Self-Tuning Control Scheme Based on the Robustness <i>σ</i>-Modification Approach

Touijer, Nabiha; Kamoun, Samira; Essounbouli, Najib; Hamzaoui, Abdelaziz

doi:https://doi.org/10.1155/2017/8653136

Journal of Electrical and Computer Engineering

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Research Article | Open Access

Volume 2017 | Article ID 8653136 | https://doi.org/10.1155/2017/8653136

Self-Tuning Control Scheme Based on the Robustness σ-Modification Approach

Nabiha Touijer,¹Samira Kamoun,¹Najib Essounbouli,²and Abdelaziz Hamzaoui²

Academic Editor: James Lam

Received24 Jul 2016

Revised03 Jan 2017

Accepted01 Feb 2017

Published08 Mar 2017

Abstract

This paper deals with the self-tuning control problem of linear systems described by autoregressive exogenous (ARX) mathematical models in the presence of unmodelled dynamics. An explicit scheme of control is described, which we use a recursive algorithm on the basis of the robustness σ-modification approach to estimate the parameters of the system, to solve the problem of regulation tracking of the system. This approach was designed with the assumptions that the norm of the vector of the parameters is well-known. A new quadratic criterion is proposed to develop a modified recursive least squares (M-RLS) algorithm with σ-modification. The stability condition of the proposed estimation scheme is proved using the concepts of the small gain theorem. The effectiveness and reliability of the proposed M-RLS algorithm are shown by an illustrative simulation example. The effectiveness of the described explicit self-tuning control scheme is demonstrated by simulation results of the cruise control system for a vehicle.

1. Introduction

Adaptive control has been known since 1950 by Caldwell [1]. Different types of adaptive controls were discussed and used to design adaptive laws of the proposed control schemes. Various studies have been focused on the development of adaptive control theory [2–4]. Stability theory was introduced. In this context, several studies have been developed [5–15].

Egardt [16] noted that the application of adaptive laws could easily be unstable in the presence of small perturbations. In the early 1980s, the robust adaptive control behavior has become much discussed [17, 18]. Several researches developed and studied the robust adaptive control [19–29]. In continuous time, Ioannou and Sun [10] developed the robust adaptive control (pole placement control and model reference control) for dynamic systems in presence of unmodelled dynamics. The different developed control scheme has been based on algorithms with different robustness approach (dead zone, normalization) to estimate the parameters of the systems. In discrete time, different robust adaptive control schemes have been developed and applied to the class of linear systems described by a mathematical model ARX in the presence of unmodelled dynamics [30–32]. Different robust adaptive control of monovariable systems have been developed on the basis of the modified recursive least squares algorithm M-RLS with approach robustness dead zone [33–36]. The stability conditions of the different proposed estimation scheme have been demonstrated. A robust explicit scheme of self-tuning regulation using the modified filtering recursive algorithm with dead zone was applied to a temperature regulation system in the building [37]. The M-RLS algorithm was extended to a multivariable system, where the stability condition of estimation scheme has been shown and a robust self-tuning control has been developed [38]. The different parametric estimation algorithms were based on the knowledge of the bounds of the unmodelled dynamics.

This paper focuses on the regulation-tracking problem for the stochastic systems described by the ARX mathematical model, in the presence of unknown unmodelled dynamics in the parameters of the system. This problem consists of developing a control law (called the corrector) allowing the output of the system to follow a time-varying reference signal while reducing the effects of disturbances acting at different locations of the system to be controlled. An explicit scheme of self-tuning control has been designed with the assumptions that the norm of the vector of the parameters is known. A quadratic criterion is proposed to develop M-RLS algorithm with σ-modification that will be used in the estimation step of control scheme. The choice of parameter σ is given. The stability condition of the proposed parametric estimation scheme is proved using the small gain theorem [39] and based on the stability condition of the RLS algorithm [40].

The remainder of his paper is structured as follows. Section 2 describes the stochastic systems by ARX mathematical model in presence of unmodelled dynamics. Section 3, firstly, treats the RLS algorithm, and, secondly, a new quadratic criterion is proposed to develop M-RLS algorithm with σ-modification. Furthermore, the choice of σ is given. The stability condition of the developed parametric estimation scheme is shown on the basis of the concepts of the small gain theorem. Section 4 presents an explicit scheme of self-tuning control using the proposed recursive algorithm M-RLS with σ-modification to estimate the parameters of the system. Section 5 provides two simulation examples. Firstly, a simulation example is given to illustrate the reliability and the effectiveness of the proposed M-RLS algorithm with σ-modification which are compared to the RLS algorithm. And secondly the simulation results of the cruise control system for vehicles are given to show the performance of the explicit scheme of self-tuning control which is compared to the explicit scheme of self-tuning control based on the RLS algorithm. Finally, concluding remarks are given in Section 6.

