Nonlinear Control of a Pneumatic Muscle Actuator System

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NONLINEAR CONTROL OF A PNEUMATIC MUSCLE ACTUATOR SYSTEM Pablo Carbonell·,l Zhong-Ping Jiang .. ,2 Daniel W. Repperger"· • Dept. of Systems Eng. and Control, Poly tech. Univ. of Valencia, Pla<;a Ferrandiz i Carbon ell, 2, E-03801 Alcoi, SPAIN •• Dept. of Electrical and Computer Eng., Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA ••• Air Force Research Laboratory, AFRL/HECP, Bid. 33, Wright-Patterson AFB, OH 45433-7022, USA

Abstract: The performance of a Pneumatic Muscle Actuator under three tracking control strategies is compared: robust backstepping, sliding-mode and gain scheduling. Robustness is assured for the three controllers in the presence of model uncertainties and external perturbations. Exponential stability is proved for the sliding-mode tracking controller, ultimate boundedness for the backstepping tracking controller and exponential stability for constant or slowly-varying reference signals for the gain scheduling controller. Computer simulations show a good performance for the tracking of a sine wave by the first two controllers, although the sliding-mode strategy yields a high-frequency switching control law. Copyright Cl} 2001 IFAC Keywords: Backstepping, sliding-mode control, gain scheduling, non linear control.

1. INTRODUCTION

Due to major advantages found in pneumatic muscle actuators (PMA) over hydraulic and electric systems, great research effort has been carried out in the past decades in the control of this highly nonlinear system. Its applications range from mobility-assistant devices to light and flexible robot actuators (Tondu and Lopez, 2000) . The complex nonlinear dynamics (Chou and Hannaford, 1994) of the PMA makes it a challenging and appealing system for modeling and control design. Caldwell and his co-workers have 1 This work was done when this author was visiting the Dept. of Electrical and Computer Engineering at Polytechnic University under a grant from the Polytechnic University of Valencia and it has been partially supported by Generalitat Valenciana under Grant GVOO-088-14. 2 Supported in part by U.S. National Science Foundation under Grants INT-9987317 and ECS--{)()93176.

developed an adaptive controller for the PMA based on a feedforward-PID regulator (Caldwell et al., 1995). Cai and Yamura proposed to employ a sliding-mode control approach (Cai and Yamaura, 1996). Other possible approaches are Hoc control (Osuka et al., 1990) and variablestructure control (Hamerlain, 1995). In this paper, three nonlinear control strategies for the PMA will be compared by MATLAB simulations. Section 2 shows a nonlinear model for the PMA. Sections 3, 4 and 5 deal with the design of a robust backstepping controller, a slidingmode controller, and a gain-schedulling controller, respectively. Some simulations are given in Section 6, with the objective to compare the performance of the three proposed controllers.

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2. DYNAMICS OF THE PNEUMATIC MUSCLE SYSTEM Several experiments reported by Repperger and his co-workers (Repperger et al., 1998) show that the dynamic model of the PMA takes the form

(1)

Xl =X2

X2 =

~ [u - it (xd -

!2(X2)

(4)

From the experimental data given in (Repperger et al. , 1998) , it is shown that the nonlinear functions it (xd and h(X2) take the form

where K(xd and B(X2) can be accurately fitted to a second order polynomial: 2

2

i=O

Assumption 1. The coefficients of polynomials in (3) are bounded by known constants:

+ d(t, x, u) ]

where Xl (t) represents the change in length; the control variable u(t) is a pressure times area variable which has units of force ; M is the driven mass; the functions it (xd and h (X2) are nonlinear functions and d(t, x, u) stands for other unmodeled nonlinearities and biased terms.

K(xd = I>ixl ,

with dynamic uncertainties). The resulting c1osedloop will show ultimate boundedness. Before we begin describing the recursive design, some assumptions have to be made on the uncertain terms in (1) .

