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Inequations
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic The International Federation of Automatic Control Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 677–680
Polynomial Control Systems: Invariant Sets Polynomial Control Systems: Invariant Polynomial Control Equations/Inequations Systems: Invariant Sets Sets given by Algebraic given by Algebraic Equations/Inequations given by Algebraic Equations/Inequations
Melanie Harms ∗∗ Christian Schilli ∗∗ Eva Zerz ∗∗∗ ∗∗ ∗ ∗∗ ∗∗∗ Melanie Harms Schilli Zerz ∗∗∗ ∗ Christian ∗∗ Eva Melanie Harms Christian Schilli Melanie Harms Christian Schilli Eva Eva Zerz Zerz ∗∗∗ ∗ ur Mathematik, RWTH Aachen University, 52062 ∗ Lehrstuhl D f¨ ∗ D f¨ u r Mathematik, RWTH Aachen University, 52062 ∗ Lehrstuhl Lehrstuhl D f¨ u RWTH Aachen, Germany (e-mail: [email protected]) Lehrstuhl D f¨ urr Mathematik, Mathematik, RWTH Aachen Aachen University, University, 52062 52062 Germany (e-mail: [email protected]) ∗∗ Aachen, Aachen, Germany (e-mail: [email protected]) Lehrstuhl B f¨ ur Mathematik, RWTH Aachen University, 52062 Aachen, Germany (e-mail: [email protected]) ∗∗ ∗∗ Lehrstuhl B f¨ u r Mathematik, RWTH Aachen University, 52062 ∗∗ Aachen, Lehrstuhl B u RWTH (e-mail: [email protected]) LehrstuhlGermany B f¨ f¨ urr Mathematik, Mathematik, RWTH Aachen Aachen University, University, 52062 52062 Aachen, Germany (e-mail: [email protected]) ∗∗∗Aachen, Germany (e-mail: [email protected]) LehrstuhlGermany D f¨ ur Mathematik, RWTH Aachen University, 52062 Aachen, (e-mail: [email protected]) ∗∗∗ ∗∗∗ Lehrstuhl D f¨ u r Mathematik, RWTH Aachen University, 52062 ∗∗∗ Lehrstuhl D u RWTH Aachen, Germany (e-mail: [email protected]) Lehrstuhl D f¨ f¨ urr Mathematik, Mathematik, RWTH Aachen Aachen University, University, 52062 52062 Aachen, Germany (e-mail: [email protected]) Aachen, Aachen, Germany Germany (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Consider a nonlinear input-affine control system x(t) ˙ = f (x(t)) + g(x(t))u(t), Abstract: Consider a nonlinear input-affine control system x(t) ˙˙ given = f (x(t)) + g(x(t))u(t), Abstract: Consider a nonlinear input-affine control system x(t) = (x(t)) + y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set algebraic equations Abstract: Consider a g,nonlinear input-affine control system x(t) ˙ given = ffby (x(t)) + g(x(t))u(t), g(x(t))u(t), y(t) = h(x(t)), where f, h are polynomial functions. Let S be a set by algebraic equations y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set given by algebraic equations and inequations (in the sense of = ). Such sets appear, for instance, in the theory of the Thomas y(t) = h(x(t)), where f, g, h are polynomial functions. Let Sinstance, be a set in given by algebraic equations and inequations (in the sense of = ). Such sets appear, for the theory of the Thomas and inequations (in the sense of = ). Such sets appear, for instance, in the theory of the Thomas decomposition, which is sense used to write a variety as a disjoint union of simpler subsets. The set S and inequations (in the of = ). Such sets appear, for instance, in the theory of the Thomas decomposition, which is used to write aa variety as aa disjoint union of simpler subsets. The set S decomposition, which is used to write variety as disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such decomposition, which is used to write a variety as a disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x ˙ = (f + gα)(x). If it is possible to achieve is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x ˙ == (f + gα)(x).then If itSis possible achieve that S invariant set closed loop x (f + If possible to achieve this with a polynomial output feedback law u(t) called to controlled that goal S is is an an invariant set of of the the closed loop system system x˙˙ = == (f β(y(t)), + gα)(x). gα)(x).then If it itSis is is possible to achieve this goal with a polynomial output feedback law u(t) β(y(t)), is called controlled this goal with a polynomial output feedback law u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties arelaw discussed and algebraically characterized, and this goal with a polynomial output feedback u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties are discussed and algebraically characterized, and and conditioned invariant. properties are discussed and algebraically characterized, and algorithms are provided forThese checking them with symbolic computation methods. and conditioned invariant. These properties are discussed and algebraically characterized, and algorithms are provided for checking them with symbolic computation methods. algorithms are provided for checking them with symbolic computation methods. algorithms are provided for checking them with symbolic computation methods. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear control systems, Multivariable polynomials, Invariance, State feedback, Keywords: Nonlinear control systems, Multivariable polynomials, Invariance, Keywords: Nonlinear controlsystems systems,theory, Multivariable polynomials, Invariance, State State feedback, feedback, Output feedback, Algebraic Computational methods. Keywords: Nonlinear control systems, Multivariable polynomials, Invariance, State feedback, Output feedback, Algebraic systems theory, Computational methods. Output feedback, Algebraic systems theory, Computational methods. Output feedback, Algebraic systems theory, Computational methods. 1. INTRODUCTION 2. SETS GIVEN BY ALGEBRAIC (IN-)EQUATIONS 1. 2. SETS GIVEN BY ALGEBRAIC (IN-)EQUATIONS 1. INTRODUCTION INTRODUCTION 2. 1. INTRODUCTION 2. SETS SETS GIVEN GIVEN BY BY ALGEBRAIC ALGEBRAIC (IN-)EQUATIONS (IN-)EQUATIONS The notions of controlled and conditioned invariance go The results of this section are easy consequences of the of section consequences The of controlled conditioned go The results results of this this section are easy consequences of of the the The notions of and controlled and conditioned invariance go The contents of (Cox et al., 1992,are Ch.easy 4, §4). back notions to Basile Marro and (1992). We studyinvariance a nonlinear The results of this section are easy consequences of the The notions of controlled and conditioned invariance go contents of (Cox et al., 1992, Ch. 4, §4). back to Basile and Marro (1992). We study a nonlinear contents of (Cox et al., 1992, Ch. 4, §4). back to of Basile and Marro Let (1992). We study study a nonlinear nonlinear version theseand concepts. K denote the field of real contents of (Cox et al., 1992, Ch. 4, §4). back to Basile Marro (1992). We a Let p1 , . . . , pk , q ∈ P be given. Consider version of concepts. K denote the of version of these these concepts. Let K K[x denote the field of real real or complex numbers and letLet P= , xn ]field denote the Let , .. .. .. ,, p , qq ∈ be Consider 1 , . . .the version of these concepts. Let K denote field of real 1 k pK ∈P P be=given. given. Consider Let p p= 1 ,, {x k ,,nq or complex numbers and let P = K[x , . . . , x ] denote the n . . . =Consider pk (x) = 0, q(x) = 0}. (1) ,p be given. p 1 n 1 (x) 1 . . .∈ k n |∈pP or complex complex numbers numbers and let P P= =over K[xK. .. ,, x denote polynomial ring in nand variables f ]] ∈ P n , gthe ∈ LetS 1 ,, .. ..Let n n | p1 (x) = . . . = pk (x) = 0, q(x) = 0}. or let K[x x denote the S = {x ∈ K (1) 1 n n | p1 (x) = . . . = pk (x) = 0, q(x) = 0}. n, g ∈ S = {x ∈ K polynomial ring in n variables over K. Let f ∈ P n×m r n, g ∈ S = {x ∈ K | p1 (x) = . . . = pk (x) = 0, q(x) = 0}. (1) (1) polynomial in n over K. Let ff polynomial ∈ P P n×m , and ring h∈P bevariables given. An input-affine polynomial ring in n variables over K. Let ∈ P , g ∈ r There is no loss of generality in assuming that there is only n×m , and h ∈ P r be given. An input-affine polynomial P n×m r be P ,, system and h ∈ P input-affine polynomial control takes thegiven. form x(t) ˙An = f (x(t)) + g(x(t))u(t), There is no loss of generality in assuming that there is only P and h ∈ P be given. An input-affine polynomial There is of there is 0assuming for 1 ≤ j that ≤ l is equivalent one inequation qj (x) =in control system takes the form ˙˙ called = f (x(t)) + g(x(t))u(t), is no no loss loss (since of generality generality assuming that there is only only control system the form x(t) = + y(t) = h(x(t)). A set M Knn x(t) is controlled invariant There (x)q = =in for ≤Let ≤ Ill is is equivalent one inequation (since control system takes takes the⊆ form x(t) ˙ = ff (x(t)) (x(t)) + g(x(t))u(t), g(x(t))u(t), jjj ≤ one inequation (since to q(x) = 0, where qqqqjjj=(x) · .000. .for · q111l ).