Adaptive Robust Output Feedback Control of a Magnetic Levitation ...
Proceedings of the 2004 IEEE International Symposium on Intelligent Control Taipei, Taiwan, September 2-4, 2004
Adaptive Robust Output Feedback Control of a Magnetic Levitation System by K-Filter Approach Zi-Jiang Yang, Kazuhiro Kunitoshi, Shunshoku Kanae, Kiyoshi Wada Department of Electrical and Electronic System Engineering Kyushu University, Fukuoka, 812-8581 JAPAN Tel: +81-92-642-3958; Fax: +81-92-642-3904 [email protected]
Abstract— This paper proposes an adaptive robust output feedback controller for position-tracking problem of a magnetic levitation system with a current feedback power amplifier. The controller is designed by a backstepping procedure with robustifying modification of the k-filter approach. The boundedness and guaranteed transient performance of the error signals are achieved by the nonlinear damping terms. And the ultimate position-tracking error is reduced by the adaptive laws. Experimental results are included to show the excellent control performance of the designed controller.
I. I NTRODUCTION Backstepping output feedback controllers are usually designed for output feedback systems, where the nonlinear terms are functions of the output and the models are in the strict feedback form. The k-filter approach is one of the typical approaches to construct the state variable observer. So far, several applications have been reported in the literature [2], [4], [5]. In our system under study, however, as described in the next section, the system model is different from the standard output feedback form, since there is an position dependent nonlinear uncertainty directly multiplied by the control input. In this work, the controller is designed by the backstepping procedure with robustifying modification of the k-filter approach. The boundedness and guaranteed transient performance of the error signals are achieved by the nonlinear damping terms. And the ultimate position-tracking error is reduced by the adaptive laws. Finally, experimental results are included to show the excellent control performance of the designed controller. II. M ODEL OF THE MAGNETIC LEVITATION SYSTEM Consider the magnetic levitation system shown in Fig. 1. If the coil current is controlled by a current feedback power amplifier as in most applications, then the system dynamics can be described in the following equations [2], [6]. x˙ 1 = x2 x˙ 2 = g + ϕ(x1 )i2 ϕ(x1 ) =
−Q 2M (X∞ + x1 )2
Diagram of the magnetic levitation system.
Our task in this study is to design an efficient and reliable output feedback controller in the case where only the position is measureable and the physical parameters are uncertain. Denoting the nominal physical parameters as g0 , M0 , Q0 and X∞0 , we have the nominal nonlinear function and its modeling error respectively as the following. ϕ0 (x1 ) = −
Q0 2M0 (X∞0 + x1 )2
(3)
∆ϕ (x1 ) = ϕ(x1 ) − ϕ0 (x1 )
(4)
Then we approximate the modeling error ∆ϕ (x1 ) by a radial basis function (RBF) network. ϕ (x1 , w t) ϕ(x 1 ) = ϕ0 (x1 ) + ∆ ϕ (x1 , w) = ∆
N
wn r (x1 − pn ) = RT w
(5)
(6)
n=1
(1)
(2)
where, x1 : air gap (vertical position) of the steel ball; x2 : velocity of the steel ball; i: coil current; g: gravity acceleration; M : mass of the steel ball; Q and X∞ : positive constants determined by the characteristics of the coil, magnetic core and steel ball.
0-7803-8635-3/04/$20.00 ©2004 IEEE
Fig. 1.
RT = [r(x1 − p1 ), r(x1 − p2 ), · · · , r(x1 − pN )] w = [w1 , w2 , · · · , wN ]T where
1 r (x1 − pn ) = exp − 2
x1 − pn σ
(7)
2 (8)
is a Gaussian basis function, in which pn is the center of the nth basis function, and σ determines its width. In this study,
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the basis functions are equidistantly located in X = {x1 | 0 ≤ x1 ≤ x1M }, i.e., the physically allowable operating region of x1 (see Fig. 1). Some assumptions are made here. Assumption 1: A sufficient number of basis functions are included into the RBF network, so that there exists a desired w∗ such that sup |ηt | ≤ εϕ
x1 ∈X
(9)
where εϕ is a sufficiently small positive real number, and ϕ (x1 , w∗ ) ηt = ∆ϕ (x1 ) − ∆
(10)
Assumption 2: The lower and upper bounds of the unknown parameters are known a priori, i.e., w ≤ w ≤ w,
0≤g≤g
(11)
Notice that an external constant (or slowly varying) disturbance can also be included into g equivalently, i.e., the gravity acceleration is biased equivalently. Assumption 3: The parameter bounds are chosen such that ϕ (x1 , w) < 0 ϕ(x 1 ) ≤ ϕ0 (x1 ) + ∆
OF THE OBSERVER BY
where x = [x1 , x2 ]T , e2 = [0, 1]T , and k = [k1 , k2 ]T , such that
−k1 1 A0 = −k2 0 = x−x . Then in is Hurwitz. Define the observer error x the ideal case we have (15)
However, since the parameters are unknown, we have to adopt a new state estimation following the K-filter approach lectured in [3]. (16)
(17)
