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If a harmonic input is applied to the considered LTI system with saturation, then the describing function can be used to compute a linear approximation of the dynamics November 26, World Scientific Book - 9in x 6in Frequency Domain Performance Analysis of LTI Systems with Saturation dynamics 55 system, which in turn can be evaluated in the frequency domain. Since it is also possible to compute an upper bound on the error between the linear approximation and the original nonlinear system, see [van den Berg et al.

Both approaches are practically validated on an experimental setup an electromechanical system and a simulation model of this setup. The outline of this paper is as follows. Section 5. Furthermore it is demonstrated by means of an example why frequency domain analysis can not be performed for nonlinear systems in general. In Section 5. The results in Sections 5. Finally, Section 5. Then, we introduce an electromechanical system within this class of systems, that will be used as a case study to practically validate the theoretical results discussed in this paper.

Finally, we show by means of an example that this system can —under certain settings— exhibit rich nonlinear dynamics, which make a frequency domain analysis virtually impossible. Based on these observations, we make some statements on the conditions that a nonlinear system should satisfy in order to allow frequency domain analysis. These statements will be elaborated in Sections 5. Rooda where Ap has one eigenvalue at 0 and the other eigenvalues if any in the open left-hand plane, i.

Although the theory that we present in Sections 5.

Dynamics and control of hybrid mechanical systems

This case is discussed in the following subsection. The hardware consists of two rotating masses connected by an element that has a certain stiffness and damping. The hardware is connected at sample rate: 1 kHz to a computer with a Matlab Simulink model Real Time Workshop , which contains a PI controller, a saturation function and a static anti-windup gain as shown in Figure 5.

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The actuator is driven by a velocity controller not shown in Figure 5. The settling time of the velocity controller is negligible, so that we can assume that the actuator exactly follows the reference velocity v. In order to perform also simulations on this case, the parameters of the electromechanical system have been identified and a simulation model has been created.

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The model is of the form 5. For periodic motions with relatively low frequencies and amplitudes that are neither too small or too large, the electromechanical system can be well approximated by the above model. In this subsection two examples are given that clearly indicate what difficulties arise when trying to make a frequency domain analysis. For the first example, consider the system 5. We evaluate the solution of this system for two initial conditions, i. The resulting rotation angle of mass 2 as a function of time is given in Figure 5.

For the second example, consider again the system 5. In this example we show that different initial conditions can not only lead to different 1periodic limit solutions, but also to multi-periodic limit solutions.

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For four initial conditions the solution of the system is evaluated using both the experimental setup and simulation. The control output u, which clearly displays the multi-periodic solutions, is shown as a function of time in Figure 5. Since frequency domain analysis is based on a one-to-one mapping from input signal e. As shown in this subsection, the output signal of a nonlinear system, however, does not necessarily satisfy these conditions, i. This motivated us to investigate whether there exist conditions under which the performance of nonlinear system 5.

Two approaches have been found that allow to find sufficient conditions under which a frequency domain analysis can be performed for system 5. These approaches will be discussed in respectively Section 5. November 26, 60 World Scientific Book - 9in x 6in dynamics R. Rooda Fig. Then, we discuss the conditions under which system 5. Finally, we show how to perform a simulation-based frequency domain analysis for the convergent system.

Furthermore, assume that f x, w satisfies some regularity conditions to guarantee the existence of local solutions x t, t0 , x0 of system 5. Definition 5. System 5.

The following statements describe some properties of this limit solution. Property 1 Pavlov et al. Property 2 Pavlov et al. Suppose system 5. November 26, 62 World Scientific Book - 9in x 6in dynamics R. Theorem 5.

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  5. Thus, we can find November 26, World Scientific Book - 9in x 6in Frequency Domain Performance Analysis of LTI Systems with Saturation dynamics 63 a kind of frequency response function if we evaluate the input-output behavior for a range of frequencies. Since the output signal is not necessarily harmonic, however, we can not obtain a typical gain and phase plot Bode plot as for linear systems. Instead, we determine a nonlinear frequency response function, as discussed in [Pavlov et al.

    As phase is not defined for nonlinear systems, only the gain as discussed above will be considered in our frequency domain analysis. Since the limit solution of a convergent system only depends on the input and is independent of the initial conditions, a single simulation run experiment suffices to determine the limit solution of the system. This simulation-based frequency domain analysis is now demonstrated for system 5. Any other desired frequency response function can be computed in a similar way. Due to the relatively high frequency in combination with the saturation function, the amplitude of the motion of the masses becomes so small that nonlinear behavior of the experimental setup becomes significant, which in turn results in a different RMS-gain.

    However, since dealing with the undesired nonlinear behavior of the experimental setup lies outside the scope of this paper, it will not be discussed further here. In the remainder of this paper, we will focus on the dynamics as described by the simulation model. Figure 5. November 26, 64 Fig.

    World Scientific Book - 9in x 6in R. Rooda Nonlinear frequency response function experiments: dots, simulations: solid Note, however, that the computed frequency response function in Figure 5. For the same reason, the superposition principle does not hold. On the other hand, computing the frequency response function for any multi-harmonic input signal is as simple as computing this function for a harmonic input signal, so the frequency response to any periodic input can be obtained by this approach. Furthermore, note that even if we were able to find a finite L2-gain for this marginally stable system, this would only be a horizontal line in Figure 5.

    Our approach based on convergence and simulation provides more detailed information on the frequency domain behavior of the system. In the following section we will consider another approach, based on the describing function method, which is much more time-efficient, but at the cost of accuracy, i.

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    Also, this approach can only be used for harmonic input signals. Then, we discuss a theorem which gives sufficient conditions for computation of a linear approximation and upper- and lower bound of the error of this approximation. Finally, we apply the theory on the system 5. Applying the fact that the saturation nonlinearity is an odd function and filling in 5. Note that the left-hand side of 5. In this Figure we plotted the left-hand side of 5.

    Left hand side of 5. Consider system 5.


    Rooda Performance analysis example To demonstrate the use of Theorem 5. From Section 5. Instead of performing many time-consuming simulations, we now simply compute the linearization and error bounds for the given range of frequencies using the approach given in Subsection 5. The result is given in Figure 5. For comparison, the results of the simulation approach, and the gain of the linear system, i.

    As one can see, the exact results as obtained with the simulation approach lie well within the error bounds of the approximation obtained by the describing function approach. Since the error bounds are relatively small for this case, the describing function approach gives a quite accurate description of the frequency domain behavior of nonlinear system 5. It can also be clearly seen that the frequency domain behavior of the system dynamics November 26, World Scientific Book - 9in x 6in Frequency Domain Performance Analysis of LTI Systems with Saturation dynamics 69 with saturation substantially differs from the system without saturation.

    Then, the simulation approach leads to an exact performance analysis, but can be time-consuming. The other approach, based on the describing function method, is computationally much faster, but at the cost of some accuracy: only an upper- and lower bound can given on the performance of the nonlinear system, although these bounds can be very close. An electromechanical system has been used as a case to demonstrate and practically validate both approaches. Analysis of periodically forced uncertain feedback systems, IEEE Transactions on circuits and systems—I: fundamental theory and applications 50, 2, pp.

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    Khalil, H. Nonlinear systems, Prentice Hall, New Jersey, third edition.

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    Pavlov, A.