Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
FrA18.3
A Direct Lyapunov Approach for Stabilization of Underactuated Mechanical Systems Warren N. White, Mikil Foss, and Xin Guo
Abstract— Control of underactuated systems is treated from a Lyapunov direct method approach. The method results in a set of three matching conditions, the solution of which is easily accomplished. The method developed is capable of treating more complicated systems than that reported in an earlier publication. The suitability of the Lyapunov candidate function is demonstrated through mathematical proofs. An application of the method to the ball and beam is presented.
U
I. INTRODUCTION
nderactuated mechanical systems provide a challenging area for control system design. Some everyday examples consist of rocket guidance, satellite and underwater vehicle orientation control, vibration damping for cargo transport using overhead cranes, and stabilization of hovering aircraft. The characterizing feature of underactuated systems is that they have fewer actuators than degrees of freedom. The control challenge stems from the fact that such systems are governed by nonlinear dynamic equations and that standard methods for controlling fully actuated mechanical systems, such as feedback linearization or Lyapunov, generally fail when applied to underactuated systems. Limited success has been achieved through linearization and state space methods. The goal of developing nonlinear controllers for underactuated mechanical systems has been approached from a variety of directions as explained in White, Foss, and Guo (2006). One particular method receiving significant attention over the past several years relies on the Lagrangian or Hamiltonian of the mechanical system. Bloch, Leonard, and Marsden (2000, 2001) have presented the controlled Lagrangian method where structured modifications are made to the uncontrolled system Lagrangian, thereby, constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop dynamics of the system. Auckly, Kapitanski, and White (2000) presented the λ method where certain nonlinear Manuscript received September 22, 2006. W. N. White is with the Mechanical and Nuclear Engineering Department of Kansas State University, Manhattan, KS 66506-5205 USA phone: 785-532-2615; fax: 785-532-7057; (e-mail:
[email protected]). M. Foss is with the Department of Mathematics at the University of Nebraska, Lincoln NE 68588-0130 USA (e-mail:
[email protected]). X. Guo received his Ph.D. in Mechanical Engineering from Kansas State University; (e-mail:
[email protected]). The support, in part, by the National Science Foundation Grants No. CMS-0556019 and CMS-0600442 of this work is gratefully acknowledged.
1-4244-0989-6/07/$25.00 ©2007 IEEE.
partial differential equations arising from so-called matching equations are recast as linear partial differential equations through a transformation; the space of admissible control laws was broad. The method of interconnection and damping assignment - passivity based control (IDA-PBC) has been presented in Ortega, Spong, Gómez-Estern, Blankenstein (2002) and Acosta, Ortega, Astolfi and Mahindrakar (2005) where a goal of the stabilizing control is to monotonically reduce the total mechanical energy. IDA-PBC is a significant development since these authors have shown that both the method of controlled Lagrangians and the λ method are special cases of IDA-PBC. All of these three controller design methods have been applied to a variety of underactuated, mechanical systems. A common limitation of these three approaches is that they all produce controllers that are quadratic in the generalized coordinate velocities and the methods become intractable as the number of degrees of freedom increases. As an illustration of the unintuitive responses created by the quadratic velocities, consider a modification of the controller presented by Kelkar, Fang, White, and Guo (2002) for the inverted pendulum cart. Figure 1 shows the cart position and pendulum angle responses for two different sets of initial conditions where each set consisted of zero initial pendulum angle and cart position but opposite signs of the initial cart velocity and pendulum angular velocity. One would expect that the different cart position responses and different pendulum angle responses to be mirror images of one another about the time axis; however, this is not the case. The application of the λ method to an underactuated system requires the solution of (m2 + n2)n linear partial differential equations where n is the number of degrees of freedom and m is the number of actuated axes. As the number of degrees of freedom increases, the solution effort increases as the cube of the number of axes. This brief discussion of the matching methods illustrates the difficulties and shortcomings they share. We note, however, that the behavior presented in Figure 1 is not always present in quadratic velocity control laws, yet the possibility of such a response always exists when using a control law of this form. These difficulties provide motivation for an alternate approach to controller design for underactuated systems. White, Foss, and Guo (2006) presented a Lyapunov approach to underactuated systems with a candidate Lyapunov function 1 (1) V (q, q& ) = q& T K q& + Φ (q )
