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Model Predictive Control for Linear and
Hybrid Systems
Robust Constrained Optimal Control
Francesco Borrelli
Department of Mechanical Engineering,University of California at Berkeley,
USA
fborrelli@berkeley.edu
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
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 3 / 35
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})
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 4 / 35
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 ].
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 5 / 35
Outline
1 Introduction
2 Robust Constrained Optimal ControlProblem Formulation. Batch ApproachProblem Formulation. Recursive Approach or Closed-Loop Predictions
3 Feasible Set Computation
4 Min-Max Constrained Robust Optimal Control
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 6 / 35
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
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 7 / 35
Constrained Robust Optimal Control. Batch Approach
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 8 / 35
Constrained Robust Optimal Control. Batch Approach
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 .
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 9 / 35
Constrained Robust Optimal Control. Batch Approach
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
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 10 / 35
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 ,
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 11 / 35
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 12 / 35
Constrained Robust Optimal Control. Closed-Loop
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 13 / 35
Constrained Robust Optimal Control. Closed-Loop
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 14 / 35
Outline
1 Introduction
2 Robust Constrained Optimal Control
3 Feasible Set ComputationBasic ResultsBatch ApproachRecursive ApproachExample
4 Min-Max Constrained Robust Optimal Control
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 15 / 35
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)
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 16 / 35
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).
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 17 / 35
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 18 / 35
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)
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 19 / 35
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?
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 20 / 35
Feasible Set Computation. Batch Approach
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 21 / 35
Feasible Set Computation. Batch Approach
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 22 / 35
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
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 23 / 35
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
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 24 / 35
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 25 / 35
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
Feasible Set Computation. Example
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]
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 27 / 35
Feasible Set Computation. Example
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].
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 28 / 35
Outline
1 Introduction
2 Robust Constrained Optimal Control
3 Feasible Set Computation
4 Min-Max Constrained Robust Optimal ControlBasic ResultRecursive Approach
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 29 / 35
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)
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 30 / 35
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)
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 31 / 35
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.
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 32 / 35
Min-Max Constrained Robust Optimal
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)
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 33 / 35
Min-Max Constrained Robust Optimal
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
Francesco Borrelli (UC Berkeley) Robust Constrained Optimal Control April 19, 2011 35 / 35
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