sliding mode control

PRINCETON/CENTRAL JERSEY SECTION OF IEEE CIRCUITS AND SYSTEMS

Sliding Mode Control with Industrial ApplicationsWu-Chung Su, Ph. D.Department of Electrical Engineering National Chung Hsing University,Taiwan, R. O. C. Department of Electrical and Computer Engineering Rutgers, The State University of New Jersey, U. S. A.Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

2008/11/17

1

PreludeA Control Engineer as an Archer

to drive the state to the target.

2008/11/17

Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

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The Old Oil-PeddlerOuyang Show(1007~1072A.D.) http://baike.baidu.com/ 2008/11/17

Princeton/Central Jersey Section of IEEE- Circuits and Systems Chapter Meeting

3

Sliding Mode: an illustration

to fill oil into a bottle through a funnel2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 4

A funnel-like domain x1 = x2 System: x2 = 2 x1 3 x2 + u Target: ( x1 , x2 ) = (0, 0)Funnel: x2 = x1 New target: s = x1 + x2 = 02008/11/17

x2

x1 s=05


A Physical ExampleCoulomb Frictionm dv = K sgn(v) + f a dt f a : applied force

For example, K = 3, f a = 2 dv = 3 2 < 0 v dt dv v < 0 m = 3 2 > 0 v dt v >0m

http://www.physics4kids.com

sliding mode: v 0 (if f a < K ).Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

2008/11/17

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Contents (1/2)Introduction to Sliding Mode Variable Structure Systems Invariance Condition (Matching Condition)

Variable Structure Systems (VSS) Design Sliding Surface Design Discontinuous Control

Discrete-Time Sliding Mode O(T ) v.s. O(T 2 ) Boundary Layer Switching v.s. Continuous Control2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 7

Contents (2/2)Minimum-Time Torque Control (SRM) Temperature Control (Plastic Extrusion P.) Rod-less Pneumatic Cylinder Servo Wireless Network Power Control Singular Perturbations Distributed Parameter SystemsPrinceton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

2008/11/17

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A funnel-like domain (cont.)disturbance x1 = x2 System: x2 x2 = 2 x1 3 x2 + u + f (t ) Target: ( x1 , x2 ) = (0, 0) Funnel: x2 = x1 New target: s = x1 + x2 = 02008/11/17

x1 s=09


Parameter x1 = x2 variations System: x2 = 2 x1 3 x2 + u + (1 x1 + 2 x2 ) x2 Target: ( x1 , x2 ) = (0, 0) Funnel: x2 = x1 New target: s = x1 + x2 = 02008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

A funnel-like domain (cont.)

x1 s=010

Introduction to Sliding mode - Variable Structure Systems x1 = x2 System : x2 = ax2 bu(K. D.Young, 1978)

Switching line : s = cx1 + x2 = 0 (Sliding Surface) x1 , if x1s > 0 (region I) Control : u = x1 , if x1s < 0 (rigion II)2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 11

Introduction to Sliding mode - Variable Structure Systems1 0 Structure I : x = x b a s 2 as + b = 0 : spiral out 0 1 Structure II : x = x b a s 2 as b = 0 : saddle point

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Introduction to Sliding mode - Invariance ConditionQ: Can the control law possibly reject the disturbance out of the system?

0 2 2 1 1 u + 1 f (t ) x = Ax + 2 1 3 0 2 2 = Ax + 1 u1 + 1 u2 + 1 f (t ) 3 2 1 2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 13

Introduction to Sliding mode - Invariance Condition 0 2 Controlsubspace: span 1 , 1 2 1 2 Disturbancesubspace: span 1 3 the disturbance subspace.2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 14

*Thedisturbancecanberejectedbythecontrollawif (andonlyif)the control subspace covers

Introduction to Sliding mode - Invariance ConditionInthisexample,2 0 2 1 = 2 1 + 1 3 2 1 u1 = v1 2 f (t ) u2 = v2 f (t )

.

d (t ) Ifisknown,thenthecontrollaw

Willleadto0 2 0 2 2 0 2 x = Ax + 1 v1 + 1 v2 + ( 2 1 1 ) f (t ) + 1 f (t ) = Ax + 1 1 v 2 1 2 1 3 2 1 2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 15

Introduction to Sliding mode - Invariance Condition(Drazenovic, 1969): The system with disturbance is invariant off (t ) in

x = Ax + Bu + Ef (t )sliding modes0

rank [ B | E ] = rank [ B ] .

iff

In the previous example,0 2 rank 1 1 2 1 2 0 2 1 = rank 1 1 = 2. 2 1 3 2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 16

A funnel-like domain (revisit)disturbance x1 = x2 System: x2 x2 = 2 x1 3 x2 + u + f (t ) Target: ( x1 , x2 ) = (0, 0) Funnel: x2 = x1 New target: s = x1 + x2 = 02008/11/17

x1 s=017


Parameter x1 = x2 variations System: x2 = 2 x1 3 x2 + u + (1 x1 + 2 x2 ) x2 Target: ( x1 , x2 ) = (0, 0) Funnel: x2 = x1 New target: s = x1 + x2 = 02008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

