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Robustness of Suboptimal Model Predictive Control

James B. Rawlings

Department of Chemical and Biological EngineeringUniversity of Wisconsin–Madison

Insitut für Systemtheorie und RegelungstechnikUniversität StuttgartStuttgart, Germany

June 24, 2011

Stuttgart – June 2011 Distributed MPC 1 / 50

Outline

1 Inherent robustness of suboptimal MPCExponential stability for difference inclusionsNominal exponential stability of suboptimal MPCRobust exponential stability of suboptimal MPCFeasibility issueMain resultsRobust stability in absence of state constraints

2 Proposal for nonlinear, distributed MPCModelObjective FunctionTerminal ControllerRemoving the terminal constraintCooperative control algorithmStability of distributed nonlinear cooperative controlExample



2/25

Why study robustness of suboptimal MPC?

Cooperative , distributed MPC is a special case of suboptimal MPC.Anything we establish about suboptimal MPC can be applied tocooperative, distributed MPC (and optimal MPC!)

Suboptimal MPC has an interesting feature: a nonunique,point-to-set control law u ∈ κN (x ).

Optimal solution of nonconvex

PN (x ) : minu∈U N

V N (x , u)

cannot be computed online for any nonlinear model. Practitionersimplement only suboptimal MPC.

We should know something about its inherent robustness properties.1

1Pannocchia, Rawlings, and Wright (2011)Stuttgart – June 2011 Distributed MPC 3 / 50

For suboptimal MPC; again, the basic MPC setup

The system modelx + = f (x , u ) (1)

State and input constraints

x (k ) ∈ X , u (k ) ∈ U for all k ∈ I≥0

Terminal constraint (and penalty)

φ(N ; x , u) ∈ Xf ⊆ X



3/25

Feasible sets

The set of feasible initial states and associated control sequences

ZN = {(x , u) | u (k ) ∈ U, φ(k ; x , u) ∈ X for all k ∈ I0:N −1,

and φ(N ; x , u) ∈ Xf }

and Xf ⊆ X is the feasible terminal set.

The set of feasible initial states is

X N = {x ∈ Rn | ∃u ∈ UN such that (x , u) ∈ ZN } (2)

For each x ∈ X N , the corresponding set of feasible input sequences is

U N (x ) = {u | (x , u) ∈ ZN }


Cost function and control problem

For any state x ∈ Rn and input sequence u ∈ UN , we define

V N (x , u) =N −1k =0

(φ(k ; x , u), u (k )) + V f (φ(N ; x , u))

(x , u ) is the stage cost; V f (x (N )) is the terminal costConsider the finite horizon optimal control problem

PN (x ) : minu∈U N

V N (x , u)



4/25

Suboptimal MPC

Rather then solving PN (x ) exactly , we consider using any(unspecified) suboptimal algorithm having the following properties.

Let u ∈ U N (x ) denote the (suboptimal) control sequence for theinitial state x , and let ũ denote a warm start for the successor initialstate x + = f (x , u (0; x )), obtained from (x , u) by

ũ := {u (1; x ), u (2; x ), . . . , u (N − 1; x ), u +} (3)

u + ∈ U is any input that satisfies the invariance conditions of Assumption 3 for x = φ(N ; x , u) ∈ Xf , i.e., u + ∈ κf (φ(N ; x , u)).


Suboptimal MPC

The warm start satisfies ũ ∈ U N (x +).

The suboptimal input sequence for any given x + ∈ X N is defined asany u+ ∈ UN that satisfies:

u+ ∈ U N (x

+) (4a)

V N (x +, u+) ≤ V N (x

+, ũ) (4b)

V N (x +

, u+

) ≤ V f (x +

) when x +

∈ r B (4c)

in which r is a positive scalar sufficiently small that r B ⊆ Xf .

Notice that constraint (4c) is required to hold only if x + ∈ r B, and itimplies that |u+| → 0 as |x +| → 0.

Condition (4b) ensures that the computed suboptimal cost is nolarger than that of the warm start.



5/25


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Basic MPC assumptions

Assumption 1 (Continuity of system and cost)

The functions f : Rn × Rm → Rn, : Rn × Rm → R≥0 and

V f : Rn → R≥0 are continuous, f (0, 0) = 0, (0, 0) = 0, and V f (0) = 0.

