On linear output regulation: data-driven, hybrid, optimal, and more! - - PowerPoint PPT Presentation

On linear output regulation: data-driven, hybrid, optimal, and more!

Sergio Galeani

joint work with

(data-driven) D. Carnevale, M. Sassano, A. Serrani HYBRID DYNAMICAL SYSTEMS: OPTIMIZATION, STABILITY AND APPLICATIONS, Trento, January 9-11, 2017

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Outline

1

The output regulation problem: Internal vs External Model

2

Data-driven External Model based regulators: the non hybrid case

3

An example for the non hybrid case

4

Conclusions

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Section 1 The output regulation problem: Internal vs External Model

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The output regulation problem

w

E

x

P

ξ

C

w u e

Problem (Output regulation)

Find, if possible, an error feedback compensator C such that the closed loop system is exponentially stable the output e is asymptotically regulated to zero Several flavours: full information vs. error feedback hybrid dynamics known vs unknown plant and/or exosystem dealing with fat plants: control allocation, optimal steady-state maps

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The output regulation problem

w

E

x

P

ξ

C

w u e

Problem (Output regulation)

Find, if possible, an error feedback compensator C such that the closed loop system is exponentially stable the output e is asymptotically regulated to zero Key observations:

1

when using error feedback, steady-state is in open-loop

2

C makes E unobservable from e

3

by linearity, stability and regulation can be studied separately

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Internal Model vs External Model

Internal Model

IM P0 E e u w K

External Model

EM P0 E Reset logic e u w K P

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Internal Model vs External Model

Internal Model

IM P0 E e u w K

External Model

EM P0 E Reset logic e u w K P

Problem (Output regulation via External Models for Unknown Plants)

Find, if possible, a data-driven compensator EM and Reset Logic such that the closed loop system is exponentially stable the output is asymptotically regulated to zero without any knowledge of the (pre-stabilized) plant P subject to arbitrary parameter uncertainties NOTE: a robust output regulation problem! Dynamics of (Feed-forward) compensator EM based on the Internal Model Principle!

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A look forward: three steps towards data-driven OR

EM P0 E Reset logic e u w K P

u = 0: learn the effect ˘ e of w

• n e

random reset of EM: learn the effect of u on ¯ e := e − ˘ e smart reset EM: achieve OR!

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Classic setting and Francis equations

LTI exosystem E: ˙ w = Sw LTI plant P0:

• ˙

x = Ax + Bu + Pw, e = Cx + Du + Qw IM P0 E e u w K

Assumptions

1

S is semi-simple with spec (S) = {0, ±ω1, . . . , ±ωr}

2

(A, B) stabilizable, (A, C) detectable

3

rank A − ωh B C D

• = n + p, h = 0, 1, . . . , r, satisfied by all (A, B, C, D)

For properly designed K, IM: Well defined steady-state solution: xss uss

• =

Π Γ

• w

Francis equations: (invariance and internal model) ΠS= AΠ + BΓ + P , 0= CΠ + DΓ + Q , ΣS= FΣ , Γ= HΣ , where (F, G, H, 0) are the matrices of the overall error feedback controller

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Section 2 Data-driven External Model based regulators: the non hybrid case

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External model - preliminary design

LTI exosystem E: ˙ w = Sw LTI plant P:

• ˙

x = Ax + Bu + Pw, e = Cx + Du + Qw LTI external model EM:

• ˙

xM = AMxM, AM = Ip ⊗ Am, yM = CMxM, CM = F(Ip ⊗ Cm), with (Am, Cm) observable and µAm(s) = µS(s)

EM P0 E Reset logic e u w K P Assumptions

1

S is semi-simple with spec (S) = {0, ±ω1, . . . , ±ωr}

2

spec (A) ⊂ Cg

3

rank A − ωh BF C DF

• = n + p, h = 0, 1, . . . , r, satisfied by all (A, B, C, D)

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External model - preliminary design

LTI exosystem E: ˙ w = Sw LTI plant P:

• ˙

x = Ax + Bu + Pw, e = Cx + Du + Qw LTI external model EM:

• ˙

xM = AMxM, AM = Ip ⊗ Am, yM = CMxM, CM = F(Ip ⊗ Cm), with (Am, Cm) observable and µAm(s) = µS(s)