B(X2) =

L aix~

(3)

i=O

Although the coefficients of these polynomials may vary with the driven mass and the state of the system, in (Repperger et al., 1998), they were reduced to only two fixed polynomials corresponding to the two main operating modes: Inflation and Deflation. The unknown function d(t , x, u) introduced in (1) represents modeling errors and exogenous disturbances. For the sake of simplicity, we assume that d(t , x , u) is bounded by a nonnegative constant . However, our control design strategies proposed in this paper can be easily extended to the more general case where d(t , x, u) is pointwise bounded by a nonlinear state-dependent function . The concepts of stability used in this paper are quite standard. For example, the notions of asymptotic stability, exponential stability and ultimate boundedness can be found in most textbooks of nonlinear system theory (see, e.g., (Khalil, 1996) ).

Assumption 2. The non linear uncertain function d(t , x,u) in (1) is bounded by a known constant: !d(t , x,u) ! $ D.

(5)

Assumption 3. The upper and lower bounds for the driven mass M in (1) are known : (6) Assumption 4. The desired tracking reference signal Xld(t) is C 2, and its first and second derivatives are bounded by known positive constants !Xld(t)! $ 81 and !Xld(t)! $ 82 for all t ~ o. An application of integrator backstepping to the PMA (1) is shown in the following two-step procedure.

Step 1. A tracking error variable is defined as = Xl - Xld . Its derivative, taking into account the model equation (1), is found to be Zl

(7)

!zr

Choosing VI = as a Lyapunov function candidate for (1), a corresponding stabilizing function is selected to be

Step 2. Defining now the error variable Z2 = X2 a(zl) = X2 + klz\ - Xld, the derivative of this new variable is given by Z2 =X2 -

(9)

Q

Q = -klZl

+ Xld

The non linear system expressed in these new variables becomes Zl

3. ROBUST BACKSTEPPING CONTROLLER DESIGN

= -klZl + Z2

Z2 =

~u - ~B(X2)X2 - ~K(xdxl M

M

1

+ Md(t, X, u) + kl (X2

In this section, a robust tracking controller will be designed employing the backstepping technique (Kokotovic, 1992) (see (Jiang and Praly, 1998) for applications to robust adaptive nonlinear control

M

(10)

- Xld) - Xld

The Lyapunov function candidate for the system (10) is chosen to be

1130

5. GAIN-SCHEDULING CONTROLLER DESING

(11)

and its time derivative is given by •

V

In this section a locally stable Gain-Scheduling controller will be designed for the Pneumatic Muscle Actuator. The scheduling variable is chosen to be the tracking signal Xld(t).

2

= -kIZI + Zl Z2 z2

+ M [u - B(X2)X2 - K(XI )XI

+ d(t, x , u)]

+Z2 [k l (X2 - XId) - XId]

In order to design the gain-scheduling controller, we have to assume smoothness between different operating regimes. In the dynamics of the PMA two operating modes can be identified: inflation (X2 > 0) , and deflation (X2 < 0). The transition between both regimes is proposed in this approach to be approximated by a smooth function d(X2(t)) between the inflation and the deflation regimes:

(12)

Theorem 1. For the Pneumatic Muscle Actuator (1) with nonlinear uncertainties and the reference signal satisfying Assumptions 1 to 4, the backstepping non linear controller _ k kl+M2[(ki-l)2+2ki+kd u - - 2Z2 Z2 2kl

-, (xg + x~ + x~ + x~ + xt + xi) Z2

(13)

where (; > O. The second order polynomials for both regimes in (3) , are gathered together in two unique functions :

with k2 > 0 and, > 0 will make the z-closed-loop system globally uniformly ultimately bounded. In particular, the ultimate bound for the tracking error Zl = Xl - Xld can be rendered arbitrarily small by picking large enough feedback gains kl and k 2 .

2

K(Xl ' 6(X2))

= ~:=raj 6(X2) + aj] ~ j=O 2

B(X2 ' 6(X2))

Proof See Appendix A.

= 2)bj6(X2) + bj]~

(18)

j=O

where

a± J

=

aj i ±ajd. b± 2' J

=

bj i ±bid

2

'

4. SLIDING-MODE CONTROLLER DESIGN

Assumption 5. The driven mass M in the PMA model (1) remains constant.

