2. [The Matrix Exponential, Laplace Transforms and Solution to Linear Systems] In this question, we use the matrix exponential function defined by The Laplace Transform defined by (b) Using the formula eAtI+At+ F(s) = L{f(t)} eAt = calculate et for the linear system = Ax with A = u(t) A²+² A³+3 + 2! 3! L-¹ (sI - A)-¹ = = =ff(t)e-stdt. system corresponding to an initial condition x(0) = xo (c) Use the formula x(t) = e(t-to) xo + 1 and the intial condition (0) = xo is as in part (b). (d) For the linear system -3 -2 0 [B] Le to calculate the solution to the linear system * = Ax+Bu where A is defined as in part (b), B = is the unit step function eA(t-T) Bu(T)dr for t ≤0 for t > 0 Also, calculate the solution to the linear x = Ax + Bu y = Cx where A and B are defined as in parts (b) and (c) and C= system and find its poles and zeros. - [b]. (² -1 ], find the transfer function of this
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![2. [The Matrix Exponential, Laplace Transforms and Solution to Linear Systems] In this question, we
use the matrix exponential function defined by
eAt = I +At+
The Laplace Transform defined by
(b) Using the formula
F(s) = L{f(t)} = f(t)e="¹dt.
eAt = L-¹ (sI
calculate et for the linear system = Ax with A =
A²t² A³4³
+
2!
3!
system corresponding to an initial condition x(0) = xo =
(c) Use the formula
x(t) = e(t-to) xo +
u(t)
=
and the intial condition x(0) = xo is as in part (b).
(d) For the linear system
-
-3
1
Le
x
y = Cx
A)-¹
to calculate the solution to the linear system = Ax+Bu where A is defined as in part (b), B =
is the unit step function
-2
0
eA(t-T) Bu(T)dT
for t ≤0
for t > 0
Ax+ Bu
Also, calculate the solution to the linear
where A and B are defined as in parts (b) and (c) and C = [₁
1
system and find its poles and zeros.
,u(t)
-1
1], find the transfer function of this](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd5e0332-0650-432d-b006-8bd12fae84dd%2F233b857f-44c5-4e8f-84cb-89be9b9f15a7%2Ft30l04_processed.png&w=3840&q=75)
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