Partial identification and control of MIMO systems via switching linear reduced-order models under weak stimulations

Abstract

In closed loop identification of an unknown control system, the stability is a big concern particularly when the system does not proceed with sufficient excitation. In this paper, under insufficient excitation of the system, identification and control are investigated by employing Evolving Linear Models (ELMs). It is explained that under weak stimulation, linear correlations between input and output signals and their derivations are occurred. Removing some correlated variables through the time, an equivalent reduced order model of the original system is appeared, which can be identified as an ELM. Defining control law based on the sliding mode control (SMC) and using appropriate adaptation rules for parameters of the model, the tracking errors converge to zero and the stability of the system is guaranteed. Then, convergence of the parameters to their true values is studied and discussed. Different simulations are given to demonstrate the efficacy of the proposed closed loop identification approach.

Appendices

Appendix a

Theorem 1

Regarding to the MIMO system described in Eq. (1) and based on A2, identification of the basic model is guaranteed. Also, the closed-loop system remains stable during identification procedures.

Proof

Consider following positive definite Lyapunov function including the sliding vector and parameters estimation errors (PEEs):

$$V\left( t \right)=\frac{1}{2}\left[ {{S^T}S+\mathop \sum \limits_{{q=1}}^{N} \tilde {\theta }_{q}^{T}\left( t \right)P_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)} \right]$$

(26)

where

$$\begin{gathered} \tilde {\theta }_{q}^{T}\left( t \right)=\left[ {\tilde {a}_{q}^{T},\tilde {b}_{q}^{T}} \right] \end{gathered}$$

(27)

Regarding to the differential equations in Eq. (2), the derivative of the Lyapunov function, formulated in Eq. (26), is computed as follows:

$$\dot {V}\left( t \right)=\frac{1}{2}\left[ {2{S^T}\dot {S}+2\mathop \sum \limits_{{q=1}}^{N} \tilde {\dot {\theta }}_{q}^{T}\left( t \right)P_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)+\mathop \sum \limits_{{q=1}}^{N} \tilde {\theta }_{q}^{T}\left( t \right)\dot {P}_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)} \right]$$

(28)

Or

$$\dot {V}\left( t \right)=\frac{1}{2}\left[ {2{S^T}\dot {S} - 2\mathop \sum \limits_{{q=1}}^{N} \hat {\dot {\theta }}_{q}^{T}\left( t \right)P_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)+\mathop \sum \limits_{{q=1}}^{N} \tilde {\theta }_{q}^{T}\left( t \right)\dot {P}_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)} \right]$$

(29)

By substituting the derivative of sliding vector from Eq. (14), the following formula is obtained:

$$\dot{V}\left( t \right) = \frac{1}{2}\left[ { - 2S^{T} \left[ {\begin{array}{*{20}c} {\tilde{b}_{1}^{T} } \\ \vdots \\ {\tilde{b}_{N}^{T} } \\ \end{array} } \right]\underline{\tilde u } + 2S^{T} \left[ {\begin{array}{*{20}c} {\tilde a _{1}^{T}~ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{1} } \\ \vdots \\ {\tilde a _{N}^{T}~ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{N} } \\ \end{array} } \right] - 2\mathop \sum \limits_{{q = 1}}^{N} \hat{\dot{\theta }}_{q}^{T} \left( t \right)P_{q}^{{ - 1}} \left( t \right)\tilde{\theta }_{q} \left( t \right) + \mathop \sum \limits_{{q = 1}}^{N} \tilde{\theta }_{q}^{T} \left( t \right)\dot{P}_{q}^{{ - 1}} \left( t \right)\tilde{\theta }_{q} (t)} \right] - S^{T} B\eta ~sign\left( S \right).$$

(30)

Equation (30) can be written in summation format as follows:

$$\dot {V}\left( t \right)=\mathop \sum \limits_{{q=1}}^{N} \left( {{s_q}\left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{q}^{T}, - ~{{\hat {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }}^T}} \right] - \left[ {\hat {\dot {a}}_{q}^{T}\left( t \right),~\hat {\dot {b}}_{q}^{T}\left( t \right)} \right]P_{q}^{{ - 1}}\left( t \right)} \right){\tilde {\theta }_q}\left( t \right)+\frac{1}{2}\mathop \sum \limits_{{q=1}}^{N} \tilde {\theta }_{q}^{T}\left( t \right)\dot {P}_{q}^{{ - 1}}\left( t \right){\tilde {\theta }_q}\left( t \right) - \mathop \sum \limits_{{q=1}}^{N} \left( {{\eta _q}\mathop \sum \limits_{{j=1}}^{N} {b_{qj}}{s_q}~sign\left( {{s_j}} \right)} \right).$$

(31)

