Lecture 2. Today s Goal. Linearization Around a Trajectory. Example - Linearization around equilibrium point. Linearization Around a Trajectory, cont.

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1 Lecture Toda s Goal Material Linearization Stabilit definitions Simulation in Matlab/Simulink To be able to linearize, bot around euilibria and trajectories, eplain definitions of local and global stabilit, ceck local stabilit and local controllabilit at euilibria simulate in Simulink, Glad& Ljung C.,., ( Kalil C.3, part of 4., and 4.3 ) (Slotine and Li, pp 4-57) Lecture slides Eample - Linearization around euilibrium point Linearization Around a Trajector Te linearization of ẍ(t)=f((t))= l sin(t) around te euilibrium =nπ is given b (t)=f ( ) (t)= l ( )n (t) Idea: Make Talor-epansion around a known solution{ (t),u (t)} Neglect small terms (i.e., keep te linear terms, as tese will locall dominate over te iger order terms). Letd /dt=f( (t),u (t)) be a known solution. How will a small deviation{,ũ} from tis solution beave? d( + ) dt =f( (t)+ (t),u (t)+ũ(t)) ( (t),u (t)) (t) ( (t)+ (t),u (t)+ũ(t)) Linearization Around a Trajector, cont. minute eercise: Linearize ẋ =4 3 +u around te solution (t)= t u (t)= Let( (t),u (t)) denote a solution toẋ=f(,u) and consider anoter solution((t),u(t))=( (t)+ (t),u (t)+ũ(t)): ẋ(t)=f( (t)+ (t),u (t)+ũ(t)) =f( (t),u (t))+ f ( (t),u (t)) (t) + f u ( (t),u (t))ũ(t)+o(,ũ ) Hint: Plug-in(t)=t + (t), epand te epressions, and finall remove iger order terms ( ) of. ( (t),u (t)) (t) ( (t)+ (t),u (t)+ũ(t)) State-space form Hence, for small(,ũ), approimatel (t)=a( (t),u (t)) (t)+b( (t),u (t))ũ(t) were (if dim=, dimu=) A( (t),u (t))= f ( (t),u (t))= B( (t),u (t))= f u ( (t),u (t))= f f f f f u f u ] ] ( (t),u (t)) ( (t),u (t)) Note tata andbare time dependent! However, if we don t linearize around a trajector but linearize around an euilibrium point( (t),u (t)) (,u ) tena andbare constant. Linearization, cont d Te linearization of te output euation (t)=((t),u(t)) around te nominal output (t)=( (t),u (t)) is given b ((t) (t))=c(t)((t) (t))+d(t)(u(t) u (t)) were (if dim= dim=, dimu=) C(t) = = (,u ) ] D(t) = u = u (,u ) u ] ( (t),u (t)) ( (t),u (t))

2 Eample: Rocket Letu (t) u >; (t) m(t) ḣ(t)=v(t) v(t)= + veu(t) m(t) ṁ(t)= u(t) (t) (t)= v (t) ; m (t)=m u t. m (t) Part II: Stabilit definitions Linearization: (t)= v eu m (t) (t)+ v e m (t) ũ(t) Definition: A norm function : R n R satisfies te following tree properties: =if and onl if (iff)=, >oterwise. a =a, for an positiveaand an signal vector. (Te triangle ineualit) + + Local Stabilit Considerẋ=f() weref()= Definition Te euilibrium=is stable if, for anr>, tere eistsr>, suc tat + + () <r = (t) <R, for allt Oterwise te euilibrium point is unstable. Definition: (Euclidean norm) =( n) / 3 = 3 (t) r R Asmptotic Stabilit Global Asmptotic Stabilit Definition Te euilibrium=is locall asmptoticall stable (LAS) if it ) is stable ) tere eistsr>so tat if () <rten (t) ast. Definition Te euilibrium is said to be globall asmptoticall stable (GAS) if it is LAS and for all() one as (t) ast. (PD-eercise: Sow tat ) does not follow from )) Lapunov s Linearization Metod Part III: Ceck local stabilit and controllabilit Teorem Assume as te linearization ẋ=f() d dt ((t) )=A((t) ) around te euilibrium point and put α(a)=ma Re(λ(A)) If α(a)<, tenẋ=f() is LAS at, If α(a)>, tenẋ=f() is unstable at, If α(a)=, ten no conclusion can be drawn. (Proof in Lecture 4)

