1 Introduction

For conventional discrete-time sampled-data systems, the input and output are sampled at a single rate and the sampling intervals are assumed to be equally spaced in time [1, 3–6]. In practice, different variables of a system may be sampled at different sampling rates [2, 22] and the sampling frequency may be varying, namely, non-equally spaced in time. The non-uniform sampling scheme has advantages over the uniform one, such as always preserving controllability and observability in discretization when a non-uniformly sampled system is described by a lifted state-space model [11, 17].

Literature on non-uniformly sampled multirate systems includes the generalized predictive control [26], the fault detection and isolation with non-uniformly sampled data [18, 19], the system reconstruction from non-uniformly sampled discrete-time systems [11], etc. Recently, the non-uniformly sampled multirate system identification has attracted much attention. Using lifting technique which is a standard tool of dealing with multirate systems, Ding et al. proposed a hierarchical identification method [11] for the lifted state-space model of the non-uniformly sampled systems [20].

The direct input–output representation is frequently considered when dealing with the non-uniformly sampled systems. Zhu et al. proposed the output error method for slowly and irregularly sampled system [35]. Ding et al. developed the partially coupled stochastic gradient algorithm for non-uniformly sampled-data systems [10]. Liu et al. proposed a recursive least squares algorithm for non-uniformly sampled systems with the aid of an auxiliary model [21]. See also [32–34] and the references therein.

Most of the existing systems can be modeled by state-space equations [12, 14], and the subspace identification methods are quite effective for the identification of state-space models of single-rate discrete-time linear systems [15, 16, 24, 27, 28]. This paper is concerned with the extension of the subspace identification from dual-rate sampled systems [25] to non-uniformly sampled multirate systems. The main purpose of this paper is to develop a subspace identification method that could cope with the causality constraints.

The rest of this paper is organized as follows. In Sect. 2, the lifted state-space model is derived by using the lifting technique, and the identification problem is discussed. Further, a subspace identification algorithm taking the causality constraints into consideration is presented in Sect. 3. In Sect. 4, a simulation example is illustrated for the proposed algorithm. Finally, some concluding remarks are offered in Sect. 5.

2 Problem Description

Consider a class of periodically non-uniformly sampled systems as depicted in Fig. 1 [11, 26], where S c is a continuous process,

$$ S_c:\ \begin{cases} \dot{\boldsymbol{x}}(t)=\boldsymbol {A}_c \boldsymbol{x}(t) +\boldsymbol {B}_cu(t),\\ y(t) = \boldsymbol {C}_c\boldsymbol{x}(t) +D_cu(t), \end{cases} $$
(1)

\(\boldsymbol{x}(t)\in\mathbb{R}^{n}\) is the state vector, \(u(t)\in\mathbb{R}\) is the control input, \(y(t)\in\mathbb{R}\) is the system output, A c ,B c ,C c ,D c the matrices with proper dimensions; \(\mathcal {H}_{T}\) and S T are the non-uniformly periodical zero-order holder and sampler with the frame period T, and with the updating and sampling intervals {τ 1,τ 2,…,τ p }, namely, the zero-order holder/sampler non-uniformly updates/samples at time t=kT+t i , i=1,2,…,p, k=0,1,2,… , where t i :=τ 1+τ 2+⋯+τ i (t 0=0), thus the frame period T:=τ 1+τ 2+⋯+τ p .

Fig. 1
figure 1

The periodically non-uniformly sampled systems

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In the kth period [kT,(k+1)T), the control input u(t) and output y(t) are non-uniformly updated at time t=kT+t i (i=0,1,2,…,p−1), the non-uniformly updating properties [10, 11] are

$$ u(t)= \begin{cases} u(kT), & kT\le t<kT+t_1,\\ u(kT+t_1), & kT+t_1\le t<kT+t_2,\\ \quad\vdots& \\ u(kT+t_{p-1}), & kT+t_{p-1}\le t<(k+1)T. \end{cases} $$
(2)

The system input and output are updated by {τ 1,τ 2,…,τ p } periodically, thus the discrete-time system from the input to output is a time-varying single-input single-output system. By the lifting technique, p inputs are grouped and p outputs are listed together to form \(\underline{\boldsymbol{u}}\) and \(\underline{\boldsymbol {y}}\), leading to a time-invariant multi-input multi-output system:

$$ S:\ \begin{cases} \boldsymbol{x}(kT+T)=\boldsymbol {A}\boldsymbol{x}(kT) +\boldsymbol {B}\underline{\boldsymbol{u}}(kT),\\ \underline{\boldsymbol {y}}(kT) = \boldsymbol {C}\boldsymbol{x}(kT) +\boldsymbol {D}\underline{\boldsymbol{u}}(kT), \end{cases} $$
(3)

with the available non-uniformly sampled data {u(kT+t i ),y(kT+t i ), i=0,1,2,…,p−1}.