≤ ⊆ equivalent P be the 1 = n is called controlled invariant y(t) = h(x(t)). A set M ⊆ K (x) for ≤ ≤ l is equivalent one inequation (since n is called y(t) = h(x(t)). A set M ⊆ K controlled invariant if there exists a state feedback law u(t) = α(x(t)) with to q(x) = 0, where q = q · . . . · q ). Let I ⊆ P be the y(t) =mh(x(t)). A state set M feedback ⊆ K is called controlled invariant to = .. .. ··Jqqll⊆ ). Let IIanother ⊆ the ideal generated by p1qq, .= k··. .. If if there exists a law u(t) = α(x(t)) with to q(x) q(x) = 0, 0, where where =.. .. qq,, 111p ). P Letis ⊆ P P be beideal, the l⊆ if there exists a state feedback law u(t) = α(x(t)) with α ∈ P such that the closed loop system x ˙ = (f + gα)(x) ideal generated by p , . p . If J P is another ideal, 1 k if there exists a state feedback law u(t) = α(x(t)) with m ideal generated by p , . . . , p . If J ⊆ P is another ideal, the ideal quotient 1 k m such that the closed loop system x α ∈ P ˙ = (f + gα)(x) ideal generated by p , . . . , p . If J ⊆ P is another ideal, 1 k m α ∈ P such that the closed loop system x ˙ = (f + gα)(x) has an invariant thatloop is, the solution initial the α ∈M P as such that the set, closed system x˙ = of (fthe + gα)(x) the ideal ideal quotient quotient has M as an invariant set, is, the solution of the initial (I : J) := {p ∈ P | pJ ⊆ I} has M as an invariant set, that is, the solution of the initial value problem with x(0) =that x0 ∈ M remains within M for the ideal quotient has M as an invariant set, that is, the solution of the initial (I : J) := {p ∈ P || pJ ⊆ I} (I value problem with x(0) = x ∈ M remains within M for 0 (I :: J) J) := := {p {p ∈ ∈P P | pJ pJ ⊆ ⊆ I} I} value problem x(0) = x M within M for all times in the with existence interval. If remains there exists an output 0 ∈ value problem with x(0) = x ∈ M remains within M for 0 is an ideal in P that contains I. If J is the principal ideal all times in the existence interval. If there exists an output all times in the existence interval. If there exists an output feedback law u(t) = β(y(t)) = β(h(x(t))) such that M is If J is the principal ideal is an ideal in P that contains I. all times in theu(t) existence interval. If there exists an output is in that contains is the the principal principal ideal ideal generated by q, we write is an an ideal ideal by in P P thatwrite contains I. I. If If J J is feedback law = β(y(t)) = β(h(x(t))) β(h(x(t))) suchMthat that M is is generated feedback law u(t) β(y(t)) = such M an invariant set of = x˙ = (f + g(β ◦ h))(x), then is called q, we feedback law u(t) = β(y(t)) = β(h(x(t))) such that M is generated (I by: q, q, we write q) := (I : q) = {p ∈ P | pq ∈ I}. generated by we write an invariant set of x ˙ = (f + g(β ◦ h))(x), then M is called an set of g(β M controlled and The then case where M is an invariant invariant setconditioned of x x˙˙ = = (f (f + +invariant. g(β ◦ ◦ h))(x), h))(x), then M is is called called (I q) := (I q) = {p ∈ P pq ∈ I}. (I ::: q) q) := := (I (I ::: q) q) = = {p {p ∈ ∈P P ||| pq pq ∈ ∈ I}. I}. and conditioned invariant. The case where M is (I controlled and conditioned invariant. The case where M is acontrolled variety (that is, the solution set of algebraic equations) controlled and conditioned invariant. The case where M is This leads to an ideal chain a variety (that is, the solution set of algebraic equations) a variety is, the solution of algebraic equations) has been (that studied (2009); Zerz et al. This leads to an ideal chain a variety (that is, in theChristopher solution set setet ofal. algebraic equations) This leads leads to to an anI ideal ideal chain ⊆ (I :chain q) ⊆ (I : q 22 ) ⊆ . . . , This has been studied in Christopher et al. (2009); Zerz et al. has been studied in Christopher et al. (2009); Zerz et al. (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). has been studied in Christopher et al. (2009); Zerz et al. I ⊆ (I : q) ⊆ (I ..., ⊆ I ⊆ (I : q) ⊆ ⊆ (I (I ::: qqq 22 ))) ⊆ (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). m I ⊆ (I : q) ⊆ .. .. .. ,, (2010); (2012); Schilli et al. (2014, 2016). The setZerz A ⊆and P m Walcher of all feedback laws making a variety in- which must become (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). stationary, since P is Noetherian. The m of all feedback laws making a variety inThe set A ⊆ P m The set A ⊆ P of all feedback laws making a variety variant has the structure of an affine module, which can inbe which must become stationary, since P is Noetherian. The The set A ⊆ P of all feedback laws making a variety inwhich must become stationary, since P is The largest ideal in this chain is called the saturation of I by q, must become stationary, since Psaturation is Noetherian. Noetherian. The variant an which can be variant has has the the structure of an affine affine module, module, which can be which determined by structure symbolic of computation methods (Gr¨ obner largest ideal in this chain is called the of I by q, variant has the structure of an affine module, which can be largest ideal in this chain is called the saturation of I by q, and is denoted by largest ideal in this chain is called the saturation of I by q, determined by symbolic symbolic computation methods (Gr¨ bner determined by (Gr¨ ooobner bases). For controlled andcomputation conditionedmethods invariance, A has and and is is denoted denoted ∞ by e determined by symbolic computation methods (Gr¨ bner by ) = {p ∈ P | ∃e ∈ N : pq ∈ I}. (I : q and is denoted by bases). For controlled and conditioned invariance, A has m bases). For conditioned invariance, to be intersected with and K[h]m , where K[h] = K[h1 , . .A hr ] ∞ ) = {p ∈ P | ∃e ∈ N : pq e e bases). For controlled controlled and conditioned invariance, A. ,has has I}. (I :: qq ∞ ∞ ) = {p ∈ P | ∃e ∈ N : pq e ∈ (I m , where K[h] = K[h1 , . . . , hr ] to be intersected with K[h] ) = {p ∈ P | ∃e ∈ N : pq ∈ ∈ I}. I}. (I : q mgenerated to be intersected with K[h] , where K[h] = K[h , . . . , h ] denotes the subalgebra of P by h , . . . , h . The 1 r 1 r to be intersected with K[h] , where K[h] = K[h , . . . , h ] We first observe that S is not changed if I is replaced by 1 r denotes the generated by ,, .. .. .. ,, by h The denotes paper the subalgebra subalgebra of Pcase generated by ish h11given hrr ...algeThe (I We first that S is not changed if I is replaced by present addresses of theP where M ∞ observe denotes the subalgebra of P generated by h , . . . , h The We first observe that S is not changed if I is replaced : q ). 1 r We first observe that S is not changed if I is replaced by by present paper addresses addresses the case case where where Msense is given given by alge- (I : q ∞ present paper the is alge∞ ). braic equations and inequations (in theM of =by ). Such present paper addresses the case where M is given by alge∞ (I : q ). Lemma 1. Let I ⊆ P be an ideal and let q ∈ P be a (I : q ). braic equations and inequations (in the sense of = ). Such braic equations and inequations the of sets appear in the of the(in Thomas decomposition, Lemma II following ⊆ an ideal qq ∈ na braic equations andtheory inequations (in the sense sense of = =). ). Such Such polynomial. Lemma 1. 1. Let Let ⊆ P P be be are an equivalent ideal and and let let ∈ xP P∈ be be a : The for any K sets appear in the theory of the Thomas decomposition, Lemma 1. Let I ⊆ P be an ideal and let q ∈ P be sets appear in theory of see e.g. B¨ achler et al. (2012). polynomial. The The following following are are equivalent equivalent for for any any x x∈ ∈K Knnn ::a sets appear in the the theory of the the Thomas Thomas decomposition, decomposition, polynomial. see e.g. B¨ a chler et al. (2012). : polynomial. The following are equivalent for any x ∈ K see e.g. B¨ a chler et al. (2012). (1) p(x) = 0 for all p ∈ I and q(x) = 0, This see e.g.work B¨ achler et al. (2012). was supported by DFG Graduiertenkolleg 1632 “Exper(1) p(x) = 0 for all p ∈ II and q(x) 0, (1) p(x) = for all p ∈ q(x) = 0, This work was supported by DFG Graduiertenkolleg 1632 “Exper (2) (I : q∞ ) and = 0. imental and Constructive Algebra”. (1) p(x) = = 000 for for all all p p∈ ∈ (I I and and = q(x) 0, = This work was supported by DFG Graduiertenkolleg 1632 “Exper This work was supported by DFG Graduiertenkolleg 1632 “Exper∞q(x) (2) p(x) :: qq ∞ )) and q(x) = 0. imental and Constructive Algebra”. ∞ (2) p(x) = 0 for all p ∈ (I and q(x) = 0. imental and Constructive Algebra”. (2) p(x) = 0 for all p ∈ (I : q ) and q(x) = 0. imental and Constructive Algebra”.
