The problem of deadzone nonlinearity has been addressed by many researches with great success by utilizing adaptive control methods to eliminate the undesirable effects on the output of a plant [1] -[5] . Demonstrated in Figure 1 is the effect of deadzone on the output of a plant for a pure sinusoidal input trajectory. The majority of earlier investigations to this problem focus on the problem where the nonlinearity is located at the input of the plant as an actuator problem [1] [2] . In an actuator deadzone, the control effort is within the span of the nonlinearity which makes it somewhat easier to reduce or eliminate its deleterious effects before it enters the dynamics of the system to be controlled. As a matter of fact, several papers present a two structure control schemes that can be designed to handle deadzone as well as other requirements for plant performance criteria [3] . On the other hand, output deadzone, which is physically inherent in some sensors that measure output signals of a plant, is a more complicated problem. The control effort has to eliminate the deleterious effect of the deadzone nonlinearity whilst going through the complicated dynamics of the plant. Therefore, whatever added control requirements enforced on the designer due to disturbances or noise affecting the plant, will further complicated the task. One of the earliest investigations of output nonlinearities such as deadzone was presented by [4] . Their proposed methodology was based on output matching control which involved the design of an adaptive deadzone inverse used to reshape the input reference trajectory to negate the effect of the deadzone. The parameters of the deadzone were adaptively estimated by designing an error function utilizing the output to observe plants states. The implementation was quiet complex in design and implemented in discrete time. In [5] , an output feedback design was analysed for robustness and was developed using input to state stability (ISS) small gain tools. The combination of observer and controller design was proved to be essential when handling output nonlinearities. An adaptive compensation scheme without constructing a dead-zone inverse was presented in [6] . The proposed adaptive method requires only the information of bounds of the deadzone slopes and treats the time-varying input coefficient as a system uncertainty. The new control scheme ensures bounded-error trajectory tracking and assures the boundedness of all the signals in the adaptive closed loop. Tian Ping et al. utilized the integral-type Lyapunov function to design an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead-zone disturbance [7] . It was demonstrated that the closed- loop control system was semi-globally uniformly bounded. In [8] , an inverse deadzone function was incorporated in control system driven from a mathematical model of a deadzone in pneumatic servo valves. Tests were performed out using controllers with and without dead zone compensation to comparison validated the efficacy of the method. In [9] , a somewhat earlier work was presented in discrete time which successfully achieved reduction

of the tracking error in plants with output deadzone nonlinearity while ensuring the global boundedness stability. The paper presented by Jing Zhoua et al. introduced a smooth approximation to the deadzone model which allowed them to employ back stepping technique [10] . In their approach, no knowledge was assumed of the uncertainty’s and the deadzone’s parameters. It is shown that the proposed controller not only can guarantee global stability, but also can achieve excellent transient performance. It is worthwhile to note that other non-classical control methods, such as fuzzy logic or neural network, have been presented by several researchers to reduce the effect of a deadzone nonlinearity [11] -[14] . For example, Wallace and Max used an adaptive fuzzy controller for nonlinear systems subject to dead-zone input. The boundedness of all closed-loop signals and the convergence properties of the tracking error are proven using Lyapunov stability theory and Barbalat’s lemma [15] .

Motivated by the success in producing successful results in handling input deadzone, we present an extended method to reduce the errors caused by output deadzone nonlinearity. The proposed method relies on the premise that by pre-shaping the input trajectory to mimic an inverse form of the deadzone nonlinearity, the combined effect will reduce if not completely eliminating the effect of output deadzone.

In this paper, a new continuous time robust adaptive output deadzone inverse controller (RAODI) is used in conjunction with a conventional model reference adaptive control to counter the distortions cause by output deadzone. The ideal deadzone inverse controller is approximated by an infinitely differentiable implementation to insure asymptotic tracking and minimized error generation. The overall stability of the system under the proposed scheme will be proven analytically and demonstrated by simulation to a practical application. The structure of the paper starts with a brief presentation of the dynamics of an output deadzone nonlinearity that defines various parameters and its effect on the output of a system are presented in Section 2. Meanwhile, the proposed control methodology is presented and its analytical proof using the Lyapunov argument is shown in Section 3. Consequently, an illustrative example of a model reference adaptive control scheme combined with the inverse control method is presented and followed by simulation results in Section 4.

