![]() ![]() However, it might be easier to see, that it is easier to construct the equations for $x_1$ and $x_2$ because their odes are similar (while constructring the ode for the spring deformation will be different).īottom Line: state representation makes much more sense in more complex systems. If you are only interested in the deformation of the string you might create a $C=$ and you are done. However, there are other reasons, which I believe you suspect as linear combination of solutions to obtain a transformed solutionĮxample: think of the following system with two mass springs you can select either the absolute $x_1$ and $x_2$ displacements of the mass.Īnother equivalent representation is $x_1$ and $x_2-x_1$ (essentially the deformation of the spring). \dot$ as a way to perform a "boil down to essentials". ![]() Where p is the equivalence transformation matrix.In the state space representation, the state equation for a linear time-invariant system is: If we have a set of matrices A, B, C and D, we can create equivalent matrices as such: The MATLAB function ss2ss can be used to apply an equivalence transformation to a system. Which in turn will satisfy the relationship:Į A t = T e D t T − 1 We can define a transformation matrix, T, that satisfies the diagonalization transformation: A diagonal matrix is a matrix that only has entries on the diagonal, and all the rest of the entries in the matrix are zero. If the matrix A has a complete set of distinct eigenvalues, the matrix can be diagonalized. If needed, we will use subscripts to differentiate between the two. The transition matrix T should not be confused with the sampling time of a discrete system. The characteristic equation of the system matrix A is given as: The remainder of this chapter will discuss eigenvalues, eigenvectors, and the ways that they affect their respective systems.Ĭharacteristic Equation Also, the eigenvalues and eigenvectors can be used to calculate the matrix exponential of the system matrix through spectral decomposition. The eigenvalues and eigenvectors of the system determine the relationship between the individual system state variables (the members of the x vector), the response of the system to inputs, and the stability of the system. Computing the eigenvalues and the eigenvectors of the system matrix is one of the most important things that should be done when beginning to analyze a system matrix, second only to calculating the matrix exponential of the system matrix. Eigenvalues and Eigenvectors have a number of properties that make them valuable tools in analysis, and they also have a number of valuable relationships with the matrix from which they are derived. The terms "Eigenvalues" and "Eigenvectors" are most commonly used. The word "eigen" comes from German and means "own" as in "characteristic", so this chapter could also be called "Characteristic values and characteristic vectors". Non-square matrices cannot be analyzed using the methods below. It is important to note that only square matrices have eigenvalues and eigenvectors associated with them. The eigenvalues and eigenvectors of the system matrix play a key role in determining the response of the system. If the system is time-variant, the methods described in this chapter will not produce valid results. Eigenvalues and Eigenvectors cannot be calculated from time-variant matrices. ![]()
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