- Email: [email protected]
Geometric Aspects of IDA-PBC
8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010
Geometric Aspects of IDA-PBC Kai H¨ offner ∗ and Martin Guay ∗ ∗
Queen’s University, Kingston, ON, Canada, K7L 3N6 (e-mail: [email protected]).
Abstract: The purpose of this work is to introduce a geometric definition of problems arising in passivity-based control design. We present a coordinate-free version of the well-known matching equations resulting from the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) design and show that feedback equivalence between control-affine systems can be used to define a family of matching problems for which an explicit solution is known. We illustrate our findings on a non-isothermal CSTR model. Keywords: port-Hamiltonian systems, feedback equivalence, matching equations, geometric methods 1. INTRODUCTION AND BACKGROUND This paper establishes a connection between the IDAPBC methodology (Ortega et al. (2002b)) and the geometric interpretation of feedback equivalence as treated by Gardner and Shadwick (1990, 1992) and collaborators. Furthermore, we show that the choice of coordinates together with an appropriate feedback law can be used to define new matching problem for which an explicit solution is known. The objective in Passivity-based Control (PBC) is to render the closed-loop system passive with respect to a desired storage function with a minimum at the desired equilibrium. IDA-PBC incorporates several passivity-based controller design methodologies, we refer the reader to the survey article by Ortega et al. (2001) for further information. Furthermore, it is a powerful control methodology for nonlinear system which has found wide range of applications. In particular, numerous contributions have been made for the class of underactuated mechanical systems, see Ortega et al. (2002a); Acosta et al. (2005) and reference therein. In the majority of reference concerned with IDA-PBC the choice of the interconnection and damping structure is guided be the choice of a specific coordinate representation. It is known that the port-Hamiltonian representation is preserved under coordinate transformation Fujimoto and Sugie (2000). Here, we focus on a coordinate free approach to allow in order to simplify the problem by a choice of the interconnection and damping structure in a specific set of coordinates. This paper is structured as follows: We review some concepts from the theory of feedback equivalence and portHamiltonian systems in the remainder of this section in order to fix our notation. Preliminary results on feedback equivalence are stated in Section 2. The main results are presented in Section 3. We illustrate the results on a nonisothermal CSTR model in Section 4. A short conclusion and future directions of research are given in Section 5. 1.1 Background Background information on feedback equivalence and port-Hamiltonian systems are presented in this section. 978-3-902661-80-7/10/$20.00 © 2010 IFAC
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Let X and U be open connected subsets of Rn and Rm , respectively with n ≥ m ≥ 1 and assume that 0m ∈ U and let f0 , g1 , . . . , gm be vector fields on X. A control-affine system is a triple Σ = (X, {f0 , g1 , . . . , gm }, U), where X is called the state space, U is the control space, the vector field f0 is called the drift vector field, and g1 , . . . , gm are called the control vector fields. Furthermore, we assume that the linear span of the control vector fields has locally constant rank m. Instead of working with vector fields, we can alternatively define a control-affine systems by a set of differential one-forms. Let {dxi }ni=1 be a coframe on X (i.e. it provides a basis for the cotangent space at each point) and let dt be a coordinate one-form on R, then we define for all u ∈ U with coordinates (u1 , . . . , um ) the following set of one-forms η j = dxj + (hf0 , dxj i + ui hgi , dxj i)dt, j = 1, . . . , n (1) where h·, ·i denotes the point-wise inner product between vector fields and one-forms on X. Furthermore, define the n-vector of one-forms η = [η 1 , . . . , η n ]> . A control-affine system is then also represented by the triple Σ = (X, η, U). The motivation for this definition is the following, we want to study the equivalence of control-affine systems by studying the equivalence of coframes which are adapted to the control-affine systems as given in (1). The rational behind choosing this type of adaption is that every trajectory x(t) of Σ under some control u(t) is in one-toone correspondence with an integral curve of the exterior differential system generated by {η j }nj=1 (see Gardner and Shadwick (1990) for further details). Next, we define when two control-affine systems are feedback equivalent. Let (x1 , . . . , xn , u1 , . . . , un , t) be coordinates on M = X × U × R and let πX : M → X be ∗ j n a projection. Notice that {πX η }j=1 1 is a set of n linear independent one-forms on M , together with the set {dt, du1 , . . . , dum } they define a coframe ωM = [η, du, dt]> on M , denoted the zero-adapted coframe. We use this construction of the coframe to define a feedback equivalence for control-affine systems. As it turns out, this 1
We omit the pull-back of the projection if it is clear from the context.
10.3182/20100901-3-IT-2016.00261
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definitions agrees with the one motivated by the idea that two systems are feedback equivalent if they generate the same trajectory under the same control input and initial conditions. In order to define feedback equivalence we need the following construction. Definition 1. Gardner (1989) Let ωU and ΩV be n-vector of one-forms on open sets U ⊂ Rn and V ⊂ Rn , respectively. Let G be a Lie subgroup of GL(n, R) , then ωU is said to be G-equivalent to ΩV if there exists a diffeomorphism Φ : U → V and a function γ : U → G satisfying Φ∗ (ΩV |Φ(u) ) = γ(u)ωU |u , for all u ∈ U .
∂ ∂ Similarly, we write R = Rij ∂x and denote by i ⊗ ∂xj R(x) the symmetric positive semi-definite matrix-valued function formed by Rij (x). With this notation, we write the drift vector field of a port-Hamiltonian system with dissipation in coordinates as ∂H > f0 (x) = (J(x) − R(x)) for all x ∈ X. ∂x x
1.3 Matching Problem
One of the main objectives in IDA-PBC is to find a feedback law such that the closed-loop system is a portHamiltonian system with desired interconnection and damping structure. A passivity argument is then used to The Lie group that we choose for this is a Lie subgroup of determined the stability of the system. In the following, GL(n + m + 1, R) which is called the structure group and we describe the matching problem introduced in the IDAis defined by PBC methodology as a feedback equivalence problem. For " # A ∈ GL(n, R) the purpose of this paper we are only concerned with the A 0 0 B C 0 ∈ GL(n + m + 1, R) B ∈ Rm×n . modification or generation of the Hamiltonian function, G0 = 0 0 1 C ∈ GL(m, R) without particular concern to stabilization. Stability of Definition 2. Let ΣX = (X, η, U) and ΣY = (Y, µ, V) be the closed-loop system is ensured by assigning a strict two control-affine systems with X, Y ⊂ Rn . Furthermore, minimum to the Hamiltonian function at the desired equilet ωM = [η, du, dt]> and ΩN = [µ, dv, dt]> be two zero- librium. We define the matching problem: adapted coframes on M = X × U × R and N = Y × V × Matching Problem: Given a control-affine system Σ = R, respectively. We say that ΣX is feedback equivalent to (X, {f , g , . . . , g }, U), a skew-symmetric 2-tensor field 0 1 m ΣY if ωM is G0 -equivalent to ΩN , i.e. there exists a G0 - J and a positive semi-definite 2-tensor field R on X d d equivalence Φ : X × U × R → Y × V × R of the form find, if possible, a function Hd such that Σ is feedback Φ(x, u, t) = (φ(x), ψ(x, u), t) ∀(x, u, t) ∈ X × U × R equivalent to the port-Hamiltonian system with dissipafor some diffeomorphism φ : X → Y, called coordinate tion Σd = (X, Jd , Rd , Hd , {g1 , . . . , gm }, U). transformation and some map ψ : X × U → V, called state Next, we describe a general technique to determine a feedback, then Φ so defined is called a feedback equivalence feedback equivalence of two control-affine systems ΣX = for ΣX and ΣY . (X, η, U) and ΣY = (Y, µ, V). The presentation of this technique is due to Atkins and Shadwick (1993). 