Overview Lecture Nonlinear Control and Servo sstems Lecture Anders Rantzer Practical information Course contents Nonlinear control sstems phenomena Nonlinear differential equations Automatic Control LTH Lund Universit Course Goal Course Material To provide students with solid theoretical foundations of nonlinear control sstems combined with good engineering abilit You should after the course be able to recognize common nonlinear control problems, use some powerful analsis methods, and use some practical design methods. Textbook Glad and Ljung, Reglerteori, flervariabla och olinjra metoder, 3, Studentlitteratur,ISBN 9--33-7 or the English translation Control Theor,, Talor & Francis Ltd, ISBN -7-8878-9. The course covers Chapters -6,8. (MPC and optimal control not covered in the other alternative textbooks.) H. Khalil, Nonlinear Sstems (3rd ed.),, Prentice Hall, ISBN -3-7-8. A good, a bit more advanced text. ALTERNATIVE: Slotine and Li, Applied Nonlinear Control, Prentice Hall, 99. The course covers chapters -3 and, and sections.7-.8, 6., 7.-7.3. Course Material, cont. Lectures and labs The lectures are given in M:E (except toda and Nov ) as follows: Handouts (Lecture notes + extra material) Exercises (can be downloaded from the course home page) Lab PMs, and 3 Home page http://www.control.lth.se/course/frtn/ Matlab/Simulink other simulation software see home page Exercise sessions and TAs The exercises are offered twice a week in lab A and lab B of Automatic Control LTH, on the ground floor in the south end of the M-building. Tue :-7: Wed :-7: Mon 3 week 9 Wed 8 week 9 Frid 8- week Lectures are given in English. The three laborator experiments are mandator. Sign-up lists are posted on the web at least one week before the first laborator session and close one da before. The Laborator PMs are available at the course homepage. Before the lab sessions some home assignments have to be done. No reports after the labs. lectures exercises The Course Martin Karlsson Kaito Ariu 3 laboratories hour exam: Januar 3, 8, 8:-3:, Sparta A-B. Open-book exam: Lecture notes but no old exams/exercises.
Course Outline Todas lecture Lecture -3 Lecture -6 Lecture 7-8 Lecture 9-3 Lecture Modelling and basic phenomena (linearization, phase plane, limit ccles) Analsis methods (Lapunov, circle criterion, describing functions)) Common nonlinearities (Saturation, friction, backlash, quantization)) Design methods (Lapunov methods, Backstepping, Optimal control) Summar Common nonlinear phenomena Input-dependent stabilit Stable periodic solutions Jump resonances and subresonances Nonlinear model structures Common nonlinear components State equations Feedback representation Linear Sstems Linear time-invariant sstems are eas to analze Different representations of same sstem/behavior u S Definitions: The sstem S is linear if S(αu) = αs(u), scaling = S(u) S(u + u ) = S(u ) + S(u ), superposition A sstem is time-invariant if delaing the input results in a delaed output: (t τ) = S(u(t τ)) ẋ(t) = Ax(t) + Bu(t), (t) = Cx(t), x() = (t) = g(t) u(t) = g(r)u(t r)dr Y (s) = G(s)U(s) Local stabilit = global stabilit: Eigenvalues of A (= poles of G(s)) in left half plane Superposition: Enough to know step (or impulse) response Frequenc analsis possible: Sinusoidal inputs give sinusoidal outputs Linear models are not alwas enough Linear models are not enough Example: Ball and beam x mg sin(φ) x = position (m) φ = angle (rad) g = 9.8 (m/s ) Can the ball move. meter from rest in. seconds? Linearization: sin φ φ for φ { ẍ(t) = gφ x() = Linear model (acceleration along beam) : Combine F = m a = m d x with F = mg sin(φ): dt ẍ(t) = g sin(φ(t)) mg φ Solving the above gives x(t) = t gφ For x(.) =., one needs φ =.. g rad Clearl outside linear region! Contact problem, friction, centripetal force, saturation How fast can it be done? (Optimal control) Warm-Up Exercise: -D Nonlinear Control Sstem Demo: Furuta pendulum ẋ = x x + u stabilit for x() = and u =? stabilit for x() = and u =? stabilit with linear feedback u = ax + b? stabilit with non-linear feedback u(x) =?
