On the PDEs arising in IDA-PBC

Transcription

1 On the PDEs arising in IDA-PBC JÁ Acosta and A Astolfi Abstract The main stumbling block of most nonlinear control methods is the necessity to solve nonlinear Partial Differential Equations In this paper we propose a methodology to replace the PDEs of Interconnection and Damping Assignment Passivity-Based Control with algebraic inequalities The idea relies on the construction of an approximating integral and a dynamic extension as an alternative to solve the PDEs Some connections with linear control theory and nonlinear cascade control techniques are also provided I INTRODUCTION Stabilization of nonlinear systems shaping their energy function, but preserving the systems structure, has attracted the attention of control researchers for several years Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is a generic name to define a controller design methodology based on the passivity property In fact, Passivity-Based Control (PBC) is a controller design methodology [6] which achieves stabilization by rendering the system passive with respect to a desired storage function (energy) which has a minimum at the desired equilibrium point PBC designs may be classified into two groups A first group where a storage function to be assigned is selected and then the controller that renders the storage function non-increasing (ie a Lyapunov-like approach) is designed In the second group, the desired structure of the closed-loop system is selected instead of fixing the closedloop storage function In the latter case, once the closed loop structure has been selected the control problem is then to characterize all assignable storage functions (energy) compatible with this structure This characterization is given in terms of the solution of Partial Differential Equations (PDEs) The most notable examples of this approach arise in the control of mechanical systems While fully-actuated mechanical systems allow an arbitrary shaping of the energy by means of feedback, and therefore stabilization to any desired equilibrium, this is in general not possible for underactuated systems In case of underactuated mechanical systems there are two main approaches: the method of Controlled Lagrangians [], [3] and IDA-PBC [7], [8] In both methodologies stabilization (of a desired equilibrium) The work of JÁ Acosta was supported by the Spanish Ministerio de Ciencia e Innovación under José Castillejo grant JC8-4 and by The Consejería de Innovación Ciencia y Empresa of The Junta de Andalucía under grants TEP563 and IAC programme JÁ Acosta is with Dept Ingeniería de Sistemas y Automática, Escuela Técnica Superior de Ingenieros, Avda de los Descubrimientos s/n, 49 Sevilla, University of Seville, Spain jaar@esiuses A Astolfi is with Electrical Engineering Department, Imperial College, Exhibition Road, London, SW7 AZ, UK and with DISP, University of Rome Tor ergata, ia del Politecnico, 33, Roma Italy aastolfi@imperialacuk is achieved identifying the class of systems, Lagrangian or Hamiltonian, respectively, that can possibly be obtained via feedback The conditions under which such a feedback law exists are called matching conditions, and consist of a set of nonlinear PDEs When these PDEs can be solved the original control system and the target dynamic system are said to match In IDA-PBC the PDEs that we have to solve are parameterized by matrices that, roughly speaking, can be simply viewed as degrees-of-freedom to enforce the required passivity property and dissipation In this paper a methodology to replace the PDEs by algebraic inequalities is proposed The idea relies on the construction of an approximating integral and a dynamic extension as an alternative to solve the PDEs The approximating integral was proposed in [4] in