- Email: [email protected]
Symbolic Analysis and Design for (Nonaffine) Nonlinear Control Systems
Copyright © 1996 IFAC 13th Triennial World Cnngrcss, San Francisco, USA
2b-242
SYMBOUC ANALYSIS AND DESIGN FOR (NONAFFINE) NONUNEAR CONTROL SYSTEMS
Bram de lager *
,.. Faculty of Mechanical Engineering, Eindhoven University of Technology, P.o. Box S13, :,600 MB Eindhoven, The Netherlands. Email: A.G.de.]agerifVw{w.wtb.rue.nl
Abstract. For the analysis and design of dynamical sys tems symbolic tools are a great help. In the past, several algOrithms for nonlinear control systems haw' been implemented in MAPLE. This implemen Lation has been extended to also handle nonafftne models and to include more algorithms. Three examples taken from the literatun.! in this area illustra le the power and limitat ions of the implemen tation. For system models that are more than a little complex symbolic com putation cannot be fully enjoyed du e to the complexHy of r h ~ algorithms that are more than polynomial in some meas ure of the problem size. Keywords. Nonlinear control systems, algorithms, lineari zation, robot conlrol, flexibl e arms, induction mown;, automotive control.
1. INTRODUCTION
so it was decided to extend the NOrt-:-CON package to alleviate this problem and some others.
SymboH c computation, also known as computer algebra. is a powerful tool for solving lOugh and intricate problems
Two approaches were used to make the pa [kag~ suitable for nonafri ne models. First. several computations. e.g., the relative degree (using a definition in (Nijmeijer and Van der Schaft, 1991)), the normal form, the zero dynamiCS, the ex· aCl lineari zation, and the inpUI -outputlinearization, were enhanced to he able 10 gracefully hand le nonaffinc models. S~(o ndl y. an option was add ed to make a nonaffine model affine by extension of the staf(' space and red efinition of its input.
in applied mathematics. With the advt:!nt and improvement of conunerciaHy available packages Ihis fool is becoming general1y available and used. It promises drasti c changes in the way problems, .in fields as diverse as one ca n imagine. are tackled. In the past a closely lied set of symbolic computation functions, together called the NON~ON package, has been presented that solves some problems in the analysis and design o f nonlinear control system, St..."C (Van Essen and De Jager, 1993; De Jager, 1993 ; De Jager, 1994a; De Jager, 1995a; De Jager, 199 5bl. The functi ons available in the package range from the computalion of the zero d ynamics of the model, that does not need \0 have a \'\!ell-defined relative degree, to computing exact Iinearizing or inputoutput I1nearizing feedbacks. All functions are based on constructi ve algorithms and implemenled in MAPLE. A model of the system must be available for the computa· lion. The model. in gene ral a set of nonlinear differential and algebraic equations in the slate. input, and output of the system, \\'as assumed to be affin e in the input. For several problems appearing in practice this was too restrictIve. see (De Jager, 19951» for an example in vehicle dynamics,
Besides Lhese enhancemen Ls to hand1e DonaITine models, the Dynamic Extension Algorilhm was added as an option to give an affi ne model a (well·defined) relative degree, by s tate extension and feedback, in those cases it did not have one yet. furthermore. the package was extended with a new and potentially more efficient algorithm to compute the local zero dynamics for affine models \"';th a well-defined relative degree that was proposc'
2687
The goal of this paper is to describe some of [he enhancements to the NOf'tCON package, to comment on several aspects of the implementation, and to provide some realistic examples of the use of the new facilities. The following Section presents some theoretical background in the fonn of (comments on) algorithms. Section 3 provides several implementation details while Section 4 covers the examples. The paper clo ~es with a discussion of the results and an indication of so me direc tions for future research.
2. ALGORITHMS The two stand ard models used are
{e\:) + g(x) u,
x
(lJ
)' = h(x)
for affine syslems and
x ~ f(x, u ),
)'
~
ht,, )
(2)
for nonaffine sys tems, vvith x c. ~n the state. u E ~ m the input. Y E [R,f' the output, r a Vt'ctor fiel d, 9 representing a se' of vector fields 9j (column-wise). and h is a colulTUl wilh slacked scalar funclions hi' Modifications [() I\\'o algorithms 1rvill bt! discussed (to compute the relative degH't' and normaJ form) that are useful in the case of nonaffine models. Also the Dynamic Extension AlgorHhm ""ill be touched up on.
2.1 Relative degree Th e relative degree of a nonaffine model of a nonlinear
system in (X ", t/ O) is, based on the defi nition in (Nij meijer and Van der Schaft, 199 [, p. 417 ). the se t of integers ri, i = 1. .. . ,p such thal.
,
C~ " l.7- hi( X, u)
-'. 0
'7 k < ri, I ::; i . .: :
r, 1 ::; j
:$ m
J
For affine models both definitions arc, however, equivalent. The two definitjons are implemented. in Ihe function
reldeg.lf no set of integers can be found that fulfills the two conditions, the relative degree is said to be not well· defined.
2.2 Dynamic extension
If a model does nol have a well-defined relative degree. several tools or algorithms are no l appli cable anyrnore, restricting the analysis and design objectives that can be achieved for this model To remedy this, a preliminary extension of the Slale space and Slate feedback are employed. The extension uf lhe state space is necessary, because lhe relative degree is invariant uncler static state feedback, necessitating the use of dynamic state feedback The recursive' algorithm presented in (lsidori, 1 989, p. 384), to which onc is referred for more details, essenlial1y iden· tifies (combinations of) inpul channels thal have to be "delayed" by" an integra ting acti on. The dynamic feedb ack from a single recursio n step i!' u ~ ",(xl +
<:
fl l(''lS
+ {h(x)v,
~ VI,
where V j is the set of "delayed" inputs and V2 the set of inputs tha t are nOI delayed. The functions IX, {J1t and Ih are computed by 1he algorithm. The algoritlun terminates if the relative degree becomes well-defined, or, \vhen this is not possible, aftcr a bounded number of steps, where the bound is at most n. A restriction of the algorithm is that only square MIMO models arc considered. Furthe rmore, additi onal states are introduced in several input-output channels, potentially introd.ucing additional phase lag that may make th~ model more difficult to conlrol. Due to Ihe increase of Ihe dimen· sjon of the state space the model becomes also less suited for symbolic computation: me mory and compute lime requiremenb may increase.
and the m a trix ti.l tlm
C
C
Lf
"h
l(X,
1
uJ
rl1
, -L( hr(x ,u) ("" Urn
is nonsingular in (x ~ , u " ). with L{ hi (X , U ) the lJe de rivative, i.e., the derivaU ve with respec t [0 x of hf(x, u) along f ch, . ( U J = -:;:.- 1" L rhiX,
'x
Remark that (he requiremen t that Adcg is 10 b e non singular is not posed in (Nijmeijer and Van der Schaft, 1991). The main differences with the definition for affine modi) eh is that th('re the term -, -L,. i!' replaced by the deMvaou.I
live along 9) : L gJ and the condiHons do not depend on
u~ .
