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A fuzzy logic compensator is designed for feedback linearizable nonlinear systems with deadzone nonlinearity. The classification property of fuzzy logic systems makes them a natural candidate for the rejection of errors induced by the deadzone, which has regions in which it behaves differently. A tuning algorithm is given for the fuzzy logic parameters, so that the deadzone compensation scheme becomes adaptive, guaranteeing small tracking errors and bounded parameter estimates. Formal nonlinear stability proofs are given to show that the tracking error is small. The fuzzy logic deadzone compensator is simulated on a one-link robot system to show its efficacy.

Very accurate control is required in mechanical devices such as xy positioning tables [

In this paper, we present the deadzone compensation method for feedback linearizable nonlinear systems. In Section 2, a brief overview on the feedback linearization theory is given [

Feedback linearization has been proved to be a powerful tool in control of nonlinear systems. The Lie derivative [

L f h = ∂ h ∂ x 1 f 1 + ⋯ + ∂ h ∂ x n f n . (1)

For g ( x ) ∈ R n , and recursively with L f 0 h = h , it follows that

L g L f h ( x ) = ∂ L f h ∂ x g (2)

L f i + 1 h = L f ( L f i h ) = ∂ L f i h ∂ x f , i = 1 , 2 , ⋯ (3)

A single input single output nonlinear system

x ˙ = f ( x ) + g ( x ) u , x ∈ R n , u ∈ R y = h ( x ) , y ∈ R (4)

has relative degree ρ at a point x 0 if 1) L g L f k h ( x ) = 0 for ∀ x ∈ N ( x 0 ) (a neighborhood of x 0 ), ∀ k = 0 , 1 , 2 , ⋯ , ρ − 2 and 2) L g L f ρ − 1 h ( x 0 ) ≠ 0 .

Assume that the system (4) has relative degree n at x and define

z = T ( x ) = [ T 1 ( x ) T 2 ( x ) T 3 ( x ) ⋮ T n ( x ) ] = [ h ( x ) L f h ( x ) L f 2 h ( x ) ⋮ L f n − 1 h ( x ) ] (5)

which is a diffeomorphism. It follows that

T ˙ k − 1 = T k , k = 2 , 3 , ⋯ , n T ˙ n = L f n h ( x ) + L g L f n − 1 h ( x ) u (6)

With the new state z = T ( x ) , the feedback control law

u = 1 L g L f n − 1 h ( x ) ( r − L f n h ( x ) ) (7)

linearizes the system (4), i.e., which leads to the system

z ˙ i = z i + 1 , i = 1 , 2 , ⋯ , n − 1 z ˙ n = r (8)

which can be put in a matrix form z ˙ = A z + b r , where r ∈ R is a new input. Here we note that T ( x ) should be a diffeomorphism in the region, which is reachable by x ( t ) driven by r ( t ) .

If the system (4) has relative degree ρ < n at x, the transformed system becomes

z ˙ 1 = z 2 , z ˙ 2 = z 3 , ⋯ , z ˙ ρ − 1 = z ρ z ˙ ρ = b ( ξ , ξ 0 ) + a ( ξ , ξ 0 ) u , ξ ˙ 0 = q ( ξ , ξ 0 ) (9)

where ξ = [ z 1 , z 2 , ⋯ , z ρ ] T and ξ 0 = [ z ρ + 1 , ⋯ , z n ] T . The zero dynamics system is ξ ˙ 0 = q ( ξ , ξ 0 ) [

u = ( 1 / a ( ξ , ξ 0 ) ) ( r − b ( ξ , ξ 0 ) ) may ensure that ξ is bounded. However, even if the system zero dynamics ξ ˙ 0 = q ( 0 , ξ 0 ) are stable, a bounded ξ may not ensure that ξ 0 is also bounded. For a tracking problem z 1 , z 2 , ⋯ , z ρ do not converge to zero. The zero dynamics subsystem is driven by the “inputs” z i , i = 1 , 2 , ⋯ , ρ , with its own states as z j , j = ρ + 1 , ρ + 2 , ⋯ , n . For bounded inputs z i , i = 1 , 2 , ⋯ , ρ , and with arbitrary initial conditions z j ( 0 ) , for z j ( t ) to be bounded, the condition is that the zero dynamics subsystem z ˙ j ( t ) = q j ( z ( t ) ) , j = ρ + 1 , ρ + 2 , ⋯ , n , is bounded input, bound-state stable [

