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Proceedings of the American Control Co nferen ceChicago , Illinois June 2000Adaptive Switching Gain for a Discrete-Time SlidingMode Controller

G. Monsees J.M.A. ScherpenG.Monsees@ITS.TUDelft.nl J.M.A.Scherpen@ITS.TUDelft.nl

Delft University of TechnologyFaculty of Information Technology and SystemsSystems and Control Engineering GroupDelft, The Netherlands

Abst rac t

Sliding Mode Control is a well-known technique capable ofmaking the closed loop system robust with respect to certainkinds of parameter variations and unmodeled dynamics. Thesliding mode control law consists of the linear control par twhich is based on the model knowledge and the discontin-uous control part which is based on the model uncertainty.This paper describes two known adaption laws for the switch-ing gain for continuous-time sliding mode controllers. Be-cause these adaption laws have some fundamental problemsin discrete-time, we introduce a new adaption law specificallydesigned for discrete-time sliding mode controllers.

1 Introduction

Sliding mode control is a well known robust control algo-rithm for linear- as well as nonlinear systems [2],[3],[5],[9].Continuous-time sliding mode control has been extensivelystud ied and have been applied to various applications. Muchless is known of discrete-time sliding mode controllers, inpractice it is often assumed that the sampling frequency issufficiently high to assume that the controller is continuous-time [12].Another possibility is to design the sliding modecontroller in discrete-time, based on a discrete-time model,however stability has not yet been assured [1],[4],[7],[10],[13].This paper focuses on an adaptive switching gain for adiscrete-time sliding mode controller. Section 2 introducestwo known adaption laws for the continuous-time slidingmode controller which are discretized in section 3 to workin discrete-time. T he first adaption law (Method I) has somesevere drawbacks, even in continuous-time, and the secondadaption law (Method 11) has some problems occurring indiscrete-time. To overcome the drawbacks these two adap-tion laws have in discrete-time, a new adaption law specif-ically designed for discrete-time sliding mode controllers isintroduced (Method 111) in section 3.4. Section 4 demon-strates the Method I1 and Method I11 adaption laws by asimple simulation example of a pendulum. Finally section 5presents the conclusions.

2 Continuous-Time Sliding Mode Control2.1 IntroductionWe consider the (single input) system:

x(t) = ( A+ AA)x(t)+ ( B+ AB)u ( t )+ d ( t ) (1)where x ( t ) E R" is the state of the system, u ( t ) E R theinput, d ( t ) E R" an unknown disturbance, (A ,B ) he nominalmodel and (AA,AB ) the modeling error. Th e disturbancesand modeling errors are assumed to be matched, i.e, theyare in the subspace spanned by B and thus can be directlycancelled by the inpu t. For simplicity we will omit th e timevariable t in the rest of this paper. We will now define theswitching surface S by :

S = A e (2)with e = x - xd and A E RIXn. After transforming thesystem to the regular form A can be determined with the aidof a classical control me thod [3].We now define a Lyapunov function V = is2which provesasymptotical stability if

v = ss 5 -qlSI (3)(q > 0) is fulfilled. Inequality (3 ) is called the Sliding Con-dition [8] r Reaching Law (41, t guarantees that the systemwill (asymptotically) reach sliding motion. If we choose aconstant K 2 q we get S = -K sign(S). S ubstituting equa-tions (I), (2 ) and e = x - d in the previous equation resultsin:U = - (A(A+AA)x+ Ad - Axd -K sign(S))A(B+ AB )

(4)where of course (AA ,AB ) and d are unknown. Th e slidingmode controller is now defined as:

(5 )-- ( A A X - Axd) ---K sign(S)AB AB7 7=U , = U d

with AB # 0 which is easily satisfied if B # 0 since A is adesign parameter. The controller part ul is called the linearcontrol part, if the system is perfectly known (and with theproper initial conditions) it keeps the system on the desiredsurface s = 0. The controller pa rt ud is called the discontin-uous control part, it drives the system towards the switching

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surface S = 0. In case A B = 0, the sliding condition (in-equality (3 )) is met if:

K Z 7+ IA(AAx + d)l (6)If the above condition is fulfilled (and the modeling error ismatched) then, once the system is in sliding mode, the sys-tems dynamics become invariant to the modeling errors [9].This rule can be extended to disturbances and uncertaintiesin B , see for example [8].One major problem in determining the switching gain in thisway is that one has to know the bound on the uncertainty.In practice this information may not always be available sothe switching gain has to be determined by experimentation(tuning) or by the use of a n adaption law for the gain. An-other disadvantage is tha t t he bound on t he error is not onlya function of th e model but also on the demands. A slowlyvarying and amplitude limited input signal could result in asmaller error th an a rapidly varying and high amplitude in-put signal. Determining th e switching gain for the worst casewill result in a relatively high switching gain which leads tohigh controller activity. It would be better to have a switch-ing gain which is automatically adap ted t o th e circumstances,two methods methods of doing this are described in the nextsections.2. 2 Adaptive Switching Gain, Method IThe most straightforward adaption mechanism for the switch-ing gain is given by Wang et. al. [ll]: .

