12.2: Similarity Transformations

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Suppose we have characterized a given dynamic system via a particular state-space representation, say with state variables \(x_{1}, x_{2}, \cdots, x_{n}\). The evolution of the system then corresponds to a trajectory of points in the state space, described by the succession of values taken by the state variables. In other words, the state variables may be seen as constituting the coordinates in terms of which we have chosen to describe the motion in the state space.

We are free, of course, to choose alternative coordinate bases - i.e., alternative state variables - to describe the evolution of the system. This evolution is not changed by the choice of coordinates; only the description of the evolution changes its form. For instance, in the LTI circuit example in the previous chapter, we could have used \(i_{L}-v_{C}\) and \(i_{L}+v_{C}\) instead of \(i_{L}\) and \(v_{C}\). The information in one set is identical with that in the other, and the existence of a state-space description with one set implies the existence of a state-space description with the other, as we now show more concretely and more generally. The flexibility to choose an appropriate coordinate system can be very valuable, and we will find ourselves invoking such coordinate changes very often.

Given that we have a state vector \(x\), suppose we define a constant invertible linear mapping from \(x\) to \(r\), as follows:

\[r=T^{-1} x \quad, \quad x=T r \ \tag{12.2}\]

Since \(T\) is invertible, this maps each trajectory \(x(k)\) to a unique trajectory \(r(k)\), and vice versa. We refer to such a transformation as a similarity transformation. The matrix \(T\) embodies the details of the transformation from \(x\) coordinates to \(r\) coordinates - it is easy to see from (12.2) that the columns of \(T\) are the representations of the standard unit vectors of \(r\) in the coordinate system of \(x\), which is all that is needed to completely define the new coordinate system.

Substituting for \(x(k)\) in the standard (LTI version of the) state-space model (10.1), we have

\[\begin{aligned}
T r(k+1) &=A(T r(k))+B u(k) \ (12.3) \\
y(k) &=C(T r(k))+D u(k) \ (12.4)
\end{aligned}

\[\begin{aligned}
r(k+1) &=\left(T^{-1} A T\right) r(k)+\left(T^{-1} B\right) u(k) \ (12.5) \\
&=\widehat{A} r(k)+\widehat{B} u(k) \ (12.6) \\
y(k) &=(C T) r(k)+D u(k) \ (12.7) \\
&=\widehat{C} r(k)+D u(k) \ (12.8)
\end{aligned}\nonumber\]

We now have a new representation of the system dynamics; it is said to be similar to the original representation. It is critical to understand, however, that the dynamic properties of the model are not at all affected by this coordinate change in the state space. In particular, the mapping from \(u(k)\) to \(y(k)\), i.e. the input/output map, is unchanged by a similarity transformation.