Switching-based Regulation of Uncertain Stable Linear Systems Affected by an Unknown Harmonic Disturbance

11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 604–609

Switching-based Regulation of Uncertain Switching-based Regulation of Uncertain Switching-based Regulation of Uncertain Stable Linear Systems Affected by an Switching-based Regulation of Uncertain Stable Linear Systems Affected by an Stable Linear Systems Affected by an Unknown Harmonic Disturbance Stable LinearHarmonic Systems Disturbance Affected by an Unknown Unknown Harmonic Disturbance Unknown Harmonic Disturbance ∗∗∗ Yang Wang ∗∗∗ Andrea Serrani ∗∗ ∗∗ ∗∗ Gilberto Pin ∗∗∗ ∗∗∗

Yang Wang ∗∗ Andrea Serrani ∗∗ Gilberto Pin ∗∗∗ ∗∗ ∗∗∗ ∗,∗∗∗∗ Yang Wang ∗ Thomas Andrea Parisini Serrani ∗∗ Gilberto Pin ∗∗∗ ∗,∗∗∗∗ ∗,∗∗∗∗ Thomas Parisini ∗,∗∗∗∗ Yang Wang Thomas Andrea Parisini Serrani ∗,∗∗∗∗ Gilberto Pin ∗ Thomas Parisini ∗,∗∗∗∗ ∗ Dept. of Electrical and Electronic Engineering, Imperial College ∗ Engineering, Imperial College Dept. of Electrical and Electronic ∗ ∗ Dept. of Electrical and Electronic Engineering, Imperial College London, UK (e-mail: yw1113@ imperial.ac.uk). London, UK (e-mail: yw1113@ imperial.ac.uk). ∗ ∗∗Dept. of Electrical and Electronic Engineering, Imperial College Dept. of Electrical and Computer Engineering, London, UK (e-mail: yw1113@ imperial.ac.uk). ∗∗ ∗∗ Dept. of Electrical and Computer Engineering, The The Ohio Ohio State State ∗∗ ∗∗ London, UK and (e-mail: yw1113@ imperial.ac.uk). University, Columbus OH, USA.(e-mail: [email protected] Dept. of Electrical Computer Engineering, The Ohio State University, Columbus OH, USA.(e-mail: [email protected] ∗∗ ∗∗∗ Dept. of Electrical and Computer Engineering, The Ohio State University, Columbus OH, USA.(e-mail: [email protected] Electrolux Italia S.p.A, Italy (e-mail: [email protected]) ∗∗∗ ∗∗∗ ElectroluxColumbus Italia S.p.A, Italy (e-mail: [email protected]) ∗∗∗∗ ∗∗∗ ∗∗∗ University, OH, USA.(e-mail: [email protected] of Engineering and Architecture, of Electrolux Italia S.p.A, Italy (e-mail:University [email protected]) ∗∗∗∗ ∗∗∗∗ Dept. of Engineering and Architecture, of Trieste, Trieste, Italy Italy ∗∗∗ ∗∗∗∗ Dept. ∗∗∗∗ Electrolux Italia S.p.A, Italy (e-mail:University [email protected]) (e-mail: [email protected]) Dept. of Engineering and Architecture, University of Trieste, Italy (e-mail: [email protected]) ∗∗∗∗ Dept. of Engineering [email protected]) Architecture, University of Trieste, Italy (e-mail: (e-mail: [email protected]) Abstract: The problem of rejection of a sinusoidal disturbance with unknown frequency acting Abstract: The problem of rejection of a sinusoidal disturbance with unknowninfrequency acting on an unknown single-input-single-output stable linear system is addressed this paper. We Abstract: The problem of rejection of a sinusoidal disturbance with unknowninfrequency acting stable linear system is addressed this paper. We on an unknown single-input-single-output Abstract: The problem of rejection of a sinusoidal disturbance unknown acting on an unknown single-input-single-output stable linear system with is the addressed infrequency this paper. present a new hybrid approach that does not require knowledge of frequency response of We the present a new hybrid approach that does not require knowledge of the frequency response of the on an unknown single-input-single-output stable linear system is the addressed in this paper. We transfer function over the range of frequencies of interest. The proposed controller reposes upon present a new hybrid approach that does not require knowledge of frequency response of the transfer afunction overapproach the rangethat of frequencies of interest. The proposed controller reposes upon present new hybrid does not require knowledge ofonthe the a switching strategy a family of linear controllers based thefrequency adaptiveresponse feedforward // transfer function overwithin the range of frequencies of interest. The proposed controller reposesof upon feedforward a switching strategy within a family of linear controllers based on the adaptive transfer function over the range of frequencies of interest. The proposed controller reposes upon ainternal switching strategy within a family of linear controllers based on the adaptive feedforward model control methodology. A dead-beat frequency estimation method is integrated/ internal model control methodology. A dead-beat estimation method is a strategy a family of controllers based the adaptive internal model control methodology. A linear dead-beat frequency estimation method is integrated integrated in switching the controller. Thewithin method also accounts for thefrequency presence ofon bounded sensorfeedforward noise as well/ in the controller. The method also accounts for the presence of bounded sensor noise as well internal model control methodology. A dead-beat frequency estimation method is integrated as imprecise frequency estimation. It is shown that, within a finite number of switchings, the in the controller. The method also accounts for