Robust MPC for fractional-order discrete-time systems

P. Sopasakisα,β, S. Ntouskasβ and H. Sarimveisβ

α Institute for Advanced Studies Lucca, Italy,β National Technical University of Athens, Greece.

June 17, 2015

Fractional-order systemsand finite-dimensional approximations

Fractional derivatives

The Grunwald-Letnikov derivative1 of order α > 0 is defined as

Dαx(t) = limh→0+

∆αhx(t)

hα(1)

where ∆αh is the fractional difference operator of order α with step size

h > 0 given by

∆αhx(t) =

∞∑j=0

(−1)j(α

j

)︸ ︷︷ ︸

cαj

x(t− kh), (2)

where (α

j

)=

j−1∏i=0

α− ii+ 1

. (3)

1S. Samko, A. Kilbas, O. Marinhev, Fractional integral and derivatives, Gordon & Breach Science Publishers, 1993;Section 20.

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Fractional derivatives in action

A few examples:

I Pharmacokinetics (Dokumetzidis et al. 2010, Magin et al. 2004)

I Semi-infinite power transmission lines (Clarke et al., 2004)

I Viscoelastic polymers (Hilfer 2000)

I Anomalous diffusion (Magin 2010, Pereira 2010)

I Electromagnetic theory (Schafer and Kruger 2006, Zelenyi andMilovanov 2004)

I Statistical mechanics (Tarasov 2005)

I and many other...

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Fractional systems

Fractional-order systems

l∑i=1

AiDαix(t) = Bu(t) (4)

Euler-type discretisation [Replace Dαi h−αi∆αih ]

l∑i=1

Ai∆αih xk+1 = Buk (5)

Notice that ∆αih is an infinite-dimensional operator!

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Finite-dimensional approximation

∆αh can be written as

∆αh = ∆α

h,ν +Rαh,ν , (6)

where

∆αh,νx(t) =

ν∑j=0

cαj x(t− kh), (7)

and Rαh,ν is a bounded operator. If we plug (6) into the original systemwe get

l∑i=1

Ai∆αih,νxk+1︸ ︷︷ ︸

Finite dimensional

+

l∑i=1

AiRαih,νxk+1︸ ︷︷ ︸

Bounded term

= Buk (8)

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LTI approximation

We can now write the system as an LTI with a bounded disturbanceterm dk as follows

xk+1 = Axk +Buk +Gdk, (9)

with state xk = (xk, xk−1, . . . , xk−ν+1) and dk ∈ Dν ⊆ Rn. Assumingxk ∈ X where X is a balanced compact set, we have

Dν =

l⊕i=1

−A−10 AiΨν(αi)X, (10)

where Ψν(α) =∑∞

j=0 |cαj | and A0 =∑l

i=1 Aicαij .

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Ψν(α) is quickly decreasing with ν

ν

50 100 150 200 250 300

Ψν

(α)

10-2

10-1

100

α=0.7

α=0.5

α=0.4

...and as a result, Dν can become arbitrarily small for adequately large ν!

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Model Predictive Controlusing the approximate system model

Why MPC?

Because...

I Optimisation-based control

I Accounts for state/input constraints xk ∈ X, uk ∈ U .

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Tube-based MPC

Nominal System

Tube-based MPC

+

-

+

+Fractional System

v

u

KeK

e

x

z

Concept: The control action u iscalculated as

uk = vk +Kek,

where ek = xk − zk and vk iscomputed by an MPC controller.

See: D.Q. Mayne, M.M. Seron, S.V. Rakovic, “Robust model predictive control of constrained linear systems withbounded disturbances,” Automatica 41(2), 219–224, 2005.

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Tube-based MPC

The set

S∞ =

∞⊕i=0

AiKGDν (11)

is robustly positive invariant for the deviation dynamics

ek+1 = AKek +Gdk (12)

Choosing z0 = x0 we have

xk ∈ {zk} ⊕ S∞, (13)

so the constraints will be satisfied if

zk ∈ X S∞, (14a)

vk ∈ U KS∞. (14b)

See: J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design. Nob Hill Publishing, 2009.

