Today s Goal. Lecture 2. Linearization Around a Trajectory, cont. Linearization Around a Trajectory. Linearization, cont d.

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1 Lecture Today s Goal To be able to linearize, bot around euilibria and trajectories eplain definitions of stability ceck local stability and local controllability at euilibria simulate in Simulink Material Glad& Ljung C.,., ( Kalil C.3, part of 4., and 4.3 ) Lecture slides Linearization Around a Trajectory Idea: Make Taylor-epansion around a known solution { (t), u (t)}. Let be a known solution. d = f How will a small deviation {, ũ} from tis solution beave? d( + ) = f( (t) + (t), u (t) + ũ(t)) Linearization Around a Trajectory, cont. Let denote a solution to ẋ = 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 + f ( (t), u (t)) (t) + f u ( (t), u (t))ũ(t) + O(, ũ ) (t) = f ( (t), u (t)) (t) + f u ( (t), u (t))ũ(t) + O(, ũ ) (t) ( (t) + (t), u (t) + ũ(t)) (t) ( (t) + (t), u (t) + ũ(t)) State-space form Hence, for small (, ũ), approimately were (if dim =, dim u = ) (t) = A(t) (t) + B(t)ũ(t) A(t) = f ( (t), u (t)) = B(t) = f u ( (t), u (t)) = [ f f [ f u u Note tat A and B are time dependent! However, if we don t linearize around a trajectory but linearize around an euilibrium point (, u ) ten A and B are constant. Linearization, cont d Te linearization of te output euation y(t) = ((t), u(t)) around te nominal output y (t) = is given by ỹ(t) = C(t) (t) + D(t)ũ(t) were (if dim y = dim =, dim u = ) C(t) = ( (t), u (t)) = D(t) = u ( (t), u (t)) = [ [ u u Eample - Linearization around euilibrium point Eample: Rocket Te linearization of ẍ(t) = g sin (t) l around te euilibrium = nπ is given by (t) m(t) ḣ(t) = v(t) v(t) = g + veu(t) m(t) ṁ(t) = u(t) (t) = g l sin(nπ + (t)) g l ( )n (t) Let u (t) u > ; (t) = (t) v (t) m (t) ; m (t) = m u t. Linearization: (t) = v eu m (t) (t) + v e m (t) ũ(t)

2 Outline Local Stability Consider ẋ = f() were f( ) = Definition Te euilibrium is stable if, for any R >, tere eists r >, suc tat () < r = (t) < R, for all t Oterwise te euilibrium point is unstable. (t) r R Asymptotic Stability Global Asymptotic Stability Definition Te euilibrium is locally asymptotically stable (LAS) if it ) is stable ) tere eists r > so tat if () < r ten (t) as t. Definition Te euilibrium is said to be globally asymptotically stable (GAS) if it is LAS and for all () one as (t) as t. (PD-eercise: Sow tat ) does not follow from )) Lyapunov s Linearization Metod Teorem Assume as te linearization ẋ = f() d ((t) ) = A((t) ) around te euilibrium point and put α(a) = ma Re(λ(A)) If α(a) <, ten ẋ = f() is LAS at, If α(a) >, ten ẋ = f() is unstable at, If α(a) =, ten no conclusion can be drawn. (Proof in Lecture 4) Eample Local Controllability Te linearization of ẋ = + + sin( ) ẋ = cos( ) 3 5 at = gives A = 3 5 Eigenvalues are given by 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 system is LAS around. Teorem Assume ẋ = f(, u) as te linearization d = A + Bũ around te euilibrium (, u ) ten te nonlinear system is locally controllable provided tat (A, B) controllable. Here local controllability is defined as follows: For every T > and ε > te set of states (T ) tat can be reaced from () =, by using controls satisfying u(t) u < ε, contains a small ball around.

3 5 minute eercise: Is te ball and beam ẍ = φ + g sin φ + r 5 φ nonlinearly locally controllable around φ = φ = = ẋ = (wit φ as input)? Remark: Tis is a bit more detailed model of te ball and beam tan we saw in Lecture. 6 IEEE telligent Veicles Symp., Gotenburg Eample An inverted pendulum wit vertically moving pivot point pi u φ(t) = (g + u(t)) sin(φ(t)), l were u(t) is acceleration, can be written as [Eveste, Ljungvist, Aeill More parking in lecture. ẋ = ẋ = l (g + u) sin( ) Eample, cont. Te linearization around = =, u = is given by ẋ = Bonus Discrete Time Many results are parallel (observability, controllability,...) ẋ = g l It is not controllable, ence no conclusion can be drawn about nonlinear controllability However, simulations sow tat te system is stabilized by 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 asymptotically stable at if te linearization f (tat is, witin te unit circle). as all eigenvalues in λ < Eample (cont d): Numerical iteration Outline to find fied point k+ = f( k ) = f( ) Wen does te iteration converge? f() f()? f() f() 3

4 Simulation Simulink Often te only metod ẋ = f() ACSL Simnon Simulink F (ẋ, ) = Omsim Dymola Modelica ( Special purpose Spice (electronics) EMTP (electromagnetic transients) Adams (mecanical systems) > matlab > > simulink Simulink, An Eample Coose Simulation Parameters File -> New -> Model Double click on Continuous Step (in Sources) Scope (in Sinks) Connect (mouse-left) Simulation->Parameters s+ Step Scope Don t forget Apply Save Results to Workspace How To Get Better Accuracy Clock t To Workspace Modify Refine, Absolute and Relative Tolerances, tegration metod Refine adds interpolation points: Signal Generator s+ y To Workspace.8.6 Refine =.8.6 Refine = Ceck Save format of output blocks ( Array instead of Structure ) > > plot(t,y) (or use Structure wic also contains te time information.) Use Scripts to Document Simulations Submodels, Eample: Water tanks Euation for one water tank: If te block-diagram is saved to stepmodel.mdl, te following Script-file simstepmodel.m simulates te system: open system( 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,y),title( y ) subplot(,,),plot(t,u),title( u ) ḣ = (u )/A = a g Corresponding Simulink model: /A f(u) s Sum Gain tegrator Fcn Make a subsystem and connect two water tanks in series. Subsystem Out Subsystem 4

5 Linearization in Simulink Linearization in Simulink, cont. Use te command trim to find e.g., stationary points to a system > > A=.7e-3;a=7e-6,g=9.8; > > % Eample to find input u for desired states/output > > [,u,y=trim( flow,[..,[,.) =.. u = e-6 y =. Use te command linmod to find a linear approimation of te system around an operating point: > > [aa,bb,cc,dd=linmod( flow,,u); > > sys=ss(aa,bb,cc,dd); > > bode(sys) Linearization in Simulink; Alternative By rigt-clicking on a signal connector in a Simulink model you can add Linearization points (inputs and/or outputs). Simulation of JAS 39 Gripen Computer eercise reference pilot T_f.s+ prefilter Kf = A+Bu y = C+Du plane dynamics 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 by Tools -> Control Design ->Linear analysis... were you can set te operating points, eport linearized model to Workspace (Model- Eport to Workspace) and muc more. Simulation Analysis of PIO using describing functions Improve design Summary, bot around euilibria and trajectories Definitions of local and global stability Ceck local stability and local controllability at euilibria Simulation tool in tis course: Simulink Net Lecture, Friday November 3 Pase plane analysis Classification of euilibria 5