K = / S J d t (7)This adaption law is based on the fact that once the switch-ing gain is sufficiently large, the system will be forced to theswitching surface S = 0. However, this adaption law hasthree major drawbacks:(1) n case of a large initial error, the switching gain k willincrease fast due to this error and not because of a model-mismatch. Thi s may result in a switching gain which is sig-nificantly larger then the desired gain.(2 ) Noise on the measurements will prevent S to be exactlyzero so the adaptive gain will continue to increase.(3) he adapt ion law can only increase the gain but never de-crease it. So if the circumstances change such that a smallerswitching gain is permitted the adaption law is not able toadapt to these new circumstances.2. 3 Adaptive Switching Gain, Method I1Another way of determining the switching gain is by the adap-tion law introduced by Lenz et. al. [6]. For this adapti on lawit is necessary to replace the switching term -&K sign(S)in the sliding control law (5 ) by:

if &K S < - @Now, u d steers the system within the boundary region IS1 0 is the adaption constant which determines thespeed of adaption . The term sign(S[k]) sign(S[k- 1) is +1 ifthe system did not cross the switching surface (gain should belarger) and -1 if the system did cross the switching surface(gain should be smaller) so the adaptive gain K is altered inth e oppropriate direction. In the next section it will be showntha t the adaptive gain K will converge to the region:

KO-y < K < KO+ y (12)where KO s the optimal gain which is the smallest switchinggain which results in:

Important to notice is that the previous equation does notsatisfy condition I1 for discrete-time sliding mode. However,th e smallest switching gain which fulfills the above equa tionwill keep th e system close to t he switching surface and t huscan be considered to be a valid (discrete-time) sliding modedefinition. I n th e remainder K O s considered to be constant.

3.4.2 Proof of Convergence: Any switching gainwhich leads to

M-

o = sign(~[kl)ign(S[k - 11) (14)k = l

(with proper initial conditions) is denoted by K,p. The low-est possible gain which fulfills this equation is called the op-timal switching gain KO (which is by assumption constant).Once again y e study th e adaption !aw (11). First it will beshown- tha t K will increasewhen K < KO nd successivelythat K will decrease when K > KO:

1. k [ k- 11< KO: f we will assume th at will reach orpass po in N I time-steps, we have to show:

1 - { k[k]-KO[ k [ k- 11- K,I} < --e (15)N1 k = lwith E = w . his leads to:

1 - { k [ k - 11- k[k]} = -e (16)N1 = lWith equation (11) this results in:

1 N 1c { y sign(S[k]) sign(S[k - ])]= E (17)k = l

As long as the system is not in discrete-time slidingmode, S[k] and S [ k - 11 will most of the time haveequal signs. The above equation implies y = E whichcan easily be satisfied since NI can be chosen a rbitrar ilylarge.

2. k [ k- 11> KO: gain we assume that I? will reach orpass KO n NZ time-steps. We have to show:

1 Nz- { k[k]-KoI - k [ k- 11- KoI} < -E (18)N 2 k = l

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with E = w . ewriting (18) gives:1 Nz {K[k]- K[k - ]} = - (19)

k = lWith equation (11) this results in:

Since the system is in discrete-time sliding mode, u[k]and o [ k- 11 will have opposite signs. T he above equa-tion then implies 7 = E which again can be easily sat-isfied since NZcan be chosen arbitrarily large.

Thus it is shown that K will always move in t he direction ofKObut because of the step sizes y the optimal gain KO annotbe reached exactly. If we assume tha t K [ k - 11 = KO 6 [ k - 11an d 6[ k- 11 approximates zero then K[k] will be:

So we may conclude tha t t he best K an do is to stay withinthe region:KO- y < B < KO+y (21)

3.4.3 Improved Convergence: The adaption con-stant y is a trade-off between switching gain accuracy andadaption speed. In the case that th e circumstances underwhich the system operates do not change this trade-off couldbe by-passed by the introduction of a non-constant adaptionconst ant. If we would st ar t with a relatively high adaptionconstant and the n decrease it, then the optimal gain KO ouldbe reached fast and accurate. Two ways of doing this are:

Time-dependent adaption constant: A straight-forward method of decreasing the adaption constant ismake it a function of time, for example by:

where 7a should be sufficiently large to be able toreach the region ( K - KO(< y[k] before y[k] ha ddecayed to much. Despite the fact th at th is methodis easy to implement it has the drawback that it ispossible that the adaption constant is decceased toofast which may prevent the switching gain K to reachthe optimal value KO as close as possible.Goal-dependent adaption constant: Anotherway of adjusting th e adaption constant is to monitorthe gain K. t is known (see section 3.4.2) hat onceI? in the region [I([.]K,J < y, t will not leave thatregion anymore. This is based on the assumption thatthe optimal gain KO is constant which will generallyyot be the case. One could howeJer monitor whetherK stays within a certain region IK[.]K,I < E, whereK. > y Ifor example IE = lay), for a certain time span.Once K stays within th e specified region for a sufficienttime, the adaption constant could be decreased slowly.

Figure 1: Simulation results for th e Method I1 adapt ion lawwith 9 = 0.5@ and @ = 0.1.

4 Applied to the PendulumIn order to demonstrate th e developed adaption law we willapply it to a driven pendulum [3],[6].The discretized MethodI1 adapt ion law (see section 3.3)will be applied aswell as theMethod I11 adaption law (see section 3.4). The dynamics ofa pendulum with friction can be described by:

.. 1 . . 10 = -- ln(8)- f8+ -U9 m12where 8 is the angle, g is the gravity constant, m the massand I the length of the pendulum. By appropriate scaling thisequation can be reduced to the normalized pendulum givenby:

which we can write in state-space form by:y +azy + a 1 s i n ( y ) = U

where z1 = y and 2 2 = y. In this example a1 an d a2 aretaken to be 0.25 and 0.1 espectively.The sliding mode controller is designed for the discrete-timelinearized model (sampling time T,= 5e-3 s) around theorigin (x = [0 OIT ) represented by equation (9) where A dand & re given by:

1.0 5.0 e- 3 ] Bd= [ 0.0 ]~d = [ -1.2 e-3 1.0 5.0 e-'

The unmodeled nonlinearity can now be considered as amatched disturbance. Th e switching surface is computed bythe use of LQ R techniques, which leads to:

S[k]= Re[k] = [-6.23 - 0.98]e[k]The control law is taken as derived in section 3 wher: theswitching gain K is now adapt ive (i.e. replaced by K[k ] ) .The system has t o track the following signal:

w sin(&)xd = [ cos(wt) ]

so the angle and t he angular velocity of t he pendulum shouldbe a perfect sine- and cosine function of the time (w = g ) .

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Method Im , , . , , , , , , 5 Conclusions

Figure 2: Simulation results for th e Method I1 adaption lawwith 9 = 0.5% and 9 = 0.01.

Figure 3: Simulation results for the Method I11 adaptionlaw with y = le-3.

The controller is tested on a (simulated) system with a,l =$ami and as2 = 2am2 (where the subscript s stands for the(simulated) system and the subscript m stands for the modelwhich is used for the controller design). Th e simulation re-sults for the Method I1 adaption law are given in figures 1and 2. Figure 1 demonstrates that the Method I1 adaptionlaw works satisfactory. The gain adapts very rapidly and th esliding variable S stays within the defined quasi sliding modeband (SI< 9. owever figure 2 shows that if the quasi slid-ing mode band is selected too small, the adaption law willbecome unstable since it is trying to force the system intothe quasi sliding mode band which is no longer achievable.The simulation results for the Method I11 adaption law aregiven in figure 3. Initially t he adap tive gain is rather largebecause of the error in the initial conditions. Once the slidingsurface is reached, the adaptive gain is reduced considerablybecause it only has to compensate for the modeling error.The results are about similar to the Method I1 results with9 = 0.1, Method 111keeps S slightly closer to zero. However,there is no danger of instability. The adaptive gain can befurther smoothened if a smaller adap tion constant y s chosen.

In this paper, a new adaption law for for the switching gainwas introduced. It has been specifically designed for discrete-time sliding mode controllers. This new method proved t ohave an important advantage over the discretized version ofthe adaptive gain introduced in [SI, namely that there is nodanger of instability of the adaption procedure because of abad choice of adaption parameters. With the latter met hoda boundary region (quasi sliding mode band) within which(discrete-time) sliding mode will take place has to be selected.This region can be chosen smaller than achievable in whichthe adaptive gain can grow unbounded. Wit h th e new adap-tion law this can no longer happen. Simulation results visu-alize the above statements.

AcknowledgementsThis work was supported by Brite-Euram under contractnumber BRPR-CT97-0508 and project number BE-97-4186.

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