Model Predictive Control for Linear and Hybrid Systems ...

Model Predictive Control for Linear and

Hybrid Systems

Robust Constrained Optimal Control

Francesco Borrelli

Department of Mechanical Engineering,University of California at Berkeley,

USA

[email protected]

April 19, 2011

Outline

1 Introduction

2 Robust Constrained Optimal Control

3 Feasible Set Computation

4 Min-Max Constrained Robust Optimal Control

Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 2 / 35

Introduction

LQR, is a robust controller, in the following sense◮ infinite gain amplification in any input,◮ 1/2 gain reduction in any input,◮ phase margin of π/3

In MPC, controller robustness means to require◮ feasibility of the controller for all time, in the presence of persistent

perturbations (disturbances).◮ convergence to a set Xf that contains the desired equilibrium point, in the

presence of persistent perturbations (disturbances).

in the next lectures we will show how to address the effect of disturbanceson a constrained systems and synthesize robust model predictive control


System Model

Consider the uncertain discrete–time dynamical system:

x(t + 1) = A(wp(t))x(t) + B(wp(t))u(t) + Ewa(t)

wherex(t) ∈ X , u(t) ∈ U , ∀t ≥ 0.

The sets X ⊆ Rn and U ⊆ R

m are polytopes.

Vectors wa(t) ∈ Rna and wp(t) ∈ R

np are unknown additive disturbancesand parametric uncertainties, respectively.

The disturbance vector is w(t) = [wa(t); wp(t)] ∈ Rnw with nw = na + np.

Only bounds on wa(t) and wp(t) are known, namely thatw ∈ W = V ×W with wa(t) ∈ V

V = conv({wa,1, . . . , wa,nV})

and wp(t) ∈ W ,W = conv({wp,1, . . . , wp,nW})


System Model

Consider the uncertain discrete–time dynamical system:

x(t + 1) = A(wp(t))x(t) + B(wp(t))u(t) + Ewa(t)

A(·), B(·) are affine functions of wp

A(wp) = A0 +

np∑

i=1

Aiwp,ic , B(wp) = B0 +

np∑

i=1

Biwp,ic

where Ai ∈ Rn×n and Bi ∈ R

n×m are given matrices

For i = 0 . . . , np and wp,ic is the i-th component of the vector wp, i.e.,

wp = [wp,1c , . . . , w

p,npc ].


Outline

1 Introduction

2 Robust Constrained Optimal ControlProblem Formulation. Batch ApproachProblem Formulation. Recursive Approach or Closed-Loop Predictions




Constrained Robust Optimal Control. Batch Approach

Define the worst case cost function

J0(x(0), U0) , maxw0,...,wN−1

[

p(xN ) +∑N−1

k=0 q(xk, uk)]

subj. to

xk+1 = A(wpk)xk + B(wp

k)uk + Ewak

wak ∈ V , w

pk ∈ W ,

k = 0, . . . , N − 1

where N is the time horizon and U0 , [u′0, . . . , u

′N−1]

′ ∈ Rs, s , mN the

vector of the input sequence.The robust optimal control problem is

J∗0 (x0) =min

U0

J0(x0, U0)

subj. to

xk ∈ X , uk ∈ Uxk+1 = A(wp

k)xk + B(wpk)uk + Ewa

k

xN ∈ Xf

k = 0, . . . , N − 1

∀wak ∈ V , w

pk ∈ W

∀k = 0, . . . , N − 1



xk denotes the state vector at time k obtained by starting from the statex0 = x(0) and applying to the system model

xk+1 = A(wpk)xk + B(wp

k)uk + Ewak

the input sequence u0, . . . , uk−1 and the disturbance sequenceswa , {wa

0 , . . . , waN−1}, wp , {wp

0 , . . . , wpN−1}.

