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Abstract

In this study, a preview repetitive control (PRC) strategy was developed for uncertain nonlinear discrete-time systems subjected to a previewable periodic reference signal. The proposed preview repetitive controller was designed such that the system output tracked a previewable periodic reference signal even with model uncertainties and nonlinear terms. An augmented two-dimensional (2D) model was constructed based on the 2D model approach and state augmented technique. Second, considering the state unmeasured and periodic tracking reference signal, a static output PRC law was designed using the linear matrix inequality (LMI) techniques. Finally, the effectiveness of the proposed controller was verified through two illustrative examples.

Keywords

Preview repetitive control augmented 2D systems nonlinear systems previewable signal output feedback LMI

1 Introduction

In a controller, tracking control is used to drive the output of the system toward the desired reference signal as accurately as possible. Preview control (PC) is an advanced control method, in which the future information of the reference signals is used to improve tracking quality.^1,2 Since PC was first proposed,³ numerous PC techniques such as optimal PC,^4–7 H₂ optimization-based PC,⁸ and H_∞ PC^9,10 have been developed. Most of these techniques are Riccati-equation-based design methods. Developing a robust PC problem using the LMI technique is the latest technology trend.^11–16

The combination of PC and other theories to solve problems has attracted considerable attention. For example, a PRC strategy of control systems has been devised^17–20 using the 2D model approach and the state augmented technique. Studies^21–24 have investigated the problem of PC for nonlinear systems based on the augmented error system and LMI techniques. A data-driven robust fault-tolerant PC scheme was designed for discrete-time systems.²⁵ A modified PC strategy was proposed²⁶ to realize wireless tracking control.

Many studies have focused on PC design and analysis for discrete-time systems or robust PC for systems with uncertain items and disturbance signals. However, the problems of PRC for discrete-time nonlinear systems with polytopic uncertainties have not been investigated. Repetitive control (RC) is a specialized control scheme for tracking and rejecting periodic signals.^27,28 Thus, investigating the feasibility of using PRC when the reference signal is periodic and previewable is critical. Therefore, we proposed a novel control strategy to address the aforementioned problems. The PRC laws design methods for nonlinear systems are special cases of this paper. In comparison with the PRC design methods given in previous studies,^17–20 the proposed results permit the system matrices to be noncommon and to exhibit nonlinear terms, which allows a wide range of objects.

This paper presents a method of PRC for uncertain nonlinear systems when the periodic reference signal is previewable. By using the 2D approach and the state augmentation technique, an augmented 2D system including previewable information was derived. An algorithm for a static output PRC and learning actions was designed. Finally, we performed numerical simulations to compare PRC and RC.

The key contribution of this paper can be summarized as follows:

An augmented 2D system without error signal is derived by a new construction method. This has two advantages: First, the order of the augmented error system and the amount of computation are reduced; Second, the output matrices of the considered uncertain system are allowed to be non-common and have uncertainties.

A robust preview repetitive controller to guarantee output tracking control performance is designed by using the construction of the augmented plant including previewable information and learning ability and the robust controller synthesis using LMIs.

Notations. $R^{n}$ and $R^{n \times m}$ denote the n -dimensional real vector space and $n \times m$ matrix space, respectively, $P > 0$ denotes that P is a positive definite, $d i a g {\dots}$ is a block-diagonal matrix, $*$ denotes the transposed elements in the symmetric positions, $A^{T}$ denotes the matrix transposition of A, $s y m (A)$ means $A + A^{T}$ , and I denotes the identity matrix appropriate dimension.

2. Problem statement

Consider the following nonlinear discrete-time system

{\begin{matrix} x (k + 1) = A (θ) x (k) + B (θ) u (k) + G (θ) g (x (k)), \\ y (k) = C (θ) x (k) + D u (k) + H (θ) h (x (k)), \end{matrix}

(1)

where

x (k) \in R^{n}

is the state vector,

u (k) \in R

is the input control vector,

y (k) \in R

is the output vector,

g (\cdot) :

R^{n} \mapsto R^{n_{g}}

and

h (\cdot) :

R^{n} \mapsto R^{n_{h}}

are known nonlinear functions, D is a constant matrix with appropriate dimensions, and

A (θ)

B (θ)

G (θ)

C (θ)

, and

H (θ)

are uncertain matrices with appropriate dimensions which are assumed to belong to the following polytopic domain:

S := (\begin{matrix} A (θ) & B (θ) & G (θ) & C (θ) & H (θ) \end{matrix}) \in Θ,

(2)

where

Θ

is a polytope as follows:

Θ := {S : S = \sum_{i = 1}^{d} θ_{i} S_{i}, \sum_{i = 1}^{d} θ_{i} = 1, θ_{i} \geq 0, i = 1, 2, \dots, d}

(3)

with

S_{i}

(i = 1, 2, \dots, d)

, being the vertices of

Θ

defined by the following:

S_{i} = (\begin{matrix} A_{i} & B_{i} & G_{i} & C_{i} & H_{i} \end{matrix}), i = 1, 2, \dots, d,

(4)

where

A_{i}

B_{i}

G_{i}

C_{i}

and

H_{i}

are given real matrices.

We made the following assumptions.

A1. Matrices $V_{g} \in R^{m_{g} \times n}$ and $V_{h} \in R^{m_{h} \times n}$ exist such that for any $x (k)$ , $z (k) \in R^{n}$ , the following holds:

{\begin{matrix} ‖ g (x (k)) - g (z (k)) ‖ \leq ‖ V_{g} (x (k) - z (k)) ‖, \\ ‖ h (x (k)) - h (z (k)) ‖ \leq ‖ V_{h} (x (k) - z (k)) ‖ . \end{matrix}

(5)

A2. Let

r (k) \in R

be the periodic reference signal with

T

. Assume that the reference signal is available from current time k to

k + M_{R}

, where

M_{R}

is the preview length and

M_{R} < T

Let

X_{R} (k) = {[\begin{matrix} r {(k)}^{T} & r {(k + 1)}^{T} & \dots & r {(k + M_{R})}^{T} \end{matrix}]}^{T} \in R^{(M_{R} + 1)} .