Copyright 2017 IFAC 700 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright 2017 IFAC 700 Copyright © 2017 IFAC 700 Peer review© of International Federation of Automatic Copyright ©under 2017 responsibility IFAC 700Control. 10.1016/j.ifacol.2017.08.118
Proceedings of the 20th IFAC World Congress 678 Melanie Harms et al. / IFAC PapersOnLine 50-1 (2017) 677–680 Toulouse, France, July 9-14, 2017
Proof: Since (I : q ∞ ) ⊇ I, the implication “(2) ⇒ (1)” is clear. For “(1) ⇒ (2)”, let p ∈ (I : q ∞ ) be given. Then pq e ∈ I for some nonnegative integer e, and thus p(x)q(x)e = 0. Since q(x) = 0, we may conclude that p(x) = 0 as desired. Due to the lemma, we may replace the given generators p1 , . . . , pk of I by generators p˜1 , . . . , p˜k˜ of (I : q ∞ ). In the following, we will always assume, without loss of generality, that this step has already been performed and that the ideal I generated by the polynomials defining the equations satisfies (I : q) = I, where q is the polynomial defining the inequation. Another important condition which we would like to impose on I is I = J (V(I)), where V(I) = {x ∈ Kn | p(x) = 0 for all p ∈ I} and J (M ) = {p ∈ P | p(x) = 0 for all x ∈ M } for arbitrary subsets I ⊆ P and M ⊆ Kn , respectively. Note that V and J are inclusion-reversing and that I ⊆ J (V(I)) and M ⊆ V(J (M )) hold for all I, M . This implies that J ◦V ◦J ≡ J and V ◦J ◦V ≡ V. The following lemma shows that the property I = J (V(I)) is not destroyed by the transition from I to (I : q ∞ ). Lemma 2. Let I ⊆ P be an ideal and let q ∈ P be a polynomial. If J (V(I)) = I holds, then we have J (V(I : q)) = (I : q). ∞
We will use the inclusion V(I : q) ⊇ V(I) \ V(q)
(2)
and the identity ( J (V(I)) : J (V(q)) ) = J (V(I) \ V(q))
(3)
(1) Let M be F -invariant and let p ∈ J (M ) be given. Then we have p(ϕ(t, x0 )) = 0
for all x0 ∈ M and t ∈ T (x0 ). Differentiation yields n ∂p Fi (ϕ(t, x0 )) = 0, ∂x i i=1 and plugging in t = 0, we get LF (p)(x0 ) = 0 for all x0 ∈ M . Thus LF (p) ∈ J (M ).
(2) Only the “if” part needs to be proven. Let J (V ) = p1 , . . . , pk for some pi ∈ P. By assumption, we have k LF (pi ) = j=1 aij pj for some aij ∈ P. Let x0 ∈ V be given. Consider yi (t) := pi (ϕ(t, x0 )). Differentiation yields y˙ i (t) = LF (pi )(ϕ(t, x0 )) =
k
aij (ϕ(t, x0 ))yj (t).
j=1
˙ = A(t)y(t) and Set Aij (t) := aij (ϕ(t, x0 )). Then y(t) y(0) = 0, which implies that y ≡ 0. Hence ϕ(t, x0 ) ∈ V(J (V )) = V(J (V(I))) = V(I) = V for all t ∈ T (x0 ), where we have used V ◦ J ◦ V ≡ V. Since x0 ∈ V was arbitrary, this implies that V is F -invariant. Z
which are standard results, for instance, see (Cox et al., 1992, Ch. 4, §4). The inclusion (I : q) ⊆ J (V(I : q)) is obvious. For the converse, let p ∈ P be such that p(x) = 0 for all x ∈ V(I : q) ⊇ V(I) \ V(q). Then p ∈ J (V(I) \ V(q)) = ( J (V(I)) : J (V(q)) )
= ( I : J (V(q)) ) ⊆ (I : q), which concludes the proof.
Proof: Let ϕ(t, x0 ) denote the solution of the initial value problem x˙ = F (x), x(0) = x0 at time t ∈ T (x0 ), where 0 ∈ T (x0 ) ⊆ R is the existence interval of the solution.
∞
By induction, this implies that J (V(I : q )) = (I : q ). Proof:
Lemma 3. (1) If M ⊆ Kn is F -invariant, then we have LF (p) ∈ J (M ) for all p ∈ J (M ). (2) Let I ⊆ P be an ideal and let V = V(I) be a variety. Then V is F -invariant if and only if LF (p) ∈ J (V ) for all p ∈ J (V ).
Summing up, we will assume (I : q) = I = J (V(I)) from now on.
Lemma 4. The Zariski closure M := V(J (M )) of a set M ⊆ Kn is F -invariant if and only if LF (p) ∈ J (M ) holds for all p ∈ J (M ). Z
Proof: “Only if”: Let M be F -invariant. According to Z part (1) of Lemma 3, we have LF (p) ∈ J (M ) for all Z p ∈ J (M ). Since J ◦ V ◦ J ≡ J , we obtain that LF (p) ∈ J (M ) for all p ∈ J (M ). “If”: Suppose that LF (p) ∈ J (M ) for all p ∈ J (M ). Using J ◦ V ◦ J ≡ J and part (2) of Lemma 3, we conclude that Z V := V(J (M )) = M is F -invariant.
Combining the previous two lemmas, we have the following immediate consequence.
3. AUTONOMOUS CASE Consider the autonomous system x˙ = F (x) with F ∈ P n . Let M ⊆ Kn be an arbitrary subset. We say that M is F invariant if the solution of the initial value problem with x(0) = x0 ∈ M remains within M , that is, x(t) ∈ M holds for all t in the existence interval. For p ∈ P, let n ∂p Fi LF (p) := ∂xi i=1 denote the Lie derivative of p along F . 701
Theorem 5. If M is F -invariant, then so is its Zariski Z closure M = V(J (M )). Now we return to semi-algebraic sets S ⊆ Kn as in Equation (1). Under the (nonrestrictive) assumptions discussed in Section 2, the F -invariance of S can be characterized in terms of the F -invariance of two varieties associated to S as follows.
Proceedings of the 20th IFAC World Congress Melanie Harms et al. / IFAC PapersOnLine 50-1 (2017) 677–680 Toulouse, France, July 9-14, 2017
Theorem 6. Let F ∈ P n and p1 , . . . , pk , q ∈ P be given. Without loss of generality, assume that the ideal I := p1 , . . . , pk satisfies (I : q) = I = J (V(I)). Then the following are equivalent: (1) S = V(I) \ V(q) is F -invariant, (2) V(I) and V(I) ∩ V(q) are F -invariant. Proof: We first note that the complement of an F invariant set is F -invariant, and that the intersection of two F -invariant sets is F -invariant. For M ⊆ Kn , let M := Kn \ M denote its complement.
