FLATNESS-BASED FEEDFORWARD AND ...

FLATNESS-BASED FEEDFORWARD AND FEEDBACK LINEARISATION OF THE BALL & PLATE LAB EXPERIMENT Veit Hagenmeyer ∗,1 Stefan Streif ∗∗ Michael Zeitz ∗∗ ∗

BASF AG Ludwigshafen, Germany [email protected] ∗∗ Institut f¨ ur Systemdynamik und Regelungstechnik, Universit¨ at Stuttgart, Germany {streif,zeitz}@isr.uni-stuttgart.de

Abstract: In this contribution, the implementation of two different flatness-based control laws to a ball & plate lab experiment is presented. Experimental results of both the exact feedback linearisation and the exact feedforward linearisation method are shown and compared with respect to their different robustness properties. Keywords: Ball and plate, ball and beam, differential flatness, exact feedback linearisation, exact feedforward linearisation, nonlinear tracking control

1. INTRODUCTION The ball & plate experiment is a nonlinear benchmark system which contains the well-known ball & beam experiment (Hauser et al., 1992) as a special case. It has the advantage that the tracking performances of different control laws are directly visualised by the movement of the ball on the plate. Moreover, the ball & plate MIMO system can be modelled as two independent ball & beam SISO systems. This allows the development of two decoupled control laws for each axis. An experimental setup for the ball & plate system was constructed in the student lab (Marquardt, 1995; Mayer, 1997). Thereafter, a flatness-based control law was developed and implemented using exact feedback linearisation (Meurer et al., 1999). In order to emphasise that flatness is not restricted to exact feedback linearisation control laws, a new flatness-based linearisation method called exact feedforward was developed in (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 1

V. Hagenmeyer was at the Universit¨ at Stuttgart when the article was written.

2003a). Furthermore, it was claimed that exact feedforward linearisation is more robust than exact feedback linearisation (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 2003b). The aim of this contribution is to implement both linearisation methods to the ball & plate system in order to compare their performance and robustness properties. Experimental results are shown which allows a comparison of both methods. The paper is structured as follows: In Sec. 2, a brief introduction to differential flatness is given. Then, the exact feedback and the exact feedforward linearisation control laws are presented. Sec. 4 deals with the setup of the ball & plate experiment and its modelling. A flatness-based analysis of the model is carried out in Sec. 5 and the two linearising control laws are derived in Sec. 6. The experimental results are shown in Sec. 7 and Sec. 8.

2. DIFFERENTIAL FLATNESS Differential flatness is a structural property of a class of nonlinear systems, for which, roughly

speaking, all system variables can be written in terms of a set of specific variables (the so-called flat outputs) and their derivatives. For sake of simplicity, only flat SISO systems ˙ x(t) = f (x(t), u(t)), x(0) = x0

(1)

with time t ∈ R, the state x ∈ Rn , the input u ∈ R are considered. The vector field f : Rn × R → Rn is considered to be smooth. The nonlinear system (1) is said to be (differentially) flat (Fliess et al., 1995; Fliess et al., 1999) iff there exists a flat output z ∈ R, such that z = F (x), x = φ(z, z, ˙ ...,z

(2) (n−1)

),

u = ψ(z, z, ˙ . . . , z (n) ),

(3)

z˙n (t) = α(z(t), u(t)), z(0) = F(x0 )

(8)

where α is also smooth with respect to its arguments and ∂α/∂u = 0.

3.1 Exact feedback linearisation Exact feedback linearisation is a well-known control design method (Nijmeijer and van der Schaft, 1990; Isidori, 1995), therefore the resulting control law is only briefly presented: Solving α(z, u) = v for u (cf. (8)) results in u = ψ(z, v). The new input v is designed as

(4)

where F , φ and ψ are smooth functions at least in an open set of R, Rn and Rn+1 . For nonlinear SISO systems the flatness property can be verified by the results of (Jakubczyk and Respondek, 1980; Charlet et al., 1989), since exact feedback linearisability is a necessary and sufficient condition for flatness of systems (Fliess et al., 1995). By virtue of the equations (3) and (4) for every given trajectory of the flat output t → z(t), the evolution of all other variables of the system t → x(t) and t → u(t) are given without integration of any differential equation. Moreover, for a sufficiently smooth desired trajectory of the flat output t → z ∗ (t) ∈ C n , eqn. (4) can be used to design the corresponding feedforward u∗ (t) directly. The trajectory u∗ (t) is called the nominal control. The family of nominal feedforwards is given by u∗ (t) = ψ(z ∗ (t), z˙ ∗ (t), . . . , z ∗(n) (t))

z˙i (t) = zi+1 (t), i ∈ {1, . . . , n − 1}

(5)

v = z˙n∗ + Λ(¯ e)