(6)

It follows from A2 and (6) that

X_{R} (k + 1) = A_{R} X_{R} (k) + B_{R} r (k + M_{R} + 1),

(7)

where

A_{R} = [\begin{matrix} 0 & 1 & 0 \\ 0 & ⋱ & ⋱ \\ ⋮ & ⋱ & ⋱ \\ 0 & \dots & \dots & 0 & 1 \\ 0 & \dots & \dots & 0 & 0 \end{matrix}] \in R^{(M_{R} + 1) \times (M_{R} + 1)}, B_{R} = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}] \in R^{(M_{R} + 1)} .

This paper aims to design a PRC law such that the output of system (1) tracks the periodic reference signal as follows:

lim_{k \to \infty} e (k) = 0,

where

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

The repetitive controller can be expressed as follows:

v (k) = {\begin{matrix} e (k), 0 \leq k < T, \\ e (k) + v (k - T), k \geq T . \end{matrix}

(8)

The linear output feedback PRC law is designed as follows:

u (k) = K_{y} y (k) + K_{r} X_{R} (k) + K_{v} v (k),

(9)

where

K_{y}

is the output feedback gain,

K_{r}

is the feedback gain of preview action, and

K_{v}

is the feedback gain of the repetitive controller.

Remark 1. The proposed PRC law integrated a preview compensations term (i.e. $K_{r} X_{R} (k)$ ), which includes the future information of $r (k)$ , and the third term (i.e. $K_{v} v (k)$ ), which contains information on the previous period. The two terms can effectively improve tracking performance.

Remark 2. For comparison purposes, let us consider the closely related published reports^21–24 which focused on PC design for nonlinear systems. However, these PC laws without learning actions did not provide satisfactory performance for periodic reference signals because of an underperforming repetitive controller. Compared with the PRC studies,^17–20 the system model considered in this paper is general.

3. Main results

In this section, we utilize the characteristics of RC and the methods of PC theory to construct an augmented system, which combines the error equation, the state equation of controlled system and the reference signal. Then, the required controller is designed by LMI technology.

3.1 Construction of the augmented 2d system

Let $k = p T + q$ , where p is a discrete-time variable belonging to the set ${0, 1, 2, \dots}$ , and the discrete-time variable $q \in {0, 1, 2, \dots, T - 1}$ .

From (1), (7), and (8), we have the following 2D representations:

x (p, q + 1) = A (θ) x (p, q) + B (θ) u (p, q) + G (θ) g (x (p, q)),

(10)

y (p, q) = C (θ) x (p, q) + D u (p, q) + H (θ) h (x (p, q)),

(11)

e (p, q) = r (p, q) - y (p, q),

(12)

v (p, q) = e (p, q) + v (p - 1, q),

(13)

X_{R} (p, q + 1) = A_{R} X_{R} (p, q) + B_{R} r (p, q + M_{R} + 1) .

(14)

Based on (9), we obtain the following expression:

u (p, q) = K_{y} y (p, q) + K_{r} X_{R} (p, q) + K_{v} v (p, q) .

(15)

Defining the following expression:

Δ ξ (p, q) = ξ (p, q) - ξ (p - 1, q) .

(16)

From (10), (11), and (16), we have the following expression:

Δ x (p, q + 1) = A (θ) Δ x (p, q) + B (θ) Δ u (p, q) + G (θ) Δ g (x (p, q)),

(17)

Δ y (p, q) = C (θ) Δ x (p, q) + D Δ u (p, q) + H (θ) Δ h (x (p, q)),

(18)

where

Δ g (x (p, q)) = g (x (p, q)) - g (x (p - 1, q)), Δ h (x (p, q)) = h (x (p, q)) - h (x (p - 1, q)) .

Through (11)–(13) and (16), we obtain the following expression:

e (p, q) = e (p - 1, q) - C (θ) Δ x (p, q) - D (θ) Δ u (p, q) - H (θ) Δ h (x (p, q)),

(19)

Δ v (p, q) = e (p, q) .

(20)

Because

Δ r (p, q + M_{R} + 1) = 0

, it yields the following expression:

Δ X_{R} (p, q + 1) = A_{R} Δ X_{R} (p, q) .

(21)

From (15), (16), and (20), the following expression is obtained:

Δ u (p, q) = K_{y} Δ y (p, q) + K_{r} Δ X_{R} (p, q) + K_{v} e (p, q) .

(22)

Using (17) and (21), the following expression is obtained:

\bar{x} (p, q + 1) = \bar{A} (θ) \bar{x} (p, q) + \bar{B} (θ) Δ u (p, q) + \bar{G} (θ) Δ g (x (p, q)),

(23)

where

\bar{x} (p, q) = [\begin{matrix} Δ x (p, q) \\ Δ X_{R} (p, q) \end{matrix}], \bar{A} (θ) = [\begin{matrix} A (θ) & 0 \\ 0 & A_{R} \end{matrix}], \bar{B} (θ) = [\begin{matrix} B (θ) \\ 0 \end{matrix}], \bar{G} (θ) = [\begin{matrix} G (θ) \\ 0 \end{matrix}] .

According to the definition of

\bar{x} (p, q)

and (19), the following expression is obtained:

e (p, q) = e (p - 1, q) - \bar{C} (θ) \bar{x} (p, q) - D Δ u (p, q) - H (θ) Δ h (x (p, q)),

(24)

where

\bar{C} (θ) = [\begin{matrix} C (θ) & 0 \end{matrix}] .

