Question

Determine a realization of the transfer matrix $$ H(s)=\left(\begin{array}{ccc} \frac{s}{s^{2}+2 s} & 0 & 0 \\ 0 & \frac{s^{2}-1}{(s+3)(s+2)} & 0 \\ 0 & 0 & \frac{s-1}{(s+1)^{3}} \end{array}\right) $$

Short answer

Realize each diagonal element separately and combine into a block-diagonal form.

Step-by-step solution

Identify State-Space Representation

To find a realization, identify state-space representation for each diagonal entry of the transfer matrix, as this matrix is already diagonal and decoupled into individual SISO systems. This suggests we can realize them separately.

Realize the First Element

For the first element of the transfer matrix, \( \frac{s}{s^2+2s} \), choose a state-space representation:- It has one pole at 0 and another at -2 leading to state matrix \( A = \begin{pmatrix} 0 & 1 \ 0 & -2 \end{pmatrix} \) - Choose \( B = \begin{pmatrix} 0 \ 1 \end{pmatrix} \), \( C = \begin{pmatrix} 1 & 0 \end{pmatrix} \), \( D = 0 \).

Realize the Second Element

For the second element \( \frac{s^2-1}{(s+3)(s+2)} \), use partial fraction decomposition:- Poles at \( s = -3 \) and \( s = -2 \) result in a state matrix of \( A = \begin{pmatrix} -3 & 1 \ 0 & -2 \end{pmatrix} \) - Choose \( B = \begin{pmatrix} 1 \ 0 \end{pmatrix} \), \( C = \begin{pmatrix} 1 & -1 \end{pmatrix} \), \( D = 0 \).

Realize the Third Element

For the third element \( \frac{s-1}{(s+1)^3} \), note it has a pole of multiplicity 3 at -1:- A suitable state-space representation would be: \[ A = \begin{pmatrix} -1 & 1 & 0 \ 0 & -1 & 1 \ 0 & 0 & -1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \]- Let \( C = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \), \( D = 0 \).

Combine the Realizations

Since the matrix is diagonal, the overall realization is block-diagonal, combining the state-space representations found:- Overall state matrix \( A \) is block-diagonal: \[ A = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & -2 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & -3 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & -2 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & -1 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & -1 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix} \]- Block-diagonal \( B \), \( C \), \( D \):\[ B = \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & -1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} \]

Key concepts

State-Space Representation

State-Space Representation is a mathematical modeling approach to represent linear time-invariant systems using matrices. This method provides a standardized framework to describe a system's dynamics by grouping them into state variables, inputs, outputs, and system equations. The state-space form uses the matrices:

• **State Matrix (A):** Captures the system's dynamics and how current states influence future states.
• **Input Matrix (B):** Describes how the input affects each state.
• **Output Matrix (C):** Connects the state to the output, showcasing how the state values are reflected in the output.
• **Feedthrough Matrix (D):** Represents the direct path from input to output (often zero in simple systems).

The state-space form of a system can be written as:\[ \dot{x}(t) = Ax(t) + Bu(t) \ y(t) = Cx(t) + Du(t) \]Here, \( x(t) \) is the state vector, \( u(t) \) is the input vector, and \( y(t) \) is the output vector. This method is particularly useful for control systems and is vital in ensuring system stability and performance.

Partial Fraction Decomposition

Partial Fraction Decomposition involves breaking down complex rational expressions into simpler, easily manageable fractions. This concept is particularly helpful in inverse Laplace transformations and solving differential equations.
In the context of control systems, we often encounter transfer functions in polynomials. To manage such functions, partial fraction decomposition helps by expressing a complex fraction into a sum of simpler fractions. For instance, if we have a transfer function:\[ H(s) = \frac{s^2 - 1}{(s+3)(s+2)} \]We decompose it into simpler terms like:\[ H(s) = \frac{A}{s+3} + \frac{B}{s+2} \]Where \( A \) and \( B \) are constants determined by solving the decomposition equations. This is beneficial as each simpler term can be individually realized in state-space form, allowing for modular analysis and design. By simplifying each component, this method grants clarity and ease in control systems analysis.

Block-Diagonal Matrix

A Block-Diagonal Matrix is a special type of square matrix where the principal diagonal might contain blocks of smaller matrices, while the non-principal diagonal elements are zeros. This structure is especially useful in the realization of systems that can be decoupled or simplified into independent subsystems.
The main advantage of a block-diagonal matrix is that it allows us to treat and analyze each block as a separate entity. In control systems, when dealing with a transfer matrix composed of multiple independent transfer functions, each diagonal element represents an independent Single Input Single Output (SISO) system. Thus, block-diagonal matrices make it straightforward to derive a combined representation from separate state-space models.
Furthermore, handling calculations and operations such as determinants, inverses, and system realizations become more efficient because each block can be addressed individually. This modularity leads to simplified mathematical analysis and is particularly beneficial when constructing or simulating large-scale systems.