Nonlinear control notes

8/3/2019 Nonlinear control notes

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Nonlinear Control System Design

. Objectiv e o f Con tr ol des ign: G iven a physic al sy stem to be con tro lledand the specif ications of i ts des ired behav io r. construc t a feedbackcon tr ol law to make the c lo sed loop sy stem d isp lay the desired behav io r.. Two bas ic type s o f con tr ol p roblems:. Regula tion: G iven a non lin ea r dynamic sy stem describ ed by

x' = f(x,u,t)

fin d a contro l law u su ch th at, sta rtin g from anywhere in a reg io n w, th estate x te nd s to 0 as t-> in f.. Track ing: G iven a non lin ea r dynamic sy stem describ ed by

x' = f(x,u,t) Y = h(x)

an d a d esired outp ut trajecto ry yd, fin d a c on tro l law fo r th e in pu t u su chthat, s ta rting f rom any in itia l sta te in a region w, th e tra ck ing e rro ry (t) -yd( t) goe s to zero, while th e who le s ta te x r emain s bounded .

. Procedure for Cont ro l Design:1 . Specify the des ired behav io r.2 . Mode l th e physic al p lant by a s et o f d if fe rentia l equations .3 . Desig n a contro l law fo r th e sy stem .4 . Ana lyze and s imu la te th e re su lting con tro l system.5 . Imp lement th e con tro l sy stem in hardwa re .

. Feedback and Feedforward.In a contro l law, th e fe ed fo rward p art is u sed to can cel th e effects o f k nowndis tu rbances and provide anti cipa tive act ions in tracking tasks.The feedback par t then s tabi li zes the tracking error dynamics.

. Many o f th e trac kin g contro lle rs c an b e w ritten in th e fo rmu =feedforward + feedback

1


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Methods for Nonl inea r Cont ro l Design .

Tri al and Error :B ased o n an aly sis meth od s, u se an aly sis to ols to g uid e th e search fo r acon trol le r. Expe ri ence and intuit ion a re c rit ical. No t sui table for complexsystems.

Feedback Linearisation:Dea ls w ith te chnique s f or tr an sfo rm ing o rig in al s ystem mode ls in tol inea r equ ival en t models o f s imp le r form.Transf orm th e non lin ea r sy stem in to (a f ully o r p artia lly ) lin ea r sy steman d u se th e well k nown an d p owerfu l lin ea r d esig n te ch niq ues tocomplete contro l design.App lic ab le to inpu t s ta te lin ea risa ble o r m in imum pha se sy stems andtypical ly requires ful l s ta te measurement .Doe s not gua rante e robu stn ess in f ac e o f p arame te r unc er ta in ty o rdisturbances.

Robust Control:

In p ure mod el b ased n on lin ea r co ntro l, co ntro l law is b ased o n anominal mode l o f th e phy sic al s ystem - uncerta in tie s a re notconsidered.

In robu st non lin ea r con tro l, c on tr olle r is d es igned based oncon sid era tio n o f both nom inal mode l and some cha ra cte riz atio n o f th emodel uncertainties.Appli cable to a speci fi c c la sses o f non linear sys tems and generallyrequire s ta te measurements .

Adapt ive Control:

Dea ls w ith unc erta in o r time vary ing systems.Adapt ive contro l i s inherently nonl inear.App lic ab le main ly to s ystems w ith known dynam ic s tr uc tu re , butunknown constant o r s lowly varying parame te rs .U sed mostly fo r S ISO systems.

2


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CS685 - Lectu re Note s, Ja na Kosecka

1 Controlling Robotic Systems

Suppose that we would like the robot for follow particular trajectory. We have already lookedat ways how to generate trajectories specified in terms of polynomials given the constraints interms of initial positions and velocities. From the trajectory generation step we will obtain desiredtrajectories B d(t) , B d (t) , B d (t)(e.g . where Bd ( t ) is a vector of joint variables of the manipulator) asfunctions of time. So one way how we can achieve the desired objective is simply generate torques7(t), by substi tuting B d(t) , B d (t) , B d (t)into equation 1 and obtain