2.3 Normal (arm and zero dyn amics
Normal forms for nonlinear models are useful for theoretical developments. because onJy a res tricted class of mod· els is to be conSidered, other classes can be transformed to thp normal form class, but are also valuable in their 0\0\'11 right, because they may highlight interesting characteristics of rhe model. One of thm:;(' characteris tics is the zero dynami cs. Rccau se the computalions involved in ob taining a norma l form are often rather tedious, it is of advantage to automa te- this. Normal forms have been defined for several c1asses of models, among them affine mod els with or without a well· defined relative degree and nonaffine models \\1th a well· defined re lative degree, but nol (yet) for nonaifine models without a well-defined relalive degree. The original computation of the normal fonn, that was only suitable for aITine models with a well-defined relative
2688
degree, could easily be adapted to incorporate nonaffine models. It was also extended to affine models without a well-defined rt'lative degree by building the "generalized" nonnal form proposed in (lsidori, 1989, p . 308), making use of the Zero D}1lamics A1gorilhm (lsidori, 1989. p. 290).
Example 1 The first exampl e is a two link mechanism with an elastic joint bet\\'een the two links. Its model is (see (lsidori, 1989. pp. 398-401) .\'5
Given the normal form, the zero dynamics can easily be derived for affine models. If Ihe mod el is nonaffine a complication may aris e because the zeroing input u zcro need nol be unique. Besides the computation of the zero dynamics by means of the normal form. another pOlerllially more efficient algorithm to compute (he local zero dynamics proposed in (f;watny and Blankenship, 1994) was implemented. Onc should consult the original paper for [he details of the algorithm. bUl \'¥l! remark that it (an only be used for affine models with a well-d efined relative degree. For models that do not have a well·defined relalive degree, that can therefore be pUI in the generalized normal fonn only, the zero d ynamics is computed directly by the Zero Dynamics Algorithm, which is possible under some conctitions. This algorithm can also be used for models with a well-defined relati \"{~ degree. circumventing the derivation via the normal form. but not for nonaffine models. There the normal form has In be computed firs!.
f =
with
r,l
=
--6s, (2x4XO + xi)" 2(1 /:"3 - 1/20x,) - (2 " 3c31(1 / 2x, - 5x:, - 353x1)
{sI = (}.I,(2'4" + x~) +
~(2 + 'IC5)(1/2"
-1/20,, )
- (2 + 3cj )( 1/ 2x2 - :lx" - :is]x~ )
hI = 3(2
I
3c,)s,(2x4X6
T
xi,) . (2 + 3cJ)(I/2x l - 1/20,,)
- 3(-1 + 2c3)( 1/ 2x 2 - >X j - 3s:'1 x~)
and 0 0 0
0 0 0
Remark that for affine models \<..'ilh a well-defined relative degree there arc three options ava ilable to compute !he zero dyn am iCS, namely by first computing th(' normal for m, and deriving the zero dynamics from this form, (2) applying the Zero Dynamics Algorithm, (3) using the proposal in (Kwatll) and Blankenship, 1994).
[{i"']
g=
2/1
- 2/1 I
,
2 /1 4(2 + 9q )/I (2 + 3e,) /I -(2 + 3e,)/1
(1)
wlule the output is defined by
Differences occur, at leas t for the implemeOlation in NoN§CoN. in !he coordinates used to represent the results.
3. IMPLEMEN TATION
The (modifications of) algorithms discussed in the previous section are rather straightforwardly implemented in MApl£ usi ng the linear algebra facilities and the procedures to solve sets of nonlinear algebraic equations and sels of nonlinea, partial differential equations (Dc Jage r. 1994b). One of the problems in the implememalion stems from the determination or selection of a suitable solution, that is not unique., for an under-determined set of algebraic equations in the Dynamic Extension Algorithm. which is not grace· fully handl ed by the standard MAPI.' fa cility solve. A work around has been implemented.
4. EXAMPLES
and the following abbreviation') are used 1 = 10 gel cos \' ] "'3 ~ sinx) .
C3 =
Here XI, •• " X3 represent the rotational coordinates of link 1. joint 2, and link 2 respective ly. and X4 ••• •• X6 represent 'he velocities. For 1he parameters appc-uring in the original equations. integers have been filled in to simplify the computations by redu ci ng the intermedia te expression swell. The follo'wing computation of the rela tive degree in the point x " = 0 shows that it is not well·defined. reldeg: the matrix Adeg turns out to be singular! reldeg: the initial state conditions for which the matrix Adeg turns out to be singular are : (all )
The three examples lo be. presented use interesting models of electro-mechanical systems, i.e .• a two link robot v.1th an elastic joint (an affine model without wen-defined relative degree taken from (fsidori, 1989)), a voltage fed Induction motor (a nonaffine model taken from (Nijmeijer and Van d er Schaft, 1991), and a four-wheel vehicle with steering (nonaffinc andlor no '.vell-defined relative degree) described in (De Jager, 1995b).
with
-2/1
2/1]
Ad., = [ (2 + 3[, )/1 -(2 + 3e,)/1 .
A well-defined relative df'grce is achieved with the Dynamic Extension Algorithm in two steps. This is illustrated with the following (condensed) output of the algorithlll-
2689
Step 1: 2
alpha: [- 6 5in(,[3)) , [6 ) x[4)
3 sin(x[l)) x[6] 2
9/ 2 x [3) + 9/ 20 , [2] - 1 si n( x[3] ) x(4) 15/ 2 cos(x[l)) x[l)
3/4 c05(x[3)) x[2)
+
2
- 9/ 2 c05 (x [3)) si n(x [ l)) x[4) ] [ )
reldeg : t he initial state conditions for which the matrix Adeg turns out to be singula r are:
[0) 2
- 5 - 9/ 2 c05(,[3 ) ) beta :
(x[4)
1
o
l/2 cos(x [3])
1
0
vector relative degree
Step 2:
total relati ve degree
0 alpha:
Adeg
0 [ [
beta :
2 (x [l) cos (u[2] ) + x[4) sin(u[2])), I
+
x [4] cos(u[2)))
+ x[3) sin Cu[ 2])jCsigma '_ 5) - x[l) si n(u [2)), 1
0
- 1 - l/2 cos(x[3])
1/16
-u [l](- x[4) sin(u[2]) / (5 i gma ls) + x[l ) sin(u(2))
Adeg:
x[l] cos(u[l])/(sigm. ls)
[4, 4 )
vector re l ative degree WELL-defined! It is easily verified that the two sets of C'( and {3 that arc computed, when co mbined in the correel' way, agree with the result in (Isidori, 1989, p. 4(1) ohilt the I'inal dynamic slale feedback (or d)'1 lamic eXlension) which gives the model a
weB-defined relative degree is u,
[
• >
[- x(4 ) cos(u[2]) / (sigma :. 5) + x[2] cos(uC>])
1
calculating Adeg
rdeg:
.
