a comparison between linear quadratic control and sliding mode control

8/3/2019 A Comparison Between Linear Quadratic Control and Sliding Mode Control

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A C O M PA R I SO N B E T W E EN L I N E A R Q U A D R A T I C C ON T R OLA N D SL I D I N G M O D E C O N T R O L

ANWER S. BASHI

A bst r ac t . Sli ding mode cont rol is quickly becoming a popular research eld

due t o t he several favorable qualit ies, including robust ness. Lin ear quadrati c

control has been one of the more popular and more traditional control tech-

niques, partially due to the ease of implementation and its optimality quality.

In this paper, we provide an introduction to sliding mode control. A tech-

nique is suggest ed t o allow gradual int roduction of a reference signal t o t he

sliding plane.Finally, an inverted pendulum system is used as an example to compare

linear quadratic regulation with sliding mode control.

Part 1. Introduction

T he majori ty of t his paper is dedicated t o introducing sliding mode control andapplying it to the invert ed pendulum problem. While the t it le indicat es no prefer-ence between linear quadratic regulation (LQR) and sliding mode control (SMC),t here are numerous volumes of mat eri al available on linear quadrati c regulation.Most of these are presented in a thorough and easy-t o-digest form. For t his reason,lit t le t ime has been spent discussing LQR.

Conversely, most sliding mode lit erature is most ly in t he form of journal art icles

and conference proceedings, and t herefore quit e diverse in approach or nomencla-t ure. Here, an att empt has been made t o capture t he general int ent of SMC, event hough t he li mit ed space does not all ow an exhaust ive r eport on avail able SMCresearch and ndi ngs.

There is st ill a great deal t o be done in SMC, and it seems that it will be a fert ileresearch area for many years t o come.

Part 2. T he Linear Quadratic Regulat or (LQR )

T he linear quadratic regulat or is an opti mal and robust t echnique for MI MOcontrol. Sources t hat discuss LQR in det ail are available in [1] and [2]. In t hisreport , we wil l primari ly discuss t ime invari ant li near quadrati c contr ol, howevert his will be briey deri ved from t he t ime-varying case.

1. T ime-V ar y in g L inear Opt imal Cont r o l

Given t he li near t ime-invari ant syst em,

_x (t) = Ax (t) + B u (t)(1)

Date: 12/ 1/ 97.

1


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2 A NW ER S. B A SHI

we wish t o minimize t he performance index given by

J =1

2

xT (T) S (T) x (T) +1

2

Z T

t 0

hxT (t) Qx (t) + u (t)T Ru (t)

idt(2)

where T denot es t he nal t ime for t he contr ol session . T he linear soluti on t hatminimizes t his index is given by some linear funct ion of t he st at es,

u (t) = K (t) x (t)(3)

where

K (t) = R 1B T S (t)(4)

_S (t) = AT S (t) + S (t) A S (t) B R 1B T S (t) + Q(5)

If S (T) is given, t hen t he Riccati equat ion ( Eq. 5) can be solved backwards intime from ti me T to time t T . T his equat ion, as well as t he time-varying gain

(Eq. 4) must be solved o-line and st ored. In t he eld, a contr oll er would uset hesest ored gains t o contr ol t he syst em.

2. T ime-I nvar iant L inear Opt imal Con t r o l

T he gains produced in a t ime-varying contr oll er generally have a long inert period, where t he gain remains const ant or near const ant. T his i s followed bya short act ive peri od where t he gains change, falling o t o zero as the contr olsession ends. T his t ime-varyi ng gain can be approximat ed simply by assuming t hatt he contr oll er wil l be in operation indenit ely, and t herefore t he contr oller i s alwaysin t he inert mode. T his is a r easonable assumpti on and gives rise to t he linearquadrati c regulator. T he st eady-st ate version of t he Riccat i equat ion i s call ed t healgebraic Ri ccat i equat ion. It can be found by assuming steady st ate condit ionshave been reached and set t ing t he rate of change of S to zero in Eq. 5. This give

us

0 = AT S + SA SB R 1B T S + Q(6)

T here is generally no analyt ical solut ion t o the algebraic Riccat i equat ion, how-ever most engineeri ng mathemat ics software packages i nclude a function t o nd anumeri cal solut ion. Since S appears quadrati call y in t he equat ion, t here are severalpossible soluti ons. T he posit ive denit e solut ion is chosen for t he calculation of t heoptimal gain

K = R 1B T S(7)

3. Ro bust ness Qua l i t ies

T he LQR is, i n general, a robust contr ol mechanism. If we assume that Q and

R are symmet ri c, t hen we can be assured ([2]) t hat t he minimum singular valuesof t he closed-loop syst em satisfy t he following two inequali t ies:

mi n [I + GC L (j w)] 1(8)

mi nh

I + GC L (j w) 1

i

12

(9)

where

GC L = K (sI A) 1 B(10)


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A C O M PA R I SO N B E T W E E N L I N E A R Q U A D R A T I C C O N T R OL A N D SL I D I N G M O D E C O N T R OL3

T his is indicative of several ext remely useful robust ness propert ies:

Upward gain margin i s innit e. T his means t hat t he syst em i s st able for anyscalar p ert urbati on of t he syst em inputs, 1 <

i

< 1 . Downward gain margin is at l east 12 . This means that t he system is stable for

any scalar pert urbati on of t he syst em i nputs, 12 < i < 1. Phase margin is at least 60o. T his means that t he syst em i s st able for any

scalar phase pert urbati on of t he syst em inputs, 60o < i < 60o.

