Question

A nonsingular coordinate transformation \(x=S \bar{x}\), (such that \(\left.A \rightarrow S^{-1} A S, C \rightarrow C S\right)\) does not destroy observability. Show this. If the observability matrix of the transformed system is denoted by \(\bar{W}\), then \(W S=\bar{W}\).

Short answer

Observability is maintained as \( WS = \bar{W} \), proving the system remains observable.

Step-by-step solution

Define Observability matrix

The observability matrix for the original system with matrix \( A \) and output matrix \( C \) is given by \( W = \begin{bmatrix} C \ CA \ CA^2 \ \vdots \ CA^{n-1} \end{bmatrix} \). This matrix helps us determine if the state of the system can be inferred from the outputs.

Apply Coordinate Transformation

Under the transformation \( x = S \bar{x} \), where \( S \) is an invertible matrix, the matrices transform as \( A \rightarrow S^{-1}AS \) and \( C \rightarrow CS \). This gives rise to the transformed observability matrix \( \bar{W} \).

Formulate the Transformed Observability Matrix

The transformed observability matrix, denoted as \( \bar{W} \), is given by \( \bar{W} = \begin{bmatrix} CS \ CSA \ (CSA)^2 \ \vdots \ (CSA)^{n-1} \end{bmatrix} \). By expanding and substituting the transformations into each power of \( A \), we can derive the equivalent expressions.

Relate Original and Transformed Observability Matrices

We want to show that \( W S = \bar{W} \). Substitute the transformed expressions into the power terms to find that each row of \( \bar{W} \) can be related back through matrix multiplication with \( W \). Essentially, performing \( WS \) equates the original and the transformed observability matrices.

Conclusion

Since \( WS = \bar{W} \), observability is maintained under this coordinate transformation as \( W \) and \( \bar{W} \) are equivalent up to a multiplication by \( S \). Thus, the transformation ensures that observability is not destroyed.

Key concepts

Coordinate Transformation

Coordinate transformation is a fundamental concept in control systems that helps in simplifying the analysis and design of systems. When dealing with complex systems, transforming the state variables into a different base can often provide more clarity and ease. In this context, a coordinate transformation is carried out using a nonsingular matrix, denoted as \( S \). If we have a system described by the state vector \( x \), the transformation becomes \( x = S \bar{x} \), where \( \bar{x} \) is the new state vector in the transformed coordinate frame.
This transformation involves mathematical operations on system matrices like the state matrix \( A \) and output matrix \( C \). The transformation rules are:

• \( A \rightarrow S^{-1}AS \)
• \( C \rightarrow CS \)

These transformations are crucial as they assure that key properties, such as observability, remain unaffected. In essence, even after the transformation, the system retains its original characteristics, just viewed in a different light.

Observability Matrix

The observability matrix is a crucial element in control systems. It determines whether the internal states of a system can be reconstructed by observing its outputs over time. For a given system with a state-space representation, the observability matrix \( W \) is constructed as follows:

• \( W = \begin{bmatrix} C \ CA \ CA^2 \ \vdots \ CA^{n-1} \end{bmatrix} \)

This matrix allows us to test for observability, signifying if we can track every state variable with output data. A system is considered observable if this matrix is full rank, meaning its rank is equal to the number of state variables.
When a coordinate transformation is applied, the observability matrix changes to \( \bar{W} = \begin{bmatrix} CS \ CSA \ (CSA)^2 \ \vdots \ (CSA)^{n-1} \end{bmatrix} \).
The beauty of linear algebra ensures that even after transformations, the essence of observability is preserved; this is demonstrated when you find \( W S = \bar{W} \). As long as the transformation matrix \( S \) is nonsingular, observability remains intact.

State-Space Representation

State-space representation is a mathematical model of a physical system represented with dynamic equations. These equations describe the system's behavior in terms of its input, output, and state variables. The main components of this representation include matrices: the state matrix \( A \), the input matrix \( B \), the output matrix \( C \), and the feedthrough matrix \( D \).
Specifically, the state-space representation for a given system is:

• State Equation: \( \dot{x} = Ax + Bu \)
• Output Equation: \( y = Cx + Du \)

This approach is advantageous as it allows modeling multi-input and multi-output systems comprehensively. It provides a unified framework to analyze and design systems using computation-friendly methods like matrix algebra.
In the context of observability, state-space representation provides deep insight by organizing system dynamics into easily manageable chunks. By interpreting the observability matrix derived from these state equations, one's ability to infer the state of a system based on outputs is assessed, making state-space representation indispensable for modern control theory.