FRICTION COMPENSATION USING LINEAR CONTROL METHODS

Friction is one of the most frequently appearing sources of problems in mechanical systems. While several controller designs exist that successfully mitigate or compensate the friction e ects in an application, they usually rely on a thoroughly identi ed model. The focus of this work is on verifying the suitability of a linear dual-loop controller for friction compensation in an arbitrary system. It does not require detailed estimation of a friction model for the design. The controller is complemented by a Luenberger observer, to improve the quality of state feedback. Based on di erent model reduction approaches, the control architecture is e ectively applied to a number of system descriptions employing various friction models. Through simulation, the versatility of the idea is reinforced. By comparing the control quality with that of approbated designs taken from recent literature, we show that the concept is competitive with model-based friction compensators. The advance with respect to robustness and disturbance rejection is shown. In preparation for an implementation, the control algorithm is discretized. To predict the practical performance of the controller, it is shown that added noise does not deteriorate the excellent results in the digital realm.


List of Figures
While, generally speaking, the term friction may refer to phenomena that involve uids, in the context of classical mechanical systems theory it describes the force counteracting the relative movement of two solid bodies whose surfaces are in contact. This force depends on various aspects like the condition of the contact surface, the mass of the moving objects or the relative velocity (see [8]).
There are dierent friction eects that can be distinguished.
A fundamental insight characterized by Richard Stribeck almost 120 years ago, is that the occurring friction force at a constant relative velocity is highly nonlinear for small velocities. While for relative movements close to the standstill the impact of friction is very large and for higher velocities it depends almost linearly on the velocity (viscous friction), there is a very distinct recession of the friction force for small velocities. This is referred to as the Stribeck eect (see [9]).
Over the years, dynamic aspects of friction have been described in more and more detail. The most obvious friction phenomenon encountered in any practical application is that the force required to overcome the standstill and start moving is much higher than the force that has to be applied to maintain a constant velocity when sliding. Ernest Rabinowicz investigated this circumstance in 1951 and discovered that the friction force reaches its maximum for small relative displace- ments (see [10]). This is known as static friction or sticking and it sparked the idea of friction being modeled as a function of the relative position of the bodies. The subsequent developments in research led to the evolution of a number of friction models with varying focuses on the emulation of certain phenomena.
All of the rst models were only static and tried to describe the basic dependence of the friction force on the velocity or the displacement. These models focus on depicting the aforementioned eects and to this end employ a characteristic curve, which usually captures the relationship between friction force and velocity.
This characteristic curve imposes a nonlinearity on the model of a mechanical sys- tem. An example for such a curve is shown in Figure 1.1. It illustrates the Stribeck eect as well as the high force to overcome static friction and the linear behavior at higher velocities.
Often the decisive factor for the improvement of a friction model is the accuracy with which the nonlinear eect of friction in a system is described. Since static models fail to capture the intricacies of friction phenomena in detail, dynamic characterization of friction has shifted to the center of attention. The model developed by Phil Dahl in 1968 laid the foundation for future dynamic models (see [11]). It has the general form (1.1) Its main advantage over previous models is that it takes the surface deformation of the bodies subject to friction into account, an idea that was inspired by the characteristics of elastic solids. This observation allows for the dynamics to accurately represent dynamic friction eects not only when sticking is predominant.
In particular, hysteresis or frictional lag can be described, a phenomenon that occurs because the friction force changes at dierent rates depending on whether the velocity decreases or increases.
Obviously, the Dahl model is dependent on the relative displacement and only the sign of the velocity. This property, which is also referred to as rate independence, makes it impossible to include the Stribeck eect or the large force necessary to overcome sticking, in the model. Accordingly, it was rened and modied over the years. There are two more recent types of models that are descendants of the Dahl approach.
The models developed by Bliman and Sorine in the nineties essentially combine multiple conventional Dahl models with dierent parameters to capture the static friction and Stribeck phenomenon next to the eects of surface deformation (see [12]). Their dynamic nature covers hysteretic phenomena of friction as well.
Another idea focuses on the idea of the deformed surface being modeled as an elastic solid.
Before sliding between two bodies can occur and actual friction eects are relevant, the force is treated as the force acting on a spring under tension. The respecitve model developed by DeWit, Olsson, Åström and Lischinsky is known as the LuGre friction model and it is widely used in engineering and science (see [13]).
Its basic concept is to think of the material surface as if it was covered with ne bristles, which act like springs. The deection z of these springs can be described by the nonlinear dierential equatioṅ z = v − |v| g(v) z, (1.2) where g(v) is used to describe the complicated eects at low velocities: Therein F C is the Coulomb force, that models the friction force present for the bodies in relative motion. F S describes the static friction, the force that has to be overcome for the relative motion to transcend from surface deformation to sliding.
Lastly v s is the Stribeck velocity, the relative velocity for which the inuence of the Stribeck eect is the strongest. The friction force then can be calculated as follows: (1.4) The parameter σ 0 is the spring stiness, σ 1 their damping coecient. σ 2 models viscous friction for the case of sliding. In the illustration in Figure 1.2 the two dierent states that the model wants to reproduce are depicted. Sticking happens as long as the relative displacement and the rate of change of the force is small enough to mainly deform the material, whereas sliding, and therefore viscous friction, happens if the breakaway force is overcome. In the rst case, the force results from the deection of the bristles that act like springs. Once the relative velocity is high enough to cause sliding, the actual friction eect entails the counteracting force.
One of the main benets of the LuGre model in the context of engineering and control in particular, is that the model is relatively simple while covering most eects that are crucial for high precision feedback control. Those  Stribeck eect and the breakaway force at low velocities, as well as the hysteretic behavior of systems subject to friction. For higher velocities the model captures viscous friction too (see [13]). An excellent overview of friction eects and the development of friction models is provided in [8].
Obviously, there are a number of dierent phenomena that are covered by the general term friction. All these phenomena have a more or less substantial impact on the proper operation of mechanical engineering applications. Particularly when high precision in control systems is required, as it is often crucial in the context of positioning and tracking, the nonlinearity that arises from the presence of friction calls for intricate measures to deal with the eects described above.

