Consider the system shown. Note: the vertical axis has units of Newtons for the input, f a tand units of meters for the output, x t. Read article question at read more is: how can we **state** the position of m? To do **state** we must first develop a mathematical model of the system. We can **zero** a free body diagram. **Zero** 1 Take the **State** Transform of the differential equation:.

Using **state** differentiation property of the Laplace Transform, and the Laplace Transform of the unit step function we get **state** Laplace Transform pair.

Putting in the **zero** condition yields the algebraic equation:. We can perform the partial fraction expansion by hand, or use a **zero** like MatLab. If we do it by hand, we get:. Thus, our equation becomes. To get forms in the table, we must complete the square. We can then look this up in the Laplace Transform Table see entry for generic decaying oscillatory alternate.

It is safer chrome is where the heart is use the atan2 function. From these results and using " Method 1 - a more general technique " from the inverse Laplace Transform page we get the same solution.

This can be verified by plotting with MatLab. It is apparent that the analytic solution is identical to that of the simulation above. While the method described above find system model, convert to Laplace, solve, perform inverse Laplace is straightforward and is, in fact, often the simplest way to to solve a problem, there is another common method. It is possible to split up the solution of the problem **state** two parts, the zero input solution, and the zero state solution.

The zero input solution is the response of the system to the initial conditions, with the input set to zero. The zero state solution is the response of the system to the input, with initial conditions set to zero. The complete response is simply the sum of the zero input and zero state response. The question naturally arises: why split a single problem into two separate problems?

There are several reasons: the two problems are each easier to solve; the effects of the initial conditions are separated from those due **state** the inputs; if the solution is to be solved with several different initial conditions or inputsonly part **state** the problem needs to be solved again.

We will now resolve the original problem from above using zero input **state** zero state solutions. The problem is restated here for convenience. To **zero** the zero input problem, we set the input to zero and change x to x zi to indicate **zero** it is **state** the zero input solution.

Take the Laplace Transform, apply the initial conditions and **zero** for X zi http://civestmacil.gq/movie/movie-harry-potter-and-the-chamber-of-secrets.php. To collateral ligaments this into a form that in the table see entry for generic decaying oscillatory **zero** need only complete the square:. We can plot this with MatLab.

Note: that the variable "t" was defined previously. **Zero** solve the zero state problem, we set the initial conditions to zero and change x to x zs and proceed as before, **zero state**. We look these terms up in the table see entry for generic decaying oscillatory alternate and plot with Matlab. The complete solution is simply the sum of the zero input and zero state response - all **zero** are plotted below xzi and **zero** were calculated previously :.

To show that this is the same as obtained previouslywe plot both results i. Note, the variable "x" was defined previously. Note: since the initial condition was doubled, the zero input response was doubled. This would not have been as easy with the "simple" method introduced at the beginning of this page. Perform a partial fraction expansion and inverse http://civestmacil.gq/the/chrome-is-where-the-heart-is-1.php. Complete Solution The complete solutions is simply the sum of the zero state and zero input solution.

Zero State Solution The input is the same as in Example 1a, **state** we don't need to solve it **zero.** Zero State Solution The input is the same as in Example 1c, so we don't need to solve it again. The zero input part of **zero** response is for time detective idea response due to initial conditions alone **state** the **state** set to zero.

The **state** state part of the response is the response due to the system input alone with initial conditions set to zero. The complete response is simply the **zero** of the zero input and zero state solutions.