# Self-Tuning Control Scheme Based on the Robustnessσ-Modification Approach

Authors: Nabiha Touijer; Samira Kamoun; Najib Essounbouli; Abdelaziz Hamzaoui
Journal: Journal of Electrical and Computer Engineering
 (2017)

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## 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 robustnessσ-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.


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## Body

## 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 withσ-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. Section2 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:(1)Aq-1yk=Bq-1uk+vk,where u(k) and y(k) represent, respectively, the input and the output of the system at the discrete-time k, v(k)=e(k)+m(k) is the noise acting on the system, where e(k) is an independent random variable with zero mean and constant variance and m(k) is unknown unmodelled dynamics, and A(q-1) and B(q-1) are polynomials, which are defined, respectively, as(2)Aq-1=1+a1q-1+⋯+anaq-na,Bq-1=b1q-1+⋯+bnbq-nb,where na and nb are the orders of the polynomials A(q-1) and B(q-1), respectively.We suppose that the ordersna and nb are known.The outputy(k) of system (1) can be given by(3)yk=-a1yk-1-⋯-anayk-na+b1uk-1+⋯+bnbuk-nb+vk.The mathematical model (3) can be written as follows:(4)yk=θTφk+vk,where θT and φT(k) are the parameters vector and the observation vector, respectively, such that(5)θT=a1⋯anab1⋯bnb,(6)φTk=-yk-1-⋯-yk-nauk-1⋯uk-nb.


## 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 vectorθ (5) are bounded; an upper bound M0 of θ is known, such that(7)0<θ≤M0.Figure 1 represents the first area of θ.Figure 1
Representation of the first area.Assumption 2.
The parameters intervening in vectorθ (5) are bounded; an upper bound and a lower bound, respectively, Mmax and Mmin, are known, such that(8)0<Mmin≤θ≤Mmax.Figure 2 represents the second area of θ with(9)θ=∑i=1naai2+∑j=1nbbj2.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 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.Figure 2
Representation of the second area.


### 3.1. Recursive Least Square Algorithm RLS

To show the advantages of the proposed recursive parametric estimation algorithm RLS withσ-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(10)θ^k=θ^k-1+Pkφkεk,Pk=Pk-1-Pk-1φkφTkPk-11+φTkPk-1φk,εk=yk-φTkθ^k-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 φ(k) 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 φ(k) are finite, and if again the adaptation gain P(k) 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 ofθ^(k-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.Step 1.
If Assumption1 (or Assumption 2) is verified, then θ^(k) is given by the RLS algorithm (10).Step 2.
If Assumption1 (or Assumption 2) is not verified, we propose to develop a parametric estimation algorithm such that(11)θ^k=fθ^k-1+Pkφkεk,where 0<f(θ^(k-1))≤M0 or Mmin≤f(θ^(k-1))≤Mmax.