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This leads to the following MPC formulation:

PN :V ?N (zk) = min

vk∈VN (zk)VN (zk,vk), (15)

where

VN (zk|k,vk)= Vf (zk+N |k)︸ ︷︷ ︸Terminal cost

+

N−1∑i=0

`(zk+i|k, vk+i|k), (16)

and for some S ⊇ S∞

VN (zk)=

v

∣∣∣∣∣∣∣∣∣∣

zk+i+1|k=Azk+i|k+Bvk+i|k,∀i∈N[0,N−1]

zk|k = zkzk+i|k ∈ X S,∀i∈N[1,N ]

vk+i|k ∈ U KS,∀i∈N[0,N−1]

zk+N |k ∈ Xf Terminal constraints

(17)

The terminal set Xf and cost function Vf are selected according to the stabilising conditions in D. Mayne, J. Rawlings,C. Rao, and P. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica 36 (6), 789–814, 2000.

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This leads to the following MPC formulation:

PN :V ?N (zk) = min

vk∈VN (zk)VN (zk,vk), (15)

where

VN (zk|k,vk)= Vf (zk+N |k)︸ ︷︷ ︸Terminal cost

+

N−1∑i=0

`(zk+i|k, vk+i|k), (16)

and for some S ⊇ S∞

VN (zk)=

v

∣∣∣∣∣∣∣∣∣∣

zk+i+1|k=Azk+i|k+Bvk+i|k,∀i∈N[0,N−1]

zk|k = zkzk+i|k ∈ X S,∀i∈N[1,N ]

vk+i|k ∈ U KS,∀i∈N[0,N−1]

zk+N |k ∈ Xf Terminal constraints

(17)

The terminal set Xf and cost function Vf are selected according to the stabilising conditions in D. Mayne, J. Rawlings,C. Rao, and P. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica 36 (6), 789–814, 2000.

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This leads to the following MPC formulation:

PN :V ?N (zk) = min

vk∈VN (zk)VN (zk,vk), (15)

where

VN (zk|k,vk)= Vf (zk+N |k)︸ ︷︷ ︸Terminal cost

+

N−1∑i=0

`(zk+i|k, vk+i|k), (16)

and for some S ⊇ S∞

VN (zk)=

v

∣∣∣∣∣∣∣∣∣∣

zk+i+1|k=Azk+i|k+Bvk+i|k, ∀i∈N[0,N−1]

zk|k = zkzk+i|k ∈ X S,∀i∈N[1,N ]

vk+i|k ∈ U KS,∀i∈N[0,N−1]

zk+N |k ∈ Xf Terminal constraints

(17)

The terminal set Xf and cost function Vf are selected according to the stabilising conditions in D. Mayne, J. Rawlings,C. Rao, and P. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica 36 (6), 789–814, 2000.

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Closed-loop properties

The solution of PN is {v?0(zk), v?1(zk), . . . , v

?N−1(zk)} and the control law

is

κN (zk) = v?0(zk), (18)

the input applied to the system is ρ(zk, xk) = κN (zk) +K(xk − zk) andthe closed-loop system (in terms of both zk and xk) becomes

xk+1 = Axk +Bρ(zk, xk) +Gdk, (19a)

zk+1 = Azk +BκN (zk). (19b)

Stability: The set S∞ × {0} is exponentially stable for (19).

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Simulation Results

Simulations I

Dynamical system:

D0.7x(t) =

0 0 10 1 00 −1 1

x(t) +

0 0−1 01 1

u(t), (20)

subject to the constraints

−

111

≤x(t) ≤

111

(21a)

−[0.10.1

]≤u(t) ≤

[0.10.1

](21b)

and let ν = 20 and N = 35.

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Simulations I

Closed-loop response...