the presence of bounded sensor noise as well as the imprecise frequency estimation. It is shown that, within a finite number of switchings, the in controller. method also It accounts forthat, the presence of bounded sensor noise as well regulation error isThe ultimately bounded a function of the norm of the noise that depends on as imprecise frequency estimation. isbyshown within a finite number of switchings, the of the noise that depends on regulation error is ultimately bounded by a function of the norm as frequency estimation. isbyshown that, within a finite switchings, regulation error ultimately bounded a function of the norm of number the noiseofthat dependsthe on theimprecise choice of theiscontroller and the It estimator gains. the choice of the controller and the estimator gains. regulation error ultimately bounded by a function the choice of theiscontroller and the estimator gains. of the norm of the noise that depends on © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. the choice of the controller and the estimator gains. Keywords: output regulation, disturbance rejection, switching algorithms, algorithms, active active noise noise control, control, rejection, switching Keywords: output regulation, disturbance Keywords: output feedforward controlregulation, disturbance rejection, switching algorithms, active noise control, feedforward controlregulation, disturbance rejection, switching algorithms, active noise control, Keywords: feedforwardoutput control feedforward control 1. PROBLEM FORMULATION for some known positive constants a ¯, ω ¯ and ω. 1. PROBLEM FORMULATION for some known positive constants a ¯, ω ¯ and ω. 1. PROBLEM FORMULATION for some known positive constants a ¯, ω ¯ and ω. The transfer function from u to y of system 1. PROBLEM FORMULATION for some known positive constants a ¯ , ω ¯ and (1) ω. is The problem of rejecting periodic disturbances of of unknown unknown The transfer function from u −1 to y of system (1) is denoted denoted The problem := C(µ)(sI − A(µ)) B(µ), where I denotes the by Wtransfer The function from u to y of system (1) is denoted −1 µ (s) The problem of rejecting periodic disturbances frequency, in of therejecting presenceperiodic of plantdisturbances uncertainty,ofisunknown a funda- by −1 B(µ), where I denotes (s) := C(µ)(sI − A(µ)) the W µ −1 µ (s)matrix. frequency, in the presence of uncertainty, is(see a fundaThe function from u −1 to yassumed of system (1) isinternally denoted := C(µ)(sI − A(µ)) B(µ), where Ibedenotes the by Wtransfer identity System (1) is to The problem periodic disturbances ofis unknown µ frequency, in of therejecting presence of plant plant uncertainty, mental issue in control theory and applications Patt µ a fundaidentity matrix. System (1) is assumed to be internally −1 mental issue in control theory and applications (see Patt (s) := C(µ)(sI − A(µ)) B(µ), where I denotes the by W identity matrix. System (1) is assumed to be internally stable, robustly with respect to µ ∈ P: µ frequency, in in the presence of is(see a fundamental issue control theory anduncertainty, applications et al. al. (2005); (2005); Esbrook et al.plant (2013); Basturk and and Krstic Patt stable, robustly with respect to µ ∈ P: et Esbrook et al. (2013); Basturk Krstic identity matrix. System (1) is assumed to be internally stable, robustly with respect to µ ∈ P: mental issue in control theory and applications (see Patt et al. (2005); Esbrook therein, et al. (2013); (2013) and references to citeBasturk but aa and few Krstic recent Assumption 2. There exist positive constants a1 , a2 and few recent 2. exist constants a11 ,, a 2 (2013) and therein, to cite but stable, robustly with respect to n×n 2 and et al. (2005); Esbrook et al. we (2013); Assumption 2. There There exist positive constants (2013) and references references therein, to citeBasturk but a and few recent a solution Px :positive Rppp µ→∈RP: contributions.) In this work, consider SISO LTIKrstic plant Assumption 1 a2 2 and n×n 0 such that the 1 n×n of the aLyapunov a such that the solution P : R → R of the Lyapunov 0 x contributions.) In this work, we consider SISO LTI plant p n×n T 0 x p n×n , a and Assumption 2. There exist positive constants a (2013) and references therein, cite but a few recent contributions.) In this work, we to consider equation P (µ)A(µ) + A (µ)P (µ) = −I satisfies a 1 models of the form a such that the solution P : R → R of the Lyapunov 0 x SISO LTI plant T x x 0 Tx equation Pxx (µ)A(µ) + AP =n×n −Iofsatisfies a2111 II ≤ ≤ models of the form p x T (µ)P x (µ) T a such that the solution : R → R the Lyapunov contributions.) In this work, we consider SISO LTI nplant models of the form P (µ) ≤ a I. Moreover, −Re{λ} ≥ a for all λ ∈ 0 x equation P (µ)A(µ) + A (µ)P (µ) = −I satisfies a I ≤ x x 1 x 2 0 x x 1 x˙ = A(µ)x + B(µ)[u(t) − d(t)] , x(0) = x0 ∈ Rnn P (µ) ≤ a I. Moreover, −Re{λ} ≥ a for all λ ∈ T x 2 0 x A(µ),P∀µ 2 ∈ P. + A (µ)Px (µ) = −I 0satisfies a1 I ≤ equation (µ)A(µ) models ofA(µ)x the form R x ˙˙ = + B(µ)[u(t) − d(t)] ,, x(0) = x 0 n spec