7 = M(B d ) B d+ C (B d , B d)B d+ N (B d , B d ) (1 )

Hence if we start with the sam e initial condition as the one we specified in our trajectory generationstep, we will indeed obtain the desired trajectories, thanks to the property of uniqueness of thesolutions of differential equations, given that the initial conditions are the same. This controls tr ategy is calledopen-loop c on t r o l and would work w ell if the model represented by our equationsis perfect. In real life the models are rarely perfect and the are many other effects which are

very difficult to model explicitly. So for example if our initial condition would be slightly off sayB (O )1 =Bd (O )we would never be able to correct to this error. In our example the error can be simplydefined as:

e = B - Bd, e =B- Bd, e =B- Bd

What we would like to do is design a control law which will have a capability to correct for the error.Hence we would to have some systematic means how to incorporate the error in the computationof the control law (in this case the expression for 7) such that the final trajectory will graduallyconverge to the desired trajectory. The type of control law is calledfe edba ck con tro l law and thesystem together with the feedback control law is referred to asclo se d lo op system . The choice of

appropriate control law will then guarantee that our system will indeed exhibit the desired behaviorin spite of the inaccuracies if the model and the errors in initial conditions. There area of controltheory offers many techniques for the design of appropriate control laws as well as quantificationwhether the control law is doing "the right thing".


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Examples Lets have a look at some examples and different components of the control law.In thefollowing examples we will assume that the output of the system y is directly the state x. Considerm ass-spring-dam per mechanism , first in the absence of any external forces

mx + ksx + kdX= 0

In the homework we had a chance to observe how the behavior of the state x= [x,xjT d ep ends onthe choice of constants ks, kd and initial conditions. Hence the open loop dynamics of the systemIS

x(t) =[ -~ -~ ] x(t)

and the control input in this case is zero. Suppose now that we will apply some external force ofthe following form Fext = -kpx - kvx which is proportional to the current position and currentvelocity of the m ass. This would yield following dynam ics equations

mx + ksx + kdX - -kpx - kvxmx + (k s+ kp)x+ (kd+ kv)x - 0

(2 )

(3 )

Note that the second equation describes the system of the same type, but by adding the externalforce terms, we effectively changed the coefficients of the system and hence change the system'sbehavior.

Example For mass-spring-damper system above, suggest the formula forFext such that thedynam ic equations of the closed-loop system will have the following have the following dynam ics,will behave as a simple point mass

x = Fext

We did this example in class(Fext will be some function onF).

Example Consider again a simple point mass system (with no damping and no friction).

x = Fext

We would like the point mass follow particular trajectory which was computed ahead of timeXd, Xd, Xd.Suppose we first apply external force

Fext = Xd

If we simply use this control law the dynamics of the system would be

x = Xd

In case we wou ld like to compensate fo r th e p ossib le in itial errors inx, x le t's consider the followingcon tro l law

Fext = Xd - kv e - kp e

with e= x - Xd and e= x - Xd. This would yield the following system dynam ics

x - Xd - kv e - kp e

e + kv e + kp e - 0

(4)

(5 )


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The above equations now describe the error dynamics. We can describe this dynamic system interms of state equations and investigate the behavior of the error as a function of time and choosethe constan ts kp, kv appropriately to yield the desired performance. In the context of robotics thiscontrol law is also referred to ascomputed to rq ue law. The Xd part of the control law is also referredto as feed-forward term and it would be sufficient if our model is perfect.

Proportional Derivative Control Given some desired trajectoryXd, Xd the sim plest controllaw which we can apply is to generate forces which are proportional to the errors between desiredand current position and velocities

Fext = -kve - kp eCompared to previous case this control law has no feed-forward term. In practice, when theobjective is to track very complex trajectories it is quite hard to achieve without the feed-forwardterm. Furthermore proportional derivative control law leaves some steady state error. In order tocompensate for steady state errors additional termin te gr al te rm can be added to the system.

Fext = -(kve + kpe + ki

Jedt)

. What is difference between closed-loop an open-loop system ?

. What is the role of feed-back in the control system?

. What is the role of feed-forward term in the control system?