1.2 Port-Hamiltonian systems 2. PRELIMINARIES Next, we define port-Hamiltonian systems. First let us recall some definitions from differential geometry. Let σ In this section we outline a procedure to determine a be a contravariant 2-tensor field on X ⊂ Rn , we associated feedback equivalence, this procedure was presented in a map σ e : T ∗ X → T X to σ by ´ Atkins (1995) and is based on the work of Elie Cartan ∗ he σ (x)(α), βi = σ(x)(α, β), for all β ∈ Tx X, x ∈ X. (Cartan (1908)). It will become clear that the projected We commonly use the same symbol for a tensor field and matching equations defined for example in Cheng et al. (2005) appear naturally in this context. the map that it is defining. Definition 3. Let X ⊂ Rn be open, H ∈ C∞ (X) and 2.1 Lifted Coframe let J be a skew-symmetric 2-tensor field on X. A portHamiltonian system is a control-affine system Σ = In the equivalence problem, as defined in the previous (X, {f0 , g1 , . . . , gm }, U) such that f0 = J(dH), where H section, we are interested in finding a diffeomorphism is called the Hamiltonian function and J is called the between M = X × U × R and N = Y × V × R such interconnection structure. We denote the system by the that its Jacobian has a predetermined form given by G0 . 5-tuple Σ = (X, J, H, {g1 , . . . , gm }, U). Proposition 5 below states that we can equivalently find Definition 4. Let R be a symmetric positive semi-definite a diffeomorphism between M × G0 and N × G0 without 2-tensor field on X ⊂ Rn . A port-Hamiltonian sys- any restrictions on its Jacobian. Let π : M × G0 → M tem with dissipation is a control-affine system Σ = be the projection onto M and define the vector of one(X, {f0 , g1 , . . . , gm }, U) with f0 = J(dH) − R(dH), R is forms ω = Sπ ∗ ωM on M × G0 , where S ∈ G0 and proceed called the damping structure. We denote the system by similarly for Ω on N ×G0 . The lifted coframe ω on M ×G0 the 6-tuple Σ = (X, J, R, H, {g1 , . . . , gm }, U). is defined as " # " #" # θ A 0 0 η 1 n Let (x , . . . , x ) be a coordinates on X, hence we have µ B C 0 du ω = Sω = = . M ∂H ∂ ∂ ∂H > i and J = J ij ∂x |x dH = ∂x dt 0 0 1 dt i dx i ⊗ ∂xj . Denote by ∂x ∂H the column vector formed by ∂x i (x) and by J(x) the The following proposition allows us to “lift” the equivaskew-symmetric matrix-valued function formed by J ij (x). lence problem to the product M × G0 . 630
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Proposition 5. (Gardner (1989)) There exists a diffeomorphism Φ : M → N satisfying Φ∗ ΩN = γN M ωM with γN M : M → G0 if and only if there exists a diffeomorphism Φ1 : M × ∗ G0 → N × G0 such that Φ1 Ω = ω. In order to solve the feedback equivalence problem we can equivalently determine if there exists a diffeomorphism ∗ Φ1 on M × G0 that solves Φ1 Ω = ω. Next, we study the structure equations (see appendix A) of the feedback equivalence problem. Since we consider only time-invariant feedback, it is clear that t is invariant for Φ and Φ1 , hence we omit this component in the structure equations. After Lie algebra compatible absorption equation (A.1) are θ α 0 θ dt ∧ Kµ d = ∧ + . (2) µ β γ µ 0 where α, β and γ are Lie algebra-valued one-forms. Furthermore, the term dt ∧ Kµ is the only non-zero intrinsic torsion. The torsion coefficient K can be expressed parametrically as K = AK0 C −1 with K0 = g, i.e. K0 is the value of K for S = e with e as the identity element of G0 . 2.2 Reduction of the Structure Group Next, we analyze how the intrinsic torsion changes under action of G0 . Instead of computing the orbit of the group action on the intrinsic torsion explicitly, we compute the infinitesimal group action to determine the reduced structure group and the adapted coframe as suggested by Gardner (1989). The infinitesimal group action on the intrinsic torsion is obtained by studying the integrability condition d◦dθ = 0. This yields in our case the congruence dK − αK + Kγ ≡ 0 mod {θ, µ, dt}, where two one-forms ω and η are said to be congruent modulo an algebraic ideal I if there exists α ∈ I such that ω = η+α, i.e. ω ≡ η mod I. We assumed that g has locally constant rank m around every point x ∈ X, hence K also has locally constant rank m. The elements A and C are required to have full-rank since G0 ⊂ GL(n + m + 1, R). We choose to normalize K matrix to the constant 0(n−m)×m KN = . Im It follows that after this normalization dKN = 0 and hence αKN ≡ KN γ mod θ, µ, dt. If we let ε α12 α= , λ α22 then we get the following congruences α12 ≡ 0 mod θ, µ, dt, α22 ≡ γ mod θ, µ, dt (3) which determine the first reduced structure group given by the isotropy group of KN . Remark 6. Parametric calculations show that the following equation AKN C −1 = KN , determines the first reduced structure group G1 . Furthermore, elements of G1 are of the form " # A1 0 0 A2 C 0 B1 B2 C where A1 ∈ GL(n − m, R), A2 ∈ Rn−m×n and B = [B1 , B2 ] ∈ Rn×m . 631
In order to define the G1 -adapted coframe we define the map βM : M → G0
g ⊥ (x) g † (x) (x, u, t) 7→ 0 0
0
0 g † (x)g(x) 0 0 1
⊥ where g ⊥ is a full-rank left annihilator of g, i.e. g g = 0 and g † is such that g † g and g ⊥ , g † have full-rank. Note that one possible choice for g † is g > . The G1 -adapted coframe is then defined by ⊥ g dx −g ⊥ f0 dt g † dx −g † (f0 + gu)dt . βM ωM = (g † g)du dt b Furthermore, define θ = [θ1 , . . . , θn−m ]> and θ¯ = [θn−m+1 , . . . , θn ]> by > S1 βM ωM = θb θ¯ µ dt , S1 ∈ G1 .
Note that θb represents the part of the control-affine system in “image of the full-rank left annihilator” and the group elements in A1 represent the degrees of freedom available in the choice of the full-rank left annihilator. We continue with the reduced equivalence problem with new structure equations for the G1 -equivalence between the G1 -adapted coframes βM ωM and βN ΩN . The basis for this is given by Proposition 15. The new structure equations are obtained by substituting the congruences (3) into the structure equations (2), which introduces new torsion. The new structure equations are " # b ¯> ε 0 0 θ ∧ Jθ¯ + dt ∧ Hθ¯ θb θ d θ¯ = λ γ 0 ∧ θ¯ + dt ∧ µ β1 β2 γ 0 µ µ where β = [β1 , β2 ] and H = [Hij ] is a matrix-valued function on M × G1 and J = [Jijk ] = −[Jikj ] are skewsymmetric matrix-valued functions on M × G1 for all i = 1, . . . , n − m. H and J are the torsion coefficients for the new structure equations. The infinitesimal group action on the torsion coefficients J and H induced by the integrability condition d ◦ dθb = 0 is determined by dJ − εJ + γ > J + Jγ ≡ 0 mod θ, dt dH − εH + Hγ ≡ 0 mod θ, µ, dt. Furthermore, we can determine the torsion coefficients parametrically by J = A1 C −> J0 C −1 and H = A1 H0 C −1 where J0 and H0 are the values of J and H at e ∈ G1 , respectively. To determine the equivalence between the control-affine systems, we repeat this process until the torsion does not explicitly depend on the group parameters. At this point there are two possibilities; either there are no free group parameters left and we have identified a canonical coframe, then this problem can be solved as an equivalence problem of coframes (see Olver (1995) for more details on equivalence of coframes). If there remain free group
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parameters, then involutivity of the defining equations must be checked. This is an existence and uniqueness question for a set of partial differential equations and is answered by the Cartan-K¨ ahler theorem, which is a geometric extension of the Cauchy-Kowalewski existence theorem (see Bryant et al. (1990)). 3. MAIN RESULT In this section we first derive the so-called matching equations associated to the matching problem and then use the transitivity property of the feedback equivalence to generate new desired interconnection and damping structures and associate a Hamiltonian function to them. 3.1 Matching equations We apply the procedure described in the previous section to the feedback equivalence of a control-affine system ΣX = (X, {f0 , g1 , . . . , gm }, U) and a port-Hamiltonian system with dissipation Σd = (X, Jd , Rd , Hd , {g1 , . . . , gm }, U). We established that any feedback equivalence Φ = (φ, ψ, Id) between ΣX and Σd has to satisfy Φ∗ ΩN = ωM , where ωM and ΩN are the adapted coframes of ΣX and Σd , respectively. Recognizing that the two control-affine systems have the same control vector fields, we get that the first torsion coefficients are the same. Hence, the first reduction of the structure group is identical for both control-affine systems, which implies that the first structure groups are equivalent. Using Proposition 15 we get that the feedback equivalence also has to satisfy Φ∗ (βN ΩN ) = αN M (βM ωM ) where αN M : M → G1 . Using the same notation as in the b previous section. The θ-component is given by ∂Hd −φ∗ g ⊥ (Jd − Rd ) = −A1 g ⊥ f0 (4) ∂x with A1 ∈ GL(n − m, R), which represents parts of the parameter of the structure group G1 . Equation (4) is the matching equation of the feedback equivalence problem, it is a partial differential equation for the unknown Hd .
