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A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems Leyla Gören Sümer* and Yaprak Yalçın** * Control Engineering Department, Istanbul Technical University Istanbul, Turkey (e-mail: leyla.goren@ itu.edu.tr). ** Control Engineering Department, Istanbul Technical University Istanbul, Turkey (e-mail: [email protected])
Abstract: In this paper, a direct discrete time design method in the sense of passivity-based control (PBC) is investigated. This method, which is known as interconnection and damping (IDA), deals with the stabilization of under-actuated mechanical systems and, it is based on the modification of both the potential and kinetic energies. In order to give a direct discrete time IDA-PBC design method, the discrete time counterpart of matching conditions is derived using an appropriate discrete gradient. The discretetime matching conditions are obtained as a set of linear partial differential equations which can be solved off-line parametrically and a set of linear equations. The unknown parameters of linear partial differential equations and the linear equations have to be solved at each sampling time, to calculate control rule. Moreover, a design procedure is given to solve these matching conditions for a class of Hamiltonian systems. To illustrate the effectiveness and the appropriateness of the proposed method, the example of pendulum on a cart is considered. 1. INTRODUCTION The port-controlled Hamiltonian (PCH) approach has been versatile not only for modeling of physical systems but also for control of a wide class of nonlinear systems (Van der Schaft, A., 2000, Ortega et al. 1998). Furthermore, the passivity-based control (PBC) is a powerful design technique for stabilizing nonlinear systems and especially for set point regulation problem both in Euler-Lagrange systems and PCH systems (Ortega and Garcia-Canseco, 2004). In continuous-time context, the PBC design is carried out in ( ) is two-steps; first the energy shaping control rule designed to assign the desired energy function as the total energy of the system, second, the damping injection control ( ) is designed to achieve asymptotic stability at rule desired equilibrium point, which corresponds to an isolated and strict minimum of the desired energy function. One can find the details of the design methodology in Ortega and Garcia-Canseco, 2004, and references therein. On the other hand, technological advancements in digital processors, and the widespread use of computer controlled systems in engineering practice demands for a theory which would inspire methods and techniques for analyzing and designing sampled-data systems and discrete-time model of non-linear systems. A framework dealing with the stabilization of sampled-data nonlinear systems using the approximate discrete time models of the system can be found in Nesic et al., 1999 and Nesic, and Teel, 2004. In the control literature, to the best of our knowledge, number of works utilizing the discrete-time models of Hamiltonian systems for control applications are limited (Laila and Astolfi, 2005, 2006a, 2006b) and (Gören-Sümer and Yalçın, 2008, Yalçın and Gören-Sümer, 2008, 2009 and 2010). 978-3-902661-93-7/11/$20.00 © 2011 IFAC
For several years now, the control researchers have been concentrated on deriving the IDA-PBC design method which is based on modifying both potential and kinetic energies for the stabilization of under actuated mechanical systems. While an arbitrary potential energy shaping stabilizes the fully actuated mechanical system at any desired equilibrium, the stabilization problem of under actuated systems is achieved by modifying the kinetic energy of the system in general. The suggestion of total energy shaping was first introduced in (Ailon and Ortega, 1993) and there are lot contributions on this subject with two main approaches: the method of controlled Lagrangians (Bloch et al., 2000) and IDA-PBC (Ortega et al. 2002). There are also closely related works (Fujimoto and Sugie, 2001) and (Viola et al., 2007) see also Ortega, and Garcia-Canseco, 2004 which contains an extensive list of references on this topic. In these methods, the stabilization of a desired equilibrium is accomplished establishing a class of systems which can be obtained using feedback. The conditions for the existence of such a feedback law are known as matching conditions which consist of a set of nonlinear partial differential equations (PDEs). Some efforts have been devoted to the solution of the matching equations Gomez-Estern et al., 2001, Acosta et al., 2005, Viola et al., 2007, and references therein. As realized from these studies, to solve these PDEs is troublesome, so the stabilization problem for Hamiltonian systems for the under actuated case is still known as a challenging problem. Therefore, considering the discrete time counterpart of the problem and then obtaining the matching conditions in discrete time setting may provide an easy technique to solve the stabilization problem for Hamiltonian systems for the under actuated case. In this study, a direct discrete time design method in the sense of IDA-PBC is investigated. In order to give a direct discrete time IDA-PBC design method, firstly an appropriate discrete gradient is proposed, which
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10.3182/20110828-6-IT-1002.01187
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
enables the derivation of a discrete time model corresponding to Hamiltonian Systems and also the determining the discrete time counterpart of matching conditions. Using this proposed discrete gradient and discrete-time model, the discrete-time matching conditions are obtained as a set of linear partial differential equations which can be solved off-line parametrically and a set of linear equations. The unknown parameters of linear partial differential equations and the linear equations have to be solved at each sampling time, to calculate control rule. Moreover, a procedure is developed to solve these matching conditions for a class of Hamiltonian systems.
( ) ( ) ( ) where ( ) The controller ( ) is obtained as a solution of the following equation, [
][
̇ [ ] ̇
( (
*
[ ( )] ( )
( )
( )
) (1)
(
[
{
The notation ( ) is used to denote the gradient vector of a scalar function ( ) with respect to (.). Furthermore, ( ) is the Hamiltonian function of the system or the energy function of the system is in the following form, )
(
)
( )
( )
( )
(2)
where ( ) and ( ) are potential and kinetic energy terms, respectively, and ( ) ( ) is the generalized ( ) inertia matrix. If , i.e. a constant matrix, the system is called a separable Hamiltonian system, and if ( ) the system is said to be fully actuated, if the system is said to be under actuated. The main idea in the method developed by Ortega et al 2002a, 2002b were to design a stabilizing controller which assigns a desired energy function, (
)
(
)
( )
( )
( )
[ ] ̇ *
( (
)
(
)) [ +
(4)
] [
]
][
(5)
]
( )
(
)
( ) }
(6)
)
( ( )
( )
( )
)
(7)
}
(8)
( )}
with a full rank left annihilator of . If the PDE s’ (7) and (8) are solvable, the energy shaping controller is derived as, (
)
{
}
(9)
Since the resulting closed loop system under this control rule is also a Hamiltonian system, the damping injection control rule which yields an asymptotically stable system is obtained as, ( )
(10)
, (Ortega et al 2002a, 2002b).
