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This paper presents a continuous-time adaptive control scheme for systems with uncertain non-symmetrical deadzone nonlinearity located at the output of a plant. An adaptive inverse function is developed and used in conjunction with a robust adaptive controller to reduce the effect of deadzone nonlinearity. The deadzone inverse function is also implemented in continuous time, and an adaptive update law is designed to estimate the deadzone parameters. The adaptive output deadzone inverse controller is smoothly differentiable and is combined with a robust adaptive nonlinear controller to ensure robustness and boundedness of all the states of the system as well as the output signal. The mismatch between the ideal deadzone inverse function and our proposed implantation is treated as a disturbance that can be upper bounded by a polynomial in the system states. The overall stability of the closed-loop system is proven by using Lyapunov method, and simulations confirm the efficacy of the control methodology.

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 [

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 [

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

where

where

where

By defining a logical switching operator

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

where

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

with

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

(A2) The deadzone parameters

(A3) Without any loss of generality the slope of the deadzone

Assumption (A1) and (A2) are the actual physical attributes of a real industrial deadzone and is adopted in [

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 ^{th} order polynomial in the states [

The desired reference model is given by

where

where

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

Hence, the states tracking error dynamics

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

where dynamics of

By defining the output tracking error

Once again, by ensuring that the plant states

where

or simply written as

where

Therefore, the deadzone effect noted by the term

where

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

Substituting the adaptive update law (7)

Utilizing Equation (23) for output tracking error

Renders the last term negative. For the third term, we utilize the general inequality

Applying this bound to (37)

By choosing the degree of freedom

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

where

friction and the electromotive force constant; and

vector

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

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

The improvement in reducing the effect of output deadzone on the output signal is demonstrated in

Systems Physical Attributes | |||
---|---|---|---|

Parameter | Value | Unit | |

1 | 40 | Gain Constant | |

2 | 13 | Gain Constant | |

3 | 100 | Gain Constant | |

4 | 1.0 | radian | |

5 | −1.0 | radian | |

6 | 1.0 | N.m/rad | |

7 | Gains | ||

8 | 1 | ||

9 | rad/s |

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