A Feedforward-Feedback Interpretation of a Sliding Mode Control Law Monsees, Govert; George, Koshy; Scherpen, Jacquelien M.A.; Verhaegen, Michel

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1 University of Groningen A Feeforwar-Feeback Interpretation of a Sliing Moe Control Law Monsees, Govert; George, Koshy; Scherpen, Jacquelien M.A.; Verhaegen, Michel Publishe in: Proceeings of the 7th Meiterranean Conference on Control an Automation (MED99) IMPORTANT NOTE: You are avise to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the ocument version below. Document Version Publisher's PDF, also known as Version of recor Publication ate: 1999 Link to publication in University of Groningen/UMCG research atabase Citation for publishe version (APA): Monsees, G., George, K., Scherpen, J. M. A., & Verhaegen, M. (1999). A Feeforwar-Feeback Interpretation of a Sliing Moe Control Law. In Proceeings of the 7th Meiterranean Conference on Control an Automation (MED99) (pp ). University of Groningen, Research Institute of Technology an Management. Copyright Other than for strictly personal use, it is not permitte to ownloa or to forwar/istribute the text or part of it without the consent of the author(s) an/or copyright holer(s), unless the work is uner an open content license (like Creative Commons). Take-own policy If you believe that this ocument breaches copyright please contact us proviing etails, an we will remove access to the work immeiately an investigate your claim. Downloae from the University of Groningen/UMCG research atabase (Pure): For technical reasons the number of authors shown on this cover page is limite to 1 maximum. Downloa ate:

2 A Feeforwar-Feeback Interpretation of a Sliing Moe Control Law Govert Monsees y Koshy George z Jacquelien M.A. Scherpen x Michel Verhaegen { Faculty of Information Technology an Systems Delft University of Technology Abstract In this paper we provie a feeforwar-feeback interpretation of a sliing moe control scheme. Given a esire trajectory, the feeforwar signal is generate using a stable inversion metho, an the feeback signal inclues the switching term of the sliing moe control law. In this manner, we introuce robustness into the stable inversion technique. This approach ismotivate by the nee to replicate time signals typically in the automobile inustry. The application of such aninterpretation to a quarter car benchmark moel yiels encouraging results. Special attention will be given to non-minimum phase systems illustrate by a simulation example of the lunar roving vehicle. 1 Introuction Sliing Moe Control is a well-known technique capable of making the close loop system robust with respect to certain kins of parameter variations an unmoele ynamics. Sliing moe controllers have proven their eectiveness in several applications iverse in their physical nature an funamental purpose. Specically, this metho provies an easy way to esign trajectory tracking control laws for a plant, linear or nonlinear. Increasingly stringent performance requirements to be satise in a variety of applications justify the fact that precision output tracking remains one of the funamental problems for control engineers. In the context of linear systems, it is well-known that perfect tracking is relatively easy to achieve in minimum phase systems. However, output tracking for non-minimum phase systems remains a challenging problem ue to the funamental limitations on the transient tracking performance characterize by the number an location of the zeros which are non-minimum phase (Qiu an Davison, 1993). For linear continuous-time systems, Francis an Wonham (1976) show that the asymptotic tracking problem is solvable if, an only if, a set of linear matrix equations is solvable. This was later generalize to nonlinear systems by replacing the linear matrix equations by a set of rst orer partial ierential equations (Isiori an Byrnes, 199). These approaches asymptotically track any member in a given family of signals This work was supporte by Brite-Euram uner contract number BRPR-CT97-58 an project number BE y g.monsees@its.tuelft.nl z k.george@its.tuelft.nl x j.m.a.scherpen@its.tuelft.nl { m.verhaegen@its.tuelft.nl 2384