From A1,

\bar{A} (θ)

and

\bar{B} (θ)

are expressed as follows:

\bar{A} (θ) = [\begin{matrix} \sum_{i = 1}^{d} θ_{i} A_{i} & 0 \\ 0 & A_{R} \end{matrix}] = \sum_{i = 1}^{d} θ_{i} [\begin{matrix} A_{i} & 0 \\ 0 & A_{R} \end{matrix}] = \sum_{i = 1}^{d} θ_{i} {\bar{A}}_{i},

(25)

\bar{B} (θ) = [\begin{matrix} \sum_{i = 1}^{d} θ_{i} B_{i} \\ 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} [\begin{matrix} B_{i} \\ 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} {\bar{B}}_{i},

(26)

\bar{G} (θ) = [\begin{matrix} \sum_{i = 1}^{d} θ_{i} G_{i} \\ 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} [\begin{matrix} G_{i} \\ 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} {\bar{G}}_{i},

(27)

\bar{C} (θ) = [\begin{matrix} \sum_{i = 1}^{d} θ_{i} C_{i} & 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} [\begin{matrix} C_{i} & 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} {\bar{C}}_{i} .

(28)

Based on (5) and (23), we obtain the following expression:

{\begin{matrix} ‖ Δ g (x (p, q)) ‖ \leq ‖ V_{g} Δ x (p, q) ‖ = ‖ {\hat{V}}_{g} \bar{x} (p, q) ‖, \\ ‖ Δ h (x (p, q)) ‖ \leq ‖ V_{h} Δ x (p, q) ‖ = ‖ {\hat{V}}_{h} \bar{x} (p, q) ‖, \end{matrix}

(29)

where

{\hat{V}}_{g} = V_{g} [\begin{matrix} I_{n} & 0_{n \times (M_{R} + 1)} \end{matrix}] and {\hat{V}}_{h} = V_{h} [\begin{matrix} I_{n} & 0_{n \times (M_{R} + 1)} \end{matrix}] .

3.2 Design of a PRC law

Substituting (18), (24), and (20) into (22), we obtain the following expression:

\begin{matrix} Δ u (p, q) = K_{y} [C (θ) Δ x (p, q) + D Δ u (p, q) + H (θ) Δ h (x (p, q))] + K_{r} Δ X_{R} (p, q) \\ + K_{v} [e (p - 1, q) - C (θ) Δ x (p, q) - D Δ u (p, q) - H (θ) Δ h (x (p, q))] . \end{matrix}

(30)

Next, we have the following expression:

\begin{aligned} Δ u (p, q) = [\begin{matrix} \frac{K_{y} - K_{v}}{1 + K_{v} D - K_{y} D} & \frac{K_{r}}{1 + K_{v} D - K_{y} D} \end{matrix}] \\ {[\begin{matrix} C (θ) & 0 \\ 0 & I \end{matrix}] [\begin{matrix} Δ x (p, q) \\ Δ X_{R} (p, q) \end{matrix}] + [\begin{matrix} H (θ) \\ 0 \end{matrix}] Δ h (x)} + \frac{K_{v}}{1 + K_{v} D - K_{y} D} e (p - 1, q) . \end{aligned}

(31)

Based on (31), the following expression is obtained

Z (p, q) = C_{Z} (θ) \bar{x} (p, q) + \bar{H} (θ) Δ h (x (p, q)),

(32)

where

C_{Z} (θ) = [\begin{matrix} C (θ) & 0 \\ 0 & I \end{matrix}], \bar{H} (θ) = [\begin{matrix} H (θ) \\ 0 \end{matrix}] .

Then, equation (31) can be expressed as follows:

Δ u (p, q) = K_{1} Z (p, q) + K_{2} e (p - 1, q),

(33)

where

K_{1} = [\begin{matrix} \frac{K_{y} - K_{v}}{1 + K_{v} D - K_{y} D} & \frac{K_{r}}{1 + K_{v} D - K_{y} D} \end{matrix}], K_{2} = \frac{K_{v}}{1 + K_{v} D - K_{y} D} .

(34)

From A1,

C_{Z} (θ)

and

\bar{H} (θ)

are expressed as follows:

C_{Z} (θ) = [\begin{matrix} \sum_{i = 1}^{d} θ_{i} C_{i} & 0 \\ 0 & I \end{matrix}] = \sum_{i = 1}^{d} θ_{i} [\begin{matrix} C_{i} & 0 \\ 0 & I \end{matrix}] = \sum_{i = 1}^{d} θ_{i} C_{Z i},

(35)

\bar{H} (θ) = [\begin{matrix} \sum_{i = 1}^{d} θ_{i} H_{i} \\ 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} [\begin{matrix} H_{i} \\ 0 \end{matrix}] = \sum_{i = 1}^{d} θ_{i} {\bar{H}}_{i} .

(36)

Replacing (23) and (24) with (33), we obtain the following expression:

\begin{matrix} \underset{ξ (p, q + 1)}{\underset{⏟}{[\begin{matrix} \bar{x} (p, q + 1) \\ e (p, q) \end{matrix}]}} = \underset{\hat{A} (θ)}{\underset{⏟}{[\begin{matrix} \bar{A} (θ) + \bar{B} (θ) K_{1} C_{Z} (θ) & \bar{B} (θ) K_{2} \\ - \bar{C} (θ) - D K_{1} C_{Z} (θ) & 1 - D K_{2} \end{matrix}]}} \underset{ξ (p, q)}{\underset{⏟}{[\begin{matrix} \bar{x} (p, q) \\ e (p - 1, q) \end{matrix}]}} \\ + \underset{\hat{D} (θ)}{\underset{⏟}{[\begin{matrix} \bar{G} (θ) & \bar{B} (θ) K_{1} \bar{H} (θ) \\ 0 & - H (θ) - D K_{1} \bar{H} (θ) \end{matrix}]}} \underset{f (x)}{\underset{⏟}{[\begin{matrix} Δ g (x (p, q)) \\ Δ h (x (p, q)) \end{matrix}]}} . \end{matrix}

(37)

From (5) and (37), we obtain:

{\begin{matrix} ‖ Δ g (x (p, q)) ‖ \leq ‖ {\hat{V}}_{g} \bar{x} (p, q) ‖ = ‖ F_{g} ξ (p, q) ‖, \\ ‖ Δ h (x (p, q)) ‖ \leq ‖ {\hat{V}}_{h} \bar{x} (p, q) ‖ = ‖ F_{h} ξ (p, q) ‖, \end{matrix}

(38)

where

F_{g} = [\begin{matrix} {\hat{V}}_{g} & 0 \end{matrix}] a n d F_{h} = [\begin{matrix} {\hat{V}}_{h} & 0 \end{matrix}] .