[1. 1]
+ 2 u Cl) (- x[3) sin(u[l]
1 0
[ 0
2
+ x[l) - x[l] sigm. ls x[l) {x [2] • ------ - - - - ------- --------------- } , x[4] s;gma ls
(x [ l] • 0, x[4] - 0)
[ [
0, x[3) • x[l) sigma ls), ( u[l) • 0),
x[4)
0
1
[
~
2
1
reldeg: the matrix Adeg turns out to be singular! Adeg:
Here Xl and X2 represent components of the stator current, x) and X4 are the corresponding flux components. The speed of the motor w is considered constant. just as all the other model parameters. The total stator flux and stator torque are the out puts. The model does have a weUdefined relative degree in (x ",u' ) ~ (O,o,x" x"uj,O), but it is not full, so L. rj < n, as shov.'ll by
~
I
x[l) cos(u[2)))
are the following.
zero dynam i cs: eta[l]dot
.
+
This result is in agreement with the onc given in (Nijmeijer and Van der Schafo, 1991, p . 419). Because the relative degree for this nonaffine model i ... well-defined, the zero dynamics can be computed, and because the model is non· affine this is done by computing the normal form with the function no rrnforrn_ Selected rc'mlts from the computation
= -(e ta[2] alpha s ;gma Ls eta[l]
· i - f,(x)+P . v, 94d:d
2 + eta [ l] eta[l] sigma Ls beta - omega eta[2]
, ~ ~
~ ~
2
RootOfC_Z
-(0: ,. . 8.);'(.1 - WX 2
~X3 +~ L~ W)q
wx"
t - - - t
(J Ls
f3x~
wxo - (e< + Il)x, - - - + _.
~
r
eta[2] ) s;gma omega eta[l] ls
+
eta[2] ) sigma beta eta[2])
VI·
Example 2 This example considers a voltage fed induction motor. The model is taken from (Nijmeijer and Van der Schaft, 1991, pp. 26 7, 419), with a correction of a printing error. It is nonaffine and given by
f
2 +
+
(JL ~ Ls -aaLsxl + U I COSU2
-CHfLsX,!
T
Ut
sm Uz
1
- RootOf(_Z
/ (etaC2) 5igma ls), eta[l)dot -
Ul cos U2 "'--'7-=
2
aLs
alpha 5igma Ls eta[l] RootOfC_Z
Ul smU2
~--;----O
a{ s
2
(3)
2 + eta[2] )
eta[l) zeroing input uzero: [ 0, locuzero[2]
2690
The interpretation of Ihe lasl line of the resull is simple. To keep the output y ~ 0 for all t, the inpul u\ should be 0, and the value of input Uz does not matter. This is eas y to check from the differential equations (3). The input u, does only occur in combination with Ul. When u, :: 0 the value of U2 is irrelevant. The dynamics of the model Wlder the constraint that y = 0 is given by the IwO differential equations for rj . The model can al so be inpul -output Iinearized with the function inoutlin. The raw results of the 10 linearizalion are voluminous. From the linearizing feedback onJy the second component of u is prese nt ed, as fo ll ows. io linearizing feedba ck ( uli n[Z) ): arctan ( - 2 v[ 2] s igma Ls x[3] 2 +
2
2 K[2] s igma 2
+ 2 x [2]
K[ ] ] al pha x[I]
Ls 2
2 +
2 omega x[ 3]
Example 3 This las l example considers a vehicle steering s tability problem. It is posed more extensively in (De lager, 19'1S b ). Refer to trus for the comple te model equations, that will not be repeated here. The ori ginal model is non· affi ne. In the present research the model is simplified by assuming rhe lateral tire forces Ft and Fr , at front and rear, to be constant and n OI functio ns of the s late and s teering angle. The reason for thi s si mplifi cation is th at the computation fo r the original model could not be completed. Several rt'suits \vill be dis cussed concerning the simplified . but s lill nonaffine. model ~i ven by the following equations
Ls
'4
",_
x(4) alpha - Ft s in( uI + x])
3
Ft- cos(:,.,
h _ [
2
-
,[3]
2
omega x[ i] sigma Ls 2
- 2 x [ 3] +
a1pha sigma Ls )([ 2]
, (2] sigma Ls ve l ] - ,[4] v(l ] )
/
/
! 2
2
2
(2 . [1]
sigm.
Ls 2
+ 2 x[l] s igma +
. ( 3] alph a 2
ls
x[l] sigma Ls vel]
x[4 ] al pha x[2] +
2 v[ 2] s;gma Ls x[4]
2
- 2 x[ 3]
a1pha si gma Ls x[ IJ
+ 2 x[3] x (2 ] sigma Ls beta x[4] - . (3J vel]
2 x[ 3] omega xCI] 5igma Ls x[4] 2
- 2 x [4J
alpha sigm. Ls x[l] 2
2 x [4]
x[l] sigma Ls bet a
+
~2 sin x3
x .j cos x~ + X ,, sin Xl - lid (X l +p COSX] )- +(Xz + p s tnx] )2
x,.
alpha s i gma Ls x [ 2]
+ 2 x[3]
Xj
]
-Ra'
where X2 represent the position of the COM (center-ofmas s) and Xl the rota tion around COM, the s tales X4 ,· .• • X6 represent the corresponding velocities. The ou tput y is defi ned in such a \'\'ay that it is Z (~ro for a curving maneuver vvith constant radiUS Rd and co nstanl forward speed component Vd . The inpul UI is the 'itccring angle and U 2 is the Iraction force. Hrst. the relative degree of the model is computed. It is we ll-defined. See the output of the funclion r e ldeg.