Assumption 2 (Properties of constraint sets)

1 The set U is compact and contains the origin. The sets X and Xf areclosed and contain the origin in their interiors, Xf ⊆ X.

2 The set U is compact and contains the origin. The sets X = Rn and

Xf = levαV f = {x ∈ Rn | V f (x ) ≤ α}, with α > 0.


Basic MPC assumptions

Assumption 3 (Basic stability assumption)

For any x ∈ Xf there exists u := κf (x ) ∈ U such that f (x , u ) ∈ Xf andV f (f (x , u )) + (x , u ) ≤ V f (x ).

Assumption 4

There exist positive constants a, a1, a2, af and r̄ , such that the costfunction satisfies the inequalities

(x , u ) ≥ a1|(x , u )|a for all (x , u ) ∈ X × U

V N (x , u) ≤ a2|(x , u)|

a for all (x , u) ∈ r̄ B

V f (x ) ≤ af |x |a for all x ∈ X



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Definition 5 (Exponential stability)

The origin of the difference inclusion z + ∈ H (z ) is exponentially stable (ES) on Z , 0 ∈ Z , if there exist scalars b > 0 and 0 < λ


8/25

Nominal exponential stability

Theorem 7 (ES)Under Assumptions 1, 2.1, 3, and 4, the origin of the closed-loop system

x + ∈ F (x ) = {f (x , u ) | u ∈ κN (x )}

is ES on (arbitrarily large) compact subsets of X N .


Robust exponential stability of suboptimal MPC

Definition 8 (RES)

The origin of the closed-loop system (6) is robustly exponentially stable (RES) on int(X N ) if there exist scalars b > 0 and 0 < λ 0, there exists δ > 0 such that for all sequences {d (k )} and{e (k )} with x (0) = x ∈ C satisfying:

maxk ≥0

|d (k )| ≤ δ, maxk ≥0

|e (k )| ≤ δ

x m(k ) = x (k ) + e (k ) ∈ X N , x (k ) ∈ X N , for all k ∈ I≥0,

it follows that

|φed (k ; x )| ≤ b λk |x | + , for all k ∈ I≥0. (8)



9/25

Robust exponential stability of suboptimal MPC

Definition 9 (SRES)

The origin of the closed-loop system (6) is strongly robustly exponentially

stable (SRES) on a compact set C ⊂ X N , 0 ∈ int(C ), if there exist scalarsb > 0 and 0 < λ 0, there exists δ > 0 such that for all sequences {d (k )} and {e (k )}satisfying

|d (k )| ≤ δ and |e (k )| ≤ δ for all k ∈ I≥0,

and all x ∈ C , we have that

x m(k ) = x (k ) + e (k ) ∈ X N , x (k ) ∈ X N , for all k ∈ I≥0, (9a)

|φed (k ; x )| ≤ b λk

|x | + , for all k ∈ I≥0. (9b)


Feasibility issue

The warm start ũ is feasible for the predicted successor state

x̃ + = f (x m, u (0; x m))

i.e., (x̃ +, ũ) ∈ ZN ,

But it may not be feasible for the measured successor state

x +m = f (x , u (0; x m)) + d + e +

(The true successor state, which is unknown, isx + = f (x , u (0; x m)) + d .)

So we have to resolve this infeasibility online. We make the followingassumption

Assumption 10

For any x , x ∈ X N and u ∈ U N (x ), there exists u ∈ U N (x

) such that|u − u| ≤ σ(|x − x |) for some K-function σ(·).



10/25

Feasibility issue

Among various options for finding p, we consider the following feasibility problem (notice that x̃ + is known):

Find p s.t. ũ + p ∈ U N (x +m ) and |p| ≤ σ(|x

+m − x̃

+|), (10)

Proposition 11

Under Assumption 10 , for any (x̃ +, ũ) ∈ ZN and x +m ∈ X N , the set of

solutions to (10) is nonempty.


Main results

Theorem 12 (SRES of suboptimal MPC)

Under Assumptions 1, 2.1, 3, 4, and 10 , the origin of the perturbed closed-loop system

x + ∈ F ed (x )

is SRES on C ρ.



