Precise tracking of periodic trajectories for periodic systems

Abstract

Stable inversion was an effective method to achieve precise trajectory tracking, for which the tracking problem of periodically time-varying systems was solved in the existing literature. However, iteration techniques were required to approach stable inversion, whose explicit formulas for this tracking problem cannot be obtained. To overcome this drawback, a special kind of stable inversion, named periodic inversion, for the periodic systems is proposed in this study. By means of the lifted technique, the tracking problem of the periodic systems is reformulated to be equivalent to that of lifted systems. In order to obtain precise tracking, the periodic inversion of the periodic systems or the lifted systems is proposed, and is analysed for the non-singular case and the singular case, each of which is further analysed for three cases: the full row rank case, the full column rank case and the rank-deficient case. Thus the periodic inversion of the periodic systems is computed. Accompanied with the optimal state transition method for the time-varying systems, precise trajectory tracking is achieved. The methodology is validated through simulations.

Keywords

dynamic inversion control linear system theory optimization periodic systems precise trajectory tracking time varying

Introduction

Output tracking is a common problem encountered in many practical applications – for example, trajectory tracking for manipulators (Wang and Deng, 2013), tracking problems for micro-electro-mechanical systems (Modirrousta et al., 2017), attitude tracking for autonomous airships (Wang et al., 2016), and others. There are many well-known contributions to trajectory tracking problems for models of practical systems (Li et al., 2017; Xu et al., 2015, Xu et al., 2016); for example, the output regulation methods (Ali et al., 2016; Isidori and Byrnes, 1990) and the differential flatness approaches (Fliess et al., 1998; Ryu and Agrawal, 2011). However, all of these methods are asymptotic output trajectory tracking methods.

In order to obtain precise tracking, various kinds of inversion techniques are proposed for different kinds of systems. The classical inversion technique for linear systems (Sain and Massey, 1969; Silverman, 1969) and nonlinear systems (Hirschorn, 1979) is restricted to the precise tracking problem of minimum phase systems, because it results in divergent solutions when the classical inversion is applied for the non-minimum phase systems.

In order to obtain precise tracking for non-minimum-phase systems, stable inversion is proposed and plays a crucial role in achieving precise tracking. For the precise tracking of linear time-invariant systems, stable inversion is applied for the discrete-time systems (George et al., 1999) and the continuous-time systems (Pallastrelli and Piazzi, 2005; Zhang and Liu, 2016). For the precise tracking of nonlinear time-invariant systems, stable inversion is also applied for the discrete-time systems (Zeng and Hunt, 2000) and the continuous-time systems (Devasia et al., 1996). Furthermore, stable inversion is extended to the continuous-time nonlinear time-varying systems (Devasia and Paden, 1998), but no explicit formulas of stable inversion can be obtained because of the iteration process of calculations. It is worth mentioning that there are no contributions to get explicit formulas for the precise tracking problem of periodic trajectories for periodic systems. Moreover, the periodic system is a bridge connecting the time-varying and time-invariant systems and appears in many practical processes. For example, the magnetic attitude control system for the polar-orbiting satellite is a periodic system, because the satellite is driven by the interaction between the magnetic moment produced by the satellite and the earth’s magnetic field, which is periodic relative to the satellite. In order to change the attitude of the satellite periodically, it is necessary to consider the precise tracking of periodic trajectories for periodic systems.

In order to obtain explicit formulas for the precise tracking of periodic trajectories for periodic systems, a special kind of stable inversion, named a periodic inversion, is proposed in this study. For the periodic systems that are not limited to square systems, different cases are considered and certain conditions are provided for the existence of the periodic inversion for these cases, then explicit formulas of the periodic inversion are offered when the periodic inversion exists in this study. In comparison with the existing literature on approximate tracking methods, this study achieves precise tracking; in comparison with stable inversion, this study not only achieves explicit formulas for stable inversion, but also deals with the non-square systems, whereas there were previously no contributions for stable inversion of discrete-time non-square systems in the literature. In addition, precise tracking was achieved by stable inversion with infinite initial time. Accompanied by the optimal state transition (OST) (Wang et al., 2012; Zhang and Liu, 2016) method for the time-varying system, this study achieves precise tracking of periodic trajectories for periodic systems by periodic inversion with finite initial time.

In summary, this study demonstrates two major achievements: (1) precise tracking of periodic trajectories for discrete-time linear periodically time-varying systems is comprehensively analysed and achieved with explicit formulas; (2) the periodic inversion of the non-square discrete-time linear time-invariant system is proposed and analysed.

In this paper, the $n \times n$ identity matrix is $I_{n}$ , or simply $I$ ; $0$ denotes the zero matrix with appropriate dimensions, $N$ denotes the set of non-negative integer numbers, $Z$ is the set of integer numbers and $R$ denotes the set of real numbers. For a matrix $B$ , its Moore–Penrose inverse is denoted by $B^{+}$ . When $B$ is the $m \times n$ zero matrix, $B^{+}$ is the $n \times m$ zero matrix. When $B$ is a full column rank matrix, $B^{+} = (B^{T} B)^{- 1} B^{T}$ and $B^{+} B = I$ . When $B$ is a full row rank matrix, $B^{+} = B^{T} (B B^{T})^{- 1}$ and $B B^{+} = I$ . For $x = [\begin{matrix} x_{1} & x_{2} & \dots & x_{n} \end{matrix}]^{T} \in R^{n}$ , $‖ x ‖_{\infty} = max_{1 ⩽ i ⩽ n} | x_{i} |$ is the standard $\infty$ -norm.

The remainder of this paper is organized as follows. The precise tracking problem for discrete-time linear periodically time-varying systems is discussed in the next section, which includes the system reformulation and the periodic inversion. Following this, simulation results are presented. The final section concludes.