Lemma 1.²⁹ For the matrix

Υ

, R, S, N, and scalar

β

Υ + S^{T} R^{T} + R S < 0

holds if the following holds:

[\begin{matrix} Υ & * \\ β R^{T} + N S & - β N - β N^{T} \end{matrix}] < 0.

Theorem 1. Let

λ_{i} > 0, i = 1, 2

be given scalars. The robust problem of the system (37) is solvable if

X (θ) > 0

and W such that

Π (θ) = [\begin{matrix} - W - W^{T} + X (θ) & * & * & * \\ \hat{A} (θ) W & - X (θ) & * & * \\ 0 & \hat{D} {(θ)}^{T} & - Λ_{1} & * \\ F W & 0 & 0 & - Λ_{2} \end{matrix}] < 0

(39)

where

$X (θ) = d i a g {X_{1} (θ), X_{2} (θ)}$ , $W = d i a g {W_{1}, W_{2}}$ , $Λ_{1} = d i a g {λ_{1} I_{n_{g}}, λ_{2} I_{n_{h}}}$ , $Λ_{2} = d i a g {1 / λ_{1} I_{m_{g}}, 1 / λ_{2} I_{m_{h}}}$ , $F = [\begin{matrix} F_{g}^{T} & F_{h}^{T} \end{matrix}]$ .

Proof. Using the relation $- W^{T} X (θ)^{- 1} W \leq X (θ) - s y m (W)$ ,³⁰ (39) can imply the following matrix:

[\begin{matrix} - W^{T} X (θ) W & * & * & * \\ \hat{A} (θ) W & - X (θ) & * & * \\ 0 & \hat{D} {(θ)}^{T} & - Λ_{1} & * \\ F W & 0 & 0 & - Λ_{2} \end{matrix}] < 0

(40)

Pre- and post-multiply (40) with

d i a g {W^{- T}, I^{T}, I^{T}, I^{T}}

, and denote

X (θ)^{- 1} = P (θ)

, to obtain the following matrix:

[\begin{matrix} - P (θ) & * & * & * \\ \hat{A} (θ) & - P {(θ)}^{- 1} & * & * \\ 0 & \hat{D} {(θ)}^{T} & - Λ_{1} & * \\ F & 0 & 0 & - Λ_{2} \end{matrix}] < 0

(41)

By applying Schur lemma³¹ and the exchange of rows and columns, (41) is equivalent to the following:

[\begin{matrix} Ψ (θ) & * \\ \hat{D} {(θ)}^{T} P (θ) \hat{A} (θ) & \hat{D} {(θ)}^{T} P (θ) \hat{D} (θ) - Λ_{1} \end{matrix}] < 0,

(42)

where

Ψ (θ) = \hat{A} (θ)^{T} P (θ) \hat{A} (θ) - P (θ) + F^{T} Λ_{2}^{- 1} F .

The positive-definite Lyapunov function is constructed as follows:

V (p, q) = V_{1} (p, q) + V_{2} (p, q),

(43)

where

V_{1} (p, q) = \hat{x} (p, q)^{T} P_{1} (θ) \hat{x} (p, q),

V_{2} (p, q) = e (p - 1, q)^{T} P_{2} (θ) e (p - 1, q) .

Next, we define

\nabla

as the difference operator

\nabla V_{1} (p, q) = \bar{x} (p, q + 1)^{T} P_{1} (θ) \bar{x} (p, q + 1) - \bar{x} (p, q)^{T} P_{1} (θ) \bar{x} (p, q),

(44)

\nabla V_{2} (p, q) = e (p, q)^{T} P_{2} (θ) e (p, q) - e (p - 1, q)^{T} P_{2} (θ) e (p - 1, q) .

(45)

Thus, from (43)–(45), we obtain the following:

\begin{matrix} \nabla V (p, q) = [\begin{matrix} \bar{x} {(p, q + 1)}^{T} & e {(p, q)}^{T} \end{matrix}] [\begin{matrix} P_{1} (θ) & 0 \\ 0 & P_{2} (θ) \end{matrix}] [\begin{matrix} \bar{x} (p, q + 1) \\ e (p, q) \end{matrix}] \\ - [\begin{matrix} \bar{x} {(p, q)}^{T} & e {(p - 1, q)}^{T} \end{matrix}] \underset{P (θ)}{\underset{⏟}{[\begin{matrix} P_{1} (θ) & 0 \\ 0 & P_{2} (θ) \end{matrix}]}} \underset{ξ (p, q)}{\underset{⏟}{[\begin{matrix} \bar{x} (p, q) \\ e (p - 1, q) \end{matrix}]}} \\ = [\hat{A} (θ) ξ (p, q) + \hat{D} (θ) f (x)]^{T} P (θ) [\hat{A} (θ) ξ (p, q) + \hat{D} (θ) f (x)] \\ - ξ (p, q)^{T} P (θ) ξ (p, q) \\ = ξ (p, q)^{T} [\hat{A} {(θ)}^{T} P (θ) \hat{A} (θ) - P (θ)] ξ (p, q) + 2 ξ (p, q)^{T} \hat{A} (θ)^{T} P (θ) \hat{D} (θ) f (x) \\ + f {(x)}^{T} Λ_{1} f (x) - f {(x)}^{T} Z (θ) f (x) \end{matrix}