11/25

Main results

u

U N

X N

S ρS

u0(x )

C ρ

ρ

ZN

x

Figure: Sketch of the main sets involved in SRES.


Robust stability in absence of state constraints

To this aim, we replace Assumption 2.1 with 2.2.

Assumption 10 will not be necessary, whereas Assumptions 1, 3 and 4(with V N (·) replaced by V

βN (·) later defined) are required.

We modify the cost function as follows:

V βN (x , u) =N −1k =0

(φ(k ; x , u), u (k )) + β V f (φ(N ; x , u))



12/25

Modified suboptimal MPC algorithm in absence of stateconstraints

u+

∈ UN

(11a)V βN (x

+, u+) ≤ V βN (x +, ũ) (11b)

V βN (x +, u+) ≤ β V f (x

+) when x + ∈ r B (11c)

Z̄ r = {(x , u) ∈ Rn × UN | V βN (x , u) ≤

V̄ ,

and V βN (x , u) ≤ β V f (x ) if x ∈ r B}

X0 = {x ∈ Rn

|∃u ∈ UN

such that (x , u) ∈ Z̄ r } (12)


Nominal exponential stability

We have the following nominal stability result.

Theorem 13 (ES without state constraints)

Under Assumptions 1, 2.2 , 3, and 4, the origin of the closed-loop system

x + ∈ {f (x , u ) | u ∈ κN (x )}

is ES on X0.



13/25


G ed (z ) = {u+ | u+ ∈ UN , V βN (x

+m , u

+) ≤ V βN (x +m , ũ),

V βN (x +m , u

+) ≤ β V f (x +m ) if x

+m ∈ r B}

V̄ ρN (z m) = maxe ∈ρB

V βN (z ) s.t. z = z m − (e , 0) (13a)

S̄ ρ = {z m ∈ Z̄ r | V̄ ρN (z m) ≤

V̄ } (13b)

C̄ ρ = {x ∈ Rn | x = x m − e , e ∈ ρB, ∃u s.t. (x m, u) ∈ S̄ ρ} (14)



Theorem 14 (SRES without state constraints)

Under Assumptions 1, 2.2 , 3 and 4, the origin of the closed-loop system

x +

∈ F ed (x )

is SRES on C̄ ρ.



14/25

The robust region of attraction

The ideal, but unachievable, result would be that the robust region of attraction under nonzero disturbances is the entire nominal feasible

set, X N .

This ideal result is approached reasonably closely, however, whenexcluding state constraints!

When |d |, |e | → 0, C̄ ρ → X0 and SRES holds over a set approachingthe admissible set of initial conditions.

These results for robustness of suboptimal nonlinear MPC all apply tocooperative, distributed MPC .


Requirements for distributed, nonlinear control

Now let’s return to the setting of distributed, nonlinear MPC, anddeal with the nonconvexity issue.

Two criteria in design:1 the optimizers should not rely on a central coordinator2 the exchange of information between the subsystems and the iteration

of the subsystem optimizations should be able to terminate beforeconvergence without compromising closed-loop properties.



15/25

Distributed nonconvex optimization

Consider the optimization

minu V (u ) s.t. u ∈U

We require approximate solutions to the following suboptimizations atiterate p ≥ 0 for all i ∈ I1:M

u p i = arg minu i ∈Ui

V (u i , u p −i )

in which u −i = (u 1, . . . , u i −1, u i +1, . . . , u M ).

Define the step υp

i

= u p

i

− u p

i

.


Algorithm

To choose the stepsize αp i , each suboptimizer initializes the stepsize2

with αi

V (u p ) − V (u p i + αp i υ

p i , u

p −i ) ≥ −σα

p i ∇i V (u

p )υp i

in which σ ∈ (0, 1).

After all suboptimizers finish the backtracking process, they exchangesteps. Each suboptimizer forms a candidate step

u p +1i = u p i + w i α

p i υ

p i ∀i ∈ I1:M

2Armijo rule: (Bertsekas, 1999, p.230)Stuttgart – June 2011 Distributed MPC 30 / 50


16/25

Algorithm

Check the following inequality, which tests if V (u p ) is convex-like

V (u p +1) ≤i ∈I

1:M

w i V (u p i + α

p i υ

p i , u

p −i ) (15)

in which

i ∈I1:M w i = 1 and w i > 0 for all i ∈ I1:M .

