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Questions and Answers
What does the differential equation order equal to?
What does the differential equation order equal to?
A system is stable when all poles of G(s) are positive.
A system is stable when all poles of G(s) are positive.
False
What is represented by the state vector X?
What is represented by the state vector X?
(x_1, x_2)
A system with numerator greater than denominator has no direct ______ through.
A system with numerator greater than denominator has no direct ______ through.
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What can be a disadvantage of using state-space representation?
What can be a disadvantage of using state-space representation?
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What do we need for a minimum realization of state variables?
What do we need for a minimum realization of state variables?
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What does a negative eigenvalue do to output?
What does a negative eigenvalue do to output?
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Study Notes
Dynamic Systems in State-Space Representation
- The order of a differential equation corresponds to the number of states, denoted as ( n \to n_{eq} ).
- Advantages include robustness in linear algebra applications, handling complex order systems, nonlinear and time-varying systems where input ( \phi \neq 0 ).
- Disadvantages revolve around the requirement for a detailed mathematical model, extensive computational science knowledge, and the necessity of mapping matrices (arrays) for control systems.
- The relationship ( y = \sum Ax ) describes the output in terms of state variables where each output corresponds to one row in the matrix.
- If the numerator exceeds the denominator in a transfer function, there is no direct feedthrough and the output response does not have jumps.
- If the numerator equals the denominator, a direct feedthrough occurs, allowing for jumps in the output.
- The notation ( P_T ) and ( P_T^2 ) refers to cases where all coefficients in the numerator are constants.
- The state vector ( X ) comprises state variables ( (x_1, x_2) ) that contain all necessary data to represent the system effectively.
- The count of state variables ( n ) is critical for achieving minimal realization of the system.
- At least ( 2n ) initial conditions are required for proper system characterization.
- Stable state-space systems ultimately converge to the point (0,0), with matrix ( A ) determining the system’s direction and strength.
- A system can transition from single-input single-output (SISO) to multiple-input multiple-output (MIMO) while maintaining the same matrix ( A ).
Solving State-Space Equations
- The solution to state-space equations is given by the formula ( x(t) = e^{At}x_0 + \int_{0}^{t} e^{A(t-\tau)} b(u(\tau))d\tau ).
- The symbol ( SI ) represents ( \sigma s ).
- The system matrix ( A ), which is an ( n \times n ) matrix, possesses ( n ) eigenvalues and ( n ) eigenvectors.
- The eigenvalues of matrix ( A ) play a crucial role in system stability.
- A system is defined as stable if all poles (represented as ( a )) of ( G(s) ) are negative, relying solely on the characteristics of matrix ( A ).
- The eigenvalue aspect of ( A ) only scales the output rather than introducing rotation.
- A negative eigenvalue in matrix ( A ) indicates that the system behavior tends to shrink toward the origin over time.
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Description
Explore the intricacies of dynamic systems using state-space representation. This quiz delves into the advantages, disadvantages, and relationships within differential equations and control systems. Test your understanding of key concepts and notation in this advanced topic.