Precise tracking for periodic systems

Problem formulation and system reformulation

Consider a multi-input, multi-output periodically time-varying discrete-time system

{\begin{matrix} x (t + 1) = A (t) x (t) + B (t) u (t) \\ y (t) = C (t) x (t) \end{matrix} t \in Z

(1)

where $u (t) \in R^{m}$ , $y (t) \in R^{p}$ and $x (t) \in R^{n}$ are the input, output and state, respectively. Furthermore, $A (t)$ , $B (t)$ , $C (t)$ are all $T$ -periodic system parameters; that is, $A (t + T) = A (t)$ , $B (t + T) = B (t), and C (t + T) = C (t)$ for a given positive integer $T ⩾ 2$ . Given a reference output trajectory $y_{d} (t) \in R^{p}$ , satisfying

y_{d} (t) = y_{d} (t + T)

(2)

we try to find bounded state $x^{★} (t)$ and input $u^{★} (t)$ to track $y_{d} (t)$ precisely for system (1) – that is, $y (t) = y_{d} (t)$ . In order to get $x^{★} (t)$ and $u^{★} (t)$ , the periodic inversion for discrete-time linear periodically time-varying systems is proposed in the following definition.

Definition 1. Given system (1) and the reference trajectory $y_{d} (t)$ , the state $x_{d} (t)$ and input $u_{d} (t)$ satisfying

x_{d} (t + 1) = A (t) x_{d} (t) + B (t) u_{d} (t)

(3.1)

y_{d} (t) = C (t) x_{d} (t)

(3.2)

y_{d} (t) = C (t) x_{d} (t)

(4.1)

u_{d} (t) = u_{d} (t + T)

(4.2)

are defined as the periodic inversion of $y_{d} (t)$ .

The periodic inversion $x_{d} (t)$ and $u_{d} (t)$ are the particular solutions of $x^{★} (t)$ and $u^{★} (t)$ , respectively. It is put forward according to the definition of stable inversion to achieve precise trajectory tracking, so it is a special kind of stable inversion. In order to obtain $x_{d} (t)$ and $u_{d} (t)$ , the lifted technique (Meyer and Burrus, 1975) is applied to transform the time-varying system (1) into an equivalent time-invariant system at time $t_{1}$ . $t_{1}$ is the initial time of the periodic inversion method. Denote

{\begin{matrix} A_{h} (i) = A (t_{1} + i) \\ B_{h} (i) = B (t_{1} + i) \\ C_{h} (i) = C (t_{1} + i) \end{matrix} i \in N

(5)

For $\forall k \in Z$ , we obtain¹

\begin{matrix} x (t_{1} + kT + T) = A_{h} (T - 1) x (t_{1} + kT + T - 1) \\ + B_{h} (T - 1) u (t_{1} + kT + T - 1) \\ ⋮ \\ = Π_{i = 1}^{T} A_{h} (T - i) x (t_{1} + kT) \\ + \sum_{j = 0}^{T - 1} Π_{i = 1}^{T - j - 1} A_{h} (T - i) B_{h} (j) u (t_{1} + kT + j) . \end{matrix}

(6)

We define

\bar{x} (k) = x (t_{1} + kT) \in R^{n}

(7)

\bar{u} (k) = [u^{T} (t_{1} + kT) \dots u^{T} (t_{1} + kT + T - 1)]^{T} \in R^{mT}

(8)

From equation (6), we get

\bar{x} (k + 1) = \bar{A} \bar{x} (k) + \bar{B} \bar{u} (k)

(10)

where

\bar{A} = Π_{i = 1}^{T} A_{h} (T - i) \in R^{n \times n},

(10)

\bar{B} = [Π_{i = 1}^{T - 1} A_{h} (T - i) B_{h} (0) \dots B_{h} (T - 1)] \in R^{n \times mT} .

(11)

It is also noted that

y (t_{1} + kT) = C_{h} (0) \bar{x} (k)

(12)

and that

\begin{matrix} y (t_{1} + kT + τ) = C_{h} (τ) Π_{i = 1}^{τ} A_{h} (τ - i) \bar{x} (k) \\ + \sum_{j = 0}^{τ - 1} C_{h} (τ) Π_{i = 1}^{τ - j - 1} A_{h} (τ - i) B_{h} (j) u (t_{1} + kT + j) \end{matrix}

(13)

for any $τ \in {1, 2, \dots, T - 1}$ . We define

{\bar{y}}_{d} (k) = [y_{d}^{T} (t_{1} + kT) \dots y_{d}^{T} (t_{1} + kT + T - 1)] .

(14)

\begin{matrix} \bar{D} = [\begin{matrix} 0 & 0 & \dots & 0 & 0 \\ L_{1, 0} & 0 & \dots & 0 & 0 \\ L_{2, 0} & L_{2, 1} & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ L_{T - 1, 0} & L_{T - 1, 1} & \dots & L_{T - 1, T - 2} & 0 \end{matrix}] \\ L_{τ, j} = C_{h} (τ) Π_{i = 1}^{τ - j - 1} A_{h} (τ - i) B_{h} (j) \in R^{p \times m} (τ > j) \\ \bar{C} = [\begin{matrix} C_{h} (0) \\ ⋮ \\ C_{h} (T - 1) Π_{i = 2}^{T} A_{h} (T - i) \end{matrix}] \in R^{pT \times n} \end{matrix}

then

\bar{y} (k) = \bar{C} \bar{x} (k) + \bar{D} \bar{u} (k) .

