3.1. Robustness issues
H ∞ (Hendless) control provides significant advantages by effectively addressing the most significant effects of unexpected noise and disturbances occurring in the system.Additionally, it is possible to design a Hendless The controller demonstrates robustness against a predetermined degree of modeling inaccuracy.Unfortunately, as the example below shows [24,25]the last alternative is not always feasible.
planned Hendless The controller’s resistance to modeling errors will be examined in the following sections.The presentation will also include efforts to build a rice-controller, and then conduct a thorough comparison of the two methods. For all simulation scenarios, procedures from the MATLAB Robust Control Toolbox will be used, specifically:
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For uncertain elements, bw1 = real(‘bw1’, 1, ‘percent’, 25)
It achieves a nominal value of 1 and a true uncertainty element ‘bw1’ varying ±25%, i.e. bw1 ranges from 0.75 to 1.25.
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To compute the limits of structured singular values, bounds = mussv(Spqf, Bl);
Where Spqf is the frd object of the system (ie frequency response output), and Bl defines the uncertainty type.
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Calculate a rice-Controller, K = dksyn(qbeam1_u, m, r);
where qbeam1_u defines an uncertain system, m and r are the number of system inputs/outputs. In this case, it is not certain that the system is created through the iconnect structure since it is more general than sysic.
The numerical models used in all simulations were implemented using three different methods:
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Through formula (16),
Then evaluate the specific k value of matrix Np and ricep.
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This is critical to the DK robust synthesis algorithm by leveraging MATLAB’s “uncertain element objects”.
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Implemented through Simulink (Figure 6a, b)
3.2. Robust analysis
Perform robust analysis using the following relationships:
(for strong stability), and,
Achieve robust performance [26,27,28].
In all subsequent simulations, the initial disturbance consists of two components: the first is the dynamic wind force (shown in Figure 7), and the second is the 10 N mechanical load imposed on the free end.The obtained H∞ controller is then subjected to a robust analysis covering specified values of mp and kp.
For the case of mp = 0 and kp = 0.9, which means that the deviation from the nominal stiffness matrix K is ±90%, and the displacement response when subjected to dynamic input is shown in Figure 8. Figure 9 depicts the boundaries of these values. The system maintains stability and robust performance, as evidenced by the fact that the upper bounds of both values are always below 1 at all relevant frequencies.
This assertion is further supported by Figure 10, which shows the applied voltage and the displacement of the free end within considerable uncertainties. For the same system, the nominal controller shows commendable performance compared to the open-loop response. Figure 10 shows the voltages at the last four nodes of the vector; blue represents the last node, the free end of the beam (one actuator), red represents the seventh node (two actuators), and green represents the sixth node ( three executors), light blue indicates for the fifth node (four executors). At all nodes, the voltage is well below 500 V, which is the limit of the piezoelectric patch.
For the case of mp = 0.9 and kp = 0: This represents a significant ±90% deviation from the nominal mass matrix M. Figure 11 visually shows the limits of these values, indicating that the system maintains its stability and operates efficiently. It is worth noting that the upper limit of these two values always remains below 1 in the relevant frequency. This assertion is further supported by Figure 12, which also shows the displacement response of the free end to the first dynamic input and the applied voltage. The nominal controller clearly performs well compared to the open loop response of the same device. Figure 12 shows the results when mp = 0.9, that is (mp) a numeric vector converted into a ±90% deviation from the nominal stiffness matrix M, i.e. the value obtained from a mass change of 1.9 relative to the initial value. See equation (16).
For the scenario of mp = 0.9 and kp = 0.9: This corresponds to a significant ±90% change in the nominal values of the mass matrix M and stiffness matrix K.Figure 12 shows the results for mp = 0.9, where mp is a numerical vector that translates to a ±90% deviation from the nominal mass matrix M = 1.9M0 (Equation (16)), that is, obtained from the relationship between the mass change of 1.9 relative to the initial value.