According to the design parameter, following inequality can be obtained:

$$\dot {V}\left( t \right) \leq \mathop \sum \limits_{{i=1}}^{N} \left( {{s_q}\left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{q}^{T},~ - {{\hat {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }}^T}} \right] - \left[ {\hat {\dot {a}}_{q}^{T}(t),~\hat {\dot {b}}_{q}^{T}(t)} \right]P_{q}^{{ - 1}}(t)} \right){\tilde {\theta }_q}\left( t \right)+\frac{1}{2}\mathop \sum \limits_{{q=1}}^{N} \hat {\theta }_{q}^{T}\left( t \right)\dot {P}_{q}^{{ - 1}}\left( t \right){\tilde {\theta }_q}\left( t \right) - \sum\nolimits_{{q=1}}^{N} \left| {{\eta _q}} \right|\left| {{s_q}} \right| \left(\left| {{b_{qq}}} \right| - \sum\nolimits_{{j=1,j \ne q}}^{N} {\left| {{b_{qj}}} \right|} \right).$$

(32)

Now, by using the given adaptation rules in Eq. (24), it is concluded that:

$$\dot {V}\left( t \right) \leq - \frac{1}{2}\mathop \sum \limits_{{q=1}}^{N} {\alpha _q}\tilde {\theta }_{q}^{T}\left( t \right)P_{q}^{{ - 1}}\left( t \right){\tilde {\theta }_q}\left( t \right) - \mathop \sum \limits_{{q=1}}^{N} \left( {{\gamma _q} - \frac{1}{2}} \right)\tilde {\theta }_{q}^{T}\left( t \right){\Phi _q}\left( t \right){\Phi _q}^{T}\left( t \right){\tilde {\theta }_q}\left( t \right) - \mathop \sum \limits_{{q=1}}^{N} \left| {{\eta _q}} \right|\left| {{s_q}} \right|\left(\left| {{b_{qq}}} \right| - \mathop \sum \limits_{{j=1,j \ne q}}^{N} \left| {{b_{qj}}} \right|\right).$$

(33)

By choosing $~{\gamma _q}>\frac{1}{2}~$, ${\alpha _q}>0$ and according to A2:

$$\dot {V}\left( t \right) \leq - \frac{1}{2}\mathop \sum \limits_{{q=1}}^{N} {\alpha _q}\tilde {\theta }_{q}^{T}\left( t \right)P_{q}^{{ - 1}}\left( t \right){\tilde {\theta }_q}\left( t \right) - \mathop \sum \limits_{{q=1}}^{N} \left( {{\gamma _q} - \frac{1}{2}} \right)\tilde {\theta }_{q}^{T}\left( t \right){\Phi _q}\left( t \right){\Phi _q}^{T}\left( t \right){\tilde {\theta }_q}\left( t \right) - \mathop \sum \limits_{{q=1}}^{N} \left| {{\eta _q}} \right|\left| {{b_{qq}}} \right|\left| {{s_q}} \right|.$$

(34)

Finally, it is concluded that:

$$\begin{gathered} \dot {V}\left( t \right) \leq - {W_3}\left( {||S||,||\tilde {\theta }||} \right)= - \left( {\frac{1}{2}a\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma } ||\tilde {\theta }|{|^2}+\delta ||S||} \right)<0 \hfill \\ {~} \hfill \\ 0<a=\mathop {\hbox{min} }\limits_{{\forall q=1, \cdots ,N}} {\alpha _q},0<~\delta =\mathop {\hbox{min} }\limits_{{\forall q=1, \cdots ,N}} \left( {\left| {{\eta _q}} \right|\left| {{b_{qq}}} \right|} \right). \hfill \\ \end{gathered}$$

(35)

The derivative of Lyapunov function remains negative definite for PEE and the sliding vector. Hence, based on the Lyapunov theory, $~{W_3}\left( {||S||,||\tilde {\theta }||} \right) \to 0,$ asymptotically. As the result, estimated parameters of the basic model converge to their true values and output signals converge asymptotically to their corresponding desired signals.

Appendix b

Theorem 1′

After switching, by using the model formulated as Eq. (9), the closed-loop system under identification remains stable. Assumptions A1, A2, A3 are stablished in this theorem.