Friction Compensation
There are numerous dierent approaches to the problem of friction compensation in feedback control. While all of them strive to reduce the disturbing inuence of the nonlinearity on the behavior of the control circuit, they do exhibit dierences depending on the respective focus on robustness, eciency, accuracy and the like. In most cases, the goal is to mitigate the inuence of friction or sometimes even suppress it to an extent where the controlled system can be treated as linear.
These objectives translate, for example, to the requirement of a small steady-state error or the prevention of limit cycles in dynamic terms.
A very rst control strategy that has been used for almost 80 years and in modied forms is still encountered to this day, is the dither (see [8]). A dither is a high frequency oscillation that is applied to the controlled plant to knock it loose. The intention of this is to overcome the static friction level at any given time of a movement, so the actual motion of the system is only governed by viscous friction, which is linear as a good approximation. Besides the obvious problem of unpredictability and the energy consumption of the dither signal, there are more aws to this approach, like the amount of modication to the system or acoustic noise, which make it rather unattractive for many applications.
Some of the more recent methods focus more on the actual model that describes the friction eects. Above all, there are friction observers or, more precisely, nonlinear friction observers (see [7] and [13], [14], [15]). Strictly speaking, an observer is not a controller. However, since a number of friction compensation approaches employ an observer, the concept is outlined here.
Various observer designs have been proposed throughout the years and even though they are wide-ranging in structure and determine the friction estimate in dierent ways, they all share the same idea. It is shown in Figure 1.3.
Analogously to the traditional idea of observers, they simulate the friction behavior based on the plant input and output. As a result, the observer estimates the current value of the friction force to be used in the control procedure. The necessity of this simulation-based approach becomes particularly apparent when dealing with dynamic friction models like the LuGre model. Since the additional dynamic state that describes the surface deection is intrinsically impossible to measure, a simulation is the only way of getting a grasp of its magnitude.
For friction observers that are based on a dynamic model, the estimation of this state is paramount. When recapitulating the LuGre model, we immediately notice that the nonlinearity directly aects the bristle deformation z. Therefore, if this state is to be calculated, the observer is consequently nonlinear itself, which may entail common problems with nonlinear systems, like diculty of stability analysis and controller design, limit cycles or unpredictable behavior.
The development of modern computers gave rise to a new class of nonlinear controllers, which implement more complicated algorithms with a higher demand on fast and precise computation. Although nonlinear controllers are still rather exotic in most elds, there are several examples of nonlinear design strategies being applied for friction compensation, like for example feedback linearization (see [4]).
The idea here is to transform the system equations in a way that allows for a nonlinear controller to be created that achieves the closed-loop behavior of a linear system. That controller naturally has to be nonlinear itself. While the linear overall system has the obvious benet of being suitable for a large number of well studied control approaches and is easy to predict and assess, the technique is prone to errors. Imprecise parameter evaluation or an inaccurate model can quickly lead to undesired behavior or even instability (see [16]).
Another nonlinear control procedure is called backstepping (see [17]). It is based on systems in strict-feedback form, namelẏ Starting from the rst equation and treating x 2 as its input, a nonlinear controller is designed that yields an asymptotically stable equilibrium for the respective subsystem. After plugging in the stabilizing input, this procedure is repeated from the inside out, hence the name backstepping, untilẋ The advantage of this technique is the systematic design procedure, while the drawback is the limited predictability of, and inuence on the control quality (see [16]).
A large class of methods is united under the name adaptive control, where the general concept is to modify the controller parameters or even its structure based on the current circumstances. Either due to unknown disturbances or to inaccurate modeling and parameter estimation, the initial controller values may be adjusted during operation. Dierent reasons can cause a change of plant parameters over time, like mechanical wear or temperature variation, which can also be alleviated by adapting the controller accordingly.
For the purpose of friction compensation, the most common use of adaptive control is made in the form of an algorithm with variable parameters that are updated online based on an observer estimation of the friction model parameters.
The strategy bypasses the need of a time intensive and laborious experimental determination of these values. Examples for adaptive friction control can be found in [2], [18], [19] and [20].
The last category of design methods that shall be discussed here uses machine learning procedures to approach the matter. Rather than trying to eradicate friction completely based on a xed model, neural networks can be used to identify the friction eects. In [21] and [22]    It is therefore desirable to develop a control method to enable friction compensation without such a detailed knowledge about a friction model. This is the main objective of the present work. The central idea of the approach studied throughout this thesis was rst considered in [1] to increase accuracy in a specic second-order tracking control system. In particular, the system that was dealt with is a belt driven translational positioning system with the control ane nonlinear state space where x 1 is the position, x 2 is the velocity and F (x 2 ) describes the velocitydependent nonlinear friction force. It can in turn be visualized by the characteristic curve in Figure 2 As the control architecture a structure similar to cascade control, with an inner velocity loop and an outer position loop, is chosen. However, the construction is referred to as nested, because rather than being cascaded the velocity loop is embedded in the position loop and the velocity state and the controller states of As can be seen from the block diagrams, both loops contain a state feedback controller and an additional dynamic part for output feedback. They are designed based on the linearized plant model in Eq. 2.3 via pole placement. This is the procedure we later want to apply to the control of an arbitrary mechanical system subject to friction.
Add. Dynamics Applying only this linearization-based control design to the friction compensation problem results in relatively poor performance, the obvious reason being that a linear dependence of the friction force can not describe the complex nature of the occurring phenomena properly. The novelty of the strategy presented in [1] consists of the addition of a linear observer to improve the control quality. Its rationale is as follows.
If in a mechanical feedback control system subject to friction the friction eect is assumed to be linear, the design procedure is based on a model that diers from reality. This discrepancy will cause the closed loop to behave unexpectedly when the controller is applied in the physical plant. The reason for that is that when the output, typically the position, is compared to the reference input, the controller will try to apply a control signal to the plant which is substantially too small to overcome the high sticking force, because linearity is assumed.
The dierence between reference and output will consequently start to build up, in an eort to overcome sticking. Once the system breaks free, the force exerted on the plant by the controller is too large, which entails an undesired acceleration. The control deviation has to be decomposed, an eect which is especially tragic if the controller contains an integrator. This process is encountered at low velocities, since nonlinear friction eects are predominant in this region of operation.
Recapitulating the additional dynamic part of the inner loop control structure discussed above, we remember that it contains an integrator. Despite its excellent inuence on the dynamic behavior of the closed loop, it does introduce the downside of having a memory. It accumulates the high control deviation, causing the dynamic state x a to build up, and applies a force to the plant which is larger than necessary. In the control system described above, the dynamic state of the velocity control loop is used for state feedback as it is, resulting in the aforementioned problem. The proposed linear observer is now used to suppress this eect.
It is a Luenberger observer, which is designed based on the linearized system  Figure 2.4: Control System with Observer in [1] description and which provides a simulation-based estimate of the additional dynamic state x a to be fed to the state feedback controller. Since it is based on the assumption of a viscous friction force, which linearly depends on the velocity, the estimatex a will be closer to the actually required magnitude of x a . This mitigates the inuence of the nonlinear friction phenomenon on the control system by inhibiting the impact of the drastically increased integrator state on state feedback.
The control system with observer is shown in Figure 2.4, where the observer is simply designed based on the linear design model of the inner loop, follows, with the observer matrix L. The paper also proposes a reduced observer, which only simulates the integrator state. Its design procedure follows the standard methodology of a reduced Luenberger observer and can, for example, be found in [26].
Rather than simulating the complete inner loop, one could argue that including the velocity is in fact unnecessary, since its estimate is not used anyway. This rationale is the onset of the theory of reduced-order observers. Starting from the linear state space model (A,B,C) the output matrix is partitioned based on a division of the state vector x into the states x 2 to be simulated by the observer and the remaining states x 1 . This gives us the form and, in addition to that, the regular coordinate transformation (2.10) By means of this transformation the system is transferred to the form which is the basis for the observer design. Equation 2.11 can be rewritten aṡ With the denition of the virtual variables u r A 21 y + B 2 u and y r ẏ − A 11 y − B 1 u (2.13) and the matrices A r A 22 , B r I and C r A 12 we can formulate the state space modelẋ (2.14) as the system to be simulated by the reduced observer. Designing a Luenberger observer for the system in Eq. 2.14 under consideration of the denitions in Eq.
2.13 yields the equatioṅ The nal step consists of the construction of the observer state vector x b x 2 − Ly, which gives us the state space representation of the reduced observer. It has the forṁ (2.16) and the estimated states can be reconstructed viâ Identically to the case of the full state observer the dynamics of the reduced observer are inuenced by modifying the poles of the matrix (A 22 − LA 12 ). This is done by performing pole placement or equivalent procedures.
What looks complicated at rst glance is genuinely easily comprehensible, if the design procedure is thoroughly studied step by step.
follows immediately and concludes the design of the reduced observer upon plugging into Eq. 2.16 and specifying the observer dynamics via pole placement. Note that, since we are dealing with a rst-order reduced observer, all matrices are actually scalar values.
By use of the nested control architecture and the linear observer, the tracking accuracy not only in simulation, but also during operation in the physical system were improved. In a direct comparison with a linear reference model, the peak error was signicantly reduced, besides reaching a considerable enhancement with respect to a gure of merit introduced in the paper. This indicator is dened by (2.19) and is obviously given in percent, where F = 100% is interpreted as perfect tracking. The integrals are taken over one period P of the reference trajectory.
Employing the reduced observer improved the control quality even further, although the advantage of the reduced observer over the full state observer was way smaller than that of the full observer over the nested controller alone. Recreating the simulation documented in [1] reveals that the peak error using the reduced observer is only eight percent smaller than that of the full state observer.
Note that the goal of this concept is to achieve a behavior of the closed control loop which is approximately linear. Without the use of an appropriate prelter the controller will only reach stationary accuracy as a response to a step input but will not track any desired reference trajectory perfectly. This has to be kept in mind for the simulations in the following.
The great performance of the controller in friction compensation indicates that the strategy could be applied to other mechanical systems subject to friction eects and that it is not limited to a particular model description. Furthermore, since the construction of the controller only requires a single viscous friction coecient, there is room for hope that the design procedure does not rely on a specic form of friction model, but can be utilized to deal with other static and dynamic eects encountered in practice.
The focus of this work is on investigating the applicability of the control method described above to an arbitrary mechanical system to achieve friction compensation and to examine whether it is competitive with nonlinear control methods. Of particular interest is the comparison with model-based strategies, to determine whether the forgoing of a completely identied friction model is worthwhile.
In light of the fact that a dierent dynamic controller may be used later and that the plant models of other physical systems will have state space representation diering from that presented in this section, the design procedure outlined here is held relatively universal. The overall approach to other problems will be identical, while certain intricacies and complications are going to be discussed along the way.

CHAPTER 3
Applying the Controller Design to Other Systems

Dierent Classes of Systems under Consideration
As already foreshadowed in Chapter 2, there is justied hope in the potential of the strategy proposed in [1]. It might be suitable for the control of an arbitrary mechanical system in which friction is to be compensated, if that system can be approximated by a linear or linearized state space representation. We are therefore trying to nd meaningful examples of system models in the literature, which were successfully used in designing friction compensation controllers or friction force estimates. As mentioned before, the vast majority of approaches uses a model for the description of the relevant friction phenomena. A very general dierentiation between the two main types of systems under consideration can be made.
Firstly, there are models which include friction based on a characteristic curve, much like that used for the derivations in Chapter 2. Another example for such a model is given in [2], where the characteristic curve is characterized by α 1 , α 2 , α 3 , β 1 , β 2 and β 3 being positive constants. This yields a characteristic curve that is very similar to that encountered in [1], but that is, moreover, continuously dierentiable. As can easily be seen upon examination of the system equations, curve-based models usually describe the dependence of the friction force on the velocity. While the modeled friction force can depend on more than one state at once, obviously complicating the system dynamics considerably, or can be based on other system states altogether, the velocity dependent friction force model is the most common one.
Secondly, and most importantly, there are friction compensation control schemes based on dynamic friction models. Since its development the LuGre friction model and its alterations have replaced almost all other approaches to friction modeling. Its superiority when it comes to grasping the complicated dynamic effects of friction, combined with the accurate reproduction of static friction eects, make it a highly capable model. There are some issues with the application of the proposed control architecture to systems employing LuGre friction models in their description that shall be discussed in the subsequent sections.

Conditions for the Applicability 3.2.1 Friction Modeling
The general idea of the cascaded dual loop is neither new nor particularly complex. While theoretically the linear plant model the controller is constructed for is arbitrary, there are some fundamental structural limitations to the applicability of the design procedure to a system.
First and foremost, the notion of compensating friction without using a friction model is slightly misleading. In the following, we will see that the controller design is not possible without any idea of the generation of friction in the respective system at hand and a rough idea about its magnitude. The advantage of the approach over traditional model-based control strategies is much rather that the friction model can be a lot less accurate, without deteriorating the control quality dramatically.
Moreover, the design is performed using one of the most common methods in linear control theory, namely pole placement of a linear state space model.
So, as a requirement for the physical system for which the controller is to be designed, we can conclude that the nonlinearity caused by the friction phenomenon has to be reducible to a linear state space representation suciently well. We will later see that, particularly when dealing with the dynamic LuGre model, this condition is by no means given, but can be achieved by taking a detour, for example over model reduction. In general however, and especially for friction models based on characteristic curves, it is fairly simple to nd a suitable linear approximation of the curve.
Recapitulating the procedure in Chapter 2, we remember that the characteristic curve in the nonlinear state space model was simplied to a viscous friction coecient corresponding to the slope of the almost linear section of the curve. In doing so, the identication of the parameters of the nonlinear curve was reduced to the estimation of a single constant value.

