Design of MRAS-based adaptive control systems

Design of MRAS-based Adaptive Control Systems Nguyen Duy Cuong, Nguyen Van Lanh, and Dang Van Huyen 

Abstract - In this paper, simple adaptive control schemes based on Model Reference Adaptive Systems (MRAS) algorithm are developed for the asymptotic output tracking problem with changing system parameters and disturbances under guaranteeing stability. The adaptive adjusting law is derived by using Liapunov theory. The direct and indirect adaptive schemes are proposed that constantly adjust the gains in the controllers and/or observers. The adaptive algorithm that is shown in this study is quite simple, robust and converges quickly. Performances of the controlled systems are studied through simulation in Matlab/Simulink environment. The effectiveness of the methods is demonstrated by numerical simulations.

I. INTRODUCTION The PID controller is an effective solution for most industrial control applications [1], [2]. It is simple, robust over a wide range of operating condition, easy adjustment and high reliability. Such that, it is often the first choice for a new controller design. The development of PID controller design went from mechanical devices to digital devices, but the control algorithm is almost the same. The PID controller is used to make decisions about changes to the control signal that drives the plant [1], [2]. With the proportional action, the controller output can be adjusted by multiplying the error by a constant proportional gain. This gain is also frequently expressed as a percentage of the proportional band. The integral action gives the controller a large gain at low frequencies that results in reducing the static error. With the derivative action, the controller output is proportional to the rate of change of the error. It can stabilize loops since it adds phase lead. And it is often used to reduce overshoot. However, we do not need to implement all these actions into a system, if not necessary [1]. The PID structure can be simplified by setting one or two of the gains to zero, which will result in for instance a PI, PD, or P control algorithm. In motion control, a PD algorithm can be considered as a state feedback controller for a second order system. The major problem with the fixed-gain PID controller is that the tracking error depends on plant parameter variations [4], [8], [9]. Because the selection of PID gains depends on the physical characteristics of the system to be controlled, there is no set of constant values that can be suited to every implementation when the dynamic characteristics are changing. Another problem with this controller is that the Paper received May 18, 2013. Nguyen Duy Cuong, Nguyen Van Lanh, and Dang Van Huyen are with Electronics Faculty, Thai nguyen University of Technology, Thai nguyen City, Viet nam (phone: +84-280-3847-092; fax: +84-280-3847-143; e-mail: [email protected]).

PID controlled system is sensitive to measurement noise. When the error is corrupted by noise, the noise content will be amplified by PID gains. These problems can be solved, for example, by using direct or indirect adaptive control systems that are designed based on MRAS. The basic philosophy behind Model Reference Adaptive Systems is to create a closed loop controller with parameters that can be adjusted based on the error between the output of the system and the desired response from the reference model [1] - [3]. The control parameters converge to ideal values that cause the plant response to track the response of the reference model asymptotically with time for any bounded reference input signal. Direct MRAS in which certain information about the plant is used directly for finding appropriate ways for convergent adaptation of the controller parameters. Direct MRAS offers a potential solution to reduce the tracking errors in the presence of uncertainties and variation in plant behavior. However, this control algorithm may fail to be robust to measurement noise [6]. Indirect MRAS in which the controller is designed based on the model of the plant. All of the parameters of the model are available for adaptation. The states and the parameters of the adjustable model converge asymptotically to those of the plant. Estimation of parameters in the model leads indirectly to adaptation of parameters in the controller. In other worlds, for indirect MRAS the adaptation mechanism modifies the system performance by adjusting the parameters of the adjustable model, by adapting the parameters of the controller. Indirect MRAS offers an effective solution to improve the control performance in the presence of parametric uncertainty and measurement noise [6], [7]. In recent decades many kinds of auto-tuning PIDs have been proposed [4], [8]. However, most PID auto-tuning methods did not pay sufficient attention to the stability of the resulting PID control systems. Such that the tuned PID parameters did not guarantee the stability of the control systems for any change [3]. In this study, design of MRAS-based adaptive control systems is developed for motion system which acts on the error to reject system disturbances and measurement noise, and to cope with system parameter changes. The adaptive laws are derived based on the Liapunov's stability theory. The structures of the direct and indirect adaptive control systems are shown with parameter calculations in more detail. The simulation results are presented and discussed. This paper is organized as follows: Direct MRAS and Indirect MRAS are abstractly introduced in Section II. The direct and indirect adaptive controller designs based on MRAS are shown in Section III with some discussions.