Notice ΩT = [Ω1 , Ω2 ]T ∈ R2×N , Ω1 , Ω2 ∈ RN ×1 , ξ, Ωg , Ω0 , Ωη ∈ R2×1 . In the sequel, the ith (i = 1, 2) elements of ξ, Ωg , Ω0 , Ωη , and the nth (n = 1, · · · , N ) elements of Ω1 , Ω2 will be denoted as ξi , Ωgi , Ω0i , Ωηi and Ω1n , Ω2n respectively. Remark 1: The last line of equation (17) related to the modeling error ηt i2 is never implemented in practice. It is written explicitly here only for covenience of theoretical analysis in the sequel. Remark 2: It is recommended to set the initial states of the filters as ξ(0) = x(0), ΩT (0) = 0 Ωg (0) = Ω0 (0) = Ωη (0) = 0
(18)
(0) = x(0), to avoid large transient error. such that x = [ε1 , ε2 ]T is readily The state estimation error ε = x − x shown to satisfy ε˙ = A0 ε
(19)
Notice that ε decays exponentially to zero. Then we have x = ξ + Ωg g + ΩT w∗ + Ω0 + Ωη + ε
Since only the position x1 is available for measurement, a nonlinear observer is first constructed here. If the parameters were known, we would design the observer
˙ = A0 x + kx1 + e2 g + ϕ0 (x1 )i2 + RT w ∗ i2 + ηt i2 x (14)
= ξ + Ωg g + Ω T w ∗ + Ω0 + Ωη x
˙ T = A0 ΩT + e2 RT i2 Ω ˙ 0 = A0 Ω0 + e2 ϕ0 (x1 )i2 Ω ˙ η = A0 Ωη + e2 ηt i2 Ω
K- FILTER
APPROACH
˙ = A0 x x
ξ˙ = A0 ξ + kx1 ˙ Ωg = A0 Ωg + e2
(12)
Assumption 4: The reference trajectory yr and it’s first and second derivatives y˙ r , y¨r are continuous, uniformly bounded, and available such that there exists a compact set Dyr = yr , y˙ r , y¨r 0 < yr < x1M , |y˙ r | ≤ y˙ r , |¨ yr | ≤ y¨r , ∃ ∃ 3 y˙ r , y¨r > 0 ⊂ R (13) III. C ONSTRUCTION
where
(20)
This equation implies that our problem is different from the standard problem setting of the k-filter approach in [3]. Since there exists the term Ωη driven by the position and input dependent modeling error ηt i2 . Therefore, extra efforts are required to construct an efficient and reliable output feedback controller. For theoretical preparation of the controller design given in the next section, we will give here some results regarding the effects of Ωη . Let h(t) is the impulse response of s + k1 H(s) = 2 s + k1 s + k2 Then we have Lemma 1: Let Assumptions 1∼3 hold, i2 > 0, and x ∈ X . If k1 and k2 are chosen such that h(t) ≥ 0, and the initial conditions are chosen such that for t ≥ 0: eT2 eA0 t Ω0 (0) + ΩT (0)w ∗ + Ωη (0) ≤ 0 t ≤ 0 eT2 eA0 t Ω0 (0) + ΩT (0)w then it can be made that the following results hold for t > 0:
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1)
A. Step 1 Define the error signals as
Ω02 + ΩT2 w∗ + Ωη2 < 0
t < 0, Ω02 + ΩT2 w
z1 = x1 − yr ,
such that
z˙1 = ξ2 − gt Ωg2 + Ωg2 gt + ε2 − y˙ r + (α1 + z2 ) (1 + ∆Ω ) (22)
t − w ∗ , and t = w where w t + Ωη2 −ΩT2 w t Ω02 + ΩT2 w
where gt = gt − g. Then we can design the virtual input α1 as follows: α1 = α10 − α11 − α12 t α10 = −c1p z1 − c1i z1 dt − ξ2 − Ωg2 gt + y˙ r 0 α11 = κ11 g0 Ω2g2 + νz1 = α11 z1 α12 = κ12 α210 + νz1 = α12 z1
2) ∆Ω −ΩT2 w + Ω t η2 1 + ∆Ω = Ω + ΩT w ∗ + Ω < ∞ 02 η2 2 3) Ωη2 0, as t → ∞
5) −ΩT w 2 t + Ωη2 0, as t → ∞
The results of boundedness of some terms listed in Lemma 1 are derived based on the fact that in each one the denominator grows as the same order as the numerator grows. These resutls will be used for design of the nonlinear damping terms and stability analysis in the following sections. IV. C ONTROLLER
(23)
where c1p , c1i , κ11 , κ12 > 0, ν = 0.01. α10 is a feedback linearization PI controller, α11 and α12 are nonlinear damping terms to counteract gt Ωg2 and ∆Ω respectively. Notice that the integral control action helps to remove the steady offset of z1 , especially in the case where the controller is fixed without parameter adaption. It will be shown in the next section that these nonlinear damping terms achieve boundedness and guaranteed transient performance of error signal z1 . The unknown parameters are updated by the following adaptive laws with projection: ⎧ ⎪ for gt = g, Ωg2 z1 < 0 ⎨0 g˙ t = 0 (24) for gt = g, Ωg2 z1 > 0 ⎪ ⎩ γ1g Ωg2 z1 otherwise
4) Ωη2 ≤ C1 εϕ , T ∗ Ω02 + Ω2 w + Ωη2
(21)
where α1 is the virtual input to stabilize z1 . From equation (20), we have subsystem S1:
Ω02 + ΩT2 w ∗ + Ωη2 = 1 + ∆Ω > 0 t Ω02 + ΩT2 w
∆Ω =
t − α1 z2 = Ω02 + ΩT2 w
DESIGN
The controller is designed through the backstepping procedure. At each step, the modeling errors are counteracted by the nonlinear damping terms which achieve boundedness and guaranteed transient performance of the error signal. Additionally, adaptive laws are introduced to achieve a small ultimate error bound.
⎧ α10 Ω2n z1 ⎪ ⎪0 for w nt = wn , 0 = 0 T ⎪ t Ω 02 + Ω2 w ⎪ ⎪ ⎪ α10 Ω2n z1 ⎪ ⎪ otherwise ⎩γ1w t Ω02 + ΩT2 w (25)
where n = 1, · · · , N , and γ1g , γ1w ≥ 0 are adaptive gains. As will be shown in the next section, the adaptive laws are employed to reduce the ultimate bound of the error signal. Remark 3: According to Lemma 2 given in the next section, subsystem S1 is stablized by α1 even when the adaptive laws are turned-off (γ1g = γ1w = 0). In this case, the controller is reduced to a fixed robust controller. Remark 4: According to the initial setting policy suggested in Remark 2, we have Ω02 (0) = 0, ΩT2 (0) = 0. Therefore, it is recommended to start the adaptive laws after the rise time of the kfilters to avoid numerical problem in w ˙ nt .
