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Stabilizability preserving quotients and partial feedback linearization
Proceedings of the Control Conference Africa, 2017 Proceedings the Conference 2017 Johannesburg, Africa, December Africa, 7-8, 2017 Proceedings of of South the Control Control Conference Africa, 2017 Proceedings of the Control Conference Africa, 2017 Johannesburg, 7-8, Available online at www.sciencedirect.com Johannesburg, South South Africa, Africa, December December 7-8, 2017 2017 Johannesburg, South Africa, December 7-8, 2017
ScienceDirect IFAC PapersOnLine 50-2 (2017) 203–208
Stabilizability preserving quotients Stabilizability preserving quotients Stabilizability preserving quotients Stabilizability preserving quotients partial feedback linearization partial feedback linearization partial feedback linearization partial feedback linearization ∗ ∗∗
and and and and
Tinashe Chingozha ∗ Otis T. Nyandoro ∗∗ ∗ Otis T. Nyandoro ∗∗ Tinashe ∗∗∗∗ Tinashe Chingozha T. Nyandoro ∗ Otis Michael A.Chingozha van Wyk ∗∗∗ John D. Ekoru∗∗ ∗∗∗ Tinashe Chingozha T.E ∗∗∗Otis ∗∗∗∗ Michael A. van Wyk John ENyandoro D. Ekoru ∗∗∗∗ Michael A. van Wyk ∗∗∗ John E D. Ekoru ∗∗∗∗ Michael A. van Wyk John E D. Ekoru ∗ ∗ University of the Witwatersrand, Johannesburg South Africa (e-mail: ∗ University of the Witwatersrand, Johannesburg South Africa (e-mail: Witwatersrand, ∗ University of the [email protected]). University of the Witwatersrand, Johannesburg Johannesburg South South Africa Africa (e-mail: (e-mail: [email protected]). ∗∗ [email protected]). of the Witwatersrand, Johannesburg South Africa. ∗∗ [email protected]). ∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗ ∗∗ University University of of the the Witwatersrand, Witwatersrand, Johannesburg Johannesburg South South Africa. Africa. ∗∗∗ of the Witwatersrand, Johannesburg South Africa. ∗∗∗University University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗ University of the Witwatersrand, Johannesburg South Africa. University of the Witwatersrand, Johannesburg South Africa. Abstract: This paper presents the problem of stabilizability preserving quotients. Given a Abstract: This paper Given aa Abstract: This paper presents the problem of stabilizability preserving quotients. Given control system its presents quotient the the problem problem of of stabilizability stabilizability preserving preserving quotients. quotients seeks Abstract: Thisand paper presents the problem of stabilizability preserving quotients. Given to a control system and its quotient the problem of stabilizability preserving quotients seeks to control system and its quotient the problem of stabilizability preserving quotients seeks to characterize all the quotients for which the system is stabilizable if and only if the quotient is control system and quotients its quotient the problem of stabilizability preserving quotients seeks to characterize all the for which the system is stabilizable if and only if the quotient is characterize all quotients for the is and if quotient is stabilizable. This paper presents mathematical formulation of thisif language characterize all the the quotients foraawhich which the system system is stabilizable stabilizable ifproblem and only onlyusing if the thethe quotient is stabilizable. This paper presents mathematical formulation of this problem using the language stabilizable. This paper presents a mathematical formulation of this problem using the language of differential geometry. Taking partial feedback linearization as an instance of quotienting of stabilizable. This paper presents a mathematical formulation of this problem using the language differential geometry. Taking partial feedback linearization as an instance quotienting of of differential geometry. Taking partial feedback linearization as of quotienting of aof control system, it is shown that quotients constructed via partial feedback of linearization of differential geometry. Taking partial feedback linearization as an an instance instance of quotientingare of a control system, it is shown that quotients constructed via partial feedback linearization are a control system, it is shown that quotients constructed via partial feedback linearization are stabilizability preserving if the “zero dynamics” are stable. In addition to this sufficient condition a control system, it is shown that quotients constructed via partial feedback linearization are stabilizability the “zero dynamics” are stable. In to this of sufficient condition stabilizability preserving if the “zero dynamics” are addition to sufficient condition another resultpreserving developed if this paper is the development of addition a construction “zero dynamics” stabilizability preserving ifin the “zero dynamics” are stable. stable. In In addition to this this of sufficient condition another result developed in this paper is the development of a construction “zero dynamics” another result developed in this paper is the development of a construction of “zero dynamics” that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by another result developed in this paper is the development of a construction of “zero dynamics” that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by the quotient map it is shown that it is possible to define an Ehresmann connection on the state that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by the quotient map it is shown that it is possible to define an Ehresmann connection on the state the quotient map it is shown that it is possible to define an Ehresmann connection on the state manifold such that the “zero dynamics” can be defined as a vertical vector field that is related the quotient map it is shown that it is possible to define an Ehresmann connection on the state manifold such that dynamics” be defined as aa vertical vector field that is related manifold such the “zero dynamics” can to the horizontal liftthe of “zero the linear system.can manifold such that that the “zero dynamics” can be be defined defined as as a vertical vertical vector vector field field that that is is related related to the horizontal lift of the linear system. to the horizontal lift of the linear system. to the horizontal lift of the linear system. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Ehresmann connections, quotient control system, zero dynamics. Keywords: Ehresmann connections, connections, quotient control control system, zero zero dynamics. Keywords: Keywords: Ehresmann Ehresmann connections, quotient quotient control system, system, zero dynamics. dynamics. 1. INTRODUCTION bundle with compact structure group the projection of the 1. INTRODUCTION INTRODUCTION bundle with with compact structure group the projection projection of the the 1. bundle the of system onto compact the basestructure space isgroup a controllability preserv1. INTRODUCTION bundle with compact structure group the projection of the system onto the base space is a controllability preservsystem onto base space is preserving reduction of the system abstractions system onto the the basesystem. space Control is aa controllability controllability preservIt is a well known fact that the analysis of control systems ing ing reduction reduction of the the system. Control system abstractions of system. Control system abstractions developed by (Pappas and Simic (2002)) generalized the It is a well known fact that the analysis of control systems ing reduction of the system. Control system abstractions It is a well known fact that the analysis of control systems and synthesis of controllers becomes increasingly difficult developed by (Pappas and Simic (2002)) generalized the It is synthesis a well known fact that the analysis of control systems developed by (Pappas and Simic (2002)) generalized the idea of projecting control systems to the case of systems and of controllers becomes increasingly difficult developed by (Pappas and Simic (2002)) generalized the and synthesis of controllers becomes increasingly difficult as the dimension/order of the control system gets bigger. idea of projecting control systems to the case of systems and synthesis of controllers becomes increasingly difficult idea of projecting control systems to the case of systems modelled on general manifolds. as the dimension/order of the control system gets bigger. idea of projecting control systems to the case of systems as the dimension/order of the control system gets bigger. This has been thecontrol fundamental in modelled modelled on general general manifolds. manifolds. as thephenomenon dimension/order of the systemmotivation gets bigger. This phenomenon has been been the fundamental motivation in modelled on on provides general manifolds. This phenomenon has fundamental motivation in the study of reduction and the decomposition techniques for Quotienting a very general framework for studyThis phenomenon has been the fundamental motivation in the study study control of reduction reduction andOfdecomposition decomposition techniques for Quotienting Quotienting providesreduction, very general general framework for studythe of and for provides aaa very studynonlinear systems. course for the techniques analysis of the ing control system in factframework all of thefor methods the study of reduction and decomposition techniques for Quotienting provides very general framework for studynonlinear control systems. Of course for the analysis of the ing control system reduction, in fact all of the methods nonlinear systems. course analysis of the ing control system reduction, fact all of the lower ordercontrol reduction to beOf of any usefor to the cited in the previous paragraphsin involve some formmethods of quononlinear control systems. Of course for the analysis of the ing control system reduction, in fact all of the methods lower order order reduction to be be of any any use to to the analysis analysis of the the cited cited in in the the previous paragraphs involve some form of of quolower reduction to use the of previous paragraphs involve some form original system the reduction process should preserve tienting (Tabauda and Papas (2005)). The construction of lower order reduction to be of of any use to the analysis of the cited in the previous paragraphs involve some form of quoquooriginal system the reduction reduction process should preserve the tienting tienting (Tabauda and Papas (2005)). The construction of original system the process should preserve the (Tabauda and Papas (2005)). The construction of property of interest. One of the earliest and most general quotients requires the definition of an equivalence relation original system the reduction process should preserve the tienting (Tabauda and Papas (2005)). The construction of propertyonof ofdecomposition interest. One One ofofthe the earliest and most most general quotients requires the the definition definition of an an equivalence equivalence relation property interest. earliest and general requires of relation studies control systems is the work quotients and its equivalence Quotients control system as property ofdecomposition interest. One of ofofthe earliest and most general quotients requires thesets. definition of anof equivalence relation studies on control systems is the work and its equivalence sets. Quotients of control system as studies on decomposition of control systems is the work and its equivalence sets. Quotients of control system as of (Krener (1977)) where it is shown that by studying developed in (Tabauda and Papas (2005)) involve constudies on decomposition ofit control systems is the work and its equivalence sets.and Quotients of control systemconas of (Krener (1977)) where is shown that by studying developed in (Tabauda Papas (2005)) involve of (Krener (1977)) where it is shown that by studying developed in (Tabauda and Papas (2005)) involve conthe structure(1977)) of the where Lie algebra the control system one structing a in quotient spaceand of the state(2005)) space and projecting of (Krener it is of shown that by studying developed (Tabauda Papas involve conthe structure of the Lie algebra of the control system one structing a quotient space of the state space and projecting the structure of algebra the control one aa quotient of the space could decompose theLie system intoof it’s subsysthe system dynamicsspace to the Theprojecting projected the structure of the the algebra theconstitutive control system system one structing structing quotient space of quotient the state state space. space and and could decompose theLie system intoof it’s constitutive subsysthe system system dynamics to the quotient space. Theprojecting projected could decompose the system into it’s constitutive subsysthe dynamics to the quotient space. The projected tems. Decompositions have also been studied as a means dynamics are a coarse approximation of the original system could decompose the system intobeen it’s constitutive subsysthe systemare dynamics to the quotient of space. The projected tems. Decompositions have also