3 y Fee Forwar + System y Fee Back Figure 1: Sliing moe control ivie in two parts. generate by an exosystem. The stable inversion approach was introuce by Devasia et al. (1996) to avoi the use of exosystems, an, in the case of non-minimum phase systems, improve the transient performance by using pre-actuation. In Section 2 we summarize this stable inversion approach for nonlinear, possibly non-minimum phase, systems. It has been suggeste (Devasia et al., 1996) that the inverse trajectory becomes a feeforwar signal use in conjunction with a more conventional feeback control law in orer to make it attractive. The stable inversion approach to output tracking is base on the asumption of perfect knowlege of the system. In the more realistic case of parametric variations an unmoele ynamics, this approach to the problem seems rather ill-pose. In Section 3, we combine the avantages of the sliing moe an inversion base control techniques to esign a control law that not only yiels perfect tracking in the case where perfect knowlege of the plant is available, but as well makes the system robust to matche parameter variations an uncertainties. By proviing a feeforwar-feeback interpretation of a typical sliing moe control law, we fulll our task in a rather convenient way. This interpretation is illustrate in Fig 1. In this gure the sliing moe control law is expresse as a sum of a feeforwar signal an a feeback signal. We compute the feeforwar signal using the inversion base technique, an the feeback signal inclues the switching part of the sliing moe controller. Thus, the feeback part of the signal attempts to compensate for the inaequacies of the nominal moel of the plant. Thereby, weintrouce robustness in a natural way to the stable inversion technique which eviently epens on the parameters of the system. We note that this paper was motivate by the nee for time waveform replication which concerns with accurate reprouction of real or synthesize target time histories. Thus complex vibration environments (such as automobile crashes) may be recreate in a test laboratory by simulating el measurements thereby saving precious resources. Other applications inclue urability tests of, for instance, automobile components, an riving comfort assessment. We emphasize that other techniques, for instance (J. Sjoberg an Ararwal, 1985; Spall an Cristion, 1998; Hjalmarsson et al., 1998), coul be use to compute the feeback signal. In each of these techniques the controller structure is typically xe a priori an the controller parameters are optimally ajuste using the knowlege of the graient of some cost function compute in ierent ways. We note that in these techniques the graient is evaluate experimentally, an therefore no knowlege of the error ynamics is require. However, several experiments are neee, an hence the convergence to the optimal parameters takes some time. On the contrary, no specic controller structure nee to be chosen to compensate for specic errors in the case of sliing moe control laws. Preliminary investigations on the application of the combine stable inversion an sliing moe techniques to a quarter car benchmark moel are encouraging (Section 4.1). Inee, the 2385

4 sliing moe is foun to cope reasonably well with nonlinearities an parameter variations espite the fact that the feeforwar signal has been compute using only the linearize moel. Simulation results for a non-minimum phase system (Section 4.2) show that although the feeforwar controller can be compute by a stable inversion technique, the close loop system (close with the sliing moe terms) still coul show an unesire oscillatory behavior. 2 Inversion-Base Control Law The stable inversion approach of Devasia et al. (1996) aims at proviing a boune inverse of nonlinear, possibly non-minimum phase systems. This boune inverse is compute by solving a two point bounary value problem obtaine via a ichotomic split of the internal ynamics of the system. This results in an acausal input in the case of non-minimum phase systems; the anti-causal part of the input sets up the esire initial conition. In this section we summarize this stable inversion approach. We note that for minimum phase systems this inverse is relate to the classical system inverse (Hirschorn, 1979). Consier the following SISO nonlinear system ane in the input: _x(t) = f(x(t)) + g(x(t))u(t) y(t) = h(x(t)) (1) where x(t) 2 IR n an f(x), g(x), an h(x) are smooth functions ene on an open set in IR n (the time variable will be omitte in the rest of the paper). The nonlinear system (1) is sai to have awell-ene relative egree r at a point x if L g L k f h(x) = for all k<r,1, an for all x in a neighborhoo of x, an L g L r,1 f h(x ) 6=. Without loss of generality, we assume x =. Given a smooth esire trajectory y 2L 1 \L1, the stable inversion problem (Devasia et al., 1996) is to n a boune u an a boune x such that u (1) = an x (1) = which satises (1) an perfect tracking y() =y ()isachieve. Let 1 = y _y y (r,1) T where y (i) enotes the ith time erivative ofy, an 2,ann,rimensional function on IR n,be chosen such that! 1 = (x) (2) 2 forms a change of coorinates. In these new coorinates, the essence of the system ynamics is represente by: The feeback control law y (r) = ( 1 ; 2 )+( 1 ; 2 )u (3) _ 2 = s 1 ( 1 ; 2 )+s 2 ( 1 ; 2 )u u =,1 ( 1 ; 2 ) y (r),( 1 ; 2 ) is well-ene in a neighborhoo of the origin an partially linearizes the system. Uner this conition, 1 = 1 = y _y y (r,1) The zero ynamics riven by the reference output trajectory is therefore T _ 2 = s 1 ( 1 ; 2 )+s 2 ( 1 ; 2 )u=s( 2 ;Y ) (4) 2386

