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Abstract

This article proposes and studies a problem of preview control for a type of discrete-time interconnected systems. First, adopting the technique of decentralized control, isolated subsystems are constructed by splitting the correlations between the systems. Utilizing the difference operator to the system equations and error vectors, error systems are built. Then, the preview controller is designed for the error system of each isolated subsystem. The controllers of error systems of isolated subsystems are aggregated as a controller of the interconnected system. Finally, by employing Lyapunov function method and the properties of non-singular M-matrix, the guarantee conditions for the existence of preview controllers for interconnected systems are given. The numerical simulation shows that the theoretical results are effective.

Keywords

Discrete-time interconnected systems preview control error system Lyapunov function non-singular M-matrix

Introduction

In many practical cases, future reference or disturbance signal of control systems is either partly or completely known, such as the flight path of aircraft, the processing path of numerically controlled machine tools, the driving path of vehicles, and so on. This future information is fully utilized to improve the quality of system, which is the preview control problem. Since Sheridan¹ put forward the concept of preview control, which was further upheld by Masayoshi Tomizuka, Tohru Katayama, and other scholars in the 1960s, preview control has attracted extensive attention in theoretical research and application, and formed a set of relatively complete theories and methods.^2–4 Preview control theory has been widely combined with various systems in recent years, which has produced many important results. In Liao et al.,⁵ preview control theory and descriptor systems are merged to investigate preview control for linear causal descriptor systems. The theory results of preview control were extended to the cooperative consensus problem of multi-agent systems, and the sufficient conditions to guarantee the achievement of cooperative preview tracking control were given.⁶ In addition, the theory of multi-rate systems preview control and random systems preview control has made progress.^7,8 At the same time, preview control has also been exploited in many engineering control problems, such as robot system, active suspension system, electromechanical servo, and aircraft.^9–11

The so-called interconnected system refers to the system with complex structure, comprehensive functions, numerous factors, and large scale. Interconnected system is also called large-scale system. The power systems, urban transportation networks, and water resources systems are the examples of actual interconnected systems, which can be seen in daily life.^12–14 For interconnected systems, if the controller is designed by centralized control method, it will be difficult to centralize and deal with a large amount of information, which makes the control difficult to achieve. Therefore, decentralized aggregation method is adopted to design the controller^15–17 from a mathematical perspective, that is, the large-scale systems decomposition method. First, the associated terms are deleted artificially to obtain several low-dimensional systems (called isolated subsystems), and controllers are designed to meet certain requirements. Then, the controller of the interconnected system is obtained through a certain method of synthesis.^18–20 So far, there have been many control theories for interconnected systems. The tracking problem of interconnected systems through decentralized iterative learning control was studied in the following literature.^21–23 Zhang and Feng²³ reported the problem of controller design for fuzzy interconnected systems, and its stability is analyzed using piecewise Lyapunov function. Furthermore, Koeln²⁴ discusses the decentralized control of interconnected systems with a special structure, and studies its application in large refrigeration and air conditioning systems.

Under the current circumstances that theories of preview control and interconnected systems have made great progress, it has important theoretical and practical significance to associate with the two theories. Until now, only Liao et al.²⁵ solved a type of preview tracking control problems related to continuous-time interconnected systems. This article designs a controller with preview effect for a type of discrete-time interconnected systems, taking into account the previewable situation of both the external disturbance signal and the reference signal.

Research contents are arranged as follows: The introduction is given in section “Introduction.” Section “Preliminaries” consist of the elementary knowledge, which gives the key concepts needed in this paper. Section “Problem formulation” presents preview control problems for a type of interconnected systems and gives fundamental assumptions. The design of error system controller of isolated subsystem and controller of interconnected systems are discussed in sections “Controller of error system of isolated subsystems” and “Preview controller design for interconnected systems,” respectively. Section “Numerical simulation” explains numerical simulation. Finally, a brief conclusion is given in section “Conclusion.”

Throughout this paper, $A \in R^{n \times m}$ represents $A$ as the real matrix of $n \times m$ ; $R > 0 (R \geq 0)$ shows the matrix $R$ is a symmetric positive definite (semi-positive definite); $B \cdot C$ indicates the Hadamard product of matrices $B$ and $C$ ; $ρ (\cdot)$ is the spectral radius of matrix; $Δ$ denotes the difference operator, which means $Δ ξ (k) = ξ (k) - ξ (k - 1)$ ; and $‖ A ‖$ means the norm of matrix $A$ derived from Euclid norm of vector.

Preliminaries

For readability, the definition and partial properties of Hadamard product and non-singular M-matrix are given here.

Definition 1

Set $B = [b_{ij}], C = [c_{ij}] \in R^{m \times n}$ . $B \cdot C$ is a matrix obtained by multiplying the corresponding element of $B$ and $C$ , that is, $B \cdot C = [b_{ij} c_{ij}]$ . Let us call $B \cdot C$ the Hadamard product of matrices $B$ and $C$ .²⁶

The following properties can be obtained instantly from Definition 1 and the definition of matrix multiplication.

Property 1

Seting $a = {[\begin{matrix} a_{1} & a_{2} & \dots & a_{n} \end{matrix}]}^{T} \in R^{n}$ , $d = {[\begin{matrix} d_{1} d_{2} \dots d_{n} \end{matrix}]}^{T} \in R^{n}$ , $z_{i} \in R (i = 1, 2, \dots, n)$ , there is

[\begin{matrix} z_{1} & z_{2} & \dots & z_{n} \end{matrix}] (a \cdot d) = a^{T} diag (z_{1}, z_{2}, \dots, z_{n}) d

Definition 2

Let $A \in R^{n \times n}$ be defined as

A = s I_{n} - B

where $s > 0$ , each element in matrix $B$ is non-negative. If $s > ρ (B)$ , then $A$ is called a non-singular M-matrix.

Lemma 1

If the non-diagonal elements of matrix $A$ are less than or equal to zero, then the necessary and sufficient condition for $A$ to be a non-singular M-matrix is that one of the following conditions must be true:

For any $α \geq 0$ , $A + α I$ is non-singular and

There is a matrix $K = diag (k_{1}, k_{2}, \dots k_{n}) > 0$ that makes $KA + A^{T} K > 0$ .

It can be proved that the non-singular M-matrix also has the following property.