A common representation of a non-symmetrical deadzone nonlinearity, shown in Figure 1, can be described as follows

where denotes the output of deadzone function, the output of a plant, m is the slope of the lines, is the width of the deadzone distance, and is the input of the plant block as shown in Figure 2. Although the width of the deadzone spacing is assumed not to be exactly known, an upper bounds on it is given by

where is a positive scalar. Output deadzone may also be written as

where represents a non-symmetrical saturation function given by

By defining a logical switching operator

Then, the dynamics of the non-symmetrical deadzone presented in (3) can be rewritten as follows

where is the. Meanwhile, thelogical indicators, can be implemented by utilizing the definition of a sign function given as

To obtain a smoothly differentiable implementation of (8), we replace it with a

with appropriately selected with high value for fast switching applications.

Hence, rewriting Equation (5) and Equation (6) as

To proceed with the design of the compensator the following assumptions are required:

(A1) The deadzone parameters and .

(A2) The deadzone parameters and are bounded as follows:

and.

(A3) Without any loss of generality the slope of the deadzone is positive and is set to 1.

Assumption (A1) and (A2) are the actual physical attributes of a real industrial deadzone and is adopted in [16] . Therefore, the saturation function given by (4) is physically bounded

Considering the following nonlinear systems with input deadzone nonlinearity described as

where the matrices A and B are given by

.

Meanwhile, the unmeasurable disturbances represented as and are assumed to be bounded by a known p^th order polynomial in the states [17] :

The desired reference model is given by

where and r is a reference signal. By reshaping the desired reference model in a way to produce a deadzone inversed version of it will reduce the effect of the deadzone. Tracking the reshaped copy of the reference model will force the output of the deadzone nonlinearity to track the original desired reference signal. The adaptive output deadzone inverse compensator can be deduced from (7) as

where is the adaptively estimated values of the exact deadzone spacing. The adaptive inverse dynamics may be determined by differentiating (16) as follows

Consequently, we can utilize (15) to construct the inverse deadzone model reference as

Hence, the states tracking error dynamics may be written as follows

where r is the desired reference signal. Equation (18) is written compactly as

where dynamics of are given by (17).

By defining the output tracking error an adaptive update law for can be written as

Once again, by ensuring that the plant states tracking will cause

where is the output mismatch error caused by the difference between the exact deadzone parameter and the estimated one is expressed as. To parameterize, we utilize Equation (7) to get

or simply written as

where the deadzone parameters estimation error is

Therefore, the deadzone effect noted by the term in (7) can be cancelled by simply ensuring that the system’s states vector track the inverse dynamics of the desired trajectory. To achieve proper tracking and global bounded stability of the overall system, we propose the following RAODI controller:

where, , and P is the positive definite symmetric solution of the Algebraic Riccati equation (ARE). Moreover, the adaptation law for is given by

The properties of the controller (25) are stated in the following theorem:

Theorem. For the plant described by (13) with input deadzone (1), and the RAODI control law (25) along with the adaptive update laws (22) and (26) will ensure the closed-loop stability and boundedness of tracking error, hence reducing the effects of deadzone on the control law driving the system dynamics and ensuresbounded output tracking.

Proof. Using the following positive definite control Lyapunov function

Differentiating along the trajectories of the system and substituting for the closed loop dynamics given by (19) yields

Applying the robust controller given in (25) into (28) gives

Collecting terms and simplifying

The first term can be simplified by solving the Algebraic Reccati Equation given by

which gives

Replacing the adaptation law (23) and replacing in (31) yields

Substituting the adaptive update law (7) makes the fourth term in

Utilizing Equation (23) for output tracking error.