### 3.2. Modified Recursive Least Square Algorithm M-RLS withσ-Modification

In order to overcome the parametric estimation problem for the considered system, we will develop a modified algorithm M-RLS withσ-modification.The following quadratic criterion is proposed to solve the parametric estimation problem for the considered system:(12)Jk=∑i=na+1kyi-θ^Tkφi22+∑i=na+1kθ^Tkζiθ^i,where ζ(i) 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 parametersθ^(i), which is given by the minimization of the quadratic criterion J(k), can be obtained by the calculation of the derived of this criterion, such that (13)∂Jk∂θ^i=-∑i=na+1kyiφi+∑i=na+1kφiφTiθ^k+∑i=na+1k-1ζiθ^i.In fact, the optimal of the vector of the estimated parameters θ^(i) corresponds to the cancellation of (13). Thus, by cancelling expression (13) derivative of the J(k) quadratic criterion considered, we can write the following expression:(14)P-1kθ^k=∑i=na+1kyiφi-∑i=na+1k-1ζiθ^i,such that(15)P-1k=∑i=na+1kφiφTi.At the discrete-time k+1, (14) is written as follows:(16)P-1k+1θ^k+1=∑i=na+1k+1yiφi-∑i=na+1kζiθ^i=∑i=na+1kyiφi-∑i=na+1k-1ζiθ^i+yk+1φk+1-ζkθ^k.Using (14) and (15) and adding and subtracting φ(k+1)φT(k+1)θ^(k) to the right member of (16), we obtain(17)P-1k+1θ^k+1=P-1k+1θ^k+φk+1εk+1-ζkθ^kwith(18)εk+1=yk+1-φTk+1θ^k.Multiplying (16) by P(k+1), the estimated vector θ^(k+1) is given as follows:(19)θ^k+1=θ^k-ζkPk+1θ^k+Pk+1φk+1εk+1.Then, the deduced recursive parametric estimation algorithm RLS with σ-modification is defined by(20)θ^k=θ^k-1-ζk-1Pkθ^k-1+Pkφkεk,Pk=Pk-1-Pk-1φkφTkPk-11+φTkPk-1φk,εk=yk-φTkθ^k-1.The determination of the following function depends on the choice of parameter ζ(k-1):(21)fθ^k-1=θ^k-1-ζk-1Pkθ^k-1.Based on Assumption 1, parameter ζ(k-1) is defined as follows:(22)0<θ^k-1-ζk-1Pkθ^k-1<M0.We propose to write matrix ζ(k-1) as follows:(23)ζk-1=σk-1P-1k.The following conditions permit determining parameter σ(k-1):(1)
If the next condition satisfies0<θ^(k-1)≤M0, then we take σ(k-1)=0.(2)
if the next condition satisfiesθ^(k-1)>M0, then we must determine a value for the parameter σ(k-1), while satisfying the following condition: (24)σk-1<1.Using (23), (22) can be written as follows:(25)0<1-σk-1θ^k-1<M0.Using the second condition and dividing (25) by θ^(k-1), (25) can be written as follows:(26)0<1-M0θ^k-1<σk<1.Then, there exists a finite scalar σ0, such that(27)σk=σ01-M0θ^k-1,if θ^(k-1)>M0, with σ0>1.Thus, the parameterσ(k-1) is defined as follows:(28)σk-1=0,ifθ^k-1≤M0σmaxk-1=σ01-M0θ^k-1,either,withσ0>1.Based on Assumption 2, the parameter σ(k-1) is defined as follows:(29)Mmin≤θ^k-1-σk-1Pkθ^k-1≤Mmax.We must consider the conditions intervening in the three following situations, in order to determine the parameter σ(k-1):(1)
If the following condition satisfiesθ^(k-1)<Mmin, then we must take (30)σk-1=σmink-1.(2)
If the following condition satisfiesMmin≤θ^(k-1)≤Mmax, then we must take (31)σk-1=0.(3)
If the following condition satisfiesθ^(k-1)>Mmax, then we must take σ(k-1)=σmax(k-1), where σ(k-1) is given by (27), with M0=Mmax.By considering the first situation, we can define the parameter asσmin(k-1) as follows:(32)θ^k-1-σmink-1θ^k-1≥Mmin.Dividing (32) by θ^(k-1), we can write the following inequality:(33)1-σmink-1>Mminθ^k-1.In (33), adding σmin(k-1) and subtracting Mmin/θ^(k-1), we obtain(34)σmink-1<1-Mminθ^k-1<0.Thus, we can affirm that there exists a finite scalar σ1, such that(35)σmink=σ11-Mminθ^k-1,where the following condition is supposed to satisfy θ^(k-1)<Mmin, with σ1>0.Thus, the parameterσ(k-1) is defined as follows:(36)σk-1=σmink-1=σ11-Mminθ^k-1ifθ^k-1<Mmin0,ifMmin≤θ^k-1≤Mmaxσmaxk-1=σ01-Mmaxθ^k-1,either,withσ1,σ0>1.Consequently, the proposed recursive parametric estimation algorithm RLS with σ-modification is defined by(37)θ^k=θ^k-1-σk-1θ^k-1+Pkφkεk,Pk=Pk-1-Pk-1φkφTkPk-11+φTkPk-1φk,εk=yk-φTkθ^k-1,where σ(k-1) is defined by (28) (or (36)).Ifσ(k-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.