The Lyapunov function stated in Eq. (15) is suitable for proving this theorem. Besides the system energy in sliding surfaces, it shows the energy of the identification error of R-OM in the input–output space. So the derivative of this function should be negative definite in order to vanish identification errors.

$$\dot {V}\left( t \right)=\frac{1}{2}\left[ {2{S^T}\dot {S} - 2\hat {\dot {\theta }}_{i}^{{'T}}\left( t \right)P_{i}^{{'-1}}\left( t \right)\tilde {\theta }_{i}^{'}\left( t \right)+\tilde {\theta }_{i}^{{'T}}\left( t \right)\dot {P}_{i}^{{' - 1}}\left( t \right)\tilde {\theta }_{i}^{'}\left( t \right) - 2\mathop \sum \limits_{{q \ne i}}^{N} \hat {\dot {\theta }}_{q}^{T}\left( t \right)P_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)+\mathop \sum \limits_{{q \ne i}}^{N} \tilde {\theta }_{q}^{T}\left( t \right)\dot {P}_{q}^{{ - 1}}\left( t \right){{\tilde {\theta }}_q}\left( t \right)} \right]$$

(36)

By substitution of the derivative of the sliding vector in Eq. (14), it is calculated:

$$\dot{V}\left( t \right) = \frac{1}{2}\left[ { - 2S^{T} \left[ {\begin{array}{*{20}c} {\tilde{b}_{1}^{T} } \\ {\begin{array}{*{20}c} \vdots \\ {\tilde{b}_{i}^{{\prime T}} } \\ \vdots \\ \end{array} } \\ {\tilde{b}_{N}^{T} } \\ \end{array} } \right]\hat{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} } + 2S^{T} \left[ {\begin{array}{*{20}c} {a_{1}^{T} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{1} } \\ {\begin{array}{*{20}c} \vdots \\ {a_{1}^{{\prime k^{T} }} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{{ik}} } \\ \vdots \\ \end{array} } \\ {a_{N}^{T} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} _{1} } \\ \end{array} } \right] - 2\hat{\dot{\theta }}_{i}^{{\prime T}} \left( t \right)P_{i}^{{\prime - 1}} \left( t \right)\tilde{\theta }_{i}^{\prime } \left( t \right) + \theta _{i}^{{\prime T}} \left( t \right)\dot{P}_{i}^{{\prime - 1}} (t)\tilde{\theta }_{i}^{\prime } (t) - 2\sum\nolimits_{{q \ne i}}^{N} {\hat{\dot{\theta }}_{q}^{T} } \left( t \right)P_{q}^{{ - 1}} \left( t \right)\tilde{\theta }_{q} \left( t \right) + \sum\nolimits_{{q \ne i}}^{N} {\tilde{\theta }_{q}^{T} } \left( t \right)\dot{P}_{q}^{{ - 1}} \left( t \right)\tilde{\theta }_{q} (t)} \right] - S^{T} B^{\prime } \eta ~\text{sign} \left( S \right).$$

(37)

By imposing adaptation rules and doing some calculations, it can be calculated:

$$\dot {V}\left( t \right) \leq - \frac{1}{2}{\alpha _i}\tilde {\theta }_{i}^{{\prime T}}\left( t \right)P_{i}^{{\prime - 1}}\left( t \right)\tilde {\theta }_{i}^{\prime }\left( t \right) - \left( {{\gamma _i} - \frac{1}{2}} \right)\tilde {\theta }_{i}^{{'T}}\left( t \right)\Phi _{i}^{\prime }\left( t \right)\Phi _{i}^{{\prime T}}\left( t \right)\tilde {\theta }_{i}^{\prime }(t) - \frac{1}{2}\sum\limits_{{q \ne i}}^{N} {{\alpha _q}\tilde {\theta }_{q}^{T}} \left( t \right)P_{q}^{{ - 1}}\left( t \right){\tilde {\theta }_q}\left( t \right) - \sum\limits_{{q \ne i}}^{N} {\left( {{\gamma _q} - \frac{1}{2}} \right)} \tilde {\theta }_{q}^{T}\left( t \right){\Phi _q}\left( t \right){\Phi _q}^{T}\left( t \right){\tilde {\theta }_q}\left( t \right) - \sum\limits_{{q \ne i}}^{N} {\left| {{\eta _q}} \right|\left| {{b_{qq}}} \right|\left| {{S_q}} \right| - \left| {{\eta _i}} \right|\left| {b_{{ii}}^{\prime }} \right|\left| {{S_i}} \right|} .$$

(38)

The whole calculations and steps of proof are the same as “Appendix a”. Even if ${\tau _2}$ is finite and identifier cannot converges to real parameters, but still the energy of the sliding surfaces decreases and the closed-loop system remains stable. It is also concluded if ${\tau _2}$ is far enough from the switching moment, parameters of identified R-OM can converge to their basic values in R-OM.

$$\dot {V}\left( t \right) \leq - {W_3}\left( {||S||,||{{\tilde {\theta }}^\prime }||} \right)= - \left( {\frac{1}{2}a{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma } }^\prime }||{{\tilde {\theta }}^\prime }|{|^2}+\delta ||S||} \right)<0$$

(39)