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Designing of Adaptive Model-Free Controller Based on Output Error and Feedback Linearization
This paper introduces a new approach to design Model-Free Adaptive Controller (MFAC) using adaptive fuzzy procedure as a feedback linearization based on output error. The basic idea is to transfer the control signal to an appropriate surface and then, depending on the output error of system, the control signal changes around this surface. Some examples are provided as well to illustrate the efficiency of the proposed approach. The obtained simulation results have shown good performances of the proposed controller.
Classical control method is based on mathematical equations of the system; however, this method suffers from some drawbacks. For example, in this method the performance of system can be affected by unmodeled dynamics of system and/or by large delays. Model-Free Adaptive Control (MFAC), as a part of modern control theory, shows superiorities compared to model-based methods. MFAC is an adaptive control method and needs no information about system model. It only uses data for controller design and therefore it is a nonlinear controller designed without the need to mathematical model of controlled system [1–3].
In 1994, Han and Wang introduced model-free topic and proved the stability of MFAC controllers. In 1993-1994, Hou Zhongsheng represented applications of these controllers. With using pseudo-Jacobian matrix, nonlinear systems are replaced to near the rail line of controlled system and then the data of controlled system are used to design MFAC controller .
In 1995, Jagannathan suggested a fuzzy stable controller for a limited class of nonlinear system in the form of , where is the unknown nonlinear function. In 1996, this controller was used for general class of nonlinear system in the form of .
Another idea is introduced in 2000 , which uses a neural network as an adaptive controller for stabilization of system. Their proposed algorithm does not need any complex tuning and can be applied to any controllable multiinput systems.
Compared with other adaptive control schemes, the MFAC approach has several advantages, which make this method suitable for control applications. First, MFAC just depends on the real-time measurement data of the controlled plant. Second, MFAC does not require any other testing signals and any training process. Third, MFAC is simple and easily implementable with small computational burden and has strong robustness. Fourth, MFAC approach does not need a specific controller for each specific process. Finally, the MFAC has been successfully implemented in many practical applications, for example, chemical industry, linear motor control, injection modeling process, PH value control, and so on .
The main contribution of this work is to introduce a new MFAC approach based on output error and feedback linearization. The proposed model has rapid monotonic tracking error convergence, robust stability, and good disturbance rejection.
The rest of the paper is organized as follows: in Section 2, adaptive fuzzy control based on feedback linearization is described. The proposed model-free adaptive controller and its Lyapunov stability are represented in Sections 3 and 4, respectively; in Section 5 simulation results are given to highlight advantages of MFAC method. Finally conclusion is stated in Section 6.
2. Feedback Linearization by Means of Adaptive Fuzzy Controller
In feedback linearization method, control law is determined in such a way that nonlinear terms eliminate the system dynamic and replace it with appropriate reference input as seen below: (for simplifying ), Conceptually, Adaptive Feedback Linearization (AFL) method is the same as nonadaptive counterpart but AFL uses adaptation law for updating controller parameter to enhance system performance. Notice that both methods should have complete description of system dynamic but our proposed method does not need any information of system dynamic. Due to the fact that it is difficult to obtain a mathematical model of the system and typically there exist some numerical approximations, the object of this paper is to control systems just with data in order to improve system performance. In order to reach this goal it is required to estimate unknown nonlinear function of system dynamic and with and or estimate control law with . Estimation algorithm in this approach utilizes basic fuzzy membership function. As an instance for estimating , we have where is the number of fuzzy roles, is membership function for th fuzzy role, also is and the result of th fuzzy role for system that is generally considered as linear combination of some series of continuous functions such as : Using (3), (4), can be rewritten as where is the approximation error of fuzzy approach. is updated by the adaptation law below: is a constant coefficient that usually is selected small. is the difference between output and reference output : It should be pointed that AFC guarantees that approaches to its desired value, .
