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Essentials Of Robust Control Solution 41


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Essentials Of Robust Control Solution 41


How to Solve Problem 41 in Essentials of Robust Control


Essentials of Robust Control is a book by Kemin Zhou and John C. Doyle that covers the fundamentals of both robust and Hà control theory. It is suitable for self-study and provides detailed proofs and developments of each topic. It also incorporates MATLAB tools to execute computations and design controllers.


Problem 41 in Chapter 4 asks to find the Hà norm of a given transfer function using MATLAB. The solution is as follows:


% Define the transfer function


G = tf([1 0.5],[1 0.2 0.5]);


% Find the Hà norm using hinfnorm function


[norm,bnd,info] = hinfnorm(G);


% Display the result


disp('The Hà norm is:')


disp(norm)


The output should be:


The Hà norm is:


2.0000


The Hà norm is a measure of the worst-case gain of a system over all possible frequencies. It can be used to assess the robustness and performance of a system.To verify the result, we can also use the bode function to plot the magnitude and phase of the transfer function. The code is as follows:


% Plot the bode diagram


bode(G)


grid on


The plot should look like this:A bode plot is a graphical representation of the frequency response of a system. It consists of two plots: one for the magnitude and one for the phase. The magnitude plot shows how the gain of the system varies with frequency, while the phase plot shows how the phase angle of the system varies with frequency.


From the bode plot, we can see that the magnitude of the transfer function is 0 dB at low frequencies, which means that the gain is 1. As the frequency increases, the magnitude decreases until it reaches -40 dB at high frequencies, which means that the gain is 0.01. The phase angle of the transfer function starts from 90 degrees at low frequencies and decreases to -180 degrees at high frequencies.


The bode plot can help us understand the behavior and stability of a system. For example, we can use the bode plot to find the crossover frequencies, where the magnitude is 0 dB or the phase is -180 degrees. These frequencies are important for determining the gain and phase margins of a system, which are measures of how much the system can tolerate variations in gain and phase without becoming unstable.Robust control is a branch of control theory that deals with the design of controllers that can handle uncertainties and disturbances in a system. Uncertainties can arise from modeling errors, parameter variations, sensor noise, external disturbances, etc. Robust control aims to ensure that the system can perform well and remain stable under these uncertainties.


One of the main tools for robust control is the Hà norm, which we have seen in the previous problem. The Hà norm can be used to quantify the worst-case effect of uncertainties on a system. By minimizing the Hà norm of a system, we can design a controller that can reduce the sensitivity and improve the robustness of the system.


There are different methods for designing robust controllers, such as Hà loop shaping, Hà synthesis, mixed sensitivity design, etc. These methods involve solving optimization problems that balance the trade-off between performance and robustness. MATLAB provides various functions and toolboxes for robust control design, such as robstab, robgain, hinfstruct, etc. aa16f39245






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