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B. Step 2 Through straightforward but tedious calculations based on some equations that appeared so far, we have α˙ 1 = F1 − F2 (ξ2 + Ωg2 g − y˙ r )
−F2 Ω02 + ΩT2 w ∗ + Ωη2 + ε2
(26)
where
F2 = c1p 1 − κ12 (α210 + ν)−0.5 α10 z1 + κ11 g0 (Ω2g2 + ν)0.5 + κ12 (α210 + ν)0.5 F1 = 1 − κ12 (α210 + ν)−0.5 α10 z1 F11 −κ11 g0 (Ω2g2
F11
−0.5
(27)
Ωg2 Ω˙ g2 z1
+ ν) ˙ g2 gt − Ωg2 g˙ t + y¨r = −c1i z1 − ξ˙2 − Ω
Then the second subsystem S2 becomes:
t ) − Ωg2 1 )i2 + F2 (Ωη2 − ΩT2 w g t + ε2 z˙2 = F3 + ϕ(x (28) where
˙ t − k2 Ω01 + ΩT1 w t F3 = ΩT2 w t − F1 g t − y˙ r + Ω02 + ΩT2 w +F2 ξ2 + Ωg2 (29) t = w t − w∗ , w
gt = gt − g
t and gt are adaptive parameters updated at step 2. w Then the control input is designed as α20 − α21 − α22 − α23 i2 = ϕ(x 1) α20 = −c2 z2 − F3 α21 = κ21 |F2 Ωg2 |g0 z2 = α21 z2 α22 = κ22 |F2 Ω02 |z2 = α22 z2 α23 = κ23 |F2 |z2 = α23 z2
˙ w nt
V. S TABILITY
ANALYSIS
For each subsystem, it will be shown that the modeling error terms are counteracted by the nonlinear damping terms which achieve boundedness and guaranteed transient performance of the error signal. Also, it will be shown for each subsystem that the ultimate error bound is reduced by the adaptive laws. A. Subsystem S1 Substituting α1 into subsystem S1, we have t z1 dt − (1 + ∆Ω ) (α11 + α12 ) z1 z˙1 = −c1p z1 − c1i
(30)
0
− gt Ωg2 + ∆Ω α10 + ε2 + z2 (1 + ∆Ω ) (34) To investigate the behaviour of z1 , we first define 2 t 1 1 V1d = z12 + c1i z1 dt 2 2 0
(31)
where c2 , κ21 , κ22 , κ23 > 0. α20 is a feedback linearization controller, α21 , α22 and α23 are nonlinear damping terms to t + Ωη2 )F2 and F2 ε2 respecgt , (−ΩT2 w counteract F2 Ωg2 tively. To furthermore improve the ultimate error bound of z2 , we adopt the following adaptive laws with projection: ⎧ ⎪ for gt = g, F2 Ωg2 z2 < 0 ⎨0 ˙ gt = 0 (32) for gt = g, F2 Ωg2 z2 > 0 ⎪ ⎩ γ2g F2 Ωg2 z2 otherwise ⎧ ⎪ ⎨0 = 0 ⎪ ⎩ γ2w F2 Ω2n z2
where n = 1, · · · , N , and γ2g , γ2w ≥ 0. Remark 5: In [3], [5], the tuning function approach is adopted to avoid the problem of overparametrization in parameter adaption. The idea is to employ some square functions of some internal signals to damp down the timevarying effects of the adaptive parameters. In our study, however, we tend to allow overparametrization to avoid involving these strong damping terms which may lead to large control effort. This is mainly due to that the levitated steel ball is relatively heavy, compared to the power of the current feedback amplifier in our set-up shown in Fig 1. Notice that it has been already pointed out in the literature that overparametrization is not always a bad thing and can be intentionally introduced to achieve similar performance with less control effort. See [1] for this topic.
nt = w , F2 Ω2n z2 < 0 for w n nt = w n , F2 Ω2n z2 > 0 for w otherwise (33)
Then through some calculations, we have c1p 2 V˙ 1d ≤ − z1 − D1 |z1 | |z1 | − µ1 2 where µ1 (t) = µ11 (t) + µ12 (t)|z2 | D1 (t) =
c1p + (1 + ∆Ω )(α11 + α12 ) 2
µ11 (t) =
|− gt Ωg2 | + |∆Ω α10 | + |ε2 | D1 (t)
µ12 (t) =
|1 + ∆Ω | D1 (t)
(35)
(36)
(37)
(38)
From Lemma 1 and equations (20) and (23), we can verify that µ11 and µ12 are uniformly bounded since in each one the denominator grows as the same order as the numerator grows, owing to the nonlinear damping terms. Therefore, if z2 (t) is continuous and uniformly bounded (z2 (t) is stabilied by the
349
controller designed at step 2), µ1 (t) becomes continuous and uniformly bounded. Therefore we have |z1 | ≥ µ1 ⇒ V˙ 1d < 0
|z1 (t)| ≤ λ0 |z1 (0)|e
∃
+ sup µ1 (τ ) 0≤τ ≤t
with respect to the following continuous and uniformly bounded function. µ1 (t) = µ11 (t) + µ12 (t)|z2 | Lemma 2 specifies the transient performance and error bound of position error z1 . The error bound can be made small by increasing the control gains. However, owing to the adaptive laws, the ultimate error bound can be further reduced. Rewriting equation (34), we have t z1 dt − (1 + ∆Ω ) (α11 + α12 ) z1 z˙1 = −c1p z1 − c1i 0
− gt Ωg2 + α10
t + Ωη2 −ΩT2 w + ε2 + z2 (1 + ∆Ω ) t Ω02 + ΩT2 w (40)
To investigate the ultimate bound of z1 which is improved by parameter adaption, we define a Lyapunov function as V1a =
Tw t w 1 2 g2 z1 + t + t , 2 2γ1g 2γ1w
γ1g , γ1w > 0
(41)
Then we have V˙ 1a ≤ − c1p + (1 + ∆Ω )(α11 + α12 ) |z1 |[|z1 | − δ1 ] (42) where δ1 (t) = δ11 (t) + δ12 (t)|z2 | Ωη2 |α10 | + |ε2 | Ω + ΩT w 02 2 t δ11 (t) = c1p + (1 + ∆Ω )(α11 + α12 ) δ12 (t) =
|z1 (t)| ≤ Cδ11 εϕ + Cδ12 |z2 |
(39)
which implies that z1 will enter a compact set. Finally, we have the following lemma. Lemma 2: Let Assumptions 1∼4 hold. If the virtual input α1 and adaptive laws (24) and (25) are applied to subsystem S1, and if z2 is continuous and uniformly bounded, then the error signal z1 is uniformly bounded and for ∃ λ0 , ∃ ρ0 > 0, we have −ρ0 t
S1, and if z2 is continuous and uniformly bounded, then the following result holds. as t → ∞
∃
where Cδ11 > 0, Cδ12 > 0 B. Subsystem S2 Substituting the designed i2 into subsystem S2, we have t + F2 Ωη2 + F2 ε2 g − F2 ΩT w z˙2 = −c2 z2 − F2 Ωg2 t
2
−κ21 |F2 |g0 z2 − κ22 |F2 Ω02 |z2 − κ23 |F2 |z2 (44) Similar to the previous analysis, we have the following results. Lemma 4: Let Assumptions 1∼4 hold. If the control input i2 and adaptive laws (32) and (33) are applied to subsystem S2, then the error signal z2 is uniformly bounded and we have |z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2 (τ ) 0≤τ ≤t