Controllability
The second condition a model has to fulll is controllability. Illustratively, this means that a dynamic system can be transferred to any given state, starting from any given state, by proper choice of the control variable. We will see that, when dealing with the LuGre model, this requirement is critically intertwined with the rst one.
It is no surprise that controllability is a prerequisite for the state space representation of a plant. Not only is it necessary mathematically, for pole placement to be possible, but also crucial from a practical standpoint, because the dynamics of the system have to be inuenced for the control quality to be satisfactory.
has to be fully controllable. According to Kalman's controllability criterion (see [26]), this is equivalent with saying that the controllability matrix 3) n being the system order, has full rank. For the system discussed in Chapter 2 and for similar approaches to friction modeling, controllability is practically guaranteed. Even if the dynamics of the inner loop should consider the state space representation of e.g. an electric motor, this condition is almost always given, which is readily shown using the aforementioned Kalman criterion.
If we take a look back at system representations including the LuGre friction model, the situation is dierent. Since it is dynamic and introduces an additional state to the state space system, problems with the controllability arise as follows.
Using the nonlinear dierential equations of the LuGre model presented and ex-  (3.6) which is given by Since the second and third row of this matrix are identical, the controllability matrix evidently has a rank deciency and the design model is therefore not controllable. This poses a signicant obstacle for the design procedure and has to be examined further.
A rst step in dealing with this issue is determining how the problematic eigenvalue becomes non-controllable. For this purpose, we have to deal with the invariant zeros of the system. Invariant zeros as dened in [27] are those complex values λ that cause a rank deciency in the Rosenbrock system matrix given by In SISO systems of order n this rank deciency can result from rank λI − A c < n, (3.9) if the system is not fully observable. Observability of a system is described analogously to controllability in [26] and means that the state of a system can be reconstructed unambiguously from the knowledge of the input and output signal in a nite time interval. In accordance to this a rank deciency in the Rosenbrock matrix can also stem from rank λI − A −b < n, (3.10) if an eigenvalue is not controllable, which is the case in our present situation.
Again illustratively speaking, invariant zeros are those complex input frequencies for which a dynamic system blocks transmission, i.e. the output is equal to zero.
The term invariant refers to the fact that these system zeros can not be inuenced by use of linear feedback or state transformations.
Both non-controllable and non-observable eigenvalues result in pole-zero cancellation with an invariant zero and do not show in a frequency domain transfer function due to the fact that their contribution to the system behavior is not inuenceable from the input and/or not noticeable at the output. This is particularly dangerous if the aected eigenvalue is unstable because of the internal dynamic eect it entails. Note that an invariant zero can be a pure transmission zero as well, which does not cancel with an eigenvalue of the system.
It is of particular interest to determine where the problem of noncontrollability arises for systems of this specic structure, since most single-mass systems have that same exact form. As shown in Appendix A, it is easy to prove that the issue at hand is indeed of structural nature and related to an unwanted pole-zero cancellation. How exactly this comes about, shall be discussed in the following.
Upon inspection of the determinant of the Rosenbrock matrix we can immediately see that the invariant zeros of the system are at λ 1,2 = 0.
Since λ = 0 causes the characteristic polynomial of the Rosenbrock matrix to be zero, this value is the invariant zero of multiplicity two. This reveals the source of the problem we are facing here: since the linearization of the plant has two invariant zeros in the origin, adding a pole at zero will inevitably cause this pole to be canceled with one of the zeros.
Apart from the proof that systems of the form described above will always exhibit two invariant zeros, which was conducted by performing structural analysis, As can be seen right away, one of the zeros remains as a pure transmission zero of the system. It does not cause any further issues. The other invariant zero, more specically the one resulting from the dynamic modeling of friction, however, cancels with the pole of the system in the origin. This is the non-controllable eigenvalue traced by the examination of the controllability matrix of the design model and it is particularly problematic, because the canceled eigenvalue is not stable.
The non-controllability of the eigenvalue in zero is caused by the introduction of the additional state z in the LuGre friction model, which describes the surface deformation. As long as the linearization of the LuGre model contains this second state next to the velocity, the resulting design model will never be fully controllable.
After discovering the root of the problem, we can try to systematically come up with a solution for it.
An attempt to modify the design procedure or the additional dynamics of the controller fails. As the only option to deal with the issues of the dynamic friction model, a reconstruction of the linearized state space representation remains. Luckily, such a modication of the system is possible by performing model reduction.
This approach shall be discussed in Section 3.3.
As a nal remark in this part of the introduction, the following has to be stated clearly. Both the LuGre friction model and the aforementioned systems incorporating characteristic curve-based friction descriptions are nonlinear. Terms used in the statements and derivations of this chapter are almost entirely limited to the description of linear systems. The notion of an eigenvalue, for example, strictly speaking, does not make any sense in the context of a nonlinear system.
However, when dealing with the linearization of nonlinear dierential equations and when examining small movements of the states around the equilibrium the linearization is performed around, the language of linear control theory is adequate. Not only does it give us a very good and visual idea of the system behavior but also approximates it decently well quantitatively.

Reduction of the LuGre Model
We discussed in Section 3.2.2 that the LuGre friction modeling philosophy requires further treatment for our control approach to even be applicable in its presence and the design procedure to be feasible. We also discovered that including the additional state of this dynamic friction model in a conventional linearization will not result in a controllable design model, rendering this intuitive approach insucient.
Another issue that has to be mentioned here is that our ultimate goal is to come up with a control method that foregoes an accurate modeling of the friction eect to be mitigated. The identication of a relatively complicated and detailed dynamic friction model just to then reduce it to its linearization would contradict our initial line of approach and essentially ridicule every eort that is made in nding a simple control algorithm.
However, the inspection of the LuGre model gives us important insights about the dynamic behavior of friction eects and the nonlinear model will later be an important building block and guide for every simulation that is performed to verify the performance of our controller. For that reason, we will dive deeper into the conversion of the LuGre model into a linear representation, qualied for the controller design. To that end, we are, as mentioned in the preceding section, going to employ model reduction techniques.
Before introducing the successful idea of using model reduction to obtain not only a controllable design model but also satisfactory control quality thereupon, some remarks about previous trials shall be made. As a very rst and naive approach, the equilibrium around which the linearization was performed was slightly shifted from the origin of the state space. In a similar fashion, the zero entries in the state space matrices of the model were tweaked, to articially produce a controllable system. The consequence in both cases, however, was that the pole placement algorithm used in the design procedure excessively exploited the falsely nonzero elements in the matrices, to achieve the desired pole location and the subsequent simulation showed unstable behavior.
Model reduction was already used very early on, the idea being that there has to be some way of depicting the friction eect as a single viscous friction parameter, since it was already possible for models based on characteristic curves.
Even though the approximation by a single constant might be signicantly worse in the presence of a dynamic model, it should still theoretically be feasible. The resulting rst order model has to give us a controllable design model, because a model of order one with nonzero input matrix can not be non-controllable.
Upon inspection of the Hankel singular values, which can be interpreted as a measure for the inuence of the single eigenvalues of a system, it became apparent right away that such a reduction was not justied with respect to properly representing the system dynamics. However, using the balred function for model reduction in Matlab yielded a design model with which a stable and satisfactory controller could be constructed.
On the other hand, using this built-in function on a LuGre model with dierent parametrization did not even result in a stable control system. This dierence showed to be explicable by the very dierent dynamic behaviors of dierent LuGre models, depending on their respective parameter values. We therefore want to determine the origin of this discrepancy and how the reduction of the linearization can be performed in order to achieve desirable control quality for all dynamic cases.
A rst step in examining the linearization of the LuGre model is to calculate its eigenvalues. Looking back at the system matrix, we can determine the characteristic polynomial and its roots, which are given by (3.14) Without further knowledge of the parameter values, no quantitative statement about the pole locations is possible. As a condition for the existence of a pole pair with an imaginary part, we quickly want to establish a relation between the parameters. For the roots of the characteristic polynomial to not be real its discriminant has to be negative: Consequently, the root locus of this ctitious system can be plotted, where the reciprocal of the mass is treated as the variable gain of a P-controller. Obviously the parameter set from [7]. The parameters could also yield eigenvalues that lie relatively close together on the real axis for this function to be applicable.
Unfortunately, this method only covers a very small portion of the root locus curve. We want to come up with an idea that includes the entirety of possible pole locations, to develop a closed theory. An option for real eigenvalues is the application of a formal model reduction algorithm. Such eigenvalue constellations are, for example, reached for the parameterizations in [28], [29] or [30]. For the reduction the modal technique proposed by Litz is used, which is described in [31].
It essentially splits the system dynamics up in a dynamically dominant part, which is typically slower, and a faster part, that is to be neglected.
Starting from an arbitrary linear state space systeṁ the procedure works as follows. The rst step is the transformation of the state space model to its modal formż using the linear transformation x = Vz. In the following, we determine the set of dominant eigenvalues by use of certain measures for the inuence of the single modes. Since in our case we just want to take the eigenvalue closer to the imaginary axis along, this step is very straight-forward. After breaking the modal coordinates z up into the states to be maintained for the reduced model, z 1 , and states z 2 to be dropped, we can formulate the state equations for the relevant part as follows: and the equations for the omitted states analogously: The most involved part of the reduction algorithm is the recreation of the inuence of the neglected states. It is achieved by introducing the reconstruction where the bar indicates the complex conjugate of the respective matrix and the matrices b 11 and b 21 are dened element-wise by In these expressions, m is the number of eigenvalues taken into account in the reduced system and λ i indicates the single eigenvalues. The matrix Q u is the diagonal matrix of squared control variable amplitudes, which are chosen as typical values of the control signal. By partitioning the Transformation matrix as wherein the top block of columns is associated with the conserved modal states z 1 , we can reconstruct the reduced model in the original coordinates: The fully reduced and reconstructed state space system is then given by the differential equationsẋ and to obtain the approximation of the controlled variable the part of the output matrix associated with the preserved states, c 1 , is used: (3.27) By using the procedure described above, we again obtain a rst-order system which approximates the behavior of a linearized LuGre model and which yields an intrinsically controllable design model. As a nal remark, it has to be said that the quality of the model reduction is of course better if the eigenvalues are far apart on the real axis. However, it has shown in simulation that even for pole positions relatively close to each other, the subsequent control quality was satisfactory.
The last case we have to consider is that of a weakly dampened eigenvalue pair in the linearization of the LuGre model, as it can be found in [6], [19] or [32].
In Thankfully, there is a very simple remedy to this that is also easy to implement. If we recall that we have already successfully approximated the friction eect of models based on characteristic curves by its behavior in the range of viscous friction, it seems natural to chose a similar approach to the LuGre model. To that end, we just drop the nonlinear dynamics of the model completely, resulting in the simple rst order approximatioṅ v = − σ 2 mṽ + 1 m u. (3.28) The remaining factor σ 2 is literally the viscous friction parameter of the model.
With this approximation in the design procedure, satisfactory tracking control in simulation is achieved. Checking back on the linearized models that were already successfully reduced, reveals that this idea seems to be able to replace the balred approach in that it results in an even better control behavior.
However, if the poles of the system are real and relatively far apart, formal model reduction is inevitable. In these cases using the modal procedure of Litz is preferable, even though considering only viscous friction does yield a stable control circuit. We can therewith cover all possible pole constellation of the linearized LuGre model and receive a reduced model, adequate for the design procedure.
It has to be said that we obviously do not have a friction model at hand before beginning the controller design and that we, frankly, do not want to use one in the strict sense. But the inspection of the dierent relevant cases for the dynamics of the LuGre model and the successive experiments based on simulations reveal two important facts.
Firstly, the approximation of a relevant friction model from literature, whether it is dynamic or not, by a rst order model is absolutely sucient for the design of an eective controller following our scheme. Since the LuGre model is known to describe friction eects very well, a simulatory control of this dynamic model with our linear controller is expected to yield meaningful results.
Secondly, and more importantly, it will become apparent quickly that the accuracy of the parameter values used in the linearization and reduction does not have to be high. It is therefore possible to compare both approaches in a physical system to be controlled based on a fairly rough identication of the three friction parameters σ 0 , σ 1 and σ 2 . After estimating these parameters, the performance of a controller based on a thoroughly reduced model and that of a controller based only on the viscous friction coecient σ 2 can easily be compared.
This allows us to think of the parameter estimation not as the identication of a friction model, but as the identication of a simple linear state space representation of the controlled system, which is one of the main objectives of the idea.