(9)

¯ where the augmented tracking error e [e0 , e1 , e2 , . . . , en ]T are defined by  t e1 (τ )dτ , e0 =

=

0

ei = zi − zi∗ , i ∈ {1, . . . , n} . n−1

The PID

(10)

feedback part is Λ(¯ e) =

n 

pi ei

(11)

i=0

where the coefficients pi , i = 1, 2, . . . , n are chosen such that the resulting characteristic polynomial is Hurwitz. Thus, the whole control structure can be denoted by  uF B = ψ z, z˙n∗ (t) +

n 

 pi ei .

(12)

i=0

that is, for each admissible trajectory z ∗ (t) ∈ C n , there is a nominal feedforward u∗ ∈ C 0 . 3.2 Exact feedforward linearisation 3. FLATNESS BASED LINEARISATION METHODS Taking into account the results of (Jakubczyk and Respondek, 1980; Charlet et al., 1989), it is easy to show, that every flat SISO system can be represented as follows. Setting z = [z, z, ˙ . . . , z (n−1) ]T = [z1 , z2 , . . . , zn ]T

(6)

the system (1) can be transformed via the well defined diffeomorphism z = F(x)

(7)

where F = φ−1 (cf. (3)), into the control normal form

In (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 2003a) it is proven, that the nominal feedforward control (5) linearises the system (1) under the knowledge of its initial condition. Therefore, the method is called exact feedforward linearisation based on differential flatness. The resulting control law consists of two parts, a feedforward part (5), and a PID-like feedback part that takes the tracking error into account. The structure of the combination of both parts is presented in the following. In (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 2003a) it is shown, that z˙n∗ plays the role of an input to a certain Brunovsk´ y form when being on the desired trajectory. Thus the nominal feedforward control (5) is combined with the same PIDn−1 feedback part as in (11) in the following

way 2 . Define the new input v like in (9) and the whole control structure can be denoted by  uF F = ψ z∗ (t), z˙n∗ (t) +

n 

 pi ei

.

(13)

i=0

This structure consists of a specific combination of a nonlinear feedforward part based on differential flatness, and a simple linear feedback part of extended PIDn−1 type. Remark that this control structure represents a truly nonlinear control. It differs from exact feedback linearisation (12) in the way, that where in (12) the state z is used, in (13) the desired trajectory z∗ (t) is applied. In (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 2003a), several structural properties of the application of (13) to (1) are discussed in detail. One of the main results consists of a stability proof for the application of (13) to (1) under the conditions that both the initial error and the velocity of the desired trajectory are bounded. In the following, the two different flatness-based control designs are applied to the ball & plate experiment. A comparison of the results is given thereafter. 4. BALL & PLATE LAB EXPERIMENT The experimental setup of the ball & plate laboratory system is depicted in Figure 1 for one plate axis as in (Meurer et al., 1999). The angular positions (φy , φx ) of the plate are directly given by two stepping motors which are displaced by 90◦ . The attached image processor (cf. also (Marquardt, 1995)) seizes the position of the ball (x, y) with an accurancy of ±2 mm and a sample time of 40 ms from a CCD-camera. The MIMO model of the ball & plate system incorporates the following states: the position of the ball (x1 , x5 ) := (x, y), the velocity of the ball (x2 , x6 ) := (x, ˙ y), ˙ the angle of the plate (x3 , x7 ) := (φx , φy ), and the states (x4 , x8 ). The states x4 and x8 correspond to integrator extensions (Mayer, 1997) at the respective stepping motors. They are required in order to counteract the offset and interference effects.

making certain assumptions and simplifications. For a detailed derivation of the equations 3 see (Streif, 2003). The stepping motor dynamics is in good approximation described as a linear first order system with the time constant T (Mayer, 1997). The dynamics of the ball & plate system is then modelled as ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ x2 0 x˙ 1 ⎢x˙ 2 ⎥ ⎢− 1 Bg sin (2x3 )⎥ ⎢x (x, u¯x )⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21 ⎥ ⎢ ⎥ ⎢x˙ 3 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ T (x4 − x3 ) ⎥ ⎢ ⎢ ⎥ ⎥ ⎢x˙ 4 ⎥ ⎢ u ¯ 0 x ⎥+⎢ ⎥ (14) ⎢ ⎥=⎢ ⎢ ⎥ ⎥ ⎢x˙ 5 ⎥ ⎢ x 0 6 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎢x˙ 6 ⎥ ⎢− Bg sin (2x7 )⎥ ⎢y (x, u¯x )⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21 ⎦ ⎣ ⎦ ⎣x˙ 7 ⎦ ⎣ 0 T (x8 − x7 ) 0 x˙ 8 u ¯y with 2 2 x1 (x3 − x4 ) 7T 2