(15)

Thus, system (1) is reformulated to the lifted system (16), as

{\begin{matrix} \bar{x} (k + 1) = \bar{A} \bar{x} (k) + \bar{B} \bar{u} (k) \\ \bar{y} (k) = \bar{C} \bar{x} (k) + \bar{D} \bar{u} (k) \end{matrix} k \in Z

(16)

which is a time-invariant system. For system (16), we define

{\bar{y}}_{d} (k) = [y_{d}^{T} (t_{1} + kT) \dots y_{d}^{T} (t_{1} + kT + T - 1)] .

(17)

The precise tracking problem of the time-varying system (1) is equivalently transformed into that of the time-invariant system (16). That is, we need to determine the bounded state ${\bar{x}}_{d} (k)$ and input ${\bar{u}}_{d} (k)$ in order to track ${\bar{y}}_{d} (k)$ precisely for system (16).

Periodic inversion for the lifted systems

According to equations (7) and (4.1), it follows that

{\bar{x}}_{d} (k) = {\bar{x}}_{d} (k + 1) .

(18)

From equations (4.2) and (8), we obtain

{\bar{u}}_{d} (k) = {\bar{u}}_{d} (k + 1) .

(19)

Then, Definition 1 is changed equivalently to Definition 2 as follows:

Definition 2. Given system (16) and the reference trajectory ${\bar{y}}_{d} (k)$ , then the state ${\bar{x}}_{d} (t)$ and input ${\bar{u}}_{d} (t)$ satisfying

{\bar{x}}_{d} (k + 1) = \bar{A} {\bar{x}}_{d} (k) + \bar{B} {\bar{u}}_{d} (k)

(20.1)

{\bar{y}}_{d} (k) = \bar{C} {\bar{x}}_{d} (k) + \bar{D} {\bar{u}}_{d} (k)

(20.2)

{\bar{x}}_{d} (k) = {\bar{x}}_{d} (k + 1)

(21.1)

{\bar{u}}_{d} (k) = {\bar{u}}_{d} (k + 1)

(21.2)

are defined as the periodic inversion of ${\bar{y}}_{d} (k)$ , which satisfies ${\bar{y}}_{d} (k) = {\bar{y}}_{d} (k + 1)$ from equations (2) and (17).

In comparison with stable inversion (Zhang and Liu, 2016) and periodic inversion, the restrictions to the state ${\bar{x}}_{d} (k)$ and input ${\bar{u}}_{d} (k)$ are different. For stable inversion, when time tends to infinity, the state and input both converge to zero; but for periodic inversion, the state and input are restricted to be constant in the $k$ -axis in equation (21).

Combining equations (20) and (21.1) leads to

(I_{n} - \bar{A}) {\bar{x}}_{d} (k) = \bar{B} {\bar{u}}_{d} (k)

(22.1)

{\bar{y}}_{d} (k) = \bar{C} {\bar{x}}_{d} (k) + \bar{D} {\bar{u}}_{d} (k)

(22.2)

Non-singular case

The matrix $(I_{n} - \bar{A})$ is non-singular. Combining (22.1) and (22.2) leads to

{\bar{y}}_{d} (k) = (\bar{C} {(I_{n} - \bar{A})}^{- 1} \bar{B} + \bar{D}) {\bar{u}}_{d} (k) = \bar{G} {\bar{u}}_{d} (k)

(23)

Case (a). $\bar{G}$ is a full row rank matrix. $\bar{G} \in R^{pT \times mT}$ , so $rank \bar{G} = pT$ .

If $m = p$ , the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained as

{\begin{matrix} {\bar{u}}_{d} (k) = {\bar{G}}^{- 1} {\bar{y}}_{d} (k) \\ {\bar{x}}_{d} (k) = {(I_{n} - \bar{A})}^{- 1} \bar{B} {\bar{G}}^{- 1} {\bar{y}}_{d} (k) \end{matrix}

(24)

If $m \neq p$ , the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained as

{\begin{matrix} {\bar{u}}_{d} (k) = {\bar{G}}^{+} {\bar{y}}_{d} (k) \\ {\bar{x}}_{d} (k) = {(I_{n} - \bar{A})}^{- 1} \bar{B} {\bar{G}}^{+} {\bar{y}}_{d} (k) \end{matrix}

(25)

According to equations (7), (8), (24) and (25), the input $u_{d} (t)$ and state $x_{d} (t)$ are obtained.

Case (b). $\bar{G}$ is a full column rank matrix, so $rank \bar{G} = mT$ , there exists an invertible matrix $P \in R^{pT \times pT}$ satisfying

\bar{G} = P [\begin{matrix} I_{mT} \\ 0 \end{matrix}] .

(26)

Substituting equation (26) into (23) leads to

P^{- 1} {\bar{y}}_{d} (k) = [\begin{matrix} I_{mT} \\ 0 \end{matrix}] {\bar{u}}_{d} (k)

(27)

Theorem 3. $(I_{n} - \bar{A})$ is non-singular, The periodic inversion ${\bar{u}}_{d} (k)$ and ${\bar{x}}_{d} (k)$ satisfying

{\begin{matrix} {\bar{u}}_{d} (k) = U_{b 1} P^{- 1} {\bar{y}}_{d} (k) \\ {\bar{x}}_{d} (k) = {(I_{n} - \bar{A})}^{- 1} \bar{B} U_{b 1} P^{- 1} {\bar{y}}_{d} (k) \end{matrix}

(28)

exist for system (16) if and only if

U_{b 2} P^{- 1} {\bar{y}}_{d} (k) = 0 for \forall k \in Z

(29)

where

U_{b 1} = [I_{mT} 0] \in R^{mT \times pT}, U_{b 2} = [0 I_{(p - m) T}] \in R^{(p - m) T \times p^{T}} .

Proof.

