L ECTURE 4: D YNAMICAL S YSTEMS 3 I NSTRUCTOR : G IANNI A. D I C ARO - PowerPoint PPT Presentation

15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 4: D YNAMICAL S YSTEMS 3 I NSTRUCTOR : G IANNI A. D I C ARO

EQUILIBRIUM A state 𝒚 " is said an equilibrium state of a dynamical system 𝒚̇ = 𝒈(𝒚) , § if and only if 𝒚 " = 𝒚 𝑢; 𝒚 " ;𝒗 𝑢 = 0 , ∀ 𝑢 ≥ 0 § If a trajectory reaches an equilibrium state (and if no input is applied) the trajectory will stay at the equilibrium state forever: internal system’s dynamics doesn’t move the system away from the equilibrium point, velocity is null : 𝒈 𝒚 " = 0 2

I S THE EQUILIBRIUM STABLE ? When a displacement (a force) is applied to an equilibrium condition: Stable equilibrium Unstable equilibrium Neutral equilibrium Metastable equilibrium § Why are equilibrium properties so important? § For the same definition of an abstract model of a (complex) real-world scenario 3

S ANDPILES , SNOW AVALANCHES AND META - STABILITY Abelian sandpile model (starting with one billion grains pile in the center) 4

L YAPUNOUV VS . S TRUCTURAL EQUILIBRIUM 𝑞 𝑞 𝑞 Structural equilibrium: is the § equilibrium persistent to (small) variations in the structure of the systems? à Sensitivity to the value of the parameters of the vector field 𝒈 Lyapunouv equilibrium: stability of § an equilibrium with respect to a small deviation from the equilibrium point 5

I S THE EQUILIBRIUM (L YAPUNOUV ) STABLE ? An equilibrium state 𝒚 " is said to be Lyapunouv stable if and only if § for any ε > 0, there exists a positive number 𝜀 𝜁 such that the inequality 𝒚 0 − 𝒚 " ≤ 𝜀 implies that 𝒚 𝑢; 𝒚 0 ,𝒗 𝑢 = 0 − 𝒚 " ≤ ε ∀ 𝑢 ≥ 0 𝑢 An equilibrium state 𝒚 " is stable (in the Lyapunouv sense) if the response § following after starting at any initial state 𝒚 0 that is sufficiently near 𝒚 " will not move the state far away from 𝒚 " 6

I S THE EQUILIBRIUM (L YAPUNOUV ) STABLE ? What is the difference between a stable and an asymptotically stable equilibrium? 7

I S THE EQUILIBRIUM ASYMPTOTICALLY STABLE ? If an equilibrium state 𝒚 " is Lyapunouv stable and every motion starting § sufficiently near to 𝒚 " converges (goes back) to 𝒚 " as 𝑢 → ∞ , the equilibrium is said asymptotically stable 𝑢 𝜁,𝜀 𝜁 → 0 as 𝑢 → ∞ 8

S OLUTION OF L INEAR ODE S The general form for an ODE: 𝒚̇ = 𝒈(𝒚) , where 𝒈 is a 𝑜 -dim vector field § The general form for a linear ODE: § 𝒚̇ = 𝐵𝒚, 𝒚 ∈ ℝ < , 𝐵 an 𝑜×𝑜 coefficient matrix A solution is a differentiable function 𝒀 𝑢 § that satisfies the vector field § Theorem: Linear combination of solutions of a linear ODE If the vector functions 𝒚 (@) and 𝒚 (A) are solutions of the linear system 𝒚̇ = 𝒈(𝒚) , then the linear combination 𝑑 @ 𝒚 (@) + 𝑑 A 𝒚 (A) is also a solution for any real constants 𝑑 @ and 𝑑 A § Corollary: Any linear combination of solutions is a solution By repeatedly applying the result of the theorem, it can be seen that every finite linear combination 𝒚 𝑢 = 𝑑 @ 𝒚 @ (𝑢) + 𝑑 A 𝒚 A (𝑢) + …𝑑 E 𝒚 E (𝑢) of solutions 𝒚 @ , 𝒚 A ,…,𝒚 E is itself a solution to 𝒚̇ = 𝒈(𝒚) 9

