"$d_web"'Neural nets/Paper reviews/240112 journal paper review- math only.txt' d_web='/media/bill/RaspPi/' local d_web='http://www.BillHowell.ca/' webSite online initial 24Jan2024 24************************24 To view this file : use a constant width font (eg Liberation Libre Mono) set tabs to 3 spaces Otherwise the [lines, symbols] will not align, which is important for debugging. This review section has my notes for going through the paper ... symbols are messed up during copy-paste. 08********08 Nomenclature, keyword notes : HFLN Howell's Flat-Line Notation - for text editors with no [super, sub]scripts etc trfm() transform operator Fixed-time control Formation control HiTL Human-in-the-loop MASs stochastic multiagent systems TVID time-varying input delay p6h1.0 : G del= (Va, EΠ) directed graph among N nodes LV Lyapunov of node set? V^ = {n1, ... nN} node set no root node is denoted by the leader nj agent j N total number of agents including the leader Nκ = {nj ∈ VΛ : (nκ,nj) ∈ EΠ} set of neighbors of node nκ (nκ,nj) ∈ EΠ signifies that agent nj can transmit information to agent nκ, nj,nk neighbors EΠ ⊆ Va × VΛ represents the edge set Vβ = [β(κ,j)] adjacency matrix β(κ,j) = 1, if (nκ, nj) ∈ EΠ β(κ,j) = 0, if (nκ, nj) !∈ EΠ Vα = diag{Q1, ..., QN} κ-th agent's degree matrix Qκ = sum[j=1 to N, j != κ: β(κ,j)] Vγ = Vα - Vβ Laplacian matrix bκ > 0 when the agent nκ obtains the leader information directed graph G has a spanning tree : assumed that there is at least one agent owning a directed path to every other nodes, i.e. the directed graph G has a spanning tree 08********08 follower dynamics +-----+ 2.2. System Description p7h0.2 Eqn (1), ith follower dynamics with time delay : (1) d(x(κ,l)) = [x(κ,l+1) + f(κ,l)(xκ)]*dt + λ(κ,l)^(-T)*dξ d(x(κ,n)) = (u(ς,κ) + f(κ,n)(xκ))*dt + λ(κ,n)^(-T)*dξ y(κ) = x(κ,1) where : l = 1, 2, ..., n-1 x(κ) = trfm[x(κ,1), x(κ,2), ..., x(κ,n)] ∈ R^n denotes the vector of states κ = 1, 2, ..., N yκ represents the output of system f(κ,l):R^n -> R and λ̄(κ,l) : R^n → R^ρ denote unknown smooth nonlinear functions ξ denotes an independent ρ-dimensional standard Brownian motion u(ς,κ) = u(κ)(t - ς(κ)) represents the control input ς(κ) being the time-varying delay p7h0.2 Eqn (2), dynamics of leader : ẋ(o) = μ(o)*x(o) + χ(o)*u(o) y(o) = x(o) where : μ(o), χ(o) are design parameters u(o) denotes a continuous bounded control input y(o) represents the system output of leader +-----+ p8h0.2 Assumption 1, The time-varying delay ς(κ) is bounded, i.e. ς(κ,min) ≤ ς(κ) ≤ ς(κ,max) with constants ς(κ,min) > 0 and ς(κ,max) > 0. Ξ(Z(c)) Gaussian basis function vector +-----+ p8h0.23 Lemma 1, For any a ∈ R, the inequality given below is accurate: 0 ≤ |a| - a*tanh(a/G) ≤ 0.2785*G where G > 0 