“(1) ⇒ (2)”: Using Equation (3) and (I : q) = I = J (V(I)), we have J (S) = J (V(I) \ V(q)) = ( J (V(I)) : J (V(q)) )
= ( I : J (V(q)) ) ⊆ (I : q) = I. Since I = J (V(I)) ⊆ J (S), we may conclude that J (S) = I. Thus
4. CONTROLLED INVARIANCE We return to control systems x(t) ˙ = f (x(t)) + g(x(t))u(t),
Using Theorem 5, we obtain the F -invariance of V(I). On the other hand, the complement of S, which is given by S = Kn \ (V(I) ∩ V(q)) = ( Kn \ V(I) ) ∪ V(q), is F -invariant as well. Therefore, also the intersection V(I) ∩ S = V(I) ∩ V(q) is F -invariant. “(2) ⇒ (1)”: The complement of V(I) ∩ V(q) is given by M := (V(I) ∩ V(q)) = V(I) ∪ V(q) and it is F -invariant. Thus the intersection V(I) ∩ M = V(I) ∩ V(q) = S
f ∈ M + im(g). If S is controlled invariant, then the set A of all α ∈ P m such that f +gα ∈ M has the structure of an affine module, that is, A = α∗ +A0 , where α∗ is a particular element of A, and A0 = {α ∈ P m | gα ∈ M} is a submodule of P m . Example: Let p = x21 + x22 − 1, q = x23 − 1 ∈ R[x1 , x2 , x3 ]. Then S is a cylinder with two circles taken out. Using the computer algebra system Singular, see Greuel and Pfister (2008) and Decker et al. (2015), we compute the module M of all vector fields F such that S is F -invariant, which is generated by the columns of the matrix p 0 0 x2 0 0 p 0 −x1 0 . 0 0 p 0 q Now let
is also F -invariant. Corollary 7. In the situation of Theorem 6, let M denote the set of all vector fields F ∈ P n such that S is F invariant. Define J := I + q . Then S is F -invariant if and only if both V(I) and V(J) = V(I) ∩ V(q) are F -invariant. Thus we have F ∈ M if and only if LF (p) ∈ I for all p ∈ I, and q ) ∈ J (V(J)) for all q¯ ∈ J (V(J)). LF (¯ In particular, M is a submodule of P n . Assume that J (V(J)) = p1 , . . . , pk , q1 , . . . , ql for suitable polynomials q1 , . . . , ql ∈ P. Then the module M can be determined as follows: First, compute the kernel N of the matrix ∂p , p I , . . . , p I 1 k k k ∂x ∂q , p1 Il , . . . , pk Il , q1 Il , . . . , ql Il ∂x
∂p ∂q where ∂x ∈ P k×n , ∂x ∈ P l×n denote the Jacobians of p = (p1 , . . . , pk )T , q = (q1 , . . . , ql )T , and Is is the identity matrix of size s. Then M is the projection of N onto the first n components.
702
(4)
where f ∈ P n , g ∈ P n×m are given. Let p1 , . . . , pk , q ∈ P be given and consider S as in Equation (1). Let M be the module of all vector fields F ∈ P n such that S is F invariant. Then S is controlled invariant if and only if there exists α ∈ P m such that f + gα ∈ M. Such an α will be called an admissible feedback law (for f, g, and S). Corollary 8. The set S is controlled invariant for the system from Equation (4) if and only if
Z
S = V(J (S)) = V(I).
679
f=
x3 x2 x1
and
g=
x1 0 0 x2 x3 0
.
Then S is controlled invariant and the set of all admissible feedback laws α is given by p 0 x22 q −x1 x3 . + im A= −x1 x3 − 1 0 p −x21 q For instance, consider the following specific solutions of the closed loop system (1 − x21 )x3 −x1 x3 x˙ = f + g = −x1 x2 x3 , −x1 x3 − 1 x1 (1 − x23 ) which has (0, ±1, 0)T ∈ S as fixed points. Choosing x0 = (1, 0, 0)T ∈ S, we get x1 (t) = 1, x2 (t) = 0, x3 (t) = tanh(t) and we note that x3 (t) ∈ (−1, 1) for all t ∈ T (x0 ) = R. Choosing x0 = (1, 0, 2)T ∈ S, we obtain x1 (t) = 1, x2 (t) = 0, x3 (t) = coth(t + 21 ln(3)) and we note that x3 (t) > 1 for all t ∈ T (x0 ) = (− 12 ln(3), ∞). These examples illustrate the invariance of S. In V(p, q), we have the fixed points (±1, 0, ±1)T . Starting in x0 = (0, 1, ±1)T ∈ V(p, q), we get the trajectory x1 (t) = ±tanh(t), x2 (t) = 1 − tanh(t)2 , x3 (t) = ±1, which illustrates the invariance of V(p, q).
Proceedings of the 20th IFAC World Congress 680 Melanie Harms et al. / IFAC PapersOnLine 50-1 (2017) 677–680 Toulouse, France, July 9-14, 2017
REFERENCES
5. CONTROLLED AND CONDITIONED INVARIANCE Finally, consider control systems x(t) ˙ = f (x(t)) + g(x(t))u(t),
y(t) = h(x(t))
(5)
where f ∈ P n , g ∈ P n×m , h ∈ P r are given. Let p1 , . . . , pk , q ∈ P be given and consider S as in Equation (1). Let M be the module of all vector fields F ∈ P n such that S is F -invariant. Let A ⊆ P m denote the set of all admissible feedback laws and let R = K[y1 , . . . , yr ]. Then S is controlled and conditioned invariant if and only if there exists β ∈ Rm such that f +g(β◦h) ∈ M, or equivalently, β ◦ h ∈ A. Recall that K[h] = K[h1 , . . . , hr ] ⊆ P denotes the subalgebra of P generated by h1 , . . . , hr . Corollary 9. The set S is controlled and conditioned invariant for the system from Equation (5) if and only if there exists α ∈ A (that is, f + gα ∈ M) and β ∈ Rm such that α = β ◦ h ∈ K[h]m . In other words, S is controlled and conditioned invariant if and only if A ∩ K[h]m = ∅. An algorithm for checking this condition, and for finding a concrete element of A ∩ K[h]m can be found in Schilli et al. (2014). An improved version of the algorithm was presented in Schilli et al. (2015). Example: Consider f, g, and S from the previous example and let h = x1 x3 . Since −x1 x3 ∈ A ∩ K[h]2 , α := −x1 x3 − 1 we conclude that S is controlled and conditioned invariant. Choosing h = (h1 , h2 )T with h1 = x21 x23 + x1 x3 − x21 , h2 = x3 , the situation is less obvious, because α ∈ / K[h1 , h2 ]2 . However, 2 p x2 q α ˜ := α − q + ∈A 0 −x21 q and α ˜=
h22 − h1 − 1 −h1 − 1
∈ K[h1 , h2 ]2 .
Thus S is still controlled and conditioned invariant. 6. CONCLUSION In this paper, the question of (controlled/conditioned) invariance of sets given by algebraic equations and inequations (in the sense of =) has been completely reduced to the known case of invariant varieties, for which algorithmic solutions have been given in Christopher et al. (2009); Zerz et al. (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). Controlled and conditioned invariant varieties have also been studied by Yuno and Ohtsuka (2014, 2015). An interesting generalization to sets given by algebraic equations and inequalities (in the sense of <) is treated in Yuno and Ohtsuka (2016). There, dynamic compensators are used instead of static feedback laws. Stability issues have not yet been addressed in this algebraic framework. This remains an open topic for future research. 703
B¨achler, T., Gerdt, V., Lange-Hegermann, M., and Robertz, D. (2012). Algorithmic Thomas decomposition of algebraic and differential systems. J. Symb. Comput., 47, 1233–1266. Basile, G. and Marro, G. (1992). Controlled and Conditioned Invariants in Linear System Theory. Prentice Hall, Englewood Cliffs. Christopher, C., Llibre, J., Pantazi, C., and Walcher, S. (2009). Inverse problems for invariant algebraic curves: explicit computations. Proc. Royal Soc. Edinburgh, 139A, 287–302. Cox, D., Little, J., and O’Shea, D. (1992). Ideals, Varieties, and Algorithms. Springer, New York. Decker, W., Greuel, G.M., Pfister, G., and Sch¨ onemann, H. (2015). Singular 4-0-2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de. Greuel, G.M. and Pfister, G. (2008). A Singular Introduction to Commutative Algebra. Springer, Berlin. Schilli, C., Zerz, E., and Levandovskyy, V. (2014). Controlled and conditioned invariant varieties for polynomial control systems. In Proc. 21st Int. Symp. Math. Theory Networks Systems (MTNS). Groningen. Schilli, C., Zerz, E., and Levandovskyy, V. (2015). Controlled and conditioned invariance for polynomial and rational feedback systems. Book chapter submitted to Advances in Delays and Dynamics (Springer). Schilli, C., Zerz, E., and Levandovskyy, V. (2016). Controlled and conditioned invariant varieties for polynomial control systems with rational feedback. In Proc. 22nd Int. Symp. Math. Theory Networks Systems (MTNS). Minneapolis. Yuno, T. and Ohtsuka, T. (2014). Lie derivative inclusion for a class of polynomial state feedback controls. Trans. Inst. Syst. Contr. Inf. Engin., 27, 423–433. Yuno, T. and Ohtsuka, T. (2015). Lie derivative inclusion with polynomial output feedback. Trans. Inst. Syst. Contr. Inf. Engin., 28, 22–31. Yuno, T. and Ohtsuka, T. (2016). Rendering a prescribed subset invariant for polynomial systems by dynamic state feedback compensator. In Proc. 10th IFAC Symp. Nonlin. Contr. Syst. Monterey. Zerz, E. and Walcher, S. (2012). Controlled invariant hypersurfaces of polynomial control systems. Qual. Theory Dyn. Syst., 11, 145–158. Zerz, E., Walcher, S., and G¨ ucl¨ u, F. (2010). Controlled invariant varieties of polynomial control systems. In Proc. 19th Int. Symp. Math. Theory Networks Systems (MTNS). Budapest.