In eect, if a dist urbance causes t he i t h input t o the syst em, ui to be of t heform

udi = i ej

i ui(11)

where12

< i < 1(12)

60o < i < 60o(13)

t hen t he LQR control law guarant ees st abilit y.

3.0.1. Note on Symmetr y. T he requirement t hat a mat ri x M is symmetric canalways be sat ised for any quadrati c functi on of t hat mat ri x

xT M x(14)

We can rewrite the matri x,

M =12

(M + M )(15)

so that

xT M x = xT12

(M + M ) x =12

xT M x + xT M x

(16)

Since xT M x is a scalar,

xT

M x =

xT

M x

T

(17)= xT M T x(18)

t herefore,

xT M x = xT

12

M + M T

x(19)

where 12M + M T

is obviously symmetric.

Part 3. Sliding M ode Contr ol (SM C)

Whil e LQ cont rol uses a single linear cont rol law t o minimize some performanceindex, sliding mode contr ol (similar t o gain scheduling) uses more t han one and i s,in general, non-linear. T he performance index is specied as a manifold of space

call ed t he sli ding surface. A sli ding mode contr oll er sends t he syst em st at es ontot he sli ding surface and keeps t hem t here. T he name sli ding mode comes fr om t heslightly inane realization that, once the system states are on the sliding surface,t hen t he syst em can be considered t o be in sliding mode.

Sliding mode control was ori ginally developed i n t he Soviet Union. A survey ofearly li t erature can be found in [3]. Recentl y, papers are emerging from a moreeclecti c group of contr ols scienti st s. A survey of current li t erature can be foundin [4]. T his new research has rejuvenat ed and, in a way, popularized t he not ion


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Table 1: A survey of publications indicates an increase of interest in SMC.

Years A pprox. N umber of Publications

94-97 1000

90-93 60086-89 100

of sliding mode control as can be seen from a tally of publications (shown in Ta-ble 1fa). The survey was conducted by searching t he on-line engineering cat alogsCOMPENDEX and COMPENDEX PLUS for ( sli ding mode OR variable st ruc-t ure ). Some publicat ions were no doubt missed, however t he result s do seem t obe indicat ive of an increasing i nterest in slidi ng mode control.

4. T he Sl id ing Sur f ace

T he slidi ng surface is generall y a pre-specied li near manifold, however someformulat ions of t he SMC problem allow for adaptive [5, 6] or non-linear [7] sliding

surfaces.Axiom 4.1. Consider two hyper-t ubes, tube and " tube, of diameters > 0," > 0. Their central axes run in the direction of t he s = 0 axis. I n general, asli ding surface can be found to exist i f, barr ing disturbance, for any x (0) the statetrajectory cannot leave the " tube after having entered i nto the tube. Usually " , however this is not always necessary.

In other words, once the state trajectory comes within of the s = 0 axis, itcannot escape t o any distance greater than " unless it is pert urbed by a dist urbance.

We will consider here only li near sli ding surfaces as t hese are generally adequat efor most purposes. Any n-dimensional l inear manifold can be expressed as

s = cT x(20)

where c is t he cost associated wit h each corr esponding stat e in x.5. Sl i d i n g M o d e C o n t r o l B a s ed o n A c k e r ma n n s G a i n

Ackermann and Ut kin (one of t he ori ginal developers and proponents of slidingmodecont rol) have proposed a t echniquet o aut omat e sli ding modedesign [8]. A ck-ermanns formula i s used t o choose t he sli ding surface based on a desired feedbackspectr um. The design procedure has been summari zed i n [8] as follows:

Choose t he desir ed feedback spectr um, f 1; : : : ; n g Obtain t he li near feedback contr ol, ua = kT x from Ackermanns formula,

kT = eT P (A)(21)

where

e

T

= (0; : : : ; 0; 1)

B ; A B ; : : : ; A

n 1

B

1

(22) P ( ) = ( 1) ( 2) : : : ( n )(23)

Design t he dynamic part of t he cont roller,

_z = B T Ax B T B

ua , z (0) = B T x (0)(24)

T here i s no clear explanat ion given concerning t he reason for adding t his


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dynamic subsyst em, however it may be seen as an augmentation t o t he orig-inal system that simpl ies control on the sli ding surface.

T he sliding surface equat ion i s found,

s = B T x + z(25)

Finally, t he discont inuous control i s designed,

u =

M (x; t) s > 0M (x; t) s < 0

(26)

where, t o ensure t he existence of a sli ding surface, we require

M (x; t) > j uaj + f 0 (x; t)(27)

T his i nequali ty sati ses t he exist ence axiom since we requir e our control t o

be great er t han any known input t o the plant as well as any dynamic t rajec-t ory deviati on t hat t he plant might init iate it self. Not e that a large enoughdist urbance may violate t his inequality and cause a st able syst em t o becomeunst able, i.e. if

j udj M (x; t) j uaj f 0 (x; t)(28)

t hen inst abilit y may result .

Example 5.1. Design a sli ding mode control using Ackermanns formula for thesystem

_x1_x2

=

1 23 0

x1x2

+

1 1

u(29)

x (0) = [0; 0]T(30)

with closed loop poles at [ 8; 6].

Looking at the eigenvalues of A, we can see that the system has poles at [3; 2]and so is inherently unstable.Using Ackermanns formula, we get

k =

5742

(31)

which allows us to design the dynamic subsystem,

_z = [1; 1]

1 2

3 0

x1

x2

[1; 1]

1

1

[57; 42]

x1

x2

(32)

_z (0) = [1; 1]

00

= 0(33)

which simpli es to

_z = [116; 82]

x1x2

(34)


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6. Sl id ing M ode Co nt r o l B ased o n Cano nica l T r ansf or mat ion

We can model t he syst em

_x (t) = Ax (t) + B u (t)by an equivalent system

_z (t) = TAT 1z (t) + TB u (t)

where T is a change of basis chosen such t hat t he equivalent system i s in controlcanonical form, i.e.

