Stability of Dynamic Systems Robert Stengel Optimal Control and Estimation, MAE 546 Princeton University, 2018 • Stability about an equilibrium point • Bounds on the system norm • Lyapunov criteria for stability • Eigenvalues • Transfer functions • Continuous- and discrete- time systems Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE546.html http://www.princeton.edu/~stengel/OptConEst.html 1 Dynamic System Stability • Well over 100 definitions of stability • Common thread: Response, x(t), is bounded as t –> ∞ • Our principal focus: Initial-condition response of NTI and LTI dynamic systems 2 Vector Norms for Real Variables • Norm = Measure of length or magnitude of a vector, x • Euclidean or Quadratic Norm 1/2 1/2 L2 norm = x = xT x = x2 + x2 ++ x2 2 ( ) ( 1 2 n ) • Weighted Euclidean Norm T 1/2 2 2 2 1/2 y 2 = (y y) = (y1 + y2 +!+ ym ) T T 1/2 xT DT Dx xT Qx = (x D Dx) = Dx 2 Q DT D = Defining matrix 3 Uniform Stability § Autonomous dynamic system § Time-invariant § No forcing input x(t) = f[x(t)] § Uniform stability about x = 0 x(t0 ) ≤ δ, δ > 0 Let δ = δ (ε) If, for every ε ≥ 0, x(t) ≤ ε, ε ≥ δ > 0, t ≥ t0 Then the system is uniformly stable § If system response is bounded, then the system possesses uniform stability 4 Local and Global Asymptotic Stability • Local asymptotic stability – Uniform stability plus x(t) ⎯t⎯→∞⎯→0 • Global asymptotic stability System is asymptotically stable for any ε • If a linear system has uniform asymptotic stability, it also is x(t) = F x(t) globally stable 5 Exponential Asymptotic Stability § Uniform stability about x = 0 plus x(t) ≤ ke−αt x(0) ; k, α ≥ 0 § –α = Lyapunov exponent § If norm of x(t) is contained within an exponentially decaying envelope with convergence, system is exponentially asymptotically stable (EAS) § Linear, time-invariant system that is asymptotically stable is EAS 6 Exponential Asymptotic Stability ∞ t ⎛ k ⎞ t ∞ k k e−α dt = − e−α = ∫ ⎜ ⎟ 0 0 ⎝ α ⎠ α Therefore, time integrals of the norm of an EAS system are bounded ∞ ∞ T 1/2 ⎛ k ⎞ ∫ x(t) dt = ∫ ⎣⎡x (t)x(t)⎦⎤ dt ≤ ⎜ ⎟ x(0) 0 0 ⎝ α ⎠ and ∞ 2 ∫ x(t) dt is bounded 0 7 Exponential Asymptotic Stability Weighted Euclidean norm and its square are bounded if system is EAS ∞ ∞ T T 1/2 ∫ Dx(t) dt = ∫ ⎣⎡x (t)D Dx(t)⎦⎤ dt is bounded 0 0 with 0 < DT D ! Q < ∞ ∞ T ∫ ⎣⎡x (t)Qx(t)⎦⎤dt is bounded 0 Conversely, if the weighted Euclidean norm is bounded, the LTI system is EAS 8 Initial-Condition Response of an EAS Linear System x(t) = Φ(t,0)x(0) = eF(t)x(0) 2 x(t) = xT (0)ΦT (t,0)Φ(t,0)x(0) is bounded • To be shown – Continuous-time LTI system is stable if all eigenvalues of F have negative real parts – Discrete-time LTI system is stable if all eigenvalues of Φ lie within the unit circle 9 Lyapunovs First Theorem • A nonlinear system is asymptotically stable at the origin if its linear approximation is stable at the origin, i.e., – for all trajectories that start close enough (in the neighborhood) – within a stable manifold (closed boundary) x(t) = f[x(t)] is stable at xo = 0 if ∂f[x(t)] Δx(t) = Δx(t) is stable ∂x xo =0 At the origin is a fuzzy concept 10 Lyapunov Function* Define a scalar Lyapunov function, a positive definite