Sliding Mode Control Part 8 ppt

Image-based control of mobile robot with central catadioptric cameras, IEEE International Conference on Robotics and Automation, pp.. A sliding mode control law for epipolar visual servo

2 4 6

0 10 20 30 40 50 60 70 80 89 0

20 40

Time (s) 0 80 160 240 320 400 480 560 640

0 60 120 180 240 300 360 420 480

u image coordinate (pixels)

(a) Path on the plane (b) Evolution of the robot state (c) Motion of the image points.

Fig 9 Robustness under image noise using a hypercatadioptric imaging system

−0.1 0 0.1 0.2 0.3

Time (s) (a) Evolution of the epipoles (b) Input velocities.

Fig 10 Performance of the reference tracking and the velocities given by the sliding modecontrol law for the servoing task of Fig 9

orientation error This is achieved in spite of the uncertainty in the distance between the

current and the target locations (d) As mentioned before, it is enough to fix this value in

the controller thanks to the robustness of the control law We claim that the second phaseregarding to depth correction may be carried out exploiting also the information provided

by the epipolar geometry This could be a way to avoid the switching to a totally differentapproach for depth correction

6 Conclusions

In this chapter, a robust control law to perform image-based visual servoing fordifferential-drive mobile robots has been presented The visual control utilizes the usualteach-by-showing strategy, in which the desired location is specified by a target imagepreviously acquired The mobile robot is driven toward the target by comparing a set ofvisual features in the current view of the onboard camera and those on the target image Thevisual features are gathered through the epipolar geometry and exploited in a sliding modecontrol law, which provides good robustness against image noise and uncertainty in cameraparameters

The major contribution of this work is the validity of the approach for generic imagingsystems This extends the applicability of the proposed control scheme given that a genericcamera allows a major maneuverability of the robot than a conventional camera becauseits wide field of view Additionally, the use of sliding mode control allows to solve theproblem of passing through a singularity induced by the epipoles, maintaining boundedinputs Furthermore, the visual control accomplishes its goal even when the robot starts on

the singularity The good performance of the approach has been evaluated through realisticsimulations using virtual images.

7 Acknowledgment

This work has been supported by project MICINN DPI 2009-08126 and grants of BancoSantander-Universidad de Zaragoza and Conacyt-Mexico

8 References

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Sastry, S (1999) Nonlinear Systems: Analysis, Stability and Control, Springer, New York.

1 Introduction

Nowadays, the major advancements in the control of motion systems are due to the automaticcontrol theory Motion control systems are characterized by complex nonlinear dynamics andcan be found in the robotic, automotive and electromechanical area, among others In suchsystems it is always wanted to impose a desired behavior in order to cope with the controlobjectives that can go from velocity and position tracking to torque and current trackingamong other variables Motion control systems become vulnerable when the output trackingsignals present small oscillations of finite frequency known as chattering The chatteringproblem is harmful because it leads to low control accuracy; high wear of moving mechanicalparts and high heat losses in power circuits The chattering phenomenon can be caused

by the deliberate use of classical sliding mode control technique This control technique ischaracterized by a discontinuous control action with an ideal infinite frequency When fastdynamics are neglected in the mathematical model such phenomenon can appear Anothersituation responsible for chattering is due to implementation issues of the sliding mode controlsignal in digital devices operating with a finite sampling frequency, where the switchingfrequency of the control signal cannot be fully implemented Despite of the disadvantagepresented by the sliding mode control, this is a popular technique among control engineerpractitioners due to the fact that introduces robustness to unknown bounded perturbationsthat belong to the control sub-space; moreover, the residual dynamic under the sliding regime,i.e., the sliding mode dynamic, can easily be stabilized with a proper choice of the slidingsurface A proof of their good performance in motion control systems can be found in thebook by Utkin et al (1999) A solution to this problem is the high order sliding mode (HOSM)technique, Levant (2005) This control technique maintains the same sliding mode properties(in this sense, first-order sliding mode) with the advantage of eliminating the chatteringproblem due to the continuous-time nature of the control action The actual disadvantage ofthis control technique is that the stability proofs are based on geometrical methods since theLyapunov function proposing results in a difficult task, Levant (1993) The work presented

in Moreno & Osorio (2008) proposes quadratic like Lyapunov functions for a special case ofsecond-order sliding mode controller, the super-twisting sliding mode controller (STSMC),making possible to obtain an explicit relation for the controller design parameters

