Fixed-Structure Discrete-Time H Controller

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Fixed-Structure Discrete-Time H∞ Controller Synthesis with HIFOO Andrey Popov1 1 Institute

of Control Systems Hamburg University of Technology Germany

Herbert Werner1

Marc Millstone2

2 Courant

Institute of Mathematical Sciences New York University USA

16.12.2010 49th IEEE Conference on Decision and Control

1 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Contents

1

Motivation

2

Discrete-time HIFOO

3

Active Suspension System

4

Conclusions

2 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Contents

1

Motivation

2

Discrete-time HIFOO

3

Active Suspension System

4

Conclusions

3 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Motivation - 1: Simple Controllers Fixed-structure controllers are + computationally efficient + economic/energy efficient + easy to implement and verify

But:

4 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Motivation - 1: Simple Controllers Fixed-structure controllers are + computationally efficient + economic/energy efficient + easy to implement and verify

But: − generally a non-convex problem − convex reformulation only for SISO systems (central polynomial approach) [Henrion, 2005], [Khatibi & Karimi, 2010] ∼ variety of methods proposed for continuous-time problems 5 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Motivation - 2: Discrete-Time Synthesis + + + −

digital controllers black-box identification high performance requirements/large sampling times T Performance is lost by continuous-time synthesis - shown in a moment

6 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Motivation - 2: Discrete-Time Synthesis + + + −

digital controllers black-box identification high performance requirements/large sampling times T Performance is lost by continuous-time synthesis - shown in a moment

Often: bilinear transformation + continuous-time synthesis z=

2 + Ts , 2 − Ts

synthesis

s=

2 z−1 T z+1

7 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Motivation - 2: Discrete-Time Synthesis + + + −

digital controllers black-box identification high performance requirements/large sampling times T Performance is lost by continuous-time synthesis - shown in a moment

Often: bilinear transformation + continuous-time synthesis z=

2 + Ts , 2 − Ts

synthesis

s=

2 z−1 T z+1

Problems − frequency warping − performance loss by constrained sampling rate 8 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Motivation - 3: Gradient-Based Techniques

Gradient-based fixed-structure H∞ synthesis [Burke et.al., 2006], [Apkarian & Noll, 2006] Offer fast convergence local optimum good results [Gumussoy et.al., 2008]

HIFOO is open source

9 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Problem Formulation Given a generalized plant G(z) find a controller K(z) ∈ K(z) such that kM (z)k∞ is minimized,

v u

where M (z) = FL (G(z), K(z))

w

G(z) e K(z) M (z)

10 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Contents

1

Motivation

2

Discrete-time HIFOO

3

Active Suspension System

4

Conclusions

11 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

HIFOO Algorithm 1

Initialization - random controllers

2

Stabilization

find K(z) ∈ K(z) s.t. M (z) = FL (G(z), K(z)) is stable

3

Closed-loop H∞ norm minimization minimize kFL (G(z), K(z))k∞ K(z)∈K(z)

12 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

HIFOO Algorithm 1

Initialization - random controllers

2

Stabilization

find K(z) ∈ K(z) s.t. M (z) = FL (G(z), K(z)) is stable

3

Closed-loop H∞ norm minimization minimize kFL (G(z), K(z))k∞ K(z)∈K(z)

13 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

HIFOO Algorithm 1

Initialization - random controllers

2

Stabilization

find K(z) ∈ K(z) s.t. M (z) = FL (G(z), K(z)) is stable

3

Closed-loop H∞ norm minimization minimize kFL (G(z), K(z))k∞ K(z)∈K(z)

14 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Stabilization Continuous-time

minimize α(A) K(z)

until α(A) < 0 α - spectral abscissa A - closed-loop system matrix

15 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Stabilization Continuous-time

minimize α(A) K(z)

until α(A) < 0 α - spectral abscissa A - closed-loop system matrix Gradient-steps 1 Given K(z), compute spec(A) and find λk at which α is attained 2 Compute gradient of α w.r.t. A 3 Compute gradient of A w.r.t. K(z) 4 Update K(z) 16 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Stabilization Continuous-time

minimize α(A) K(z)

until α(A) < 0 α - spectral abscissa A - closed-loop system matrix

Discrete-time

minimize ρ(A) K(z)

until ρ(A) < 1 ρ - spectral radius

Gradient-steps 1 Given K(z), compute spec(A) and find λk at which α is attained 2 Compute gradient of α w.r.t. A 3 Compute gradient of A w.r.t. K(z) 4 Update K(z) 17 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Stabilization Continuous-time

minimize α(A) K(z)

until α(A) < 0 α - spectral abscissa A - closed-loop system matrix

Discrete-time

minimize ρ(A) K(z)

until ρ(A) < 1 ρ - spectral radius

Gradient-steps 1 Given K(z), compute spec(A) and find λk at which ρ is attained 2 Compute gradient of ρ w.r.t. A 3 Compute gradient of A w.r.t. K(z) 4 Update K(z) 18 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Computation Let A(t) = A0 + P t, A ∈ Rn×n and small t. Let λk ∈ spec(A) has algebraic multiplicity one and eigenvectors y (left) and x (right). y?P x dλk = ? dt y x

