Impulse Control Inputs and the Theory of Fast Feedback Control A. - - PowerPoint PPT Presentation

Impulse Control Inputs and the Theory of Fast Feedback Control

• A. N. Daryin and A. B. Kurzhanski

Moscow State (Lomonosov) University Faculty of Computational Mathematics and Cybernetics

IFAC World Congress, 2008

Introduction

Impulse controls: instantaneous control actions (”hits“) trajectories with discontinuities, jumps, resets, etc. Mechanical systems: Using ordinary δ-functions: gives jump in velocity Using higher derivatives of δ-functions: gives reset of all coordinates.

Introduction

The emphasis in this paper is on: Higher-order distributions as control inputs (δ-function and its derivatives) Fast controls. Feedback control.

The Impulse Control Problem

˙ x(t) = A(t)x(t) + B(t)u(t) t ∈ [t0, t1] — fixed time interval Problem (1, a Mayer–Bolza Analogy) Minimize J(U(·)) = Var

[t0,t1] U(·) + ϕ(x(t1 + 0))

• ver U(·) ∈ BV [t0, t1] with x(t) generated by control input

u(t) = dU dt starting from x(t0 − 0) = x0.

The Impulse Control Problem

Known result (N. N. Krasovski [1957], L. W. Neustadt [1964]): u(t) =

n

• i=1

hiδ(t − τi) Important particular case: ϕ(x) = I (x | {x1}) — steer from x0 to x1 on [t0, t1]. I (x | A) =

• 0,

x ∈ A; +∞, x ∈ A.

The Value Function

Definition The minimum of J(U(·)) with fixed initial position x(t0 − 0) = x0 is called the value function: V (t0, x0) = V (t0, x0; t1, ϕ(·)). How to find the value function? Integrate the HJB equation. An explicit representation (convex analysis).

The Dynamic Programming Equation

The value function V (t, x; t1, ϕ(·)) satisfies the Principle of Optimality V (t0, x0; t1, ϕ(·)) = V (t0, x0; τ, V (τ, ·; t1, ϕ(·))), τ ∈ [t0, t1] The value function it is the solution to the Hamilton–Jacobi–Bellman variational inequality: min {H1(t, x, Vt, Vx), H2(t, x, Vt, Vx)} = 0, V (t1, x) = V (t1, x; t1, ϕ(·)). H1 = Vt+Vx, A(t)x, H2 = min

u∈S1 Vx, B(t)u+1 = −

• BT(t)Vx
• +1.

The Control Structure

(t, x) H1(t, x) = 0 H2(t, x) = 0 jump U(τ) = α · d · χ(τ − t) dU(t) = 0 wait choose jump direction d = −BTVx choose jump amplitude min α 0 : H1(t, x + αd) = 0

The Explicit Formula

V (t0, x0) = inf

x1∈Rn

• ϕ(x1) + sup

p∈Rn

p, x1 − X(t1, t0)x0 p[t0,t1]

• .

The value function is convex and its conjugate equals V ∗(t0, p) = ϕ∗(X T(t0, t1)p) + I

• X T(t0, t1)p
• B·[t0,t1]
• where p[t0,t1] =
• BT(·)X T (t1, ·)p
• C[t0,t1] and

∂X(t, τ) = A(t)X(t, τ), X(τ, τ) = I. See (Daryin, Kurzhanski, and Seleznev, 2005).

The Generalized Impulse Control Problem

Problem (2) Minimize J(u) = ρ∗[u] + φ(x(t1 + 0))

• ver distributions u ∈ Dk[α, β], (α, β) ⊇ [t0, t1] where x(t) is the

trajectory generated by control u starting from x(t0 − 0) = x0. Here ρ∗[u] is the conjugate norm to the norm ρ on C k[α, β]: ρ[ψ] = max

t∈[α,β]

• ψ(t)2 + ψ′(t)2 + · · · +
• ψ(k)(t)
• 2.

u(t) =

n

• i=1

h(0)

i

δ(t − τi) + h(1)

i

δ′(t − τi) + · · · + h(k)

i

δ(k)(t − τi).

Reduction to the “Ordinary” Impulse Control Problem

How to deal with higher-order derivatives δ(j)(t)? Reduce to problem with ordinary δ-functions, but for a more complicated system. General form of distributions u ∈ Dk: u = dU0 dt + d2U1 dt2 + · · · + dkUk dtk Uj ∈ BV Problem 2 reduces to a particular case of Problem 1 for the system ˙ x = A(t)x + B(t)u, B(t) =

• L0(t)

L1(t) · · · Lk(t)

• and the control u = dU

dt , U(t) =    U0(t) . . . Uk(t)   , with L0(t) = B(t), Lj(t) = A(t)Lj−1(t) − L′

j−1(t).