In this section a fuzzy logic precompensator is designed for the non-symmetric deadzone nonlinearity. It is shown that the fuzzy logic approach includes and subsumes approaches based on switching logic and indicator functions [

If u, v are scalars, the nonsymmetric deadzone nonlinearity, shown in

u = N ( v ) = { v + d − , v < − d − 0 , − d − ≤ v < d + v − d + , d + ≤ v (10)

The parameter vector d = [ d + d − ] T characterizes the width of the system deadband. In practical control systems the width of the deadzone is unknown, so that compensation is difficult. Most compensation schemes cover only the case of symmetric deadzones where d − = d + .

The nonsymmetric deadzone may be written as

u = N ( v ) = v − s a t d ( v ) (11)

where the nonsymmetric saturation function is defined as

s a t d ( v ) = { − d − , v < − d − u , − d − ≤ v < d + d + , d + ≤ v (12)

To offset the deleterious effects of deadzone, one may place a precompensator as illustrated in

to cause the composite throughput from w to u to be unity. The power of fuzzy logic systems is to that they allow one to use intuition based on experience to design control systems, then provide the mathematical machinery for rigorous analysis and modification of the intuitive knowledge, for example, through learning or adaptation, to give guaranteed performance, as will be shown in Section 4. Due to the fuzzy logic classification property, they are particularly powerful when the nonlinearity depends on the region in which the argument v of the nonlinearity is located, as in the non-symmetric deadzone.

A deadzone precompensator using engineering experience would be discontinuous and depend on the region within which w occurs. It would be naturally described using the rules

If (w is positive ) then ( v = w + d ^ + )

If (w is negative) then ( v = w − d ^ − ) (13)

where d ^ = [ d ^ + d ^ − ] T is an estimate of the deadzone width parameter vector d.

To make this intutive notion mathematically precise for analysis define the membership function’s

X + ( w ) = { 0 , w < 0 1 , 0 ≤ w X − ( w ) = { 1 , w < 0 0 , 0 ≤ w (14)

One may write the precompensator as

v = w + w F (15)

where w F is given by the rule base.

If ( w ∈ X + ( w ) ) then ( w F = d ^ + ).

If ( w ∈ X − ( w ) ) then ( w F = − d ^ − ). (16)

The output of the fuzzy logic system with this rule base is given by

w F = d ^ + X + ( w ) − d ^ − X − ( w ) X + ( w ) + X − ( w ) . (17)

The estimates d ^ + , − d ^ − are, respectively, the control representive value of X + ( w ) and X − ( w ) . This may be written (note X + ( w ) + X − ( w ) = 1 ) as

w F = d ^ T X ( w ) (18)

where the fuzzy logic basis function vector given by

X ( w ) = [ X + ( w ) − X − ( w ) ] (19)

is easily computed given any value of w.

It should be noted that the membership functions (14) are the indicator functions and X ( w ) is similar to the regressor [

u = N ( v ) = N ( w + w F ) = w + [ w F − s a t d ( w + w F ) ] . (20)

The fuzzy logic compensator may be expressed as follows

v = w + w F = w + d ^ T X ( w ) (21)

where d ^ is estimated deadzone widths.

Given the fuzzy logic compensator with rulebase (16), the throughput of the compensator plus deadzone is given by

u = w + d ˜ T X ( w ) − d ˜ T δ (22)

where the deadzone width estimation error is given by

d ˜ = d − d ^ (23)

and the modeling mismatch term δ is bounded so that | δ | < δ M for some scalar δ M .