Theorem 1

If $A$ is a non-singular M-matrix, $G = (I - A) (I + A)^{- 1}$ , then there is diagonal matrix $K > 0$ , such that $K - G^{T} KG > 0$ .

Proof

From Lemma 1, there is a diagonal matrix $K > 0$ , so that $KA + A^{T} K > 0$ . We know from the obvious equality

\begin{matrix} 2 (KA + A^{T} K) & = A^{T} KA + KA + A^{T} K + K \\ - (A^{T} KA - KA - A^{T} K + K) \\ = (I + A)^{T} K (I + A) \\ - (I - A)^{T} K (I - A) \overset{def}{=} Φ \end{matrix}

that $Φ$ is a positive definite matrix. Since $(I + A)^{- 1}$ exist, there is

{[{(I + A)}^{- 1}]}^{T} Φ {(I + A)}^{- 1} = K - G^{T} KG

which means $K - G^{T} KG$ and $Φ$ are congruent. Because the congruent matrix has the same positivity, $K - G^{T} KG > 0$ . Theorem 1 is proved.

Problem formulation

Consider discrete-time interconnected system

{\begin{matrix} [\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \\ ⋮ \\ x_{N} (k + 1) \end{matrix}] = [\begin{matrix} A_{1} & A_{12} & \dots & A_{1 N} \\ A_{21} & A_{2} & \dots & A_{2 N} \\ ⋮ & ⋮ & ⋮ \\ A_{N 1} & A_{N 2} & \dots & A_{N} \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \\ ⋮ \\ x_{N} (k) \end{matrix}] \\ + [\begin{matrix} B_{1} & 0 & \dots & 0 \\ 0 & B_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & B_{N} \end{matrix}] [\begin{matrix} u_{1} (k) \\ u_{2} (k) \\ ⋮ \\ u_{N} (k) \end{matrix}] + [\begin{matrix} E_{1} & 0 & \dots & 0 \\ 0 & E_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & E_{N} \end{matrix}] [\begin{matrix} d_{1} (k) \\ d_{2} (k) \\ ⋮ \\ d_{N} (k) \end{matrix}] \\ y (k) = [\begin{matrix} C_{1} & C_{2} & \dots & C_{N} \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \\ ⋮ \\ x_{N} (k) \end{matrix}] \end{matrix}

(1)

Clearly, system (1) is able to be equivalently expressed as

{\begin{matrix} x_{i} (k + 1) = A_{i} x_{i} (k) + \sum_{\binom{j = 1}{j \neq i}}^{N} A_{ij} x_{j} (k) + B_{i} u_{i} (k) + E_{i} d_{i} (k), i = 1, 2, \dots, N \\ y (k) = \sum_{i = 1}^{N} y_{i} (k) = \sum_{i = 1}^{N} C_{i} x_{i} (k) \end{matrix}

(2)

Here, $x_{i} (k) \in R^{n_{i}}$ , $u_{i} (k) \in R^{m_{i}}$ , $d_{i} (k) \in R^{q_{i}}$ , $y_{i} (k) \in R^{p}$ , $A_{i} \in R^{n_{i} \times n_{i}}$ , $A_{ij} \in R^{n_{i} \times n_{j}}$ , $B_{i} \in R^{n_{i} \times m_{i}}$ , $E_{i} \in R^{n_{i} \times q_{i}}$ , $C_{i} \in R^{p \times n_{i}}$ are constant matrices; $A_{ij} (i \neq j)$ is the correlation matrix.

First, the basic assumptions are given as the following:

Assumption 1. $(A_{i}, B_{i})$ is stabilizable and $rank [\begin{matrix} A_{i} - I & B_{i} \\ C_{i} & 0 \end{matrix}] = n_{i} + p (i = 1, 2, \dots, N)$ .

Assumption 2. $(C_{i}, A_{i})$ is observable and $A_{i}$ is invertible $(i = 1, 2, \dots, N)$ .

Assumption 3. The reference signal $r (k)$ is M_r-steps previewable, that is, at the current time $k$ , the reference signals $r (k)$ , $r (k + 1)$ , $r (k + 2)$ , …, $r (k + M_{r})$ are available. And there is

r (k + j) = r (k + M_{r}) (j > M_{r})

Assumption 4

The disturbance signal $d_{i} (k)$ is $M_{d_{i}} - steps$ previewable, that is, at the current time $k$ , the disturbance signals $d_{i} (k)$ , $d_{i} (k + 1)$ , $d_{i} (k + 2)$ , …, $d_{i} (k + M_{d_{i}})$ are available. And there is

\begin{matrix} d_{i} (k + j) = d_{i} (k + M_{d_{i}}) \\ (j > M_{d_{i}}, i = 1, 2, \dots, N) \end{matrix}

Remark 1

Assumption 1 and Assumption 2 are fundamental assumptions for the original system. In the design of the controller, it is necessary to build the error system (10) and take the performance index function of system (11). The controller of interconnected system is obtained under the conditions where $(Φ_{i}, G_{i})$ is stabilizable and $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable. Naturally, it is necessary to give conditions that the original system satisfies, so that $(Φ_{i}, G_{i})$ is stabilizable and $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable. According to Katayama et al.,⁴Assumption 1 can guarantee that $(Φ_{i}, G_{i})$ is stabilizable. Assumption 2 can guarantee that $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable. Assumption 3 and Assumption 4 are the standard assumptions for preview control. In fact, by the characteristics of the control systems, only a period of previewable information has obvious impact on quality of the systems. The value beyond previewable steps has a little effect on the characteristics of the systems. Therefore, in the theory of preview control, the future value outside the previewable information is usually assumed to be a constant.²

The tracking error $e (k)$ can be defined as follows

e (k) = r (k) - y (k)

(3)

The aim of this article is to adopt optimal control theory to design a previewable controller that allows the output $y (k)$ of the system (2) to asymptotically track $r (k)$ . In other words, $lim_{k \to \infty} e (k) = lim_{k \to \infty} [r (k) - y (k)] = 0$ .

Controller of error system of isolated subsystems

For the sake of designing the controller, the method of decentralized control is utilized. First, we cut off the linkage between subsystems to form isolated subsystems and design controller for each isolated subsystem. Then, the controllers of the isolated subsystems are combined to get the controller of the interconnected systems. Finally, by discussing the stability of interconnected system, the constraints of associated terms are obtained.