Stabilit Can Depend on Amplitude Stabilit Can Depend on Amplitude r + Motor Valve Process? s (s+) r + Motor Valve Process s (s+) Valve characteristic f(x) =??? Step changes of amplitude, r =., r =.68, and r =.7 Valve characteristic f(x) = x Step changes of amplitude, r =., r =.68, and r =.7 Step Responses Output.. r =. 3 Output r =.68 Output 3 3 r =.7 Stabilit depends on amplitude! Stable Periodic Solutions Example: Motor with back-lash Video: JAS Gripen crash Constant Sum P controller s +s Motor Backlash Motor: G(s) = s(+s) Controller: K = Stable Periodic Solutions Output for different initial conditions: Rela Feedback Example Period and amplitude of limit ccle are used for autotuning Output.. 3. Σ PID Rela A T u Process Output. 3. Output. 3 u Frequenc and amplitude independent of initial conditions! Several sstems use the existence of such a phenomenon Time [ patent: T Hgglund and K J strm] 3
Jump Resonances Jump Resonances u =. sin(.3t), saturation level =. Sine Wave Sum Saturation s +s Motor Two different initial conditions 6 Response for sinusoidal depends on initial condition Problem when doing frequenc response measurement Output 6 3 give two different amplifications for same sinusoid! Jump Resonances Measured frequenc response (man-valued) New Frequencies Example: Sinusoidal input, saturation level a sin t Saturation Magnitude saturated linear saturated Amplitude a = Frequenc (Hz) 3 3 Frequenc [rad/s] Amplitude a = Frequenc (Hz) 3 New Frequencies Example: Electrical power distribution THD = Total Harmonic Distortion = k= energ in tone k energ in tone Nonlinear loads: Rectifiers, switched electronics, transformers Important, increasing problem Guarantee electrical qualit Standards, such as T HD < % New Frequencies Example: Mobile telephone Effective amplifiers work in nonlinear region Introduces spectrum leakage Channels close to each other Trade-off between effectivit and linearit Subresonances When is Nonlinear Theor Needed? Example: Duffing s equation ÿ + ẏ + 3 = a sin(ωt) a sin ωt.. 3.. 3 Hard to know when - Tr simple things first! Regulator problem versus servo problem Change of working conditions (production on demand, short batches, man startups) Mode switches Nonlinear components How to detect? Make step responses, Bode plots Step up/step down Var amplitude Sweep frequenc up/frequenc down
Todas lecture Some Nonlinearities Common nonlinear phenomena Static dnamic Input-dependent stabilit Stable periodic solutions Jump resonances and subresonances u Abs e u Math Function Saturation Nonlinear model structures Common nonlinear components Sign Dead Zone Look Up Table State equations Feedback representation Rela Backlash Coulomb & Viscous Friction Nonlinear Differential Equations Finite escape time Problems No analtic solutions Existence? Uniqueness? etc Example: The differential equation has solution Finite escape time dx dt = x, x() = x x(t) = x x t, t < x t f = x x(t). 3. 3... Finite Escape Time Finite escape time of dx/dt = x Uniqueness Problems Example: The equation ẋ = x, x() = has man solutions: { (t C) x(t) = / t > C t C x(t)... 3 3 Local Existence and Uniqueness Compare with water tank: dh/dt = a h, h : height (water level) Change to backward-time: If I see it empt, when was it full? ) Global Existence and Uniqueness For R >, let Ω R denote the ball Ω R = {z : z a R}. Theorem If, f is Lipschitz-continuous in Ω R, i.e., then f(z) f() K z, for all z, Ω R, has a unique solution { ẋ(t) = f(x(t)) x() = a Theorem If f is Lipschitz-continuous in R n, i.e., then f(z) f() K z, for all z, R n, has a unique solution ẋ(t) = f(x(t)), x() = a x(t), t. where C R = max x ΩR f(x) x(t), t < R/C R,
State-Space Models Transformation to Autonomous Sstem State vector x Input vector u Output vector general: f(x, u,, ẋ, u, ẏ,...) = explicit: ẋ = f(x, u), = h(x) affine in u: ẋ = f(x) + g(x)u, = h(x) linear time-invariant: ẋ = Ax + Bu, = Cx Nonautonomous: ẋ = f(x, t) Alwas possible to transform to autonomous sstem Introduce x n+ = time ẋ = f(x, x n+ ) ẋ n+ = Transformation to First-Order Sstem Equilibria (=singular points) highest derivative of [ Introduce x = d dt... Assume dk dt k Example: Pendulum x = [ θ θ ] T gives d k dt k ] T MR θ + k θ + MgR sin θ = Put all derivatives to zero! General: f(x, u,,,,,...) = Explicit: f(x, u ) = Linear: Ax + Bu = (has analtical solution(s)!) ẋ = x ẋ = k MR x g R sin x Multiple Equilibria A Standard Form for Analsis Example: Pendulum Transform to the following form MR θ + k θ + MgR sin θ = Equilibria given b θ = θ = = sin θ = = θ = nπ Alternativel, gives x =, sin(x ) =, etc ẋ = x ẋ = k MR x g R sin x Nonlinearities G(s) Example, Closed Loop with Friction Next Lecture F Friction u C G v Linearization Stabilit definitions Simulation in Matlab Friction G +CG 6