I think an example of a non-affine in control system is:

dx/dt = f(x,u)

It means that 'u' appears in a nonlinear format in ' f ', for example f(x,u)=0.5x^2+tanh(u).

Search the word "non-affine-in control" in the literature!

Hi Mohammad!

Thanks for replying. I kind of understand what your saying. I found this paper that talks about controlling "not affine in control" and system being described is something like:

dx1/dt = f(x1) + x2 + g(u)

dx2/dt = u

where u is the input and x1 and x2 are the state variables.

So if I simplify this, affine means that the input/control is multiplied by a function when it enters the state equation like:

dx/dt = f(x) + g(x)u

and non affine means it goes through a function like:

dx/dt = f(x,u)

or

dx/dt = f(x) + g(u)

Did I get that right? Sorry for asking again.

Affine means the system dynamics is linear with the control.

Affine simply means "linear", and nonaffine means "nonlinear", which are used in the context of nonlinear systems (with respect to the state). Therefore, a nonlinear system in which the control(s) appear(s) linearly is called a control/input-affine nonlinear system (or simply control/input-affine system, where the nonlinearity with respect to the state is automatically implied), whereas a system that has nonlinearities both in the state and in the control(s) is called a control/input-nonaffine nonlinear system (or simply control/input-nonaffine system).

The general form of an input-affine system is represented by the vector differential equation:

dx/dt = f(x) + g(x)u

whereas the general form of a system which is nonaffine in the control(s) is given by:

dx/dt = f(x) + g(x,u).

A control-nonaffine system can be brought into a control-affine form by introducing integral control so that the input, which appears is nonlinearly, is defined as a (new) state variable, resulting in an augmented system with the new state x_new/dt = [dx/dt du/dt] and new control u_new, where

dx/dt = f(x) + g(x,u)

du/dt = Cu + Du_new

In its simplest form C=0 and D=Identity.

Dear Tayfun,

Thank you so much. Your answer made it very clear. Does that mean that with u becoming a state in the system, statefeedback techniques can be used?

Is there a book that you can suggest for introductory topics on nonlinear control. Sometimes the problem I find is that I'm not familiar with the technical terms in nonlinear control. Is the book by Isidori a good start?

That is correct Luis.

I wouldn't recommend Isidori's book unless you have a very strong mathematical background in differential geometry. I suggest that you make a fresh start with "Applied Nonlinear Control" (by Slotine and Li) and, only when you feel comfortable with the topics therein, then progress to the book "Nonlinear Systems" (by Khalil), which is undoubtedly the best book in the area.

Thank you so much for giving references! I will definitely look into the first book you suggested. I didn't really know where to start.

Once again thank you very much!

I definetely agree with Mr. Tayfun Cimen, in my university, when I took nonlinear system course, the references were same

Thanks everyone!

I have another question that's kind of related and since everyone except myself seem to have some experience with nonlinear systems and nonlinear control, I'll ask it here first.

I'm trying to control a system that has a configuration in the shape of

ddx1 = f1(x1,dx1) + g1(x1)u

ddx2 = f2(x1,dx1) + g2(x1)u

"dd" means the second derivative.

"d" means the first derivative.

So after reading the replies I got I began to understand that

ddx1 = f1(x1,dx1) + g1(x1)u

of system I am trying to control is nonlinear through "f1" and affine in the input "u". And the output

ddx2 = f2(x1,dx1) + g2(x1)u

that I'm hoping to be able to control is defined by another nonlinear function "f2" and has a what I think is a "feedforward term" from "u" linearly entering through the nonlinear function "g2". I am pretty new to these kinds of systems and would really appreciate a clue to what kind of control can be used.

So, defining your states as y1 = x1, y2 = x2, y3 = dx1, y4 = dx2, in state-space form your system can be represented by the fourth-order model:

dy1 = y3

dy2 = y4

dy3 = f1(y1,y3) + g1(y1)u

dy4 = f2(y1,y3) + g2(y1)u

which is thus in the control-affine form

dy/dt = f(y) + g(y)u

since "u" enters the system linearly in the above state equations.

To control (regulate or track) any or all of the states, you can use any state-feedback linear or nonlinear control method. In order to apply linear state-space methods, linearize the above dynamics at the equilibrium (or operating) point of interest, and design a linear feedback controller using the linear(ized) model at that point, and then apply the resulting design to your original nonlinear dynamics. For a large operating envelope, you can linearize the nonlinear system at several operating points, design feedback controllers for each point, and then combine the resulting designs using the gain scheduling approach, which is one of the most widely used nonlinear control methods in practice. There are a variety of other nonlinear control methods which are all viable approaches for controlling such systems. Some of these are feedback linearization, backstepping, sliding-mode control, state-dependent Riccati equation (SDRE) control, etc.

Hello again Tayfun,

I have tried linearizing the system near the equilibrium point and using LQR to calculate the gains. I then applied Adaptive LQR to the system by estimating the A and B matrices of the linearized system then calculating the gains online. Although this does work and it does extend the envelope, I was hoping to improve the transients of the system.