4817
2
D
FrA18.3
where V (q,q& ) : ℜ2n → ℜ is the candidate Lyapunov function, n
q, q& ∈ ℜ are generalized coordinate and velocity vectors, respectively, Φ(q): ℜn → ℜ is a potential function, and KD ∈ℜn×n is a symmetric, positive definite matrix defined as (2) K D = P(t )M (q)
(a)
present an orderly solution process regardless of the complexity of the mechanical system. The third matching condition usually consists of simultaneous linear PDEs and the ability to solve these equations for a variety of systems needs to be demonstrated. The Lyapunov approach did not exploit any particular property of the dynamic equations of motion which provides the opportunity to apply this technique to other types of systems. Another contribution of this work is that it shows how Lyapunov, a method usually used to demonstrate stability, can be used for underactuated systems control law design. This paper presents the application of a direct Lyapunov technique to more complicated systems which will necessitate augmentation of the previous developments. The matrix P(t) is determined by the differential equations of the first matching condition and plays a significant influence in the controller produced dynamic response. In order for the response of the system to be invariant in time, the elements of P(t) must return to prescribed values as t increases. Achieving this desired behavior for P(t) will be achieved through the introduction of equal and opposite terms to the first two matching conditions. A simulation to be presented later will demonstrate the suitability of this approach. II. THE LYAPUNOV FORMULATION AND MATCHING CONDITIONS
(b) Figure 1: Influence of Initial Conditions Signs with Control Laws Quadratic in Velocities where M(q) ∈ ℜn×n is the symmetric, positive definite mass/inertia matrix and P(t) ∈ ℜn×n is to be determined so that KD has the stated properties. White, Foss, and Guo (2006) applied the technique to a particular system class where all of the dynamic nonlinear terms included only one of the generalized coordinates. The Lyapunov approach resulted in three matching conditions where the first two matching conditions consisted of either linear ordinary differential or linear algebraic equations while the third matching condition consisted of either linear ODEs or linear PDEs. The attractiveness of this approach is that the ODEs of the first matching condition can be solved numerically as part of the feedback and, for the class of system condidered, the ODEs of the third matching condition can also be solved as part of the feedback. A significant advantage of the matching equations is that the resulting control laws are linear in the generalized velocities and the first two matching conditions
Before developing the Lyapunov direct method, attention will be directed at the dynamic equations of motion of the mechanical system. The dynamic equations of motion governing the behavior of mechanical systems with holonomic constraints are determined from (3) M(q)q&& + C(q, q&)q& + CDq& + G(q) = τ . n The right-hand side of (3), specified as τ ∈ ℜ , consists of the actuations. For an underactuated system, only m of the inputs are nonzero where m < n. It is assumed that the degrees of freedom are ordered so that the first m elements of τ contain the nonzero actuations. The term C (q, q& )q& ∈ ℜn consists of centripetal and Coriolis forces and/or moments, the symmetric matrix CD ∈ ℜn×n contains viscous damping coefficients, and G(q) ∈ ℜn consists of forces and/or moments stemming from gradients of conservative fields. The matching conditions are determined from the time derivative of the candidate Lyapunov function, which is 1 (4) V& = q& T K q&& + q& T K& q& + q& T ∇Φ (q ) = −q& T K q& D
2
D
n×n
V
where the matrix KV ∈ ℜ is symmetric and at least positive semi-definite and ∇Φ(q) is the gradient of the potential with respect to the generalized positions. Owing to the nature of the right hand side of (4), LaSalle’s theorem, as discussed by Khalil (2002), will be necessary to demonstrate asymptotic stability, however the right side of (4) is similar to the Hamiltonian formulations. Substituting for the accelerations of the generalized coordinates from (3) into (4) produces
4818
FrA18.3 F 1 V& = q& T K D M −1 (− C (q, q& ) − C D ) + K& D q& + K D M −1 2 0
(5)
F + q& T − K D M −1Gm q + ∇Φ (q ) = q& T K D M −1 − Gm q 0 + q& T ∇Φ (q ) + q& T K D M −1 (− H 1 − H 2 + H1 + H 2 )q&
(
)
1 + q& T K D M −1 (− C (q, q& ) − C D ) + K& D q& = −q& T K v q&. 2
