Closed-Loop Impulse Control of Oscillating Systems A. N. Daryin and - - PowerPoint PPT Presentation

Closed-Loop Impulse Control

• f Oscillating Systems
• A. N. Daryin and A. B. Kurzhanski

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

Periodic Control Systems, 2007

Outline

1

Problem

2

Dynamic Programming Approach

3

Numerical Algorithm

4

Ellipsoidal Approximation

5

Asymptotic Solution (∆t → ∞)

6

Unilateral Impulses

7

Double Constraint Approach

8

Generalized Impulse Control Problem

Oscillating System

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

Oscillating System

           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 )

N → ∞

ρ(ξ)wtt(t, ξ) = [Y (ξ)wξ(t, ξ)]ξ, t > t0, 0 < ξ < L w(t, 0) = 0, wξ(t, L) = u(t)/Y (L), t t0 w(t0, ξ) = w0(ξ), wt(t0, ξ) = ˙ w0(ξ), 0 ξ L w(t, ξ) — displacement from the equilibrium u(t) = dU

dt — impulse control

ρ(ξ) — mass density Y (ξ) — Young modulus

Oscillating System

Normalized matrix form: dx(t) = Ax(t)dt + BdU(t) x(t) = w(t) ˙ w(t)

• w(t) =

   w1(t) . . . wN(t)    This system is completely controllable.

Impulse Control Problem

Problem (1) Minimize J(U(·)) = Var

[t0,t1] U(·) + ϕ(x(t1 + 0))

• ver U(·) ∈ BV [t0, t1] where x(t) is the trajectory generated by

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

2N

• i=1

hiδ(t − τi) Important particular case: ϕ(x) = I (x | {0}) — completely stop oscillations on fixed time interval [t0, t1].

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(·)). V (t0, x0) = inf

x1∈Rn

• ϕ(x1) + sup

p∈Rn

• p, x1 − e(t1−t0)Ax0
• BTe(t1−·)AT p
• C[t0,t1]
• .

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

• e(t0−t1)AT p
• B·[t0,t1]
• where p[t0,t1] =
• BTe(t1−·)AT p
• C[t0,t1].

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 quasi-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, Ax, H2 = min

u∈S1 Vx, Bu + 1 = −

• BTVx
• + 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

Numerical Algorithm

The value function is V (t0, x0) = max

p∈Rn ‚ ‚ ‚BT e(t1−t)AT p ‚ ‚ ‚1 ∀t∈[t0,t1]

p, x0. Replace

• BTe(t1−t)AT p
• 1 by a finite number of linear

inequalities, and [t0, t1] with a finite number of time instants: ˆ V (t0, x0) = max

p∈Rn D qi,BT e(t1−t)AT p E 1,i=1,M t=θ1,θ2,...,θK

p, x0 which is a LP problem.

Numerical Algorithm

Finding control for given (t, x) is a LP ranging problem. The error estimate is V (t, x) ˆ V (t, x) V (t, x)

• 1 + O
• K −1

,

Ellipsoidal Approximation

Xν[t] — backward reach set under condition Var U ν V (t, x) = min {ν | x ∈ Xν[t]} We look for an approximation of Xν[t]. Ellipsoids: E (q, Q) =

• x
• x − q, Q−1(x − q)
• 1
• ρ(ℓ | E (q, Q)) = ℓ, q + ℓ, Qℓ

1 2

(see Kurzhanski and V´ alyi, 1997)

Ellipsoidal Approximation

Ellipsoidal approximation is derived through comparison principle for Hamilton–Jacobi equations (Kurzhanski, 2006): X −

ν [t] = E (0, (ν − k(t))Z(t))

˙ Z = AZ + ZAT − η(t)BBT ˙ k = − 1

4η(t)

• Z(t1)

= 0 k(t1) = 0 Here η(t) 0 is a parameter function Xν[t] = cl

• ν(·)

X −

ν [t]

Ellipsoidal Approximation

−1.5 −1 −0.5 0.5 1 1.5 −1 −0.5 0.5 1 x9 x10

Asymptotic Solution (∆t → ∞)

¨ hi = −ω2

i hi + biu,

i = 1, N.

5 10 15 −3 −2 −1 1 Time t h1, dh1/dt 5 10 15 −1 1 Time t h2, dh2/dt 5 10 15 −1 1 Time t h3, dh3/dt

Asymptotic Solution (∆t → ∞)

cl

• tt1

X1[t] = C =

N

• j=1

Cj X1[t] — backward reach set under condition Var U 1 Cj =

• (h, ˙

h)

• ω2

j h2 j + ˙

h2

j b2 j

• — ellipsoids

V →

t→−∞ V = max j=1,N

• ω2

j h2 j + ˙

h2

j

b2

j

(*) Control strategy: “Optimal”: jump if

• BTVx
• = 1 ⇐

⇒ hj = 0 for all maximizers j in (*). Useless after first jump. ε-optimal: jump if

• BTVx
• 1 − ε:

Var U(·) V 1 − ε

Unilateral Impulses

Additional constraint: dU 0 (dU 0). General case: KdU 0 (K — matrix) or dU ∈ K (K — cone). The minimum number of impulses is the same — 2N. Numerical procedures apply with minor modifications. Asymptotic solution does not change. The problem may be not solvable on small time intervals.