Since solving the PDEs given in (7) and (8) is not straightforward, we will consider the discrete time counterpart of the problem and then obtain the matching conditions in discrete time setting. 2.2 Discrete-time model of Hamiltonian Systems In this study, a direct discrete time IDA-PBC design method for the system given (1) for the under-actuated case will be developed. To fulfill this, the gradient based discrete model presented and used in (Gören-Sümer and Yalçın, 2008) and (Yalçın and Gören-Sümer, 2009 and 2010) will be used in slightly modified manner as the discrete time model of the Hamiltonian systems. In general, a discrete gradient is defined in (Gonzalez and Simo, 1996), restated below. Definition 1: Let, ( ) be a differentiable scalar function in then ̅ ( ) is a discrete gradient of ( ) if it is continuous in and,
(3)
) of the closed which has an isolated equilibrium point ( system. The stabilization problem for Hamiltonian systems for the under actuated case is known as a challenging problem since it needs the appropriate choice of the desired energy function and also assignment of a new interconnection matrix. In literature, the desired system is considered as, ̇
(
with
]
[
or the following equivalent constraints,
( )
) + )
where ( ) is an 2n - dimensional manifold, ( ) is system output, ( ) is the control input, ( ) is input force matrix and is the standard skew-symmetric matrix, namely,
(
( )
{
{
Consider the following continuous-time Hamiltonian systems given in standard coordinates,
( )
[ ( )]
( ) and In case the matrix ( ) is full column rank, the ( ) holding the following constraint should be constructed,
2. PRELIMINARIES and PROBLEM FORMULATION 2.1 Continuous-time Hamiltonian Systems
]
̅
(
)( ̅
) (
in which the gradient of ( )
*
)
(
) (
(
)
)
(11a) (11b)
( ) defined as, +
[
]. □
(12)
The discrete gradient proposed by Yalçın and Gören-Sümer, 2009 has been constructed using the Quadratic Approximation Lemma, (Diewert, 1976) and it has been shown that there exists a discrete gradient which exactly satisfies the conditions (11) if the energy function ( ) has the general quadratic form. Consequently, the second order Taylor approximation of any ( ) might be used to define a discrete gradient as follows,
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
̅ (
( ̃(
)
̃(
)
))
where ̃( ) is the second order Taylor approximation of ( ), namely, ̃( )
(
)
(
)(
)
(
)
(
)
for . As a consequence of the above analysis the following definition is presented. Definition 2: Consider a differentiable function in given as ( ), then the discrete gradient of a ( ) is defined as, ̅ (
)
in which, function of
( ) is the Hessian matrix of the energy ( ).□
(
)
(
)(
)
(16)
Approximating to derivatives of state variables of (1) by Forward Euler with sampling period , and replacing the gradient term in (1) with the discrete gradients ̅ and ̅ the gradient-based discrete-time description of the system (1) can be obtained as follows, [
]
*
̅ ̅
* ( )+ ( )
+
( )̅
()
(
)
(17) Furthermore, the similar expressions can also be obtained for the discrete-time description of the desired system as follows, [
]
( ( )
[
(
( )) *
̅ ̅
+
(18)
)]
(19)
where, ( ) ( )
*
(
)
)
(
(
)
( ( ) (
)
( )
) +
(20)
If the right hand side of (17) is equated to the right hand side of (18), the discrete time control rule responsible for energy shaping is obtained as follows in terms of discrete gradients, ( )
(
{̅
)
̅
̅
} (21)
and for the resulting closed-loop Hamiltonian system the damping injection control rule can be written as, ̅
( )
(
)(
)
(
)
for discrete case using (18) as following, (
)
(
)
(
̅
(
)
(
)
(
)
̅
̅
and using the property of Taylor series expansion, (
)
(
)
̃( ̅ ̃(
)
̃(
)
̅
(
)̅
̃) ̅ ̃
(24)
̅ ̅ ̅ ̃ ̃ ̅ ̃. ( ) where This relation implies that the discrete model of the open loop system provides an extra energy or extra dissipation ) according to the sign of ( . Obviously, for this extra term tends to zero. As a consequence of the above discussion the following Remark presented in (GörenSümer and Yalçın, 2008) can be slightly modified to obtain the stabilizability property for the discrete time control rules.
