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Subspace-Based Deterministic Identification of MIMO Linear State-Delayed Systems

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Abstract

A subspace-based deterministic identification method is proposed for MIMO linear systems with time delay in their state variables. This method can estimate the matrices of the state-delayed system, its order and the amount of time delay. First, by extending the orthogonal decomposition subspace method for state-delayed systems, the augmented system matrices which include the original system matrices are identified. Then, an algorithm is presented to find a similarity transformation that converts the augmented matrices to a special form through which matrices of the state-delayed system can be extracted. The values of the time delay and model order are estimated by introducing appropriate criteria. Finally, three examples are provided to validate the proposed identification method.

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Authors and Affiliations

  1. Department of Electrical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

    Mahboubeh Ahmadipour, Saeid Reza Seydnejad & Mojtaba Barkhordari-Yazdi

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  1. Mahboubeh Ahmadipour
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  2. Saeid Reza Seydnejad
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  3. Mojtaba Barkhordari-Yazdi
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Correspondence to Mojtaba Barkhordari-Yazdi.

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Appendices

Appendix 1

Assume similarity transformation matrix \( {\varvec{S}} \in {\mathbb{R}}^{(\tau + 1)n \times (\tau + 1)n} \) converts the estimated augmented matrices \( {\varvec{\hat{\bar{A}}}} \), \( {\varvec{\hat{\bar{B}}}} \), \( {\varvec{\hat{\bar{C}}}} \) to the following desired form,

$$ \begin{aligned} & {\varvec{\hat{\bar{A}}}}_{{{\varvec{desired}}}} = \left[ {\begin{array}{*{20}c} {\varvec{0}} & {\varvec{I}} & {\varvec{0}} & \ldots & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{I}} & {} & {\varvec{0}} & {\varvec{0}} \\ {} & {} & {} & {\varvec{O}} & {} & {} \\ {} & {} & {} & {} & {\varvec{I}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {} & {\varvec{0}} & {\varvec{I}} \\ {{\hat{\varvec{A}}}_{{\varvec{2}}}^{{\varvec{d}}} } & {\varvec{0}} & {\varvec{0}} & \ldots & {\varvec{0}} & {{\hat{\varvec{A}}}_{{\varvec{1}}}^{{\varvec{d}}} } \\ \end{array} } \right], \, \quad {\varvec{\hat{\bar{B}}}}_{{{\varvec{desired}}}} = \left[ {\begin{array}{*{20}c} {\varvec{0}} \\ {\varvec{0}} \\ \vdots \\ {\varvec{0}} \\ {\varvec{0}} \\ {{\hat{\varvec{B}}}^{{\varvec{d}}} } \\ \end{array} } \right], \\ & \hat{{\bar{\varvec{C}}}}_{{{\varvec{desired}}}} = \left[ {\begin{array}{*{20}c} {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & \ldots & {\varvec{0}} & \hat{\varvec{C}}^{{\varvec{d}}} \\ \end{array} } \right], \\ \end{aligned} $$
(49)

where \( {\hat{\varvec{A}}}_{{\varvec{1}}}^{{\varvec{d}}} , \, {\hat{\varvec{A}}}_{{\varvec{2}}}^{{\varvec{d}}} \in {\mathbb{R}}^{n \times n} \), \( {\hat{\varvec{B}}}^{{\varvec{d}}} \in {\mathbb{R}}^{n \times m} \) and \( {\hat{\varvec{C}}}^{{\varvec{d}}} \in {\mathbb{R}}^{p \times n} \). Therefore, we have,

$$ {\varvec{\hat{\bar{A}}}}_{{{\varvec{desired}}}} {\varvec{S}} = {\varvec{S\hat{\bar{A}}}}{ ,} $$
(50)
$$ {\varvec{\hat{\bar{B}}}}_{{{\varvec{desired}}}} = {\varvec{S\hat{\bar{B}}}}, $$
(51)
$$ {\varvec{\hat{\bar{C}}}}_{{{\varvec{desired}}}} = {\varvec{\hat{\bar{C}}S}}^{ - 1} . $$
(52)

Let us represent matrix \( {\varvec{S}} \) as

$$ {\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{S}}_{{\varvec{1}}} } \\ {{\varvec{S}}_{{\varvec{2}}} } \\ \vdots \\ {{\varvec{S}}_{{{\varvec{\tau + 1}}}} } \\ \end{array} } \right], $$
(53)

where \( {\varvec{S}}_{{\varvec{1}}} , \, {\varvec{S}}_{{\varvec{2}}} , \, \ldots { , }{\varvec{S}}_{{{\varvec{\tau + 1}}}} \in {\mathbb{R}}^{n \times (\tau + 1)n} \). Then, Eq. (50) implies

$$ {\varvec{S}}_{{{\varvec{i + 1}}}} = {\varvec{S}}_{{\varvec{1}}} {\varvec{\hat{\bar{A}} }}^{i} { , }i = 1,2, \ldots ,\tau , $$
(54)
$$ \left[ {\begin{array}{*{20}c} {{\hat{\varvec{A}}}_{{\varvec{1}}}^{{\varvec{d}}} } & {\varvec{0}} & \ldots & {\varvec{0}} & {{\hat{\varvec{A}}}_{{\varvec{2}}}^{{\varvec{d}}} } \\ \end{array} } \right]{\varvec{S}} = {\varvec{S}}_{{{\varvec{\tau + 1}}}} {\varvec{\hat{\bar{A}}}}. $$
(55)