In this section we show how to provide fuzzy logic deadzone compensation for deadzone in feedback linearizable nonlinear system. The proposed control structure is shown in

The system has a deadzone nonlinearity N ( ⋅ ) with control input v, that is

x ˙ = f ( x ) + g ( x ) u , u = N ( v ) y = h ( x ) (24)

From the (24) and (22)

x ˙ = f ( x ) + g ( x ) ( w + d ˜ X ( w ) − d ˜ T δ ) . (25)

The control task now is to design a feedback row of w and a fuzzy logic logic tuning law for d ˜ to ensure desired closed loop system properties.

1) Design with exact feedback linearization [

z ˙ = ∂ T ∂ x ( f ( x ) + g ( x ) w + g ( x ) ( d ˜ T X ( w ) − d ˜ T δ ) ) . (26)

Choose the feedback linearizing control input w as

w = 1 L g T n ( r − L f T n ) = α − 1 ( x ) ( r − β ( x ) ) (27)

where T n ( x ) is the last row of T ( x ) , α ( x ) = L g T n , and β ( x ) = L f T n . This control law has the properties

∂ T ∂ x ( f ( x ) − g ( x ) α − 1 ( x ) β ( x ) ) = A z (28)

∂ T ∂ x g ( x ) α − 1 ( x ) = b (29)

where A and b are from (8).

With the controller (27), we can rewrite (26) as

z ˙ = A z + b r + ∂ T ∂ x g ( x ) ( d ˜ T X ( w ) − d ˜ T δ ) . (30)

Then we choose the linear feedback control law

r = − F z + r d (31)

where F ∈ R 1 × n is such that A − b F has desired stable eigenvalues and r d is a reference input, and introduce the reference system as

z ˙ m = ( A − b F ) z m + b r d . (32)

For the tracking error z ˜ = z − z m , it follows from (30)-(32) that

z ˜ ˙ = z ˙ − z ˙ m = ( A − b F ) z ˜ + ∂ T ∂ x g ( x ) ( d ˜ T X ( w ) − d ˜ T δ ) (33)

Theorem 1: Given the system (24), select the tracking control (31) and deadzone compensator (21), where X ( w ) is given by (19). The estimated deadzone widths be provided by the fuzzy logic system tuning algorithm

d ^ ˙ ( t ) = { α ( x ) Γ X ( w ) z ˜ T P b if δ = 0 α ( x ) Γ ( X ( w ) z ˜ T − k d ^ ‖ z ˜ ‖ ) P b if δ ≠ 0 (34)

where the scalar k > 0 and Γ is chosen dialgonal with positive diagonal elements. If δ = 0 , then all closed loop signals are bounded. If δ ≠ 0 , The tracking error evolves with a practical bound

‖ z ˜ ‖ ≤ α ( x ) P b c 0 2 4 Q min k . (35)

where Q min , minimum singular value of Q.

Proof: Define a Lyapunov function candidate for error dynamics (33) as

V ( z ˜ , d ˜ ) = 1 2 ( z ˜ T P z ˜ + d ˜ T Γ − 1 d ˜ ) (36)

where P ∈ R n × n with P = P T > 0 satisfies the Lyapunov equation P ( A − b F ) + ( A − b F ) T P = − 2 Q for a chosen n × n matrix Q = Q T > 0 .

Differentiating (36) and using (29) and (33) yields

V ˙ ( t ) = − z ˜ T ( t ) Q z ˜ ( t ) + α ( x ) z ˜ T ( t ) P b ( d ˜ T X ( w ) − d ˜ T δ ) + d ˜ T Γ − 1 d ˜ ˙ . (37)

If δ = 0 , the tuning algorithm (34) results in V ˙ = − z ˜ T ( t ) Q z ˜ ( t ) ≤ 0 . This means that V ( t ) is bounded and z ˜ = z − z m ∈ L 2 , i.e., z ˜ and d ˜ are bounded, and so z and d. Since z = T ( x ) is a diffeomorphism. x ( t ) is also bounded, and are w and z ˜ ˙ , which implies that lim t → ∞ ( z ( t ) − z m ( t ) ) = 0 .