(46)

where

Z (θ) = Λ_{1} - \hat{D} (θ)^{T} P (θ) \hat{D} (θ)

Inequality (42) implies that

Z (θ) > 0

, and

[Z {(θ)}^{- 1} \hat{D} {(θ)}^{T} P (θ) A (θ) ξ (p, q) - f (x)]^{T} Z (θ) [Z {(θ)}^{- 1} \hat{D} {(θ)}^{T} P (θ) A (θ) ξ (p, q) - f (x)] \geq 0

leads to

\begin{aligned} ξ (p, q)^{T} \hat{A} (θ)^{T} P (θ) \hat{D} (θ) Z (θ)^{- 1} \hat{D} (θ)^{T} P (θ) A (θ) ξ (p, q) \\ \geq 2 ξ (p, q)^{T} \hat{A} (θ)^{T} P (θ) \hat{D} (θ) f (x) - f {(x)}^{T} Z (θ) f (x) . \end{aligned}

(47)

Furthermore, based on (38), the following expression is obtained:

\begin{matrix} f {(x)}^{T} Λ_{1} f (x) = λ_{1} ‖ g_{1} (Δ x (p, q)) ‖ + λ_{2} ‖ h (Δ x (p, q)) ‖ \\ \leq λ_{1} ‖ F_{g} ξ (p, q) ‖ + λ_{2} ‖ F_{h} ξ (p, q) ‖ \\ = ξ (p, q)^{T} F^{T} Λ_{2}^{- 1} F ξ (p, q), \end{matrix}

(48)

Thus, combining (46)–(48) reveals:

Δ V (ξ) \leq ξ (p, q)^{T} Ξ (θ) ξ (p, q),

(49)

where

Ξ (θ) = \hat{A} (θ)^{T} P (θ) \hat{A} (θ) - P (θ) + F^{T} Λ_{2}^{- 1} F + \hat{A} (θ)^{T} P (θ) \hat{D} (θ) Z (θ)^{- 1} \hat{D} (θ)^{T} P (θ) \hat{A} (θ) .

According to Schur's complement,

Ξ (θ) < 0

is equivalent to (42), and (42) is satisfied if and only if (39) holds. Thus, from (39),

\nabla V < 0

, which indicates system (37) is asymptotically stable.

Based on Theorem 1, an LMI method is presented in the next theorem to design a PRC law as (33) to solve the PC problem for polytopic uncertain Lipschitz system (37).

Theorem 2. For the given scalar $λ_{1}$ , $λ_{2}$ , and $β$ and the matrix $Q_{1}$ , $Q_{2}$ , the robust stabilization problem of system (37) is solvable if $X_{1} (θ) > 0$ , $X_{2} (θ) > 0$ , and the matrix L, U, $Y_{2}$ , $W_{1}$ , $W_{2}$ such that

Ω (θ) = [\begin{matrix} Ω_{11} (θ) & * & * & * & * \\ Ω_{21} (θ) & - X (θ) & * & * & * \\ 0 & Ω_{32} (θ) & - Λ_{1} & * & * \\ Ω_{41} & 0 & 0 & - Λ_{2} & * \\ Ω_{51} (θ) & Ω_{52} (θ) & Ω_{53} (θ) & 0 & - β U - β U^{T} \end{matrix}] < 0,

(50)

where

Ω_{11} (θ) = [\begin{matrix} - W_{1} - W_{1}^{T} + X_{1} (θ) & 0 \\ 0 & - W_{2} - W_{2}^{T} + X_{2} (θ) \end{matrix}], X (θ) = [\begin{matrix} X_{1} (θ) & 0 \\ 0 & X_{2} (θ) \end{matrix}],

Ω_{21} (θ) = [\begin{matrix} \bar{A} (θ) W_{1} + \bar{B} (θ) L Q_{1} & \bar{B} (θ) Y_{2} \\ - \bar{C} (θ) W_{1} - D L Q_{1} & W_{2} - D Y_{2} \end{matrix}],

Ω_{32} (θ) = [\begin{matrix} \bar{G} {(θ)}^{T} & 0 \\ Q_{2}^{T} L^{T} \bar{B} {(θ)}^{T} & - H {(θ)}^{T} - Q_{2}^{T} L^{T} D^{T} \end{matrix}], Ω_{41} = [\begin{matrix} {\hat{V}}_{g} W_{1} & 0 \\ {\hat{V}}_{h} W_{1} & 0 \end{matrix}],

Ω_{51} (θ) = [\begin{matrix} C_{Z} (θ) W_{1} - U Q_{1} & 0 \end{matrix}], Ω_{52} (θ) = β [\begin{matrix} L^{T} \bar{B} {(θ)}^{T} & - L^{T} D^{T} \end{matrix}],

Ω_{53} (θ) = [\begin{matrix} 0 & \bar{H} (θ) - U Q_{2} \end{matrix}] .

Proof. To facilitate the use of Lemma 1, we modify the matrix to the following:

Υ = [\begin{matrix} Ω_{11} (θ) & * & * & * \\ Ω_{21} (θ) & - X (θ) & * & * \\ 0 & Ω_{32} (θ) & - Λ_{1} & * \\ Ω_{41} & 0 & 0 & - Λ_{2} \end{matrix}], N = U,

R = {[\begin{matrix} 0 & [\begin{matrix} L^{T} \bar{B} {(θ)}^{T} & - L^{T} D^{T} \end{matrix}] & 0 & 0 \end{matrix}]}^{T},

S = U^{- 1} [\begin{matrix} [\begin{matrix} C_{Z} (θ) W_{1} - U Q_{1} & 0 \end{matrix}] & 0 & [\begin{matrix} 0 & \bar{H} (θ) - U Q_{2} \end{matrix}] & 0 \end{matrix}] .

