd(t, σs) + a3Te(ws,Θ)us (49)

whereVs, ws, σs and us are the slow components of thevariablesV , w, σ and u, respectively,a2 = −(Bw +BeK2

g )/(Jw + JeK2g ), b2 = −re/(Jw + JeK2

g ), a3 =Kg/(Jw + JeK2

g ).

3.2 Sliding Mode Dynamic Spark Advance Controller

The engine RPM and the throttle input are assumed to beavailable measurements whereas an analytical expression forthe tire force/relative slip curve as well as the optimal slip

are unknown. The original control problem of robust opera-tion around the optimal slip is formulated as an optimizationproblem of an analytically unknown criterion using the op-timization method summarized in Section 2.1. The controldesign is carried out on the slow system and the tire force isobtained by the estimator of Section 2.2.

The sliding surface is selected as follows:s = e +∫ t0 Λ(e(τ))dτ where e = Fd − F r

d (t), F rd (t) is a user-

specified explicit time function andΛ(e) is to be chosen. Ifs can be kept constant, the constrained motion satisfies

dedt

+ Λ(e) = 0 (50)

and the tire force behaves as desired with proper selectionsof F r

d (t) andΛ(e). The error variable is governed by

dedt

=1V

∂Fd

∂σ[A(w, V, Fd) + B(w, Θ)u] +

∂Fd

∂t− dF r

d

dt(51)

where

A(w, V, Fd) = a2w + b2Fd − a1wV − b1FdwV

(52)

B(w, Θ) = a3Te(w, Θ) (53)

Let A(•) = A + ∆A, 0 < Bmin < B(•) < Bmax whereArepresents the nominal part ofA whereas the unknown term∆A is bounded according to|∆A| ≤ δA with δA known andB =

√BminBmax for which β−1 ≤ (B/B) ≤ β where

β = (Bmax/Bmin)1/2.The control law is selected asu = −B−1[A + γ Φ(s)]

whereγ = βδA + (β − 1)|A|∞ + M with anM > 0 andΦ(s) = sgn sin(2πs/α) is the periodic switching function[1]. This selection guarantees thats is kept atkα for somek which depends on the system and the initial conditions, ifthe following sliding mode existence condition is satisfied:

1V

∣

∣

∣

∣

∂Fd

∂σ

∣

∣

∣

∣

Mβ−1 >∣

∣

∣

∣

∂Fd

∂t− dF r

d

dt+ Λ(e)

∣

∣

∣

∣

(54)

If F rd is chosen as a constant,Λ(e) = λe with a λ > 0 and

also assume that explicit time dependence of the tire force isnegligible, the sliding mode existence condition becomes

1V

∣

∣

∣

∣

∂Fd

∂σ

∣

∣

∣

∣

Mβ−1 > λ |e| (55)

In sliding mode, ifF rd can be reached the tire force con-

verges exponentially towards it with a rate dependent uponλ . On the other hand, ifF r

d > Fd,max the tire force be-haves as before until it enters the region where the gradientis too small such that the sliding mode existence conditionof Eq. (55) can no longer be guaranteed. After that, the sys-tem becomes uncontrollable and the tire force behaves ar-bitrarily. However, the controller creates a region of attrac-tion around the maximum point whose width can be con-trolled by M . Consider the region|∂Fd/∂σ| ≤ ∆. Forany controller parameterM > 0, there exists a∆ given byM > Mδ = V λ|Fd − F r

d |max/∆ such that the tire force isguaranteed to be kept in this region.

3.3 Optimal Sliding Mode DOFSAC

The original DOFSAC ([6]) controls the engine torque out-put via dynamic spark advance based on filtered engine RPMmeasurement. Engine RPM is filtered with a band pass filterfor practical differentiation purposes and then it is comparedto a constant threshold value. If the filter output is greaterthan the threshold, the spark timing is retarded proportionalto the error. This decouples the high energy terms of theengine from the wheel so that the likelihood of wheel spinreduces. However, the threshold value needs to be selectedand originally it is tuned in advance for different conditionsthrough simulation studies and experimental tests.