is a design parameter. >> hand check with only a calculator : tanh(x) varies from (-1, 1) |tanh(x)| <= |x| (between line x=y, y = 0) |tanh(a/G)| <= |a/G| = |a|/G |tanh(a/G)/a| <= 1/G but sign(a) = sign(a/G) = sign(tanh(a/G)) because G>0 tanh(a/G)/a <= 1/G 0 <= a*tanh(a/G) <= 1.00 let Q = |a| - a*tanh(a/G) let P = |tanh(a/G)| as a -> 0 P -> 1, as a -> infinity P -> 0 Q = |a| - a *tanh(a/G) = |a| - |a|*P = |a|*(1 - P) so as a -> 0 Q -> 0, as a -> infinity Q -> 0?, maximum in middle somewhere? by calculator max(tanh(a/G)/a) 0.1 0.99667994625 0.5 0.92423431452 1e6 1 P = |tanh(a/G)| Q = |a| - a*tanh(a/G) = |a| - P G a tanh(a) P a*P Q Q/G 0.1 1 0.761 0.9999 0.9999 0.0001 0.0010 15 1 0.761 0.0665 0.0665 0.9335 0.0044 0.1 0.1 0.0996 0.7615 0.0761 0.0239 0.239 15 0.1 0.0996 0.0067 0.0007 0.0993 0.0066 0.1 0.2 0.1974 0.9640 0.1928 0.0072 0.072 15 0.2 0.1974 0.0133 0.0027 0.1973 0.0131 0.1 10 0.9999 1 10 0 0 15 10 0.9999 0.5827 5.827 4.173 0.2782 >> too much trouble by hand calculator, but seems plausible This is a very [loose, conservative] criteria! It could be much tighter, but that would complicate subsequent inequalities. I later checked with a very simple computer script. It's nice to see an area of output : grid show qnial> a := 0.2 * ((count 21) - 11) -2.0000 -1.8000 -1.6000 -1.4000 -1.2000 -1.0000 -0.8000 -0.6000 -0.4000 -0.2000 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000 >> leave same qnial> G := 0.1 * ((count 10) + 10) 1.1000 1.2000 1.3000 1.4000 1.5000 1.6000 1.7000 1.8000 1.9000 2.0000 qnial> aG := cart a G >> OK, big table qnial> a_tneqDG := EACH tanh_neqDivG aG >> G still not large enough to see level-off or max qnial> G := 0.2 * ((count 10) + 10) qnial> aG := cart a G qnial> a_tneqDG := EACH tanh_neqDivG aG a ,G | 2.2000 2.4000 2.6000 2.8000 3.0000 3.2000 3.4000 3.6000 3.8000 4.0000 -----+---------------------------------------------------------------------- -2.0 | 0.2539 0.2648 0.2719 0.2762 0.2781 0.2784 0.2773 0.2752 0.2723 0.2689 -1.8 | 0.2667 0.2736 0.2773 0.2785 0.2778 0.2757 0.2727 0.2689 0.2647 0.2601 -1.6 | 0.2753 0.2781 0.2782 0.2763 0.2731 0.2689 0.2642 0.2590 0.2536 0.2480 -1.4 | 0.2785 0.2770 0.2736 0.2689 0.2634 0.2574 0.2512 0.2448 0.2385 0.2323 -1.2 | 0.2743 0.2689 0.2625 0.2554 0.2480 0.2406 0.2333 0.2262 0.2193 0.2126 -1.0 | 0.2611 0.2525 0.2436 0.2348 0.2262 0.2179 0.2100 0.2025 0.1955 0.1888 -0.8 | 0.2369 0.2262 0.2159 0.2062 0.1972 0.1888 0.1809 0.1736 0.1668 0.1605 -0.6 | 0.2001 0.1888 0.1784 0.1691 0.1605 0.1528 0.1456 0.1391 0.1332 0.1277 -0.4 | 0.1491 0.1391 0.1304 0.1226 0.1157 0.1095 0.1039 0.0988 0.0942 0.0900 -0.2 | 0.0827 0.0764 0.0710 0.0663 0.0622 0.0586 0.0554 0.0525 0.0499 0.0475 0.0 | 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2 | 