State Space to Transfer Function Examples. A. Introduction. For a linear, time-invariant, continuous-time system, the state and output equations are. ·x(t) = Ax(t) +. Request PDF | Transfer function matrices of state-space models | This paper presents a new algorithm for computing the transfer function from state equations for. domain by a differential equation or from its transfer function representation. Both cases will be considered in this section. Four state space forms—the phase.

- Transformation: Transfer Function ↔ State Space
- Electrical Engineering
- Transfer Function to State Space
- University of Hawaii
- State Space to Transfer Function

Two of the most powerful and common ways to represent systems are the transfer function form and the state space form. This page describes how to transform a transfer function to a state space representation, and vice versa.

## Transformation: Transfer Function ↔ State Space

Converting from state space form to a transfer function is straightforward because the transfer function form is unique. Converting from transfer function to state space is more involved, largely because there are many state space forms to describe a system.

Now, take the Laplace Transform with zero initial conditions since we are finding a transfer function :. We want to solve for the ratio of Y s to U sso we need so remove Q s from the output equation. We start by solving the state equation for Q s.

Now we put this into the output equation. Note that although there are many state space representations of a given system, all of those representations will result in the same transfer function i. Details are here.

Rules for inverting a 3x3 matrix are here. To make this task easier, MatLab has a command ss2tf for converting from state space to transfer function.

Recall that state space models of systems are not unique ; a system has many state space representations. Therefore we will develop a few methods for creating state space models of systems. Before we look at procedures for converting from a transfer function to a state space model of a system, let's first examine going from a differential equation to state space.

## Electrical Engineering

We'll do this first with a simple system, then move to a more complex system that will demonstrate the usefulness of a standard technique. First we start with an example demonstrating a simple way of converting from a single differential equation to state space, followed by a conversion from transfer function to state space. Consider the differential equation with no derivatives on the right hand side. We'll use a third order equation, thought it generalizes to n th order in the obvious way.

For such systems no derivatives of the input we can choose as our n state variables the variable y and its first n-1 derivatives in this case the first two derivatives. Note: For an nth order system the matrices generalize in the obvious way A has ones above the main diagonal and the differential equation constants for the last row, B is all zeros with b 0 in the bottom row, C is zero except for the leftmost element which is one, and D is zero.

## Transfer Function to State Space

Consider the transfer function with a constant numerator note: this is the same system as in the preceding example. Taking the derivatives we can develop our state space model which is exactly the same as when we started from the differential equation. Note: For an nth order system the matrices generalize in the obvious way A has ones above the main diagonal and the coefficients of the denominator polynomial for the last row, B is all zeros with b 0 the numerator coefficient in the bottom row, C is zero except for the leftmost element which is one, and D is zero.

If we try this method on a slightly more complicated system, we find that it initially fails though we can succeed with a little cleverness.

The method has failed because there is a derivative of the input on the right hand, and that is not allowed in a state space model. However we can make use of the fact:. The process described in the previous example can be generalized to systems with higher order input derivatives but unfortunately gets increasingly difficult as the order of the derivative increases. When the order of derivatives is equal on both sides, the process becomes much more difficult and the variable "D" is no longer equal to zero.

Clearly more straightforward techniques are necessary. Two are outlined below, one generates a state space method known as the "controllable canonical form" and the other generates the "observable canonical form the meaning of these terms derives from Control Theory but are not important to us.

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. To see how this method of generating a state space model works, consider the third order differential transfer function:. We also convert back to a differential equation.

In this case, the order of the numerator of the transfer function was less than that of the denominator. If they are equal, the process is somewhat more complex. A result that works in all cases is given below; the details are here.

## University of Hawaii

For a general n th order transfer function:. Another commonly used state variable form is the "observable canonical form. To understand how this method works consider a third order system with transfer function:.

We can convert this to a differential equation and solve for the highest order derivative of y:. Now we integrate twice the reason for this will be apparent soonand collect terms according to order of the integral:.

There are many other forms that are possible. Now we can find the transfer function.

## State Space to Transfer Function

Key Concept: Transforming from State Space to Transfer Function Given a state space representation of a system the transfer function is give by and the characteristic equation i.

Example: Differential Equation to State Space simple Consider the differential equation with no derivatives on the right hand side. For such systems no derivatives of the input we can choose as our n state variables the variable y and its first n-1 derivatives in this case the first two derivatives Taking the derivatives we can develop our state space model Note: For an nth order system the matrices generalize in the obvious way A has ones above the main diagonal and the differential equation constants for the last row, B is all zeros with b 0 in the bottom row, C is zero except for the leftmost element which is one, and D is zero Repeat Starting from Transfer Function Consider the transfer function with a constant numerator note: this is the same system as in the preceding example.

Example: Differential Equation to State Space harder Consider the differential equation with a single derivative on the right hand side. We can try the same method as before: The method has failed because there is a derivative of the input on the right hand, and that is not allowed in a state space model.

Fortunately we can solve our problem by revising our choice of state variables.

MargConversion Between State Space and Transfer Function Representations in State Space Representation (noise free linear systems) z State Space form A.

MiramarExample 2: Find the state-space representation of the following transfer function sys- tem (13) in the diagonal canonical form. G(s) = 2s + 3 s2 +.

YolarDr. Radhakant Padhi, AE Dept., IISc-Bangalore. 2. State Space Representation. (noise free linear systems). ○ State Space form. ○ Transfer Function form.