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

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1 Lecture Today s Goal Linearization Stability definitions Simulation in Matlab/Simulink 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) ( (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 Local Stability Consider ẋ = f() were f( ) = Definition Te euilibrium is stable if, for any R >, tere eists r >, suc tat Part II: Stability definitions () < 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 Part III: Ceck local stability and controllability 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. However... Eample An inverted pendulum wit vertically moving pivot point pi u Bosc 8 (Automatic parking assistance) Multiple turns parking lot > car lengt + 8 cm More parking in lecture φ(t) = (g + u(t)) sin(φ(t)), l were u(t) is acceleration, can be written as 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 to find fied point k+ = f( k ) = f( ) Often te only 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 Dymola ttp:// Modelica ttp:// Special purpose Spice (electronics) EMTP (electromagnetic transients) Adams (mecanical systems) 3

4 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 Clock t To Workspace Signal Generator s+ y To Workspace Ceck Save format of output blocks ( Array instead of Structure ) > > plot(t,y) (or use Structure wic also contains te time information.) Don t forget Apply How To Get Better Accuracy Use Scripts to Document Simulations Modify Refine, Absolute and Relative Tolerances, tegration metod Refine adds interpolation points: Refine = Refine = 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 ) Submodels, Eample: Water tanks Linearization in Simulink Euation for one water tank: 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 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 =. 4

5 Linearization in Simulink, cont. Linearization in Simulink; Alternative By rigt-clicking on a signal connector in a Simulink model you can add Linearization points (inputs and/or outputs). Use te command linmod to find a linear approimation of te system around an operating point: put Point /A Gain s tegrator a*srt(*g*u[) Fcn Output Point > > [aa,bb,cc,dd=linmod( flow,,u); > > sys=ss(aa,bb,cc,dd); > > bode(sys) 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. Computer eercise Summary Simulation of JAS 39 Gripen Linearization, bot around euilibria and trajectories reference pilot T_f.s+ prefilter Kf = A+Bu y = C+Du plane dynamics Cteta teta pitc angle Definitions of local and global stability Ceck local stability and local controllability at euilibria command upilot L states Simulation tool: Simulink Clock t time Net Lecture Simulation Analysis of PIO using describing functions Improve design Pase plane analysis Classification of euilibria 5