This is an up-dated addendum/erratum to the second edition of the book Dissipative Systems Analysis and Control, Theory and Applications, Springer-Verlag London, 2nd Edition, 2007

Brogliato, Bernard

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HAL Id: hal-01338369https://hal.inria.fr/hal-01338369Submitted on 28 Jun 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Dissipative Systems Analysis and Control, Theory andApplications: Addendum/ErratumBernard BrogliatoTo cite this version:Bernard Brogliato. Dissipative Systems Analysis and Control, Theory and Applications: Adden-dum/Erratum. [Research Report] INRIA Grenoble - Rhone-Alpes. 2016. ￿hal-01338369￿Addendum-Erratum to Dissipative Systems Analysis and Control,Springer-Verlag London, 2nd Edition, 2007(the first figure indicates the page number)• Notation section: λ(A): an eigenvalue of A ∈ IRn×n.• Notation section: σ(A): the set of eigenvalues of A ∈ IRn×n (i.e. the spectrum ofA).• 17, 18, 28: line 2 after (2.22), and also in (2.24), (2.29), (2.30), line 3 in the equationabove (2.75): change uT (t) to ut(τ), and uT (jω) to ut(jω).• 17: line 8, the integration is done over τ ∈ [0, t]• 32: obviously the expression for P (ωe) under (2.95) is meaningless. Please contactProf. O. Egeland for details.• 59, lemma 2.55: rational• 81, Lemma 3.11: µmin(A) denotes the minimum eigenvalue of the symmetrizedmatrix 12(A+ AT ).• 81: σmin(B) > 0 is equivalent to Ker(B) = {0}, and to m < n and B of full rank m.• 81, last line: it is H(jω − µ), not T (jω − µ)• 89 line 6: it is H(jω), not T (jω)• 88, line -3: it is H(jω), not T (jω)• 101: in the framed set of implications/equivalences, the last implication has to bereversed and in the last line this is P = P T ≥ 0 (hence the last condition is reallysufficient for PRness as indicated in the paragraph in between the two framed sets).Reference [444] is also worth looking at.• 119: in (3.134) it is∂Vf∂x(x)[Ax+ Bu] + w(x, u) ≥ 0• 121, line 5 after remark 3.38: Thus H(−jω, jω) is the condition....• 123, left hand side of (3.148): P is G• 124, line 2: P is G• 125: line 5 after Theorem 3.44: numerically• Section 3.9 (Lur’e problem and absolute stability): it’s also worth reading the surveyby R. Shorten et al, ”Stability Criteria for Switched and Hybrid Systems”, SIAMreview, vol.49, no 4, pp.545-592, 2007, and in particular section 6 therein. TheirTheorem 6.2 is close to Theorem 2.49 page 57 in the book.• 147, example 3.75: it is Ax = 0 if x < 0. The notation x+ means max(0, x).• 149, the dissipativity of linear complementarity systems is throughly investigatedin Camlibel, Iannelli, Vasca, ”Passivity and complementarity”, Mathematical Pro-gramming A, 2013, DOI: 10.1007/s10107-013-0678-4 . Closely related results con-cerning Lur’e set-valued systems are in Brogliato and Goeleven, ”Existence, unique-ness of solutions and stability of nonsmooth multivalued Lur’e dynamical systems”,Journal of Convex Analysis, vol.20, no 3, 2013, and ”Well-posedness, stability andinvariance results for a class of multivalued Lur’e dynamical systems”, NonlinearAnalysis: Theory, Methods and Applications, vol.74, pp.195-212, 2011.• 167, in Remark 3.96 this is [0,∞) (see Theorem 3.91)• 175, preservation of dissipativity after time-discretization has recently been tackledin S. Greenhalg, V. Acary, B. Brogliato, ”Preservation of the dissipativity propertiesof a class of nonsmooth dynamical systems with the (θ, γ)-algorithm”, NumerischeMathematik, 125(4), 2013. See several references that concern this issue in this arti-cle. The issue that is tackled concerns the preservation of the strorage function, thesupply rate, and the dissipation function, and therefore yields much more stringentconditions than the mere preservation of dissipativity (with possible different supplyrate, dissipation function after discretization).• 170, about the Popov’s line. In the book by Aiserman and Gantmacher from 1965it is pointed outy that Popov criterion also holds in the case of a negative slope 1r.• 183, Fact 8: the inequality is strict in (1 ≤ p < +∞), because obviously this doesnot work for the infinity norm (take f(·) as a constant non zero function).