Sliding Mode Control of Electric Drives

CONTROL OF ELECTRIC DRIVESUSING SLIDING MODE CONTROL

Wg Cdr D Viswanath

Guide: Dr.Dipankar Deb

09 Nov 2011

Part I

Introduction

Wg Cdr D Viswanath CONTROL OF DRIVES Nov 2011 2/ 1

Introduction

I Mathematical model of a physical system is the basisof control system design. (Model Accuracy)

I Parameters of their physical representation or theensuing mathematical model need to be measured oraccurately estimated for feedback.(Sensors/Estimators)

I Processes are dynamic and change with time. Hencesystem dynamics experience unpredictable parametervariations as the control operation goes on.(Uncertainty/Disturbance)


Introduction





Introduction





Introduction





Introduction

Examples

Figure: Process Control


Introduction

Examples

Figure: Robot Manipulation


Introduction

Examples

Figure: Steering Ship


Introduction

Examples

Figure: Aircraft Control


Introduction

ScopeI Formulation of DC Motor Speed Control Problem.

I Discussion on Sliding Mode Control and applications.

I Implementation of Sliding Mode Control in SpeedControl Problem.


Introduction





Introduction





Part II

Control of DC Electric Motors


IntroductionDirect current (DC) machines are ubiquitous in their applica-tions and find extensive use in industry. Speed control of DCmachines has been an area of interest by various researchersparticularly with respect to non linear control outlook forprecision applications and robust control. Sliding mode con-trol (SMC) is one such non linear control technique. SMCis a class of variable structure control systems which is wellknown for its robustness to uncertainties, non-linearities andexternal disturbances. SMC has been well-researched andhas found a wide variety of applications including electricdrives


Drives

Speed Control of DC Motors

θm(t)

Rm Lm

Tm(t)

Im(t)

Eemf(t)

+

-

+

-

Vm(t)

Figure: Armature Circuit of DC Motor


Dynamics of DC MotorConsider the linear time-invariant (LTI) model of a simpleDC drive control system representing the dynamics of shaftangular speed and phase current.

ω(t) = −h

Jω(t) +

kt

Ji(t) −

1

JTl(t) (1)

i(t) = −ke

lω(t) −

r

li(t) +

1

lv(t) (2)

where ω(t) is the shaft angular speed, i(t) is the phase cur-rent, v(t) is the supply voltage, Tl(t) is the load torque,h, kt, r, l, ke and J are the motor parameters. Assume thatthe load conditions and motor parameters are uncertain.


Dynamics of DC MotorThe aim of the controller is to force the actual angular speedw(t) to track a reference angular speed ω∗(t) inspite of un-certainties and load disturbances.Choosing the system states as x1 = ω(t), x2 = i(t) andu = v(t), the above dynamics can be represented as

x1 = −(a11 + ∆a11)x1 + (a12 + ∆a11)x2 + w(t) (3)

x2 = −(a21 + ∆a11)x1 − (a22x2 + ∆a11)x2 (4)

+ (b + ∆b)u

where a11 = hJ , a12 = kt

J , a21 = ke

l , a22 = rl , b = 1

l ,w(t) =

−1J Tl(t) and ∆a11,∆a12,∆a21,∆a22 and ∆b are the uncer-

tainties in motor parameters.


Dynamics of DC MotorA simple matrix representation will be

x = (A + ∆A)x + (B + ∆B)u + W (5)

Clubbing all the uncertainties together as the lumped uncer-tainty E = ∆Ax + ∆Bu + W, the above equation can berewritten as

x = Ax + Bu + E (6)


Part III

Sliding Mode Control


IntroductionFeedback control system design is based on certain specifi-cations.

CONTROL SYSTEM DESIGN

SPECIFICATIONSTIME DOMAIN

SPECIFICATIONS

FREQUENCY DOMAIN

-rise time

-settling time

-peak overshoot

-stability

-gain margin

-phase margin

under-damped

critically damped

over-damped

stable

conditionally stable

unstable

critically damped

ζ

ωn

Figure: Control System Design


IntroductionFeedback control system design is based on certain specifi-cations.