with respect to the following continuous and uniformly bounded function. t + Ωη2 + |ε2 | |F2 | |Ωg2 gt | + −ΩT2 w µ2 (t) = c2 + α21 + α22 + α23 2 Lemma 5: Let Assumptions 1∼4 hold. If the control input i2 and adaptive laws (32) and (33) are applied to subsystem S2, then the following result holds. |z2 (t)| ≤ Cδ2 εϕ
as t → ∞
∃
where Cδ2 > 0 C. The overall error system The overall error system is a cascade of the two subsystems characterized by Lemmas 2 and 4 respectively. Then along the same line of the proof of Lemma C.4 in [3], we have the following results. Theorem 1: Let Assumptions 1∼4 hold. If the initial states and the design parameters are chosen appropriately, all the internal signals are bounded and the following results hold. 1) There exists a compact set D such that [x1 , i2 ]T ∈ D ⊂ F. 2) The error signals are uniformly bounded and the transient performance is guaranteed as |z(t)| ≤ β|z(0)|e−ρt + 2 sup µ11 (τ ) 0≤τ ≤t
(43)
+(2β12 + 1) sup µ2 (τ ) 0≤τ ≤t
|(1 + ∆Ω )| c1p + (1 + ∆Ω )(α11 + α12 )
Based on the results of Lemma 1 and equations (20) and (23), we have Lemma 3: Let Assumptions 1∼4 hold. If the virtual input α1 and adaptive laws (24) and (25) are applied to subsystem
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√ where β = 2max(λ0 , 2β12 + 1), β12 = µ12 ∞ . 3) The ultimate bound of |z(t)| can be made sufficiently small by the adaptive laws such that |z(t)| ≤ Cεϕ ∃
with C > 0
VI. E XPERIMENTAL
RESULTS
Extensive experiments have been performed on the setup shown in Fig. 1. The physically allowable operating region of the steel ball shown in Fig. 1 is limited to 0[m] ≤ x1 ≤ 0.013[m]. To mimic a noisy position sensor, a unifomly distributed stochastic noise between −0.2[mm] and +0.2[mm] was added to the measured position x1 , so that it is unacceptable to calculate the velocity x2 through numericl differentiation. The control algorithm is coded in Borland C++ language and discretized with a sampling interval of T = 0.5[ms]. The following nominal system parameters with considerable errors were used to verify the robust performance of our controllers. M0 = 0.30[kg], g0 = 9.0[m/s2 ] X∞0 = 0.0020[m], Q0 = 0.0003[Hm]
(45)
Five Gaussian basis functions were used and the designed controller parameters are shown as follows. k1 = 200, k2 = 10000 c1p = 40, c1i = 202 , c2 = 40 κ11 = κ12 = 3, κ21 = κ22 = κ23 = 1 γ1g = 2 × 104 , γ1w = 2 × 105 γ2g = 2 × 103 , γ2w = 2 × 103
(46)
x [mm]
15
used for feedback control, it is illustrated here to show how noisy it is. It can be seen that x2 is very noisy due to the high position measurement noise. In spite of the considerable large errors of the physical parameters, the controller achieves boundedness and guaranteed transient performance of the error signals. This is owing to the nonlinear damping terms employed in the controllers. Also, we can find that thanks to the parameter adaption mechanisms, the error signals become smaller gradually. Notice that the results reflect the theoretical results of Theorem 1 quite well, i.e., the adaptive robust output feedback controller achieves guaranteed transient performance and a sufficiently small ultimate error bound. This is quite important for practical use of adaptive controllers, since an adaptive controller without guaranteed transient performance may make the strongly unstable steel ball oscillate roughly and even hit the electromagnet. VII. C ONCLUSIONS In this paper, we proposed an adaptive robust output feedback controller for position-tracking problem of a magnetic levitation system when a noisy position sensor is equipped. The system model is different from the standard output feedback form, since there is an position dependent nonlinear uncertainty directly multiplied by the control input. In this work, the controller was designed by the backstepping procedure with robustifying modification of the k-filter approach. And theoretical analysis results were verified through experimental studies.
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VIII. REFERENCES
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z1[mm]
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z2[m/s]
0.1 0 −0.1
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2 1 0
Fig. 2. Experimental results of the adaptive robust output feedback controller.
A. Comments on the experimental results The results are shown in Fig. 2, where from the top to the bottom are respectively the position x1 , velocity x2 , control input i, error signals of subsystems S1 and S2, i.e., z1 and z2 . The velocity x2 is measured by pseudo-differentiation of the measured position x1 as sx1 /(0.004s + 1). x2 is never
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[1] W. Chen and C. Xu, Adaptive nonlinear control with partial overparametrization, System & control Letters, Vol. 44, 13/24, 2001. [2] S. A. Green and K. C. Craig, Robust, digital, nonlinear control of magnetic-levitation systems, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 120, 488/495, 1998. [3] M. L. Krstic, Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995. [4] R. Milman and S. A. Bortoff, Observer-based adaptive control of a variable reluctance motor: experimental results, IEEE. Trans. Control Systems Technology, Vol. 7, 613/621,1999. [5] L. Xu and B. Yao, Output feedback adaptive robust precision motion control of linear motors, Automatica, Vol. 7, 1029/1039, 2001. [6] Z. J. Yang and M. Tateishi, Adaptive robust nonlinear control of a magnetic levitation system, Automatica, Vol. 7, 1125/1131, 2001.
