a comparison between linear quadratic control and sliding mode control
TRANSCRIPT
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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.
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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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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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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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18 A NW ER S. B A SHI
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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20 A NW ER S. B A SHI
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
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22 A NW ER S. B A SHI
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
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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.
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24 A NW ER S. B A SHI
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.
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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.
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26 A NW ER S. B A SHI
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.
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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
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28 A NW ER S. B A SHI
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.
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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.
Fig ur e 23. The LQR is designed for a pendulum length of 0.36,and t est ed on one wit h l engt h 0.18.
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30 A NW ER S. B A SHI
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.
Fig ur e 25. The LQR is designed for a pendulum length of 0.36,and t est ed on one wit h l engt h 0.18.
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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" .
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32 A NW ER S. B A SHI
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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