−1 where (Γ−1 : U → G denotes the inverse of Γ−1 W V ◦ Ψ) WV ◦ Φ(u) ∈ G for all u ∈ U , it follows that αW U , (Γ−1 WV ◦ Ψ)−1 γV U defines a G-equivalence for ωU and ηW . Corollary 8. Let ΣY = (Y, µ, V) be a control-affine system and let Jd and Rd be the desired interconnection and damping structure on Y, respectively. Assume there exists a solution to the matching problem define by ΣY , Jd and Rd . Let ΣX = (X, η, U) be a control-affine system that is feedback equivalent to ΣY with feedback equivalence given by Φ : M → N , then there exists a feedback equivalence for ΣX and Σd .
Proof. Let ωM and ΩN be the adapted coframe associated to ΣX and ΣY , respectively. Define the portHamiltonian system with dissipation Σd = (Y, µd , V) = (Y, Jd , Rd , Hd , {g1 , . . . , gm }, V) and its adapted coframe d ωM . By assumption Σd is feedback equivalent to ΣY , which is that there exists a G0 -equivalence for ΣY and Σd . Hence, d by Proposition 7, ωM is G0 -equivalent to ωM . Next, we establish that the feedback equivalence guaranteed to exists by Corollary 8 allows us to define new interconnection and damping structures on Y. Proposition 9. Under the assumptions of Corollary 8, ¯ : M → N, Φ ¯ = there exists a local diffeomorpism Φ ¯ ¯ (φ, ψ, Id) such that the matching problem defined by ΣX , ¯ ¯ d = φ¯∗ Rd has a solution H ¯ = H ◦ φ. J¯d = φ¯∗ Jd and R Proof. Let Ψ : N → N be a the feedback equivalence for ¯ = Ψ ◦ Φ is a feedback equivalence for ΣY and Σd , then Φ ΣX and Σd such that d ¯ ∗ (ωM Φ ) = SωM (5) where S : M → G0 . Note that (5) written in compo¯ ∗ µi = Sij η j . Let us chosen a coordinate nent gives Φ d charts such that the local representative of φ¯ is such that y i = φ¯i (x1 , . . . , xn ), then φ¯∗ dy i = Sij (y)dxj implies that ¯ ∂(y i ◦Φ) ∂xj
i
∂y = ∂x = Sij (x) for all i, j = 1, . . . , n. Define j A : M → GL(n, R) such that Aij = Sij for i = 1, . . . , n and j = 1, . . . , n. We assumed that the feedback time¯ ∗ (dt) = dt which implies that the funcinvariant, hence Φ ¯ ∗ (µd ) and Aij η j must agree. tions multiplying dt in Φ Hence,
3.2 Transitivity Next, a result that allows to define solutions to an equivalence problem by utilizing known solutions is presented. We first give a general result for arbitrary equivalence of coframes and then apply this result to the special case of the matching problem. Proposition 7. Let ωU , ΩV , ηW be coframes on U, V, W ⊂ Rn respectively and let G ⊂ GL(n, R) be a Lie group. Assume there exists a G-equivalence Φ for ωU and ΩV and a G-equivalence Ψ for ΩV and ηW , then there exists a G-equivalence for ωU and ηW . Proof. Since ωU and ΩV are G-equivalent, there exists γV U : U → G such that Φ∗ (ΩV ) = γV U ωU and there exists ΓW V : V → G such that Ψ∗ (ηW ) = ΓW V ΩV . Hence, ∗ Φ∗ (ΩV ) = γV U ωU ⇔ Φ∗ (Γ−1 W V Ψ (ηW )) = γV U ωU ∗ ⇔ Γ−1 W V ◦ Φ(Ψ ◦ Φ) (ηW ) = γV U ωU −1 ⇔ (Ψ ◦ Φ)∗ (ηW ) = (Γ−1 γV U ωU W V ◦ Φ)
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¯ ∗ (Jd (dy i , dHd ) − Rd (dy i , dHd ) + hdy i , gj iv j ) Φ = A(x)(hdxi , f0 i + hdxi , gj iuj ), where f0 is the drift vector field of ΣX and g1 , . . . , gm are ¯ ∗ Jd (dy i , dHd ) = its control vector fields. Also, we have Φ ¯ ∗ Rd (dy i , dHd ) = φ¯∗ Rd (dy i , dHd ), φ¯∗ Jd (dy i , dHd ) and Φ since their component functions do not depend on v by assumption. Next, we show that these functions define the ¯ d = φ¯∗ Rd on M . We only 2-tensors J¯d = φ¯∗ Jd and R consider J¯d and note that calculations are identical for ¯ d . Using the definition of the pull-back, we get R ∂Hd ¯ φ¯∗ (Jd (dy i , dHd )) = (Jd (dy i , dHd )) ◦ φ¯ = Jdik k ◦ φ. ∂y Furthermore, we have that ∗ kl −1 ∂ −1 ∂ ¯ ¯ ¯ ¯ φ Jd = Jd ◦ φ φ∗ ⊗ φ∗ ∂y k ∂y l which we use to compute
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φ¯∗ Jd (dxi , φ¯∗ dHd ) ∗ −1 ∂ kl i −1 ∂ ¯ ¯ ¯ ¯ φ dHd φ∗ = Jd ◦ φ dx φ∗ ∂y k ∂y l kl i −1 ∂ −1 ∂ ¯ ¯ ¯ ¯ = Jd ◦ φ dx φ∗ dHd φ∗ φ∗ ∂y k ∂y l i ∂y ∂Hd ¯ = Jdkl ◦ φ¯ k ◦φ ∂x ∂y l i ∂y kl ∂Hd = J ◦ φ¯ d ∂xk ∂y l for all i = 1, . . . , n. Furthermore, we have that i ∂y ∗ i ¯∗ ¯ φ Jd (dx , φ dHd ) = φ¯∗ (Jd (dy k , dHd )). ∂xk Hence, we have defined a port-Hamiltonian system with dissipation on M with interconnection and damping struc¯ d . Furthermore, the Hamiltonian function of ture J¯d and R ¯ ¯ d = Hd ◦ φ. this system is H 4. EXAMPLE In order to illustrate the application of the results presented above, we apply the IDA-PBC methodology to a non-isothermal CSTR studied in Guay (2002). Here we adopt the following perspective; the first matching problem that is posed is a simple energy balancing of a generic simple mechanical control system under full actuation with the objective to stabilize a generic equilibrium. Then we establish a feedback equivalence between the generic simple mechanical control system and the non-isothermal CSTR. This equivalence is guaranteed by differentiable flatness of both systems. Using the transitivity property, the resulting matching problem for the CSTR system is then easily solved. The dynamics of the non-isothermal CSTR are governed by the equations C˙ A = rA (CA , CB , T ) + (CA,In − CA )u1 /V C˙ B = rB (CA , CB , T ) + (CB,In − CB )u1 /V T˙ = h(CA , CB , T ) + α(TJ − T )/V + (Tin − T )u1 /V T˙J = β(T − TJ )/Vj + λ(TJ0 − TJ )u2 /V, where CA and CB are the concentrations of the components A and B, T is the reactor temperature, TJ is the jacket temperature, rA and rB are arbitrary chemical kinetic expression, h(CA , CB , T ) is an arbitrary function of the heat of reaction, V is the reactor volume, VJ is the jacket volume, α, β and λ are constants. It is easy to see that the system has the required control-affine form. Let us denote this system by ΣX . The control objective is ∗ ∗ to stabilize a desired state (CA , CB , T ∗ , TJ∗ ) using the inlet volumetric flow rate u1 and the jacket volumetric flow rate u2 as the control input. Matching equations We consider a second control-affine control system, which is feedback equivalent to ΣX , and desired interconnection and damping structures such that the matching problem defined by these elements can be solved easily. For this purpose we chose a simple mechanical control system with full actuation denoted by ΣY . Let Q be the configuration space with coordinates q = (q 1 , q 2 ) and let (q, p) = (q 1 , q 2 , p1 , p2 ) be coordinates on T ∗ Q. The Euler-Lagrange equations are 633
> ∂H q˙ 0 I ∂q > + 0 τ, = B p˙ −I 0 ∂H ∂p where B ∈ R2×2 is constant and non-singular and the generalized torque τ is the control input. The total energy H(q, p) = 12 p> M −1 p + V (q) is the sum of kinetic energy, with constant inertia matrix M = M > > 0, and potential energy function V , which is bounded from below with global minimum q ∗ . Since the system is fully actuated, we can shape the potential energy of the system without any constraints. In terms of our notation this translates into a simple feedback equivalence Ψ = (Id, Id, KB > ∂H ∂p +u). The desired interconnection and damping structure is given in coordinates by 0 0 0 I and Rd = . Jd = −I 0 0 BKB > Transitivity Next, we establish the feedback equivalence for ΣX and ΣY . It has been shown in Guay (2002) that the system ΣX is state-feedback linearizable with linearizing outputs CB CA,In − CA CB,In CB − CB,In h= TIn CB − T CB,In . CB − CB,In Let us denote by Φ1 the feedback equivalence that transforms ΣX into Brunovsky normal form, i.e. Φ1 = (h, Lf h, L2f h + Lg Lf hu). The simple mechanical control system ΣY is also feedback linearizable with linearizing output y = q. Let us denote by Φ2 the feedback equivalence that transforms ΣY into Brunovsky normal form, > ∂H > + Bτ ), i.e. Φ2 (q, p, τ ) = (q, M −1 p, − ∂H ∂q − BKB ∂p >