Jorge Rivera1, Luis Garcia2, Christian Mora3,

1,2,3,4Centro Universitrio de Ciencias Exactas e Ingenierías, Universidad de Guadalajara

5Centro de Investigación y Estudios Avanzados del I.P.N Unidad Guadalajara

México

Super-Twisting Sliding Mode

in Motion Control Systems

13

In this chapter, two motion control problems will be addressed First, a position trajectorytracking controller for an under-actuated robotic system known as the Pendubot will bedesigned Second, a rotor velocity and magnetic rotor flux modulus tracking controller will

be designed for an induction motor

The Pendubot (see Spong & Vidyasagar (1989)) is an under-actuated robotic system,characterized by having less actuators than links In general, this can be a natural designdue to physical limitations or an intentional one for reducing the robot cost The control ofsuch robots is more difficult than fully actuated ones The Pendubot is a two link planarrobot with a dc motor actuating in the first link, with the first one balancing the secondlink The purpose of the Pendubot is research and education inside the control theory ofnonlinear systems Common control problems for the Pendubot are swing-up, stabilization

Pendubot is designed for trajectory tracking, where the proper choice of the sliding functioncan easily stabilize the residual sliding mode dynamic A novel Lyapunov function is used for

a rigorous stability analysis of the controller here designed Numeric simulations verify thegood performance of the closed-loop system

In the other hand, induction motors are widely used in industrial applications due to itssimple mechanical construction, low service requirements and lower cost with respect to DCmotors that are also widely used in the industrial field Therefore, induction motors constitute

a classical test bench in the automatic control theory framework due to the fact that represents

a coupled MIMO nonlinear system, resulting in a challenging control problem It is worthmentioning that there are several works that are based on a mathematical induction motormodel that does not consider power core losses, implying that the induction motor presents

a low efficiency performance In order to achieve a high efficiency in power consumptionone must take into consideration at least the power core losses in addition to copper losses;then, to design a control law under conditions obtained when minimizing the power coreand copper losses With respect to loss model based controllers, there is a main approachfor modeling the core, as a parallel resistance In this case, the resistance is fixed in parallel

stationary reference frame, Levi et al (1995) In this work, one is compelled to design arobust controller-observer scheme, based on the super-twisting technique A novel Lyapunovfunction is used for a rigorous stability analysis In order to yield to a better performance ofinduction motors, the power core and copper losses are minimized Simulations are presented

in order to demonstrate the good performance of the proposed control strategy

The remaining structure of this chapter is as follows First, the sliding mode control will

be revisited Then, the Pendubot is introduced to develop the super-twisting controllerdesign In the following part, the induction motor model with core loss is presented, andthe super-twisting controller is designed in an effort of minimizing the power losses Finallysome comments conclude this chapter

2 Sliding mode control

The sliding mode control is a well documented control technique, and their fundamentalscan be founded in the following references, Utkin (1993), Utkin et al (1999), among others.Therefore in this section, the main features of this control technique are revisited in order tointroduce the super-twisting algorithm

The first order or classical sliding mode control is a two-step design procedure consisting

appears explicitly in ˙S), and a discontinuous control action that ensures a sliding regime or a

sliding mode When the states of the system are confined in the sliding mode, i.e., the states

of the system have reached the surface, the convergence happens in a finite-time fashion,moreover, the matched bounded perturbations are rejected From this time instant the motion

of the system is known as the sliding mode dynamic and it is insensitive to matched boundedperturbations This dynamic is commonly characterized by a reduced set of equations Atthe initial design stage, one must predict the sliding mode dynamic structure and then todesign the sliding surface in order to stabilize it It is worth mentioning that the slidingmode dynamic (commonly containing the output) is commonly asymptotically stabilized.This fact is sometimes confusing since one can expect to observe the finite-time convergence

at the output of the system, but as mentioned above the finite-time convergence occurs atthe designed surface The main disadvantage of the classical sliding mode is the chatteringphenomenon, that is characterized by small oscillations at the output of the system that canresult harmful to motion control systems The chattering can be developed due to neglectedfast dynamics and to digital implementation issues