Based on the output of system (2), we rewrite $e (k)$ as follows

e (k) = \sum_{i = 1}^{N} e_{i} (k) = \sum_{i = 1}^{N} [α_{i} r (k) - y_{i} (k)]

(4)

where $α_{i} (i = 1, 2, \dots, N)$ are constant and satisfy $\sum_{i = 1}^{N} α_{i} = 1$ . According to equation (4), if for any $e_{i} (k) = α_{i} r (k) - y_{i} (k) (i = 1, 2, \dots, N)$ , there is $lim_{k \to \infty} e_{i} (k) = 0$ , then $lim_{k \to \infty} e (k) = 0$ .

Remark 2

We can think of $y_{i} (k)$ as the output of ith subsystem. Equation (4) means that if output $y_{i} (k)$ of ith subsystem tracks $α_{i} r (k) (i = 1, 2, \dots, N)$ , then the output $y (k) = \sum_{i = 1}^{N} y_{i} (k)$ of interconnected system (1) can track $r (k)$ . The parameter $α_{i} (i = 1, 2, \dots, N)$ gives us the freedom of choice. For example, we can choose $α_{1} = α_{2} = \dots = α_{N} = (1 / N)$ , which means that all $y_{i} (k)$ keep track of $(1 / N) r (k)$ . If $α_{i} = 0$ , it indicates that the output of the ith subsystem tracks the zero vector, and the task of tracking $r (k)$ is completed by the output of other subsystems, and so on.

The equation of ith isolated subsystem is

{\begin{matrix} x_{i} (k + 1) = A_{i} x_{i} (k) + B_{i} u_{i} (k) + E_{i} d_{i} (k) \\ y_{i} (k) = C_{i} x_{i} (k) \end{matrix}

(5)

Currently, the error system is constructed for the isolated subsystem by the method of usually preview control. As a result, the tracking problem of isolated subsystem is turned into the error system regulation problem. Since the error system of interconnected system is still needed in the construction of the controller of the interconnected system, the error system (2) is constructed first to avoid the repetition calculations. Then, the correlation term is cut off to obtain the error systems of the isolated subsystems.

The $Δ$ is applied on both ends of the state equation of system (2) to get

\begin{matrix} Δ x_{i} (k + 1) & = A_{i} Δ x_{i} (k) + \sum_{\binom{j = 1}{j \neq i}}^{N} A_{ij} Δ x_{j} (k) + B_{i} Δ u_{i} (k) \\ + E_{i} Δ d_{i} (k) (i = 1, 2, \dots, N) \end{matrix}

(6)

Utilizing $Δ$ to both sides of $e_{i} (k + 1) = α_{i} r (k + 1) - y_{i} (k + 1) (i = 1, 2, \dots, N)$ , we get

\begin{matrix} Δ e_{i} (k + 1) & = α_{i} Δ r (k + 1) - Δ y_{i} (k + 1) \\ = α_{i} Δ r (k + 1) - C_{i} Δ x_{i} (k + 1) \\ (i = 1, 2, \dots, N) \end{matrix}

(7)

Notice that $Δ e_{i} (k + 1) = e_{i} (k + 1) - e_{i} (k)$ , then substitute equation (6) into equation (7) to get

\begin{matrix} e_{i} (k + 1) & = e_{i} (k) + α_{i} Δ r (k + 1) - C_{i} A_{i} Δ x_{i} (k) \\ - \sum_{\binom{j = 1}{j \neq i}}^{N} C_{i} A_{ij} Δ x_{j} (k) - C_{i} B_{i} Δ u_{i} (k) \\ - C_{i} E_{i} Δ d_{i} (k) (i = 1, 2, \dots, N) \end{matrix}

(8)

Combine equations (6) and (8) to get

\begin{matrix} X_{i} (k + 1) = & Φ_{i} X_{i} (k) + \sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k) + G_{i} Δ u_{i} (k) \\ + G_{r_{i}} Δ r (k + 1) + G_{d_{i}} Δ d_{i} (k) \\ (i = 1, 2, \dots, N) \end{matrix}

(9)

Here

\begin{matrix} X_{i} (k) = [\begin{matrix} e_{i} (k) \\ Δ x_{i} (k) \end{matrix}], Φ_{i} = [\begin{matrix} I & - C_{i} A_{i} \\ 0 & A_{i} \end{matrix}], \\ Φ_{ij} = [\begin{matrix} 0 & - C_{i} A_{ij} \\ 0 & A_{ij} \end{matrix}], G_{i} = [\begin{matrix} - C_{i} B_{i} \\ B_{i} \end{matrix}], \\ G_{r_{i}} = [\begin{matrix} α_{i} I \\ 0 \end{matrix}], G_{d_{i}} = [\begin{matrix} - C_{i} E_{i} \\ E_{i} \end{matrix}] \end{matrix}

System (9) is the error system of interconnected system (2).

Noted that $e_{i} (k)$ is a partial vector of $X_{i} (k)$ , so if there is $lim_{k \to \infty} X_{i} (k) = 0$ in system (9), there is $lim_{k \to \infty} e_{i} (k) = 0 (i = 1, 2, \dots, N)$ . In this way, $y (k)$ of the interconnected system (2) can track $r (k)$ asymptotically.

The error system of isolated subsystems is collected by cutting off the correlation item in system (9). The error system of the ith $(i = 1, 2, \dots, N)$ isolated subsystem is

\begin{matrix} X_{i} (k + 1) = & Φ_{i} X_{i} (k) + G_{i} Δ u_{i} (k) \\ + G_{r_{i}} Δ r (k + 1) + G_{d_{i}} Δ d_{i} (k) \end{matrix}

(10)

For the sake of utilizing the results of optimal control, a quadratic performance index function is defined for error system (10)

{\tilde{J}}_{i} = \sum_{k = 1}^{\infty} [X_{i}^{T} (k) Q_{i} X_{i} (k) + Δ u_{i}^{T} (k) H_{i} Δ u_{i} (k)]

(11)

where $Q_{i} = [\begin{matrix} Q_{e_{i}} & 0 \\ 0 & Q_{x_{i}} \end{matrix}] \in R^{(p + n_{i}) \times (p + n_{i})}$ , $Q_{e_{i}} > 0$ , $Q_{x_{i}} \geq 0$ , $H_{i} \in R^{m_{i} \times m_{i}}$ , $H_{i} > 0$ .