If δ ≠ 0 , substitution the tuning algorithm (34) gives

V ˙ ( t ) = − z ˜ T ( t ) Q z ˜ ( t ) + α ( x ) z ˜ T ( t ) P b ( d ˜ T X ( w ) − d ˜ T δ ) + d ˜ T Γ − 1 { − α ( x ) Γ ( X ( w ) z ˜ T − k d ^ ‖ z ˜ ‖ ) P b } . (38)

where d ˜ = d − d ^ and d ˙ = 0 by deadzone widths is constant, d ˜ ˙ = − d ^ ˙ . Therefore,

V ˙ ( t ) = − z ˜ T ( t ) Q z ˜ ( t ) − α ( x ) z ˜ P b d ˜ T δ + d ˜ T α ( x ) k ( d − d ˜ ) ‖ z ˜ ‖ P b ≤ − Q | z ˜ | 2 − α ( x ) ‖ z ˜ ‖ P b ‖ d ˜ ‖ δ M + ‖ d ˜ ‖ α ( x ) k d M ‖ z ˜ ‖ P b − α ( x ) k ‖ d ˜ ‖ 2 ‖ z ˜ ‖ P b (39)

where δ < δ M , d < d M for some scalar d M , δ M respectively, and bounding properties were used. Therefore

V ˙ ( t ) ≤ − ‖ z ˜ ‖ [ Q min ‖ z ˜ ‖ − α ( x ) P b c 0 ‖ d ˜ ‖ + α ( x ) k ‖ d ˜ ‖ 2 P b ] (40)

where c o = k d M − δ M .

This is negative as long as the quantity in the brace is positive. To determine conditions for this, complete the square to see that V ˙ is negative as long as either

‖ z ˜ ‖ > α ( x ) P b c 0 2 4 Q min k (41)

or

‖ d ˜ ‖ > c 0 k . (42)

According to the standard Lyapunov theorem, the tracking error decreases as long as the error is bigger than the right-hand side of Equation (41). This implies Equation (43) gives a practical bound on the tracking error

‖ z ˜ ‖ ≤ α ( x ) P b c 0 2 4 Q min k . (43)

Also, Lyapunov extension shows that the deadzone width bound, ‖ d ˜ ‖ , is bounded to a neighborhood of the right hand side of Equation (42).

2) Design for systems with zero dynamics: When the system relative degree is ρ < n , the system can only be partially feedback linearized. Let z = T ( x ) ∈ R n be a diffeomorphism. In this case, there are two parts of z = T ( x ) : ξ = T c ( x ) = [ z 1 , ⋯ , z ρ ] T , which is the feedback linearization part, and ξ 0 = T z ( x ) = [ z ρ + 1 , ⋯ , z n ] T , which is the zero dynamics. Similar to (26), we can obtain

ξ ˙ = ∂ T f ∂ x ( f ( x ) + g ( x ) w + g ( x ) ( d ˜ T X ( w ) − d ˜ T δ ) ) = A ρ ξ + b ρ r + ∂ T c ∂ x g ( x ) ( d ˜ T X ( w ) − d ˜ T δ ) (44)

by choosing a control law w = α − 1 ( x ) ( r − β ( x ) ) , which has the properties

∂ T f ∂ x ( f ( x ) − g ( x ) α − 1 ( x ) β ( x ) ) = A ρ ξ

∂ T f ∂ x g ( x ) α − 1 ( x ) = b ρ (45)

The zero dynamics system then is ξ ˙ 0 = q ( ξ , ξ 0 ) . The reference model is similar to that in (32) but z m ∈ R ρ . Defining the new tracking error z ˜ = ξ − z m ; the fuzzy logic tuning algorithm is of the same form as that in (34) which ensures that ξ and d ^ are bounded. If the zero dynamics system is bounded input, bounded-state stable, the system state z is bounded, and so is x.