Vehicle

SLIDING MODE

OPTIMIZER

(s)DifferentiatorK1- u

Engine Torque

Multiplier

Throttle Angle

Engine RPM

Set Point

0.5

0.5

Σ+

−

Figure 1: Optimized DOFSAC (from [11], [25])

This section summarizes an optimal sliding mode DOF-SAC design where basically an additional loop is de-vised for the threshold so as to maximize the tire force(Figure 1). To this end, DOFSAC is first modeled byu = 1−K(n− ρ) = ρ−Kn whereρ denotes the thresh-old andK is selected such that the errorK(n − ρ) is prop-erly mapped to a value in the admissible control domain toproduce a control within its limits. Using the singular per-turbation theory, the order of the complete system with thecontrol in the loop can also be reduced as in Section 3.2 asfollows:

Vs = a1V 2s + b1Fd(t, σs)

ws = a2ws + b2Fd(t, σs) + a3Te(ws,Θ)ρs (56)

with state dependenta2, b2 anda3. Repeating the design ofSection 3.2, the set point is obtained fromρ = −B−1[A +γ Φ(s)] wheres = e +

∫ t0 Λ(e)dτ , Λ(e) is to be chosen and

γ = βδA + (β − 1)|A|∞ + M . With a sufficiently largeM , this set point selection forms a positively invariant re-gion around the optimal slip defined by|∂Fd/∂σ| ≤ ∆. Insliding mode,e + Λ(e) = 0 and the tire force converges tothis region as desired with properF r

d andΛ(e). The size ofthis region can also be controlled byM . Onceρ has been de-terminedρ can be computed usingρ = K−1(ρ− 1). Furtherdetails of the presented traction control designs as well as thesimulation results can be found in [11].

4 Friction Compensation for the PositionControl of a Throttle System

The system involves a plant which is driven by an actua-tor with faster dynamics. The plant has inherent coulomb-viscous friction and stiff position feedback which are the twosources of stiction in the state space.

+ +1 1 11f ( )

xx.

z z+

2

h( x , z )

u ω ωω

Coulomb Friction

Stiff Position Feedback

Plant

Actuator..

s s

f (h(.), )3 ω

s

f ( x )

Figure 2: The system to be controlled (from [14])

4.1 The Plant Model

Consider the plant depicted in Figure 2. The state space rep-resentation is given by

x = (1/Kg)f1(ω) (57)

ω = (1/J) (h(x, z)− (C/Kg)ω − (1/Kg)f3(h(x, z), ω))

z = (1/L) (−Rz −Ktω + u)

where

f1(ω) = deadzone(ω,±δ, 1) (58)

h(x, z) = Ktz − (1/Kg) (f2(x) + K (180x/π − θo)) (59)

f2(x) = γsat ((180x/π − θo) α) (60)

f3(h(.), ω) = βsat (f1(ω)/δ) + βsat(Kgh(.)f4(f1(ω))) (61)

f4(f1(ω)) = 1− (relaywdzn(f1(ω),±δ, 1))2 (62)

wherex is the position,ω is the angular velocity, andz isthe auxiliary state variable that describes the dynamics of thefirst order actuator. Due to the physical limits of the systemthe position variablex(t) should be between7◦ and85◦. Thereferred nonlinearities, the desired tracking signals and theparametric values can be found in [14].

4.2 An Approximate Model for Control Design

Although friction has been modeled in details so as to includethe Stribeck effects as well as stick-slip behavior, it is con-cluded that a simpler model suffices to describe the motionof the system with good precision while easing the controllerdesign phase. Consider,

x = a12ω (63)

ω = a21(x− xo) + a22ω − κsgn (x− xo)

−µsgn(ω) + a23z

z = a32ω + a33z + bu

Figure 3: The pneumatic throttle system (from [16])

where a12 = (1/Kg), a21 = −(K/KgJ), a22 =−(C/KgJ), a23 = (Kt/J), a32 = −(Kt/L), a33 =(R/L), b = (1/L), xo = (πθo/180), κ = (γ/KgJ) andµ = (β/KgJ). The unforced system (whenu = 0) con-verges the stable equilibrium point given by(x, ω, z)eq =(x∗o, 0, 0) wherex∗o = {x ∈ R : |x − xo| ≤ ζ} in thesliding mode sense starting from any initial conditions dueto the existence of the discontinuous terms on the right handside of the state space representation in Eq. (63). Based onthe numerical data, it has been observed that the coupling onthe third equation is weak so that thez term in the secondequation can be replaced withz = −(a32/a33)ω accordingto the singular perturbation theory.