EM P0 E Reset logic e u w K P

Ignoring the Reset logic: Well defined steady-state solution: xss uss

• =

ΠM CM

• xM +

Πw

• w

ΠMAM= AΠM + BCM , ΠwS= AΠw + P , Steady-state error: ess = MxM + Nw where M = CΠM + DCM, N = CΠw + Q e ≡ 0 iff xM(0) = Ψw(0) with Ψ such that ΨS = AMΨ 0= MΨ + N

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A coordinate transformation

For the interconnection of EM, E and P, change coordinates to expose: the offset z between x and its steady-state value xss: z = x − ΠMxM − Πww the offset η between xM and its ideal value Ψw: η = xM − Ψw

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A coordinate transformation

For the interconnection of EM, E and P, change coordinates to expose: the offset z between x and its steady-state value xss: z = x − ΠMxM − Πww the offset η between xM and its ideal value Ψw: η = xM − Ψw It can be easily computed that     ˙ w ˙ z ˙ η e     =     Sw Az AMη Cz + Mη    , since: ˙ z = ˙ x − ΠM ˙ xM − Πw ˙ w = (Ax + BCMxM + Pw) − ΠMAMxM − ΠwSw = Az + (AΠM + BCM − ΠMAM)xM + (AΠw + P − ΠwS)w = Az, ˙ η = ˙ xM − Ψ ˙ w = AMxM − ΨSw = AMη + (AMΨ − ΨS)w = AMη, e = · · · = Cz + Mη.

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A coordinate transformation

For the interconnection of EM, E and P, change coordinates to expose: the offset z between x and its steady-state value xss: z = x − ΠMxM − Πww the offset η between xM and its ideal value Ψw: η = xM − Ψw It can be easily computed that     ˙ w ˙ z ˙ η e     =     Sw Az AMη Cz + Mη    , since: ˙ z = ˙ x − ΠM ˙ xM − Πw ˙ w = (Ax + BCMxM + Pw) − ΠMAMxM − ΠwSw = Az + (AΠM + BCM − ΠMAM)xM + (AΠw + P − ΠwS)w = Az, ˙ η = ˙ xM − Ψ ˙ w = AMxM − ΨSw = AMη + (AMΨ − ΨS)w = AMη, e = · · · = Cz + Mη. Hence, for t large enough, it holds that e(t) ≈ Mη(t) Regulation with a single reset is achieved by “inverting” this relation!

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Inverting the error relation, when M is known

Let T, τ0 ∈ R>0 with τ0 ≪ T, and AMD = eAMT , AMI = e−AM τ0, AD = eAT , AI = e−Aτ0. For h ∈ N, define the extended error ˆ e(t) ˆ e(t) =     e(t) e(t − τ0) · · · e(t − hτ0)     = CDz(t − T) + MDη(t − T), CD :=     C CAI · · · CAh−1

I

    AD, MD :=     M MAMI · · · MAh−1

MI

    AMD, where limT→+∞ AD = limT→+∞ eAT = 0 and limT→+∞ CD = 0

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Inverting the error relation, when M is known

Let T, τ0 ∈ R>0 with τ0 ≪ T, and AMD = eAMT , AMI = e−AM τ0, AD = eAT , AI = e−Aτ0. For h ∈ N, define the extended error ˆ e(t) ˆ e(t) =     e(t) e(t − τ0) · · · e(t − hτ0)     = CDz(t − T) + MDη(t − T), CD :=     C CAI · · · CAh−1

I

    AD, MD :=     M MAMI · · · MAh−1

MI

    AMD, where limT→+∞ AD = limT→+∞ eAT = 0 and limT→+∞ CD = 0 Large T and t > T imply ˆ e(t) ≈ MDη(t − T) with MD left invertible, so that η(t − T) ≈ M♯

D ˆ

e(t), η(t) ≈ AMDη(t − T)

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Inverting the error relation, when M is known

Let T, τ0 ∈ R>0 with τ0 ≪ T, and AMD = eAMT , AMI = e−AM τ0, AD = eAT , AI = e−Aτ0. For h ∈ N, define the extended error ˆ e(t) ˆ e(t) =     e(t) e(t − τ0) · · · e(t − hτ0)     = CDz(t − T) + MDη(t − T), CD :=     C CAI · · · CAh−1