Necessity. The periodic inversion exists, then equation (27) is changed as

[\begin{matrix} U_{b 1} P^{- 1} {\bar{y}}_{d} (k) \\ U_{b 2} P^{- 1} {\bar{y}}_{d} (k) \end{matrix}] = [\begin{matrix} {\bar{u}}_{d} (k) \\ 0 \end{matrix}] .

(30)

From equation (30), we get $U_{b 2} P^{- 1} {\bar{y}}_{d} (k) = 0$ for $\forall k \in Z$ .

Sufficiency. If $U_{b 2} P^{- 1} {\bar{y}}_{d} (k) = 0$ , construct the input $u^{*} (k)$ and state $x^{*} (k)$ as

u^{*} (k) = U_{b 1} P^{- 1} {\bar{y}}_{d} (k)

(31.1)

x^{*} (k) = {(I_{n} - \bar{A})}^{- 1} \bar{B} U_{b 1} P^{- 1} {\bar{y}}_{d} (k) .

(31.2)

Combining equations (31.1) and (31.2) leads to

(I_{n} - \bar{A}) x^{*} (k) = \bar{B} u^{*} (k);

(32)

moreover, combining equations (31.1) and (29) leads to

\begin{matrix} {\bar{y}}_{d} (k) = P [\begin{matrix} u^{*} (k) \\ 0 \end{matrix}] = \bar{G} u^{*} (k) \\ = \bar{C} x^{*} (k) + \bar{D} u^{*} (k) \end{matrix}

(33)

According to equations (32) and (33), the input $u^{*} (k)$ and state $x^{*} (k)$ are not only the solutions of equation (22), but also the solutions of equations (20) and (21); that is, they are the periodic inversion as ${\bar{u}}_{d} (k) = u^{*} (k)$ and ${\bar{x}}_{d} (k) = x^{*} (k)$ .

According to equations (8), (7) and (28), the input $u_{d} (t)$ and state $x_{d} (t)$ are obtained.

Case (c). $\bar{G}$ is rank deficient and $rank \bar{G} = r$ , then $\bar{G}$ can be decomposed as

\bar{G} = R [\begin{matrix} I_{r} & 0 \\ 0 & 0 \end{matrix}] S,

(34)

where $R \in R^{pT \times pT}$ and $S \in R^{mT \times mT}$ are invertible matrices. Substituting equation (34) into (23) leads to

R^{- 1} {\bar{y}}_{d} (k) = [\begin{matrix} I_{r} & 0 \\ 0 & 0 \end{matrix}] S {\bar{u}}_{d} (k) = [\begin{matrix} I_{r} & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} {\bar{u}}_{1} (k) \\ {\bar{u}}_{2} (k) \end{matrix}] .

(35)

Theorem 4. $(I_{n} - \bar{A})$ is non-singular. The periodic inversion ${\bar{u}}_{d} (k)$ and ${\bar{x}}_{d} (k)$ in equation (39) exist if and only if

U_{c 2} R^{- 1} {\bar{y}}_{d} (k) = 0 for \forall k \in Z,

(36)

where

U_{c 1} = [I_{r} 0] \in R^{r \times pT}, U_{c 2} = [0 I_{pT - r}] \in R^{(pT - r) \times pT} .

The proof process is omitted for brevity. When equation (36) holds, the input component ${\bar{u}}_{1} (k)$ is obtained as

{\bar{u}}_{1} (k) = U_{c 1} R^{- 1} {\bar{y}}_{d} (k)

(37)

and the other input component ${\bar{u}}_{2} (k)$ is chosen, satisfying

\sum_{k = - \infty}^{\infty} {‖ {\bar{u}}_{2} (k) ‖}_{\infty} < \infty .

(38)

Then the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained as

{\begin{matrix} {\bar{u}}_{d} (k) = S^{- 1} [\begin{matrix} U_{c 1} R^{- 1} {\bar{y}}_{d} (k) \\ {\bar{u}}_{2} (k) \end{matrix}] \\ {\bar{x}}_{d} (k) = {(I_{n} - \bar{A})}^{- 1} \bar{B} S^{- 1} [\begin{matrix} U_{c 1} R^{- 1} {\bar{y}}_{d} (k) \\ {\bar{u}}_{2} (k) \end{matrix}] \end{matrix}

(39)

According to equations (8), (7) and (39), the input $u_{d} (t)$ and state $x_{d} (t)$ are obtained.

Singular case

The matrix $(I_{n} - \bar{A})$ is singular. We denote

\bar{L} = [\begin{matrix} \bar{C} & \bar{D} \\ \bar{A} - I_{n} & \bar{B} \end{matrix}], {\bar{v}}_{d} (k) = [\begin{matrix} {\bar{x}}_{d} (k) \\ {\bar{u}}_{d} (k) \end{matrix}], {\bar{Y}}_{d} (k) = [\begin{matrix} {\bar{y}}_{d} (k) \\ 0 \end{matrix}] .

${\bar{Y}}_{d} (k) \in R^{n + pT}$ , equation (22) is rewritten as

\bar{L} {\bar{v}}_{d} (k) = {\bar{Y}}_{d} (k)

(40)

Case (a). $\bar{L}$ is a full row rank matrix. We denote

\begin{matrix} ρ = n + pT, η = n + mT, \\ H_{1} = [I_{n} 0] \in R^{n \times η}, H_{2} = [0 I_{mT}] \in R^{mT \times η} \end{matrix}

then $\bar{L} \in R^{ρ \times η}$ and $rank \bar{L} = ρ$ .