1 2 2 − 2 x1 (x3 − x4 ) −1 T cos2 x3

1 1 − (x3 − x4 ) x1 − 2x2 tan x3 T T 1 − x1 u¯x tan x3 T

x (x, u¯x ) = −

and 2 2 x5 (x7 − x8 ) 7T 2

1 2 2 − 2 x5 (x7 − x8 ) − 1 T cos2 x7

1 1 − (x7 − x8 ) x5 − 2x6 tan x7 T T 1 − x5 u ¯y tan x7 , T B = 57 , T = 0.15s, and the gravitational acceleration g = 9.81 sm2 . ¯y ) = − y (x, u

The simplified model of the ball & plate system (14) is a decoupled and symmetric description 3 Note that (Meurer et al., 1999) modelled the ball & plate system considering a slipping body. Here a rolling ball is considered in order to stay close to (Hauser et al., 1992).

Ball position (x, y)

g

4.1 Model of the ball & plate system

Ball

The equations of motion of the ball can be derived using the Lagrangian equations 2. type and by x

φx 2 Note that in (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 2003a) partial state feedback using a PIDk 0 ≤ k ≤ n − 1 feedback part has been proven to be possible. That is, depending on the system, using simple PID control combined with a flatness-based feedforward can lead to good tracking performances.

Control Stepping motor interface for each axis

Plate Mechanical setup and sensing of ball position

Image processing and control computer

Fig. 1. Experimental setup of the ball & plate lab system (Meurer, 1999).

for the two subsystems with states {x1 , x2 , x3 , x4 } ¯y plus input u¯x and {x5 , x6 , x7 , x8 } plus input u respectively.

4.2 Model of the ball & beam system In order to use the experimental ball & plate setup for the ball & beam experiments, the ball & plate setup is modified by using a track. This track forces the ball to roll only in x-direction and fixes thus the y-coordinate of the ball. Furthermore, the corresponding stepping motor is disabled which fixes also the φy angular position of the plate. This leads to the description ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ x2 0 x˙ 1 ⎢x˙ 2 ⎥ ⎢− 1 Bg sin (2x3 )⎥ ⎢x (x, u¯x )⎥ ⎥+⎢ ⎢ ⎥ = ⎢ 21 ⎥ (15) ⎦ ⎣ ⎣x˙ 3 ⎦ ⎣ ⎦ 0 T (x4 − x3 ) x˙ 4 0 u ¯x with 2 2 x1 (x3 − x4 ) 7T 2

1 2 − 2 x1 (x3 − x4 )2 − 1 T cos2 x3

1 1 − (x3 − x4 ) x1 − 2x2 tan x3 T T 1 − x1 u¯x tan x3 . T

¯x ) = − x (x, u

5. FLATNESS ANALYSIS OF THE MODEL The model of the ball & plate system consists of two independent subsystems for each axis. For the sake of readability, the flatness-based controller design is described in the following for the ball & beam experiment (15) only. The controller design for the ball & plate experiment is thereafter undertaken in the same way for each subsystem.

leads to (cf. eq. (8)) 4 z˙i = zi+1 , i ∈ {1, . . . , 3}, (18)    2 1 4T z3 z4 2 2 u ¯x (Bg) − 4z3 + z4 + z˙4 = − 2 T (Bg) − 4z32 The parameterisation of the state (3) are then given by ⎤ ⎡ ⎤ ⎡ z1 x1 ⎥ z2 ⎢x2 ⎥ ⎢ ⎥ ⎢ ⎥=⎢ 2z3 1 ⎥ (19) ⎢ − 2 arcsin Bg ⎣x3 ⎦ ⎣ ⎦ 2z3 T z 1 4 − 2 arcsin Bg − √ x4 2 2 (Bg) −4z3