Referring to the method in [11] and discretizing (3) yields

$$\begin{aligned} \boldsymbol{x}(kT+T) =&\mathrm {e}^{\boldsymbol {A}_cT}\boldsymbol{x}(kT)+\int^{(k+1)T}_{kT}\mathrm {e}^{\boldsymbol {A}_c((k+1)T-\tau)} \boldsymbol {B}_cu(\tau)\,\mathrm {d}\tau \end{aligned}$$
(4)
$$\begin{aligned} =:&\boldsymbol {A}\boldsymbol{x}(kT)+\sum _{i=1}^{p}\boldsymbol {B}_iu(kT+t_{i-1}), \end{aligned}$$
(5)
$$\begin{aligned} =:&\boldsymbol {A}\boldsymbol{x}(kT)+\boldsymbol {B}\underline{\boldsymbol{u}}(kT), \end{aligned}$$
(6)

where

$$\begin{aligned} \boldsymbol {A} :=&\mathrm {e}^{{\scriptsize \boldsymbol {A}}_cT}\hbox{$\in{\mathbb{R}}^{n\times n}$}, \end{aligned}$$
(7)
$$\begin{aligned} \boldsymbol {B} :=&[\boldsymbol {B}_1, \boldsymbol {B}_2, \ldots, \boldsymbol {B}_p]\in{\mathbb{R}}^{n\times p}, \end{aligned}$$
(8)
$$\begin{aligned} \boldsymbol {B}_i :=&\mathrm {e}^{{\scriptsize \boldsymbol {A}}_c(T-t_i)}\int ^{\tau_i}_0\mathrm {e}^{{\scriptsize \boldsymbol {A}}_ct}\,\mathrm {d}t \boldsymbol {B}_c, \end{aligned}$$
(9)
$$\begin{aligned} \underline{\boldsymbol{u}}(kT) :=& \bigl[u(kT),\ u(kT+t_1),\ \ldots,\ u(kT+t_{p-1}) \bigr]^{\tiny \mathrm {T}} \in{\mathbb{R}}^{p}. \end{aligned}$$
(10)

Because of the non-uniformly zero-order holder in system (1), it is easy to obtain

$$\begin{aligned} \boldsymbol{x}(kT+t_i) =&\mathrm {e}^{\boldsymbol {A}_ct_i} \boldsymbol{x}(kT)+\int^{kT+t_i}_{kT} \mathrm {e}^{\boldsymbol {A}_c(kT+t_i-\tau)}\boldsymbol {B}_cu(\tau)\,\mathrm {d}\tau \\ =&\mathrm {e}^{\boldsymbol {A}_ct_i}\boldsymbol{x}(kT) \\ &{}+[\boldsymbol {B}_1, \boldsymbol {B}_2,\ldots ,\boldsymbol {B}_i] \bigl[u(kT), u(kT+t_1), \ldots, u(kT+t_{i-1})\bigr]^{\tiny \mathrm {T}}. \end{aligned}$$
(11)

The output equation is given by

$$\begin{aligned} y(kT+t_i) =&\boldsymbol {C}_c \boldsymbol{x}(kT+t_i)+D_cu(kT+t_i) \\ =&\boldsymbol {C}_c\mathrm {e}^{\boldsymbol {A}_ct_i}\boldsymbol{x}(kT)+[ \boldsymbol {C}_c\boldsymbol {B}_1,\boldsymbol {C}_c\boldsymbol {B}_2, \ldots ,\boldsymbol {C}_c\boldsymbol {B}_i]\underline{\boldsymbol{u}}(kT) +D_cu(kT+t_i) \\ =:&\boldsymbol {C}_i\boldsymbol{x}(kT)+[D_1,D_2, \ldots,D_i, D_c] \left[ \begin{array}{c} {u(kT)}\\ {u(kT+t_1)}\\ {\vdots}\\ {u(kT+t_{i-1})}\\ {u(kT+t_i)} \end{array}\right], \end{aligned}$$
(12)