The matrices Hi ∈ ℜn×n are added to and subtracted from the motion equations so that the dynamics are unchanged. These matrices will be determined as part of the matching equation solution. The matrix Gm ∈ ℜn×n is defined such that Gmq consists of the conservative field produced forces and/or torques and the matrix is formed by factoring the vector q from the force/torque vector. The matching conditions arise from equating groups of terms in (5) either to its right hand side or to zero. The system input is divided into three parts, namely, (6) F = F1 (q& , q ) + F2 (q, q& ) + F3 (q ) where Fi ∈ ℜm. The input Fi will be used with the ith matching condition. The first matching condition consists of (7) F 1 q& T K D M −1 (− C (q, q& ) − H1 − H 2 ) + K& D q& + K D M −1 1 = 0 2
0
while the second matching condition is F q& T K D M −1 (− C D + H1 + H 2 )q& + K D M −1 2 = −q& T KV q& 0
(8)
and, finally, the third matching equation is F q& T K D M −1 3 − Gm q + ∇Φ (q ) = 0. 0
(9)
A. Solving the First Matching Condition Examination of the first matching condition shows that a differential equation in KD will be involved in the solution; however, before proceeding further, there are several aspects of (7) to be considered: 1) If KD changes in time, it is desired that the matrix remains both positive definite and symmetric. 2) As it will be seen, the matrix KD (and, thus, P) plays a salient role in determining the controller created dynamics of the system response and KD must return to some nominal form as the system state approaches equilibrium if the system response is to be replicable 3) As pointed out earlier, it is undesirable for the control F1 to contain terms that are quadratic in the velocities and, as a result, is set to zero. The part of (7) that remains is 1 q& T − K D M −1 (C (q, q& ) + H1 + H 2 ) + K& D q& = 0 2
(10)
which can be satisfied provided the matrix making up the portion in the parentheses in (10) is either zero or skew symmetric. The matrices H1 and H2 are chosen as (11) H1 = β o M and
−1
H 2 = − β o MK D K DF
(12)
where KDF is the symmetric, final value of the matrix KD and, as will be seen, βo is a negative constant. Substituting (11)
and (12) into (10) produces 1 . q& T − K D M −1C (q, q& ) − β o K D + β o K DF + K& D q& = 0 2
(13)
Since KDF and KD together with its time derivative are symmetric matrices, the solution of (13) is found from (14) K& D − 2(K D M −1C (q, q& ))S − 2β o K D = −2β o K DF where subscript S denotes the matrix symmetric part. The attractive feature of (14) is that the evaluation of the solution can be done as part of the feedback algorithm. If the system is close to the equilibrium, the generalized velocities will approach zero and (14) becomes (15) K& D − 2 β o K D = −2 β o K DF which is an equation having the steady state solution of KD = KDF provided the constant βo < 0. The differential equation in (14) consists of n(n+1)/2 equations for the same number of unknowns in the matrix KD and the problem of solving the first matching equation can be considered well posed. B. Solving the Second Matching Condition The second matching condition solution method will be presented. The second matching condition, as stated in (8), can be simplified, with the definitions in (11) and (12), to get F (16) K M −1 − C q& + 2 + (β K − β K )q& = − K q& D
0
D
o
D
o
DF
V
where the equation has been stripped of the q& T . Using the definition of P from (2) and re-arranging (16), we have F2 (17) = P −1 (− K − β K + β K )q& + C q& 0
V
o
D
o
DF
D
where the first m rows determine the control law contribution F2. Since F2 is expressed in terms of q& , this vector can be factored from each side of the equation, thus, providing n(n-m) equations (derived from the lower n-m rows of (17)) for the n(n+1)/2 unknown elements of KV. If there are more unknowns in KV than equations then n(2m-n+1)/2 of the unknowns can be freely picked. The case where there are more equations than unknowns is indicated by the quantity 2m-n+1 being negative. How to handle the over-determined case is a research issue. C. Solving the Third Matching Equation The third matching condition does not involve differential equations in time nor is it made exclusively of algebraic equations, but rather ordinary or partial differential equations in the coordinates. The final matching condition results in (18) F3 −1 0 = G m q − P ∇Φ ( q )
where the first m equations are used to determine the control law contribution F3. The remaining n – m equations are all first order, linear, partial differential equations used to determine the potential Φ(q). Note that the PDE number depends linearly on the difference n – m. The potential Φ(q) must consist of n – m different functions so that the system of
