Classical Control Topics covered: Modeling. ODEs. Linearization. Laplace transform. Transfer functions. Block diagrams. Mason’s Rule. Time resppponse specifications. Effects of zeros and poles. Stability via Routh-Hurwitz. Feedback: Disturbance rejection, Sensitivity, Steady -state tracking. PID controllers and Ziegler-Nichols tuning procedure. Actuator saturation and integrator wind-up. Root locus. Frequency response--Bode and Nyquist diagrams. Stability Margins . Design of dynamic compensators. Classical Control – Prof. Eugenio Schuster – Lehigh University 1 Classical Control Text: Feedback Control of Dynamic Systems, 4th Edition, G .F . Franklin , J .D . Powel and A. Emami-Naeini Prentice Hall 2002. Classical Control – Prof. Eugenio Schuster – Lehigh University 2 What is control? For any analysis we need a mathematical MODEL of the system Model → Relation between gas pedal and speed: 10 mph change in speed per each degree rotation of gas pedal Disturbance → Slope of road: 5 mph change in speed per each degree change of slope Block diagram for the cruise control plant: w Slope (degrees) 0.5 y =10(u − 0.5w) Control - Output speed (degrees) (mph) 10 u + y Classical Control – Prof. Eugenio Schuster – Lehigh University 3 What is control? Open-loop cruise contltrol: w r u = PLANT 10 0.5 yol =10(u − 0.5w) - r 1/10 10 =10( − 0.5w) r u + yol 10 Reference = r − 5w (mph) r = 65,w = 0 ⇒ eol = 0 eol = r − yol = 5w r = 65,w =1⇒ eol = 5mph,eol = 7.69% r − yol w eol[%] = = 500 OK when: r r 1- Plant is known exactly 2- There is no disturbance Classical Control – Prof. Eugenio Schuster – Lehigh University 4 What is control? Close d-loop cruise contltrol: w u = 20(r − ycl ) PLANT 0.5 ycl =10(u − 0.5w) 200 5 - + = r − w 10 201 201 1/10 u + r - ycl 1 5 1 ecl = r − ycl = r + w r = 65,w = 0 ⇒ e = % = 0.5% 201 201 cl 201 1 5 5 r − y 1 5 w r = 65,w =1⇒ e = + = 0.69% e [%] = cl = + cl 201 20165 cl r 201 201 r Classical Control – Prof. Eugenio Schuster – Lehigh University 5 What is control? Feedback control can help: • reference following (tracking) • disturbance rejection • changing dynamic behavior LARGE gain is essential but there is a STABILITY limit “TheThe issue of how to get the gain as large as possible to reduce the errors due to disturbances and uncertainties without making the system become unstable is what much of feedback control design is all about” First step in this design process: DYNAMIC MODEL Classical Control – Prof. Eugenio Schuster – Lehigh University 6 Dynamic Models MECHANICAL SYSTEMS: F = ma Newton’s law m&x& = u − bx& v = x& velocity a = v& = &x& acceleration b u Vo 1 m TfTransfer FtiFunction v& + v = ⎯⎯⎯st ⎯⎯⎯st → = v=Voe ,u=Uoe d m m Uo s + b m → s dt Classical Control – Prof. Eugenio Schuster – Lehigh University 7 Dynamic Models MECHANICAL SYSTEMS: F = Iα Newton’s law 2 ml θ&& = −lmg sinθ + Tc ω = θ& angular velocity α = ω& = θ&& angular acceleration 2 I = ml moment of inertia g T g T θ&&+ sinθ = c ⎯⎯⎯→θ&&+ θ = c Linearization l ml 2 sinθ ≈θ l ml 2 Classical Control – Prof. Eugenio Schuster – Lehigh University 8 Dynamic Models g T θ&&+ θ = c l ml 2 Reduce to first order equations: x1 = x2 x1 = θ & g Tc x2 = θ& x = − x + &2 l 1 ml 2 ⎡ x ⎤ T ⎡ 0 1⎤ ⎡0⎤ x ≡ 1 ,u ≡ c ⇒ x = ⎢ g ⎥x + u State Variable ⎢x ⎥ ml 2 & − 0 ⎢1⎥ Representation ⎣ 2 ⎦ ⎣⎢ l ⎦⎥ ⎣ ⎦ General case: x& = Fx + Gu y = Hx + Ju Classical Control – Prof. Eugenio Schuster – Lehigh University 9 Dynamic Models ELECTRICAL SYSTEMS: Kirchoff’s Current Law (KCL): The alge bra ic sum of currents entering a node is zero at every instant Kirchoff’s Voltage Law (KVL) The algebraic sum of voltages around a loop is zero at every instant Resistors: vR (t) = RiR (t) ⇔ iR (t) = GvR (t) i i R iC L Capacitors: + + + t v v dv (t) 1 R vC L i (t) = C C ⇔ v (t) = i (τ )dτ + v (0) C C ∫ C C dt C 0 Inductors: di (t) 1 t v (t) = L L ⇔ i (t) = v (τ )dτ + i (0) L L ∫ L L dt L 0 Classical Control – Prof. Eugenio Schuster – Lehigh University 10 Dynamic Models ELECTRICAL SYSTEMS: OP AMP: vO = A(vp − vn ), A → ∞ i + p + vp RO iO v vp = vn RI + O + in + i = i = 0 v A(vp-vn) p n n - - To work in the linear mode we need FEEDBACK!!! Classical Control – Prof. Eugenio Schuster – Lehigh University 11 Dynamic Models ELECTRICAL SYSTEMS: KCL: R2 R 1 C dvO 1 1 v vO = vO − vI 1 dt R C R C - 2 1 + 1 t R = ∞ (OC) ⇒ v ()t = v (0 )− v (μ)dμ 2 O O ∫0 I R1C v v 1 1 K ∫ O K = − RC Inverting integrator Classical Control – Prof. Eugenio Schuster – Lehigh University 12 Dynamic Models ELECOCTRO-MECCSSSCCHANICAL SYSTEMS: DC Motor torque armature current T = Ktia e = Keθ&m emf shaft velocity Jmθ&&m = −bθ&m + T di − v + R i + L a + e = 0 a a a dt Obtain the State Variable Representation Classical Control – Prof. Eugenio Schuster – Lehigh University 13 Dynamic Models HEAT-FLOW: Heat Flow Temperature Difference 1 q = (T −T ) R 1 2 1 T& = q C Thermal capacitance Thermal resistance 1 ⎛ 1 1 ⎞ T&I = ⎜ + ⎟(To −TI ) CI ⎝ R1 R2 ⎠ Classical Control – Prof. Eugenio Schuster – Lehigh University 14 Dynamic Models FLUID-FLOW: Mass rate Mass Conservation law m& = win − wout Inlfllet mass flow Outlfllet mass flow 1 m = ρAh& ⇒ h& = ()w − w & ρA in out A: area of the tank ρ: density of fluid h: height of water Classical Control – Prof. Eugenio Schuster – Lehigh University 15 Linearization Dynamic System: x& = f (x,u) 0 = f (xo ,uo ) Equilibrium Denote δx = x − xo ,δu = u − uo δx& = f (xo + δx,uo + δu) Taylor Expansion ∂f ∂f δx ≈ f (x ,u )+ δx + δu & o o ∂x ∂u xo ,uo xo ,uo ∂f ∂f F ≡ ,G ≡ ⇒ δx ≈ Fδx + Gδu ∂x ∂u & xo ,uo xo ,uo Classical Control – Prof. Eugenio Schuster – Lehigh University 16 Linearization δx& ≈ Fδx + Gδu ⎡∂f1 ∂f1 ⎤ ⎡ ∂f1 ∂f1 ⎤ ⎢∂x L ∂x ⎥ ⎢∂u L ∂u ⎥ ∂f ⎢ 1 n ⎥ ∂f ⎢ 1 m ⎥ F ≡ = ⎢ M M ⎥ ,G ≡ = ⎢ M M ⎥ ∂x x ,u ∂u x ,u o o ⎢∂fn ∂fn ⎥ o o ⎢∂fn ∂fn ⎥ ⎢ L ⎥ ⎢ L ⎥ ∂x1 ∂xn ∂u1 ∂um ⎣ ⎦ xo ,uo ⎣ ⎦ xo ,uo k g Example: Pendulum with friction θ&&+ θ& + sinθ = 0 m l ⎡ 0 1 ⎤ x = g k x & ⎢− cos x − ⎥ ⎢ 1 ⎥ ⎣ l m⎦ xo Classical Control – Prof. Eugenio Schuster – Lehigh University 17 Laplace Transform • Function f(t) of time bt – Piecewise continuous and exponential order f (t) < Ke ∞ α + j∞ −st −1 1 st F(s) = ∫ f (t)e dt L []F()s = f (t) = ∫ F(s)e ds π 0- 2 j α − j∞ – 0- limit is used to capture transients and discontinuities at t=0 – s is a complex variable (σ+jω) • There is a need to worryyg about regions of conver gence of the inte gral – Units of s are sec-1=Hz • A frequency – If f(t ) il()his volts (amps) then F(s ) ilis volt-second(ds (amp-second)ds) Classical Control – Prof. Eugenio Schuster – Lehigh University 18 Laplace transform examples • Step function – unit Heavyside Function ⎧0, for t < 0 • After Oliver Heavyside (1850-1925) u(t) = ⎨ ⎩1, for t ≥ 0 ∞ ∞ ∞ ∞ e−st e−(σ + jω)t 1 F(s) = ∫u(t)e−stdt = ∫ e−stdt = − = − = if σ > 0 s σ + jω s 0− 0− 0 0 • Exponential function • After Oliver Exponential (1176 BC- 1066 BC) ∞ ∞ ∞ e−(s+α )t 1 F(s) = ∫∫e−αte−stdt = e−(s+α )tdt = − = if σ > α s +α s +α 00 0 • Delta (impulse) function δ(t) ∞ st F(s) = ∫δ (t)e− dt =1 for all s 0− Classical Control – Prof. Eugenio Schuster – Lehigh University 19 Laplace Transform Pair Tables Signal Waveform Transform impulse δ (t) 1 step u(t) 1 s 1 ramp tu(t) s2 1 exponential e−αtu(t) s+α t 1 damped ramp te−α u(t) (s+α )2 β sine sin(βt)u(t) s2+β 2 s cosine cos(βt)u(t) s2+β 2 damped sine t β e−α sin()βt u(t) (s+α )2+β 2 damped cosine s+α e−αt cos(βt)u(t) (s+α )2+β 2 Classical Control – Prof. Eugenio Schuster – Lehigh University 20 Laplace Transform Properties Linearity: (absolutely critical property) L{Aff1(t) + Bff2(t)}= AL{f1 (t)}+ BL{f2 (t)}= AF1(s) + BF2(s) ⎧t ⎫ F(s) Integration property: L⎨∫ f (τ )dτ ⎬ = ⎩0 ⎭ s ⎧df (t)⎫ Differentiation property: L⎨ ⎬ = sF(s) − f (0−) ⎩ dt ⎭ ⎧ 2 ⎫ ⎪d f (t)⎪ 2 L = s F(s) − sf (0−) − f ′(0−) ⎨ 2 ⎬ ⎩⎪ dt ⎭⎪ ⎪⎧d m f (t)⎪⎫ = m F(s) − sm−1 f (0−) − sm−2 f ′(0−) − − f (m) (0−) L⎨ m ⎬ s L ⎩⎪ dt ⎭⎪ Classical Control – Prof. Eugenio Schuster – Lehigh University 21 Laplace Transform Properties Translation properties: s-domain translation: L{e−αt f (t)} = F(s +α) −as t-domain translation: L{f (t − a)u(t − a)}= e F(s) for a > 0 Initial Value Property: lim f (t) = lim sF(s) t→0+ s→∞ Final Value Property: lim f (t) = lim sF(s) t→∞ s→0 If all poles of F(s) are in the LHP Classical Control – Prof. Eugenio Schuster – Lehigh University 22 Laplace Transform Properties 1 s Time Scaling: L{ f (at)} = F( ) a a dF(s) Multiplication by time: L{tf (t)} = − ds t Convolution: L{ f (τ )g(t −τ )dτ} = F(s)G(s) ∫0 1 σ + jω Time product: L{ f (t)g(t)} = F(s)G(s − λ)dλ 2πj ∫σ − jω Classical Control – Prof.