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The Scenario Approach: Robust Optimization and Application to Control M.C. Campi University of Brescia E-Mail: campi@ing.unibs.it

A general fact: • convex optimization is easy but • robust convex optimization is hard min c T x subject to: f ( x, δ ) ≤ 0 , ∀ δ ∈ ∆

Example (stability)  x = Ax ˙ P ≻ 0     A T P + PA ≺ 0 LMI − convex   x S

Uncertainty - robust -0.8 nominal -1 D -1.2 -1 -1.3 -0.7 x = A ( δ ) x ˙  P ≻ 0     A ( δ ) T P + PA ( δ ) ≺ 0 ∀ δ ∈ ∆   infinite number of constraints!!!

A 1 A 5 A 3 A 2 � A ( δ ) = i δ i A i A 6 (convex: 0 ≤ δ i ≤ 1 � i δ i = 1) A 4 A 7 A 8   P ≻ 0         A T  1 P + PA 1 ≺ 0   . .  .         A T n P + PA n ≺ 0   

Towards generality {A( δ )} relaxation   P ≻ 0    A ( δ ) T P + PA ( δ ) ≺ 0 QS - Quadratic Stability   − P = P 0 + δ 1 P 1 + · · · + δ m P m AQS - Affine Quadratic Stability − P ( z, δ ) linear in z GQS - Generalized Quadratic Stability − P ( δ ) general case

Other problems in control • state-feedback stabilization • H ∞ control • H 2 control • LPV control . . .

Robust Convex Optimization min c T x subject to: f ( x, δ ) ≤ 0 , ∀ δ ∈ ∆

Uncertainty -1 x S -1

Violation set violation x set satisfaction set X ∆ ,P r Pr (violation set) ≤ ǫ • chance-constrained optimization

The ”Scenario” Paradigm (1) δ (2) δ (3) δ . . (4) δ * x N . (N) δ X ∆ SCP N = scenario convex program • SCP N is a standard finite convex optimization problem • x ∗ N is superoptimal

Fundamental how feasible is x ∗ N ? question:

Example c T = [ − 1 − 1] c T x RLP: min a T a T subject to 1 x ≤ 2 , 1 = [1 0] + ρ 1 δ 1 , ρ 1 = 0 . 1 , | δ 1 | ≤ 1 a T a T 2 x ≤ 1 , 2 = [1 0] + ρ 2 δ 2 , ρ 2 = 0 . 15 , | δ 2 | ≤ 1 a T a T 3 x ≤ 0 , 3 = [ − 1 0] a T a T 4 x ≤ 0 , 4 = [0 − 1]

boundary of nominal feasible set optimal solution (nominal) x 2 1 0.8 0.6 objective direction 0.4 0.2 0 0 0.5 1 1.5 2 x 1

boundary of nominal feasible set optimal solution (nominal) x 2 1 0.8 boundary of robust feasible set 0.6 optimal solution (robust) objective direction 0.4 0.2 0 0 0.5 1 1.5 2 x 1

randomly drawn linear constraints x 2 1 0.8 0.6 0.4 optimal solution of randomized LP (is `close' to robust optimal) 0.2 0 x 1 0 0.5 1 1.5 2

Fundamental how feasible is x ∗ N ? question:

generalization = ⇒ need for structure Good news: the structure we need is convexity • double role of convexity: - practice (computation) - theory (generalization)

Theorem Fix ǫ ∈ (0 , 1) (violation parameter) β ∈ (0 , 1) (confidence parameter) If N ≥ N ( ǫ, β ) . = 2 ǫ ln 1 β + 2 n x + 2 n x ǫ ln 2 ǫ , then, with probability ≥ 1 − β , x ∗ N is ǫ -level robustly feasible.

≤ε * x N satisfaction set X ∆ ≤β bad set (1) , δ (2) ,..., δ (N) ) ( δ ∆ N

Extensions: • SCP N is unfeasible • x ∗ N is not unique • SCP N is feasible, but x ∗ N does not exist

Comments: N ≥ 2 ǫ ln 1 β + 2 n x + 2 n x ǫ ln 2 ǫ • N usually tractable by standard solvers • N easy to compute • N independent of Pr • permits to address problems otherwhise intractable Ex : stability of A ( δ ) P ( z, δ ) GQS • even when RCP is tractable, SCP N gives a way to trade probability of violation for performance → ǫ = tuning knob

Example (stability-synthesis)     0 . 5 δ 2 1 + δ 1  10  x +  u x = ˙  − (1 + δ 1 ) 2 2(0 . 1 + 0 . 5 δ 2 )(1 + δ 1 ) 15 | δ 1 | ≤ 1 , | δ 2 | ≤ 1 Goal: design u = Kx such that the closed-loop is quadratically stable

A cl ( δ ) = A ( δ ) + BK Lyapunov condition: B T + B KP PA T ( δ ) + A ( δ ) P + PK T ≺ 0 ∀ δ ∈ ∆ � �� � ���� =: Y T =: Y K = Y P − 1 min P,Y,γ γ    − P 0  � γI, subject to − I � ∀ δ PA T ( δ )+ A ( δ ) P + Y T B T + BY 0

 = 0 . 05 ǫ   → N = 1174 = 0 . 001 β   0 . 0273 − 0 . 0212 � � P = Y = − 0 . 1620 − 0 . 2280   − 0 . 0212 0 . 4852 K = [ − 6 . 5162 − 0 . 7550] γ ∗ < 0

A-posteriori: Monte-Carlo analysis Uncertainty space 1 δ 2 0.8 0.6 0.4 0.2 0 Violation set -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 δ 1 N = 100 , 000 ˆ ǫ = 0 . 0096

Other problems in systems theory • construction of interval models for prediction 1.5 1 0.5 y(k) 0 −0.5 −1 −1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 u(k) • min-max identification y ) 2 ]) min M max d ( S, M ) ( e.g. d ( S, M ) = E [( y − ˆ S

Conclusions • Finite convex optimization is simple, but semi-infinite convex optimization is hard in gen- eral • The scenario approach offers a viable way to solve semi-infinite convex optimization problems in a risk-adjusted sense, based on a generalization result valid for all convex problems • ǫ trades robustness for performance

References G. Calafiore and M.C. Campi. The Scenario Approach to Robust Control Design. IEEE Trans. on Automatic Control , to appear (May or June, 2006). G. Calafiore and M.C. Campi. Uncertain convex programs: randomized solutions and confidence levels. Mathematical Programming , 102, no.1: 25-46, 2005.