Remark 1: When the gradient based discrete model proposed here is used to design a control rule to stabilize the nonseparable sampled-data Hamiltonian system, the extra term due to discrete model of the open loop system does not effect ) stabilizability condition of system, if ( , i.e., the model has more energy than the system. On the other hand, if ̃ ( ) , namely, the discrete model of the open loop system has an extra dissipation, the control rule responsible for dissipation injection must be designed considering this fact, especially when slow sampling is used. As it can be obviously followed from (24), the stabilizability condition of continuous Hamiltonian system by discrete-time control remains same, i.e. stability can be achieved by adding an extra dissipation. Furthermore, it can be realized from (24) that this extra term does not have an effect on the procedure deriving the control rule which is responsible for energy shaping. □ 3. DIRECT DISCRETE TIME IDA-PBC DESIGN METHOD
(22)
It is known that in IDA-PBC stabilization is achieved by assigning a desired total energy function to the closed loop system and dissipation injection for the continuous case. On the other hand, the analogy between continuous and discrete cases would give rise to a similar energy relation for continuous case as in, ̇ ( )
under the assumption such that there exist a discrete gradient, ), ( ) namely ̅ ( , which satisfies the conditions in Definition 1. Therefore, the stability of the discrete system given in (18), namely closed loop system, can be guaranteed by the total energy function of the system and dissipation injection with . In the design method proposed here, it will be assured that the discrete gradient of the desired energy function holds the conditions given in Definition 1, but the discrete gradient of the energy function of open loop system may not satisfy these conditions. In this case, the energy relation for the open loop system can be written as follows,
In order to develop a direct discrete time IDA-PBC design method for the system given (1) for the under-actuated case, the matching conditions should be derived in discrete time setting, firstly. Using the expression (16) in the Definition 2, the following relations can be written for the discrete gradients according to the variable , present in (17-22),
(23)
̅ ̅
)̅
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( (
( (
)
(
)(
)) (25a)
)
(
)(
)) (25b)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
̅
(
)
̅
(
(
)
)(
)
(
(25c) ) (25d)
)(
where , and are the Hessian ( ) matrices - according to the variable - of ( ) ( ) and ( ), respectively; and for the discrete gradients according to the variable are written as, ̅
(
)
(
( ̅
)(
(
)(
)(
{
(26b)
) and use the following
̅
̅
( )̅
(27)
for the discrete gradient terms given in (25), then the following relations are obtained for the discrete gradients of the kinetic energies, namely for (25a, b), ̅
([
(
)]
(
̅ ̅
([
(
(
)
(
)]
)̅
(
̅
(
(28)
)̅
)
(29) )̅
)
)
(
)̅
)
[
(
)]
(31)
(
)
[
(
)]
(
(32)
)
) (33)
(
{ ( )
and for the discrete gradients of the potential energies, ̅
(
̅
)
(
(
)
(
) (
)̅ (
)
)̅
(34) (35)
(
)
)̅
(
(
)
(
̅
) (
( )
(
)(
)
( (
(
)
(
)
(
)
(
)}
(38)
( )}
( )
(
( )
(
)) }
(39)
(
)
(
)
(
)
(
)(
)
(40)
simultaneously, in which,
(
)
(
)
(
)
(
)
.
Using the matrices , and which are calculated the previous steps, the control rules for energy shaping,
))
) }
)
)
and
( )
(36)
)
{̅
( )
(37)
These conditions can be regarded as the discrete time matching conditions for the solution of the IDA-PBC of Hamiltonian system given in (1) for under actuated case. The method suggested in this study is based on obtaining the parameters of the desired potential energy function ( )
(
̅
̅
}
and for dissipation injection,
and {
)
(
) (
(
)̅ }
and using the relations derived in (28-35) the equivalent conditions are obtained as follows, { (
(
and the discrete gradient property,
Furthermore, the matching condition which is obtained from equation (13) and (14) can be constructed as follows, {̅
)
Since the desired inertia matrix determined in the previous step is not a matrix function of the , but it is only a matrix sequence depended on the value of the ( ) at each sampling time, the vector ̅ ( ) or ( ) equivalently the matrix ( ), and the matrix ( ) must be calculated at each sampling time all satisfying the kinetic energy matching condition (36), namely,
(30)
where, (
(
( )
( ) ( ) is are analytically solved by assuming a constant matrix, then a function ( ) with coefficients depended on the values of the elements of ( ) the matrix is found, such that . After then the Hessian matrix of this ( ) is calculated and a matrix function is generated in terms of the elements of , satisfying ( ) , if it is possible. It should be noted that the class is determined by the existence of a ( ) matrix . Since the matching condition (38) has been solved analytically, the matching conditions (38) so (37) hold for the solution of ( ), therefore the ( ) can be made only assuring choice of ( ) and .
)
)
(
The following linear PDEs, which are equivalent the discrete time potential energy matching condition (37),
(26a)
(
(
The main idea of the method proposed here may be best explained by giving a design procedure - which determines the class of Hamiltonian systems- step by step as follows;
)
)
)
Let’s define, ̅ equation,
)(
( ) ( ) with and , which can be solved ( ) ) off-line, and the matrix sequences and ( such that these conditions hold at each sampling time.
̅
are evaluated at each sampling time. 4. THE CART and PENDULUM SYSTEM In this paper, a well-known under actuated Hamiltonian system; The Cart and Pendulum system is considered in
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
discrete time. The dynamic equation of the cart and pendulum on a cart are given by (1) with, ( ) ( )
( )
( )
],
[
(
⁄
( (
[ ⁄
(
)⁄(
(
)
( (
) ) )
( (
) )
) )
]
)
and , where is the pendulum angle with the upright vertical, is the cart position as shown in Figure 1, is the mass of the pendulum, is the length of the pendulum, is the mass of the cart and is the gravity acceleration. The equilibrium to be stabilized is the upward position of the pendulum with the cart placed in any desired position, which corresponds to and any arbitrary .
⁄ ( )) must It can easily be shown that, the term ( be positive for all , in order to satisfy the condition ( ) ( ) . Therefore, the matrices ( )
satisfying chosen as,
( )
at each sampling time can be
[
( ) ] ( )
( )
in which, ( )
( )
( )
( )
with and ( ) is the ( arbitrary real numbers and ( ) which is “ ( ) ” here. Fig. 1. The cart and pendulum system
4.1 Solving the Matching Conditions The solution procedure given in previous section will be carried out for the cart and pendulum system as follows; Step 1: The potential energy matching condition in discrete time setting is, {
(
( )
)
(
)
(
( )
)
(
(
( )
(
( (
)
where ( ( )) is an arbitrary differentiable function satisfying the condition ( ( )) for and arbitrary , in this study it is taken as ( ) ( ⁄ ) with . The gradient and Hessian of the desired ( ) are potential energy, namely ( ) and obtained as follows, (
( ) [
)
( (
*(
) )
)
*( )
( (
( (
) )
+
) )
+ ]
̅
̅ (
(
)
(
)
(
,
)̅
)
)(
the control rule responsible for energy shaping is found by the relation, ( )
) )
̅ ̅
( )
)
( ( ))
)
Step 3: Using the following relations and the matrices and ( ) which are found in previous steps, ̅
The general solution of this PDE is given in terms of ( ) and ( ) as, ( )
( ) and Step 2: In order to determine the matrices ( ) ( ), the three linear non-homogenous equations given in (39) and (40) with five unknowns are solved for each sampling time using the matrices ( ) and ( ) which are found in previous step.