studied as a means dynamics a coarse approximation the original system tems. Decompositions have also been studied as a means dynamics are a coarse approximation of the original system of achieving disturbance decoupling and noninteracting as they only capture the gross motion of the system from tems. Decompositions havedecoupling also been and studied as a means dynamics are capture a coarse the approximation of of thethe original system of achieving disturbance noninteracting as they only gross motion system from of achieving disturbance decoupling and noninteracting as they only capture the gross motion of the system from control (Isidori (1995)), these decompositions rely on the one equivalence set to another. All the previously menof achieving disturbance decoupling and noninteracting as they only capture the gross motion of the system from control (Isidori (Isidori (1995)), these these decompositions rely on the the one one equivalence equivalence set to another. another. Allsome the notion previously mencontrol (1995)), decompositions rely on set to All the previously menexistence of a controlled invariant distribution (Nijmeijer tioned reduction methods involve of quoticontrol (Isidori (1995)), these decompositions rely on the one equivalence set to another. All the previously menexistence of aaSchaft controlled invariant distribution (Nijmeijer tioned reduction reduction methods involve some notion of quotiquotiexistence of controlled invariant distribution (Nijmeijer methods some of and van der (1982)). In physics it is known that tioned enting, for symmetry based involve reduction the notion equivalence sets existence of aSchaft controlled invariant distribution (Nijmeijer tioned reduction methods involve some notion of quotiand van der (1982)). In physics it is known that enting, for symmetry based reduction the equivalence sets and van der Schaft (1982)). In physics it is known that enting, for symmetry based reduction the equivalence sets systems that admit symmetries possess conserved quanare the orbits of the symmetry Lie group, for controlled and van that der Schaft (1982)). In physics itconserved is knownquanthat enting, for symmetry based reduction the equivalence sets systems admit symmetries possess are the orbits of the symmetry Lie group, for controlled systems that admit symmetries possess conserved quanare the orbits of the symmetry Lie group, for controlled tities, this is admit the celebrated Noether’s (Frankel based reduction the Lie equivalence sets are the systems that symmetries possesstheorem conserved quan- invariance are the orbits of the symmetry group, for controlled tities, this is the celebrated Noether’s theorem (Frankel invariance based reduction the equivalence sets are the tities, this is (Frankel based reduction equivalence sets are (2004)). Conserved quantitiesNoether’s represent theorem redundant infor- invariance maximal integral the controlled invariance tities, this is the the celebrated celebrated Noether’s theorem (Frankel invariance based submanifolds reduction the theof equivalence sets are the the (2004)). Conserved quantities represent redundant informaximal integral submanifolds of the controlled invariance (2004)). Conserved quantities redundant integral submanifolds of the invariance mation which can be factoredrepresent out resulting in a inforlower maximal distribution and for abstractions the controlled equivalence sets are (2004)). Conserved quantities represent redundant informaximal integral submanifolds of the controlled invariance mationsystem, which such can be be factored out resulting in(van lower distribution and for projection abstractions the equivalence equivalence sets are are mation which can aaa lower for abstractions the order an factored approachout wasresulting taken inin der distribution the level setsand of the map. mation which such can be factored out resulting lower distribution for projection abstractions the equivalence sets sets are order system, system, an approach was taken in inin(van (van der the the level level sets setsand of the the map. order such an approach was taken der of projection map. Schaft (1981)) where it was applied to Hamiltonian control order such an approach was taken in (van der As the previously level sets of the projection map.methods are only useSchaft system, (1981)) where it was was appliedwas to Hamiltonian Hamiltonian control mentioned reduction Schaft (1981)) where it applied to control systems. This notion of symmetry extended to general As previously mentioned reduction methods are only useSchaft (1981)) where it was applied to Hamiltonian control As mentioned methods only usesystems.systems This notion notion of symmetry symmetry was extended to general general in control design/analysis if they preserveare the system As previously previously mentioned reduction reduction methods are only usesystems. This of extended to control in (Grizzle and was Marcus (1985)) where ful ful in control design/analysis if they preserve the system systems. This notion of symmetry was extended to general ful in control design/analysis if they preserve the system control systems systemstechniques in (Grizzle (Grizzle anddeveloped Marcus (1985)) (1985)) where properties relevant to the control design/analysis. For sysful in control design/analysis if they preserve the system control in and Marcus where decomposition were based on the properties relevant to the control design/analysis. For syscontrol systemstechniques in (Grizzle anddeveloped Marcus (1985)) where properties to the control design/analysis. sysdecomposition were based on the tems with relevant a state space endowed with a principalFor bundle properties relevant to the control design/analysis. For sysdecomposition techniques were developed based on the existence of symmetries. In (Martin and Crouch (1984)) it tems with a state space endowed with a principal bundle decomposition techniques were developed based on the with aa state space endowed with aa principal existence ofthat symmetries. In (Martin (Martin and Crouch (1984)) it tems structure (Martin and Crouch (1984)) showed thatbundle if the tems with state space endowed with principal bundle existence of symmetries. In and Crouch (1984)) it was shown for systems modelled on a principal fibre structure (Martin (Martin and and Crouch Crouch (1984)) (1984)) showed showed that that if if the the existence symmetries. In (Martin and (1984)) it structure was shown shownofthat for systems systems modelled onCrouch a principal fibre was was shown that that for for systems modelled modelled on on aa principal principal fibre fibre structure (Martin and Crouch (1984)) showed that if the Copyright © 2017, 2017 IFAC 203 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 203 Copyright ©under 2017 responsibility IFAC 203Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 203 10.1016/j.ifacol.2017.12.037
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principal bundle has compact structure group and if the system is accessible then controllability of the projected system on the base manifold is a necessary and sufficient condition for controllabillity of the system. In (Pappas and Simic (2002)) a complete theory of controllability preserving quotients was developed for general nonlinear control systems. As an application of the controllablity preserving quotients of (Pappas and Simic (2002)) a hierarchical controllability algorithm for linear systems was developed in (Pappas et al. (2000)). This same approach was brought to bear on the problem of stabilizability preserving quotients for linear systems in (Pappas and Lafferriere (2001)) where stabilizability preserving linear quotients of linear systems were fully characterized. To our best knowledge no such general results exist for stabilizability preserving reductions of nonlinear systems. Of course this can be attributted to the fact that unlike the case for linear systems no simple test exists for deciding stabilizability of general nonlinear systems. This work therefore presents a step towards trying to develop a framework for stabilizability preserving quotients for nonlinear systems. The rest of the paper is organized as follows section 2 contains a brief summary of the prerequiste mathematical concepts, section 3 presents the mathematical formulation of the problem. In section 4 the connection of feedback linearization with stabilizability preserving quotients is made, the main result of this paper is developed in this section. An application of these results is illustrated through an example in section 5, concluding remarks are given in section 7. 2. MATHEMATICAL PRELIMINARIES This section presents a brief survey of the differential geometric concepts that will be used in the main sequel. For a more comprehensive coverage of the concepts presented here the reader may consult (S.Kobayshi and Nomizu (1963)). All manifolds and all maps considered will be C ∞ . Let M be a n-dimensional manifold and p ∈ M , the set of all vectors at p is called the tangent space at p and will be denoted Tp M . The disjoint union of all tangent spaces at all points in M has a manifold structure, this manifold is called the tangent bundle, it is 2n-dimensional and will be denoted T M . Let M and N be manifolds, consider the map ψ : M → N , this map induces a linear map Tp ψ : Tp M → Tψ(p) N called the tangent map.
A fiber bundle is a manifold that locally looks like a product space. Formally we define a fiber bundle as follows. Definition 1. (SteenRod (1960)) A fiber bundle is the 4tuple (E, M, π, U ) where E and M are smooth manifolds called the total space and base space respectively, π is a surjective map from E to M , π : E → M which is called the projection map . For some x in M the set π −1 (x) is called the fiber over the point x in M where π −1 (x) is required to be homeomorphic to the smooth manifold U . U is called the typical fiber over M . The tangent bundle introduced above is an example of a vector bundle(i.e fiber bundle where the fibers are homeomorphic to a vector space), the manifold M is the base space, πM is the projection map which maps a tangent vector to its base point. When T M is viewed as a vector 204
bundle vector fields can be intepreted as sections of the vector bundle. Consider a smooth vector field on the total space X ∈ Γ∞ (T E) where Γ∞ (T E) is the set of all smooth sections of the tangent bundle T E over E, the vector field X is called projectable if there exists a vector field on the base space Y ∈ Γ∞ (T M ) such that the following holds Tx π ◦ X(x) = Y ◦ π(x). (1) It can easily be shown that if the curve γ(t) in E is an an integral curve of the vector field X then the curve π ◦ γ(t) in M is an integral curve of the vector field Y . Since π is surjective the projection operation is a many to one operation, thus there is no canonical way of relating a vector field (smooth curve) on the base space back to a vector field (smooth curve) on the total space. To be able to uniquely relate a vector field (smooth curve) on the base space back to a vector field (smooth curve) on the total space requires the specification of extra structure and this extra structure comes in the form of a connection. Definition 2. Let (E, M, π, U ) be a fiber bundle, the vertical bundle VE is the sub-bundle of the tangent bundle T E defined as V E = {vx ∈ Tx E|Tx π(vx ) = 0, ∀x ∈ E} = ker(T π). (2) The vertical bundle contains all those tangent vectors that get projected to the zero vector, this means that the vertical bundle contains all those vectors which are tangent to the fibers. A connection on a fiber bundle (E, M, π, U ) is a sub-bundle of T E which is complementary to the vertical bundle, this sub-bundle is called the horizontal sub-bundle (Cushman and Bates (2015)). This is expressed formally as: Definition 3. A connection on the fiber bundle (E, M, π, U ) is a smooth sub-bundle HE ⊂ T E such that T E = HE ⊕ V E. The connection makes it possible to lift objects from the base space back to the total space. Consider a vector Yp ∈ Tp M on the base space, the horizontal lift of Yp denoted Horx (Yp ) ∈ Tx E is a vector on the total space which gets projected to the vector Yp . Thus there is a horizontal lift map Horx : Tp M → Tx E, where π(x) = p (3) This horizontal lift map satisfies the property that Tx π ◦ Horx = idTp M . For a curve γ(t) ∈ M the horizontally lifted curve γ˜ (t) ∈ E is such that, d γ˜ (t) = Horγ˜ (t) (γ(t)) ˙ (4) dt ¯ ∈ Tx E which get projected to the Given two vectors X, X ¯ the difference between same vector i.e Tx π(X) = Tx π(X), ¯ ∈ Vx E). these two vectors is a vertical vector (X − X 3. PROBLEM STATEMENT We shall make use of the fiber bundle description developed in (Brockett (1977)), a control system is defined by the 5-tuple (E, M, π, U, F ) where (E, M, π, U ) defines a fiber bundle and a smooth map F : E → T M which satisfies πT M ◦ F = π where πT M is the tangent bundle projection. This situation is described by the following commutative diagram.
