3.1. Robustness issues
planned H_{endless} The controller’s resistance to modeling errors will be examined in the following sections.The presentation will also include efforts to build a ricecontroller, 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:

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.

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.

Calculate a riceController, 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:

Through formula (16),
medium=medium_{0} + meters_{0}(I + m_{p}d_{medium size})
medium size_{0} is the initial mass matrix
K_{0} is the initial stiffness matrix:
K=K_{0} +K_{0}(I+k_{p}d_{K})
D = D_{0} +0.0005[K_{0}k_{p}I_{2n×2n}δ_{K} + M_{0}m_{p}I_{2n×2n}δ_{M}]Then evaluate the specific k value of matrix N_{p} and rice_{p}.

This is critical to the DK robust synthesis algorithm by leveraging MATLAB’s “uncertain element objects”.

Implemented through Simulink (Figure 6a, b)
3.2. Robust analysis
$$\underset{\mathsf{oh}\epsilon \mathbb{right}}{\mathrm{s}\mathrm{you}\mathrm{p}}{\mathsf{rice}}_{\mathsf{D}}\left({\mathrm{nitrogen}}_{11}\left(\mathrm{j}\mathsf{oh}\right)\right)<1$$
(for strong stability), and,
$$\underset{\mathsf{oh}\epsilon \mathbb{right}}{\mathrm{s}\mathrm{you}\mathrm{p}}{\mathsf{rice}}_{{\mathsf{D}}_{\mathrm{A}}}\left(\mathrm{nitrogen}\left(\mathrm{j}\mathsf{oh}\right)\right)<1$$
Achieve robust performance [26,27,28].
By exploiting the structural uncertainty of real plants, riceAnalysis can improve the accuracy of singular value functions of closedloop systems.The socalled DK iteration can be used in riceComprehensive 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 H_{endless}Based 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 H_{endless}Based 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: μcontroller
For the case of m_{p} = 0 and k_{p} = 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.
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.