Since \( {\hat{\varvec{A}}}_{{\varvec{1}}}^{{\varvec{d}}} \) and \( {\hat{\varvec{A}}}_{{\varvec{2}}}^{{\varvec{d}}} \) are full matrices, Eq. (55) only causes the condition

$$ {\varvec{S}}_{{{\varvec{\tau + 1}}}} {\varvec{\hat{\bar{A}}S}}^{ - 1} (:,n + 1:\tau n) = {\varvec{0}}_{n,(\tau - 1)n} . $$
(56)

Now, we investigate Eq. (51). It is obvious from Eq. (51) that

$$ \left[ {\begin{array}{*{20}c} {{\varvec{S}}_{{\varvec{1}}} } \\ {{\varvec{S}}_{{\varvec{2}}} } \\ \vdots \\ {{\varvec{S}}_{{\varvec{\tau}}} } \\ \end{array} } \right]{\varvec{\hat{\bar{B}}}} = {\varvec{0}}_{\tau n \times m} . $$
(57)

Equations (57) and (54) imply that

$$ {\varvec{S}}_{{\varvec{1}}} [\begin{array}{*{20}c} {{\varvec{\hat{\bar{B}}}}} & {{\varvec{\hat{\bar{A}}\hat{\bar{B}}}}} & \ldots & {{\varvec{\hat{\bar{A}}}}^{{{\varvec{\tau - 1}}}} {\varvec{\hat{\bar{B}}}}} \\ \end{array} ] = {\varvec{0}}_{n \times \tau m} . $$
(58)

It can be concluded from Eq. (58) that the row vectors of matrix \( {\varvec{S}}_{{\varvec{1}}} \) are in the space spanned through the row vectors of \( \left( {{\text{null}}\{ [\begin{array}{*{20}c} {{\varvec{\hat{\bar{B}}}}} & {{\varvec{\hat{\bar{A}}\hat{\bar{B}}}}} & \ldots & {{\varvec{\hat{\bar{A}}}}^{\tau - 1} {\varvec{\hat{\bar{B}}}}} \\ \end{array} ]^{T} \} } \right)^{T} \), where \( {\text{null}}\{ .\} \) is the null space of a matrix. Let us define,

$$ {\varvec{F}} = \left[ {\begin{array}{*{20}c} {{\varvec{F}}_{{\varvec{1}}} } \\ {{\varvec{F}}_{{\varvec{2}}} } \\ \vdots \\ {{\varvec{F}}_{{\varvec{q}}} } \\ \end{array} } \right]\mathop = \limits^{\Delta } \left( {{\text{null}}\{ [\begin{array}{*{20}c} {{\varvec{\hat{\bar{B}}}}} & {{\varvec{\hat{\bar{A}}\hat{\bar{B}}}}} & \cdots & {{\varvec{\hat{\bar{A}}}}^{\tau - 1} {\varvec{\hat{\bar{B}}}}} \\ \end{array} ]^{T} \} } \right)^{T} , $$
(59)

in which, \( q = (\tau + 1)n{\text{ - columnrank}}\{ [\begin{array}{*{20}c} {{\varvec{\hat{\bar{B}}}}} & {{\varvec{\hat{\bar{A}}\hat{\bar{B}}}}} & \ldots & {{\varvec{\hat{\bar{A}}}}^{\tau - 1} {\varvec{\hat{\bar{B}}}}} \\ \end{array} ]\} \) and \( {\varvec{F}}_{{\varvec{1}}} , { }{\varvec{F}}_{{\varvec{2}}} , \, \ldots { , }{\varvec{F}}_{{\varvec{q}}} \in {\mathbb{R}}^{1 \times (\tau + 1)n} \) are the row vectors of matrix \( {\varvec{F}} \). Then, \( {\varvec{S}}_{{\varvec{1}}} \) must be as,

$$ {\varvec{S}}_{{\varvec{1}}} = {\varvec{ZF}}, $$
(60)

where \( {\varvec{Z}} \in {\mathbb{R}}^{n \times q} \).