4.3 Controller Design

Let ex = x − xr wherexr is the reference to be followed.From (63), one obtains,

x = a12 (a21(x− xo) + (a22/a12)x + · · · (64)

· · · −κsgn(x− xo)− µsgn (x/a12) + a23z)

Following the design method of Section 2.4, the controllaw is governed by

zfl = zfl + (w/a12a23) (65)

zfl =1

a12a23[xr + ξ1xr + ξ2xr + a12a21xo+ (66)

· · · − (a12a21 + ξ2)x− (a22 + ξ1)x +

· · ·+ a12κsgn(x− xo) + a12µsgn(x/a12)]

sw = ex + C1ex

w = (ξ1 − C1)ex + ξ2ex − Msgnk(sw)sz = z − zfl

u = −Msgn(sz)

(67)

whereC1 > 0, M > |Φ(·)| with Φ(·) being the lumpeduncertainty originating from the bounded uncertaintiesin the plant parameters,M is sufficiently large positivenumber so as to induce a sliding motion onsz = 0. Thespeed information required for the control implementation

is obtained by an equivalent control based observer whosedesign idea has been presented in Section 2.3. The detailsof the overall control design summarized above and thesimulation results can be found in [14].

5 Position Control of a Throttle System

This section presents a previous throttle angle position con-troller developed at the Ohio State University as a part ofan Intelligent Vehicles and Highway Systems study. The ex-isting pneumatic throttle actuator of a 1992 Honda Accordstation-wagon was controlled for vehicle speed control pur-poses.

5.1 Throttle Actuator Model

Figure 3 shows a simplified diagram of the throttle actuatorsystem including the throttle actuator, throttle cable, and thethrottle plate connection. The throttle actuator is a pneumaticcylinder which creates a force proportional to the ratio of thecylinder’s internal air pressure to the external (atmospheric)air pressure. The internal air pressure is controlled using twovalves which allow either the engine’s intake manifold pres-sure or the atmospheric pressure be applied to the input ofthe air cylinder. One cable connects the pneumatic cylinderto the accelerator pedal while a second cable connects theaccelerator pedal to the throttle plate. The actuator’s internalpressure is controlled using three solenoid actuated valveswhich control the air flow in and out of the pneumatic cylin-der. The throttle angle is controlled by opening or closing thevent and vacuum valves until the internal air pressure that isneeded to move the throttle angle to the desired position isachieved.

For control law derivation purposes, two separate secondorder linear models were experimentally determined, one ofwhich is valid when the vent valve is open and the other whenin full vacuum mode. If both the vent and vacuum valvesare closed the system is assumed to behave according to theunforced vacuum model.

5.2 Sliding Mode Control of the Throttle Angle

Due to the nature of the control input to the throttle actu-ator system being either full vacuum or full vent, a slidingmode design was adopted. The sliding surface was definedass = e + ke with e = θ − θdes, θ is the throttle angle indegrees andθdes was the desired throttle angle.u takes twovalues,+1 and−1, which represent full vacuum or full vent,respectively, depending ons.

The region in which the sliding mode existence conditioncan be guaranteed to hold with the available control author-ity was determined by considering the worst-case scenarios.This region is clearly affected by the value ofk. For im-plementation,k was selected to have a reasonable decay insliding mode by giving up the global sliding mode existencealthough it was possible to select ak which would providethe control objective globally. In order to implement the con-trol, θ, which was not directly available, was estimated by alinear Kalman filter. Since there were two possible linearsystems, the filter parameters were determined individuallyand switched according to the input. The further details ofthe design as well as the experimental results were reportedin [16].

6 Concluding Remarks

The usage of sliding mode estimation, optimization and com-pensation methods in automotive control problems have beendemonstrated on there different examples: a traction controldesign for anti-spin acceleration, a tracking control designfor a throttle system subject to stiction nonlinearities and aposition tracking control design for a pneumatic throttle sys-tem of an internal combustion engine. The theoretical back-ground and the relevant literature on the sliding mode designmethods used have also been reported for an easy reference.

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