Remark 3

Obviously, the input $Δ u_{i} (k)$ of the system (10) is used in system (11). For original system (2), it is to quote $Δ u_{i} (k) (not u_{i} (k))$ in the performance index function. This causes the controller to include integrators, which helps to eliminate static errors.^2,4

From the known conclusion in Katayama et al.,⁴ Theorem 2 can be proved directly.

Theorem 2

Let us assume that $(Φ_{i}, G_{i})$ is stabilizable, $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable, and Assumption 3 and Assumption 4 hold, then the controller of the system (10), which minimizes the performance index function of system (11), has the form of

\begin{array}{l} Δ u_{i} (k) = F_{i} X_{i} (k) + \sum_{j = 1}^{M_{r}} F_{r_{i}} (j) Δ r (k + j) \\ + \sum_{j = 0}^{M_{d_{i}}} F_{d_{i}} (j) Δ d_{i} (k + j) \\ = F_{e_{i}} e_{i} (k) + F_{x_{i}} Δ x_{i} (k) + \sum_{j = 1}^{M_{r}} F_{r_{i}} (j) Δ r (k + j) \\ + \sum_{j = 0}^{M_{d_{i}}} F_{d_{i}} (j) Δ d_{i} (k + j) \end{array}

(12)

where

\begin{matrix} F_{i} & = [\begin{matrix} F_{e_{i}} & F_{x_{i}} \end{matrix}] = - {[H_{i} + G_{i}^{T} P_{i} G_{i}]}^{- 1} G_{i}^{T} P_{i} Φ_{i}, \\ F_{r_{i}} (j) & = - {[H_{i} + G_{i}^{T} P_{i} G_{i}]}^{- 1} G_{i}^{T} (ξ_{i}^{T})^{j - 1} P_{i} G_{r_{i}} (j = 1, 2, \dots, M_{r}) \\ F_{d_{i}} (j) & = - {[H_{i} + G_{i}^{T} P_{i} G_{i}]}^{- 1} G_{i}^{T} (ξ_{i}^{T})^{j} P_{i} G_{d_{i}} (j = 0, 1, \dots, M_{d_{d}}) \\ ξ_{i} & = Φ_{i} + G_{i} F_{i} \end{matrix}

$P_{i}$ is the positive definite solution of the following Riccati equation

P_{i} = Q_{i} + Φ_{i}^{T} P_{i} Φ_{i} - Φ_{i}^{T} P_{i} G_{i} {[H_{i} + G_{i}^{T} P_{i} G_{i}]}^{- 1} G_{i}^{T} P_{i} Φ_{i}

(13)

Preview controller design for interconnected systems

The vector

Δ u (k) = [\begin{matrix} Δ u_{1} (k) \\ Δ u_{2} (k) \\ ⋮ \\ Δ u_{N} (k) \end{matrix}]

(14)

is constructed, where $Δ u_{i} (i = 1, 2, \dots, N)$ is determined by equation (12). $Δ u (k)$ is utilized as the controller of the error system (9). Let us discuss the conditions under which the correlation term satisfies, so that the state vector $X_{i} (k) (i = 1, 2, \dots, N)$ of the closed-loop system (9) tends to zero asymptotically.

By substituting equation (14) into system (9), the closed-loop system

\begin{matrix} X_{i} (k + 1) = & ξ_{i} X_{i} (k) + \sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k) + Θ_{i} (k) \\ (i = 1, 2, \dots, N) \end{matrix}

(15)

can be obtained, here

\begin{matrix} Θ_{i} (k) = & \sum_{j = 1}^{M_{r}} G_{i} F_{r_{i}} (j) Δ r (k + j) \\ + \sum_{j = 0}^{M_{d_{i}}} G_{i} F_{d_{i}} (j) Δ d_{i} (k + j) + G_{r_{i}} Δ r (k + 1) \\ + G_{d_{i}} Δ d (k) (i = 1, 2, \dots, N) \end{matrix}

Next, a sufficient condition is given to assure system (15) asymptotically approaches the zero vector.

Theorem 3

Suppose

$(Φ_{i}, G_{i})$ is stabilizable and $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable $(i = 1, 2, \dots, N)$

Assumption 3 and Assumption 4 hold

Matrix $L = (T - S) (T + S)^{- 1}$ is a non-singular M-matrix

then, the state vector $X_{i} (k) (i = 1, 2, \dots, N)$ of system (15), that is, the closed-loop system (9) tends to zero asymptotically.

Here

S = [\begin{matrix} s_{11} & s_{12} & \dots & s_{1 N} \\ s_{21} & s_{22} & \dots & s_{2 N} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ s_{N 1} & s_{N 2} & \dots & s_{NN} \end{matrix}], T = diag (γ_{1}, γ_{2}, \dots, γ_{N}),

where $s_{ij} = ‖ Φ_{ij} ‖ (i, j = 1, 2, \dots, N, i \neq j)$ , $s_{ii} = ‖ ξ_{i} ‖$ , $γ_{i} = (s_{ii}^{2} + η_{i} υ_{i}^{- 1})^{1 / 2}$ , and $υ_{i} = ‖ P_{i} ‖$ , $η_{i} = - λ_{max} [ξ_{i}^{T} P_{i} ξ_{i} - P_{i}] (i, j = 1, 2, \dots, N, i \neq j)$ .

Proof. First, it is proved that the zero solution of the homogeneous system

X_{i} (k + 1) = ξ_{i} X_{i} (k) + \sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k) (i = 1, 2, \dots, N)

(16)

corresponding to system (15) is asymptotically stable.