If $m = p$ , ${\bar{v}}_{d} (k) = {\bar{L}}^{- 1} {\bar{Y}}_{d} (k)$ , the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained as

{\begin{matrix} {\bar{u}}_{d} (k) = H_{2} {\bar{L}}^{- 1} {\bar{Y}}_{d} (k) \\ {\bar{x}}_{d} (k) = H_{1} {\bar{L}}^{- 1} {\bar{Y}}_{d} (k) \end{matrix}

(41)

If $m \neq p$ , ${\bar{v}}_{d} (k) = {\bar{L}}^{+} {\bar{Y}}_{d} (k)$ , the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained as

{\begin{matrix} {\bar{u}}_{d} (k) = H_{2} {\bar{L}}^{+} {\bar{Y}}_{d} (k) \\ {\bar{x}}_{d} (k) = H_{1} {\bar{L}}^{+} {\bar{Y}}_{d} (k) \end{matrix}

(42)

Case (b). $\bar{L}$ is a full column rank matrix, then $rank \bar{L} = η$ . There exists an invertible matrix $Q \in R^{ρ \times ρ}$ satisfying

\bar{L} = Q [\begin{matrix} I_{η} \\ 0 \end{matrix}] .

(43)

Substituting equation (43) into (40) results in

[\begin{matrix} {\bar{v}}_{d} (k) \\ 0 \end{matrix}] = Q^{- 1} {\bar{Y}}_{d} (k) .

(44)

Theorem 5. $(I_{n} - \bar{A})$ is singular. The periodic inversion ${\bar{u}}_{d} (k)$ and ${\bar{x}}_{d} (k)$ satisfying

{\begin{matrix} {\bar{u}}_{d} (k) = H_{2} V_{b 1} Q^{- 1} Y_{d} (k) \\ {\bar{x}}_{d} (k) = H_{1} V_{b 1} Q^{- 1} Y_{d} (k) \end{matrix}

(45)

exist for system (16) if and only if

V_{b 2} Q^{- 1} Y_{d} (k) = 0 for \forall k \in Z,

(46)

where

V_{b 1} = [I_{η} 0] \in R^{η \times ρ}, V_{b 2} = [0 I_{ρ - η}] \in R^{(ρ - η) \times ρ} .

The proof process is omitted for brevit.According to equations (8), (7) and (45), the input $u_{d} (t)$ and state $x_{d} (t)$ are obtained.

Case (c). $\bar{L}$ is rank deficient and $rank \bar{L} = c$ . then $\bar{L}$ can be decomposed as

\bar{L} = E [\begin{matrix} I_{c} & 0 \\ 0 & 0 \end{matrix}] F,

(47)

where $E \in R^{ρ \times ρ}$ and $F \in R^{η \times η}$ are invertible matrices. Substituting equation (47) into (40) leads to

E^{- 1} {\bar{Y}}_{d} (k) = [\begin{matrix} I_{c} & 0 \\ 0 & 0 \end{matrix}] F {\bar{v}}_{d} (k) = [\begin{matrix} I_{c} & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} {\bar{v}}_{1} (k) \\ {\bar{v}}_{2} (k) \end{matrix}] .

(48)

Theorem 6. $(I_{n} - \bar{A})$ is singular. The periodic inversion ${\bar{u}}_{d} (k)$ and ${\bar{x}}_{d} (k)$ in equation (52) exist for system (16) if and only if

V_{c 2} E^{- 1} {\bar{Y}}_{d} (k) = 0 for \forall k \in Z,

(49)

where

V_{c 1} = [I_{c} 0] \in R^{c \times ρ}, V_{c 2} = [0 I_{ρ - c}] \in R^{(ρ - c) \times ρ} .

The proof process is omitted for brevity. When equation (49) holds, the input component ${\bar{v}}_{1} (k)$ is obtained as

{\bar{v}}_{1} (k) = V_{c 1} E^{- 1} {\bar{Y}}_{d} (k)

(50)

and the other input component ${\bar{v}}_{2} (k)$ is chosen, satisfying

\begin{matrix} \sum_{k = - \infty}^{\infty} {‖ {\bar{v}}_{2} (k) ‖}_{\infty} < \infty . \end{matrix}

(51)

Then the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained as

{\begin{matrix} {\bar{u}}_{d} (k) = H_{2} F^{- 1} [\begin{matrix} V_{c 1} E^{- 1} {\bar{Y}}_{d} (k) \\ {\bar{v}}_{2} (k) \end{matrix}] \\ {\bar{x}}_{d} (k) = H_{1} F^{- 1} [\begin{matrix} V_{c 1} E^{- 1} {\bar{Y}}_{d} (k) \\ {\bar{v}}_{2} (k) \end{matrix}] \end{matrix}

(52)

According to equations (8), (7) and (52), the input $u_{d} (t)$ and state $x_{d} (t)$ are obtained.

Finally, for the periodically time-varying system (3), when the input ${\bar{u}}_{d} (k)$ and state ${\bar{x}}_{d} (k)$ are obtained, the input $u_{d} (t)$ is computed by

[u_{d}^{T} (t_{1} + kT) \dots u_{d}^{T} (t_{1} + kT + T - 1)]^{T} = {\bar{u}}_{d} (k),

(53)

the desired state $x_{d} (t_{1})$ is computed by

\begin{matrix} x_{d} (t_{1}) = {\bar{x}}_{d} (0) . \end{matrix}

(54)

In order to obtain the desired state $x_{d} (t_{1})$ in the actual situation, the following OST method for time-varying systems is proposed.

Optimal state transition for time-varying systems

In the previous sections, the input $u_{d} (t)$ and state $x_{d} (t)$ for $t \in Z$ are obtained. However, in the actual situation, the control input functions from a limited initial time $t_{0} \in Z$ . In order to obtain precise tracking, the initial state $x (t_{0})$ is required to be transferred to the desired state $x_{d} (t_{1})$ $(t_{1} > t_{0})$ by the optimality of the corresponding state transition, then the control input of the periodic inversion is applied.

In order to achieve the desired state $x_{d} (t_{1})$ obtained by the periodic inversion, the following assumption holds.

