Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics

Nabiha Touijer Samira Kamoun

Nabiha Touijer and Samira Kamoun. Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics. International Journal of Automation and Computing, vol. 13, no. 3, pp. 277-284, 2016. doi: 10.1007/s11633-015-0921-y
Citation: Nabiha Touijer and Samira Kamoun. Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics. International Journal of Automation and Computing, vol. 13, no. 3, pp. 277-284, 2016. doi: 10.1007/s11633-015-0921-y

doi: 10.1007/s11633-015-0921-y
基金项目: 

This work was partially funded by the Australian Research Council . (No.DP110102076)

Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics

Funds: 

This work was partially funded by the Australian Research Council . (No.DP110102076)

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出版历程
  • 收稿日期:  2013-09-27
  • 修回日期:  2014-07-30
  • 刊出日期:  2016-06-01

Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics

doi: 10.1007/s11633-015-0921-y
    基金项目:

    This work was partially funded by the Australian Research Council . (No.DP110102076)

English Abstract

Nabiha Touijer and Samira Kamoun. Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics. International Journal of Automation and Computing, vol. 13, no. 3, pp. 277-284, 2016. doi: 10.1007/s11633-015-0921-y
Citation: Nabiha Touijer and Samira Kamoun. Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics. International Journal of Automation and Computing, vol. 13, no. 3, pp. 277-284, 2016. doi: 10.1007/s11633-015-0921-y
    • The problem of self-tuning regulators of linear systems has been solved,in the absence of unmodelled dynamics in [1, 2]. The self-tuning regulators were based on the recursive least squares (RLS) parametric estimation algorithm. Patete et al.[3-6] demonstrated the stability of the implicit scheme of generalized minimum variance self-tuning control for linear systems with/without noise. Robust adaptive control for time-varying systems in the presence of unmodelled dynamics was developed on the basis of the sliding mode control by using the gradient parametric estimation algorithm in [7, 8]. Note that,it is possible to introduce a control parameter in order to give some robustness to the control system[9]. Reference [10] developed a robust adaptive control for linear time-varying systems in the presence of unmodelled dynamics and disturbances on the basis of the RLS parametric estimation algorithm with dead zone. Reference [11] developed a robust self-tuning control for multi-input multi-output (MIMO) linear systems in the presence of unmodelled dynamics. The stability conditions of the developed estimation scheme are established on the basis of the Lyapunov method.

      Considering these different researches,we propose to develop an explicit scheme of generalized minimum variance $\alpha l $-equivalent robust self-tuning control based on the discrete-time sliding mode control for the linear timevarying (LTV) systems,which can be described by the auto-regressive exogenous (ARX) mathematical model in the presence of unmodelled dynamics. A new criterion is proposed including the two parameters a and l. The choice of these two parameters will be demonstrated in this paper. We can use the RLS parametric estimation algorithm with dead zone given in [10] to estimate the parameters of the considered system. But,we found,in case of time-varying parameters of the system,that this parametric estimation algorithm has a disadvantage. This disadvantage is that the estimated parameters in the discrete-time k takes the same estimated values in the discrete-time k-1 if the prediction error is less than a known upper bound of unmodelled dynamics and disturbances. To solve this problem,we propose to use the RLS parametric estimation algorithm with forgetting factor to estimate the system parameters if the prediction error is less than a known upper bound of unmodelled dynamics and disturbances. The proposed parametric estimation algorithm is called modified m-RLS with dead zone and forgetting factor. The stability conditions of this algorithm are established on the basis of the Lyapunov function.

      This paper is organized as follows: In Section 2,an LTV system is described by a discrete-time ARX mathematical model in the presence of unmodelled dynamics. Section 3 presents the proposed m-RLS parametric estimation algorithm with dead zone and forgetting factor. Lyapunov function is used to analyze the stability of this algorithm. The explicit scheme of generalized minimum variance self-tuning control based on discrete-time sliding mode control,which can be applied to the LTV system described by the ARX mathematical model in the absence of unmodelled dynamics,is the subject of Section 4. In Section 5,we develop the proposed explicit schema of control GMVRSTC-al for the considered LTV system in Section 2,and we demonstrate the choice of the parameters a and l. A numerical simulation example was given in Section 6 to prove the performances of the proposed explicit schema of control GMVRSTC- al. The conclusion is elaborated in Section 7.

