optimal nonlinear control for time delay...

Optimal nonlinear control fortime delay system

Liliam Rodríguez-Guerrero, Omar Santos and Sabine MondiéAutomatic Control Department

CINVESTAV-IPNMexico

International WorkshopDelsys 2013

November 22th, 2013

. . (International WorkshopDelsys 2013)Optimal nonlinear control for time delay system November 22th, 2013 1 / 21

Content

Inverse optimality for delay free nonlinear systems.

Problem for a class of nonlinear time delay systems.

Inverse optimality for a class of nonlinear time delaysystems.

Dehydration process.

. . (CINVESTAV)Optimal nonlinear control for time delay system November 22th, 2013 2 / 21

Inverse optimality for free nonlinear systems

The dificulty in delay free nonlinear systems is to solve theHamilton-Jacobi-Bellman (HJB) equation and to propose theappropriate functional.

This problem is avoided by the approach known as inverse optimality(Freeman and Kokotovic, 1996) using Control Lyapunov Functions(CLF). In this approach it is possible to obtain an optimal control lawwithout solving the HJB´s equation, by defining a specificperformance index that depends on the proposed CLF.

For applying this approach, it is necesary to prove first that thefunctional is a CLF.


Control Lyapunov Function (CFL) (Freeman andKokotovic,1996).

DefinitionA positive definite continuously diifferentiable function V (x) is a ControlLyapunov Function (CFL) of the affi ne system

x(t) = f0(x) + f1(x)u (1)

if there exist a control law u such that the time derivative along thesolutions of system (1) satisfies

dV (x)dt

∣∣∣∣(1)= ∇xV (x)T · x(t) = ψ0(x) + ψT1 (x)u(t) < 0 (2)

where: ψ0(x) = ∇xV (x)T f0(x) and ψT1 (x) = ∇xV (x)T f1(x).


From the previous definition of CLF, it is clear that:

If ψT1 (x) = 0 when x 6= 0 (the solution has not converged yet).

It is only necessary to prove that ψ0(x) < 0 in order to satisfy inequality(2).

If this condition is satisfied the asymptotic stability of the system isguaranteed.


Inverse optimality approach

Freeman and Kokotovic proposed to solve the inverse optimality problemapplying the CLF approah for nonlinear affi ne systems (1). Suppose thatV (x) is a CLF for this system and the derivative along its trajectories is

dV (x)dt

∣∣∣∣(1)= ψ0(x) + ψT1 (x)u(t)

The positive definite functions are

q(x) =[ΨT1 (xt )Ψ1(xt )

]+

√[Ψ0(xt )]

2 +[ΨT1 (xt )Ψ1(xt )

]2,

r(x) =14

[ΨT1 (xt )Ψ1(xt )

]ΨT1 (xt )Ψ1(xt ) +Ψ0(xt ) +

√[Ψ0(xt )]

2 +[ΨT1 (xt )Ψ1(xt )

]2 ,. . (CINVESTAV)Optimal nonlinear control for time delay system November 22th, 2013 6 / 21

Optimal control law

The performance index is

J =∫ ∞

0

[q(x) + r(x)uT u

]dt

The HJB´s equation associated to the system (1) and to the performanceindex is

minu

(dV (x)dt

∣∣∣∣(1)+ q(x) + r(x)uT u

)= 0

which guarantees asymptotic stability.

If we guarantee that V (x) is a CLF it is posible to synthesize the optimalcontrol law. We compute the first derivative with respect to u and weobtain the control law of the form:

u = −12

Ψ1(xt )r(x)


Inverse optimality for a class of nonlinear time delaysystems

Inverse optimality is an open problem for time delay systems, so themain idea in this work is to extend this approach to a class ofnonlinear time delay systems, those which have a stable nominallinear part and a nonlinear part which satisfies some properties.

In contrast with others approaches, our proposal is constructive. Infact, it is based in the complete type functional approach (Kharitonovand Zhabko, 2003).


We consider the class of nonlinear systems with state delay

x(t) = A0x(t) + A1x(t − h) + F (xt ) + Bu(t), (3)

where:

the state x(t) ∈ Rn,with inicial condition x(θ) = ϕ(θ), θ ∈ [−h, 0],ϕ ∈ PC([−h, 0],Rn), and a constant delay h > 0.the nominal system matrices are A0, A1 ∈ Rn×n.

the nominal system

x(t) = A0x(t) + A1x(t − h) (4)

is stable (if not, a preliminary stabilyzing control is applied).

F (xt ) ∈ Rn, F (xt ) = F (x(t), x(t − h)) is a nonlinear function. It isknown and satisfies the Lipschitz´s condition

‖F (x)‖ ≤ α ‖x(t)‖+ β ‖x(t − h)‖ .

B ∈ Rn×m , and the control law is u(t) ∈ Rm .


Inverse optimality approach apply to nonlinear affi ne timedelay systems

Consider the system

x(t) = A0x(t) + A1x(t − h) + F (xt )︸︷︷︸f0(xt )

+ B︸︷︷︸f1(xt )

u(t).

We first consider consider the complete type functional (Kharitonov andZhabko, 2003) associated to the nominal system (4),

V (xt ) = xT (t)U(0)x(t) + 2xT (t)∫ 0

−hU(−h− θ)A1x(t + θ)dθ (5)

+∫ 0

−hxT (t + θ) [W1 + (h+ θ)W2] x(t + θ)dθ

+∫ 0

−h

∫ 0

−hxT (t + θ1)AT1 U(θ2 − θ1)A1x(t + θ2)dθ1dθ2.


