Formulario esercizi d'esame Control of network systems

TROVARE IL FEEDBACK GAIN

Given the open loop system G(s) and considering the closed loop system with negative unitary feedback , find

the state feedback gain [K1 K2] such that the damping factor is sqrt(2)/2 and the settling time at 2% is 16s

G(s) = 0.03 (s+1)/(s*(s+0.01)) = (0.03*s + 0.03)/(s^2+0.01*s)

% DEFINIZIONE DEL SISTEMA G(S)

n = 0.03 * [1 1]; % Numeratore di G(s)

d = [1 0.01 0]; % Denominatore di G(s)

G = tf(n,d)

G =

0.03 s + 0.03

-------------

s^2 + 0.01 s

Continuous-time transfer function.

Model Properties

% Conversione a rappresentazione in spazio di stato

[A, B, C, D] = tf2ss(n, d);

% SPECIFICHE DI PROGETTO

delta = sqrt(2)/2; % Fattore di smorzamento

Ts = 16; % Tempo di assestamento (2%) in secondi

wn = 4 / (delta * Ts); % Frequenza naturale

% Calcolo dei poli desiderati

p1 = - delta * wn + 1i * delta * wn;

p2 = - delta * wn - 1i * delta * wn;

poles = [p1, p2];

% Calcolo del guadagno di feedback dello stato [K1 K2]

K = -place(A, B, poles)

K = 1×2

-0.4900 -0.1250

TROVARE LO STATE FEEDBACK E L'INPUT DELL'OBSERVER

Given the following system, find the state feedback u=kx and the observer input -Ly such that

1. the feedback gain K sets the system poles in [-1-i,-1+i,-3];

2. the observer gain L sets the observer poles in [-3-3i,-3+3i,-6]

3. Find the transfer function of the controller

4. Find the closed-loop transfer function of this two systems

G(s) = 12 / s(s−2)(s+6) 2

n = 12;

d = [1 4 -12 0];

[A,B,C,~]=tf2ss(n,d);

G = tf(n,d)

G = 12

------------------

s^3 + 4 s^2 - 12 s

Continuous-time transfer function.

Model Properties

p = [-1-1i,-1+1i,-3];

K=-place(A, B, p)

K = 1×3

-1.0000 -20.0000 -6.0000

po=[-3-3i,-3+3i,-6];

L = -place(A', C', po);

L=L'

L = 3×1

-5.6667

-2.8333

-0.6667

[nc, dc] = ss2tf(A+B*K+L*C, -L , K, 0)

nc = 1×4 0 -66.3333 -407.0000 -54.0000

dc = 1×4

1.0000 13.0000 82.0000 308.0000

G_C = tf(nc,dc);

G_CL = feedback(G*G_C, 1, +1) 3

G_CL = -796 s^2 - 4884 s - 648

----------------------------------------------------------

s^6 + 17 s^5 + 122 s^4 + 480 s^3 + 1044 s^2 + 1188 s + 648

Continuous-time transfer function.

Model Properties

G_CL2 = feedback(G, G_C, +1)

G_CL2 = 12 s^3 + 156 s^2 + 984 s + 3696

----------------------------------------------------------

s^6 + 17 s^5 + 122 s^4 + 480 s^3 + 1044 s^2 + 1188 s + 648

Continuous-time transfer function.

Model Properties SISTEMA ECONOMICO NAZIONALE

The national economic system can be represented by the following scheme. G_P(s) is the transfer function

describing the production; G_C(s) is the transfer function describing the national consumption; K_T represents

the government's taxation. Moreover, R(s) is the government's spending and Y(s) is the national income. Let

K_P=1.2, K_C=3, K_T=0.35, tau1=2 and tau2=3. Find a realization of the system G(s)=Y(s)/R(s), i.e. matrices

A,B,C and D.

% Parametri

K_P = 1.5;

K_C = 3;

K_T = 0.3;

tau1 = 1;

tau2 = 2;

% Definizione delle funzioni di trasferimento

s = tf('s');

Ge = (K_P *(1+tau2*s))/((1+tau2*s)*(1+tau1*s)-K_C*K_P*(1-K_T));

% Conversione a rappresentazione stato-spazio

[A, B, C, D] = tf2ss(Ge.Numerator{1}, Ge.den{1})

A = 2×2

-1.5000 1.0750

1.0000 0

B = 2×1

1

0

C = 1×2

1.5000 0.7500

D =

0

Consider the national economic system found in the previous exercise whose input is now integrated. Knowing

that the values of the parameters are the same as the previous exercise, find the state feedback u=kx and the

observer input -Ly such that: 4