A funnel-like domain (revisit)

x1 s=018

VSS Design x1 = x2 System : x2 = ax2 bu(K. D.Young, 1978)

Switching line : s = cx1 + x2 = 0 (Sliding Surface) x1 , if x1s > 0 (region I) Control : u = x1 , if x1s < 0 (rigion II)2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 19

VSS DesignExistence of sliding mode i. Reaching condition (sufficient, global)s < sgn( s), > 0

ii. Sliding condition (sufficient, local)x(0)s 0

lim s < 0, +u+

s 0

lim s > 0

Sliding Surface: s = 0

u

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VSS DesignFinite time reaching: s < sgn( s), > 0Lyapunov functionV = s2 dV = 2 ss = 2 s < 0 dt s( x) 0

Reaching time

T 0 satisfied if f (t ) < f max .2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 23

VSS DesignTwo steps in VSS design1. 2.

x = f ( x, u, t ) s( x ) = 0

Choose sliding surface Design discontinuous control to render sliding mode f ( x, u + , t ) = f + , s > 0 x= f ( x, u , t ) = f , s < 0 2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 24

VSS Design - Sliding Surfacex = A x + B u + D f (t )

Eigenspace Approach v.s. Lyapunov ApproachNonsingular transformation Canonical form x1 A11 x = A 2 21

Stabilizing feedback Lyapunov functionPAs + AsT P = Q x = A x B K x + D f (t )

0( n m )m x1 TB = , Tx = x B2 2A12 x1 0 0 + u + f (t ) A22 x2 B2 D2 I mm ] Tx

Sliding surface

x2 = Kx1 , or s( x ) = [ K

Sliding surface2008/11/17

1 T V ( x ) = x Px 2s ( x ) = BT Px25


VSS Design - Discontinuous Controlx = A x + B u + D f (t ) slid in g su rfa ce: s ( x ) = C x = 0 s = C A x + C B u + C D f (t )

Signum function(Single-input case)u = (CB) 1 CAx (CB) 1 ( + d max ) sgn( s )

v.s. Unit Control(Multi-input case)u = (CB) 1 CAx (CB) 1 (d max + ) 2008/11/17

s s26


Discrete-Time Sliding ModeContinuous-timex = Ax + Bu + Df (t ) Sliding surface: s (t ) = Cx (t ) = 0

Sampled-dataxk +1 = xk + uk + d k Sliding surface: sk = Cxk = 0AT T 0 ( k +1)T

= e , = e A d B,T

dk =

kT

e A(t ) Df ( )d = e A Df ((k + 1)T )d 02008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 27

Discrete-Time Sliding Mode = e A d B, d k = e A Df ((k + 1)T )d T

xk +1 = xk + uk + d kT 0

Issues on disturbance rejection: 1. Matching condition fails d k range() However, d k = f (kT ) + O(T 2 ) 2. Non-causal disturbances However, d k = d k 1 + O(T 2 )d k 1 = xk xk 1 uk 1

0

Ultimate achievable accuracy:O(T 2 )2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 28

Discrete-Time Sliding ModeQuasi-sliding mode sk = O(T ), s (t ) = O(T )Discontinuous control

sk = 0, s (t ) = O(T 2 ) Discrete-time sliding modeContinuous control

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Discrete-Time Sliding ModeUkins discrete-time equivalent control (Continuous control law)sk +1 = C xk + C uk + Cd k = 0eq k 1

u = (C ) C (xk + d k )

Discrete-time O(T 2 ) sliding mode control (Modification of ukeq )uk = (C ) 1 C (xk + d k 1 ) sk +1 = C ( d k d k 1 ) = O (T 2 )2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 30

Contents (1/2)Introduction to Sliding Mode Variable Structure Systems Invariance Condition (Matching Condition)

Variable Structure Systems (VSS) Design Sliding Surface Design Discontinuous Control

Discrete-Time Sliding Mode O(T ) v.s. O(T 2 ) Boundary Layer Switching v.s. Continuous Control2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 31


2008/11/17

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Switched Reluctance Motors

Rotor2008/11/17

StatorPrinceton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 33

Switched Reluctance MotorsA A1 2 3 4 6 5

4-phase, 8/6-pole SRM2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 34

Switched Reluctance Motors Stator Rotor

reluctance force

Magnetic flux

Reluctance force

L ( )

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Switched Reluctance MotorsR aL

(m H ) La 30

Lb

Lc

Ld

a'

10 7 .5 1 5 2 2 .5 3 0 3 7 .5 4 5 5 2 .5 6 0 (D eg ree)

V j = L j ( )

di j dt

+

dL j ( ) dt2

i j + R j i j , j = A, B, C , D

1 dL j ( ) 2 Torque i : T j = ij 2 d2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 36

Minimum-Time Torque ControlVdc Q1 Sensor 1 A C B Sensor 2 D Q3

System dynamics: di = A( , )i + B ( )V (t ) dtSliding surface: s = i ir = 0Minimum-Time VSC: Vdc , if i (t ) > ir V (t ) = Vdc , if i (t ) < irV* = K i (t 0 ) ir

