2010 Electronics, Robotics and Automotive Mechanics Conference
Discrete-time modeling and control of an under-actuated robotic system Jorge Rivera, Luis Garc´ıa, Susana Ortega and Juan Raygoza Department of Electronic Engineering DIVEC CUCEI, University of Guadalajara 44430, Guadalajara, Jalisco, M´exico
[email protected]
Abstract
pled dynamics of the system. This problem has motivated in the last decades the work of various researchers, with the aim of improving the control performance by designing the controller directly on the basis of the digital model [6], [7]. The problem of sampling continuous time systems is not trivial. In fact, in general, a sampled closed representation of the sampled dynamics does not exist; whereas for linear systems, a sampled model in closed form can be easily obtained [1], for nonlinear systems in general, the sampled data representations are given in the form of infinite series [6]. Hence, in practice, one uses truncated models of desired approximation [3]. This difficulty motivates the following possible solutions: 1) the design of the control law in the continuous time setting and its implementation by means of zero-order holder; 2) the use of much simpler discretization methods, such as those due to Euler (explicit or implicit), Tustin, etc., and the design of the control law in the digital setting. The first solution has the drawback of possible poor performance of the resulting sampled controller. The second solution [3] has the disadvantage that the accuracy of the resulting approximate discrete-time system decreases as the sampling period increases. An alternative to these approaches for mechanical systems are the variational integrator as the symplectic Euler method, where the advantages of the symplectic Euler method are explained over Euler (explicit or implicit) methods in [11]. Therefore, in this work one designs a discrete-time controller for trajectory tracking of the Pendubot based on a sampled model obtained by means of the symplectic Euler method. The control technique is based on the discrete-time sliding mode technique. Moreover, an observer is designed for the estimation of state variables in order to eliminate the encoder in the first link. The rest of this work is organized as follows: In Section 2 the explicit, implicit and symplectic Euler methods are revisited. The mathematical model and problem formulation for the Pendubot are shown in Section 3. In Section 4 the discrete-time modeling and control for the Pendubot are carried on. Simulation results are shown in Section 5 and
In this work a discrete-time model for the Pendubot is proposed based on the symplectic Euler method. Then, a discrete-time sliding mode controller and observer are designed. A simulation study is carried on in order to verify the good performance of the proposed modeling and control scheme when compared to the case of using explicit Euler method for discrete-time modeling.
1. Introduction Under-actuated robotic systems [5] are characterized by consisting of less actuators than links. This can be a natural design due to physical limitations or an intentional one for reducing the robot cost. The control of such robots is more difficult than fully actuated ones. There is very popular under-actuated robotic system known as the Pendubot [10]. The Pendubot is a two link planar robot with a dc motor actuating in the first link, while the second link is balanced by the first one. The purpose of the Pendubot is research and education inside the control theory of nonlinear systems. Common control problems for the Pendubot are swing-up, stabilization and trajectory tracking. There are several works about the control of the Pendubot: output regulation [8], discontinuous output regulation [9], sliding modes [13], Takagi-Sugeno [2], passivity [4] among others. It is worth to mention that all of the above control techniques are designed in the continuous-time domain and have been real-time implemented in digital devices. The implementation of control laws designed for these devices is nowadays generally made in discrete time. In fact, the recent advancements in digital microprocessor technology have rendered cheaper, simpler, and more flexible the discrete-time implementation of controllers designed using continuous-time techniques. The main problem here is the degradation of the expected performance, due to the fact that the design of the control law is not made using the sam-
978-0-7695-4204-1/10 $26.00 © 2010 IEEE DOI 10.1109/CERMA.2010.120
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finally some comments conclude the work in Section 6.
sampled when deriving the Euler-Lagrange equation instead of directly sampling such equation. Although the Explicit and Symplectic methods require the same amount of computations, the symplectic method yields to better results, and better than the implicit method. This fact will be illustrated in a subsequent section.