( )}
( ) ( ) and and if the matrix is defined as the gradient of the potential energy of the system obtained as ] is used, then the condition is ( ) [ written as,
being any ) element of
(
)
{̅
̅
̅
}
and finally the control rule responsible dissipation injection is ̅ ( ) calculated as follows, at each sampling time. In simulations, the system parameters are taken as , , and the discrete time controller design parameters are chosen as, and . The simulations are carried out for the sampling period , the initial conditions ( ( ) ( ) ) ( ) and the desired cart position . The simulation results are presented in Figure 2, 3 and 4 which illustrate time domain responses under the direct discretetime control. These results reveal that the direct discrete time controller design method proposed in this study yields a good performance for sampled data Hamiltonian systems. 5. DISCUSSION The results reported in this work are the first results on the direct discrete time IDA-PBC design for under actuated Hamiltonian systems according to the best of the authors’ knowledge. Surely, the method reported here should be developed especially in determining a domain of attraction
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
for the equilibrium. Also the values of design parameters which ensure the boundedness of all trajectories starting in the domain of attraction can be determined. These two issues are our current research.
Fig 2 The pendulum angle versus time.
Fig 3 The cart position versus time.
Fig 4 Control signal versus time
REFERENCES Acosta, J. Á., Ortega, R., Astolfi, A. ve Mahindrakar, A. D., 2005: Interconnection and Damping Assignment Passivity-Based Control of Mechanical Systems with Underactuation Degree One, IEEE Transactions on Automatic Control, 50, 12, 1936-1954. Ailon, A. and Ortega, R., 1993: An observer-based controller for robot manipulators with flexible joints, System and Control Letters, 21, 329–335. Bloch, A.M., Leonard, N.E. and Marsden, J.E., 2000: Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Trans. Automatic Control, 45, 12, 2253–2269. Diewert, W. E., 1976: Exact and Superlative Index Numbers, Journal of Econometrics, 4, 2, 115-145. Gomez-Estern, F., Ortega, R., Rubio, F. and Aracil, J., 2001: Stabilization of a class of underactuated mechanical systems via total energy shaping, IEEE Conf. on Decision and Control, Orlando, FL, USA.
Gonzalez, O. and Simo, J.C., 1996: On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry, Computer Methods in Applied Mechanics and Engineering, 134, 3-4, 197–222. Gören-Sümer, L. and Yalçın, Y., 2008: Gradient Based DiscreteTime Modeling and Control of Hamiltonian Systems, 17th IFAC World Congress, Korea, Seoul, 2008. Fujimoto, K. and Sugie, T., 2001: Canonical transformations and stabilization of generalized Hamiltonian systems, System and Control Letters, 42, 3, 217–227. Laila, D. S. and Astolfi, A., 2005: Discrete-time IDA-PBC design for separable Hamiltonian systems, 16th IFAC World Congress, Prague. Laila, D. S. and Astolfi, A., 2006a: Direct Discrete-time Design for Sampled-data Hamiltonian Control Systems, 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nogoya, 57-62. Laila, D. S. and Astolfi, A., 2006b: Discrete-time IDA-PBC design for underactuated Hamiltonian control systems. American Control Conference, USA, Minnesota. Nesic, D., Teel, A.R. and Kokotovic, P. V., 1999: Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, System and Control Letters, 38, 4-5, 259-270. Nesic, D. and Angeli, D., 2002: Integral versions of ISS for sampled-data nonlinear systems via their approximate discretetime models, IEEE Trans. Automatic Control, 47, 2033-2038. Nesic, D. and Laila, D.S., 2002: A note on input-to-state stabilization for nonlinear sampled-data systems, IEEE Trans. Automatic Control, 47, 1153-1158. Nesic, D., and Teel, A.R., 2004: A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models, IEEE Trans. on Automatic Control, 49, 7, 1103-1122. Ortega, R., Loría Perez, J.A., Nicklasson, P.J. and Sira-Ramirez, H.J. 1998: Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications, Springer-Verlag. Ortega, R., Van der Schaft, A.J., Maschke, B. ve Escobar, G., 2002a: Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, 38, 4, 585-596. Ortega, R., Spong, M.W., Gómez-Estern, F. and Blankenstein, G., 2002b: Stabilization of a Class of Underactuated Mechanical Systems Via Interconnection and Damping Assignment, IEEE Trans. on Automatıc Control, 47, 8, 1218-1233. Ortega, R. and Garcia-Canseco, E., 2004: Interconnection and Damping Assignment Passivity-Based Control: A survey, European Journal of Control, 10, 5, 432-450. Van der Schaft, A., 2000: L2-Gain and Passivity Techniques in Nonlinear Control, Springer, London. Viola, G., Ortega, R., Banavar, R., Acosta, J. A. and Astolfi, A., 2007: Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations via Coordinate Changes, IEEE Trans. Automatic Control, 52, 6, 1093-1099. Yalçın, Y., and Gören-Sümer, L., 2008: Disturbance Attenuation in Hamiltonian Systems via Direct Discrete Time Design, 17th IFAC World Congress, Korea, Seoul. Yalçın, Y., and Gören-Sümer, L, 2009: Robust Disturbance Attenuation in Hamiltonian Systems via Direct Digital Control, European Control Conference, Budapest. Yalçın, Y., and Gören-Sümer, L., 2010: Direct Discrete-Time Control of Port Controlled Hamiltonian Systems, Turkish Journal of Electrical Engineering and Computer Sciences, 18, 5, 913-924.