Notice that the initial values of can be chosen as or any selected value by information that is acquired from system dynamic or even can be determined from another available control approach; for instance, the columns of can be taken as ( is obtained from state feedback method ) or, for example, if we want to estimate function, the columns of are better to contain coefficients of system states of . Fuzzy rules are expressed as follows: is related to th row of matrix. So the unknown nonlinear function is estimated as by fuzzy approach.
Adaptive-fuzzy control method has been divided to two parts: direct and indirect approachs; this paper uses indirect method.
2.1. Indirect Adaptive Fuzzy Controller (IAFC)
First, consider the affine system equations given below: are known parts of system dynamic (which are obtained from experimental and numerical methods), and , are system equations that must be identified. Considering the above discussions, and can be obtained as , are approximated errors between real system and fuzzy system with upper bound values , (, are known values). So estimation of system parameters can be rewritten as follows: are matrices that will be updated adaptively with the following updating laws: and finally the control law is described as follows:
2.2. Direct Adaptive Fuzzy Controller (DAFC)
Control law in feedback linearization method can be obtained as follows: The goal is to estimate , with consideration of basic fuzzy function as seen below: where is the estimated signal and is the error between the described by fuzzy method and desired (appropriate) ; this error is restricted by upper known bound ; since practically it is difficult to define , so its value is specified and adjusted in designing process repetition; this adjustment procedure continues until the performance of controller indicates that comes close to proper value. Finally updating law is obtained as follows: The initial value of may be considered as .
Remark 1. This method is suitable for minimum phase plants and also has the following limitation for :
3. Proposed Approach
In the previous section, an adaptive fuzzy controller using fuzzy basis functions is described for stabilizing the controlled system. In our introduced method, the same approach has been employed but our method does not need to utilize fuzzy basis functions for estimating system parameters. Instead, three proposed rules based on output errors are considered to control the plant.
Consider a system with general form At first, nonlinear functions and must be estimated in such a way that controller does not rely on system dynamic. It should be noted that in this paper, the system is considered continues time but the output is sampled and these sampled points are used in proposed controller. The estimations of system dynamic can be defined as And the updating values of , are given by where , are constant coefficients with respect to , . It should also be noted that usually these parameters do not need to be changed for different system dynamics. The value of will be defined in (27); is the so-called “filtered tracking error” which is the sum of errors as defined below: where , are coefficients of the predictive error. And is the desired output. For eliminating nonlinear terms of system dynamic, feedback linearization method has been exploited; for this purpose, is chosen as follows: where is defined by In (23), is the tracking error coefficient. By choosing appropriate , is converged into zero and at last it causes to be restricted. In other words, if the bigger is opted, better performance of output and quick tendency to the desired value are achieved while increasing the control signal ; it should be noted that if is chosen very large, it can make the system unstable.
As it is seen from (22), the problem of singularity occurs when is near zero and the control signal can become unbounded. To solve this problem, separate controller comprising two parts is proposed in the following form: In the above equation, is a constant value that its amount impacts on control signal . are defined as where is robust control signal coefficient that the decreasing or increasing of this coefficient affects quantity of final control signal , and also is a constant value that its small amount decreases the effect of on final control signal ; that is, will become more effective factor for determining ; and is given by the equation below: Here, is lower bound of . Proper selection of prevents to be singular.
Remark 2. , and are designing parameters with the following conditions : .
Remark 3. If is greater than , it is assumed to be equal to .
The major principles of proposed MFA controller are constructed based on the three following experimental rules, which make controller produce appropriate control signal: where is threshold value for changing final control signal in the above rules. These laws utilize appropriate last level control signal (if error becomes less than specified value that is called ) for using at next level. Indeed, if error at time and becomes less than and also error at time is greater than the previous time, the controller applies negative value of previous control signal for current time.
Regarding adaptive fuzzy controllers [5, 7] that estimate unknown nonlinear functions of the system by fuzzy algorithm and when new reference input is applied to the system, this estimation becomes better. But the sensitivity of proposed controller on estimation of nonlinear functions is less than that of AFC method. This is because output error has significant role in control effort rather than significant role of estimation in AFC.