2. Symplectic Euler method The symplectic Euler method belongs to the family of variational integrators. As stated in [11], these integrators often perform better than their non-variational counterparts (Explicit Euler, Implicit Euler, among others) because they preserve geometric properties of the physical system. As a consequence the integrators are guaranteed to be symplectic implying a good energy behavior, rather than perpetual damping or blowing up. Also, they are guaranteed to preserve discrete momenta of the system, implying a good simulation performance at a low computational burden. Let us consider the following double integrator mechanical system x˙
= v
v˙
= f (x, v) + g(x, v)u
3. Mathematical model of the Pendubot and problem formulation In this section the continuous-time model of the Pendubot and the problem formulation are presented.
where x ∈ and v ∈ can represent linear or angular displacement and velocity respectively, u ∈ is an input force, f (x, v) and g(x, v) are smooth functions. With first order approximations one can easily determine equivalent discrete-time systems. Making use of a forward difference approximation one obtains the following discrete-time system: xk+1 vk+1
= =
xk + δvk vk + δ(f (xk , vk ) + g(xk , vk )uk )
where xk = x(kδ), vk = v(kδ), uk = u(kδ), with δ as the sampling period and k = 0, 1, 2, . . . as the sampling time index. This time integrator is called the Explicit Euler method. Another representation is obtained when using backward differences, in this case the discrete-time system results as follows: xk+1
=
xk + δvk+1
vk+1
=
vk + δ(f (xk+1 , vk+1 ) + g(xk+1 , vk+1 )uk+1 )
Figure 1. Schematic diagram of the Pendubot. The equation of motion for the Pendubot (see Figure 1) can be described by the following general equation [10]: D(q)¨ q + C(q, q) ˙ + G(q) + F (q) ˙ =τ
where q = [q1 , q2 ]T ∈ n is the vector of joint variables (generalized coordinates), q1 ∈ m represents the actuated joints, and q2 ∈ (n−m) represents the unactuated ones. D(q) is the n × n inertia matrix, C(q, q) ˙ is the vector of Coriolis and centripetal torques, G(q) contains the gravitational terms, F (q) ˙ is the vector of viscous frictional terms, and τ is the vector of input torques. For the Pendubot system, the dynamic model (1) is particularized as τ1 C1 G1 F1 D11 q¨1 + D12 q¨2 = + + + G2 F2 0 D12 q¨1 + D22 q¨2 C2 (2) 2 2 where D11 (q2 ) = m1 lcl + m2 (l12 + lc2 + 2l1 lc2 cos q2 ) + 2 I1 + I2 , D12 (q2 ) = m2 (lc2 + l1 lc2 cos q2 ) + I2 , D22 = 2 + I2 , C1 (q2 , q˙1 , q˙2 ) = −2m2 l1 lc2 q˙1 q˙2 sin q2 − m2 lc2 m2 l1 lc2 q˙22 sin q2 , C2 (q2 , q˙1 ) = m2 l1 lc2 q˙12 sin q2 ,
this time integrator is known as the Implicit Euler method and requires to be solved in each time step in order to determine xk+1 and vk+1 . An alternative representation is a mix of Explicit and Implicit Euler. The velocity equation is updated explicitly while the position is updated implicitly, resulting in the following discrete-time system xk+1 vk+1