Adaptive Robust Output Feedback Control of a Magnetic Levitation System by K-Filter Approach Zi-Jiang Yang, Kazuhiro Kunitoshi, Shunshoku Kanae, Kiyoshi Wada Department of Electrical and Electronic System Engineering Kyushu University, Fukuoka, 812-8581 JAPAN Tel: +81-92-642-3958; Fax: +81-92-642-3904 [email protected]
Abstract— This paper proposes an adaptive robust output feedback controller for position-tracking problem of a magnetic levitation system with a current feedback power amplifier. The controller is designed by a backstepping procedure with robustifying modification of the k-filter approach. The boundedness and guaranteed transient performance of the error signals are achieved by the nonlinear damping terms. And the ultimate position-tracking error is reduced by the adaptive laws. Experimental results are included to show the excellent control performance of the designed controller.
I. I NTRODUCTION Backstepping output feedback controllers are usually designed for output feedback systems, where the nonlinear terms are functions of the output and the models are in the strict feedback form. The k-filter approach is one of the typical approaches to construct the state variable observer. So far, several applications have been reported in the literature [2], [4], [5]. In our system under study, however, as described in the next section, the system model is different from the standard output feedback form, since there is an position dependent nonlinear uncertainty directly multiplied by the control input. In this work, the controller is designed by the backstepping procedure with robustifying modification of the k-filter approach. The boundedness and guaranteed transient performance of the error signals are achieved by the nonlinear damping terms. And the ultimate position-tracking error is reduced by the adaptive laws. Finally, experimental results are included to show the excellent control performance of the designed controller. II. M ODEL OF THE MAGNETIC LEVITATION SYSTEM Consider the magnetic levitation system shown in Fig. 1. If the coil current is controlled by a current feedback power amplifier as in most applications, then the system dynamics can be described in the following equations [2], [6]. x˙ 1 = x2 x˙ 2 = g + ϕ(x1 )i2 ϕ(x1 ) =
−Q 2M (X∞ + x1 )2
Diagram of the magnetic levitation system.
Our task in this study is to design an efficient and reliable output feedback controller in the case where only the position is measureable and the physical parameters are uncertain. Denoting the nominal physical parameters as g0 , M0 , Q0 and X∞0 , we have the nominal nonlinear function and its modeling error respectively as the following. ϕ0 (x1 ) = −
Q0 2M0 (X∞0 + x1 )2
(3)
∆ϕ (x1 ) = ϕ(x1 ) − ϕ0 (x1 )
(4)
Then we approximate the modeling error ∆ϕ (x1 ) by a radial basis function (RBF) network. ϕ (x1 , w t) ϕ(x 1 ) = ϕ0 (x1 ) + ∆ ϕ (x1 , w) = ∆
N
wn r (x1 − pn ) = RT w
(5)
(6)
n=1
(1)
(2)
where, x1 : air gap (vertical position) of the steel ball; x2 : velocity of the steel ball; i: coil current; g: gravity acceleration; M : mass of the steel ball; Q and X∞ : positive constants determined by the characteristics of the coil, magnetic core and steel ball.
0-7803-8635-3/04/$20.00 ©2004 IEEE
Fig. 1.
RT = [r(x1 − p1 ), r(x1 − p2 ), · · · , r(x1 − pN )] w = [w1 , w2 , · · · , wN ]T where
1 r (x1 − pn ) = exp − 2
x1 − pn σ
(7)
2 (8)
is a Gaussian basis function, in which pn is the center of the nth basis function, and σ determines its width. In this study,
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the basis functions are equidistantly located in X = {x1 | 0 ≤ x1 ≤ x1M }, i.e., the physically allowable operating region of x1 (see Fig. 1). Some assumptions are made here. Assumption 1: A sufficient number of basis functions are included into the RBF network, so that there exists a desired w∗ such that sup |ηt | ≤ εϕ
x1 ∈X
(9)
where εϕ is a sufficiently small positive real number, and ϕ (x1 , w∗ ) ηt = ∆ϕ (x1 ) − ∆
(10)
Assumption 2: The lower and upper bounds of the unknown parameters are known a priori, i.e., w ≤ w ≤ w,
0≤g≤g
(11)
Notice that an external constant (or slowly varying) disturbance can also be included into g equivalently, i.e., the gravity acceleration is biased equivalently. Assumption 3: The parameter bounds are chosen such that ϕ (x1 , w) < 0 ϕ(x 1 ) ≤ ϕ0 (x1 ) + ∆
OF THE OBSERVER BY
where x = [x1 , x2 ]T , e2 = [0, 1]T , and k = [k1 , k2 ]T , such that
−k1 1 A0 = −k2 0 = x−x . Then in is Hurwitz. Define the observer error x the ideal case we have (15)
However, since the parameters are unknown, we have to adopt a new state estimation following the K-filter approach lectured in [3]. (16)
(17)