F UNDAMENTAL AND G ENERAL S OLUTION OF L INEAR ODE S § Theorem: Linearly independent solutions If the vector functions 𝒚 @ , 𝒚 A ,…, 𝒚 < are linearly independent solutions of the 𝑜 -dim linear system 𝒚̇ = 𝒈(𝒚) , then, each solution 𝒚(𝑢) can be expressed uniquely in the form: 𝒚 𝑢 = 𝑑 @ 𝒚 @ (𝑢) + 𝑑 A 𝒚 A (𝑢) + …𝑑 < 𝒚 < (𝑢) § Corollary: Fundamental and general solution of a linear system If solutions 𝒚 @ , 𝒚 A ,…, 𝒚 < are linearly independent (for each point in the time domain), they are fundamental solutions on the domain, and the general solution to a linear 𝒚̇ = 𝒈(𝒚) , is given by: 𝒚 𝑢 = 𝑑 @ 𝒚 @ (𝑢) + 𝑑 A 𝒚 A (𝑢) + …𝑑 < 𝒚 < (𝑢) 10

G ENERAL SOLUTIONS FOR LINEAR ODE S Corollary: Non-null Wronskian as condition for linear independence § The proof of the theorem uses the fact that if 𝒚 @ , 𝒚 A ,…, 𝒚 < are linearly independent (on the domain), then det 𝒀 𝑢 ≠ 0 𝑦 @@ (𝑢) ⋯ 𝑦 @< (𝑢) ⋮ ⋱ ⋮ 𝒀(𝑢) = Wronskian 𝑦 <@ (𝑢) ⋯ 𝑦 << (𝑢) Therefore, 𝒚 @ , 𝒚 A ,…, 𝒚 < are linearly independent if and only if W[𝒚 @ , 𝒚 A ,…, 𝒚 < ](𝑢) ≠ 0 § Theorem: Use of the Wronskian to check fundamental solutions If 𝒚 @ , 𝒚 A ,…, 𝒚 < are solutions, then the Wroskian is either identically to zero or else is never zero for all 𝑢 § Corollary: To determine whether a given set of solutions are fundamental solutions it suffices to evaluate W[𝒚 @ , 𝒚 A ,…,𝒚 < ](𝑢) at any point 𝑢 11

S TABILITY OF L INEAR M ODELS § Let’s start by studying stability in linear dynamical systems … The general form for a linear ODE: § 𝒚̇ = 𝐵𝒚, 𝒚 ∈ ℝ < , 𝐵 an 𝑜×𝑜 coefficient matrix Equilibrium points are the points of the Null space / Kernel of matrix 𝐵 § 𝐵𝒚 = 𝟏, 𝑜×𝑜 homogeneous system § Invertible Matrix Theorem, equivalent facts: 𝐵 is invertible ⟷ det 𝐵 ≠ 0 § The only solution is the trivial solution, 𝒚 = 𝟏 § Matrix 𝐵 has full rank § < det 𝐵 = ∏ 𝜇 U § , all eigenvalues are non null UV@ § … § In a linear dynamical system, solutions and stability of the origin depends on the eigenvalues (and eigenvectors) of the matrix 𝐵 12

R ECAP ON E IGENVECTORS AND E IGENVALUES Geometry: Eigenvectors: Directions 𝒚 that the linear transformation 𝐵 § doesn’t change. The eigenvalue 𝜇 is the scaling factor of the transformation § along 𝒚 (the direction that stretches the most) Algebra: § Roots of the characteristic equation 𝑄 𝜇 = 𝜇𝑱 − 𝐵 𝒚 = 0 → det 𝜇𝑱 − 𝐵 = 0 § For 2×2 matrices: det 𝜇𝑱 − 𝐵 = 𝜇 A − 𝜇 tr 𝐵 + det 𝐵 § Algebraic multiplicity 𝒐 : each eigenvalue can be repeated 𝑜 ≥ 1 times § (e.g., (𝜇 − 3) A , 𝑜 = 2 ) Geometric multiplicity 𝒏 : Each eigenvalue has at least one or 𝑛 ≥ 1 § eigenvectors, and only 1 ≤ 𝑟 ≤ 𝑛 can be linearly independent § An eigenvalue can be 0, as well as can be a real or a complex number 13