0 ∈ n P a x (µ) ≤ x 2 I. Moreover, −Re{λ} ≥ a0 0 for all λ ∈ x 2 ∈ R x = A(µ)x + B(µ)[u(t) − d(t)] x(0) = x 0 spec A(µ), ∀µ ∈ P. yy = C(µ)x , y = y + ν (1) 0 d ≤ ∀µ a2 I.∈ P. Moreover, −Re{λ} ≥ a0 for all λ ∈

= A(µ)x C(µ)x ,+ B(µ)[u(t) ydd = y + − ν d(t)] , x(0) = x0 ∈ Rn (1) P spec A(µ),

x (µ) x ˙y = n , = C(µ)x yddstate =y+ νan internally stable uncertain (1) As the disturbance is generated by the LTI exosystem is the of where x ∈ R spec A(µ), ∀µ ∈ P.

n n of νan internally stable uncertain where x∈ Rnn is, theydstate y= C(µ)x = ythe + (1) As the disturbance is∗ generated by the LTI2 exosystem state of an internally stable uncertain where x∈ R uis the plant model, ∈ R is control input, y ∈ R is the As the disturbance is generated by the LTI exosystem T w, w(0) = w ∈ R w ˙ = ω ∗ 2 0 plant ∈ Rstate is the control input, y ∈ uncertain R is the As the disturbance w, w(0)by =w R22 exosystem w˙ = ωis∗∗∗ Tgenerated the LTI 0 ∈ of an internally stable where model, x ∈ output, Rn u regulated R the plant model, uis the ∈ yyRd is∈ input, y output ∈ R is and the = Γw wd˙ = ω ∗ T w, w(0) = w000 ∈ R22 (3) regulated output, ∈ the R is iscontrol the measured measured output and d d plant model, u ∈ R is the control input, y ∈ R is the d = Γw (3) ν ∈ R is a bounded noise, satisfying regulated output, yadditive is the measured output and d ∈ R measurement wd˙ = = Γw ω T w, w(0) = w0 ∈ R d (3) νν(·) ∈ R is a bounded additive measurement noise, satisfying where regulated output, y∞ ∈ R is the measured output and ν ∈ R∞ is a≤bounded additive measurement noise, satisfying ν ¯ < for some given constant ν ¯ > 0. d where   d = Γw (3) ν(·) ≤ ν ¯ < ∞ for some given constant ν ¯ > 0. ∞ where ∞   ν ∈ R∞ is a(1) additive measurement noise, satisfying  0 1 System is a sinusoidal disturbance ν(·) ≤bounded ν¯ affected < ∞ forby some given constant ν¯ > (of 0. ∞     System (1) is affected by a sinusoidal disturbance (of 00 11 , Γ = 1 0 , T ν(·) ≤ ν¯ affected < ∞amplitude forbysome given constant ν¯ > (of 0. where unknown frequency, and ∞ (1) System is a sinusoidal T = = −1 , Γ= 1 0 , unknown frequency, amplitude and phase) phase)disturbance T = System (1) is affected by a sinusoidal disturbance (of ∗ 0 100 , Γ = 1 0 , unknown frequency, amplitude and −1 d(t) = a sin(ω φ ) (2) ∗ t +phase) 0 ∗ −1 0 T = , Γ = as1 follows: d(t) = a sin(ωand φ00 ) (2) the problem can 0 , unknown frequency, amplitude ∗ ∗ t +phase) be formally d(t) (2)3 p= a sin(ω t + φ0 ∗ 0) −1 0 stated the problem can be formally stated as follows: and µ , φ ) ∈ R The vectors µ ∈ R ∗ := col(a, ω ∗ p 3 d 0 p ∗ 3 a sin(ω + φcol(a, µd t:= ω ∗∗ , φ00 ) ∈ (2) R33 the The vectors µ ∈d(t) Rp=and problem can be formally stated as follows: 0) Problem 1. Regulation Problem) For collect the uncertain of the plant model 1. (Output (Output Regulation Problem) For system system (1), (1), and µddd := col(a, ω∗ , φ R3 Problem The vectors µ ∈ Rppparameters 0 0 ) ∈ and the problem can be formally stated as follows: collect the uncertain parameters of the plant model and design a dynamic output-feedback controller of the form Problem 1. (Output Regulation Problem) For system (1), and µIt := col(a, ωthat ,φ ) ranges ∈ and R Thedisturbance, vectors µ ∈ R parameters the respectively. assumed d is 0µ design a dynamic output-feedback controller of the form collect the uncertain of the plant model the disturbance, respectively. It isofassumed that µ ranges Problem 1. (Output Regulation Problem) For (1), m ofsystem p design a dynamic output-feedback controller the form ˙ collect the uncertain parameters the plant model and on given known compact set, ⊂ Rpp . Moreover, the ξ˙ = fa (ξ, yd ) , ξ(0) = ξ0 ∈ Rm the adisturbance, respectively. It isPassumed that µ ranges m of the form design a dynamic output-feedback controller on a given known compact set, P ⊂ R . Moreover, the ξ = f (ξ, y ) , ξ(0) = ξ ∈ R a d 0 m p the respectively. It isPassumed that µ ranges on adisturbance, given known compact set, ⊂ assumption: Rp . Moreover, the disturbance d(t) the following ξ˙ = h faaaa(ξ, ydd ) , ξ(0) = ξ00 ∈ Rm u (ξ, (4) disturbance d(t) satisfies satisfies the set, following assumption: p uξ˙ = = h (ξ, yyyddddd))) , ξ(0) = ξ00 ∈ Rm (4) a on a given known compact P ⊂ R . Moreover, the = f (ξ, a a disturbance d(t) satisfies the following assumption: ∗ Assumption 1. The unknown amplitude a and frequency u = h (ξ, y ) (4) a d a d such that, for all µ ∈ P and ω ∈ [ω, ω ¯ ], the trajectories ∗ Assumption 1. The unknown amplitude a and frequency ∗ disturbance d(t) satisfies the following assumption: ∗ ω ¯ ], the trajectories u all = hµ (ξ, yd )and (4) such that, for ∈ P ω ∈ [ω, ∗ Assumption 1. The unknown amplitude a and frequency of the disturbance d(t) are bounded respectively by ω asystem ∗(3) ∗ ∗ such that, for all µ ∈ P and ω ∈ [ω, ω ¯ ], the trajectories of the closed-loop (1), and (4) are bounded, the disturbance d(t) are amplitude bounded respectively by ω ∗ of the closed-loop system (1), ∗ of ∗(3) and (4) are bounded, Assumption 1. The unknown a and frequency ∗ of the disturbance d(t) are bounded respectively by ω such that, for all µsystem ∈ P and ω (3) ∈ [ω, ω ¯ ], sup thet→∞ trajectories 0 a ¯ < and the output satisfies lim ω ¯ |y(t)| of the (1), and bounded, 0≤ ≤a a≤ ≤ a ¯,, areω ωbounded <ω ω ∗∗∗ ≤ and theclosed-loop output of of the the plant plant satisfies lim(4) supare ≤ respectively ω ¯ |y(t)| ≤ ≤ d(t) by ω ∗ of the disturbance t→∞ t→∞ of (1), (3) and bounded, 0≤a≤a ¯, ω < ω ∗∗ ≤ ω andthe theclosed-loop output of system the plant satisfies lim(4) supare ¯ t→∞ t→∞ |y(t)| ≤ 0 ≤ a(International ≤a ¯, ω
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605

r(ν∞ ), where r(·) is a class-N function 1 that depends on the controller (4).  Remark 1.1. Assumption 1 implies that the initial condition w0 is taken over a compact set W ⊂ R2 . Without loss of generality, we take W to be an arbitrary, but fixed, compact set that is invariant under the flow of (3).