4. Lyapunov Stability for the Proposed Model-Free Adaptive Controller
Let the Lyapunov function candidate be given by 
(a) For the first region, .
At first we should obtain the difference of Lyapunov function in and level. So general Lyapunov function can be acquired as follows: If we prove that , it will demonstrate that , and are bounded and they are able to stabilize system. , and are obtained as follows: With combination of the above three Lyapunov functions, (32) is established By simplifying (32) and using the below definition, Inequality (34) is acquired as follows: with assuming that is the maximum value of and by resimplifying (34), inequality (35) is stated as given below: where and are described as By considering Lemma A.4 in Appendix A, And with substituting inequality (37) in (35), With simplifying the above inequality, the below equation is obtained: where , and are defined as By assuming that , the below statement becomes negative: As regards , becomes positive too and finally totally; the first term in (39) becomes nonpositive, if it obeys the below condition: in which is acquired from the below inequality: Thus, both terms of inequality (39) have nonpositive values and finally .
(b) For the second region, .
In this region the proposed Lyapunov functions , and are obtained like (30) and they are stated as follows: By combination of the above three equations (and definition of ), the statement is demonstrated below: with the consideration of Lemma A.4 By substituting inequality (46) in (45), in which , and are as follows: By the assumption that , the statement below becomes negative: As regards , becomes positive too and finally . The first term in (39) becomes nonpositive, if it obeys the below condition: in which is obtained from the relation below: As a result, both terms of inequality (47) have nonpositive values which means . And finally if Lyapunov function becomes negative.
5. Implementation of MFA Controller
Example 4. Consider MIMO system with 2 input-2 output system state equations which are given below :
Select control parameters as follows:
Simulation results of MIMO system without disturbances are demonstrated in Figure 1. In Figure 2 external disturbances are applied to the system with power , and finally, the system with existence of white Gaussian measurement noise (power 0.0001) and external disturbance is shown in Figure 3; also desired inputs are assumed: ,.
The aim of using this MIMO system which is brought by  (along with two control signals) is surveying coupling effect in multiinput multioutput systems. As it is understood from the result of Figure 1 caused by MFA controller and the result of AFC in , it can be said that system outputs of MFA controller do not have any overshoot in its response but in AFC, this problem is observed. As it is seen from Figure 2, MFA controller rejects disturbance without any noticeable effort in control signals and in the outputs. Also it can be deduced from Figure 3 that the measurement noise has intense effect on system outputs and control signals (approximately became twice) but it should be noted that the proposed controller is able to track desired output and yet remains stable. As it is demonstrated in Figures 1, 2, and 3, the disadvantage of this controller is oscillating control signal. Profits of MFAC in addition of good (also rapid) tracking and system stabilizing are decreasing settling time of output response.
Example 5. The second system under study is Maglev train which is shown in Figure 4. Using the notations given in Figure 4, the vertical dynamics is described by .
System state equations are described as follows: Here is the distance between rail and train, is the current of windings, is the train mass, is the force of disturbance, is the permeability coefficient which equals , the number of turns in winding, is the section surface of windings, and is the wire resistance in windings.
Consider the state vector as below; thus the above equations convert to new state equations:
Maglev trains work in one of two ways; both methods are based on the same concept but involve different approaches.(1)Electromagnetic Suspension is based on magnetic attraction; it is very complex and somewhat unstable.(2)Electrodynamic Suspension is based on the repulsion of magnets. The magnetic levitation force balances the weight of the car at a stable position. Controlling EMS is more difficult than EDS train, because normality of this dynamic state is unstable.