Taking into account only the second equation of the nonlinear system (10) , and choosing a Lyapunov function candidate for this equation as

V2(Z2)

= ~z~

Assumption 6. The non linear function d(t , x , u) in the PMA model (1) is identically zero for all t ~ O.

(14)

2

For any constant value of the reference XIdO , and for the nominal value of the driven mass M, an equilibrium point (X(XIdO) ,U(X1do))) can be obtained from Equation (1) :

its derivative will be given as:

. Z2 V2 = M [u - B(X2)X2 - K(Xt}XI

+ d(t , x , u)]

+Z2 [kl(X2 - XId) - Xld]

(15) X(XIdO)

Theorem 2. For the Pneumatic Muscle Actuator (1) with nonlinear uncertainties and the reference signal satisfying Assumptions 1 to 4 the sliding mode controller u

U(XIdO) = Uo = !l(XIdO)

(19)

Linearizing around the equilibrium point (19) leads to the following linear system:

[ii21x213 + iiIlx21 2 + aolX21 3 2 +b2l x II +bllxll +bo lxll+b.+M2k 1 Ix21

=-

6x = A(XIdO)6x + B(XIdO)6u

-

+M2k1 161 1+ M21621+ M2b]sgn(z2)

= Xo = [XIdO 0] T

Yli = C(XIdO)6x

(16)

with state matrices

with b > 0 will lead the system to the sliding surface Z2 = 0 in finite time, and the closedloop system will become globally asymptotically stable.

Proof See Appendix B.

113 I

(20)

where t5x = X-Xo, t5u = U-Uo and YlI = y-XldO. Note that the output matrix C(XldO), which maps only the first state Xl (t), has been defined in order to have an integral controller, as will be shown below. Now, given any equilibrium point and its linear system associated, we can design a linear controller employing any of the standard techniques. In this paper, we propose to use the standard LQR design (Friedland, 1986), with an integral term, used for the correction of equilibrium estimation. The control law is given by

t5u = kl (XldO)t5x if

This Lispchitz constant can be employed as a measure of time-variations (Shamma and Athans, 1990):

+ k2(xldO)a

= Yo

J(~TQ~+uTRu)df,; ~= 00

[t5x af(23)

° The existence of the controller is assured as long as for all XldO the pair [A(XldO), B(XldO)] is controllable, the pair [A(XldO), C(XldO)] is observable and the matrix [A(XldO) B(XldO)] has rank = 3 C(XldO) 0 (see (Khalil, 1996)) . It is straightforward to check these properties for (20). The solution will be given as

K=-R-IB'P

(29)

(22)

where k = [kl(XldO) k 2(XldO)] is the feedback gain that minimizes the cost index

J=

Using Assumption 7 and the smoothness of hand we can easily prove the following statement: The closed-loop matrix Ac/(t) = A(t) - B(t)K(t) is bounded and locally Lipschitz continuous in a neighborhood r ~ ~3 . That is, there exist constants kA ~ 0 and LA ~ 0 such that 'r:It E ~+ and ~ E r,

12,

(24)

where P is the solution of the Algebraic Riccati Equation

and A, B denote the extended equation state matrixes for ~ = A~ + Bu. The nonlinear ~-state system can be expressed as a linear scheduled system under a non linear pert ur bat ion :

~ = A(XldO)~ + B(XldO)t5u + t5f(~,XldO) (26) where t5f(~, XldO) stands for the residual nonlinear terms of the linearization. It is easy to check that only the second element in the vector field t5f(~, XldOh, which corresponds to the second state t5x2 of ~, is not identically equal to zero, as long as the other states are linear in their formulation .