2
x[4]
t! cos
co~ Xl
aF( cos i~~_ bFr j,
+ 2 x[4] omega x[2] s; gma Ls x[l]
- 2
+ XJ) +
F,. sin x3 + Uz
M
2 x[ 4] x[l] s igma Ls beta x[3 ]
- 2 omega ,[4]
3
, [4] + 2 omega x[4] ))
Due to the complexit\' of the outpul, it is difficult to verify the resull by simple inspection.
x[2] sigm a Ls beta - 2 omega x[3]
2 x[ 3]
omega x[2] s igma Ls
2
sigma 2
+
2
- 2 x[4]
vector r el ati ve degree - [ 1. 2 ] total rel at ive degree
- 3
The zero dynamics can be computed successfull y, but it is complicated. making it tedious to derive its characteristi cs, c.g.• to decide if the zero dynamics is s table. Second, sta te extension (an integra tor in the firs t input channel) and inpul redefinition have been used to get a model tha t is affine. It appears that the affine model does not have a well-defined relative degree, so the Dynamic Ex· tension Al gorithm is applied, gh·i ng an affine model with well·defined relative degree, that is of rugher order than th e original onf'. The computation of the normal fo rm for this ex tended model could not be completed due to interme· dia te express ion swell. The zero dyn amic could therefore not be computed. The vehicle model presented in (Kwatny and Bla nkensrup, 1994) has also been investigated. The algorithms to corn· pUle the zero d)-llamics were able to finish their computation for this mod el. So it appeared tha t it is m uch easier to handle.
2691
The results of the first two examples are satisfactory. For the last example, which has a quite complicated model description, the results are disappointing due to ., the complexity of the algorithm s, • limitations of the compuler algebra system. Based on (he M>\PlE resuJt s for several examples, not all presented here, the follm"ing conclusions can b e drawn. • To make a model affine and to give it a well·defined relative degree are relatively simple problems, that can be solved rather eaSily by symbolic computation tools, making a non affine model affine is nol always advantageou·s, the affine model of larger dimension may be more involved for compulational purposes, ., the: nc:wly proposed algorithm 10 compute the zero dynamics is more efficientl han the other IWO methods presented, • the full vehicle zero dynamics problem can be solved by none of the algorHhms prest:nt.1y available, mod el simplifications are mandatory.
s. m SClfSSlON The extensions to the NONKON package enable one to treat a larger class of models, including those that occur in practice for frequently encountered systems. The extended NOr-.GCON package is sw.:cessful for low order
models that do nOI contain large intricate expressions. For large order models the package is less successful, among othe rs due to an intrinsic limitation in MAPLE for the han dling of large numbers 01· obj~c t s. To il\·oirl this limitation, and so 10 fully exploit the po\,,rer of symbolic computation, onc can hardly avoid to become- familiar with the peculiar· ities of the program used for implementation. Ideillly one would want to avoid this. This shou ld be impro\,ed. Some other problems stem from {he ne(~d to solve intricate sets of nonlinear «(partial) diffcrentiall equations. For these kind of problems - very diffif.:ull ones - no powerful, builtin, read y to use, functions are a,·ajJable. Therefore, some ad-hoc facilities were implemented to improve the capability of MAPLE in thi s respect, see, e.g., (l) e .lager, 1994b). This should be improved also. The main trouble lies, however, in the fact Ihal the algorithms (and their implementations) are rather complex. While in numencaJ computations onc is ready 10 accept a computational complexily growth rate that is a cube of the problem size, in s}mbol1c computations one should be ready to accept a double exponential relatio n, see (DeJager, 19940). This means that slightly larger problems lban those used in this paper will readily become intractable, gathering dark clouds at the horizon for practical symbolic computation.
Acknowledgements The extensions to NO~CON were coded by Gose Fisher starling from the code by Harm van Essen (1992).
(I ~94),
6. REFERENCES Blankenship, G. L, R. Ghanadan, H. G. Kwatny, C. LaVigna and Y. Polyakov (1995). Tools for integrated modeling, design, and nonlinear control IEEE Control Sy.... terns /1ag. 15(2), 65-79. de Jager, Hram (1993). Zero d)1Iamics and input·output ex· aCI Iincarization for systems withoul a relative degree using Maple. In: Proc. of the 199.3 Inlernat. Symp. on Nonlin ea r Theory and its Applications. VoJ. 4. IElCE. Honolulu, Hawaii. pp. 120:;-1208. de Jager. Bram (19940). Computer aided hybrid analysiS and design 01" nonlinear control sys tems . In: .4ppli· calion o/" Mullivariable ;)"stem Techniques (AMST94) (R. Whalley, Ed.). IMC. London: MEP. Bradford, UK . pp. 247-254. de Jager, Rram (l994b). Symbolic solution for a class of partial differential equations. In: Prm. :. of the Rhine Wurkshop on Computer Alqebra Oacques Cal met, Ed.). Karlsruhe, (;ermany. pp. 84-88. de Jager, Bram (l ~g5a). Symbo lics for control: Maple used in solving Ihe exact linea rization problem. ]n: Computer Algebra in Industry 2: Problem Solving in Prac· lice (Arjeh M. Cohen, Leenderl van Gaslcl and Sjoerd Verduyn Lunel, Eds.). pp. 29 1-31 J. Jo hn Wile)" & Sons, Lld .. Chichester. de .lager, Bram (l995b). The use of symbolic t:umputation in nonlinear control: Is it Viable'? IEEE Tran s. Automat. ControI40(J), 84- 89.
Fisher, J. P. j. (Gose) (1994). NONLlNCON: Symbolic analy' sis and deSign package for nonlinear conlrol systems. fo.·ISc th{~sis. Eindhoven University of Technology. Fac. of Mechanical Engineering. Report WFW 94.09S. Isidori, A. (1989). Nonlinear Con trol Systems: An Inlroduc· tion. 2nd ed .. Springer·Yerlag. Berlin. Kwatny, H. G. and G. L. Hlankens hip (1994). Symbolic tools fur variable S{fucture conI rol system design: the zero dynamics. In: Proe. the IEFI' Workshop on Robust
or
Control via Variable Structure & l.yapunov Techniques.
IEEE. Benevento, Italy. Meller-Pcdersen• .lens .md Martin Pagh Pererscn (1995). Control of nonlinear plan1s - Volumes I and 11. MSc thesis. Mathematical Institute, Technical University of Denmark. Lyngby. Denmark. van der Schaft (1991). Nonlinear Dy. Nijmeijer, H. and namical Contml Systems. SpJinger·Yerlag. New York. Corrected 2nd printing. Post, B. A. (19<)0). Computer algebra analysis of nonlin· ear discrete-time systems. MSc rhesis. Dept. of Applied Mathematics, University of Twente. Enschede, The Netherlands. van Essen, H. (l99l). Symbols s peak loudet than numbers: Analysis and design of nonlinear control systems with the symbolic computation system MAPLE. MSc thesiS. Eindhoven University of Technology. Fae. of Mechani· cal Engineering. Report WFW 92.061. van Essen. Harm and Bram de Jager (1993). Analysis and design of nonlinear control systems vvith the symbolic computation sys tem Maple. In: Proc. of the second European Control Con( 0. W. Nieuwenhuis, C. Praagman and H. L. Trentelman. Eds.). Vol. 4. Groningen, The Netherlands. pp. 208 1-208i.