where KB > ∂H represents an additional damping injec∂p tion feedback. Hence, one possible transformation that establishes the feedback equivalence for ΣX and ΣY is given by Φ = Φ−1 2 ◦ Φ1 . The coordinate transformation on the state space is given by φ(CA , CB , T, TJ ) = (h, M Lf h). A new matching problem is defined by ΣX and the desired interconnection and damping structure φ∗ Jd and φ∗ Rd . Its solution is given by 1 1 H ◦ φ = (Lf h)> M Lf h + (h − q ∗ )> P (h − q ∗ ), 2 2 if we chose a quadratic potential energy function with P = P > > 0. Note that this choice of interconnection and damping structure is not obvious in the natural coordinates of ΣX . Hence, we can stabilize the desired ∗ ∗ equilibrium by choosing q ∗ to be h−1 (CA , CB , T ∗ , TJ∗ ). 5. CONCLUSION We have presented a geometric interpretation for some problems in the IDA-PBC methodology. We showed how feedback equivalence can be utilized to generate new matching problems for which an explicit solution is known. Furthermore, we have seen in the example that the transitivity property of the feedback equivalence can be used to define a simpler matching problem for feedback equivalent system. Future work will consider how the set of matching
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problems, which can be generated by an existing feedback equivalence, can be characterized.
Appendix A. CARTAN’S METHOD In this appendix some essential results of Cartan’s method are stated (see Gardner (1989); Olver (1995); Ivey and Landsberg (2003) for proofs and further details). We define the structure equations and intrinsic torsion, and state results that allows us to reduce the equivalence problem to an equivalence problem with reduced structure group.
REFERENCES Acosta, J., Ortega, R., Astolfi, A., and Mahindrakar, A. (2005). Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Trans. Automat. Control, 50(12), 1936–1955. Atkins, R.D.J. (1995). Normal forms for nonlinear systems in n states and n − 1 controls and singularities. J. Differential Equations, 117(1), 56–66. Atkins, R.D.J. and Shadwick, W.F. (1993). Invariant Functions of Control-linear Systems. Systems & Control Letters, 21(5), 389–395. Bryant, R.L., Chern, S.S., Gardner, R., Goldschmidt, H., and Griffiths, P. (1990). Exterior Differential Systems. Springer-Verlag. ´ (1908). Les sous-groupes des groupes continCartan, E. ues de transformations. Ann. Ec. Norm., 25, 57–194. Oeuvres Completes II, 719–856. Cheng, D., Astolfi, A., and Ortega, R. (2005). On feedback equivalence to port controlled hamiltonian systems. Systems & Control Letters, 54(9), 911–917. Fujimoto, K. and Sugie, T. (2000). Canonical transformation and stabilization of generalized Hamiltonian systems. Systems & Control Letters, 42(3), 217–227. Gardner, R.B. and Shadwick, W.F. (1990). Feedback equivalence for general control systems. Systems & Control Letters, 15(1), 15–23. Gardner, R. and Shadwick, W. (1992). The GS algorithm for exact linearization to Brunovsky normal forms. IEEE Trans. Automat. Control, 37(2), 224–230. Gardner, R.B. (1989). The Method of Equivalence and Its Applications, volume 58 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA. Guay, M. (2002). On the linearizability of nonisothermal continuous stirred-tank reactors. Automatica, 38(2), 269 – 278. Ivey, T.A. and Landsberg, J.M. (2003). Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, volume 61 of Graduate Studies in Mathematics. AMS, Philadelphia, PA. Olver, P.J. (1995). Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge. Ortega, R., Spong, M., Gomez-Estern, F., and Blankenstein, G. (2002a). Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Automat. Control, 47(8), 1218–1233. Ortega, R., van der Schaft, A.J., Mareels, I., Maschke, B.M., and LSS-Supelec, G. (2001). Putting energy back in control. IEEE Control Systems Magazine, 21(2), 18– 33. Ortega, R., van der Schaft, A.J., Maschke, B., and Escobar, G. (2002b). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica, 38(4), 585–596. 634
Let G be a Lie subgroup of GL(n, R), U ⊂ Rn , ωU a coframe on U and S : U → G a smooth map. For our purpose it is beneficial to consider Rn as an n-dimensional vector space V with basis {ei }ni=1 , and denote by {f i }ni=1 the basis of V ∗ dual to {ei }ni=1 . Furthermore, the duality pair between elements ofPV and V ∗ is denoted by h·, ·i. i Hence, we write ωU = ωU ei . The lift of ωU is then ∗ i j ω = Sπ ωU = Sj ωU ei . Let g denote the Lie algebra of G, suppose {εα } is a basis for g and {π β } right-invariant oneP j forms on G dual to {εα }, then we have εl = ail ej ⊗ f i . Proposition 10. The exterior derivative of ω satisfies the structure equations X X i dω i ei = aijl π l ∧ ω j + γjk ω j ∧ ω k ei (A.1) i are called the torsion coefficients where the functions γjk X i i j −1 γjk = (cjk ◦ π)hf , S · er ihf k , S −1 · es ihS −1 · f t , er i P i j with cijk given by dω i = cjk ω ∧ ω k .
Next we define the intrinsic torsion. Define the linear P l map L : g ⊗P V ∗ → g ⊗ V ∗ ' V ⊗ ∧2 V ∗ by ν = νk εl ⊗ f k 7→ − (aijl νkl − aikl νjl )ei ⊗ f j ∧ f k . The intrinsic torsion P i is defined by τU (u, S) = γjk (u, S)ei ⊗ f j ∧ f k + Im L. Furthermore, we define a representation of G. Definition 11. A representation of a group G is a group homorphism ρ : G → Aut(V ) from G to the space of invertible linear transformation of a vector space V . The dual of ρ is denoted by ρ† : G → Aut(V ∗ ). Lemma 12. Assume there exists an equivalence Φ1 : U × G → V × G, then the intrinsic torsion is invariant, i.e. τU = τV ◦ Φ1 . Lemma 13. If the equivalence problem is of first-order constant type (Gardner (1989)), then for each u ∈ U there exists a C(u) ∈ G such that (ρ⊗∧2 ρ† )(C(u))τU (u, e) = τ0 , where τ0 is the normalized value of the intrinsic torsion and e denotes the identity element of G. For a normalized value τ0 of τU we can define the isotropy group Gτ0 , {C ∈ G | ρ ⊗ ∧2 ρ† (C)τ0 = τ0 }. Further computations show that τU−1 (τ0 ) , {(u, s) ∈ U × G | u ∈ U and s ∈ Gτ0 (C(u))} = {(u, Gτ0 C(u)) | u ∈ U } is a submersed submanifold of U . Definition 14. A τ0 -modified coframe is a section of τU−1 (τ0 ) which gives rise to a local section Γ(p) = (p, βM (p)) of U × G, where βU : U → G satisfies τU (p, βU (p)) = τU (Γ(p)) = τ0 . Finally, we state one of the important results in Cartan’s method. Proposition 15. A map Φ : U → V induces a Gequivalence if and only if Φ induces a Gτ0 -equivalence between τ0 -modified coframes given by βU ωU and βV ΩV .