From Katayama et al.,⁴ if $({\bar{Φ}}_{i}, {\bar{G}}_{i})$ is stabilizable, $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable, Assumption 3 and Assumption 4 are established, then there is a unique positive definite solution matrix $P_{i}$ for Riccati equation (13).²⁸ Using $P_{i}$ to construct $V_{i} (X_{i}) = X_{i}^{T} P_{i} X_{i}$ , it is a positive definite quadratic form of $X_{i}$ . Take difference to $V_{i} (X_{i})$ along the system (16) trajectory to obtain

\begin{matrix} Δ V_{i} |_{(16)} & = X_{i}^{T} (k + 1) P_{i} X_{i} (k + 1) - X_{i}^{T} (k) P_{i} X_{i} (k) \\ = [X_{i}^{T} (k) ξ_{i}^{T} + \sum_{\binom{j = 1}{j \neq i}}^{N} X_{j}^{T} (k) Φ_{ij}^{T}] \\ P_{i} [ξ_{i} X_{i} (k) + \sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k)] - X_{i}^{T} (k) P_{i} X_{i} (k) \\ = X_{i}^{T} (k) [ξ_{i}^{T} P_{i} ξ_{i} - P_{i}] X_{i} (k) \\ + 2 X_{i}^{T} (k) ξ_{i}^{T} P_{i} \sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k) + [\sum_{\binom{j = 1}{j \neq i}}^{N} X_{j}^{T} (k) Φ_{ij}^{T}] \\ P_{i} [\sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k)] \end{matrix}

Notice $η_{i} = - λ_{max} [ξ_{i}^{T} P_{i} ξ_{i} - P_{i}]$ , further to

\begin{matrix} Δ V_{i} |_{(16)} & \leq - η_{i} X_{i}^{T} (k) X_{i} (k) + 2 X_{i}^{T} (k) ξ_{i}^{T} P_{i} \sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k) \\ + {[\sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k)]}^{T} P_{i} [\sum_{\binom{j = 1}{j \neq i}}^{N} Φ_{ij} X_{j} (k)] \end{matrix}

Continuously, use the properties of norms to obtain

\begin{matrix} Δ V_{i} |_{(16)} & \leq - η_{i} ‖ X_{i} (k) ‖^{2} + 2 ‖ X_{i}^{T} (k) ‖ ‖ ξ_{i}^{T} ‖ ‖ P_{i} ‖ \sum_{\binom{j = 1}{j \neq i}}^{N} ‖ Φ_{ij} ‖ ‖ X_{j} (k) ‖ + {[\sum_{\binom{j = 1}{j \neq i}}^{N} ‖ Φ_{ij} ‖ ‖ X_{j} (k) ‖]}^{2} ‖ P_{i} ‖ \\ = - η_{i} ‖ X_{i} (k) ‖^{2} + 2 s_{ii} υ_{i} ‖ X_{i}^{T} (k) ‖ \sum_{\binom{j = 1}{j \neq i}}^{N} s_{ij} ‖ X_{j} (k) ‖ + {[\sum_{\binom{j = 1}{j \neq i}}^{N} s_{ij} ‖ X_{j} (k) ‖]}^{2} υ_{i} \\ = - {[{(s_{ii}^{2} υ_{i} + η_{i})}^{1 / 2} ‖ X_{i} (k) ‖]}^{2} + υ_{i} {[s_{ii} ‖ X_{i} (k) ‖ + \sum_{\binom{j = 1}{j \neq i}}^{N} s_{ij} ‖ X_{j} (k) ‖]}^{2} \\ = {[\sqrt{υ_{i}} \sum_{j = 1}^{N} s_{ij} ‖ X_{j} (k) ‖]}^{2} - {[{(s_{ii}^{2} υ_{i} + η_{i})}^{1 / 2} ‖ X_{i} (k) ‖]}^{2} \\ = [\sqrt{υ_{i}} \sum_{j = 1}^{N} s_{ij} ‖ X_{j} (k) ‖ - {(s_{ii}^{2} υ_{i} + η_{i})}^{1 / 2} ‖ X_{i} (k) ‖] [\sqrt{υ_{i}} \sum_{j = 1}^{N} s_{ij} ‖ X_{j} (k) ‖ + {(s_{ii}^{2} υ_{i} + η_{i})}^{1 / 2} ‖ X_{i} (k) ‖] \\ = - [\sqrt{υ_{i}} (γ_{i} ‖ X_{i} (k) ‖ - \sum_{j = 1}^{N} s_{ij} ‖ X_{j} (k) ‖)] [\sqrt{υ_{i}} (γ_{i} ‖ X_{i} (k) ‖ + \sum_{j = 1}^{N} s_{ij} ‖ X_{j} (k) ‖)] \end{matrix}

Let $i = 1, 2, \dots, N$ to get

\begin{matrix} [\begin{matrix} Δ V_{1} |_{(16)} \\ Δ V_{2} |_{(16)} \\ ⋮ \\ Δ V_{N} |_{(16)} \end{matrix}] \leq - [O (T - S) [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}]] \cdot \\ [O (T + S) [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}]] \end{matrix}

(17)

where $O = diag (\sqrt{υ_{1}}, \sqrt{υ_{2}}, \dots, \sqrt{υ_{N}})$ .

According to 3, $L = (T - S) (T + S)^{- 1}$ is a non-singular M-matrix. From Theorem 1, it is well known that there exists $K = diag (k_{1}, k_{2}, \dots, k_{N}) > 0$ such that $K - G^{T} KG > 0$ , here

G = (I - L) (I + L)^{- 1}

(18)

Substitute $L$ into equation (18) to get

\begin{matrix} G & = [I - (T - S) (T + S)^{- 1}] [I + (T - S) (T + S)^{- 1}]^{- 1} \\ = [(T + S) (T + S)^{- 1} - (T - S) (T + S)^{- 1}] \\ [(T + S) (T + S)^{- 1} + (T - S) (T + S)^{- 1}]^{- 1} \\ = 2 S (T + S)^{- 1} [2 T (T + S)^{- 1}]^{- 1} \\ = S T^{- 1} \end{matrix}

So, $K - (S T^{- 1})^{T} K (S T^{- 1}) > 0$ .

Using $K$ and $O$ , the Lyapunov function of system (16) is taken as positive definite quadratic

V = \sum_{i = 1}^{N} \frac{k_{i}}{υ_{i}} V_{i} = [\begin{matrix} \frac{k_{1}}{υ_{1}} & \frac{k_{2}}{υ_{2}} & \dots & \frac{k_{N}}{υ_{N}} \end{matrix}] [\begin{matrix} V_{1} \\ V_{2} \\ ⋮ \\ V_{N} \end{matrix}]

(19)

Then, the difference of $V$ along system (16) trajectory is

Δ V |_{(16)} = [\begin{matrix} \frac{k_{1}}{υ_{1}} & \frac{k_{2}}{υ_{2}} & \dots & \frac{k_{N}}{υ_{N}} \end{matrix}] [\begin{matrix} Δ V_{1} |_{(16)} \\ Δ V_{2} |_{(16)} \\ ⋮ \\ Δ V_{N} |_{(16)} \end{matrix}]