We denote with XOLi ⊆ X the set of states xi for which the robust

optimal control problem is feasible, i.e.,

XOLi = {xi ∈ X such that ∃(ui, . . . , uN−1) such that

xk ∈ X , uk ∈ U , k = i, . . . , N − 1, xN ∈ Xf∀ wak ∈ V , w

pk ∈ W

k = i, . . . , N − 1 where xk+1 = A(wpk)xk + B(wp

k)uk + Ewak}.

The reason for including constraints in the minimization problem and notin the maximization problem is that wa

j and wpj are free to act regardless

of the state constraints. On the other hand, the input uj has the duty ofkeeping the state within the constraints for all possible disturbancerealization.



The optimal control problem looks for the worst value J(x0, U) of theperformance index and the corresponding worst sequences wp∗, wa∗ as afunction of x0 and U0.

It minimizes such a worst performance subject to the constraint that theinput sequence must be feasible for all possible disturbance realizations.

Note that worst sequences wa∗, wp∗for the performance are notnecessarily worst sequences in terms of constraints satisfaction.

The min-max formulation is based on an open-loop prediction and thusreferred to as Constrained Robust Optimal Control with open-looppredictions (CROC-OL).

The optimal control problem can be viewed as a deterministic zero-sumdynamic game between two players: the controller U and the disturbanceW .



or Open-Loop Predictions

The player U plays first. Given the initial state x(0), U chooses his actionover the whole horizon {u0, . . . , uN−1}, reveals his plan to the opponentW , who decides on his actions next {wa

0 , wp0 , . . . , wa

N−1, wpN−1}.

For this reason the player U has the duty of counteracting any feasibledisturbance realization with just one single sequence {u0, . . . , uN−1}.

This prediction model does not consider that at the next time step, thepayer can measure the state x(1) and “adjust” his input u(1) based onthe current measured state.

By not considering this fact, the effect of the uncertainty may grow overthe prediction horizon and may easily lead to infeasibility of the minproblem


Constrained Robust Optimal Control. Recursive

Approach or Closed-Loop Predictions

The constrained robust optimal control problem based on closed-looppredictions (CROC-CL) is defined as:

J∗j (xj) , min

uj

Jj(xj , uj)

s.t.

{xj ∈ X , uj ∈ UA(wp

j )xj + B(wpj )uj + Ewa

j ∈ Xj+1

}

∀waj ∈ V , w

pj ∈ W

Jj(xj , uj) , maxwa

j ∈V, wpj ∈W

{q(xj , uj) + J∗

j+1(A(wpj )xj + B(wp

j )uj + Ewaj )

},

for j = 0, . . . , N − 1 and with boundary conditions

J∗N (xN ) = p(xN )

XN = Xf ,


Constrained Robust Optimal Control. Closed-Loop

Predictions

Xj denotes the set of states x for which the CROC-CL is feasible

Xj = {x ∈ X such that ∃u ∈ Us.t. A(wp)x + B(wp)u + Ewa ∈ Xj+1 ∀wa ∈ V , wp ∈ W}.

The reason for including constraints in the minimization problem and notin the maximization problem is that wa

j and wpj are free to act regardless

of the state constraints. On the other hand, the input uj has the duty ofkeeping the state within the constraints for all possible disturbancerealization.



Predictions

The optimal control problem can be viewed as a deterministic zero-sumdynamic game between two players: the controller U and the disturbanceW .

The game is played as follows. At the generic time j player U observes xj

and responds with uj(xj). Player W observes (xj , uj(xj)) and respondswith wa

j and wpj .

Note that player U does not need to reveal his action uj to player W (thedisturbance). This happens for instance in games where U and W play atthe same time, e.g. rock-paper-scissors.

The player W will always play the worst case action only if it hasknowledge of both xj and uj(xj). In fact, wa

j and wpj are a function of xj

and uj .

If U does not reveal his action to player W , then we can only claim thatthe player W might play the worst case action.

Robust constraint satisfaction and worst case minimization will always beguaranteed.



