multi parametric with references

8/6/2019 Multi Parametric With References

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Part III:

MPC for Constrained Linear Systems

An Explicit Solution

Miroslav Baric

Johan Lofberg

Helfried Peyrl

Ioannis Fotiou

Manfred Morari


2/29

MPC and Receding Horizon Policy

PLANT

inputref

Measurement

output

constrained

finite timeoptimal control

xt|t

rt ytut|t

At time instance t:

solve Constrained Finite Time Optimal

Control problem to obtain open-loop

solution:

U =

u

T

t|t uTt+1|t . . . u

Tt+N1|t

T apply only u

t|t ,

repeat the optimization at the next time

instance for a shifted prediction horizon

Receding horizon policy introduces feedback


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Motivation

MPC requires solving a constrained optimizationproblem (e.g. linear or quadratic) on-line.

Drawbacks

the optimization problem may not be solved

within a prescribed time due to a highsampling rate limited to slow processes

(e.g. chemical plants)

it is difficult to estimate strict (and tight)

upper bound on the time needed for the

optimization procedure

on-line optimization usually implies

expensive and large computational

infrastructure inconvenient for embedded

control applications

Remedy

Move the optimization effort off-line . How?


4/29

Parametric Programming - Part I

Before providing the answer to the question, letus consider the following example:

Example: Perturbed quadratic program

f(x) = minz

f(z, x) = 12z

2 + 2xz

subj. to z 1 + x,

x R is an unknown parameter.

The goals:

1. find z(x) = argminz f(z, x),

2. find all x for which the problem has a

solution

3. compute the value function f(x)

Define Lagrangian:

L(z , x , ) = f(z, x) + (z x 1)

KKT conditions:

z + 2x + = 0, (1)

z x 1 0, (2)

(z x 1) = 0, (3)

0. (4)


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Parametric Programming - Part II

Consider two strictly complementary cases(from (3)):

1.z x 1 = 0

> 0

z(x) = x + 1,

x < 13f(x) = 52x

2 + 3x + 12

2.z x 1 < 0

= 0

z(x) = 2x,

x > 13f(x) = 2x2

-2 -1 0 1 2

-2

-1

0

1

2

x

z, f

x= 13

f(x)

z(x)

z(x) =

x + 1, ifx 1

3,

2x, ifx 13

The optimization problem is solved for all

admissible values of parameter x.

The optimal solution for a given x amounts

to simple evaluation of z(x) and f(x).


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Parametric Programming - General

General formulation:

f(x) = infz

f(z, x),

subj. to g(z, x) 0,

where z Z Rm is the optimization variable

and x X Rn

is a vector of parameters.

x Rn, n > 1 multi-parametric program

closely related to sensitivity analysis:

sensitivity analysis: local behavior of the

solution for small perturbations

parametric program: solution for the full

range of parameters x

We consider two important classes:

1. multi-parametric LP (mpLP)

2. multi-parametric QP (mpQP)


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mpLP

FormulationPRIMAL PROBLEM (PP) DUAL PROBLEM (DP)

J(x) = minz

J(z, x) = cTz,

subj. to Gz W + Sx

max

(W + Sx)T,

subj. to GT = c,

0

where c Rm, G Rqm, S Rqn, W Rq.

Feasible Set: X X is the set of all x for

which a solution to the PP & DP.

Given constraint indices I = {1, . . . , q}:

Set of Active Constraints:

A(x) = {i I | z : J(z, x) = J(x)

Giz Six Wi = 0} ,

Set of Inactive Constraints:

N(x) = {i I | z : J(z, x) = J(x)

Giz Six Wi < 0} ,

Critical region: for a given set A I

CRA = {x X | A(x) = A}


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mpLP: Geometric Algorithm (I)

exploration of the parameter space X usinggeometric methods

requires an LP solver

for simplicity, assume: no degeneracy

Optimality Conditions

primal feasibility:

Gz W + Sx

dual feasibility:

GT

= c, 0

complementarity slackness:

i(Giz Six Wi) = 0, i I

Step 1: solve an LP

For an initial parameter vector x0 X:

solve PP & DP z(x0) and (x0),

obtain A0 = A(x0) and N0 = N(x0) and

matrices:

{GA0, SA0, WA0} = {Gi, Si, Wi | i A(x0)}

{GN0, SN0, WN0} = {Gi, Si, Wi | i N(x0)}


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mpLP: Geometric Algorithm (II)

Step 2: determine CRA0, z0(x) and J0(x)

From the primal feasibility conditions:

GA0z0(x) = WA0 + SA0x,

GN0z0(x) < WN0 + SN0x.

compute:

optimizer z0(x):

z0(x) = G1A0

SA0x + G1A0

WA0 = F0x + g0

critical region CRA0:

CRA0 =

x | (GN0F0 SN0)x < WN0 GN0g0

Using the solution to the DP compute the value

function J0(x) (remember, strong duality holds):

J0(x) = (SA0x + WA0)TA0

Simplifying notation: CRAj CRj


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mpLP: Geometric Algorithm (III)

Critical region CR0 is defined by strictinequalities an open polyhedral set

x0

x1

x2

CR0 X

A = A(x) uniquely determines CRA

critical regions do not overlap

the optimizer z(x) is affine over CR0

Step 3: explore the rest of X

replace CR0 by its closure CR0 (change


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mpLP: Geometric Algorithm (IV)

I) Reversing inequalities

x1

x2

CR0

XR1

R2

R3

R4

CR0 = {x X | Hx K}

Ri = {x X | Hix Ki

Hjx Kj, j < i}

X \ CR0 =i

Ri

NOTE: regions Ri are not critical regions

proceed recursively: repeat the whole

procedure for each Ri guaranteed: set X explored in finite number

of iterations

problem: critical regions can be artificially

divided among different Ri

x1

x2

CR0

CR1

XR1

R2

R3

R4CR1 is split between R2and R3 keep track of

the critical regions already

discovered


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mpLP: Geometric Algorithm (V)

II) Crossing the facets

x1 x1

x2 x2

CR0 CR0CR4 CR4

CR1 CR1

CR2 CR2

CR3 CR3

X X

for each of the facets of CR0 a point outside

the region but close to the facet is selected

and the procedure is repeated

critical regions are computed in one piece,

no artificial splitting

heuristics: how far to step over the facet?

no formal proof that whole X is covered

however: in practice usually outperforms the

strategy based on reversing inequalities


13/29

mpLP: Solution Properties

The following theorem summarizes theproperties of the solution to the mpLP:

Theorem 1

i) Feasible set X is closed and convex ,

ii) If the optimal solution z is unique x X,

the optimizer function z(x) : X Rm is:

continuous

polyhedral piecewise affine (PPWA) over

X , affine in each CRi

Otherwise, it is always possible to choosea continuous and PPWA optimizer function

z(x).

iii) The value function J(x) : X R is:

convex

PPWA over X , affine in each CRi


14/29

mpLP: Example

minz

3z1 8z2,

s.t.

z1 + z2 13 + x15z1 4z2 208z1 + 22z2 121 + x24z1 z2 8z1 0z2 0

Critical regions and the optimizer

20 10 0 10 20 30 40 5060

40

20

0

20

40

60

x1

x2

CR1 CR2

CR3

A1 = {1,3} A2 = {2,3}

A3 = {1,4}

100

1020

3040

50

50

0

50

2

4

6

8

10

12

14

PPWA optimizer function

x1x2

z

(x)

z(x) =

0.733 0.03330.267 0.0333

x +

5.507.50

ifx CR1

0 0.0513

0.0641

x +

11.8

9.81

ifx CR2

0.333 0

1.33 0

x +

1.67

14.7

ifx CR3


15/29

mpQP

Formulation

We consider multi-parametric quadratic

programs of the form:

J(x) = minz J(z, x) =

12z

THz ,subj. to Gz W + Sx

where z Z Rm, and x X Rn, G Rqm.