Finally, investigating Eq. (52) implies,

$$ {\varvec{\hat{\bar{C}}S}}^{ - 1} (:,1:\tau n) = {\varvec{0}}_{p \times \tau n} . $$
(61)

Therefore, according to Eqs. (53), (54), (56), (60) and (61), the similarity transformation matrix \( {\varvec{S}} \) which transforms the identified augmented matrices to the desired form (49) can be considered as,

$$ {\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{ZF}}} \\ {{\varvec{ZF\hat{\bar{A}}}}} \\ \vdots \\ {{\varvec{ZF\hat{\bar{A}}}}^{\tau } } \\ \end{array} } \right] $$
(62)

where \( {\varvec{Z}} \in {\mathbb{R}}^{n \times q} \) is a matrix such that the conditions

$$ \begin{aligned} & {\varvec{ZF\hat{\bar{A}}}}^{\tau + 1} {\varvec{S}}^{ - 1} (:,n + 1:\tau n) = {\varvec{0}}_{n \times (\tau - 1)n} , \\ & {\varvec{\hat{\bar{C}}S}}^{ - 1} (:,1:\tau n) = {\varvec{0}}_{p \times \tau n} , \\ \end{aligned} $$
(63)

be satisfied.

Now, we find a solution \( {\varvec{Z}} \in {\mathbb{R}}^{n \times q} \) for the special case \( \tau = 1 \). In this case, the similarity transformation is as

$$ {\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{ZF}}} \\ {{\varvec{ZF\hat{\bar{A}}}}} \\ \end{array} } \right] \in {\mathbb{R}}^{2n \times 2n} $$
(64)

where \( {\varvec{Z}} \in {\mathbb{R}}^{n \times q} \) is a matrix such that \( {\varvec{\hat{\bar{C}}S}}^{ - 1} (:,1:n) = {\varvec{0}}_{p \times n} \). Therefore, the row vectors of matrix \( {\varvec{\hat{\bar{C}}}} \) must be in the space spanned through the row vectors of \( \left( {{\text{null}}\{ ({\varvec{S}}^{ - 1} (:,1:n))^{T} \} } \right)^{T} \). From \( {\varvec{SS}}^{ - 1} = {\varvec{I}} \), we have \( {\varvec{ZF\hat{\bar{A}}S}}^{ - 1} (:,1:n) = {\varvec{0}}_{n \times n} \). Since \( {\text{rank}}({\varvec{ZF\hat{\bar{A}}}}) = n \), \( {\varvec{ZF\hat{\bar{A}}}} \) can be considered as \( \left( {{\text{null}}\{ ({\varvec{S}}^{ - 1} (:,1:n))^{T} \} } \right)^{T} \). In summary,

$$ {\varvec{\hat{\bar{C}}}} = {\varvec{\beta ZF\hat{\bar{A}}}} , $$
(65)

where \( {\varvec{\beta}} \in {\mathbb{R}}^{p \times n} \).

Assume

$$ {\varvec{\hat{\bar{C}}}}_{{\varvec{s}}} = {\varvec{\beta}}_{{\varvec{1}}} {\varvec{ZF\hat{\bar{A}}}} , $$
(66)

where \( {\varvec{\beta}}_{{\varvec{1}}} \in {\mathbb{R}}^{l \times n} \) and \( {\varvec{\hat{\bar{C}}}}_{{\varvec{s}}} \in {\mathbb{R}}^{l \times 2n} \) contains the independent rows of \( {\varvec{\hat{\bar{C}}}} \), then Eq. (62) is satisfied. Without loss of generality, we consider \( {\varvec{\beta}}_{{\varvec{1}}} = [\begin{array}{*{20}c} {{\varvec{I}}_{{\varvec{l}}} } & {{\varvec{0}}_{{{\varvec{l \times (n - l)}}}} } \\ \end{array} ] \), then \( {\varvec{Z}}(1:l,:) = {\varvec{\hat{\bar{C}}}}_{{\varvec{s}}} /({\varvec{F\hat{\bar{A}}}}) \). There is no constraint on \( {\varvec{Z}}(l + 1:n,:) \in {\mathbb{R}}^{(n - l) \times q} \) unless it makes \( {\varvec{Z}} \) and \( {\varvec{S}} \) full rank.