PredictionsConsider the system

xk+1 = xk + uk + wk

where x, u and w are state, input and disturbance, respectively.

Let uk ∈ {−1, 0, 1} and wk ∈ {−1, 0, 1} be feasible input and disturbance.

Here {−1, 0, 1} denotes the set with three elements: -1, 0 and 1.

Let x(0) = 0 be the initial state. The objective for player U is to play twomoves in order to keep the state x2 at time 2 in the set [−1, 1].

If U is able to do so for any possible disturbance, then he will win thegame.

The open-loop formulation is infeasible. In fact, in open-loop U canchoose from nine possible sequences: (0,0), (1,1), (-1,-1), (-1,1) (1,-1),(-1,0), (1,0), (0,1) and (0,-1). For any of those sequence there will alwaysexist a disturbance sequence w0, w1 which will bring x2 outside thefeasible set [-1,1].

The closed-loop formulation is feasible and has a simple solution:uk = −xk. In this case the system becomes xk+1 = wk and x2 = w1 lies inthe feasible set [−1, 1] for all admissible disturbances w1.


Outline

1 Introduction


3 Feasible Set ComputationBasic ResultsBatch ApproachRecursive ApproachExample



Basic Results

Proposition

Let g : Rnz × R

n × Rnw → R

ng be a function of (z, x, w) convex in w for each(z, x). Assume that the variable w belongs to the polytope W with vertices{wi}

nW

i=1. Then, the constraint

g(z, x, w) ≤ 0 ∀w ∈ W (1)

is satisfied if and only if

g(z, x, wi) ≤ 0, i = 1, . . . , nW . (2)


Basic Results

Proposition

Assume g(z, x, w) = g1(z, x) + g2(w). Then, the constraint (1) can be replaced

by g1(z, x) ≤ −g, where g ,[g1, . . . , gng

]′is a vector whose i-th component is

gi = maxw∈W

g2i (w), (3)

and g2i (w) denotes the i-th component of g2(w).


Basic Results

Consider the second order autonomous system

x(t + 1) = A(wp(t))x(t) + wa(t) =

[0.5 + wp(t) 0

1 −0.5

]

x(t) + wa(t)

(4)subject to the state constraints

x(t) ∈ X =

{

x s.t.

[−10−10

]

≤ x ≤

[1010

]}

, ∀t ≥ 0

wa(t) ∈ Wa =

{

wa s.t.

[−1−1

]

≤ wa ≤

[11

]}

, ∀t ≥ 0

wp(t) ∈ Wp = {wp s.t. 0 ≤ wp ≤ 0.5} , ∀t ≥ 0.

(5)

Let w = [wa; wp] and W = Wa ×Wp.


Basic Results

The set X is a polytope and it can be represented as an H-polytope

X = {xs.t. Hx ≤ h}, (6)

The set Pre(X ,W) can be rewritten as

Pre(X ,W) ={x s.t. Hfa(x, w) ≤ h, ∀w ∈ W

}(7)

= {x s.t. HA(wp)x ≤ h − Hwa, ∀wa ∈ Wa, wp ∈ Wp} .

(8)

By using the propositions on previous slides we obtain

x ∈ Pre(X ,W) = {x ∈ Rn s.t.

[HA(0)

HA(0.5)

]

x ≤

[h

h

]

} (9)

withhi = min

wa∈Wa(hi − Hiw

a). (10)


Feasible Set Computation. Batch Approach

Consider XOLi ⊆ X the set of states xi for which the robust optimal control

problem is feasible, i.e.,

XOLi = {xi ∈ X such that ∃(ui, . . . , uN−1) such that

xk ∈ X , uk ∈ U , k = i, . . . , N − 1, xN ∈ Xf , ∀ wak ∈ V , w

pk ∈ W

k = i, . . . , N − 1 where xk+1 = A(wpk)xk + B(wp

k)uk + Ewak}.