Assumption 7. The discrete-time Gramian matrix

\begin{matrix} G_{(t_{0}, t_{1})} = \sum_{j = t_{0}}^{t_{1} - 1} Π_{i_{2} = 1}^{t_{1} - j - 1} A (t_{1} - i_{2}) B (j) \\ \times R^{- 1} (j) B^{T} (j) {(Π_{i_{3} = 1}^{t_{1} - j - 1} A (t_{1} - i_{3}))}^{T} \in R^{n \times n} \end{matrix}

(55)

is non-singular, where $R (t)$ is an symmetric positive definite matrix and $G_{(t_{0}, t_{1})}$ shares the same form with that of the linear system theory (Chen, 1986).

Definition 8. The method applied to transfer the initial state $x (t_{0})$ $(t_{0} < t_{1})$ to the desired state $x_{d} (t_{1})$ with minimum input energy for the periodic system (1) is named as the OST method.

Our goal is to find the control sequence to obtain the desired state $x_{d} (t_{1})$ . The associated performance index is a quadratic function defined as

J = \frac{1}{2} \sum_{t = t_{0}}^{t_{1} - 1} u^{T} (t) R (t) u (t) .

(56)

We begin with the Hamiltonian function (Lewis et al., 2012)

\begin{matrix} H (t) = \frac{1}{2} u^{T} (t) R (t) u (t) \\ + λ^{T} (t + 1) (A (t) x (t) + B (t) u (t)) . \end{matrix}

(57)

The state equation is obtained as

x (t + 1) = \frac{\partial H (t)}{\partial λ (t + 1)} = A (t) x (t) + B (t) u (t);

(58)

the costate equation is obtained as

λ (t) = \frac{\partial H (t)}{\partial x (t)} = A^{T} (t) λ (t + 1);

(59)

the stationarity condition is obtained as

\frac{\partial H (t)}{\partial u (t)} = R (t) u (t) + B^{T} (t) λ (t + 1) = 0;

(60)

from equation (60), it results in

\begin{matrix} u (t) = - R^{- 1} (t) B^{T} (t) λ (t + 1); \end{matrix}

(61)

combining equation (58) with (61) leads to

\begin{matrix} x (t + 1) = A (t) x (t) - B (t) R^{- 1} (t) B^{T} (t) λ (t + 1) . \end{matrix}

(62)

By some calculations, equation (59) can be changed as

\begin{matrix} λ (t) = Π_{i = 0}^{t_{1} - t - 1} A^{T} (t + i) λ (t_{1}), \end{matrix}

(63)

and combining equation (62) with (63), such that

\begin{matrix} x (t + 1) = A (t) x (t) \\ - B (t) R^{- 1} (t) B^{T} (t) Π_{i = 1}^{t_{1} - t - 1} A^{T} (t + i) λ (t_{1}), \end{matrix}

(64)

then by some calculations, equation (64) can be deduced as

\begin{matrix} x (t) = Π_{i_{1} = 1}^{t - t_{0}} A (t - i_{1}) x (t_{0}) - \sum_{j = t_{0}}^{t - 1} Π_{i_{2} = 1}^{t - j - 1} A (t - i_{2}) \\ \times B (j) R^{- 1} (j) B^{T} (j) {(Π_{i_{3} = 1}^{t_{1} - j - 1} A (t_{1} - i_{3}))}^{T} λ (t_{1}) . \end{matrix}

(65)

Therefore, when $t = t_{1}$ , equation (65) is changed as

\begin{matrix} x (t_{1}) = Π_{i_{1} = 1}^{t_{1} - t_{0}} A (t_{1} - i_{1}) x (t_{0}) - G_{(t_{0}, t_{1})} λ (t_{1}), \end{matrix}

(66)

where

\begin{matrix} G_{(t_{0}, t_{1})} = \sum_{j = t_{0}}^{t_{1} - 1} Π_{i_{2} = 1}^{t_{1} - j - 1} A (t_{1} - i_{2}) B (j) \\ \times R^{- 1} (j) B^{T} (j) {(Π_{i_{3} = 1}^{t_{1} - j - 1} A (t_{1} - i_{3}))}^{T} . \end{matrix}

(67)

$G_{(t_{0}, t_{1})}$ is just the discrete-time Gramian matrix, which is non-singular from Assumption 7. $λ (t_{1})$ can be obtained from equation (66), then substitute $λ (t_{1})$ into (59) to obtain

\begin{matrix} λ (t) = Π_{i = 0}^{t_{1} - t - 1} A^{T} (t + i) G_{(t_{0}, t_{1})}^{- 1} \\ \times (Π_{i_{1} = 1}^{t_{1} - t_{0}} A (t_{1} - i_{1}) x (t_{0}) - x (t_{1})) . \end{matrix}

(68)

Then, combining equation (61) and (68), the control sequence is obtained as

\begin{matrix} u (t) = - R^{- 1} (t) B^{T} (t) Π_{i = 1}^{t_{1} - t - 1} A^{T} (t + i) G_{(t_{0}, t_{1})}^{- 1} \\ \times (Π_{i_{1} = 1}^{t_{1} - t_{0}} A (t_{1} - i_{1}) x (t_{0}) - x (t_{1})) . \end{matrix}

(69)

For equation (69), $x (t_{0}) = 0$ is the initial state for system (1) while $x (t_{1}) = x_{d} (t_{1})$ is the desired state; the desired state $x_{d} (t_{1})$ is obtained by the periodic inversion.

Simulation

In order to validate the precise tracking performances of the proposed methods, three numerical simulations are given: the non-singular case (a), the non-singular case (b) and the singular case (b). As seen in the Figures 1 –3, the x-axis is in seconds and the y-axis is in meters. Our goal is to achieve precise trajectory tracking of the reference trajectory $y_{d} (t)$ for $t ⩾ - 11$ , $t \in Z$ .

