Robust Attenuation of Frequency Varying Disturbances

Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and \mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of Recent Advances in Robust Control give a selective overview of recent theoretical developments and present selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as well as different application areas. The first volume covers selected problems in the theory of robust control and its application to robotic and electromechanical systems. The second volume is dedicated to special topics in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable reference for those interested in the recent theoretical advances and for researchers working in the broad field of robotics and mechatronics.


Introduction
Systems described by differential equations with time-periodic coefficients have a long history in mathematical physics. Applications cover a wide area of systems ranging from helicopter blades, rotor-bearing systems, mechanics of structures, stability of structures influenced by periodic loads, applications in robotics and micro-electromechanical systems etc. (Rao, 2000;Sinha, 2005). Processes characterized by linear time-invariant or time-varying dynamics corrupted by sinusoidal output disturbance belong to this class of systems. Robust and adaptive analysis and synthesis techniques can be used to design suitable controllers, which fulfill the desired disturbance attenuation and other performance characteristics of the closed-loop system. Despite of the fact that LTP (Linear Time Periodic) system theory has been under research for years (Deskmuhk & Sinha, 2004;Montagnier et al., 2004) the analysis on LTPs with experimental data has been seriously considered only recently (Allen, 2007). The importance of new innovative ideas and products is of utmost importance in modern industrial society. In order to design more accurate and more economical products the importance of model-based control, involving increasingly accurate identification schemes and more effective control methods, have become fully recognized in industrial applications. An example of the processes related to the topic is vibration control in electrical machines, in which several research groups are currently working. Active vibration control has many applications in various industrial areas, and the need to generate effective but relatively cheap solutions is enormous. The example of electrical machines considered concerns the dampening of rotor vibrations in the so-called critical speed declared by the first flexural rotor bending resonance. In addition, the electromagnetic fields in the air-gap between rotor and stator may couple with the mechanic vibration modes, leading to rotordynamic instability. The vibration caused by this resonance is so considerable that large motors often have to be driven below the critical speed. Smaller motors can be driven also in super-critical speeds, but they have to be accelerated fast over the critical speed. Active vibration control would make it possible to use the motor in its whole operation range freely, according to specified needs given by the load process. Introducing characteristics of this kind for the electric drives of the future would be a major technological break-through, a good example of an innovative technological development.

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www.intechopen.com In practice, the basic electromechanical models of electrical machines can be approximated by linear time-invariant models with a sinusoidal disturbance signal entering at the so-called critical frequency. That frequency can also vary which makes the system model time-variable. The outline of the article is as follows. Two test processes are introduced in Section 2. A systematic and generic model structure valid for these types of systems is presented in Section 3. Three types of controllers for active vibration control are presented in Section 4 and their performance is verified by simulations and practical tests. Specifically the extension to the nonlinear control algorithm presented in Section 4.4 is important, because it extends the optimal controller to a nonlinear one with good robustness properties with respect to variations in rotation frequency. Conclusions are given in Section 5.

Problem statement
The control algorithms described in the paper were tested by two test processes to be discussed next. In electrical motors both radial and axial vibration modes are of major concern, because they limit the speed at which the motor can be run and also shorten the lifetime of certain parts of the motor. The fundamental vibration forces are typically excited at discrete frequencies (critical frequencies), which depend on the electrodynamics of the rotor and stator (Inman, 2006). In some machines the critical frequency can be passed by accelerating the rotor speed fast beyond it, but specifically in larger machines that is not possible. Hence these machines must be run at subcritical frequencies. It would be a good idea to construct an actuator, which would create a separate magnetic field in the airgap between the stator and rotor. That would cause a counterforce, which would attenuate the vibration mode of the rotor. Running the rotor at critical speeds and beyond will need a stable and robust vibration control system, because at different speeds different vibration modes also wake.

An electric machine
In Fig.1 a 30 kW induction machine is presented, provided with such a new actuator, which is a coil mounted in the stator slots of the machine (b). The electromechanical actuator is an extra winding, which, due to the controlled current, produces the required counter force to damp the rotor vibrations. The actuator is designed such that the interaction with the normal operation of the machine is minimal. More on the design and modelling of the actuator can be found in (Laiho et al., 2008). Some of the machine parameters are listed in Table 1. The vibration of the rotor is continuously measured in two dimensions and the control algorithm is used to calculate the control current fed into the coil. The schema of the control arrangement is shown in Fig.2. The idea is to