A. .r.
2692
2b-242
SYMBOUC ANALYSIS AND DESIGN FOR (NONAFFINE) NONUNEAR CONTROL SYSTEMS
Bram de lager *
,.. Faculty of Mechanical Engineering, Eindhoven University of Technology, P.o. Box S13, :,600 MB Eindhoven, The Netherlands. Email: A.G.de.]agerifVw{w.wtb.rue.nl
Abstract. For the analysis and design of dynamical sys tems symbolic tools are a great help. In the past, several algOrithms for nonlinear control systems haw' been implemented in MAPLE. This implemen Lation has been extended to also handle nonafftne models and to include more algorithms. Three examples taken from the literatun.! in this area illustra le the power and limitat ions of the implemen tation. For system models that are more than a little complex symbolic com putation cannot be fully enjoyed du e to the complexHy of r h ~ algorithms that are more than polynomial in some meas ure of the problem size. Keywords. Nonlinear control systems, algorithms, lineari zation, robot conlrol, flexibl e arms, induction mown;, automotive control.
1. INTRODUCTION
so it was decided to extend the NOrt-:-CON package to alleviate this problem and some others.
SymboH c computation, also known as computer algebra. is a powerful tool for solving lOugh and intricate problems
Two approaches were used to make the pa [kag~ suitable for nonafri ne models. First. several computations. e.g., the relative degree (using a definition in (Nijmeijer and Van der Schaft, 1991)), the normal form, the zero dynamiCS, the ex· aCl lineari zation, and the inpUI -outputlinearization, were enhanced to he able 10 gracefully hand le nonaffinc models. S~(o ndl y. an option was add ed to make a nonaffine model affine by extension of the staf(' space and red efinition of its input.
in applied mathematics. With the advt:!nt and improvement of conunerciaHy available packages Ihis fool is becoming general1y available and used. It promises drasti c changes in the way problems, .in fields as diverse as one ca n imagine. are tackled. In the past a closely lied set of symbolic computation functions, together called the NON~ON package, has been presented that solves some problems in the analysis and design o f nonlinear control system, St..."C (Van Essen and De Jager, 1993; De Jager, 1993 ; De Jager, 1994a; De Jager, 1995a; De Jager, 199 5bl. The functi ons available in the package range from the computalion of the zero d ynamics of the model, that does not need \0 have a \'\!ell-defined relative degree, to computing exact Iinearizing or inputoutput I1nearizing feedbacks. All functions are based on constructi ve algorithms and implemenled in MAPLE. A model of the system must be available for the computa· lion. The model. in gene ral a set of nonlinear differential and algebraic equations in the slate. input, and output of the system, \\'as assumed to be affin e in the input. For several problems appearing in practice this was too restrictIve. see (De Jager, 19951» for an example in vehicle dynamics,
Besides Lhese enhancemen Ls to hand1e DonaITine models, the Dynamic Extension Algorilhm was added as an option to give an affi ne model a (well·defined) relative degree, by s tate extension and feedback, in those cases it did not have one yet. furthermore. the package was extended with a new and potentially more efficient algorithm to compute the local zero dynamics for affine models \"';th a well-defined relative degree that was proposc'
2687
The goal of this paper is to describe some of [he enhancements to the NOf'tCON package, to comment on several aspects of the implementation, and to provide some realistic examples of the use of the new facilities. The following Section presents some theoretical background in the fonn of (comments on) algorithms. Section 3 provides several implementation details while Section 4 covers the examples. The paper clo ~es with a discussion of the results and an indication of so me direc tions for future research.
2. ALGORITHMS The two stand ard models used are
{e\:) + g(x) u,
x
(lJ
)' = h(x)
for affine syslems and
x ~ f(x, u ),
)'
~
ht,, )
(2)
for nonaffine sys tems, vvith x c. ~n the state. u E ~ m the input. Y E [R,f' the output, r a Vt'ctor fiel d, 9 representing a se' of vector fields 9j (column-wise). and h is a colulTUl wilh slacked scalar funclions hi' Modifications [() I\\'o algorithms 1rvill bt! discussed (to compute the relative degH't' and normaJ form) that are useful in the case of nonaffine models. Also the Dynamic Extension AlgorHhm ""ill be touched up on.
2.1 Relative degree Th e relative degree of a nonaffine model of a nonlinear
system in (X ", t/ O) is, based on the defi nition in (Nij meijer and Van der Schaft, 199 [, p. 417 ). the se t of integers ri, i = 1. .. . ,p such thal.
,
C~ " l.7- hi( X, u)
-'. 0
'7 k < ri, I ::; i . .: :
r, 1 ::; j
:$ m
J
For affine models both definitions arc, however, equivalent. The two definitjons are implemented. in Ihe function
reldeg.lf no set of integers can be found that fulfills the two conditions, the relative degree is said to be not well· defined.
2.2 Dynamic extension
If a model does nol have a well-defined relative degree. several tools or algorithms are no l appli cable anyrnore, restricting the analysis and design objectives that can be achieved for this model To remedy this, a preliminary extension of the Slale space and Slate feedback are employed. The extension uf lhe state space is necessary, because lhe relative degree is invariant uncler static state feedback, necessitating the use of dynamic state feedback The recursive' algorithm presented in (lsidori, 1 989, p. 384), to which onc is referred for more details, essenlial1y iden· tifies (combinations of) inpul channels thal have to be "delayed" by" an integra ting acti on. The dynamic feedb ack from a single recursio n step i!' u ~ ",(xl +
<:
fl l(''lS
+ {h(x)v,
~ VI,
where V j is the set of "delayed" inputs and V2 the set of inputs tha t are nOI delayed. The functions IX, {J1t and Ih are computed by 1he algorithm. The algoritlun terminates if the relative degree becomes well-defined, or, \vhen this is not possible, aftcr a bounded number of steps, where the bound is at most n. A restriction of the algorithm is that only square MIMO models arc considered. Furthe rmore, additi onal states are introduced in several input-output channels, potentially introd.ucing additional phase lag that may make th~ model more difficult to conlrol. Due to Ihe increase of Ihe dimen· sjon of the state space the model becomes also less suited for symbolic computation: me mory and compute lime requiremenb may increase.
and the m a trix ti.l tlm
C
C
Lf
"h
l(X,
1
uJ
rl1
, -L( hr(x ,u) ("" Urn
is nonsingular in (x ~ , u " ). with L{ hi (X , U ) the lJe de rivative, i.e., the derivaU ve with respec t [0 x of hf(x, u) along f ch, . ( U J = -:;:.- 1" L rhiX,
'x
Remark that (he requiremen t that Adcg is 10 b e non singular is not posed in (Nijmeijer and Van der Schaft, 1991). The main differences with the definition for affine modi) eh is that th('re the term -, -L,. i!' replaced by the deMvaou.I
live along 9) : L gJ and the condiHons do not depend on
u~ .