Section IV introduces a direct discrete adaptive control system in practice. Conclusions are given in Section V. II. MODEL REFERENCE ADAPTIVE SYSTEMS Model Reference Adaptive Systems (MRAS) are one of the main approaches to adaptive control [1] - [3]. The desired performance is expressed in terms of a reference model (a model that describes the desired input-output properties of the closed loop system). When the behavior of the controlled process differs from the “ideal behavior”, which is determined by the reference model, the process is modified, either by adjusting the parameters of a controller, or by generating an additional input signal for the process based on the error between the reference model output and the system output. The aim is to let parameters converge to ideal values that result in a plant response that tracks the response of the reference model. As can be seen from Fig 1, there are two loops, namely an inner (primary) loop, which provides the ordinary feedback control, and an outer (secondary) loop, which adjusts the parameters in the inner loop. The inner loop is assumed to be faster than the outer loop.

Fig 1: Block diagram of direct MRAS

MRAS can be classified into two main classes [1] - [3]. The first is indirect or explicit adaptive control, in which online estimates of the plant parameters are used for control law adjustment. The second is direct, or implicit adaptive control (see Fig 1), in which no effort is made to identify the plant parameters, that is, the control law is directly adjusted to minimize the errors between the reference outputs and the process outputs. Fig 2 shows one type the block diagrams of the indirect MRAS, which combines an adaptive observer and an adaptive Linear Quadratic Regulator (LQR) [3]. The control scheme consists of two phases at each time step. The first phase consists of identifying the process dynamics by adjusting the parameters of the model. In the second phase, the adaptive LQR design is implemented, not from a fixed mathematical model of the process, but from the identified model [6], [8]. III. DESIGN OF DIRECT AND INDIRECT ADAPTIVE CONTROL SYSTEMS We try to design adaptive controllers for a simple system and we will encounter the problems which require more theoretical background. Simple and generally applicable adaptive laws can be found when we use the suitable Liapunov function. In Fig 3 a block diagram of the process is given which will be used as example throughout this paper. A. Direct adaptive PD control system The structure depicted in Fig 3 can be used as an adaptive PD controlled system. A second-order process is controlled with the aid of a PD-controller. The parameters of this controller are and . Variations in the process parameters and can be compensated for by variations in and . We are going to find the form of the adjustment laws for and .

MIT rule is one of the basic techniques of adaptive control. It can be embedded into a general scheme of circuit with MRAS structure. However, the drawback of MIT rule based MRAS design is that there is no guarantee that the resulting closed loop system will be stable [3], [8]. To overcome this difficulty, the Lyapunov theory based MRAS can be designed, which ensures that the resulting closed loop system is stable.

Fig 3: A process, a reference model and an adaptive controller

Fig 2: Block diagram of adaptive LQG

The following steps are thus necessary to design an adaptive controller with the method of Liapunov [2], [3], [5]:

Step 1: Determine the differential equation for The desired performance of the complete feedback system is described by the transfer function:

=

(1)

The description of the process is: ̇ = (2) ̇ =− . − + . + . . (3) This can be rewritten in state variables: 0 1 0 ̇ [ ] (4) = − . + − + . . ̇ For the design of this adaptive system it is easier to describe the system with the aid of the state variables and , where = − (5) After the state variable is introduced we get = − → ̇ = ̇ − ̇ = − (6) ̇ = . − + . (7) The process in Fig 3 can be described in state variables: ̇ = + (8) where 0 −1 0 = ; = ; = . . − + . 0 By the same way, the description of the reference model is: = − → ̇ = ̇− ̇ =− (9) ̇ = . −2 . (10) It can be rewritten as ̇ = + (11) where 0 −1 0 = ; = ; = ; −2 0 and , , , , , , , , , , , and are defined in Fig 3. Subtracting eq. (8) from eq. (11) yields = − (12) ̇= ̇ − ̇ = + − − = − +( − ) +( − ) (13) = + + where 0 −1 = − = −2 0 −1 − . − + . 0 0 = (14) − . −2 + + . 0 0 0 − = , ̇ = , 0 0 0 =[ ], = − , = − . (15) Step 2: Choose a Liapunov function ( ) Simple adaptive laws are found when we use the Liapunov function ( )= + + where is an arbitrary definite positive symmetrical matrix; and are vectors which contain the non-zero elements of the =(

−

)=

and matrices; and are diagonal matrices with positive elements which determine the speed of adaptation. Step 3: Determine the conditions under which ̇ ( ) is definite negative ̇ = ̇ + ̇ +2 ̇ +2 ̇ = + + + + + +2 ̇ +2 ̇ =( + =(

)

+ )+ +2 ̇ ) + +2 ( +2

+( +

(

) (

)

+2 ̇ ( ) + + 2 ̇ + )+2 ̇

(16)

Fig 4: Adaptive system designed with Liapunov

Let: + =− (17) Because the matrix belongs to a stable system, it follows the theorem of Malkin that is a definite positive matrix. This implies that the first part of eq (16): ( ) = − + (18) is definite negative. The last part of eq. (16) gets zero, stability of the system can be guaranteed if the second part + ̇ =0 (19) where 0 ], ̇ = , = [ ], = =[ , 0 0 0 = , = , and = . After some mathematical manipulations, this yields: ( . + . ). ̇ = ̇

=

(

.