Renders the last term negative. For the third term, we utilize the general inequality the third term in (37) can be bounded as

Applying this bound to (37)

By choosing the degree of freedom satisfying the condition and choosing to be greater than ensures all terms of negative.

To illustrate the efficacy of the proposed compensator a second order sinusoidal desired reference model is selected for tracking. Simulations of the system in (22) under the adaptive control law (23) and (24) have been performed for a sinusoidal reference trajectory given by represented by a second order model. The actual plant is also chosen to be a second order system simulating a rotational gear with deadzone resulting form the spacing between its meshing teeth.

where represent the driving motor angle and velocity respectively; represent the viscous

friction and the electromotive force constant; and represents the output load angle. By defining the state

vector to represent, then the system under investigation can be represented in space state form as

where the matrices A and B along with the gain k are given by

Meanwhile, the desired reference model to be tracked at the output for the overall system may be rewritten as

where used to insure the stability of the desired tracked model. In the case of meshing gears, the deadzone spacing parameter can be easily predetermined and measured. The reference point is chosen to be at the center of the deadzone spacing. Hence, define with being the adaptation that estimatesits value as given by Equation (20). Therefore, the adaptive deadzone inverse trajectory written as follows

The proposed controller is given by

where the first term is the conventional PD-controller, the second term is the robust adaptive controller, and the third term is the adaptive deadzone inverse one.

Meanwhile, the initial value of is set to be zero and no prior knowledge of its values is needed. The exact value of the simulated deadzone parameter is set to. For all other simulated parameters refer to Table 1.

Figure 3 shows the output trajectory for the system under RAODI control is presented and is compared to the trajectory tracking of the system under adaptive without the inverse (in dotted blue), and a PD-con- troller (dashed red). The system performance is shown with the black solid line while the performance of a regular PD controller is shown in dotted red line. Clearly, the output of the system under RAODI outperforms the system with a conventional PD controller. The deadzone spacing effect is practically eliminated and the tracking error is held to a small negligible amount.

The improvement in reducing the effect of output deadzone on the output signal is demonstrated in Figure 4 where the error is plotted in solid line as apposed to the same error for the system under a PD controller plotted in dotted red line. In addition, in Figure 4, the dashed blue line reflects the output tracking error for the system without the use of inverse deadzone modifier. The error without the deadzone inverter is much larger than the improved performance due to RAODI controller.The system state tracking performance (solid) verses the deadzone inverted trajectory for the system under RAODI control is presented in Figure 5, with Figure 6 demonstrating the state tracking error for the system under the proposed control scheme. The second state tracking performance and its error are presented in Figure 7 and Figure 8, respectively. In addition, Figure 9 and Figure 10 show the evolution of the adaptations and confirming their bounded stability. Meanwhile, the adaptive controller effort is shown in Figure 11.

In this paper, an adaptive inverse deadzone controller is compared with a robust adaptive controller for systems with output deadzone nonlinearity. Both controllers have been shown to effectively stabilize a second order system, and achieve bounded input bounded output (BIBO) tracking. The proposed deadzone inverse controller has greatly improved the performance of the system over the robust controller. The deadzone inverse controller was implemented in continuous time and was used to modify a desired model reference to mimic an inverse deadzone trajectory. The RAODI is smoothly differentiable and can easily be combined with any of the advanced control methodologies. The stability of the closed-loop system has been proven by using Lyapunov arguments and simulations results confirm the efficacy of the control methodology.

This work is supported by the Public Authority for Applied Education and Training (PAAET) Kuwait grant number TS-14-03.

Nizar J.Ahmad,Ebraheem K.Sultan,Mohammed Q.Qasem,Hameed K.Ebraheem,Jasem M.Alostad, (2015) Adaptive Control for a Class of Systems with Output Deadzone Nonlinearity. Intelligent Control and Automation,06,215-228. doi: 10.4236/ica.2015.64021