= =
(1)
xk + δvk+1 vk + δ(f (xk , vk ) + g(xk , vk )uk )
this integrator structure is called Symplectic Euler method. This method is obtained when the least action principle is
509
G1 (q1 , q2 ) = m1 glc1 cos q1 + m2 gl1 cos q1 + m2 glc2 cos (q1 + q2 ), G2 (q1 , q2 ) = m2 glc2 cos (q1 + q2 ), F1 (q˙1 ) = μ1 q˙1 , F2 (q˙2 ) = μ2 q˙2 , with m1 and m2 as the mass of the first and second link of the Pendubot respectively, l1 is the length of the first link , lc1 and lc2 are the distance to the center of mass of link one and two respectively, g is the acceleration of gravity, I1 and I2 are the moment of inertia of the first and second link respectively about its centroids, and μ1 and μ2 are the viscous drag coefficients. Choosing T T x = x1 x2 x3 x4 = q1 q2 q˙1 q˙2 as the state vector, u = τ1 as the input. The description of the system can be given in state space form as: x(t) ˙ y
= f (x) + g(x)u(t) = x2
4.1 Discrete-time model For the Pendubot presented as a nonlinear affine system in (3), its corresponding discrete-time approximation by means of the Symplectic Euler method outlined in Section 2 results as follows:
(3)
f (x)
⎜ = ⎜ ⎝ ⎛
g(x)
⎜ = ⎜ ⎝
⎞ ⎛ x3 f1 (x3 ) ⎟ ⎜ f2 (x4 ) x4 ⎟=⎜ ⎠ ⎝ b3 (x2 )p1 (x) f3 (x) f4 (x1 , x2 , x3 ) b4 (x2 )p2 (x) ⎞ ⎛ 0 b1 ⎜ 0 ⎟ b2 ⎟ ⎜ D22 = ⎜ 2 (x ) b3 (x2 ) ⎠ ⎝ D11 (x2 )D22 −D12 2 −D12 (x2 ) b4 (x2 ) D (x )D −D2 (x ) 11
p1 (x)
p2 (x)
2
22
12
=
x1,k + δx3,k+1
x2,k+1 x3,k+1
= =
x2,k + δx4,k+1 x3,k + δ(b3,k p1,k + b3,k uk )
x4,k+1
=
x4,k + δ(b4,k p2,k + b4,k uk )
(4)
where xi,k = xi (kδ), i = 1, 2, 3, 4, b3,k = b3 (x2,k ), b4,k = b4 (x2,k ), p1,k = p1 (xk ), p2,k = p2 (xk ), xk = x(kδ). Replacing the third and fourth equations in the first and second of (4) and sampling the output y results in
where y is the output, ⎛
x1,k+1
⎞ ⎟ ⎟, ⎠
x1,k+1 x2,k+1
= =
x1,k + δx3,k + δ 2 (b3,k p1,k + b3,k uk ) x2,k + δx4,k + δ 2 (b4,k p2,k + b4,k uk )
x3,k+1 x4,k+1
= =
x3,k + δ(b3,k p1,k + b3,k uk ) x4,k + δ(b4,k p2,k + b4,k uk )
yk
=
x2,k .
(5)
This model will be used for the design of a control law.
⎞
4.2 Discrete-time control law design
⎟ ⎟ ⎟, ⎠
Let us define a nonlinear transformation x ¯k = ¯2,k , x ¯3,k , x ¯4,k )T = ψ(xk ) of the following form (¯ x1,k , x
2
D12 (x2 ) = (C2 (x2 , x3 ) + G2 (x1 , x2 ) + F2 (x4 )) D22 −C1 (x2 , x3 , x4 ) − G1 (x1 , x2 ) − F1 (x3 ), D11 (x2 ) = (C2 (x2 , x3 ) + G2 (x1 , x2 ) + F2 (x4 )) D12 −C1 (x2 , x3 , x4 ) − G1 (x1 , x2 ) − F1 (x3 ).
x¯1,k
=
x1,k − b3,k−1 b−1 4,k−1 x2,k
x¯2,k x¯3,k
= =
x2,k x3,k − b3,k−1 b−1 4,k−1 x4,k
x¯4,k
=
x4,k .
(6)
The discrete-time model of the Pendubot (5) is now shown in the new variables by taking one step ahead in (6) x¯1,k+1
The problem is to obtain a discrete-time model of the dynamic of the Pendubot that preserves its geometric properties, and then, based on this model, to design a discrete-time control law that forces the output to track a desired trajectory.