0.0827 0.0764 0.0710 0.0663 0.0622 0.0586 0.0554 0.0525 0.0499 0.0475 0.4 | 0.1491 0.1391 0.1304 0.1226 0.1157 0.1095 0.1039 0.0988 0.0942 0.0900 0.6 | 0.2001 0.1888 0.1784 0.1691 0.1605 0.1528 0.1456 0.1391 0.1332 0.1277 0.8 | 0.2369 0.2262 0.2159 0.2062 0.1972 0.1888 0.1809 0.1736 0.1668 0.1605 1.0 | 0.2611 0.2525 0.2436 0.2348 0.2262 0.2179 0.2100 0.2025 0.1955 0.1888 1.2 | 0.2743 0.2689 0.2625 0.2554 0.2480 0.2406 0.2333 0.2262 0.2193 0.2126 1.4 | 0.2785 0.2770 0.2736 0.2689 0.2634 0.2574 0.2512 0.2448 0.2385 0.2323 1.6 | 0.2753 0.2781 0.2782 0.2763 0.2731 0.2689 0.2642 0.2590 0.2536 0.2480 1.8 | 0.2667 0.2736 0.2773 0.2785 0.2778 0.2757 0.2727 0.2689 0.2647 0.2601 2.0 | 0.2539 0.2648 0.2719 0.2762 0.2781 0.2784 0.2773 0.2752 0.2723 0.2689 qnial> a_tneqDG EACHLEFT < 0.2785 llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll llllllllll >> OK, all fit criteria, and I now see a decline with increasing [a, G] >> The above are not proofs, but I ran out of time to differentiate for the maximum. +--+ p8h0.4 Lemma 2, Define a continuous function f(Z) over the compact set U. The following radial basis function NNs transpose(Π∗)*Ξ(Z) can be utilized to approximate f(Z) such that f(Z) - transpose(Π∗)*Ξ(Z) ≤ h where : Π = transpose[Π1, ..., Πo] represents the ideal weight vector o = number of neural networks Z ⊂ R^Y is the input vector Y input dimension of NNs |h1| ≤ h3 network reconstruction error h3 positive constant Ξ(Z) = transpose[Ξ1(Z), ..., Ξo(Z)] basis function vector Π∗ = arg min{Π∈Ro: sup[Z∈Λ: |f(Z) - transpose(Π)*Ξ(Z)|]} optimal weight vector Ξυ(Z) = exp[-transpose(Z - αυ)*(Z - αυ) / βυ^2], υ = 1, ...,o Gaussian function where : αυ center of the receptive field while βυ width of the Gaussian function >> this appears to be a standard Gaussian kernel function approximation sort of arrangement (not an SVP?) +--+ p9h0.0 Lemma 3: Ξ(Z(c)) = transpose[ Ξ1(Z(c)), . . . , Ξo(Z(c)) ] basis function vector with : Z(c) = transpose[x1, ..., xc] there exist any integers c ≥ v > 0 such that ∥Ξ(Z(c))∥^2 ≤ ∥Ξ(Z(v))∥^2 >> looks straightforward +--+ p9h0.25 Lemma 4 [29]: For any p ≥ 0 and w > 0, the following inequality holds (p(w - p))^s ≤ 1/(s+1)*(w^(1+s) - p^(1+s)) where s denotes a positive constant. >> I wanted to learn Isabelle/HOL theorem prover software with thins, but ran out of time. But mjor [crash, recovery] of my main computer might have been due to Isabelle? The segmentation faults had not happened before. +--+ p9h0.4 Lemma 5 [30]: For any υ1 > 0 and υ2 ≤ υ1 , the following inequality holds (υ1 - υ2)^z ≥ υ2^z - υ1^z where z > 1 is a positive constant. >> seems standard from the past, but I didn't check simplest case: z=2 (υ1 - υ2)^2 = u1^2 -2*u1*u2 + u2^2 therefore (υ1 - υ2)^2 - (υ2^2 - υ1^2) = u1^2 -2*u1*u2 + u2^2 - υ2^2 + υ1^2 = 2*u1^2 - 2*u1*u2 but given υ2 ≤ υ1, the expression on the right is aways >= 0 case of z is R, need much more general proof >> I wanted to learn Isabelle/HOL theorem prover software with this, but ran out of time. +--+ p9h0.6 Lemma 6: There exist variables A ∈ R and K ∈ R and constants g > 0, d > 0, p > 0, such that |A|^g*|K|^d ≤ g/(g + d)*p|A|^(g+d) + d/(g + d)*p^(=g/d)*|K|^(g+d) +--+ p9h0.80 (3) dx = f(x,u)*dt + transpose(λ̄(x,u))*dξ where x represents the system state vector. f and λ̄ denote unknown functions u is the system input ξ is the Brownian motion +--+ p9h0.85 Definition 1 [8, 48]: For system (3), there exists a Lyapunov function V(x) ∈ C^2 such that the differential operator L of V(x) can be defined as LV = ∂V/∂x*f + 1/2*Tr{ transpose(λ̄)*∂^2V/∂x^2*λ̄} where Tr{·} denotes a matrix trace. +--+ p10h0.15 Definition 2 [36]: For system (3), there is a positive constant k and a settling time T ≤ T max with T max being bounded, such that E(|x(t)|^2) < k, then the equilibrium x = 0 can be said to be semiglobal practical fixed-time stability (SPFTS) in mean square. +--+ p10h0.3 Lemma 7 [29]: For ∀ε ∈ [0, t] and the system ẋ(t) = f(x(t), u), there exist two K∞-functions M1(∥x∥) and M2(∥x∥), and a function Ψ(x(t)) ∈ C^2 such that: M1(∥x∥) ≤ Ψ(x(t)) ≤ M2(∥x∥) Ψ̇(x(t)) ≤ -A*Ψ(x(t))^ι - B*Ψ(x(t))^ρ + Λ_ideal with Ψ(x(t)) - Ψ(x(ε)) ≤ - A*∫[ε to t: Ψ(x(ω))^ι*dω + Λ_ideal(t - ε) ] - B*∫[ε to t: Ψ(x(ω))^ρ*dω where : ι > 1, 0 < ρ < 1 A and B are positive constants. 0 < Λ∗ < I with I = min{(1 - m)*A , (1 - m)*B} For ∀t ≥ T , ∥x(t)∥ < Θ is hold with T and Θ being positive constants. Then the system is practical fixed-time stable, and the settling time T can be written as T ≤ T max := 1/m/B(ι - 1) + 1/m/A(1 - ρ) where m ∈ (0, 1). The proof of settling time T can be founded in [29]. 08********08 2. Preliminaries and Problem Formulation +-----+ Lyapunov functions : p9h0.85 Definition 1 [8, 48]: For system (3), there exists a Lyapunov function V(x) ∈ C^2 such that the differential operator L of V(x) can be defined as LV = ∂V/∂x*f + 1/2*Tr{ transpose(λ̄)*∂^2V/∂x^2*λ̄} where Tr{·} denotes a matrix trace. V(x) ∈ C^2 p13h0.2 node n1 (agent) : leader V(κ,1) = ϖ(κ,1)^4/4 + Υ̃(κ,1)^2/2/λ(κ,1)^2* p15h1.0 node nl (agent) : intermediate V(κ,ℓ) = V(κ,ℓ-1) + ϖ(κ,ℓ)/4 + Υ̃(κ,ℓ)^2/2/λ(κ,ℓ) + ψ(κ,ℓ)^2/2 + B̃(κ,ℓ)^2/2/Q(κ,ℓ) p19h0.5 node nN (agent) : last node (agent) V(κ,n) = V(κ,n-1) + ϖ(κ,n)^4/4 + Υ̃(κ,n)^2/2/λ(κ,n) + ψ(κ,n)^2/2 + B̃(κ,n)^2/2/Q(κ,n) +-----+ differential operator of ϖ : View this Section in mono-spaced font (eg Liberation Serif Regular), 10 pixel height, full-screen mode (~230 characters/line), tab=3 spaces. This is important to easily see some alignment of terms for [comparison, error-checking]. I have re-ordered some terms for alignment. p12h0.85 differential operator of ϖ(κ,1) dϖ(κ,1) = [ Ωκ*{ ϖ(κ,2) + α(κ,2) + f(κ,1) - ζ̇κ } - bκ*(µo*xo + χo*uo) - ∑[j=1 to N: β(κ,j)*{x(j,2) + f(j,1) - ζ̇j}] + φ(κ,1)*sgn(δ(κ,1)) + P(κ,1)*δ(κ,1)^3 ]*dt +[ Ωκ*trfm(λ(κ,1)) - ∑[j=1 to N: β(κ,j)*trfm(λ(κ,1))] ] *dξ >> in original equation, - Ωκ*δ(κ,2) term cancels δ(κ,2) term after Ωκ between {} (or "(...)" in original) >> please check as I may have made a mistake p15h0.6 Step ℓ (2 ≤ ℓ ≤ n − 1): differential operator of ϖ(κ,ℓ) dϖ(κ,ℓ) = [ ϖ(κ,ℓ+1) + α(κ,ℓ+1) + f(κ,ℓ) + ψ(κ,ℓ)*(1/τ(κ,ℓ) + 1/2) + {ψ(κ,ℓ) /τ_ideal(κ,ℓ)} + B̂(κ,ℓ)*tanh{B̂(κ,ℓ)*ψ(κ,ℓ))/mκ} + φ(κ,ℓ)*sgn(δ(κ,ℓ)) + P(κ,ℓ)*δ(κ,ℓ)^3 + δ(κ,ℓ−1) ]*dt + trfm(λ̄(κ,ℓ)) *dξ p19h0.2 Step n differential operator of ϖ((κ,n)) dϖ(κ,n) = [ uκ + + f(κ,n) + ψ(κ,n)*(1/τ(κ,n) + 1/2) + ψ(κ,n)^(2*t-1)/τ_ideal(κ,n) + B̂(κ,n)*tanh{B̂(κ,n)*ψ(κ,n)/mκ} + φ(κ,n)*sgn(δ(κ,n)) + P(κ,n)*δ(κ,n)^3 + δ(κ,n−1) + S(κ,n)*δ(κ,n) ]*dt + trfm(λ̄(κ,n)) *dξ +-----+ LV: Lyapunov infinitesimal operator p19h0.7 LV: Lyapunov infinitesimal operator >> I ran out of time, didn't list [initial, intermediate] forms. Young's inequality, Lemmas [1, 4, 5, 6] were used. p23h0.25 LV: Lyapunov infinitesimal operator, Eq (52) LV((κ,1)) ≤ + sum[κ=1 to N: sum[r=1 to n: - ϕ(κ,r)*ϖ(κ,r)^(4*ι) - { ϵ(κ,r)/2 /λ(κ,r)*Y(κ,r)^2 }^p - ϑ(κ,r)*ϖ(κ,r)^(4*p) - Γ(κ,r)/2/ι/λ(κ,r)*Y(κ,r)^(2*ι) ] ] + Λ + sum[κ=1 to N: sum[r=2 to n: - {W(κ,r)/2/Q(κ,r)*B(κ,r)^2 }^p - ψ(κ,r)^(2*ι)/τ_ideal(κ,r) - E(κ,r)/2/Q(κ,r)*B(κ,r)^(2*ι) - { ψ(κ,r)^2 /τ(κ,r) }^p ] ] where : Λ = + sum[κ=1 to N: sum[r=1 to n: - ϵ(κ,r)/2 /λ(κ,r)*Y(κ,r)^2 - ϑ(κ,r)*ϖ(κ,r)^(4*p) - Γ(κ,r) /ι/λ(κ,r)*Y(κ,r)^(2*ι) ] ] + N*(1 - p)*(3*n - 1)*p^(p/(1 - p)) + sum[κ=1 to N: sum[r=2 to n: - W(κ,r)/2/Q(κ,r)*B(κ,r)^2 - E(κ,r)/ι/Q(κ,r)*B(κ,r)^(2*ι) ] ] + sum[κ=1 to N: Λ(κ,n)] 08********08 p11h0.0 Section 3. Controller Design +-----+ 5.1. Numerical Example : p27h0.3 Figure 1: Directed communication graph p27h0.7 Figure 2: Trajectories [yo, yκ] yo leader yκ (κ = 1, 2, 3, 4, 5) followers p28h0.0 Figure 3: Formation errors ℘κ,1 with different methods ℘κ,1 (κ = 1, 2, 3, 4, 5) >> this is important to seeing the impact of the auxilliary systems p28h0.4 Figure 4: Curves of filtering errors ψκ,2 ψκ,2 (κ = 1, 2, 3, 4, 5) p29h0.0 Figure 5: Curves of control signals uκ uκ (κ = 1, 2, 3, 4, 5) +-----+ 5.2. Practical Example : p29h0.5 Figure 6: Trajectories [yo, yκ] yo leader yκ (κ = 1, 2, 3, 4, 5) followers p30h0.0 Figure 7: Curves of formation errors ϖκ,1 π(κ,1) (κ = 1, 2, 3, 4, 5) p30h0.5 Figure 8: Curves of filtering errors ψκ,2 ψ(κ,2) (κ = 1, 2, 3, 4, 5) p31h0.0 Figure 9: Curves of control signals uκ uκ (κ = 1, 2, 3, 4, 5) 08********08 4. Stability Analysis