From Lemma 1, (50) can guarantee

\begin{aligned} [\begin{matrix} Ω_{11} (θ) & * & * & * \\ Ω_{21} (θ) & - X (θ) & * & * \\ 0 & Ω_{32} (θ) & - Λ_{1} & * \\ Ω_{41} & 0 & 0 & - Λ_{2} \end{matrix}] \\ + s y m ([\begin{matrix} 0 \\ [\begin{matrix} \bar{B} (θ) \\ - D \end{matrix}] \\ 0 \\ 0 \end{matrix}] L U^{- 1} [\begin{matrix} [\begin{matrix} C_{Z} (θ) W 1 - U Q 1 & 0 \end{matrix}] & 0 \\ [\begin{matrix} 0 & \bar{H} (θ) - U Q_{2} \end{matrix}] & 0 \end{matrix}]) \\ = [\begin{matrix} Ω_{11} (θ) & * & * & * \\ {\hat{Ω}}_{21} (θ) & - X (θ) & * & * \\ 0 & {\hat{Ω}}_{32} (θ) & - Λ_{1} & * \\ Ω_{41} & 0 & 0 & - Λ_{2} \end{matrix}] \\ + s y m ([\begin{matrix} 0 \\ [\begin{matrix} \bar{B} (θ) \\ - D \end{matrix}] \\ 0 \\ 0 \end{matrix}] L U^{- 1} [\begin{matrix} [\begin{matrix} C_{Z} (θ) W_{1} - U Q_{1} + U Q_{1} & 0 \end{matrix}] & 0 \\ [\begin{matrix} 0 & \bar{H} (θ) - U Q_{2} + U Q_{2} \end{matrix}] & 0 \end{matrix}]) \\ < 0 \end{aligned}

(51)

where

{\hat{Ω}}_{21} (θ) = [\begin{matrix} \bar{A} (θ) W_{1} & \bar{B} (θ) Y_{2} \\ - \bar{C} (θ) W_{1} & W_{2} - D (θ) Y_{2} \end{matrix}], {\hat{Ω}}_{32} (θ) = [\begin{matrix} \bar{G} {(θ)}^{T} & 0 \\ 0 & - H {(θ)}^{T} \end{matrix}] .

Let $K_{1} = L U^{- 1}$ , then (51) is

\begin{matrix} [\begin{matrix} Ω_{11} (θ) & * & * & * \\ {\hat{Ω}}_{21} (θ) & - X (θ) & * & * \\ 0 & {\hat{Ω}}_{32} (θ) & - Λ_{1} & * \\ Ω_{41} & 0 & 0 & - Λ_{2} \end{matrix}] \\ + s y m ([\begin{matrix} 0 \\ [\begin{matrix} \bar{B} (θ) \\ - D \end{matrix}] \\ 0 \\ 0 \end{matrix}] K_{1} [\begin{matrix} [\begin{matrix} C_{Z} (θ) W_{1} & 0 \end{matrix}] & 0 & [\begin{matrix} 0 & \bar{H} (θ) \end{matrix}] & 0 \end{matrix}]) \\ < 0 \end{matrix}

that is,

[\begin{matrix} Ω_{11} (θ) & * & * & * \\ Π_{21} (θ) & - X (θ) & * & * \\ 0 & Π_{32} (θ) & - Λ_{1} & * \\ Ω_{41} & 0 & 0 & - Λ_{2} \end{matrix}] < 0,

(52)

where

Π_{21} (θ) = [\begin{matrix} (\bar{A} (θ) + \bar{B} (θ) K_{1} C_{Z} (θ)) W_{1} & \bar{B} (θ) Y_{2} \\ (- \bar{C} (θ) - D K_{1} C_{Z} (θ)) W_{1} & W_{2} - D Y_{2} \end{matrix}],

Π_{32} (θ) = [\begin{matrix} \bar{G} {(θ)}^{T} & 0 \\ \bar{H} {(θ)}^{T} K_{1}^{T} \bar{B} {(θ)}^{T} & - H {(θ)}^{T} - \bar{H} {(θ)}^{T} K_{1}^{T} D^{T} \end{matrix}] .

In theorem 1, letting

X (θ) = d i a g {X_{1} (θ), X_{2} (θ)}, W = d i a g {W_{1}, W_{2}},

\hat{A} (θ) = [\begin{matrix} \bar{A} (θ) + \bar{B} (θ) K_{1} C_{Z} (θ) & \bar{B} (θ) K_{2} \\ - \bar{C} (θ) - D K_{1} C_{Z} (θ) & 1 - D (θ) K_{2} \end{matrix}], \hat{D} (θ) = [\begin{matrix} \bar{G} (θ) & \bar{B} (θ) K_{1} \bar{H} (θ) \\ 0 & - H (θ) - D K_{1} \bar{H} (θ) \end{matrix}],

K_{2} W_{2} = Y_{2}, F = {[\begin{matrix} F_{g}^{T} & F_{h}^{T} \end{matrix}]}^{T} .

Next, we can obtain (39) from (52). Thus, it follows from Theorem 1 that Theorem 2 holds.