Unilateral Impulses

10

1

10

2

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 ∆ t Minimal Control Norm Unilateral Impulses Bilateral Impulses

Impulse vs Bang-Bang Controls

10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 ∆ t Minimal Control Norm Bang−Bang Control Impulse Control

Double Constraint Approach

Problem (2) Minimize J(u) = t1

t0

|u(t)| dt + ϕ(x(t1))

• ver controls u(t) satisfying |u(t)| µ, where x(t) is the

trajectory generated by control u starting from x(t0) = x0. Here controls are bounded functions. Optimal controls only take values −µ, 0, µ. Vµ(t, x) is the value function for Problem 2. 0 Vµ(t, x) − V (t, x) = O(µ−1) for each (t, x)

Double Constraint Approach

−5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5

u = 0 u = 0 u = −µ u = µ Not Solvable Not Solvable

x1 x2 mu = 5

Double Constraint Approach

−25 −20 −15 −10 −5 5 10 15 20 25 −25 −20 −15 −10 −5 5 10 15 20 25

u = −µ u = µ u = 0 u = 0

x1 x2

Generalized Impulse Control Problem

Problem (3) Minimize J(u) = ρ∗[u] + ϕ(x(t1 + 0))

• ver distributions u ∈ D1[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 1[α, β]: ρ[ψ] = max

t∈[α,β]

• ψ(t)2 + ψ′(t)2.

u(t) =

2N

• i=1

h(0)

i

δ(t − τi) + h(1)

i

δ′(t − τi).

Reduction to Impulse Control Problem

u ∈ D1 : u = dU0 dt + d2U1 dt2 U0, U1 ∈ BV Problem 3 reduces to a particular case of Problem 1 for the system ˙ x = Ax + Bu, B =

• B

AB

• and the control

u = dU dt , U(t) = U0(t) U1(t)

• .

Error bound for numerical algorithm: V (t, x) ˆ V (t, x) V (t, x)

• 1 + O
• K −1 + M−2

,

Examples

Chain of 5 springs String (10 elements)

References

Bellman, R. (1957). Dynamic Programming. Princeton Univ. Press. Bensoussan, A. and J.-L. Lions (1982). Contrˆ

• le impulsionnel et

in´ equations quasi variationnelles. Paris. Crandall, M. G. and P.-L. Lions (1983). Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277, 1–41. Daryin, A. N., A. B. Kurzhanski and A. V. Seleznev (2005). A dynamic programming approach to the impulse control synthesis problem. In:

• Proc. Joint 44th IEEE CDC-ECC 2005. IEEE. Seville.

Demyanov, V. F. (1974). Minimax: Directional Derivates. Nauka. Moscow. Dykhta, V. A. and O. N. Sumsonuk (2003). Optimal impulsive control with applications. Fizmatlit. Moscow. Gusev, M. I. (1975). On optimal control of generalized processes under non-convex state constraints. In: Differential Games and Control

• Problems. UNC AN SSSR. Sverdlovsk.

Kalman, R. E. (1960). On the general theory of control systems. In:

• Proc. 1st IFAC Congress. Vol. 1. IFAC. Butterworths. London.

Krasovski, N. N. (1957). On a problem of optimal regulation. Prikl.

• Math. & Mech. 21(5), 670–677.

References

Krasovski, N. N. (1968). The Theory of Motion Control. Nauka. Moscow. Kurzhanski, A. B. (1975). Optimal systems with impulse controls. In: Differential Games and Control Problems. UNC AN SSSR. Sverdlovsk. Kurzhanski, A. B. and I. V´ alyi (1997). Ellipsoidal Calculus for Estimation and Control. SCFA. Birkh¨

• auser. Boston.

Kurzhanski, A. B. and Yu. S. Osipov (1969). On controlling linear systems through generalized controls. Differenc. Uravn. 5(8), 1360–1370. Miller, B. M. and E. Ya. Rubinovich (2003). Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer. N.Y. Motta, M. and F. Rampazzo (1995). Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differential and Integral Equations 8, 269–288. Neustadt, L. W. (1964). Optimization, a moment problem and nonlinear

• programming. SIAM J. Control 2(1), 33–53.

Rockafellar, R. T. (1970). Convex Analysis. Vol. 28 of Princeton Mathematics Series. Princeton University Press.

Continuous and Smooth Controls

It is required to use continuous or smooth controls. Control force is produced by an integrator F(t) = t

t0

τν

t0

· · · τ2

t0

• ν times

u(τ1) dτ1 · · · dτν u(t) is the new control variable. Hard bound (geometrical constraint) on control: u(t) ∈ P = [−µ, µ] Examples: Continuous Control Smooth Control