How can you convert a transfer function to a state-space model?

If you are an electrical engineer, you may have encountered the problem of converting a transfer function to a state-space model. A transfer function is a mathematical expression that relates the output of a system to its input, usually in the frequency domain. A state-space model is a representation of a system that uses a set of variables, called state variables, that describe the state of the system at any given time, usually in the time domain. State-space models are useful for analyzing the stability, controllability, and observability of a system, as well as for designing feedback controllers. In this article, you will learn how to convert a transfer function to a state-space model using two methods: the controllable canonical form and the observable canonical form.

Find expert answers in this collaborative article

Selected by the community from 5 contributions. Learn more

1 Controllable canonical form

The controllable canonical form is a state-space model which has a structure that makes it easy to determine if the system is controllable, meaning that you can guide the system to any desired state by applying an appropriate input. To obtain the controllable canonical form, you must write the transfer function in the form of G(s) = b(s)/a(s) , where a(s) and b(s) are polynomials in s . Additionally, you need to identify the coefficients of a(s) and b(s) , and arrange them in vectors a = [a_n, a_(n-1), ..., a_1, a_0] and b = [b_n, b_(n-1), ..., b_1, b_0] , where n is the order of a(s) . Afterward, you have to construct the state matrix A , input matrix B , output matrix C , and feedforward matrix D . Finally, you will have the state-space model in the controllable canonical form given by

x' = Ax + Bu
y = Cx + Du

where x is the state vector, <

Add your perspective

Ehsan Mokhtari

Electrical Engineer ⚡ | Oil & Gas Expert ⛽ | Green Hydrogen Advocate 🌍 | Passionate About Sustainable Energy Solutions 🌿
Report contribution
Probably the most straightforward method for converting from the transfer function of a system to a state space model is to generate a model in "controllable canonical form." This term comes from Control Theory but its exact meaning is not important to us.

Like
Ehsan Mokhtari

Electrical Engineer ⚡ | Oil & Gas Expert ⛽ | Green Hydrogen Advocate 🌍 | Passionate About Sustainable Energy Solutions 🌿
Report contribution
Some mathematic problems help students to have an active mind to solve upcoming matters. They empowered engineers to have a solution for each case.

Like
Muhammad Maaz

PEC Registered Electrical Engineer | Major in Computer Engineering |
Report contribution
The controllable canonical form is particularly useful for control system designers who need to ensure that they can steer the system's state as required. By representing the transfer function as a matrix in state-space, engineers can apply linear control theory to design controllers that guarantee system controllability. This form explicitly shows the connection between state variables and input, making it easier to design input signals that achieve desired state transitions. It's a fundamental concept for modern control systems and indispensable for robust controller design.

Like

2 Observable canonical form

The observable canonical form is a state-space model with a special structure that allows you to determine if the system is observable - meaning the state of the system can be reconstructed by observing its output. To obtain this form, you must first write the transfer function in the form of G(s) = b(s)/a(s), where a(s) and b(s) are polynomials in s. Then, identify the coefficients of a(s) and b(s), and arrange them in vectors a = [a_n, a_(n-1), ..., a_1, a_0] and b = [b_n, b_(n-1), ..., b_1, b_0], where n is the order of a(s). Next, construct the state matrix A by placing -a in the first column, and shifting it to the bottom by one row in each column to the right. Additionally, construct the input matrix B by placing b in the first row, and filling the rest of the matrix with zeros. Likewise, construct the output matrix C by taking the first element of b, and placing it in the first element. The feedforward matrix D should be constructed by taking the constant term of a(s), and placing it in the first element. After these steps have been completed, you will have obtained a state-space model in an observable canonical form given by x' = Ax + Bu and y = Cx + Du; where x is the state vector, u is the input vector, y is the output vector, and x' is its derivative.

Add your perspective