3 System model in general Dynamical systems modeled

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 3 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ System model in general Dynamical systems, modeled by a finite number of coupled first-order NL ODE’s: t : time, : state variables : input variables Using vector notation: STATE SPACE REPRESENTATION The measurements eq. : y : the vector of output variables. ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 5 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Equilibrium Point • The system state is at equilibrium point x* if ▫ start at x* => remains at it OR ▫ states reach to x* at steady state • For an autonomous system x* is the real root of • Nonlinear function may have �No roots �Many roots No Eq. Point. Multiple Eq. Points. • Isolated Equilibrium Point There are no other Eq. point at vicinity of x* ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 8 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Phase Plane Analysis • Is: to generate motion trajectories of a 2 nd order system in state space phase portrait • How? ▫ 2 nd order autonomous system ▫ x 1(t) , x 2(t): the solution to the above equations ▫ Eliminate time and plot x 1 vs x 2 phase trajectory ▫ Plot several phase trajectories for different initial conditions • Why important in analysis? ▫ Visualization ▫ No need to solve NL ODEs ▫ No matter what kind of NL it is (hard, …) ▫ Dominant dynamics of many real systems is 2 nd order ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 10 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Singular Points • An Eq. point is called a singular point !!? • Why? ▫ The slope of curves at any point: ▫ phase portrait has only one specific slope at any definite point �phase portraits does not intersect � They can intersect only at eq. or singular points ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 13 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Phase Portrait Construction • Analytic method • Method of isoclines • • Delta method Lienard’s method Pell’s method Method of adjoint solutions • Numerical Simulation Reference: Nonlinear Control Systems Z. Vukic, L. Kuljaca, D. Donlagic, S. Tesnjak 2003 ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 17 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Numerical Simulation • Numerically solve and plot the phase portrait for different initial conditions • remarks: ▫ Selection of limits ▫ Selection of the initial points �Near stable equilibrium points: �Solve the state space equation reverse in time �Near Unstable equilibrium points: �Solve the state space equation forward in time ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 23 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Method of isoclines • Isocline : locus of the points with a given tangent slope ▫ Example: unit mass spring �System state eq. Trajectory slopes �Isoclines: (constant slope a) �Plot isoclines for different a ’s �Plot the trajectory using the isoclines ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 27 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Phase plane analysis, introduction Phase Portrait Construction ü Analytic method ü Numerical solution ü Method of isoclines →Q & A q. Phase plane analysis for Lin. Systems q. Phase plane analysis for NL. Systems o Limit cycle study using phase portrait ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 28 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Review of the previous lecture • Autonomous = TI • equilibrium point: ▫ start at it => remain at it OR reach to it at steady state ▫ For autonomous: • phase trajectory, phase portrait • singular point = Eq. point, Why? unique slope except at sing. • Isocline : locus of the points with a given tangent slope ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ

ﺗﺤﻠﻴﻞ ﺩﺭ ﻧﻤﺎﻱ ﻓﺎﺯ : ﺟﻠﺴﻪ ﺩﻭﻡ 29 ﺹ ﺳﺠﺎﺩ ﺍﺯگﻠﻲ - ﻛﻨﺘﺮﻝ ﻏﻴﺮ ﺧﻄﻲ Introduction to Phase plane analysis Phase Portrait Construction ü Analytic method ü Numerical solution ü Method of isoclines →Phase P. Anal. for Lin. Sys. q. Phase plane analysis for NL. Systems o Behavior in vicinity of equilibria o Multiple equilibria o Limit cycle, Definition, Types, Existence Theorems ﺩﺍﻧﺸگﺎﻩ ﺗﺮﺑﻴﺖ ﻣﺪﺭﺱ ، ﺩﺍﻧﺸﻜﺪﻩ ﺑﺮﻕ ﻭ ﻛﺎﻣپﻴﻮﺗﺮ ، پﺎﻳﻴﺰ ﻫﺸﺘﺎﺩ ﻭ ﻧﻪ