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Proof. Let γ M (t) be a trajectory of Σ by definition there exists a curve γ E (t) such that π ◦ γ E (t) = γ M (t). From the fiber preserving property we have:
E F
TM
π
π ˜ ◦ ψ ◦ γ E (t) = φ ◦ π ◦ γ E (t)
πT M
M Within this framework of control system representation trajectories of a control system are given by the following definition from (Tabauda and Papas (2005)). Definition 4. Let Σ = (E, M, π, U, F ) be a control system, the smooth curve γ M (t) : R → M is called a trajectory of Σ if there exists a curve γ E (t) : R → E such that: (1) π ◦ γ E (t) = γ M (t) d M γ (t) = F ◦ γ E (t) (2) dt
ψ
π
M
˜ E π ˜
φ
N
˜ N, π ˜ ) are fiber bundles. ConNote (E, M, π, U ) and (E, ˜, U sider the two control systems Σ = (E, M, π, U, F ) and ˜ = (E, ˜ N, π ˜ , F˜ ), Σ ˜ is a quotient of Σ if there exists a Σ ˜, U ˜ surjective submersion map φ : M → N , a map ψ : E → E such that the pair (φ, ψ) is a fiber preserving map and given a smooth curve γ M (t) ∈ M which is a trajectory of ˜ Σ the curve φ(γ M (t)) ∈ N is a trajectory of Σ.
Proposition 1. Consider the two control systems Σ = ˜ = (E, ˜ N, π ˜ , F˜ ). Σ ˜ is a quotient (E, M, π, U, F ) and Σ ˜, U of Σ if there exists the bundle morphism (φ, ψ) where φ is a surjective submersion which makes the following diagram commute. ψ
E TM
Tφ
F˜
TN
π ˜ πT N
πT M
M
d d (φ ◦ γ M (t)) = T φ ◦ (γ M (t)) dt dt = T φ ◦ F ◦ γ E (t) = F˜ ◦ ψ ◦ γ E (t)
φ
Thus the problem of stabilizability preserving quotient can be phrased in the following way. Problem Statement 1. Given the smooth control system ˜ = (E, ˜ N, π ˜ , F˜ ), Σ = (E, M, π, U, F ) and it’s quotient Σ ˜, U where the quotient is defined by the fiber morphism (φ, ψ) which satisfies proposition 1. Characterize the fiber bundle ˜ is morphisms for which Σ is stabilizable if and only if Σ stabilizable In the following section this problem is studied for systems that are feedback linearizable where it is shown that the “zero dynamics” play an essential role in answering the problem. 4. STABILIZABILITY PRESERVING QUOTIENTS OF FEEDBACK LINEARIZABLE SYSTEMS The main idea behind feedback linearization is the realization that at times non-linearities are a product of the choice of coordinate system used to represent the control system. As such feedback linearization prescribes a method of constructing coordinate transformations and feedback transformations such that in the new coordinates the system is a controllable linear system. For the sake of brevity we shall focus on affine single input single output systems. Consider an affine n-dimensional control system x˙ = f (x) + g(x)u,
x ∈ Rn ,
u∈R
(5)
The system is said to have a relative degree r ∈ N at some point x0 ∈ Rn if there exists a function λ : Rn → R such that (1) Lg Lkf λ(x) = 0 k < r − 1 (2) Lg Lr−1 λ(x0 ) = 0 f If r = n the affine control system can be exactly linearized, the case that will be of interest is when r < n this is the partial linearization case (Isidori (1995)). For the case of r < n the state transformation φ(x) : Rn → Rr is constructed as follows T φ = λ(x), Lf λ(x), · · · , Lr−1 λ(x) (6) f
˜ E
F π
π ˜ ◦ (ψ ◦ γ E )(t) = φ ◦ γ M (t) This proves the first part of the trajectory definition. For the second part differentiate the curve φ ◦ γ M .
= F˜ ◦ (ψ ◦ γ E )(t)
Let M be a n-dimensional smooth manifold assume there exists an equivalence relation on M such that the equivalence sets on M are all of the same dimension q. The equivalence sets can be viewed as leaves of a foliation of M , from the Haefliger cocycle representation of foliations we know that this foliation can be represented as level sets of a surjective submersion map (Moerdijk and Mrcun (2003)). Thus the quotienting of the state space of the control system can be compactly represented by simply choosing the appropriate surjective submersion map φ : M → N where N is the quotient space. Since the control system is modelled on a fiber bundle we will require the map that characterizes the quotienting procedure to preserve the fiber bundle structure. Thus we require also a map ˜ where E is the total space over M and ψ : E → E, ˜ E is total space over N such that the following diagram commutes. E
205
N
and there exists a feedback transformation u = α(x) + β(x)v such that in these new coordinates the system is a r-dimensional linear controllable system. Within the 205
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quotient framework developed in the previous section this situation can be described as follows. For a n-dimensional control system Σ = (E, M, π, U, F ) which in fibered coordinates (x, u) of E is affine and has a relative degree r < n, there exists a fiber bundle morphism (φ, ψ) : (E, M, π, U ) → (Rr+1 , Rr , π ˜ , R) where φ is defined as is shown in equation (6) above and ψ : E → Rr+1 is defined as u − α(x) ψ : (x, u) → (z = φ(x), v = ) (7) β(x) ˜ , R) is a fiber bundle with total space Note that (Rr+1 , Rr , π Rr+1 , base space Rr and projection map π ˜ : Rn+1 → Rr . The fiber bundle morphism defines a linear controllable ˜ = (Rr+1 , Rr , π quotient control system Σ ˜ , R, F˜ ) with F˜ : r+1 r R → R defined as F˜ (z, v) = T φ ◦ F (φ−1 (z), α(φ−1 (z)) + β(φ−1 (z))v) (8) ˜ is stabilizable since it is The quotient control system Σ controllable by construction. Let x0 ∈ M be an equilibrium point of Σ and also assume that φ(x0 ) = 0. Stabiliz˜ guarantees that there exists a linear feedback ability of Σ controller such that for all z ∈ Rr in the neighbourhood of the origin there is a trajectory γ˜ : R → Rr which starts at z i.e. γ˜ (0) = z and asymptotically gets close to the origin as time tends to infinite. Stabilizability of the linear quotient system therefore guarantees that the original system is asymptotically stabilizable to the the submanifold φ−1 (0). The triple (M, Rr , φ) is a fiber bundle with the equivalence sets of the surjective submersion map φ being the fibers. Now consider the submanifold φ(x) = 0, the system dynamics on this manifold are given by differentiating φ(x) from which we get, dφ(x) = T φ ◦ F (x, u) = 0. dt
(9)
That is γ H (t) is a trajectory of the horizontally lifted system. Consider the time derivative of the curve φ ◦ γ H (t) ∈ Rr , d d (φ ◦ γ H (t)) = Tγ H (t) φ ◦ γ H (t) (12) dt dt = Tγ H (t) φ ◦ Horγ H (t) ◦ F˜ ◦ (φ, ψ) · · · (13) ◦ (γ H (t), uH (t)) = F˜ ◦ (φ, ψ) ◦ (γ H (t), uH (t)).
(14)
(15)
This shows that the trajectories of the horizontally lifted system get projected to the trajectories of the original system. Consider the control system defined as the difference between the original system dynamics and the horizontally lifted linear system dynamics, F Z (x, u) = F (x, u) − Horx ◦ F˜ ◦ (φ, ψ) ◦ (x, u).