x¯2,k+1 x¯3,k+1 x¯4,k+1 yk
4. Discrete-time modeling and control of the Pendubot
= x ¯1,k + (ρk−1 − ρk )(¯ x2,k + δ¯ x4,k ) + δ¯ x3,k + δ 2¯b3,k (¯ p1,k − p¯2,k ) = x ¯2,k + δ¯ x4,k + δ 2¯b4,k (¯ p2,k + uk ) = x ¯3,k + (ρk−1 − ρk )¯ x4,k + δ¯b3,k (¯ p1,k − p¯2,k ) = x ¯4,k + δ¯b4,k (¯ p2,k + uk ) = x ¯2,k
¯ where ρk = ¯b3,k¯b−1 x2,k ), ¯b4,k = b4 (¯ x2,k ), 4,k , b3,k = b3 (¯ p¯1,k = p1 (¯ xk ), p¯2,k = p2 (¯ xk ). Now, the steady-state for system (4), xr,k = (x1,r,k , x2,r,k , x3,r,k , x4,r,k )T , will be determined. For that, we consider the following exosystem that will generate a sinusoidal shape output reference signal:
In this section, a discrete-time model of the Pendubot is derived by means of the Symplectic Euler method, then, based on this model, a control law is designed using a discrete-time sliding mode control technique, finally an observer is proposed for the estimation of the unmeasured states.
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w1,k+1
=
cos(αδ)w1,k + sin(αδ)w2,k
w2,k+1
=
− sin(αδ)w1,k + cos(αδ)w2,k ,
When the sliding mode occurs, i. e., Sk = 0, one can calculate z4,k from (9) of the following form:
where α is the frequency of the generated signals and if the initial conditions are chosen as w1,0 = w2,0 , then, the √ amplitude is w1,0 . The steady state for the output is assigned as x2,r,k = w2,k . Making use of a natural steady-state constraint given in [9], that states that, the sum of the two angles, q1 and q2 equals π/2, one can easily determine the steady-state for x1,k as x1,r,k = π/2 − x2,r,k . Finally, the steadystate values for x3,k and x4,k can be determined by using the first two equations in (4), in the form of difference equations, i. e., x3,r,k+1 = (x1,r,k+1 − x1,r,k )/δ and x4,r,k+1 = (x2,r,k+1 − x2,r,k )/δ. Transforming xr,k through the diffeomorphism (6) rex1,r,k , x ¯2,r,k , x ¯3,r,k , x¯4,r,k )T , where sults in x ¯r,k = (¯ x ¯1,r,k
=
x1,r,k − b3,r,k−1 b−1 4,r,k−1 x2,r,k
x ¯2,r,k x ¯3,r,k
= =
x2,r,k x3,r,k − b3,r,k−1 b−1 4,r,k−1 x4,r,k
x ¯4,r,k
=
x4,r,k .
z4,k = −k1 z1,k − k2 z2,k − k3 z3,k ,
then, by replacing (10) and (11) in the first three equations of (8) yields to the sliding mode dynamic 1 zk+1
ψk1
z¯k+1 = ψk =
+ φ2,k (·) = − φ3,k (·) = + φ4,k (·) =
(8)
+ γ (z2,k , x ¯2,r,k )ueq,k |Sk =0 .
zk1 , zˆk2 )T = (ˆ z1,k , zˆ2,k , zˆ3,k , zˆ4,k )T , and νk = where zˆ¯k = (ˆ (ρ1 v2,k , ρ2 v2,k , ρ3 v2,k , ρ4 v2,k )T . Now, one defines the estimation error as ˜z¯k = z¯k − ˆz¯k and by taking one step ahead, the estimation error dynamic is obtained: ˜z¯k+1 = φ˜ − νk
(14)
with φ˜ = φ(zk1 , zk2 , x¯r,k ) − φ(ˆ zk1 , zˆk2 , x ¯r,k ). In order to find proper values for the observer gains ρ1 , ρ2 , ρ3 , ρ4 , we linearly approximate the estimation error system (14) resulting in the following linear system: ⎛˜ ⎞ ⎞⎛˜ ⎞ ⎛ z¯1,k+1 z¯1,k a11 a12 a13 a14 ˜ ˜ a a a a z ¯ z ⎜ 2,k+1 ⎟ ⎜ 21 22 23 24 ⎟ ⎜ ¯2,k ⎟ ⎝˜ ⎠ = ⎝ ⎠⎝˜ ⎠ a31 a32 a33 a34 z¯3,k+1 z¯3,k a41 a42 a43 a44 z˜¯4,k+1 z˜¯4,k ⎞ ⎛ v1,k ⎜ v2,k ⎟ (15) − ⎝ ⎠, v3,k v4,k
z4,k + x ¯4,r,k + δ¯b4,k p¯2,k − x¯4,r,k+1 .
