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 Sergio Galeani Linear Output Regulation OptHySYS 1 / 19
Outline The output regulation problem: Internal vs External Model 1 Data-driven External Model based regulators: the non hybrid case 2 An example for the non hybrid case 3 Conclusions 4 Sergio Galeani Linear Output Regulation OptHySYS 2 / 19
Section 1 The output regulation problem: Internal vs External Model Sergio Galeani Linear Output Regulation OptHySYS 3 / 19
The output regulation problem E w w u e C P ξ x 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 Sergio Galeani Linear Output Regulation OptHySYS 4 / 19
The output regulation problem E w w u e C P ξ x 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: when using error feedback, steady-state is in open-loop 1 C makes E unobservable from e 2 by linearity, stability and regulation can be studied separately 3 Sergio Galeani Linear Output Regulation OptHySYS 4 / 19
Internal Model vs External Model Internal Model External Model E E w w P u e u e E M P 0 K I M P 0 K Reset logic Sergio Galeani Linear Output Regulation OptHySYS 5 / 19
Internal Model vs External Model Internal Model External Model E E w w P u e u e E M P 0 K I M P 0 K Reset logic Problem (Output regulation via External Models for Unknown Plants) Find, if possible, a data-driven compensator E M 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 E M based on the Internal Model Principle! Sergio Galeani Linear Output Regulation OptHySYS 5 / 19
A look forward: three steps towards data-driven OR E w P u e E M P 0 K Reset logic u = 0: learn the effect ˘ e of w on e random reset of E M : learn the effect of u on ¯ e := e − ˘ e smart reset E M : achieve OR! Sergio Galeani Linear Output Regulation OptHySYS 6 / 19
Classic setting and Francis equations E LTI exosystem E : ˙ w = Sw w � x = Ax + Bu + Pw , ˙ u e K I M P 0 LTI plant P 0 : e = Cx + Du + Qw Assumptions S is semi-simple with spec ( S ) = { 0 , ± ω 1 , . . . , ± ω r } 1 ( A , B ) stabilizable, ( A , C ) detectable 2 � A − ω h � B = n + p , h = 0 , 1 , . . . , r , satisfied by all ( A , B , C , D ) 3 rank C D For properly designed K , I M : � x ss � � Π � Well defined steady-state solution: = w u ss Γ Francis equations: (invariance and internal model) Π S = A Π + B Γ + P , Σ S = F Σ , 0= C Π + D Γ + Q , Γ= H Σ , where ( F , G , H , 0) are the matrices of the overall error feedback controller Sergio Galeani Linear Output Regulation OptHySYS 7 / 19
Section 2 Data-driven External Model based regulators: the non hybrid case Sergio Galeani Linear Output Regulation OptHySYS 8 / 19
External model - preliminary design E LTI exosystem E : ˙ w = Sw w � x = Ax + Bu + Pw , ˙ P LTI plant P : u e e = Cx + Du + Qw E M P 0 � x M = A M x M , ˙ A M = I p ⊗ A m , K LTI external model E M : y M = C M x M , C M = F ( I p ⊗ C m ) , Reset with ( A m , C m ) observable and µ A m ( s ) = µ S ( s ) logic Assumptions S is semi-simple with spec ( S ) = { 0 , ± ω 1 , . . . , ± ω r } 1 spec ( A ) ⊂ C g 2 � A − ω h � BF rank = n + p , h = 0 , 1 , . . . , r , satisfied by all ( A , B , C , D ) 3 C DF Sergio Galeani Linear Output Regulation OptHySYS 9 / 19
External model - preliminary design E LTI exosystem E : ˙ w = Sw w � x = Ax + Bu + Pw , ˙ P LTI plant P : u e e = Cx + Du + Qw E M P 0 � x M = A M x M , ˙ A M = I p ⊗ A m , K LTI external model E M : y M = C M x M , C M = F ( I p ⊗ C m ) , Reset with ( A m , C m ) observable and µ A m ( s ) = µ S ( s ) logic Ignoring the Reset logic : � x ss � � Π M � � Π w � Well defined steady-state solution: = x M + w u ss C M 0 Π M A M = A Π M + BC M , Π w S = A Π w + P , Steady-state error: e ss = Mx M + Nw where M = C Π M + DC M , N = C Π w + Q e ≡ 0 iff x M (0) = Ψ w (0) with Ψ such that Ψ S = A M Ψ 0= M Ψ + N Sergio Galeani Linear Output Regulation OptHySYS 9 / 19
