4. Coordinate Transformation

After having introduced the basic formulations of discrete-time invariant linear systems and before going in depth in the narrative of the Eigensystem Realization Algorithm, some results concerning coordinate transformation are introduced in this section.

In many problems, analysts need to use different coordinate systems in order to describe different quantities. A well-chosen coordinate system allows position and direction in space to be represented in a very convenient manner and allow sometimes to have some insights about the system itself. After all, two independent observers might well choose coordinate systems with different origins, and different orientations of the coordinate axes. The dynamic behavior of a mechanical system strongly depends upon its mathematical representation. This is due to the fact that nonlinearity is not an inherent attribute of a physical system, but rather dependent upon the mathematical description of the system’s behavior. Ideally, one has an infinity of coordinate choices to represent the same physical system. In the study of celestial mechanics, the quest to find a judicious coordinate system led to the development of various canonical transformations. An extended discussion will be conducted in section concerning coordinate systems and transformations. This section only presents a few important results for linear discrete-time invariant systems.

Let a new state vector be defined such that

\begin{align} \label{Eq: tilde x = Tx} \tilde{\boldsymbol{x}}(k) &= T\boldsymbol{x}(k), \end{align}

where \(T\) is a nonsingular square matrix. Substitution of Eq. \eqref{Eq: tilde x = Tx} into the difference equations of a linear system yields

\begin{align} \left\lbrace\begin{array}{ll} \tilde{\boldsymbol{x}}(k+1) &\hspace{-0.7em}= TAT^{-1}\tilde{\boldsymbol{x}}(k)+TB\boldsymbol{u}(k)\\ \boldsymbol{y}(k) &\hspace{-0.7em}= CT^{-1}\tilde{\boldsymbol{x}}(k)+\tilde{D}\boldsymbol{u}(k) \end{array}\right. \end{align}

The effect of the input \(\boldsymbol{u}(k)\) on the output \(\boldsymbol{y}(k)\) is unchanged for the system. Thus the matrices \(\left\lbrace TAT^{-1}, TB, CT^{-1}, D\right\rbrace\) describe the same input-output relationship as the matrices \(\left\lbrace A, B, C, D\right\rbrace\):

\begin{align} \left\lbrace\begin{array}{ll} \boldsymbol{x}(k+1) &\hspace{-0.7em}= A\boldsymbol{x}(k)+B\boldsymbol{u}(k)\\ \boldsymbol{y}(k) &\hspace{-0.7em}= C\boldsymbol{x}(k)+D\boldsymbol{u}(k) \end{array}\right. \ \ \Leftrightarrow \ \ \left\lbrace\begin{array}{ll} \tilde{\boldsymbol{x}}(k+1) &\hspace{-0.7em}= \tilde{A}\tilde{\boldsymbol{x}}(k)+\tilde{B}\boldsymbol{u}(k)\\ \boldsymbol{y}(k) &\hspace{-0.7em}= \tilde{C}\tilde{\boldsymbol{x}}(k)+\tilde{D}\boldsymbol{u}(k) \end{array}\right. \end{align}


\begin{align} \tilde{\boldsymbol{x}}(k) &= T\boldsymbol{x}(k),\\ \tilde{A} &= TAT^{-1},\\ \tilde{B} &= TB,\\ \tilde{C} &= CT^{-1},\\ \tilde{D} &= D. \end{align}

This transformation is called a similarity transformation. The fact that the input-output relationship remains unchanged should also indicate that the pulse sequence, or Markov parameters, is invariant through coordinate transformation. Indeed, for \(i \geq 1\),

\begin{align} \tilde{h}_i = \tilde{C}\tilde{A}^{i-1}\tilde{B} = CT^{-1}\left(TAT^{-1}\right)^{i-1}TB = CT^{-1}TA^{i-1}T^{-1}TB = CA^{i-1}B = h_i. \end{align}

As a result, there are an infinite number of state-space representations that produce the same input-output description. Additionally, because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator (their rank in particular).