    • Let us consider an LTV system,which can be described by the following discrete-time ARX mathematical model:

      \label{eq1} A_c (q^{-1},k)y(k)=q^{-d}B_c (q^{-1},k)u(k)+e(k)

      (1)

      where u(k) and y(k) represent the input and the output of the system at the discrete-time k,respectively,e(k) is a white noise with zero mean and constant variance acting on the system,d is the dead-time (is an integral number of sample intervals),and Ac (q-1,k) and Bc (q-1,k) are polynomials of degree naand nb,respectively,which are defined by

      \label{eq2} A_c (q^{-1},k)=A(q^{-1},k)+\varsigma _A (q^{-1},k)

      (2)

      and

      \label{eq3} B_c (q^{-1},k)=B(q^{-1},k)+\varsigma _B (q^{-1},k)

      (3)

      where

      \label{eq4} A(q^{-1},k)=1+a_1 (k)q^{-1}+\cdots +a_{n_a } (k)q^{-n_a }

      (4)

      \label{eq5} B(q^{-1},k)=1+b_1 (k)q^{-1}+\cdots +b_{n_b } (k)q^{-n_b }

      (5)

      \label{eq6} \varsigma _A (q^{-1},k)=\varsigma _{a_1 } (k)q^{-1}+\cdots +\varsigma _{a_{n_a } } (k)q^{-n_a }

      (6)

      and

      \label{eq7} \varsigma _B (q^{-1},k)=\varsigma _{b_1 } (k)q^{-1}+\cdots +\varsigma _{b_{n_b } } (k)q^{-n_b }

      (7)

      $\varsigma _A (q^{-1},k) and \varsigma _B (q^{-1},k)$ are the unmodelled dynamics presented in the considered system.

      The considered system can be described by the following mathematical model:

      \label{eq8} A(q^{-1},k)y(k)=q^{-d}B(q^{-1},k)u(k)+v(k)

      (8)

      where v(k) represents noise,which is defined as follows:

      \label{eq9} v(k)=\varsigma ^{\rm T}(k)\varphi (k)+e(k).

      (9)

      The output $y(k)$ can be given by

      \label{eq10} y(k)=\theta ^{\rm T}(k)\varphi (k)+v(k)

      (10)

      where

      \label{eq11} \theta ^{\rm T}(k)=[a_1 (k)\;\cdots \;a_{n_a } (k)\;b_1 (k)\;\cdots \;b_{n_b } (k)]

      (11)

      \label{eq12} \varsigma ^{\rm T}(k)=[\varsigma _{a_1 } (k)\;\cdots \;\varsigma _{a_{n_a } } (k)\;\varsigma _{b_1 } (k)\;\cdots \;\varsigma _{b_{n_b } } (k)]

      (12)

      and

      $\eqalign{ & {\varphi ^{\text{T}}}(k) = [ - {y^{\text{T}}}(k - 1) \cdots - {y^{\text{T}}}(k - {n_a})\qquad \cr & \qquad {u^{\text{T}}}(k - d - 1) \cdots {u^{\text{T}}}(k - d - {n_b})]. \cr} $

      (13)

      For the formulation of the self-tuning control problem of the considered system,which is described by the ARX mathematical model (1),we retain the following assumptions: 1) the parameters d,naand nbare known; 2) the upper bound p of ς(k) is known; 3) the noise e(k) is bounded and known with upper bound mo.

      The upper bound of the noise v(k) can be defined by a relative dead zone dz(k),such that

      \label{eq14} d_z (k)=\rho \left\| {\varphi (k)} \right\|+m_o

      (14)

      Nothing that the dead zone dz(k) is used in the recursive parametric estimation algorithm.