I initially thought that backstepping would work for the system but the feedforward term to the output term doesn't seem to fit into applicable shapes of the backstepping and so I hit a bump there.

Sorry for asking so many questions, I really don't have anyone to ask.

If it is the transient response you would like to improve, performance-oriented nonlinear control methods, such as adaptive LQR, will generally yield better performance than stability-oriented nonlinear design methods, such as backstepping and feedback linearization. I suggest that you try other performance-based nonlinear control methods such as state-dependent Riccati equation or model predictive control.

I see... I'll try looking into model predictive control just because state-dependent riccati equation seems to be similar to adaptive LQR but I'll still give it a try.

Thank you so much for all your answers and thoughts.

I think you don't have to linearize the model of the plant.. Just keep into nonlinear model and then you solve it through sliding mode control.. since this controller can be designed directly for nonlinear control system

Hi Ramadhani,

Sliding mode would be one solution to the problem and it might even help with the uncertainty in the system. I'll definitely have a look at it. Do you have any suggestions about solving the chattering that comes with sliding mode?

There are so many techniques to reduce chattering, simple to say, you can replace sign function to saturation function or tanh function.

Hi Arghya,

Could you be more clear to what you mean about the curve? Do you mean the response of the system?

Dear Arghya,

Your interpretation, I believe, is related to affine spaces in the context of "Affine Geometry", but NOT input-affine nonlinear systems within the context of "Control Theory". Otherwise, all control systems would be nonaffine according to such a definition.

Regarding the definitions of control-affine/linear, the difference is in the drift term, not the *linearity* of the input. For example,

1) dx/dt=f(x)+g(x)u

is control-affine because the input u is *affine*, not linear.

2) dx/dt=g(x)u

is control-linear because the input is indeed linear

3) dx/dt=f(x,u)

is a general non-linear system.

Thus non-affine systems are systems *not* of the form of 1. It is called control-affine, because equations of the type:

y=ax+b

are called affine equations (not linear), whereas

y=ax

are called linear.

Also, the term f(x) in 1 is called the drift.

You are trying the pick on the semantics George...

The point is that a "real" control system is represented in form (1) or in form (3), never in form (2)... which is pure mathematical formulation, not respresenting the reality of a control system "design".

Additionally, y = ax+b is also called a "linear equation" George - it represents a straight line !!! (this is Math 101).

I think the person asking the original question has already clearly understood the difference between affine and nonaffine "control systems". So what novelty does your answer offer to understanding "control systems" (and I don't mean mathematical expressions) ? Like I said, semantics....

Tayfun, definitions are not semantics, and are needed in order to know what we are talking about. If Luis starts diving into non-linear control, definitions will save him alot of time and trouble. As for your comments on the equations, i don't know what *real* systems are. It seems to me that all systems are mathematical constructs, unless you can suggest a way to model a system with infinite precision. Although you are indeed correct about systems in the form of 1, many systems are in the form of 2, nearly all kinematic systems.

As for the *line equation*, this is misuse of terminology. The linear equation is called, as you say, linear, because it expresees the graph of a line. Not because it is *linear*. It is *affine*. As you may know, a fundamental property of linearity is superposition, which the *linear* equations do not conform to. Thus, the affine in the control-affine, is because the input is affine, not linear.

I don't think this discussion is going to go anywhere worthwhile George! The literature about affine and nonaffine nonlinear systems in the context of control systems is clear! Yet, you are still talking about misuse of terminology. The terminology has been set as said, whether you like it or not! The point of the previous discussion was to guide Luis in understanding this terminology with its relevance to control systems design (with a post of more than 20 answers, which I'm sure you have read). So, what exactly is your disagreement from these previous posts? Apparently misuse of terminolgy, as you very elegantly put it... Again, my question as to "what your contribution is in making Luis better understand the use of the terminology in the literature in the context of control systems" is unclear ! The question in this post has already been answered by many...

With regard to *real* systems generally being in form (1) and (3) when designing feedback control systems - I do not have the time to enlighten you - you can refer to the literature on the subject of nonlinear feedback control design with practical ("real"-life that is) applications.

Tayfun, i can't understand your need to comment on my answer (defending what exactly?), making disingenious comments like the equations of line are linear, or about semantics etc. My main point is that the *affine* in "control-affine" is because the equation is...affine! Not linear. Do you disagree? Thus, non-affine is just equations not in the specific form. Again, do you disagree?

Anyway, there is no need to start a flame or whatever. Let Luis decide is my answer helps him or not.

I think simply George Moustris is right. "Affine in something" has a precise meaning in mathematics, "Linear" also. There is not something like "Affine means this and that if you are in the context of Control Theory".

Coming to the contents: What Tayfun means saying that "real"control systems cannot be of the form dx/dt=g(x)u? There are plenty of "real"systems which are modelled as dx/dt=g(x)u (that is "linearly in the control).

Ok, here it goes once again for (what I sincerely hope is) the very last time to clarify the correct use of the terminology "affine" in the context of CONTROL SYSTEMS (I cannot possibly stress more on this).