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IFAC PapersOnLine 50-1 (2017) 677–680
Polynomial Control Systems: Invariant Sets Polynomial Control Systems: Invariant Polynomial Control Equations/Inequations Systems: Invariant Sets Sets given by Algebraic given by Algebraic Equations/Inequations given by Algebraic Equations/Inequations
Melanie Harms ∗∗ Christian Schilli ∗∗ Eva Zerz ∗∗∗ ∗∗ ∗ ∗∗ ∗∗∗ Melanie Harms Schilli Zerz ∗∗∗ ∗ Christian ∗∗ Eva Melanie Harms Christian Schilli Melanie Harms Christian Schilli Eva Eva Zerz Zerz ∗∗∗ ∗ ur Mathematik, RWTH Aachen University, 52062 ∗ Lehrstuhl D f¨ ∗ D f¨ u r Mathematik, RWTH Aachen University, 52062 ∗ Lehrstuhl Lehrstuhl D f¨ u RWTH Aachen, Germany (e-mail: [email protected]) Lehrstuhl D f¨ urr Mathematik, Mathematik, RWTH Aachen Aachen University, University, 52062 52062 Germany (e-mail: [email protected]) ∗∗ Aachen, Aachen, Germany (e-mail: [email protected]) Lehrstuhl B f¨ ur Mathematik, RWTH Aachen University, 52062 Aachen, Germany (e-mail: [email protected]) ∗∗ ∗∗ Lehrstuhl B f¨ u r Mathematik, RWTH Aachen University, 52062 ∗∗ Aachen, Lehrstuhl B u RWTH (e-mail: [email protected]) LehrstuhlGermany B f¨ f¨ urr Mathematik, Mathematik, RWTH Aachen Aachen University, University, 52062 52062 Aachen, Germany (e-mail: [email protected]) ∗∗∗Aachen, Germany (e-mail: [email protected]) LehrstuhlGermany D f¨ ur Mathematik, RWTH Aachen University, 52062 Aachen, (e-mail: [email protected]) ∗∗∗ ∗∗∗ Lehrstuhl D f¨ u r Mathematik, RWTH Aachen University, 52062 ∗∗∗ Lehrstuhl D u RWTH Aachen, Germany (e-mail: [email protected]) Lehrstuhl D f¨ f¨ urr Mathematik, Mathematik, RWTH Aachen Aachen University, University, 52062 52062 Aachen, Germany (e-mail: [email protected]) Aachen, Aachen, Germany Germany (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Consider a nonlinear input-affine control system x(t) ˙ = f (x(t)) + g(x(t))u(t), Abstract: Consider a nonlinear input-affine control system x(t) ˙˙ given = f (x(t)) + g(x(t))u(t), Abstract: Consider a nonlinear input-affine control system x(t) = (x(t)) + y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set algebraic equations Abstract: Consider a g,nonlinear input-affine control system x(t) ˙ given = ffby (x(t)) + g(x(t))u(t), g(x(t))u(t), y(t) = h(x(t)), where f, h are polynomial functions. Let S be a set by algebraic equations y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set given by algebraic equations and inequations (in the sense of = ). Such sets appear, for instance, in the theory of the Thomas y(t) = h(x(t)), where f, g, h are polynomial functions. Let Sinstance, be a set in given by algebraic equations and inequations (in the sense of = ). Such sets appear, for the theory of the Thomas and inequations (in the sense of = ). Such sets appear, for instance, in the theory of the Thomas decomposition, which is sense used to write a variety as a disjoint union of simpler subsets. The set S and inequations (in the of = ). Such sets appear, for instance, in the theory of the Thomas decomposition, which is used to write aa variety as aa disjoint union of simpler subsets. The set S decomposition, which is used to write variety as disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such decomposition, which is used to write a variety as a disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x ˙ = (f + gα)(x). If it is possible to achieve is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x ˙ == (f + gα)(x).then If itSis possible achieve that S invariant set closed loop x (f + If possible to achieve this with a polynomial output feedback law u(t) called to controlled that goal S is is an an invariant set of of the the closed loop system system x˙˙ = == (f β(y(t)), + gα)(x). gα)(x).then If it itSis is is possible to achieve this goal with a polynomial output feedback law u(t) β(y(t)), is called controlled this goal with a polynomial output feedback law u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties arelaw discussed and algebraically characterized, and this goal with a polynomial output feedback u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties are discussed and algebraically characterized, and and conditioned invariant. properties are discussed and algebraically characterized, and algorithms are provided forThese checking them with symbolic computation methods. and conditioned invariant. These properties are discussed and algebraically characterized, and algorithms are provided for checking them with symbolic computation methods. algorithms are provided for checking them with symbolic computation methods. algorithms are provided for checking them with symbolic computation methods. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear control systems, Multivariable polynomials, Invariance, State feedback, Keywords: Nonlinear control systems, Multivariable polynomials, Invariance, Keywords: Nonlinear controlsystems systems,theory, Multivariable polynomials, Invariance, State State feedback, feedback, Output feedback, Algebraic Computational methods. Keywords: Nonlinear control systems, Multivariable polynomials, Invariance, State feedback, Output feedback, Algebraic systems theory, Computational methods. Output feedback, Algebraic systems theory, Computational methods. Output feedback, Algebraic systems theory, Computational methods. 1. INTRODUCTION 2. SETS GIVEN BY ALGEBRAIC (IN-)EQUATIONS 1. 2. SETS GIVEN BY ALGEBRAIC (IN-)EQUATIONS 1. INTRODUCTION INTRODUCTION 2. 1. INTRODUCTION 2. SETS SETS GIVEN GIVEN BY BY ALGEBRAIC ALGEBRAIC (IN-)EQUATIONS (IN-)EQUATIONS The notions of controlled and conditioned invariance go The results of this section are easy consequences of the of section consequences The of controlled conditioned go The results results of this this section are easy consequences of of the the The notions of and controlled and conditioned invariance go The contents of (Cox et al., 1992,are Ch.easy 4, §4). back notions to Basile Marro and (1992). We studyinvariance a nonlinear The results of this section are easy consequences of the The notions of controlled and conditioned invariance go contents of (Cox et al., 1992, Ch. 4, §4). back to Basile and Marro (1992). We study a nonlinear contents of (Cox et al., 1992, Ch. 4, §4). back to of Basile and Marro Let (1992). We study study a nonlinear nonlinear version theseand concepts. K denote the field of real contents of (Cox et al., 1992, Ch. 4, §4). back to Basile Marro (1992). We a Let p1 , . . . , pk , q ∈ P be given. Consider version of concepts. K denote the of version of these these concepts. Let K K[x denote the field of real real or complex numbers and letLet P= , xn ]field denote the Let , .. .. .. ,, p , qq ∈ be Consider 1 , . . .the version of these concepts. Let K denote field of real 1 k pK ∈P P be=given. given. Consider Let p p= 1 ,, {x k ,,nq or complex numbers and let P = K[x , . . . , x ] denote the n . . . =Consider pk (x) = 0, q(x) = 0}. (1) ,p be given. p 1 n 1 (x) 1 . . .∈ k n |∈pP or complex complex numbers numbers and let P P= =over K[xK. .. ,, x denote polynomial ring in nand variables f ]] ∈ P n , gthe ∈ LetS 1 ,, .. ..Let n n | p1 (x) = . . . = pk (x) = 0, q(x) = 0}. or let K[x x denote the S = {x ∈ K (1) 1 n n | p1 (x) = . . . = pk (x) = 0, q(x) = 0}. n, g ∈ S = {x ∈ K polynomial ring in n variables over K. Let f ∈ P n×m r n, g ∈ S = {x ∈ K | p1 (x) = . . . = pk (x) = 0, q(x) = 0}. (1) (1) polynomial in n over K. Let ff polynomial ∈ P P n×m , and ring h∈P bevariables given. An input-affine polynomial ring in n variables over K. Let ∈ P , g ∈ r There is no loss of generality in assuming that there is only n×m , and h ∈ P r be given. An input-affine polynomial P n×m r be P ,, system and h ∈ P input-affine polynomial control takes thegiven. form x(t) ˙An = f (x(t)) + g(x(t))u(t), There is no loss of generality in assuming that there is only P and h ∈ P be given. An input-affine polynomial There is of there is 0assuming for 1 ≤ j that ≤ l is equivalent one inequation qj (x) =in control system takes the form ˙˙ called = f (x(t)) + g(x(t))u(t), is no no loss loss (since of generality generality assuming that there is only only control system the form x(t) = + y(t) = h(x(t)). A set M Knn x(t) is controlled invariant There (x)q = =in for ≤Let ≤ Ill is is equivalent one inequation (since control system takes takes the⊆ form x(t) ˙ = ff (x(t)) (x(t)) + g(x(t))u(t), g(x(t))u(t), jjj ≤ one inequation (since to q(x) = 0, where qqqqjjj=(x) · .000. .for · q111l ).