5 where Y () represents 1 () an y (r) (), an u =,1 ( 1 ; 2 ) y (r),( 1 ; 2 ) The inversion problem is thus reuce to ning boune Caratheoory solutions (Hale, 198) to the possibly unstable non-linear ierential equations _ 2 = s( 2 ;Y ) subject to the bounary conitions 2 (1) =. It is shown (Devasia et al., 1996; Devasia an Paen, 1998) that boune solutions exist provie certain regularity conitions on the Jacobian linearization A s of s( 2 ;Y ) are met: s satises a local Lipschitz conition, the ierence between s an A s is boune, an A s is hyperbolic in the sense that it has no eigenvalues on the imaginary axis. Uner these conitions, starting from an initial guess 2,we iteratively solve for 2 as follows: o i+1 2 (t) = 2();Y i (),A s 2() i Z1,1 (t, ) n s Here is the boune state transition matrix associate with A s. We note that the convergence of the sequence f2 i g is guarantee by the regularity conitions state earlier. Once the esire 2 is compute, the esire state trajectory is obtaine by the inverse coorinate transformation x =,1 ( 1 ; 2 ) an the esire input trajectory by u = b,1 (x ) y (r), a(x ) where a(x) =( (x)) an b(x) =( (x)). We note that a major rawback of the inversion base approach is that it is applicable for systems that o not have zeros on the imaginary axis, an oes not take into account the noise in target time histories. A metho that overcomes these problems is suggeste in (George et al., 1999) for linear iscrete time systems. However, the feeforwar controller for the case of nonminimum phase systems generates a signal that is anti-causal, an therefore, such schemes are applicable where trajectory preview is permitte, or in situations wherein the performance with a truncate anti-causal feeforwar signal is reasonable. In this section we have summarize the stable inversion technique for nonlinear systems. The necessary fee-forwar input can be compute by obtaining rst the esire state trajectory. We emphasize that the stable inversion technique yiels perfect tracking for systems where there are no parametric variations or unmoele ynamics. In the next section we introuce robustness through a sliing moe control law. 3 Sliing Moe Controller In this section we present a feeforwar-feeback interpretation of a sliing moe controller. We introuce a sliing moe controller which turns out to be a combination of the inversion-base control law ene in the previous section an a feeback term (Section 3.1). Despite the fact that we have a stable inversion technique for inverting non-minimum phase systems, Section 3.2 points out that this oes not entirely solve the problems for output-base sliing moe controllers in relation to non-minimum phase systems. 3.1 Sliing Moe Control Law We now esign a sliing moe control law for the system (1). Several enitions for the sliing surface exist. In (Slotine an Li, 1991) the sliing surface is ene as the weighte sum of the (5) 2387

