Robust Model Predictive Control - University of Sheffield/file/...Robust Model Predictive Control Colloquium on Predictive Control University of Sheffield, April 4, 2005 David Mayne

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