CHAPTER 4 Application to Specic System Models
Throughout the research leading up to this thesis, a number of dierent models from recent literature were researched and the proposed system models underwent treatment with our control approach. All considered models were developed within the last 20 years, with a considerable portion being less than ten years old. While a number of the examples contain a single-mass system like the one we have already encountered in Chapter 2, there are some that deal with more complicated dynamic models. Some of the models explicitly take motor dynamics into account, yielding an inner loop model of higher order.
There are also mechanical system models subject to friction in which the position signal is fed back to the calculation of the acceleration. This system structure renders the proposed nested control with an inner velocity loop that is designed separately, impossible to apply. It is shown, however, that in a similar fashion, we can design a dual position loop for these systems, which yields satisfactory control quality.
Finally, there are systems where the modeling of nonlinearities is not limited to friction eects. We will see that our control strategy lends itself to the mitigation of backlash in one particular case, which in turn sparks hope that the idea of a nested control architecture might not be limited to friction compensation.

Simple Single-Mass System
The most elementary system that could be thought of, is an either translatory or rotatory single-mass system with mass m or inertia J. Its general structure is given by the block diagram in

Characteristic Curve-Based Model
For the model description in [2], which already functioned as an example for friction modeling using a characteristic curve, the complete nonlinear dynamics are given byẋ Therein, the state x 1 is the angle and x 2 denotes the angular velocity. In contrast to the considerations in the paper, disturbances and uncertainties are neglected for the time being, since the purpose of this chapter is to present an overview over the design procedure rather than a thorough assessment of the control performance.
To illustrate the application of the nested structure to an arbitrary system, we want to perform the design procedure step by step. Consequently, we rst reduce the dierential equation to its linear part, which contains only a viscous friction coecient. We directly obtain the linear state space model wherein the equation for the angular velocity is given bẏ ready for the construction of the inner loop. Plugging these values into our design model for the inner loop and using a simple integrator as the additional dynamics, yields essentially the same matrices as for the model treated in [1]: for the augmented inner loop. A design model of the outer loop is constructed analogously to that of the inner loop: Herein, the reference input is symbolized by r and the controlled variable x 1 is renamed θ, the angle of the rotatory system. The state space representation of the additional dynamics are given by A a = 0 and b a = 1 for the outer and inner loop alike.
This reference model derived above is used in the subsequent simulation to assess the ability of the linear controller to mitigate friction in the system to an extent where the closed loop can be treated like a linear system as a good approximation. Because the matching is obviously not going to be perfect due to the nonlinearity that has been neglected throughout the design process, we want to examine how good the tracking quality can be.
The last step in the design process is the construction of the linear observer, which estimates the states of the inner loop, the angular velocity and the additional integrator state, based on the output of the inner loop and its input, the virtual control variable u c . As we remember, the observer is designed according to the linear model of the inner loop, i.e. the nonlinear friction eect is neglected.
The idea of an observer in the conventional sense is to estimate the state of a dynamic system by modeling the system dynamics in an online simulation. Since The observer vector L is readily obtained using pole placement. As the observer poles, we choose S obs = {−97.272 ± 56.16 j} , (4.12) the scaled Bessel poles that are shifted in the left complex half plane by a factor of three with respect to the poles of the inner loop. This is done so the observer error decreases much faster than control actions due to reference variable changes or disturbances are executed.
Instead of using the complete state of the inner loop for control, only the estimate of the additional integrator statex ai is fed back into the loop. With the controller and observer complete, we can turn towards the simulation, which is set up as follows.
In analogy with the paper which the friction and system model stem from, the reference signal is a sine wave of low amplitude. The rationale behind this is that we want to challenge the control system with a reference trajectory which causes the velocity to be near zero, where the inuence of the nonlinear friction eect is  Furthermore, we assume that the system has unit moment of inertia, therefore The experiments are performed in Simulink using the ode45 solver with automatic step size selection, where the block diagrams from Chapter 2 were directly transferred to the simulation environment. Because we are lacking a physical system, the control performance is studied based on a plant model which incorporates the respective nonlinear friction model from the literature source the parameters were gathered from. This is also the approach of other upcoming simulations.
With a sine wave of amplitude 0.01 rad at a frequency of 1Hz a rst test is to be made as to how well the control circuit performs in tracking the reference signal. To that end, the observer is omitted for the time being and the additional dynamic state is used for state feedback in the simulation. That yields the output value and reference model output progression displayed in the plot in Figure 4.2.
We can directly see that the tracking performance is fairly good. Even though a thorough comparison with the control quality presented in the paper is inadequate due to the missing parameter values, we can conclude that, at rst glance, When speaking of the tracking error in this and the following simulations, the deviation of the output of the control system with the nonlinear plant from the output of the linear reference model is meant. This comparison is more meaningful than contrasting the controlled variable to the system input because there is, naturally, a delay between the input and output and the controller still lacks a prelter, which could ensure the correct amplitude for an arbitrary frequency.
Bear in mind that we are using the term reference for both the desired trajectory the system is supposed to track and the benchmark system the closed control loop is compared to.
The error reaches its peak right after the zero crossing of the velocity and exhibits a maximum of eight percent of the reference signal amplitude. In general though and especially given the fact that we are requiring extremely small movements from the system, the tracking quality is good. This particularly shows when  Right o the bat, it can be seen that by use of the estimated integrator state during operation, the peak error is reduced to less than ve percent of the input amplitude. Moreover, the gure of merit value is now up to F = 99.8%. These results prove what we have initially assumed: the control concept is not only applicable to other systems from literature, but also yields promising tracking quality in simulation.
Two nal remarks have to be made. Firstly, the impact the observer has on the control performance depends critically on the system description and friction model at hand. We will see later that, in particular when dynamic friction models are studied, the eect of the observer is typically much more substantial and can indeed reduce the peak error by a factor of two to three. Secondly, tuning the

LuGre Friction Model
The logical next step in our investigation is answering the question whether the control strategy is suitable to replace model-based controller designs based on dynamic friction models. In recent publications on friction compensation, the predominant choice is the LuGre model, hence why it gets so much attention in this work. As we can easily see from Section 3.3, the design procedure for LuGrebased approaches is more laborious than the ones discussed so far. However, we will nd out quickly that for the most part the construction of the controller is executed identically.
We want to use the nonlinear system model described in [3] and design a linear controller in the same way as in Section 4.  Therefore, we go down the path of simply dropping the nonlinear dynamics and the additional state z from the linearization to receive our controllable design model.
This design model is constructed in analogy with the procedure described in Section 4.1.1. With our model descriptioṅ which then, combined with an integrator as the additional dynamics, yields the inner loop design model The rest of the design process is completely identical to the one described in  We therefore arrive at a closed inner loop with the state space representation closely resembling the structure of the inner loop of the previous model. The inner loop is subsequently augmented by an integrator to receive the position: which again gives us the design model for the outer loop,  the matrices of the observer subsystem again take the same exact shape as the ones encountered before: How the design procedure is carried out in a programming environment to calculate the controller and observer gains for operation, is shown in Appendix B.
Note that the code also covers the case where modal model reduction has to be carried out.
The simulations performed in the following illustrate the rst advancement over the initial inspiration in [1]. Not only do they show that the presented control concept is applicable beyond the exemplary system model from the paper, but they also prove that it is successful in compensating friction eects when a dynamic friction model is underlying the simulation. In contrast to the paper studied in  Apart from the peaks, which occur when the system approaches zero velocity, the controlled system matches the behavior of the linear reference almost perfectly.
The maximum error remains smaller than ten percent of the signal amplitude and  As we can see, the maximum error is reduced by more than a factor of two and signicantly narrowed. This shows in the increased merit F = 99.9%. As an added bonus, the control energy is thereby reduced to less than a half by use of the observer.
A nal remark shall be made about the reduced-order observer mentioned in [1]. While it did yield a small, yet noticeable, improvement over the performance of a full state observer in estimating the additional controller state, this enhancement The conclusion we can draw from the simulations in this section is that our goal of constructing a simple and versatile control method for mechanical systems is within a grasp. Even though the comparison of a pure simulation is always lacking compared to actual physical experiments, the results are promising and can be examined in more depth in the next chapters. Before we dive into the analysis of the control quality, however, we want to take a look at more complex models and the adaption of the design procedure to those systems.

More Complex System Dynamics
A logical next question is whether a more detailed system model would render the control architecture useless for application in a physical plant. We therefore want to try to incorporate our controller in a rotatory electromechanical system, which also includes the model description of an electric motor. The model, which can be found in [28] with the set of parameters collected from [4], has the following nonlinear state space representation: It can now simply be plugged into the design model and the rest of the design procedure can be carried out in perfect analogy to the systems considered before.
Since we are now treating a system of higher order, only a few slight changes result for the process. We need to specify an additional real eigenvalue for the pole placement of the inner loop. It is, motivated by the selection of poles in Obviously, the rst part of the inner loop feedback vector K v is now no longer a scalar but rather a two-dimensional vector k v,1 , consisting of its rst two entries. Apart from a few dierences in dimensionality, the construction of the control architecture is completely identical.
In accordance with the simulation results found in the paper, a rectangular wave of low amplitude (0.01 rad) is used as the reference trajectory. Without the observer in the loop, the controller achieves the tracking error plotted in Figure   4.8, which is satisfactory.
Again, introducing the observer yields a noticeable dierence in tracking quality. Looking at the plot in Figure 4.9 we can see right away that the maximum error is reduced to less than a half. The overall tracking result compared to the linear reference can be inferred from Figure 4.10.
A remarkable fact is that, for dierent reference signals, the observer shines even more. As an example, if a sine wave reference is used, the state estimation reduces the peak error by a factor of three. This goes to show how eective and, ultimately, versatile this linear observer-based control concept seems to be. As a last side note before focusing on the implementation and merit of our controller, we want to see if its signicance is limited to a specic mechanical system model  description. We will therefore be looking at plants of a dierent structure.

