Robust Model PredictiveControl

Colloquium on Predictive Control

University of Sheffield, April 4, 2005

David Mayne

(with Maria Seron and Sasa Rakovic)

Imperial College London

IC – p.1/25

ContentsI. Robust control problem

II.Conventional model predictive control

III. Disturbance/uncertainty

IV. Feedback model predictive control

V. Tube model predictive control

VI. Novel robust model predictive control

VII. Conclusions

IC – p.2/25

Robust controlproblem

Uncertain System

x+ = f(x, u, w) = Ax + Bu + w

Constraints :

x ∈ X, u ∈ U, w ∈ W

φ(k; x,u,w), solution of x+ = f(x, u, w) attime k

u , {u0, u1, . . . , uN−1}; also w.

Control objectives: stabilization andperformance IC – p.3/25

Conventional RHCSystem:z+ = f(x, v) = Az + Bv,

Constraints: z ∈ Z, v ∈ V.

FH OC Pb is: PN (z):

minu

V (z,v) ,

N−1∑

i=0

`(zi, vi) + Vf(zN)

zi ∈ Z, vi ∈ V, zN ∈ Zf

zi = φ(i; z,v), solution of z+ = Az + Bv attime k.

IC – p.4/25

Conventional RHCSolution: v

0(z) = {v00(z), v0

1(z), . . . , v0N−1

(z)}.

Value f’n: V 0N(z) = VN(z,v0(z)), domain ZN .

RH control law: κN(z) , v0(0; z).

IF A: Vf(·) is local CLF in sense:

minv∈V{Vf(f(z, u)) + `(z, u) | f(z, u) ∈Zf} ≤ Vf(z) for all z ∈ Zf ,THEN:

V 0N(f(z, κN (z)) + `(z, κN (z)) ≤ V 0

N(z) ∀z ∈ ZN

and V 0N(z) ≤ Vf(z) ∀z ∈ Zf =⇒ V 0

N(·) is aLyapunov fn for controlled system.

IC – p.5/25

Value function V 0N(·)

Proposition 1 Assume that `(·) and Vf(·) arequadratic and pos. def. and that (stabilizing)assumption A is satisfied. Then

V 0

N(z) ≥ c1|z|2, ∀z ∈ ZN

V 0

N(f(z, κN (z)) ≤ V 0

N(z) − c1|z|2, ∀z ∈ ZN

V 0

N(z) ≤ c2|z|2, ∀z ∈ Zf

Theorem 1 The origin is exp.stable for nominalsystem with MPC law κN(·) (if ZN is bounded).Solution same as that obtained with DP. IC – p.6/25

Nominal robustnessCan use nominal controller u = κN(x) tocontrol uncertain system x+ = Ax + Bu + w

Under some conditions, get ‘ultimateboundedness’.

PSfrag replacements

Zf

z

ZNLV (c2)

LV (c1)

IC – p.7/25

‘Feedback’ RHCConservative. State constraints may renderx+ = f(x, κN (x)) non-robust (Teel)

better to design controller to be robust.

predicting effect of uncertainty

hence, use feedback RHC (in which decisionvariable is policy π = {µ0(·), µ1(·), . . . , µN−1(·)}(sequence of control laws).

rather than u = {u0, u1, . . . , uN−1} (sequenceof control actions ).

IC – p.8/25

Control policy

PSfrag replacements

µi(x)

x

Figure 1: π = {µ0(·), µ1(·), . . . , µN−1(·)}

IC – p.9/25

Tube

Whether control is ol (u) or fb (π):

obtain tube of predicted trajectories

Tube more ‘compact’ with fb policy π

Satisfaction of constraints easier (possible)

Optimizing over control sequences yieldsworse result than with DP

Optimizing over control policies yields sameresult as with DP

IC – p.10/25

Tube {X0, X1, . . . , XN}

PSfrag replacements

z

x

t

X0

X1X2 XN

(a) OL u

PSfrag replacements

z

x

t

X0

X1

X2

XN

(b) FB π

IC – p.11/25

Simplification

Exact tube {X0, X1, . . . , XN} complex: aN .

Policy π complex ...inf dimensional

Outer approx’n: Xi = zi ⊕ S .

zi = φ(i; z, v(·)), solution at time i of:

nominal system: z+ = f(z, v) = Az + Bv.

S robust pos. invariant for x+ = AKx + w

AKS ⊕ W ⊂ S, AK , A + BK stable.