the context of nonlinear observers The approach is applied to IDA- PBC methodology and some connections with linear systems control theory and nonlinear cascade control techniques are discussed The outline of the paper is as follows In section II the main idea and procedure of the IDA-PBC control methodology is described; the main theoretical result is given in Section III; in Sections I and the applicability of the method to systems in lower and upper triangular form, respectively, is discussed; and finally a conclusion section is given II BACKGROUND In [8] IDA-PBC has been introduced as a procedure to control physical systems described by a generalized Hamiltonian structure (see also []) Consider a nonlinear system, affine in control, given by ẋ = f(x) + g(x)u, () with x R n, u R m, m < n, and f, g smooth The IDA- PBC methodology relies on matching the system () with a generalized Hamiltonian structure ẋ = F (x) H(x), () for some smooth scalar function H(x) and a full-rank matrix F (x) R n n satisfying F (x) + F (x) Throughout the paper all vectors are column vectors, even the gradient x( ) := ( ) When clear from the context the subindex of the operator x and the arguments of the functions are omitted For a mapping h : R n R n, we define the Jacobian matrix as h(x) := [ h (x),, h n(x)] If x R n are the energy variables then H(x) represents the stored energy

2 Assume there exist matrices g (x), F (x) and a scalar function H(x) that verify the PDEs g(x) f(x) = g(x) F (x) H(x), (3) resulting from the matching of systems () and (), where g(x) is a full-rank left annihilator of g(x), ie g(x) g(x) =, and H(x) is such that x = argmin H(x), (4) with x R n an equilibrium to be stabilized We assume without loss of generality that x = Then, the open-loop system () with the feedback u(x) = (g(x) g(x)) g(x) ( f(x) + F (x) H(x)), (5) takes the form () and x is a (locally) stable equilibrium It is asymptotically stable if, in addition it is proved that x is attractive, possibly invoking LaSalle s Invariance Principle The key step in the design procedure relies on solving the PDEs given by (3) which is, usually, a difficult task (see [] and [] for some examples) III MAIN RESULT In this section an approximating integral and a dynamic extension are designed as an alternative to solve the PDEs in (3) The integral design is based on solving algebraically (3) for H(x) For, define the mapping h : R n R n, h(x) := [h (x), h (x),, h n (x)], which solves the algebraic equations (AEs) given by g(x) f(x) = g(x) F (x)h(x), (6) and define the map H : R n R n R as n xi H(x, ˆx) := h i (ˆx) i= ˆxi=s ds, (7) and the vector of errors e := ˆx x In what follows we describe the system in (x, e)-coordinates Note that, the partial derivative of (7) with respect to x is given by x H(x, e) = h(x) + x H(x, e) h(x), (8) }{{} =:δ(x,e)e where h(x) = x H(x, e) e= and the matrix δ(x, e) R n n is defined via (8) Consider now the closed-loop dynamics generated by (7), replacing in (5) the gradient H(x) with x H(x, e), namely ẋ = f(x) + g(x)u = f(x) + G(x)( f(x) + F (x) x H(x, e)) = f(x) + G(x)( f(x) + F (x)h(x)) + G(x)F (x)δ(x, e)e = F (x)h(x) + S(x, e)e, (9) where G(x) := g(x)(g(x) g(x)) g(x) and S(x, e) := G(x)F (x)δ(x, e) The following assumption is instrumental to derive the main result Assumption : The mapping x h(x) is a diffeomorphism in a domain D (possibly D R n ) containing x = The Assumption and the closed-loop dynamics (9) identify the equilibrium as ẋ e= = F (x)h(x) and since F (x) is a full-rank matrix then the equilibrium is at h(x) = Proposition : Suppose there exist symmetric and positive definite matrices Γ and K and, a full-rank matrix F (x), such that Γ h(x)f (x) + (Γ h(x)f (x)), x D, () with h(x) a solution of (6) satisfying the Assumption Then