Fig. 2. Rotor vibration control by a built-in new actuator
generate a control force to the rotor through a new actuator consisting of extra windings mounted in the stator slots. An adaptive model-based algorithm controls the currents to the actuator thus generating a magnetic field that induces a force negating the disturbance force exited by the mass imbalance of the rotor. The configuration in the figure includes an excitation force (disturbance) consisting of rotation harmonics and harmonics stemming from the induction machine dynamics. The control force and the disturbance exert a force to the rotor, which results in a rotor center displacement. If the dynamic compensation signal is chosen cleverly, the rotor vibrations can be effectively reduced. In practical testing the setup shown in Fig.3 has been used. The displacement of the rotor in two dimensions (xy) is measured at one point with displacement transducers, which give a voltage signal proportional to the distance from sensor to the shaft. A digital tachometer at the end of the rotor measures the rotational frequency. The control algorithms were programmed in Matlab/Simulink model and the dSpace interface system and the Real-Time Workshop were used to control the current fed to the actuator winding.

An industrial rolling process
The second tests were made by a rolling process consisting of a reel, hydraulic actuator and force sensor. The natural frequency of the process was 39 Hz, and the hydraulic actuator acts both as the source of control forces and as a support for the reel. The actuator is connected to the support structures through a force sensor, thus providing information on the forces acting on the reel. The test setup is shown in Fig.4 and the control schema is presented in Fig.5

Modeling and identification
Starting from the first principles of electromagnetics (Chiasson, 2005;Fuller et al., 1995) and structure mechanics, the vibration model can for a two-pole cage induction machine be written in the form (Laiho et al., 2008) where q denotes the states (real and complex) of the system, v is the control signal of the actuator, f ex is the sinusoidal disturbance causing the vibration at the critical frequency, and u rc is the radial rotor movement in two dimensions. The matrices A, B, G and C are constant.
The constant parameter values can be identified by the well-known methods (Holopainen et al., 2004;Laiho et al., 2008;Repo & Arkkio, 2006). The results obtained  by using finite-element (FE) model as the "real" process have been good and accurate (Laiho et al., 2007), when both prediction error method (PEM) and subspace identification (SUB) have been used. Since the running speed of the motor was considered to be below 60 Hz, the sampling rate was chosen to be 1 kHz. A 12th order state-space model was used as the model structure (four inputs and two outputs corresponding to the control voltages, rotor displacements and produced control forces in two dimensions). The model order was chosen based on the frequency response calculated from the measurement data, from which the approximate number of poles and zeros were estimated. In identification a pseudo random (PSR) control signal was used in control inputs. That excites rotor dynamics on a wide frequency range, which in limited only by the sampling rate. However, because the second control input corresponds to the rotor position and has a big influence on the produced force a pure white noise signal cannot be used here. Therefore the model output of the rotor position added with a small PSR signal to prevent correlation was used as the second control input. After identification the model was validated by using independent validation data. The fit was larger than 80 per cent, which was considered to be adequate for control purposes. The results have later been confirmed by tests carried out by using the real test machine data, and the results were found to be equally good.
The model structure is then as shown in Fig.6, where the actuator model and electromechanic model of the rotor have been separated, and the sinusoidal disturbance term is used to model the force that causes the radial vibration of the rotor. In Fig.6a the models of the actuator and rotor have been separated and the disturbance is modelled to enter at the input of the rotor model. The internal feedback shown is caused by the unbalanced magnetic pull (UMP), which means that the rotor when moved from the center position in the airgap causes an extra distortion in the magnetic field. That causes an extra force, which can be taken into consideration in the actuator model. However, in practical tests it is impossible to separate the models of the actuator and rotor dynamics, and therefore the model in Fig.6b has been used in identification. Because the models are approximated by linear dynamics, the sinusoidal disturbance signal can be moved to the process output, and the actuator and rotor models can be combined. In Fig. 6a the 4-state dynamical (Jeffcott) model for the radial rotor dynamics iṡ where y r is the 2-dimensional rotor displacement from the center axis in xy-coordinates, and u r is the sum of the actuator and disturbance forces. The actuator model iṡ where y a are the forces generated by the actuator, and u are the control voltages fed into the windings. The self-excited sinusoidal disturbance signal is generated by (given here in two dimensions)ẋ where ω d is the angular frequency of the disturbance and d(t) denotes the disturbance forces in xy-directions. The initial values of the state are chosen such that the disturbance consists of two sinusoidal signals with 90 degree phase shift (sine and cosine waves). The initial values are then where A is the amplitude of the disturbance. The models of the actuator, rotor and disturbance can be combined into one state-space representatioṅ with As mentioned, the actuator and rotor model can be combined and the disturbance can be moved to enter at the output of the process (according to Fig. 6b). The state-space representation of the actuator-rotor model is theṅ x ar (t)=A ar x ar (t)+B ar u(t) y ar (t)=C ar x ar (t) where u is a vector of applied control voltages and y ar is vector of rotor displacements. The whole system can be modeled aṡ with The process was identified with a sampling frequency of 1 kHz, which was considered adequate since the running speed of the motor was about 60 Hz and therefore well below 100 Hz. Pseudorandom signals were used as control forces in both channels separately, and the prediction error method (PEM) was used (Ljung, 1999) to identify a 12th order state-space representation of the system. The identified process model is compared to real process data, and the results are shown in Figs.7 and 8, respectively. The fit in x and y directions were calculated as 72.5 % and 80.08 %, which is considered to be appropriate. From the frequency domain result it is seen that for lower frequency the model agrees well with response obtained form measured data, but in higher frequencies there is a clear difference. That is because the physical model used behind the identification is only valid up to a certain frequency, and above that there exist unmodelled dynamics.