6 errors between the esire states an the actual states. In this paper we use the enition given in (Lin, 1994): s = e (r,1) + c r,1e (r,2) + +c 1 e+c Z et (6) with e = y, y (y being the actual system output an y the esire system output) an the constants c i are such that r r + c r,1 r,1 r + +c 1 +c is a Hurwitz polynomial. The (assume to be) constant r is the relative egree of the system (1) which was introuce in the previous section. The avantage of a sliing surface as ene in (6) compare to one ene in terms of the states is that it is a function of the known output of the system, instea of the generally unknown states. The erivative of sis given by: _s = e (r) + c r,1e (r,1) + +c 1 _e+c e By substituting equation (3) (with a(x) =( (x)) an b(x) =( (x))) with e (r) = y (r), y (r) in the above equation we get: If we nowchoose u to be: _s = a(x)+b(x)u,y (r) u = b,1 (x) ( y (r), a(x), +c r,1e (r,1) + +c e {z } = P r,1 i= c ie (i) r,1 X i= c i e (i), ksgn(s) ) (7) we fulll the sliing conition 1 s 2,jsj (8) 2 t if k>. If we have a closer look at (7) we recognize that one part of it is the inversion-base control law given in equation (5 ). Therefore: u = u inv + u sli (9) where u inv = b,1 (x) ny (r) u sli =,b,1 (x) ( o, a(x) ksgn(s)+ r,1 X i= c i e (i) ) (1) (11) The expression u inv is ientical to the expression for u in the previous section (equation 5) so the same techniques can be applie to etermine this control law. The expression for u sli represents the sliing moe strategy. The switching term,b,1 (x)k sgn(s) rives the system to the sliing surface s an the sliing term,b,1 (x) P r,1 i= c ie (i) \slies" the system along the R sliing surface to the esire (time-varying) position [y (r,1) y t] T. The switching term,k sgn(s) will introuce high frequency components to the control signal which may excite high-frequency ynamics which were neglecte in the moeling proceure 2388

7 (Slotine an Li, 1991). One way of reucing this eect is by \softening" the switching law, by for example a saturation function as ene below: u s =,b,1 (x)ksat( s )= 8 > < >:,b,1 (x)k s>,b,1 (x)k s jsj< b,1 (x)k s<, The high frequency components in the control action are now reuce but stability is only guarantee outsie the region (Slotine an Li, 1991). Also the eect of chattering is reuce by the use of this \softene" switching law. The switching gain k is a trae-o between accuracy an actuator emans. The gain shoul be as high as possible to be able to compensate for large moeling errors. On the other han the gain shoul be as small as possible to relax the controller emans. For this reason we want the gain k to be large enough to satisfy the sliing conition (we call this value the optimal gain k o ). Three ways to etermine k are: Known maximum error: If we assume that there is a moeling error in a(x), we can reene a(x) by: a(x) =^a(x)+a(x) where ^a(x) is the nominal value of a(x) an a(x) is boune to some constant D (thus ja(x)j < D). k shoul then satisfy k + D to fulll the sliing conition (8) (Lin, 1994). This rule is extene to uncertainties in b(x) in (Slotine an Li, 1991). By Experimentation: In practice, where we generally o not have a known boun on the moel-error, the parameter k can be tune in a few experiments. Aaptive Gain: In (Lenz et al., 1998) an aaptive gain k is introuce. The switching term b,1 (x)k sgn(s) is replace by the term b,1 (x)k sat( s ) an the gain k is given by: k = Z (jsj, ) t where is constant. In the formula for k one can see that jsj takes care of the increase of k an takes care of the ecrease of k. In (Lenz et al., 1998) the proof is given that k will converge to the optimal gain k o provie that < < (Lenz et al., 1998). A sliing moe controller can compensate for any matche uncertainty (uncertainties are \matche" if they lie in the subspace spanne by a basis of g(x)). However, in the presence of unmatche uncertainties, it is not possible for the sliing control law to steer the system in such a way that the variable s will be arbitrarily small. The aaption mechanism will break own in these circumstances since the gain will increase until s will stay within the ene limits which might not happen. 3.2 Non-minimum phase systems Sliing moe controllers base on output information cannot be applie to non-minimum phase systems (Ewars an Spurgeon, 1998),(Lin, 1994). One reason for this limitation lies in the fact that output base sliing moe controllers have (as we have exploite in this paper) a 2389