2. System Description

This section describes a stochastic system by ARX mathematical model with unknown parameters and in the presence of unmodelled dynamics.

Let us consider a linear stochastic system, which can be described by the following discrete-time ARX mathematical model:where and represent, respectively, the input and the output of the system at the discrete-time , is the noise acting on the system, where is an independent random variable with zero mean and constant variance and is unknown unmodelled dynamics, and and are polynomials, which are defined, respectively, aswhere and are the orders of the polynomials and , respectively.

We suppose that the orders and are known.

The output of system (1) can be given byThe mathematical model (3) can be written as follows:where and are the parameters vector and the observation vector, respectively, such that

3. Parametric Estimation Algorithm

This section concerns solving the parametric estimation problem for the considered stochastic system (1) on the basis of the two following assumptions.

Assumption 1. The parameters intervening in vector (5) are bounded; an upper bound of is known, such thatFigure 1 represents the first area of .

Assumption 2. The parameters intervening in vector (5) are bounded; an upper bound and a lower bound, respectively, and , are known, such thatFigure 2 represents the second area of withThe aim of this section is the development of a robust recursive parametric estimation algorithm for uncertain dynamic system. Thus, we propose to use, in the parametric estimation algorithm RLS, a parameter of robustness which is known in the literature by σ-modification. The developed algorithm is called modified recursive least square (M-RLS) algorithm with σ-modification. However, before the formulation of this algorithm, we present, in the following subsection, the recursive least square algorithm RLS.

3.1. Recursive Least Square Algorithm RLS

To show the advantages of the proposed recursive parametric estimation algorithm RLS with σ-modification to be proposed later, the RLS algorithm is given to compare its performance to the performance of the proposed parametric estimation scheme, which is described in this subsection.

The recursive parametric estimation algorithm RLS is given by

Theorem 3 (see [40]). Consider a linear system which can be described by input-output mathematical model (3) (without unmodelled dynamic). The estimation of the parameters intervening in the mathematical model can be made by using RLS algorithm (10). If the components of the vectors and are finite, then the convergence of the RLS algorithm is ensured.

Lemma 4 (see [40]). Let us consider the RLS algorithm (10) to estimate the parameters intervening in (3). If the components of vectors and are finite, and if again the adaptation gain is decreasing, then the convergence of this algorithm is ensured.

In the presence of unmodelled dynamics, the inconvenient of the RLS algorithm is that, at the computing of , the corresponding norm can exceed certain threshold. Then the effectiveness of this algorithm is not ensured.

The proposed key idea is based on the two following steps.

Step 1. If Assumption 1 (or Assumption 2) is verified, then is given by the RLS algorithm (10).

Step 2. If Assumption 1 (or Assumption 2) is not verified, we propose to develop a parametric estimation algorithm such thatwhere or .

3.2. Modified Recursive Least Square Algorithm M-RLS with σ-Modification

In order to overcome the parametric estimation problem for the considered system, we will develop a modified algorithm M-RLS with σ-modification.

The following quadratic criterion is proposed to solve the parametric estimation problem for the considered system:where is a symmetrical matrix, whose choice is to give certain robustness to the developed estimation scheme with respect to the unmodelled dynamics.

The optimal of the estimated vector of parameters , which is given by the minimization of the quadratic criterion , can be obtained by the calculation of the derived of this criterion, such that In fact, the optimal of the vector of the estimated parameters corresponds to the cancellation of (13). Thus, by cancelling expression (13) derivative of the quadratic criterion considered, we can write the following expression:such thatAt the discrete-time , (14) is written as follows:Using (14) and (15) and adding and subtracting to the right member of (16), we obtainwithMultiplying (16) by , the estimated vector is given as follows:Then, the deduced recursive parametric estimation algorithm RLS with σ-modification is defined byThe determination of the following function depends on the choice of parameter :Based on Assumption 1, parameter is defined as follows:We propose to write matrix as follows:The following conditions permit determining parameter :(1)If the next condition satisfies , then we take .(2)if the next condition satisfies , then we must determine a value for the parameter , while satisfying the following condition: Using (23), (22) can be written as follows:Using the second condition and dividing (25) by , (25) can be written as follows:Then, there exists a finite scalar , such thatif , with .