Control design
In the following sections different control methods are presented for vibration control of single or multiple disturbances with a constant or varying disturbance frequencies. Two of the methods are based on the linear quadratic gaussian (LQ) control, and one belongs to the class of higher harmonic control algorithms (HHC), which is also known as convergent control.I ft h e sinusoidal disturbance frequency signal varies in frequency, the algorithms must be modified by combining them and using direct frequency measurement or frequency tracking.

Direct optimal feedback design
In this method the suppressing of tonal disturbance is posed as a dynamic optimization problem, which can be solved by the well-known LQ theory. The idea is again that the model generating the disturbance is embedded in the process model, and that information is then automatically used when minimizing the design criterion. That leads to a control algorithm which inputs a signal of the same amplitude but opposite phase to the system thus canceling the disturbance. The problem can be defined in several scenarios, e.g. the disturbance can be modelled to enter at the process input or output, the signal to be minimized can vary etc.
Starting from the generic model the control criterion is set where z is a freely chosen performance variable and Q ≥ 0, R > 0 are the weighing matrices for the performance variable and control effort. By inserting z(t)=C z x(t) the criterion changes into the standard LQ form The disturbance dynamics can be modelled aṡ where A dn = 01 −ω 2 dn −ε , i = 1, 2, ..., n and the initial values According to the formalism a sum of n sinusoidal disturbance components (angular frequencies ω dn ) enter the system. The very small number ǫ is added in order the augmented system to be stabilizable, which is needed for the solution to exist. The damping of the resulting sinusoidal is so low that it does not affect the practical use of the optimal controller.
The constant b can be used for a constant bias term in the disturbance. Compare the disturbance modelling also to that presented in equations (4) and (5).
To minimize of sinusoidal disturbances the following performance variable can be chosen which leads to the cost function (10) The solution of the LQ problem can now be obtained by standard techniques (Anderson & Moore, 1989) as where S is the solution of the algebraic Riccati equation It is also possible to choose simply z(t)=x(t) in (9). To force the states approach zero it is in this case necessary to introduce augmented states The system to which the LQ design is used is theṅ In this design the weights in Q corresponding to the augmented states should be set to considerably high values, e.g. values like 10 5 have been used. Usually a state observer must be used to implement the control law. For example, in the configuration shown in Fig.6a (see also equation (5)) that has the forṁx The gain in the estimator can be chosen based on the duality between the LQ optimal controller and the estimator. The state error dynamicsx(t)=x(t) −x(t) follows the dynamicṡx which is similar toẋ . The weighting matrix K N can be determined by minimizing where the matrices Q obs and R obs contain the weights for the relative state estimation error and its convergence rate. The optimal control law (13) can now be combined with the observer model (17). Including the augmented states (15) the control law can be stated aṡ where y r is the rotor displacement, u LQ is the optimal control signal, and A LQ , B LQ and C LQ are the parameters of the controller.

Convergent controller
The convergent control (CC) algorithm (also known as instantaneous harmonic control (IHC) is a feedforward control method to compensate a disturbance at a certain frequency (Daley et al., 2008). It is somewhat similar to the well-known least means squares compensator (LMS), (Fuller et al., 1995;Knospe et al., 1994) which has traditionally been used in many frequency compensating methods in signal processing. A basic schema is presented in Fig.9. The term r is a periodic signal of the same frequency as d, but possibly with a different 0 ( ) Fig. 9. Feedforward compensation of a disturbance signal amplitude and phase. The idea is to change the filter parameters h i such that the signal u compensates the disturbance d. The standard LMS algorithm that minimizes the squared error can be derived to be as where α is a tuning parameter (Fuller et al., 1995;Tammi, 2007). In the CC algorithm the process dynamics is presented by means of the Fourier coefficients as where G F is the complex frequency response of the system and the symbols E F , U F and D F are the Fourier coefficients of the error, control and disturbance signals. For example where N is the number of samples in one signal period, and n is the number of the spectral line of the corresponding frequency. If the sampling time is T s ,thent = kT s . The criterion to be minimized is J = E * F E F which gives where * denotes the complex transpose. The pseudoinverse is used if necessary when calculating the inverse matrix. In terms of Fourier coefficients the Convergent Control Algorithm can be written as where α and β are tuning parameters. It can be shown (Daley et al., 2008;Tammi, 2007) that the control algorithm can be presented in the form of a linear time-invariant pulse transfer function where Y(z) is the sampled plant output and U(z) is the sampled control signal. The convergent controller can operate like an LMS controller in series with the plant, by using a reference signal r proportional to the disturbance signal to be compensated. The 'plant' can here mean also the process controlled by a wide-frequency band controller like the LQ controller for instance.