    • The following modified recursive least square estimation algorithm[10] is adopted to estimate the parameters contained in the vector θ(k),which is given by (11)

      $\eqalign{ . \hat \theta (k) = \hat \theta (k - 1) + \delta (k)L(k)\zeta (k) \cr . L(k) = \frac{{P(k - 1)\varphi (k)}}{{\mu (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}} \cr . P(k) = P(k - 1) - \delta (k)\frac{{P(k - 1)\varphi (k){\varphi ^{\text{T}}}(k)P(k - 1)}}{{\mu (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}} \cr . \zeta (k) = y(k) - {{\hat \theta }^{\text{T}}}(k - 1)\varphi (k) \cr} $

      (15)

      with

      $\delta \left( k \right)=\left\{ \begin{align} & 0,if\left| \zeta \right.\left. \left( k \right) \right|\le \beta {{d}_{z}}\left( k \right) \\ & \gamma ,otherwise \\ \end{align} \right.$

      (16)

      where $\gamma \in \left[{\sigma _0 ,\frac{1}{\beta}-\sigma _0 } \right]$,$\sigma _0 \in \left[{0,1} \right]$,$\beta >\frac{2}{(1+\sigma _0 )}$and $\mu (k)\in \left[{0,1} \right]$ .

      The constant $\sigma _0 \in \left[{0,1} \right]$ and $\beta >\frac{2}{(1+\sigma _0 )}$ are arbitrarily chosen by the designer such that y >0,e.g. as special case $\beta =2,0<{{\sigma }_{0}} <\frac{1}{4}$ And the dead zone width depends on the choice of β selected by the designer.

      The property of u(k) contained in the observation vector φ T(k) given by (13) is referred to as persistent excitation (PE) and is crucial in many adaptive schemes where parameter convergence is one of the objectives.

      Discussion. If $\left| {\zeta (k)} \right|\le \beta d_z (k)$ , then \hat {\theta }(k)\approx \hat {\theta }(k-1)$ . Else,we will use the recursive parametric estimation algorithm (15) where δ(k)=γ. But,the parameters of the considered systems are variable. Then this estimation algorithm cannot assure the convergence of the estimated parameters.

      To remedy this problem,we propose a modified recursive parametric estimation algorithm to estimate the parameters involved in the vector θ(k),which is given by (11).

    • The idea of the proposed recursive parametric estimation algorithm is the development of a new recursive estimator that tracks the parametric variations if$\left| {\zeta (k)} \right| \leqslant \beta {d_z}(k)$ . The formulation of the modified RLS recursive parametric estimation algorithm with dead zone is based on the recursive least squares techniques with forgetting factor. This algorithm permits to compensate the unmodelled dynamics and the external disturbances. Then,we propose the following recursive parametric estimation algorithm:

      $\eqalign{ & \hat \theta (k) = \hat \theta (k - 1) + (1 - \delta (k))L(k)\zeta (k) \cr & L(k) = \frac{{P(k - 1)\varphi (k)}}{{\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}} \cr & P(k) = \frac{{P(k - 1)}}{{\lambda (k)}} - \frac{{1 - \delta (k)}}{{\lambda (k)}}\frac{{P(k)\varphi (k){\varphi ^{\text{T}}}(k)P(k - 1)}}{{\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}} \cr & \zeta (k) = y(k) - {{\hat \theta }^{\text{T}}}(k - 1)\varphi (k) \cr} $

      (17)

      with

      \label{eq18} \delta (k)=\left\{ {\begin{array}{l}0,\;{\rm if}\;\left| {\zeta (k)} \right|\le \beta d_z (k),\;(\beta >1) \\\gamma ,\;otherwise \\\end{array}} \right.

      (18)

      λ(k) is an exponential forgetting factor,which can be defined by the following equation:

      \lambda (k)=\lambda _{o} \lambda (k-1)+\lambda ^{o}(1-\lambda _{o} )

      where $0 < \lambda \left( k \right) < 1,0 < {\lambda _ \circ } < 1,0 < {\lambda _ \circ } < 1.$

      In the following,we will take

      \label{eq19} \delta _1 (k)=1-\delta (k).