function of the state in the region of interest V ⎣⎡x(t)⎦⎤ ≥ 0 Examples mV 2 E E V 2 V = E = + mgh; = = + h 2 mg weight 2g 1 1 V = xT x; V = xT Px 2 2 T 2 V = ⎣⎡x (t )Px(t )⎦⎤ * Who was Lyapunov? see http://en.wikipedia.org/wiki/Aleksandr_Lyapunov 11 V ⎣⎡x(t)⎦⎤ ≥ 0 Lyapunovs V[0] = 0 for t ≥ 0 Second Theorem Evaluate the time derivative of the Lyapunov function dV ∂V ∂V ∂V ∂V = + x! = + f ⎡x(t)⎤ dt ∂t ∂x ∂t ∂x ⎣ ⎦ ∂V = f ⎣⎡x(t)⎦⎤ for autonomous systems ∂x dV • If < 0 in the neighborhood of the origin, dt the origin is asymptotically stable • Sufficient but not necessary condition for stability • Lyapunov function is not unique 12 Quadratic Lyapunov Function for LTI System Lyapunov function Linear, Time-Invariant System V ⎡x(t)⎤ = xT (t)Px(t) x(t) = F x(t) ⎣ ⎦ Rate of change for quadratic Lyapunov function dV ∂V = x! = xT (t)Px! (t) + x! T (t)Px(t) dt ∂x = xT (t)(PF + FT P)x(t) " −xT (t)Qx(t) 13 Lyapunov Equation The LTI system is stable if the Lyapunov equation is satisfied with positive-definite P and Q PF + FT P = −Q with P > 0, Q > 0 14 Lyapunov Stability: 1st-Order Example Δx(t) = aΔx(t) , Δx(0) given 1st-order initial- condition response F = a, P = p, Q = q Unstable, a > 0 t t Δx(t) = ∫ Δx!(t)dt = ∫ aΔx(t)dt Stable, a < 0 0 0 at = e Δx(0) PF + FT P = −Q PF + FT P = −Q with p > 0, a < 0 with p > 0, a > 0 2 pa < 0 and q > 0 2 pa < 0 and q < 0 ∴ system is stable ∴ system is unstable 15 Lyapunov Stability and the HJB Equation T V ⎣⎡x(t)⎦⎤ = x (t)Px(t) Lyapunov stability Dynamic programming optimality dV ∂V * < 0 = − min H dt ∂t u(t ) Rantzer, Sys.Con. Let., 2001 16 Lyapunov Stability for Nonautonomous (Time-Varying) Systems x!(t) = f[x(t),t]; f[0,t] = 0 for t ≥ 0 Time-varying Lyapunov function bounded above and below by time-invariant Lyapunov functions V ⎣⎡x(t),t ⎦⎤ ≥ Va ⎣⎡x(t)⎦⎤ > 0 in neighborhood of x(0) = 0 For asymptotic stability of time-varying system V ⎣⎡x(t),t ⎦⎤ ≤ Vb ⎣⎡x(t)⎦⎤ > 0 for large values of x(t) Time-derivative of Lyapunov function must be negative-definite dV ⎣⎡x(t ),t ⎦⎤ ⎪⎧∂V ⎣⎡x(t ),t ⎦⎤ ⎪⎫ ⎪⎧∂V ⎣⎡x(t ),t ⎦⎤ ⎪⎫ = ⎨ ⎬ + ⎨ f ⎣⎡x(t )⎦⎤⎬ < 0 in neighborhood of x = 0 dt ⎩⎪ ∂t ⎭⎪ ⎩⎪ ∂x ⎭⎪ 17 Laplace Transforms and Linear System Stability 18 Fourier Transform of a Scalar Variable ∞ ω = frequency, rad / s F [x(t)] = x( jω ) = x(t)e− jωt dt ∫ j ! i ! −1 −∞ x(t) x(t) : real variable x( jω ) : complex variable = a(ω ) + jb(ω ) = A(ω )e jϕ (ω ) x( jω) = a(ω) + jb(ω) A : amplitude ϕ : phase angle 19 Laplace Transforms of Scalar Variables Laplace transform of a scalar variable is a complex number s is the Laplace operator, a complex variable ∞ L[x(t)] = x(s) = ∫ x(t)e−st dt, s = σ + jω 0 Laplace transformation is a linear operation Multiplication by a constant x(t) : real variable x(s) : complex variable L[a x(t)] = a x(s) = a(ω ) + jb(ω ) jϕ (ω ) Sum of Laplace transforms = A(ω )e L[x1(t) + x2 (t)] = x1(s) + x2 (s) 20 Laplace Transforms of Vectors and Matrices Laplace transform of a Laplace transform of a vector variable matrix variable ⎡ ⎤ x1(s) ⎡ a (s) a (s) ... ⎤ ⎢ ⎥ ⎢ 11 12 ⎥ L[x(t)] = x(s) = ⎢ x2 (s) ⎥ L[A(t)] = A(s) = ⎢ a21(s) a22 (s) ... ⎥ ⎢ ... ⎥ ⎢ ... ... ... ⎥ ⎣ ⎦ ⎣ ⎦ Laplace transform of a derivative w.r.t. time ⎡dx(t)⎤ L = sx(s) − x(0) ⎢ dt ⎥ ⎣ ⎦ Laplace transform of an integral over time ⎡ ⎤ x(s) L x(τ )dτ = ⎢∫ ⎥ s ⎣ ⎦ 21 Transformation of the LTI System Equations Time-Domain System Equations x!