22.3: Modal Aspects

The following result begins to make the connection of reachability with modal structure. 

Corollary \(\PageIndex{22.1}\)

The reachable subspace \(\mathbb{R}\) is \(A\)-invariant, i.e. \(x \in \mathbb{R} \Longrightarrow A x \in \mathbb{R}\). We write this as \(A \mathbb{R} \subset \mathbb{R}\)

Proof

We first show

\[R a\left(A R_{n}\right) \subset R a\left(R_{n}\right)\label{22.12}\]

For this, note that

\[A R_{n}=\left[\begin{array}{l|l|l|l}
A^{n} B & A^{n-1} B & \cdots & A B
\end{array}\right]\nonumber\]

The last \(n - 1\) blocks are present in \(R_{n}\), while the Cayley-Hamilton theorem allows us to write \(A^{n}B\) as a linear combination of blocks in \(R_{n}\). This establishes (22.12). It follows that \(x=R_{n} \alpha \Longrightarrow A x=A R_{n} \alpha=R_{n} \beta \in \mathbb{R}\).

Some feel for how this result connects to modal structure may be obtained by considering what happens if the subspace \(\mathbb{R}\) is one-dimensional. If \(v(\neq 0)\) is a basis vector for R , then Corollary 22.1 states that

\[A v=\lambda v\label{22.13}\]

for some \(\lambda\), i.e. \(\mathbb{R}\) is the space spanned by an eigenvector of \(A\). More generally, it is true that any \(A\)-invariant subspace is the span of some eigenvectors and generalized eigenvectors of \(A\). (It turns out that \(\mathbb{R}\) is the smallest \(A\)-invariant subspace that contains \(Ra(B)\), but we shall not pursue this fact.)

Modal Reachability Tests

An immediate application of the standard form is to prove the following modal test for (un)reachability

Theorem \(\PageIndex{22.2}\)

The system (22.1) is unreachable if and only if \(w^{T} B=0\) for some left eigenvector \(w^{T}\) of \(A\). We say that the corresponding eigenvalue \(\lambda\) is an unreachable eigenvalue.

Proof

If \(w^{T} B=0\) and \(w^{T} A=\lambda w^{T}\) with \(w^{T} \neq 0\), then \(w^{T} A B=\lambda w^{T} B=0\) and similarly \(w^{T} A^{k} B=0\), so \(w^{T} R_{n}=0\), i.e. the system is unreachable.

Conversely, if the system is unreachable, transform it to the standard form (22.14). Now let \(w_{2}^{T}\) denote a left eigenvector of \(A_{2}\), with eigenvalue \(\lambda\). Then \(w^{T}=\left[\begin{array}{ll}
0 & w_{2}^{T}
\end{array}\right]\) is a left eigenvector of the transformed \(A\) matrix, namely \(\bar{A}\), and is orthogonal to the (columns of the) transformed \(B\), namely \(\bar{B}\).

An alternative form of this test appears in the following result.

Corollary \(\PageIndex{22.2}\)

The system (22.1) is unreachable if and only if \(\left[\begin{array}{l|l}
z I-A & B
\end{array}\right]\) loses rank for some \(z = \lambda\). This \(\lambda\) is then an unreachable eigenvalue

Proof

The matrix \(\left[\begin{array}{l|l}
z I-A & B
\end{array}\right]\) has less than full rank at \(z = \lambda\) iff \(w^{T}\left[\begin{array}{ll}
s I-A & B
\end{array}\right]=0\) for some \(w^{T} \neq 0\). But this is equivalent to having a left eigenvector of \(A\) being orthogonal to (the columns of ) \(B\).

Example \(\PageIndex{22.2}\)

Consider the system

\[x(k+1)=\underbrace{\left[\begin{array}{cc}
3 & 0 \\
0 & 3
\end{array}\right]}_{A} x(k)+\underbrace{\left[\begin{array}{c}
1 \\
1
\end{array}\right]}_{B} u\tag{k}\]

Left eigenvectors of \(A\) associated with its eigenvalue at \(\lambda=3\) are \(w_{1}^{T}=\left[\begin{array}{ll}
1 & 0
\end{array}\right]\) and \(w_{=}^{T}\left[\begin{array}{ll}
0 & 1
\end{array}\right]\), neither of which is orthogonal to \(B\). However, \(w_{0}^{T}=\left[\begin{array}{ll}
1 & -1
\end{array}\right]\) is also a left eigenvector associated with \(\lambda=3\), and is orthogonal to \(B\). This example drives home the fact that the modal unreachability test only asks for some left eigenvector to be orthogonal to \(B\).

Jordan Chain Interpretation

Recall that the system (22.1) may be thought of as having a collection of "Jordan chains" at its core. Reachability, which we first introduced in terms of reaching target states, turns out to also describe our ability to independently "excite" or drive the Jordan chains. This is the implication of the reachable subspace being an \(A\)-invariant subspace, and is the reason why the preceding modal tests for reachability exist.

The critical thing for reachability is to be able to excite the beginning of each chain; this excitation can then propagate down the chain. An additional condition is needed if several chains have the same eigenvalue; in this case, we need to be able to independently excite the beginning of each of these chains. (Example 22.2 illustrates that reachability is lost otherwise; with just a single input, we are unable to excite the two identical chains independently.) With distinct eigenvalues, we do not need to impose this independence condition; the distinctness of the eigenvalues permits independent motions.

Some additional insight is obtained by considering the distinct eigenvalue case in more detail. In this case, \(A\) in (22.1) is diagonalizable, and \(A=V \Lambda W\), where the columns of \(V\) are the right eigenvectors of \(A\) and the rows of \(W\) are the left eigenvectors of \(A\). For \(x(0) = 0\) we have

\[x(k)=\sum_{\ell=1}^{n} v_{\ell} w_{\ell}^{T} B g_{\ell}(k)\label{22.18}\]

where

\[g_{\ell}(k)=\sum_{i=0}^{k-1} \lambda_{\ell}^{k-i-1} u(i)\label{22.19}\]

If \(w_{j}^{T} B=0\) for some \(j\), then (22.18) shows that \(x(k)\) is confined to the span of \(\left\{v_{\ell}\right\}_{\ell \neq j}\) i.e. the system is not reachable. For example, suppose we have a second-order system (n = 2), and suppose \(w_{1}^{T} B=0\). Then if \(x(0) = 0\), the response to any input must lie along \(v_{2}\). This means that \(v_{2}\) spans the reachable space, and that any state which has a component along \(v_{1}\) is not reachable.