Canonical Forms of State-Space Models

Hello! Welcome to your next lesson in modeling dynamic systems.

In our last session, we learned how to linearize nonlinear models to obtain a standard linear state-space representation, $x ˙ = A x + B u$ . This is a crucial step for applying linear control theory to real-world systems like your rotary inverted pendulum.

Today, we address a fundamental concept: the state-space representation of a system is not unique. By choosing a different set of state variables, we can arrive at a different set of matrices $(A, B, C, D)$ that describe the exact same input-output behavior. Our goal is to learn how to transform a state-space representation into a canonical form.

These canonical, or standard, forms are special matrix structures that make it much easier to analyze system properties and, more importantly for our goal, to design controllers.

Why Do We Need Canonical Forms?

Imagine you have a state-space model for a system, but the matrices $A$ and $B$ are dense and don't have an obvious structure. Designing a controller to place the system's poles at desired locations can be a complex task.

However, if the system is in a special structure called Controllable Canonical Form (CCF), the controller design becomes remarkably straightforward. This is a powerful motivator for the transformations we'll learn today.

State space feedback 3 - transformation to a canonical form

To see why these forms are so useful, let's watch the beginning of the video 'State space feedback 3 - transformation to a canonical form' by John Rossiter.

Watch the first 1 minute and 33 seconds. The video explains that pole placement is simple for systems in control canonical form and introduces the idea of using a transformation to get there.

The Structure of Canonical Forms

The two most common canonical forms are the Controllable Canonical Form (CCF) and the Observable Canonical Form (OCF). Their structure is directly related to the coefficients of the system's transfer function.

Let's consider a general nth-order transfer function:
$G (s) = \frac{b _{n - 1} s ^{n - 1} + \dots + b _{1} s + b _{0}}{s ^{n} + a _{n - 1} s ^{n - 1} + \dots + a _{1} s + a _{0}}$

Controllable Canonical Form (CCF)

In this form, the coefficients of the denominator polynomial appear in the last row of the $A$ matrix, and the input matrix $B$ has a very simple structure.

Controllable Canonical Form Explanation — This block diagram provides an intuitive feel for the structure of the controllable canonical form. The chain of integrators (1/s) represents the states, with feedback paths corresponding to the coefficients of the characteristic equation.

The state-space matrices for CCF are:

A_{cc f} = 00 ⋮ 0 - a_{0} 10 ⋮ 0 - a_{1} 01 ⋮ 0 - a_{2} \dots \dots ⋱ \dots \dots 00 ⋮ 1 - a_{n - 1}, B_{cc f} = 00 ⋮ 01

C_{cc f} = [b_{0} b_{1} b_{2} \dots b_{n - 1}], D_{cc f} = [0] (for strictly proper systems)

Notice how the coefficients from the transfer function denominator $(a_{i})$ and numerator $(b_{i})$ populate the matrices in a structured way.

Observable Canonical Form (OCF)

The OCF has a structure that is, in a sense, the "transpose" of the CCF. The denominator coefficients are in the last column of the $A$ matrix.

A_{oc f} = 010 ⋮ 0 001 ⋮ 0 \dots \dots \dots ⋱ \dots 000 ⋮ 1 - a_{0} - a_{1} - a_{2} ⋮ - a_{n - 1}, B_{oc f} = b_{0} b_{1} b_{2} ⋮ b_{n - 1}

C_{oc f} = [00 \dots 01], D_{oc f} = [0]

There is a powerful relationship between these two forms known as duality:

$A_{oc f} = A_{cc f}^{T}$
$B_{oc f} = C_{cc f}^{T}$
$C_{oc f} = B_{cc f}^{T}$

This means if you can derive one form, you can find the other through simple matrix transpositions.

Examples On Controllable Canonical Form And Observable Canonical Form | GATE Control System

The video 'Examples On Controllable Canonical Form And Observable Canonical Form' by Ekeeda GATE & ESE provides a very direct, procedural walkthrough of how to write down these matrices from a transfer function.

Watch from the beginning to 11:54. The video works through two examples (a 2nd-order and a 3rd-order system). Focus on how the narrator directly maps the transfer function coefficients into the A, B, and C matrices for both CCF and OCF. Pay special attention to the demonstration of the transpose relationship between the forms.

The Transformation Process

We've just seen how to construct a canonical form from a transfer function. But what if you already have a state-space model $(A, B)$ that is not in a canonical form? How do you transform it?

This is done using a similarity transformation. We define a new set of state variables $z$ that are a linear combination of the original state variables $x$ , related by an invertible transformation matrix $T$ :
$z = T x ⟹ x = T^{- 1} z$

Substituting this into our original state equation $x ˙ = A x + B u$ yields the new state equation for $z$ :
$z ˙ = (T A T^{- 1}) z + (T B) u$
So, our new system matrices are $A^= T A T^{- 1}$ and $B^= T B$ . Our goal is to find the specific matrix $T$ that transforms our original $(A, B)$ into the desired canonical form $(A^, B^)$ .