The time derivative along the trajectories of system (3) is

dV (xt )dt

∣∣∣∣(3)= −ω0(xt ) + 2 [F (xt ) + Bu(t)]

T ω1(xt ),

where

ω0(xt ) = xT (t)W0x(t) + xT (t − h)W1x(t − h)

+∫ 0

−hxT (t + θ)W2x(t + θ)dθ

and

ω1(xt ) = U(0)x(t) +∫ 0

−hU(−h− θ)A1x(t + θ)dθ.


Equivalently,

dV (xt )dt

∣∣∣∣(3)= Ψ0(xt ) +ΨT

1 (xt )u(t),

where:

Ψ0(xt ) = −ω0(xt ) + 2ωT1 (xt )F (xt )

ΨT1 (xt ) = 2ωT

1 (xt )B.


After appropriate majorizations we get

dV (xt )dt

∣∣∣∣(3)≤ −ηTEη,

where:ηT =

[x(t) x(t − h)

∫ 0−h S(θ)x(t + θ)dθ

],

S(θ) = U(−h− θ)A1 ∈ Rn×n, θ ∈ [−h, 0],

E =

W0 − αIn 0 00 W1 − βIn 00 0 1

hs2W2 − (α+ β) In

,

where α = 2α ‖U(0)‖+ β ‖U(0)‖+ α, β = β ‖U(0)‖+ β,s = supθ∈[−h,0] ‖S(θ)‖ .


Suffi cient conditions

Proposition:

Consider the nonlinear delay system (3), suppose that the nominal lineardelay system is stable and that the nonlinear function F (xt ) satisfies aLipschitz´s condition.

If there exist positive definite matrices Wi ∈ Rn×n, i = 0, 1, 2, whichsatisfy W = W0 +W1 + hW2, and positive scalars α and β such that thematrix E is positive definite then the complete type functional V (xt ) givenby (5) is a Control Lyapunov Function for system (3).


Illustrative example

Consider the chemical refining process with transport lag (Ross, 1971),described by the model

x(t) = A0x(t) + A1x(t − 1) + F (xt ) + Bu(t)

A0 =

−4.93 −1.01 0 0−3.2 −5.3 −12.8 06.4 0.347 −32.5 −1.040 0.833 11.0 −3.96

,

A1 =

1.92 0 0 00 1.92 0 00 0 1.87 00 0 0 0.724

, B =

1 00 10 00 0

,with inicial condition x(θ) = ϕ(θ), θ ∈ [−h, 0], whereϕ(θ) = {0.1, 0.01, 0.01, 0.01}. The nonlinear function is given by

F (xt ) = 0.178 sin(x(t)) + 0.042 sin(x(t − 1))


We compute the optimal control law as

u(t) =

{− 12

Ψ1(xt )r (x ) , Ψ1(xt ) 6= 0

0, Ψ1(xt ) = 0

0 1 2 3 4 5 60.04

0.02

0

0.02

0.04

0.06

0.08

0.1

time (s)

x1(t)x2(t)x3(t)x4(t)

0 1 2 3 4 5 60.4

0.2

0

0.2

time (s)

cont

rol

u1(t)u2(t)

.. . (CINVESTAV)Optimal nonlinear control for time delay system November 22th, 2013 16 / 21

Dehydration process

We want to apply our theoretical approach to dehydration process. In thisprocess, the high consumption of energy for the temperature control,justifies the use of optimal control techniques.

x(t) = a0x(t) + a1x(t − h) + f (xt ) + bu(t − τ)


In this process the distance between the heat source and the productinduces a transport delay.

In previous contributions, an experimental comparison betwen twocontrol laws is reported (Santos and et al, 2012) and(Rodríguez-Guerrero and et al, 2012).

A linear optimal control law with delay compensation and anindustrial PID controller were applied to the dehydration process.

The performance, in terms of power consumption of the optimalcontrol law outperformed the PID controller.


Future work

Our current work includes the synthesis of the optimal control law usinginverse optimality approach and its application to a dehydration process.

Our objective is to study the power consumption and its effects on theproduct quality :

lycopene

phenols

vitamin C

color in the case of tomato.

We want to minimize power consumption and loss of nutrients usingvariational calculus techniques.


References

Ross, D. W. Controller desing for time lag systems via a quadraticcriterion, IEEE Transaction on Automatic Control 16 (6) (1971)664-672.

Kharitonov, V.L. and A.P. Zhabko, Lyapunov-Krasovskii approach forrobust stability of time delay systems, Automatica, 39:15-20, 2003.

Santos, O., Rodríguez-Guerrero, L. & López-Ortega, O. Experimentalresults of a control time delay system using optimal control„OptimalControl, Applications and Methods, Wiley Inter-Science. 2012, 33 (1):100-113..

Rodríguez-Guerrero L., López-Ortega O. & Santos O. Object-orientedoptimal controller for a batch dryer system, International Journal ofAdvanced Manufacturing Technology. 2012, 58 (1-4): 293-307.

Freeman, R. A., Kokotovic, P. (1996). Robust Nonlinear ControlDesign.State-Space and Lyapunov Techniques .Birkhäuser.


Thank you


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