Note that the names of the terms actually correlate with the way the arrows are drawn in thesystem.

. The proportional control law at each instance of tim e responds to the current error in position.How fast should it respond is specified by constantkp which is called proportional gain. Ifthe gain is too high the system can overshoot and eventually lead to oscillations. Dampingcan be used to prevent the oscillations.

. The derivative control is used to correct momentum of the system depending how far awayit is from the goal. It is proportional to the derivative of position (or error).

. The integral control provides another improvement to the control law since it integrates thesteady state errors over time to compensate for errors.

1.1 Mobile agent control strategies - low level behaviorsIn the context of mobile robotics, in many instances the task cannot be clearly specified in termof trajectories, unless the agent is involved in some precise maneuvering, such as lane change inthe autonomous driving, or parallel parking. However there are many useful low level strategiesone can develop in order for the robot to move around in the environment and accomplish someelementary tasks. The design of these strategies follow the same principles, but are more tied to theenvironment the robot resides in. Before we review some basic strategies we will discuss differentways how to represent environment and address the problem of finding a path from current todesired configuration.


27/54

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IL ~\ + \5\l .

yVl. wa..t\k h writ ~ V .-U J .ekr(U 1 vcuJ CA.-bLU~ L-Jo-rlNWV ~c9\. ..rk.2. IV cm P Ko!~ I ~..u ~ IA-tVr..m "ipet?Jt.i~ 1, ' (1 '< J " ' fW 1 . J )V ~~

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45/54

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t:o\:;

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48/54

r~eA 2-

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~t

fL.oCz:)J Lbo

1(I ~ - LI.b => j


51/54

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l~J"C c:= I t:o

tM:::

f [x(t:)l" [NDtr ) - ~ (NaCt:) - " ' ,sJn (?l(t:))l d -Z :c:h: ~t:o

n n

+ ~(t)NC)("l:)- ?t ( ~)~oLb3) - .KtC:\ x(~)' t- LKG1 \X ( t o..t

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l= /

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t) ()

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52/54

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1E rror s~~te.rn dwdopn1e.nr:

l, ,

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... e = - e - rf t 'l,

'l- = -rf - (K1-t\)tt.T e - e.+

~= e t~ +,:~t-1(")l'~=M (? l) (Y q- 2.r4- ef) -)(:1 Mex) ~ t f b{ J X) - LA

f'It(?lJrl= -I'-z.M (~)YL +-N (?(, x',r) - M (x) (2.rft~f)-U -

wh.tA..t. N (.) = M ('){) X~ T f (-;r/X),. ,

Ld" "'d(t) ~ 1'J'?


53/54

N :: : tV - N .t- M (? t ) (2 . r1T ~ f) + -iM e x )YL

/IN U .< ~C}I"Z-H)n~1\ z == ' [~ ~~ r t- 'L1

I5bM ;.li~ tU1~$i.6:

r

'L 'Z.. 1 "L

1'Z..

V:: 1.1'1 ( ?l)rt.. +- 1. e~ +- r~ +- e2.- 2- r :z . r 2-'1 ,.."v =

~ M (X)'t + M1[- + 'l(I


54/54

V III ,: ; - (' - ~ '(~: Ii)) I1 z\1 '2 .

6J ),In!rl:..l' ~ a,t- P, .nu. f'l t ek. hO\..J bo b~ ~~ ~.t-

p.:: Zb - ~ l~"t~ +~~ tet)(Nd - ~IS~()(~te~) dt:~t

:: ib - f(y t'f) (tJ~-K, s~n (Y ) dt

~

'; K 't -= - \ N di\+ 1 tJcjI tru...n P:r 0t'\

2-1,:: L kit. J e(ta )/- e (t;e NJ (bo)L:'I

v ~ - (, - ! '2 .(\\~It) ) n-z .u 'Z..1 ~'t (~ Vl).~) )V ~ - BlIz1\'2 . ~ Kt'I:> ~2. ( f , \"fll7))\)

v >0V :5 -6(t) a

(t) ~ 0

(r) ~ B 2 ~'T~ ~(t) ~ I,m

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Nonlinear control notes

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