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The state space or the internal description of the system still involves a relationship between the input and output signals, what are the additional set of variables it also involves?
Explanation: Although the state space or the internal description of the system still involves a relationship between the input and output signals, it also involves an additional set of variables, called State variables.
State variables provide information about all the internal signals in the system.
Explanation: The state variables provide information about all the internal signals in the system. As a result, the statespace description provides a more detailed description of the system than the inputoutput description.
Which of the following gives the complete definition of the state of a system at time n0?
Explanation: We define the state of a system at time n0 as the amount of information that must be provided at time n0, which, together with the input signal x(n) for n≥n0 determines output signal for n≥n0.
From the definition of state of a system ,the system consists of only one component called memory less component.
Explanation: According to the definition of state of a system, the system consists of two components called memory component and memory less component.
If we interchange the rows and columns of the matrix F, then the system is called as:
Explanation: The transpose of the matrix F is obtained by interchanging its rows and columns, and it is denoted by FT. The system thus obtained is known as Transposed system.
A single inputsingle output system and its transpose have identical impulse responses and hence the same inputoutput relationship.
Explanation: If h(n) is the impulse response of the single inputsingle output system, and h1(n) is the impulse response of the transposed system, then we know that h(n)=h1(n). Thus, a single inputsingle output system and its transpose have identical impulse responses and hence the same inputoutput relationship.
A closed form solution of the state space equations is easily obtained when the system matrix F is:
Explanation: A closed form solution of the state space equations is easily obtained when the system matrix F is diagonal. Hence, by finding a matrix P so that F^{1}=PFP^{1} is diagonal, the solution of the state equations is simplified considerably.
What is the condition to call a number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector?
Explanation: A number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector if
FU= λU
Thus, we obtain (F λI)U=0.
The determinant F λI=0 yields the characteristic polynomial of the matrix F.
Explanation: We know that (F λI)U=0
The above equation has a nonzero solution U if the matrix F λI is singular, which is the case if the determinant of (F λI) is zero. That is, F λI=0.
This determinant yields the characteristic polynomial of the matrix F.
The parallel form realization is also known as normal form representation.
Explanation: The parallel form realization is also known as normal form representation, because the matrix F is diagonal, and hence the state variables are uncoupled.
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