### 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 Figure3, where H1 and H2 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.Figure 3
General closed-loop system.Theorem 5 (small gain theorem [39]).
Consider the closed-loop system Figure3, where the operators H1 and H2 are bounded. Let the gains of the systems H1 and H2 are γ1 and γ2, respectively. If γ1γ2<1, then the closed-loop system is input-output stable.The a posteriori prediction errorε∘(k) is given by(38)ε∘k=yk-φTkθ^k=-φTkθ~kwith(39)θ~k=θ^k-θ.Subtracting θ of the first equation in (37) and based on (39), (39) can be given by(40)θ~k=θ~k-1+Pkφkε∘k-σk-1θ^k-1.Using (40), the a posteriori prediction error ε∘(k) is given by(41)ε∘k=-φTkPkφkε∘k+θ~k-1-σk-1θ^k-1.Let us consider parameter w(k), which is defined as follows:(42)wk=-ε∘k.Using (41) and (42), the closed-loop system is shown in Figure 4.Figure 4
Closed-loop system of the parametric estimation scheme.Based on Lemma4, we assume that the gain matrix P(k) is decreasing and bounded and that the components of vector φ(k) are finite. If P(k)φ(k) and φ(k) are bounded, then there exists γ1≥0, γ2≥0, β1 and β2, such that(43)Pkφkε∘k≤γ1ε∘k+β1,φkθ~k≤γ2θ~k+β2.Based on the closed-loop system shown in Figure 4, ε∘(k) and θ~(k) can be written, respectively, as follows:(44)ε∘k=-φTkθ~k+U1=-φTkθ~k,θ~k=Pkφkε∘k+U2kwith(45)U1=0,U2k=θ~k-1-σk-1θ^k-1.Based on (43) and using (44), we can write(46)ε∘k≤γ2θ~k+β2,(47)θ~k≤U2k+γ1ε∘k+β1.If(48)γ1γ2<1,then (49)ε∘k≤11-γ1γ2γ2U2k+γ2β1+β2,(50)θ~k≤11-γ1γ2U2k+γ1β2+β1.So, if U2(k) is bounded, then the stability condition of the recursive parametric estimation scheme is ensured, such that(51)U2k=θ~k-1-σk-1θ^k-1=θ^k-1-θ-σk-1θ^k-1=θ^k-11-σk-1-θ.The norm of U2(k) is given by(52)U2k=θ^k-11-σk-1-θ≤θ^k-11-σk-1+θ.Based on (7) (or (8)) and (25), U2(k) is bounded, such that(53)U2k≤2M0.Based on (47), (48) and (49) are given by, respectively,(54)ε∘k≤11-γ1γ22γ2M0+γ2β1+β2,θ~k≤11-γ1γ22M0+γ1β2+β1.Theorem 6.
The closed-loop system of the proposed recursive parametric estimation scheme shown in Figure4 is stable; if the operator P(k)φ(k) and φ(k) are bounded and have positive gain, respectively, γ1 and γ2 are defined, such that γ1γ2<1.


## 3.1. Recursive Least Square Algorithm RLS

To show the advantages of the proposed recursive parametric estimation algorithm RLS withσ-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(10)θ^k=θ^k-1+Pkφkεk,Pk=Pk-1-Pk-1φkφTkPk-11+φTkPk-1φk,εk=yk-φTkθ^k-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 φ(k) 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 φ(k) are finite, and if again the adaptation gain P(k) 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 ofθ^(k-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.Step 1.
If Assumption1 (or Assumption 2) is verified, then θ^(k) is given by the RLS algorithm (10).Step 2.
If Assumption1 (or Assumption 2) is not verified, we propose to develop a parametric estimation algorithm such that(11)θ^k=fθ^k-1+Pkφkεk,where 0<f(θ^(k-1))≤M0 or Mmin≤f(θ^(k-1))≤Mmax.