If the condition above is not satisfied, then we find the direction withthe worst cost improvement

i max = arg maxi

{V (u p i + αp i υ

p i , u

p −i )}

and eliminate this direction by setting w i max to zero and repartitioning

the remaining w i so that they sum to 1.At worst, condition (15) is satisfied with one direction only.


Distributed nonconvex optimization — Properties

Lemma 15 (Feasibility)

Given a feasible initial condition, the iterates u p are feasible for all p ≥ 0.

Lemma 16 (Objective decrease)

The objective function decreases at every iterate, that is,V (u p +1) ≤ V (u p ).

Lemma 17 (Convergence)

Every accumulation point of the sequence {u p } is stationary.



17/25

Distributed nonconvex optimization

0

1

2

3

4

0 1 2 3 4

u 2

u 1

Figure: Nonconvex function optimized with Distributed nonconvex optimizationalgorithm


Distributed nonlinear cooperative control

We have the following local models

x +1 = f 1(x 1, x 2, u 1, u 2) x +2 = f 2(x 1, x 2, u 1, u 2) (16)

Collect these models to form the plantwide model

x + = f (x 1, x 2, u 1, u 2) = f (x , u )

in which

x =

x 1x 2

u =

u 1u 2

f (x , u ) =

f 1(x 1, x 2, u 1, u 2)f 2(x 1, x 2, u 1, u 2)

for which x ∈ Rn, u ∈ Rm, and f : Rn × Rm → Rn.

At each time step k , we require the inputs to satisfy

u 1(k ) ∈ U1 u 2(k ) ∈ U2 k ∈ I0:N −1

in which each Ui ∈ Rmi is compact, convex, and contains the origin

in its interior.



18/25


The objective function for each subsystem i ∈ I1:2 is defined

V i

x (0), u1, u2

=

N −1

k =0

i

x i (k ), u i (k )

+ V if

x (N )

We define plantwide objective

V

x 1(0), x 2(0), u1, u2

= ρ1V 1

x (0), u1, u2

+ ρ2V 2

x (0), u1, u2

To simplify notation we use V (x , u) for the plantwide objective.

Assumption 18

For each i ∈I

1:2, there exists a K ∞ function αi (·) such that

i (x i , u i ) ≥ αi (|x i |) ∀(x i , u i ) ∈ Rni × Ui (17)



Denote the plantwide terminal penalty V f (x ) = ρ1V 1f (x ) + ρ2V 2f (x ).

We define the terminal region Xf to be a sublevel set of V f . Fora > 0, define

Xf = {x | V f (x ) ≤ a}

Assumption 19

The plantwide terminal penalty V f (·) satisfies

αf (|x |) ≤ V f (x ) ≤ γ f (|x |) ∀x ∈ Xf

in which αf (·) and γ f (·) are K ∞ functions.



19/25


Defining (x , u ) = ρ11(x 1, u 1) + ρ22(x 2, u 2), we require the followingstability assumption.

Assumption 20

The terminal cost V f (·) satisfies

min(u 1,u 2)∈U1×U2

V f

f (x , u 1, u 2)

+ (x , u )

s.t. f (x , u 1, u 2) ∈ Xf

≤ V f (x ) ∀x ∈ Xf

This assumption implies that there exists a κif (x ) ∈ Ui for all i ∈ I1:2such that

V f

f (x , κ1f (x ), κ2f (x ))

+

x , κ1f (x ), κ2f (x )

≤ V f (x ) (18)

s.t. f (x , κ1f (x ), κ2f (x )) ∈ Xf


Removing the terminal constraint

To avoid coupled input constraints, must remove the terminal stateconstraint.