For any initial state xi ∈ XOLi there exists a feasible sequence of inputs

Ui , [u′i, . . . , u

′N−1] which keeps the state evolution in the feasible set X

at future time instants k = i + 1, . . . , N − 1 and forces xN into Xf at timeN for all feasible disturbance sequence wa

k ∈ V , wpk ∈ W , k = i, . . . , N − 1.

Clearly XOLN = Xf .

How to compute XOLi for i = 0, . . . , N − 1?



Let the state and input constraint sets X , Xf and U be the H-polyhedraAxx ≤ bx, Afx ≤ bf , Auu ≤ bu.

Recall that the disturbance sets are defined as V = conv{wa,1, . . . , wa,nV }and W = conv{wp,1, . . . , wp,nW}.

Define Ui , [u′i, . . . , u

′N−1] and the polyhedron Pi of robustly feasible

states and input sequences at time i, defined as

Pi = {(Ui, xi) ∈ Rm(N−i)+ns.t. GiUi − Eixi ≤ Wi}.

Gi, Ei and Wi are obtained by collecting all the following inequalities:◮ Input Constraints

Auuk ≤ bu, k = i, . . . , N − 1

◮ State Constraints

Axxk ≤ bx, k = i, . . . , N−1 for all wal ∈ V, wp

l ∈ W, l = i, . . . , k−1. (11)

◮ Terminal State Constraints

AfxN ≤ bf , for all wal ∈ V, wp

l ∈ W, l = i, . . . , N − 1. (12)

Constraints are enforced for all feasible disturbance sequences.



To enforce constraints for all feasible disturbance sequences. rewrite themas a function of xi and the input sequence Ui.

Since we assumed that that A(·), B(·) are affine functions of wp and sincethe composition of a convex constraint with an affine map generates aconvex constraint we can use the Propositions presented in “basic results”and impose the constraints at all the vertices of the sets V × V × . . . × V

︸︷︷︸

i,...,N−1

and W ×W × . . . ×W︸︷︷︸

i,...,N−1

.

Note that the resulting constraints are now linear in xi and Ui.

Once the matrices Gi, Ei and Wi have been computed, the set XOLi is a

polyhedron and can be computed by projecting the polyhedron Pi on thexi space.


Feasible Set Computation. Revursive ApproachIn the recursive approach we have

Xi = {x ∈ X s.t. ∃u ∈ U such that A(wpi )x + B(wp

i )u + Ewai ∈ Xi+1

∀wai ∈ V , w

pi ∈ W}, i = 0, . . . , N − 1

XN = Xf . (13)

The definition of Xi is recursive and it requires that for any feasible initialstate xi ∈ Xi there exists a feasible input ui which keeps the next stateA(wp

i )x + B(wpi )u + Ewa

i in the feasible set Xi+1 for all feasibledisturbances wa

i ∈ V , wpi ∈ W .

Recall that X0 is different from XOL0 .

Let Xi be the H-polyhedron AXix ≤ bXi

.

The set Xi−1 is the projection of the following polyhedron on the xi space

Au

0AXi

B(wpi )

ui +

0Ax

AXiA(wp

i )

xi ≤

bu

bx

bXi− Ewa

i

(14)

for all wai ∈ {wa,i}nV

i=1, wpi ∈ {wp,i}nW

i=1


Feasible Set Computation. ExampleConsider the system

xk+1 = xk + uk + wk (15)

Let uk ∈ [−1, 1] and wk ∈ [−1, 1] be the feasible input and disturbance. Theobjective for player U is to play two moves in order to keep the state at timethree x3 in the set Xf = [−1, 1].Batch approach

We rewrite the terminal constraint as

x3 = x0 + u0 + u1 + u2 + w0 + w1 + w2 ∈ [−1, 1]for all w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1]