(16)
By construction the original system dynamics and the horizontally lifted dynamics are horizontal thus the difference between the two is vertical. Therefore F Z (x, u) restricted to the submanifold φ−1 (0) represents the system dynamics on this submanifold. Now assume that there exists a linear feedback v = Kz that stabilizes the quotient linear controllable system. Under the action of this feedback the closed loop dynamics of the original system when restricted to the submanifold φ−1 (0) are, F (x, u) − Horx ◦ F˜ ◦ (φ(x), ψ(x, u))|u=α(x)+β(x)Kφ(x) (17) If the closed loop dynamics given by (17) are stable then the quotient map φ defined in equation (6) is a stabilizability preserving quotient map. This result is stated formally as follows. Result 1. For a control system Σ = (E, M, π, U, F ) assume there exists a function λ : M → R such that Σ has relative degree r < dim(M ). The fiber bundle morphism (φ, ψ) : (E, M, π, U ) → (Rr+1 , Rr , π ˜ , R) defined as
Thus the system dynamics on the submanifold φ(x) = 0 lie in the kernel of the tangent map which means that the vector fields induced by the control system on this submanifold are vertical. By equipping the fiber T bundle (M, Rr , φ) with a connection HM the vertical φ(x) = λ(x), Lf λ(x), · · · , Lr−1 λ(x) (18) f component of the system dynamics can be calculated as the difference between the original system dynamics and u − α(x) ψ(x, u) = (φ(x), ) (19) the horizontal component. Given a connection on the fiber β(x) bundle (M, Rr , φ) define the horizontal lift of the linear is a stabilizability preserving quotient if the dynamics controllable system as follows. r+1 r ˜ ˜ Definition 5. Given the linear system Σ = (R , R , π ˜ , R, F ) given by H F (x, u) − Horx ◦ F˜ ◦ (φ(x), ψ(x, u))|u=α(x)+β(x)Kφ(x) (20) the horizontally lifted system is Σ = (E, M, π, U, F˜ H ) H where the map F˜ : E → T M is defined as are stable. F˜ H (x, u) = Horx ◦ F˜ ◦ (φ, ψ)(x, u).
(10)
To show that this definition of the horizontally lifted control system is proper consider the curve (γ H (t), uH (t)) ∈ E such that, d H γ (t) = Horγ H (t) ◦ F˜ ◦ (φ, ψ) ◦ (γ H (t), uH (t)). dt
(11) 206
The dynamics given by F Z (x, u) coincide with the notion of “zero dynamics” defined in (Krener and Isidori (1980)) where the zero dynamics are given by the dynamics of the system when the input is chosen such that the output stays zero. Finally let us see how the following proposition from (Isidori (1995)) fits into the notion of zero dynamics developed in this paper. Proposition 2. Suppose an n-dimensional system has relative degree r < n at x0 . Set λi = Li−1 f h(x) for i =
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1, · · · , r, it is always possible to find n − r functions λr+1 (x), · · · , λn (x) such that the mapping Λ(x) =
λ1 (x) ··· λn (x)
η1 (x) + 2η1 (x)η2 (x)η4 (x) − η2 (x)η3 (x) (29) 1 + η4 (x)η2 (x) η3 (x) − η4 (x)η1 (x) . (30) a21 = 1 + η4 (x)η2 (x)
a11 (x) =
The horizontal lift of the linear controllable quotient system becomes
is a diffeomorphism.
From the diffeomorphism Λ let Ω1 and Ω2 be two cotangent subbundles defined as follows. Ω1 = [dλ1 (x), · · · , dλr (x)]
T
Ω2 = [dλr+1 (x), · · · , dλn (x)]
(21) T
(22)
The vertical and horizontal sub-bundles can be defined as the annihilators of the cotangent bundles Ω1 and Ω2 ⊥ respectivley(V M = Ω⊥ 1 , HM = Ω2 ). Thus the choice of n−r maps in the above proposition can be understood as a prescription of a flat connection on the state manifold. To show how these ideas can be applied consider the following example taken from (Isidori (1995)). 5. EXAMPLE Consider the SISO system x2 e −x1 1 u x˙ = x1 x2 + 0 x2 y = h(x) = x3
(23) (24)
For this system let λ(x) = h(x) then the surjective submersion φ : R3 → R2 is given by φ(x) = (h(x), Lf h(x)) = (z1 = x3 , z2 = x2 ). (25) Together with the feedback transformation ψ(x, u) = v = u + x1 x2 the pair (φ, ψ) forms a fiber bundle morphism which defines a controllable linear quotient control system defined as z˙1 = z2 z˙2 = v.
207
(26)
The vertical bundle denoted as V is the kernel of T φ ∂ and is the subspace V = span{ ∂x }. Possible choices of 1 the horizontal subspace denoted H are subspaces of the following form. ∂ ∂ ∂ ∂ + η2 (x) + , η3 (x) ··· ∂x1 ∂x2 ∂x3 ∂x1 ∂ ∂ + η4 (x) } (27) + ∂x2 ∂x3
H = span{η1 (x)
Where ηi : R3 → R for i = 1, · · · , 4. The corresponding horizontal lift map is given as a 3 × 2 matrix of the form a11 (x) a21 (x) 0 1 Horx = . (28) 1 0 With the functions a11 (x) and a12 (x) being defined as,
207
a11 (x)z2 + a21 (x)v v z2
(31) z=φ(x),v=ψ(x,u)
The linear feedback v = K1 z1 + K2 z2 , K1 , K2 < 0 renders the quotient control system asymptotically stable. Using this feedback to control the original system will render the submanifold φ(x) = 0 asymptotically stable. The resulting system dynamics on this submanifold are given by the difference between the original system dynamics and the horizontally lifted closed loop dynamics. These dynamics have the form
−x1 + uex2 − a11 (x)x2 + a12 (x)(u + x1 x2 ) 0 0
(32)
For u = x1 x2 +K1 x3 +K2 x2 . Note that on the submanifold φ(x) = 0 the coordinates are of the form (x1 , 0, 0) thus the above dynamics take the form x˙1 = −x1 + u(1 + a12 (x)), which is an asymptotically stabilizable dynamical system. This shows that the surjective submersion map φ(x) is a stabilizability preserving quotient map. Now compare the approach that has been set out with the typical feedback linearisation approach as set out in (Isidori (1995)). By proposition 2 the map φ(x) is extended to a diffeomorphism Λ : R3 → R3 which is of the form: x3 h(x) z1 x2 (33) Φ(x) = z2 = Lf h(x) = φ3 (x) 1 + x1 − exp(x2 ) z3 The choice of the map φ3 (x) is equivalent to a choice of an Ehressman connection on the state space. This connection is defined as the kernel of the tangent map of φ3 (x) : R3 → R which gives the horizontal subspace H = span{
∂ ∂ ∂ , ex 2 + } ∂x3 ∂x1 ∂x2
(34)
Comparing this horizontal subspace to the general horizontal subspace gives the following η1 = η2 = η4 = 0, η3 = ex2 which gives the following horizontal lift map
Horx =
0 ex2 0 1 1 0
.
From which the zero dynamics are calculated as: −x1 (1 + x2 exp(x2 )) Z 0 f = 0
(35)
(36)
The zero dynamics are parameterised by (x2 , x3 ) which means that the dynamics vary depending on the fiber
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however the linear feedback constrains the system dynamics to evolve on the submanifold φ−1 (0)this implies that (x2 = 0, x3 = 0), the dynamics therefore become x˙ 1 = −x1 . (37) Which agrees with the key result in (Isidori (1995)) which serves to verify the validity of the approach presented in this paper. 6. SUMMARY OF RESULT The question posed in this paper deals with the existence of necessary and sufficient conditions for the existence of stabilizability preserving quotients. The procedure developed in this paper shows that having a finite relative degree and stable zero dynamics is sufficient for a feedback linearizable system to admit stabilizability preserving quotients. It should be noted that the main advantage of the notion of ”zero dynamics” developed in this paper is that it can be applied to general nonlinear systems which are not necessarily feedback linearizable. However this result can be seen to very restrictive from two view points. Firstly the set of systems that are feedback linearizable is very small (Respondek and Tall (2002)) secondly this approach requires the feedback controller of the lower dimensional to be able to stabilize the higher dimensional system. Future work will therefore focus on relaxing this restrictions by looking at general nonlinear quotient system and also by developing a lifting procedure of the feedback controller for the lower dimensional quotient system. This lifting procedure will aim to add fiber dependent elements to the low dimensional controller so as to render the higher dimensional system stable. 7. CONCLUSION Quotient control systems provide a very general way of dealing with control system reductions. In this paper we have presented the notion of stabilizability preserving quotients and how the problem of their construction can be dealt with in a differential geometric framework. As a special instance of quotients of control systems the case of partial feedback linearization is studied. This quotienting process is stabilizability preserving if the “zero dynamics” are stable. An alternative interpretation of the notion of zero dynamics is developed where it is shown that, by utilising the fiber bundle structure of the state space manifold a connection can be defined on the state space manifold. This connection enables the lifting of the linearized dynamics from the base space to the total space. The zero dynamics are then defined as the difference between the original dynamics and the lifted linear dynamics. It then follows from the construction that the zero dynamics represent the motion of the system along the fibres while the linearized dynamics represent the motion of the system from fiber to fiber. REFERENCES Brockett, R. (1977). Control theory and analytical mechanics. In C. Martin and R. Hermann (eds.), Geometric Control Theory. Math. Sci Press. 208
Cushman, R.H. and Bates, L.M. (2015). Global Aspects of Classical Integrable Systems. Springer. Frankel, T. (2004). The Geometry of Physics An Introduction. Cambridge University Press, second edition. Grizzle, J.W. and Marcus, S.I. (1985). The structure of nonlinear control systems possessing symmetries. IEEE Transactions on Automatic Control, AC-30(3), 248–258. Isidori, A. (1995). Nonlinear Control Systems. Springer. Krener, A. and Isidori, A. (1980). Nonlinear zero distributions. In IEEE Conference on Decision and Control, volume 20, 665–668. Krener, A.J. (1977). A decomposition theory for differentiable systems. SIAM Journal of Control and Optimization, 15(5), 813–829. Martin, L.S. and Crouch, P. (1984). Controllability on principal fibre bundles with compact structure group. Systems and Control Letters, 5, 35–40. Moerdijk, I. and Mrcun, J. (2003). Introduction to Foliations and Lie Groupoids. Cambridge University Press. Nijmeijer, H. and van der Schaft, A. (1982). Controlled invariance for nonlinear systems. IEEE Transactions on Automatic Control, AC-27(4), 904–914. Pappas, G.J. and Lafferriere, G. (2001). Hierarchies of stabilizability preserving linear systems. In 40th IEEE Conference on Decision and Control, 2081–2085. Pappas, G.J., Lafferriere, G., and Sastry, S. (2000). Hierarchically consistent control systems. IEEE Transaction on Automatic Control, 45, 1144–1160. Pappas, G.J. and Simic, S. (2002). Consistent abstractions of affine control systems. IEEE Transactions on Automatic Control, 47(5), 745–756. Respondek, W. and Tall, I.A. (2002). Nonlinearizable single input control systems do not admit stationary symmetries. Systems and Control Letters, (46), 1 – 16. S.Kobayshi and Nomizu, K. (1963). Foundations of Differential Geometry Volume 1. InterScience Publishers. SteenRod, N. (1960). The Topology of Fibre Bundles. Princeton University Press. Tabauda, P. and Papas, G.J. (2005). Quotients for fully nonlinear control systems. SIAM Journal of Control and Optimization, 43(5), 1844–1860. van der Schaft, A. (1981). Symmetries and conservation laws for hmailtonian systems with inputs and outputs: A generalization of noether’s theorem. Systems and Control Letters, 1(2), 108–115.