Magnetic trains have two important issues, levitation and propulsion; the target of controller is first: goes up train to desired level along with guarantee stabilizing against some uncertainty such as wind and changing train mass in boltroads. And second adjust train speed at the working frequency of magnets. In this paper just levitation part (important section of train in controlling) is discussed. The train in this example goes from 10 mm to 16 mm level. By considering these assumption values for train and controller as follows,
The result of simulating without considering disturbance has been displayed in Figure 5; in Figure 6, input step as external disturbance with amplitude is applied to the system; for Figure 7, sinusoidal measurement noise with amplitude .1 mm in presence of disturbance is considered. Also white Gaussian noise in existence of external disturbance is applied to the system with power (it is considered small because amplitude is small) which is shown in Figure 8. The purpose of presenting this system is studying on sort of complicated practical system dynamics such as Maglev in which they have some nonlinear terms like existence of first state in denominator (that affect directly the destabilizing system), and so forth. As it is demonstrated in Figure 5(a), the system output, that is distance between rail and train, has no overshoot; after about .25 second it goes from 10 to 16 mm smoothly. Such as the last example, it is acquired from Figure 6 that disturbance did not have any effect on the output, control effort, and settling time. In Figure 7, it is illustrated that sinusoidal measurement noise with no changes appears into output but no noticeable alternation in control signal has been seen. And finally in Figure 8, an impalpable effect of white Gaussian measurement noise on output and control signal can be shown; in this example some repeated benefits which are seen in Maglev system response are expressed, such as appropriate system stabilizing, good and rapid tracking, disturbance rejection, and small settling time.
Model-free adaptive controller is an adaptive controller that just uses system outputs and does not require another system states (so does not need observer) and with this, as model-free controller, it performs good and also has some merits such as good stability, appropriate tracking, robust against uncertainty, disturbance rejection, and good decoupling and the biggest advantage of MFA controller is no requiring to system dynamic and even does not need to any prior experiment about system.
A. Some Requirements
Lemma A.1. For each time , is bounded by which is , called “tracking error,” and are computable positive constants.
Proof (see [8, Lemma 3.2]). Let , in which is a compact subset of . Assume that ; that is, is a smooth function , so that the Taylor series expansion of exists. Then using the bound on , express the function on a compact set as linear form of parameters. Thus, the upper bound for can be obtained as where and are constant matrices .
Remark A.2. , are estimation errors of , , respectively, and is external disturbance. These errors are bounded and have these definite bounds
Remark A.3. According to (27), if then , so is bounded and if then , but by considering the mentioned condition at Remark 3 in Section 5, in which if is greater than , it is assumed to be equal . Thus, it can be deduced that in both regions and in all of times, is bounded and its bounds are shown as follows:
Lemma A.4. With regards to that, the proposed control algorithm is divided into two regions and . We will prove the following relationship which is established in each region:
Proof. (a) First region: or and .
In consideration of Lemma A.1, it is demonstrated that , where , are constant matrices: By using the result of relation (A.5), the following inequality is obtained:
As regards , By considering (A.6), Also by utilizing Remark 3, (A.9) is stated as follows: By using (A.8) and (A.9) the following inequality is acquired: Thus, the assumption is proved in the first region.
(b) Second region: or and and the mentioned condition for the proposed MFA controller in Remark 3 in Section 5 is stated as follows: In consideration of Lemma A.1, it is demonstrated that , where are constant matrices. By using the result of relation (A.5) and the condition that is given above, the following inequality is obtained: As regards , By considering (B.5), By using (A.9) and (A.14), the following inequality is obtained: Finally, the assumption is proved in the second region. Thus, inequality (A.4) is correct in both regions.
In consideration of error definition in (21), (I), (II), (III)Thus, by substituting (I), (II), and (III) in the above equation, Equation (25) can be rewritten as follows: By using (23), (25), and (B.2), Also by considering the above equation, the tracking errors, and , are acquired as follows: where . By the following definitions about the , , , and , (B.5) is rewritten as follows: with substitution of in (B.6); final relation of tracking error is demonstrated as follows:
Remark B.2. By utilizing (20) in this region, and are acquired as follows:
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