2.3 Normal (arm and zero dyn amics
Normal forms for nonlinear models are useful for theoretical developments. because onJy a res tricted class of mod· els is to be conSidered, other classes can be transformed to thp normal form class, but are also valuable in their 0\0\'11 right, because they may highlight interesting characteristics of rhe model. One of thm:;(' characteris tics is the zero dynami cs. Rccau se the computalions involved in ob taining a norma l form are often rather tedious, it is of advantage to automa te- this. Normal forms have been defined for several c1asses of models, among them affine mod els with or without a well· defined relative degree and nonaffine models \\1th a well· defined re lative degree, but nol (yet) for nonaifine models without a well-defined relalive degree. The original computation of the normal fonn, that was only suitable for aITine models with a well-defined relative
2688
degree, could easily be adapted to incorporate nonaffine models. It was also extended to affine models without a well-defined rt'lative degree by building the "generalized" nonnal form proposed in (lsidori, 1989, p . 308), making use of the Zero D}1lamics A1gorilhm (lsidori, 1989. p. 290).
Example 1 The first exampl e is a two link mechanism with an elastic joint bet\\'een the two links. Its model is (see (lsidori, 1989. pp. 398-401) .\'5
Given the normal form, the zero dynamics can easily be derived for affine models. If Ihe mod el is nonaffine a complication may aris e because the zeroing input u zcro need nol be unique. Besides the computation of the zero dynamics by means of the normal form. another pOlerllially more efficient algorithm to compute (he local zero dynamics proposed in (f;watny and Blankenship, 1994) was implemented. Onc should consult the original paper for [he details of the algorithm. bUl \'¥l! remark that it (an only be used for affine models with a well-d efined relative degree. For models that do not have a well·defined relalive degree, that can therefore be pUI in the generalized normal fonn only, the zero d ynamics is computed directly by the Zero Dynamics Algorithm, which is possible under some conctitions. This algorithm can also be used for models with a well-defined relati \"{~ degree. circumventing the derivation via the normal form. but not for nonaffine models. There the normal form has In be computed firs!.
f =
with
r,l
=
--6s, (2x4XO + xi)" 2(1 /:"3 - 1/20x,) - (2 " 3c31(1 / 2x, - 5x:, - 353x1)
{sI = (}.I,(2'4" + x~) +
~(2 + 'IC5)(1/2"
-1/20,, )
- (2 + 3cj )( 1/ 2x2 - :lx" - :is]x~ )
hI = 3(2
I
3c,)s,(2x4X6
T
xi,) . (2 + 3cJ)(I/2x l - 1/20,,)
- 3(-1 + 2c3)( 1/ 2x 2 - >X j - 3s:'1 x~)
and 0 0 0
0 0 0
Remark that for affine models \<..'ilh a well-defined relative degree there arc three options ava ilable to compute !he zero dyn am iCS, namely by first computing th(' normal for m, and deriving the zero dynamics from this form, (2) applying the Zero Dynamics Algorithm, (3) using the proposal in (Kwatll) and Blankenship, 1994).
[{i"']
g=
2/1
- 2/1 I
,
2 /1 4(2 + 9q )/I (2 + 3e,) /I -(2 + 3e,)/1
(1)
wlule the output is defined by
Differences occur, at leas t for the implemeOlation in NoN§CoN. in !he coordinates used to represent the results.
3. IMPLEMEN TATION
The (modifications of) algorithms discussed in the previous section are rather straightforwardly implemented in MApl£ usi ng the linear algebra facilities and the procedures to solve sets of nonlinear algebraic equations and sels of nonlinea, partial differential equations (Dc Jage r. 1994b). One of the problems in the implememalion stems from the determination or selection of a suitable solution, that is not unique., for an under-determined set of algebraic equations in the Dynamic Extension Algorithm. which is not grace· fully handl ed by the standard MAPI.' fa cility solve. A work around has been implemented.
4. EXAMPLES
and the following abbreviation') are used 1 = 10 gel cos \' ] "'3 ~ sinx) .
C3 =
Here XI, •• " X3 represent the rotational coordinates of link 1. joint 2, and link 2 respective ly. and X4 ••• •• X6 represent 'he velocities. For 1he parameters appc-uring in the original equations. integers have been filled in to simplify the computations by redu ci ng the intermedia te expression swell. The follo'wing computation of the rela tive degree in the point x " = 0 shows that it is not well·defined. reldeg: the matrix Adeg turns out to be singular! reldeg: the initial state conditions for which the matrix Adeg turns out to be singular are : (all )
The three examples lo be. presented use interesting models of electro-mechanical systems, i.e .• a two link robot v.1th an elastic joint (an affine model without wen-defined relative degree taken from (fsidori, 1989)), a voltage fed Induction motor (a nonaffine model taken from (Nijmeijer and Van d er Schaft, 1991), and a four-wheel vehicle with steering (nonaffinc andlor no '.vell-defined relative degree) described in (De Jager, 1995b).
with
-2/1
2/1]
Ad., = [ (2 + 3[, )/1 -(2 + 3e,)/1 .
A well-defined relative df'grce is achieved with the Dynamic Extension Algorithm in two steps. This is illustrated with the following (condensed) output of the algorithlll-
2689
Step 1: 2
alpha: [- 6 5in(,[3)) , [6 ) x[4)
3 sin(x[l)) x[6] 2
9/ 2 x [3) + 9/ 20 , [2] - 1 si n( x[3] ) x(4) 15/ 2 cos(x[l)) x[l)
3/4 c05(x[3)) x[2)
+
2
- 9/ 2 c05 (x [3)) si n(x [ l)) x[4) ] [ )
reldeg : t he initial state conditions for which the matrix Adeg turns out to be singula r are:
[0) 2
- 5 - 9/ 2 c05(,[3 ) ) beta :
(x[4)
1
o
l/2 cos(x [3])
1
0
vector relative degree
Step 2:
total relati ve degree
0 alpha:
Adeg
0 [ [
beta :
2 (x [l) cos (u[2] ) + x[4) sin(u[2])), I
+
x [4] cos(u[2)))
+ x[3) sin Cu[ 2])jCsigma '_ 5) - x[l) si n(u [2)), 1
0
- 1 - l/2 cos(x[3])
1/16
-u [l](- x[4) sin(u[2]) / (5 i gma ls) + x[l ) sin(u(2))
Adeg:
x[l] cos(u[l])/(sigm. ls)
[4, 4 )
vector re l ative degree WELL-defined! It is easily verified that the two sets of C'( and {3 that arc computed, when co mbined in the correel' way, agree with the result in (Isidori, 1989, p. 4(1) ohilt the I'inal dynamic slale feedback (or d)'1 lamic eXlension) which gives the model a
weB-defined relative degree is u,
[
• >
[- x(4 ) cos(u[2]) / (sigma :. 5) + x[2] cos(uC>])
1
calculating Adeg
rdeg:
.