[Horn & Johnson, 1985]   Let λk = R + jI; `k = R I . p ρ = |λk | = R2 + I 2 = |`k | h ∂ρ = √RR 2 +I 2 ∂`k

√ I R2 +I 2

i

=

Hence dρ(A) = Re dt

`k |λk |

⇔

∂ρ λk = ∂λk |λk |

 ¯  λk y ? P x |λk | y ? x 19 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Computation Let A(t) = A0 + P t, A ∈ Rn×n and small t. Let λk ∈ spec(A) has algebraic multiplicity one and eigenvectors y (left) and x (right). dλk y?P x = ? dt y x

[Horn & Johnson, 1985]   Let λk = R + jI; `k = R I . p ρ = |λk | = R2 + I 2 = |`k | h ∂ρ = √RR 2 +I 2 ∂`k

√ I R2 +I 2

i

=

Hence dρ(A) = Re dt

`k |λk |

⇔

∂ρ λk = ∂λk |λk |

 ¯  λk y ? P x |λk | y ? x 20 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Computation Let A(t) = A0 + P t, A ∈ Rn×n and small t. Let λk ∈ spec(A) has algebraic multiplicity one and eigenvectors y (left) and x (right). dλk y?P x = ? dt y x

[Horn & Johnson, 1985]   Let λk = R + jI; `k = R I . p ρ = |λk | = R2 + I 2 = |`k | h ∂ρ = √RR 2 +I 2 ∂`k

√ I R2 +I 2

i

=

Hence dρ(A) = Re dt

`k |λk |

⇔

∂ρ λk = ∂λk |λk |

 ¯  λk y ? P x |λk | y ? x 21 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Computation Let A(t) = A0 + P t, A ∈ Rn×n and small t. Let λk ∈ spec(A) has algebraic multiplicity one and eigenvectors y (left) and x (right). dλk y?P x = ? dt y x

[Horn & Johnson, 1985]   Let λk = R + jI; `k = R I . p ρ = |λk | = R2 + I 2 = |`k | h ∂ρ = √RR 2 +I 2 ∂`k

√ I R2 +I 2

i

=

Hence dρ(A) = Re dt

`k |λk |

⇔

∂ρ λk = ∂λk |λk |

 ¯  λk y ? P x |λk | y ? x 22 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Computation Let A(t) = A0 + P t, A ∈ Rn×n and small t. Let λk ∈ spec(A) has algebraic multiplicity one and eigenvectors y (left) and x (right). dλk y?P x = ? dt y x

[Horn & Johnson, 1985]   Let λk = R + jI; `k = R I . p ρ = |λk | = R2 + I 2 = |`k | h ∂ρ = √RR 2 +I 2 ∂`k

√ I R2 +I 2

i

=

Hence dρ(A) = Re dt

`k |λk |

⇔

∂ρ λk = ∂λk |λk |

 ¯  λk y ? P x |λk | y ? x 23 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Computation Let A(t) = A0 + P t, A ∈ Rn×n and small t. Let λk ∈ spec(A) has algebraic multiplicity one and eigenvectors y (left) and x (right). dλk y?P x = ? [Horn & Johnson, 1985] dt y x   Let λk = R + jI; `k = R I . p ρ = |λk | = R2 + I 2 = |`k | Theorem For a matrix A with ρ(A) = |λk | holds   λk yx? ∇A ρ = Re |λk | y ? x 24 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Steps Stabilization 1

Compute spec(A) and find λk at which ρ is attained

2

Compute gradient of ρ w.r.t. A

3

Compute gradient of A w.r.t. K(z)

4

Update K(z)

25 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Gradient Steps Stabilization 1

Compute spec(A) and find λk at which ρ is attained

2

Compute gradient of ρ w.r.t. A

3

Compute gradient of A w.r.t. K(z)

4

Update K(z)

H∞ norm minimization 1

Compute f = kM (z)k∞ = σ ¯ (M )
−1 x jω x where Mx = C Ie −A B+D

2

Compute the gradient of f w.r.t. Mx

3

Compute the gradient of Mx w.r.t. K(z)

4

Update K(z) 26 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Contents

1

Motivation

2

Discrete-time HIFOO

3

Active Suspension System

4

Conclusions

27 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Active Suspension Benchmark System d