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A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems Leyla Gören Sümer* and Yaprak Yalçın** * Control Engineering Department, Istanbul Technical University Istanbul, Turkey (e-mail: leyla.goren@ itu.edu.tr). ** Control Engineering Department, Istanbul Technical University Istanbul, Turkey (e-mail: [email protected])
Abstract: In this paper, a direct discrete time design method in the sense of passivity-based control (PBC) is investigated. This method, which is known as interconnection and damping (IDA), deals with the stabilization of under-actuated mechanical systems and, it is based on the modification of both the potential and kinetic energies. In order to give a direct discrete time IDA-PBC design method, the discrete time counterpart of matching conditions is derived using an appropriate discrete gradient. The discretetime matching conditions are obtained as a set of linear partial differential equations which can be solved off-line parametrically and a set of linear equations. The unknown parameters of linear partial differential equations and the linear equations have to be solved at each sampling time, to calculate control rule. Moreover, a design procedure is given to solve these matching conditions for a class of Hamiltonian systems. To illustrate the effectiveness and the appropriateness of the proposed method, the example of pendulum on a cart is considered. 1. INTRODUCTION The port-controlled Hamiltonian (PCH) approach has been versatile not only for modeling of physical systems but also for control of a wide class of nonlinear systems (Van der Schaft, A., 2000, Ortega et al. 1998). Furthermore, the passivity-based control (PBC) is a powerful design technique for stabilizing nonlinear systems and especially for set point regulation problem both in Euler-Lagrange systems and PCH systems (Ortega and Garcia-Canseco, 2004). In continuous-time context, the PBC design is carried out in ( ) is two-steps; first the energy shaping control rule designed to assign the desired energy function as the total energy of the system, second, the damping injection control ( ) is designed to achieve asymptotic stability at rule desired equilibrium point, which corresponds to an isolated and strict minimum of the desired energy function. One can find the details of the design methodology in Ortega and Garcia-Canseco, 2004, and references therein. On the other hand, technological advancements in digital processors, and the widespread use of computer controlled systems in engineering practice demands for a theory which would inspire methods and techniques for analyzing and designing sampled-data systems and discrete-time model of non-linear systems. A framework dealing with the stabilization of sampled-data nonlinear systems using the approximate discrete time models of the system can be found in Nesic et al., 1999 and Nesic, and Teel, 2004. In the control literature, to the best of our knowledge, number of works utilizing the discrete-time models of Hamiltonian systems for control applications are limited (Laila and Astolfi, 2005, 2006a, 2006b) and (Gören-Sümer and Yalçın, 2008, Yalçın and Gören-Sümer, 2008, 2009 and 2010). 978-3-902661-93-7/11/$20.00 © 2011 IFAC
For several years now, the control researchers have been concentrated on deriving the IDA-PBC design method which is based on modifying both potential and kinetic energies for the stabilization of under actuated mechanical systems. While an arbitrary potential energy shaping stabilizes the fully actuated mechanical system at any desired equilibrium, the stabilization problem of under actuated systems is achieved by modifying the kinetic energy of the system in general. The suggestion of total energy shaping was first introduced in (Ailon and Ortega, 1993) and there are lot contributions on this subject with two main approaches: the method of controlled Lagrangians (Bloch et al., 2000) and IDA-PBC (Ortega et al. 2002). There are also closely related works (Fujimoto and Sugie, 2001) and (Viola et al., 2007) see also Ortega, and Garcia-Canseco, 2004 which contains an extensive list of references on this topic. In these methods, the stabilization of a desired equilibrium is accomplished establishing a class of systems which can be obtained using feedback. The conditions for the existence of such a feedback law are known as matching conditions which consist of a set of nonlinear partial differential equations (PDEs). Some efforts have been devoted to the solution of the matching equations Gomez-Estern et al., 2001, Acosta et al., 2005, Viola et al., 2007, and references therein. As realized from these studies, to solve these PDEs is troublesome, so the stabilization problem for Hamiltonian systems for the under actuated case is still known as a challenging problem. Therefore, considering the discrete time counterpart of the problem and then obtaining the matching conditions in discrete time setting may provide an easy technique to solve the stabilization problem for Hamiltonian systems for the under actuated case. In this study, a direct discrete time design method in the sense of IDA-PBC is investigated. In order to give a direct discrete time IDA-PBC design method, firstly an appropriate discrete gradient is proposed, which
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enables the derivation of a discrete time model corresponding to Hamiltonian Systems and also the determining the discrete time counterpart of matching conditions. Using this proposed discrete gradient and discrete-time model, the discrete-time matching conditions are obtained as a set of linear partial differential equations which can be solved off-line parametrically and a set of linear equations. The unknown parameters of linear partial differential equations and the linear equations have to be solved at each sampling time, to calculate control rule. Moreover, a procedure is developed to solve these matching conditions for a class of Hamiltonian systems.
( ) ( ) ( ) where ( ) The controller ( ) is obtained as a solution of the following equation, [
][
̇ [ ] ̇
( (
*
[ ( )] ( )
( )
( )
) (1)
(
[
{
The notation ( ) is used to denote the gradient vector of a scalar function ( ) with respect to (.). Furthermore, ( ) is the Hamiltonian function of the system or the energy function of the system is in the following form, )
(
)
( )
( )
( )
(2)
where ( ) and ( ) are potential and kinetic energy terms, respectively, and ( ) ( ) is the generalized ( ) inertia matrix. If , i.e. a constant matrix, the system is called a separable Hamiltonian system, and if ( ) the system is said to be fully actuated, if the system is said to be under actuated. The main idea in the method developed by Ortega et al 2002a, 2002b were to design a stabilizing controller which assigns a desired energy function, (
)
(
)
( )
( )
( )
[ ] ̇ *
( (
)
(
)) [ +
(4)
] [
]
][
(5)
]
( )
(
)
( ) }
(6)
)
( ( )
( )
( )
)
(7)
}
(8)
( )}
with a full rank left annihilator of . If the PDE s’ (7) and (8) are solvable, the energy shaping controller is derived as, (
)
{
}
(9)
Since the resulting closed loop system under this control rule is also a Hamiltonian system, the damping injection control rule which yields an asymptotically stable system is obtained as, ( )
(10)
, (Ortega et al 2002a, 2002b).
