Hi,

In tracking example, you take sliding variable s as a tracking error. Assume that the relative degree = r, then you can stabilise a r-integrator chain.

the surface defined by s=s(1)=... = s(r-1)=0 is the sliding surface.

In HOSM, It's not necessary to define a sliding surface as hurwitz Polynomiale (contrary to SMC). you can check to my CDC12/IEEE CST 2014 papers. 

Regards

Hi,

In tracking example, you take sliding variable s as a tracking error. Assume that the relative degree = r, then you can stabilise a r-integrator chain.

the surface defined by s=s(1)=... = s(r-1)=0 is the sliding surface.

In HOSM, It's not necessary to define a sliding surface as hurwitz Polynomiale (contrary to SMC). you can check to my CDC12/IEEE CST 2014 papers. 

Regards

The sliding surface dictates the system dynamics in the sliding phase, and it is usually designed to full fill the desired control objectives. 

There is no general method for selection of the sliding surface.

however, a linear combination of the state variables  is a choice.

(if one of the state variables is system output, this variable must be used.)

the value of the state-variable that are used in sliding surface must be zero during steady-state.

it is completely usual to add integral and derivative of the important stat variables to sliding surface.

and ...

good luck

For the first order sliding mode controller, the easy way for the choice of the sliding surface is the combination between the controlled state and its derivative. 

The sliding surface specifies the relative degree of a system, and, therefore from the tracking error and its derivative we can also define the relative degree from the expression of J.J. Sloting it is used in my paper:

https://www.researchgate.net/publication/269762247_A_Simple_and_Efficient_Maximized_Power_Control_of_DFIG_Variable_Speed_Wind_Turbine?ev=prf_pub

https://www.researchgate.net/publication/269810463_Maximized_Power_Control_based_on_High_Order_Sliding_Mode_for_Sensorless_DFIG_Variable_Speed_Wind_Turbine?ev=prf_pub

For the first order the control vector appears from the error, so, this is the first order sliding mode control, the sliding surface is the error

For the second order the control vector appears from the derivative of the  error, so, this is the second order sliding mode control, the sliding surface is the error and the derivative of the error.

Article A Simple and Efficient Maximized Power Control of DFIG Varia...

Article Maximized Power Control based on High Order Sliding Mode for...

There is no generalized mechanism for designing sliding surfaces. However there are two necessary points to be kept in consideration while taking/designing a sliding surface.

1) The sliding mode dynamics produced must be stable.

2) The surface must satisfy the relative degree requirement of the sliding mode controller i.e. A conventional first order SMC can't enforce sliding modes if the surface it uses has a relative degree other than 1, like wise a real twisting algorithm will be in-effective if the surface has a relative degree other than 2.

Moreover a sliding surface should reflect the design specifications/objectives, when the sliding modes are established. 

Why Sliding Mode Controller is not called as tracking mode controller?

Good!

Fist, the state variables of the system should be redefined to have zero steady-state value.

In fact, the system in new coordinate should go to origin after the transient response.

Then, usually it is possible to define the sliding surface by linear combination of the new variable.

Good Luck 

The core consideration in choosing a sliding manifold is to check whether it supports the existence of "SLIDING MODES" in the given system's state space. Also you must verify whether the sliding mode dynamics does perform according to your application specifications.

GOOD LUCK

Dear Farzin Piltan,

The choice of sliding variable sets the desired dynamics of the system under sliding: s=0

Hi to everyone,

I have a question related to the method of finding a sliding surface which leads to a certain convergence rate to the origin. Attached you can find the question 3 (sliding mode control). I really appreciate it if you can help me how to do this.  

Hi!

Sliding surface is both dynamics and place to be! Concept of sliding surface will be more clear if we start thinking about it in terms of place to be! For instance in simple stabilization problems, if we can enforce Vdot to be strictly negative for all x, except at x=0, then x=0 becomes the invariant set, because by enforcing Vdot to be strictly negative definite we are essentially taking out all the possibilities where the trajectories can get stuck-- for instance in a limit cycle.

Similarly in trajectory tracking problem, we fabricate a variable 's' such that single derivative of that variable yields u and when s->0, e->0 as well. The reason why we "want" u to appear in single differentiation of s, is because the way we construct the lyapunov function and hence the Vdot. It is actually the Vdot where we want u to appear! Because by "choosing" the u, we can 'force' the Vdot to be strictly negative definite.

In essence by introducing this variable 's' we are converting trajectory tracking problem into a stabilization problem!

so the best way to select s is s=((d/dt+c)^(n-1))*e

As before we "enforce" Vdot to be strictly negative definite in order to preclude all the possibilities where trajectories can get stuck and thus s=0 becomes the only invariant set or trajectories converge to s=0.

In a more generalized MIMO setting, sliding surface is selected as s=[s1,s2,s3...sn]^T.

Where s1,s2...sn are sliding surfaces for each outputs. One needs to design sliding surface for each output depending the relative degree of that particular output.

s=((d/dt+c)^(n-1))*e or in integral form-

s=((d/dt+c)^n)*Integral(e).

In this case, the invariant set happens to be the intersection of these individual sliding surfaces.

@kasra

Your question is fairly easy, we need some simple transformation to transform your problem-

You need to multiply your system by b inverse, lets call it p, now your system becomes px' =p*f(x)+u.

Let our desired trajectory be xd=sin(t), and hence sliding surface will be s=ce and lyapunov function will be: V=1/2*|p|*s^2. In case of simple sliding mode control whether you take V=1/2*p*s^2 or V=1/2*|p|*s^2, the expression for u will be same as I have derived below except that it will have p term associated with -eta*sign(s) and -k*s.

u=-p*f(x)+p*xd'-eta*sign(s)*p-k*s*p,

so, ueq= -p*f(x)+p*xd'-k*s*p

The expression for u as usual will consist of a continuous part that is responsible for maintaining the trajectory on the hyperplane and a discontinuous part that comes into play when the trajectory goes above or below the hyperplane.

In the graph, test.eps you can see perfect tracking is achieved for problem (b).

PS- Kindly note I have used constant plus proportional rate reaching law, for reaching phase. You can choose anything as long as it guarantees the strict negative definiteness of your Vdot.

In order to choose a sliding surface for a sliding mode control design, following points must be kept in mind.

• The sliding surface thus selected is stable. A control law designed on the basis of unstable sliding surface won't be able to steer the system to stability.
• To check whether the reach ability condition or existence condition is met. This proof certifies the stability of the surface and the controller as well. You can prove it with different choices of reaching laws (assuming sliding mode occurs), as constant rate reaching law, exponential reaching law, power rate reaching law etc. Hence a stability analysis.
• The relative degree of the system must be kept into consideration while choosing a sliding surface. If a system has a relative degree "r", then the sliding surface has a relative degree of r-1. It must not exceed the system's order, in a sense.

what is range of Ueq value? Between 0 and 1?

Hi,

For systems described by phase variable canonical form, the following classical sliding variable is used (becomes sliding surface when it equal to zero)

S=xn+cn-1*xn-1+......+c1*x1

For general linear or nonlinear systems, the situation is quite different and you can say that this is a crucial step (the selection of sliding variable).

The transformation to the Regular Form is a way to transform the system dynamics into a form that can be used to select the sliding variable as above. you can refer to Utkin book "Sliding Mode Control in Electro-Mechanical Systems" chapter three. In addition, the transformation to the regular form was applied in order to select the sliding variable in the following paper "Sliding Mode Controller Design for a Crane Container System"

Regards