A coordinate transformation For the interconnection of E M , E and P , change coordinates to expose: the offset z between x and its steady-state value x ss : z = x − Π M x M − Π w w the offset η between x M and its ideal value Ψ w : η = x M − Ψ w Sergio Galeani Linear Output Regulation OptHySYS 10 / 19
A coordinate transformation For the interconnection of E M , E and P , change coordinates to expose: the offset z between x and its steady-state value x ss : z = x − Π M x M − Π w w the offset η between x M and its ideal value Ψ w : η = x M − Ψ w w ˙ Sw z ˙ Az It can be easily computed that = , since: η ˙ A M η e Cz + M η z = ˙ ˙ x − Π M ˙ x M − Π w ˙ w = ( Ax + BC M x M + Pw ) − Π M A M x M − Π w Sw = Az + ( A Π M + BC M − Π M A M ) x M + ( A Π w + P − Π w S ) w = Az , η = ˙ ˙ x M − Ψ ˙ w = A M x M − Ψ Sw = A M η + ( A M Ψ − Ψ S ) w = A M η, e = · · · = Cz + M η. Sergio Galeani Linear Output Regulation OptHySYS 10 / 19
A coordinate transformation For the interconnection of E M , E and P , change coordinates to expose: the offset z between x and its steady-state value x ss : z = x − Π M x M − Π w w the offset η between x M and its ideal value Ψ w : η = x M − Ψ w w ˙ Sw z ˙ Az It can be easily computed that = , since: η ˙ A M η e Cz + M η z = ˙ ˙ x − Π M ˙ x M − Π w ˙ w = ( Ax + BC M x M + Pw ) − Π M A M x M − Π w Sw = Az + ( A Π M + BC M − Π M A M ) x M + ( A Π w + P − Π w S ) w = Az , η = ˙ ˙ x M − Ψ ˙ w = A M x M − Ψ Sw = A M η + ( A M Ψ − Ψ S ) w = A M η, e = · · · = Cz + M η. Hence, for t large enough, it holds that e ( t ) ≈ M η ( t ) Regulation with a single reset is achieved by “inverting” this relation! Sergio Galeani Linear Output Regulation OptHySYS 10 / 19
Inverting the error relation, when M is known Let T , τ 0 ∈ R > 0 with τ 0 ≪ T , and A MD = e A M T , A MI = e − A M τ 0 , A D = e AT , A I = e − A τ 0 . For h ∈ N , define the extended error ˆ e ( t ) e ( t ) e ( t − τ 0 ) e ( t ) = ˆ = C D z ( t − T ) + M D η ( t − T ) , · · · e ( t − h τ 0 ) C M CA I MA MI C D := A D , M D := A MD , · · · · · · CA h − 1 MA h − 1 I MI where lim T → + ∞ A D = lim T → + ∞ e AT = 0 and lim T → + ∞ C D = 0 Sergio Galeani Linear Output Regulation OptHySYS 11 / 19
Inverting the error relation, when M is known Let T , τ 0 ∈ R > 0 with τ 0 ≪ T , and A MD = e A M T , A MI = e − A M τ 0 , A D = e AT , A I = e − A τ 0 . For h ∈ N , define the extended error ˆ e ( t ) e ( t ) e ( t − τ 0 ) e ( t ) = ˆ = C D z ( t − T ) + M D η ( t − T ) , · · · e ( t − h τ 0 ) C M CA I MA MI C D := A D , M D := A MD , · · · · · · CA h − 1 MA h − 1 I MI where lim T → + ∞ A D = lim T → + ∞ e AT = 0 and lim T → + ∞ C D = 0 Large T and t > T imply ˆ e ( t ) ≈ M D η ( t − T ) with M D left invertible, so that η ( t − T ) ≈ M ♯ D ˆ e ( t ) , η ( t ) ≈ A MD η ( t − T ) Sergio Galeani Linear Output Regulation OptHySYS 11 / 19
Inverting the error relation, when M is known Let T , τ 0 ∈ R > 0 with τ 0 ≪ T , and A MD = e A M T , A MI = e − A M τ 0 , A D = e AT , A I = e − A τ 0 . For h ∈ N , define the extended error ˆ e ( t ) e ( t ) e ( t − τ 0 ) e ( t ) = ˆ = C D z ( t − T ) + M D η ( t − T ) , · · · e ( t − h τ 0 ) C M CA I MA MI C D := A D , M D := A MD , · · · · · · CA h − 1 MA h − 1 I MI where lim T → + ∞ A D = lim T → + ∞ e AT = 0 and lim T → + ∞ C D = 0 Large T and t > T imply ˆ e ( t ) ≈ M D η ( t − T ) with M D left invertible, so that η ( t − T ) ≈ M ♯ D ˆ e ( t ) , η ( t ) ≈ A MD η ( t − T ) Reset x M ( t + ) := x M ( t − ) − A MD M ♯ e ( t ) to achieve x M ( t + ) ≈ Ψ w ( t ) since D ˆ x M ( t + ) ≈ x M ( t − ) − η ( t − ) = x M ( t − ) − ( x M ( t − ) + Ψ w ( t )) = Ψ w ( t ) A single reset yields practical regulation, for known M Sergio Galeani Linear Output Regulation OptHySYS 11 / 19
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