Theorem 3. For the given $λ_{1}$ , $λ_{2}$ , $β$ and the matrix $Q_{1}$ , $Q_{2}$ . System (37) is asymptotically stable, matrices $X_{1 i} > 0$ , $X_{2 i} > 0$ , $Y_{2}$ , $W_{1}$ , $W_{2}$ , L, and U exist such that

Ω_{i} = [\begin{matrix} Ω_{11 i} & * & * & * & * \\ Ω_{21 i} & - X_{i} & * & * & * \\ 0 & Ω_{32 i} & - Λ_{1} & * & * \\ Ω_{41} & 0 & 0 & - Λ_{2} & * \\ Ω_{51 i} & Ω_{52 i} & Ω_{53 i} & 0 & - β U - β U^{T} \end{matrix}] < 0,

(53)

Where

Ω_{11 i} = [\begin{matrix} - W_{1} - W_{1}^{T} + X_{1 i} & 0 \\ 0 & - W_{2} - W_{2}^{T} + X_{2 i} \end{matrix}], X_{i} = [\begin{matrix} X_{1 i} & 0 \\ 0 & X_{2 i} \end{matrix}],

Ω_{21 i} = [\begin{matrix} {\bar{A}}_{i} W_{1} + {\bar{B}}_{i} L Q_{1} & {\bar{B}}_{i} Y_{2} \\ - {\bar{C}}_{i} W_{1} - D L Q_{1} & W_{2} - D Y_{2} \end{matrix}], Ω_{32 i} = [\begin{matrix} {\bar{G}}_{i}^{T} & 0 \\ Q_{2}^{T} L^{T} {\bar{B}}_{i}^{T} & - H_{i}^{T} - Q_{2}^{T} L^{T} D^{T} \end{matrix}],

$Ω_{41} = [\begin{matrix} {\hat{V}}_{g} W_{1} & 0 \\ {\hat{V}}_{h} W_{1} & 0 \end{matrix}], Ω_{51 i} = [\begin{matrix} C_{Z i} W_{1} - U Q_{1} & 0 \end{matrix}], Ω_{52 i} = β [\begin{matrix} L^{T} {\bar{B}}_{i}^{T} & - L^{T} D^{T} \end{matrix}], Ω_{53 i} = [\begin{matrix} 0 & {\bar{H}}_{i} - U Q_{2} \end{matrix}] .$

Next, the gain matrices are expressed by the following expression:

K_{1} = L U^{- 1} a n d K_{2} = Y_{2} W_{2}^{- 1},

(54)

Proof. Because LMIs are linear in $Ω_{i}$ , multiplying the matrix (53) by $θ_{i}$ for $1 \leq i \leq d$ and summing the elements, we have the following expression:

\sum_{i = 1}^{d} θ_{i} Ω_{i} < 0.

(55)

Based on (53), we determine that $\sum_{i = 1}^{d} θ_{i} Ω_{i} = Ω (θ) < 0$ , which reveals that (53) holds. Thus, the proof is complete.

Suppose that LMIs (53) have a feasible solution. Next, $K_{1}$ and $K_{2}$ in (34) can be obtained and defined as follows:

K_{1} = [\begin{matrix} K_{11} & K_{12} \end{matrix}] = L U and K_{2} = Y_{2} W_{2}^{- 1}

(56)

From (34) and derivation, the gain matrices are obtained as follows:

K_{y} = \frac{K_{11} + K_{2}}{1 + D K_{11}}, K_{r} = \frac{K_{12}}{1 + D K_{11}}, K_{v} = \frac{K_{2}}{1 + D K_{11}} .

Numerical example

Example 1

A flexible-link robot model is cited in.^[22] The state-space model of the flexible-link robot control system (F-LRCS) is expressed as follows:

\dot{x} (t) = A x (t) + B u (t) + f (x (t))

(57)

where

x (t) = {[\begin{matrix} θ_{m} (t) & w_{m} (t) & θ (t) & w (t) \end{matrix}]}^{T},

A = [\begin{matrix} 0 & 1 & 0 & 1 \\ - 48.6 & - 1.25 & 48.6 & 0 \\ 0 & 0 & 0 & 1 \\ 1.95 & 0 & - 1.95 & 0 \end{matrix}], B = [\begin{matrix} 0 \\ 21.6 \\ 0 \\ 0 \end{matrix}], f (x) = [\begin{matrix} 0 \\ 0 \\ 1 \\ - 0.33 \sin (x_{3}) \end{matrix}] .

where $θ_{m} (t)$ , $t \geq 0$ is the angular position of the motor; $w_{m} (t)$ , $t \geq 0$ is the angular velocity of the motor; $θ (t)$ , $t \geq 0$ is the angular position of the link; and $w (t)$ , $t \geq 0$ is the angular velocity of the link.

The output equation is expressed as follows:

y (t) = [\begin{matrix} 2 & 0 & 1 & 0 \end{matrix}] x (t) + B u (t) .

By the sampling period $T = 0.01 s$ , the following discrete-time model is obtained:

{\begin{matrix} x (k + 1) = [\begin{matrix} 1 & 0.01 & 0 & 0 \\ - 0.486 & 0.9875 & 0.486 & 0 \\ 0 & 0 & 1 & 0.01 \\ 0.195 & 0 & - 0.195 & 1 \end{matrix}] x (k) + [\begin{matrix} 0 \\ 0.216 \\ 0 \\ 0 \end{matrix}] u (k) \\ + [\begin{matrix} 0 \\ 0 \\ 0 \\ - 0.0333 \end{matrix}] g (x_{3} (k)), \\ y (k) = [\begin{matrix} 2 & 0 & 1 & 0 \end{matrix}] x (k) + u (k) . \end{matrix}

(58)

For comparison and analysis, RC and PRC were applied to F-LRCS. For the given $λ_{1} = λ_{2} = 1$ , $β = 1$ and $Q_{1} = C_{Z 1} + C_{Z 2}$ , $Q_{2} = {\bar{H}}_{1} + {\bar{H}}_{2}$ , the gain matrices are obtained by solving the condition of Theorem 3.

When $M_{R} = 3$ , the gain matrices of PRC law are as follows:

K_{y} = 8789.51, K_{v} = - 18438.398, K_{r} = 10^{4} [\begin{matrix} - 1.36368 & 0.09098 & 0.04784 \end{matrix}] .

The gain matrices of RC law are as follows:

K_{y} = 237 .31, K_{v} = - 182.88 .

Let the reference signal be as follows:

r (k) = 1.5 + 0.9 \sin (0.1 π k) + 0.7 \sin (0.2 π k) + 0.1 \sin (0.3 π k) .