Q2

Q4

Drive Circuit

CPLD XC9536

Drive

encoder current sensor

V * = Ki (t 0 )

i

Control System Setup2008/11/17


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Minimum-Time Torque Control

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Temperature Control - A Plastic Extrusion Process

System setup: ITRI (Industrial Technology Research Institute), Taiwan, R.O.C.Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

2008/11/17

39

Temperature Control - A Plastic Extrusion ProcessDynamics of Channel temperature:dy hA hA 1 y+ y + w = dt cV cV cV

Rate of change of energy input:

a + ( w, u , t ), u 0 dw = a ( w, u , t ) = dt a ( w, u , t ), u < 0

Single-channel dynamics:

y = my + v a ( w, u , t ) v= + cV2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 40

Temperature Control - A Plastic Extrusion ProcessSingle-channel dynamics:

y = my + v

a ( w, u, t ) v= + cVSliding surface (PI control):v = K P e K I e or V ( s ) = ( K P + KI )( ydesired ( s ) Y ( s )) s

Output feedback sliding mode controls = e + ( K P + m)e + K I e sk a1sk 1 + b0 ek + b1ek 12008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 41

Temperature Control - A Plastic Extrusion Process heat, sk < 0 Switching control law: uk = water, sk > 0

2008/11/17


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Rod-less Pneumatic Cylinder Servo

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Rod-less Pneumatic Cylinder ServoEquation of motion: M LY = uY c sgn Y + AT P Pressure dynamics: P = 1 ( P , P2 , Y , X ) X + 2 ( P , P2 , Y , Y , X ) 1 1

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Rod-less Pneumatic Cylinder ServoSliding surface: M 1 = P L Yd + k1 (Yd Y ) + k2 (Yd Y ) + uY + c sgn Y A AT T

Variable Structure Control: X = X v sgn( )

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Wireless Network Power Control

Zoran Gajic, Plenary Lecture 2007 CACS International Automatic Control Conference National Chung Hsing University, Taichung, Taiwan, November 9-11, 20072008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 46

,

Wireless Network Power ControlSignal-to-interference ratio (SIR) for the ith usergii (t ) pi (t ) gii (t ) pi (t ) i (t ) = = , i = 1, 2, I i (t ) gij (t ) p j (t ) + i (t )j i

, N.

Control objective: i (t ) i

tar

Logarithmic scale representation

i = gii + pi I i ( pi )

System dynamics:

pi (t ) = ui (t ), i = 1, 2,

N47

i (t ) = pi (t ) + fi (t )2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

Wireless Network Power ControlSliding surface si (t ) = i (t ) Discrete-time representationtar i

=0

si (k + 1) = si (k ) + Tui (k ) + d (k ) + v(k )

Discrete-time SMC (continuous) Stability analysis 1 si (k + 1) u (k + 1) = 1 i Tz 1 det 1 T

1 1 ui (k ) = si (k ) (d (k 1) + v(k 1)) T TT si (k ) 1 1 (d (k ) + v(k )) 1 ui (k ) 2 T Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 48

T = z2 = 0 z + 1 2008/11/17


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Extended Research ProblemsSingular Perturbationsx(0)

u+

uSingularly Perturbed Systems Quasi-steady state (slow mode) Fast mode Continuous control Infinite-time reaching (boundary layer)

Sliding Surface: s = 0VSS Sliding mode Reaching phase Discontinuous control Finite-time reaching

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Extended Research ProblemsDistributed Parameter SystemsHeat Conduction System : U t ( x, t ) = U xx ( x, t ) + U ( x, t ) Boundary conditions U (0, t ) = 0 U (l , t ) = Q(t ) + f (t )x=0x=l

Kernel function k ( x, y )

wt = wxx cwU ( x, t )

w( x, t )

w(0, t ) = 0 w(l , t ) = Q(t ) + f w (t )Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 51

2008/11/17

Extended Research ProblemsDistributed Parameter SystemsSliding surface: S (t ) = x (l , t ) = 0Kernel function : k ( x, y ) = y I1x=0

x=l

(

( x2 y 2 ) ( x2 y 2 )

)

wt = wxx cwU ( x, t )

w( x, t )

w(0, t ) = 0 w(l , t ) = Q(t ) + f w (t )Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

2008/11/17

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Extended Research ProblemsDistributed Parameter SystemsSliding surface: S (t ) = wx (l , t ) = 0S (t ) = U x (l , t ) k (l , l )U (l , t ) k x (l , y )U ( y, t ) dy0 l

Sliding Mode Control:Q(t ) = K m sgn( S ( ))d0

t

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ConclusionsMerits of VSS Robustness (against disturbances and system parameter variations) Reduced-order system design Simple control structure Fits switching power electronics controlPrinceton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting

2008/11/17

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ConclusionsChattering Reduction (Elimination) Discrete-Time Sliding Mode Filtering methods

Interesting Problems Output feedback Singular Perturbations Infinite-dimensional systems Systems with delays (e.g. wireless network)2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 55

Thank You!

http://www.chneic.sh.cn/2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 56

sliding mode control

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