where \(\boldsymbol {C}_{i}=:\boldsymbol {C}_{c}\mathrm {e}^{\boldsymbol {A}_{c}t_{i}}\), D i =:C c B i , i=1,2,…,p−1. Thus, we obtain the lifted state-space model in (3) for the multirate system, where

$$\begin{aligned} \underline{\boldsymbol {y}}(kT) =& \bigl[y(kT),\ y(kT+t_1), \ldots, y(kT+t_{p-1}) \bigr]^{\tiny \mathrm {T}}\in{\mathbb{R}}^{p}, \end{aligned}$$
(13)
$$\begin{aligned} \boldsymbol {C} =& \left[ \begin{array}{c} {\boldsymbol {C}_c}\\ {\boldsymbol {C}_1}\\ { \boldsymbol {C}_2}\\ {\vdots}\\ {\boldsymbol {C}_{p-1}} \end{array}\right]\in {\mathbb{R}}^{p\times n} \end{aligned}$$
(14)
$$\begin{aligned} \boldsymbol {D} =& \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}D_c & 0 & \ldots& \ldots& 0\\ D_1 & D_c & & & \vdots\\ D_1 & D_2 & \ddots& & \vdots\\ \vdots& & \ddots& D_c & 0\\ D_1 & D_2 & \ldots& D_{p-1} & D_c \end{array} \right ]\in{\mathbb{R}}^{p\times p}. \end{aligned}$$
(15)

Replacing the lifted output \(\underline{\boldsymbol {y}}(kT)\) by the lifted noise-contaminated one \(\underline{\boldsymbol{z}}(kT)\) and omitting the frame period T yields

$$ \begin{cases}\boldsymbol{x}(k+1)=\boldsymbol {A}\boldsymbol{x}(k) +\boldsymbol {B}\underline{\boldsymbol{u}}(k),\\ \underline{\boldsymbol{z}}(k) = \boldsymbol {C}\boldsymbol{x}(k) +\boldsymbol {D}\underline{\boldsymbol{u}}(k)+\underline{\boldsymbol{v}}(k), \end{cases} $$
(16)

with \(\underline{\boldsymbol{v}}(k):=[v(k), v(k+t_{1}), \ldots, v(k+t_{p-1})]^{\tiny \mathrm {T}}\in{\mathbb{R}}^{p}\) the lifted noise vector.

3 Subspace Identification Method

Given the periodically non-uniformly sampled data {u(kT+t i ),z(kT+t i ), i=0,1,2,…,p−1}, the lifted input and output data are \(\{\underline{\boldsymbol{u}}(k), \underline{\boldsymbol{z}}(k)\}\), while the input and output block Hankel matrices can be defined as

$$\begin{aligned} \boldsymbol {U}_{0|l-1} :=& \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c}\underline{\boldsymbol{u}}(0)& \underline {\boldsymbol{u}}(1) & \dots& \underline{\boldsymbol{u}}(N-1)\\ \underline{\boldsymbol{u}}(1) & \underline{\boldsymbol{u}}(2) &\dots& \underline{\boldsymbol{u} }(N)\\ \vdots& \vdots& &\vdots\\ \underline{\boldsymbol{u}}(l-1) & \underline{\boldsymbol{u}}(l) & \ldots& \underline{\boldsymbol{u}}(l+N-2) \end{array} \right ]\in{\mathbb{R}}^{lp\times N}, \end{aligned}$$
(17)
$$\begin{aligned} \boldsymbol {Z}_{0|l-1} :=& \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \underline{\boldsymbol{z}}(0)& \underline{\boldsymbol{z}}(1) & \ldots& \underline{\boldsymbol{z}}(N-1)\\ \underline{\boldsymbol{z}}(1) & \underline{\boldsymbol{z}}(2) &\ldots& \underline{\boldsymbol{z}} (N)\\ \vdots& \vdots& &\vdots\\ \underline{\boldsymbol{z}}(l-1) & \underline{\boldsymbol{z}}(l) & \ldots& \underline{\boldsymbol{z}}(l+N-2) \end{array} \right ]\in{\mathbb{R}}^{lp\times N}, \end{aligned}$$
(18)

where l is strictly greater than the dimension n of state vector, N is sufficiently large, the indices 0 and l−1 denote the arguments of the upper-left and lower-left elements, respectively.