Geometric Aspects of IDA-PBC Kai H¨ offner ∗ and Martin Guay ∗ ∗
Queen’s University, Kingston, ON, Canada, K7L 3N6 (e-mail: [email protected]).
Abstract: The purpose of this work is to introduce a geometric definition of problems arising in passivity-based control design. We present a coordinate-free version of the well-known matching equations resulting from the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) design and show that feedback equivalence between control-affine systems can be used to define a family of matching problems for which an explicit solution is known. We illustrate our findings on a non-isothermal CSTR model. Keywords: port-Hamiltonian systems, feedback equivalence, matching equations, geometric methods 1. INTRODUCTION AND BACKGROUND This paper establishes a connection between the IDAPBC methodology (Ortega et al. (2002b)) and the geometric interpretation of feedback equivalence as treated by Gardner and Shadwick (1990, 1992) and collaborators. Furthermore, we show that the choice of coordinates together with an appropriate feedback law can be used to define new matching problem for which an explicit solution is known. The objective in Passivity-based Control (PBC) is to render the closed-loop system passive with respect to a desired storage function with a minimum at the desired equilibrium. IDA-PBC incorporates several passivity-based controller design methodologies, we refer the reader to the survey article by Ortega et al. (2001) for further information. Furthermore, it is a powerful control methodology for nonlinear system which has found wide range of applications. In particular, numerous contributions have been made for the class of underactuated mechanical systems, see Ortega et al. (2002a); Acosta et al. (2005) and reference therein. In the majority of reference concerned with IDA-PBC the choice of the interconnection and damping structure is guided be the choice of a specific coordinate representation. It is known that the port-Hamiltonian representation is preserved under coordinate transformation Fujimoto and Sugie (2000). Here, we focus on a coordinate free approach to allow in order to simplify the problem by a choice of the interconnection and damping structure in a specific set of coordinates. This paper is structured as follows: We review some concepts from the theory of feedback equivalence and portHamiltonian systems in the remainder of this section in order to fix our notation. Preliminary results on feedback equivalence are stated in Section 2. The main results are presented in Section 3. We illustrate the results on a nonisothermal CSTR model in Section 4. A short conclusion and future directions of research are given in Section 5. 1.1 Background Background information on feedback equivalence and port-Hamiltonian systems are presented in this section. 978-3-902661-80-7/10/$20.00 © 2010 IFAC
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Let X and U be open connected subsets of Rn and Rm , respectively with n ≥ m ≥ 1 and assume that 0m ∈ U and let f0 , g1 , . . . , gm be vector fields on X. A control-affine system is a triple Σ = (X, {f0 , g1 , . . . , gm }, U), where X is called the state space, U is the control space, the vector field f0 is called the drift vector field, and g1 , . . . , gm are called the control vector fields. Furthermore, we assume that the linear span of the control vector fields has locally constant rank m. Instead of working with vector fields, we can alternatively define a control-affine systems by a set of differential one-forms. Let {dxi }ni=1 be a coframe on X (i.e. it provides a basis for the cotangent space at each point) and let dt be a coordinate one-form on R, then we define for all u ∈ U with coordinates (u1 , . . . , um ) the following set of one-forms η j = dxj + (hf0 , dxj i + ui hgi , dxj i)dt, j = 1, . . . , n (1) where h·, ·i denotes the point-wise inner product between vector fields and one-forms on X. Furthermore, define the n-vector of one-forms η = [η 1 , . . . , η n ]> . A control-affine system is then also represented by the triple Σ = (X, η, U). The motivation for this definition is the following, we want to study the equivalence of control-affine systems by studying the equivalence of coframes which are adapted to the control-affine systems as given in (1). The rational behind choosing this type of adaption is that every trajectory x(t) of Σ under some control u(t) is in one-toone correspondence with an integral curve of the exterior differential system generated by {η j }nj=1 (see Gardner and Shadwick (1990) for further details). Next, we define when two control-affine systems are feedback equivalent. Let (x1 , . . . , xn , u1 , . . . , un , t) be coordinates on M = X × U × R and let πX : M → X be ∗ j n a projection. Notice that {πX η }j=1 1 is a set of n linear independent one-forms on M , together with the set {dt, du1 , . . . , dum } they define a coframe ωM = [η, du, dt]> on M , denoted the zero-adapted coframe. We use this construction of the coframe to define a feedback equivalence for control-affine systems. As it turns out, this 1
We omit the pull-back of the projection if it is clear from the context.
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definitions agrees with the one motivated by the idea that two systems are feedback equivalent if they generate the same trajectory under the same control input and initial conditions. In order to define feedback equivalence we need the following construction. Definition 1. Gardner (1989) Let ωU and ΩV be n-vector of one-forms on open sets U ⊂ Rn and V ⊂ Rn , respectively. Let G be a Lie subgroup of GL(n, R) , then ωU is said to be G-equivalent to ΩV if there exists a diffeomorphism Φ : U → V and a function γ : U → G satisfying Φ∗ (ΩV |Φ(u) ) = γ(u)ωU |u , for all u ∈ U .
∂ ∂ Similarly, we write R = Rij ∂x and denote by i ⊗ ∂xj R(x) the symmetric positive semi-definite matrix-valued function formed by Rij (x). With this notation, we write the drift vector field of a port-Hamiltonian system with dissipation in coordinates as ∂H > f0 (x) = (J(x) − R(x)) for all x ∈ X. ∂x x
1.3 Matching Problem
One of the main objectives in IDA-PBC is to find a feedback law such that the closed-loop system is a portHamiltonian system with desired interconnection and damping structure. A passivity argument is then used to The Lie group that we choose for this is a Lie subgroup of determined the stability of the system. In the following, GL(n + m + 1, R) which is called the structure group and we describe the matching problem introduced in the IDAis defined by PBC methodology as a feedback equivalence problem. For " # A ∈ GL(n, R) the purpose of this paper we are only concerned with the A 0 0 B C 0 ∈ GL(n + m + 1, R) B ∈ Rm×n . modification or generation of the Hamiltonian function, G0 = 0 0 1 C ∈ GL(m, R) without particular concern to stabilization. Stability of Definition 2. Let ΣX = (X, η, U) and ΣY = (Y, µ, V) be the closed-loop system is ensured by assigning a strict two control-affine systems with X, Y ⊂ Rn . Furthermore, minimum to the Hamiltonian function at the desired equilet ωM = [η, du, dt]> and ΩN = [µ, dv, dt]> be two zero- librium. We define the matching problem: adapted coframes on M = X × U × R and N = Y × V × Matching Problem: Given a control-affine system Σ = R, respectively. We say that ΣX is feedback equivalent to (X, {f , g , . . . , g }, U), a skew-symmetric 2-tensor field 0 1 m ΣY if ωM is G0 -equivalent to ΩN , i.e. there exists a G0 - J and a positive semi-definite 2-tensor field R on X d d equivalence Φ : X × U × R → Y × V × R of the form find, if possible, a function Hd such that Σ is feedback Φ(x, u, t) = (φ(x), ψ(x, u), t) ∀(x, u, t) ∈ X × U × R equivalent to the port-Hamiltonian system with dissipafor some diffeomorphism φ : X → Y, called coordinate tion Σd = (X, Jd , Rd , Hd , {g1 , . . . , gm }, U). transformation and some map ψ : X × U → V, called state Next, we describe a general technique to determine a feedback, then Φ so defined is called a feedback equivalence feedback equivalence of two control-affine systems ΣX = for ΣX and ΣY . (X, η, U) and ΣY = (Y, µ, V). The presentation of this technique is due to Atkins and Shadwick (1993). 