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Stabilizability preserving quotients Stabilizability preserving quotients Stabilizability preserving quotients Stabilizability preserving quotients partial feedback linearization partial feedback linearization partial feedback linearization partial feedback linearization ∗ ∗∗
and and and and
Tinashe Chingozha ∗ Otis T. Nyandoro ∗∗ ∗ Otis T. Nyandoro ∗∗ Tinashe ∗∗∗∗ Tinashe Chingozha T. Nyandoro ∗ Otis Michael A.Chingozha van Wyk ∗∗∗ John D. Ekoru∗∗ ∗∗∗ Tinashe Chingozha T.E ∗∗∗Otis ∗∗∗∗ Michael A. van Wyk John ENyandoro D. Ekoru ∗∗∗∗ Michael A. van Wyk ∗∗∗ John E D. Ekoru ∗∗∗∗ Michael A. van Wyk John E D. Ekoru ∗ ∗ University of the Witwatersrand, Johannesburg South Africa (e-mail: ∗ University of the Witwatersrand, Johannesburg South Africa (e-mail: Witwatersrand, ∗ University of the [email protected]). University of the Witwatersrand, Johannesburg Johannesburg South South Africa Africa (e-mail: (e-mail: [email protected]). ∗∗ [email protected]). of the Witwatersrand, Johannesburg South Africa. ∗∗ [email protected]). ∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗ ∗∗ University University of of the the Witwatersrand, Witwatersrand, Johannesburg Johannesburg South South Africa. Africa. ∗∗∗ of the Witwatersrand, Johannesburg South Africa. ∗∗∗University University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗ University of the Witwatersrand, Johannesburg South Africa. ∗∗∗∗ University of the Witwatersrand, Johannesburg South Africa. University of the Witwatersrand, Johannesburg South Africa. Abstract: This paper presents the problem of stabilizability preserving quotients. Given a Abstract: This paper Given aa Abstract: This paper presents the problem of stabilizability preserving quotients. Given control system its presents quotient the the problem problem of of stabilizability stabilizability preserving preserving quotients. quotients seeks Abstract: Thisand paper presents the problem of stabilizability preserving quotients. Given to a control system and its quotient the problem of stabilizability preserving quotients seeks to control system and its quotient the problem of stabilizability preserving quotients seeks to characterize all the quotients for which the system is stabilizable if and only if the quotient is control system and quotients its quotient the problem of stabilizability preserving quotients seeks to characterize all the for which the system is stabilizable if and only if the quotient is characterize all quotients for the is and if quotient is stabilizable. This paper presents mathematical formulation of thisif language characterize all the the quotients foraawhich which the system system is stabilizable stabilizable ifproblem and only onlyusing if the thethe quotient is stabilizable. This paper presents mathematical formulation of this problem using the language stabilizable. This paper presents a mathematical formulation of this problem using the language of differential geometry. Taking partial feedback linearization as an instance of quotienting of stabilizable. This paper presents a mathematical formulation of this problem using the language differential geometry. Taking partial feedback linearization as an instance quotienting of of differential geometry. Taking partial feedback linearization as of quotienting of aof control system, it is shown that quotients constructed via partial feedback of linearization of differential geometry. Taking partial feedback linearization as an an instance instance of quotientingare of a control system, it is shown that quotients constructed via partial feedback linearization are a control system, it is shown that quotients constructed via partial feedback linearization are stabilizability preserving if the “zero dynamics” are stable. In addition to this sufficient condition a control system, it is shown that quotients constructed via partial feedback linearization are stabilizability the “zero dynamics” are stable. In to this of sufficient condition stabilizability preserving if the “zero dynamics” are addition to sufficient condition another resultpreserving developed if this paper is the development of addition a construction “zero dynamics” stabilizability preserving ifin the “zero dynamics” are stable. stable. In In addition to this this of sufficient condition another result developed in this paper is the development of a construction “zero dynamics” another result developed in this paper is the development of a construction of “zero dynamics” that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by another result developed in this paper is the development of a construction of “zero dynamics” that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by the quotient map it is shown that it is possible to define an Ehresmann connection on the state that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by the quotient map it is shown that it is possible to define an Ehresmann connection on the state the quotient map it is shown that it is possible to define an Ehresmann connection on the state manifold such that the “zero dynamics” can be defined as a vertical vector field that is related the quotient map it is shown that it is possible to define an Ehresmann connection on the state manifold such that dynamics” be defined as aa vertical vector field that is related manifold such the “zero dynamics” can to the horizontal liftthe of “zero the linear system.can manifold such that that the “zero dynamics” can be be defined defined as as a vertical vertical vector vector field field that that is is related related to the horizontal lift of the linear system. to the horizontal lift of the linear system. to the horizontal lift of the linear system. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Ehresmann connections, quotient control system, zero dynamics. Keywords: Ehresmann connections, connections, quotient control control system, zero zero dynamics. Keywords: Keywords: Ehresmann Ehresmann connections, quotient quotient control system, system, zero dynamics. dynamics. 1. INTRODUCTION bundle with compact structure group the projection of the 1. INTRODUCTION INTRODUCTION bundle with with compact structure group the projection projection of the the 1. bundle the of system onto compact the basestructure space isgroup a controllability preserv1. INTRODUCTION bundle with compact structure group the projection of the system onto the base space is a controllability preservsystem onto base space is preserving reduction of the system abstractions system onto the the basesystem. space Control is aa controllability controllability preservIt is a well known fact that the analysis of control systems ing ing reduction reduction of the the system. Control system abstractions of system. Control system abstractions developed by (Pappas and Simic (2002)) generalized the It is a well known fact that the analysis of control systems ing reduction of the system. Control system abstractions It is a well known fact that the analysis of control systems and synthesis of controllers becomes increasingly difficult developed by (Pappas and Simic (2002)) generalized the It is synthesis a well known fact that the analysis of control systems developed by (Pappas and Simic (2002)) generalized the idea of projecting control systems to the case of systems and of controllers becomes increasingly difficult developed by (Pappas and Simic (2002)) generalized the and synthesis of controllers becomes increasingly difficult as the dimension/order of the control system gets bigger. idea of projecting control systems to the case of systems and synthesis of controllers becomes increasingly difficult idea of projecting control systems to the case of systems modelled on general manifolds. as the dimension/order of the control system gets bigger. idea of projecting control systems to the case of systems as the dimension/order of the control system gets bigger. This has been thecontrol fundamental in modelled modelled on general general manifolds. manifolds. as thephenomenon dimension/order of the systemmotivation gets bigger. This phenomenon has been been the fundamental motivation in modelled on on provides general manifolds. This phenomenon has fundamental motivation in the study of reduction and the decomposition techniques for Quotienting a very general framework for studyThis phenomenon has been the fundamental motivation in the study study control of reduction reduction andOfdecomposition decomposition techniques for Quotienting Quotienting providesreduction, very general general framework for studythe of and for provides aaa very studynonlinear systems. course for the techniques analysis of the ing control system in factframework all of thefor methods the study of reduction and decomposition techniques for Quotienting provides very general framework for studynonlinear control systems. Of course for the analysis of the ing control system reduction, in fact all of the methods nonlinear systems. course analysis of the ing control system reduction, fact all of the lower ordercontrol reduction to beOf of any usefor to the cited in the previous paragraphsin involve some formmethods of quononlinear control systems. Of course for the analysis of the ing control system reduction, in fact all of the methods lower order order reduction to be be of any any use to to the analysis analysis of the the cited cited in in the the previous paragraphs involve some form of of quolower reduction to use the of previous paragraphs involve some form original system the reduction process should preserve tienting (Tabauda and Papas (2005)). The construction of lower order reduction to be of of any use to the analysis of the cited in the previous paragraphs involve some form of quoquooriginal system the reduction reduction process should preserve the tienting tienting (Tabauda and Papas (2005)). The construction of original system the process should preserve the (Tabauda and Papas (2005)). The construction of property of interest. One of the earliest and most general quotients requires the definition of an equivalence relation original system the reduction process should preserve the tienting (Tabauda and Papas (2005)). The construction of propertyonof ofdecomposition interest. One One ofofthe the earliest and most most general quotients requires the the definition definition of an an equivalence equivalence relation property interest. earliest and general requires of relation studies control systems is the work quotients and its equivalence Quotients control system as property ofdecomposition interest. One of ofofthe earliest and most general quotients requires thesets. definition of anof equivalence relation studies on control systems is the work and its equivalence sets. Quotients of control system as studies on decomposition of control systems is the work and