Lecture 2 – Linear Systems This lecture: EE263 material recap + some controls motivation • Continuous time (physics) • Linear state space model • Transfer functions • Black-box models; frequency domain analysis • Linearization EE392m - Spring 2005 Control Engineering 2-1 Gorinevsky Modeling and Analysis Input signal System Model Output signal (control) [internal states] (observation) state evolution This lecture considers • Linear models. More detail on modeling in Lecture 7 • Simulation: computing state evolution and output signal • Stability: does the solution diverge after some time? • Approximate linear models EE392m - Spring 2005 Control Engineering 2-2 Gorinevsky Linear Models • Model is a mathematical representations of a system – Models allow simulating the system – Models can be used for conceptual analysis – Models are never exact • Linear models – Have simple structure – Can be analyzed using powerful mathematical tools – Can be matched against real data using known procedures – Many simple physics models are linear – They are just models, not the real systems EE392m - Spring 2005 Control Engineering 2-3 Gorinevsky State space model • Generic state space model Example: F16 Longitudinal Model – is described by ODEs from Aircraft Control and Simulation – e.g., physics-based system by Stevens and Lewis model θ α u = δ – state vector x e V – observation vector y x – control vector u δ dx y = 57.3θ z e = f (x,u,t) state evolution dt = = ⋅ -2 + α − θ − + y g(x,t) observation V& -1.93 10 V 8.82 32.2 0.58q 0.17δe α = ⋅ -4 − α + − ⋅ -3δ x1 - velocity V [ft/sec] & -2.54 10 V 1.02 0.91q 2.15 10 e x2 - angle of attack α [rad] θ& = q x3 - pitch angle θ [rad] q = 2.95 ⋅10-12V + 0.82α −1.08q − 0.18δ x4 - pitch rate q [rad/sec] & e δ e - elevator deflection [deg] EE392m - Spring 2005 Control Engineering 2-4 Gorinevsky Linear state space model • Linear Time Invariant (LTI) state space model: dx = Ax + Bu state evolution dt y = Cx observations • Can be integrated analytically or numerically (simulation) • Can be well analyzed: stability, response Example: F16 Longitudinal Model ⎡−1.93⋅10−2 8.82 − 32.2 − 0.48⎤ ⎡ 0.17 ⎤ ⎢ −4 ⎥ ⎢ −3 ⎥ dx − 2.54 ⋅10 −1.02 0 0.91 − 2.15⋅10 = ⎢ ⎥ ⋅ x + ⎢ ⎥ ⋅ u dt ⎢ 0 0 0 1 ⎥ ⎢ 0 ⎥ ⎢ −12 ⎥ ⎢ − ⎥ ⎣ 2.95⋅10 0.82 0 −1.08⎦ ⎣ 0.18 ⎦ y = []0 0 57.3 0 ⋅ x EE392m - Spring 2005 Control Engineering 2-5 Gorinevsky Integrating linear autonomous system dx Example: = Ax >>A = [-1.93e-2 8.82 -32.2 -0.58; dt -2.54e-4 -1.02 0 0.91; 0 0 0 1; • Matrix exponential 2.95e-12 0.82 0 -1.08]; – Can be computed in Matlab >> expm(A) as expm(A) ans = 0.9802 3.1208 -31.8830 -10.2849 – The definition corresponds -0.0002 0.5004 0.0031 0.3604 -0.0000 0.2216 1.0002 0.6709 to integrating the ODE by -0.0001 0.3239 0.0007 0.4773 Euler method x(T + ∆t) ≈ x(T ) + ∆tAx(t) = ()I + ∆tA x(T ) x(T + n∆t) = ()I + ∆tA n x(T ) → exp( An∆t)x(T ) ∆ exp( At) = lim ()I + ∆tA t / t ∆t→0 EE392m - Spring 2005 Control Engineering 2-6 Gorinevsky Integrating linear autonomous system 3 dx = Ax, x(0) = x 2 dt 0 1 0 5 10 15 20 • Initial condition response 0.02 = 0 x(t) exp( At)x0 -0.02 -0.04 0 5 10 15 20 0.02 Example: 0 %Take A, B, C from the F16 example -0.02 -0.04 >> k = 0.02 -0.06 >> G = A + k*B*C; -0.08 >> x0 = [0.1700; 0 5 10 15 20 -0.0022; 0.05 0; 0 -0.1800]; -0.05 >> -0.1 for j = 1:length(t); -0.15 x(:,j)=expm(G*t(j))*x0; end; 0 5 10 15 20 EE392m - Spring 2005 Control EngineeringTIME 2-7 Gorinevsky Eigenvalues and Stability Example: % take A from % the F16 example • Consider eigenvalues of the matrix A >> eig(A) ans = λ = λ − = -1.9125 { j } eig( A); det( I A) 0 -0.1519 + 0.1143i -0.1519 - 0.1143i 0.0970 • Suppose A has all different and nonzero eigenvalues, then = ⋅ λ ⋅ −1 A V diag{ j } V λ − exp( At) = V ⋅ diag{e jt } ⋅V 1 λ < • The system solution is exponentially stable if Re j 0 • If A has eigenvalues with multiplicity more than 1, things are a bit more complicated: Jordan blocks, polynomials in t λ < • Still the condition of exponentially stability is Re j 0 EE392m - Spring 2005 Control Engineering 2-8 Gorinevsky Input-output models • Black-box models – describe system P as an operator u P y input signal output signal x internal state (hidden) • Historically (50 years ago) – Black-box models EE – State-space models ME, AA EE392m - Spring 2005 Control Engineering 2-9 Gorinevsky Linear System (input-output) • Linearity ⋅ ⎯⎯→P ⋅ ⋅ ⎯⎯→P ⋅ u1( ) y1( ) u2 ( ) y2 ( ) ⋅ + ⋅ ⎯⎯→P ⋅ + ⋅ au1( ) bu2 ( ) ay1( ) by2 ( ) • Linear Time-Invariant systems - LTI u(⋅ − T ) ⎯⎯→P y(⋅ − T ) u y ⎯⎯→P t t EE392m - Spring 2005 Control Engineering 2-10 Gorinevsky Convolution representation • Convolution integral t signal y(t) = ∫ h(t −τ )u(τ )dτ processing y = h * u −∞ notation • Impulse response u(t) = δ (t) ⇒ y(t) = h(t) • Step response: u = 1 for t > 0 t t d g(t) = ∫ h(t −τ )dτ =∫ h(τ )dτ h(t) = g(t) 0 0 dt EE392m - Spring 2005 Control Engineering 2-11 Gorinevsky Impulse Response for State Space Model dx Example: = + Ax Bu state evolution >> A = [-0.3130 56.7000 0; dt -0.0139 -0.4260 0; y = Cx observation 0 56.7000 0]; >> B = [0.232; 0.0203; 0]; >> C = [0, 0, 1]; • Impulse response for the state x 1 x(∆t) ≈ ∆tAx(0) + B∆tu 0.5 { 0 1 -0.5 x(∆t) ≈ B∆tu = B 0 10 20 0.04 x(t) = exp( At)B 0.02 0 -0.02 0 10 20 • System impulse response 1 h(t) = C exp( At)B 0.5 0 0 10 20 TIME EE392m - Spring 2005 Control Engineering 2-12 Gorinevsky Formal transfer function • Rational transfer function = IIR (Infinite Impulse Response) model – Broad class of input-output linear models d m y d m−1 y d n u d n−1u a + a + + a + y = b + b + + b + u 1 dt m 2 dt m−1 K m 1 1 dt n 2 dt n−1 K n 1 d • Differentiation operator → s dt • Formal transfer function – rational function of s N (s) N (s) = a s m + + a s + a y = H (s) ⋅ u = ⋅ u 1 K m m+1 D(s) = n + + + D(s) b1s K bn s bn+1 • For a causal system m ≤ n EE392m - Spring 2005 Control Engineering 2-13 Gorinevsky Poles, Impulse Response N (s) N (s) = 0 - zeros y = ⋅ u D(s) D(s) = 0 - poles = − M 1 ⋅ ⋅ − M K • Expand D(s) (s p1 ) K (s pK ) N (s) N (s) Then y = 1 ⋅ u + + K ⋅ u − M 1 K − M K (s p1 ) (s p K ) • Quasi-polynomial impulse response – see a textbook − − M 1 1 p1t M K 1 pK t h(t) = (c t + + c + )e + + (c t + + c + )e 1,1 K 1,M 1 1 K K ,1 K K ,M K 1 2 d y = 1 ⋅ •Example: = u Transfer function: y 2 u dt 2 s = + Impulse response: h(t) v0t x0 EE392m - Spring 2005 Control Engineering 2-14 Gorinevsky Transfer