In order to overcome the chattering phenomenon, the high-order sliding mode concept

and consists locally of Filippov trajectories The motion on the set above mentioned is said

to be discontinuous or non-existent Therefore the high-order sliding mode removes therelative-degree restriction and can practically eliminate the chattering problem

controllers are used to zero the outputs with relative degree two or to avoid chattering while

sub-optimal controller, the terminal sliding mode controllers, the twisting controller and the

super-twisting controller In particular, the twisting algorithm forces the sliding variable S

of relative degree two in to the 2-sliding set, requiring knowledge of ˙S The super-twisting algorithm does not require ˙S, but the sliding variable has relative degree one Therefore,

the super-twisting algorithm is nowadays preferable over the classical siding mode, since iteliminates the chattering phenomenon

The actual disadvantage of HOSM is that the stability proofs are based on geometricalmethods, since the Lyapunov function proposal results in a difficult task, Levant (1993) Thework presented in Moreno & Osorio (2008) proposes quadratic like Lyapunov functions forthe super-twisting controller, making possible to obtain an explicit relation for the controllerdesign parameters In the following lines this analysis will be revisited

Let us consider the following SISO nonlinear scalar system

˙

and chattering elimination is given by

u = − k1



| σ | sign(σ) +v

System (1) closed by control (2) results in



4k2+k2− k1

,Its time derivative along the solution of (3) results as follows:

q T1 =2k2+1

2k2 −1

2k1.Applying the bounds for the perturbations as given in Moreno & Osorio (2008), the expressionfor the derivative of the Lyapunov function is reduced to

In this case, if the controller gains satisfy the following relations

k1>2δ, k2> k15δk1+4δ2

2(k1−2δ),

3 STSMC for an under-actuated robotic system

In this section a super-twisting sliding mode controller for the Pendubot is designed ThePendubot is schematically shown in Figure 1

Fig 1 Schematic diagram of the Pendubot.

3.1 Mathematical model of the Pendubot

The equation of motion for the Pendubot can be described by the following generalEuler-Lagrange equation Spong & Vidyasagar (1989):

D(q)¨q+C(q, ˙q) +G(q) +F(˙q) =τ (4)

n × n inertia matrix, C(q, ˙q)is the vector of Coriolis and centripetal torques, G(q)contains the

torques For the Pendubot system, the dynamic model (4) is particularized as

l1l c2 cos q2) +I2, D22 = m2l c22 +I2, C1(q2, ˙q1, ˙q2) = − 2m2l1l c2 ˙q1˙q2sin q2− m2l1l c2 ˙q22sin q2,

C2(q2, ˙q1) =m2l1l c2 ˙q2sin q2, G1(q1, q2) =m1gl c1 cos q1+m2gl1cos q1+m2gl c2cos(q1+q2),

G2(q1, q2) = m2gl c2cos (q1+q2), F1(˙q1) = μ1˙q1, F2(˙q2) = μ2˙q2, with m1 and m2 as the

=q1q2 ˙q1 ˙q2T

where e(x, w)is output tracking error, w= (w1, w2)T , and w2as the reference signal generated

by the known exosystem (6),

are needed, but in general this is a difficult task, that it is commonly solved proposing anapproximated solution as in Ramos et al (1997) and Rivera et al (2008) Thus, one proposes

the following approximated solution forπ1(w)

π1(w) =a0+a1w1+a2w2+a3w21+a4w1w2+a5w22+a6w31

π3(w) =0.3a1w2− 0.3a2w1+0.6a3w1w2+0.3a4w2− 0.3a4w21− 0.6a5w2w1+0.9a6w21w2

+0.6a7w1w2− 0.3a7w31+0.3a8w3− 0.6a8w2w2− 0.9a9w2w1+O4( w 1) (14)