its equivalence sets. Quotients of control system as of (Krener (1977)) where it is shown that by studying developed in (Tabauda and Papas (2005)) involve constudies on decomposition ofit control systems is the work and its equivalence sets.and Quotients of control systemconas of (Krener (1977)) where is shown that by studying developed in (Tabauda Papas (2005)) involve of (Krener (1977)) where it is shown that by studying developed in (Tabauda and Papas (2005)) involve conthe structure(1977)) of the where Lie algebra the control system one structing a in quotient spaceand of the state(2005)) space and projecting of (Krener it is of shown that by studying developed (Tabauda Papas involve conthe structure of the Lie algebra of the control system one structing a quotient space of the state space and projecting the structure of algebra the control one aa quotient of the space could decompose theLie system intoof it’s subsysthe system dynamicsspace to the Theprojecting projected the structure of the the algebra theconstitutive control system system one structing structing quotient space of quotient the state state space. space and and could decompose theLie system intoof it’s constitutive subsysthe system system dynamics to the quotient space. Theprojecting projected could decompose the system into it’s constitutive subsysthe dynamics to the quotient space. The projected tems. Decompositions have also been studied as a means dynamics are a coarse approximation of the original system could decompose the system intobeen it’s constitutive subsysthe systemare dynamics to the quotient of space. The projected tems. Decompositions have also studied as a means dynamics a coarse approximation the original system tems. Decompositions have also been studied as a means dynamics are a coarse approximation of the original system of achieving disturbance decoupling and noninteracting as they only capture the gross motion of the system from tems. Decompositions havedecoupling also been and studied as a means dynamics are capture a coarse the approximation of of thethe original system of achieving disturbance noninteracting as they only gross motion system from of achieving disturbance decoupling and noninteracting as they only capture the gross motion of the system from control (Isidori (1995)), these decompositions rely on the one equivalence set to another. All the previously menof achieving disturbance decoupling and noninteracting as they only capture the gross motion of the system from control (Isidori (Isidori (1995)), these these decompositions rely on the the one one equivalence equivalence set to another. another. Allsome the notion previously mencontrol (1995)), decompositions rely on set to All the previously menexistence of a controlled invariant distribution (Nijmeijer tioned reduction methods involve of quoticontrol (Isidori (1995)), these decompositions rely on the one equivalence set to another. All the previously menexistence of aaSchaft controlled invariant distribution (Nijmeijer tioned reduction reduction methods involve some notion of quotiquotiexistence of controlled invariant distribution (Nijmeijer methods some of and van der (1982)). In physics it is known that tioned enting, for symmetry based involve reduction the notion equivalence sets existence of aSchaft controlled invariant distribution (Nijmeijer tioned reduction methods involve some notion of quotiand van der (1982)). In physics it is known that enting, for symmetry based reduction the equivalence sets and van der Schaft (1982)). In physics it is known that enting, for symmetry based reduction the equivalence sets systems that admit symmetries possess conserved quanare the orbits of the symmetry Lie group, for controlled and van that der Schaft (1982)). In physics itconserved is knownquanthat enting, for symmetry based reduction the equivalence sets systems admit symmetries possess are the orbits of the symmetry Lie group, for controlled systems that admit symmetries possess conserved quanare the orbits of the symmetry Lie group, for controlled tities, this is admit the celebrated Noether’s (Frankel based reduction the Lie equivalence sets are the systems that symmetries possesstheorem conserved quan- invariance are the orbits of the symmetry group, for controlled tities, this is the celebrated Noether’s theorem (Frankel invariance based reduction the equivalence sets are the tities, this is (Frankel based reduction equivalence sets are (2004)). Conserved quantitiesNoether’s represent theorem redundant infor- invariance maximal integral the controlled invariance tities, this is the the celebrated celebrated Noether’s theorem (Frankel invariance based submanifolds reduction the theof equivalence sets are the the (2004)). Conserved quantities represent redundant informaximal integral submanifolds of the controlled invariance (2004)). Conserved quantities redundant integral submanifolds of the invariance mation which can be factoredrepresent out resulting in a inforlower maximal distribution and for abstractions the controlled equivalence sets are (2004)). Conserved quantities represent redundant informaximal integral submanifolds of the controlled invariance mationsystem, which such can be be factored out resulting in(van lower distribution and for projection abstractions the equivalence equivalence sets are are mation which can aaa lower for abstractions the order an factored approachout wasresulting taken inin der distribution the level setsand of the map. mation which such can be factored out resulting lower distribution for projection abstractions the equivalence sets sets are order system, system, an approach was taken in inin(van (van der the the level level sets setsand of the the map. order such an approach was taken der of projection map. Schaft (1981)) where it was applied to Hamiltonian control order such an approach was taken in (van der As the previously level sets of the projection map.methods are only useSchaft system, (1981)) where it was was appliedwas to Hamiltonian Hamiltonian control mentioned reduction Schaft (1981)) where it applied to control systems. This notion of symmetry extended to general As previously mentioned reduction methods are only useSchaft (1981)) where it was applied to Hamiltonian control As mentioned methods only usesystems.systems This notion notion of symmetry symmetry was extended to general general in control design/analysis if they preserveare the system As previously previously mentioned reduction reduction methods are only usesystems. This of extended to control in (Grizzle and was Marcus (1985)) where ful ful in control design/analysis if they preserve the system systems. This notion of symmetry was extended to general ful in control design/analysis if they preserve the system control systems systemstechniques in (Grizzle (Grizzle anddeveloped Marcus (1985)) (1985)) where properties relevant to the control design/analysis. For sysful in control design/analysis if they preserve the system control in and Marcus where decomposition were based on the properties relevant to the control design/analysis. For syscontrol systemstechniques in (Grizzle anddeveloped Marcus (1985)) where properties to the control design/analysis. sysdecomposition were based on the tems with relevant a state space endowed with a principalFor bundle properties relevant to the control design/analysis. For sysdecomposition techniques were developed based on the existence of symmetries. In (Martin and Crouch (1984)) it tems with a state space endowed with a principal bundle decomposition techniques were developed based on the with aa state space endowed with aa principal existence ofthat symmetries. In (Martin (Martin and Crouch (1984)) it tems structure (Martin and Crouch (1984)) showed thatbundle if the tems with state space endowed with principal bundle existence of symmetries. In and Crouch (1984)) it was shown for systems modelled on a principal fibre structure (Martin (Martin and and Crouch Crouch (1984)) (1984)) showed showed that that if if the the existence symmetries. In (Martin and (1984)) it structure was shown shownofthat for systems systems modelled onCrouch a principal fibre was was shown that that for for systems modelled modelled on on aa principal principal fibre fibre structure (Martin and Crouch (1984)) showed that if the Copyright © 2017, 2017 IFAC 203 Hosting by Elsevier Ltd. 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principal bundle has compact structure group and if the system is accessible then controllability of the projected system on the base manifold is a necessary and sufficient condition for controllabillity of the system. In (Pappas and Simic (2002)) a complete theory of controllability preserving quotients was developed for general nonlinear control systems. As an application of the controllablity preserving quotients of (Pappas and Simic (2002)) a hierarchical controllability algorithm for linear systems was developed in (Pappas et al. (2000)). This same approach was brought to bear on the problem of stabilizability preserving quotients for linear systems in (Pappas and Lafferriere (2001)) where stabilizability preserving linear quotients of linear systems were fully characterized. To our best knowledge no such general results exist for stabilizability preserving reductions of nonlinear systems. Of course this can be attributted to the fact that unlike the case for linear systems no simple test exists for deciding stabilizability of general nonlinear systems. This work therefore presents a step towards trying to develop a framework for stabilizability preserving quotients for nonlinear systems. The rest of the paper is organized as follows section 2 contains a brief summary of the prerequiste mathematical concepts, section 3 presents the mathematical formulation of the problem. In section 4 the connection of feedback linearization with stabilizability preserving quotients is made, the main result of this paper is developed in this section. An application of these results is illustrated through an example in section 5, concluding remarks are given in section 7. 2. MATHEMATICAL PRELIMINARIES This section presents a brief survey of the differential geometric concepts that will be used in the main sequel. For a more comprehensive coverage of the concepts presented here the reader may consult (S.Kobayshi and Nomizu (1963)). All manifolds and all maps considered will be C ∞ . Let M be a n-dimensional manifold and p ∈ M , the set of all vectors at p is called the tangent space at p and will be denoted Tp M . The disjoint union of all tangent spaces at all points in M has a manifold structure, this manifold is called the tangent bundle, it is 2n-dimensional and will be denoted T M . Let M and N be manifolds, consider the map ψ : M → N , this map induces a linear map Tp ψ : Tp M → Tψ(p) N called the tangent map.