Non-singular case (a)

Consider the discrete-time linear periodically time-varying system (1) with a period of $T = 3$ and

\begin{matrix} A (0) = [\begin{matrix} 1 & - 0.2 & 0.4 \\ 1.0 & 0 & 1 \\ 0.8 & - 0.4 & 0.2 \end{matrix}], A (1) = [\begin{matrix} 0.5 & - 0.2 & 0.4 \\ 1.0 & 0 & 1 \\ 0.8 & - 0.4 & 0.8 \end{matrix}], \\ A (2) = [\begin{matrix} 0.8 & - 0.2 & 0.4 \\ 1.0 & 0 & 1 \\ 0.8 & - 0.4 & 0.2 \end{matrix}], B (0) = {[\begin{matrix} 2 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}]}^{T}, \\ B (1) = {[\begin{matrix} 1.2 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}]}^{T}, B (2) = {[\begin{matrix} 1 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}]}^{T}, \\ C (0) = [\begin{matrix} 2.5 & 0.2 & 2.5 \end{matrix}], C (1) = [\begin{matrix} 2.5 & 1.25 & 2.5 \end{matrix}], \\ C (2) = [\begin{matrix} 2.5 & 0.1 & 2.5 \end{matrix}] . \end{matrix}

The initial time $t_{0} = - 14$ and $t_{1} = - 11$ , the reference trajectory $y_{d} (t)$ for $t ⩾ - 11$ is given as

\begin{matrix} y_{d} (t) = {\begin{matrix} 1, t = 3 s + 1, s \in Z \\ 0, t = 3 s + 2, s \in Z \\ - 1, t = 3 s + 3, s \in Z \end{matrix} \end{matrix}

(70)

The reference trajectory for $t < - 11$ , $t \in Z$ is $y_{d} (t) = 0$ . Owing to the fact that the reference trajectory $y_{d} (t)$ is defined for $t ⩾ - 11$ , so $y_{d} (t)$ for $t < - 11$ does not appear in Figures 1 –3.

The desired state $x_{d} (t_{1})$ at $t_{1} = - 11$ is obtained by the OST technique as

x (- 11) = x_{d} (- 11) = {[\begin{matrix} \frac{309}{2168} & \frac{20146}{55789} & \frac{980}{12741} \end{matrix}]}^{T} .

(71)

The inputs of the OST technique – that is $u_{d} (t)$ for $t \in {- 14, - 13, - 12}$ – are obtained as

\begin{matrix} u_{d} (- 12) = [\frac{451}{24166} \frac{256}{3479}]^{T}, u_{d} (- 13) = [\frac{125}{6159} - \frac{101}{2457}]^{T}, \\ u_{d} (- 14) = [- \frac{17}{48941} - \frac{692}{18197}]^{T} . \end{matrix}

The input $u_{d} (t)$ for $t ⩾ - 11, t \in Z$ is obtained as

\begin{matrix} u_{d} (t) = {\begin{matrix} {[\begin{matrix} - \frac{457}{10822} & \frac{107}{4669} \end{matrix}]}^{T}, t = 3 s + 1, s \in Z \\ {[\begin{matrix} - \frac{395}{5049} & - \frac{848}{12313} \end{matrix}]}^{T}, t = 3 s + 2, s \in Z \\ {[\begin{matrix} \frac{213}{2351} & \frac{961}{7058} \end{matrix}]}^{T}, t = 3 s + 3, s \in Z \end{matrix} \end{matrix}

(72)

The output tracking performance is illustrated in Figure 1. In the starting stage within $- 14$ to $- 12$ time range, the OST technique functions to transfer the zero initial state to $x_{d} (- 11)$ in equation (71), which is calculated by the periodic inversion. In the second stage after $- 11$ , the control input of the periodic inversion functions with the initial state of $x (- 11)$ . As seen in Figure 1, clearly, precise tracking is achieved for $t ⩾ - 11$ , $t \in Z$ .

Non-singular case (b)

Consider the discrete-time linear periodically time-varying system (1) with a period of $T = 3$ and

\begin{matrix} A (0) = [\begin{matrix} 1 & - 0.2 & 0.4 \\ 1.0 & 0 & 1 \\ 0.8 & - 0.4 & 0.2 \end{matrix}], A (1) = [\begin{matrix} 0.5 & - 0.2 & 0.4 \\ 1.0 & 0 & 1 \\ 0.8 & - 0.4 & 0.8 \end{matrix}] \\ A (2) = [\begin{matrix} 0.8 & - 0.2 & 0.4 \\ 1.0 & 0 & 1 \\ 0.8 & - 0.4 & 0.2 \end{matrix}], B (0) = [\begin{matrix} 2.5 \\ 0.2 \\ 2.5 \end{matrix}], \\ B (1) = {[\begin{matrix} 2.5 & 1.25 & 2.5 \end{matrix}]}^{T}, B (2) = {[\begin{matrix} 2.5 & 0.1 & 2.5 \end{matrix}]}^{T}, \\ C (0) = [\begin{matrix} 2 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}], C (1) = [\begin{matrix} 1.2 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}], \\ C (2) = [\begin{matrix} 1 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}] . \end{matrix}

The initial time $t_{0} = - 14$ and $t_{1} = - 11$ , the reference trajectory $y_{d} (t)$ for $t ⩾ - 11$ is given as

y_{d} (t) = {\begin{matrix} {[\begin{matrix} \frac{1583}{4468} & \frac{417}{2606} \end{matrix}]}^{T}, t = 3 s + 1, s \in Z \\ {[\begin{matrix} \frac{205}{478} & \frac{1025}{1257} \end{matrix}]}^{T}, t = 3 s + 2, s \in Z \\ {[\begin{matrix} - \frac{79}{38786} & - \frac{34}{21507} \end{matrix}]}^{T}, t = 3 s + 3, s \in Z \end{matrix}

(73)

The desired state $x_{d} (t_{1})$ at $t_{1} = - 11$ is obtained by the OST technique as

x (- 11) = x_{d} (- 11) = {[\begin{matrix} \frac{223}{1400} & \frac{63}{345304} & \frac{3352}{20591} \end{matrix}]}^{T} .