3 Te linearization of Eample ẋ = + +sin( ) ẋ = cos( ) 3 5 at = givesa= 3 5 Eigenvalues are given b te caracteristic euation =det(λi A)=(λ+)(λ+5)+3 Tis gives λ={, 4}, wic are bot in te left alf-plane, ence te nonlinear sstem is LAS around. Teorem Assume as te linearization Local Controllabilit ẋ=f(,u) d dt ((t) )=A((t) )+B(u(t) u ) around te euilibrium(,u ) ten (A,B) controllable f(,u) nonlinear locall controllable Here nonlinear locall controllable is defined as: For evert> and ε> te set of states(t) tat can be reaced from()=, b using controls satisfing u(t) u <ε, contains a small ball around. 5 minute eercise: Is te ball and beam 7 φ 5ẍ= + sinφ+ r φ 5 nonlinearl locall controllable around φ= φ==ẋ=(wit φ as input)? Remark: Tis is a bit bit more detailed model of te ball and beam tan we saw in Lecture. However... Eample An inverted pendulum wit verticall moving pivot point pi u Bosc 8 (Automatic parking assistance) Multiple turns parking lot>car lengt + 8 cm More parking in lecture φ(t)= l ( +u(t))sin(φ(t)), wereu(t) is acceleration, can be written as ẋ = ẋ = l ( +u)sin( ) Eample, cont. Te linearization around = =,u=is given b ẋ = Bonus Discrete Time Man results are parallel (observabilit, controllabilit,...) ẋ = l It is not controllable, ence no conclusion can be drawn about nonlinear controllabilit However, simulations sow tat te sstem is stabilized b if ω is large enoug! u(t)=εω sin(ωt) Demonstration We will come back to tis eample later. Eample: Te difference euation k+ =f( k ) is asmptoticall stable at if te linearization f (tat is, witin te unit circle). as all eigenvalues in λ < 3

4 Eample (cont d): Numerical iteration k+ =f( k ) to find fied point =f( ) Often te onl metod ẋ=f() Part IV: Simulation ACSL Simnon Simulink Wen does te iteration converge? f() f() f()? f() F(ẋ,)= Omsim ttp:// cace/omsim.tml Dmola ttp:// Modelica ttp:// Special purpose Spice (electronics) EMTP (electromagnetic transients) Adams (mecanical sstems) Simulink Simulink, An Eample > matlab > > simulink File -> New -> Model Double click on Continuous Step (in Sources) Scope (in Sinks) Connect (mouse-left) Simulation->Parameters Step s+ Scope Coose Simulation Parameters Save Results to Workspace t Clock To Workspace Signal Generator s+ To Workspace Ceck Save format of output blocks ( Arra instead of Structure ) > > plot(t,) (or use Structure wic also contains te time information.) Don t forget Appl How To Get Better Accurac Points were correct, onl te plot was bad Modif Refine, Absolute and Relative Tolerances, tegration metod Refine adds interpolation points: Refine = Refine =

5 Use Scripts to Document Simulations If te block-diagram is saved to stepmodel.mdl, te following Script-file simstepmodel.m simulates te sstem: open_sstem( stepmodel ) set_param( stepmodel, RelTol, e-3 ) set_param( stepmodel, AbsTol, e-6 ) set_param( stepmodel, Refine, ) tic sim( stepmodel,6) toc subplot(,,),plot(t,),title( ) subplot(,,),plot(t,u),title( u ) Linearization in Simulink Submodels, Eample: Water tanks Euation for one water tank: ḣ = (u )/A = a Corresponding Simulink model: /A f(u) s Sum Gain tegrator Fcn Make a subsstem and connect two water tanks in series. Subsstem Out Subsstem Linearization in Simulink, cont. Use te command trim to find e.g., stationar points to a sstem > > A=.7e-3;a=7e-6,g=9.8; > > % Eample to find input u for desired states/output > >,u,]=trim( flow,..],],.) =.. u = e-6 =. Use te command linmod to find a linear approimation of te sstem around an operating point: > > aa,bb,cc,dd]=linmod( flow,,u); > > ss=ss(aa,bb,cc,dd); > > bode(ss) Linearization in Simulink; Alternative B rigt-clicking on a signal connector in a Simulink model ou can add Linearization points (inputs and/or outputs). Simulation of JAS 39 Gripen Computer eercise reference pilot T_f.s+ prefilter Kf = A+Bu = C+Du plane dnamics Cteta teta pitc angle put Point /A Gain s tegrator a*srt(*g*u]) Fcn Output Point command upilot L states Clock t time Start a Control and Estimation Tool Manager to get a linearized model b Tools -> Control Design ->Linear analsis... were ou can set te operating points, eport linearized model to Workspace (Model-> Eport to Workspace) and muc more. Simulation Analsis of PIO using describing functions Improve design Toda Linearization, bot around euilibria and trajectories, Definitions of local and global stabilit, How to ceck local stabilit and local controllabilit at euilibria Simulation tool: Simulink, Net Lecture Pase plane analsis Classification of euilibria 5