• 183: Fact 7: in the proof, we compute the smallest lower-bound.• 192, in (4.18) all T are t• 197: second line of definition 4.26: u(t) ∈ U .• 197-198: the role of the additive constants β which basically accounts for initialstored energy, has also been investigated in D.H. Hill: ’Dissipative nonlinear systems:basic properties and stability analysis”, Proc. of IEEE CDC, pp.3259-3264, 16-18December 1992.• 204, Theorem 4.43: It is supposed that there exists a storage function V (·) suchthat V (x⋆) = 0, so that Vr(x⋆) = 0. From the definition of the required supplyin Definition 4.36, this also assumes that the system is reachable from x⋆. This isTheorem 2 in [510].• 202: more on reversibility and its relationship with reciprocity, for LTI systems, isin [511,sections 8 and 9]. In particular Theorem 8 of [51] provides a way to checkreversibility.• 209 and 210: in (4.45) the last term is yT (s)Su(s), and same mistake in line 4 afterTheorem 4.56.• 230 and 116: Lemma 3.36 page 116, and Lemma 4.91 page 230, are taken fromthe book referenced 145 by Faurre, Clerget and Germain. In fact Lemma 3.36 ispresented in the book by Faurre et al as a corollary of Lemma 4.91.• 230, Lemma 4.91: in (4.94) this is V (t, x).• p.247, line 10: alll should be all.• 248 and 249, replace all (4.108) by (4.163)• 251, Theorem 4.111, line 2: IRm×m• Section 5.1: Most of the results presented in this section have been stated in twoarticles by D.J. Hill and P. Moylan: ”Stability results for nonlinear feedback sys-tems”, Automatica, vol.13, pp.377-382, 1977, and ”General instability results forinterconnected systems”, SIAM J. Control and Optimization, vol.21, no 2, pp.256-279, 1983. The errata for these papers include the following. In the SIAM paper,the first condition that appears in equation (12), Theorem 7, is redundant. In theAutomatica article, condition (iii) of Theorem 4 can be removed, while condition(ii) should be stated with TĀ(·) a nonnegative real valued function.• 254, 255: in (4.185) replace ∂u∂xin the first term of the left-hand side by ∂u∂t. Lineafter lemma 4.115: do the same for the equality after “One has”, and delete the“is”. Also the power 2 is missing in the integrand. In (4.188) multiply the first termin the left-hand side by 12. These two Lemmas are taken from H. Brézis, AnalyseFonctionnelle, Théorie et Applications, Masson, Paris, 1983, sections X.1 and X.3.• 267: from Lemma 5.13 and Corollary 3.4 it may be deduced that the solutionsP = P T of the KYP Lemma LMI, are > 0 if (C,A) is observable.• 298: in Theorem 5.68, this is ”...with input u(·)...”• 305: line before section 5.9.4: Bounded• 306: Theorem 5.71 is taken from M.R. James et al, SIAM J. Control Optimization,vol.43, no 5, pp.1535-1582, 2005.• 339: In definition 6.37: a symmetric positive definite matrix R(x); in equation(6.95), this is (J(x)−R(x))∂H0∂x+ ....• 366: generally speaking BV functions do not satisfy that for any t, there exists aσ such the function is continuous on [t, t + σ). This is because at t a BV functionmay possess an accumulation of jumps on the left, and an accumulation of jumpson the right. But in mechanical systems with frictionless unilateral constraints andpiecewise analytic data, accumulations of impacts on the right do not exist (result ofBallard in 2000). So in this particular case the generalized velocity has this property.• 368: line -8, it is −∂ψV (x).• 382: ξ in the line after (7.10) refers to the ξ in (6.184)• section 3.9.4: Notice that for a maximal monotone operator F : IRn → IRn, x 7→y = F (x), with F = ∂f where f(·) is convex, proper, lower semi continuous, thenthe “input-output” product satisfies xTy = f(x) + f ∗(y), where f ∗(·) is the dualfunction of f(·).• 409: CTCE means Cross Terms Cancellation Equality• 431 line 2 section 7.8.3: (6.162) is (6.156), and λz1 is as in (6.169)• 133, line -2 before (3.176): Pe• 426, line 6: actuator dynamics• 431, line 1, (7.169): λzd is λd• 531: Theorem A.61 extends to positive semidefinite matrices, see proposition 8.2.3in [Bernstein 2005, Matrix Mathematics].• 533, Lemma A.65: the last condition is S(I − R†R) = 0 or equivalently (I −RR†)ST = 0.• 536, line 11 in the proof of Lemma A.71: realization• 564, reference [470]: 1992