R ECAP ON E IGENVECTORS AND E IGENVALUES 14

L INEAR M ULTI -D IMENSIONAL M ODELS For the case of linear (one dimensional) growth model, 𝑦̇ = 𝑏𝑦, solutions § were in the form: 𝑦 𝑢 = 𝑦 c 𝑓 ef § The sign of a would affect stability and asymptotic behavior: x = 0 is an asymptotically stable solution if a < 0, while x = 0 is an unstable solution if a > 0, since other solutions depart from x = 0 in this case. Does a multi-dimensional generalization of the form 𝒚 𝑢 = 𝒚 c 𝑓 𝑩f hold? § What about operator 𝑩 ? § A two-dimensional example: 𝒚 = 𝑦 @ −4 −3 𝑦̇ @ = −4𝑦 @ − 3𝑦 A 𝐵 = 𝒚 (0) = (1,1) 2 3 𝑦̇ A = 2𝑦 @ + 3𝑦 A 𝑦 A Eigenvalues and Eigenvectors of 𝐵 : § 1 3 𝜇 @ = 2, 𝒗 @ = 𝜇 A = −3, 𝒗 A = −2 −1 (real, negative) (real, positive) 15

S OLUTION ( EIGENVALUES , EIGENVECTORS ) The eigenvector equation: 𝐵𝒗 = 𝜇𝒗 § Let’s set the solution to be 𝒚 𝑢 = 𝑓 hf 𝒗 and lets’ verify that it satisfies § the relation 𝒚̇ 𝑢 = 𝐵𝒚 Multiplying by 𝐵 : 𝐵𝒚(𝑢) = 𝑓 hf 𝐵𝒗 , but since 𝒗 is an eigenvector: § 𝐵𝒚 𝑢 = 𝑓 hf 𝐵𝒗 = 𝑓 hf (𝜇𝒗 ) 𝒗 is a fixed vector, that doesn’t depend on 𝑢 → if we take 𝒚 𝑢 = 𝑓 hf 𝒗 § and differentiate it: 𝒚̇ 𝑢 = 𝜇𝑓 hf 𝒗 , which is the same as 𝐵𝒚 𝑢 above Each eigenvalue-eigenvector pair ( 𝜇 , 𝒗 ) of 𝐵 leads to a solution of 𝒚̇ 𝑢 = 𝐵𝒚 , taking the form: 𝒚 𝑢 = 𝑓 hf 𝒗 § The general solution to the linear ODE is obtained by the linear combination of the 𝒚 𝑢 = 𝑑 @ 𝑓 h i f 𝒗 @ + 𝑑 A 𝑓 h j f 𝒗 A individual eigenvalue solutions (since 𝜇 @ ≠ 𝜇 A, 𝒗 𝟐 and 𝒗 𝟑 are linearly independent) 16

S OLUTION ( EIGENVALUES , EIGENVECTORS ) 𝒚 𝑢 = 𝑑 @ 𝑓 h i f 𝒗 @ + 𝑑 A 𝑓 h j f 𝒗 A 𝑦 A 𝒚 0 = (1,1) (1,1) 1,1 = 𝑑 @ (1,−2) + 𝑑 A (3,−1) 𝒗 𝟑 à 𝑑 @ = −4/5 𝑑 A = 3/5 𝑦 @ 𝒗 @ 𝒚 𝑢 = −4/5𝑓 Af 𝒗 @ + 3/5𝑓 opf 𝒗 A 𝑦 @ 𝑢 = − 4 5 𝑓 Af + 9 5 𝑓 opf 𝑦 A 𝑢 = 8 5 𝑓 Af − 3 5 𝑓 opf Saddle equilibrium (unstable) Except for two solutions that approach the origin along the direction of the § eigenvector 𝒗 A = (3 , - 1), solutions diverge toward ∞ , although not in finite time Solutions approach to the origin from different direction, to after diverge from it § 17