and the parameterisation of the input (4) is   4z3 z4 2 −T T1 z4 + (Bg) ˙4 2 −4z 2 + z 3  u¯x = . (20) (Bg)2 − 4z32 6. TWO LINEARISATION CONTROL LAWS FOR THE BALL & BEAM EXPERIMENT In the following, both flatness-based linearisation methods are derived for the approximated ball & beam model (16). Following the presentation in Sec. 3 and with the results of Sec. 5, the exact feedback linearising control law reads as   4z z42 −T T1 z4 + (Bg)23−4z ˙4∗ + Λ(e) 2 + z 3  , u ¯F B = (Bg)2 − 4z32 (21) and the exact feedforward linearising control law is   4z ∗ z ∗ 2 −T T1 z4∗ + (Bg)23−44 z∗ 2 + z˙4∗ + Λ(e) 3  u¯F F = . 2 (Bg) − 4 z3∗ 2 (22) Thereby, the tracking error in the feedback part Λ(e) =

4 

pi ei

(23)

i=1

Due to (Jakubczyk and Respondek, 1980; Charlet et al., 1989), it can be shown that the model (15) is not flat. However, by neglecting the small term x (x, u¯x ) in (15), the model becomes flat. The approximated model can be described by ⎤ ⎡ ⎤ ⎡ x2 x˙ 1 ⎢x˙ 2 ⎥ ⎢− 1 Bg sin(2x3 )⎥ ⎥ . ⎢ ⎥ = ⎢ 21 (16) ⎦ ⎣x˙ 3 ⎦ ⎣ T (x4 − x3 ) x˙ 4 u¯x

of (9) is defined as in (10). Note that an integral control is not necessary due to the integrator extension at the input (cf. Sec. 4).

A flat output of this system is z = x1 = x. Applying (6) to (16), the diffeomorphism (7) ⎡ ⎤ ⎡ ⎤ x1 z1 ⎢z2 ⎥ ⎢ ⎥ x2 ⎢ ⎥=⎢ ⎥ (17) 1 ⎣z3 ⎦ ⎣ ⎦ − 2 Bg sin(2x3 ) z4 − Bg T (x4 − x3 ) cos(2x3 )

is constructed for the ball & beam experiment as introduced in (Rothfuß, 1997). Thereby, f is the

Since both control laws (21) and (22) use full state information and only the position of the ball can be measured (z = x), a nonlinear tracking observer ˆ˙ = f (ˆ ˆ (0) = x ˆ0 (24) x x, u ¯) + l(t)(z − zˆ), x

4



Note that (Bg)2 − 4z32 /(Bg) = cos 2x3 . Therefore . The the parameterisation becomes singular at x3 = ± π 4 desired trajectories are planned in Sec. 7 and Sec. 8 such that these singularities are avoided.

Position of the ball 0.2 0.15

e [m]

0.1 0

0

−0.2 0

5

10

−0.05 0

15

5

t [s]

10

15

t [s]

Fig. 2. The ball & beam experiment with exact feedback linearisation control law (21). Position of the ball

Tracking error 0.3

which depends on x∗3 (t), x˙ ∗3 (t), x¨∗3 (t) and the coefficients q3 and q4 .

0.2 0.25

0.1 0 −0.1

As a benchmark scenario for the two different control laws (21) and (22), a cosine trajectory as in (Hauser et al., 1992) is chosen. To this end, the desired trajectory for the flat output z = x is defined as t ∗ x (t) = 0.2m cos . (27) s Note that the desired trajectory x∗ (t) is chosen such that its velocity is bounded. This is a necessary condition for robust stability of both exact feedback and exact feedforward linearisation (Hagenmeyer, 2003). The coefficients pi , i = 1, . . . , 4 in (23) are chosen such that the characteristic polynomial has all poles at −3 1s . The coefficients qi , i = 1, . . . , 4 for the observer (24) are chosen such that all poles are at −11 1s . For both experiments, the system’s and the observer’s initial conditions, i.e. x(0) and ˆ (0), are approximately at the origin. x Independent of the respective control law, the results of the ball & beam experiment turned out to be highly correlated to the changing quality of the position measurement provided by the image processing tool connected to the CCDcamera. Thus, the measurement of the position is afflicted by noise; for great measurement errors the experiments become easily unstable. The results of the ball & beam experiment using the control law (21) and the control law (22) are depicted in Fig. 2 and Fig. 3. According to slight differences of the system’s initial conditions for both experiments, the control errors at time t = 0 are not equal. Nevertheless, both control laws achieve a good tracking versus the desired trajectory and the results are more or less equal. However, the exact feedforward linearisation law leads to a smoother tracking than the exact feedback linearisation law.

