State Transition Matrix, Determining States

In summary, the state of the given LTI system with a unit-step input and initial conditions x1(0) = 1 and x2(0) = 0 is represented by the following equations in the Laplace domain: X1(s) = (s/(s^2 + s - 0.5)) + (0.5/(s^2 + s - 0.5))X2(s) = 1/(s^2 + s - 0.5)
  • #1
jegues
1,097
3

Homework Statement



An LTI system is given in state-space form,

[itex]\left( \begin{array}{cc}
\dot{x_{1}} \\
\dot{x_{2}}
\end{array} \right)
=
\left( \begin{array}{cc}
-1 & 0.5 \\
1 & 0
\end{array} \right)
\left( \begin{array}{cc}
x_{1} \\
x_{2}
\end{array} \right)
+
\left( \begin{array}{cc}
0.5 \\
0
\end{array} \right)
u
[/itex]

A unit-step signal is applied to the input of the system. If,

[itex]x_{1}(0) = 1, \quad x_{2}(0) = 0[/itex]

determine the state of the system after t = 0.1 sec.

Homework Equations


The Attempt at a Solution



[itex]\underline{\dot{x}} = \underline{A} \underline{x} + \underline{B}u[/itex]

[itex]\mathcal{L} \Rightarrow s \underline{X(s)} - \underline{x(0)} = \underline{A}\underline{X(s)} + \underline{B} U(s)[/itex]

[itex]\Rightarrow \underline{X(s)} = (s\underline{I} - \underline{A})^{-1} \underline{x(0)} + (s\underline{I} - \underline{A})^{-1}\underline{B}U(s)[/itex]

Working through the simplification I obtain,

[itex]\left( \begin{array}{cc}
X_{1}(s) \\
X_{2}(s)
\end{array} \right)
=
\left( \begin{array}{cc}
\frac{s}{s^{2}+s-0.5}\\
\frac{1}{s^{2}+s-0.5}
\end{array} \right)
+
\left( \begin{array}{cc}
\frac{0.5}{s^{2}+s-0.5}\\
\frac{0.5}{s(s^{2}+s-0.5)}
\end{array} \right)
[/itex]
Thus,
[itex]
X_{1}(s) = \frac{s}{s^{2}+s-0.5} + \frac{0.5}{s^{2}+s-0.5}
[/itex]
[itex]
X_{2}(s) = \frac{1}{s^{2}+s-0.5} + \frac{0.5}{s(s^{2}+s-0.5)}
[/itex]

How can I put this in a form that I can easily pull out of the s-domain back into time domain using inverse Laplace transform? If weren't for the -0.5 in the denominator I think I could work something out by reworking it into one of the following two forms,
[itex]
\frac{\omega}{(s+a)^{2} + \omega^{2}} \quad \text{ or } \quad \frac{s+a}{(s+a)^{2} + \omega^{2}}
[/itex]

Any ideas? Did I make a mistake in my simplification perhaps?
 
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  • #2
1.) what are the initial conditions for the system? i am assuming x(0) equals zero. If x(0) equals zero you need to remove those second sections from your equations for X1 and X2.

2.)
below is one of the forms you were talking about
e^(-at)*sin(wt) = invlap{ω /( (s+a)^2+ω^2)}

why don't you try this one?
e^(-at)*sinh(wt) = invlap{ω /( (s+a)^2-ω^2)}
 
  • #3
or thise^(at)*sin(wt) = invlap{ω /( (s-a)^2+ω^2)}
 
  • #4
or thise^(at)*sinh(wt) = invlap{ω /( (s-a)^2-ω^2)}
 
  • #5
donpacino said:
1.) what are the initial conditions for the system?

They are provided in the original post.
 
  • #6
jegues said:
[itex]
X_{1}(s) = \frac{s}{s^{2}+s-0.5} + \frac{0.5}{s^{2}+s-0.5}
[/itex]
[itex]
X_{2}(s) = \frac{1}{s^{2}+s-0.5} + \frac{0.5}{s(s^{2}+s-0.5)}
[/itex]

My apologies. For some reason I though that was the input vector.

due to the fact that X2(0)=0, your expression for X2(s) is not correct. Eliminate the second part of the statement
 

FAQ: State Transition Matrix, Determining States

What is a State Transition Matrix?

A State Transition Matrix is a mathematical tool used to represent the change in state of a system over time. It is typically represented as a square matrix with the number of rows and columns equal to the number of states in the system.

How is a State Transition Matrix determined?

A State Transition Matrix is determined by analyzing the behavior of a system and identifying the different states it can be in. The matrix is then constructed based on the probability of transitioning from one state to another over time.

What is the purpose of using a State Transition Matrix?

The purpose of using a State Transition Matrix is to model and predict the behavior of a system over time. It is commonly used in fields such as engineering, economics, and biology to analyze and understand complex systems.

What are the key components of a State Transition Matrix?

The key components of a State Transition Matrix include the state space, transition probabilities, and initial state probabilities. The state space represents all possible states of the system, the transition probabilities define the likelihood of transitioning between states, and the initial state probabilities represent the starting state of the system.

How is a State Transition Matrix used in real-world applications?

A State Transition Matrix is used in a variety of real-world applications, such as forecasting stock market trends, predicting traffic flow, and analyzing population dynamics. It is also used in the design and control of complex systems, such as aircraft and computer networks.

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