TAT 1 =

2

666664

a1 a2 an 1 an1 0 0 00 1 0 0...

.... . .

......

0 0 1 0

3

777775

(35)

TB =

2

66664

10000

3

77775

(36)

We may t hen choose t he sliding surface as in [9] t o be

s = Gt z

where t he rate of convergence on t he slidi ng surface is specied by t he choice ofGt . Gt contains t he coe cients of t he polynomial wit h t he desired convergencespectrum of t he equivalent syst em t o t he sli ding surface,

Gt = [v1; v2; : : : ; vn ](37)

where v1; v2; : : : ; vn can be found by using t hebinomial t heorem [10] for an (n 1)t h

polynomial,

v1zn 1 + v2zn 2 + : : : + vn = (z )n 1

(38)

Since t his spectrum is wit h respect t o t he t ransformed syst em, we can also t rans-form t he basis of t he sli ding surface,

s = Gt z = Gt Tx = Gx

T he contr ol l aw t hen becomes [11]

u (t) = (K eq + K sw ) x (t)

where

K eq = (GB ) 1 G (A I )(39)

K sw;i =

M (x; t) GBsx i > 0M (x; t) GBsx i < 0

(40)

M (x; t) is usually chosen t o be t he maximum all owable input t o try t o meet t herequir ement t hat

M (x; t) > j uj + f 0 (x; t)(41)


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Example 6.1. Design a sli ding mode controller using canonical transformati on forthe system

_x1_x2

=

1 23 0

x1x2

+

1 1

u(42)

x (0) = [0; 0]T(43)

with closed loop poles at [ 8].

T he T required to convert this system to control canonical form can be found t obe

T =

1 2 1 4

We may now nd G by solving the polynomial equation and transforming by T ,

(z + 8) (z + 6) = z2 + 14z + 48(44)

Gt = [8; 1](45)G = Gt T = [7; 12](46)

T herefore, the sli ding mode controller is dened by:

u (t) = (K eq + K sw ) x (t)(47)

K eq = (GB ) 1 G (A I ) = [ 1:895; 1:3684](48)

K sw;i =

M (x; t) (133x1 128x2) x i > 0M (x; t) (133x1 128x2) x i < 0

(49)

7. Co nt inuous S l id ing M ode Con t r o l

Sliding mode contr ol requires an innit ely f ast controller t o insure it s robust nessand dist urbance reject ion charact eri st ics. In pract ice, t his i s of course impossibleand i s replace wit h t he highest frequency t he cont roller can manage. T he number oft ransit ions required becomes especially bad once t he syst em is on the sli ding mode.Let us assume t hat t he syst em is at s = " 1 where " k > 0 is some small real number.T he controller wi ll respond by blast ing t he syst em wit h a short bang unt il itreaches s = " 2. Since the syst em is now below t he sli ding surface, t he contr ollerwill blast it the ot her way, to s = " 3, and so on. What we end up wit h i s aneect call ed chat t eri ng, where t he cont roll er is swit ching at very high fr equencies.Needless t o say, t his i s undesirable for most systems since chat t eri ng induces wearin t he control element and could also cause undesir able high-frequency dynamicsin t he syst em t o surface.

Replacing the ideal swit ching charact eri st ics,

u =

M (x; t) s > 0M (x; t) s < 0

with a dead zone,

u =

8 "0 j sj "M (x; t) s < "

9=

;


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reduces t he chat t er somewhat . However, if t his approximat ion is made, t hen t hest at es can no longer be expected to converge t o t he sliding mode, but only t o wit hinsome hyper-t ube centered on t he sli ding mode. T his t ube may be made narrower

by making " small er, however t his comes at t he expense of requiring more contr olleraction.

A vari able sli ding gain M (x; t) is suggest ed in [12] t hat decreases t he controlact ion as t he st at e t raject ory approaches t he sliding surface. T his has been t ermedconti nuous sli ding mode contr ol, since t he contr ol act ions t hus generated are dif-ferenti able near t he sli ding surface. In [9], t he foll owing heurist ic is chosen fordampening controller action:

M (x; t) =M j sj

k xk + "

where M is t he init ial sliding gain value, and " is a small posit ive const ant.

Example 7.1. Dampen the control ler generated in Example 6.1.

We may arbitrarily set " = 0:1. I f simulati on shows that a di erent value of "provides better result s, t hen " is changed appropriately. Assuming our system canprovide 10V input, our control acti on becomes

M (x; t) =10j 133x1 128x2j

px21 + x

22 + 0:1

(50)

which is implemented in the sliding mode controller,

u (t) = (K eq + K sw ) x (t)(51)

K eq = [ 1:895; 1:3684](52)

K sw;i =

8 010j 133x 1 128x 2jp

x 21

+ x 22

+ 0:1(133x1 128x2) x i < 0

9=

;(53)

8. O t h er F o r ms o f Sl id i n g M o d e C o n t r o l

Since SMC is st il l a developing eld, each researcher enteri ng t his eld bri ngswit h him or her expert ise from anot her eld. For t his reason, t here are manydierent approaches to t he SMC problem. A few of t he more common approachesare noted here.