Since solving the PDEs given in (7) and (8) is not straightforward, we will consider the discrete time counterpart of the problem and then obtain the matching conditions in discrete time setting. 2.2 Discrete-time model of Hamiltonian Systems In this study, a direct discrete time IDA-PBC design method for the system given (1) for the under-actuated case will be developed. To fulfill this, the gradient based discrete model presented and used in (Gören-Sümer and Yalçın, 2008) and (Yalçın and Gören-Sümer, 2009 and 2010) will be used in slightly modified manner as the discrete time model of the Hamiltonian systems. In general, a discrete gradient is defined in (Gonzalez and Simo, 1996), restated below. Definition 1: Let, ( ) be a differentiable scalar function in then ̅ ( ) is a discrete gradient of ( ) if it is continuous in and,
(3)
) of the closed which has an isolated equilibrium point ( system. The stabilization problem for Hamiltonian systems for the under actuated case is known as a challenging problem since it needs the appropriate choice of the desired energy function and also assignment of a new interconnection matrix. In literature, the desired system is considered as, ̇
(
with
]
[
or the following equivalent constraints,
( )
) + )
where ( ) is an 2n - dimensional manifold, ( ) is system output, ( ) is the control input, ( ) is input force matrix and is the standard skew-symmetric matrix, namely,
(
( )
{
{
Consider the following continuous-time Hamiltonian systems given in standard coordinates,
( )
[ ( )]
( ) and In case the matrix ( ) is full column rank, the ( ) holding the following constraint should be constructed,
2. PRELIMINARIES and PROBLEM FORMULATION 2.1 Continuous-time Hamiltonian Systems
]
̅
(
)( ̅
) (
in which the gradient of ( )
*
)
(
) (
(
)
)
(11a) (11b)
( ) defined as, +
[
]. □
(12)
The discrete gradient proposed by Yalçın and Gören-Sümer, 2009 has been constructed using the Quadratic Approximation Lemma, (Diewert, 1976) and it has been shown that there exists a discrete gradient which exactly satisfies the conditions (11) if the energy function ( ) has the general quadratic form. Consequently, the second order Taylor approximation of any ( ) might be used to define a discrete gradient as follows,
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
̅ (
( ̃(
)
̃(
)
))
where ̃( ) is the second order Taylor approximation of ( ), namely, ̃( )
(
)
(
)(
)
(
)
(
)
for . As a consequence of the above analysis the following definition is presented. Definition 2: Consider a differentiable function in given as ( ), then the discrete gradient of a ( ) is defined as, ̅ (
)
in which, function of
( ) is the Hessian matrix of the energy ( ).□
(
)
(
)(
)
(16)
Approximating to derivatives of state variables of (1) by Forward Euler with sampling period , and replacing the gradient term in (1) with the discrete gradients ̅ and ̅ the gradient-based discrete-time description of the system (1) can be obtained as follows, [
]
*
̅ ̅
* ( )+ ( )
+
( )̅
()
(
)
(17) Furthermore, the similar expressions can also be obtained for the discrete-time description of the desired system as follows, [
]
( ( )
[
(
( )) *
̅ ̅
+
(18)
)]
(19)
where, ( ) ( )
*
(
)
)
(
(
)
( ( ) (
)
( )
) +
(20)
If the right hand side of (17) is equated to the right hand side of (18), the discrete time control rule responsible for energy shaping is obtained as follows in terms of discrete gradients, ( )
(
{̅
)
̅
̅
} (21)
and for the resulting closed-loop Hamiltonian system the damping injection control rule can be written as, ̅
( )
(
)(
)
(
)
for discrete case using (18) as following, (
)
(
)
(
̅
(
)
(
)
(
)
̅
̅
and using the property of Taylor series expansion, (
)
(
)
̃( ̅ ̃(
)
̃(
)
̅
(
)̅
̃) ̅ ̃
(24)
̅ ̅ ̅ ̃ ̃ ̅ ̃. ( ) where This relation implies that the discrete model of the open loop system provides an extra energy or extra dissipation ) according to the sign of ( . Obviously, for this extra term tends to zero. As a consequence of the above discussion the following Remark presented in (GörenSümer and Yalçın, 2008) can be slightly modified to obtain the stabilizability property for the discrete time control rules.