CONTROL SYSTEM DESIGN

SPECIFICATIONSTIME DOMAIN

SPECIFICATIONS

FREQUENCY DOMAIN

-rise time

-settling time

-peak overshoot

-stability

-gain margin

-phase margin

under-damped

critically damped

over-damped

stable

conditionally stable

unstable

critically damped

ζ

ωn

Figure: Control System Design


SMC

Variable structure control originated in the early 1950’s in theSoviet Union. In their pioneer works, Emelyanov and severalco-researchers considered the plant as a linear second-ordersystem modeled in phase variable form. The most importantproperty of VSC is its ability to result in very robust controlsystems; in many cases invariant control systems. Invariancemeans that the system is completely insensitive to paramet-ric uncertainty and external disturbances. Successful resultshave been reported in terms of eliminating disturbances, ad-dressing nonlinearities, and achieving acceptable control inthe presence of modeling errors.


SMC

Variable structure control (VSC) results in high performancesystems that are robust to parameter uncertainties and noise.Design of such systems includes two steps:-

(a) Choosing a set of switching surfaces that representsome sort of a desired motion and

(b) Designing a discontinuous control law that guaranteesthe attractiveness of the switching surfaces andensures convergence to the switching surfaces.


SMC

Sliding Mode Control (SMC) is best explained with a sec-ond order system whose phase plane plot clearly shows theresponse of the system when in sliding mode. Consider asystem represented by

x1 = x2 (7)

x2 = −a1x1 − a2x2 + bu

where a1 and a2 and b are parameters that are not exactlyknown but some knowledge about their range is known i.e.,

ai < ai < ai

where i = 1, 2, and

0 < b < b < b < 1


SMC

The control is to be designed such that the system trajectoryis attracted to the sigma line and once the trajectory inter-sects it, keep switching the control to keep the trajectory onthe sigma line so that it does not leave it. As a result aftera certain finite time, the system is governed by the equationof the sigma line only and not on the system equation. Thusthe dynamics will be independent of the system parameters.


SMC

One such trajectory is shown in the phase plane plot of figurebelow.

Figure: Sliding Mode Control Phase Plot


SMC

Along the curve PQ the system is affected by system equa-tion . This is called the reaching phase. Along the line QOthe system is affected by sliding mode equation .

σ = c1x1 + x2 = 0 (8)

This is called the sliding mode (SM) phase. The control ini-tially applied in equation helps the system trajectory to movetowards the sigma line. Thereafter the control is switchedso that the trajectory does not leave the sigma line.


SMC

The switching action can be explained as follows: -

(a) When the trajectory (consider curve PQ) crosses thesigma line, sigma becomes greater than zero. We nowuse a control that makes the rate of change of sigmabecome less than zero so that the trajectory is broughtback to the sigma line.

(b) If now the trajectory crosses the sigma line again butfrom the opposite direction, sigma becomes less thanzero. We use a control that makes the rate of changeof sigma to become greater than zero so that thetrajectory is again brought back to the sigma line.


Part IV

Simulation Results


Initial DataThe reference input um is chosen as a square waveform ofamplitude 100 V with a frequency of 0.1 Hz. The load torqueapplied is 5 N−m which is represented in the E matrix. Thesliding surface is chosen as S = [4 1]T. The adaptationgains are chosen as λ0 = 0.01 and λ1 = 0.01.


Error PlotFigure ?? shows that the error between the reference andplant states goes to zero inspite of a constant load torqueof 5N − m applied..

Figure: Error


Figure ?? shows that the plant state x1 which represents thespeed follows that of the reference model.

Figure: Reference and Plant Output States x1.


Figure ?? shows that the plant state x2 which represents thephasing current follows that of the reference model.

Figure: Reference and Plant Output States x2.


QUESTIONS PLEASE?


THANK YOU


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Sliding Mode Control of Electric Drives