Assumption 7. There exists a constant ko > 0 such that

Theorem 3. For the Pneumatic Muscle Actuator defined by equations (1) and (18) with nonlinear uncertainties bounded under Assumptions 5 to 7, the gain scheduling controller (22) optimally designed for the cost index (23) will provide local ultimate boundedness in a region r ~ ~ for sufficiently slow time-variations in Ac/, i.e. for sufficient small value of /'i, in (29) . Proof See Appendix C. 6. SIMULATION RESULTS In Figure 1 the tracking performance of a sine wave of amplitude 1 and frequency 0.1 Hz is compared for the three controllers, starting with an initial condition of (Xl (0) = 1, X2(0) = 0) . The coefficients have been taken as in (Repperger et al., 1998). In three cases, a perturbation signal d(t) of amplitude 0.1 and frequency 10 Hz has been injected. The experiments were carried on for a nominal mass of M = 10 lb. As expected, the sliding-mode controler attains exponential stability. However, its switching behavior shown in Figure 2 suggests that the application will result in high energy consuming as long as the pressure valves have to be closed and opened very fast . The undesirable chattering can be attenuated by replacing the signum function by a saturation function, although the resulting system will become ultimately bounded (Khalil, 1996). The response obtained by the backstepping controller shows a slight persistent error, which, according to Theorem 1, can be reduced by increasing the feedback gains. Again, the saturation at the actuators will limit the maximum gain attainable at the control signal. Nonetheless, because of the presence of d(t, X, u) and of incomplete knowledge about h (xJ) and h(X2), it is unlikely that we can attain perfect tracking via smooth feedback control without imposing additional assumptions on the PMA system. Our results -

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falling into the category of practical tracking seem to meet the need from applications involving PMA. In contrast to the sliding-mode controller, the main advantage of the backstepping controller is the disappearance of the chattering and a smoother behaviour in the control signal, as shown in Figure 2. Finally, the gain-scheduling controller gives the worst response. That is due to the operating nature of gain-scheduling controllers, that is, they are designed for handling constant or slowlychanging reference signals. As can be seen in Figure 3, the tracking error becomes apparently smaller when the reference signal frequency is decreased .

7. CONCLUSIONS The nonlinear internal structure of the PMA requires a nonlinear control strategy in order to achieve robust tracking with better performance. This conclusion has been corroborated by simulations results, where it was shown how the gainscheduling regulator, although nonlinear in formulation, has strong dependency on the operating point. With the backstepping and the slidingmode, the results obtained are compliant with the objectives. The main drawback of the slidingmode control is its nonsmoothness and its chattered response, the backstepping otherwise can overcome this limitation, but only at the expense of achieving robust global ultimate boundedness. Further research should be pursued in the direction of working with a better nonlinear model of the PMA. Experimental results will be reported separately.

8. REFERENCES

Fig. 1. Sine wave tracking comparison.

,::~ ..:.... H.... H.H.· .H7 ~"':--''' : ..... , .. :... ~

0

"

_10

:......

•..••• ; .

• •• ~..

.:......

.....

. .. :-

':

- 1000~~,--!".--!-.-!-,---;,~ , ~,;-,-7:";---;;,,-7;,,---;,,

~

.~ -

I~

.1l~

___

I~ I ~

.......

i I

~I I '

""'Ill

.......

,;:~ o

2

4

,

•

10

u_ (_I

U

14

16

11

2.

Fig. 2. Comparison between control signals u(t) .

Fig. 3. Slowly-varying sine wave tracking for the gain-scheduling controller.

Cai, D. and H. Yamaura (1996) . A robust controller for manipulator driven by artificial muscle actuator. In: Proc. of the IEEE Con/. on Control Applications. pp. 540-545. Caldwell, D.G., G.A. MedranoCerda and M. Goodwin (1995) . Control of pneumatic muscle actuators. IEEE Control Systems Magazine 15(1), 40-48. Chou, C .P. and B. Hannaford (1994). Static and dynamic characteristics of mckibben pneumatic artificial muscles. In: Proc. of the IEEE Con/. on Robotics and Automation. pp. 281286. Friedland, B. (1986). Control System Design. McGraw-Hill. New York, NY. Hamerlain, M. (1995). An anthropomorphic robot arm driven by artificial muscles using a variable structure control. In: Proc. of the IEEE/RSJ Conf. on Intelligent Robots and Systems. pp. 550-555. Jiang, Z.P. and L. Praly (1998) . Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica 34(7), 825-840. Khalil, H.K (1996) . Nonlinear Systems. second ed .. Prentice-Hall. Upper Saddle River, NJ . Kokotovic, P.V. (1992) . The joy of feedback : nonlinear and adaptive. IEEE Control Systems Magazine 12(3), 7-17. Osuka, K, T. Kimura and T . Ono (1990) . Hoo control of a certain nonlinear actuator. In: Proc. of the IEEE Con/. on Decision and Control. Vol. 1. pp. 370-371. Repperger, D.W., KR. Johnson and C.A. Phillips (1998) . A vsc position tracking system involving a large scale pneumatic muscle actuator.