≤ ⊆ equivalent P be the 1 = n is called controlled invariant y(t) = h(x(t)). A set M ⊆ K (x) for ≤ ≤ l is equivalent one inequation (since n is called y(t) = h(x(t)). A set M ⊆ K controlled invariant if there exists a state feedback law u(t) = α(x(t)) with to q(x) = 0, where q = q · . . . · q ). Let I ⊆ P be the y(t) =mh(x(t)). A state set M feedback ⊆ K is called controlled invariant to = .. .. ··Jqqll⊆ ). Let IIanother ⊆ the ideal generated by p1qq, .= k··. .. If if there exists a law u(t) = α(x(t)) with to q(x) q(x) = 0, 0, where where =.. .. qq,, 111p ). P Letis ⊆ P P be beideal, the l⊆ if there exists a state feedback law u(t) = α(x(t)) with α ∈ P such that the closed loop system x ˙ = (f + gα)(x) ideal generated by p , . p . If J P is another ideal, 1 k if there exists a state feedback law u(t) = α(x(t)) with m ideal generated by p , . . . , p . If J ⊆ P is another ideal, the ideal quotient 1 k m such that the closed loop system x α ∈ P ˙ = (f + gα)(x) ideal generated by p , . . . , p . If J ⊆ P is another ideal, 1 k m α ∈ P such that the closed loop system x ˙ = (f + gα)(x) has an invariant thatloop is, the solution initial the α ∈M P as such that the set, closed system x˙ = of (fthe + gα)(x) the ideal ideal quotient quotient has M as an invariant set, is, the solution of the initial (I : J) := {p ∈ P | pJ ⊆ I} has M as an invariant set, that is, the solution of the initial value problem with x(0) =that x0 ∈ M remains within M for the ideal quotient has M as an invariant set, that is, the solution of the initial (I : J) := {p ∈ P || pJ ⊆ I} (I value problem with x(0) = x ∈ M remains within M for 0 (I :: J) J) := := {p {p ∈ ∈P P | pJ pJ ⊆ ⊆ I} I} value problem x(0) = x M within M for all times in the with existence interval. If remains there exists an output 0 ∈ value problem with x(0) = x ∈ M remains within M for 0 is an ideal in P that contains I. If J is the principal ideal all times in the existence interval. If there exists an output all times in the existence interval. If there exists an output feedback law u(t) = β(y(t)) = β(h(x(t))) such that M is If J is the principal ideal is an ideal in P that contains I. all times in theu(t) existence interval. If there exists an output is in that contains is the the principal principal ideal ideal generated by q, we write is an an ideal ideal by in P P thatwrite contains I. I. If If J J is feedback law = β(y(t)) = β(h(x(t))) β(h(x(t))) suchMthat that M is is generated feedback law u(t) β(y(t)) = such M an invariant set of = x˙ = (f + g(β ◦ h))(x), then is called q, we feedback law u(t) = β(y(t)) = β(h(x(t))) such that M is generated (I by: q, q, we write q) := (I : q) = {p ∈ P | pq ∈ I}. generated by we write an invariant set of x ˙ = (f + g(β ◦ h))(x), then M is called an set of g(β M controlled and The then case where M is an invariant invariant setconditioned of x x˙˙ = = (f (f + +invariant. g(β ◦ ◦ h))(x), h))(x), then M is is called called (I q) := (I q) = {p ∈ P pq ∈ I}. (I ::: q) q) := := (I (I ::: q) q) = = {p {p ∈ ∈P P ||| pq pq ∈ ∈ I}. I}. and conditioned invariant. The case where M is (I controlled and conditioned invariant. The case where M is acontrolled variety (that is, the solution set of algebraic equations) controlled and conditioned invariant. The case where M is This leads to an ideal chain a variety (that is, the solution set of algebraic equations) a variety is, the solution of algebraic equations) has been (that studied (2009); Zerz et al. This leads to an ideal chain a variety (that is, in theChristopher solution set setet ofal. algebraic equations) This leads leads to to an anI ideal ideal chain ⊆ (I :chain q) ⊆ (I : q 22 ) ⊆ . . . , This has been studied in Christopher et al. (2009); Zerz et al. has been studied in Christopher et al. (2009); Zerz et al. (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). has been studied in Christopher et al. (2009); Zerz et al. I ⊆ (I : q) ⊆ (I ..., ⊆ I ⊆ (I : q) ⊆ ⊆ (I (I ::: qqq 22 ))) ⊆ (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). m I ⊆ (I : q) ⊆ .. .. .. ,, (2010); (2012); Schilli et al. (2014, 2016). The setZerz A ⊆and P m Walcher of all feedback laws making a variety in- which must become (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). stationary, since P is Noetherian. The m of all feedback laws making a variety inThe set A ⊆ P m The set A ⊆ P of all feedback laws making a variety variant has the structure of an affine module, which can inbe which must become stationary, since P is Noetherian. The The set A ⊆ P of all feedback laws making a variety inwhich must become stationary, since P is The largest ideal in this chain is called the saturation of I by q, must become stationary, since Psaturation is Noetherian. Noetherian. The variant an which can be variant has has the the structure of an affine affine module, module, which can be which determined by structure symbolic of computation methods (Gr¨ obner largest ideal in this chain is called the of I by q, variant has the structure of an affine module, which can be largest ideal in this chain is called the saturation of I by q, and is denoted by largest ideal in this chain is called the saturation of I by q, determined by symbolic symbolic computation methods (Gr¨ bner determined by (Gr¨ ooobner bases). For controlled andcomputation conditionedmethods invariance, A has and and is is denoted denoted ∞ by e determined by symbolic computation methods (Gr¨ bner by ) = {p ∈ P | ∃e ∈ N : pq ∈ I}. (I : q and is denoted by bases). For controlled and conditioned invariance, A has m bases). For conditioned invariance, to be intersected with and K[h]m , where K[h] = K[h1 , . .A hr ] ∞ ) = {p ∈ P | ∃e ∈ N : pq e e bases). For controlled controlled and conditioned invariance, A. ,has has I}. (I :: qq ∞ ∞ ) = {p ∈ P | ∃e ∈ N : pq e ∈ (I m , where K[h] = K[h1 , . . . , hr ] to be intersected with K[h] ) = {p ∈ P | ∃e ∈ N : pq ∈ ∈ I}. I}. (I : q mgenerated to be intersected with K[h] , where K[h] = K[h , . . . , h ] denotes the subalgebra of P by h , . . . , h . The 1 r 1 r to be intersected with K[h] , where K[h] = K[h , . . . , h ] We first observe that S is not changed if I is replaced by 1 r denotes the generated by ,, .. .. .. ,, by h The denotes paper the subalgebra subalgebra of Pcase generated by ish h11given hrr ...algeThe (I We first that S is not changed if I is replaced by present addresses of theP where M ∞ observe denotes the subalgebra of P generated by h , . . . , h The We first observe that S is not changed if I is replaced : q ). 1 r We first observe that S is not changed if I is replaced by by present paper addresses addresses the case case where where Msense is given given by alge- (I : q ∞ present paper the is alge∞ ). braic equations and inequations (in theM of =by ). Such present paper addresses the case where M is given by alge∞ (I : q ). Lemma 1. Let I ⊆ P be an ideal and let q ∈ P be a (I : q ). braic equations and inequations (in the sense of = ). Such braic equations and inequations the of sets appear in the of the(in Thomas decomposition, Lemma II following ⊆ an ideal qq ∈ na braic equations andtheory inequations (in the sense sense of = =). ). Such Such polynomial. Lemma 1. 1. Let Let ⊆ P P be be are an equivalent ideal and and let let ∈ xP P∈ be be a : The for any K sets appear in the theory of the Thomas decomposition, Lemma 1. Let I ⊆ P be an ideal and let q ∈ P be sets appear in theory of see e.g. B¨ achler et al. (2012). polynomial. The The following following are are equivalent equivalent for for any any x x∈ ∈K Knnn ::a sets appear in the the theory of the the Thomas Thomas decomposition, decomposition, polynomial. see e.g. B¨ a chler et al. (2012). : polynomial. The following are equivalent for any x ∈ K see e.g. B¨ a chler et al. (2012). (1) p(x) = 0 for all p ∈ I and q(x) = 0, This see e.g.work B¨ achler et al. (2012). was supported by DFG Graduiertenkolleg 1632 “Exper(1) p(x) = 0 for all p ∈ II and q(x) 0, (1) p(x) = for all p ∈ q(x) = 0, This work was supported by DFG Graduiertenkolleg 1632 “Exper (2) (I : q∞ ) and = 0. imental and Constructive Algebra”. (1) p(x) = = 000 for for all all p p∈ ∈ (I I and and = q(x) 0, = This work was supported by DFG Graduiertenkolleg 1632 “Exper This work was supported by DFG Graduiertenkolleg 1632 “Exper∞q(x) (2) p(x) :: qq ∞ )) and q(x) = 0. imental and Constructive Algebra”. ∞ (2) p(x) = 0 for all p ∈ (I and q(x) = 0. imental and Constructive Algebra”. (2) p(x) = 0 for all p ∈ (I : q ) and q(x) = 0. imental and Constructive Algebra”.