Policy π = {µi(·)},

µi(x; z, v(·)) = vi + K(x − zi) (K & R). IC – p.12/25

Simplification

x ∈ X0 implies xi = φ(i; x, π,w) ∈ Xi ∀i,w .(Xi → {zi} as W → {0})

PSfrag replacements

z

x

t

X0

X1

X2

XNxi

zi

Xi = zi + S

Figure 2: Tube

IC – p.13/25

Nominal OC PbNominal system is z+ = Az + Bv.

φ(i; z,v) is sol’n at i of nominal system.

Nominal OC Pb: same as before with tighterconstraints:

vi ∈ V , U KS (W small enough)

zi ∈ Z , X S (W small enough)

zN ∈ Zf ⊂ X S, where

UN(z) = {v satisfying all constraints , z(0) = z}

(C&R&Z, M&L)

IC – p.14/25

Control policy

Consider policy π0 = π0(z) constructed fromsolution to modified nominal OC Pb:

π0(z) , {µ0(·; z), µ1(·; z), . . . , µN−1(·; z)}

µi(x; z) , v0i (z) + K(x − z0

i (z))

Proposition 2 Policy π0(z) steers any any initialstate of x+ = Ax + Bu + w lying in X0(x) = z ⊕ Sto Xf = Zf ⊕ S ⊆ X in N steps satisfying allconstraints for all admissible disturbancesequences. (C&R&Z, M&L)

IC – p.15/25

Control policyPSfrag replacements

xz0

t

X0

X1

X2

XNxi

zi

Xi = zi + S

Zf Xf

Figure 3: TubeIC – p.16/25

Receding horizon

Above ... single shot

Receding horizon ... two factors:

Controlling tube ...not a trajectory

Control to set S (not origin) ... S is robust‘origin’

Prob. 1: Assumption equivalent to A for casewhen terminal ‘state’ is a set X rather than apoint x?

Prob. 2: Value function that is zero in S (theorigin)?

IC – p.17/25

Terminal Set

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5−3

−2

−1

0

1

2

3

4

5

x1

x2

PSfrag replacements

Xf = Zf ⊕ S

Zf

z ⊕ S

S

IC – p.18/25

Terminal Set

Let Xf , {z ⊕ S | z ∈ Zf}.

Suppose (i) S is robust pos. invariant forx+ = AKx + w, and, (ii) Zf is positively invariantfor z+ = AKz (don’t need same K ’s).

IF: X ∈ Xf

THEN: X+ = AKX ⊕ W ∈ Xf

IC – p.19/25

New Optimal ControlProblem

To get Lyapunov function zero in S (origin)

propose new finite horizon O.C Pb:

OC Pb: P∗N(x):

V ∗N(x) = min

z,v{VN(z,v) | v ∈ UN(z), x ∈ z ⊕ S}

= minz{V 0

N(z) | x ∈ z ⊕ S}

Solution:

(z∗(x), v∗(x))

v∗(x) = {v∗0(x), v∗1(x), . . . , v∗N−1(x)}

IC – p.20/25

Model predictivecontrol law

Implicit model predictive control law:

κ∗N(x) = v∗0(x) + K(x − z∗(x))

PSfrag replacementsz∗(x)

x ⊕ S

x

IC – p.21/25

Exponential stability

PSfrag replacementsz∗i

zi+1

z∗i+1

xi

xi+1

z∗i ⊕ S

zi

i i + 1

IC – p.22/25

Exponential stabilityz∗i → zi+1 → z∗i+1,

V 0N(zi+1) ≤ V 0

N(z∗i ) − `(z∗i , κN(z∗i )).

V 0N(z∗i+1) ≤ V 0

N(zi+1)

=⇒ zi → 0 exponentially

But xi ∈ zi ⊕ S

=⇒ xi → S (rob) exponentially

Theorem 2 The set S is rob. exponentially stable(if X∗

N is bounded) for x+ = Ax + Bκ∗N(x) + w.

The domain of attraction is XN = ZN ⊕ S.IC – p.23/25

Exponential stability

PSfrag replacements

x

k

IC – p.24/25

ConclusionsHave presented novel version of modelpredictive control

Uses feedback mpc and bounding tube

And initial state as a decision variable

Simple online optimization problem (QP)

V ∗N(·) zero in set S (origin)

Set S is exponentially stable

Moral : control tube, not individualtrajectories.

IC – p.25/25