# Poles and Zeros of Linear Systems

This period I follow a course in System and Control Theory. This is all about linear systems

$$\frac{dx}{dt}= Ax + Bu$$ $$y = Cx + Du$$ where A,B,C,D are matrices, and x, u and y are vectors.

To describe my (little) background, I should know things like linearization, Impulse/Step Response, Equillibrium points, (asymptotic) stability (routh, lyapunov), controllability, observability, stabilizability, realization, hankel, etc.

Given my background, I would like to ask some questions about poles and zeros of linear systems. If you know the answer of just one of the questions, please dont hesitate to write it down :-)

1.) What are poles and zeros of linear system {A,B,C,D} exactly? What does it mean for a system to have a pole at a certain value, or a zero at certain value?

2.) The author writes about 'poles of a transfer function matrix H(s)'. What is a transfer function matrix? The only thing I know is how to compute it and that it describes some relation between input/output of the system. But why do we need tranfser function matrices?

3.) To calculate the poles and zeros, the author says that we need the Smith and Smith-McMillan Forms. These are matrices that have only diagonal entries. What is exactly the algorithm to calculate the Smith-(McMillan)-form of a transfer matrix?

4.) What is the relation between the poles of a system and the controllability, observability, stability and stabilizability ? The same for a zero ?

5.) What is an invariant zero polynomial of the system {A,B,C,D} ?

6.) What is 'a realization of a system'?

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You are asking an awful lot, almost a mini course in basic control systems. All of the above can be found in most modern control system texts. – copper.hat Dec 27 '12 at 18:08
Maybe somebody can try to give some short/simple answers just to get me started. Most important question is why we use poles/zeros and why is it interesting to know them. Second most important is what a transfer function of a system really is. – Applied mathematician Dec 27 '12 at 21:42
Have you taken a basic course in SISO systems? The transfer function is the most basic element, it relates the Laplace transforms of the outputs to the inputs for linear, time-invariant, finite dimensional systems. The poles govern the dynamic behavior of the system (cf. residue theorem), the zeros place limits on the dynamic behavior of closed loop systems. But there are no easy one-liner answers here. – copper.hat Dec 27 '12 at 21:55
Agree with copper.hat. I'd vote to close it as "Overly broad". But the (slightly absurd IMO) bounty prevents it. – leonbloy Jan 3 '13 at 2:48
This is very broad question that would (due to the wide scope) be hard to answer even if the author was familiar with the basic prerequisite mathematical tools, e.g., Laplace transforms, transfer functions, controllability/observability, etc. But alas, he/she isn't. I'm not sure any answer I took the time to type could be comprehended. I believe your time would be better spent with your text or a book like Brogan's Modern Control Theory. – JohnD Jan 3 '13 at 2:56

1.) What are poles and zeros of linear system {A,B,C,D} exactly? What does it mean for a system to have a pole at a certain value, or a zero at certain value?

Intuitively, I do not know exactly what poles or zeros are. All I know is that the poles are roots of the denominator of the transfer function, or the eigenvalues of the $A$ matrix, like the one in your question. Poles show up explicitly in the solutions of ordinary differential equations, and an example of this can be seen here:

So what kind of question can we answer using information about poles?

i) Is the system stable?

ii) If it is stable, is the response of the system oscillatory, is it like a rigid body?

iii) If it is unstable, is it possible to stabilize this system using output feedback? (you need information about zeros here)

Now, let's talk about zeros. Zeros show up in literature because it has an effect on the behavior of control systems.

i) They impose fundamental limitations on the performance of control systems.

iii) They tell you about the "internal stability" of a control system.

As far as I can tell, zeros are more subtle than poles. I cannot say I fully understand them.

2.) The author writes about 'poles of a transfer function matrix H(s)'. What is a transfer function matrix? The only thing I know is how to compute it and that it describes some relation between input/output of the system. But why do we need tranfser function matrices?

Taking the Laplace transform of a differential equation that has a single-input and a single-output yields a transfer function. An example of this is in the link above. A transfer function describes the relationship between a single output and a single input. So if you have a system of differential equations that has, say, 2 inputs and 3 outputs, then a transfer matrix is a matrix of transfer functions that contains 6 elements. Each individual element describing the relationship between one of the inputs and one of the outputs. (The superposition principle plays a big role here)

But why would one want a transfer matrix. I believe it is because calculating zeros for a multi-input multi-output system is not easy. Here is an article that talks about all the different kinds of zeros and why they are important:

http://www.smp.uq.edu.au/people/YoniNazarathy/Control4406/resources/HoaggBernsteinNonMinimumPhaseZero.pdf

3.) To calculate the poles and zeros, the author says that we need the Smith and Smith-McMillan Forms. These are matrices that have only diagonal entries. What is exactly the algorithm to calculate the Smith-(McMillan)-form of a transfer matrix?

Sorry. I don't have much on this one.

4.) What is the relation between the poles of a system and the controllability, observability, stability and stabilizability ? The same for a zero ?