8.1. Quadratic Control. In [13], Ut kin suggest s a quadrati c cost for SMC para-met er design identi cal t o the LQR cost function,

J = 12

xT (T) F (T) x (T) + 12

Z T

t 0

xT (t) Qx (t) + u (t)T Ru (t) dt(54)

T his proposed cost , due t o di culty of adaptati on t o general SMC has notbecome very popular.

8.2. PI D Co n t ro l . In [14], a proport ional plus deri vat ive (PD) SMC controlleris proposed which is used as t he basis for automat ic opti mizat ion wit h genet ic


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algori t hms in [15]. T he general st ructure optimized in [15] makes use of a hard-swit ched proport ional plus int egral plus deri vat ive sli ding mode cont roller (PI D-SMC) contr oll er such as t hat given by

u = P e IZ

edt Ddedt

where

=

1 s < 02 s > 0

P =

1 es < 0 2 es > 0

I =

1 s < 02 s > 0

D =

1 _es < 02 _es > 0

with e representi ng t he (negative) t racking err or. T his development allows a sim-pler t ransit ion t o SMC for pract icing engineers t hat are already familiar wit h PI Dcontrol law.

8.3. Fuzzy Control. Several papers have pr oposed fuzzy logic SMC controll ers,including [16] and [17]. Non-lineari t ies are dealt wit h very well by ut il izing fuzzylogic control and control chatt er is done away wit h, however t he design pr ocedurebecomes signicantly more complicated. In [18], genet ic algori t hms are employedquit e successful ly t o design near-opti mal fuzzy controll ers for non-li near systems.

9. Ref er ence I nput

We have not yet discussed t he introduct ion of a r eference input t o t he SMC.

T his is generally an easy t ask due to t he nat ure of t he sliding surface. Basicall y,t he sli ding surface can be dened such t hat

s0 = cT (x xd)(55)

where xd is t hedesir ed syst em st at es. T heint roduct ion of a reference input, howevermay invali dat e t he convergence propert ies of t he controlled developed sliding modecontroller sincewe are intr oducing dynamicsinto t he sliding surface t hat cannot , ingeneral, be modeled wit hout predening t he reference input . Consider producing asliding mode controller for t he syst em

_x1_x2

=

1 23 0

x1x2

+

1 1

u(56)

with closed loop poles at 8. Our choice of poles is based on t he fact t hat t hesystem has poles at [3; 2]. If t he syst em is in sliding mode wit h respect t o s0

and t he reference st ates are suddenly changed, t hen t he controll er experiences ahigh-frequency t ransient t hat may not have been modeled. If t he cont roller wasnot designed t o deal wit h such high frequency t ransient s, t he st abili ty r equirementthat

M (x; t) > j uaj + f 0 (x; t)(57)

may be t emporari ly violated. For t his reason, considerat ion of t he reference signal isrequired when designing the controller. A t echnique for condit ioning the reference


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signal such t hat st abil it y is maintained is suggest ed below. T he intr oduction of areference input is a subtopic of SMC t hat has received very li t t le recognit ion.

9.1. Slew-Lim it ing Pr elt er t o M aintain St abilit y. The basic problem canbe considered t o be one of sliding surface adaptation. By using a reference input incalculati ng t he sliding surface, we eecti vely disrupt normal sli ding mode operationby changing t he surface as t he st at es t ravel in t he sliding t raject ory. T his is notnormally a problem unless we change t he surface fast er t han the SMC can foll ow it .For t his reason, we propose a slewrate l imi t ing lt er t o pr eprocess t he referenceinput such t hat sliding surface does not drift t oo quickly for st able control.

We wish t o l imi t high frequency surface changes, so we can create a low passlt er in t he frequency domain,

R0i (s) =pNi

(s + pi )N Ri (s)(58)

where N is t he order and is chosen t o ensure a st eep f requency roll -o. Choosing a

high order for N may be necessary since increasing t he reference input s amplit udewil l also increase t he high f requency components. If t he lt er order is higher,and t herefore more l ike t he ideal brick-wall lt er, t hen t he references amplit udebecomes less signi cant .

A good choice of t he pole, pi might be equal t o the closed loop pole of t he SMCcontroller corresponding to the i t h st ate.

9.2. R educed O r der P r elt er . Whil e it would be nice to employ an 1 -orderlt er, it is not generally t he pract ice of engineers t o squander electronic part s orpower pointl essly. We would like t o be able t o det ermine t he minimum ordernecessary for some reference input ,

Umax < xd (t) < Umax(59)

In eect, if we specify bounds for our reference input, as if it were a dist urbancet o t he sli ding surface, t hen we can design a lt er t o ensure t hat t he frequencycomponents hi gher t han pi are su cientl y damped. A t heoretical deri vat ion forlt er order may be proposed in a fut ure paper, or an experimental heurist ic found.

The aim of this paper, however is primarily to give the reader an introductiont o sliding mode control and compare it s performance wit h linear quadrat ic controlby solving the invert ed pendulum problem. T he aut hor suggest s t he t ri al and errorengineeri ng met hod. If t he sliding mode cont roller i s designed, it is a tr ivial t ask t ochange t he number of lt er st ages between t he reference input and t he st ate vector.

Part 4. Extended Example: The Inverted Pendulum

In order t o ill ust rate t he performance of SMC, we will design a conti nuous SMC

for t he invert ed pendulum problem. It s performance will be compared to t hat ofan LQR under various condit ions and degrees of mismatch.