Simulations and test runs
The controller performance was tested in two phases. Firstly, extensive simulations by using a finite element (FE) model of the electrical machine and actuator were carried out. Secondly, the control algorithms were implemented in the test machine discussed in Section 2.1 by using a dSpace system as the program-machine interface. The disturbance frequency was 49.5 Hz, and the controller was discretized with the sampling frequency 1 kHz. Time domain simulations are shown in Figs. 12 and 13. The damping is observed to be about 97 per cent, which is a good result. The critical frequency of the 30 kW test motor was 37.7 Hz. However, due to vibrations the rotor could not be driven at this speed in open loop, and both the identification and initial control tests were performed at 32 Hz rotation frequency. In the control tests the LQ controller was used alone first, after which the CC controller was connected, in order to verify the performance of these two control configurations. Both controllers were discretized at 5 kHz sampling rate. The test results are shown in Figs. 14-17. In Fig.14 the control signal and rotor vibration amplitude are shown, when the machine was driven at 32.5 Hz. The LQ controller was used first alone, and then the CC controller was connected. It is seen that the CC controller improves the performance somewhat, and generally the vibration damping is good and well comparable to the results obtained by simulations. The same can be noticed from the xy-plot shown in Fig.15. Next, the operation speed was increased to the critical frequency 37.5 Hz. Controller(s) tuned for this frequency could be driven without any problems at this speed. Similar results as above are shown in Figs.16 and 17. It is remarkable that now connecting the CC controller on improved the results more than before. So far there is no clear explanation to this behaviour.

Nonlinear controller
If the frequency of the disturbance signal is varying, the performance of a controller with constant coefficients deteriorates considerably. An immediate solution to the problem involves the use of continuous gain scheduling, in which the controller coefficients are modified according to the current disturbance frequency. To this end the disturbance frequency (usually the rotating frequency) has to be measured of tracked (Orivuori & Zenger, 2010;. The state estimator can be written in the forṁ where it has been assumed that the model topology is as in Fig.6b and the disturbance model is included in the system matrix A. The matrix K changesasafunctionoffrequencyas where f i are suitable functions of frequency. Solving the linear optimal control problem in a frequency grid gives the values of K, which can be presented like in Fig.18 The functions f i Fig. 18. Projections of the hypersurface to the elements of K can be chosen to be polynomials, so that the feedback gain has the form where a ij are the polynomial coefficients and The optimal control gain L(ω hz ) can be computed similarly.
The controller was tested with the industrial rolling process presented in Section 2.2. A sinusoidal sweep disturbance signal was used, which corresponds to a varying rotational speed of the reel with constant width. The rotation frequency ranged over the frequency range 5 Hz..50 Hz. Before the practical tests the theoretical performance of the controller was analyzed. The result is shown in Fig.19, which shows a neat damping of the vibration near the critical frequency 39 Hz. Simulation and practical test results are shown in Figs.20 and Fig. 19. Theoretical damping achieved with the nonlinear controller 21, respectively. The controller turns out to be effective over the whole frequency range the damping ratio being 99 per cent in simulation and about 90 per cent in practical tests. The result of the good performance of the nonlinear controller is further verified by the output spectrum of the process obtained with and without control. The result is shown in Fig.22.

Conclusion
Vibration control in rotating machinery is an important topic of research both from theoretical and practical viewpoints. Generic methods which are suitable for a large class of such processes are needed in order to make the analysis and controller design transparent and straightforward. LQ control theory offers a good and easy to learn model-based control technique, which is effective and easily implemented for industrial processes. The control algorithm can be extended to the nonlinear case covering systems with varying disturbance frequencies.
The performance of such an algorithm has been studied in the paper, and the performance has been verified by analysis, simulation and practical tests of two different processes. The vibration control results have been excellent. In future research it is investigated, how the developed methods can be modified to be used is semi-active vibration control. That is important, because active control has its risks, and all industrial users are not willing to use active control methods.