I

    AD, MD :=     M MAMI · · · MAh−1

MI

    AMD, where limT→+∞ AD = limT→+∞ eAT = 0 and limT→+∞ CD = 0 Large T and t > T imply ˆ e(t) ≈ MDη(t − T) with MD left invertible, so that η(t − T) ≈ M♯

D ˆ

e(t), η(t) ≈ AMDη(t − T) Reset xM(t+) := xM(t−) − AMDM♯

D ˆ

e(t) to achieve xM(t+) ≈ Ψw(t) since xM(t+) ≈ xM(t−) − η(t−) = xM(t−) − (xM(t−) + Ψw(t)) = Ψw(t) A single reset yields practical regulation, for known M

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Asymptotic regulation with M unknown

Let η[k] = η(kT), ηk = η(kT −), etc Define the reset rule for xM as xM,[k] = xM,k + Lˆ ek, where ˆ ek = CDz[k−1] + MDη[k−1] It can be shown that z[k+1] η[k+1]

• =

AD − ΠMLCD −ΠMLMD LCD AMD + LMD z[k] η[k]

• (1)

where limT→+∞ AD = 0 and limT→+∞ CD = 0

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Asymptotic regulation with M unknown

Let η[k] = η(kT), ηk = η(kT −), etc Define the reset rule for xM as xM,[k] = xM,k + Lˆ ek, where ˆ ek = CDz[k−1] + MDη[k−1] It can be shown that z[k+1] η[k+1]

• =

AD − ΠMLCD −ΠMLMD LCD AMD + LMD z[k] η[k]

• (1)

where limT→+∞ AD = 0 and limT→+∞ CD = 0

Proposition

As T → +∞, the spectrum of the matrix in (1) approaches the set {0} ∪ spec (AMD + LMD) Moreover, let ˆ MD be such that limT→+∞( ˆ MD − MD) = 0. If L is such that (AMD + L ˆ MD) is Schur, then there exists T ∗

1 > 0 such that for any T > T ∗ 1 the equilibrium (zeq, ηeq) = (0, 0)

• f (1) is globally exponentially stable.

A periodic reset yields asymptotic regulation, for unknown M

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Estimating M (when w is not there)

Consider a scalar input u, the system

• ˙

x = Ax + Bu, ¯ e = Cx + Du, and

• ˙

xm = Amxm, ym = Cmxm = u, Choose τ1 > 0 such that τ1 = 2π ωi − ωj , ∀i, j ∈ {0, 1, . . . , r}, where ω0 = 0 Choose z ∈ N, z ≥ q and define Ma(t) := ¯ e(t) ¯ e(t − τ1) ¯ e(t − 2τ1) · · · ¯ e(t − zτ1) , ˙ Mb(t) := AmMb(t), Mb(0) :=

• xm0

˜ A−1

m xm0

˜ A−2

m xm0

· · · ˜ A−z

m xm0

• ,

ˆ M(t) := Ma(t)M♯

b(t),

where ˜ Am := eAmτ1 and xm0 is such that (Am, xm0) is reachable Recalling that z := x − xss, it is easy to see that Mb(t) = xm(t) xm(t − τ1) xm(t − 2τ1) · · · xm(t − zτ1) Ma(t) = MMb(t) + C z(t) z(t − τ1) z(t − 2τ1) · · · z(t − zτ1)

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Estimating M (when w is not there)

Consider a scalar input u, the system

• ˙

x = Ax + Bu, ¯ e = Cx + Du, and

• ˙

xm = Amxm, ym = Cmxm = u, Choose τ1 > 0 such that τ1 = 2π ωi − ωj , ∀i, j ∈ {0, 1, . . . , r}, where ω0 = 0 Choose z ∈ N, z ≥ q and define Ma(t) := ¯ e(t) ¯ e(t − τ1) ¯ e(t − 2τ1) · · · ¯ e(t − zτ1) , ˙ Mb(t) := AmMb(t), Mb(0) :=

