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In adaptive inverse control (AIC), adaptive inverse of the plant is used as a feed-forward controller. Majority of AIC schemes estimate controller parameters using the indirect method. Direct adaptive inverse control (DAIC) alleviates the adhocism in adaptive loop. In this paper, we discuss the stability and convergence of DAIC algorithm. The computer simulation results are presented to demonstrate the performance of the DAIC. Laboratory scale experimental results are included in the paper to study the efficiency of DAIC for physical plants.

Adaptive inverse control (AIC) is a well-established adaptive tracking methodology [

Discrete time plants for which one or more zeros lie outside the unit circle are called non-minimum phase plants. Similarly continuous time plants in which one or more zeros lie on the right hand side of the

Right inverse

Left inverse

A direct adaptive inverse technique based on NLMS for control of discrete time linear plants to alleviate the adhocism in adaptive loop is proposed in [

The rest of the paper is organized as follows. Section

Let us consider

Control scheme for linear Single Input Single Output (SISO) plants that uses IAIC proposed in [

Indirect control scheme for non-minimum phase plants [

Right inverse

AIC based on linear and nonlinear adaptive filtering discussed in [

Indirect AIC structure for linear SISO plants [

Indirect adaptive tracking schemes discussed above prove good for stable or stabilized plant. IAIC schemes estimate

DAIC structure for controlling stable or stabilized minimum/non-minimum phase linear SISO plants [

The online estimation of

Adaptive plant model

The mismatch error

Output obtained from the second step

In this algorithm, the parameters of the controller

Direct adaptive inverse control scheme.

DAIC is much simpler as compared to methods presented in [

Mean square error (MSE) between desired output and plant output for non-minimum phase plants can be made small by incorporating the delay

The parameter

Using

In this section, estimation algorithms for linear SISO systems are developed. Parameter estimation is developed based on NLMS algorithm.

The parameters of the plant model

Parameters of the plant model should be updated in the direction of negative gradient as

NLMS is self-normalized version of LMS. Convergence of NLMS is faster than LMS [

Parameters of controller are obtained by minimizing the performance index

Since NLMS is used, weight updating for controller is given by

Further,

Using triangle difference inequality [

Controller output will remain bounded if

Subtracting (

Using (

Equation (

Computer simulations of DAIC and IAIC schemes are presented to show effectiveness of DAIC. Two linear non-minimum phase systems are chosen, one without disturbance and other with disturbance.

A disturbance free discrete time non-minimum phase linear plant is chosen having

This is a stable non-minimum phase plant having zero at −1.2000 and poles at −0.2500

Tracking desired output: first 1 sec.

Tracking desired output: amplitude −1.2~1.5.

Tracking error: first 1 sec.

Tracking error: amplitude −1~1.

Mean square error: first 2 sec.

Mean square error: amplitude 0~0.5.

Control input.

Model identification error in DAIC.

A disturbance

This is a stable non-minimum phase plant having zeros at 1.5000

Disturbance.

Control input in DAIC.

Desired output tracking in DAIC.

Desired output tracking in IAIC.

Control input is depicted in Figure

The proposed scheme is implemented on laboratory scale temperature control of a heating process, speed, and position tracking of direct current motor. The temperature control of a process is a non-minimum phase system while the speed and position control of a DC motor is a minimum phase system. To accomplish the adaptive tracking, the proposed DAIC does not require a prior information of the system phase. In these experiments, a standard IBM PC-type Pentium IV is used for the computation in real time. Data acquisition is accomplished by National Instrument card NI-6024E and the controller is implemented in SIMULINK real-time windows target environment. The computations are performed in floating-point format and the sampling interval is selected as 0.001 sec.

In this experiment, we use process trainer PT326 manufactured by Feedback Ltd., UK. This process is composed of a blower, a heating grid, tube, and temperature sensor (bead thermistor). A variable power supply provides power to the heater. This power can be controlled by initiating an appropriate controlling signal from the computer. The process can be considered as a second-order time delay system. This is a non-minimum phase system. Input of the process is power and output is the temperature of air at some desired location in the process tube. This is a time delay system. In this experiment 10 parameters are selected for the plant estimation and 30 for inverse of the plant. The proposed DAIC does not provide a procedure for the selection of optimal number of the plant and the controller parameters. However, the control input, plant output, and the tracking error remain bounded for any selected number of these parameters. Experimental and simulations studies show that large number of parameters achieves better tracking at the cost of computational burden. Figure

Temperature control of heating process.

Control input to heating process.

In this experiment, we use modular servo system (MSS) manufactured by Feedback Ltd., UK. All the modules used in this experiment are parts of MSS. MT150F is a module containing DC motor and tacho-generator. PS150E is the power supply and SA-150D is a servo amplifier. Control input to the servo amplifier is through a preamplifier PA150C. In the speed control experiment the speed signal is measured by the tacho-generator. This generator measures the speed in

Speed control of DC motor.

Control input to speed control system.

Position control of Dc motor.

Control input to position control system.

Stability and convergence of DAIC are discussed in detail. Simulation results show that DAIC performs better than IAIC in terms of mean square tracking error and disturbance rejection. The stability of the closed loop is discussed in detail. The convergence of the error to zero and the boundedness of the controller parameters are proved. However, an algorithm to determine the optimal number of the estimated plant and the controller parameters is needed. Laboratory scale experiments show that DAIC accomplishes tracking of the plant output to the desired smooth trajectory. The synthesized control input in simulations and experiments remains smooth and bounded.

The authors declare that they have no conflicts of interest.