By substituting system (17) and using Property 1, we get

\begin{matrix} Δ V |_{(16)} & \leq - [\begin{matrix} \frac{k_{1}}{υ_{1}} & \frac{k_{2}}{υ_{2}} & \dots & \frac{k_{N}}{υ_{N}} \end{matrix}] {[O (T - S) [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}]] \cdot [O (T + S) [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}]]} \\ = - [\begin{matrix} ‖ X_{1} (k) ‖ & ‖ X_{2} (k) ‖ & \dots & ‖ X_{N} (k) ‖ \end{matrix}] (T - S)^{T} O^{T} diag (\frac{k_{1}}{υ_{1}}, \frac{k_{2}}{υ_{2}}, \dots, \frac{k_{N}}{υ_{N}}) O (T + S) [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}] \\ = - [\begin{matrix} ‖ X_{1} (k) ‖ & ‖ X_{2} (k) ‖ & \dots & ‖ X_{N} (k) ‖ \end{matrix}] (T - S)^{T} O^{T} [\begin{matrix} \frac{k_{1}}{υ_{1}} & 0 & \dots & 0 \\ 0 & \frac{k_{2}}{υ_{2}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & \frac{k_{2}}{υ_{2}} \end{matrix}] O (T + S) [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}] \\ = - [\begin{matrix} ‖ X_{1} (k) ‖ & ‖ X_{2} (k) ‖ & \dots & ‖ X_{N} (k) ‖ \end{matrix}] [T^{T} KT - S^{T} KS] [\begin{matrix} ‖ X_{1} (k) ‖ \\ ‖ X_{2} (k) ‖ \\ ⋮ \\ ‖ X_{N} (k) ‖ \end{matrix}] \end{matrix}

(20)

Transform $T^{T} KT - S^{T} KS$ using positive definite diagonal matrix $T^{- 1}$ to get

\begin{matrix} (T^{- 1})^{T} [T^{T} KT - S^{T} KS] T^{- 1} = K - T^{- 1} S^{T} KS T^{- 1} \\ = K - (S T^{- 1})^{T} K (S T^{- 1}) > 0 \end{matrix}

then, $T^{T} KT - S^{T} KS > 0$ . When $λ$ is utilized to represent the minimum eigenvalue of $T^{T} KT - S^{T} KS$ , there must be $λ > 0$

Δ V |_{(16)} \leq - λ \sum_{i = 1}^{N} {‖ X_{i} (k) ‖}^{2} = - λ ‖ X (k) ‖^{2}

can be proved by substituting equation (20), and $X (k) = [X_{1}^{T} (k) X_{2}^{T} (k) \dots X_{N}^{T} (k)]^{T}$ is the state vector of the system (16). Therefore, the difference of $V$ along with system (16) trajectory is negative definite. As a result, the zero solution of system (16) is asymptotically stable.

Then, we prove that $lim_{k \to \infty} Θ_{i} (k) = 0 (i = 1, 2, \dots, N)$ . According to Assumption 3 and Assumption 4, when $k$ tends to infinity, $Δ r (k)$ and $Δ d_{i} (k)$ tend to zero vectors. In view of the relationship between $Θ_{i} (k)$ and $Δ r (k)$ , $Δ d_{i} (k)$ , it is easy to get $lim_{k \to \infty} Θ_{i} (k) = 0 (i = 1, 2, \dots, N)$ .

Because system (16) is asymptotically stable and $lim_{k \to \infty} Θ_{i} (k) = 0 (i = 1, 2, \dots, N)$ , according to Theorem 1 of Chen,²⁹ there is $lim_{k \to \infty} X_{i} (k) = 0 (i = 1, 2, \dots, N)$ in system (15). Hence, Theorem 3 is proved.

Below, we use the relevant parameters of interconnected system (2) to give the conditions, which ensure that $(Φ_{i}, G_{i})$ can be stabilized and $({Q_{i}}^{1 / 2}, Φ_{i})$ can be observed.

According to Katayama et al.,⁴ the sufficient and necessary condition for $(Φ_{i}, G_{i})$ to be stabilizable is that $rank [\begin{matrix} A_{i} - I_{n_{i}} & B_{i} \\ C_{i} & 0 \end{matrix}] = n_{i} + p$ and $(A_{i}, B_{i})$ is stabilizable $(i = 1, 2, \dots, N)$ . If $A_{i}$ is invertible and $(C_{i}, A_{i})$ is observable, then $({Q_{i}}^{1 / 2}, Φ_{i})$ is observable $(i = 1, 2, \dots, N)$ .

To sum up, one of the main theorems in this paper is as follows.

Theorem 4

Suppose

Assumption 1 to Assumption 4 hold;

$L = (T - S) (T + S)^{- 1}$ is a non-singular M-matrix;

$Q_{e_{i}} > 0$ , $H_{i} > 0$ $(i = 1, 2, \dots, N)$ ;

Let $u_{i} (k) = 0$ , $r (k) = 0$ , $x_{i} (k) = 0$ , $d_{i} (k) = 0 (i = 1, 2, \dots, N)$ for $k < 0$ ;

then the controller with preview effect, which enables the output signal of system (2) to track the reference signal asymptotically, is

u (k) = {[\begin{matrix} u_{1}^{T} (k) & u_{2}^{T} (k) & \dots & u_{N}^{T} (k) \end{matrix}]}^{T}

(21)

where

\begin{matrix} u_{i} (k) = & u_{i} (0) + F_{e_{i}} \sum_{j = 1}^{k} e_{i} (j) + F_{x_{i}} [x_{i} (k) - x_{i} (0)] \\ + \sum_{j = 1}^{M_{r}} F_{r_{i}} (j) [r (k + j) - r (j)] \\ + \sum_{j = 0}^{M_{d_{i}}} F_{d_{i}} (j) [d_{i} (k + j) - d_{i} (j)] \\ (i = 1, 2, \dots, N) \end{matrix}

(22)

When steps 1–3 of this theorem are true, all the conditions of Theorem 3 are satisfied, so the conclusion of Theorem 3 is true. The controller of system (1) can be achieved by solving $u_{i} (k) (i = 1, 2, \dots, N)$ from equation (14) or equation (12).