Reduction to the “Ordinary” Impulse Control Problem

B(t) =

• L0(t)

L1(t) · · · Lk(t)

• What does it look like?

For example: A = 0: B(t) =

• B(t)

−B′(t) B′′(t) · · · (−1)kB(k)(t)

• A, B = const:

B(t) =

• B

AB A2B · · · Ak−1B

Fast Controls

With rank B = n, system can be steered from x0 to x1 in zero time by an ideal control u(t) = h(0)δ(t − t0) + h(1)δ′(t − t0) + · · · + h(k)δ(k)(t − t0). i.e. x1 − x0 = L0(t0)h(0) + L1(t0)h(1) + · · · + Lk(t0)h(k). Approximations of ideal zero-time controls are Fast Controls. They steer the system in arbitrary small “fast” time (“nano” time).

Fast Controls

−4 −2 2 4 −1 −0.5 0.5 1 t Approximation of δ(t) −4 −2 2 4 −1 −0.5 0.5 1 t Approximation of δ(1)(t) −4 −2 2 4 −2 −1 1 2 t Approximation of δ(2)(t) −4 −2 2 4 −2 2 t Approximation of δ(3)(t)

Fast Controls

Problem with Fast Controls reduces to an impulse control problem ˙ x = A(t)x+Mσ(t)u, Mσ(t) =

• M(0)

σ (t)

M(1)

σ (t)

· · · M(k)

σ (t)

• with

M(j)

σ (t) =

t+kσ

t

X(t + kσ, τ)B(τ)∆j

σ(τ − t)dτ

∆0

σ(t) = 1

σ 1[0,σ](t), ∆j

σ(t) = 1

σ(∆j−1

σ

(t) − ∆j−1

σ

(t − σ)) We have Mσ(t) → B(t) as σ → 0.

Examples — Oscillating Systems

k1 k2 m1 m2 w1 w2 kN mN−1 mN wN−1 wN F L1 C1 L2 C2 LN VCN

Examples — Oscillating Systems

           m1 ¨ w1 = k2(w2 − w1) − k1w1 mi ¨ wi = ki+1(wi+1 − wi) − ki(wi − wi−1) mν ¨ wν = kν+1(wν+1 − wν) − kν(wν − wν−1) + u(t) mN ¨ wN = −kN(wN − wN−1) wi = wi(t) — displacements from the equilibrium mi — masses of the loads ki — stiffness coefficients u(t) = dU

dt — impulse control (U ∈ BV )

This system is completely controllable. For N = 20 springs, the dimension of the system is 2N = 40. Feedback control (all wi and ˙ wi measured).

Feedback Control Structure for N = 1

−10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10

jump down jump up wait wait

x1 x2 −10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10 x1 x2

Chain, N = 3, Control with Second Derivatives

Chain, N = 5, Control with Second Derivatives

String, N = 20, Ordinary Impulse Control

String, N = 20, Control with Second Derivatives

Application: Formalization of Hybrid Systems

• ˙

x = A(t, z)x + B(t, z)u + Iu0 ˙ z = ud

• x

∈ Rn z ∈ {0, 1, . . . , N} u = u(t, x) — the online control u0 = u0(t, x, z) — resetting the state space vector u0(t, x, z) =

n−1

• j=0

αj(t, x, z(t − 0))δ(i)(f (x, z)) ud = ud(x, z) = β(x, z(t − 0))δ(fd(x, z)) — resetting the subsystem from k′ to k′′ (β(x, z(t − 0)) = (k′′ − k′)) f0(x, z) = 0 fd(x, z) = 0 — switching surfaces

State Space of a Hybrid System

References

Bensoussan, A. and J.-L. Lions. Contrˆ

• le impulsionnel et in´

equations quasi-variationnelles. Dunod, Paris, 1982. Daryin, A. N. and A. B. Kurzhanski. Generalized functions of high order as feedback controls. Differenc. Uravn., 43(11), 2007. Daryin, A. N., A. B. Kurzhanski, and A. V. Seleznev. A dynamic programming approach to the impulse control synthesis problem. In Proc. Joint 44th IEEE CDC-ECC 2005, Seville, 2005. IEEE. Dykhta, V. A. and O. N. Samsonuk. Optimal impulsive control with

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