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

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 wellknown.Anewquadratic criterion is proposed to develop amodified recursive least squares (M-RLS) algorithmwithσ-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 proposedM-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.

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][20][21][22][23][24][25][26][27][28][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][31][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][34][35][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 2 Journal of Electrical and Computer Engineering 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 -modification that will be used in the estimation step of control scheme.The choice of parameter  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 modification.Furthermore, the choice of  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 selftuning control using the proposed recursive algorithm M-RLS with -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 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.

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 , V() = () + () 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 ( −1 ) and ( −1 ) are polynomials, which are defined, respectively, as where   and   are the orders of the polynomials ( −1 ) and ( −1 ), respectively.We suppose that the orders   and   are known.The output () of system (1) can be given by The mathematical model (3) can be written as follows: where   and   () are the parameters vector and the observation vector, respectively, such that

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  0 of  is known, such that Figure 1 represents the first area of .
Assumption 2. The parameters intervening in vector  (5) are bounded; an upper bound and a lower bound, respectively,  max and  min , are known, such that Figure 2 represents the second area of  with The 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 -modification.The developed algorithm is called modified recursive least square (M-RLS) algorithm with -modification.However, before the formulation of this algorithm, we present, in the following subsection, the recursive least square algorithm RLS.

Recursive Least Square Algorithm RLS.
To show the advantages of the proposed recursive parametric estimation algorithm RLS with -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 θ () = θ ( − 1) +  ()  ()  () ,  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 θ(0) 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 θ(0) 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 θ(−1), 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.

Modified Recursive Least Square Algorithm M-RLS with
-Modification.In order to overcome the parametric estimation problem for the considered system, we will develop a modified algorithm M-RLS with -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.
If ( − 1) = 0, 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.

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  1 and  2 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 closedloop system Figure 3, where the operators  1 and  2 are bounded.Let the gains of the systems  1 and  2 are  1 and  2 , respectively.If  1  2 < 1, then the closed-loop system is inputoutput stable.
The a posteriori prediction error  ∘ () is given by with Subtracting  of the first equation in (37) and based on ( 39), ( 39) can be given by Using (40), the a posteriori prediction error  ∘ () is given by Let 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  1 ≥ 0,  2 ≥ 0,  1 and  2 , such that Based on the closed-loop system shown in Figure 4,  ∘ () and θ() can be written, respectively, as follows: with  Based on (43) and using ( 44 If then So, if  2 () is bounded, then the stability condition of the recursive parametric estimation scheme is ensured, such that Based 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,  1 and  2 are defined, such that  1  2 < 1.

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 ( +  + 1) is used to design the controller: where   ( +  + 1) represents the desired output signal, () is the control law, and ( −1 ) and ( −1 ) are two polynomials, such that (56) Note that the orders   and   of the polynomials ( −1 ) and ( −1 ), respectively, are chosen by the designer.The derivate of the criterion (++1), which is described by (55), is given by with where ( −1 , ) and ( −1 , ) are solutions of the following polynomial equation: The polynomials ( −1 , ) and ( −1 , ) are given by Thus, the optimal control law () can be written by where the polynomials ( −1 , ) and ( −1 , ) are given by, respectively, (62) 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).

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  min = 0.85 and  max = 1.05.
In simulation, the nominal values of the uncertain parameters of system are defined by the following.
The objective of this simulation example is the demonstration of the performance of the robust recursive algorithm for parameter estimation M-RLS with -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  2 = 0.2, and the gain matrix (0) = 1000.
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  2  () and Figure 7 shows the evolution curve of the norm of the vector of the estimated parameters ‖ θ()‖.
We use the proposed recursive algorithm M-RLS with modification (37) to estimate the parameters involved in (67).Figure 8 represents the evolution curve of the variance of the prediction error  2  () and Figure 9 represents the evolution curve of the norm of the vector of the estimated parameters ‖ θ()‖.
Based on the simulation results, we conclude that the proposed recursive algorithm M-RLS with -modification (37) is more robust than the recursive algorithm defined by (10).

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 ( −1 ) relating to (70) can be defined as follows, such that the used sampling period is   = 0.02 sec: The  For the system to be stable, the closed-loop poles or the roots of the following characteristic equation 1 +  ( −1 ) = 0 (72) must lie within the unit circle.
To ensure this condition of the stability of the system in closed loop, the parameter  1 () must be defined as follows: −0.054 <  1 () < 1.941. (73) This system can be described by the following mathematical model ARX: where () represents the velocity of vehicle, () represents the input force,  1 and  1 () are unknown parameters, and V() is noise which can be given by the following equation: in which the element () designates the unmodelled dynamics related to the parameter  1 ().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  1 = −0.9418, 1 () = 2 + 0.1 sin(0.2),() = 0.05 sin(0.2)(− 1).
(6) The evolution curve of the reference velocity   () is shown in Figure 11.
The tracking error is defined by Using the same initial conditions, we will compare the numerical simulation results of the explicit scheme of selftuning based on the recursive algorithm RLS (10) (control scheme (1)) and of the robust explicit scheme of self-tuning control based on the robust recursive algorithm M-RLS with -modification (37) (control scheme (2)).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 (1) based on the recursive algorithm RLS (10) to estimate the parameters involved in (79) with considering (77).
In control scheme (2), 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 â1 (), Figure 17 shows the evolution curve of the estimated parameter b1 (), Figure 18 shows the evolution curve of the norm of the vector of the estimated parameters ‖ θ()‖, Figure 19 shows the evolution curve of the variance of the tracking error  2  (), and Figure 20 shows the evolution curve of the variance of the prediction error  2  ().The different illustrated simulation results in  show the performance of the developed robust explicit self-tuning control scheme on the basis of the proposed M-RLS algorithm with the robustness -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.

Conclusion
In this paper, we have proposed the M-RLS algorithm with the robustness -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 -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 -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.

Figure 1 :Figure 2 :
Figure 1: Representation of the first area.

Figure 4 :
Figure 4: Closed-loop system of the parametric estimation scheme.

Figure 5 :
Figure 5: Evolution curve of the norm of the vector of the nominal values of parameters ‖()‖.

Figure 6 :
Figure 6: Evolution curve of the variance of the prediction error  2  ().

Figure 7 :
Figure 7: Evolution curve of the norm of the vector of the estimated parameters ‖ θ()‖.

Figure 8 :
Figure 8: Evolution curve of the variance of the prediction error  2  ().

Figure 9 :Figure 10 :
Figure 9: Evolution curve of the norm of the vector of the estimated parameters ‖ θ()‖.

Figure 15 :
Figure 15: Evolution curve of the control law ().

Figure 19 :
Figure 19: Evolution curve of the variance of the tracking error  2  ().

Figure 20 :
Figure 20: Evolution curve of the variance of the prediction error  2  ().