For a given $i (i = 1, 2, \dots, N)$

\begin{matrix} u_{i} (s) - u_{i} (s - 1) = & F_{e_{i}} e_{i} (s) + F_{x_{i}} [x_{i} (s) - x_{i} (s - 1)] \\ + \sum_{j = 1}^{M_{r}} F_{r_{i}} (j) [r (s + j) - r (s - 1 + j)] \\ + \sum_{j = 0}^{M_{d_{i}}} F_{d_{i}} (j) [d_{i} (s + j) - d_{i} (s - 1 + j)] \end{matrix}

(23)

is obtained from equation (12). In equation (23), taking $s = 1, 2, \dots, k$ , adding the two sides and moving $u_{i} (0)$ to the right side of equation to get equation (22), then combining $u_{i} (k) (i = 1, 2, \dots, N)$ , the controller of the interconnected system (2) is attained, that is, equation (21).

Remark 4

In equation (22), $F_{x_{i}} x_{i} (k)$ is the state feedback, $F_{e_{i}} \sum_{j = 1}^{k} e_{i} (j)$ is the integrator, $\sum_{j = 1}^{M_{r}} F_{r_{i}} (j) [r (k + j) - r (j)]$ is the preview feed-forward of reference information, $\sum_{j = 0}^{M_{d_{i}}} F_{d_{i}} (j) [d_{i} (k + j) - d_{i} (j)]$ is the preview feed-forward of disturbance information, $u_{i} (0)$ is the initial value of input, $F_{x_{i}} x_{i} (0)$ is the compensation of initial value.

Numerical simulation

Two examples are given to illustrate the effectiveness of the designed controller in this section.

Example 1

Consider interconnected system with two subsystems (i.e. $N = 2$ ), $n_{1} = 3$ , $n_{2} = 2$ and the coefficient matrices are

\begin{matrix} A_{1} = [\begin{matrix} 0.5 & 2 & 1.5 \\ 0 & - 0.5 & 1 \\ - 1.5 & 0 & - 5 \end{matrix}] A_{12} = [\begin{matrix} 0.005 & 0.003 \\ 0.002 & 0.004 \\ 0.002 & - 0.005 \end{matrix}] \\ B_{1} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] E_{1} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] C_{1} = [\begin{matrix} 0.5 & 1 & 1 \end{matrix}] \\ A_{2} = [\begin{matrix} 1 & 3 \\ 0 & - 5 \end{matrix}] A_{21} = [\begin{matrix} 0.001 & 0.0015 & - 0.001 \\ 0.001 & - 0.002 & 0.001 \end{matrix}] \\ B_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}] E_{2} = [\begin{matrix} 1 \\ 1 \end{matrix}] C_{2} = [\begin{matrix} 1 & 1 \end{matrix}] \end{matrix}

By adopting the Popov-Belevitch-Hautus (PBH) rank criterion, it is known that $(A_{i}, B_{i}) (i = 1, 2)$ is controllable and $(A_{i}, C_{i}) (i = 1, 2)$ is observable. In addition, it is easy to verify $rank [\begin{matrix} A_{i} - I & B_{i} \\ C_{i} & 0 \end{matrix}] = n_{i} + p (i = 1, 2)$ , and $A_{i} (i = 1, 2)$ is an invertible matrix.

Let the weight matrix of the performance index function of system (11) be

Q_{1} = [\begin{matrix} Q_{e_{1}} & 0 & 0 & 0 \\ 0 & 30 & 0 & 0 \\ 0 & 0 & 30 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], Q_{e_{1}} = 30, H_{1} = 2

Q_{2} = [\begin{matrix} Q_{e_{2}} & 0 & 0 \\ 0 & 30 & 0 \\ 0 & 0 & 0 \end{matrix}], Q_{e_{2}} = 20, H_{2} = 3

The solution of Riccati equation of two isolated subsystems and the feedback gain matrix of the controller are calculated using MATLAB

\begin{matrix} P_{1} = [\begin{matrix} 76.2583625100376 & - 7.1354483853848 & - 10.9509238062926 & - 615934591770557 \\ - 7.1354483853848 & 44.0598854799177 & 31.1938011523749 & 60.3567253364750 \\ - 10.9509238062926 & 31.1938011523749 & 171.4334406413428 & 66.2669915269203 \\ - 61.5934591770557 & 60.3567253364750 & 66.2669915269203 & 431.9922105508578 \end{matrix}] \\ P_{2} = [\begin{matrix} 51.0269298948671 & - 24.5802244149587 & - 71.2879879528538 \\ - 0.245802244149587 & 87.1434148884782 & 165.1725946429253 \\ - 71.2879879528538 & 165.1725946429253 & 551.4440506041630 \end{matrix}] \\ F_{e_{1}} = 0.217624980452534 \\ F_{x_{1}} = [\begin{matrix} 1.387583158168247 & - 0.260959136238370 & 4.321642830283796 \end{matrix}] \\ F_{e_{2}} = 0.163512352801475, F_{x_{2}} = [\begin{matrix} - 0.417176668167282 & 3.728417778359146 \end{matrix}] \end{matrix}

Besides

L = [\begin{matrix} 0.001558426283261 & - 0.001166519362468 \\ - 0.000531636745513 & 0.000529082708424 \end{matrix}]

The eigenvalues of $L$ are 0.001984526309942 and 0.000102982681743, which means that $L$ is a non-singular M-matrix.

First, considering the non-disturbance situation, that is, $d_{i} (k) = 0 (i = 1, 2)$ , $r (k)$ can be chosen as

r (k) = {\begin{matrix} 0, & k \leq 15 \\ 0.05 (k - 15), & 15 < k \leq 35 \\ 1, & k > 35 \end{matrix}

(24)

Let the initial state be $x_{1} (0) = {[\begin{matrix} 0 & 0.01 & 0 \end{matrix}]}^{T}$ , $u_{1} (0) = 0$ , $x_{2} (0) = {[\begin{matrix} 0.01 & 0 \end{matrix}]}^{T}$ , $u_{2} (0) = 0$ . Selecting $α_{1} = 0.4$ , $α_{2} = 0.6$ , the three cases of $M_{r} = 0$ , $M_{r} = 1$ , and $M_{r} = 5$ are numerically simulated.

The tracking effect of interconnected system (2) is shown in Figure 1. We can see that the output of interconnected system (2) is able to track $r (k)$ asymptotically with the increase of time $k$ . Thus, it is clear that the tracking speed can be accelerated using a larger number of preview steps.

Figure 1.

The output response of the interconnected systems without disturbance.