Notation Throughout the paper, λi (M ), i = 1, . . . , n, denotes eigenvalues of the matrix M ∈ Rn×n , whereas λmax (M ) and λmin (M ) denote the maximum and minimum eigenvalues of M , respectively. We denote with  ·  both the Euclidean vector norm and the corresponding induced matrix norm.

The majority of techniques proposed in the literature to solve the given problem assume prior knowledge and persistence of the sign of either the real part or the imaginary part of the frequency response (i.e. the sign of Wµ (jω) does not change) over the range of the frequencies of interest. This assumption is known as SPR-like conditions in the literature (Bodson et al. (1994); Marino and Tomei (2015).) Under the assumption that ω ∗ is known, the recent work Wang et al. (2016, 2018) have proposed a multiple-model adaptive control scheme and a switchedbased strategy, respectively, that remove the necessity of SPR-like conditions. In the presence of uncertainties on the frequency of the disturbance, Battistelli et al. (2017) proposes a robust switching solution for discrete-time linear system given a nominal plant model. Liu et al. (2009) and Marino and Tomei (2011) address similar problems under the assumption that the system is minimum-phase with a known relative degree, which is relaxed in Marino and Tomei (2017), but an SPR-like condition is still required.

2. STATE-SHARING MULTI-CONTROLLER

In this paper, we seek a novel switching-based solution to eliminate the need for SPR-like conditions in presence of model uncertainty for both the plant and the exosystem. A family of four certainty-equivalence candidate controllers is designed off-line, ensuring that at least one of the candidate controllers provides stability of the closedloop system. Given the availability of frequency estimate (possibly not equal to ω ∗ ), a switching logic is devised, which is capable of removing from the family of admissible controllers those that display destabilizing performance. It is shown that the switching sequence terminates in finite time, and the resulting controller is able to reject the disturbance completely if the frequency estimate is accurate. If not, the frequency estimation error will be corrected in finite time by employing the deadbeat method presented in Pin et al. (2017). A distinctive feature of this solution is that a reliable disturbance rejection (or attenuation in the presence of sensor noise) is achieved within a finite number of switchings. Another advantage of the presented approach is its modularity: the design of the frequency estimator can be completely decoupled from the design of the switching logic. In this paper, we employ a non-asymptotic estimator that provides finite time estimation, but any of many existing robust frequency estimation techniques can in principle be employed in the proposed architecture. This paper is organized as follows: In Section 2, we introduce the overall architecture of the supervisory switching control systems and the candidate controller family. Our main contribution, a state-norm-estimator-based switching control strategy, is presented and analyzed in Section 3. An illustrative example is provided in Section 4.

A class-N function r(·) : R+ →  R+ is non-negative, continuous and strictly increasing, but does not necessarily satisfy r(0) = 0.

1

The overall control architecture follows the general paradigm of supervisory control architecture proposed in Morse (1995). The measured output yd of the plant drives a bank of controllers, each one generating a candidate feedback signal dˆi , which is an estimate of the disturbance. The control signal applied to the plant is ˆ := dˆσ(t) (t) (5) u(t) = d(t) where σ : [0, ∞) → I is a piecewise-constant switching signal taking values in the index set of the family of the candidate controllers I := {1, 2, 3, 4}. The system that generates the switching signal is referred to as the supervisory system. The candidate controllers C i , i ∈ I, are designed as w ˆ˙ i = ω ˆT w ˆ i − kφi yd , w ˆ i (0) = w ˆ i ∈ R2 0

ˆi , dˆi = Γw

i∈I

(6)

where ω ˆ is a constant estimate of ω and φ ∈ R , i ∈ I, are constant vectors given by         1 0 −1 0 1 2 3 4 φ = ,φ = ,φ = ,φ = . (7) 0 −1 0 1 ∗

i

2

In Marino and Tomei (2015), it has been shown that one of the dynamic feedback controllers in (6) solves Problem 1 given a suitable controller gain k > 0 and an accurate frequency estimate, i.e. the estimation error ω ˜ := ω ˆ − ω ∗ = 0. In this paper, we will first show that at least one of candidate controllers in (6) is capable of stabilizing the closed-loop system even with a non-zero frequency estimation error. The dimension of the overall controller can be reduced by employing a shared-state parametrization. Consider a coordinate change w ˆci := Mci w ˆ i , i ∈ I, where

Mci := φi1 I − φi2 T and φij , j = 1, 2 represents the j-th element of φi . This yields the set of controllers (equivalent to (6)) w ˆ˙ c = ω ˆT w ˆc − kGc yd , w ˆc (0) = w ˆc0 ∈ R2

ˆc , i∈I (8) dˆi = ϕiT w Remark 2.1. Although the state-sharing form (8) is used for the implementation of the algorithm, in the forthcoming analysis for the sake of simplicity we will use its equivalent form (6). 3. STATE NORM ESTIMATOR-BASED SWITCHING We first present the development of a norm estimatorbased supervisor system that solves the following problem: Problem 2. (Stabilizing Problem) Given compact sets X ⊂ Rn , W ⊂ R2 , find a selection for the gain k > 0 and a design for the supervisory system such that for all ω ˆ ∈ [ω, ω ¯ ] trajectories of the closed-loop system(1), (3), (5) and

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(6), originating from any x0 ∈ X and w ˆ0i ∈ W are bounded and satisfy

START : Initialization J, $,̅ $% , &(0)

(a) limt→∞ |y(t)| = 0 if ν(·)∞ = 0 and |˜ ω | = 0;

ω |}) if ν(·)∞ = (b) lim supt→∞ |y(t)| ≤ r (max{ν∞ , |˜ 0 and/or |˜ ω | = 0.

