design of control system

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design of control

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it is lab assignment in design of control system and you need to use the matlab to solve question in the file down and you need to start with your introduction first then solve after that conclusion.

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University of the District of Columbia Control Systems Lab Experiment #3 Laplace transform L = laplace(F) is the Laplace transform of the sym F with default independent variable t. The default return is a function of s. If F = F(s), then laplace returns a function of z: L = L(z). By definition, L(s) = int(F(t)*exp(-s*t),t,0,inf). L = laplace(F,z) makes L a function of z instead of the default s: laplace(F,z) L(z) = int(F(t)*exp(-z*t),t,0,inf). L = laplace(F,w,u) makes L a function of u instead of the default s (integration with respect to w). laplace(F,w,u) L(u) = int(F(w)*exp(-u*w),w,0,inf). Examples: syms a t y f = exp(-a*t); h1=laplace(f) pretty(h) 1 ----a + s laplace(f, y) syms t s h2=laplace(dirac(t - 3), t, s) syms a s t w x F(t) laplace(t^5) laplace(exp(a*s)) returns 120/s^6 returns -1/(a-z) laplace(sin(w*x),t) returns w/(t^2+w^2) laplace(cos(x*w),w,t) returns t/(t^2+x^2) laplace(x^(3/2),t) returns (3*pi^(1/2))/(4*t^(5/2)) laplace(diff(F(t))) returns s*laplace(F(t),t,s) - F(0) Inverse Laplace transform in MATLAB F = ilaplace(L) is the inverse Laplace transform of the sym L with default independent variable s. The default return is a function of t. If L = L(t), then ilaplace returns a function of x: F = F(x). By definition, F(t) = int(L(s)*exp(s*t),s,c-i*inf,c+i*inf) where c is a real number selected so that all singularities of L(s) are to the left of the line s = c, i = sqrt(-1), and the integration is taken with respect to s. F = ilaplace(L,y) makes F a function of y instead of the default t: ilaplace(L,y) F(y) = int(L(y)*exp(s*y),s,c-i*inf,c+i*inf). F = ilaplace(L,y,x) makes F a function of x instead of the default t: ilaplace(L,y,x) F(y) = int(L(y)*exp(x*y),y,c-i*inf,c+i*inf), integration is taken with respect to y. Examples: syms s f1 = 1/s^2; h3=ilaplace(f1) t syms s f2=1/(s*(s+1)^2) h4=ilaplace(f2) 1 - t*exp(-t) - exp(-t) syms s f3=(3*s+2)/(s^2+2*s+10) h5=ilaplace(f3) pretty(h5) / sin(3 t) \ exp(-t) | cos(3 t) - -------- | 3 \ 9 / syms s t w x y f(x) ilaplace(1/(s-1)) returns exp(t) ilaplace(1/(t^2+1)) returns sin(x) ilaplace(t^(-5/2),x) returns (4*x^(3/2))/(3*pi^(1/2)) ilaplace(y/(y^2 + w^2),y,x) returns cos(w*x) ilaplace(laplace(f(x),x,s),s,x) returns f(x) ASSIGNMENT Find the Laplace transform corresponding to problems of 3.3, 3.4 as well as the inverse Laplace corresponding to problems 3.7 and 3.8 of the text book. charles L. phillips fourth edition 3.3. (a) Draw a simulation diagram for the system described by the transfer function s Y(S) = G(S) = U(s) $3+1 (b) From the simulation diagram write a set of state equations for the system of (a). 3.4. (a) Consider the closed-loop control system in Figure P3.4(a), and its simulation diagram in part (b) of the figure. Write the state equations for the control system, from the simulation diagram (b) Find G (s), the compensator transfer function, and Gy(s), the plant transfer function, directly from the simulation diagram. (c) Find the closed-loop transfer function Y(s)/U(s). (d) Show that the denominator of the closed-loop transfer function is equal to det(sI - A). using A from (a). (e) Verify the results of (d) by finding the A of Mason's gain formula. Note that, in this prob- lem the denominator of the closed-loop transfer function has been calculated by three dif- ferent procedures. (f) Modify the MATLAB program of Example 3.4 to verify the closed-loop transfer function. Compensator U(s) Plant Gels) Y\s) Gols) (a) Compensator - Gls) Plant - Gols) 2 U(s) 1 4 2 1 Yls) 3 -5 (b) Figure P3.4 Control system for Problem 3.4. transfer functions in (e). 3.7. Consider the motor in the steel-rolling system of Example 3.3. In this problem we use the same motor for a position control system, that is, the motor is not used for speed control. (a) Choosing the state variables to be x1 = 1, X2 = 0, X3 = d0/dt, and the output y = 0, write the state equations for the motor. (b) Draw a simulation diagram for (a). Note that when using the laws of physics, the state model is neither the control canonical nor the observer canonical form. (c) Using Mason's gain formula and the simulation diagram of (b), verify the transfer function (s)/E,(s) developed in Section 2.7. 3.8. The equations for the circuit of Figure P3.8 are developed in Example 2.2, Section 2.2. (a) Express these equations in the state variable format, using the variables x(t) = (t) = dt, u(t) = v. (1), and y(t) = v2 (1). ſiceydt. (b) Draw a simulation diagram for these state equations. (e) Verify the transfer function calculated in Example 2.3, using Mason's gain formula that is, show that Y(s) R, Cs + 1 U(s) (R+R) Cs +1 (d) Verify the transfer funtion in (c) using (3-42). iſt) R w R2 } V,(t) Vat) Figure P3.8
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Lab Report - Design of Control System
Course’s Name
Student’s Name
Professor’s Name
Institution
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Lab Report - Design of Control System
Introduction
We change over elements of time into capacities that are mathematical in the
unpredictable factors. We supplant separation and necessary tasks by mathematical activities all
including the unpredictable variable. We enable the utilization of graphical techniques to foresee
framework execution without comprehending the differential conditions of the framework. These
incorporate reaction, enduring state conduct, and transient conduct. At long last, we
fundamentally utilized as a part of control framework examination and outline.
Solution to Problems
Problem 3.3:
a) It’s given that
Y s

s2
 G s  2
U s
s 1

The simulation diagram for the system described by the given transfer function is drawn below:

b) A set of state equation can be written as follow:
 0 1 0
0


x  t    0 0 1  x  t    0  u  t 
 1 0 0 
1 
y  t    0 1 0 y  ...


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