To find $T$ , we need a tool called the controllability matrix. For a system $(A, B)$ , the controllability matrix $M_{c}$ is defined as:
$M_{c} = [B A B A^{2} B \dots A^{n - 1} B]$
We will study the physical meaning of this matrix and the concept of controllability in Module 2. For today, we will use it as a computational tool.

The key insight is that the controllability matrices of the original system ( $M_{c, x}$ ) and the transformed system ( $M_{c, z}$ ) are related by the transformation matrix $T$ :
$M_{c, z} = T \cdot M_{c, x}$
This gives us a direct way to calculate $T$ :
$T = M_{c, z} \cdot (M_{c, x})^{- 1}$
This inverse exists only if the system is controllable—a condition we'll formalize later.

The following video walks through this entire process with excellent clarity.

State space feedback 3 - transformation to a canonical form

Let's return to John Rossiter's video, which derives and applies this exact transformation algorithm.

Watch from 01:59 to 08:59. The video first explains the similarity transformation and then derives the crucial relationship between the controllability matrices. Finally, it outlines a clear 5-step algorithm and applies it to an example.

The Full Algorithm: A Summary

Let's consolidate the steps from the video into a clear workflow for transforming a given system $(A, B)$ to Control Canonical Form $(A^_{cc f}, B^_{cc f})$ :

Find the System's Transfer Function: Given the original system $(A, B, C, D)$ , calculate its transfer function $G (s) = C (s I - A)^{- 1} B + D$ . (We will cover this in detail in the next lesson, but for now, you can use MATLAB's ss2tf or accept it as a given step).
Define the Target CCF: From the transfer function's coefficients, write down the target CCF matrices $A^_{cc f}$ and $B^_{cc f}$ by inspection.
Compute Controllability Matrices:
- Calculate $M_{c, x} = [B A B \dots A^{n - 1} B]$ for the original system.
- Calculate $M_{c, z} = [B^_{cc f} A^_{cc f} B^_{cc f} \dots A^_{cc f}^{n - 1} B^_{cc f}]$ for the target CCF system.
Find the Transformation Matrix T: Calculate $T = M_{c, z} \cdot (M_{c, x})^{- 1}$ .
Verify (Optional but Recommended): Check that your transformation works: $A^_{cc f} = T A T^{- 1}$ and $B^_{cc f} = T B$ .

Test your understanding!

A system has the transfer function:
$G (s) = \frac{s + 5}{s ^{2} + 3 s + 2}$

Write down the state-space model for this system in Controllable Canonical Form (CCF).
Using the duality principle, write down the state-space model in Observable Canonical Form (OCF).

Show answer

From the transfer function, we identify the coefficients:

Denominator: $a_{1} = 3$ , $a_{0} = 2$
Numerator: $b_{1} = 1$ , $b_{0} = 5$

Controllable Canonical Form (CCF):
$A_{cc f} = [0 - a_{0} 1 - a_{1}] = [0 - 2 1 - 3]$ $B_{cc f} = [01]$ $C_{cc f} = [b_{0} b_{1}] = [51]$ $D_{cc f} = [0]$
Observable Canonical Form (OCF):
We use the duality rules: $A_{oc f} = A_{cc f}^{T}$ , $B_{oc f} = C_{cc f}^{T}$ , $C_{oc f} = B_{cc f}^{T}$ .
$A_{oc f} = [01 - 2 - 3]$ $B_{oc f} = [51]$ $C_{oc f} = [01]$ $D_{oc f} = [0]$

Other Canonical Forms

Besides CCF and OCF, another important structure is the Normal (or Diagonal) Canonical Form. This form is particularly insightful because it decouples the system's modes.

7.4. Canonical Forms — EGLM03 Modern Control Systems

For a broader perspective, let's read about the Normal Form in the online textbook 'EGLM03 Modern Control Systems'.

Read section 7.4.8, 'Normal Canonical Form'. Focus on the main idea: if you use partial fraction expansion on the transfer function, you can create a state-space model where the A matrix is diagonal. The diagonal elements are the system's poles (eigenvalues).

In the Normal Form, the state equations become a set of decoupled first-order differential equations, which is a very intuitive representation, especially given your background in signal processing and system dynamics.

Conclusion

In this lesson, we have demystified the process of transforming state-space models. We learned that while a system's input-output behavior is fixed, we can change our internal perspective—the state variables—to simplify our analysis and design tasks.

Key Takeaways:

State-space representations are not unique. A similarity transformation $z = T x$ can change the system matrices while preserving the input-output dynamics.
Canonical forms are standard structures. We focused on the Controllable (CCF) and Observable (OCF) forms, which are directly related to the coefficients of the system's transfer function.
We can transform an arbitrary state-space model $(A, B)$ to CCF by computing a transformation matrix $T = M_{c, z} \cdot (M_{c, x})^{- 1}$ , where $M_{c}$ is the controllability matrix.
Other forms like the Normal (Diagonal) Form exist, which make the system's modes (poles) explicit by diagonalizing the $A$ matrix.

Preview of the Next Lesson:

A key step in our transformation algorithm today was converting from state-space to a transfer function. In our next lesson, we will formally tackle this and the reverse process. We will master the techniques to convert between state-space representations and transfer function representations, solidifying the link between these two fundamental views of a system.

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