8 similar structure as inversion base controllers (Lin, 1994). If a non-minimum phase system is straightforwarly inverte then the zeros of the system become the poles of the controller an hence we have esigne an unstable controller. This problean be solve by the use of stable inversion techniques. The question is now of course whether we have solve the limitation of sliing moe controllers to minimum phase systems or not. The answer seems to be no, which is also emonstrate with the simulation results for the lunar roving vehicle (Section 4.2) which is a nonminimum phase system. By applying the stable inversion technique we have esigne a stable controller. But unless the feeforwar controller completely cancels the non-minimum phase behavior of the system, the sliing moe feeback term will still act on a non-minimum phase system an therefore the use of the sliing moe controllers shoul still be limite to minimum phase systems. Interesting to note is that (as can be seen in Section 4.2) the close loop system is marginally stable, the close loop system will oscillate but oes not become unstable. 4 Simulation Results The feeforwar-feeback interpretation will be emonstrate by the use of two examples. The rst example is the quarter car moel which escribes one quarter of a car place on a testbench. The secon example is the angle control of the lunar roving vehicle which shows a non-minimum phase behavior. 4.1 Application to the Quarter Car Moel In this section we will apply the feeforwar-feeback interpretation of a sliing moe controller introuce in the previous section to a special application in the inustry of automobiles. To improve reproucibility of test proceures for cars as well as urability tests of new evelope cars, one woul like tohave a test setup where a car is place on four shakers (actuators) calle the base, which simulate the behavior of the car on the roa. In other wors, the shakers shoul be controlle in such away that the car vibrates in the esire way. This \esire way" is given as a reference signal which coul be measure on a car uring riving conitions. In this paper we only look to one quarter of the car (which is the reason for the name \Quarter Car Moel"), only one wheel is consiere. A full-car moel woul be compose of four (or in the case of trucks even more) Quarter Car moels which are connecte in a fairly complex way to represent the interepenencies between the \Quarters". We moel the tire bya simple spring (with stiness c w ) an the connection of the wheel to the chassis by a combination of a (nonlinear) spring an a (nonlinear) amper (stiness c c an amping coecient c ), gure 2 emonstrates this. The variables x c, x w an x b represent the car, wheel an base isplacement respectively. The input to the system is the base isplacement (x b ), the acceleration of the car (x c ) is the output. Where: _x = f(x; u) = Ax + ^f(x)+bu (12) y = h(x) = Cx + ^h(x) (13) x = [x c x w _x c _x w ] T 239

9 m c x c c c c m w x w c w x b Figure 2: Mechanical iagram of the Quarter Car Moel. A = ^f(x) = B = C = , cc c c m w c c, cc+cw m w, c c m w c, c m w 3 7 5, cc (x c, x w ) 2, c (_x c, _x w ) 2 cc (x c,x w ) 2 + c (_x c, _x w ) 2, c c c c c T w m w, c c ^h(x) =, c c (x c,x w ) 2, c (_x c, _x w ) 2 Note that the parameters an etermine the importance of the (quaratic) nonlinearities. If = an = the moel becomes a linear moel. The erivative of yis obtaine by ierentiating (13) once with respect to time: _y = CAx + C ^f(x)+ _^h(x) {z } = a(x) + {z} CB u = b(x) where it is interesting to note that in this case b(x) is a constant, i.e. b(x) =b. From the above we can conclue that the Quarter Car has relative egree r = 1 so we ene the sliing surface as: s = e + c with e = y, y an c>to satisfy the requirement that + c is a Hurwitz polynomial. For the erivative of swe can write: Z et _s = _e + ce = a(x)+bu, _y + ce

10 acceleration [m/s 2 ] y an y, feeforwar Sliing variable s, feeforwar acceleration [m/s 2 ] y an y, sliing moe Sliing variable s, sliing moe.1.1 s s x 1 3 Control input x b x 1 3 Control input x b 1 1 x b x b time [s] time [s] Figure 3: Simulation results with switching gain k = 5. The left gures represent the situation without feeback, the right gures represent the propose strategy. The top gures isplay the esire an the actual system output (y an y), the mile gures present the sliing variable (s) an the lower gures isplay the input (u = uinv + usli). 2392