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equations is not over determined. One possible way to include different functions in Φ(q) is to write Φ(q) as a quadratic matrix function. If there are more functions than equations, then the additional functions can be conveniently chosen to accomplish perhaps desired potential shape or solution ease. If the potential Φ(q) satisfying (18) does not have the proper shape (such as the potential is a function of all of the generalized coordinates except one), one option is to write Φ(q) = Λ(q) + Γ(q) where Γ(q) is a specified potential that provides the proper overall shape and where Λ(q) is now the potential the matching equations determine. The PDEs for the potential Φ(q) are solvable by use of the method of characteristics where the solution is expressed as an integral evaluated along the characteristic. The elements of P appear in (18). Since the elements of P are functions of time, they are treated as parameters when solving for Φ(q). However, the evaluation of the potential is not necessary since the control law requires only the gradient of Φ(q). In calculating the gradient of Φ(q), care must be exercised. A short example will illustrate the counter-intuitive mistakes that can occur. Consider the linear PDE ∂Λ ∂Λ ∂Λ (19) a +a +a + 2a K θ + 2a K ψ + 2 a K φ = 0 1
∂θ
2
∂ψ
3
∂φ
1
1
2
3
3
3
where the ai and Ki are real constants, Λ(θ,ψ,φ) is the solution, and θ, ψ, and φ are real coordinates. The solution is (20) 1 Λ(θ ,ψ , φ ) =
a1
∫
(− a1 2 K1θ − a2 2K 2ψ − a3 2 K 3φ )dθ
which is evaluated along the characteristics of C1 = a2θ – a1ψ and C2 = a3ψ– a2φ where the Ci are constants. Note, for brevity, the homogeneous solution of (20) has been set to zero. Young (1972) provides the details behind (20). Computing partials of (20) with respect to ψ and φ produces 2a 2a ∂ ∂ (21) Λ(θ ,ψ , φ ) = − 2 K θ , and Λ(θ ,ψ , φ ) = − 3 K θ ∂ψ
a1
∂φ
2
3
a1
which is easy to compute by inspection of (20). It would seem the partial derivative of (20) with respect to θ would simply be the integrand of (20), however, if the partial derivatives of (21) are substituted into (19) and the resulting expression is solved for the partial with respect to θ then (22) 2a 2 2a 2a 2 2a ∂ Λ(θ ,ψ , φ ) = −2 K1θ + 22 K 2θ + 23 K 3θ − 2 K 2ψ − 3 K 3φ ∂θ
a1
a1
a1
a1
where it is seen that the partial with respect to θ is not simply the integrand of (20). Note that (22) can be demonstrated by partial differentiation of Λ(θ,ψ,φ) with respect to θ, where Λ(θ,ψ,φ) is a2 2a a2 2a (23) Λ (θ ,ψ , φ ) = − K θ 2 + 2 K θ 2 + 3 K θ 2 − 2 K ψθ − 3 K φθ . 1
a12
2
a12
3
a1
2
a1
3
The point to be made here is that the solution Φ(q) does not need to be computed. Only the gradient of Φ(q) is needed for the control law. This gradient can be determined by computing the gradient of the solution expressed as an integral along the characteristics such as (20). If, after computing the gradient, components of the gradient remain as
integral expressions, then these expressions may be evaluated numerically. Provided the gradient is correctly calculated, the final determination of the control law is easily achieved. Also, note that in computing the gradient components, it must be demonstrated that in an equilibrium neighborhood the Hessian of Φ(q) has positive eigenvalues. III. COMPARISON TO IDA-PBC This section provides a brief comparison between the Direct Lyapunov Approach and IDA-PBC, which was mentioned earlier. The main objective in IDA-PBC is to design a control law that renders the associated energy function into a desired form. This is essentially the same objective for the Direct Lyapunov Approach. Following Acosta, Ortega, Astolfi and Mahindrakar (2005), the new total energy function for the system is 1 T −1 p M d (q ) p + Vd (q ) where p is the vector of generalized 2 momenta. Clearly, since the mass matrix M(q) is invertible, one can identify IDA-PBC energy functional with the candidate Lyapunov function presented above. In the IDA-PBC approach, additional flexibility in controller design is made possible via the introduction of a free skew-symmetric matrix J2. Such a free matrix could also be incorporated in the Direct Lyapunov Approach by allowing a non-symmetric matrix KD; the anti-symmetric part of KD would correspond to the free matrix J2. The primary distinction between the Direct Lyapunov Approach and IDA-PBC is the manner in which the matching equations for the kinetic portion of the new energy functional are handled. For the IDA-PBC approach, a judicious choice is made for the structure of Md and J2, thereby simplifying the matching equation to something that can be readily solved. In the Direct Lyapunov Approach, the matrix KD is simply treated as a matrix function of time, so the matching equation is reduced to a system of ordinary differential equations. In its current state, the Lyapunov Direct Approach presented here appears not as flexible a method as IDA-PBC because the second and third matching equations are solved using an algebraic approach instead of a projective method.