Now, consider the situation with disturbance signals as follow

d_{1} (k) = {\begin{matrix} 0, & k < 50 \\ 0.2, & k \geq 50 \end{matrix}, d_{2} (k) = {\begin{matrix} 0, & k < 70 \\ 0.3, & k \geq 70 \end{matrix}

The reference signal is still in the form of equation (24). At this time, the output curve of the interconnected system is depicted in Figure 2.

Figure 2.

Closed-loop responses of the interconnected system with disturbance.

The tracking error of interconnected system is depicted by Figure 3. It can be clearly seen from Figures 2 and 3 that the output of interconnected system is able to track the reference signal asymptotically, even if there are disturbance signals. Moreover, the controller with preview effect can apparently decrease the tracking error and the overshoot caused by disturbance.

Figure 3.

The tracking error of the interconnected system with disturbance.

Example 2

Consider interconnected system (2), where $N = 2$ , $n_{1} = n_{2} = 2$

\begin{matrix} \begin{matrix} A_{1} = [\begin{matrix} 0.1 & 1 \\ 0 & 0.2 \end{matrix}] A_{12} = [\begin{matrix} 0.1 & 0.05 \\ 0.02 & 0.1 \end{matrix}] \\ B_{1} = [\begin{matrix} - 0.2 \\ 0 \end{matrix}] E_{1} = [\begin{matrix} - 1.5 \\ 0 \end{matrix}] C_{1} = [\begin{matrix} - 3 & 1 \end{matrix}] \\ A_{2} = [\begin{matrix} 0.1 & - 1 \\ 0 & - 0.3 \end{matrix}] A_{21} = [\begin{matrix} 0.02 & 0.03 \\ 0.05 & 0.1 \end{matrix}] \\ B_{2} = [\begin{matrix} 0.2 \\ 0 \end{matrix}] E_{2} = [\begin{matrix} 2 \\ 1 \end{matrix}] C_{2} = [\begin{matrix} 1.5 & 0 \end{matrix}] \end{matrix} \end{matrix}

After verification, $(A_{i}, B_{i}) (i = 1, 2)$ can be stabilized (but not controllable), $(A_{i}, C_{i}) (i = 1, 2)$ can be observed, and the matrix $[\begin{matrix} A_{i} - I & B_{i} \\ C_{i} & 0 \end{matrix}] (i = 1, 2)$ is of full row rank. Let

Q_{x_{1}} = [\begin{matrix} 20 & 0 \\ 0 & 0 \end{matrix}], Q_{e_{1}} = 30, H_{1} = 2

Q_{x_{2}} = [\begin{matrix} 20 & 0 \\ 0 & 0 \end{matrix}], Q_{e_{2}} = 30, H_{2} = 3

Similarly, the solution of Riccati equation for two isolated subsystems and the feedback gain matrix of the controller are obtained

\begin{matrix} P_{2} = [\begin{matrix} 36.304268573061421 & 1.360259638323270 & 12.574782360793279 \\ 1.360259638323270 & 20.438323123878305 & 4.214645954245819 \\ 12.574782360793279 & 4.214645954245819 & 42.131239677202679 \end{matrix}] \\ P_{2} = [\begin{matrix} 52.575702989830198 & - 2.756669712831776 & 24.738103635858831 \\ - 2.756669712831776 & 20.496692403145200 & - 4.544892821345821 \\ 24.738103635858831 & - 4.544892821345821 & 44.533491801009028 \end{matrix}] \\ F_{e_{1}} = 1.360259638323257, F_{x_{1}} = [\begin{matrix} 0.438323123878289 & 4.214645954245810 \end{matrix}] \\ F_{e_{2}} = 1.837779808554515, F_{x_{2}} = [\begin{matrix} - 0.331128268763466 & 3.029928547563879 \end{matrix}] \end{matrix}

By calculation, there is

L = [\begin{matrix} 0.337474647269200 & - 0.237820233607235 \\ - 0.129901485275353 & 0.093322541770793 \end{matrix}]

$L$ is a non-singular M-matrix.

Reference signal and disturbance signals are selected as

Selecting $α_{1} = 0.3$ , $α_{2} = 0.7$ , the initial value is set as $x_{1} (0) = {[\begin{matrix} 0 & 0.02 \end{matrix}]}^{T}$ , $u_{1} (0) = 0$ , $x_{2} (0) = {[\begin{matrix} 0.01 & 0 \end{matrix}]}^{T}$ , $u_{2} (0) = 0$ . We carried out numerical simulations for two cases $M_{r} = 0$ , $M_{d_{1}} = 0$ , $M_{d_{2}} = 0$ , and $M_{r} = 6$ , $M_{d_{1}} = 4$ , $M_{d_{2}} = 3$ .

Figure 4 indicated that the $r (k)$ can be tracked by the $y (k)$ of the interconnected system without static error, and the controller with preview effect can shorten the adjustment time and restrain the disturbance signal to a certain extent.

Figure 4.

Step response of interconnected system in Example 2.

Conclusion

This paper investigates the previewable controller for a type of discrete-time interconnected systems. Initially, using the basic scheme of preview control, the previewable controller is designed for the error system of each isolated subsystem. Then, the controller of the error system of the isolated subsystem is combined as the controller of error interconnected system. In resort of Lyapunov function and the properties of non-singular M-matrix, the stability of error interconnected system is discussed, then the criterion to ensure its stability is given. Finally, the guarantee conditions for the existence of the preview controller, then the controller for the original interconnected system are derived. The theoretical results and numerical simulation show that the designed controller is able to make the output of the system to track reference signal without static error regardless of the existence of the disturbance signal, and the tracking performance is improved with the increase of the preview steps.

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 Oriented Award Foundation for Science and Technological Innovation, Inner Mongolia Autonomous Region, China (Grant no. 2012) and National Key R&D Program of China (Grant no. 2017YFF0207401).

ORCID iD

Fucheng Liao

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Design of preview controller for a type of discrete-time interconnected systems

Abstract

Keywords

Introduction

Preliminaries

Definition 1

Property 1

Definition 2

Lemma 1

Theorem 1

Proof

Problem formulation

Assumption 4

Remark 1

Controller of error system of isolated subsystems

Remark 2

Remark 3

Theorem 2

Preview controller design for interconnected systems

Theorem 3

Theorem 4

Remark 4

Numerical simulation

Example 1

Example 2

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References