Overview of Linear Algebra computations

Let consider a state space representation of a linear time-invariant (LTI) system:

We take our starting time to be 0, without loss of generality (since we are dealing with time-invariant models). Consider the response of the undriven system, i.e., the response with u(t) ≡ 0 for t ≥ 0, but with some nonzero initial condition x(0). This is the zero-input-response (ZIR) of the system, and is a solution of the undriven (or unforced or homogeneous) system

It is natural when analyzing an undriven LTI system to look for a solution in exponential form (essentially because exponentials have the unique property that shifting them is equivalent to scaling them, and undriven LTI systems are characterized by invariance to shifting and scaling of solutions). We accordingly look for a nonzero solution of the form

where each state variable is a scalar multiple of the same exponential eλt , with these scalar multiples assembled into the vector v. (The boldface 0 at the end of the preceding equation denotes an L-component column vector whose entries are all 0 — we shall use 0 for any vectors or matrices whose entries are all 0, with the correct dimensions being apparent from the context. Writing v ≠ 0 signifies that at least one component of v is nonzero.)

Substituting results in the equation

from which we can conclude that the vector v and scalar λ must satisfy

where I denotes the identity matrix, in this case of dimension L × L. The above equation has a nonzero solution v if and only if the coefficient matrix (λI − A) is not invertible, i.e., if and only if its determinant is 0:

In summary, the undriven system has a solution of the assumed exponential form if and only if λ equals some characteristic value or eigenvalue of A, and the nonzero vector v is an associated characteristic vector or eigenvector.

Consequently a more general solution for the zero-input response with distinct eigenvalues is

The contribution to the modal decomposition from a conjugate pair of eigenvalues λi = σi + jωi and λi* = σi − jωi, with associated complex conjugate eigenvectors vi = ui + jwi and vi = u*i − jwi respectively, is a real term of the form

With a little algebra, the real expression can be reduced to the form

It is often useful to examine the behavior of a state-space system by rewriting the original description in terms of a transformed set of variables. A particularly important case involves the transformation of the state vector x(t) to a new state vector z(t) that decomposes the behavior of the system into its components along each of the eigenvectors vi:

where the constant matrix M is chosen to be invertible. The state space representation becomes:

What makes the modal transformation interesting and useful is the fact that the state evolution matrix A transforms to a diagonal matrix Λ:

Then β = M-1B is the shape factor vector and ξ = CM is the participation factor vector.

The transfer function H(s) of the model describes the zero-state input-output relationship in the Laplace transform domain, and is straight­forward to find because the equations are totally decoupled. Taking the Laplace transforms of those equations, with zero initial conditions, results in

This is also the transfer function of the transformed model, as similarity transformations do not change transfer functions. The resulting expression is

which can be rewritten as