Observability

Introduction

Consider a linear time-invariant system: \[ \begin{align*} \begin{split} \mathbf{\dot{x}} &= \mathbf{Ax+Bu} \\ \mathbf{y} &= \mathbf{Cx+Du} \end{split} \end{align*} \]

Observability

The discussion of controllability and observability is introduced from Kalman, []

\[ \mathbf{A}^n + a_{n-1}\mathbf{A}^{n-1} + \cdots + a_1\mathbf{A} + a_0\mathbf{I}_n = \mathbf{0}_n \]

• Some Remarks
  • Cayley-Hamilton’s theorem simplifies the calculation of \(\mathbf{A}^{p}, p>n\), since any \(\mathbf{A}^{p}\) can be simplified as the sum of \(\mathbf{A}^{n-1}, \cdots \mathbf{A}^{1}, \mathbf{I}_{n}\).
  • Based on the dimension theorem of vector spaces, the number of bases of vector space \((\mathbf{M}_n, +, \cdot)\) is \(n^2\), where \(\mathbf{M}_n\) denotes the set of \(n\times n\) matrices. Meaning, it is guaranteed that any sum of more than \(n^2\) matrices are linearly dependent. Cayley-Hamilton’s theorem provides a tighter bound by showing that there exists a sum of \(n+1\) matrices that are linearly dependent.
  • Note that we can formulate the matrix equations as: \[ \mathbf{A}^n + a_{n-1}\mathbf{A}^{n-1} + \cdots + a_1\mathbf{A} + a_0\mathbf{I}_n = (\mathbf{A}-\lambda_1\mathbf{I})^{m_1}(\mathbf{A}-\lambda_2\mathbf{I})^{m_2}\cdots (\mathbf{A}-\lambda_k\mathbf{I})^{m_k} \] where \(m_i\) is the algebraic multiplicity of eigenvalue \(\lambda_i\). This matrix equation is parallel with the scalar version equation. [1]

References

[1]

B. Friedland, Control system design: An introduction to state-space methods. Courier Corporation, 2012, pp. 209–211.