Figure 13 illustrates the limits of these values. Clearly, given that the upper limit of both values for all relevant frequencies always remains below 1, the system maintains stability and exhibits robust behavior.
By exploiting the structural uncertainty of real plants, rice-Analysis can improve the accuracy of singular value functions of closed-loop systems.The so-called DK iteration can be used in rice-Comprehensive improved controller, considering structured singular value functions. Weighting factors and controllers were developed iteratively using this process. This method still works even if the joint optimization or DK iteration is not convex, and global convergence is not guaranteed.The purpose of this study is to prove HendlessBased on control design strategies, providing reliable stability and minimum performance. Many nominal performance and strong stability parameters will be provided as they are critical to the controller.The purpose of this work is to propose a HendlessBased on control design methods, providing nominal performance and reliable stability. Since standards specifying both types are critical to controller design, many nominal performance and robust stability characteristics will be provided. However, the problem is more difficult given that the identification process results in a nominal model of the inverted pendulum. To select strong stability and nominal performance criteria, various design options are provided. Although meeting the necessary high stability criteria, in order to ensure nominal and reliable performance, the developed controller employs DK iterations in the synthesis.
3.3. Robust synthesis: μ-controller
A rice-Controllers can be designed using the DK iteration technique discussed earlier.As mentioned before, this method approximates rice-Value and quote range [29,30,31,32]. To facilitate comparison with the controller, we will apply equivalent constraints on the uncertainty. In all simulations, we apply a second mechanical force of 10 N at the free end of the beam.
For the case of mp = 0 and kp = 0.9: This means that the stiffness matrix K deviates from the nominal value by ±90%.
As mentioned earlier, the commands required to perform this process in MATLAB are:
beam_u = ss(A0_u, eye(2 × nd), C, Zeros(nd/2, 2 × nd));
M = connection;
nn = icsignal(4);
d = icsignal(8);
u = icsignal(4);
y = icsignal(4);
M.Equation1 = eqate(y, beam_u × [B0_u × u + G0_u × Wd × d]);
M.Input = [d; nn; u];
M.Output = [We × y; Wu × u; y + Wn × nn];
qbeam_w_o = M.System;
[K, qbeam_w_c_m, gam_miu] = dksyn(qbeam_w_o, m, r);
Among them, G0_u, B0_u and A0_u are uncertainty matrix objects.
Executing this command will produce a robust controller of order 42. However, although this is acknowledged in the literature, it has not been fully addressed and is in fact a limitation.As far as we know, there is no easy way to reduce orders short of laborious and cumbersome manual methods [33,34,35].Figure 14 illustrates rice-The calculated value of the controller.It is obvious that the controller exhibits robustness over a wide frequency range [36].
In Figure 15, the performance of different devices is compared rice– Free-end controller and H∞ controller covering overall performance.It is obvious that the H∞ controller performs better than rice-Controller, although at the cost of tighter control work. Figure 16 supports this observation, which shows that the H∞ controller performs more efficiently at extreme values.This change may stem from digital challenges during rice-Controller calculations due to low condition number of the plant. Orders from high-level controllers may also contribute to this discrepancy.For the scenario of mp = 0.9 and kp = 0.9: This corresponds to a significant ±90% change in the nominal values of the mass matrix M and stiffness matrix K (equation (16)), which means M = 1.9 × M0K = 1.9 × K0 or M = 0.1 × M0K = 0.1×K0.
This paper provides a novel approach to incorporating uncertainty into simulation models and damping structural vibrations using mass and stiffness matrices. A major technical novelty is the ability to suppress oscillations even with extremely significant modifications of the starting matrix of the model. The starting mass and stiffness vary within a range of plus or minus 90% of the nominal values, which means that the model changes too much. However, the oscillations are suppressed within the resistance limits of the piezoelectric piece. This difference may be the result of model failure and modeling uncertainty.
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