      (19)

      Remark 1.

      1) if $\left| {\zeta (k)} \right|\le \beta d_z (k)$ ,then we have δ(k) ≈ 0.. The used recursive parametric estimation algorithm is similar to the RLS algorithm with forgetting factor;

      2) if $\left| {\zeta (k)} \right|>\beta d_z (k)$ ,then we have δ(k) = γ. We will determine the necessary condition on the parameter γ,which ensures the stability of the recursive parametric estimation scheme.

    • The parametric estimation error is written as follows:

      \label{eq20} \tilde {\theta }(k)=\hat {\theta }(k)-\theta (k).

      (20)

      The prediction error ζ(k) (17) can be rewritten as follows:

      \label{eq21} \zeta (k)=v(k)-\varphi ^{\rm T}(k)\tilde {\theta }(k-1)

      (21)

      We have

      \label{eq22} \hat {\theta }(k)=\hat {\theta }(k-1)+\delta _1 (k)L(k)\zeta (k).

      (22)

      Using (17),(21) and (22),the parametric estimation error $\tilde {\theta }(k)$ becomes:

      \label{eq23} \tilde{\theta }(k)=\lambda(k)P(k)P^{-1} (k-1)\tilde{\theta }(k-1)+\delta _1 (k)L(k)v(k).

      (23)

      Let us consider the followings Lyapunov function:

      \label{eq24} V(k)=\tilde {\theta }^{\rm T} (k)P^{-1} (k)\tilde {\theta }(k).

      (24)

      Using(23),the term P-1 $k){{\tilde{\theta }}_{(k)}}$ can be defined by

      $\eqalign{ & {P^{ - 1}}(k)\tilde \theta (k) = \lambda (k){P^{ - 1}}(k - 1)\tilde \theta (k - 1) + \cr & \qquad {\delta _1}(k){P^{ - 1}}(k)L(k)v(k). \cr} $

      (25)

      Using (23) and (25),the Lyapunov function becomes:

      $\eqalign{ & V(k) = \lambda (k){{\tilde \theta }^{\text{T}}}(k - 1)P(k){P^{ - 1}}(k - 1){P^{ - 1}}(k - 1)\tilde \theta (k - 1) + \quad \cr & 2\lambda (k){\delta _1}(k){L^{\text{T}}}(k){P^{ - 1}}(k - 1)\tilde \theta (k - 1)v(k) + \quad \cr & \delta _1^2(k){L^{\text{T}}}(k){P^{ - 1}}(k)L(k){v^2}(k). \cr} $

      (26)

      By using the following expression

      $\eqalign{ & P(k){P^{ - 1}}(k - 1){P^{ - 1}}(k - 1) = \left( {\frac{1}{{\lambda (k)}}} \right)({P^{ - 1}}(k - 1) - \cr & {\delta _1}(k)\frac{{\varphi (k){\varphi ^{\text{T}}}(k)}}{{\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}}) \cr} $

      (27)

      we can rewrite (26) as follows:

      $\eqalign{ & V(k) = \lambda (k){{\tilde \theta }^{\text{T}}}(k - 1){P^{ - 1}}(k - 1)\tilde \theta (k - 1) - \quad \cr & {\delta _1}(k)\frac{{{{\tilde \theta }^{\text{T}}}(k - 1)\varphi (k){\varphi ^{\text{T}}}(k)\tilde \theta (k - 1)}}{{\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}} + \quad \cr & 2\lambda (k){\delta _1}(k)\frac{{{\varphi ^{\text{T}}}(k)\tilde \theta (k - 1)}}{{\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}}v(k) + \quad \cr & \delta _1^2(k){L^{\text{T}}}(k){P^{ - 1}}(k)L(k){v^2}(k). \cr} $

      (28)

      The following expressions on the parameters L1(k) and L2 (k) are used

      ${L_1}(k) = \lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)$

      (29)

      and

      ${L_2}(k) = \lambda (k) + (1 - {\delta _1}(k)){\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k).$