## 3.2. Modified Recursive Least Square Algorithm M-RLS withσ-Modification

In order to overcome the parametric estimation problem for the considered system, we will develop a modified algorithm M-RLS withσ-modification.The following quadratic criterion is proposed to solve the parametric estimation problem for the considered system:(12)Jk=∑i=na+1kyi-θ^Tkφi22+∑i=na+1kθ^Tkζiθ^i,where ζ(i) 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 parametersθ^(i), which is given by the minimization of the quadratic criterion J(k), can be obtained by the calculation of the derived of this criterion, such that (13)∂Jk∂θ^i=-∑i=na+1kyiφi+∑i=na+1kφiφTiθ^k+∑i=na+1k-1ζiθ^i.In fact, the optimal of the vector of the estimated parameters θ^(i) corresponds to the cancellation of (13). Thus, by cancelling expression (13) derivative of the J(k) quadratic criterion considered, we can write the following expression:(14)P-1kθ^k=∑i=na+1kyiφi-∑i=na+1k-1ζiθ^i,such that(15)P-1k=∑i=na+1kφiφTi.At the discrete-time k+1, (14) is written as follows:(16)P-1k+1θ^k+1=∑i=na+1k+1yiφi-∑i=na+1kζiθ^i=∑i=na+1kyiφi-∑i=na+1k-1ζiθ^i+yk+1φk+1-ζkθ^k.Using (14) and (15) and adding and subtracting φ(k+1)φT(k+1)θ^(k) to the right member of (16), we obtain(17)P-1k+1θ^k+1=P-1k+1θ^k+φk+1εk+1-ζkθ^kwith(18)εk+1=yk+1-φTk+1θ^k.Multiplying (16) by P(k+1), the estimated vector θ^(k+1) is given as follows:(19)θ^k+1=θ^k-ζkPk+1θ^k+Pk+1φk+1εk+1.Then, the deduced recursive parametric estimation algorithm RLS with σ-modification is defined by(20)θ^k=θ^k-1-ζk-1Pkθ^k-1+Pkφkεk,Pk=Pk-1-Pk-1φkφTkPk-11+φTkPk-1φk,εk=yk-φTkθ^k-1.The determination of the following function depends on the choice of parameter ζ(k-1):(21)fθ^k-1=θ^k-1-ζk-1Pkθ^k-1.Based on Assumption 1, parameter ζ(k-1) is defined as follows:(22)0<θ^k-1-ζk-1Pkθ^k-1<M0.We propose to write matrix ζ(k-1) as follows:(23)ζk-1=σk-1P-1k.The following conditions permit determining parameter σ(k-1):(1)
If the next condition satisfies0<θ^(k-1)≤M0, then we take σ(k-1)=0.(2)
if the next condition satisfiesθ^(k-1)>M0, then we must determine a value for the parameter σ(k-1), while satisfying the following condition: (24)σk-1<1.Using (23), (22) can be written as follows:(25)0<1-σk-1θ^k-1<M0.Using the second condition and dividing (25) by θ^(k-1), (25) can be written as follows:(26)0<1-M0θ^k-1<σk<1.Then, there exists a finite scalar σ0, such that(27)σk=σ01-M0θ^k-1,if θ^(k-1)>M0, with σ0>1.Thus, the parameterσ(k-1) is defined as follows:(28)σk-1=0,ifθ^k-1≤M0σmaxk-1=σ01-M0θ^k-1,either,withσ0>1.Based on Assumption 2, the parameter σ(k-1) is defined as follows:(29)Mmin≤θ^k-1-σk-1Pkθ^k-1≤Mmax.We must consider the conditions intervening in the three following situations, in order to determine the parameter σ(k-1):(1)
If the following condition satisfiesθ^(k-1)<Mmin, then we must take (30)σk-1=σmink-1.(2)
If the following condition satisfiesMmin≤θ^(k-1)≤Mmax, then we must take (31)σk-1=0.(3)
If the following condition satisfiesθ^(k-1)>Mmax, then we must take σ(k-1)=σmax(k-1), where σ(k-1) is given by (27), with M0=Mmax.By considering the first situation, we can define the parameter asσmin(k-1) as follows:(32)θ^k-1-σmink-1θ^k-1≥Mmin.Dividing (32) by θ^(k-1), we can write the following inequality:(33)1-σmink-1>Mminθ^k-1.In (33), adding σmin(k-1) and subtracting Mmin/θ^(k-1), we obtain(34)σmink-1<1-Mminθ^k-1<0.Thus, we can affirm that there exists a finite scalar σ1, such that(35)σmink=σ11-Mminθ^k-1,where the following condition is supposed to satisfy θ^(k-1)<Mmin, with σ1>0.Thus, the parameterσ(k-1) is defined as follows:(36)σk-1=σmink-1=σ11-Mminθ^k-1ifθ^k-1<Mmin0,ifMmin≤θ^k-1≤Mmaxσmaxk-1=σ01-Mmaxθ^k-1,either,withσ1,σ0>1.Consequently, the proposed recursive parametric estimation algorithm RLS with σ-modification is defined by(37)θ^k=θ^k-1-σk-1θ^k-1+Pkφkεk,Pk=Pk-1-Pk-1φkφTkPk-11+φTkPk-1φk,εk=yk-φTkθ^k-1,where σ(k-1) is defined by (28) (or (36)).Ifσ(k-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.