Re-initialize $,̅

%$Update J, $,̅ $% Norm-Estimator

where r (·) is a class-N function that depends on the tuning parameters of the controller and the switching mechanism. 

$ ≤ $?̅

In the light of the setup of Problem 2, the results will be valid in a semi-global sense, that is, on the basis of an arbitrary (but fixed) choice of compact sets for the initial conditions of the plant, the controller and the exoystem.

Update &(>) No

Yes No

5 =5+1

$% ≤ .E ?

Update Frequency Estimation

Yes : reaches steady state

3.1 Switching Logic The proposed supervisor is a cascade connection of two subsystems: a scheduling logic Σm and a routing function β(·) : {1, 2, 3, · · · } → I. The output of the Σm subsystem m(·) : [0, +∞) → Z+ , termed the switching sequence, is a piecewise-constant signal to be determined. The routing function β(·) employed here is constructed to satisfy the revisitation property (Morse (1995)) β(σ) := mod(m + σ(0) − 1, 4) + 1, where σ(0) ∈ I is the initial selection for the active controller. The idea behind the switching logic is that the supervisory system keeps adjusting σ through the index set I along a pre-specified path β(σ) until the output yd is small in a suitable sense. The scheduling logic Σm exploits three auxiliary signals to make decisions: a state-norm estimator that generates the ¯ a performance piecewise-continuous switching threshold J, index signal J obtained through an auxiliary filter and a monotonic decreasing signal Jε that represents the normbound of the transient term. The flow chart of the switching logic Σm is given in Figure 1, where N > 1 and εy , εd , ε0 are constants will be determined later. After the algorithm is initiated, the performance index J evolves according to J˙ = −δJ + δy 2 (t), J(0) = 0. (9) d

where δ > 0 is a forgetting factor.

3.2 Generation of the switching threshold J¯ Let the sequence {Tm }∞ m=1 denote the set of time instants at which switching takes place, where it is assumed that the same controller is kept in the interval [Tm , Tm+1 ). Consider the changes of coordinates ζ := w ˆ σ − w and n×2 z := x − Π(µ)ζ, where Π(µ) ∈ R is the unique solution of the Sylvester equation ω ˆ Π(µ)T = A(µ)Π(µ) + B(µ)Γ. Then, the dynamics of the interconnection of the plant (1), the exosystem (3) and the active candidate controller (6) can be written as ζ˙ = ω ˆ T ζ − kφσ yd + ω ˜ T w , ζ(Tm ) = ζ0 ∈ R2 z˙ = A(µ)z + kΠ(µ)φσ yd + ω ˜ Π(µ)T w , z(Tm ) = z0 ∈ Rn yd = C(µ)z + ϑT (µ, ω ˆ )ζ + ν

(10)

Initialize clock: :;< 0=0, 43 ≔ = ,- ≤ ./ ?

3 For 0 ∈ [0, 4) Yes

5=0

No No

@ ≤ .- ? |A|

3 For 0 ∈ [0, 4)

Yes

BCD

Fig. 1. Flowchart of the switching logic. where

  ˆ ) := C(µ)Π(µ) = Re{Wµ (j ω ϑT (µ, ω ˆ )}, Im{Wµ (j ω ˆ )} .

Since the plant model is unknown, the parameter ϑ ∈ R2 is unknown, but assumed to range in a known set. Specifically, let the set Θ ⊂ R2 be the annular region defined, for given real numbers 0 < δ1 < δ2 , as   Θ := ϑ ∈ R2 | δ12 ≤ ϑ21 + ϑ22 ≤ δ22 (11) and consider the following assumption that replaces the typical SPR-like conditions : Assumption 3. The unknown parameter vector ϑ satisfies ϑ(µ, ω ˆ ) ∈ int Θ for all µ ∈ P and all ω ˆ ∈ [ω, ω ¯ ].  Define η := col(ζ, z) ∈ Rn+2 , and rewrite (10) as η˙ = Eσ η + ω ˜ Hw + kF ν (12) yd = Lη + ν with   ω ˆ T − kφσ ϑT −kφσ C Eσ = , kΠφσ ϑT A + kΠφσ C     T −φσ H := , F := Πφσ ΠT   and L := ϑT C . Note that the dependence of matrices on µ has been dropped for the sake of notational simplicity. For future use, let X denotes the maximum value of the matrix norm of matrix X(µ), defined by X := maxµ∈P ||X(µ)||. Two subsets Ik and Ik∗ , which depends on the choice of the gain k > 0 are defined as follows: (13) Ik := {σ ∈ I : Re{λmax (Eσ )} < 0} Ik∗ := {σ ∈ I : Re{λmax (Eσ )} ≤ −α(k)} (14)

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where α(k) is a class-K function to be determined. Note that Ik∗ ⊆ Ik ⊆ I for all k > 0.

