Some basic classical and quantum stochastic processes

• 17. Classical stochastic processes: Brownian motion, Poisson process and more generally Levy processes.
• 18.Ito’s formula for Brownian motion and Poisson processes.
• 19. Quantum Brownian motion and quantum Poisson processes: Noncom-mutative generalizations of classical BM and classical PP.
• 20. Realizing classical stochastic processes using families of operators in Boson Fock space in a coherent state.
• 21. Bernoulli random variables in classical and quantum probability.
• 22. Stochastic differential equations driven by Brownian motion: Existence and Uniqueness.
• 23. Stochastic differential equations driven by Levy processes.
• 24. The generalized Ito-Doleans Dade-Meyer differential rule for discontinuous Martingales.

Kolmogorov’s Martingale inequality, Doob’s Martingale inequality, convergence theory of Martingales based on the down-crossing theorem, construction of the stochastic integral w.r.t L² Martingales. Exponential Martingales and the Martingale version of the change of measure theorem.

25. Girsanov’s formula for the change of drift in an sde.

Consider two diffusion processes xi(t),X2(t) with same diffusion coefficient cr(x) but different drift coefficients ¡q (x), («) respectively. Thus, they satisfy the sde’s

dxk(t) = bk(x(t))dt + k = 1, 2

Let Pk, k = 1,2 denote the probability measures induced by them on the spaces C[0, T] starting at a fixed point rr(O). Then, we have from intuitive considerations, after defining a(x) = a(x)a(x)^T ,b(x) = bi(x) — bi(it’),

cIP. fT

⁼ = ^ea:P((-l/2) / [(dx(t)-b₂(x(t))dt)^T(a(x(t))dt)~¹(dx(t)-b2(x(t))dt)

“¿1 Jo

—(d®(t) - b₁(x(t))dt)^T(a(x(t))dt)~¹(dx(t) - d,(a:(t))(it)])

_rT

= exp( / [dx(t)^Ta(x(t))~^lb(x(t))-{l/2)(b-2(x(t)^Ta(x(t)pⁱb‘2(x(t))-bi(x(t))^Ta(x(t))~^lbi(x(t))dt]) Jo

It should be noted that if x is a diffusion process with drift-diffusion coefficient pair w.r.t the probability measure Pi, then a; is a diffusoin process

with drift-diffusion coefficient pair (b^x), f(a;)) w.r.t the probability measure R.Pi = P?. In particular, suppose a(x) = I and bi(x) = 0. Then, Fi is the Wiener measure, bz(x) = b(x) and writing B(t) = x(t), we have that B(t) is a Brownian motion process w.r.t Pi while

T T

R(B) = exp( f b(B(t))^TdB(t) - (1/2) f || d(B(t)) ||² dt) Jo Jo

with B(t) being a Brownian motion with drift b(.) w.r.t P% = R-Pi, ie, B(t) — Jo b(B(s))ds is a Brownian motion process w.r.t R.Pi. This is Girsanov’s formula for diffusion process.

Some applications of classical probability to engineering systems

26. Disturbance observer models and disturbance rejection methods.

Abstract: Here, we describe the general kind of disturbance observer used in nonlinear systems and more specifically in robotics. The idea in the design of the disturbance observer is that its governing differential equations should be based on only the instantaneous angular position and velocity of the robot or equivalently only on the current state of the nonlinear system governed by the state variable dynamics. It should not involve the acceleration of the robot or equivalently, the derivative of the state of the nonlinear system. Without taking into account random noise in the state dynamics, we evaluate the rate of change of the Lyapunov energy of the disturbance estimation error and derive conditions under which this error converges to zero as time goes to oc under the condition that in this asymptotic limit, the disturbance converges to a constant de level. After taking random noise into account, we evaluate the performance in the sense of mean value of the rate of change of the Lyapunov energy and study conditions under which this value is asymptotically bounded.

Consider the robot system

Af (q)q" + A^r(g, q) = r(t) + d(t) + w(t) where q is the angular position vector of the links, t is the external torque, d is the disturbance and w is WGN. Consider the equations

= z(t)+p(q'(t)),

z'(t) = L(q(t), q'- N(q(t), q'(t)) - d(t))

Then, we have

d'(t) = L(N - t - d) + piq'^q"

= L(N -r-d) + p'tq^Mtq)-¹^ -N + d + w)

I^p'^M^

we get

d'(t) =L(d-d+w)

Assume that

wi = d — d

is WGN. Then writing e = wi +w, it follows that e is also WGN. So our system of equations for the state and disturbance observer after disturbance rejection becomes

M(q)q" + N(q, q') = r + d- d + w

= t + e,

d! = L(q,q)e

and the EKF can be applied to this system. More generally, consider a system of the form

Other models for disturbance rejection in other kinds of systems.

x'(t) = + d(t) + w(t)

where a;(t), d(t), w(t) € K'¹. The explicit dependence of f on time arises because of a known input, d is the disturbance and w is WGN. We design the disturbance observer as follows:

d(t) = z(t) + p(x(t))

where

?(t) = -L(t,a;(t))(/(t,a;(t)) + d(t))

Then, we get

d'(t) = p'(x(t)((f(t, x(t)) + d(t) + w(t)) - L(t, a;(t))(/(t, x(t)) + d(t))

Assume that

=p(a;(t))

Then,

d'(t) = - d(t) + w(t))

which is appropriate for a disturbance observer provided that p^z(a;(t)) is positive definite.

Lyapunov energy based disturbance observer: Let w(t) = 0 and define

V(t) = (l/2)(d(t) - d(t))^T J(x(t))(d(t) - d(t))

Then,

V'(i) = -(l/2)(d(i)-d(t))'^r[/<(_a;(t))^rJ(_a;(t))+J(_a;(i))A'(x(i))-£.7_i(a:(t))a_;'(i)](d(t)-d(i))

whereand we neglect d' (t), ie, at large times, the disturbance is nearly a de disturbance. Let = v and mini^x'^t) = u. Then, if we can ensure that J(x) is

positive definite and simultaneously

K(x)^T J(x) + J(x)K(x) - £

i

is positive definite where each v, is either v or u, then our disturbance observer will converge to the true disturbance. More precisely, for the rate of change of the Lyapunov energy to be negative, it is sufficient that

K(x)^TJ(x) + J(x)K(x) - u₀(£ II /<(*) II)/ > 0 i

where || . || is the spectral norm of matrices and vq = |xy (t)|. Taking random noise into account:

V'(t) = -(l/2)(d(t)-d(t)+w(t))^T[K(x(t))^TJ(x(t))+J(_a!(t))K(_a;(t))

-£j,(_3!(t))_a!'(i)l(rf(i)-d(t)+«-W) i

and hence, if we assume that d — d = wi is WGN and define e = u>+wi.

Conclusions: We have based on the Lyapunov energy method designed a general kind of disturbance observer based on the instantaneous state of a nonlinear dynamical system and have applied to to the robot problem. The disturbance observer guarantees that in the absence of noise, and the condition that the disturbance converges to a constant de value, the disturbance error will converge to zero. In the presence of random noise, we need to make an analysis of the ensemble averaged Lyapunov energy rate of change. That will be the subject of a future paper.