## 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 Figure3, where H1 and H2 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.Figure 3
General closed-loop system.Theorem 5 (small gain theorem [39]).
Consider the closed-loop system Figure3, where the operators H1 and H2 are bounded. Let the gains of the systems H1 and H2 are γ1 and γ2, respectively. If γ1γ2<1, then the closed-loop system is input-output stable.The a posteriori prediction errorε∘(k) is given by(38)ε∘k=yk-φTkθ^k=-φTkθ~kwith(39)θ~k=θ^k-θ.Subtracting θ of the first equation in (37) and based on (39), (39) can be given by(40)θ~k=θ~k-1+Pkφkε∘k-σk-1θ^k-1.Using (40), the a posteriori prediction error ε∘(k) is given by(41)ε∘k=-φTkPkφkε∘k+θ~k-1-σk-1θ^k-1.Let us consider parameter w(k), which is defined as follows:(42)wk=-ε∘k.Using (41) and (42), the closed-loop system is shown in Figure 4.Figure 4
Closed-loop system of the parametric estimation scheme.Based on Lemma4, we assume that the gain matrix P(k) is decreasing and bounded and that the components of vector φ(k) are finite. If P(k)φ(k) and φ(k) are bounded, then there exists γ1≥0, γ2≥0, β1 and β2, such that(43)Pkφkε∘k≤γ1ε∘k+β1,φkθ~k≤γ2θ~k+β2.Based on the closed-loop system shown in Figure 4, ε∘(k) and θ~(k) can be written, respectively, as follows:(44)ε∘k=-φTkθ~k+U1=-φTkθ~k,θ~k=Pkφkε∘k+U2kwith(45)U1=0,U2k=θ~k-1-σk-1θ^k-1.Based on (43) and using (44), we can write(46)ε∘k≤γ2θ~k+β2,(47)θ~k≤U2k+γ1ε∘k+β1.If(48)γ1γ2<1,then (49)ε∘k≤11-γ1γ2γ2U2k+γ2β1+β2,(50)θ~k≤11-γ1γ2U2k+γ1β2+β1.So, if U2(k) is bounded, then the stability condition of the recursive parametric estimation scheme is ensured, such that(51)U2k=θ~k-1-σk-1θ^k-1=θ^k-1-θ-σk-1θ^k-1=θ^k-11-σk-1-θ.The norm of U2(k) is given by(52)U2k=θ^k-11-σk-1-θ≤θ^k-11-σk-1+θ.Based on (7) (or (8)) and (25), U2(k) is bounded, such that(53)U2k≤2M0.Based on (47), (48) and (49) are given by, respectively,(54)ε∘k≤11-γ1γ22γ2M0+γ2β1+β2,θ~k≤11-γ1γ22M0+γ1β2+β1.Theorem 6.
The closed-loop system of the proposed recursive parametric estimation scheme shown in Figure4 is stable; if the operator P(k)φ(k) and φ(k) are bounded and have positive gain, respectively, γ1 and γ2 are defined, such that γ1γ2<1.


## 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 criterionJ(k+d+1) is used to design the controller:(55)Jk+d+1=Sq-1yk+d+1-yrk+d+12+Qq-1uk2,where yr(k+d+1) represents the desired output signal, u(k) is the control law, and S(q-1) and Q(q-1) are two polynomials, such that(56)Sq-1=1+s1q-1+⋯+snsq-ns,Qq-1=q0+q1q-1+⋯+qnqq-nq.Note that the orders ns and nq of the polynomials S(q-1) and Q(q-1), respectively, are chosen by the designer.The derivate of the criterionJ(k+d+1), which is described by (55), is given by(57)∂Jk+d+1∂uk=2b1Sq-1yk+d+1-yrk+d+1+2q0Qq-1ukwith(58)Sq-1yk+d+1=qBq-1Fq-1,kuk+Gq-1,kyk+Fq-1,kvk+d+1,where F(q-1,k) and G(q-1,k) are solutions of the following polynomial equation:(59)Sq-1=Aq-1Fq-1,k+q-d-1Gq-1,k.The polynomials F(q-1,k) and G(q-1,k) are given by(60)Fq-1,k=1+F1kq-1+⋯+Fdkq-d,Gq-1,k=G0k+G1kq-1+⋯+Gna-1kq1-na.Thus, the optimal control law u(k) can be written by(61)uk=1Zq-1,k-Gq-1,kyk+1b1Sq-1yrk+d+1,where the polynomials H(q-1,k) and Z(q-1,k) are given by, respectively,(62)Zq-1,k=Hq-1,k+1b1q0Qq-1,Hq-1,k=qBq-1Fq-1,k.