U l|2l−1 and Z l|2l−1 can be defined in a similar way. The block Hankel matrices U 0|l−1 and Z 0|l−1 are usually called the past inputs and outputs, respectively, whereas the block Hankel matrices U l|2l−1 and Z l|2l−1 are called the future inputs and outputs, respectively. Define , the LQ decomposition of the input and output block Hankel matrices can be performed as

$$ \left[ \begin{array}{c} {\boldsymbol {U}_{l|2l-1}}\\ {\boldsymbol {W}_p}\\ { \boldsymbol {Z}_{l|2l-1}}\end{array}\right]= \left [ \begin{array}{c@{\quad }c@{\quad }c}\boldsymbol {R}_{11}& 0 & 0\\ \boldsymbol {R}_{21} & \boldsymbol {R}_{22} & 0\\ \boldsymbol {R}_{31} & \boldsymbol {R}_{32} & 0 \end{array}\right] \left[ \begin{array}{c} {\boldsymbol {Q}^{\tiny \mathrm {T}}_1}\\ { \boldsymbol {Q}^{\tiny \mathrm {T}}_2}\\ {\boldsymbol {Q}^{\tiny \mathrm {T}}_3} \end{array}\right] $$
(19)

where \(\boldsymbol {R}_{11}\in{\mathbb{R}}^{lp\times lp}\), \(\boldsymbol {R}_{22}\in {\mathbb{R}}^{2lp\times2lp}\), \(\boldsymbol {Q}_{1}, \boldsymbol {Q}_{3}\in{\mathbb{R}}^{N\times lp}\), \(\boldsymbol {Q}_{2}\in{\mathbb{R}}^{N\times2lp}\).

Defining ξ as the oblique projection of Z l|2l−1 onto W p along U l|2l−1, with the above LQ decomposition, we have

$$ \boldsymbol {\xi }=\boldsymbol {R}_{32}\boldsymbol {R}^{\dagger}_{22} \boldsymbol {W}_p, $$
(20)

† denoting the pseudo inverse. The details are referred to Theorem 6.3 in [16], and thus omitted here.

Let the SVD of ξ be

$$ \boldsymbol {\xi }=[\boldsymbol {U}_1, \boldsymbol {U}_2] \left[ \begin{array}{c@{\quad }c}{ \boldsymbol {\varSigma }_1}& {\bf0}\\ {\bf0}& {\bf0} \end{array}\right] \left[ \begin{array}{c} {\boldsymbol {V}^{\tiny \mathrm {T}}_1}\\ {\boldsymbol {V}^{\tiny \mathrm {T}}_2}\end{array}\right]= \boldsymbol {U}_1\boldsymbol {\varSigma }_1 \boldsymbol {V}^{\tiny \mathrm {T}}_1. $$
(21)

Defining the state sequence X l :=[x(l),x(l+1),…,x(l+N−1)], we have the estimated state sequence

$$ \hat{\boldsymbol {X}}:=\bigl[\hat{\boldsymbol{x}}(l), \hat{\boldsymbol{x}}(l+1), \ldots, \hat{\boldsymbol{x} }(l+N-1)\bigr]\in{\mathbb{R}}^{n\times N}. $$
(22)

By defining

$$\begin{aligned} \hat{\boldsymbol {X}}_{l+1} :=&\bigl[\hat{\boldsymbol{x}}(l+1), \hat{\boldsymbol{x}}(l+2), \ldots, \hat{\boldsymbol{x} }(l+N-1)\bigr]\in{\mathbb{R}}^{n\times(N-1)}, \end{aligned}$$
(23)
$$\begin{aligned} \hat{\boldsymbol {X}}_l :=&\bigl[\hat{\boldsymbol{x}}(l), \hat{\boldsymbol{x}}(l+1), \ldots, \hat{\boldsymbol{x} }(l+N-2)\bigr]\in{\mathbb{R}}^{n\times(N-1)}, \end{aligned}$$
(24)
$$\begin{aligned} \boldsymbol {U}_{l|l} :=&\bigl[\underline{\boldsymbol{u}}(l), \underline{\boldsymbol{u}}(l+1),\ldots, \underline{\boldsymbol{u}}(l+N-2)\bigr]\in{\mathbb{R}}^{p\times (N-1)}, \end{aligned}$$
(25)
$$\begin{aligned} \boldsymbol {Z}_{l|l} :=&\bigl[\underline{\boldsymbol{z}}(l), \underline{\boldsymbol{z}}(l+1),\ldots, \underline{\boldsymbol{z}}(l+N-2)\bigr]\in{\mathbb{R}}^{p\times(N-1)}, \end{aligned}$$
(26)