Plants with Position Feedback
A large impact on the system behavior is caused by adding feedback of the position state back to the calculation of the acceleration. This structure is encountered, for example, in [5] or [15] and the biggest letdown for us is that our control architecture does not seem to be applicable to these cases anymore. The obvious reason for that is that the velocity state, which is supposed to be the output of the inner loop, can no longer be treated in an isolated manner, since it depends on the position itself. Having a nested velocity and position loop is no longer feasible.
However, modifying the controller composition slightly to arrive at a fully cascaded dual-loop structure allows us to apply the general idea of the design and results in a properly functioning controller. Instead of an inner velocity loop and an outer position loop, both loops now have the position as their output. We want to take a closer look at the model presented in [5]. The plant at hand is a servo system that is driven by two separate motors via a gearbox. It is particularly interesting, because not only is the state space model considerably more complex than the ones encountered so far, but it also considers, aside from the friction eect, a model for backlash in the gears. Treating this nonlinear model will hopefully show us if the idea of our controller design is expandable.
The dynamics of the two driving electric motors and the gearing are given by the nonlinear dierential equationṡ while the description of the load motor iṡ The friction torque T f is only relevant in the load motor dynamics due to the fact that its impact is negligible at the comparatively high velocities of the driving motors. It is, again, based on the LuGre friction model. As can be seen, this system has two input variables, the two motor voltages u 1 and u 2 . The parameters J i (i = 1,2,m) stand for the driving motors' and load motor's moments of inertia and the value w is a bias torque used to maintain contact of the gears in the transmission during operation. The state vector To cope with the backlash nonlinearity, we assume that the dead zone is narrow enough to neglect it. Assuming α ≈ 0, we can formulate a linear plant model as a groundwork for our controller design. It is given by the following system equations:   Performing the rst step in the design procedure results in the closed inner loop described by the same model structure that was already derived in the previous sections and that was discussed in Chapter 2. It has the form   In line with the previous tests, we rst exclude the observer to examine the capabilities of the controller rst.
Recreating the reference signal used in the paper, which is a sine wave of amplitude 0.5 rad and frequency 0.2 Hz, the tracking performance is investigated.
As we can see from the plot in Figure 4.11, the control quality is already decent.
A distinct dierence to the previous tests lies in the articulate peak of the output progression at the beginning of the simulation, while the output of the linear reference model is smooth. This deviation is caused by the initial gap between the gears in the transmission that has to be closed before the controller can function properly. To that end, a small constant bias torque w = 50 Nm is applied to the driving motors, which ensures contact between the gears. Without the bias torque the tracking quality deteriorates considerably, as can be told from the curves in As we would expect, the addition of the observer improves the merit of the control. Not only does the estimation of the additional dynamic state of the inner loop decrease the peak error caused by the friction nonlinearity from around seven percent of the reference signal amplitude to less than three, it also mitigates the initial error caused by the gearing backlash. Furthermore, with the reduced error, the control quality becomes competitive with the documented results from the original paper. The conclusion we can draw from this is that the slight modication we made to apply our control strategy to systems of the structure at hand, where the position is fed back into the calculation of the acceleration, is promising with respect to the tracking quality. There are, however, some important remarks to be made at this point.
First of all, the design procedure took a lot of tinkering with the settling time that is demanded from the closed loop. Basing the design procedure on the Even though the simulation runs smoothly and the closed loop is robust across a sucient bandwidth, given the correct T s was determined, this sensitivity to the settling time is a new issue this system structure brings along. Since a similar eect was encountered during tests based on the model in [15], we have to assume that this is a problem linked to the structure of the plant, which poses a signicant drawback compared to the models without position feedback.
Another obstacle is the newly added dead zone nonlinearity. While the bias torque successfully reduces the inuence of backlash on the control loop, the settling time has to be tuned appropriately to ensure a sensible trade-o between the compensation of friction and overcoming backlash in the gearbox. This makes the design process more elaborate than that in previous cases.
Yet, all in all, the adaption of our control architecture to accommodate a dierent plant structure and even another form of nonlinear eect was successful.
Under some more attention the control concept could be rened to apply to this

Stability of the Control Loop
Concluding this chapter, a nal remark about an important issue has to be made. Throughout the controller design, we never examined the stability of the closed loop including the friction nonlinearity. Even though the design based on the linearized plant is of course geared towards producing a stable and dynamically benecial behavior, this does not formally prove stability for the controller in the loop with a nonlinear plant. This holds true, in particular, for this case, where we are constructing a controller based on a signicantly reduced system model.
To thoroughly prove stability for our controller design in a conventional sense, one would have few alternatives to applying Lyapunov theory to the system. It is outlined in [16]. Similar approaches were already chosen, for example, in [6], [19] and [29], either to ensure or to demonstrate stability of a control concept. There are, however, some fundamental issues with a formal proof of stability, which make it dicult to perform and rather pointless for our purpose.
First and foremost, the stability theory of nonlinear systems, which revolves mostly around Lyapunov's theorems, requires relatively well-behaved system dynamics to be easily applicable. Since in our case the model equations are nondierentiable, it is dicult or even impossible to nd a Lyapunov function, which has to be continuously dierentiable. Generally a proof of Lyapunov stability is not trivial to conduct and relies heavily on the patience and experience of the control engineer.
On top of that, Lyapunov's stability theorem is sucient, not necessary. Just because there is no suitable candidate for a Lyapunov function, Lyapunov stability is by no means precluded. These issues make the proof complicated, but there are some other aspects, which plainly make it superuous for our case.
In contrast to the stability theory of linear systems, the stability of a nonlinear systems is less expressive. Since there is no such concept as pole positions in the strict sense, aspects like eigenfrequency, overshoot or settling time have no analytical equivalent in the nonlinear realm. More importantly though, Lyapunov theory does not reveal a stability region in a straight-forward manner. This is the biggest issue, because even if a proof of stability existed, it would be dicult to determine what state trajectories would yields stable system behavior.
The bottom line of this section is that, from a practical standpoint, formal stability analysis for our controller design is relatively unsuitable. Instead, we want to carefully inspect the system behavior and its deterioration in the face of uncertainties and disturbances. This is an essential facet of Chapter 5.

CHAPTER 5 Comparison with Nonlinear Control Concepts
After ascertaining that our linear controller can cope with static and dynamic friction eects in system models taken from the literature, an important next step is to compare its performance with the merit of nonlinear control strategies. As discussed thoroughly in Chapter 1, there is a number of dierent ways to address friction compensation. In applications that require high accuracy, these almost always involve nonlinear system theory in the form of a state observer or the actual controller. To show that our idea is a competitive alternative to these approaches, we want to take a closer look at the construction of the controllers and contrast their performance with that of ours.
Similarly to the methodology in Chapter 4, we are trying to get a grasp of models from literature, for which a nonlinear controller has been designed that successfully compensates friction. Finding sources that suciently document the design procedure and present controllers that we can recreate with reasonable eort is a challenge in and of itself. There are, however, two methods presented in the studied material that provide a good foundation for a comparison, because they meet those requirements. On top of that, the systems under consideration choose fairly dierent strategies to deal with friction compensation. They shall be complemented by a third nonlinear controller we want to design from ground up, to get a better understanding of problems arising from nonlinear controllers.

Output Feedback Controller with Nonlinear Friction Compensator 5.1.1 Controller Design
The rst controller we want to examine is an output feedback controller with superimposed friction compensator. It employs a nonlinear friction observer to es-timate the unmeasureable friction force as well as an additional high-gain observer to determine the state variables. Its design can be found in [6].
Starting from the plant model of a simple rotatory single-mass system, where J is the moment of inertia, M f is the moment generated by friction and k c is the gain of the input voltage u, the rst objective is to construct a controller based on output feedback, which achieves the dynamic requirements. For the design of that controller, friction is neglected. Given the reference trajectory r, the error dynamics are dened asė and the goal is to achieveė 2 + 2ηω 0 e 2 + ω 2 0 e 1 ≈ 0, where η and ω 0 are adjusted appropriately. To that end, the controller is chosen The parametersĴ andk c result from the identication of the plant, while the estimates of the state variablesx 1 andx 2 originate from the high-gain observeṙ wherein > 0 is another design parameter to be tuned properly. The design goal in Eq. 5.3 is attained employing the controller in Eq. 5.4. If friction is considered in the process, however, the error is no longer small enough to comply with the requirements. The nonlinear friction eect is described by a modied form of the LuGre model, namely , which now depends on the direction of movement and is therefore dened as: To deal with these nonlinear friction phenomena, [6] proposes a friction compensator that relies on the estimate of a nonlinear observer. This observer is described by the equationsM As can be seen, the observer is complemented by the additional nonlinear component K(v,e 1 ,ê 2 ). This function stems from the stability-based design procedure and is chosen as    It is perfectly evident that the control algorithm proposed in [6] relies on the rst and second time derivative of the reference trajectory. A rst test with actual dierentiator blocks in Simulink failed due to stability issues, a problem that is plausible, because of the approximating nature of numerical dierentiation. Instead, the reference progression is assumed to be known in an advance and its derivatives are simply fed into the system analytically.
Furthermore, comparing the tracking quality of this approach to a linear reference model is impossible, since not only the controlled plant, but also the controller and observer are inherently nonlinear. We have chosen this comparison as a benchmark test so far, because it is our goal to achieve approximately linear behavior in the closed control loop and there is also an unavoidable time delay in any physical system. It makes the direct comparison with a reference trajectory less meaningful than the comparison with a linear reference model.
The main concept the simulations are supposed to capture, however, is that the control concepts compensate friction. The performance of the controller in Eq. 5.11 is therefore compared to a closed control loop without friction (M f = 0). It is controlled with the initial algorithm described by Eq. 5.4, which was designed under the assumption that no friction is present. This reference system also involves the high-gain observer and the analytical derivative of the reference trajectory. By comparing these two systems, we recreate the framework used in our previous simulations to focus on the ability of the controller to suppress friction eects.