(74)

The inputs of the OST technique – that is $u_{d} (t)$ for $t \in {- 14, - 13, - 12}$ – are obtained as

u_{d} (- 14) = - \frac{18}{22039}, u_{d} (- 13) = - \frac{95}{50698}, u_{d} (- 12) = \frac{119}{1772} .

The input $u_{d} (t)$ for $t ⩾ - 11, t \in Z$ is obtained as

\begin{matrix} u_{d} (t) = {\begin{matrix} - \frac{125}{1518}, t = 3 s + 1, s \in Z \\ \frac{137}{3674}, t = 3 s + 2, s \in Z \\ \frac{337}{5610}, t = 3 s + 3, s \in Z \end{matrix} \end{matrix}

(75)

The output tracking performance is illustrated in Figure 2. Clearly, precise tracking is achieved for $t ⩾ - 11$ , $t \in Z$ .

Singular case (b)

Consider the discrete-time linear periodically time-varying system (1) with a period of $T = 3$ and

\begin{matrix} A (0) = [\begin{matrix} 1 & - 0.8 & 0.4 \\ 0 & 0 & 1 \\ 0 & 0 & 0.8 \end{matrix}], A (1) = [\begin{matrix} 1 & - 0.2 & 0.4 \\ 0 & 0 & 1 \\ 0 & 0 & 0.8 \end{matrix}] \\ A (2) = [\begin{matrix} 1 & - 0.4 & 0.4 \\ 0 & 0 & 1 \\ 0 & 0 & 0.8 \end{matrix}], B (0) = [\begin{matrix} 2.5 \\ 0.2 \\ 2.5 \end{matrix}] \\ B (1) = {[\begin{matrix} 2.5 & 1.25 & 2.5 \end{matrix}]}^{T}, B (2) = {[\begin{matrix} 2.5 & 0.1 & 2.5 \end{matrix}]}^{T}, \\ C (0) = [\begin{matrix} 2 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}], C (1) = [\begin{matrix} 1.2 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}], \\ C (2) = [\begin{matrix} 1 & 2 & 1 \\ 1 & 4 & 0 \end{matrix}] . \end{matrix}

The initial time $t_{0} = - 14$ and $t_{1} = - 11$ , the reference trajectory $y_{d} (t)$ for $t ⩾ - 11$ is given as

\begin{matrix} y_{d} (t) = {\begin{matrix} {[\begin{matrix} \frac{274}{1825} & - \frac{291}{878} \end{matrix}]}^{T}, t = 3 s + 1, s \in Z \\ {[\begin{matrix} \frac{1240}{2533} & \frac{1278}{1709} \end{matrix}]}^{T}, t = 3 s + 2, s \in Z \\ {[\begin{matrix} - \frac{643}{2558} & \frac{223}{2990} \end{matrix}]}^{T}, t = 3 s + 3, s \in Z \end{matrix} \end{matrix}

(76)

The desired state $x_{d} (t_{1})$ at $t_{1} = - 11$ is obtained by the OST technique as

\begin{matrix} x (- 11) = x_{d} (- 11) = {[\begin{matrix} \frac{387}{2692} & - \frac{443}{3729} & \frac{82}{381} \end{matrix}]}^{T} . \end{matrix}

(77)

The inputs of the OST technique – that is $u_{d} (t)$ for $t \in {- 14, - 13, - 12}$ – are obtained as

u_{d} (- 14) = - \frac{87}{7352}, u_{d} (- 13) = - \frac{187}{3843}, u_{d} (- 12) = \frac{238}{1795} .

The input $u_{d} (t)$ for $t ⩾ - 11, t \in Z$ is obtained as

\begin{matrix} u_{d} (t) = {\begin{matrix} - \frac{117}{2393}, t = 3 s + 1, s \in Z \\ - \frac{77}{1039}, t = 3 s + 2, s \in Z \\ \frac{238}{1795}, t = 3 s + 3, s \in Z \end{matrix} \end{matrix}

(78)

The output tracking performance is illustrated in Figure 3. Clearly, precise tracking is achieved for $t ⩾ - 11$ , $t \in Z$ .

Conclusion

In order to achieve precise tracking of periodic trajectories for discrete-time linear periodically time-varying systems, the periodic inversion is proposed. The periodic inversion of the periodically time-varying system can be equivalently transformed into that of the lifted time-invariant system. This research provides explicit solutions to the periodic inversion for the lifted time-invariant systems involving all kinds of rank cases. Accompanied by the OST method for time-varying systems, precise trajectory tracking is achieved. However, there is a limitation in that the periodic system and trajectory have the same period. Moreover, only the linear discrete-time system is considered in this study; this work may be improved to fit linear systems. Finally, the precise tracking of periodic trajectories for nonlinear periodically time-varying systems is also a future direction of research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant Nos. NSFC: 51405430, 61473258, 61433013, and U1509210), the Public Welfare Technology Application Research Plan of Zhejiang (2016C33G2010137), the Fundamental Research Funds for the Central Universities (2017QNA5011) and the Science Fund for Creative Research Groups of NSFC (61621002) and National Innovation (1716311XJ00100501).

ORCID iD

Youling Zhang

Notes

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