+

.

From eq. (14) it follows that: = − . → ̇ = −2 + + . This yields:

). = − . ̇ → ̇ =

(20) (21)

. ̇

(22) (23)

=

∫(

+

)

=−

∫(

+

)

from

+

Step 4: Solve Let

+

(0)

+

(24)

(0)

(25)

=−

=

(26)

which yields the following matrix equation: 0 0 + −1 −2

−1 −2

= − This can be rewritten as: ( + ) − −2 +

(27) − + . −2 − − − 4

= −

(28)

This yields

=

;

=

. . .

(29)

Fig 5 shows the corresponding responses for the system depicted in Fig 4. The obtained responses showed that the control structure was robust in the presence of sudden process parameter changes. It can be seen that when a parameter variation is added at = 6 [s], after a short time, the tracking error = − converges rapidly to a small value. The conventional PD controller cannot do that. In other words, with respect to representative parameter variations, the direct adaptive PD algorithm is more robust than the PD algorithm with fixed parameters. When the designer has limited knowledge of the plant parameters, it may be desirable to utilize MRAS to adjust the control law on-line in order to reduce the effects of the unknown parameters. Low frequency disturbances of the input of the process with amplitude can be compensated by an additional input with gain , which may be considered as a kind of adaptive integrating action. In the stationary state = should hold (see Fig 6). ) + (0) = ∫( + (31)

Based on eq. (24) and eq. (25) the adaptive system designed with Liapunov in Fig 3 is redrawn as in Fig 4. The following numerical values are chosen: 10 15 = 1, = 0.7, = 68, = 2500, = ; 15 10 = 100; = 250. The settings result = 0.12;

= 0.5.

(30)

Fig 6: Adaptive

Fig 5: Responses of process, reference model, tracking error and adaptation of and with a process parameter change is added at = 6 [s]

For simple systems can be found manually, for higher order systems, the computer can be used.

applied to suppression of low-frequency disturbances in process

B. Indirect Adaptive Control System Adjustable model The reference model, in this case referred to as the “adjustable model”, will follow the response of the process. In the following discussions the terms ‘adjustable model’ and “adaptive observer” are used interchangeably. The goal in process identification is to obtain a satisfactory model of a real process by observing the process input-output behavior (see Fig 7). Identification of a dynamic process contains four basic steps [1], [2]. The first step is structural identification, which allows us to characterize the structure of the mathematical model of the process to be identified. This can be done from the phenomenological analysis of the process. Next, we determine the inputs and outputs. Third step is parameter identification. This step allows us to determine the

parameters of the mathematical model of the process. Finally, the identified model is validated. When the parameters of the identified model and the process are supposed to be ‘identical’, the model states can be considered as estimates of the process states. When the states of the process are corrupted with noise, the structure of the adaptive observer can be used to get filtered estimates of the process states. When the input signal itself is not very noisy, the model states will also be almost free of noise. It is important to notice that in this case the filtering is realized with minimum phase lag [3]. However, this adjustable is also able to deal with unknown or timevarying parameters [6].

= 0.15; = 0.35 The following speeds of adaptation are chosen = 100; = 250. Linear Quadratic Regulator In the theory of optimal control, the Linear Quadratic Regulator (LQR) is a method of designing state feedback control laws for linear systems that minimize a given quadratic cost function [1]. In the so called Linear Quadratic Regulator, the term “Linear” refers to the system dynamics which are described by a set of linear differential equations and the term “Quadratic” refers to the performance index which is described by a quadratic functional. The aim of the LQR algorithm is finding an appropriate state feedback controller. The design procedure is implemented by choosing the appropriate positive semi-definite weighting matrix and positive definite weighting matrix . The advantage of the control algorithm is that it provides a robust system by guaranteeing stability margins. In the control design, the feedback gains will be determined based on the , and matrices of the adjustable model which follows continously the process at different load conditions.

Fig 7: Indirect adaptive controlled system

We are going to find the form of the adjustment laws for and . By the same procedure that indicated in Part – Section III, the adjustable parameters of the model are [5]: =

∫(

=

+

∫(

+

) ̇

+

(0)

(32)

)

+

(0)

(33)

where , , ̇ , , , and ̇ are defined in Fig 7. , and are elements of the matrix, obtained from the solution of the Liapunov equation indicated in eq (34) + =− (34) In this equation is a positive definite matrix; matrix belongs to the process model. The following setting were used =

10 15 , 15 10

=

68 0 , 1 0

= 68,

(0) = 0,

= 2500.

