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A Global Optimization Approach to Structured Regulation Design under - - PowerPoint PPT Presentation

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Introduction Global optimization approach: Branch and Bound H norm of MISO systems Example Conclusion A Global Optimization Approach to Structured Regulation Design under H Constraints Dominique Monnet, Jordan Ninin, Beno t Cl

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

A Global Optimization Approach to Structured Regulation Design under H∞ Constraints

Dominique Monnet, Jordan Ninin, Benoˆ ıt Cl´ ement LAB-STICC, UMR 6285 / ENSTA-Bretagne December 12, 2016

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Plan

1 Introduction 2 Global optimization approach: Branch and Bound 3 H∞ norm of MISO systems 4 Example 5 Conclusion

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Structured H∞ synthesis problem

w P z K y u P interconnected with K : z = Tw→zw Structured H∞ synthesis problem Find K ∈ Ks such that ||Tw→z||∞ ≤ γ and K stabilizes Tw→z

  • r
  • min

K∈Ks||Tw→z||∞

subject to K stabilizes Tw→z

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Problem solved?

K ||Tw→z||∞ Convex approaches (LMI). Non convex approaches based on local optimization. Non convex approaches based on global optimization→

  • nly for SISO (polynomial formulation).

Global optimization for MIMO?

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Almost possible

r K G y d n + − + + e u W1 W2 W3 ˜ e ˜ u ˜ y P Constraints: ||Tw→˜

e||∞ ≤ 1.

||Tw→˜

u||∞ ≤ 1.

||Tw→˜

y||∞ ≤ 1.

||Tw→z||∞ is an over approximation: max(||Tw→˜

e||∞, ||Tw→˜ u||∞, ||Tw→˜ y||∞) ≤ ||Tw→z||∞

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Almost possible

K(k, s) is a parametric rational function. PID example: K(k, s) = kp + ki

s + kds, k = (kp, ki, kd)

Problem Find k ∈ k such that ||Tw→zj(k)||∞ ≤ γ, j ∈ {1, ..., p} and K stabilizes Tw→z

  • r
  • min

k∈k{max j ||Tw→zj(k)||∞}

subject to K stabilizes Tw→z The global optimization algorithm provides: A guaranteed enclosure [lb, ub] of the minimum. A good controller K. And a certificate of unfeasibility if lb > γ.

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Plan

1 Introduction 2 Global optimization approach: Branch and Bound 3 H∞ norm of MISO systems 4 Example 5 Conclusion

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Branch and Bound Algorithm

Minimization problem min

k∈k

  • max

j ||Tw→zj(k)||∞

  • k

max

j ||Tw→zj||∞

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Branch and Bound Algorithm

k max

j ||Tw→zj||∞

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Branch and Bound Algorithm

k max

j ||Tw→zj||∞

ub lb

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Branch and Bound Algorithm

k max

j ||Tw→zj||∞

ub lb

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Branch and Bound Algorithm

k max

j ||Tw→zj||∞

ub lb

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Enclosure of infinity norm of MISO

Problem Find binf and bsup such that: binf ≤ max

j ||Tw→zi(k)||∞ ≤ bsup, ∀k ∈ k

k k bsup binf

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Plan

1 Introduction 2 Global optimization approach: Branch and Bound 3 H∞ norm of MISO systems 4 Example 5 Conclusion

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Expression of ||.||∞ of MISO systems

Tw→z(k, s) is a matrix of rational functions which maps w to z: Tw→z(k, s) =    Tw1→z1(k, s) ... Twn→z1(k, s) . . . . . . Tw1→zp(k, s) ... Twn→zp(k, s)    =    Tw→z1(k, s) . . . Tw→zp(k, s)    Tw→zj(k, s) =

  • Tw1→zj(k, s)

... Twn→zj(k, s)

  • Proposition

||Tw→zj(k)||∞ = sup

ω≥0

n

l=1 |Twl→zj(k, iω)|2

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Demonstration

||Tw→zj(k, s)||∞ = sup

ω≥0

{σmax(Tw→zj(k, iω))} = sup

ω≥0

  • λmax(Tw→zj(k, iω)Tw→zj(k, iω)∗)

= sup

ω≥0

  • λmax

  

  • Tw1→zj(k, iω), ...Twn→zj(k, iω)

  Tw1→zj(k, iω) . . . Twn→zj(k, iω)       = sup

ω≥0

n

l=1 |Twl→zj(k, iω)|2

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Maximization problem

Problem Find binf and bsup such that: binf ≤ max

j ||Tw→zi(k)||∞ ≤ bsup, ∀k ∈ k

max

j ||Tw→zj(k)||∞ = max j

  sup

ω

  • n
  • l=1

|Twl→zj(k, iω)|2    = sup

ω

  max

j

  • n
  • l=1

|Twl→zj(k, iω)|2    Use another branch and bound! → Finite frequency range

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

2nd Branch and Bound

ω Tw→z1 Tw→z2 n

l=1 |Twl→zj(k, iω)|2

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

2nd Branch and Bound

ω max

j

n

l=1 |Twl→zj(k, iω)|2

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

2nd Branch and Bound

ω max

j

n

l=1 |Twl→zj(k, iω)|2

bsup binf

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Min-Max formulation

The problem has a Min-Max formulation: Min-Max Problem min

k∈k

  • sup

ω

  • max

j

n

l=1 |Twl→zj(k, iω)|2

  • s.t. K stabilizes the system: P(k) ≤ 0

Complexity of a branch and bound: exponential. → Branch and bound inside a branch and bound: Tricky problem.

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Plan

1 Introduction 2 Global optimization approach: Branch and Bound 3 H∞ norm of MISO systems 4 Example 5 Conclusion

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Example

w kp + ki

s + kds 1+s 1 s2+1.4s+1

y

10s+100 1000s+1 10s+1 s+10 100s+1 s+10

z1 z2 z3 + − e u kp ∈ [−10, 10], ki ∈ [−10, 10], kd ∈ [−10, 10] Method Cpu (s) ||Tw→z||∞ max

j (||Tw→zj||∞)

H∞ full 2 1.0258 1.0161 H∞ struct 73 (500 rand start) 1.0411 1.0411 H∞ hifoo 88 (30 rand start) 1.0349 1.0348 H∞ systune 73 (2.1e6 rand start) 1.0986 0.9912 GO struct 80 1.0811 [0.9496,0.9947]

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Plan

1 Introduction 2 Global optimization approach: Branch and Bound 3 H∞ norm of MISO systems 4 Example 5 Conclusion

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Introduction Global optimization approach: Branch and Bound H∞ norm of MISO systems Example Conclusion

Conclusion

Analytic expression of ||.||∞ of MISO system (square root

  • f a rational function).

Lower bound on the minimum = ⇒ we can prove that the CSP is not feasible. Another step towards structured robust synthesis? (yes). G(∆, s) → sup

ω,∆

{max

j

  • n
  • l=1

|Twl→zj(k, ∆, iω)|2}

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