1.2 Port-Hamiltonian systems 2. PRELIMINARIES Next, we define port-Hamiltonian systems. First let us recall some definitions from differential geometry. Let σ In this section we outline a procedure to determine a be a contravariant 2-tensor field on X ⊂ Rn , we associated feedback equivalence, this procedure was presented in a map σ e : T ∗ X → T X to σ by ´ Atkins (1995) and is based on the work of Elie Cartan ∗ he σ (x)(α), βi = σ(x)(α, β), for all β ∈ Tx X, x ∈ X. (Cartan (1908)). It will become clear that the projected We commonly use the same symbol for a tensor field and matching equations defined for example in Cheng et al. (2005) appear naturally in this context. the map that it is defining. Definition 3. Let X ⊂ Rn be open, H ∈ C∞ (X) and 2.1 Lifted Coframe let J be a skew-symmetric 2-tensor field on X. A portHamiltonian system is a control-affine system Σ = In the equivalence problem, as defined in the previous (X, {f0 , g1 , . . . , gm }, U) such that f0 = J(dH), where H section, we are interested in finding a diffeomorphism is called the Hamiltonian function and J is called the between M = X × U × R and N = Y × V × R such interconnection structure. We denote the system by the that its Jacobian has a predetermined form given by G0 . 5-tuple Σ = (X, J, H, {g1 , . . . , gm }, U). Proposition 5 below states that we can equivalently find Definition 4. Let R be a symmetric positive semi-definite a diffeomorphism between M × G0 and N × G0 without 2-tensor field on X ⊂ Rn . A port-Hamiltonian sys- any restrictions on its Jacobian. Let π : M × G0 → M tem with dissipation is a control-affine system Σ = be the projection onto M and define the vector of one(X, {f0 , g1 , . . . , gm }, U) with f0 = J(dH) − R(dH), R is forms ω = Sπ ∗ ωM on M × G0 , where S ∈ G0 and proceed called the damping structure. We denote the system by similarly for Ω on N ×G0 . The lifted coframe ω on M ×G0 the 6-tuple Σ = (X, J, R, H, {g1 , . . . , gm }, U). is defined as " # " #" # θ A 0 0 η 1 n Let (x , . . . , x ) be a coordinates on X, hence we have µ B C 0 du ω = Sω = = . M ∂H ∂ ∂ ∂H > i and J = J ij ∂x |x dH = ∂x dt 0 0 1 dt i dx i ⊗ ∂xj . Denote by ∂x ∂H the column vector formed by ∂x i (x) and by J(x) the The following proposition allows us to “lift” the equivaskew-symmetric matrix-valued function formed by J ij (x). lence problem to the product M × G0 . 630
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Proposition 5. (Gardner (1989)) There exists a diffeomorphism Φ : M → N satisfying Φ∗ ΩN = γN M ωM with γN M : M → G0 if and only if there exists a diffeomorphism Φ1 : M × ∗ G0 → N × G0 such that Φ1 Ω = ω. In order to solve the feedback equivalence problem we can equivalently determine if there exists a diffeomorphism ∗ Φ1 on M × G0 that solves Φ1 Ω = ω. Next, we study the structure equations (see appendix A) of the feedback equivalence problem. Since we consider only time-invariant feedback, it is clear that t is invariant for Φ and Φ1 , hence we omit this component in the structure equations. After Lie algebra compatible absorption equation (A.1) are θ α 0 θ dt ∧ Kµ d = ∧ + . (2) µ β γ µ 0 where α, β and γ are Lie algebra-valued one-forms. Furthermore, the term dt ∧ Kµ is the only non-zero intrinsic torsion. The torsion coefficient K can be expressed parametrically as K = AK0 C −1 with K0 = g, i.e. K0 is the value of K for S = e with e as the identity element of G0 . 2.2 Reduction of the Structure Group Next, we analyze how the intrinsic torsion changes under action of G0 . Instead of computing the orbit of the group action on the intrinsic torsion explicitly, we compute the infinitesimal group action to determine the reduced structure group and the adapted coframe as suggested by Gardner (1989). The infinitesimal group action on the intrinsic torsion is obtained by studying the integrability condition d◦dθ = 0. This yields in our case the congruence dK − αK + Kγ ≡ 0 mod {θ, µ, dt}, where two one-forms ω and η are said to be congruent modulo an algebraic ideal I if there exists α ∈ I such that ω = η+α, i.e. ω ≡ η mod I. We assumed that g has locally constant rank m around every point x ∈ X, hence K also has locally constant rank m. The elements A and C are required to have full-rank since G0 ⊂ GL(n + m + 1, R). We choose to normalize K matrix to the constant 0(n−m)×m KN = . Im It follows that after this normalization dKN = 0 and hence αKN ≡ KN γ mod θ, µ, dt. If we let ε α12 α= , λ α22 then we get the following congruences α12 ≡ 0 mod θ, µ, dt, α22 ≡ γ mod θ, µ, dt (3) which determine the first reduced structure group given by the isotropy group of KN . Remark 6. Parametric calculations show that the following equation AKN C −1 = KN , determines the first reduced structure group G1 . Furthermore, elements of G1 are of the form " # A1 0 0 A2 C 0 B1 B2 C where A1 ∈ GL(n − m, R), A2 ∈ Rn−m×n and B = [B1 , B2 ] ∈ Rn×m . 631
In order to define the G1 -adapted coframe we define the map βM : M → G0
g ⊥ (x) g † (x) (x, u, t) 7→ 0 0
0
0 g † (x)g(x) 0 0 1
⊥ where g ⊥ is a full-rank left annihilator of g, i.e. g g = 0 and g † is such that g † g and g ⊥ , g † have full-rank. Note that one possible choice for g † is g > . The G1 -adapted coframe is then defined by ⊥ g dx −g ⊥ f0 dt g † dx −g † (f0 + gu)dt . βM ωM = (g † g)du dt b Furthermore, define θ = [θ1 , . . . , θn−m ]> and θ¯ = [θn−m+1 , . . . , θn ]> by > S1 βM ωM = θb θ¯ µ dt , S1 ∈ G1 .
Note that θb represents the part of the control-affine system in “image of the full-rank left annihilator” and the group elements in A1 represent the degrees of freedom available in the choice of the full-rank left annihilator. We continue with the reduced equivalence problem with new structure equations for the G1 -equivalence between the G1 -adapted coframes βM ωM and βN ΩN . The basis for this is given by Proposition 15. The new structure equations are obtained by substituting the congruences (3) into the structure equations (2), which introduces new torsion. The new structure equations are " # b ¯> ε 0 0 θ ∧ Jθ¯ + dt ∧ Hθ¯ θb θ d θ¯ = λ γ 0 ∧ θ¯ + dt ∧ µ β1 β2 γ 0 µ µ where β = [β1 , β2 ] and H = [Hij ] is a matrix-valued function on M × G1 and J = [Jijk ] = −[Jikj ] are skewsymmetric matrix-valued functions on M × G1 for all i = 1, . . . , n − m. H and J are the torsion coefficients for the new structure equations. The infinitesimal group action on the torsion coefficients J and H induced by the integrability condition d ◦ dθb = 0 is determined by dJ − εJ + γ > J + Jγ ≡ 0 mod θ, dt dH − εH + Hγ ≡ 0 mod θ, µ, dt. Furthermore, we can determine the torsion coefficients parametrically by J = A1 C −> J0 C −1 and H = A1 H0 C −1 where J0 and H0 are the values of J and H at e ∈ G1 , respectively. To determine the equivalence between the control-affine systems, we repeat this process until the torsion does not explicitly depend on the group parameters. At this point there are two possibilities; either there are no free group parameters left and we have identified a canonical coframe, then this problem can be solved as an equivalence problem of coframes (see Olver (1995) for more details on equivalence of coframes). If there remain free group
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parameters, then involutivity of the defining equations must be checked. This is an existence and uniqueness question for a set of partial differential equations and is answered by the Cartan-K¨ ahler theorem, which is a geometric extension of the Cauchy-Kowalewski existence theorem (see Bryant et al. (1990)). 3. MAIN RESULT In this section we first derive the so-called matching equations associated to the matching problem and then use the transitivity property of the feedback equivalence to generate new desired interconnection and damping structures and associate a Hamiltonian function to them. 3.1 Matching equations We apply the procedure described in the previous section to the feedback equivalence of a control-affine system ΣX = (X, {f0 , g1 , . . . , gm }, U) and a port-Hamiltonian system with dissipation Σd = (X, Jd , Rd , Hd , {g1 , . . . , gm }, U). We established that any feedback equivalence Φ = (φ, ψ, Id) between ΣX and Σd has to satisfy Φ∗ ΩN = ωM , where ωM and ΩN are the adapted coframes of ΣX and Σd , respectively. Recognizing that the two control-affine systems have the same control vector fields, we get that the first torsion coefficients are the same. Hence, the first reduction of the structure group is identical for both control-affine systems, which implies that the first structure groups are equivalent. Using Proposition 15 we get that the feedback equivalence also has to satisfy Φ∗ (βN ΩN ) = αN M (βM ωM ) where αN M : M → G1 . Using the same notation as in the b previous section. The θ-component is given by ∂Hd −φ∗ g ⊥ (Jd − Rd ) = −A1 g ⊥ f0 (4) ∂x with A1 ∈ GL(n − m, R), which represents parts of the parameter of the structure group G1 . Equation (4) is the matching equation of the feedback equivalence problem, it is a partial differential equation for the unknown Hd .