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In: Proc. of the IEEE Con/. on Decision and Control. Vo!. 4. pp. 4302-4307. Shamma, J. S. and M. Athans (1990). Analysis of gain scheduled control for nonlinear plants. IEEE Trans. on Automatic Control 35(8), 898-907. Tondu, B. and P. Lopez (2000) . Modeling and control of mckibben artificial muscle robot actuators. IEEE Control Systems Magazine 20(2), 15-38.

Substituting the control law (16) in this last expression, we arrive at

Appendix A. PROOF OF THEOREM 1

From (B.2), it follows that, starting at any initial condition, the state Z2(t) will be zero in finite time and stay zero afterwards. After this critical time instant, because of (10), the tracking error Zl converges exponentially to zero as t -+ 00.

.

+M2k1 181 1+ M21821+ M2 b]

.

V2

First of all, we state some useful lemmas. Lemma 1. For any x and y in positive real number E, we have:

)Rn,

IZ21 [

Z2

V2 :::; MU + M Ih(xdl + Ih(X2)1 +.6. + M2kllx21

:::;

IZ21 -MM 2 b:::; -blz21

(B.1)

(B.2)

and for any Appendix C. PROOF OF THEOREM 3

(A .l)

The proof will be only outlined here. For a detailed recount, see (Shamma and Athans, 1990).

Lemma 2. The function f(lxl) = -,lxI2 + alxl, with " a > 0 attains its maximum value ~~ at

Proposition 1. The closed-loop linear time-variant system on (20), with the control law (22), that is

Ixl =

;"1 '

~

=

A(XldO)~ + B(XldO)K(XldO)~

(C.l)

will show exponential stability, as long as timevariations,.. are kept small, i.e. the reference signal Xld is slowly-varying.

By using Lemma 1, the derivative of the Lyapunov function (12) can be bounded as

So, we can always find positive constants A and m such that any solution ~(t) of (C.1) verifies 1~(t)1

V(~,t) = supe-Y'\(1'-t)14{T,~,t)1

I

1

where cP( 1',~, t) is the solution of (26) with initial state ~ at time t.

MI

3

+M [ 1

-2

+ -b2] i

a 2:4;=1

i

'

(C.3)

1'~t

. kl 2 k2 2 V< - - z - -z2 2

(C.2)

Furthermore, in (Shamma and Athans, 1990) a Lyapunov functional for system (C.l) is proposed: for any, E (0,1) :

Substituting in this last expression the input u(t) by the control law (13), and taking into account (3) and its bounds (4, and finally from Lemma 2, this last expression can be bounded by

-

:::; me-'\(t-to)I~(to)1

X"2

Id +2+

Using (Shamma and Athans, 1990, Theorem 3.3) we can conclude that there exists some,' E (0,1) such that, for any initial state with

A 2 ....

2M (A.3) 1

From this last equation, it is shown that the system becomes globally ultimately bounded and that the ultimate bound for the tracking error zl = Xl - xld can be made arbitrarily small by selecting large enough values of kl and k 2 .

I~(to)1

,A,

:::; ~k m 0'

(C.4)

the solution ~(t) of the nonlinear system (26) satisfies the inequality

1~(t)l:::; Appendix B. PROOF OF THEOREM 2

A 'k ,', t mo

~ to

which leads to ultimate boundedness. The derivative of the Lyapunov function candidate for the Z2 equation (10), can be bounded by

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(C.5)