Function for State Space • Formal transfer function for a state space model − = + y = ()sI − A 1 B ⋅ u d → sx Ax Bu s − dt y = Cx H (s) = C()sI − A 1 B • Characteristic polynomial N (s) = det()sI − A = 0 Poles det()sI − A = 0 eigenvalues • Poles are the same as eigenvalues of the state-space matrix A λ • For stability we need Re pk = Re k < 0 EE392m - Spring 2005 Control Engineering 2-15 Gorinevsky Laplace transform ∞ • Laplace integral transform: x(t) → xˆ(s) = ∫ x(t)e st dt 0 • Laplace transform of the convolution integral yields ∞ Imag s yˆ(s) = H (s)uˆ(s) H (s) = ∫ h(t)est dt 0 • Transfer function: H (s) Re s – function of complex variable s – analytical in a right half-plane Re s ≥ a – for a stable system a ≤ 0 N(s) dx –for an IIR model H(s) = → sxˆ(s) D(s) dt EE392m - Spring 2005 Control Engineering 2-16 Gorinevsky Frequency decomposition • Sinusoids are eigenfunctions of an LTI system: y = H (s)u LTI Plant iωt iωt ω e H (i )e ω d ω ω s()ei t → ei t = iωei t dt • Frequency domain analysis ω ω i kt → = ω i kt ∑uk e y ∑uk H(i k )e EE392m - Spring 2005 Control Engineering 2-17 Gorinevsky Frequency domain description • Frequency domain analysis 1 − ω 1 − ω u = u~(ω)e i t dω ⇒ y = H(iω)u~(ω)e i t dω 2π ∫ 2π ∫ 142 43 ~y (ω) ω Packet ei t Packet eiωt u of u~(ω) H(iω) of ~y(ω) y sinusoids sinusoids • Fourier transform – numerical analysis • Laplace transform – complex analysis u(t) = 0, for t < 0 ∞ ∞ u~(ω) = u(t)eiωt dω = ⎛ u(t)est dω⎞ = uˆ(iω) −∞ ⎜ −∞ ⎟ ∫ ⎝∫ ⎠s=iω EE392m - Spring 2005 Control Engineering 2-18 Gorinevsky Continuous systems in frequency domain ∞ ~x(ω) = ∫ x(t)eiωt dt • Fourier transform −∞ [−∞,∞] → [−∞,∞] ∞ 1 − ω x(t) = ~x(ω)e i t dω • Inverse Fourier transform π ∫ 2 −∞ t y(t) = ∫ h(t −τ )u(τ )dt • I/O impulse response model −∞ ∞ H (s) = ∫ h(t)est dt • Transfer function 0 ~ ω = ω ~ ω y( ) H (i )u( ) • System frequency response EE392m - Spring 2005 Control Engineering 2-19 Gorinevsky Frequency domain description M (ω) = H(iω) • Bode plots: ϕ(ω) = arg H(iω) u = eiωt Bode Diagram 15 iωt = ω y H(i )e 10 • Example: 5 1 0 Magnitude (dB) H(s) = -5 s − 0.7 0 -45 •|H| is often measured -90 in dB -135 – [dB] = 20 log M (deg) Phase 10 -180 -2 -1 0 10 10 10 Frequency (rad/sec) EE392m - Spring 2005 Control Engineering 2-20 Gorinevsky Model Approximation • Model structure – physics, computational • Determine parameters from data • Step/impulse responses are close the input/output models are close • Example – fit step response • Linearization of nonlinear model EE392m - Spring 2005 Control Engineering 2-21 Gorinevsky Black-box model from data • Linear black-box model can be determined from the data, e.g., step response data, or frequency response STEP RESPONSE 1 0.8 0.6 0.4 HEAT FLUX 0.2 0 0 20 40 60 80 100 TIME • Example problem: fit an IIR model of a given order • This is called model identification • Considered in more detail in Lecture 8 EE392m - Spring 2005 Control Engineering 2-22 Gorinevsky Linear PDE models • Include functions of spatial variables – electromagnetic fields – mass and heat transfer – fluid dynamics – structural deformations • Example: sideways heat equation x ∂ ∂2 T = T k 2 ∂t ∂x Tinside=u Toutside=0 T (0) = u; T (1)