كنترل غير خطي جلسه سوم : ادامة بحث نماي فاز (phase plane) سجاد ازگلي.

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كنترل غير خطي جلسه سوم : ادامة بحث نماي فاز (phase plane) سجاد ازگلي

جلسه سوم: تحليل در نماي فاز  Introduction to Phase plane analysis  Phase Portrait Construction  Phase plane analysis for Lin. Systems پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي o Multiple equilibria o Limit cycle, Definition, Types, Existence Theorems ص 2ص 2 ص 2ص 2

جلسه سوم: تحليل در نماي فاز Lin. ↔ NL systems Linear systems: six different type of equilibria (associated with the type of eigenvalues): Stable or unstable Node, Saddle point, Stable or unstable focus, center. “global” qualitative behavior determined by the type of eq. point. Nonlinear system: Only “local” behavior (vicinity of eq. p.) determined by the type of eq. point. NL. Only behaviors:  Multiple equilibria  limit cycle پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 3ص 3 ص 3ص 3

جلسه سوم: تحليل در نماي فاز NL Sys. Behavior, Vicinity of Eq. ▫ Let p =( p 1, p 2 ) be an eq. point of ▫ Taylor series expansion: ▫ where, پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 4ص 4 ص 4ص 4

جلسه سوم: تحليل در نماي فاز NL Sys. Behavior, Vicinity of Eq. ▫ p is an eq. point => ▫ Change of variables to and Neglect higher order terms => ▫ Note: A is the Jacobian matrix Trajectories of the NL vicinity of p are close to the trajectories of the linearized system. پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 5ص 5 ص 5ص 5

جلسه سوم: تحليل در نماي فاز Example: Tunnel Diode  Jacobian  where,  Evaluate equilibrium points Stable Node Saddle point Stable Node NL Sys. Behavior, Vicinity of Eq. پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 6ص 6 ص 6ص 6

جلسه سوم: تحليل در نماي فاز NL Sys. Behavior, Vicinity of Eq. Example: pendulum  محاسبه ماتريس ژاكوبي  Evaluate equilibrium points Stable Focus Saddle point پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 7ص 7 ص 7ص 7

جلسه سوم: تحليل در نماي فاز Hyperbolic Equilibrium Point  The general observation above is only true if No eigenvalues of the Jacobian is on the imaginary axis. Example: non-hyperbolic) Inconclusive from linearization(  eq. point is at (0,0).  eq. point center!  Using polar coordinates:  Intuitively پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 8ص 8 ص 8ص 8

جلسه سوم: تحليل در نماي فاز NL Sys. Behavior, Vicinity of Eq. Example: non-hyperbolic (cont.) ▫ Use polar coordinates: ▫ Intuitively Stable Focus Unstable Focus پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 9ص 9 ص 9ص 9

جلسه سوم: تحليل در نماي فاز  Introduction to Phase plane analysis  Phase Portrait Construction  Phase plane analysis for Lin. Systems  Phase plane analysis for NL. Systems پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي o Limit cycle, Definition, Types, Existence Theorems ص 10

جلسه سوم: تحليل در نماي فاز Multiple Equilibria Example: Tunnel Diode State Equations Parameter values Numerical polynomial fit to h ( x ): Solve numerically for eq. points پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 11

جلسه سوم: تحليل در نماي فاز Multiple Equilibria Example: Tunnel Diode (cont.) phase-portrait پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 12 Stable Node Saddle point

جلسه سوم: تحليل در نماي فاز Multiple Equilibria Example: pendulum phase-portrait پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 13 Stable Node Saddle point

جلسه سوم: تحليل در نماي فاز  Introduction to Phase plane analysis  Phase Portrait Construction  Phase plane analysis for Lin. Systems  Phase plane analysis for NL. Systems Behavior in vicinity of equilibria Multiple equilibria پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 14

جلسه سوم: تحليل در نماي فاز Limit cycle Oscillation: Nontrivial periodic solution ▫ Periodic: ▫ Nontrivial: exclude constant and zero solutions Cycle: Oscillation in phase portrait پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 15

جلسه سوم: تحليل در نماي فاز Every Oscillation is a Cycle but not a Limit Cycle ▫ Example: Mass-spring OR LC circuit  Linear System with Eigenvalues on j  axis  Center: sustained oscillation  Amplitude of Oscillation: x o  Depending on the initial condition  No limiting Action Just a Cycle but not a Limit Cycle! Limit cycle پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي A Cycle ص 16

جلسه سوم: تحليل در نماي فاز Limit cycle specifications پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي A Limit Cycle ص 17 ▫ Nonlinear oscillator is structurally stable ▫ It attracts all nearby trajectories ▫ The oscillation amplitude does not depend on the initial conditions ▫ For any nearby initial conditions the trajectories converges to the Limit Cycle. ▫ Isolated and Limiting Closed Curve.