(t) = Fx(t) + Gu(t) Dynamic Equation y(t) = H x(t) + H u(t) x u Output Equation Laplace Transforms of System Equations sx(s) − x(0) = Fx(s) + Gu(s) Dynamic Equation y(s) = Hxx(s) + Huu(s) Output Equation 22 Laplace Transform of State Vector Response to Initial Condition and Control Rearrange Laplace Transform of Dynamic Equation sx(s) − Fx(s) = x(0) + Gu(s) [sI − F]x(s) = x(0) + Gu(s) 1 x(s) = [sI − F]− [x(0) + Gu(s)] The matrix inverse is −1 Adj(sI − F) [sI − F] = (n x n) sI − F Adj(sI − F) : Adjoint matrix (n × n) Transpose of matrix of cofactors sI − F = det(sI − F) : Determinant (1×1) 23 Characteristic Polynomial of a Dynamic System Matrix Inverse −1 Adj(sI − F) [sI − F] = (n x n) sI − F Characteristic matrix of the system ⎛ ⎞ (s − f11 ) − f12 ... − f1n ⎜ ⎟ ⎜ − f21 (s − f22 ) ... − f2n ⎟ (sI − F) = ⎜ ⎟ (n x n) ⎜ ... ... ... ... ⎟ ⎜ f f ... s f ⎟ ⎝ − n1 − n2 ( − nn ) ⎠ Characteristic polynomial of the system sI − F = det(sI − F) n n−1 24 ≡ Δ(s) = s + an−1s + ...+ a1s + a0 Eigenvalues 25 Eigenvalues of the LTI System Characteristic equation of the system n n−1 Δ(s) = s + an−1s + ...+ a1s + a0 = 0 = (s − λ1 )(s − λ2 )(...)(s − λn ) = 0 Eigenvalues, λi , are solutions (roots) of the polynomial, Δ(s) = 0 λi = σ i + jωi * λ i = σ i − jωi s Plane 26 Factors of a 2nd-Degree Characteristic Equation (s − f11 ) − f12 sI − F = ! Δ(s) − f21 (s − f22 ) 2 = s − ( f11 + f22 )s + ( f11 f22 + f12 f21 ) = (s − λ1 )(s − λ2 ) = 0 [real or complex roots] δ = cos−1 ζ = s2 + 2ζω s +ω 2 with complex-conjugate roots n n λ1 = σ1, λ2 = σ 2 λ1 = σ 1 + jω1 λ2 = σ 1 − jω1 s Plane ω n : natural frequency, rad/s ζ : damping ratio, dimensionless 27 z Transforms and Discrete-Time Systems 28 Application of Dirac Delta Function to Sampling Process § Periodic sequence of numbers Δxk = Δx(tk ) = Δx(kΔt) § Dirac delta function ⎧ ⎪ ∞, (t0 − kΔt) = 0 δ (t0 − kΔt)=⎨ 0, t − kΔt ≠ 0 ⎩⎪ ( 0 ) (t0 − kΔt)+ε δ (t0 − kΔt)dt = 1 ∫(t0 − kΔt)−ε § Periodic sequence of scaled delta functions Δx kΔt δ t − kΔt ( ) ( 0 ) 29 Laplace Transform of a Periodic Scalar Sequence § Periodic sequence of numbers Δxk = Δx(tk ) = Δx(kΔt) § Periodic sequence of scaled delta Δx kΔt δ t − kΔt functions ( ) ( ) § Laplace transform of the delta function sequence ∞ L [Δx(kΔt)δ (t − kΔt)] = Δx(z) = ∫ Δx(kΔt)δ (t − kΔt)e−sΔtdt 0 ∞ ∞ = ∫ Δx(kΔt)e−skΔt dt ! ∑Δx(kΔt)z−k 0 k=0 30 z Transform of the Periodic Sequence z transform is the Laplace transform of the delta function sequence ∞ ∞ L [Δx(kΔt)δ (t − kΔt)] = ∑Δx(kΔt)e−skΔt ! ∑Δx(kΔt)z−k k=0 k=0 z Transform (time-shift) Operator z ! esΔt [advance by one sampling interval] z−1 ! e−sΔt [delay by one sampling interval] 31 z Transform of a Discrete-Time Dynamic System System equation in sampled time domain Δxk +1 = Φ Δxk + Γ Δuk + ΛΔwk Laplace transform of sampled-data system equation (z Transform) zΔx(z) − Δx(0) = Φ Δx(z) + ΓΔ u(z) + ΛΔw(z) 32 z Transform of a Discrete-Time Dynamic System Rearrange zΔx(z) − Φ Δx(z) = Δx(0) + ΓΔ u(z) + ΛΔw(z) Collect terms