2 4 6 8 10 12 14 16 18 20−0.2

−0.1

0

0.1

0.2

State

xk

2 4 6 8 10 12 14 16 18 20

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Control action

Time

uk

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Simulations II

Dynamical system:

D0.7x(t) =

[1 0.9−0.9 0.2

]x(t) +

[01

]u(t), (22)

subject to the constraints

−[33

]≤x(t) ≤

[33

](23a)

−0.5 ≤u(t) ≤ 0.5 (23b)

The open-loop response of this system in unstable.

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Simulations IIClosed-loop reponse...

time

0 5 10 15 20

x1

-0.5

0

0.5

1

1.5

2

ν=5

ν=10

ν=20

ν=50

time

0 5 10 15 20

x2

-3

-2

-1

0

ν=5

ν=10

ν=20

ν=50

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Simulations II – Runtime

Computation time as a function of ν for N = 35

ν

20 40 60 80 100 120

MO

SE

K t

ime

(m

s)

50

100

150

200

250

MOSEK can be found at https://www.mosek.com/

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Conclusions

Conclusions

I Fractional-order discrete-time systems are infinite-dimensional,

I We can approximate such systems as LTI with bounded additivedisturbance,

I The order of approximation controls the magnitude of thedisturbance,

I The approach is computationally tractable.

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Ongoing

I Under what conditions is the controlled system asymptotically stable?

I Application: controlled drug administration (reference tracking).

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Thank you for your attention

Acknowledgement: This work was funded by project 11ΣYN.10.1152, which is cofinanced by the European Union and Greece,Operational Program “Competitiveness & Entrepreneurship”, NSFR 2007-2013 in the context of GSRT National action

“Cooperation”.

References

1. T. Clarke, B. N. Achar, and J. W. Hanneken, “MittagLeffler functions and transmission lines,” J. Molec. Liq.,114(1–3), 159–163, 2004. Diffusion and Relaxation in Disordered Fractal Systems. Proceedings from the meeting onDiffusion and Relaxation in Disordered Fractal Systems.

2. A. Dokoumetzidis, R. Magin, and P. Macheras, “Fractional kinetics in multi-compartmental systems,” J. Pharmacok.Pharmacod., 37(5), 507–524, 2010.

3. R. Hilfer, Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000.

4. R. Magin, “Fractional calculus models of complex dynamics in bio- logical tissues,” Comp. & Math. with Appl.,59(5), 1586–1593, 2010. Fractional Differentiation and Its Applications.

5. R. Magin, M. D. Ortigueira, I. Podlubny, and J. Trujillo, “On the fractional signals and systems,” Signal Processing,91(3), 350–371, 2011. Advances in Fractional Signals and Systems.

6. L. Pereira, “Fractal pharmacokinetics,” Comput Math Methods Med., 11(2), 161–184, 2010.

7. S. Samko, A. Kilbas, O. Marinhev, Fractional integral and derivatives, Gordon & Breach Science Publishers, 1993.

8. L.M. Zelenyi, A.V. Milovanov, “Fractal topology and strange kinetics: from percolation theory to problems in cosmicelectrodynamics,” Physics Uspekhi 47 (2004) 749–788.

9. I. Schafer and K. Kruger, “Modelling of coils using fractional derivatives,” J. Magnetism & Magnetic Materials,307(1), 91–98, 2006.

What are the stabilising conditions for MPC?

The following conditions are assumed for Xf and Vf :

1. Xf ⊆ X = X × . . .×X, Xf is closed and 0 ∈ Xf ,

2. κN (z) ∈ U for all z ∈ Xf ,

3. Az +BκN (z) ∈ Xf for all z ∈ Xf ,

4. Vf (z+)− Vf (z) ≤ `(z, κN (z)), where z+ = Az +BκN (z), for allz ∈ Xf .

These conditions can be found in D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, “Constrained model predictive control:Stability and optimality,” Automatica 36 (6), 789–814, 2000.

Is the controlled system asymptotically stable?

We have shown that the following condition entails asymptotic stabilityproperties for the controlled system:⊕

j∈NAjKG

⊕i∈N[1,l]

−A−10 AiΨν(αi)Bn ⊆ Bnε ,

for some ε > 0.