Thus, the parameter is defined as follows:Based on Assumption 2, the parameter is defined as follows:We must consider the conditions intervening in the three following situations, in order to determine the parameter :(1)If the following condition satisfies , then we must take (2)If the following condition satisfies , then we must take (3)If the following condition satisfies , then we must take , where is given by (27), with .

By considering the first situation, we can define the parameter as as follows:Dividing (32) by , we can write the following inequality:In (33), adding and subtracting , we obtainThus, we can affirm that there exists a finite scalar , such thatwhere the following condition is supposed to satisfy , with .

Thus, the parameter is defined as follows:Consequently, the proposed recursive parametric estimation algorithm RLS with σ-modification is defined bywhere is defined by (28) (or (36)).

If , then the RLS algorithm has been used.

In the next, the convergence condition of the RLS algorithm is used to demonstrate the sufficient condition of stability of the proposed estimation scheme.

3.3. Stability Analysis of the Proposed Parametric Estimation Scheme

Based on the small gain theorem, the stability analysis of the proposed parametric estimation scheme is established.

Consider the closed-loop system Figure 3, where and are causal operators. The small gain theorem gives a sufficient condition for stability of the closed-loop system below, using the notion of the gain operator defined later.

Theorem 5 (small gain theorem [39]). Consider the closed-loop system Figure 3, where the operators and are bounded. Let the gains of the systems and are and , respectively. If , then the closed-loop system is input-output stable.

The a posteriori prediction error is given bywithSubtracting of the first equation in (37) and based on (39), (39) can be given byUsing (40), the a posteriori prediction error is given byLet us consider parameter , which is defined as follows:Using (41) and (42), the closed-loop system is shown in Figure 4.

Based on Lemma 4, we assume that the gain matrix is decreasing and bounded and that the components of vector are finite. If and are bounded, then there exists , , and , such thatBased on the closed-loop system shown in Figure 4, and can be written, respectively, as follows:withBased on (43) and using (44), we can writeIfthen So, if is bounded, then the stability condition of the recursive parametric estimation scheme is ensured, such thatThe norm of is given byBased on (7) (or (8)) and (25), is bounded, such thatBased on (47), (48) and (49) are given by, respectively,

Theorem 6. The closed-loop system of the proposed recursive parametric estimation scheme shown in Figure 4 is stable; if the operator and are bounded and have positive gain, respectively, and are defined, such that .

4. Explicit Scheme of Self-Tuning Control

This section discusses the regulation-tracking problem for the considered system, where an explicit scheme of self-tuning control will be developed. The following quadratic criterion is used to design the controller:where represents the desired output signal, is the control law, and and are two polynomials, such thatNote that the orders and of the polynomials and , respectively, are chosen by the designer.

The derivate of the criterion , which is described by (55), is given bywithwhere and are solutions of the following polynomial equation:The polynomials and are given byThus, the optimal control law can be written bywhere the polynomials and are given by, respectively,

4.1. Explicit Scheme of Self-Tuning Control

The recursive algorithm of the explicit robust self-tuning control scheme is formulated by the following steps.

Step 1. Estimate the parameters intervening in the ARX mathematical model (1) using the M-RLS algorithm with -modification (37).

Step 2. Calculate the parameters intervening in the polynomials and by solving the polynomial equation defined as follows:

Step 3. Calculate the control law given by the following equation:Note that if , then we take .

5. Simulation Results

5.1. Simulation Example 1

Let us consider that the dynamic system can be described by the following mathematical model ARX:where and are the output and the input of the second-order system with time delay being one and is white noise acting on the system.

The output of the system can be given as follows:with where and represent, respectively, the vector of the parameters and the vector of the observations.

The bounds of unmodelled dynamic presented in the system are unknown, but the norm of the vector of the parameters is given by the following inequality:with and .

In simulation, the nominal values of the uncertain parameters of system are defined by the following.

, , , and . The evolution curve of the norm of the vector of the nominal values of parameters is given Figure 5.

The objective of this simulation example is the demonstration of the performance of the robust recursive algorithm for parameter estimation M-RLS with σ-modification (37). A comparative study between the recursive algorithm RLS (10) and the proposed recursive algorithm (37) is treated. The more robust algorithm is the algorithm which can estimate the parameters such that the norm of the vector of the estimated parameters is inside the desired area.