Dissipative Systems Analysis and Control, Theory and Applications: Addendum/Erratum

Hal - Université Grenoble Alpes

Dissipative Systems Analysis and Control, Theory andApplications: Addendum/ErratumBernard BrogliatoTo cite this version:Bernard Brogliato. Dissipative Systems Analysis and Control, Theory and Applications: Ad-dendum/Erratum. [Research Report] INRIA Grenoble - Rhone-Alpes. 2016. <hal-01338369>HAL Id: hal-01338369https://hal.inria.fr/hal-01338369Submitted on 28 Jun 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Addendum-Erratum to Dissipative Systems Analysis and Control,Springer-Verlag London, 2nd Edition, 2007(the first figure indicates the page number)• Notation section: λ(A): an eigenvalue of A ∈ IRn×n.• Notation section: σ(A): the set of eigenvalues of A ∈ IRn×n (i.e. the spectrum ofA).• 17, 18, 28: line 2 after (2.22), and also in (2.24), (2.29), (2.30), line 3 in the equationabove (2.75): change uT (t) to ut(τ), and uT (jω) to ut(jω).• 17: line 8, the integration is done over τ ∈ [0, t]• 32: obviously the expression for P (ωe) under (2.95) is meaningless. Please contactProf. O. Egeland for details.• 59, lemma 2.55: rational• 81, Lemma 3.11: µmin(A) denotes the minimum eigenvalue of the symmetrizedmatrix 12(A+ AT ).• 81: σmin(B) > 0 is equivalent to Ker(B) = {0}, and to m < n and B of full rank m.• 81, last line: it is H(jω − µ), not T (jω − µ)• 89 line 6: it is H(jω), not T (jω)• 88, line -3: it is H(jω), not T (jω)• 101: in the framed set of implications/equivalences, the last implication has to bereversed and in the last line this is P = P T ≥ 0 (hence the last condition is reallysufficient for PRness as indicated in the paragraph in between the two framed sets).Reference [444] is also worth looking at.• 119: in (3.134) it is∂Vf∂x(x)[Ax+ Bu] + w(x, u) ≥ 0• 121, line 5 after remark 3.38: Thus H(−jω, jω) is the condition....• 123, left hand side of (3.148): P is G• 124, line 2: P is G• 125: line 5 after Theorem 3.44: numerically• Section 3.9 (Lur’e problem and absolute stability): it’s also worth reading the surveyby R. Shorten et al, ”Stability Criteria for Switched and Hybrid Systems”, SIAMreview, vol.49, no 4, pp.545-592, 2007, and in particular section 6 therein. TheirTheorem 6.2 is close to Theorem 2.49 page 57 in the book.• 147, example 3.75: it is Ax = 0 if x < 0. The notation x+ means max(0, x).• 149, the dissipativity of linear complementarity systems is throughly investigatedin Camlibel, Iannelli, Vasca, ”Passivity and complementarity”, Mathematical Pro-gramming A, 2013, DOI: 10.1007/s10107-013-0678-4 . Closely related results con-cerning Lur’e set-valued systems are in Brogliato and Goeleven, ”Existence, unique-ness of solutions and stability of nonsmooth multivalued Lur’e dynamical systems”,Journal of Convex Analysis, vol.20, no 3, 2013, and ”Well-posedness, stability andinvariance results for a class of multivalued Lur’e dynamical systems”, NonlinearAnalysis: Theory, Methods and Applications, vol.74, pp.195-212, 2011.• 167, in Remark 3.96 this is [0,∞) (see Theorem 3.91)• 175, preservation of dissipativity after time-discretization has recently been tackledin S. Greenhalg, V. Acary, B. Brogliato, ”Preservation of the dissipativity propertiesof a class of nonsmooth dynamical systems with the (θ, γ)-algorithm”, NumerischeMathematik, 125(4), 2013. See several references that concern this issue in this arti-cle. The issue that is tackled concerns the preservation of the strorage function, thesupply rate, and the dissipation function, and therefore yields much more stringentconditions than the mere preservation of dissipativity (with possible different supplyrate, dissipation function after discretization).• 170, about the Popov’s line. In the book by Aiserman and Gantmacher from 1965it is pointed outy that Popov criterion also holds in the case of a negative slope 1r.• 183, Fact 8: the inequality is strict in (1 ≤ p < +∞), because obviously this doesnot work for the infinity norm (take f(·) as a constant non zero function).• 183: Fact 7: in the proof, we compute the smallest lower-bound.