optimalctrl sp
Post on 01-Jun-2018
218 Views
Preview:
TRANSCRIPT
-
8/9/2019 OptimalCtrl SP
1/6
Optimal control of an oscillator system
J.C.C. Henriques
2015/02/06
1 Optimal control
1.1 Problem statement
Let x(t ) the state of the system with input u(t ) that satisfies the state equation [1]
ẋ = F (x, u) , (1)
with
x (0) = x0 (2)
u∈ (3)
t ∈ [0, T ] (4)
with T fixed. The set shows the constraints applied to u.
The goal is to determine u, defined in [0, T ] which maximizes the cost functional J defined as
J (u) = Ψ (xT ) +
T 0
(u, x) dt (5)
where
xT = x (T ) . (6)
Ψ (xT ) is the cost associated with the terminal state xT . The Lagrangian (u, x) is the function to
be maximized in [0, T ].
1.2 Pontriagyn maximum principle
Along optimal path for (x, u,λ(t )) it is verified the following necessary conditions for maxi-
mizing J
ẋ = F (x, u) ,
with
x (0) = x0
u∈
-
8/9/2019 OptimalCtrl SP
2/6
t ∈ [0, T ] (7)
and
− λ̇ = λT ∇F (x, u) +∇ (x, u) (8)
subjected to the terminal conditionλ (T ) =∇Ψ (T ) . (9)
The vector λ(t ) is designated by co-state and Eq. ( 8) by adjoint equation.
For each t , the Hamiltonian defined by
(x, u,λ) = λT F (x, u) + (x, u) (10)
is maximum for the optimal input u. If the maximum is for an u in the interior of then∂
∂ u i= 0. (11)
In the case that the optimum is in the boundary of then Eq. ( 11) does not apply.
2 Oscillator system
Let as consider a simple oscillator system comprising a spring, k , a energy extraction damper,c ,
and control damper, G, with damping as function of, u , see Fig. 1. The equation that describes the
motion of this system is
m ¨ z + c ̇ z + g u ˙ z + kz = f cos(Ωt ) (12)
where the dot denotes time derivative, m is the block mass, f and Ω are the modulus are angular
frequency of the external force. The initial conditions are z (0) = z0 and ˙ z(0) = v0.
The extracted energy, e , can be computed using the instantaneous power of the damper c which
is given by
ė = c ˙ z2. (13)
The initial condition is e (0) = 0.
The system of equations ( 12) and ( 13) can be written as a first-order system in vectorial form as
ẋ = F (x, u) , (14)
defining
x =
v z e T
= x1 x2 x3
T , (15)
2
-
8/9/2019 OptimalCtrl SP
3/6
Figure 1: Oscillating system with spring, k , energy extraction damper,c , and control damper, g .
and
F (x, u) =
F 1
F 2
F 3
=
( f cos(Ωt )− c x1− g ux1− k x2)/m
x1
c x21
, (16)
with initial conditions
x0 =
v0 z0 0T
. (17)
We want to determine u (t ) for 0≤ t ≤ T such that
J (u) = e (T ), (18)
is maximum subjected to the constraint 0 ≤ u ≤ 1.
In the present case Ψ (xT ) = e (T ) and (x, u) = 0, see Eq. ( 5). Using this, the adjoint equationis given by
− λ̇ = λT ∇F (x, u) . (19)
The gradient of F is given by
∇F =
∂ F 1∂ x1
∂ F 1∂ x2
∂ F 1∂ x3
∂ F 2∂ x1
∂ F 2∂ x2
∂ F 2∂ x3
∂ F 3∂ x1
∂ F 3∂ x2
∂ F 3∂ x3
=
− (c + g u)/m −k/m 0
1 0 0
2c x1 0 0
, (20)
resulting in
λ̇1
λ̇2
λ̇3
=−
−λ1 (c + g u)/m + λ2 + 2λ3c x1
−λ1k/m
0
. (21)
The final condition of λ is λ(T ) =∇Ψ (T ), givingλ1(T )
λ2(T )λ3(T )
=
∂ Ψ
∂ x1∂ Ψ ∂ x2∂ Ψ ∂ x3
x = xT
=
0
01
. (22)
3
-
8/9/2019 OptimalCtrl SP
4/6
For each t , the Hamiltonian
(x, u,λ) = λT F =λ1 λ2 λ3
( f cos(Ωt )− c x1− g ux1− k x2)/m
x1
c x21
= λ1 ( f cos(Ωt )− c x1− g ux1− k x2)/m +λ2 x1 +λ3c x
21
(23)
is maximum for the optimal input u. The conditions to maximize are
u(t ) =
1, if (−λ1(t ) x1(t ))≥ 0
0, otherwise(24)
The usual solution of this problem is:
1. Set u (t ) = 0.
2. Compute ( 14) using the ( 17).
3. Compute ( 21) backward (from T to 0) to impose the “final” condition ( 22).
4. Determine u (t ) using ( 24).
5. If not converged go to step 2.
A Demonstration of the Pontriagyn maximum principle
Let us define the functional as
= J −
T 0
λT [ẋ−F (x, u)] dt . (25)
The vector λ is introduced so is optimal and satisfies the system of ordinary differential equa-
tions ( 1) for all t ∈ [0, T ]. In other words,
T
0
λT [ẋ−F (x, u)] dt , (26)
is a constraint that is zero for the optimal solution1. The vector λ is called co-state. Introducing
Eq. ( 5) in Eq. ( 28) we get
= Ψ (xT ) +
T 0
(u, x) dt −
T 0
λT [ẋ−F (x, u)] dt
= Ψ (xT ) +
T
0
(u, x) +λT F (x, u)dt −
T
0
λT ẋ dt .
(27)
1Similar to a Lagrange multiplier.
4
-
8/9/2019 OptimalCtrl SP
5/6
Defining the Hamiltonian function
(x, u,λ) = λT F (x, u) + (x, u) , (28)
we get
= Ψ (xT ) +
T
0
(u, x,λ) dt −
T
0
λT ẋ dt . (29)
Let us assume that uopt is the function which maximizes then functional J then a small pertur-
bation δu results in a decrease of . The system will follow another path x +δx and the variation
of the objective function is negative
δ = (x +δx, v)− x, uopt
< 0, (30)
where v = uopt +δu. Using Eq. ( 29) we get
δ = Ψ (xT +δxT )−Ψ (xT ) +
T 0
[ (v, x +δx,λ)− (u, x,λ)] dt −
T 0
λT δẋ dt . (31)
Integrating by parts the last term of Eq. ( 31)
T 0
λT δẋ dt = λT (T )δx (T )−λT (0)δx (0)−
T 0
λ̇T δx dt (32)
Since the optimal control does not change x (0) we have δx (0). As a result,
δ =Ψ (xT +δxT )−Ψ (xT )−λT (T )δx (T )
+
T 0
[ (v, x +δx,λ)− (u, x,λ)] dt +
T 0
λ̇T δx dt .
(33)
Performing a first order Taylor series expansion of the terms on δx we get
Ψ (xT + δxT )≈Ψ (xT ) +∇Ψ (xT )δxT , (34)
and
(v, x +δx,λ)≈ (v, x,λ) +∇ (v, x,λ)δx (35)
Replacing both expansions in Eq. ( 33)
δ =∇Ψ (xT )−λ
T (T )δx (T ) +
T 0
∇ (u, x,λ) + λ̇
T δx dt
+
T 0
[ (v, x,λ)− (u, x,λ)] dt .
(36)
We can choose
λ̇T
=−∇ (u, x,λ) , (37)
5
-
8/9/2019 OptimalCtrl SP
6/6
satisfying
λT (T ) =∇Ψ (xT ) , (38)
to assure that the first term of the right-hand-side of Eq. ( 36) is zero. This results in
δ = T
0[ (v, x,λ)− (u, x,λ)] dt . (39)
The states x and co-states λ are computed for uopt and independently of v. If uopt is the optimum
then
uopt, x,λ
≥ (v, x,λ) (40)
∀v∈ . This needs to be proved because we compute uopt by maximizing (u, x,λ).
Suppose that there is an time instant, t 1 where a function w satisfies
(w (t 1) , x (t 1) ,λ (t 1)) > uopt (t 1) , x (t 1) ,λ (t 1)
(41)
Since is a continuous function, there is an interval [ t 1− ε, t 1 + ε] where this inequality holds.
Let w = uopt except in this interval. Using this choice, the variation is
δ =
t 1+εt 1−ε
(w (t ) , x (t ) ,λ (t ))−
uopt (t ) , x (t ) ,λ (t )
dt > 0 (42)
However, this contradicts the hypothesis that uopt is the optimal control.
References
[1] J. L. M. Lemos, Lectures on Optimal Control, Instituto Superior Técnico, 2012.
6
top related