(16)

which becomes−1 ≤ x0 + u0 + u1 + u2 + 3 ≤ 1−1 ≤ x0 + u0 + u1 + u2 + 1 ≤ 1−1 ≤ x0 + u0 + u1 + u2 − 1 ≤ 1−1 ≤ x0 + u0 + u1 + u2 − 3 ≤ 1

(17)

which by removing redundant constraints becomes the (infeasible) constraint

2 ≤ x0 + u0 + u1 + u2 ≤ −2


Feasible Set Computation. Example

The set XOL0 is the projection on the x0 space of the polyhedron P0

P0 = {(u0, u1, u2, x0) ∈ R4s.t.

1 0 0−1 0 00 1 00 −1 00 0 10 0 −11 1 1−1 −1 −1

u0

u1

u2

+

0000001−1

x0 ≤

111111−2−2

(18)which is empty since the terminal state constraint is infeasible.


Feasible Set Computation. ExampleRecursive approach

By using the recursive we have X3 = Xf = [−1, 1].Rewrite the terminal constraint as

x3 = x2 + u2 + w2 ∈ [−1, 1] for all w2 ∈ [−1, 1] (19)

which becomes−1 ≤ x2 + u2 + 1 ≤ 1−1 ≤ x2 + u2 − 1 ≤ 1

(20)

which by removing redundant constraints becomes

0 ≤ x2 + u2 ≤ 0

The set X2 is the projection on the x2 space of the polyhedron

1 0−1 01 1−1 −1

[u2, x2] ≤

1100

(21)

which yields X2 = [−1, 1]. Since X2 = X3 one can conclude that X2 is themaximal controllable robust invariant set and X0 = X1 = X2 = [−1, 1].Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 26 / 35


Batch approach with closed-loop predictions

The horizon is three and therefore we consider the disturbance set overthe horizon 3-1=2, W ×W = [−1, 1]× [−1, 1].

Such a set has four vertices: {w10 = 1, w1

1 = 1}, {w20 = −1, w2

1 = 1},{w1

0 = 1, w31 = −1}, {w2

0 = −1, w41 = −1}.

We introduce an input sequence u0, ui1, u

j2, where the index i ∈ {1, 2} is

associated with the vertices w10 and w2

0 and the index j ∈ {1, 2, 3, 4} isassociated with the vertices w1

1 , w21 , w3

1 , w41 .

The terminal constraint is thus rewritten as

x3 = x0 +u0+ ui1+ u

j2+wi

0 +wj1 +w2 ∈ [−1, 1], i = 1, 2, j = 1, . . . , 4, (22)

∀ w2 ∈ [−1, 1]



which becomes

−1 ≤ x0 + u0 + u11 + u1

2 + 0 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]−1 ≤ x0 + u0 + u2

1 + u22 + 0 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]

−1 ≤ x0 + u0 + u11 + u3

2 + 2 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]−1 ≤ x0 + u0 + u2

1 + u42 − 2 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]

−1 ≤ u0 ≤ 1−1 ≤ u1

1 ≤ 1−1 ≤ u2

1 ≤ 1−1 ≤ u1

2 ≤ 1−1 ≤ u2

2 ≤ 1−1 ≤ u3

2 ≤ 1−1 ≤ u4

2 ≤ 1

(23)

The set X0 can be obtained by using Proposition 2 for the polyhedron (23)and projecting the resulting polyhedron in the(x0, u0, u1

1, u21, u1

2, u22, u3

2, u42)-space on the x0 space. This yields

X0 = [−1, 1].


Outline

1 Introduction



4 Min-Max Constrained Robust Optimal ControlBasic ResultRecursive Approach


Basic Result

Lemma

Let f : Rs × R

n × Rnw → R and g : R

s × Rn × R

nw → Rng be functions of

(z, x, w) convex in w for each (z, x). Assume that the variable w belongs to thepolyhedron W with vertices {wi}

NW

i=1 . Then the min-max multiparametricproblem

J∗(x) = minz maxw∈W f(z, x, w)subj. to g(z, x, w) ≤ 0 ∀w ∈ W

(24)

is equivalent to the multiparametric optimization problem

J∗(x) = minµ,z µ

subj. to µ ≥ f(z, x, wi), i = 1, . . . , NW

g(z, x, wi) ≤ 0, i = 1, . . . , NW .