Calculating from (10) c(w ) = − p2(π(w )) − α2w2/b4(π2(w)), and using it along with (14) inequation (9) and performing a series Taylor expansion of the right hand side of this equationaround the equilibrium point(π/2, 0, 0, 0)T , then, one can find the values a i(i=0, , 9)if thecoefficients of the same monomials appearing in both side of such equation are equalized In

worth mentioning that there is a natural steady-state constraint (12) for the Pendubot (see

simulated yielding to the same results when using the approximate manifold, which is to

be expected if the motion of the pendubot is forced only along the geometric constraints

Then, the variable z=x − π(w) =z1, z2T

˙

φ(w, z) =b4(z2+π2)p2(z+π ) − ∂π4

∂w s(w) +k1(z3+π3− ∂π1

∂w s(w))+k2(z4+π4− ∂π2

∂w s(w)) +k3(b3(z2+π2)p1(z+π ) − ∂π3

∂w s(w)),

γ(w, z) =b4(z2) +π2+k3b3(z2+π2),

outlined in the previous section

For a proper choice of such constant parameters one can linearize the sliding mode dynamic

˙z1=A sm(κ)z1

equalized with the ones related with p d(s), i e., det(sI − A sm(κ)) = p d(s), in such manner

objective

80 85 90 95 100

In order to show the performance of the control methodology here proposed, simulations

due to possible measurement errors, therefore, the mass of the second link is considered as

super-twisting controller is put in evidence

4 STSMC for induction motors with core loss

4.1 Induction motor model with core loss

In this section a super-twisting sliding mode controller for the induction motor is designedfor copper and core loss minimization Now we show the nonlinear affine representation

Rivera Dominguez et al (2010):

frame by means of the following change of coordinates

The field oriented or(d, q)model of the induction motor with core loss is now shown

˙

θ ψ=n p ω+η4L m i qL m

ψ d dω

time load torque rejection The control problem will be solved in a subsequent subsection bymeans of a super-twisting sliding mode controller

4.2.1 Optimal rotor flux calculation

The copper and core losses are obtained by the corresponding resistances and currents.Therefore, the power lost in copper and core are expressed as follows:

can be considered as a cost function and then to be minimized with any desired variables, inthis case the most suitable is the rotor flux, i e.,

In order to solve the posed control problem using the super-twisting sliding mode approach,

which are the output which we want to force to zero The error tracking dynamic for the rotorvelocity results as

˙

z1=η0ψ d i q − Tl

Proposing a desired dynamic for z1of the following form

following trackng error

control law is proposed of the following form:

i q=ξ2+i qr

and when substituting it along with (25) in (24) yields to

˙

z1=k1z1+η0ψ d ξ2.Finally, collecting the equations

and that with a proper choice of k1, one can lead to z1=0.

˙

z2= − η4ψ d+η4L m i dL m − ψ˙dr, (30)

control law is proposed of the following form:

outlined in the previous section Now, from (33) one can write

i d =ξ1+i dr

and replacing it in (31) along with (32) yields to

˙

z3=k3z3+η4η2L m ξ1.Finally, collecting the equations

a classical Luemberger observer is designed

The proposed sliding mode observer for rotor fluxes and magnetization currents is proposedbased on (22) as follows:

d ˆ ψ α

dt = − η4ψˆα − N pω ˆψ β+η4L m ˆi α,Lm+ρ α ν α

d ˆ ψ β

dt = − η4ψˆβ+N pω ˆψ α+η4L m ˆi β,Lm+ρ β ν β dˆi α,Lm

dt = −( η1+η2)ˆi α,Lm+ η1

L m

ˆ

ψ α+η2i α+λ α ν α dˆi β,Lm

dt = −( η1+η2)ˆi β,Lm+ η1

L m

ˆ

ψ β+η2i β+λ β ν β dˆi α

dt = −( R s η3+η5)ˆi α − η1η3ψˆα+ (η5+η1η3L m)ˆi α,Lm+η3v α+ν α dˆi β

dt = −( R s η3+η5)ˆi β − η1η3ψˆβ+ (η5+η1η3L m)ˆi β,Lm+η3v β+ν β

inputs to the observer that will be defined in the following lines Now one defines the