11 acceleration [m/s 2 ] y an y, feeforwar Sliing variable s, feeforwar acceleration [m/s 2 ] y an y, sliing moe Sliing variable s, sliing moe.2.2 s s x 1 3 Control input x b x 1 3 Control input x b x b x b time [s] time [s] Figure 4: Simulation results with switching gain k = 5 for a perio of 1 secons. The left gures represent the situation without feeback, the right gures represent the propose strategy. The top gures isplay the esire an the actual system output (y an y), the mile gures present the sliing variable (s) an the lower gures isplay the input (u = uinv + usli). 2393

12 We nowchose u = u inv + u sli where u inv is given by (1) an u sli by (11). The parameters use in simulation are: = 2=19, m w = 33=3, c c = 9=8, c w = 2=22, c = 12=13, w = =, = 5= an = 5= (where the rst value is use for the system an the secon value is use for the inversion-base control law). One can see in the parameters that the inversion-base controller is esigne on a linear moel with perturbe parameters. Since there are unmatche uncertainties we cannot use the aaptive gain mechanism as was also iscovere in simulations (results are not shown for brevity), so we have selecte a constant gain. To reuce the loa on the actuators we have selecte the saturation function (sat( s )) instea of the signum function. The parameters for the controller are: c =1, = :1 an k = 5=5 (rst/secon/thir experiment). Figure 3 (result for k = 5) shows a remarkable improvement of the tracking performance, things look goo. Figure 4 shows the results for k = 5 (i.e. same circumstances as in gure 3) for a longer time perio, here again some peaks in s appear. This coul be the result of the fact that k is still to small but it coul also be cause by the unmatche uncertainties. The gures also show the input signals for both the open loop case (only inversion-base control law) as the propose control law. These gures look quite similar, there are no substancial high frequency terms on top of the inversion-base control law. So we may conclue that even in the case where the inversion-base controller is esigne on a linear moel with perturbe parameters, the systems tracks rather well together with the sliing moe feeback term. The use of the saturation function has resulte in a smooth control signal. A higher gain increases the performance but the error cannot be mae arbitrarily small since there are unmatche uncertainties. 4.2 Lunar Roving Vehicle We will now apply the feeforwar-feeback interpretation of a sliing moe controller to the angle control of a moon vehicle. This systean be escribe by the transfer function (Dorf an Bishop, 1995): G(s) = 2e,:1s :2s+1 After approximating the elay by a secon orer Pae approximation the following state-space moel can be foun: _x = Ax + Bu y = Cx where x 2 R 3, y 2 R, u 2 R an A = 2 6 4,65,15, B = C = [1, 6 12] The poles of this system are at s =,5 an s =,3 17:3i so the open loop system is stable. The zeros of the system are at s =317:3i so the system is non-minimum phase. Figure 5 shows the simulation results for the (stable inversion base) feeforwar controller to the vehicle uner ieal circumstances. Since the relative egree of the system is r =1we ene the switching surface s like in the quarter car example by: s = e + c with e = y, y an c> to satisfy the requirement that + c is a Hurwitz polynomial. Z et 2394

13 1.2 Feeforwar 1.8 y an y u f time [s] Figure 5: Simulation results for the Lunar Roving Vehicle in feeforwar conguration uner ieal circumstances (no noise an no moel mismatch). The top gure presents the esire an the actual output. The lower gure presents the (boune) input signal. Figure 6 shows the simulation results for the feeforwar only an the sliing moe controller conguration (k = :9, c = 1 an = 1e,3 ) where the controllers are esigne on the above moel but they are applie to the linear system with the matrices A s = A + A, B s = B + B an C s = C where: A = :5, B = 4,:1 The gures emonstrate that although the stable inversion technique generates a boune input signal which woul give perfect tracking uner ieal circumstances, the feeforwar-feeback implementation of a sliing moe controller oes not yiel the esire robustness. In fact, the close loop system will start oscillating aroun the esire position. In this example it can harly be a surprise that the sliing moe controller runs into problems. A sliing moe controller expects instantaneous switching which is in a system with a elay of course not possible. 5 Conclusions It was emonstrate that one can easily ivie a sliing moe control law in a feeforwar term an a feeback term. The feeforwar term is ientical to an inversion-base control law an the feeback terontains the aitional sliing moe terms. A metho was emonstrate to etermine the inversion-base control law for nonlinear an possibly non-minimum phase systems. The sliing moe feeback law was then ae to improve the robustness of the controller. Results were emonstrate in simulation for an inversion-base control law esigne for a linear moel an applie to a nonlinear system, for this purpose a Quarter Car moel was use. The tracking performance is consierably improve with the ae sliing moe feeback terms. Also simulation results were shown for the lunar roving vehicle which is an example of a non-minimum phase system. The results showe that although the stable inversion technique