it follows that

$$ \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l+1}}\\ {\boldsymbol {Z}_{l|l}}\end{array}\right]= \left[ \begin{array}{c@{\quad }c} {\boldsymbol {A}}& {\boldsymbol {B}}\\ {\boldsymbol {C}}& {\boldsymbol {D}} \end{array}\right] \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l}}\\ { \boldsymbol {U}_{l|l}}\end{array}\right], $$
(27)

then the system matrices can be estimated by using the least-squares technique,

$$ \left[ \begin{array}{c@{\quad }c} {\hat{\boldsymbol {A}}}& {\hat{\boldsymbol {B}}}\\ {\hat{\boldsymbol {C}}}& {\hat{\boldsymbol {D}}}\end{array}\right]= \left\{ \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l+1}}\\ {\boldsymbol {Z}_{l|l}}\end{array}\right]^{\tiny \mathrm {T}} \left[ \begin{array}{c} {\hat{ \boldsymbol {X}}_{l+1}}\\ {\boldsymbol {Z}_{l|l}}\end{array}\right] \right\}^{-1} \left[ \begin{array}{c} {\hat{ \boldsymbol {X}}_{l+1}}\\ {\boldsymbol {Z}_{l|l}}\end{array}\right]^{\tiny \mathrm {T}} \left[ \begin{array}{c} {\hat{ \boldsymbol {X}}_{l}}\\ {\boldsymbol {U}_{l|l}}\end{array}\right]. $$
(28)

Note that the upper triangular blocks in D are zero, namely, the zero-entries of this upper triangular block in D do not need to be identified, but the upper triangular blocks may not equal zero in \(\hat{\boldsymbol {D}}\). In order to tackle this causality constraint for the lifted model, we propose a two-stage way to estimate the matrices (A,B,C,D).

From (27), one can get the estimates of (A,B) by solving the following least-squares form:

$$ \hat{\boldsymbol {X}}_{l+1}=[\boldsymbol {A}, \boldsymbol {B}] \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l}}\\ {\boldsymbol {U}_{l|l}}\end{array}\right]. $$
(29)

To obtain the non-zero subblock matrices in D, we decompose the matrix Z l|l in (26) and U l|l in (25) into p row vectors according to their row dimension,

$$ \boldsymbol {Z}_{l|l}:= \left[ \begin{array}{c} {\boldsymbol {Z}_1}\\ { \boldsymbol {Z}_2}\\ {\vdots}\\ {\boldsymbol {Z}_p}\end{array}\right],\qquad \boldsymbol {U}_{l|l}:= \left[ \begin{array}{c} {\boldsymbol {U}_1}\\ {\boldsymbol {U}_2}\\ {\vdots}\\ {\boldsymbol {U}_p}\end{array}\right], $$
(30)

From Equation (14) and

$$ \boldsymbol {Z}_{l|l}=[\boldsymbol {C}, \boldsymbol {D}] \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l}}\\ {\boldsymbol {U}_{l|l}}\end{array}\right], $$
(31)

we have

$$\begin{aligned} \boldsymbol {Z}_1 =&[\boldsymbol {C}_c, D_c] \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l}}\\ {\boldsymbol {U}_1}\end{array}\right], \end{aligned}$$
(32)
$$\begin{aligned} \boldsymbol {Z}_2 =&[\boldsymbol {C}_1, D_1, D_c] \left[ \begin{array}{c} {\hat{\boldsymbol {X}}_{l}}\\ {\boldsymbol {U}_1}\\ {\boldsymbol {U}_2}\end{array}\right], \end{aligned}$$
(33)
$$\begin{aligned} \vdots& \\ \boldsymbol {Z}_p =&[\boldsymbol {C}_{p-1}, D_1, D_2,\ldots, D_{p-1}, D_c] \left[ \begin{array}{c} {\hat {\boldsymbol {X}}_{l}}\\ {\boldsymbol {U}_1}\\ {\boldsymbol {U}_2}\\ {\vdots}\\ {\boldsymbol {U}_p}\end{array}\right]. \end{aligned}$$
(34)

Note that D c can be estimated by solving (32), thus it can be used to estimate D 1 in (33), and the rest unknown entries in D can be estimated in a similar way.

