Question

EGM 312: System Dynamics 5. Recall that the height deviation in the tanks of following system can be modeled as a second order system with the following form: State Equations: R, State Matrices R,4 R,A, RA, R,A 0 0 H,(s) a) Use MATLAB to determine the transfer function ) Q (s) for the following system parameters: rho-1000 쏙 : g-9.81 m/s' From the transfer function, determine the dominant root and determine the approximate time required for tank 2 to reach its new steady state deviation height. IN b) Plot the response of the system to the following input: Q(0-Su,().Assume that system starts at equilibrium (zero initial conditions) c) confirm your answer from a) using the plot from b)

Answer 1

R1400: R2700; A12; rho = 1000; g = 9.81; A = [ (-g/ (R1 *A1) ) (g/ (R1 *A1) ) ; (g/ (R1 A2)) (g/ (R1 A2)g/ (R2 A2))1: B-[1/(rho A1):01 C = [1 0:0 11: D[0:01 ay 33 (A, B, C, D) ; % Converting State Space to Transfer Functions [NUM, DEN] = 332tf(A, B,C,D,1); % NUM will contain two rows (2 TF) as there are % 2 outputs B2 = tf (NUM (2.. ) , DEN) ; dominant-root max (pole (H2 )); % dominant pole 13 pole close to imaginary axis fprintf("\n Dominant Root :8.4f ", dominant-root); Creating Transfer function (2nd) H2(S) /Qmi (3) from 2nd NUM (close to 0) S stepinfo (H2); % Obtaining Step Info on H2(3) /Qmi (3) tsS.SettlingTime; fprintf("\n Time Required for Tank 2 to reach steady state Getting Settling Time %. 3f m3",ts) ; Simulating Step Response of H2 (S)/Qmí (S) opt stepData0ptions (' StepAmplitude', 5); % Creating Step of amplitude 5 H2,tstep (H2, opt) plot (t, H2) xlabel ("Time [ms]"):ylabel ("Height [m]"): title ("Step Response of Tank 2"); grid on

Time Required for Tank 2 to reach steady state -2928.143 ms Step Response of Tank 2 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Time [ms]

Text :-

R1 = 400;   R2 = 700;
A1 = 2;     A2 = 8;
rho = 1000; g = 9.81;

A = [(-g/(R1*A1))   (g/(R1*A1));
     (g/(R1*A2))    (-g/(R1*A2) - g/(R2*A2))];
B = [1/(rho*A1);0];
C = [1 0;0 1];
D = [0;0];

sys = ss(A,B,C,D);

% Converting State Space to Transfer Functions
[NUM,DEN] = ss2tf(A,B,C,D,1);   % NUM will contain two rows(2 TF) as there are
                                % 2 outputs

H2 = tf(NUM(2,:),DEN);          % Creating Transfer function(2nd) H2(S)/Qmi(s) from 2nd NUM
dominant_root = max(pole(H2));  % dominant pole is pole close to imaginary axis (close to 0)
fprintf("\n Dominant Root :%.4f ",dominant_root);

S = stepinfo(H2);   % Obtaining Step Info on H2(s)/Qmi(s)
ts = S.SettlingTime;    % Getting Settling Time
fprintf("\n Time Required for Tank 2 to reach steady state = %.3f ms",ts);

% Simulating Step Response of H2(S)/Qmi(S)
opt = stepDataOptions('StepAmplitude',5);   % Creating Step of amplitude 5
[H2,t] = step(H2,opt);
plot(t,H2)
xlabel("Time [ms]");    ylabel("Height [m]"); title("Step Response of Tank 2");
grid on