Comparison with the Novel Control Architecture
A rst question we want to answer, is whether our control algorithm yields a tracking quality that is competitive with that achieved by the controller described in [6]. To this end, we assume that both approaches have perfect knowledge of the plant model, which means that there are no parameter uncertainties and that there is no disturbance in the system.
Orienting on the experiments conducted in the paper, we choose the angle progression r (t) = sin (πt) (5.17) as the reference signal. Subject to this input, the nonlinear controller shows the Evidently, the control system needs around a third of the simulation time to settle in and function properly. In this respect, the simulation deviates from the experimental results documented in the paper, which do not exhibit this behavior.
The discrepancy could be due to improper tuning of some of the design parameters or initial conditions of the control system. We want to focus on the latter phase of the control operation, since it will yield the most signicant information. If we consider the error between the system subject to friction and the unperturbed reference model, we can see that the nonlinearity causes a peak error of around 6.5 × 10 −3 rad in this phase. The error progression is shown in Figure 5.3.
We now want to compare the performance of this controller with that of our own, in simulation. In the following investigations the observer is permanently switched on, since its inuence has proven to be consistently benecial. Applying the same reference signal to the control system yields a maximum error of 5 ×  The control variable for both controllers has a periodic progression which resembles a sinusoid in phases of higher velocity, since in these intervals the nonlinear friction eect only requires minor controller action. In either case, the control sig- nal reaches a maximum amplitude of about 0.4, but the nonlinear controller shows a small but extremely narrow peak when the system crosses zero velocity.
As a next step, we want to make an assessment of the control quality subject to parameter uncertainties and disturbances. Even the best controller will fail in practice, if it relies too heavily on a perfectly accurate plant model, so robustness is an important requirement for any application. Disturbance rejection is another concern. Mechanical systems like, for example, electric motors are always exposed to rapidly varying loads and therefore have to react quickly to unpredictable excitation. We want to address both issues separately.
Based on our experience with the LuGre friction model, a sensible prediction about the robustness of our controller design should be that parameter mismatches do not spoil the tracking quality. After we went through the whole process of linearizing the model and then reducing the resulting state space representation, the degree of approximation is high enough to allow for more discrepancy between the reference model and the true system without any dramatic consequences.
Unfortunately, we are dealing with nonlinear systems and the classical theory of robust control is not applicable in this setting. Instead, we want to systematically expose the controller to a faulty system identication. This is achieved by keeping the parameter values that are used in the controller design xed and modifying the true plant parameters.
Judging the robustness of our controller design by its performance rather than by conventional measures of robustness is in line with our overall approach. It will become obvious soon that it takes a lot for the control loop to actually become unstable due to uncertainties. A more interesting question for us to answer is, how much the tracking quality is impaired under these circumstances.
Right away, we can see that the nonlinear controller in Eq. 5.11 is highly susceptible to errors due to parameter mismatches. Exposing the controller to the already yields a visible decline in tracking quality. The peak tracking error is increased almost tenfold, which can be inferred from the plot in Figure 5.5.
On the other hand, if our linear controller has to deal with the imprecise parameter identication above, the control quality is much better. Since the controller is designed based on the values of J, k c and σ 2 alone, the slight modication in the true parameters goes unnoticed in the simulation. Only when the viscous friction parameter is changed by more than an order of magnitude, or the measurement of the moment of inertia is erroneous, the discrepancy becomes larger and tracking becomes compromised. This proves our initial guess that knowing the plant model with high accuracy is not necessary to achieve excellent control quality.
The nal question we want to clarify in our investigations is, which system copes better with unforeseen disturbances. To achieve such an eect, the parameterization is, again, assumed to be known perfectly. However, the moment of inertia is periodically increased to 500 times its original value, to pretend that the load varies rapidly. Mathematically speaking the moment of inertia is now a time-dependent function that consists of the constant original value and a pulse wave with amplitude 500 J, period 0.9 s and a duty cycle of ve percent: If this disturbance is applied to the system at hand, with the nonlinear controller given in Eq. 5.11 in the loop, the tracking quality is severely compromised.
Instead of the high accuracy in the rst test, the error now reaches peak values of up to ten percent of the reference trajectory. This can be inferred from the contrast between the reference progression and the actual output in Figure 5.6. In contrast to that, our novel controller design is, again, almost unaected by the load disturbance, which can be seen in Figure 5 There is a point to be made for a disturbance of this dimension being unrealistically large. However, even if the system will never have to face unpredictable load uctuations like this during operation, it illustrates the immense capabilities of our control approach in the face of disturbances. In this regard, the simulations reveal another advantage of the novel design over the nonlinear controller in [6].
As a nal statement, we can conclude that our linear control algorithm yields superior tracking quality compared to the observer-based controller in Eq. 5.11.
Especially in the face of parameter uncertainties and disturbances, the novel approach performs signicantly better and exhibits predictable behavior, in contrast On the other hand, the approach in [6] relies heavily on a suciently accurate parameterization of the system. A nal remark has to be made about the design procedure. While dimensioning the nonlinear controller requires an elaborate process of identifying a set of ten parameters fairly accurately and tuning another six, our linear approach is decently simple. One only needs to measure three plant parameter values in this particular instance, to arrive at a highly capable controller designed by well-known procedures.
Note however, that we are still in the realm of simulations. Problems that may arise in the practical implementation of such a controller can not be considered in this context. In light of the very promising results of the experiments, these issues are not anticipated to be dramatic.

Friction Observer and State Feedback Control 5.2.1 Controller Design
We want to take the examination of our proposed controller design one step further, by comparing it to another concept from literature. The model in question is presented in [7]. Its concept is based on a linear observer for the state variables that is accompanied by another model-based nonlinear observer, which is used to estimate the friction force. From the linear observer, the forecasted variables are used in a state feedback controller.
As the plant, a simple translatory single-mass system subject to the LuGre friction model is considered. Given the nonlinear equationṡ (5.21) with the state vector x T = [p, v, z], the state-dependent model is formulated. Based on this system model, the paper proposes the equations of a nonlinear observer. Extended by the representation of the friction eect,F f , and an additional correctional term dF f , this yields the form of a descriptor system:  (5.24) r being the reference signal which, is also amplied by the constant gain k 1 .
Essentially, the control strategy presented in the paper is fairly simple. An  The system is therewith set up for simulation. Following the same rationale as in Section 5.1.1, a reference system is constructed in which friction is omitted.
In contrast to the preceding investigations in Section 5.1.2, this reference system is now linear. It has the simple form for the plant and controller. For the observer, we arrive at the equally plain

Comparison with the Novel Control Architecture
According to the paper, the observer-based controller was tested in simulation not only with a sinusoid of frequency 2 Hz and amplitude 0.1 m, but also with a step reference of height 0.002 m. We want to address both cases. In line with  Similarly, if the system is subject to the step input, it shows an error as well. In this case, however, it is even more minute. The only interesting realization we can infer from this experiment relates to the consumed control variable. For both the observer-based controller proposed in [7] and our novel linear control concept, the control variable progression required to track the sine reference is almost identical.
It is a periodic signal with a very distinct peak around zero velocity. A kindred correlation holds for the step input, although the nonlinear design peaks at a higher value due to the slightly higher demanded settling time. This is not fundamentally surprising. Because both controllers try to compensate friction occurring in the same plant and both yield good or even phenomenal results in doing so, the control signal they are providing ought to be very similar.
The next step now is to investigate how well the controllers cope with parameter uncertainties and disturbance. Again, we examine these aspects in isolation.
Simulations with a modied parameter set are documented in [7], where the true parameters F c , σ 0 and σ 1 are increased by 20 percent compared to the values used in the controller design. We want to start from there. Examining this plot, we can see that the tracking quality is severely impaired.
This results in a peak error of around 2 × 10 −3 m, more than four times larger than the error in the rst test. Meanwhile, our controller is almost unaected by the modied plant parameters. In fact, the peak error is not noticeably larger than it was with the initial modications to F c , σ 0 and σ 1 . Similarly, the error of the controller in Eq. 5.24 when exposed to the step reference reaches a peak of more than 2 × 10 −4 m, while our novel controller design yields a maximum error We can therefore conclude that, in simulation, our control approach performs better in the face of parameter uncertainties. This advantage in robustness is particularly astonishing when we remind ourselves that the whole controller design is based on only two individual parameters which, on top of that, are decently simple to identify. On the other hand, it is no surprise that the tracking performance of our controller does not deteriorate any further if parameters are modied that are not used in the design anyway.
Of course we are assuming large uncertainties in the parameterization here.
One could argue that, even in a worst-case scenario, the plant could be identied better. However, this examination is supposed to show just how well both controllers can deal with parameter mismatches. It has to be emphasized that the observer-based controller does by no means preform badly. Having said that, given more severe uncertainty in the identication of the plant, our linear controller yields superior tracking quality. This leaves us with a test of disturbance rejection.
To test the controller's reaction to unpredictable disturbances, we again simu- Inspecting the error curve reveals that the peak error in both cases is almost identical. The more important insight here lies in the location of the maximum deviation of the output of the controlled plant from the reference model behavior.
As can easily be seen in either of the plots, the largest error caused by the load steps is caused when load steps occur close to zero velocity. This is the case for Summing up, we can say that our novel linear controller again shows a considerable improvement over the approach originally proposed to deal with the friction eect. The observer-based design in [7] yields perfect tracking, as long as the nonlinear plant subject to friction is perfectly identied. However, its performance quickly deteriorates and becomes worse than that of our novel controller, once the plant parameters are not known without aws anymore. Although exhibiting great robustness, the tracking error of the controller is larger than that achieved by our controller. When it comes to disturbance rejection, both controllers yield similar results. Tracking errors are particularly dramatic around zero velocity.

Input-Output Linearization Controller
Inspired by the idea of compensation in the previous section, we want to design our very own nonlinear controller, to highlight the issue of plant identication. Not only do we want to get a grasp of the eect of erroneous parameter estimation, but also take a look at another controller design, to gain a better understanding of the whole procedure by performing it oursleves. As a naive idea for a nonlinear approach, input-output linearization comes to mind. The rationale here is to suppress all nonlinearities occuring in the plant, while imprinting an arbitrary linear behavior on the system. For a detailed description of the controller design and the source for the follwing outline see [16].
Starting from the control ane nonlinear plant equations of a single-mass system including the LuGre friction model, , (5.31) the rst task is to determine the relative degree of the system. To this end we calculate the Lie derivative, which is dened as the gradient of a multivariable function multiplied with a vector eld, therefore For Lie derivatives of higher degree, the abbreviatory notation is used, which can be understood as the repeated concatenation of a Lie derivative with itself.
The relative degree now is given by the order of the time derivative of the output equation, Eq. 5.31, which rst depends on the control variable. To determine the time derivatives of the output function c (x), we employ the Lie derivatives of higher order: Not only is the relative degree essentially the order of the linear system dynamics we can demand from the closed control loop, but it is also the order of the nonlinear system dynamics that are controllable and observable from the outside.
Because the relative degree δ = 2 is smaller than the system order, there is a rst-order part of the system, which is not visible from the output or can not be inuenced from the input. This is obviously particularly problematic if these internal dynamics, as they are called, are not stable. Since we will not be able to aect the internal dynamics, we have to keep their stability in mind. Even though their eect might not show at the output, they do have physical inuence on the system, leading to destruction or failure of the surrounding structure at worst.
Furthermore, their behavior can be determined by the complete system dynamics in unpredictable ways.
We want to treat the external system dynamics and the internal dynamics separately. As we have already found, the relative degree allows us to specify desired linear dynamics of order two. In general, these dynamics are given by the dierential equationÿ +ẏa 1 + ya 0 = V w (5.35) and can be achieved by using the control rule and prelter  Furthermore, the internal dynamics must not depend on the control variable, so in this case.
All of this can be achieved by simply choosing t 3 (x) = z, which fullls the requirements and yields the internal dynamicṡ z. With the linearizing control law in Eq. 5.36 derived and the internal dynamics sorted out, we are ready to turn our attention to experiments. Before doing so, however, some important remarks have to be made. The predominant goal of this section is to display the importance of accurate system identication for the proper functioning of nonlinear controllers. Our controller above, that was designed from scratch, does not claim to perform well in comparison with any tried and tested control concept. Rather than trying to improve its merit using common approaches to make input-output linearization more robust towards modeling mismatches, we want to leave it as it is and point out the issues pertaining to robustness.
The specic plant model, the experiments are based on, can be found in [7].
It was already the foundation of the comparison in Section 5. It turns out that a slight change in the functional form, for example does not increase the error dramatically. The maximum deviation of the actual controlled variable and the reference system output is only 10 −7 m. Surprisingly though, even seemingly more severe interference with this function, as in For the sake of brevity and especially to keep the results presented in this thesis as meaningful as possible, these experiments are not discussed here. Nonetheless, it has to be said that in all cases our controller design successfully compensated friction and yielded a tracking quality that was at least competitive with the respective nonlinear control concepts presented for the various models. Particularly the LuGre friction model was studied exhaustively. The reason for that is not only the relevance of this dynamic model, but also the wide spectrum of dierent behaviors it can exhibit depending on its parameterization.
A total of thirteen dierent plant models were studied with unanimously good results. In all cases our controller achieved at least competitive results or even out-performed the existing controller, while requiring similar control energy. Especially when facing parameter uncertainties or disturbances, our approach yielded excellent results.
Without going into detail, the control loops were also examined using various dierent reference signals like sine waves, ramps, steps and rectangular waves.
Regardless of the input progression, the controller managed to compensate the occurring friction eect well and at most needed some tuning with respect to the settling time. The tracking accuracy is satisfactory across a wide spectrum of reference amplitudes and over a bandwidth of up to two orders of magnitude without a signicant deterioration of quality. We now want to take our idea one step closer to a practical implementation of the controller. For the actual application to a physical problem, the algorithm has to be discretized.