For some β ≥ 1, define the modified objective function

V β(x , u) =

N −1k =0

(x (k ), u (k )) + β V f (x (N )) (19)

Define the set of admissible initial (x , u) pairs as

Z0 = {(x , u) ∈ X × UN | V β(x , u) ≤ V , φ(N ; x , u) ∈ Xf } (20)

The set of initial states X0 is the projection of Z0 onto X

X0 = {x ∈ X | ∃u such that (x , u) ∈ Z0}



20/25


Proposition 21 (Terminal constraint satisfaction)

Let {(x (k ), u(k )) | k ∈ I≥0} denote the set of states and control sequences generated by the suboptimal system. There exists a β > 1 suchthat for all β ≥ β , if (x (0), u(0)) ∈ Z0, then (x (k ), u(k )) ∈ Z0 withφ(N ; x (k ), u(k )) ∈ Xf for all k ∈ I≥0.

For the remainder of the talk, we replace the plantwide objective with themodified objective V (·) ← V β(·) and hence the terminal constraint issatisfied.


Cooperative control algorithm

Let ũ ∈ U be the initial condition for the cooperative MPC algorithmsuch that φ(N ; x (0), ũ) ∈ Xf .

At each iterate p , an approximate solution of the followingoptimization problem is found

minu

V x 1(0), x 2(0), u1, u2 (21a)s.t. x +1 = f 1(x 1, x 2, u 1, u 2) (21b)

x +2 = f 2(x 1, x 2, u 1, u 2) (21c)

ui ∈ UN i ∀i ∈ I1:2 (21d)

|ui | ≤ δ i (|x i (0)|) if x (0) ∈ B r ∀i ∈ I1:2 (21e)



21/25

Cooperative control algorithm

Let the input sequence returned by distributed nonconvexoptimization algorithm be up̄ (x , ũ).

The first input of this sequence κ¯p

(x (0)) = u ̄p

(0; x (0), ũ) is injectedinto the plant and the state is moved forward.

To reinitialize the algorithm at the next sampling time, we define thewarm start

ũ+1 = {u 1(1), u 1(2), . . . , u 1(N − 1), κ1f

x (N )

}

ũ+2 = {u 2(1), u 2(2), . . . , u 2(N − 1), κ2f

x (N )

}

in which x (N ) = φ(N ; x (0), u1, u2).


Nominal stability results

Theorem 22 (Asymptotic stability)

Let Assumptions 18 -20 hold and let V (·) ← V β(·) by Proposition 21.Then for every x (0) ∈ XN , the origin is asymptotically stable for the closed-loop system x + = f (x , κp̄ (x )).



22/25

A nonlinear example

Consider the unstable nonlinear system

x +1 = x 21 + x 2 + u

31 + u 2

x +2 = x 1 + x 22 + u 1 + u

32

with initial condition (x 1, x 2) = (3, −3).

For this example, we use the stage cost

1(x 1, u 1) =1

2(x 1Q 1x 1 + u

1R 1u 1)

2(x 2, u 2) =1

2(x 2Q 2x 2 + u

2R 2u 2)

For the simulation we choose the parameters

Q = I R = I N = 2 p = 3 Ui = [−2.5, 2.5] ∀i ∈ I1:2



-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10

x

k

x 1

x 2

Figure: State trajectory (p = 3)

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10

x

k

x 1

x 2

Figure: Centralized state trajectory(p = 10)



23/25


-3

-2

-1

0

1

2

3

0 2 4 6 8 10

u

k

u 1

u 2

Figure: Input trajectory (p = 3)

-3

-2

-1

0

1

2

3

0 2 4 6 8 10

u

k

u 1

u 2

Figure: Centralized input trajectory(p = 10)



10−10

10−810−6

10−4

10−2

100

102

104

0 2 4 6 8 10

V (x (k ), ū (k ))

k

p̄ = 3p̄ = 10

Figure: Open-loop cost to go versus time on the closed-loop trajectory fordifferent numbers of iterations.



24/25


25/25

Further Reading I

D. P. Bertsekas. Nonlinear Programming . Athena Scientific, Belmont,MA, second edition, 1999.

G. Pannocchia, J. B. Rawlings, and S. J. Wright. Conditions under whichsuboptimal nonlinear MPC is inherently robust. Sys. Cont. Let., 2011.Accepted for publication.


Acknowledgments

The author is indebted to Luo Ji of the University of Wisconsin andGanzhou Wang of the Universität Stuttgart for their help in organizing thematerial and preparing the overheads.

06 mpc jbr stanford

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