10. P endul um M odel

T he invert ed pendulum is often used as a benchmark for contr oll ers of all kinds.It is a nonlinear, unst able syst em, which makes it a challenge to contr ol. Manydierent models have been developed for t he invert ed pendulum problem, including[9], [1], [19], and [20].


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A C O M P A R I S O N B E T W E E N L I N E A R Q U A D R A T I C C O N T R O L A N D S L I D I N G M O D E C O N T R O L11

F i g u r e 1 . Force diagram of t he invert ed pendulum device.

F i g u r e 2 . Free body diagram of t he cart and pendulum.

T he invert ed pendulum we are using has a very l ight pole (compared to t he massof t he pendulum bob), and a relati vely fr iction-free pivot for t he pole. Because oft his, pole frict ion and pole mass aect t he overall model deli ty very li t t le. Sincethey make the model considerably more complicated we chose to go with a modelwhich ignores them. T he foll owing model is formulated as presented in [21]. T hemodel is shown i n gs. 1 and 2.


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In the horizontal direct ion, we have for t he cart

F = M s + b_s + N

N = F M s b_s(60)and for t he pendulum

N = ms + ml cos ml_2

sin (61)

Subst it uting eq. 60 into eq. 61,

F M s b_s = ms + ml cos ml_2

sin

F = ms + ml cos ml_2

sin + M s + b_s

F = (M + m) s +

cos _2

sin

ml + b_s(62)

We have two unknown states, s and . T heir deri vat ives, _s, s, _, and mayeasil y be solved for i f s and are known. So far, we only have one equat ion for t wounknowns. Our second equat ion may be found by summing the forces in t he planeperpendicular t o t he pendulum. T his choice for t he axes saves us a lot of algebra,giving t he following equat ion:

P sin + N cos mg sin = ml + ms cos(63)

We can furt her simplify t his equation by summing t he moments about t he cen-t roid of t he pendulum

Pl sin N l cos = I

l (P sin + N cos) = I

P sin + N cos = I l

and substituting for (P sin + N cos) in eq. 63, we get

I l

mg sin = ml + ms cos

I mgl sin = ml 2 + ml s cosI + ml 2

+ mgl sin = ml s cos(64)

And so, our nonlinear invert ed pendulum model is described by eqs. 62 and 64:

(M + m) s +

cos _2

sin

ml + b_s = FI + ml 2

+ mgl sin = ml s cos

10.1. Linearization. T he non-linearit y of t he problem forces us t o eit her use

t ri cky, nonli near control t echniques, or quasi-nonli near control t echniques, wherea nonlinear syst em is modied in some way so as t o allow l inear contr ol scheme t obe used. One of t hese techniques is l ineari zat ion, where a l inear approximati on oft he functi on is used.

If a function, f () has n deri vat ives at = 0, t hen t he polynomial expansion,nX

i = 0

f ( i ) (0)i !

( 0)i


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isthe nt h order seri es for f () at 0. If a rst order approximat ion is used, we havea rst order polynomial, or a l ine. In many cases t he higher order t erms can beconsidered negligible if t he vari able is close to 0.

In our inverted pendulum problem however, we are concerned only with valuesof t hat are close t o (i.e. pendulum in t he upright posit ion). It is simple toreplace with + , 0,

sin(+ )

cos(+ ) 1

d (+ )2

d2t=

d2

d2t

d (+ )dt

2 0

T he pendulum equat ions become:

(M + m) s + b_s ml = u(65)I + ml 2

mgl = ml s(66)

10.2. System Equations. If wewant t o obtain the li nearized pendulum equat ionsin st at e space format, t hen we need t o obt ain eqs. 65 and 66 in t erms of t he st at evector, x,

x =

2

664

s_s_

3

775 ; _x =

2

664

_ss_

3

775

T his is achieved easil y enough by arr anging t he equat ions we have. Rearranging,eq. 66 becomes

=ml s + mgl

I + ml 2

We plug t his value for int o eq. 65:

u = (M + m) s + b_s mlml s + mgl

I + ml 2

(M + m) s = b_s + mlml s + mgl

I + ml 2+ u

s = b_s + ml m l s+ mglI + m l 2 + u

M + m

=ml (ml s + mgl)

I + ml 2

b_s +

I + ml 2

u

(M + m) (I + ml 2)

=m2l 2s + m2gl2

I + ml 2

b_s +

I + ml 2

u

(M + m) (I + ml 2)


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Taking all the s to the left, we get

1 m2l 2

(M + m) (I + ml2

)

s =m2gl2

I + ml 2

b_s +

I + ml 2

u

(M + m) (I + ml2

)M I + M ml 2 + mI + m2l2 m2l 2

(M + m) (I + ml 2)s =

m2gl2 I + ml 2

b_s +

I + ml 2

u

(M + m) (I + ml 2)

M I + M ml 2 + mI(M + m) (I + ml 2)

s =m2gl2

I + ml 2

b_s +

I + ml 2

u

(M + m) (I + ml 2)

s =m2gl2

I + ml 2

b_s +

I + ml 2

u

(M + m) I + M ml 2

or, in state format,

_s = _s

s =

I + ml 2

b

(M + m) I + M ml 2_s +

m2gl2

(M + m) I + M ml 2 +

I + ml 2

(M + m) I + M ml 2u

Similarly, if we st art wit h eq. 65,

s =u b_s + ml

M + mPlugging into eq. 66,

I + ml 2

mgl = mlu b_s + ml

M + m

=ml u b_s+ m l

M + m + mgl

I + ml 2

=mlu ml b_s + m2l2 + (M + m) mgl

(M + m) (I + ml 2)

Taking all the t o the left side, we get1

m2l 2

(M + m) (I + ml 2)