[1. 1]
+ 2 u Cl) (- x[3) sin(u[l]
1 0
[ 0
2
+ x[l) - x[l] sigm. ls x[l) {x [2] • ------ - - - - ------- --------------- } , x[4] s;gma ls
(x [ l] • 0, x[4] - 0)
[ [
0, x[3) • x[l) sigma ls), ( u[l) • 0),
x[4)
0
1
[
~
2
1
reldeg: the matrix Adeg turns out to be singular! Adeg:
Here Xl and X2 represent components of the stator current, x) and X4 are the corresponding flux components. The speed of the motor w is considered constant. just as all the other model parameters. The total stator flux and stator torque are the out puts. The model does have a weUdefined relative degree in (x ",u' ) ~ (O,o,x" x"uj,O), but it is not full, so L. rj < n, as shov.'ll by
~
I
x[l) cos(u[2)))
are the following.
zero dynam i cs: eta[l]dot
.
+
This result is in agreement with the onc given in (Nijmeijer and Van der Schafo, 1991, p . 419). Because the relative degree for this nonaffine model i ... well-defined, the zero dynamics can be computed, and because the model is non· affine this is done by computing the normal form with the function no rrnforrn_ Selected rc'mlts from the computation
= -(e ta[2] alpha s ;gma Ls eta[l]
· i - f,(x)+P . v, 94d:d
2 + eta [ l] eta[l] sigma Ls beta - omega eta[2]
, ~ ~
~ ~
2
RootOfC_Z
-(0: ,. . 8.);'(.1 - WX 2
~X3 +~ L~ W)q
wx"
t - - - t
(J Ls
f3x~
wxo - (e< + Il)x, - - - + _.
~
r
eta[2] ) s;gma omega eta[l] ls
+
eta[2] ) sigma beta eta[2])
VI·
Example 2 This example considers a voltage fed induction motor. The model is taken from (Nijmeijer and Van der Schaft, 1991, pp. 26 7, 419), with a correction of a printing error. It is nonaffine and given by
f
2 +
+
(JL ~ Ls -aaLsxl + U I COSU2
-CHfLsX,!
T
Ut
sm Uz
1
- RootOf(_Z
/ (etaC2) 5igma ls), eta[l)dot -
Ul cos U2 "'--'7-=
2
aLs
alpha 5igma Ls eta[l] RootOfC_Z
Ul smU2
~--;----O
a{ s
2
(3)
2 + eta[2] )
eta[l) zeroing input uzero: [ 0, locuzero[2]
2690
The interpretation of Ihe lasl line of the resull is simple. To keep the output y ~ 0 for all t, the inpul u\ should be 0, and the value of input Uz does not matter. This is eas y to check from the differential equations (3). The input u, does only occur in combination with Ul. When u, :: 0 the value of U2 is irrelevant. The dynamics of the model Wlder the constraint that y = 0 is given by the IwO differential equations for rj . The model can al so be inpul -output Iinearized with the function inoutlin. The raw results of the 10 linearizalion are voluminous. From the linearizing feedback onJy the second component of u is prese nt ed, as fo ll ows. io linearizing feedba ck ( uli n[Z) ): arctan ( - 2 v[ 2] s igma Ls x[3] 2 +
2
2 K[2] s igma 2
+ 2 x [2]
K[ ] ] al pha x[I]
Ls 2
2 +
2 omega x[ 3]
Example 3 This las l example considers a vehicle steering s tability problem. It is posed more extensively in (De lager, 19'1S b ). Refer to trus for the comple te model equations, that will not be repeated here. The ori ginal model is non· affi ne. In the present research the model is simplified by assuming rhe lateral tire forces Ft and Fr , at front and rear, to be constant and n OI functio ns of the s late and s teering angle. The reason for thi s si mplifi cation is th at the computation fo r the original model could not be completed. Several rt'suits \vill be dis cussed concerning the simplified . but s lill nonaffine. model ~i ven by the following equations
Ls
'4
",_
x(4) alpha - Ft s in( uI + x])
3
Ft- cos(:,.,
h _ [
2
-
,[3]
2
omega x[ i] sigma Ls 2
- 2 x [ 3] +
a1pha sigma Ls )([ 2]
, (2] sigma Ls ve l ] - ,[4] v(l ] )
/
/
! 2
2
2
(2 . [1]
sigm.
Ls 2
+ 2 x[l] s igma +
. ( 3] alph a 2
ls
x[l] sigma Ls vel]
x[4 ] al pha x[2] +
2 v[ 2] s;gma Ls x[4]
2
- 2 x[ 3]
a1pha si gma Ls x[ IJ
+ 2 x[3] x (2 ] sigma Ls beta x[4] - . (3J vel]
2 x[ 3] omega xCI] 5igma Ls x[4] 2
- 2 x [4J
alpha sigm. Ls x[l] 2
2 x [4]
x[l] sigma Ls bet a
+
~2 sin x3
x .j cos x~ + X ,, sin Xl - lid (X l +p COSX] )- +(Xz + p s tnx] )2
x,.
alpha s i gma Ls x [ 2]
+ 2 x[3]
Xj
]
-Ra'
where X2 represent the position of the COM (center-ofmas s) and Xl the rota tion around COM, the s tales X4 ,· .• • X6 represent the corresponding velocities. The ou tput y is defi ned in such a \'\'ay that it is Z (~ro for a curving maneuver vvith constant radiUS Rd and co nstanl forward speed component Vd . The inpul UI is the 'itccring angle and U 2 is the Iraction force. Hrst. the relative degree of the model is computed. It is we ll-defined. See the output of the funclion r e ldeg.