4 Example

Consider a continuous process model described by

$$G(s)=\frac{1}{100s^2+10s+1}, $$

its canonical state space form being

$$S_c:\ \begin{cases}\dot{\boldsymbol{x}}(t)=\left[ \begin{array}{c@{\quad }c} {-0.1}&{-0.1}\\ {1}&{0}\end{array}\right]\boldsymbol{x}(t) + \left[ \begin{array}{c} {1}\\ {0}\end{array}\right]u(t),\\ z(t) = [0,0.01]\boldsymbol{x}(t)+v(t). \end{cases} $$

Taking p=2, τ 1=0.618 s, τ 2=0.382 s, hence, t 1=τ 1=0.618 s, t 2=τ 1+τ 2=T=1 s. Then the corresponding lifted state-space model is

$$\begin{aligned} \boldsymbol{x}(kT+T) =&\boldsymbol {A}\boldsymbol{x}(kT) +\boldsymbol {B}\underline{\boldsymbol{u}}(kT) \\ =&\left[ \begin{array}{c@{\quad }c} {0.9002}& {-0.0095}\\ {0.9500}& {0.9952}\end{array}\right]\boldsymbol{x}(kT) + \left[ \begin{array}{c@{\quad }c} {0.5753} &{0.37470}\\ {0.4113}& {0.07203}\end{array}\right] \left[ \begin{array}{c} {u(kT)}\\ {u(kT+t_1)}\end{array}\right] \\ \left[ \begin{array}{c} {z(kT)}\\ {z(kT+t_1)}\end{array}\right] =& \left[ \begin{array}{c@{\quad }c} {0}& {0.01}\\ {0.005989}& {0.009981}\end{array}\right]\boldsymbol{x}(kT) \\ & +\left[ \begin{array}{c@{\quad }c} {0}& {0}\\ {0.004113}& {0}\end{array}\right] \left[ \begin{array}{c} {u(kT)}\\ {u(kT+t_1)}\end{array}\right] + \left[ \begin{array}{c} {v(kT)}\\ {v(kT+t_1)}\end{array}\right]. \end{aligned}$$

The input signals u(kT) and u(kT+t 1) are taken as two random signal sequences with zero mean and unit variances and two uncorrelated noise sequences with zero mean and variances σ 2=0.102. The noise terms are independent of the inputs.

With the non-uniformly sampled input and output data, we apply the modified subspace identification method respectively to the above lifted model and to the following single-rate model, as follows.

Taking T=1 s for a single-rate sampled system yields the discrete-time state-space model

$$ S_d:\ \begin{cases}\boldsymbol{x}(kT+T)= \left[ \begin{array}{c@{\quad }c} {0.9783}&{-0.0095}\\ {0.95}&{0.9952}\end{array}\right]\boldsymbol{x}(kT) +\left[ \begin{array}{c} {0.95}\\ {0.4833}\end{array}\right]u(kT),\\ z(kT) = [0,0.01]\boldsymbol{x}(kT)+v(kT). \end{cases} $$

The step responses of the identified lifted system and single-rate system are shown in Figs. 2–3: The lifted model can capture the actual system dynamics better than the single-rate model does. The estimated poles of the lifted model and the single-rate model are listed in Table 1: the estimated poles of the lifted model are closer to the actual system poles than that of the single-rate model.

Fig. 2
figure 2

The step responses of the actual system and the estimated model under non-uniform sampling

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Fig. 3
figure 3

The step responses of the actual system and the estimated model under single-rate sampling

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Table 1 The estimated poles of the lifted model and the single-rate model
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Furthermore, the Bode diagrams of the actual system and the estimated systems are shown in Figs. 4–5. This indicates that the estimated lifted model can achieve satisfactory results.

Fig. 4
figure 4

The Bode diagrams of the actual system and the estimated system

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Fig. 5
figure 5

The Bode diagrams of the actual system and the estimated single-rate system

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5 Conclusions

We have discussed the identification methods for periodically non-uniformly sampled system. By using the lifting technique, we propose a two-stage subspace identification method to identify the lifted state-space models, the advantages of the proposed method lie in that:

  • The lifted system can be estimated by using non-uniformly sampled data directly, thus it can achieve better performance than the single-rate one.

  • The developed algorithm can tackle the casuality constraints in the lifted state-space model.

The proposed method can be extended to other linear or nonlinear systems [7–9, 13, 23, 29–31].