Now, following a discrete-time sliding mode procedure [12], one defines the sliding mode function as follows: (9)
In order to force the states of (8) to the sliding manifold (9) one can make use of the equivalent control method. The equivalent control ueq,k is calculated from Sk+1 = 0, therefore, one can calculate it as: −k1 φ1 − k2 φ2 − k3 φ3 − φ4 . δ¯b4,k (1 + k2 δ)
1
ˆz¯k+1 = ψˆk = φ(ˆ zk1 , zˆk2 , x ¯r,k ) + γ(z2,k , x ¯2,r,k )uk + νk (13)
x ¯2,r,k+1 , z3,k + x ¯3,r,k + (ρk−1 − ρk )(z4,k + x ¯4,r,k ) ¯ δ b3,k (¯ p1,k − p¯2,k ) − x ¯3,r,k+1 ,
ueq,k =
φ
(12)
(zk1 , z4,k , x ¯r,k )
In this subsection we design a discrete-time sliding mode observer based on system (8) and the measurement of z2,k . The proposed observer is as follows:
z1,k + x ¯1,r,k + (ρk−1 − ρk )(z2,k + x ¯2,r,k ) ¯4,r,k ) + δ(z3,r + x ¯3,r,k ) δ(ρk−1 − ρk )(z4,k + x 2¯ δ b3,k (¯ p1,k − p¯2,k ) − x ¯1,r,k+1 , z2,k + x ¯2,r,k + δ(z4,k + x ¯4,r,k ) + δ 2¯b4,k p¯2,k
Sk = z4,k + k1 z1,k + k2 z2,k + k3 z3,k .
1
4.3 Discrete-time observer design
where φ(·) = (φ1 (·), φ2 (·))T with φ1 (·) = T (φ1 (·), φ2 (·), φ3 (·)) , φ2 (·) = φ4 (·) and γ(·) = (γ 1 (·), γ 2 (·))T where γ 1 (·) = (0, δ 2¯b4,k , 0)T , γ 2 (·) = δ¯b4,k , with φ1,k (·) = +
=
ψk1
where Asm (κ) = ∂ψk1 /∂zk1 |zk1 =0 , with κ = (k1 , k2 , k3 ). In order to choose the design parameters, a polynomial with desired poles is proposed, pd (z) = (z−λ1 )(z−λ2 )(z−λ3 ), such that, the coefficients of the characteristic equation that results from the matrix Asm are equalized with the ones related with pd (z), i. e., det(zI − Asm ) = pd (z), in such manner one can find explicit relations for κ. In this case limk→∞ z¯k = 0, accomplishing with the control objective.
(7)
+ γ(z2,k , x ¯2,r,k )uk
=
The sliding function parameters k1 , k2 and k3 should stabilize the sliding mode dynamic (12). For a proper choice of such constant parameters one can linearize the sliding mode dynamic 1 = Asm (κ)zk1 zk+1
where b3,r,k = b3 (x2,r,k ), b4,r,k = b4 (x2,r,k ). Now, one introduces the error variable z¯k = (zk1 , zk2 )T = x ¯k − x¯r,k , with zk1 = (z1,k , z2,k , z3,k )T and zk2 = z4,k . Taking one step ahead on z¯k , yields to the following error system: φ(zk1 , zk2 , x ¯r,k )
(11)
where ai,j = ∂ φ˜i /∂ z˜¯j,k |z˜¯k =0 , v1,k = ρ1 v2,k , v3,k = ρ3 v2,k and v3,k = ρ3 v2,k ; v2,k will be defined in the
(10)
511
following lines. One can choose the sliding function as So,k = z˜¯2,k . Here, the equivalent control method can not be applied to observer designs since some variables are not measurable. One must introduce a robust function to unknown terms. Therefore one proposes the following slidz¯2,k |1/2 sign(˜ z¯2,k ). In order ing mode function v2,k = ρ2 |˜ to investigate the convergence of the state z˜¯k to the sliding manifold So,k = 0, one proposes the Lyapunov function Vk = (|˜ z¯2,k |1/2 sign(˜ z¯2,k ))2 where its increment is given by (16) ΔV = Vk+1 − Vk = |So,k+1 | − |So,k |
At this point, a polynomial with desired poles is proposed, pod (z) = (z − λo1 )(z − λo2 )(z − λo3 ), such that, the coefficients of the characteristic equation that results from the matrix Aosm are equalized with the ones related with pod (z), i. e., det(zI − Aosm ) = pod (z), in such manner one can find explicit relations for ρ1 , ρ3 and ρ4 . In this case limk→∞ ˆz¯k = z¯k .