جلسه سوم: تحليل در نماي فاز Limit cycle study using Ph.P. Stable Limit Cycle ▫ All trajectories in the vicinity of the limit cycle converges to it as ▫ Example: ▫ Use Polar coordinates:  IF پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 18

جلسه سوم: تحليل در نماي فاز Limit cycle study using Ph.P. UnStable Limit Cycle ▫ All trajectories in the vicinity of the limit cycle diverges from it as ▫ Example: ▫ Use Polar coordinates:  IF پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 19

جلسه سوم: تحليل در نماي فاز Limit cycle study using Ph.P. Semi-Stable Limit Cycle ▫ Some of the trajectories in the vicinity of the limit cycle converges to it, while others diverge from it as ▫ Example: پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 20

جلسه سوم: تحليل در نماي فاز Limit Cycle, Existence theorems Relation between L.C. and Eq.points N: The No. of nodes, centers and foci enclosed by a L.C. S: The No. of saddle points enclosed by a L.C. => The Limit cycle must enclose at least one eq. point Eq. point, limit cycle, and trajectory پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 21

جلسه سوم: تحليل در نماي فاز Limit Cycle, Existence theorems Existence of Limit Cycles Proof: By Contradiction For any trajectory (including a Limit Cycle) Thus, along a closed curve L (a limit cycle): Hence the integrand must vanish or at least change sign. پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي Using Green’s Theorem ص 22

جلسه سوم: تحليل در نماي فاز  …  Phase plane analysis for Lin. Systems  Phase plane analysis for NL. Systems Behavior in vicinity of equilibria Multiple equilibria Limit cycle study using Phase Portrait پاييز هشتاد و نه، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي  Lyapunov theory ص 23

كنترل غير خطي جلسه چهارم : پایداری لياپانوف (Lyapunov stability) سجاد ازگلي

جلسه سوم: تحليل در نماي فاز Review the previous lecture پاييز هشتاد و هشت، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 25 Linear systems: six different type of equilibria Stable or unstable Node, Saddle point, Stable or unstable focus, center. Nonlinear systems: “local” behavior (vicinity of eq. p.) NL. Only behaviors:  Multiple equilibria  limit cycle Lim.Cycle Existence Theorems

جلسه سوم: تحليل در نماي فاز  Introduction to Phase plane analysis  Phase Portrait Construction  Phase P. Anal. for Lin. Sys.  Phase plane analysis for NL. Systems Limit Cycle پاييز هشتاد و هشت، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي  Stability Definitions (in the sense of Lyaounov)  Stability analysis o Linearization method o Lyapunov direct method ص 26

جلسه سوم: تحليل در نماي فاز Stability Stability definitions: equilibrium point(xeq=0)is stable if ▫ start from any initial condition adequately close to it => remain in the neighborhood of it. In the sense of lyapunove: بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 27

جلسه سوم: تحليل در نماي فاز بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي  Different states of stability: o Asymptotic stability o Exponential stability o Marginally stability ص 28 Unstable# stable

جلسه سوم: تحليل در نماي فاز Asymptotic stability: X(eq) is asymptotically stable is stable + x(0) є B d lim x(t)=0 The Stability is global if: Starting from any initial state X eq is stable. Other wise the stability is locally. بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 29 t→∞ Domain of attraction

جلسه سوم: تحليل در نماي فاز Exponential stability: X eq =0 is exponentially stable if: Convergence rate of all states to this point is greater than an exponential function: r X (0) Є Br d,λ э ||≤X(t)|| ≤α e ||X(0)|| FOR LINEAR SYSTEMS: If is asymptotically stable is exponentially stable too. بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 30 -λt-λt

جلسه سوم: تحليل در نماي فاز Methods for investigating the stability: Linearization(Lyapunove's first method) Direct method Invariant set theorem. پاييز هشتاد و هفت، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 31

جلسه سوم: تحليل در نماي فاز Linearization: بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 32 Kind of stability of linearized sys ≡ site of eigen values of matrix(A) Kind of stability of equilibrium point’s of NL SYS is obtained.

جلسه سوم: تحليل در نماي فاز Linearization i Re{λ i } < 0 strictly stable. i Re{λ i } > 0 unstable. i λ i є { LHP} U { JOA} j Re{λ j } = 0 بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 33 unknown

جلسه سوم: تحليل در نماي فاز Linearization: Example F FB u=tan (x)+x cos(x)+x cos(x) Lin cL x+0+(0+1)u =0 u=x+0+x CL Lin λ+λ+1=0 fihv88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص 34 X+5X+(sin(X)+1)u= X + X+ X=

جلسه سوم: تحليل در نماي فاز Linearization: Example λ=(-1+j√3)/2 Asympt 2) X1=X X2=X A= λ i (A)=(-1+j√3)/2 Asympt بهار88، بخش برق، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي ص

جلسه سوم: تحليل در نماي فاز  …  Phase plane analysis for Lin. Systems  Phase plane analysis for NL. Systems Behavior in vicinity of equilibria Multiple equilibria Limit cycle study using Phase Portrait پاييز هشتاد و هشت، دانشكده برق و كامپيوتر، دانشگاه تربيت مدرس كنترل غير خطي - سجاد ازگلي  Lyapunov theory ص 36