Notice ΩT = [Ω1 , Ω2 ]T ∈ R2×N , Ω1 , Ω2 ∈ RN ×1 , ξ, Ωg , Ω0 , Ωη ∈ R2×1 . In the sequel, the ith (i = 1, 2) elements of ξ, Ωg , Ω0 , Ωη , and the nth (n = 1, · · · , N ) elements of Ω1 , Ω2 will be denoted as ξi , Ωgi , Ω0i , Ωηi and Ω1n , Ω2n respectively. Remark 1: The last line of equation (17) related to the modeling error ηt i2 is never implemented in practice. It is written explicitly here only for covenience of theoretical analysis in the sequel. Remark 2: It is recommended to set the initial states of the filters as ξ(0) = x(0), ΩT (0) = 0 Ωg (0) = Ω0 (0) = Ωη (0) = 0
(18)
(0) = x(0), to avoid large transient error. such that x = [ε1 , ε2 ]T is readily The state estimation error ε = x − x shown to satisfy ε˙ = A0 ε
(19)
Notice that ε decays exponentially to zero. Then we have x = ξ + Ωg g + ΩT w∗ + Ω0 + Ωη + ε
Since only the position x1 is available for measurement, a nonlinear observer is first constructed here. If the parameters were known, we would design the observer
˙ = A0 x + kx1 + e2 g + ϕ0 (x1 )i2 + RT w ∗ i2 + ηt i2 x (14)
= ξ + Ωg g + Ω T w ∗ + Ω0 + Ωη x
˙ T = A0 ΩT + e2 RT i2 Ω ˙ 0 = A0 Ω0 + e2 ϕ0 (x1 )i2 Ω ˙ η = A0 Ωη + e2 ηt i2 Ω
K- FILTER
APPROACH
˙ = A0 x x
ξ˙ = A0 ξ + kx1 ˙ Ωg = A0 Ωg + e2
(12)
Assumption 4: The reference trajectory yr and it’s first and second derivatives y˙ r , y¨r are continuous, uniformly bounded, and available such that there exists a compact set Dyr = yr , y˙ r , y¨r 0 < yr < x1M , |y˙ r | ≤ y˙ r , |¨ yr | ≤ y¨r , ∃ ∃ 3 y˙ r , y¨r > 0 ⊂ R (13) III. C ONSTRUCTION
where
(20)
This equation implies that our problem is different from the standard problem setting of the k-filter approach in [3]. Since there exists the term Ωη driven by the position and input dependent modeling error ηt i2 . Therefore, extra efforts are required to construct an efficient and reliable output feedback controller. For theoretical preparation of the controller design given in the next section, we will give here some results regarding the effects of Ωη . Let h(t) is the impulse response of s + k1 H(s) = 2 s + k1 s + k2 Then we have Lemma 1: Let Assumptions 1∼3 hold, i2 > 0, and x ∈ X . If k1 and k2 are chosen such that h(t) ≥ 0, and the initial conditions are chosen such that for t ≥ 0: eT2 eA0 t Ω0 (0) + ΩT (0)w ∗ + Ωη (0) ≤ 0 t ≤ 0 eT2 eA0 t Ω0 (0) + ΩT (0)w then it can be made that the following results hold for t > 0:
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1)
A. Step 1 Define the error signals as
Ω02 + ΩT2 w∗ + Ωη2 < 0
t < 0, Ω02 + ΩT2 w
z1 = x1 − yr ,
such that
z˙1 = ξ2 − gt Ωg2 + Ωg2 gt + ε2 − y˙ r + (α1 + z2 ) (1 + ∆Ω ) (22)
t − w ∗ , and t = w where w t + Ωη2 −ΩT2 w t Ω02 + ΩT2 w
where gt = gt − g. Then we can design the virtual input α1 as follows: α1 = α10 − α11 − α12 t α10 = −c1p z1 − c1i z1 dt − ξ2 − Ωg2 gt + y˙ r 0 α11 = κ11 g0 Ω2g2 + νz1 = α11 z1 α12 = κ12 α210 + νz1 = α12 z1
2) ∆Ω −ΩT2 w + Ω t η2 1 + ∆Ω = Ω + ΩT w ∗ + Ω < ∞ 02 η2 2 3) Ωη2 0, as t → ∞
5) −ΩT w 2 t + Ωη2 0, as t → ∞
The results of boundedness of some terms listed in Lemma 1 are derived based on the fact that in each one the denominator grows as the same order as the numerator grows. These resutls will be used for design of the nonlinear damping terms and stability analysis in the following sections. IV. C ONTROLLER
(23)
where c1p , c1i , κ11 , κ12 > 0, ν = 0.01. α10 is a feedback linearization PI controller, α11 and α12 are nonlinear damping terms to counteract gt Ωg2 and ∆Ω respectively. Notice that the integral control action helps to remove the steady offset of z1 , especially in the case where the controller is fixed without parameter adaption. It will be shown in the next section that these nonlinear damping terms achieve boundedness and guaranteed transient performance of error signal z1 . The unknown parameters are updated by the following adaptive laws with projection: ⎧ ⎪ for gt = g, Ωg2 z1 < 0 ⎨0 g˙ t = 0 (24) for gt = g, Ωg2 z1 > 0 ⎪ ⎩ γ1g Ωg2 z1 otherwise
4) Ωη2 ≤ C1 εϕ , T ∗ Ω02 + Ω2 w + Ωη2
(21)
where α1 is the virtual input to stabilize z1 . From equation (20), we have subsystem S1:
Ω02 + ΩT2 w ∗ + Ωη2 = 1 + ∆Ω > 0 t Ω02 + ΩT2 w
∆Ω =
t − α1 z2 = Ω02 + ΩT2 w
DESIGN
The controller is designed through the backstepping procedure. At each step, the modeling errors are counteracted by the nonlinear damping terms which achieve boundedness and guaranteed transient performance of the error signal. Additionally, adaptive laws are introduced to achieve a small ultimate error bound.
⎧ α10 Ω2n z1 ⎪ ⎪0 for w nt = wn , 0 = 0 T ⎪ t Ω 02 + Ω2 w ⎪ ⎪ ⎪ α10 Ω2n z1 ⎪ ⎪ otherwise ⎩γ1w t Ω02 + ΩT2 w (25)
where n = 1, · · · , N , and γ1g , γ1w ≥ 0 are adaptive gains. As will be shown in the next section, the adaptive laws are employed to reduce the ultimate bound of the error signal. Remark 3: According to Lemma 2 given in the next section, subsystem S1 is stablized by α1 even when the adaptive laws are turned-off (γ1g = γ1w = 0). In this case, the controller is reduced to a fixed robust controller. Remark 4: According to the initial setting policy suggested in Remark 2, we have Ω02 (0) = 0, ΩT2 (0) = 0. Therefore, it is recommended to start the adaptive laws after the rise time of the kfilters to avoid numerical problem in w ˙ nt .