      (30)

      Now,using the matrix inversion lemma,we can write P-1(k) as follows:

      ${P^{ - 1}}(k) = \lambda (k){P^{ - 1}}(k - 1) + \frac{{{\delta _1}(k)\varphi (k){\varphi ^{\text{T}}}(k)}}{{{L_2}(k)}}.$

      (31)

      Based on (31),we have

      ${L^{\text{T}}}(k){P^{ - 1}}(k)L(k) = \lambda (k)\frac{{{\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)}}{{{L_1}(k){L_2}(k)}}.$

      (31)

      Thus,we can write the Lyapunov function $V(k)$ as follows:

      $\eqalign{ & V(k) - \lambda (k)V(k - 1) = \lambda (k)\left[{\frac{{\delta _1^2(k){L^{\text{T}}}(k)\varphi (k){v^2}(k)}}{{{L_2}(k)}}} \right. + \cr & \quad \left. {\frac{{{\delta _1}(k){\varphi ^{\text{T}}}(k)\tilde \theta (k - 1)}}{{{L_1}(k)}}(2v(k) - {\varphi ^{\text{T}}}(k)\tilde \theta (k - 1))} \right] \cr} $

      (33)

      with

      \label{eq34} V(k-1)=\tilde {\theta }^{\rm T} (k-1)P^{-1} (k-1)\tilde {\theta }(k-1).

      (34)

      We can obtain the following two equations:

      $\eqalign{ & 2v(k){\varphi ^{\text{T}}}(k)\tilde \theta (k - 1) - {[{\varphi ^{\text{T}}}(k)\tilde \theta (k - 1)]^2} = \quad \cr & - {[v(k) - {\varphi ^{\text{T}}}(k)\tilde \theta (k - 1)]^2} + {v^2}(k) = \quad \cr & - {\zeta ^2}(k) + {v^2}(k) \cr} $

      (35)

      then

      $V(k) - \lambda (k)V(k - 1) = - \lambda (k){\delta _1}(k)\left[{\frac{{{\zeta ^2}(k)}}{{{L_1}(k)}} - \frac{{{v^2}(k)}}{{{L_2}(k)}}} \right].$

      (36)

      Multiplying and dividing the term $\frac{\delta _1 (k)v^2(k)}{L_2 (k)}$ by the parameter &β,we get

      $\eqalign{ & V(k) - \lambda (k)V(k - 1) = \quad \cr & - \lambda (k){\delta _1}(k)\left[{\frac{{[{\zeta ^2}(k) - \beta {v^2}(k)]}}{{\beta [\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)]}}} \right. + \quad \cr & \frac{{[(\beta - 1)\lambda (k) + [(\beta - 1) - \beta {\delta _1}(k)]{\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)]}}{{\beta [\lambda (k) + {\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)]}}\quad \cr & \left. {\frac{{{\zeta ^2}(k)}}{{[\lambda (k) + [1 - {\delta _1}(k)]{\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)]}}} \right]. \cr} $

      (37)

      Then,for obtaining the following inequality

      $V(k) - \lambda (k)V(k - 1) < 0$

      (38)

      we must have

      $\eqalign{ & 0 < \frac{{{\zeta ^2}(k) - \beta {v^2}(k)}}{{\beta {L_1}(k)}} + \frac{{{\zeta ^2}(k)[(\beta - 1)\lambda (k)]}}{{\beta {L_1}(k){L_2}(k)}} + \quad \cr & \frac{{{\zeta ^2}(k)\left[{[\beta - 1 - \beta {\delta _1}(k)]{\varphi ^{\text{T}}}(k)P(k - 1)\varphi (k)} \right]}}{{\beta {L_1}(k){L_2}(k)}} \cr} $

      (39)

      such that β > 1 and δ1(k) > 0.