Remark 1: When the gradient based discrete model proposed here is used to design a control rule to stabilize the nonseparable sampled-data Hamiltonian system, the extra term due to discrete model of the open loop system does not effect ) stabilizability condition of system, if ( , i.e., the model has more energy than the system. On the other hand, if ̃ ( ) , namely, the discrete model of the open loop system has an extra dissipation, the control rule responsible for dissipation injection must be designed considering this fact, especially when slow sampling is used. As it can be obviously followed from (24), the stabilizability condition of continuous Hamiltonian system by discrete-time control remains same, i.e. stability can be achieved by adding an extra dissipation. Furthermore, it can be realized from (24) that this extra term does not have an effect on the procedure deriving the control rule which is responsible for energy shaping. □ 3. DIRECT DISCRETE TIME IDA-PBC DESIGN METHOD
(22)
It is known that in IDA-PBC stabilization is achieved by assigning a desired total energy function to the closed loop system and dissipation injection for the continuous case. On the other hand, the analogy between continuous and discrete cases would give rise to a similar energy relation for continuous case as in, ̇ ( )
under the assumption such that there exist a discrete gradient, ), ( ) namely ̅ ( , which satisfies the conditions in Definition 1. Therefore, the stability of the discrete system given in (18), namely closed loop system, can be guaranteed by the total energy function of the system and dissipation injection with . In the design method proposed here, it will be assured that the discrete gradient of the desired energy function holds the conditions given in Definition 1, but the discrete gradient of the energy function of open loop system may not satisfy these conditions. In this case, the energy relation for the open loop system can be written as follows,
In order to develop a direct discrete time IDA-PBC design method for the system given (1) for the under-actuated case, the matching conditions should be derived in discrete time setting, firstly. Using the expression (16) in the Definition 2, the following relations can be written for the discrete gradients according to the variable , present in (17-22),
(23)
̅ ̅
)̅
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( (
( (
)
(
)(
)) (25a)
)
(
)(
)) (25b)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
̅
(
)
̅
(
(
)
)(
)
(
(25c) ) (25d)
)(
where , and are the Hessian ( ) matrices - according to the variable - of ( ) ( ) and ( ), respectively; and for the discrete gradients according to the variable are written as, ̅
(
)
(
( ̅
)(
(
)(
)(
{
(26b)
) and use the following
̅
̅
( )̅
(27)
for the discrete gradient terms given in (25), then the following relations are obtained for the discrete gradients of the kinetic energies, namely for (25a, b), ̅
([
(
)]
(
̅ ̅
([
(
(
)
(
)]
)̅
(
̅
(
(28)
)̅
)
(29) )̅
)
)
(
)̅
)
[
(
)]
(31)
(
)
[
(
)]
(
(32)
)
) (33)
(
{ ( )
and for the discrete gradients of the potential energies, ̅
(
̅
)
(
(
)
(
) (
)̅ (
)
)̅
(34) (35)
(
)
)̅
(
(
)
(
̅
) (
( )
(
)(
)
( (
(
)
(
)
(
)
(
)}
(38)
( )}
( )
(
( )
(
)) }
(39)
(
)
(
)
(
)
(
)(
)
(40)
simultaneously, in which,
(
)
(
)
(
)
(
)
.
Using the matrices , and which are calculated the previous steps, the control rules for energy shaping,
))
) }
)
)
and
( )
(36)
)
{̅
( )
(37)
These conditions can be regarded as the discrete time matching conditions for the solution of the IDA-PBC of Hamiltonian system given in (1) for under actuated case. The method suggested in this study is based on obtaining the parameters of the desired potential energy function ( )
(
̅
̅
}
and for dissipation injection,
and {
)
(
) (
(
)̅ }
and using the relations derived in (28-35) the equivalent conditions are obtained as follows, { (
(
and the discrete gradient property,
Furthermore, the matching condition which is obtained from equation (13) and (14) can be constructed as follows, {̅
)
Since the desired inertia matrix determined in the previous step is not a matrix function of the , but it is only a matrix sequence depended on the value of the ( ) at each sampling time, the vector ̅ ( ) or ( ) equivalently the matrix ( ), and the matrix ( ) must be calculated at each sampling time all satisfying the kinetic energy matching condition (36), namely,
(30)
where, (
(
( )
( ) ( ) is are analytically solved by assuming a constant matrix, then a function ( ) with coefficients depended on the values of the elements of ( ) the matrix is found, such that . After then the Hessian matrix of this ( ) is calculated and a matrix function is generated in terms of the elements of , satisfying ( ) , if it is possible. It should be noted that the class is determined by the existence of a ( ) matrix . Since the matching condition (38) has been solved analytically, the matching conditions (38) so (37) hold for the solution of ( ), therefore the ( ) can be made only assuring choice of ( ) and .
)
)
(
The following linear PDEs, which are equivalent the discrete time potential energy matching condition (37),
(26a)
(
(
The main idea of the method proposed here may be best explained by giving a design procedure - which determines the class of Hamiltonian systems- step by step as follows;
)
)
)
Let’s define, ̅ equation,
)(
( ) ( ) with and , which can be solved ( ) ) off-line, and the matrix sequences and ( such that these conditions hold at each sampling time.
̅
are evaluated at each sampling time. 4. THE CART and PENDULUM SYSTEM In this paper, a well-known under actuated Hamiltonian system; The Cart and Pendulum system is considered in
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
discrete time. The dynamic equation of the cart and pendulum on a cart are given by (1) with, ( ) ( )
( )
( )
],
[
(
⁄
( (
[ ⁄
(
)⁄(
(
)
( (
) ) )
( (
) )
) )
]
)
and , where is the pendulum angle with the upright vertical, is the cart position as shown in Figure 1, is the mass of the pendulum, is the length of the pendulum, is the mass of the cart and is the gravity acceleration. The equilibrium to be stabilized is the upward position of the pendulum with the cart placed in any desired position, which corresponds to and any arbitrary .
⁄ ( )) must It can easily be shown that, the term ( be positive for all , in order to satisfy the condition ( ) ( ) . Therefore, the matrices ( )
satisfying chosen as,
( )
at each sampling time can be
[
( ) ] ( )
( )
in which, ( )
( )
( )
( )
with and ( ) is the ( arbitrary real numbers and ( ) which is “ ( ) ” here. Fig. 1. The cart and pendulum system
4.1 Solving the Matching Conditions The solution procedure given in previous section will be carried out for the cart and pendulum system as follows; Step 1: The potential energy matching condition in discrete time setting is, {
(
( )
)
(
)
(
( )
)
(
(
( )
(
( (
)
where ( ( )) is an arbitrary differentiable function satisfying the condition ( ( )) for and arbitrary , in this study it is taken as ( ) ( ⁄ ) with . The gradient and Hessian of the desired ( ) are potential energy, namely ( ) and obtained as follows, (
( ) [
)
( (
*(
) )
)
*( )
( (
( (
) )
+
) )
+ ]
̅
̅ (
(
)
(
)
(
,
)̅
)
)(
the control rule responsible for energy shaping is found by the relation, ( )
) )
̅ ̅
( )
)
( ( ))
)
Step 3: Using the following relations and the matrices and ( ) which are found in previous steps, ̅
The general solution of this PDE is given in terms of ( ) and ( ) as, ( )
( ) and Step 2: In order to determine the matrices ( ) ( ), the three linear non-homogenous equations given in (39) and (40) with five unknowns are solved for each sampling time using the matrices ( ) and ( ) which are found in previous step.