### 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 σ-modification (37).Step 2.
Calculate the parameters intervening in the polynomialsF(q-1,k) and G(q-1,k) by solving the polynomial equation defined as follows:(63)Sq-1=A^q-1Fq-1,k+q-d-1Gq-1,k.Step 3.
Calculate the control lawu(k) given by the following equation:(64)uk=1z1k-∑r=2nb+d-1zrkuk-r+1-∑t=0na-1gtkyk-t+1b^1kyrk+d+1-j+∑j=1nsstb^1kyrk+d+1-j.Note that if b^1(k)=0, then we take h1(k)=0.01.


## 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 σ-modification (37).Step 2.
Calculate the parameters intervening in the polynomialsF(q-1,k) and G(q-1,k) by solving the polynomial equation defined as follows:(63)Sq-1=A^q-1Fq-1,k+q-d-1Gq-1,k.Step 3.
Calculate the control lawu(k) given by the following equation:(64)uk=1z1k-∑r=2nb+d-1zrkuk-r+1-∑t=0na-1gtkyk-t+1b^1kyrk+d+1-j+∑j=1nsstb^1kyrk+d+1-j.Note that if b^1(k)=0, then we take h1(k)=0.01.


## 5. Simulation Results

### 5.1. Simulation Example 1

Let us consider that the dynamic system can be described by the following mathematical model ARX:(65)yk=-a1kyk-1-a2kyk-2+b1kuk-2+b2kuk-3+ek,where y(k) and u(k) are the output and the input of the second-order system with time delay being one and e(k) is white noise acting on the system.The output of the system can be given as follows:(66)yk=θTkφkwith (67)θTk=a1ka2kb1kb2k,(68)φTk=-yk-1-yk-2uk-2uk-3,where θ(k) and φ(k) 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:(69)Mmin≤θ∗≤Mmaxwith Mmin=0.85 and Mmax=1.05.In simulation, the nominal values of the uncertain parameters of system are defined by the following.a
1
(
k
)
=
0.79
+
0.1
sin
⁡
(
0.1
k
), a2(k)=0.2+0.1cos⁡(0.1k), b1(k)=0.4+0.1sin⁡(0.1k), and b2(k)=0.3+0.1cos⁡(0.1k). The evolution curve of the norm of the vector of the nominal values of parameters is given Figure 5.Figure 5
Evolution curve of the norm of the vector of the nominal values of parametersθ(k).The objective of this simulation example is the demonstration of the performance of the robust recursive algorithm for parameter estimation M-RLS withσ-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,e(k) is a sequence of random variables with zero mean and variance σ2=0.2, and the gain matrix P(0)=1000I.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 σε2(k) and Figure 7 shows the evolution curve of the norm of the vector of the estimated parameters θ^(k).Figure 6
Evolution curve of the variance of the prediction errorσε2(k).Figure 7
Evolution curve of the norm of the vector of the estimated parametersθ^(k).We use the proposed recursive algorithm M-RLS withσ-modification (37) to estimate the parameters involved in (67). Figure 8 represents the evolution curve of the variance of the prediction error σε2(k) and Figure 9 represents the evolution curve of the norm of the vector of the estimated parameters θ^(k).Figure 8
Evolution curve of the variance of the prediction errorσε2(k).Figure 9
Evolution curve of the norm of the vector of the estimated parametersθ^(k).Based on the simulation results, we conclude that the proposed recursive algorithm M-RLS withσ-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. Figure10 represents the scheme of this vehicle, as considered by Sam Fadali [41], in which U is the input force, V is the velocity of this vehicle, and b is the coefficient of viscous friction.Figure 10