the equilibrium (x, e) = (, ) of the system [ẋ ] [ ] [ ] F (x) S(x, e) h(x) = ė S(x, e) h(x), () Γ K e is (locally) stable in (x, e) D R n, with Lyapunov function Proof Note that (x, e) := h(x) Γh(x) + () = h(x) Γ h(x)ẋ + e ė = h(x) Γ h(x)(f (x)h(x) + S(x, e)e) + e ė = h(x) ( Γ h(x)f (x) + (Γ h(x)f (x)) ) h(x) + e (S(x, e) h(x) Γh(x) + ė), (3) and choosing the error dynamics as in () yields = h(x) ( Γ hf + (Γ hf ) ) h(x) e Ke (4) Therefore (4) is negative (semi) definite if condition () is satisfied A Linear systems In this subsection we apply the main result of Proposition to linear systems This yields a Lyapunov-like interpretation of condition () Let f(x) = Ax and g(x) = b, for some A R n n and b R n m In this case the general solution of (6) is given by h(x) := Qx for some constant and full-rank matrices Q and F Thus (6) reads b (A F Q)x =, for all x D The function (7) has a closed form given by H(x, ˆx) = x diag(q)x + x (Q diag(q))ˆx, and the Lyapunov function () proposed for the nonlinear case becomes (x, e) = (Qx) Γ (Qx) + Therefore, the stability condition () reads Γ(QF ) + (QF ) Γ, (5)

3 which is a Lyapunov inequality and hence, if the matrix (QF ) is Hurwitz there is Γ > solving (5) Remark : The previous result for linear systems includes the case of the standard IDA-PBC approach In fact, selecting Q = Q > and defining Γ := Q then inequality (5) becomes F + F, which is the main assumption of the IDA-PBC approach The case of linear systems, but in another context, was else thoroughly discussed in [] B A simple example Consider a nonlinear system of the form () with [ ] [ ] x + x f(x) =, g(x) =, (6) x + x x R and u R The algebraic equation (6) reads Let x + x = f h (x) + f h (x) (7) [ ] f f F (x) =, f (x) f (x) with f and f constants Thus, the function (7) becomes x H(x, ˆx) = h (ˆx) ds + x x=s Picking h (ˆx) = ˆx and f yields ˆx + ˆx f h (ˆx) ds f x=s H(x, ˆx) = x + x f + ˆx f ˆx f x, (8) which is not positive definite, although H(x, ˆx) ˆx=x is locally positive definite 3 In this case it is enough to select Γ = I and [ ] F (x) = x + ρ x (x + ), (9) for some ρ > /4 Note that F + F < The condition () for stability becomes [ ] ρ x <, for all x R, () hence the equilibrium (x, e) = (, ) is globally asymptotically stable In Fig and Fig a batch of simulations for random initial conditions (x(), e()) are shown Fig and Fig show the time histories of the norm of the errors (top) and of the Lyapunov function (bottom) Remark : The solution of the PDE (3) for f and f constants is ( H(x) = x3 + x + Ψ x f ) x 3f f f Therefore, we cannot construct a radially unbounded Lyapunov function selecting a particular H, hence the IDA-PBC methodology can only be used for a local design 3 It is interesting to underscore that H(x, ˆx) ˆx=x is not a solution of the PDE (3) Fig Batch of simulations Random initial conditions (x(), e()) [, ] [, ] Fig Batch of simulations Random initial conditions (x(), e()) [, ] [, ] I LOWER-TRIANGULAR SYSTEMS The procedure of Section IIIB can be used to design control laws for a class of strict-feedback systems (see [5] and [9]) This class is usually controlled by backstepping technique [5] Consider the class of systems given by ẋ = f (x, x ) ẋ = f (x, x, x 3 ) ẋ n = f n (x,, x n ) ẋ n = a(x) + b(x)u, () where x := (x,, x n ) R n, f i := x i+ + f i (x,, x i+ ) with f i () = for i =,, n, b() and u R