      Using the definition of the parameters β and λ(k),we have

      ${\zeta ^2}(k) - \beta {v^2}(k) > 0{\text{ }}$

      (40)

      and

      $(\beta - 1)\lambda (k){\zeta ^2}(k) > 0.$

      (41)

      Now,we determine the following condition,such that

      $\beta - 1 - \beta {\delta _1}(k) > 0.$

      (42)

      Then,the necessary condition on the parameter $\delta _1 (k)$ to have (42) is

      $0{\text{ < }}{\delta _1}(k) < 1 - \frac{1}{\beta }.$

      (43)

      Using (19) and remarks,the necessary condition on the parameterγ can be written as follows:

      $\gamma \in \left[{\frac{1}{\beta }\;,\;1} \right]$

      (44)

      where β is arbitrarily chosen by the designer.

      Hence,the stability condition of the parametric estimation scheme (17) is established.

    • This section presents a self-tuning control,which can be applied for the LTV non-minimum phase dynamic systems. Let us suppose that the considered LTV dynamic system can be described by the following ARX mathematical model in the absence of unmodelled dynamics:

      $A({q^{ - 1}},k)y(k) = {q^{ - d}}B({q^{ - 1}},k)u(k) + e(k)$

      (45)

      where u(k),y(k) and e(k) are defined in Section 2,and the polynomials A(q-1,k)and B(q-1,k) are described by (4) and (5),respectively.

      The designed self-tuning controller based on sliding mode control permits to minimize the following variable:

      $\eqalign{ & s(k + d + 1) = S({q^{ - 1}})[y(k + d + 1) - {y_r}(k + d + 1)] + \quad \cr & Q({q^{ - 1}})u(k) \cr} $

      (46)

      where the output y(k+d+1) and the weight polynomials S(q-1) and Q(q-1) are defined by

      $y(k + d + 1) = {q^{ - d}}\frac{{B({q^{ - 1}},k)}}{{A({q^{ - 1}},k)}}u(k) + e(k + d + 1)$

      (47)

      $S({q^{ - 1}}) = 1 + {s_1}{q^{ - 1}} + \; \cdots \; + {s_{{n_s}}}{q^{ - {n_s}}}$

      (48)

      and

      $Q({q^{ - 1}}) = {q_0} + {q_1}{q^{ - 1}} + \; \cdots \; + {q_{{n_q}}}{q^{ - {n_q}}}$

      (49)

      where ns and nq are the orders of the polynomials S(q-1) and Q(q-1),respectively,which are chosen by the designer.

      The criterion (46) is rewritten as

      $\eqalign{ & s(k + d + 1) = H({q^{ - 1}},k)u(k) + G({q^{ - 1}},k)y(k) - \quad \cr & S({q^{ - 1}}){y_r}(k + d + 1) + F({q^{ - 1}},k)e(k + d + 1) \cr} $

      (50)

      where the polynomial H(q-1,k) is given by

      $H({q^{ - 1}},k) = qB({q^{ - 1}},k)F({q^{ - 1}},k) + Q({q^{ - 1}})$

      (51)

      and the polynomials F(q-1,k) and G(q-1,k) are solutions of the following polynomial equation:

      $S({q^{ - 1}}) = A({q^{ - 1}},k)F({q^{ - 1}},k) + {q^{ - d - 1}}G({q^{ - 1}},k).$

      (52)

      The polynomials degrees of F(q-1,k),G(q-1,k) and H(q-1,k) are,respectively,nf ,ng and nh,where nf =d,ng =max (na +d,ns)-d-1 and nh =max (nb +d+1,nq ).

      The control law which minimizes s(k+d+1) is given by

      $\eqalign{ & u(k) = - {H^{ - 1}}({q^{ - 1}},k)[G({q^{ - 1}},k)y(k) - \qquad \cr & \quad S({q^{ - 1}}){y_r}(k + d + 1)] \cr} $

      (53)

      The explicit scheme of self-tuning control based on the generalized minimum variance strategy is carried out in the following three steps:

      Step 1. Estimation of the system parameters using the RLS recursive parametric estimation algorithm.

      Step 2. Calculation of the parameters of the control law resolving the polynomial equation (53).