The input signal is a square signal with amplitude that equals two and a period that equal 100, is a sequence of random variables with zero mean and variance , and the gain matrix .

We use the recursive algorithm RLS (10) to estimate the parameters involved in (67). Figure 6 shows the evolution curve of the variance of the prediction error and Figure 7 shows the evolution curve of the norm of the vector of the estimated parameters .

We use the proposed recursive algorithm M-RLS with σ-modification (37) to estimate the parameters involved in (67). Figure 8 represents the evolution curve of the variance of the prediction error and Figure 9 represents the evolution curve of the norm of the vector of the estimated parameters .

Based on the simulation results, we conclude that the proposed recursive algorithm M-RLS with σ-modification (37) is more robust than the recursive algorithm defined by (10).

5.2. Simulation Example 2: The Vehicle

We treat here an example of numerical simulation which is related to the control of a vehicle of laboratory, by using the described algorithm of the explicit scheme of the self-tuning control. Figure 10 represents the scheme of this vehicle, as considered by Sam Fadali [41], in which is the input force, is the velocity of this vehicle, and is the coefficient of viscous friction.

Sam Fadali [41] determined the following transfer function in open loop, such that describes the dynamic behavior of the vehicle:The discrete transfer function relating to (70) can be defined as follows, such that the used sampling period is sec:For the system to be stable, the closed-loop poles or the roots of the following characteristic equationmust lie within the unit circle.

To ensure this condition of the stability of the system in closed loop, the parameter must be defined as follows:This system can be described by the following mathematical model ARX:where represents the velocity of vehicle, represents the input force, and are unknown parameters, and is noise which can be given by the following equation:in which the element designates the unmodelled dynamics related to the parameter .

The output of the vehicle can be defined as follows:where the vectors of the parameters and of the observation are given by the following expression, respectively:Thus, the data of the explicit scheme of the proposed robust self-tuning control are as follows:(1)The different values of the parameters involved in (77) are chosen such that , , .(2)The sequence of noise is composed of independent random variables with zero mean and constant variance .(3)We will take and (where is an identity matrix).(4)The application of the recursive algorithm M-RLS with σ-modification is based on the knowledge of the region where , such that(5), , and .(6)The evolution curve of the reference velocity is shown in Figure 11.

The tracking error is defined byUsing the same initial conditions, we will compare the numerical simulation results of the explicit scheme of self-tuning based on the recursive algorithm RLS (10) (control scheme ) and of the robust explicit scheme of self-tuning control based on the robust recursive algorithm M-RLS with σ-modification (37) (control scheme ). Control laws are applied to the example of the vehicle in the presence of unmodelled dynamics.

Figure 12 show the evolution curve of the variance of the tracking error and Figure 13 show the evolution curve of the variance of the prediction error in the control scheme based on the recursive algorithm RLS (10) to estimate the parameters involved in (79) with considering (77).

In control scheme , Figure 14 shows the evolution curve of the velocity , Figure 15 shows the evolution curve of the input force (or the control law), Figure 16 shows the evolution curve of the estimated parameter , Figure 17 shows the evolution curve of the estimated parameter , Figure 18 shows the evolution curve of the norm of the vector of the estimated parameters , Figure 19 shows the evolution curve of the variance of the tracking error , and Figure 20 shows the evolution curve of the variance of the prediction error .

The different illustrated simulation results in Figures 11–20 show the performance of the developed robust explicit self-tuning control scheme on the basis of the proposed M-RLS algorithm with the robustness σ-modification approach. This control scheme is robust, in the presence of unknown unmodelled dynamics, and allows the output to follow the desired velocity while reducing the effects of disturbances acting at different locations in the system. In addition, the estimated parameters are within the desired region.

6. Conclusion

In this paper, we have proposed the M-RLS algorithm with the robustness σ-modification approach. This approach was designed assuming that the bound of desired system parameters norm is known. The stability condition of the parametric estimation scheme was established using the concepts of the small gain theorem. A numerical simulation example has shown the effectiveness and the performance of M-RLS algorithm with σ-modification.

An explicit scheme of self-tuning control was developed to solve the regulation-tracking problem for the linear systems in the presence of unknown unmodelled dynamics. This control scheme was based on the proposed M-RLS algorithm with σ-modification approach. The robustness of the proposed control scheme for the stochastic system, in the presence of unknown unmodelled dynamics, is shown using the simulation results of the cruise control system for the vehicle.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2017 Nabiha Touijer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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