Fix, arbitrarily, a positive constant r0 > 0, such that X := {x0 ∈ Rn : |x0 | ≤ r0 }. The following property of Lyapunov function candidates is instrumental in the ensuing analysis: Property 1. There exist a scalar k1 > 0 and constants c3 ≥ c2 > c1 > 0 such that the solution Po : (ϑ, ε, ω ˆ ) → R2×2 of the parametrized family of Lyapunov equations    T ˆ − ε ϑϑT + wT ˆ − ε ϑϑT Po = −ε ϑT ϑI (15) Po wT satisfies c1 I ≤ Po ≤ c2 I and Po  ≤ c3 for all (ϑ, ε, ω ˆ) ∈ Θ × (0, k1 ] × [ω, ω ¯ ].  The next lemma establishes the fact that at least one candidate controller defined in (6) solves Problem 2 for the closed-loop system (12) with a proper selection of k. Lemma 3. If Assumptions 1-3 hold, there exists a constant k ∗ > 0, such that for any k ∈ (0, k ∗ ] and ω ˆ ∈ [ω, ω ¯ ], Ik and kδ 2 1 Ik∗ are a non-empty sets with α(k) := min{ 8√2δ1 c , 16a }. 2 2 2

The proofs of Property 1 and Lemma 3 are omitted. For any k ∈ (0, k ∗ ] and for all ω ˆ ∈ [ω, ω ¯ ], if the active controller satisfies σ ∈ Ik , then by virtue of Lemma 3, the output of the plant satisfies yd (t) = yss (t) + ytr (t) + ν(t) for all t ∈ [Tm , Tm+1 ), where yss (t) is the steadystate response and ytr (t) is an exponentially decaying term representing the transient response. Remark 3.1. The strategy behind the proposed switching mechanism is to develop a threshold based on an upper bound of yd (t) under the assumption that σ ∈ Ik∗ . Then, the fact that the performance index J(t) violates the threshold J¯ implies that σ(t) ∈ / Ik∗ . On the other hand, boundedness of the output suggests a stabilizing property of the active controller. However, a pathological case may still exist where the active controller introduces a pair of eigenvalues on the imaginary axis, while boundedness is preserved. In that case, the output may still be bounded by the threshold, but the active controller is not a stabilizing one. We will show that this case can be ruled out by ˆ suitable checking on the input signal d. Recast (12) as a parallel interconnection of two subsystems  ˜ Hw, η1 (0) = 0 η˙ 1 = Eσ η1 + ω Σ1 : (16) y1 = Lη1  η˙ 2 = Eσ η2 + kF ν, η2 (0) = η(Tm ) (17) Σ2 : y2 = Lη2 + ν

with η(t, η(Tm )) = η1 (t, 0) + η2 (t, η(Tm )) and yd (t) = y1 (t) + y2 (t). Subsystem Σ1 generates the response of the plant with respect to w(t). Using integration by parts, one can easily obtain the output of Σ1 as (18) y1 (t) = y1,ss (t) + y1,tr (t) with   y1,ss (t) := −˜ ω L[Eσ2 + ω ∗ 2 I]−1 Eσ H + ω ∗ HT ) w(t) and  ˜ L[Eσ2 + ω ∗ 2 I]−1 Eσ eEσ (t−Tm ) H y1,tr (t) := ω  + ω ∗ eEσ (t−Tm ) HT w(t),

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where I is an identity matrix of suitable dimension. Thanks to Assumption 1 and Lemma 3, the terms y1,ss (t) and y1,tr (t) are norm-bounded by |y1,ss (t)| ≤ κ1 |˜ ω |, |y1,tr (t)| ≤ κ1 e−α(k)(t−Tm ) |˜ ω | (19) ¯ )¯ a and respectively, with κ1 = L κ0 (α)(Eσ H + H ω √ ω 2 + 2Eσ ) 2n/2 n + 2(¯ κ0 (α) := . α2n+4 The constant κ0 is the norm bound of the matrix ||[Eσ2 + ω ∗2 I]−1 ||, which can be determined using the results in Piazza and Politi (2002). The output y2 (t) is the forced response y2,f o (t) to the measurement noise ν, and it is bounded by  t eEσ (t−τ ) kF ν(τ )dτ + ν| |y2,f o (t)| = |L 0

≤ k||L|| ||F || || − Eσ−1 || eEσ t (e−Eσ t − 1)|ν|

≤ kL F || − Eσ−1 || (1 − eEσ t )¯ ν + ν¯ ≤ (κ2 + 1)¯ ν (20) for all t ≥ 0, where κ2 = kL F /α(k). The computation of κ2 makes use of the identity λmax (−Eσ−1 ) = 1/λmin (−Eσ ) ≤ 1/α(k). The free response y2,f r (t) from the initial state at time Tm satisfies |y2,f r (t)| = |LeEσ (t−Tm ) η(Tm )|

(21) ≤ L e−α(k)(t−Tm ) ||η(Tm )||. Recalling the coordinate change η = col(ζ, z), z = x − Πζ and ζ = w ˆ σ − w, one obtains ||η(Tm )|| ≤ ||ζ(Tm )|| + ||Π ζ(Tm )|| + ||x(Tm )||

a + ||ˆ ω σ(Tm ) (Tm )||) + ||x(Tm )||. (22) ≤ (Π + 1)(¯ All variables in (22) are available except for the last term ||x(Tm )||, which is estimated by the norm-estimator ˙ = − 1 ξ(t) + 2a2 2 (d(t) ˆ 2+a ¯2 ) (23) Σnorm : ξ(t) 2 B 2a2 where ξ(0) = 0 ∈ R and a ¯, a2 are defined in Assumption 1 and 2, respectively. Here, we restrict the initial condition to be zero just for simplicity of the analysis (Krichman et al. (2001).) Lemma 4. If Assumptions 1 and 2 hold and x0 ∈ X then the state of the plant is norm-bounded by ξ(t) a2 r0 − 2a1 t + e 2 , t≥0 (24) ||x(t)||2 ≤ a1 a1 where ξ(t) is given by (23).