[Landau et.al. 2003]

28 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Active Suspension Benchmark System

d

−

K(z)

u

P (z)

y

29 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

sensitiv. [dB]

Design Requirements 5 0 −5

control sens. [dB]

−10 20 0 −20 −40 −60 −80

0

50

100

150

200

250

300

350

400

0

50

100

150 200 250 frequency [Hz]

300

350

400

30 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Active Suspension System - 4-Block Design

v1

V1 (z)

v2

V2 (z)

W2 (z)

w2

W1 (z)

w1

y −

K(z)

P (z)

[Hol et. al. 2003]

31 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Active Suspension System - 4-Block Design

v1

V1 (z)

v2

V2 (z)

−

K(z)

Pˆ (z)

W2 (z)

w2

W1 (z)

w1

y

[Hol et. al. 2003] Cont. gain of zero at 0.5Fs

⇒

Pˆ (z) = P (z) z+1 z 32 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Comparison Controller design approach Full order Balanced reduction Curved line search Cone complement. Hybrid Evol.-Algebr. Hybrid Evol.-Algebr. HIFOO

- continuous

HIFOO

- discrete

k

γ

27 5 5 5 5 2 5 2 1 5 2 1

2.476 3.405 2.506 2.630 2.589 2.596 2.611 2.473 2.611 2.466 2.470 2.611

Computation time load 11.9 s 12.8 s 4h23m45s 14m33s 2m42s 1m25s 22.3 s 41.4 s 20.6 s 1m33s 28.2 s 7.7 s

1 1.07 1329.00 73.36 13.61 7.16 11.69 21.68 10.82 51.87 14.75 4.17 33 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Comparison Controller design approach Full order Balanced reduction Curved line search Cone complement. Hybrid Evol.-Algebr. Hybrid Evol.-Algebr. HIFOO

- continuous

HIFOO

- discrete

k

γ

27 5 5 5 5 2 5 2 1 5 2 1

2.476 3.405 2.506 2.630 2.589 2.596 2.611 2.473 2.611 2.466 2.470 2.611

Computation time load 11.9 s 12.8 s 4h23m45s 14m33s 2m42s 1m25s 22.3 s 41.4 s 20.6 s 1m33s 28.2 s 7.7 s

1 1.07 1329.00 73.36 13.61 7.16 11.69 21.68 10.82 51.87 14.75 4.17 34 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Comparison Controller design approach Full order Balanced reduction Curved line search Cone complement. Hybrid Evol.-Algebr. Hybrid Evol.-Algebr. HIFOO

- continuous

HIFOO

- discrete

k

γ

27 5 5 5 5 2 5 2 1 5 2 1

2.476 3.405 2.506 2.630 2.589 2.596 2.611 2.473 2.611 2.466 2.470 2.611

Computation time load 11.9 s 12.8 s 4h23m45s 14m33s 2m42s 1m25s 22.3 s 41.4 s 20.6 s 1m33s 28.2 s 7.7 s

1 1.07 1329.00 73.36 13.61 7.16 11.69 21.68 10.82 51.87 14.75 4.17 35 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

sensitiv. [dB]

Performance

27th order

0 −5 −10

control sens. [dB]

Requir.

5

20 0 −20 −40 −60 −80

0

50

100

150

200

250

300

350

400

0

50

100

150 200 250 frequency [Hz]

300

350

400 36 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

sensitiv. [dB]

Performance 5

Requir.

0

1st order

−5 −10

control sens. [dB]

27th order

20 0 −20 −40 −60 −80

0

50

100

150

200

250

300

350

400

0

50

100

150 200 250 frequency [Hz]

300

350

400 37 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

sensitiv. [dB]

Performance 5

Requir.

0

1st order 2nd order

−5 −10

control sens. [dB]

27th order

20 0 −20 −40 −60 −80

0

50

100

150

200

250

300

350

400

0

50

100

150 200 250 frequency [Hz]

300

350

400 38 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Contents

1

Motivation

2

Discrete-time HIFOO

3

Active Suspension System

4

Conclusions

39 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Conclusions

Discrete-time fixed-structure H∞ synthesis Allows direct structural restrictions Better results than previous techniques

40 / 41

Motivation

Discrete-time HIFOO

Active Suspension System

Conclusions

Software HIFOO +d

http://bitbucket.org/andrey.popov/hifoo-d Discrete-time Replicated/repeated controller blocks:   K1 (z) K1 (z)  K2 (z) K(z) ∈  K2 (z)

redundant elements symmetric systems multi-agent systems [Popov & Werner, 2010] µ-synthesis [Apkarian, 2010]

features, not available in M ATLAB 2010b or HIFOO 3.0 GPL 41 / 41