14 feeforwar sliing moe angle [ra] angle [ra] s s Control input u.5 u u time [s] time [s] Figure 6: Simulation results for the sliing moe controller applie to the Lunar Roving Vehicle with switching gain k = :5. The left column of gures represent the feeforwar controller only an the right column of gures represent the sliing moe controller results. The top two gures represent the esire an the actual output, the mile gures represent the variables s an the lower gures represent the total input signal in sliing moe (utot = uinv + usli). 2396

15 results in a boune input signal which yiels perfect tracking uner ieal circumstances (perfect moel an no isturbances), the close loop system suers from a limit cycle behavior. Further research is require to solve this problem. Acknowlegments At this point we woul like to thank C. Ewars for the fruitful iscussions about outputbase sliing moe controllers applie to non-minimum phase systems an J. e Cuyper for the iscussions on splitting sliing moe controllers into feeforwar an feeback parts. References Devasia, S., D. Chen, an B. Paen (1996). \Nonlinear inversion-base output tracking," IEEE Transactions on Automatic Control, Vol. 41, no. no. 7, pp. pp. 93{942. Devasia, S. an B. Paen (1998). \Stable inversion for nonlinear nonminimum-phase timevarying systems," IEEE Transactions on Automatic Control, 43, no. 2, pp. 283{288. Dorf, R. an R. Bishop (1995). Moern Control Systems, Aison-Wasley, 7th en. Ewars, C. an S. Spurgeon (1998). Sliing Moe Control, Theory an Applications, Taylor & Francis Lt. Francis, B. an W. M. Wonham (1976). \The internal moel principle of control theory," Automatica, 12, pp. 457{465. George, K., M. Verhaegen, an J. M. A. Scherpen (1999). \Stable inversion of MIMO linear iscrete time non-minimum phase systems," Submitte to The 7th IEEE Meiterranean Conference on Control an Automation. Hale, J. (198). Orinary Dierential Equations, Robert E. Krieger Publishing Company, New York, USA, 2n en. Hirschorn, R. M. (1979). \Invertibility of nonlinear control systems," SIAM Journal of Control an Optimization, 17, no. 2, pp. 289{297. Hjalmarsson, H., M. Gevers, S. Gunnarsson, an O. Lequin (1998). \Iterative feeback tuning: Theory an applications," IEEE Control Systems. Isiori, A. an C. Byrnes (199). \Output regulation of nonlinear systems," IEEE Transactions on Automatic Control, 35, no. 2, pp. 131{14. J. Sjoberg an M. Ararwal (1985). \Moel-free controller tuning for non-linear systems," Preprint. Lenz, H., R. Berstecher, an M. Lang (1998). \Aaptive sliing-moe control of the absolute gain," in Nonlinear Control Systems Design Symposium Lin, C. F. (1994). Avance Control Systems Design, Prentice Hall. Qiu, L. an E. J. Davison (1993). \Performance limitations of non-minimum phase systems in the servomechanism problem," Automatica, 29, no. 2, pp. 337{

16 Slotine, J. an W. Li (1991). Applie Nonlinear Control, Prentice Hall. Spall, J. an J. Cristion (1998). \Moel-free control of nonlinear stochastic systems with iscretetime measurements," IEEE transactions on automatic control, Vol. 43, pp. pp. 1198{