# Talk:Sliding mode control

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## Existence is not correct

\dot V < 0 only implies asymtotic movement to the sliding surface. A more strong condition like \dot V \leq \alpha o \dot V \leq V^{1/p}, p>1 is required for the finite time entrance to the sliding surface.

Cheers.

Marco Tulio Angulo —Preceding unsigned comment added by 189.136.148.79 (talk) 15:38, 4 November 2008 (UTC)

I'm not an expert in sliding mode control, but I've gone through and added comments that the system CAN reach the surface in finite time because the control need not be continuous. I think that will be enough to satisfy most people. It would be great if an expert could make things more precise. —TedPavlic | (talk) 18:29, 29 November 2008 (UTC)
Just about to add a section making this precise. —TedPavlic (talk) 16:28, 5 June 2009 (UTC)

## Feldbaum

The best perfomance of idea in sliding mode controll achieved prof. Feldbaum. The main idea of his method is finding the optimal phase curves. And after some modification we are may find hyperplane - sliding hyperplane. This is the best knowing method for the controlling of the object.

Then add a reference. It's common to discuss sliding mode control schemes in optimal control classes. In fact, this topic may need its OWN Wikipedia article. —TedPavlic | (talk) 18:30, 29 November 2008 (UTC)

## Region of attraction?

The set in Theorem 2 is empty. This definition clearly is incorrect. Is a dot missing? --TedPavlic 19:58, 25 May 2007 (UTC)

Someone fixed this sometime. I tried to make it a little better now. —TedPavlic | (talk) 18:31, 29 November 2008 (UTC)

## Plain English

Anyone care to add an introduction in plain English?

- What is the problem to be solved

- How does Sliding mode solve the problem

- Possibly an analogy or an example

Thanks

Davide Andrea (talk) 00:45, 22 February 2008 (UTC)

Better? If not, what more would help? —TedPavlic | (talk) 18:31, 29 November 2008 (UTC)
Quite good, but
The state-feedback control law is not a continuous function of time; it switches from one smooth condition to another. That is, the structure of the control law changes based on the position of the state trajectory; hence, sliding mode control is a variable structure control method because it switches from one smooth control law to another.
seems to be redundant. However I am not sure and therefore cannot solve the problem.--Hfst (talk) 19:58, 8 May 2009 (UTC)
Better now? —TedPavlic (talk) 20:42, 8 May 2009 (UTC)
I think the answer is "Yes". Thanks.--Hfst (talk) 14:40, 10 May 2009 (UTC)

Could I suggest adding a phase portrait to the introduction to illustrate a sliding action? I find these really help when you are first learning about sliding mode (even if it can only apply to second order systems). If nobody objects then I could provide an image - Adlab (talk) 17:29, 15 January 2009 (UTC)

I think that's a great idea. It might be a good idea to use a phase portrait for one of the commonly-used examples already given on the page. Go ahead and provide one. Be WP:BOLD. —TedPavlic (talk) 22:27, 15 January 2009 (UTC)
I've generated a plot but it seems that I have to make ten edits before I can upload images (so will upload it asap) - Adlab (talk) 17:29, 19 January 2009 (UTC)

## Moved away from article on April 6th, 2009

Sliding mode control is sometimes critiqued as being a blunt instrument when compared to other forms of nonlinear control that have more moderate control action.[citation needed] In particular, because actuators have delays and other imperfections, the hard sliding-mode-control action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics. Continuous control design methods are not as susceptible to these problems and can be made to mimic sliding-mode controllers.

Reason: Unsourced that the apparant "brutality" of the control algoritm is a disadvantage. If used with switching power converters the choice of sliding mode controller seems very natural. (Source an further explanation added to the article). The socalled chattering problem should be treated in the article, however.

Correct me if I'm wrong. User:Nillerdk (talk) 07:40, 6 April 2009 (UTC)

If I recall correctly, Khalil has an entire section devoted to continuous alternatives to sliding mode control. SMC has been treated as a panacea in industrial control (often using the same justification that you use -- the future is in switching). However, the mathematics needed to justify comments about the robustness (and safety) of the control method are far more sophisticated than what you need with continuous control. VSC needs to be applied with care and should not be a hammer applied to every nail. This is not a new criticism. It's just that VSC has become the PID of industrial nonlinear control; it evolves on its own without any informed help. (All of that being said, I'll revert your change and massage it to not be as "brutal") —TedPavlic (talk) 20:27, 8 May 2009 (UTC)
When I learned about sliding mode control about 2000 chattering (high frequency switching) was a topic although I cannot recall the correct bibliographical data of that utkin-paper. Without mentioning sliding mode control "Föllinger, Nichtlineare Regelung I & II" described the effect as deleted from the article. However, he does not discuss the topic "low-pass behaviour of the system" which makes in my opinion the problem negletable in sytems with order higher as lets say 2 or 3. However I do not agree to the words above, because chattering also happens under ideal conditions unless you have a zero-band.--Hfst (talk) 14:28, 9 June 2009 (UTC)
I think that's why the original statement says not as susceptible. Of course, chattering can occur. In fact, in real systems, even when there is no delay, unmodeled nonlinearities from quantization error and friction produce the same discontinuities you'd see in a sliding mode controller. Hence, chattering is just a fact of life. However, a sliding mode controller nearly depends on it; it's not a maybe, it's a given. Sliding mode control is a blunt instrument that gains its robustness from its brutality. It's also primitive when you consider all of the recent work in hybrid systems (see, for example, the terrific review from IEEE CSM this year: doi:10.1109/MCS.2008.931718). All of that being said, please review the bottom of the second paragraph and the third paragraph. I think they're a good unbiased summary of the state of the art in SMC application. —TedPavlic (talk) 15:05, 9 June 2009 (UTC)
I just made a change that changes the order so that my request (about the bottom of the second paragraph) makes no sense. See the last three paragraphs in the introduction. See if that's satisfactory. They now include a note about the optimality of sliding mode in certain contexts. —TedPavlic (talk) 21:24, 9 June 2009 (UTC)