      Step 3. Calculation of the control law $u(k)$ using the following equation:

      $\eqalign{ & u(k) = \frac{1}{{{h_0}(k)}}\left[{\sum\limits_{i = 1}^{nh} {{h_i}(k)u(k - i) + } \sum\limits_{j = 0}^{ng} {{g_j}(k)y(k - ng)} } \right. - \;\;\quad \cr & \quad \left. {{y_r}(k + d + 1) - \sum\limits_{t = 1}^{ns} {{s_t}{y_r}(k + d + 1 - t)} } \right]. \cr} $

      (54)

      It should be noted that it may happen in the calculation of the control law at time k0,to have h0 (k0)=0. To overcome this problem,one can do a test on the value of the parameter h0 (k); and if h0 (k0 )=0,then take h0 (k0 )=ξ with ξ a relatively small parameter to be chosen a priori.

    • In this section,we develop a robust self-tuning control for the considered LTV system with unknown parameters,which is described by the ARX mathematical model in the presence of unmodelled dynamics,as given by (8). The formulation of this robust self-tuning control is based on the minimization of a proposed criterion.

      The explicit scheme of generalized minimum variance self-tuning control is studied in the next.

      The development of the robust adaptive control is based on the minimization of the criterion (46),which corresponds to the minimization of the variance of the regulation error of the considered system. In this case,we propose to use two parameters a and l to compensate the sliding mode error and the model deviations caused by the presence of unmodelled dynamics[8].

      We propose to minimize the following criterion to formulate the control law for the considered system,which is described by the ARX mathematical model (8)

      $\eqalign{ & J(k + d + 1) = s(k + d + 1) + ls(k + d) + \quad \cr & \alpha (s(k + d) - s(k + d - 1)) \cr} $

      (55)

      where the term s(k+d+1) is given by (46).

      The optimal control law $u(k)$,which corresponds to the minimization of the criterion (55),is given by

      $\eqalign{ & u(k) = {H^{ - 1}}({q^{ - 1}},k)\left[{S({q^{ - 1}}){y_r}(k + d + 1)} \right. - G({q^{ - 1}},k)y(k) - \qquad \cr & \quad ls(k + d)\left. { - \alpha [s(k + d) - s(k + d - 1)]} \right] \cr} $

      (56)

      Noting that the determination of the parameters included in the polynomials H(q-1,k) and G(q-1,k) is made using (51) and (52).

      We propose the following supplementary control law us (k) to ameliorate the robustness of the system,such that

      ${u_s}(k) = - \frac{1}{{{h_0}(k)}}\left[{ls(k + d) + \alpha [s(k + d) - s(k + d - 1)]} \right].$

      (57)

      We will call the proposed control as the generalized minimum variance al-equivalent self-tuning control (GMVSTC- al).

    • The explicit scheme of the generalized minimum variance al-equivalent self-tuning control,which can be applied to the considered LTV dynamic system,is carried out in the following three steps:

      Step 1. Estimation of the parameters contained in the vector parameters (11) using the proposed recursive parametric estimation algorithm m-RLS (17).

      Step 2. Calculation of the parameters of control law.

      Step 3. Calculation of the control law $u(k)$,which is given by

      $\eqalign{ & u(k) = - \frac{1}{{{h_0}(k)}}[\sum\limits_{i = 1}^{nh} {{h_i}(k)u(k - i)} + \sum\limits_{i = 0}^{ng} {{g_i}(k)y(k - i)} - \qquad \cr & \sum\limits_{i = 1}^{ns} {{s_i}(k){y_r}(k + d + 1 - i)}] + {u_s}(k) \cr} $

      (58)

      where the supplementary control law us (k) is defined by (57) such that at time k0,if h0 (k0)=0,then h0 (k0 )=ξ.

    • The stability condition of the considered LTV dynamic system,where the control law u(k) given by (58) is applied,depends on the suitable choice of the parameters a and l.