4 Select the matrix F (x) as f f f F (x) := f 3 f 3 f 33, () f n f nn and the vector h(x) as i h i = f i f (i j)j h j, i =,, n (3) j= with h (ˆx) := ˆx This choice provides a lower-triangular matrix hf (with Γ = I n ) and allows to cancel all the off-diagonal terms with the n(n )/ free off-diagonal parameters of the matrix F solving the n(n )/ equations ( hf ) (i+)i =, i =,, n, (4) ( hf ) ij =, i = 3,, n; j =,, i (5) Thus, it only remains to select the diagonal terms f ii (x) <, i =,, n, such that the matrix hf, now a diagonal matrix, is negative definite A Example Consider a 3-dimensional system with x + x f(x) = x 3, g(x) = (6) x + x 3 Define the matrix F (x) as f f F (x) := f (x) f (x) f 3, (7) f 3 (x) f 3 (x) f 33 (x) with f, f and f 3 constants The AEs (6) read x + x = f h (x) + f h (x) (8) x 3 = f h (x) + f h (x) + f 3 h 3 (x) (9) Thus, the function (7) becomes x H(x, ˆx) = h (ˆx) ds + + x x3 x=s ˆx + ˆx f h (ˆx) f ds x=s ˆx + ˆx f h (ˆx) f h (ˆx) f 3 ds x3=s To show the flexibility of the approach we solve this example using the example in Section IIIB as a partial solution and then using partially the system of equations (4) (5) proposed in this section Thus, pick h (ˆx) = ˆx and let F (x) = x + ρ x (x + ), (3) f 3 (x) f 3 (x) f 33 (x) for some ρ > /4 Notice that we still have f 3, f 3 and f 33 as free parameters to complete the design Moreover, solving (4) (5) for f 3 and f 3 yields α(x) := 4x + 4x + ρ + + x f 3 = (4x + ) + (x + ) (α(x) + f (x )) f 3 = (4x + ) (x + )α(x) + f (x ), and setting Γ = I 3 the condition () becomes ρ x < (3) f (x ) + f 33 (x) The condition (3), together with F + F from (3), allows to select the parameter f 33 as with β(x) := [ f3 + f 3 f 33 := β(x) ρ + x + f < (3) ] [ ] [ ] f x + f3 > x + + f 3 Therefore, since (3) holds for all x R 3 the equilibrium (x, e) = (, ) is glogally asymptotically stable Remark 3: We underscore an important difference with respect to the original IDA-PBC approach In IDA-PBC the fact that the full-rank matrix F (x) is such that F + F is instrumental In the example, the selection (3) implies the condition F + F, which is not necessary, since the only condition for stability is () Hence, in this example the free parameter f 33 can be chosen to be negative and such that f 33 (x) < f (x ) (instead of (3)) to satisfy only (3) For example we could select f 33 := γf <, (33) for some γ > Obviously, the design without the requirement F + F is simpler than the previous one (just compare (33) with (3)) Remark 4: In this example the original set of PDEs (3) for any f and f (not necessarily constants) and f 3 constant is inconsistent, ie there is no function H(x) that satisfies the PDEs We display the result of a batch of simulations for random initial conditions in (x(), e()) Figures 3, 4, 5 and 6 show the time histories of the norm of the errors (top) and the Lyapunov function (bottom) In Fig 3 and Fig 4 the function f 33 is given by (3), and in Fig 5 and Fig 6 the function f 33 is given by (33) UPPER-TRIANGULAR SYSTEMS The approach proposed allows to solve control problems for classes of strict-feedforward systems [9] To illustrate this issue consider a system with x + x 3 3 f(x) = x 3, g(x) =, (34)

5 Fig 3 Batch of simulations Random initial conditions (x(), e()) [, ] [, ], and f 33 given by (3) Fig 5 Batch of simulations Random initial conditions (x(), e()) [, ] [, ] and f 33 given by (33) Fig 4 Batch of simulations Random initial conditions (x(), e()) [, ] [, ] and f 33 given by (3) Fig 6 Batch of simulations Random initial conditions (x(), e()) [, ] [, ] and f 33 given by (33) x R 3 and u R The AEs (6) read x + x 3 3 = f h (x) + f h (x) + f 3 h 3 (x) (35) x 3 = f h (x) + f h (x) + f 3 h 3 (x) (36) In this example we propose directly an explicit solution for the equations (35) (36) For, define F (x) as x 3 F (x) :=, (37) ρ which is full-rank for ρ Thus, with this selection a solution of (35) (36) is h(x) = [x, x + x, x 3 ] In this case, we first check that hf is Hurwitz for all x and then solve the Lyapunov inequality for Γ > Note that x 3 h(x)f (x) = + x 3, ρ with the characteristic polynomial p(λ) = λ 3 + λ + ρ( + x 3)λ + ρ d Note that dλp(λ) > for ρ > 4/3, ie p(λ) is strictly monotonic; and that p() = ρ and p( ) = ρ( + x 3) This means that p(λ) has a real root for λ [, ] and this is the only real root 4 A detailed and straightforward analysis shows that the bound ρ > 4/3 obtained for x 3 = is enough 4 This can be seen using the discriminant of p(λ)