5. Simulations
with So,k+1 = a2,1 z˜¯1,k + a2,2 z˜¯2,k + a2,3 z˜¯3,k + a2,4 z˜¯4,k − v2,k , (17) moreover, from So,k+1 = 0 one can calculate the equivalent control v2,eq,k as follows v2,eq,k = a2,1 z˜¯1,k + a2,3 z˜¯3,k + a2,4 z˜¯4,k ,
(18)
and then, to suppose that the term v2,eq,k is globally bounded by |v2,eq,k | ≤ v2,m |So,k |1/2 for some constant v2,m > 0. Now replacing (17) in (16) can result in the following expressions: ΔV = |a2,1 z˜¯1,k +a2,2 z˜¯2,k +a2,3 z˜¯3,k +a2,4 z˜¯4,k −v2,k |−|So,k |. Making some simplifications ΔV
= |a2,2 z˜¯2,k + v2,eq,k − ρ2 |z˜¯2,k |1/2 sign(z˜¯2,k )| − |So,k | ≤ |a22 So,k + v2,m |So,k |1/2 − ρ2 |So,k |1/2 sign(So,k )| − |So,k | < |a22 + v2,m − ρ2 ||So,k |1/2 − |So,k |1/2 < −|So,k |1/2 (1 − |a22 + v2,m − ρ2 |) ,
and by choosing (a22 + v2,m + 1) > ρ2 > (a22 + v2,m − 1) the increment of the Lyapunov function is negative definite, i. e., ΔV < −|So,k |1/2 , and the sliding manifold is locally stable. When the motion of system (15) is confined to the sliding mode, i.e., So,k = 0, it is known as the sliding mode dynamic characterized by a reduced order system. Applying the equivalent control (18) for analysis purposes one can express such dynamic as follows ⎞ ⎛ ⎞ ⎛ z˜¯1,k z˜¯1,k+1 ⎝ z˜¯3,k+1 ⎠ = Aosm ⎝ z˜¯3,k ⎠ z˜¯4,k z˜¯4,k+1 where Aosm
⎛
a11 − ρ1 a2,1 = ⎝ a31 − ρ3 a2,1 a41 − ρ4 a2,1
a13 − ρ1 a2,3 a33 − ρ3 a2,3 a43 − ρ4 a2,3
⎞ a14 − ρ1 a2,4 a34 − ρ3 a2,4 ⎠ a44 − ρ4 a2,4 512
In order to show the performance of the control methodology here proposed, simulations are carried out. The nominal values of the parameters of the Pendubot are defined as follows: m1 = 0.8293, m2 = 0.3402, l1 = 0.2032, lc1 = 0.1551, lc2 = 0.1635125, g = 9.81, I1 = 0.00595035, I2 = 0.00043001254, μ1 = 0.00545, μ2 = 0.00047. The constant parameters have been chosen for the controller as α = 0.3, w1,0 = w2,0 = 0.09, λ1 = 0.9941 + 0.0030j, λ2 = 0.9941 − 0.0030j and λ3 = 0.9978. The vector κ depends on the different values assigned to δ and therefore it is only shown for the particular value of δ = 0.001, resulting in k1 = 41.1463, k2 = 14.4040, k3 = 9.2852. The constant parameters related to the observer have been chosen as ρ2 = 0.002, λo,1 = 0.90, λo,2 = 0.91 and λo,3 = 0.92. Also, the observer gains are only shown in the case of δ = 0.001, resulting in the following values: ρ1 = −553.4245, ρ2 = −10944.4278 and ρ3 = 246.3188. We first show the open-loop performance of the Pendubot and two sampled models without friction in Figure 2. The continuous-time model has been simulated with a sampling period of 0.1ms, and sampled models obtained via symplectic and explicit Euler methods, both using a sampling period of 1ms. In the explicit Euler method, the motion of the Pendubot amplifies over time, i. e., its energy amplifies over time, rather than being conserved. Thus, the solution becomes unstable as time increases. On the other hand the symplectic model performs similar to the continuous-time one, i. e., it is stable and oscillates with constant amplitudes. This is a superior method for modeling mechanical systems in the discrete-time setting, with the advantage of no additional numerical operations needed in order to retrieve the correct behavior. Now in Figure 3 it is shown a closed-loop phase-portrait