−1 where (Γ−1 : U → G denotes the inverse of Γ−1 W V ◦ Ψ) WV ◦ Φ(u) ∈ G for all u ∈ U , it follows that αW U , (Γ−1 WV ◦ Ψ)−1 γV U defines a G-equivalence for ωU and ηW . Corollary 8. Let ΣY = (Y, µ, V) be a control-affine system and let Jd and Rd be the desired interconnection and damping structure on Y, respectively. Assume there exists a solution to the matching problem define by ΣY , Jd and Rd . Let ΣX = (X, η, U) be a control-affine system that is feedback equivalent to ΣY with feedback equivalence given by Φ : M → N , then there exists a feedback equivalence for ΣX and Σd .
Proof. Let ωM and ΩN be the adapted coframe associated to ΣX and ΣY , respectively. Define the portHamiltonian system with dissipation Σd = (Y, µd , V) = (Y, Jd , Rd , Hd , {g1 , . . . , gm }, V) and its adapted coframe d ωM . By assumption Σd is feedback equivalent to ΣY , which is that there exists a G0 -equivalence for ΣY and Σd . Hence, d by Proposition 7, ωM is G0 -equivalent to ωM . Next, we establish that the feedback equivalence guaranteed to exists by Corollary 8 allows us to define new interconnection and damping structures on Y. Proposition 9. Under the assumptions of Corollary 8, ¯ : M → N, Φ ¯ = there exists a local diffeomorpism Φ ¯ ¯ (φ, ψ, Id) such that the matching problem defined by ΣX , ¯ ¯ d = φ¯∗ Rd has a solution H ¯ = H ◦ φ. J¯d = φ¯∗ Jd and R Proof. Let Ψ : N → N be a the feedback equivalence for ¯ = Ψ ◦ Φ is a feedback equivalence for ΣY and Σd , then Φ ΣX and Σd such that d ¯ ∗ (ωM Φ ) = SωM (5) where S : M → G0 . Note that (5) written in compo¯ ∗ µi = Sij η j . Let us chosen a coordinate nent gives Φ d charts such that the local representative of φ¯ is such that y i = φ¯i (x1 , . . . , xn ), then φ¯∗ dy i = Sij (y)dxj implies that ¯ ∂(y i ◦Φ) ∂xj
i
∂y = ∂x = Sij (x) for all i, j = 1, . . . , n. Define j A : M → GL(n, R) such that Aij = Sij for i = 1, . . . , n and j = 1, . . . , n. We assumed that the feedback time¯ ∗ (dt) = dt which implies that the funcinvariant, hence Φ ¯ ∗ (µd ) and Aij η j must agree. tions multiplying dt in Φ Hence,
3.2 Transitivity Next, a result that allows to define solutions to an equivalence problem by utilizing known solutions is presented. We first give a general result for arbitrary equivalence of coframes and then apply this result to the special case of the matching problem. Proposition 7. Let ωU , ΩV , ηW be coframes on U, V, W ⊂ Rn respectively and let G ⊂ GL(n, R) be a Lie group. Assume there exists a G-equivalence Φ for ωU and ΩV and a G-equivalence Ψ for ΩV and ηW , then there exists a G-equivalence for ωU and ηW . Proof. Since ωU and ΩV are G-equivalent, there exists γV U : U → G such that Φ∗ (ΩV ) = γV U ωU and there exists ΓW V : V → G such that Ψ∗ (ηW ) = ΓW V ΩV . Hence, ∗ Φ∗ (ΩV ) = γV U ωU ⇔ Φ∗ (Γ−1 W V Ψ (ηW )) = γV U ωU ∗ ⇔ Γ−1 W V ◦ Φ(Ψ ◦ Φ) (ηW ) = γV U ωU −1 ⇔ (Ψ ◦ Φ)∗ (ηW ) = (Γ−1 γV U ωU W V ◦ Φ)
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¯ ∗ (Jd (dy i , dHd ) − Rd (dy i , dHd ) + hdy i , gj iv j ) Φ = A(x)(hdxi , f0 i + hdxi , gj iuj ), where f0 is the drift vector field of ΣX and g1 , . . . , gm are ¯ ∗ Jd (dy i , dHd ) = its control vector fields. Also, we have Φ ¯ ∗ Rd (dy i , dHd ) = φ¯∗ Rd (dy i , dHd ), φ¯∗ Jd (dy i , dHd ) and Φ since their component functions do not depend on v by assumption. Next, we show that these functions define the ¯ d = φ¯∗ Rd on M . We only 2-tensors J¯d = φ¯∗ Jd and R consider J¯d and note that calculations are identical for ¯ d . Using the definition of the pull-back, we get R ∂Hd ¯ φ¯∗ (Jd (dy i , dHd )) = (Jd (dy i , dHd )) ◦ φ¯ = Jdik k ◦ φ. ∂y Furthermore, we have that ∗ kl −1 ∂ −1 ∂ ¯ ¯ ¯ ¯ φ Jd = Jd ◦ φ φ∗ ⊗ φ∗ ∂y k ∂y l which we use to compute
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φ¯∗ Jd (dxi , φ¯∗ dHd ) ∗ −1 ∂ kl i −1 ∂ ¯ ¯ ¯ ¯ φ dHd φ∗ = Jd ◦ φ dx φ∗ ∂y k ∂y l kl i −1 ∂ −1 ∂ ¯ ¯ ¯ ¯ = Jd ◦ φ dx φ∗ dHd φ∗ φ∗ ∂y k ∂y l i ∂y ∂Hd ¯ = Jdkl ◦ φ¯ k ◦φ ∂x ∂y l i ∂y kl ∂Hd = J ◦ φ¯ d ∂xk ∂y l for all i = 1, . . . , n. Furthermore, we have that i ∂y ∗ i ¯∗ ¯ φ Jd (dx , φ dHd ) = φ¯∗ (Jd (dy k , dHd )). ∂xk Hence, we have defined a port-Hamiltonian system with dissipation on M with interconnection and damping struc¯ d . Furthermore, the Hamiltonian function of ture J¯d and R ¯ ¯ d = Hd ◦ φ. this system is H 4. EXAMPLE In order to illustrate the application of the results presented above, we apply the IDA-PBC methodology to a non-isothermal CSTR studied in Guay (2002). Here we adopt the following perspective; the first matching problem that is posed is a simple energy balancing of a generic simple mechanical control system under full actuation with the objective to stabilize a generic equilibrium. Then we establish a feedback equivalence between the generic simple mechanical control system and the non-isothermal CSTR. This equivalence is guaranteed by differentiable flatness of both systems. Using the transitivity property, the resulting matching problem for the CSTR system is then easily solved. The dynamics of the non-isothermal CSTR are governed by the equations C˙ A = rA (CA , CB , T ) + (CA,In − CA )u1 /V C˙ B = rB (CA , CB , T ) + (CB,In − CB )u1 /V T˙ = h(CA , CB , T ) + α(TJ − T )/V + (Tin − T )u1 /V T˙J = β(T − TJ )/Vj + λ(TJ0 − TJ )u2 /V, where CA and CB are the concentrations of the components A and B, T is the reactor temperature, TJ is the jacket temperature, rA and rB are arbitrary chemical kinetic expression, h(CA , CB , T ) is an arbitrary function of the heat of reaction, V is the reactor volume, VJ is the jacket volume, α, β and λ are constants. It is easy to see that the system has the required control-affine form. Let us denote this system by ΣX . The control objective is ∗ ∗ to stabilize a desired state (CA , CB , T ∗ , TJ∗ ) using the inlet volumetric flow rate u1 and the jacket volumetric flow rate u2 as the control input. Matching equations We consider a second control-affine control system, which is feedback equivalent to ΣX , and desired interconnection and damping structures such that the matching problem defined by these elements can be solved easily. For this purpose we chose a simple mechanical control system with full actuation denoted by ΣY . Let Q be the configuration space with coordinates q = (q 1 , q 2 ) and let (q, p) = (q 1 , q 2 , p1 , p2 ) be coordinates on T ∗ Q. The Euler-Lagrange equations are 633
> ∂H q˙ 0 I ∂q > + 0 τ, = B p˙ −I 0 ∂H ∂p where B ∈ R2×2 is constant and non-singular and the generalized torque τ is the control input. The total energy H(q, p) = 12 p> M −1 p + V (q) is the sum of kinetic energy, with constant inertia matrix M = M > > 0, and potential energy function V , which is bounded from below with global minimum q ∗ . Since the system is fully actuated, we can shape the potential energy of the system without any constraints. In terms of our notation this translates into a simple feedback equivalence Ψ = (Id, Id, KB > ∂H ∂p +u). The desired interconnection and damping structure is given in coordinates by 0 0 0 I and Rd = . Jd = −I 0 0 BKB > Transitivity Next, we establish the feedback equivalence for ΣX and ΣY . It has been shown in Guay (2002) that the system ΣX is state-feedback linearizable with linearizing outputs CB CA,In − CA CB,In CB − CB,In h= TIn CB − T CB,In . CB − CB,In Let us denote by Φ1 the feedback equivalence that transforms ΣX into Brunovsky normal form, i.e. Φ1 = (h, Lf h, L2f h + Lg Lf hu). The simple mechanical control system ΣY is also feedback linearizable with linearizing output y = q. Let us denote by Φ2 the feedback equivalence that transforms ΣY into Brunovsky normal form, > ∂H > + Bτ ), i.e. Φ2 (q, p, τ ) = (q, M −1 p, − ∂H ∂q − BKB ∂p >