0.15 0.1 0.05 0

−0.2 0

7. EXPERIMENTAL RESULTS OF THE BALL & BEAM EXPERIMENT

0.2

e [m]

(26)

x (-),x∗ (−.) [m]

 − 2T 2 x ¨∗3 tan(2x∗3 )

0.1 0.05

−0.1

3

where s(t) is 1  s = 2 1 − q4 T + q3 T 2 − 4T 2 x˙ ∗3 2 T + 2T (−1 + q4 T )x˙ ∗3 tan(2x∗3 )

Tracking error 0.2

x (-),x∗ (−.) [m]

right hand side of the full model (14) and the timevariant gain vector l(t) is given by ⎤ ⎡ q4 − T1 + 4x˙ ∗3 tan(2x∗3 ) ⎥ ⎢ s ⎥ ⎢ (25) l = ⎢ − −1+q4 T −q3 T 2 +q2 T 3 ⎥ ⎦ ⎣ BgT 3 cos(2x∗ 3) q1 T − Bg cos(2x ∗) ,

5

10

t [s]

15

−0.05 0

5

10

15

t [s]

Fig. 3. The ball & beam experiment with exact feedforward linearisation control law (22). 8. EXPERIMENTAL RESULTS OF THE BALL & PLATE EXPERIMENT Since the model of the ball & plate MIMO system (14) is equal to two orthogonal models of the ball & beam SISO system (15), the controller design for the ball & plate experiment is undertaken in the same way for each subsystem as presented for the ball & beam experiment in Sec. 6. As a benchmark scenario for both control laws (21) and (22), the following desired trajectory for the flat outputs z 1 = x1 = x and z 2 = x5 = y is considered:  ∗    x (t) sin( st ) = 0.15m . (28) y ∗ (t) cos( st ) Note again that the desired trajectory (x∗ (t), y ∗ (t))T is chosen such that its velocity is bounded. This is a necessary condition for robust stability of both exact feedback and exact feedforward linearisation (Hagenmeyer, 2003). For both subsystems and control methods the pi coefficients in (23) and the qi -coefficients for the observer (24) are chosen as in Sec. 7. Again, for both experiments, the system’s and the observer’s ˆ (0), are approxiinitial conditions, i.e. x(0) and x mately at the origin. The results of the ball & plate experiment using the control law (21) and the control law (22) are depicted in Fig. 4 and Fig. 5. Both control laws achieve good tracking performance versus the desired trajectory. However, the tracking of the desired circular trajectory is more accurate using the exact feedforward linearising control law (22) than by use of the exact feedback linearising control law (21). Comparing (21) and (22) makes this difference explicable: it stems from the fact, that in (21) the state is injected not only for correcting the tracking error, but also for

Tracking error

Position of ball in x, y-plane e1,1 (−), e2,1 (−−) [m]

0.2 0.15

y [m]

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.2

−0.1

0

0.1

0.2

0.15

0.1

0.05

0

−0.05 0

5

10

15

t [s]

x [m]

Fig. 4. The ball & plate experiment with exact feedback linearisation control law (21). Tracking error

Position of ball in x, y-plane e1,1 (−), e2,1 (−−) [m]

0.2 0.15

y [m]

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.2

−0.1

0

0.1

0.2

0.15

0.1

0.05

0

−0.05 0

x [m]

5

10

15

t [s]

Fig. 5. The ball & plate experiment with exact feedforward linearisation control law (22). cancelling the respective non-linearities. This implies, that modelling errors related to the derived model (14) and measurement noise lead to less robustness than in the case of exact feedforward linearisation (Hagenmeyer, 2003; Hagenmeyer and Delaleau, 2003b). 9. CONCLUSIONS The experimental ball & plate setup is used as a benchmark problem for two different flatnessbased control strategies. To this end, the model of the plant is derived as two independent ball & beam subsystems, which can be approximated by flat subsystems. On the basis of the flat models, both an exact feedback linearisation control law and an exact feedforward linearisation control law is applied to the plant by making use of a nonlinear tracking observer. It turns out that exact feedforward linearisation leads to smoother tracking performances and exact feedforward linearisation is more robust than exact feedback linearisation against modelling errors and measurement noise. In (Streif, 2003), additionally the robustness of both control laws is analysed theoretically. The outcome of the analysis confirms the experimental results. 10. ACKNOWLEDGEMENT The authors are very grateful to Thomas Meurer for his advise and help during the experimental phase.

11. REFERENCES Charlet, B., J. L´evine and R. Marino (1989). On dynamic feedback linearization. Systems and Control Letters 13, 143–151.

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