## Good article

I think the article is a good introduction to SMC. Thanks for correcting my corrections. I have a terminology question. I am thinking of the "sliding surface" as the supspace/place where the motion - the "sliding mode" - takes place. I think the article confuses the place and the (kind of) motion, but I might be wrong. For example, I can say "the system trajectories cross the sliding surface, but sliding mode does not occur" but if I swap "sliding surface" and "sliding mode" the sentance makes no sense anymore. Do I have a point? I'll look for some sources myself too.User:Nillerdk (talk) 07:11, 3 July 2009 (UTC)

I'm going to argue that it's sloppy to say that a "sliding mode does not occur." However, I concede your point. The issue is that I can create a switching function ${\displaystyle \sigma }$, and there is a surface corresponding to ${\displaystyle \sigma =0}$. However, that surface is not necessarily a sliding mode of the system – I have no guarantee that a sliding mode exists. That being said, if the sliding mode exists, then it is totally appropriate to substitute one term for the other. The reason why "mode" and "surface" are so easily swapped is that systems in which they do not match are of no interest to us. That is, we're building a control that generates a sliding mode (i.e., a crease in the vector field) precisely where the zero-switching surface is. We need to be cognizant that it may be impossible to create a sliding mode along our sliding hypersurface, but I don't think we need to gild the lily. That being said, it's certainly fine to make things more precise. This article still needs a lot of work. —TedPavlic (talk/contrib/@) 13:41, 3 July 2009 (UTC)

## High frequency switching control ?

From my point of view, the sentence in the introduction " ... is a nonlinear control method that alters the dynamics of a nonlinear system by application of a high-frequency switching control" is really ambiguous. In continuous time, the sliding mode control given by the solution in the sense of Filippov is not an highly swithing signal. Since the state is on the sliding surface, the control has the same regularity as the external input. In a discrete implementation, when an explicit Euler method or an explicit ZOH is used, we get a high frequency switching signal. This is not inherent to the sliding mode control but to an poor disctrization scheme or to the actuator's technology. When an implicit discretization is performed and the actuators is able to deliver the whole convex hull of the systems right-hand-side, the control does not switch anymore.

I suggest to formulate this remark somewhere in this introduction.

What are your opinions ? Vacary (talk) 10:04, 26 May 2011 (UTC)

These are minor technicalities that bridge into the realm of interpretation. I concede your point that switching is more an artifact of the real world than of the idealization of the control method. Perhaps the control method should be presented in a more pure form for motivation and then something should be added about implementation details... or realizations of sliding mode controllers. Perhaps all of this could be added to the first sentence or two of the article and the rest could remain. If you have a suggestion on new wording, you should give it. I think changes should be made with great care though; there is danger that the discussion can actually add more ambiguity for the casual reader (and face it, most people who use SMC in industry are going to view it as a switching control... or perhaps the asymptotic limit of very high gain control with the understanding that real-world effects lead to chattering). —TedPavlic (talk/contrib/@) 17:06, 26 May 2011 (UTC)
I've made some changes. How is that? —TedPavlic (talk/contrib/@) 17:43, 26 May 2011 (UTC)
It is fine for me. I agree that the introduction should keep its generality. I will work on a new section on the realization/discretization of sliding modes controllers if I have some time to show how the infinitely switching control is not mandatory. Vacary (talk) 08:56, 27 May 2011 (UTC)

## Removed new section on regular form

I removed the following section on an SMC regular form which was added by User:jdsanch1 (talk). It's a mess. It was apparently lifted from someone's LaTeX manuscript (notice all of the \ref's and \cite's). Consequently, I worry that its source might have been an unpublished manuscript that was rejected(?). Regardless, if it is to be added here, all of the LaTeX \refs and \cites must be replaced with the equivalent Wikipedia versions. Moreover, the \cite targets must actually be added to the references. Finally, it should be proof read to remove typos (e.g., change "for" to "form" in the first sentence).