( )}
( ) ( ) and and if the matrix is defined as the gradient of the potential energy of the system obtained as ] is used, then the condition is ( ) [ written as,
being any ) element of
(
)
{̅
̅
̅
}
and finally the control rule responsible dissipation injection is ̅ ( ) calculated as follows, at each sampling time. In simulations, the system parameters are taken as , , and the discrete time controller design parameters are chosen as, and . The simulations are carried out for the sampling period , the initial conditions ( ( ) ( ) ) ( ) and the desired cart position . The simulation results are presented in Figure 2, 3 and 4 which illustrate time domain responses under the direct discretetime control. These results reveal that the direct discrete time controller design method proposed in this study yields a good performance for sampled data Hamiltonian systems. 5. DISCUSSION The results reported in this work are the first results on the direct discrete time IDA-PBC design for under actuated Hamiltonian systems according to the best of the authors’ knowledge. Surely, the method reported here should be developed especially in determining a domain of attraction
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for the equilibrium. Also the values of design parameters which ensure the boundedness of all trajectories starting in the domain of attraction can be determined. These two issues are our current research.
Fig 2 The pendulum angle versus time.
Fig 3 The cart position versus time.
Fig 4 Control signal versus time
REFERENCES Acosta, J. Á., Ortega, R., Astolfi, A. ve Mahindrakar, A. D., 2005: Interconnection and Damping Assignment Passivity-Based Control of Mechanical Systems with Underactuation Degree One, IEEE Transactions on Automatic Control, 50, 12, 1936-1954. Ailon, A. and Ortega, R., 1993: An observer-based controller for robot manipulators with flexible joints, System and Control Letters, 21, 329–335. Bloch, A.M., Leonard, N.E. and Marsden, J.E., 2000: Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Trans. Automatic Control, 45, 12, 2253–2269. Diewert, W. E., 1976: Exact and Superlative Index Numbers, Journal of Econometrics, 4, 2, 115-145. Gomez-Estern, F., Ortega, R., Rubio, F. and Aracil, J., 2001: Stabilization of a class of underactuated mechanical systems via total energy shaping, IEEE Conf. on Decision and Control, Orlando, FL, USA.
Gonzalez, O. and Simo, J.C., 1996: On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry, Computer Methods in Applied Mechanics and Engineering, 134, 3-4, 197–222. Gören-Sümer, L. and Yalçın, Y., 2008: Gradient Based DiscreteTime Modeling and Control of Hamiltonian Systems, 17th IFAC World Congress, Korea, Seoul, 2008. Fujimoto, K. and Sugie, T., 2001: Canonical transformations and stabilization of generalized Hamiltonian systems, System and Control Letters, 42, 3, 217–227. Laila, D. S. and Astolfi, A., 2005: Discrete-time IDA-PBC design for separable Hamiltonian systems, 16th IFAC World Congress, Prague. Laila, D. S. and Astolfi, A., 2006a: Direct Discrete-time Design for Sampled-data Hamiltonian Control Systems, 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nogoya, 57-62. Laila, D. S. and Astolfi, A., 2006b: Discrete-time IDA-PBC design for underactuated Hamiltonian control systems. American Control Conference, USA, Minnesota. Nesic, D., Teel, A.R. and Kokotovic, P. V., 1999: Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, System and Control Letters, 38, 4-5, 259-270. Nesic, D. and Angeli, D., 2002: Integral versions of ISS for sampled-data nonlinear systems via their approximate discretetime models, IEEE Trans. Automatic Control, 47, 2033-2038. Nesic, D. and Laila, D.S., 2002: A note on input-to-state stabilization for nonlinear sampled-data systems, IEEE Trans. Automatic Control, 47, 1153-1158. Nesic, D., and Teel, A.R., 2004: A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models, IEEE Trans. on Automatic Control, 49, 7, 1103-1122. Ortega, R., Loría Perez, J.A., Nicklasson, P.J. and Sira-Ramirez, H.J. 1998: Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications, Springer-Verlag. Ortega, R., Van der Schaft, A.J., Maschke, B. ve Escobar, G., 2002a: Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, 38, 4, 585-596. Ortega, R., Spong, M.W., Gómez-Estern, F. and Blankenstein, G., 2002b: Stabilization of a Class of Underactuated Mechanical Systems Via Interconnection and Damping Assignment, IEEE Trans. on Automatıc Control, 47, 8, 1218-1233. Ortega, R. and Garcia-Canseco, E., 2004: Interconnection and Damping Assignment Passivity-Based Control: A survey, European Journal of Control, 10, 5, 432-450. Van der Schaft, A., 2000: L2-Gain and Passivity Techniques in Nonlinear Control, Springer, London. Viola, G., Ortega, R., Banavar, R., Acosta, J. A. and Astolfi, A., 2007: Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations via Coordinate Changes, IEEE Trans. Automatic Control, 52, 6, 1093-1099. Yalçın, Y., and Gören-Sümer, L., 2008: Disturbance Attenuation in Hamiltonian Systems via Direct Discrete Time Design, 17th IFAC World Congress, Korea, Seoul. Yalçın, Y., and Gören-Sümer, L, 2009: Robust Disturbance Attenuation in Hamiltonian Systems via Direct Digital Control, European Control Conference, Budapest. Yalçın, Y., and Gören-Sümer, L., 2010: Direct Discrete-Time Control of Port Controlled Hamiltonian Systems, Turkish Journal of Electrical Engineering and Computer Sciences, 18, 5, 913-924.
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