A fiber bundle is a manifold that locally looks like a product space. Formally we define a fiber bundle as follows. Definition 1. (SteenRod (1960)) A fiber bundle is the 4tuple (E, M, π, U ) where E and M are smooth manifolds called the total space and base space respectively, π is a surjective map from E to M , π : E → M which is called the projection map . For some x in M the set π −1 (x) is called the fiber over the point x in M where π −1 (x) is required to be homeomorphic to the smooth manifold U . U is called the typical fiber over M . The tangent bundle introduced above is an example of a vector bundle(i.e fiber bundle where the fibers are homeomorphic to a vector space), the manifold M is the base space, πM is the projection map which maps a tangent vector to its base point. When T M is viewed as a vector 204
bundle vector fields can be intepreted as sections of the vector bundle. Consider a smooth vector field on the total space X ∈ Γ∞ (T E) where Γ∞ (T E) is the set of all smooth sections of the tangent bundle T E over E, the vector field X is called projectable if there exists a vector field on the base space Y ∈ Γ∞ (T M ) such that the following holds Tx π ◦ X(x) = Y ◦ π(x). (1) It can easily be shown that if the curve γ(t) in E is an an integral curve of the vector field X then the curve π ◦ γ(t) in M is an integral curve of the vector field Y . Since π is surjective the projection operation is a many to one operation, thus there is no canonical way of relating a vector field (smooth curve) on the base space back to a vector field (smooth curve) on the total space. To be able to uniquely relate a vector field (smooth curve) on the base space back to a vector field (smooth curve) on the total space requires the specification of extra structure and this extra structure comes in the form of a connection. Definition 2. Let (E, M, π, U ) be a fiber bundle, the vertical bundle VE is the sub-bundle of the tangent bundle T E defined as V E = {vx ∈ Tx E|Tx π(vx ) = 0, ∀x ∈ E} = ker(T π). (2) The vertical bundle contains all those tangent vectors that get projected to the zero vector, this means that the vertical bundle contains all those vectors which are tangent to the fibers. A connection on a fiber bundle (E, M, π, U ) is a sub-bundle of T E which is complementary to the vertical bundle, this sub-bundle is called the horizontal sub-bundle (Cushman and Bates (2015)). This is expressed formally as: Definition 3. A connection on the fiber bundle (E, M, π, U ) is a smooth sub-bundle HE ⊂ T E such that T E = HE ⊕ V E. The connection makes it possible to lift objects from the base space back to the total space. Consider a vector Yp ∈ Tp M on the base space, the horizontal lift of Yp denoted Horx (Yp ) ∈ Tx E is a vector on the total space which gets projected to the vector Yp . Thus there is a horizontal lift map Horx : Tp M → Tx E, where π(x) = p (3) This horizontal lift map satisfies the property that Tx π ◦ Horx = idTp M . For a curve γ(t) ∈ M the horizontally lifted curve γ˜ (t) ∈ E is such that, d γ˜ (t) = Horγ˜ (t) (γ(t)) ˙ (4) dt ¯ ∈ Tx E which get projected to the Given two vectors X, X ¯ the difference between same vector i.e Tx π(X) = Tx π(X), ¯ ∈ Vx E). these two vectors is a vertical vector (X − X 3. PROBLEM STATEMENT We shall make use of the fiber bundle description developed in (Brockett (1977)), a control system is defined by the 5-tuple (E, M, π, U, F ) where (E, M, π, U ) defines a fiber bundle and a smooth map F : E → T M which satisfies πT M ◦ F = π where πT M is the tangent bundle projection. This situation is described by the following commutative diagram.
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Proof. Let γ M (t) be a trajectory of Σ by definition there exists a curve γ E (t) such that π ◦ γ E (t) = γ M (t). From the fiber preserving property we have:
E F
TM
π
π ˜ ◦ ψ ◦ γ E (t) = φ ◦ π ◦ γ E (t)
πT M
M Within this framework of control system representation trajectories of a control system are given by the following definition from (Tabauda and Papas (2005)). Definition 4. Let Σ = (E, M, π, U, F ) be a control system, the smooth curve γ M (t) : R → M is called a trajectory of Σ if there exists a curve γ E (t) : R → E such that: (1) π ◦ γ E (t) = γ M (t) d M γ (t) = F ◦ γ E (t) (2) dt
ψ
π
M
˜ E π ˜
φ
N
˜ N, π ˜ ) are fiber bundles. ConNote (E, M, π, U ) and (E, ˜, U sider the two control systems Σ = (E, M, π, U, F ) and ˜ = (E, ˜ N, π ˜ , F˜ ), Σ ˜ is a quotient of Σ if there exists a Σ ˜, U ˜ surjective submersion map φ : M → N , a map ψ : E → E such that the pair (φ, ψ) is a fiber preserving map and given a smooth curve γ M (t) ∈ M which is a trajectory of ˜ Σ the curve φ(γ M (t)) ∈ N is a trajectory of Σ.
Proposition 1. Consider the two control systems Σ = ˜ = (E, ˜ N, π ˜ , F˜ ). Σ ˜ is a quotient (E, M, π, U, F ) and Σ ˜, U of Σ if there exists the bundle morphism (φ, ψ) where φ is a surjective submersion which makes the following diagram commute. ψ
E TM
Tφ
F˜
TN
π ˜ πT N
πT M
M
d d (φ ◦ γ M (t)) = T φ ◦ (γ M (t)) dt dt = T φ ◦ F ◦ γ E (t) = F˜ ◦ ψ ◦ γ E (t)
φ
Thus the problem of stabilizability preserving quotient can be phrased in the following way. Problem Statement 1. Given the smooth control system ˜ = (E, ˜ N, π ˜ , F˜ ), Σ = (E, M, π, U, F ) and it’s quotient Σ ˜, U where the quotient is defined by the fiber morphism (φ, ψ) which satisfies proposition 1. Characterize the fiber bundle ˜ is morphisms for which Σ is stabilizable if and only if Σ stabilizable In the following section this problem is studied for systems that are feedback linearizable where it is shown that the “zero dynamics” play an essential role in answering the problem. 4. STABILIZABILITY PRESERVING QUOTIENTS OF FEEDBACK LINEARIZABLE SYSTEMS The main idea behind feedback linearization is the realization that at times non-linearities are a product of the choice of coordinate system used to represent the control system. As such feedback linearization prescribes a method of constructing coordinate transformations and feedback transformations such that in the new coordinates the system is a controllable linear system. For the sake of brevity we shall focus on affine single input single output systems. Consider an affine n-dimensional control system x˙ = f (x) + g(x)u,
x ∈ Rn ,
u∈R
(5)
The system is said to have a relative degree r ∈ N at some point x0 ∈ Rn if there exists a function λ : Rn → R such that (1) Lg Lkf λ(x) = 0 k < r − 1 (2) Lg Lr−1 λ(x0 ) = 0 f If r = n the affine control system can be exactly linearized, the case that will be of interest is when r < n this is the partial linearization case (Isidori (1995)). For the case of r < n the state transformation φ(x) : Rn → Rr is constructed as follows T φ = λ(x), Lf λ(x), · · · , Lr−1 λ(x) (6) f
˜ E
F π
π ˜ ◦ (ψ ◦ γ E )(t) = φ ◦ γ M (t) This proves the first part of the trajectory definition. For the second part differentiate the curve φ ◦ γ M .
= F˜ ◦ (ψ ◦ γ E )(t)
Let M be a n-dimensional smooth manifold assume there exists an equivalence relation on M such that the equivalence sets on M are all of the same dimension q. The equivalence sets can be viewed as leaves of a foliation of M , from the Haefliger cocycle representation of foliations we know that this foliation can be represented as level sets of a surjective submersion map (Moerdijk and Mrcun (2003)). Thus the quotienting of the state space of the control system can be compactly represented by simply choosing the appropriate surjective submersion map φ : M → N where N is the quotient space. Since the control system is modelled on a fiber bundle we will require the map that characterizes the quotienting procedure to preserve the fiber bundle structure. Thus we require also a map ˜ where E is the total space over M and ψ : E → E, ˜ E is total space over N such that the following diagram commutes. E
205
N
and there exists a feedback transformation u = α(x) + β(x)v such that in these new coordinates the system is a r-dimensional linear controllable system. Within the 205
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quotient framework developed in the previous section this situation can be described as follows. For a n-dimensional control system Σ = (E, M, π, U, F ) which in fibered coordinates (x, u) of E is affine and has a relative degree r < n, there exists a fiber bundle morphism (φ, ψ) : (E, M, π, U ) → (Rr+1 , Rr , π ˜ , R) where φ is defined as is shown in equation (6) above and ψ : E → Rr+1 is defined as u − α(x) ψ : (x, u) → (z = φ(x), v = ) (7) β(x) ˜ , R) is a fiber bundle with total space Note that (Rr+1 , Rr , π Rr+1 , base space Rr and projection map π ˜ : Rn+1 → Rr . The fiber bundle morphism defines a linear controllable ˜ = (Rr+1 , Rr , π quotient control system Σ ˜ , R, F˜ ) with F˜ : r+1 r R → R defined as F˜ (z, v) = T φ ◦ F (φ−1 (z), α(φ−1 (z)) + β(φ−1 (z))v) (8) ˜ is stabilizable since it is The quotient control system Σ controllable by construction. Let x0 ∈ M be an equilibrium point of Σ and also assume that φ(x0 ) = 0. Stabiliz˜ guarantees that there exists a linear feedback ability of Σ controller such that for all z ∈ Rr in the neighbourhood of the origin there is a trajectory γ˜ : R → Rr which starts at z i.e. γ˜ (0) = z and asymptotically gets close to the origin as time tends to infinite. Stabilizability of the linear quotient system therefore guarantees that the original system is asymptotically stabilizable to the the submanifold φ−1 (0). The triple (M, Rr , φ) is a fiber bundle with the equivalence sets of the surjective submersion map φ being the fibers. Now consider the submanifold φ(x) = 0, the system dynamics on this manifold are given by differentiating φ(x) from which we get, dφ(x) = T φ ◦ F (x, u) = 0. dt
(9)
That is γ H (t) is a trajectory of the horizontally lifted system. Consider the time derivative of the curve φ ◦ γ H (t) ∈ Rr , d d (φ ◦ γ H (t)) = Tγ H (t) φ ◦ γ H (t) (12) dt dt = Tγ H (t) φ ◦ Horγ H (t) ◦ F˜ ◦ (φ, ψ) · · · (13) ◦ (γ H (t), uH (t)) = F˜ ◦ (φ, ψ) ◦ (γ H (t), uH (t)).