• xm0

˜ A−1

m xm0

˜ A−2

m xm0

· · · ˜ A−z

m xm0

• ,

ˆ M(t) := Ma(t)M♯

b(t),

where ˜ Am := eAmτ1 and xm0 is such that (Am, xm0) is reachable Recalling that z := x − xss, it is easy to see that Mb(t) = xm(t) xm(t − τ1) xm(t − 2τ1) · · · xm(t − zτ1) Ma(t) = MMb(t) + C z(t) z(t − τ1) z(t − 2τ1) · · · z(t − zτ1)

Proposition

For any εM > 0 there exists T ∗

3 > 0 such that if T > T ∗ 3 then ˆ

M(T) − M has norm less than εM. If u is not scalar, M is computed iteratively (columnwise)

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Offsetting the disturbance

Original plant and exosystem      ˙ w = Sw, ˙ x = Ax + Bu + Pw, e = Cx + Du + Qw, Equivalent plant and exosystem      ˙ w = Sw, S = Ip ⊗ Am, ˙ x = Ax + Bu, P = 0, e = Cx + Du + Qw, Q = Ip ⊗ Cm, The steady-state of e for u ≡ 0 is “matched” by the equivalent output disturbance generator: ˙ xed(t) = Aedxed(t) + δ(t)Led(e(t) − ˘ e(t)), Aed = Ip ⊗ Am, Led = Ip ⊗ Lm, ˘ e(t) = Cedxed(t), Ced = Ip ⊗ Cm, Lm : spec (Am − LmCm) ⊂ Cg, [chosen for fast decay of the estimation error] δ(t) =

• 0,

if maxt∈[t−T,t] |e(t) − ˘ e(t)| < εν, 1,

• therwise,

Generate the “(almost) disturbance free” output as ¯ e = e − ˘ e

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Offsetting the disturbance

Original plant and exosystem      ˙ w = Sw, ˙ x = Ax + Bu + Pw, e = Cx + Du + Qw, Equivalent plant and exosystem      ˙ w = Sw, S = Ip ⊗ Am, ˙ x = Ax + Bu, P = 0, e = Cx + Du + Qw, Q = Ip ⊗ Cm, The steady-state of e for u ≡ 0 is “matched” by the equivalent output disturbance generator: ˙ xed(t) = Aedxed(t) + δ(t)Led(e(t) − ˘ e(t)), Aed = Ip ⊗ Am, Led = Ip ⊗ Lm, ˘ e(t) = Cedxed(t), Ced = Ip ⊗ Cm, Lm : spec (Am − LmCm) ⊂ Cg, [chosen for fast decay of the estimation error] δ(t) =

• 0,

if maxt∈[t−T,t] |e(t) − ˘ e(t)| < εν, 1,

• therwise,

Generate the “(almost) disturbance free” output as ¯ e = e − ˘ e

Proposition

For any εν > 0 there exist T ∗

2 > 0 such that T > T ∗ 2 implies |e(t) − ˘

e(t)| < εν, for all t ≥ T.

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Section 3 An example for the non hybrid case

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A numerical example: data

Consider the plant described by the matrices: A =   −2 1 −1 −2 −2 −1   , B =   1 1 1   , C = 1 1 1 −1

• ,

D =

• ,

P =   1 1 1 −3 −3 −3   , Q = 2 1 1 1 −1

• ,

The exosystem is characterized by ω0 = 0, ω1 = π 4 , ω2 = 3π 5 and initial state q(0) = −1 1 −1 0′. None of these plant data is used for control design, except the values ωh, h = 0, 1, 2.

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A numerical example: practical regulation

Two Inputs, Two Outputs case with ε = 10−4. Evolution of the two regulated outputs (top) and control inputs (bottom).

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A numerical example: practical regulation

Single Input, Single Output case with ε = 10−4. Evolution of the first regulated output e1 (top) and first control input u1 (bottom).

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A numerical example: practical regulation

Two Inputs, Single Output case with ε = 10−4. Evolution of the first regulated output e1 (top) and two control inputs (bottom). Smaller input amplitudes are used (with respect to the SISO case).

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A numerical example: asymptotic regulation

Asymptotic regulation via periodic resets with ε = 0.5. Evolution of the regulated output e (top) and control input u (bottom). The state of the exosystem is reset after 38 time units.

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Conclusions Still lots of interesting (and fun!) things going on in linear regulation Data-driven external model based for hybrid output regulation: in CDC2016 Hybrid output regulation still largely to understand even in very simple cases

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