where KB > ∂H represents an additional damping injec∂p tion feedback. Hence, one possible transformation that establishes the feedback equivalence for ΣX and ΣY is given by Φ = Φ−1 2 ◦ Φ1 . The coordinate transformation on the state space is given by φ(CA , CB , T, TJ ) = (h, M Lf h). A new matching problem is defined by ΣX and the desired interconnection and damping structure φ∗ Jd and φ∗ Rd . Its solution is given by 1 1 H ◦ φ = (Lf h)> M Lf h + (h − q ∗ )> P (h − q ∗ ), 2 2 if we chose a quadratic potential energy function with P = P > > 0. Note that this choice of interconnection and damping structure is not obvious in the natural coordinates of ΣX . Hence, we can stabilize the desired ∗ ∗ equilibrium by choosing q ∗ to be h−1 (CA , CB , T ∗ , TJ∗ ). 5. CONCLUSION We have presented a geometric interpretation for some problems in the IDA-PBC methodology. We showed how feedback equivalence can be utilized to generate new matching problems for which an explicit solution is known. Furthermore, we have seen in the example that the transitivity property of the feedback equivalence can be used to define a simpler matching problem for feedback equivalent system. Future work will consider how the set of matching
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
problems, which can be generated by an existing feedback equivalence, can be characterized.
Appendix A. CARTAN’S METHOD In this appendix some essential results of Cartan’s method are stated (see Gardner (1989); Olver (1995); Ivey and Landsberg (2003) for proofs and further details). We define the structure equations and intrinsic torsion, and state results that allows us to reduce the equivalence problem to an equivalence problem with reduced structure group.
REFERENCES Acosta, J., Ortega, R., Astolfi, A., and Mahindrakar, A. (2005). Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Trans. Automat. Control, 50(12), 1936–1955. Atkins, R.D.J. (1995). Normal forms for nonlinear systems in n states and n − 1 controls and singularities. J. Differential Equations, 117(1), 56–66. Atkins, R.D.J. and Shadwick, W.F. (1993). Invariant Functions of Control-linear Systems. Systems & Control Letters, 21(5), 389–395. Bryant, R.L., Chern, S.S., Gardner, R., Goldschmidt, H., and Griffiths, P. (1990). Exterior Differential Systems. Springer-Verlag. ´ (1908). Les sous-groupes des groupes continCartan, E. ues de transformations. Ann. Ec. Norm., 25, 57–194. Oeuvres Completes II, 719–856. Cheng, D., Astolfi, A., and Ortega, R. (2005). On feedback equivalence to port controlled hamiltonian systems. Systems & Control Letters, 54(9), 911–917. Fujimoto, K. and Sugie, T. (2000). Canonical transformation and stabilization of generalized Hamiltonian systems. Systems & Control Letters, 42(3), 217–227. Gardner, R.B. and Shadwick, W.F. (1990). Feedback equivalence for general control systems. Systems & Control Letters, 15(1), 15–23. Gardner, R. and Shadwick, W. (1992). The GS algorithm for exact linearization to Brunovsky normal forms. IEEE Trans. Automat. Control, 37(2), 224–230. Gardner, R.B. (1989). The Method of Equivalence and Its Applications, volume 58 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA. Guay, M. (2002). On the linearizability of nonisothermal continuous stirred-tank reactors. Automatica, 38(2), 269 – 278. Ivey, T.A. and Landsberg, J.M. (2003). Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, volume 61 of Graduate Studies in Mathematics. AMS, Philadelphia, PA. Olver, P.J. (1995). Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge. Ortega, R., Spong, M., Gomez-Estern, F., and Blankenstein, G. (2002a). Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Automat. Control, 47(8), 1218–1233. Ortega, R., van der Schaft, A.J., Mareels, I., Maschke, B.M., and LSS-Supelec, G. (2001). Putting energy back in control. IEEE Control Systems Magazine, 21(2), 18– 33. Ortega, R., van der Schaft, A.J., Maschke, B., and Escobar, G. (2002b). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica, 38(4), 585–596. 634
Let G be a Lie subgroup of GL(n, R), U ⊂ Rn , ωU a coframe on U and S : U → G a smooth map. For our purpose it is beneficial to consider Rn as an n-dimensional vector space V with basis {ei }ni=1 , and denote by {f i }ni=1 the basis of V ∗ dual to {ei }ni=1 . Furthermore, the duality pair between elements ofPV and V ∗ is denoted by h·, ·i. i Hence, we write ωU = ωU ei . The lift of ωU is then ∗ i j ω = Sπ ωU = Sj ωU ei . Let g denote the Lie algebra of G, suppose {εα } is a basis for g and {π β } right-invariant oneP j forms on G dual to {εα }, then we have εl = ail ej ⊗ f i . Proposition 10. The exterior derivative of ω satisfies the structure equations X X i dω i ei = aijl π l ∧ ω j + γjk ω j ∧ ω k ei (A.1) i are called the torsion coefficients where the functions γjk X i i j −1 γjk = (cjk ◦ π)hf , S · er ihf k , S −1 · es ihS −1 · f t , er i P i j with cijk given by dω i = cjk ω ∧ ω k .
Next we define the intrinsic torsion. Define the linear P l map L : g ⊗P V ∗ → g ⊗ V ∗ ' V ⊗ ∧2 V ∗ by ν = νk εl ⊗ f k 7→ − (aijl νkl − aikl νjl )ei ⊗ f j ∧ f k . The intrinsic torsion P i is defined by τU (u, S) = γjk (u, S)ei ⊗ f j ∧ f k + Im L. Furthermore, we define a representation of G. Definition 11. A representation of a group G is a group homorphism ρ : G → Aut(V ) from G to the space of invertible linear transformation of a vector space V . The dual of ρ is denoted by ρ† : G → Aut(V ∗ ). Lemma 12. Assume there exists an equivalence Φ1 : U × G → V × G, then the intrinsic torsion is invariant, i.e. τU = τV ◦ Φ1 . Lemma 13. If the equivalence problem is of first-order constant type (Gardner (1989)), then for each u ∈ U there exists a C(u) ∈ G such that (ρ⊗∧2 ρ† )(C(u))τU (u, e) = τ0 , where τ0 is the normalized value of the intrinsic torsion and e denotes the identity element of G. For a normalized value τ0 of τU we can define the isotropy group Gτ0 , {C ∈ G | ρ ⊗ ∧2 ρ† (C)τ0 = τ0 }. Further computations show that τU−1 (τ0 ) , {(u, s) ∈ U × G | u ∈ U and s ∈ Gτ0 (C(u))} = {(u, Gτ0 C(u)) | u ∈ U } is a submersed submanifold of U . Definition 14. A τ0 -modified coframe is a section of τU−1 (τ0 ) which gives rise to a local section Γ(p) = (p, βM (p)) of U × G, where βU : U → G satisfies τU (p, βU (p)) = τU (Γ(p)) = τ0 . Finally, we state one of the important results in Cartan’s method. Proposition 15. A map Φ : U → V induces a Gequivalence if and only if Φ induces a Gτ0 -equivalence between τ0 -modified coframes given by βU ωU and βV ΩV .