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B. Step 2 Through straightforward but tedious calculations based on some equations that appeared so far, we have α˙ 1 = F1 − F2 (ξ2 + Ωg2 g − y˙ r )
−F2 Ω02 + ΩT2 w ∗ + Ωη2 + ε2
(26)
where
F2 = c1p 1 − κ12 (α210 + ν)−0.5 α10 z1 + κ11 g0 (Ω2g2 + ν)0.5 + κ12 (α210 + ν)0.5 F1 = 1 − κ12 (α210 + ν)−0.5 α10 z1 F11 −κ11 g0 (Ω2g2
F11
−0.5
(27)
Ωg2 Ω˙ g2 z1
+ ν) ˙ g2 gt − Ωg2 g˙ t + y¨r = −c1i z1 − ξ˙2 − Ω
Then the second subsystem S2 becomes:
t ) − Ωg2 1 )i2 + F2 (Ωη2 − ΩT2 w g t + ε2 z˙2 = F3 + ϕ(x (28) where
˙ t − k2 Ω01 + ΩT1 w t F3 = ΩT2 w t − F1 g t − y˙ r + Ω02 + ΩT2 w +F2 ξ2 + Ωg2 (29) t = w t − w∗ , w
gt = gt − g
t and gt are adaptive parameters updated at step 2. w Then the control input is designed as α20 − α21 − α22 − α23 i2 = ϕ(x 1) α20 = −c2 z2 − F3 α21 = κ21 |F2 Ωg2 |g0 z2 = α21 z2 α22 = κ22 |F2 Ω02 |z2 = α22 z2 α23 = κ23 |F2 |z2 = α23 z2
˙ w nt
V. S TABILITY
ANALYSIS
For each subsystem, it will be shown that the modeling error terms are counteracted by the nonlinear damping terms which achieve boundedness and guaranteed transient performance of the error signal. Also, it will be shown for each subsystem that the ultimate error bound is reduced by the adaptive laws. A. Subsystem S1 Substituting α1 into subsystem S1, we have t z1 dt − (1 + ∆Ω ) (α11 + α12 ) z1 z˙1 = −c1p z1 − c1i
(30)
0
− gt Ωg2 + ∆Ω α10 + ε2 + z2 (1 + ∆Ω ) (34) To investigate the behaviour of z1 , we first define 2 t 1 1 V1d = z12 + c1i z1 dt 2 2 0
(31)
where c2 , κ21 , κ22 , κ23 > 0. α20 is a feedback linearization controller, α21 , α22 and α23 are nonlinear damping terms to t + Ωη2 )F2 and F2 ε2 respecgt , (−ΩT2 w counteract F2 Ωg2 tively. To furthermore improve the ultimate error bound of z2 , we adopt the following adaptive laws with projection: ⎧ ⎪ for gt = g, F2 Ωg2 z2 < 0 ⎨0 ˙ gt = 0 (32) for gt = g, F2 Ωg2 z2 > 0 ⎪ ⎩ γ2g F2 Ωg2 z2 otherwise ⎧ ⎪ ⎨0 = 0 ⎪ ⎩ γ2w F2 Ω2n z2
where n = 1, · · · , N , and γ2g , γ2w ≥ 0. Remark 5: In [3], [5], the tuning function approach is adopted to avoid the problem of overparametrization in parameter adaption. The idea is to employ some square functions of some internal signals to damp down the timevarying effects of the adaptive parameters. In our study, however, we tend to allow overparametrization to avoid involving these strong damping terms which may lead to large control effort. This is mainly due to that the levitated steel ball is relatively heavy, compared to the power of the current feedback amplifier in our set-up shown in Fig 1. Notice that it has been already pointed out in the literature that overparametrization is not always a bad thing and can be intentionally introduced to achieve similar performance with less control effort. See [1] for this topic.
nt = w , F2 Ω2n z2 < 0 for w n nt = w n , F2 Ω2n z2 > 0 for w otherwise (33)
Then through some calculations, we have c1p 2 V˙ 1d ≤ − z1 − D1 |z1 | |z1 | − µ1 2 where µ1 (t) = µ11 (t) + µ12 (t)|z2 | D1 (t) =
c1p + (1 + ∆Ω )(α11 + α12 ) 2
µ11 (t) =
|− gt Ωg2 | + |∆Ω α10 | + |ε2 | D1 (t)
µ12 (t) =
|1 + ∆Ω | D1 (t)
(35)
(36)
(37)
(38)
From Lemma 1 and equations (20) and (23), we can verify that µ11 and µ12 are uniformly bounded since in each one the denominator grows as the same order as the numerator grows, owing to the nonlinear damping terms. Therefore, if z2 (t) is continuous and uniformly bounded (z2 (t) is stabilied by the
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controller designed at step 2), µ1 (t) becomes continuous and uniformly bounded. Therefore we have |z1 | ≥ µ1 ⇒ V˙ 1d < 0
|z1 (t)| ≤ λ0 |z1 (0)|e
∃
+ sup µ1 (τ ) 0≤τ ≤t
with respect to the following continuous and uniformly bounded function. µ1 (t) = µ11 (t) + µ12 (t)|z2 | Lemma 2 specifies the transient performance and error bound of position error z1 . The error bound can be made small by increasing the control gains. However, owing to the adaptive laws, the ultimate error bound can be further reduced. Rewriting equation (34), we have t z1 dt − (1 + ∆Ω ) (α11 + α12 ) z1 z˙1 = −c1p z1 − c1i 0
− gt Ωg2 + α10
t + Ωη2 −ΩT2 w + ε2 + z2 (1 + ∆Ω ) t Ω02 + ΩT2 w (40)
To investigate the ultimate bound of z1 which is improved by parameter adaption, we define a Lyapunov function as V1a =
Tw t w 1 2 g2 z1 + t + t , 2 2γ1g 2γ1w
γ1g , γ1w > 0
(41)
Then we have V˙ 1a ≤ − c1p + (1 + ∆Ω )(α11 + α12 ) |z1 |[|z1 | − δ1 ] (42) where δ1 (t) = δ11 (t) + δ12 (t)|z2 | Ωη2 |α10 | + |ε2 | Ω + ΩT w 02 2 t δ11 (t) = c1p + (1 + ∆Ω )(α11 + α12 ) δ12 (t) =
|z1 (t)| ≤ Cδ11 εϕ + Cδ12 |z2 |
(39)
which implies that z1 will enter a compact set. Finally, we have the following lemma. Lemma 2: Let Assumptions 1∼4 hold. If the virtual input α1 and adaptive laws (24) and (25) are applied to subsystem S1, and if z2 is continuous and uniformly bounded, then the error signal z1 is uniformly bounded and for ∃ λ0 , ∃ ρ0 > 0, we have −ρ0 t
S1, and if z2 is continuous and uniformly bounded, then the following result holds. as t → ∞
∃
where Cδ11 > 0, Cδ12 > 0 B. Subsystem S2 Substituting the designed i2 into subsystem S2, we have t + F2 Ωη2 + F2 ε2 g − F2 ΩT w z˙2 = −c2 z2 − F2 Ωg2 t
2
−κ21 |F2 |g0 z2 − κ22 |F2 Ω02 |z2 − κ23 |F2 |z2 (44) Similar to the previous analysis, we have the following results. Lemma 4: Let Assumptions 1∼4 hold. If the control input i2 and adaptive laws (32) and (33) are applied to subsystem S2, then the error signal z2 is uniformly bounded and we have |z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2 (τ ) 0≤τ ≤t