Assumptions:

H 0 no dual degeneracy

no primal degeneracy

Optimality conditions (KKT)

Hz + GT = 0, (5)

i (Giz

Six Wi) = 0, i = 1, . . . , q (6)

0, (7)

Gz W + Sx (8)


16/29

mpQP: Geometric Algorithm (I)

the algorithm is conceptually identical to thegeometric algorithm for mpLP

required: QP solver

Step 1: Solve QP

For an initial parameter vector x0 solve QP and: obtain the optimizer z(x0),

identify indices of active and inactive

constraints A = A(x0) and N = N(x0) and

the corresponding matrices {G,S,W}A and

{G,S,W}N

Step 2: Compute z(x), (x) and CRA

from (5):

z = H1GT (9)

from (9) and (6) (N = 0):

A(x) = (GAH1GTA)

1(WA + SAx), (10)

the optimizer function: from (9) and (10):

z(x) = H1GTA(GAH1GTA)

1(WA+SAx)


17/29

mpQP: Geometric Algorithm (II)

critical region: put

(x) and z

(x) into (7)and (8):

CRA = {x | Ax b}

A = GH1GTA(GAH

1GTA)1SA S

(GAH1GT

A

)1SA b =

W + GH1GTA(GAH

1GTA)1WA

(GAH1GTA)

1WA

NOTE: we immediately use the closure of the

critical region

Also note:

the value function z(x) is a uniquely defined

affine function over the critical region CRA

the critical region is a polyhedral set in the

x-space

Step 3: explore the rest of X

exploration strategies as in mpLP

propagation of active constraints: no

degeneracy active set of the neighboring

critical region can be computed without

solving a QP


18/29

mpQP: Solution Properties

Under the assumption that no degeneracy

occurs, the following theorem summarizes the

global properties of the solution to the mpQP:

Theorem 2 i) feasible set X is closed and

convex

ii) the optimizer function z(x) : X Rm is:

continuous

PPWA over the critical regions CRi

iii) the value function J(x) : X R is:

continuous

convex

polyhedral piecewise quadratic (PPWQ)

(piecewise quadratic over polyhedra), in

particular over the critical regions CRi


19/29

mpQP: Example

minz

12(z21 + z22),

s.t.

z1 + z2 13 + x15z1 4z2 208z1 + 2z2 10 + x24z1 z2 8z1 0z2 0

Critical regions and the optimizer

20 10 0 10 20 30 40 50 6060

40

20

0

20

40

60

x1

x2

CR1

CR2

CR3

CR4CR5

0

20

4050

0

50

2

2.5

3

3.5

4

4.5

5

PPWA optimizer function

x1 x2

z

(x)

z(x) =

0 00 0

x +

1.880.47

ifx CR1

0.3333 0

1.3333 0

x +

1.67

14.67

ifx CR2

0 0.0625

0 0.25

x +

0.38

6.5

ifx CR3

0 0.125

0 0 x +

1.25

0 ifx CR4

0 0.1818

0 0.2273

x +

3.64

9.55

ifx CR5


20/29

MPC based on Linear Norms (I)

Consider optimal control problem:

minu

N1k=0

||Qxt+k|t||p + ||Rut+k|t||p

s.t. xt+k+1|t = Axt+k|t + But+k, k 0xt+k|t Xut+k|t U

(11)

where p {1, }, X and U are polyhedral sets.

Recall vectorized notation:

X = xTt|t x

Tt+1|t . . . xt+N1|t

T

U =

uTt|t u

Tt+1|t . . . u

Tt+N1|t

T

The predicted states and outputs can be

conveniently written as

X = Axt|t + BU

where

A =

I

A

A2

...

AN1

, B =

0 0 0 . . . 0B 0 0 . . . 0

AB B 0 . . . 0... . . . . . . . . . ...

AN2B . . . AB B 0


21/29

MPC based on Linear Norms (II)

Polyhedral sets X and U are given by:

U U = {U : EuU Fu}

X X = {X : ExX Fx}

(for suitably defined matrices Eu, Ex, Fu and Fx).

Consider p = 1 (sum of absolute values):

Introduce decision vectors Zx and Zu to model

absolute values of QX and RU.

Zx QX ZxZu RU Zu

Inserting the definition of X, the problem (11)

can be written as:

minU,Zx,Zu

1TZx + 1TZu

s.t.