6 to guarantee that the real part of the two imaginary roots are negative for all x 3 R and then, the design is completed Thus, since for x 3 = the matrix hf is constant Γ is obtained from the solution of (5) Finally, we can conclude that the equilibrium (x, e) = (, ) is asymptotically stable Fig 7 Batch of simulations Random initial conditions (x(), e()) [, ] [, ] Fig 8 Batch of simulations Random initial conditions (x(), e()) [, ] [, ] We display a batch of simulations for random initial conditions in (x(), e()) Figures 7 and 8 show the time histories of the norm of the errors (top) and of the Lyapunov function (bottom) The matrix Γ has been computed for ρ =, and (5) yields 3 5/ 3/ Γ = 5/ 6 3/ 5/4 Remark 5: In this example the original set of PDEs (3) is again inconsistent hence there is no function H(x) that satisfies the PDEs (3) I CONCLUSIONS A methodology to replace nonlinear PDEs by algebraic inequalities in the context of IDA-PBC is proposed To this end an approximating integral together with a dynamic extension are designed Main differences with the standard IDA-PBC methodology are highlighted through examples Some connections with linear control theory and nonlinear cascade control techniques are also provided II ACKNOWLEDGMENTS This work was developed and completed during the stay of JÁ Acosta as member of the Control & Power Group in the Electrical Engineering Department of Imperial College London JÁ Acosta would like to thank the Spanish Ministerio de Ciencia e Innovación for the grant José Castillejo to support the stay during 8/9 REFERENCES [] JÁ Acosta, R Ortega, A Astolfi and AD Mahindrakar, Interconnection and damping assignment passivity based control of mechanical systems with underactuation degree one, IEEE Transaction Automatic Control, vol 5, no, 5 [] D Auckly, L Kapitanski and W White, Control of nonlinear underactuated systems, Comm Pure Appl Math vol 3, pp , [3] A Bloch, N Leonard and J Marsden, Controlled Lagrangians and the stabilization of mechanical systems I The first matching theorem, IEEE Trans Automatic Control, vol 45, no, pp 53-7, Dec [4] D Karagiannis and A Astolfi, Observer design for a class of nonlinear systems using dynamic scaling with application to adaptive control, IEEE Conference on Decision and Control, pp 34-39, 9- Dec 8 [5] M Krstić, I Kanellakopoulos and P Kokotović, Nonlinear Adaptative Control Design, John Wiley & Sons, 995 [6] R Ortega and M Spong, Adaptive motion control of rigid robots: A tutorial, Automatica, vol 5, no 6, pp , 989 [7] R Ortega, M Spong, F Gómez and G Blankenstein, Stabilization of a underactuated mechanical systems via interconnection and damping assignment, IEEE Transaction Automatic Control, vol 47, no 8, pp 8-33, [8] R Ortega, A van der Schaft, B Maschke and G Escobar, Interconnection and damping assignment passivity based control of portcontrolled Hamiltonian systems, Automatica vol 38, no 4, pp , [9] R Sepulchre, M Janković and P Kokotović, Constructive Nonlinear Control, Springer-erlag, London 997 [] S Prajna, A J van der Schaft and G Meinsma, An LMI approach to stabilization of linear port-controlled Hamiltonian systems, Systems & Control Letters, vol 45, pp , [] A J van der Schaft, L Gain and Passivity Techniques in Nonlinear Control,Springer-erlag, Berlin, 999 [] G iola, R Banavar, JÁ Acosta and A Astolfi, Total Energy Shaping Control of Mechanical Systems: Simplifying the matching equations via coordinate changes, IEEE Transactions on Automatic Control, vol 5,no 6, pp 93-99, 7