with respect to the following continuous and uniformly bounded function. t + Ωη2 + |ε2 | |F2 | |Ωg2 gt | + −ΩT2 w µ2 (t) = c2 + α21 + α22 + α23 2 Lemma 5: Let Assumptions 1∼4 hold. If the control input i2 and adaptive laws (32) and (33) are applied to subsystem S2, then the following result holds. |z2 (t)| ≤ Cδ2 εϕ
as t → ∞
∃
where Cδ2 > 0 C. The overall error system The overall error system is a cascade of the two subsystems characterized by Lemmas 2 and 4 respectively. Then along the same line of the proof of Lemma C.4 in [3], we have the following results. Theorem 1: Let Assumptions 1∼4 hold. If the initial states and the design parameters are chosen appropriately, all the internal signals are bounded and the following results hold. 1) There exists a compact set D such that [x1 , i2 ]T ∈ D ⊂ F. 2) The error signals are uniformly bounded and the transient performance is guaranteed as |z(t)| ≤ β|z(0)|e−ρt + 2 sup µ11 (τ ) 0≤τ ≤t
(43)
+(2β12 + 1) sup µ2 (τ ) 0≤τ ≤t
|(1 + ∆Ω )| c1p + (1 + ∆Ω )(α11 + α12 )
Based on the results of Lemma 1 and equations (20) and (23), we have Lemma 3: Let Assumptions 1∼4 hold. If the virtual input α1 and adaptive laws (24) and (25) are applied to subsystem
350
√ where β = 2max(λ0 , 2β12 + 1), β12 = µ12 ∞ . 3) The ultimate bound of |z(t)| can be made sufficiently small by the adaptive laws such that |z(t)| ≤ Cεϕ ∃
with C > 0
VI. E XPERIMENTAL
RESULTS
Extensive experiments have been performed on the setup shown in Fig. 1. The physically allowable operating region of the steel ball shown in Fig. 1 is limited to 0[m] ≤ x1 ≤ 0.013[m]. To mimic a noisy position sensor, a unifomly distributed stochastic noise between −0.2[mm] and +0.2[mm] was added to the measured position x1 , so that it is unacceptable to calculate the velocity x2 through numericl differentiation. The control algorithm is coded in Borland C++ language and discretized with a sampling interval of T = 0.5[ms]. The following nominal system parameters with considerable errors were used to verify the robust performance of our controllers. M0 = 0.30[kg], g0 = 9.0[m/s2 ] X∞0 = 0.0020[m], Q0 = 0.0003[Hm]
(45)
Five Gaussian basis functions were used and the designed controller parameters are shown as follows. k1 = 200, k2 = 10000 c1p = 40, c1i = 202 , c2 = 40 κ11 = κ12 = 3, κ21 = κ22 = κ23 = 1 γ1g = 2 × 104 , γ1w = 2 × 105 γ2g = 2 × 103 , γ2w = 2 × 103
(46)
x [mm]
15
used for feedback control, it is illustrated here to show how noisy it is. It can be seen that x2 is very noisy due to the high position measurement noise. In spite of the considerable large errors of the physical parameters, the controller achieves boundedness and guaranteed transient performance of the error signals. This is owing to the nonlinear damping terms employed in the controllers. Also, we can find that thanks to the parameter adaption mechanisms, the error signals become smaller gradually. Notice that the results reflect the theoretical results of Theorem 1 quite well, i.e., the adaptive robust output feedback controller achieves guaranteed transient performance and a sufficiently small ultimate error bound. This is quite important for practical use of adaptive controllers, since an adaptive controller without guaranteed transient performance may make the strongly unstable steel ball oscillate roughly and even hit the electromagnet. VII. C ONCLUSIONS In this paper, we proposed an adaptive robust output feedback controller for position-tracking problem of a magnetic levitation system when a noisy position sensor is equipped. The system model is different from the standard output feedback form, since there is an position dependent nonlinear uncertainty directly multiplied by the control input. In this work, the controller was designed by the backstepping procedure with robustifying modification of the k-filter approach. And theoretical analysis results were verified through experimental studies.
1
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VIII. REFERENCES
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z1[mm]
2 0 −2
z2[m/s]
0.1 0 −0.1
i[A]
2 1 0
Fig. 2. Experimental results of the adaptive robust output feedback controller.
A. Comments on the experimental results The results are shown in Fig. 2, where from the top to the bottom are respectively the position x1 , velocity x2 , control input i, error signals of subsystems S1 and S2, i.e., z1 and z2 . The velocity x2 is measured by pseudo-differentiation of the measured position x1 as sx1 /(0.004s + 1). x2 is never
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[1] W. Chen and C. Xu, Adaptive nonlinear control with partial overparametrization, System & control Letters, Vol. 44, 13/24, 2001. [2] S. A. Green and K. C. Craig, Robust, digital, nonlinear control of magnetic-levitation systems, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 120, 488/495, 1998. [3] M. L. Krstic, Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995. [4] R. Milman and S. A. Bortoff, Observer-based adaptive control of a variable reluctance motor: experimental results, IEEE. Trans. Control Systems Technology, Vol. 7, 613/621,1999. [5] L. Xu and B. Yao, Output feedback adaptive robust precision motion control of linear motors, Automatica, Vol. 7, 1029/1039, 2001. [6] Z. J. Yang and M. Tateishi, Adaptive robust nonlinear control of a magnetic levitation system, Automatica, Vol. 7, 1125/1131, 2001.