For me, poles and zeros are important to transfer functions, which describe the relationship between inputs and outputs, and they can tell you about stabilizability and stability. However, concepts like controllability and observability are state space concepts (At least for me). If you write a transfer function in state space form, as you have written in your question, then there is a very simple test for controllability and observability. You can find more about this in almost any course, for example in Stephen Boyd's introductory control course at Stanford.edu.

5.) What is an invariant zero polynomial of the system {A,B,C,D} ?

A SISO system just has one kind of zero. A MIMO system has many kinds of zeros, one of which is an invariant zero. The roots of the invariant zero polynomial gives you invariant zeros. It makes me kind of sad that I do not know very much about zeros of MIMO systems.

6.) What is 'a realization of a system'?

Let's say you start off with a differential equation. Then you take its Laplace transform, and obtain a transfer function. Then, for this transfer function, there are an infinite number of state space representations. That is, there are an infinite number of matrices $A, B, C, D$ that yield the same input-output relationship as the original transfer function. These representations are called realizations. We can go from one realization to another using "Similarity Transformations".

If you are a mathematician, then you should probably look for a more mathematical text on control systems. Most engineers use a classical control book ( like the one by Ogata ) in undergrad, which is mostly about transfer functions, zeros, poles, and various stability tests. Then, in grad school, engineers take a course called "Linear Systems Theory", where they learn about State Space theory of control systems. The book I used was by "Chen", but I did not like it very much.

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Great. Thank you for answers! I upvoted and gave you the 50 points – Applied mathematician Jan 3 '13 at 16:07

I will provide some references for you to peruse and see if you can get through some of your questions.

There are Open Courseware classes and notes, for example MIT.

Another example at MIT, and I am sure you can find others (free lectures from top schools from around the world).

You might enjoy reviewing these nice State Space Methods lecture notes.

You should also review the Wiki.

Here is a Wiki Book on the matter.

There are entire books written on the matter and here are some excellent ones.

$\bullet$ Modern Control Theory (3rd Edition), William L. Brogan (This is an amazing book!)

$\bullet$ Control System Design: An Introduction to State-Space Methods (Dover Books on Electrical Engineering), Bernard Friedland (Price is great!)

I would recommend reviewing the notes and references above and see if they answer your questions, then look into those books (peruse them online).

I can provide further details and review of your questions if needed, but wanted to get you started. Please respond if you want further insights and response.

Regards

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Thanks. To be honest, I don't have time to read books and lectures about this subject. I have two weeks to absorb the text and understand it :-) – Applied mathematician Dec 27 '12 at 15:21
So, maybe you can just give me 'simple' answers, that would help me at this moment I think. – Applied mathematician Dec 27 '12 at 16:55
Did you at least review the State Space Methods notes I linked to in my response? Those answer at least 80% of your questions in rapid fashion? After reviewing those, please tell us if you are still having questions. – Amzoti Dec 28 '12 at 2:02
Thanks I will look at it. I like the colors etc :-) – Applied mathematician Dec 28 '12 at 11:48
Great resources, them all! – amWhy May 10 '13 at 1:19

This is not a complete answer, but... If you are familiar with the theory of linear ordinary differential equations with constant coefficients, the general solution of such equations can be determined by determining the "characteristic equation" (a polynomial) and solving for the roots. The roots of the characteristic equation determines the set of functions (generally of the form of complex exponentials) that make up the general solution the differential equation and thus the behavior of the physical system that the equation represents. What are referred to as "poles" in control theory are the roots of this "characteristic equation".

In control theory, the characteristic equation (polynomial) is arrived at by applying the Laplace transform to the underlying equations and solving what is now a set of algebraic equations. In modern control systems theory, differential equations are almost entirely hidden by formal math. The Formal math is the application of the Laplace transform to all dynamic equations and the structuring of the resulting equations into matrix form. As a result, a control system can be represented by something called a "transfer matrix". The poles of the system are equal to the eigenvalues of the transfer matrix. The poles are complex numbers and which can be represented on the complex plane.

The location of poles on the complex plane provides a great deal of information about the behavior of a linear system. For example if the poles lie on the left hand side of the plane, the system is said to be stable. This is because all of the functions that make up the general solution to the underlying differential equations are of a form that decay over time. If any poles lie on the imaginary axis, the system will oscillate. If any poles lie on the right side of the complex plane, the system output will grow without bounds. For both of these latter cases, the system is said to be unstable.

If a system is unstable, it can be modified (at least sometimes) to stabilize the system (an important topic in control theory). System realization implies that the system represented by equations will be constructed as a real, physical system. Some mathematical structures cannot be realized. For example a system that has a single complex pole cannot be represented by a real physical system. Complex poles must always be paired so the underlying equations yield all real coefficients. The coefficients of such equations generally represent physical values (or combinations of such values) of the components that make up the system. More (and much better) info can be found by reviewing basic control system theory.

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Structuring your answer into paragraphs would make it much easier to read. – Andy Brandi Jan 3 '13 at 3:50