IV. THE EVOLUTION OF KD In the following, a series of lemmas and proofs will be presented concerning the evolution of the matrix KD where it will be demonstrated that it remains symmetric and positive definite. Let βo ≤ 0 and a symmetric, positive-definite matrix It is desired to find a symmetric, KDF be given. positive-definite matrix-valued function KD satisfying (24) K& D − 2(K D M −1C + βo K D − βo K DF ) = E for some skew-symmetric E. Lemma 1. Suppose that KD, E: [0,+∞) → ℜn×n is continuously differentiable and satisfies KD(0) = KD(0)T. Further, suppose that E satisfies E(t) = -E(t)T for all t ∈ [0,
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FrA18.3
+∞). Then KD is symmetric and satisfies (24) for all t ∈ (0, +∞) if and only if T T (25) K& D − (K D M −1C + C T M −1 K D + β o K D + β o K D − 2β o K DF ) = 0
for all z ∈ ℜn, then KD is uniformly positive-definite for all t∈ [0, +∞). In particular, each eigenvalue λ(t) of KD satisfies (36) λ (t ) ≥ −2 β oνe 2 ( β 0 − A ) .
for all t ∈ (0, +∞). Remark. Since (25) implies that KD is symmetric, then (25) can be rewritten as T K& D − (K D M −1C + C T M −1 K D + 2β o K D − 2β o K DF ) = 0 . (27)
Remark. The condition in (34) is satisfied if, for example, M C is uniformly bounded. Proof. The proof is the same for t ∈ [0, 1), so we only establish the result for t ≥ 1. With x, λ, and ϕ defined as in the proof of Lemma 2, it was shown that the solution to (31) was stated by (33). Hence t (37) λ (t ) ≥ eϕ (t ) λ − 2β ν e −ϕ (t )ds .
The proof of Lemma 1 is essentially the same as one for an analogous Lemma in White, Foss and Guo (2006). It is now shown that if KD is positive definite at t = 0, then it is always positive definite. Lemma 2. Suppose that KD : [0,+∞) → ℜn×n is continuously differentiable and satisfies KD(0) = KD(0)T. If (24) holds for all t ∈ (0, +∞), then KD is positive definite for all t ∈ (0, +∞) provided that KD(0) is positive definite. Proof. By lemma 1, it is concluded that KD is symmetric and that (27) holds for all t ∈ [0, +∞). Let x : [0, +∞) → ℜn be a normalized eigenvector of KD with λ : [0, +∞) → ℜ the corresponding eigenvalue. Since KD is symmetric and continuously differentiable, it follows that x and λ are continuously differentiable. Since KD(t)x(t) = λ(t)x(t) for all t ∈ [0, +∞), it is seen that d (28) (K D x ) = K& D x + K D x& = λ&x + λx& . dt
Also KD(t)]is symmetric and x (t ) = 1 for all t ∈ [0, +∞), so 2 2 x ⋅ (K& D x ) = λ& x + λx ⋅ x& − x ⋅ (K D x& ) = λ& x .
