Question: Consider a linear time-invariant (LTI) system described by the transfer function . Using MATLAB, design a robust controller using the H-infinity optimization technique to achieve a maximum sensitivity of 0.1 at the crossover frequency of 2 rad/s.
Answer: To design a robust controller using the H-infinity optimization technique for the given transfer function and achieve a maximum sensitivity of 0.1 at the crossover frequency of 2 rad/s, follow these steps in MATLAB:
Step 1: Define the plant transfer function.
s = tf('s');
G = (2*s + 3)/(s^2 + 5*s + 10);
Step 2: Define the closed-loop transfer function with the controller as a variable.
K = tf('K');
T = feedback(K*G, 1);
Step 3: Set up the H-infinity optimization problem.
wc = 2; % Crossover frequency
gamma = 0.1; % Maximum sensitivity
PerfWeight = makeweight(1/gamma, wc, 1);
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Wu = tf(1); % Unweighted plant uncertainty
\delta = ultidyn('Delta', [1 1]);
Wt = [1 0; 0 1]; % Weights for plant states
[K, CL, gamma, info] = hinfsyn(G, 1, 1, PerfWeight, Wu, Wt);
Step 4: Extract the controller transfer function.
Controller = tf(K);
Controller = tf(K); The resulting robust controller, given by the transfer function Controller, can be used to stabilize and achieve the desired robust performance for the given system.
Question: A flexible mechanical structure is modeled by a set of ordinary differential equations. Using MATLAB, develop a robust control strategy based on the H-inf control design technique to stabilize the system and achieve a decay rate of -5 rad/s.
Answer: To develop a robust control strategy based on the H-inf control design technique in MATLAB for stabilizing a flexible mechanical structure modeled by a set of ordinary differential equations and achieving a decay rate of -5 rad/s, follow these steps:
Step 1: Define the state-space representation of the flexible mechanical structure.
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A = [0 1; 0 -5];
B = [0; 1];
C = [1 0];
D = 0;
sys = ss(A, B, C, D);
Step 2: Set up the H-infinity optimization problem.
gamma = 1; % H-infinity performance level
[K, CL, gamma, info] = hinfsyn(sys, 1, 1, gamma);
Step 3: Extract the controller transfer function.
Controller = tf(K);
The resulting robust controller, given by the transfer function Controller, can be used to stabilize the flexible mechanical structure and achieve the desired decay rate of -5 rad/s.
Question: An uncertain linear time-invariant system is described by the transfer function with an uncertainty range of ±10%. Design a robust controller using MATLAB's robust control toolbox to ensure stability and achieve a maximum peak gain of 5.
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Answer: To design a robust controller in MATLAB's Robust Control Toolbox for an uncertain linear time-invariant system described by the transfer function
with an uncertainty range of ±10%, ensuring stability and a maximum peak gain of 5, follow these steps:
Step 1: Define the uncertain system.
s = tf('s');
G0 = 10/(s^2 + 3*s + 7);
deltaG = ultidyn('DeltaG', [1 1], 'Bound', 0.1);
G = G0*(1 + deltaG);
Step 2: Set up the robust control problem.
w = logspace(-2, 2, 100); % Frequency grid for analysis
S = tf('s');
Wp = 5*(S + 1)/(S + 5); % Weighting function for peak gain
RobustSynth = ucover(G, Wp, [], w);
Step 3: Extract the controller transfer function.
Controller = RobustSynth.Fopt; The resulting robust controller, given by the transfer function Controller, will ensure stability and a maximum peak gain of 5 for the uncertain system.
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Question: Consider a multi-input multi-output (MIMO) system with two inputs and two outputs, described by the transfer function matrix
Using MATLAB, design a robust decentralized controller to achieve desired closed-loop poles at -1 ± j2 and -2 ± j3.
Answer: To design a robust decentralized controller in MATLAB for a multi-input multi-output (MIMO) system with two inputs and two outputs, described by the transfer function matrix and achieve desired closed-loop poles at -1 ± j2 and -2 ± j3, follow these steps:
Step 1: Define the plant transfer function matrix.
s = tf('s');
G = [ (s + 1)/(s^2 + 3*s + 5), 2*s/(s^2 + 2*s + 3);
3*s/(s^2 + s + 4), (4*s + 2)/(s^2 + 2*s + 7) ];
Step 2: Set up the decentralized control problem. poles = [-1+2j, -1-2j, -2+3j, -2-3j]; % Desired closed-loop poles
L = place(G, zeros(2, 2), poles);
Step 3: Extract the controller transfer function matrix.
Controller = ss(L);
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The resulting robust decentralized controller, given by the transfer function matrix Controller, will achieve the desired closed-loop poles for the given MIMO system.
Question: An unstable linear time-invariant system is described by the transfer function Using MATLAB, design a robust observer-based controller to stabilize the system and achieve a settling time of 0.5 seconds.
Answer: To design a robust observer-based controller in MATLAB for stabilizing an unstable linear time-invariant system described by the transfer function achieving a settling time of 0.5 seconds, follow these steps:
Step 1: Define the unstable plant transfer function.
s = tf('s');
G = 1/(s^2 + 2*s + 10);
Step 2: Design an observer for the plant.
observerPoles = [-10, -20]; % Desired observer poles
L = place(G', eye(2), observerPoles)';
Step 3: Set up the robust control problem.
Ts = 0.01; % Sampling time
sysd = c2d(G, Ts, 'foh'); % Convert plant to discrete-time
desiredSettlingTime = 0.5; % Desired settling time
refModel = tf(1, [1, 2*zeta*desiredSettlingTime, desiredSettlingTime^2]);
Wp = tf([1], [1], Ts); % Weighting function for performance
RobustSynth = looptune(sysd, refModel, Wp);
Step 4: Extract the controller transfer function.
Controller = RobustSynth.K; The resulting robust observer-based controller, given by the transfer function
Controller, will stabilize the unstable plant and achieve the desired settling time of 0.5 seconds.
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