Eu

ExB

U

Fu

Fx ExAxk|k

Zx Q(Axt|t + BU) ZxZu RU Zu


22/29

MPC based on Linear Norms (III)

Written in a more compact form, the problem

becomes:

minz

cTz,

subj. to Gz W + Sxt|t(12)

where z =

UT, ZTx , ZTu

T.

The problem (12) can be solved on-line as an

LP for a particular xt|t or off-line for all xt|t as

an mpLP, with xt|t as the parameter vector.


24/29

Explicit Solution to MPC

For constrained linear systems and quadraticcost or cost based on linear norms:

MPC is a continuous, piecewise

affine feedback law.

The computational burden is moved off-line:

off-line: solve mpLP(QP)

on-line: detect which critical region xt|t

belongs to and evaluate the corresponding

affine feedback law LOOK-UP TABLE

EXPLICIT FEEDBACK LAW

LOOKUP TABLEOPTIMIZATION PROBLEM

(mpLP, mpQP, ...)

PLANT

OFFLINE

u(xt|t)

u(xt|t)

rt

yt

xt|t

xt|t

u = Fixt|t + gi

Receding horizon policy: extract ut|t from

the optimizer vector z(xt|t) and apply it to the

plant


25/29

Example: Double Integrator (II)

G(s) = 1s2

ZOH, Ts=0.1s= xt+1 =

1 0.10 1

xt +

0.005

1

ut

Constraints:

|u| 1

x 5

-norm based cost

J(x, u) =

10k=0

xt+k|t + ut+k|t

5 4 3 2 1 0 1 2 3 4 55

4

3

2

1

0

1

2

3

4

5

Position

Speed

Control partition with 116 regions

5

0

5

5

0

5

1

0.5

0

0.5

1

Value of the control action over 116 regions

x1x2

u

(x)

0 1 2 3 4 5 6 7 8 9 102

1

0

1

2

3

4

x1

,x2

0 1 2 3 4 5 6 7 8 9 101

0.5

0

0.5

1

u * t

|t

time [s]


26/29

Example: Double Integrator (II)

Quadratic cost

J(x, u) =

10k=0

xt+k|t

22 + u

2t+k|t

5 4 3 2 1 0 1 2 3 4 55

4

3

2

1

0

1

2

3

4

5

Position

Speed

Control partition with 17 regions

5

0

55

0

5

1

0.5

0

0.5

1

Value of the control action over 17 regions

x1x2

u

(x)

0 1 2 3 4 5 6 7 8 9 102

1

0

1

2

3

4

x1

,x2

0 1 2 3 4 5 6 7 8 9 101

0

1

u * t

|t

time [s]


27/29

As a conclusion ...

Explicit solution to MPC for linear constrained

systems:

enables application of MPC to systems with

fast sampling rates

suitable for embedded applications better insight into MPC properties

(performance, stability, feasibility)

Caveat: The complexity of the explicit solution

(the number of regions) typically grows with

the dimension of the state space may easily

outgrow any practical application

Extensions

time optimal control

complexity reduction using suboptimal

solutions

infinite horizon solution

robust min-max controller

This (and much more) can be found in ...


28/29

The One and Only ...

MULTI-PARAMETRIC

TOOLBOX

http://control.ee.ethz.ch/~mpt


29/29

References

[1] A. Bemporad, M. Morari, V. Dua, and E.N. Pistikopoulos.The explicit linear quadratic regulator for constrained sys-tems. Automatica, 38(1):320, January 2002.

[2] F. Borrelli. Constrained Optimal Control Of Linear And Hy-brid Systems, volume 290 of Lecture Notes in Control andInformation Sciences. Springer, 2003.

[3] M. Kvasnica, P. Grieder, M. Baotic, and M. Morari. MultiParametric Toolbox (MPT). In Hybrid Systems: Compu-tation and Control, Lecture Notes in Computer Science.Springer Verlag, 2003. http://control.ee.ethz.ch/~mpt .

[4] R.D.C. Monteiro and I. Adler. A geometric view of paramet-ric linear programming. Algoritmica, 8(2):161176, 1992.

[5] P. Tndel, T.A. Johansen, and A. Bemporad. An algo-rithm for multiparametric quadratic programming and ex-plicit mpc solutions. Automatica, 39(3):489497, 2003.

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