Copyright 2017 IFAC 700 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright 2017 IFAC 700 Copyright © 2017 IFAC 700 Peer review© of International Federation of Automatic Copyright ©under 2017 responsibility IFAC 700Control. 10.1016/j.ifacol.2017.08.118
Proceedings of the 20th IFAC World Congress 678 Melanie Harms et al. / IFAC PapersOnLine 50-1 (2017) 677–680 Toulouse, France, July 9-14, 2017
Proof: Since (I : q ∞ ) ⊇ I, the implication “(2) ⇒ (1)” is clear. For “(1) ⇒ (2)”, let p ∈ (I : q ∞ ) be given. Then pq e ∈ I for some nonnegative integer e, and thus p(x)q(x)e = 0. Since q(x) = 0, we may conclude that p(x) = 0 as desired. Due to the lemma, we may replace the given generators p1 , . . . , pk of I by generators p˜1 , . . . , p˜k˜ of (I : q ∞ ). In the following, we will always assume, without loss of generality, that this step has already been performed and that the ideal I generated by the polynomials defining the equations satisfies (I : q) = I, where q is the polynomial defining the inequation. Another important condition which we would like to impose on I is I = J (V(I)), where V(I) = {x ∈ Kn | p(x) = 0 for all p ∈ I} and J (M ) = {p ∈ P | p(x) = 0 for all x ∈ M } for arbitrary subsets I ⊆ P and M ⊆ Kn , respectively. Note that V and J are inclusion-reversing and that I ⊆ J (V(I)) and M ⊆ V(J (M )) hold for all I, M . This implies that J ◦V ◦J ≡ J and V ◦J ◦V ≡ V. The following lemma shows that the property I = J (V(I)) is not destroyed by the transition from I to (I : q ∞ ). Lemma 2. Let I ⊆ P be an ideal and let q ∈ P be a polynomial. If J (V(I)) = I holds, then we have J (V(I : q)) = (I : q). ∞
We will use the inclusion V(I : q) ⊇ V(I) \ V(q)
(2)
and the identity ( J (V(I)) : J (V(q)) ) = J (V(I) \ V(q))
(3)
(1) Let M be F -invariant and let p ∈ J (M ) be given. Then we have p(ϕ(t, x0 )) = 0
for all x0 ∈ M and t ∈ T (x0 ). Differentiation yields n ∂p Fi (ϕ(t, x0 )) = 0, ∂x i i=1 and plugging in t = 0, we get LF (p)(x0 ) = 0 for all x0 ∈ M . Thus LF (p) ∈ J (M ).
(2) Only the “if” part needs to be proven. Let J (V ) = p1 , . . . , pk for some pi ∈ P. By assumption, we have k LF (pi ) = j=1 aij pj for some aij ∈ P. Let x0 ∈ V be given. Consider yi (t) := pi (ϕ(t, x0 )). Differentiation yields y˙ i (t) = LF (pi )(ϕ(t, x0 )) =
k
aij (ϕ(t, x0 ))yj (t).
j=1
˙ = A(t)y(t) and Set Aij (t) := aij (ϕ(t, x0 )). Then y(t) y(0) = 0, which implies that y ≡ 0. Hence ϕ(t, x0 ) ∈ V(J (V )) = V(J (V(I))) = V(I) = V for all t ∈ T (x0 ), where we have used V ◦ J ◦ V ≡ V. Since x0 ∈ V was arbitrary, this implies that V is F -invariant. Z
which are standard results, for instance, see (Cox et al., 1992, Ch. 4, §4). The inclusion (I : q) ⊆ J (V(I : q)) is obvious. For the converse, let p ∈ P be such that p(x) = 0 for all x ∈ V(I : q) ⊇ V(I) \ V(q). Then p ∈ J (V(I) \ V(q)) = ( J (V(I)) : J (V(q)) )
= ( I : J (V(q)) ) ⊆ (I : q), which concludes the proof.
Proof: Let ϕ(t, x0 ) denote the solution of the initial value problem x˙ = F (x), x(0) = x0 at time t ∈ T (x0 ), where 0 ∈ T (x0 ) ⊆ R is the existence interval of the solution.
∞
By induction, this implies that J (V(I : q )) = (I : q ). Proof:
Lemma 3. (1) If M ⊆ Kn is F -invariant, then we have LF (p) ∈ J (M ) for all p ∈ J (M ). (2) Let I ⊆ P be an ideal and let V = V(I) be a variety. Then V is F -invariant if and only if LF (p) ∈ J (V ) for all p ∈ J (V ).
Summing up, we will assume (I : q) = I = J (V(I)) from now on.
Lemma 4. The Zariski closure M := V(J (M )) of a set M ⊆ Kn is F -invariant if and only if LF (p) ∈ J (M ) holds for all p ∈ J (M ). Z
Proof: “Only if”: Let M be F -invariant. According to Z part (1) of Lemma 3, we have LF (p) ∈ J (M ) for all Z p ∈ J (M ). Since J ◦ V ◦ J ≡ J , we obtain that LF (p) ∈ J (M ) for all p ∈ J (M ). “If”: Suppose that LF (p) ∈ J (M ) for all p ∈ J (M ). Using J ◦ V ◦ J ≡ J and part (2) of Lemma 3, we conclude that Z V := V(J (M )) = M is F -invariant.
Combining the previous two lemmas, we have the following immediate consequence.
3. AUTONOMOUS CASE Consider the autonomous system x˙ = F (x) with F ∈ P n . Let M ⊆ Kn be an arbitrary subset. We say that M is F invariant if the solution of the initial value problem with x(0) = x0 ∈ M remains within M , that is, x(t) ∈ M holds for all t in the existence interval. For p ∈ P, let n ∂p Fi LF (p) := ∂xi i=1 denote the Lie derivative of p along F . 701
Theorem 5. If M is F -invariant, then so is its Zariski Z closure M = V(J (M )). Now we return to semi-algebraic sets S ⊆ Kn as in Equation (1). Under the (nonrestrictive) assumptions discussed in Section 2, the F -invariance of S can be characterized in terms of the F -invariance of two varieties associated to S as follows.