Sliding mode control based on the Regular Form
In this section a particular canonical for, the so-called Regular Form is presented to obtain a convenient interpretation of the reduced order dynamics provided by sliding mode control, decoupling the system into two subsystems of lower dimensions.
${\displaystyle \mathbf {{\dot {x}}=f} \left(x\right)\mathbf {+B} \left(x\right)\mathbf {u} }$
where ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$, ${\displaystyle \mathbf {u} \in \mathbb {R} ^{m}}$, ${\displaystyle \mathbf {B} \in \mathbb {R} ^{n\times m}}$, and ${\displaystyle {\text{rank}}\left(\mathbf {B} \right)=m. Following the regular form design approach, a nonlinear transformation should be found such that the system is decoupled into two subsystems of lower dimensions ${\displaystyle (n-m)}$ and ${\displaystyle m}$:
${\displaystyle \left\{{\begin{array}{l}\mathbf {\dot {x}} _{1}=\mathbf {f} _{1}\left(\mathbf {x} _{1}\mathbf {,x} _{2}\right)\\\mathbf {\dot {x}} _{2}=\mathbf {f} _{2}\left(\mathbf {x} _{1}\mathbf {,x} _{2}\right)+\mathbf {B} _{2}\mathbf {\left(x_{1},x_{2}\right)u} \end{array}}\right.}$
where ${\displaystyle \mathbf {x} _{1}\in \mathbb {R} ^{n-m}}$, ${\displaystyle \mathbf {x} _{2}\in \mathbb {R} ^{m}}$, ${\displaystyle \mathbf {u} \in \mathbb {R} ^{m}}$ and ${\displaystyle \det \left(\mathbf {B} _{2}\right)\neq 0}$. The system, where the dimension of the lower equation coincides with that of the control input ${\displaystyle \mathbf {u} }$ and the upper equation does not depend on the real control, is referred to as a Regular Form \cite{Loukyanov1981}.
The idea of transformation is formulated in the following way: Let ${\displaystyle y^{T}=\left[y_{1}^{T},y_{2}^{T}\right]}$ be a vector of new state variables defined by the nonlinear transformation
${\displaystyle y_{1}=\phi \left(\mathbf {x} \right),\quad y_{2}=\mathbf {x} _{2},}$
where the vector function ${\displaystyle \phi \left(\mathbf {x} \right)}$ is continuous and continuously differentiable with respect to ${\displaystyle \mathbf {x} }$. The equations with respect to ${\displaystyle y_{1}}$,
${\displaystyle {\dot {y}}_{1}={\frac {\partial \phi \left(\mathbf {x} \right)}{\partial \mathbf {x} }}f\left(\mathbf {x} \right)+{\frac {\partial \phi \left(\mathbf {x} \right)}{\partial \mathbf {x} }}\mathbf {B} \left(\mathbf {x} \right)\mathbf {u,} }$
will be independent of the control if the vector function ${\displaystyle \phi \left(\mathbf {x} \right)}$ is a solution to the matrix partial differential equation
${\displaystyle {\frac {\partial \phi \left(\mathbf {x} \right)}{\partial \mathbf {x} }}\mathbf {B} \left(x\right)=0.}$
Necessary and sufficient conditions for solving the equation (\ref{BCPFM05}) may be found based on the theory of Pfaffian's form in the text book of Rashevskii \cite{Rashevskii1947}. It should be noticed that partial differential equations of this type need strong solvability conditions. In \cite{Loukyanov1981} authors have investigated this problem and proposed a design regularization algorithm. It has been established that the regularization problem is solvable only for one class of systems which fulfills the Frobenius' theorem conditions. The reader is referred to \cite{Utkin1992} for a complete overview of this approach applied to both single-input and multiple-input.
According to system (\ref{BCPFM02}), the sliding mode control approach assumes that ${\displaystyle \mathbf {u} }$ is a discontinuous control enforcing sliding mode in the manifold ${\displaystyle S\left(\mathbf {x} \right)=0}$ with ${\displaystyle m}$ selected switching surfaces denoted by the vector ${\displaystyle S\left(\mathbf {x} \right)=\left[s_{1}\left(\mathbf {x} \right),s_{2}\left(\mathbf {x} \right),\ldots ,s_{m}\left(\mathbf {x} \right)\right]^{T}}$ . After sliding mode occurs on ${\displaystyle S(\mathbf {x} )=0}$, ${\displaystyle m}$ components of the state vector are function of the remaining ${\displaystyle (n-m)}$ ones: ${\displaystyle \mathbf {x} _{2}=S_{0}\left(\mathbf {x} _{1}\right)}$. As a result, the sliding mode equation along the manifold ${\displaystyle S(\mathbf {x} )=\mathbf {x} _{2}-S_{0}\left(\mathbf {x} _{1}\right)=0}$ is
${\displaystyle \mathbf {\dot {x}} _{1}=\mathbf {f} _{1}\left(\mathbf {x} _{1},S_{0}\left(\mathbf {x} _{1}\right)\right)}$
In other words, the evolution of the upper subsystem in (\ref{BCPFM02}) is determined by equation (\ref{BCPFM06}). The desired dynamics of sliding mode can be designed by a proper choice of the function ${\displaystyle S_{0}\left(\mathbf {x} _{1}\right)}$ which takes part of the reduced order system dynamics(\ref{BCPFM06}).

TedPavlic (talk/contrib/@) 16:28, 2 November 2011 (UTC)