Proof Consider the Lyapunov candidate V (x) = xT Px x, whose derivative along trajectories of (1) reads as ˆ − d(t)). V˙ (x) = −xT x + 2xT Px B(d(t)

Applying Young’s inequality, it follows that 1 ˆ 2 + d(t)2 ) V˙ (x) ≤ − xT x + 2||Px ||2 ||B||2 (d(t) 2 1 ˆ 2+a V (x) + 2a22 2B (d(t) ¯2 ). (25) ≤− 2a2 ˆ 2 +¯ ˙ 1 ξ from Substituting the identity 2a2 2 (d(t) a2 ) = ξ+ 2 B

(23) into (25), one obtains 1 (V (x) − ξ) V˙ (x) − ξ˙ ≤ − 2a2 which implies

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1

1

V (x) ≤ ξ(t) + e− 2a2 t (V (x0 ) − ξ(0)) ≤ ξ(t) + e− 2a2 t a2 r0 where the assumption x0 ∈ X has been used. Due to the fact that ||x(t)||2 ≤ V (t)/a1 , (24) follows. 

Substituting the value of ||x(Tm )|| given by (24) into (22), one obtains the norm-bound of y2,f r (t) as |y2,f r (t)| ≤ κ3 e

−α(k)(t−Tm )

with κ3 := L [(Π + 1)(¯ a + ||ˆ ω

σ(Tm−1 )

(Tm )||) +

1

(26) ( ξ(t) a1

+

a2 r0 − 2a2 Tm 12 ) a1 e (26), if σ ∈ Ik∗

]. In summary, by virtue of (19), (20) and then the output yd is norm-bounded as |yd (t)| ≤ |y1 (t)| + |y2 (t)| ¯ + (κ2 + 1)¯ ν + Jε (t) := y¯ (27) ≤ κ1 ω where Jε (t) is the norm bound of the transient terms ¯ + κ3 )e−α(k)(t−Tm ) ≥ |y1,tr | + |y2,f r | (28) Jε (t) := (κ1 ω Applying (27) to (9), one obtains the dynamics of J¯ as ¯ m ) = J0 ¯˙ = −δ J(t) ¯ + δ y¯2 (t), J(T (29) J(t) for t ∈ [Tm , Tm+1 ), where J0 > 0 can be chosen arbitrarily. 3.3 Development of thresholds ε0 , εy and εd Thanks to Lemma 3 and (9), the threshold (29) is valid for all the candidate controllers σ ∈ Ik∗ with k ∈ (0, k ∗ ]. Hence, there exists a positive integer m ¯ ≤ 4 such that the switching stops after Tm . As J (t) is an exponentially ¯ ε decaying signal, for any fixed ε0 there exists Tss > 0 such that Jε (t) ≤ ε0 , ∀t ≥ Tm ¯ + Tss . For practical implementation, we consider the system to have reached steady state when Jε is smaller than or equal to a given tolerance ε0 > 0. Using (19) and (20), it follows that if σ ∈ Ik∗ then the output of the plant reaches its steady state after t ≥ Tm ¯ + Tss and satisfies ω | + (κ2 + 1)¯ ν + ε0 . |yd (t)| ≤ |y1,ss (t)| + |y2,f o (t)| + ε0 ≤ κ1 |˜ Given ε0 , set εy := κ1 γ0 (¯ ν ) + (κ2 + 1)¯ ν + ε0 , (30) where γ0 (·) is a class-K function that quantifies the inaccuracy of the frequency estimate provided by the estimator 2 . If |yd (t)| ≤ εy does NOT hold for all t ∈ [Tm ¯ + Tss , Tm ¯ + Tss + T¯], 3 , two cases for the closed-loop system may hold: a) It is internally stable but |˜ ω | > γ0 (¯ ν );

b) It has a neutrally stable mode.

As shown in Fig. 1, we distinguish case a) and b) by ˆ on a time interval of length T¯ > 0 checking the value of d(t) seconds. This is possible due to the band-pass-filtering feature of the controllers in (6) stated in the following lemma (proof omitted due to space limitation): 2

In this study, we employ the deadbeat estimator of Pin et al. (2017). This type of estimator yields in finite time an accurate estimation of the frequency of a sinusoidal signal. In the presence of additive noise, the estimated frequency ω ˆ provided by the deadbeat estimator enters into a neighborhood of the true value ω ∗ in finite time and the frequency estimation error ω ˜ satisfies an asymptotic bound with respect to the magnitude of the noise. 3 T ¯ := 2N π/ω, N ≥ 1, is a multiple of the largest period of any sinusoidal signal with frequency satisfying Assumption 1.

Lemma 5. There exist a class-KL function γk (k, |˜ ω |), a class-K function γv (¯ ν ) and constant k ∗∗ ∈ (0, k ∗ ] such that ˆ ≤ γk (k, |˜ |d| ω |) + γv (¯ ν ), ∀t ≥ Tm (31) ¯ + Tss . for all k ∈ (0, k ∗∗ ] and all σ ∈ Ik .