2
x[4]
t! cos
co~ Xl
aF( cos i~~_ bFr j,
+ 2 x[4] omega x[2] s; gma Ls x[l]
- 2
+ XJ) +
F,. sin x3 + Uz
M
2 x[ 4] x[l] s igma Ls beta x[3 ]
- 2 omega ,[4]
3
, [4] + 2 omega x[4] ))
Due to the complexit\' of the outpul, it is difficult to verify the resull by simple inspection.
x[2] sigm a Ls beta - 2 omega x[3]
2 x[ 3]
omega x[2] s igma Ls
2
sigma 2
+
2
- 2 x[4]
vector r el ati ve degree - [ 1. 2 ] total rel at ive degree
- 3
The zero dynamics can be computed successfull y, but it is complicated. making it tedious to derive its characteristi cs, c.g.• to decide if the zero dynamics is s table. Second, sta te extension (an integra tor in the firs t input channel) and inpul redefinition have been used to get a model tha t is affine. It appears that the affine model does not have a well-defined relative degree, so the Dynamic Ex· tension Al gorithm is applied, gh·i ng an affine model with well·defined relative degree, that is of rugher order than th e original onf'. The computation of the normal fo rm for this ex tended model could not be completed due to interme· dia te express ion swell. The zero dyn amic could therefore not be computed. The vehicle model presented in (Kwatny and Bla nkensrup, 1994) has also been investigated. The algorithms to corn· pUle the zero d)-llamics were able to finish their computation for this mod el. So it appeared tha t it is m uch easier to handle.
2691
The results of the first two examples are satisfactory. For the last example, which has a quite complicated model description, the results are disappointing due to ., the complexity of the algorithm s, • limitations of the compuler algebra system. Based on (he M>\PlE resuJt s for several examples, not all presented here, the follm"ing conclusions can b e drawn. • To make a model affine and to give it a well·defined relative degree are relatively simple problems, that can be solved rather eaSily by symbolic computation tools, making a non affine model affine is nol always advantageou·s, the affine model of larger dimension may be more involved for compulational purposes, ., the: nc:wly proposed algorithm 10 compute the zero dynamics is more efficientl han the other IWO methods presented, • the full vehicle zero dynamics problem can be solved by none of the algorHhms prest:nt.1y available, mod el simplifications are mandatory.
s. m SClfSSlON The extensions to the NONKON package enable one to treat a larger class of models, including those that occur in practice for frequently encountered systems. The extended NOr-.GCON package is sw.:cessful for low order
models that do nOI contain large intricate expressions. For large order models the package is less successful, among othe rs due to an intrinsic limitation in MAPLE for the han dling of large numbers 01· obj~c t s. To il\·oirl this limitation, and so 10 fully exploit the po\,,rer of symbolic computation, onc can hardly avoid to become- familiar with the peculiar· ities of the program used for implementation. Ideillly one would want to avoid this. This shou ld be impro\,ed. Some other problems stem from {he ne(~d to solve intricate sets of nonlinear «(partial) diffcrentiall equations. For these kind of problems - very diffif.:ull ones - no powerful, builtin, read y to use, functions are a,·ajJable. Therefore, some ad-hoc facilities were implemented to improve the capability of MAPLE in thi s respect, see, e.g., (l) e .lager, 1994b). This should be improved also. The main trouble lies, however, in the fact Ihal the algorithms (and their implementations) are rather complex. While in numencaJ computations onc is ready 10 accept a computational complexily growth rate that is a cube of the problem size, in s}mbol1c computations one should be ready to accept a double exponential relatio n, see (DeJager, 19940). This means that slightly larger problems lban those used in this paper will readily become intractable, gathering dark clouds at the horizon for practical symbolic computation.
Acknowledgements The extensions to NO~CON were coded by Gose Fisher starling from the code by Harm van Essen (1992).
(I ~94),
6. REFERENCES Blankenship, G. L, R. Ghanadan, H. G. Kwatny, C. LaVigna and Y. Polyakov (1995). Tools for integrated modeling, design, and nonlinear control IEEE Control Sy.... terns /1ag. 15(2), 65-79. de Jager, Hram (1993). Zero d)1Iamics and input·output ex· aCI Iincarization for systems withoul a relative degree using Maple. In: Proc. of the 199.3 Inlernat. Symp. on Nonlin ea r Theory and its Applications. VoJ. 4. IElCE. Honolulu, Hawaii. pp. 120:;-1208. de Jager. Bram (19940). Computer aided hybrid analysiS and design 01" nonlinear control sys tems . In: .4ppli· calion o/" Mullivariable ;)"stem Techniques (AMST94) (R. Whalley, Ed.). IMC. London: MEP. Bradford, UK . pp. 247-254. de Jager, Rram (l994b). Symbolic solution for a class of partial differential equations. In: Prm. :. of the Rhine Wurkshop on Computer Alqebra Oacques Cal met, Ed.). Karlsruhe, (;ermany. pp. 84-88. de Jager, Bram (l ~g5a). Symbo lics for control: Maple used in solving Ihe exact linea rization problem. ]n: Computer Algebra in Industry 2: Problem Solving in Prac· lice (Arjeh M. Cohen, Leenderl van Gaslcl and Sjoerd Verduyn Lunel, Eds.). pp. 29 1-31 J. Jo hn Wile)" & Sons, Lld .. Chichester. de .lager, Bram (l995b). The use of symbolic t:umputation in nonlinear control: Is it Viable'? IEEE Tran s. Automat. ControI40(J), 84- 89.
Fisher, J. P. j. (Gose) (1994). NONLlNCON: Symbolic analy' sis and deSign package for nonlinear conlrol systems. fo.·ISc th{~sis. Eindhoven University of Technology. Fac. of Mechanical Engineering. Report WFW 94.09S. Isidori, A. (1989). Nonlinear Con trol Systems: An Inlroduc· tion. 2nd ed .. Springer·Yerlag. Berlin. Kwatny, H. G. and G. L. Hlankens hip (1994). Symbolic tools fur variable S{fucture conI rol system design: the zero dynamics. In: Proe. the IEFI' Workshop on Robust
or
Control via Variable Structure & l.yapunov Techniques.
IEEE. Benevento, Italy. Meller-Pcdersen• .lens .md Martin Pagh Pererscn (1995). Control of nonlinear plan1s - Volumes I and 11. MSc thesis. Mathematical Institute, Technical University of Denmark. Lyngby. Denmark. van der Schaft (1991). Nonlinear Dy. Nijmeijer, H. and namical Contml Systems. SpJinger·Yerlag. New York. Corrected 2nd printing. Post, B. A. (19<)0). Computer algebra analysis of nonlin· ear discrete-time systems. MSc rhesis. Dept. of Applied Mathematics, University of Twente. Enschede, The Netherlands. van Essen, H. (l99l). Symbols s peak loudet than numbers: Analysis and design of nonlinear control systems with the symbolic computation system MAPLE. MSc thesiS. Eindhoven University of Technology. Fae. of Mechani· cal Engineering. Report WFW 92.061. van Essen. Harm and Bram de Jager (1993). Analysis and design of nonlinear control systems vvith the symbolic computation sys tem Maple. In: Proc. of the second European Control Con( 0. W. Nieuwenhuis, C. Praagman and H. L. Trentelman. Eds.). Vol. 4. Groningen, The Netherlands. pp. 208 1-208i.
A. .r.
2692