(29)
Multiplying (27) by x yields K& D x − K D M −1Cx − C T M −1 K D x − 2β o K D x + 2β o K DF x = 0 ⇒ K& x − K M −1Cx − C T M −1 K x − 2β λx = −2β K x. D
D
D
o
o
(30)
DF
Taking the inner product of the above equation with x, using (29) and the symmetry of KD, it is found that 2 2 x λ& − λx ⋅ (M −1Cx ) − λx ⋅ (C T M −1 x ) − 2 β o λ x = (31)
(
(
-1
t −1
t * e −ϕ (t ) = ∫ e −ϕ (t )ds.
that
(38)
t −1
It follows from (37), (34), and (35) and the definition of ϕ that * * λ (t ) ≥ eϕ (t ) λo − 2β oνe −ϕ (t ) ≥ −2β oνeϕ (t )−ϕ (t )
{
}
t ≥ −2 β oν exp 2 β o − 2 ∫ M −1C dr t −1 2(β o − A) ≥ −2 β oνe .
(39)
Lemma 4. Suppose that KD : [0, +∝) → ℜn×n is continuously differentiable and satisfies KD(0) = KD(0)T. Further suppose that (24) holds for all t ∈ (0, +∝). If there exists an A < +∝ such that (34) is true for each t ≥ 1, then KD is bounded for each t ∈ [0, +∝). In particular, each eigenvalue λ(t) of KD satisfies β K (40) λ (t ) ≤ λ + o DF e 4 ( A−β )t .
o
o
A − β 0
Proof. Again, the proof is the same for t ∈ [0, 1), so the result is only established for t ≥ 1. With x, λ, and ϕ defined as in the proof of Lemma 3 it was shown that the solution to (31) was stated by (33). Thus
))
t
0 ≤ λ (t ) ≤ e 2( A−βo )t λo − 2 β o K DF
∫e
2 ( A− β o ) s
(41)
ds
0
ϕ (t ) := 2∫ {βo + x(r ) ⋅ (M −1Cx(r ))}dr.
≤ e 2( A− βo )t λo +
(32)
t
o
The general solution to (31) may be written as
}
λ (t ) = eϕ (t ) λo − 2 β o ∫ x (s ) ⋅ ( K DF x (s ))e −ϕ (t )ds . t
0
(33)
Since KD(0) is positive definite, we necessarily have that λ(0) > 0. Also KDF is positive-definite, so x⋅(KDFx) > 0. Recall that βo ≤ 0, so it can be concluded that λ(t) > 0 for all t ∈ [0, +∞). Hence KD(t) is positive-definite if KD(0) is. Lemma 3. Suppose that KD : [0,+∞) → ℜn×n is continuously differentiable and satisfies KD(0) = KD(0)T. Further suppose that (24) holds for all t ∈ (0, +∞). If there exists a non-negative A < + ∝ such that
∫ (
∫
By the Mean Value Theorem, there exists a t ∈ [t-1,t] such
Define the function ϕ : [0, +∝) → ℜ by
t
o
*
− 2 β o x ⋅ ( K DF x ) ⇒ λ& − 2 β o + x ⋅ M −1Cx λ = −2 β o x ⋅ K DF x .
{
o
(34)
M −1C dr ≤ A
max 0 , t −1)
z ⋅ ( K DF z ) ≥ ν z , 2
(35)
o
)t
β K DF ≤ λo + o A − β0
4( A− βo )t e .
