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Phase-lead compensator with specified centre frequency and lead.

%% phlead_example
% MATLAB script demonstrating the use of phlead


%% Problem Statement
% Imagine we have an unstable plant, and we want to design a control
% system for it that gives an approximate bandwidth of 10 rad/s and a phase
% margin of 60 degrees.

P = zpk([], [0 0], 1);
wc = 10;
Pm = 60;

%% Gain Calculation
% Firstly, let's work out how much we need to scale it to get crossover at
% the desired frequency.

crossoverGain = abs(freqresp(P, wc));
Kp = 1/crossoverGain;
Pw = P*Kp;

%% Phase Calculation
% Next, work out the phase lead we need to add at crossover. (We could have
% got this from the freqresp earlier, but this syntax is more general.)
% Note that we can safely ignore warnings about stability: we expect the
% stability properties to be poor until we add a phase lead compensator.

warning off Control:analysis:MarginUnstable
[~,ThisPm] = margin(Pw);
warning on Control:analysis:MarginUnstable
PhaseLead = Pm - ThisPm;
Clead = phlead(wc, PhaseLead);

%% Combine the gain and phase lead to get a single compensator
% We combine the compensator, then calculate the closed-loop properties of
% our system.

C = Clead * Kp;

L = P * Clead * Kp;
T = feedback(L, 1);
S = 1 - T;
Y = feedback(C, P); % actuator gain, a.k.a. the Youla parameter

%% Plot the results
bodemag(S, T, Y,'m--', ss(1),'k:')

bottomLeft = 3;
legend('Sensitivity', 'Complementary sensitivity', ...
    'Actuator gain', bottomLeft);


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