of the designed controller based on the symplectic model and compared with a similar design based on the explicit model. It can be appreciated that both controllers perform well with a sampling period of 1ms. It is worth to mention that these simulations and the following ones are made using the continuous-time model of the Pendubot. With a sampling period of 2.77ms the explicit based controller becomes unstable meanwhile the symplectic based controller still performs well as can be seen in Figure 4. Figure (5) shows the simulation of the symplectic based controller for several sampling periods,
0.2 Reference Symplectic Euler Explicit Euler 0.15
0.1
0.05
0
−0.05
−0.1
Figure 2. Open-loop phase-portraits in the x4 − x2 plane. a)Continuous-time system simulated with δ = 0.1ms. b)Sampled system with Symplectic Euler with δ = 1ms. c)Sampled system with Explicit Euler with δ = 1ms.
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Figure 3. Closed-loop phase-portraits in the x4 − x2 plane with δ = 1ms.
models obtained by using variational integrators will result in excellent energy behavior, rather than perpetual damping or blowing up, and simulations using these integrators usually have great physical fidelity with a reduced computational burden. On the basis of such sampled data model, an observer-based controller, which guarantees asymptotic output reference tracking has been designed. The proposed control strategy is verified by means of numeric simulations where the closed-loop performance of the Pendubot is better when compared with a controller based on the explicit Euler method. Some interesting issues, such as the robustness of the controller with respect to parameter variations, determination of the sampling period, and modeling of the actuators dynamics, are currently under study.
while Figure 6 is a zoomed version of Figure 5. A quantification of the tracking errors is presented in Table 1. Table 1. Quantification of tracking error Label in δ Sampling period Tracking Figure (6) (ms) increment error b 1.00 2.11% c 1.40 40 % 4.10 % d 1.80 80 % 6.95 % e 2.20 120 % 11.31 % f 2.60 160 % 18.07 % g 3.00 200 % 29.45 % h 3.40 240 % 50.88 % i 3.80 280 % 100.27 %
References [1] K. J. Astr¨om and B. Wittenmark. Computer-Controlled Systems. Prentice-Hall, Englewood Cliffs, NJ, 1990. [2] O. Begovich, E. Sanchez, and M. Maldonado. Takagisugeno fuzzy scheme for real-time trajectory tracking of an underactuated robot. Control Systems Technology, IEEE Transactions on, 10(1):14 –20, jan 2002. [3] M. N. Cirstea and A. Dinu. A vhdl holistic modeling approach and fpga implementation of a digital sensorless induction motor control scheme. IEEE Trans. Ind. Electron., 54(4):1853–1864, 2007. [4] I. Fantoni, R. Lozano, and M. Spong. Passivity based control of the pendubot. In American Control Conference, 1999.
6. Conclusions In this paper, a sampled data representation of the Pendubot has been obtained. This sampled data model has been determined by means of a variational integrator. Variational integrators often perform better than explicit or implicit Euler methods because they preserve the underlying geometry of the physical system. As a consequence sampled data
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