V. BALL AND BEAM EXAMPLE The control law design method was applied to a ball and beam system. The system geometry and the dynamic equations of motion are shown in Figure 2 with definitions of the physical parameters. The physical parameters are I = 0.4 Kg m2, m = 1.5 Kg, Ro = 0.02 m, Cd = 0.16 Nsec./m, and g = 9.81 m/sec2. Chosen constants are K = 100, b = 100, βo = =-100 sec.-1 The system poles at equilibrium were specified as -1 ± j, -12 ± 12 j. The potential Γ(θ,r) was chosen as (42) Γ (θ , r ) = K (aθ 2 + br 2 ) and the complete potential Φ(θ,r) is found to be Φ (θ , r ) = Kb
for each t ≥ 1and a real, positive ν such that
β o K DF 2( A−β e A − βo
(P θ 2 11
2
)
(P P − P21P12 ) mg cos(θ ) + 2 P11 P21rθ + P 212r 2 + 11 22 P212 P21
(43)
where the constant “a” does not appear in the potential and
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the value of P21 remains negative so that the second term in the potential is concave upward. Note that the numerator in the second term of (43) is the determinant of P which remains positive since both KD and M are positive definite. The homogeneous part of the potential was not used in (43). 7 7 2 2 I + 5 mRo + mr − 5 mRo θ&& 7 7 m &r& − mRo 5 5 r m r& r mθ& θ& + & − m rθ Cd r& r mg cos(θ ) − Ro mg sin (θ ) τ + = 0 mg sin (θ )
I
θ r m Ro Cd
Y
Ro r
τ
θ
X
= the mass moment of inertia of the beam = angle of inclination of the beam = radial position of ball center relative to beam center = the mass of the ball = the radius of the ball = viscous damping coefficient of the rolling ball
The First Matching Theorem,” IEEE Transactions on Automatic Control, v 45, n 12, Dec, 2000, p 2253-2270. [4] Bloch, A.M., Dong Eui Chang, N.E. Leonard, J.E. Marsden; “Controlled Lagrangians and the stabilization of mechanical systems. II. Potential Shaping,” IEEE Transactions on Automatic Control, Volume: 46, Issue: 10 , Oct. 2001, p. 1556 – 1571. [5] Kelkar, Atul G., Bo Fang, Warren White, Xin Guo; “Feedback Stabilization of Underactuated Nonlinear Pendulum Cart System Using Matching Conditions,” Proceedings of the American Control Conference, v 6, 2002, p 4696-4701. [6] Khalil, Hassan K.; Nonlinear Systems, Upper Saddle River, NJ: Prentice-Hall, 2002. [7] Ortega, Romeo, Mark W. Spong, Fabio Gómez-Estern, and Guido Blankenstein: “Stabilization of a Class of Underactuated Mechanical Systems Via Interconnection and Damping Assignment,” IEEE Trans. Automat. Contr., Vol. 47, No. 8, Aug. 2002, p. 1218-1233. [8] White, Warren N., Mikil Foss, and Xin Guo; “A Direct Lyapunov Approach for a Class of Underactuated Mechanical Systems,” Proceedings of the American Control Conference, Minneapolis, Minnesota, June 14 – 16, 2006, p 103-110. [9] Young, Eutiquio C., Partial Differential Equations, An Introduction, Boston, Massachusetts: Allyn and Bacon, 1972.
Figure 2: Ball and Beam System In the example, the initial angle of the beam was 1.5 rad. (almost vertical) and the initial velocity of the ball was 2 m/sec. Both the initial angular velocity and the ball position were zero. The matrix KD was evaluated through numerical integration of (14). Figure 3 illustrates the ball position and the beam angle as a function of time. Figure 4 shows the Lyapunov function value and its first time derivative. The behavior shown in Figures 3 and 4 demonstrates the validity of the Lyapunov candidate function. VI. CONCLUSION
Figure 3: Generalized Coordinate Time Histories
A direct Lyapunov method for underactuated systems has been presented. The matching method results in three equations. The first matching condition results in linear ODEs for the matrix KD that can be solved as part of the feedback and it was mathematically proved that the eigenvalues of KD remain positive and bounded for all time. In this new formulation, KD approaches a constant matrix as time increases assuring consistent performance. The first matching equation improvement also results in a modified second matching condition that still consists of linear algebraic equations. The third matching condition produces n – m linear PDEs, the number of which is linear in the number of degrees of freedom and the number of actuators. Simulations demonstrate expected performance. Figure 4: Lyapunov Time History
VII. REFERENCES [1] Acosta, J. A., R. Ortega, A. Astolfi and A. D. Mahindrakar: “Interconnection and damping assignment passivity–based control of mechanical systems with underactuation degree one,” IEEE Trans. Automat. Contr., Vol. 50, No. 12, December 2005, p. 1936-1955. [2] Auckly, D, L. Kapitanski, and W. White; “Control of Nonlinear Underactuated Systems,” Comm. Pure Appl. Math, 53, number 3, 2000, pp.354-369. [3] Bloch, Anthony M., Naomi Ehrich Leonard, and Jerrold E. Marsden; “Controlled Lagrangians and the Stabilization of Mechanical Systems. I:
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