In this section, we illustrate the effectiveness of fuzzy logic deadzone compensation with feedback linearization by computer simulation. Consider the control of a single link manipulator with joint flexibility and damping as well as with deadzone nonlinearity in

I q ¨ 1 + M g L sin q 1 + k e ( q 1 − q 2 ) + b ( q ˙ 1 − q ˙ 2 ) = 0 J q ¨ 2 − k e ( q 1 − q 2 ) − b ( q ˙ 1 − q ˙ 2 ) = u , u = N ( v ) (46)

where q 1 , q 2 are the angular positions of the link and motor, I and J are the inertia, k e is the elasticity constant of the joint spring, M and L represent the mass and the position of the center of gravity of the link, and u is the torque applied at the motor. With the state variables x 1 = q 1 , x 2 = q ˙ 1 , x 3 = q 2 , x 4 = q ˙ 2 , this system can be expressed as that in (24).

For the system (46) with flexibility only, i.e., k e > 0 , b = 0 , as in [

z 1 = T 1 ( x ) = x 1 z 2 = T 2 ( x ) = x 2 z 3 = T 3 ( x ) = − M g l I sin x 1 − k e I ( x 1 − x 3 ) z 4 = T 4 ( x ) = − M g l I x 2 cos x 1 − k e I ( x 2 − x 4 ) (47)

and the feedback linearizing control is

w = ( 1 / L g T 4 ) ( r − L f T 4 ) = I J k e ( r − M g l I sin x 1 ( x 2 2 + M g l I cos x 1 + k e I ) + k e I ( x 1 − x 3 ) ( k e I + k e J + M g l I cos x 1 ) ) (48)

For the system (46) with joint flexibility and damping, i.e., k e > 0 , b > 0 , it can be verified that it has a relative degree of three and can only be partially feedback linearized. With

z 1 = T 1 ( x ) = x 1 z 2 = T 2 ( x ) = x 2 z 3 = T 3 ( x ) = − M g l I sin x 1 − k e I ( x 1 − x 3 ) − b I ( x 2 − x 4 ) (49)

the partial feedback linearization control input is

w = I J b ( r + M g l I x 2 cos x 1 − b I ( k e I + k e J ) ( x 1 − x 3 ) + ( k e I − b I ( b I + b J ) ) ( x 2 − x 4 ) ) (50)

which results in the partial linear system z ˙ 1 = x 2 , z ˙ 2 = z 3 , z ˙ 3 = r . With z 4 = T 4 ( x ) = x 3 and T ( x ) = [ z 1 , z 2 , z 3 , z 4 ] T , ∂ T ( x ) / ∂ x is nonsingular, i.e., T ( x ) is diffeomorphism. It then follows that

z ˙ 4 = − k e b z 4 + I b z 3 + M g l b sin z 1 + k e b z 1 + z 2 (51)

which is a bounded input, bounded state stable zero dynamic system.

In both cases, the control signal w is applied to a fuzzy logic deadzone compensator for the deadzone nonlinearity u = N ( v ) . Some simulation results for fuzzy logic deadzone compensation of the system (46) with deadzone widths d + = 0.2 , d − = − 0.2 are given in

A fuzzy logic deadzone compensator has been proposed for feedback linearizable nonlinear systems. The classification property of fuzzy logic systems makes them a natural candidate for offsetting this sort of actuator nonlinearity having a

strong dependence on the region in which the arguments occur. It was shown how to tune the fuzzy logic parameters so that the unknown deadzone parameters are learned on line, resulting an adaptive deadzone compensator. Using nonlinear stability techniques, the bound on tracking error is derived from the tracking error dynamics. Simulation results show that significantly improved system performance can be achieved by the proposed adaptive fuzzy logic compensation. The future research is to get the results of experiment on robot manipulators.

The author declares that there is no conflict of interest regarding the publication of this paper.

Jang, J.O. (2019) Fuzzy Logic Deadzone Compensation with Feedback Linearization of Nonlinear Systems. Applied Mathematics, 10, 87-99. https://doi.org/10.4236/am.2019.103008