Schematic representation of the cruise control system for a vehicle.Sam Fadali [41] determined the following transfer function G(s) in open loop, such that describes the dynamic behavior of the vehicle:(70)Gs=Ks+3.The discrete transfer function G(q-1) relating to (70) can be defined as follows, such that the used sampling period is Te=0.02 sec:(71)Gq-1=b1kq-11-0.9418q-1.For the system to be stable, the closed-loop poles or the roots of the following characteristic equation(72)1+Gq-1=0must lie within the unit circle.To ensure this condition of the stability of the system in closed loop, the parameterb1(k) must be defined as follows:(73)-0.054<b1k<1.941.This system can be described by the following mathematical model ARX:(74)yk=-a1yk-1+b1kuk-1+vk,where y(k) represents the velocity of vehicle, u(k) represents the input force, a1 and b1(k) are unknown parameters, and v(k) is noise which can be given by the following equation:(75)vk=ek+mk,in which the element m(k) designates the unmodelled dynamics related to the parameter b1(k).The output of the vehicle can be defined as follows:(76)yk=θTkφk+vk,where the vectors of the parameters θ(k) and of the observation φ(k) are given by the following expression, respectively:(77)θTk=a1b1k,(78)φk=-yk-1uk-1.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 a1=-0.9418, b1(k)=2+0.1sin⁡(0.2k), m(k)=0.05sin⁡(0.2k)u(k-1).(2)
The sequence of noise{e(k)} is composed of independent random variables with zero mean and constant variance σ2=0.04.(3)
We will takeθ^(0)=0 and P(0)=100I (where I is an identity matrix).(4)
The application of the recursive algorithm M-RLS withσ-modification is based on the knowledge of the region Σ where θ∈Σ, such that(79)Σ=θ∈Rn,0<θk<M0=0.9433∪a1∈Rn,b1k>0,Mmin=0.9433<θk<Mmax=2.1628.(5)
s1=0.51, q0=2.6, and q1=-2.1.(6)
The evolution curve of the reference velocityyr(k) is shown in Figure 11.Figure 11
Evolution curve of the reference velocityyr(k).The tracking error is defined by(80)υk=yrk-yk.Using 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 (1)) and of the robust explicit scheme of self-tuning control based on the robust recursive algorithm M-RLS with σ-modification (37) (control scheme (2)). Control laws are applied to the example of the vehicle in the presence of unmodelled dynamics.Figure12 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).Figure 12
Evolution curve of the variance of the tracking errorσυ2(k) (control scheme (1)).Figure 13
Evolution curve of the variance of the prediction errorσε2(k) (control scheme (1)).In control scheme(2), Figure 14 shows the evolution curve of the velocity y(k), Figure 15 shows the evolution curve of the input force u(k) (or the control law), Figure 16 shows the evolution curve of the estimated parameter a^1(k), Figure 17 shows the evolution curve of the estimated parameter b^1(k), Figure 18 shows the evolution curve of the norm of the vector of the estimated parameters θ^(k), Figure 19 shows the evolution curve of the variance of the tracking error συ2(k), and Figure 20 shows the evolution curve of the variance of the prediction error σε2(k).Figure 14
Evolution curve of the output signaly(k) and of the reference velocity yr(k).Figure 15
Evolution curve of the control lawu(k).Figure 16
Evolution curve of the estimated parametera^1(k).Figure 17
Evolution curve of the estimated parameterb^1(k).Figure 18
Evolution curve of the norm of the vector of the estimated parametersθ^(k).Figure 19
Evolution curve of the variance of the tracking errorσυ2(k).Figure 20
Evolution curve of the variance of the prediction errorσε2(k).The different illustrated simulation results in Figures11–20 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.