      Using the control law u(k),as given by (56),(8) can be rewritten as follows:

      $\eqalign{ & \Gamma ({q^{ - 1}},k)s(k + d + 1) = \tilde A({q^{ - 1}},k)y(k + d + 1) - \qquad \cr & \quad q\tilde B({q^{ - 1}},k)u(k) - {{\tilde b}_0}(k)v(k + d + 1) \cr} $

      (59)

      such that

      $\Gamma ({q^{ - 1}},k) = - 1 - (l + \alpha ){q^{ - 1}} + \alpha {q^{ - 2}}$

      (60)

      ${\tilde b_0}(k) = \frac{{{h_0}(k)}}{{{b_0}(k)}}$

      (61)

      $\tilde A({q^{ - 1}},k) = {\tilde b_0}(k)A({q^{ - 1}},k) - \hat A({q^{ - 1}},k)F({q^{ - 1}},k)$

      (62)

      and

      $\tilde B({q^{ - 1}},k) = {\tilde b_0}(k)B({q^{ - 1}},k) - \hat B({q^{ - 1}},k)F({q^{ - 1}},k).$

      (63)

      Noting that the parameters a,l and si,i = 1,... ,ns, are chosen such that the characteristic roots of polynomials Γ(q−1) and S(q−1) are inside the unit circle.

    • Let us consider an LTV dynamic system with a dead-time d=1,which can be described by the following mathematical model:

      $\eqalign{ & y(k) = [1.26 + 0.02\sin (0.02k)]u(k - 2) - \qquad \cr & \quad [- 0.5 + 0.02\sin (0.06k)]y(k - 1) + v(k) \cr} $

      (64)

      where v(k) is given by

      $\eqalign{ & v(k) = 0.1\sin (0.02k)u(k - 2) - \qquad \cr & \quad 0.1\cos (0.02k)y(k - 1) + e(k). \cr} $

      (65)

      We suppose that e(k) is a white noise with zero mean and constant variance σ2 = 0.01. The reference output is a square signal with amplitude 1 and period 100. We take: S(q−1) = 1+0.9q−1 and Q(q−1) = 0.9 − 0.89q−1.

      The variances of the prediction error $\sigma _{pe}^2 $ and of the regulation error $\sigma _{te}^2 $ are given in Table 1,for different values of λ(k),a and l,to show the improved performance of the proposed m-RLS and the GMVSTC-al.

      Figure 1.  Evolution curve of the output y(k) in control scheme (1)

      We will present the results of numerical simulation for the GMVSTC ((α; l) = (0; 0)) on the basis of the RLS algorithm (λ = 1),by the solid line (control scheme (1)), whereas the results of simulation for GMVSTC-αl((α; l) = (−0.68; 0.3)),on the basis of the m-RLS algorithm (0 < λ < 1),are represented by the dashed line (control scheme (2)).

      Figs. 1 and 2 show,respectively,the evolution curve of the output y(k) in control scheme (1) and in control scheme (2). Figs. 3 and 4 show,respectively,the evolution curves of the control law $u(k)$ in control scheme (1) and in control scheme (2). Fig. 5 shows the evolution curve of the prediction error variance. Fig. 6 shows the evolution curve of the regulation error variance.

      Figure 2.  Evolution curve of the output y(k), (control scheme (2))

      Figure 3.  Evolution curve of the control law u(k), (control scheme (1))

      Figure 4.  Evolution curve of the control law u(k), (control scheme (2))

      Figure 5.  Evolution curve of the prediction error variance

      Figure 6.  Evolution curve of the regulation error variance

      From Figs. 1-6 and Table 1,we can get the following conclusions:

      Table 1.  Prediction error variance value and regulation error variance value

      1) The outputs of the system track the desired signals;

      2) The prediction error variances and the regulation error variances become small in the discrete-time and converge to

      3) The control scheme (2) can overcome the effect of transitional regime;

      4) The developed control schema based on the m-RLS parametric estimation algorithm with dead zone and forgetting factor is robust.

    • In this paper,we have proposed a robust explicit scheme of generalized minimum variance $\alpha l $-equivalent self-tuning control for the LTV dynamic systems,which can be described by the ARX mathematical models in the presence of unmodelled dynamics.

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