The setting results in the steady state values

(0) = 0,

Fig 8: Responses of process, adjustable model, errors and adaptation of and with a process parameter change is added at t = 6 [s].

For a continuous-time linear adjustable model described by ̇ = + (35) with a cost functional defined as

∞

) =∫ ( + (36) the feedback control law that minimizes the value of the cost is =− (37) where is given by = (38) and is found by solving the continuous time algebraic Riccati equation + − + =0 (39) The following parameters are used in the simulation: 0 = = ; = = , 1 0 0 100 200 = ; = 0.0001. (40) 200 100 These values results in the following stationary feedback controller gains ] = [1.05 0. 85] =[ (41) The results of the combination of the adaptive LQR and the adjustable model are illustrated in Fig 7. The gain of the integrator can be tuned manually, or it can be included in the solution of the Riccati equation [6]. The obtained responses show that the control structure was robust in the presence of process parameter changes. IV. DISCRETE ADAPTIVE CONTROL SYSTEMS Fig 9 shows control configuration of the discrete adaptive direct PID controlled system can be used in practice.

For implementing MRAS when only input and output measurements are available, State Variable Filters (SVFs) can be used, allowing us to obtain filtered derivatives. Moreover, the SVFs offer a beneficial solution to reduce the influence of the measurement noise if the noise spectrum is principally located outside the band pass of the SVFs. However, SVFs cause phase lags. The phase lags can be reduced by means of increasing the omega of the SVF. In practice, the choice of the omega is a compromise between the phase lag and the sensitivity for noise [3]. Discussion -The stability conditions found with the method of Liapunov are sufficient conditions, they are not necessary. - The speed of adaptation, which can be varied by the adaptive gains 1 ; 1 , may in priciple be chosen freely. In a practical system the adaptive gains are limited. - The structure of the adjustable model depends on the chosen order, which is used for the identification. V. CONCLUSION This paper presents direct and indirect parameter adaptive control systems and their performances with a second order processes. The adaptive laws are derived based on the Liapunov's stability theory. The fast adaptive schemes are proposed that continuously adjust the parameters in the controllers and/or observers. They have the advantages of the adaptive systems - quickly compensating the disturbances that can appear in the system. They are robust to changing system parameters. The proposed adaptive control systems were tested through simulation in Matlab/Simulink environment. REFERENCES [1]

[2] [3] [4]

Fig 9: Discrete direct MRAS – Parameter adaptive system

[5]

In discrete time the adjustable parameters of the controller are given by [3], [6]: [6]

∑[(

=

+

)( −

)] .

+

(0) (42)

(0)

(43)

[7]

= =

∑[( ∑[(

+ +

)] .

+

) ̇ ].

+

(0)

(44)

where , , , , and ̇ are defined in Fig 9, is the sampling interval, and are elements of the matrix, obtained from the solution of the Liapunov equation: ( )= + + . + . (45)

[8]

[9]

Astrom, K. J., Wittenmark, B., Computer-Controlled Systems - Theory and Design, Third Edition, Prentice Hall Information and System sciences Series, Prentice Hall, Upper Saddle River, 1997. Landau, Y. D., Control and Systems Theory - Adaptive Control - The Model Reference Approach, Marcel Dekker, 1979. Van Amerongen, J., Intelligent Control (part 1)-MRAS, Lecture notes, University of Twente, The Netherlands, March 2004. R. Burkan, “Modelling of bound estimation laws and robust controllers for robustness to parametric uncertainty for control of robot manipulators”, Journal of Intelligent and Robotic Systems, Vol. 60, pp. 365–394, 2010. Pankaj, K., Kumar, J.S. and Nema, R.K. Comparative Analysis of MIT Rule and Lyapunov Rule in Model Reference Adaptive Control Scheme, Innovative Systems Design and Engineering, 2(4) , pp 154162, 2011. Nguyen Duy Cuong, Advanced Controllers for Electromechanical Motion Systems, PhD thesis, University of Twente, Enschede, The Netherlands, 2008. D. Simon, “Kalman filtering with state constraints- a survey of linear and nonlinear algorithm”, Control Theory and Application, IET, Vol. 4, No. 8, pp. 1303 – 1318, 2010. P. Swarnkar, S. Jain, R. K. Nema, “Application of Model Reference Adaptive Control Scheme To Second Order System Using MIT Rule”, International Conference on Electrical Power and Energy Systems (ICEPES-2010), MANIT, Bhopal, India, 2010. Kuo-Kai Shyu, Ming-Ji Yang, Yen-Mo Chen, Yi-Fei Lin, “Model Reference Adaptive Control Design for A Shunt Active-Power-Filter System”, IEEE Transactions on Industrial Electronics, Vol. 55, No. 1, pp. 97-106, 2008.