CHAPTER 6
Discretization of the Control Algorithm

Design of the Digital Controller
Up until now, all simulations were conducted in continuous time, whereas in reality, the control algorithm would have to be implemented on a digital computer.
Transferring the controller to discrete time introduces a number of new complications that have to be kept in mind to maintain the excellent performance of our controller.
The goal of this section is to construct a discrete-time control algorithm for a continuous-time plant. We do this based on the plant model presented in [3].
Its state space representation was already presented in Section 4.1.2. While the linear controller we are about to design is going to be digital, the plant used in the subsequent simulation is still continuous, to establish conditions that are close to reality. To make the discretization procedure as universal as possible, we try to refrain from using specic values as much as possible.
As the outset for the discretization of our controller architecture, we choose the complete linearized plant model, which contains the position and the reduced equation for the velocity, The rst step in determining the discrete controller is transferring the plant model into a discrete-time expression as well. This is achieved by following the procedure described in [34]. Based on the discrete plant the controller will be designed.
It is well known that the solution to the dierential equation 6.1 is of the form (6.2) if the state space matrices are labeled in the conventional way and the system is  Another substitution (k + 1)T − τ = ν, and the consequential modication to the integration operator dτ = −dν, allow us to formulate the dierence equation . . (6.5) Given our simple second-order plant model, this is fairly straight-forward: gives us the discrete system matrix A d and upon plugging the result into the integral and solving it, we also receive the input matrix Therefore, the dierence equations describing the system in Eq. 6.1 at the sampling times are . (6.8) Note that determining the transition matrix can be signicantly more complicated, especially for systems of higher order. Matlab provides the built-in function c2d, which determines the discrete equivalent of a continuous-time state space model for a certain sampling time. It can also be used to obtain the exact same matrices derived above.
The following controller design directly follows the procedure that was already discussed thoroughly throughout this thesis. However, some obvious modications The eigenvalues of this matrix are placed in the locations specied by the set Z i = {0.3968 ± 0.2006 j} (6.13) using the place command. They result from the second order Bessel poles, scaled by the desired settling time T s = 0.2 s, and the sampling time T = 0.01 s using the relationship in Eq. 6.10. We arrive at the closed inner loop  (6.15) where the state vector is x T 1 = p v x a,i . Based on these equations and the additional dynamics, the design model of the outer loop follows directly: Note that the settling time of the observer subsystem is considerably smaller than it was during the tests with the continuous system. It results from a systematic tuning procedure, which was performed to ensure the highest possible tracking accuracy.
The continuous control concept is successfully transferred to the discrete domain and therefore ready to be implemented on a digital computer. To test the suitability of the system for the control of a physical plant, we want to deploy it in a simulation with a continuous-time model. Before presenting the ndings of said simulation, however, some important modications to the previous setup have to be explained.
The most obvious change happens to the additional dynamics of the respective loop. Continuous integrators with their poles in the origin of the complex plane have to be replaced by discrete integrators with the transfer function G I (z) = T z − 1 (6.19) in the z domain. Another alteration that has to be kept in mind when controlling a continuous plant with a digital algorithm, is the addition of a sample and hold element. Because the discrete controller deals with sequences of numbers rather than continuous progressions, it has to be fed with both a sampled version of the plant output, the controlled position, and the reference input. To achieve this, we append a sample and hold block to the output of the continuous plant and add another one of these blocks to the input. They have to be triggered, which is taken care of by a pulse generator with pulse length equal to the sampling time T .
A last modication pertains to determining the velocity signal. Up until this point, we have always assumed that a measurement of the velocity is available for control. In reality this assumption is relatively problematic, because measuring the velocity is generally dicult to realize in practice. Instead, in most implementations, the velocity has to be calculated from the position. There are various ways to go about this. Simulink provides a numerical dierentiator, which approximates the necessary mathematical operation of forming the time derivative by using a dierence quotient: From a theoretical standpoint, using the numerical approximation introduces the evident problem of pretending to have knowledge about future events. Since the value p(k + 1) will never be known beforehand, the dierence quotient can not determine the true value of the velocity from the position with absolute certainty.
The element that would be required to do so, a true dierentiator, is not causal.
On top of that, the numerical derivative causes a drastic amplication of noise, which is a practical issue that has to be dealt with.
Another option we have, to come up with an approximation for the velocity signal, is a dierentiating lowpass lter. It uses a second-order dierence quotient to calculate the velocity, but in addition to the zeros this produces in the transfer function, introduces two fast poles: While at low frequencies the dierentiation is almost unaected, the lter does roll o at higher frequencies, to mitigate the eect of noise on the control behavior.
As good an idea the dierentiating lter is, in practice it does not surpass the performance of the straight-forward derivative block. For the approximation of the derivative by G f (z) to reach the quality of G d (z), the cuto frequency of the lter has to be chosen so high that the noise amplication acts almost in its entirety. As an approximate dierentiator the lowpass lter is not a good choice in this context.
Since we have already successfully used an observer to estimate the additional dynamic state of the inner loop controller, applying the same concept to this problem seems to suggest itself. Similarly to the dierentiating lter, however, the observer poles have to be chosen close to the origin of the complex z−plane for the quality to be competitive to the numerical derivative in Eq. 6.20. In doing so, the observer obtains a distinctly noise-amplifying behavior, which renders an observer useless for our purpose.
We therefore opt for the standard Simulink dierentiator in our simulations.
Equipped with these modications, we turn our attention to the experimental investigations. The block diagram of the continuous plant, controlled by the digital controller, is shown in Figure 6.1.

Performance of the Discrete Controller
To see how well the control quality is maintained in the discrete implementa- On the other hand, a problem in any practical application is associated with noise of various sources. It can be caused by electronic components in the signal chain, by encoder jitter or it can be picked up from the environment. Either way, noise is a source of disturbance that is encountered in any system and that has not been considered up until this point. We want to take a closer look at the consequences of noise in the control circuit.

Impact of Sampling on the Control Behavior
A rst elementary test is to whether sampling the reference signal and controlled variable drastically disturbs the good results we previously achieved with the continuous controller. To that end, we compare the discrete linear controller and observer designed above, in the loop with the continuous plant (see Figure   6.1), to a continuous reference system without friction. It is designed based on the same settling times. This is of course oriented on the experiments that were conducted with the continuous controller. As the reference position, we choose a sinusoid of amplitude 0.01 m and frequency 0.5 Hz.
From Figure 6.2, we can immediately tell that the actual position visibly deviates from the continuous reference progression. The largest dierence is still reached around zero velocity, where the nonlinear LuGre friction eect has the biggest inuence on the control behavior. This is in line with our experiences so far.
On top of that, however, the digital controller entails a staircase-like progression of the controlled variable. Instead of smoothly following the reference trajectory, the course of the position exhibits distinct steps. They are likely caused by the piecewise constant control signal, which forces the system to approach zero velocity in each sampling interval and therewith entails a noticeable impact of Unfortunately though, the signicant steps in the progression of the position are unacceptable for the overall performance of the control circuit. To improve the tracking quality, the sampling time has to be decreased. By the choice of a sampling rate twice as high than the previous one, namely T = 0.005 s, the output progression is smoothed out and we achieve the behavior plotted in Figure 6 We therefore settle with a trade-o again, which is not only dictated by the optimal for System with Discrete Controller and Shorter Sampling Interval tracking quality, but also by the required control energy.
The control variable to achieve the progression in Figure 6.3 is displayed in the plot of Figure 6.4, while the required control variable for a settling time of T s = 0.1 s instead of 0.2 s is shown in Figure 6.5. We can see that the progression in the former plot is relatively smooth and reaches a maximum of around 0.25 V, while it peaks at more than 0.65 V in the latter. Furthermore, the plot in Figure   6.5 exhibits an extreme chatter, which an actuator might not be able to provide in reality, or which could entail mechanical wear in a real-life system. Concluding, we can say that a larger settling time combined with a moderate sampling rate is preferable in this case.
Another aspect we want to highlight again in this section, is the robustness and disturbance rejection in the face of parameter uncertainties and load changes, respectively. Rather than systematically testing a number of dierent parameter sets, we want to simulate faulty system identication based on our experiences from Chapter 5 and in addition to that disturb the system with load steps of the form encountered in said chapter. Since the only information about the plant that is  for Disturbed and Uncertain System used here is the system mass and the viscous friction coecient, we limit ourselves to modifying those two parameters and opt for the severely dierent values m true = 4.5 kg and σ 2,true = 50 kg s . (6.22) Even though the plant parameters are assumed to be far from the true values and the disturbance of the load is signicant in this simulation, the tracking quality is still surprisingly good. Figure 6.6 shows the progression of the controlled variable. The plot shows that the tracking quality is largely unaected with the exception of the instances where load steps occur around zero velocity (e.g. at t = 2.7 s). At these particular times, the peak error is increased by around 50 percent, also causing a rapid increase of the required control variable. All in all, however, the closed-loop behavior is satisfying.
This section has shown that the promising results of the rst chapters translate almost seamlessly into the realm of digital control. Our hope that the control concept proposed in [1] is suitable for experiments in the presence of dynamic friction phenomena is therewith conrmed and the idea is t for practical trials.
By examining the inuence of noise, we want to eliminate another potential source of problems in a physical system.