## 5.1. Simulation Example 1

Let us consider that the dynamic system can be described by the following mathematical model ARX:(65)yk=-a1kyk-1-a2kyk-2+b1kuk-2+b2kuk-3+ek,where y(k) and u(k) are the output and the input of the second-order system with time delay being one and e(k) is white noise acting on the system.The output of the system can be given as follows:(66)yk=θTkφkwith (67)θTk=a1ka2kb1kb2k,(68)φTk=-yk-1-yk-2uk-2uk-3,where θ(k) and φ(k) 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:(69)Mmin≤θ∗≤Mmaxwith Mmin=0.85 and Mmax=1.05.In simulation, the nominal values of the uncertain parameters of system are defined by the following.a
1
(
k
)
=
0.79
+
0.1
sin
⁡
(
0.1
k
), a2(k)=0.2+0.1cos⁡(0.1k), b1(k)=0.4+0.1sin⁡(0.1k), and b2(k)=0.3+0.1cos⁡(0.1k). The evolution curve of the norm of the vector of the nominal values of parameters is given Figure 5.Figure 5
Evolution curve of the norm of the vector of the nominal values of parametersθ(k).The objective of this simulation example is the demonstration of the performance of the robust recursive algorithm for parameter estimation M-RLS withσ-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,e(k) is a sequence of random variables with zero mean and variance σ2=0.2, and the gain matrix P(0)=1000I.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 σε2(k) and Figure 7 shows the evolution curve of the norm of the vector of the estimated parameters θ^(k).Figure 6
Evolution curve of the variance of the prediction errorσε2(k).Figure 7
Evolution curve of the norm of the vector of the estimated parametersθ^(k).We use the proposed recursive algorithm M-RLS withσ-modification (37) to estimate the parameters involved in (67). Figure 8 represents the evolution curve of the variance of the prediction error σε2(k) and Figure 9 represents the evolution curve of the norm of the vector of the estimated parameters θ^(k).Figure 8
Evolution curve of the variance of the prediction errorσε2(k).Figure 9
Evolution curve of the norm of the vector of the estimated parametersθ^(k).Based on the simulation results, we conclude that the proposed recursive algorithm M-RLS withσ-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. Figure10 represents the scheme of this vehicle, as considered by Sam Fadali [41], in which U is the input force, V is the velocity of this vehicle, and b is the coefficient of viscous friction.Figure 10
Schematic representation of the cruise control system for a vehicle.Sam Fadali [41] determined the following transfer function G(s) in open loop, such that describes the dynamic behavior of the vehicle:(70)Gs=Ks+3.The discrete transfer function G(q-1) relating to (70) can be defined as follows, such that the used sampling period is Te=0.02 sec:(71)Gq-1=b1kq-11-0.9418q-1.For the system to be stable, the closed-loop poles or the roots of the following characteristic equation(72)1+Gq-1=0must lie within the unit circle.To ensure this condition of the stability of the system in closed loop, the parameterb1(k) must be defined as follows:(73)-0.054<b1k<1.941.This system can be described by the following mathematical model ARX:(74)yk=-a1yk-1+b1kuk-1+vk,where y(k) represents the velocity of vehicle, u(k) represents the input force, a1 and b1(k) are unknown parameters, and v(k) is noise which can be given by the following equation:(75)vk=ek+mk,in which the element m(k) designates the unmodelled dynamics related to the parameter b1(k).The output of the vehicle can be defined as follows:(76)yk=θTkφk+vk,where the vectors of the parameters θ(k) and of the observation φ(k) are given by the following expression, respectively:(77)θTk=a1b1k,(78)φk=-yk-1uk-1.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 a1=-0.9418, b1(k)=2+0.1sin⁡(0.2k), m(k)=0.05sin⁡(0.2k)u(k-1).(2)
The sequence of noise{e(k)} is composed of independent random variables with zero mean and constant variance σ2=0.04.(3)
We will takeθ^(0)=0 and P(0)=100I (where I is an identity matrix).(4)
The application of the recursive algorithm M-RLS withσ-modification is based on the knowledge of the region Σ where θ∈Σ, such that(79)Σ=θ∈Rn,0<θk<M0=0.9433∪a1∈Rn,b1k>0,Mmin=0.9433<θk<Mmax=2.1628.(5)
s1=0.51, q0=2.6, and q1=-2.1.(6)
The evolution curve of the reference velocityyr(k) is shown in Figure 11.Figure 11
Evolution curve of the reference velocityyr(k).The tracking error is defined by(80)υk=yrk-yk.Using 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 (1)) and of the robust explicit scheme of self-tuning control based on the robust recursive algorithm M-RLS with σ-modification (37) (control scheme (2)). Control laws are applied to the example of the vehicle in the presence of unmodelled dynamics.Figure12 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).Figure 12
Evolution curve of the variance of the tracking errorσυ2(k) (control scheme (1)).Figure 13
Evolution curve of the variance of the prediction errorσε2(k) (control scheme (1)).In control scheme(2), Figure 14 shows the evolution curve of the velocity y(k), Figure 15 shows the evolution curve of the input force u(k) (or the control law), Figure 16 shows the evolution curve of the estimated parameter a^1(k), Figure 17 shows the evolution curve of the estimated parameter b^1(k), Figure 18 shows the evolution curve of the norm of the vector of the estimated parameters θ^(k), Figure 19 shows the evolution curve of the variance of the tracking error συ2(k), and Figure 20 shows the evolution curve of the variance of the prediction error σε2(k).Figure 14
Evolution curve of the output signaly(k) and of the reference velocity yr(k).Figure 15
Evolution curve of the control lawu(k).Figure 16
Evolution curve of the estimated parametera^1(k).Figure 17
Evolution curve of the estimated parameterb^1(k).Figure 18
Evolution curve of the norm of the vector of the estimated parametersθ^(k).Figure 19
Evolution curve of the variance of the tracking errorσυ2(k).Figure 20
Evolution curve of the variance of the prediction errorσε2(k).The different illustrated simulation results in Figures11–20 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 robustnessσ-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 withσ-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.



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*Source: 8653136-2017-03-08.xml*