(25)


Basic Result

Corollary

If f is convex and piecewise affine in (z, x), i.e. f(z, x, w) = maxi=1,...,nf

{Li(w)z + Hi(w)x + Ki(w)} and g is linear in (z, x) for all w ∈ W,g(z, x, w) = Kg(w) + Lg(w)x + Hg(w)z (with Kg(·), Lg(·), Hg(·), Li(·), Hi(·),Ki(·), i = 1, . . . , nf , convex functions), then the min-max multiparametricproblem (24) is equivalent to the mp-LP problem

J∗(x) = minµ,z µ

subj. to µ ≥ Kj(wi) + Lj(wi)z + Hj(wi)x,

i = 1, . . . , NW , j = 1, . . . , nf

Lg(wi)x + Hg(wi)z ≤ −Kg(wi), i = 1, . . . , NW

(26)


Min-Max Constrained Robust Optimal

Control-Recursive Approach

Theorem

There exists a state-feedback control law u∗(k) = fk(x(k)),fk : Xk ⊆ R

n → U ⊆ Rm, solution of the CROC-CL with cost based on one or

infinity norm which is time-varying, continuous and piecewise affine onpolyhedra

fk(x) = F ikx + gi

k if x ∈ CRik, i = 1, . . . , Nr

k (27)

where the polyhedral sets CRik = {x ∈ R

n s.t. Hikx ≤ Ki

k}, i = 1, . . . , Nrk are a

partition of the feasible polyhedron Xk. Moreover fi, i = 0, . . . , N − 1 can befound by solving N mp-LPs.



Control-Recursive Approach

Consider the first step j = N − 1 of dynamic programming

J∗N−1(xN−1) , min

uN−1

JN−1(xN−1, uN−1) (28)

subj. to

FxN−1 + GuN−1 ≤ f

A(wpN−1)xN−1 + B(wp

N−1)uN−1 + EwaN−1 ∈ Xf

∀waN−1 ∈ V , w

pN−1 ∈ W

(29)

JN−1(xN−1, uN−1) , maxwa

N−1∈V, wp

N−1∈W

‖QxN−1‖p + ‖RuN−1‖p++‖P (A(wp

N−1)xN−1++B(wp

N−1)uN−1 + EwaN−1)‖p

.

(30)



Control-Recursive ApproachThe cost function in the maximization problem (30) is piecewise affineand convex with respect to the optimization vector wa

N−1, wpN−1 and the

parameters uN−1, xN−1.

The constraints in the minimization problem linear in (uN−1, xN−1) forall vectors wa

N−1, wpN−1.

Therefore, by “basic Results” J∗N−1(xN−1), u∗

N−1(xN−1) and XN−1 arecomputable via the mp-LP:

J∗N−1(xN−1) , min

µ,uN−1

µ (31a)

subj. to µ ≥ ‖QxN−1‖p + ‖RuN−1‖p +

+ ‖P (A(wph)xN−1 + B(wp

h)uN−1 + Ewai )‖p (31b)

FxN−1 + GuN−1 ≤ f (31c)

A(wph)xN−1 + B(wp

h)uN−1 + Ewai ∈ XN (31d)

∀i = 1, . . . , nV , ∀h = 1, . . . , nW .

The convexity and linearity arguments still hold for j = N − 2, . . . , 0 and theprocedure can be iterated backwards in time j.Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 34 / 35

We Will not Cover

Robust Invariant Set

Robust invariant Set Evolution

Robust MPC Theorem

Robust Estimation


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