Inuence of Noise on the Tracking Quality
It is well known that noise is an obstacle in physical environments, which is impossible to avoid. Even though there are numerous ways of coping with noise from dierent sources, reaching from proper shielding of electronics to signal ltering, it is a disturbance which can never be fully avoided. In our case, noise is a particularly delicate topic. Since we are using a dierentiator block to obtain the velocity, any form of noise will be amplied in an unduly manner. Therefore, it has to be made absolutely certain that noise will not deteriorate the control behavior to an extent where the tracking quality becomes useless.
We, again, assume that the system is free of other sources of uncertainty or disturbance and set up another simulation of our discrete linear controller.
The only modication that is made to the experimental assembly is the addition of a noise source, which superimposes a randomly generated number onto the sampled position signal. Instead of white Gaussian noise a uniformly distributed random number is chosen, since it oers a more intuitive interpretation of its range.
Tests with a Gaussian noise source resulted in the same tendencies and showed no supercial dierence to the case considered here.
If the noise samples are uniformly distributed on the interval [−10 −4 , 10 4 ] and the controller is otherwise unaltered, the progression of the position plotted in A bigger problem could be caused with respect to the control variable. It exhibits a high-frequency chatter, similar to that already observed in the plot of Figure 6.5. Without any knowledge about the actuator dynamics, it is dicult to make assumptions to how well a physical system would hold up in the face of noisy measurements, but it is a factor that has to be kept in mind. The problem here is not that an actuator might not be able to provide the required energy, but rather that the rapid changes of the control variable might be unfeasible or could cause unreasonable mechanical wear.
We want to neglect the control variable for the time being and take a closer look at the magnitude of the noise, to verify that our control architecture can in fact operate in the presence of it. If we consider noise of lower amplitude than that used in the simulation above, say uniformly distributed on [−10 −5 , 10 5 ], it has virtually no impact on the tracking quality. However, if we increase the noise level, the deterioration in accuracy and the consumption of control energy it entails, is so large that the controller would be rendered useless for most practical situations. At rst glance, the noise resistance of the controller therefore seems really rather poor. Since noise samples equal to ±10 −3 m already cause a severe degradation in control quality, one could expect that the controller is not practical at all. If we demand high accuracy from our controller, however, we have to assume high accuracy in the hardware as well. A measuring error that reaches values of up to ten percent of the actual value, which would be the case here, is simply not acceptable in the rst place.
Conversely, if we can base our considerations on an encoder providing a measurement within a tolerance of only a few percent, the tracking quality is acceptable and the controller design is applicable in practice. On top of that, if we respect the signal to noise ratio, the noise amplitude is no longer very unrealistic anyway. The output trajectory is a sinusoid of amplitude A = 0.01 m as a good approximation and therefore has a mean square of E These numbers are by no means far fetched when it comes to hardware requirements for a high-accuracy system. We can therefore conclude that the system employing our novel linear controller design yields satisfying tracking quality, if reasonable demands to the noise amplitude are made. It has shown to work really well in simulation and deed not only parameter uncertainties, but also disturbance in the form of load variations and noise.
On top of that, a practical application is within a grasp, since the controller was successfully transferred to the discrete domain. This concludes our examination of the linear control concept and its aptitude for friction compensation.

CHAPTER 7 Summary and Conclusion
To reect on the ndings of the research, we want to go over the results again and point out the essential benets of our approach. We set o with the goal of proving the applicability of the controller design in [1] to arbitrary mechanical systems subject to friction. Our basic condition was that the design is meant to be based on a simple linear approximation of the nonlinear plant. By matching the exceptional results reported in [1] in the context of other systems, we wanted to show that the idea is apt for practical implementation on a larger scale.
As a rst step on the way of dealing with arbitrary friction models, we examined the reconditioning of the plant equations to receive a controllable design model. This was necessary to construct our control concept based on alternative system descriptions. Particularly for the case of the very important LuGre model, this step took some creative handling of the linearized state space representation, as can be seen from Section 3.3. Proven by the extraordinary results in the following simulations, we found that dierent model reduction techniques are suitable for the treatment of the plant equations. This insight constitutes the rst major progress of the work at hand, since it makes the design applicable to a wide variety of systems that are studied in literature.
Throughout Chapter 4 the controller was successfully used for tracking in a number of models. Besides the most elementary single-mass systems, the design also allowed for friction compensation if motor dynamics are taken into account and if the system equations are considerably more complex. On top of that, the control architecture was modied to accommodate for a dierent plant structure in which no distinct velocity loop can be constructed. We have reinforced the hope that the idea is indeed suitable for many system models and can even be adapted to systems of divergent structure. This proves the potential of the approach.
Chapter 5 proved not only that the linear controller design is competitive with existing nonlinear approaches, but also that our design procedure does not rely on an exact model identication. This was another elementary goal of this work. It was shown that the controller is more robust in the presence of parameter uncertainties and disturbances than nonlinear model-based alternatives. Systems and controllers from literature were used as a benchmark.
Finally, and perhaps most importantly, the control algorithm was transferred to the discrete domain, to prepare it for implementation on a digital computer.
In this process, the promising results from the preceding chapters were preserved, which paves the way to an application with high demands on accuracy. Additionally, the inuence of noise on the control quality was studied to determine whether this practical aspect could become an obstacle. Despite the dierentiator in the control loop, noise of reasonable amplitude does not impair the tracking quality drastically and the controller operates with satisfying accuracy.
Thereby, the design was brought as close to a practical application as possible, leaving only an implementation to a physical system as a next step. We can conclude that using the linear controller with the observer for the inner loop has good prospects for friction compensation in mechanical systems. Especially if high accuracy is paramount, the linear approach seems to be promising.

Future Work on the Topic
Despite the highly promising results of this work, there are still some unanswered questions and topics to cover in order to nalize an approbated control concept. The most obvious item on this list is the implementation to a physical system. Through the tests performed and documented in this thesis and the discretization of the control law, the way to such a practical application was paved.
However, no simulation will ever show all problems that could occur during operation with a physical plant. Although the control architecture was successfully used in the context of one particular system, as shown in [1], tests with other models would be indispensable to verify the tness of this controller.
The next potential issue with the cascaded controller is related to the control variable consumption. As mentioned in Section 6.2.2, the rapid chatter that has to be provided by the actuator might not be feasible or could be harmful for the hardware, if a mechanical actuator is used. While the amplitude of the control signal itself should not be a showstopper, rapid changes in its progression might prevent the control circuit from functioning properly. Taking the actuator dynamics into account would be a sensible step to prevent a deterioration of performance in a real-life scenario.
Note that, in our case, this problem has foreshadowed only in the presence of noise, so there might be a dierent way of avoiding it using proper ltering or alternative noise reduction techniques.
Another important supplement to the existing structure of the controller would be a prelter, which allows the controller to reach the desired reference signal amplitudes. It was briey discussed at the end of Chapter 2 that the controller only compensates friction and can not achieve accurate tracking of an arbitrarily fast reference trajectory. This could be mended, however, by employing a linear dynamic prelter.
On top of these facets of the controller, there are some levers in the design process which could have the potential of enhancing the quality even further. One is the location of the required closed-loop poles of the system. The choice of Bessel poles in [1] and throughout this thesis was made because they ensure optimal reproduction of the reference input. These eigenvalues have proven to preserve the Lastly, the control architecture bears opportunities as well. After successfully handling friction in a nonlinear setting, there is hope that the controller might be able to compensate other forms of nonlinearities as well. We saw in Section 4.3 that backlash, modeled by a deadzone in the system dynamics, was mitigated upon proper tuning of the desired settling time, even on top of the already existing LuGre friction eect. This gives rise to hope that our controller might not even be limited to the compensation of friction, but could also deal with other nonlinear phenomena in dynamic systems.

APPENDIX A Appendix A: Structural Analysis of Invariant Zeros
Apart from computing the controllability matrix of a state space model, there is a plainer and more visual way of examining the structural controllability of a dynamic system. It is presented in [35] and comprises a graph-theoretical approach to calculating the number of invariant zeros of a system.
To perform the analysis of invariant zeros of a system and to nd out where they are caused, the following graph is constructed to represent the Rosenbrock matrix. Each state variable of the system corresponds to a node in the graph. An oriented edge of this graph from a state node n i to another one denoted by n j exists, if in the state space representation the corresponding entry of the matrix A, namely [A] ji , is dierent from zero.
For the sake of simplicity and suciently for our purpose, these edges are not weighted with specic factors. Note that, because the graph depicts the Rosenbrock matrix, each state node receives an edge leading back to itself, which is weighted by the variable λ. Both the input and output receive an extra node marked with I and O, respectively. Connecting edges from the input node to state nodes are put in place, if the respective entries in the input matrix b are nonzero and, analogously, from state nodes to the output node if the entries in the output matrix c are dierent from zero.
Finally, for the analysis, the input and output node are connected, completing what is referred to as the structural graph of a system. It is worth noting that in systems with multiple inputs and outputs every output is supposed to be connected to every input. For the system at hand and for systems of equal structure, for that matter, this procedure results in the graph in Figure  Visually speaking, the search for 1-factors means nding the shortest path from an input node to an output node through a sequence of state nodes. Note that the 1-factor of a graph does by no means have to be unique. For the present system structure, the determination of the single minimal 1factor is very straight-forward. The edges of the particular subgraph of interest are marked blue in Figure A.1. To obtain the number of invariant zeros, all that is left to do is to determine the number of loops in the 1-factor, which means nding the number of state nodes in the subgraph that have an edge connecting them to themselves. Since this is clearly the case for the nodes representing z and x a , the system has two invariant zeros. Due to the fact that they arise in loops weighted solely by the factor λ, we can conclude that they are located in the origin of the complex plane.
They are given the name structural invariant zeros because they are a structural property of a dynamic system. This becomes evident when we recall the fact that their identication was carried out without taking the values of model parameters into account, solely considering the mathematical from of the system.
The great advantage of this procedure shows here: we can easily include additional dynamic properties of the system description, like for example motor dynamics, into the analysis. Not only is the algorithm easy to execute and adapt for our purpose, it also shows the same tendency of the linearization of the LuGre friction model to cause an unwanted invariant zero for more complicated system matrices alike.
Analysis of more detailed system models including motor dynamics reveal the same issue due to an invariant zero caused by the dynamic LuGre model.