Appendix 2

Let the characteristic polynomial of \( {\bar{\varvec{A}}} \) be given as

$$ \left| {\lambda {\varvec{I}} - {\bar{\varvec{A}}}} \right| = \lambda^{{n_{1} }} + \alpha_{1} \lambda^{{n_{1} - 1}} + \cdots \alpha_{{n_{1} }} , $$
(67)

and

$$ {\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot } \mathop = \limits^{\Delta } \left[ {\begin{array}{*{20}c} {\alpha_{{n_{1} }} {\varvec{I}}_{{\varvec{p}}} } & \ldots & {\alpha_{1} {\varvec{I}}_{{\varvec{p}}} } & {{\varvec{I}}_{{\varvec{p}}} } & {{\varvec{0}}_{{{\varvec{p \times p}}}} } & \ldots & {{\varvec{0}}_{{{\varvec{p \times p}}}} } \\ \end{array} } \right]^{T} \in {\mathbb{R}}^{kp \times p} . $$
(68)

Then, using the Cayley–Hamilton theorem, we derive

$$ {{\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot }}^{T} {\varvec{O}}_{{\varvec{k}}} = {\varvec{C}}(\alpha_{{n_{1} }} {\varvec{I}} + \cdots + \alpha_{{_{1} }} {\bar{\varvec{A}}}^{{n_{1} - 1}} + {\bar{\varvec{A}}}^{{n_{1} }} ) = {\varvec{0}}. $$
(69)

Premultiplying Eq. (7) with \( {{\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot }}^{T} \) implies

$$ {{\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot }}^{T} {\varvec{y}}_{{\varvec{k}}}^{{\varvec{d}}} (t) = {{\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot }}^{T} {\varvec{\psi}}_{{\varvec{k}}} {\varvec{u}}_{k} (t), $$
(70)

which can be rewritten as

$$ {\varvec{y}}_{{\varvec{d}}} (t + n_{1} ) = - \sum\limits_{i = 1}^{{n_{1} }} {\alpha_{i} {\varvec{y}}_{{\varvec{d}}} (t + n_{1} - i} ) + {{\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot }}^{T} {\varvec{\psi}}_{{\varvec{k}}} {\varvec{u}}_{{\varvec{k}}} (t). $$
(71)

Define

$$ {\hat{\varvec{y}}}_{{\varvec{d}}} (t) = - \sum\limits_{i = 1}^{{\hat{n}_{1} }} {\hat{\alpha }_{i} {\varvec{y}}_{{\varvec{d}}} (t - i )} + \mathop{{\hat{\varvec{\Gamma }}}_{{{\varvec{CP}}}}^{ \bot }}\nolimits^{T} {\hat{\varvec{\psi }}}_{{\varvec{k}}} {\varvec{u}}_{{\varvec{k}}} (t - \hat{n}_{1} ), $$
(72)

where \( \hat{\alpha }_{i} , \, i = 1, \ldots ,\hat{n}_{1} \) are the coefficients of the characteristic polynomial of \( {\varvec{\hat{\bar{A}}}} \), and \( \mathop{{\hat{\varvec{\Gamma }}_{{\varvec{CP}}}^{ \bot }}}\nolimits^{T} \) and \( {\hat{\varvec{\psi }}}_{{\varvec{k}}} \) are constructed similar to \( {{\varvec{\Gamma}}_{{{\varvec{CP}}}}^{ \bot }}^{T} \) and \( {\varvec{\psi}}_{{\varvec{k}}} \), respectively, but with the coefficients \( \hat{\alpha }_{i} \) and the estimated matrices \( {\varvec{\hat{\bar{B}}}} \), \( {\varvec{\hat{\bar{C}}}} \) and \( {\hat{\varvec{D}}}^{{\varvec{d}}} \).

Equations (71) and (72) concludes,

$$ {\varvec{e}}_{{\varvec{d}}} (t) = {\varvec{y}}_{{\varvec{d}}} (t) + \sum\limits_{i = 1}^{{\hat{n}_{1} }} {\hat{\alpha }_{i} {\varvec{y}}_{{\varvec{d}}} (t - i} ) - \mathop{{\hat{\varvec{\Gamma }}}_{{{\varvec{CP}}}}^{ \bot }}\nolimits^{T} {\hat{\varvec{\psi }}}_{{\varvec{k}}} {\varvec{u}}_{{\varvec{k}}} (t - \hat{n}_{1} ) = \mathop{{\hat{\varvec{\Gamma }}}_{{{\varvec{CP}}}}^{ \bot }}\nolimits^{T} \left[ {\begin{array}{*{20}c} {\varvec{I}} & { - {\hat{\varvec{\psi }}}_{{\varvec{k}}} } \\ \end{array} } \right] \, \left[ {\begin{array}{*{20}c} {{\varvec{y}}_{{\varvec{k}}}^{{\varvec{d}}} (t - \hat{n}_{1} )} \\ {{\varvec{u}}_{{\varvec{k}}} (t - \hat{n}_{1} )} \\ \end{array} } \right]. $$
(73)

Appendix 3

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Ahmadipour, M., Seydnejad, S.R. & Barkhordari-Yazdi, M. Subspace-Based Deterministic Identification of MIMO Linear State-Delayed Systems. Circuits Syst Signal Process 39, 4067–4093 (2020). https://doi.org/10.1007/s00034-020-01355-4

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