(14)
(15)
This shows that the trajectories of the horizontally lifted system get projected to the trajectories of the original system. Consider the control system defined as the difference between the original system dynamics and the horizontally lifted linear system dynamics, F Z (x, u) = F (x, u) − Horx ◦ F˜ ◦ (φ, ψ) ◦ (x, u).
(16)
By construction the original system dynamics and the horizontally lifted dynamics are horizontal thus the difference between the two is vertical. Therefore F Z (x, u) restricted to the submanifold φ−1 (0) represents the system dynamics on this submanifold. Now assume that there exists a linear feedback v = Kz that stabilizes the quotient linear controllable system. Under the action of this feedback the closed loop dynamics of the original system when restricted to the submanifold φ−1 (0) are, F (x, u) − Horx ◦ F˜ ◦ (φ(x), ψ(x, u))|u=α(x)+β(x)Kφ(x) (17) If the closed loop dynamics given by (17) are stable then the quotient map φ defined in equation (6) is a stabilizability preserving quotient map. This result is stated formally as follows. Result 1. For a control system Σ = (E, M, π, U, F ) assume there exists a function λ : M → R such that Σ has relative degree r < dim(M ). The fiber bundle morphism (φ, ψ) : (E, M, π, U ) → (Rr+1 , Rr , π ˜ , R) defined as
Thus the system dynamics on the submanifold φ(x) = 0 lie in the kernel of the tangent map which means that the vector fields induced by the control system on this submanifold are vertical. By equipping the fiber T bundle (M, Rr , φ) with a connection HM the vertical φ(x) = λ(x), Lf λ(x), · · · , Lr−1 λ(x) (18) f component of the system dynamics can be calculated as the difference between the original system dynamics and u − α(x) ψ(x, u) = (φ(x), ) (19) the horizontal component. Given a connection on the fiber β(x) bundle (M, Rr , φ) define the horizontal lift of the linear is a stabilizability preserving quotient if the dynamics controllable system as follows. r+1 r ˜ ˜ Definition 5. Given the linear system Σ = (R , R , π ˜ , R, F ) given by H F (x, u) − Horx ◦ F˜ ◦ (φ(x), ψ(x, u))|u=α(x)+β(x)Kφ(x) (20) the horizontally lifted system is Σ = (E, M, π, U, F˜ H ) H where the map F˜ : E → T M is defined as are stable. F˜ H (x, u) = Horx ◦ F˜ ◦ (φ, ψ)(x, u).
(10)
To show that this definition of the horizontally lifted control system is proper consider the curve (γ H (t), uH (t)) ∈ E such that, d H γ (t) = Horγ H (t) ◦ F˜ ◦ (φ, ψ) ◦ (γ H (t), uH (t)). dt
(11) 206
The dynamics given by F Z (x, u) coincide with the notion of “zero dynamics” defined in (Krener and Isidori (1980)) where the zero dynamics are given by the dynamics of the system when the input is chosen such that the output stays zero. Finally let us see how the following proposition from (Isidori (1995)) fits into the notion of zero dynamics developed in this paper. Proposition 2. Suppose an n-dimensional system has relative degree r < n at x0 . Set λi = Li−1 f h(x) for i =
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1, · · · , r, it is always possible to find n − r functions λr+1 (x), · · · , λn (x) such that the mapping Λ(x) =
λ1 (x) ··· λn (x)
η1 (x) + 2η1 (x)η2 (x)η4 (x) − η2 (x)η3 (x) (29) 1 + η4 (x)η2 (x) η3 (x) − η4 (x)η1 (x) . (30) a21 = 1 + η4 (x)η2 (x)
a11 (x) =
The horizontal lift of the linear controllable quotient system becomes
is a diffeomorphism.
From the diffeomorphism Λ let Ω1 and Ω2 be two cotangent subbundles defined as follows. Ω1 = [dλ1 (x), · · · , dλr (x)]
T
Ω2 = [dλr+1 (x), · · · , dλn (x)]
(21) T
(22)
The vertical and horizontal sub-bundles can be defined as the annihilators of the cotangent bundles Ω1 and Ω2 ⊥ respectivley(V M = Ω⊥ 1 , HM = Ω2 ). Thus the choice of n−r maps in the above proposition can be understood as a prescription of a flat connection on the state manifold. To show how these ideas can be applied consider the following example taken from (Isidori (1995)). 5. EXAMPLE Consider the SISO system x2 e −x1 1 u x˙ = x1 x2 + 0 x2 y = h(x) = x3
(23) (24)
For this system let λ(x) = h(x) then the surjective submersion φ : R3 → R2 is given by φ(x) = (h(x), Lf h(x)) = (z1 = x3 , z2 = x2 ). (25) Together with the feedback transformation ψ(x, u) = v = u + x1 x2 the pair (φ, ψ) forms a fiber bundle morphism which defines a controllable linear quotient control system defined as z˙1 = z2 z˙2 = v.
207
(26)
The vertical bundle denoted as V is the kernel of T φ ∂ and is the subspace V = span{ ∂x }. Possible choices of 1 the horizontal subspace denoted H are subspaces of the following form. ∂ ∂ ∂ ∂ + η2 (x) + , η3 (x) ··· ∂x1 ∂x2 ∂x3 ∂x1 ∂ ∂ + η4 (x) } (27) + ∂x2 ∂x3
H = span{η1 (x)
Where ηi : R3 → R for i = 1, · · · , 4. The corresponding horizontal lift map is given as a 3 × 2 matrix of the form a11 (x) a21 (x) 0 1 Horx = . (28) 1 0 With the functions a11 (x) and a12 (x) being defined as,
207
a11 (x)z2 + a21 (x)v v z2
(31) z=φ(x),v=ψ(x,u)
The linear feedback v = K1 z1 + K2 z2 , K1 , K2 < 0 renders the quotient control system asymptotically stable. Using this feedback to control the original system will render the submanifold φ(x) = 0 asymptotically stable. The resulting system dynamics on this submanifold are given by the difference between the original system dynamics and the horizontally lifted closed loop dynamics. These dynamics have the form
−x1 + uex2 − a11 (x)x2 + a12 (x)(u + x1 x2 ) 0 0
(32)
For u = x1 x2 +K1 x3 +K2 x2 . Note that on the submanifold φ(x) = 0 the coordinates are of the form (x1 , 0, 0) thus the above dynamics take the form x˙1 = −x1 + u(1 + a12 (x)), which is an asymptotically stabilizable dynamical system. This shows that the surjective submersion map φ(x) is a stabilizability preserving quotient map. Now compare the approach that has been set out with the typical feedback linearisation approach as set out in (Isidori (1995)). By proposition 2 the map φ(x) is extended to a diffeomorphism Λ : R3 → R3 which is of the form: x3 h(x) z1 x2 (33) Φ(x) = z2 = Lf h(x) = φ3 (x) 1 + x1 − exp(x2 ) z3 The choice of the map φ3 (x) is equivalent to a choice of an Ehressman connection on the state space. This connection is defined as the kernel of the tangent map of φ3 (x) : R3 → R which gives the horizontal subspace H = span{
∂ ∂ ∂ , ex 2 + } ∂x3 ∂x1 ∂x2
(34)
Comparing this horizontal subspace to the general horizontal subspace gives the following η1 = η2 = η4 = 0, η3 = ex2 which gives the following horizontal lift map
Horx =
0 ex2 0 1 1 0
.
From which the zero dynamics are calculated as: −x1 (1 + x2 exp(x2 )) Z 0 f = 0
(35)
(36)
The zero dynamics are parameterised by (x2 , x3 ) which means that the dynamics vary depending on the fiber
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however the linear feedback constrains the system dynamics to evolve on the submanifold φ−1 (0)this implies that (x2 = 0, x3 = 0), the dynamics therefore become x˙ 1 = −x1 . (37) Which agrees with the key result in (Isidori (1995)) which serves to verify the validity of the approach presented in this paper. 6. SUMMARY OF RESULT The question posed in this paper deals with the existence of necessary and sufficient conditions for the existence of stabilizability preserving quotients. The procedure developed in this paper shows that having a finite relative degree and stable zero dynamics is sufficient for a feedback linearizable system to admit stabilizability preserving quotients. It should be noted that the main advantage of the notion of ”zero dynamics” developed in this paper is that it can be applied to general nonlinear systems which are not necessarily feedback linearizable. However this result can be seen to very restrictive from two view points. Firstly the set of systems that are feedback linearizable is very small (Respondek and Tall (2002)) secondly this approach requires the feedback controller of the lower dimensional to be able to stabilize the higher dimensional system. Future work will therefore focus on relaxing this restrictions by looking at general nonlinear quotient system and also by developing a lifting procedure of the feedback controller for the lower dimensional quotient system. This lifting procedure will aim to add fiber dependent elements to the low dimensional controller so as to render the higher dimensional system stable. 7. CONCLUSION Quotient control systems provide a very general way of dealing with control system reductions. In this paper we have presented the notion of stabilizability preserving quotients and how the problem of their construction can be dealt with in a differential geometric framework. As a special instance of quotients of control systems the case of partial feedback linearization is studied. This quotienting process is stabilizability preserving if the “zero dynamics” are stable. An alternative interpretation of the notion of zero dynamics is developed where it is shown that, by utilising the fiber bundle structure of the state space manifold a connection can be defined on the state space manifold. This connection enables the lifting of the linearized dynamics from the base space to the total space. The zero dynamics are then defined as the difference between the original dynamics and the lifted linear dynamics. It then follows from the construction that the zero dynamics represent the motion of the system along the fibres while the linearized dynamics represent the motion of the system from fiber to fiber. REFERENCES Brockett, R. (1977). Control theory and analytical mechanics. In C. Martin and R. Hermann (eds.), Geometric Control Theory. Math. Sci Press. 208
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