Unlike my previous comments in this thread, this time I use the following book as supporting evidence of all my statements and definitions:

Eduardo D. Sontag. Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd Edition, 1998.

For those in the field of control, Prof. Sontag needs no introductions! Others can visit his personal webpage at http://www.math.rutgers.edu/~sontag/. Interested researches can also download an online version of the aforementioned book from this website.

********************************************************************************

DEFINITION OF "SYSTEMS AFFINE IN CONTROL" FROM SONTAG:

From page 57:

"By a system affine in controls (or control-affine) we mean a system over R for which f is affine in u, so that its equations take the form

dx/dt = f(x) + G(x)u.

This includes in particular linear systems."

From pages 154 and 155:

"An especially interesting class of systems, very common in applications, is that of control-affine systems, for which U is a subset of Rm, f(x, u) is affine in u:

dx/dt = f(x) + G(x)u. ...(4.18)"

"Remark 4.3.5 Any time-invariant linear system dx/dt = Ax+Bu is control-affine, because it has the form (4.18) with f(x) = the linear vector field Ax..."

From page 158:

"A very interesting class of strongly reversible systems is obtained from those control-affine systems for which f(x)=0, that is, systems of the form

dx/dt = G(x)u.

... These are called systems without drift (because, when u = 0, the state does not “drift” but, instead, remains constant) and they arise as kinematic models of mechanical systems."

********************************************************************************

First, let us revisit the question originally asked in this thread, which was "What does "affine in the control" mean when used to describe a system?"

The question is *NOT* asking what an affine equation means in MATHEMATICS - it is asking the meaning of the terminology used in the CONTEXT OF CONTROL THEORY for describing *SYSTEMS AFFINE IN CONTROL*!!!

So let me restate the correct use of the terminology "affine in control" in the context of CONTROL THEORY, based on the above definitions:

A system is "affine in control" when it can be represented as a LINEAR function of u.

Accordingly, a system is "nonaffine in control" when it becomes a NONLINEAR function of u.

When the equations of motion of dynamical systems (emphasis on the word "dynamical" here, and NOT "kinematical") can be represented in the form

dx/dt = f(x) + G(x)u,

such SYSTEMS are called CONTROL-AFFINE. These inlcude linear systems (as mentioned in the above reference); in this case we simply call them LINEAR SYSTEMS.

In the case of kinematical systems that can be represented in the form

dx/dt = G(x)u,

such SYSTEMS are also called CONTROL-AFFINE - *NOT* linear-affine!!! (once again see above reference) - we are talking about control-affine systems, NOT affine equations!!!

It is misleading to young researchers in the control community when other individual interpretations of this terminology (in CONTROL !!!) are portrayed as facts. These are false claims! In the context of CONTROL SYSTEMS, therefore, I strongly recommend the correct use of the terminology referenced here (as is well established in the control community), instead of futile attempts at trying to reinvent its meaning...

Well my friend Sontag says exactly what I and Moustris have affirmed. I do not see your point.

I guess that is the problem my friend; you really don't !

This is getting ridiculous. Systems in the form of 2 ( dx/dt=g(x)u ) are called either control-linear or driftless control-affine, as Sontag mentions. They are, of course, not affine in the control (input). They are linear. Apparently, they are a subclass of control-affine systems, with the drift being zero, hence the second name. If you search sciencedirect for "control-linear", or "input-linear", your find many papers, who call driftless control-affine, as control-linear (how dare they!). Anyway, there's no point in continuing this. I think Luis got the picture.

First of all, I do not see any mention of control-linear in Sontag! Sontag mentions control-affine ONLY! - Additionally, he mentions (2) are indeed ALSO driftless.

I think everyone's got the picture George!

In general all control systems are nonlinear and not affine in control, i.e.

dx/dt = f(x, u, d) (1)

where,

x - state

u - control input

d - disturbance

A control system that is affine in control, i.e

dx/dt = f(x) + g(x)u

is a subset of (1). Complex nonlinear systems such as a helicopter or a car can be described as (1).

If the input appears linearly in the system then it is affine in input.

consider a nonlinear system

dx/dt = f(x) + g(x)u

1. dx/dt = f(x) + 2u (affine)

2. dx/dt = f(x) + u^2 (non-affine)

I had the chance to look into few answers over this thread, it contains good information for starters regarding non-linear control :)

Affine systems are nonlinear systems that are linear in the input. When the system is affine, some control methods such as backstepping can be applied.

what about the affine matrix!!??

What dose “Z(x) is affine in x” mean?

It means its in the form of Z(x)=A+Bx.

Thanks, George P. Moustris

I have a question George P. Moustris.Thanks a lot!

For the multivariable input affine system xdot=f(x)+g(x)u, u=[u_1,u_2,...,u_n] is the input vector, and each input element has a range [u_i_min, u_i_max] . So when the system state reaches the maximum (minimum) value, does it mean that each input element takes its maximum or minimum value?

@Zhidong Lu for xdot=cos(x)u doesn't seem to be the case. So no.