=

ml u mlb_s + (M + m) mgl(M + m) (I + ml 2)

M I + M ml 2 + mI(M + m) (I + ml 2)

=ml u mlb_s + (M + m) mgl

(M + m) (I + ml 2)

=ml u mlb_s + (M + m) mgl

(M + m) I + M ml 2

or in state format,

_ = _

= mlb

(M + m) I + M ml 2_s +

(M + m) mgl(M + m) I + M ml 2

+ml

(M + m) I + M ml 2u

Our st ate space system equat ion becomes:

2

664

_ss_

3

775 =

2

6664

0 1 0 0

0 (I + m l2 )b

(M + m ) I + M m l2m 2 gl 2

(M + m ) I + M m l 2 00 0 0 10 m l b(M + m ) I + M m l2

(M + m )m gl(M + m ) I + M m l 2 0

3

7775

2

664

x_s_

3

775 +

2

6664

0(I + m l 2)

(M + m ) I + M m l 2

0m l

(M + m ) I + M m l 2

3

7775

u

(67)


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10.2.1. Numerical Values. Assume t hat our pendulum has t he following character-istics:

M = 2:4kg

m = 0:23kg

b = 0:05N

m si = 0:0099kg m2

l = 0:36m

g = 9:81ms2

Fmax = 24N

Fmi n = 24N

Plugging i nto eq. 67, we get

A =

2664

0 1 0 00 0:0203 0:6893 00 0 0 10 0:0424 21:8933 0

3775

B =

2

664

00:4069

00:8486

3

775

T hese are t he values we wil l use for A and B in calculating the LQR and SMCcontroller gains. Al so, t he maximum allowable input f orce t o t he pendulum is 24N .

10.3. Tr ansfer Funct ion. If we take the Laplace transform of the equations of

mot ion for t his syst em (assuming the init ial conditi ons are close enough t o zero t oignore), we get

(M + m) X s2 + bX s mls2 = U(68)I + ml 2

s2 mgl = mlX s2(69)

where the X ;; U are t he Laplace t ransforms of t he displacement, s, t he angle, ,and the input u. We do not use S for the Laplace transform of the displacementdue to t he confusion t hat might ari se between it and s, t he L aplace operator.Rearr anging eq. 68 in order t o obtain t he relation of X t o ,

X =

" I + ml 2

ml

gs2

#

and subst it uti ng back i nto eq. 69, we have (after a lit t le rearranging, fact ori ng, andcancell ati on):

U

=m lq s

s3 + b( I + m l2 )

q s2

(M + m )m glq s

bmglq

where

q =h

(M + m)I + ml 2

(ml )2

i


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F i g u r e 3 . SMC set up for t he invert ed pendulum.

11. Sl id ing M ode Con t r o l l er f or t he I nver t ed P endul um

Fig. 3 shows our SMC set up for t he invert ed pendulum. The output and dist ur-bance are t he same as in gs. 9 and 10.

We wil l use an A ckermann SMC ( sec. 5) wi t h cont inuous contr ol (sec. 6). Wend t hrough t ri al and err or t hat t he contr ol spectrum given by

= [ 4; 4; 8; 8]

produces desirable result s. In order t o observe t he eects of init ial condit ions, weset

x (0) =

2

664

0:10:10:10:1

3

775

T he procedure shown i n sec. 5 can be followed, or t he Matl ab function ackermay be used to obt ain k (see program listi ng for zrat e.m ). T he augmented syst emequat ion is found t o be:

_z = [ 108:9473; 81:7105; 273:3339; 64:2349] x

z (0) = 0:1256

giving

s = B T x + z

u =

M B T x + z > 0M B T x + z < 0

We chose M = 24, since t his is the maximum control force we can apply t o t hependulum apparat us.


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F i g u r e 4 . T he complet e continuous sliding mode controller.

11.1. Continuous Control. We chose " = 0:1, resulting in the damped controlinput

u =8 0

24j B T x + zjk x k + 0:1 B

T x + z < 0

9=

;

T he CSMC is shown in g. 4.

11.2. Reference Input. In t he discussion of sec. 9, we suggest t hat suddenchanges in t he sliding surface might cause inst abilit y. To t est t his out , we willobserve t he behavior of t he syst em wit h and without a slew-liming lt er. Since t heslowest pole in the SMC is at s = 4, t he lt er would t ake the form

R0

(s) =4N

(s + 4)NR (s)

where N is t he order of the lt er. T he referencemodule wit h lt er in place is shownin g. 5. A reference is being intr oduced int o t he posit ion st at e t hrough a 2n d orderslew-limi t ing lt er.

12. St at e Est imat ed Sl idin g M od e Co nt r ol

T he est imat or used for t he SMC is t he same as t hat used for t he LQR. Ourcombined SMC-est imator can be seen i n g. 6.


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F i g u r e 5 . Reference i ntroduced t o t he posit ion t hrough a 2n d

order slew-limit ing lt er.

Fig ur e 6. A combined SMC-est imator is developed t o provide t hemissing states.

As in t he LQR case, we will assume that t he est imator has access t o t he t rueinput t o t he syst em; t he controller input plus t he dist urbance input, as shown ing. 7.


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F i g u r e 7 . T he SMC-est imator syst em wit h correct input dat aavailable t o the est imator.