• 192, in (4.18) all T are t• 197: second line of definition 4.26: u(t) ∈ U .• 197-198: the role of the additive constants β which basically accounts for initialstored energy, has also been investigated in D.H. Hill: ’Dissipative nonlinear systems:basic properties and stability analysis”, Proc. of IEEE CDC, pp.3259-3264, 16-18December 1992.• 204, Theorem 4.43: It is supposed that there exists a storage function V (·) suchthat V (x⋆) = 0, so that Vr(x⋆) = 0. From the definition of the required supplyin Definition 4.36, this also assumes that the system is reachable from x⋆. This isTheorem 2 in [510].• 202: more on reversibility and its relationship with reciprocity, for LTI systems, isin [511,sections 8 and 9]. In particular Theorem 8 of [51] provides a way to checkreversibility.• 209 and 210: in (4.45) the last term is yT (s)Su(s), and same mistake in line 4 afterTheorem 4.56.• 230 and 116: Lemma 3.36 page 116, and Lemma 4.91 page 230, are taken fromthe book referenced 145 by Faurre, Clerget and Germain. In fact Lemma 3.36 ispresented in the book by Faurre et al as a corollary of Lemma 4.91.• 230, Lemma 4.91: in (4.94) this is V (t, x).• p.247, line 10: alll should be all.• 248 and 249, replace all (4.108) by (4.163)• 251, Theorem 4.111, line 2: IRm×m• Section 5.1: Most of the results presented in this section have been stated in twoarticles by D.J. Hill and P. Moylan: ”Stability results for nonlinear feedback sys-tems”, Automatica, vol.13, pp.377-382, 1977, and ”General instability results forinterconnected systems”, SIAM J. Control and Optimization, vol.21, no 2, pp.256-279, 1983. The errata for these papers include the following. In the SIAM paper,the first condition that appears in equation (12), Theorem 7, is redundant. In theAutomatica article, condition (iii) of Theorem 4 can be removed, while condition(ii) should be stated with TA¯(·) a nonnegative real valued function.• 254, 255: in (4.185) replace ∂u∂xin the first term of the left-hand side by ∂u∂t. Lineafter lemma 4.115: do the same for the equality after “One has”, and delete the“is”. Also the power 2 is missing in the integrand. In (4.188) multiply the first termin the left-hand side by 12. These two Lemmas are taken from H. Bre´zis, AnalyseFonctionnelle, The´orie et Applications, Masson, Paris, 1983, sections X.1 and X.3.• 267: from Lemma 5.13 and Corollary 3.4 it may be deduced that the solutionsP = P T of the KYP Lemma LMI, are > 0 if (C,A) is observable.• 298: in Theorem 5.68, this is ”...with input u(·)...”• 305: line before section 5.9.4: Bounded• 306: Theorem 5.71 is taken from M.R. James et al, SIAM J. Control Optimization,vol.43, no 5, pp.1535-1582, 2005.• 339: In definition 6.37: a symmetric positive definite matrix R(x); in equation(6.95), this is (J(x)−R(x))∂H0∂x+ ....• 366: generally speaking BV functions do not satisfy that for any t, there exists aσ such the function is continuous on [t, t + σ). This is because at t a BV functionmay possess an accumulation of jumps on the left, and an accumulation of jumpson the right. But in mechanical systems with frictionless unilateral constraints andpiecewise analytic data, accumulations of impacts on the right do not exist (result ofBallard in 2000). So in this particular case the generalized velocity has this property.• 368: line -8, it is −∂ψV (x).• 382: ξ in the line after (7.10) refers to the ξ in (6.184)• section 3.9.4: Notice that for a maximal monotone operator F : IRn → IRn, x 7→y = F (x), with F = ∂f where f(·) is convex, proper, lower semi continuous, thenthe “input-output” product satisfies xTy = f(x) + f ∗(y), where f ∗(·) is the dualfunction of f(·).• 409: CTCE means Cross Terms Cancellation Equality• 431 line 2 section 7.8.3: (6.162) is (6.156), and λz1 is as in (6.169)• 133, line -2 before (3.176): Pe• 426, line 6: actuator dynamics• 431, line 1, (7.169): λzd is λd• 531: Theorem A.61 extends to positive semidefinite matrices, see proposition 8.2.3in [Bernstein 2005, Matrix Mathematics].• 533, Lemma A.65: the last condition is S(I − R†R) = 0 or equivalently (I −RR†)ST = 0.• 536, line 11 in the proof of Lemma A.71: realization• 564, reference [470]: 1992

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