Proceedings of the 20th IFAC World Congress Melanie Harms et al. / IFAC PapersOnLine 50-1 (2017) 677–680 Toulouse, France, July 9-14, 2017
Theorem 6. Let F ∈ P n and p1 , . . . , pk , q ∈ P be given. Without loss of generality, assume that the ideal I := p1 , . . . , pk satisfies (I : q) = I = J (V(I)). Then the following are equivalent: (1) S = V(I) \ V(q) is F -invariant, (2) V(I) and V(I) ∩ V(q) are F -invariant. Proof: We first note that the complement of an F invariant set is F -invariant, and that the intersection of two F -invariant sets is F -invariant. For M ⊆ Kn , let M := Kn \ M denote its complement.
“(1) ⇒ (2)”: Using Equation (3) and (I : q) = I = J (V(I)), we have J (S) = J (V(I) \ V(q)) = ( J (V(I)) : J (V(q)) )
= ( I : J (V(q)) ) ⊆ (I : q) = I. Since I = J (V(I)) ⊆ J (S), we may conclude that J (S) = I. Thus
4. CONTROLLED INVARIANCE We return to control systems x(t) ˙ = f (x(t)) + g(x(t))u(t),
Using Theorem 5, we obtain the F -invariance of V(I). On the other hand, the complement of S, which is given by S = Kn \ (V(I) ∩ V(q)) = ( Kn \ V(I) ) ∪ V(q), is F -invariant as well. Therefore, also the intersection V(I) ∩ S = V(I) ∩ V(q) is F -invariant. “(2) ⇒ (1)”: The complement of V(I) ∩ V(q) is given by M := (V(I) ∩ V(q)) = V(I) ∪ V(q) and it is F -invariant. Thus the intersection V(I) ∩ M = V(I) ∩ V(q) = S
f ∈ M + im(g). If S is controlled invariant, then the set A of all α ∈ P m such that f +gα ∈ M has the structure of an affine module, that is, A = α∗ +A0 , where α∗ is a particular element of A, and A0 = {α ∈ P m | gα ∈ M} is a submodule of P m . Example: Let p = x21 + x22 − 1, q = x23 − 1 ∈ R[x1 , x2 , x3 ]. Then S is a cylinder with two circles taken out. Using the computer algebra system Singular, see Greuel and Pfister (2008) and Decker et al. (2015), we compute the module M of all vector fields F such that S is F -invariant, which is generated by the columns of the matrix p 0 0 x2 0 0 p 0 −x1 0 . 0 0 p 0 q Now let
is also F -invariant. Corollary 7. In the situation of Theorem 6, let M denote the set of all vector fields F ∈ P n such that S is F invariant. Define J := I + q . Then S is F -invariant if and only if both V(I) and V(J) = V(I) ∩ V(q) are F -invariant. Thus we have F ∈ M if and only if LF (p) ∈ I for all p ∈ I, and q ) ∈ J (V(J)) for all q¯ ∈ J (V(J)). LF (¯ In particular, M is a submodule of P n . Assume that J (V(J)) = p1 , . . . , pk , q1 , . . . , ql for suitable polynomials q1 , . . . , ql ∈ P. Then the module M can be determined as follows: First, compute the kernel N of the matrix ∂p , p I , . . . , p I 1 k k k ∂x ∂q , p1 Il , . . . , pk Il , q1 Il , . . . , ql Il ∂x
∂p ∂q where ∂x ∈ P k×n , ∂x ∈ P l×n denote the Jacobians of p = (p1 , . . . , pk )T , q = (q1 , . . . , ql )T , and Is is the identity matrix of size s. Then M is the projection of N onto the first n components.
702
(4)
where f ∈ P n , g ∈ P n×m are given. Let p1 , . . . , pk , q ∈ P be given and consider S as in Equation (1). Let M be the module of all vector fields F ∈ P n such that S is F invariant. Then S is controlled invariant if and only if there exists α ∈ P m such that f + gα ∈ M. Such an α will be called an admissible feedback law (for f, g, and S). Corollary 8. The set S is controlled invariant for the system from Equation (4) if and only if
Z
S = V(J (S)) = V(I).
679
f=
x3 x2 x1
and
g=
x1 0 0 x2 x3 0
.
Then S is controlled invariant and the set of all admissible feedback laws α is given by p 0 x22 q −x1 x3 . + im A= −x1 x3 − 1 0 p −x21 q For instance, consider the following specific solutions of the closed loop system (1 − x21 )x3 −x1 x3 x˙ = f + g = −x1 x2 x3 , −x1 x3 − 1 x1 (1 − x23 ) which has (0, ±1, 0)T ∈ S as fixed points. Choosing x0 = (1, 0, 0)T ∈ S, we get x1 (t) = 1, x2 (t) = 0, x3 (t) = tanh(t) and we note that x3 (t) ∈ (−1, 1) for all t ∈ T (x0 ) = R. Choosing x0 = (1, 0, 2)T ∈ S, we obtain x1 (t) = 1, x2 (t) = 0, x3 (t) = coth(t + 21 ln(3)) and we note that x3 (t) > 1 for all t ∈ T (x0 ) = (− 12 ln(3), ∞). These examples illustrate the invariance of S. In V(p, q), we have the fixed points (±1, 0, ±1)T . Starting in x0 = (0, 1, ±1)T ∈ V(p, q), we get the trajectory x1 (t) = ±tanh(t), x2 (t) = 1 − tanh(t)2 , x3 (t) = ±1, which illustrates the invariance of V(p, q).
Proceedings of the 20th IFAC World Congress 680 Melanie Harms et al. / IFAC PapersOnLine 50-1 (2017) 677–680 Toulouse, France, July 9-14, 2017
REFERENCES
5. CONTROLLED AND CONDITIONED INVARIANCE Finally, consider control systems x(t) ˙ = f (x(t)) + g(x(t))u(t),
y(t) = h(x(t))
(5)
where f ∈ P n , g ∈ P n×m , h ∈ P r are given. Let p1 , . . . , pk , q ∈ P be given and consider S as in Equation (1). Let M be the module of all vector fields F ∈ P n such that S is F -invariant. Let A ⊆ P m denote the set of all admissible feedback laws and let R = K[y1 , . . . , yr ]. Then S is controlled and conditioned invariant if and only if there exists β ∈ Rm such that f +g(β◦h) ∈ M, or equivalently, β ◦ h ∈ A. Recall that K[h] = K[h1 , . . . , hr ] ⊆ P denotes the subalgebra of P generated by h1 , . . . , hr . Corollary 9. The set S is controlled and conditioned invariant for the system from Equation (5) if and only if there exists α ∈ A (that is, f + gα ∈ M) and β ∈ Rm such that α = β ◦ h ∈ K[h]m . In other words, S is controlled and conditioned invariant if and only if A ∩ K[h]m = ∅. An algorithm for checking this condition, and for finding a concrete element of A ∩ K[h]m can be found in Schilli et al. (2014). An improved version of the algorithm was presented in Schilli et al. (2015). Example: Consider f, g, and S from the previous example and let h = x1 x3 . Since −x1 x3 ∈ A ∩ K[h]2 , α := −x1 x3 − 1 we conclude that S is controlled and conditioned invariant. Choosing h = (h1 , h2 )T with h1 = x21 x23 + x1 x3 − x21 , h2 = x3 , the situation is less obvious, because α ∈ / K[h1 , h2 ]2 . However, 2 p x2 q α ˜ := α − q + ∈A 0 −x21 q and α ˜=
h22 − h1 − 1 −h1 − 1
∈ K[h1 , h2 ]2 .
Thus S is still controlled and conditioned invariant. 6. CONCLUSION In this paper, the question of (controlled/conditioned) invariance of sets given by algebraic equations and inequations (in the sense of =) has been completely reduced to the known case of invariant varieties, for which algorithmic solutions have been given in Christopher et al. (2009); Zerz et al. (2010); Zerz and Walcher (2012); Schilli et al. (2014, 2016). Controlled and conditioned invariant varieties have also been studied by Yuno and Ohtsuka (2014, 2015). An interesting generalization to sets given by algebraic equations and inequalities (in the sense of <) is treated in Yuno and Ohtsuka (2016). There, dynamic compensators are used instead of static feedback laws. Stability issues have not yet been addressed in this algebraic framework. This remains an open topic for future research. 703
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