13. L i n e a r Q u a d r at i c R e g u l at o r f o r t h e I n v e r t ed P e n d u l u m

Fig. 8 shows a typical LQR set up for t he invert ed pendulum. Several of t heout puts of t he syst em are saved t o t he workspacefor later analysis and display (gs.8 and 9). We assume here t hat our C mat rix allows complet e st at e observat ion,

A =

2

664

0 1 0 00 0:0203 0:6893 00 0 0 1

0 0:0424 21:8933 0

3

775

B =

2

664

00:4069

00:8486

3

775

C = I 4 4

This is a requirement of both LQR and SMC. Later, we will design a st at eest imat or t o provide est imat es of t he st at es from t he act ual observables of t heinverted pendulum, s and .

In order t o design an L QR controller, we rst have t o decide upon our weight ingmatri ces, Q and R. Since we have only one input, force, we can immediat ely(arbitrarily) set

R = 1

For t he st at es, we will set

Q =

2

664

x 0 0 00 x 0 00 0 y 00 0 0 y

3

775


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F i g u r e 8 . LQR set up for t he invert ed pendulum.

F i g u r e 9 . Several of t he vari ables are recorded t o t he workspacefor later display.

Af t er experi mental t ri al and error, we nd t hat good values for x and y are

x = 100

y = 60

T hese values gives us t he fast est set t ling t ime wit hout exceeding the limit set ont he pendulum i nput (t oo oft en). T he LQR gains can now be calculated using theMatlab funct ion lqr :

K = lqr (A;B;Q;R)

which turn out to be

K = [ 10:0000; 17:0407; 118:3489; 26:8734]


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F i g u r e 1 0. T hree dierent types of angular dist urbance are in-t roduced into t he pendulum.

13.1. Refer ence I nput . See [1] for det ail s on i ntr oducing a r eference input t o aregulated syst em. However, in our case, t he simpli city of having a single inputallows us to forgo t he calculati ons normally required. If our regulator gain for t hereference st ate (t he posit ion, s) is -10, we can simply mult iply our reference inputfor t hat st at e by -10. T his would result in a zero error when s equals t he reference.We call t his mult iplicat ion factor N bar ;

N bar = 10

13.2. Disturbance. We want t he dist urbance entering t he syst em t o aect t heangle of t he pendulum, but none of t he ot her st ates. T his sit uat ion is more li kelyin actual t rials. We achieve t his by augmenting both t he B matr ix and the input,

B =

2

664

0 b10 b21 b30 b4

3

775

ua =

wu

where b1:: :4 are the 1st : : : 4t h elements of the original B matrix, w is the angulardisturbance, and ua is t he augmented input vector.

T hree dierent types of dist urbances are int roduced into t he system (see g. 10):Brownian, Gaussian, and St ep. T he Brownian dist urbance is a gradual dri ft , of the

type produced by st rong wind hit t ing the pendulum. T he Gaussian dist urbance issimi lar t o noise produced by sensor err ors, and is generall y accept ed as normalnoise. T hest ep dist urbance is of t hetype generated by, for example, a ri val graduat est udent t ryi ng to knock t he pendulum over when t he inst ruct ors t urns his back.

Whil e it might seem l ike overki ll t o include t hree kinds of dist urbance, each hascharact eri st ics that reveal dierent features in t he cont roller. For example, t heGaussian noise is zero mean for each ensemble, whil e t he Brownian noise is zeromean only over many ensembles (i t is not zero mean over each single ensemble). I f


22/33


F i g u r e 1 1. A combined li near quadrati c regulator-est imator isdeveloped t o provide t he missing st ates.

a contr oll er can only handle zero mean noise, t hen t his wil l be revealed. Simil arly,t he st ep distur bance shows how fast a cont roll er can recover from an unmodell eddiscont inuous dynamic.

14. St at e Est imat ed Qu adr at ic Reg ul at ion

We have assumed t hat all t he pendulum st ates are available for measure. Inactuality, only s and are available, i.e.

C =

1 0 0 0

0 0 1 0

We will design a st ate observer t o est imate t he missing st at es.See [1] for det ails on development of stat e observers (or st ate est imat ors). Ba-

sicall y, a state observer is t he dual problem t o t he linear regulator problem. Wechoose our observer poles t o be at least twi ce as fast as t he pendulum dynamics.

eig (A) =

2

664

0 0:0194:6784

4:6797

3

775

We choose poles at P = [ 10; 11; 12; 13]. Using the Matlab function, place ,

L = place([A B K ]0

; C0

; P )0

where0

denot es t he t ransposit ion operat or.Our combined LQ regulat or-est imat or can be seen in gs. 11 and 12.T his combined regulator-est imator is closer t o what would be impl emented i n

act uali ty, and it can be shown t hat t he observer does not considerably change t hesyst em dynamics (if t he observer poles are mush fast er t han t he syst em dynamics).However, if t he input being supplied t o t he observer is incorrect (due t o error)t hen t he performance of the regulator-est imator wi ll be considerably changed. T he


23/33


Fig ur e 12. A st ate est imator is shown. T he st at e-space modulecalculat es: _x = (A LC) x + I 4u; y = x

Figur e 13. A fudged syst em might provide more insight intot he propert ies of the cont roller.

eect t his is t hat t he performance of t he observer might mask t he performance oft he regulat or.

Instead of r emaining st ri ctly correct, we wil l assume t hat t he observer has accesst o sensors t hat measure t he angle of t he pendulum (wit h noise). T his result s in t he

system shown in g. 13.

15. R esul t s

15.1. Reference Conditioning. First ly, we test our t heory about t he int roduc-t ion of a reference input. T he sliding mode controller is simulat ed wit hout t heslew-limi t ing lt er (g. 14). T he CSMC becomes unst able as soon as the sli dingsurface is (di scontinuously) changed.