(59)

Output responses of F-LRCS (with PRC and RC) are displayed in Figure 1. Figure 2 plots the tracking error. From Figures 1–2, the tracking error decreases, and the output can track the reference signal faster with PRC.

Figure 1.

Output of F-LRCS with PRC with RC.

Figure 2.

Tracking error of F-LRCS with PRC with RC.

Figure 3 shows the control input of F-LRCS. It illustrates that once the output vector of the system tracks the reference signal, the input vector basically remains unchanged or changes periodically, which is the same as the actual situation.

Figure 3.

Control input of F-LRCS with PRC with RC.

The reference signal is as follows:

r (k) = \sin (0.1 π k) .

(60)

To see the tracking performance clearly, we investigate the effect of PRC and RC on the F-LRCS system. The simulation results have been illustrated in Figures 4–6. By comparison with the RC, Figures 4 and 5 reveal that the PRC can track the desired signal faster than RC.

Figure 4.

Output response of F-LRCS for signal (60).

Figure 5.

Tracking error of F-LRCS for signal (60).

Figure 6.

Control input of F-LRCS for signal (60).

Example 2

In system (1),

A_{1} = [\begin{matrix} 0.9 & - 0.4 & 0.8 \\ 1 & 0 & 0 \\ 0 & 1 & - 0.4 \end{matrix}], A_{2} = [\begin{matrix} 1 & - 0.6 & 0.7 \\ 1 & 0 & 0 \\ 0 & 1 & 0.5 \end{matrix}],

B_{1} = [\begin{matrix} 1 \\ 0.3 \\ 0 \end{matrix}], B_{1} = [\begin{matrix} 1 \\ 1.1 \\ 0 \end{matrix}], G_{1} = [\begin{matrix} 0.2 \\ - 0.1 \\ 0 \end{matrix}], G_{1} = [\begin{matrix} 0.3 \\ - 0.1 \\ 0.1 \end{matrix}],

C_{1} = [\begin{matrix} 1 & 0 & 0 \end{matrix}], C_{2} = [\begin{matrix} 1 & 0 & 0 \end{matrix}], D = 1, H_{1} = 0.6, H_{2} = 0.4,

g (x) = \sin [0.3 x_{2} (k) - 0.2 x_{1} (k)], h (x) = \sin [0.3 x_{1} (k) + 0.2 x_{2} (k)] .

Since

d = 2

holds, we have

θ_{2} = 1 - θ_{1}

, where

θ_{1} \in [0, 1]

. Thus, the uncertainty can be expressed by

θ_{1}

. In the simulation,

θ_{1}

is a random number between

[0, 0.5]

. The other parameters are displayed in example 1. According to Theorem 3, the corresponding control gains can be obtained as follows:

The gain matrices of PRC law with $M_{R} = 8$ are as follows:

K_{y} = - 0.48098, K_{v} = 0.84944,

\begin{aligned} K_{r} = [\begin{matrix} 0.62126 & 0.32955 & - 0.08252 & 0.04175 & 0.04223 & - 0.01947 \end{matrix} \\ \begin{matrix} 0.00362 & 0.00851 & - 0.00668 \end{matrix}] . \end{aligned}

The gain matrices of PRC law with

M_{R} = 2

K_{y} = - 0 .59141, K_{v} = 0 .86923, K_{R} = [\begin{matrix} 0.65414 & 0.36121 & - 0.19045 \end{matrix}] .

The gain matrices of RC law (i.e.

M_{R} = 0

) are as follows:

K_{x} = - 0 .46708, K_{v} = 0.75754 .

The system output (with PRC and RC) and

r (k)

are displayed in Figure 7, Figure 8 reveals the tracking error, and Figure 8 shows the control input. From Figures 7–9, the output tracks the reference signal faster with PRC. The proposed PRCR with

M_{R} = 8

exhibits superior performance to that the case with

M_{R} = 2

Figure 7.

Comparison of the output of system (1) with PRC with RC.

Figure 8.

Tracking error of system (1) with PRC with RC.

Figure 9.

Control input of system (1) with PRC with RC.

Furthermore, we investigated the effect of PRC and RC on the closed-loop system for signal (60). The simulation results are illustrated in Figures 10–12. Through the comparison with the RC (i.e. no preview), Figures 10 and 11 reveal that the output with PRC and RC can track the desired signal faster.

Figure 10.

Comparison of the output response between PRC and RC.

Figure 11.

Tracking error of system (1) between PRC and RC.

Figure 12.

Control input of system (1) between PRC and RC.

6. Conclusion

This paper presented a method of designing a PRC law for a class of uncertain nonlinear discrete-time systems. The design problem of PRC is converted into the stability analysis of a nonlinear discrete-time 2D system by using the augmented model approach. Next, the Lyapunov functions method was used to derive sufficient conditions for the existence of a PRC law such that the tracking error converges to zero. The effectiveness of the proposed method was verified using two illustrative examples.

In future, one of the important research topics is to extend the present result to the discrete-time descriptor systems, which is more challenging due to the special structure of descriptor systems. And the issue of the PRC based on the techniques of sliding mode will be interesting and of great significance.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China [61903130]; Key Scientific Research Project of Hubei Department of Education [D20192202]; Hubei University of Economics Research And Development Project [PYZD202005, PYYB202007]; Shandong Provincial Natural Science Foundation [ZR2020QA036].

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 Shandong Provincial Natural Science Foundation (grant number ZR2020QA036); the National Natural Science Foundation of China [61903130]; Key Scientific Research Project of Hubei Department of Education [D20192202]; Hubei University of Economics Research And Development Project [PYZD202005, PYYB202007].

ORCID iD

Chen Jia

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Preview repetitive control for polytopic nonlinear discrete-time systems

Abstract

Keywords

1 Introduction

2. Problem statement

3.1 Construction of the augmented 2d system

Numerical example

Example 1

Example 2

Footnotes

Acknowledgments

Declaration of Conflicting Interests

Funding

ORCID iD

References