ˆ ≤ ν )) + γv (¯ ν ). If the condition |d(t| Set εd := γk (k, γ0 (¯ εd for all t ≥ Tm ¯ + Tss is verified for a sufficiently small εd , then yss (t) is guaranteed to contain only one harmonic component with frequency equal to ω ∗ . Otherwise, case b) occurs. In that case (which is highly unlikely in practice), the controller that is currently active needs to be deselected and switched to the next one in the family of candidate controllers. Problem 2 is readily solved by the proposed switching mechanism with the selection k ∈ (0, k ∗∗ ]. However, if |yd (t)| > εy for t ≥ Tm ¯ + Tss , the output regulation problem (Problem 1) has not been solved yet. The next step is to obtain a new ω ˆ via the deadbeat estimator and repeat the stabilizing controller selection procedure. In this regard, it follows that if the estimator provides a bound on the estimation error of the form |˜ ω | ≤ γ0 (¯ ν ), then Problem 1 is solved with r(¯ v ) := εy = κ1 γ0 (¯ ν ) + (κ2 + 1)¯ ν + ε0 . Remark 3.2. It worth mentioning that, in absence of sensor noise, the ultimate bound of the output signal only depends on the value of ε0 , which, in theory, can be chosen arbitrarily small. Remark 3.3. The state-norm-estimator-based switching logic presented in this section is a pseudo pre-routed switching. The feature that distinguishes the proposed one from the pre-routed switching logic presented in Morse (1995) is that the proposed switching mechanism guarantees that the stabilizing controller will be selected within at most 4 switchings. According to Morse (1995), if the cardinality of I is large, the performance of the pre-routed switching will be significantly degrades. This is completely avoided in the proposed switching mechanism, thanks to the small cardinality 4 of I and the guaranteed termination of the switching sequence. 4. ILLUSTRATIVE EXAMPLE In this section, a numerical example is presented to demonstrate the effectiveness and robustness of the proposed method. Consider the stable non-minimum phase plant model described by the transfer function 2(s − 1) (32) W1 (s) = 2 s + 2s + 5 and disturbance signal given by d(t) = 5 sin(ω ∗ t) with ω ∗ = 3. The same setting of the controller gain, k = 0.25, has been used in all simulations. The Runge-Kutta integration method has been employed for all simulations with fixed sampling interval Ts = 10−4 s. The two frequency estimates used in the simulation are listed in Table 4, together with the corresponding frequency response parameter and set of stabilizing controllers. The initial frequency estimation error is |˜ ω | = 2 and the initial controller C 2 is not a stabilizing one. The additive noise ν(t) is selected as a random noise with uniform distribution 4 For supervisory systems with more than four candidate controllers, the same thresholds can be used but the transient performance may not necessarily improve.

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Frequency

ϑT

ω1 = 1 [rad/s]

(−0.2, 0.6)

ω2 = 3 [rad/s]

(0.85, −0.23)

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capable of attenuating the disturbance and confining the output of the plant to a small residual set, all with a finite number of switches.

Ik∗

{4}

{1,2}

REFERENCES

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Fig. 3. Filtered norm-bound of output (top), norm-bound ˆ of output (center) and norm-bound of input d(t) (bottom). within the interval [−0.05, 0.05]. The thresholds are chosen as εy = εd = 0.5 and ε0 = 1 × 10−2 .

The results of the simulations are shown in Fig. 2 and Fig. 3. After 2 switches the supervisor selects the stabilizing controller C 4 , and the closed-loop system reaches steady state in about 300 seconds. Then, the deadbeat estimator provides a frequency estimate with small mismatch, ω ˆ  3 at about t = 1000 seconds. For this new estimate, the family of stabilizing controller comprises C 1 and C 2 . The supervisory system is automatically reactivated, and asymptotic regulation of the plant output to zero is achieved with the new stabilizing controller C 1 . Under the effect of the estimation error, the proposed approach is

Basturk, H.I. and Krstic, M. (2013). Adaptive wave cancelation by acceleration feedback for ramp-connected air cushion-actuated surface effect ships. Automatica, 49(9), 2591–2602. Battistelli, G., Selvi, D., and Tesi, A. (2017). Robust switching control: Stability analysis and application to active disturbance attenuation. IEEE Transactions on Automatic Control, 62(12), 6369–6376. Bodson, M., Sacks, A., and Khosla, P. (1994). Harmonic generation in adaptive feedforward cancellation schemes. IEEE Transactions on Automatic Control, 39(9), 1939–1944. Esbrook, A., Tan, X., and Khalil, H.K. (2013). An indirect adaptive servocompensator for signals of unknown frequencies with application to nanopositioning. Automatica, 49(7), 2006–2016. Krichman, M., Sontag, E.D., and Wang, Y. (2001). Inputoutput-to-state stability. SIAM Journal on Control and Optimization, 39(6), 1874–1928. Liu, L., Chen, Z., and Huang, J. (2009). Parameter convergence and minimal internal model with an adaptive output regulation problem. Automatica, 45(5), 1306– 1311. Marino, R. and Tomei, P. (2011). An adaptive learning regulator for uncertain minimum phase systems with undermodeled unknown exosystems. Automatica, 47(4), 739–747. Marino, R. and Tomei, P. (2015). Output regulation for unknown stable linear systems. IEEE Transactions on Automatic Control, 60(8), 2213–2218. Marino, R. and Tomei, P. (2017). Hybrid adaptive multisinusoidal disturbance cancellation. IEEE Transactions on Automatic Control, 62(8), 4023–4030. Morse, A.S. (1995). Control using logic-based switching. In Trends in Control. A European Perspective, 69–113. Springer Verlag, London, UK. Patt, D., Liu, L., Chandrasekar, J., Bernstein, D.S., and Friedmann, P.P. (2005). Higher-harmonic-control algorithm for helicopter vibration reduction revisited. Journal of Guidance, Control, and Dynamics, 28(5), 918– 930. Piazza, G. and Politi, T. (2002). An upper bound for the condition number of a matrix in spectral norm. Journal of Computational and Applied Mathematics, 143(1), 141–144. Pin, G., Chen, B., and Parisini, T. (2017). Robust finitetime estimation of biased sinusoidal signals: A Volterra operators approach. Automatica, 77, 120 – 132. Wang, Y., Pin, G., Serrani, A., and Parisini, T. (2018). Switching-based sinusoidal disturbance rejection for uncertain stable linear systems. In 2018 Annual American Control Conference (ACC), 4502–4507. Wang, Y., Pin, G., Serrani, A., and Parisini, T. (2016). Removing SPR-like conditions in adaptive feedforward control of uncertain systems. In Proceedings of the 55th IEEE Conference on Decision and Control CDC 2016, 4728–4733. Las Vegas, NV.

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