24/33


Figur e 14. The CSMC becomes unstable as soon as the slidingsurface is suddenly changed.

Applying a 2n d order lt er t o t he referencei nput, t he CSMC f oll ows t he referencewell , maintaining i t s st abil it y as seen i n g. 15.

15.2. Standard Sliding M ode Control. It would be interest ing at t his point tosee how a st andard SMC would perform for t his problem. SMC assumes innit eswit ching speed, however in practice t his is never t he case. An SMC was designedfor t he invert ed pendulum (using A ckermanns gain) wit h fast swit ching act ion.It s performance is bet t er t han t hat of t he CSMC when t he reference changes (g.16). However, i t is immediately obvious t hat t his comes at t he cost of much greatercontrol authori ty. A slower swit ching SMC is designed which performs considerablyworse (g. 17).

15.3. Perform ance wit h M inor Di st urbance. We simulat e bot h controllerswit h t he foll owing angular dist urbance (in radians):

Brownian = 0:05

Gaussian = 0:2

Step = 0:1

T he result s of t he LQR are shown in g. 18. T he CSMC result s are shown ing. 19.

We see t hat the CSMC follows t he reference a lit tl e more smoot hly.


25/33


F i g u r e 1 5. T he CSMC follows t he reference successfull y aft ercondit ioning the reference.

Fig ur e 16. An SMC with fast switching action.


26/33


Fig ur e 17. An SMC with slow switching action.

Fig ur e 18. Result s of t he LQR wit h mi nor dist urbance.


27/33


Fig ur e 19. Result s of t he CSMC wit h minor dist urbance.

15.4. Perform ance wit h M ajor D ist urbance. We simulat e bot h controllers ont he pendulum using t he following disturbance:

Brownian = 0:2

Gaussian = 0:5

Step = 0:5

T he result s of t he LQR are shown in g. 18. T he CSMC result s are shown ing. 19.

Neit her of t he contr ollers are able t o converge complet ely t o t he posit ion ref-erence. T his i s because of t he considerable disturbance being applied t o t he pen-dulums angle. If we visualized t he disturbance as coming from a nger pushingagainst t he pendulum bob, we can see t hat if a controller insist s on maint aining t hependulum posit ion wit h no compromise, t he result wil l be the pendulum t opplingover. We can see, however t hat t he CSMC cont roller achieves a much closer delit ybetween t he pendulums posit ion and t he desired posit ion.

15.5. Contr oller M ismatch. T wo t est s are performed where t he controller ismismat ched t o t he actual syst em. T his is done by designing t he contr oller for t hependulum, and t hen changing t he lengt h of t he pendulum. In t he rst t est , t helengt h of t he pendulum is increased from 0.36 to 0.54. In t he second, it is reducedt o 0.18. T he result s for t he LQR are shown in gs. 22 and 23, t hose for t he CSMCare shown in gs. 24 and 25.

Both contr ollers perform exceptionall y well . In order t o discriminate betweent he two controllers, we will inst ead t ry t o mismat ch t hem i n t he anot her way. T he


28/33


Fig ur e 20. Result s of t he LQR wit h major dist urbance.

Fig ur e 21. Result s of t he CSMC wit h major dist urbance.


29/33


Fig ur e 22. The LQR is designed for a pendulum length of 0.36,and t est ed on one wit h l engt h 0.54.



30/33


Fig ur e 24. T he CSMC is designed for a pendulum lengt h of 0.36,and t est ed on one wit h l engt h 0.54.



31/33


Fig ur e 26. T he is increased unt il t he LQR is no longer able t operform. = 1:902

controll er is designed for t he system

_x = Ax + B u

and t est ed on t he syst em

_x = A x + B uThe st able range for is t hen found. In t he case of t he LQR ( g. 26),

0 < L QR < 1:902

For t he CSMC (g. 27),

0 < CSM C < 3:360

It is int erest ing t o not e that, just before the controllers fail, t hey seem t o beoperating quit e well . T he degradat ion i n performance comes suddenly.

16. Observat ions

Both the LQR and the CSMC were able to control the inverted pendulum in arobust fashion.

T he presence of a slew-limi t ing prelt er wasseen to be necessary. I n it s absence,t he CSMC became unst able when t he sli ding surface changed t oo suddenly.

T heCSMC had bet t er dist urbance rejection qualit ies t han t heL QR. Wesee t hat ,in all t he sit uat ions t est ed, t he CSMC generally maint ained bett er deli ty betweent he pendulum st at es and t he desired referencestat es t han t he LQR. However, it canalso be seen t hat t he CSMC demanded more controll er act ion once the st ate wasclose t o t he desired r eference. T his contr oller acti on could be reduced by r educing" .


32/33


Fig ur e 27. T he is increased until the CSMC is no longer ableto perform. = 3:360

Based on t he amount of mi smat ch t hat t he contr oll ers could suer wit hout inst a-bil it y arising, t he CSMC seemed t o be t he more robust contr oller for t he invert edpendulum.

Several design decisions were made based on human judgement, such as t he

choice of Q for t he LQR or t he choice of cont roller poles for t he CSMC. Carewas t aken t o maintain impart ialit y in design, and both contr ollers were renedseparately in order t o ensure performance t hat was close t o opti mal. For t hisreason, it i s reasonable t o expect t heperformance of the contr ollers wit h t heinvert edpendulum t o be indicat ive of t he control algorit hms charact eristics.

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http:/ / rclsgi.eng.ohio-state.edu/ matlab/ index.html.

a comparison between linear quadratic control and sliding mode control

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