Partial eigenstructure assignment for descriptor high-order linear systems via proportional plus derivative state feedback: A parametric approach

Abstract

In this paper, we present a parametric method to design proportional plus derivative (PD) state feedback for the problem of partial eigenstructure assignment (PESA) in a type of descriptor high-order linear time-invariant (LTI) systems. By dividing the original eigenstructure into replaced and remaining parts and using the solutions to the high-order generalized Sylvester equations (HGSEs), complete parameterized expressions of PD feedback controller and the replaced part of the right eigenvector matrix are established. With the proposed approach, the unsatisfactory eigenstructure in the open-loop system is changed into the expected eigenstructure in the closed-loop system, while the satisfactory part in the open-loop system is retained. Meanwhile, the regularity of the closed-loop system can also be well-guaranteed by the selection of arbitrary parameters. Finally, a numerical example and a flexible-joint robot system are presented to verify the feasibility and effectiveness of the proposed method, respectively.

Keywords

Partial eigenstructure assignment descriptor high-order LTI systems parametric approach generalized Sylvester equations (GSEs)

Introduction

Descriptor linear system, which is also called singular or generalized system, plays a significant role in the design of control systems and has been widely considered by scholars in the control field during the past three decades (Lewis, 1986; Zhou et al., 2013). In the practical world, descriptor linear system has wide physical backgrounds in practical engineering, such as power systems (Freitas et al., 2008), circuit systems (Reis, 2010), high-speed railway systems (Wu et al., 2017), bioeconomic systems (Zhang and Zhang, 2007), and other applications (Chen et al., 2009; Tavazoei and Haeri, 2010). When constructing a model of systems using some physical laws, such as Kirchhoff’s Law or Newton’s Law, a high-order model is obtained. Therefore, it is worthwhile and meaningful to study the control strategy of descriptor high-order systems.

The stability of a linear time-invariant (LTI) system depends on the eigenvalues of the closed-loop system and the performance characteristics of the system are determined by the eigenstructure (eigenvalues and eigenvectors) of the closed-loop system (Sobel et al., 1994; White, 1995). Hence, as an important means of control, eigenstructure assignment for normal systems (Li and Lam, 2016; Yu and Duan, 2009), descriptor systems (Duan and Patton, 1997), and descriptor quasi-linear systems (Gu and Zhang, 2020, 2021; Gu et al., 2019) have become the focus of many scholars and been applied in related practical fields (Wang et al., 2022). In addition, many other problems of control system design have been solved effectively by using eigenstructure assignment theory (Gu and Wang, 2022; Gu et al., 2022a; Padula et al., 2021).

The traditional eigenstructure assignment problem is to replace all the eigenvalues and eigenvectors of the closed-loop system, also known as “entire eigenstructure assignment.” However, in some cases, we only need to replace a part of the unsatisfied eigenstructure while keeping the rest in the open-loop system (Liu and Xu, 2017; Mao and Dai, 2013), thus the partial eigenstructure assignment problem (PESA) arises. The main superiority of PESA is that it can reduce computation such that the controller can be economically designed and the extra freedom can be used to meet the specific requirements of the system as well as enhance its performance. This problem has various applications in many actual systems. For example, in a second-order spring-damped system, it is only necessary to eliminate the undesirable eigenvalues (or eigenstructures) to avoid resonance (Yu, 2022). Besides, there are other applications of the PESA problem like flight control system design of aircraft model (Satoh and Sugimoto, 2004), which mostly have high-order or descriptor high-order form. In the problem of PESA of normal systems, some achievements have been made in the study of first-order, second-order, and high-order systems (Cai et al., 2012; Gu et al., 2021; Xu and Qian, 2008; Zhang et al., 2014, 2016). Recently, Yu et al. (2022a, 2022b, 2022c) have deeply studied the PESA problem of various control systems and proposed a controller calculation algorithm with the minimum norm and robust stability. Although the above achievements are fruitful, the research on descriptor systems is still relatively few. Duan and Wang (2003) proposed the complete parametric method for the PESA problem of first-order and descriptor systems by circulation algorithm, which provides all degrees of design freedom.

It can be seen that the above researchers are mostly aimed at normal low-order systems, but few results on descriptor systems, let alone on descriptor high-order systems. This is because compared with the normal system, the dimension of the state space of the system is greater than the dynamic order of the system, which indicates that the system is subject to linear constraints in the process of operation. Therefore, there are new problems different from the conventional linear problems, such as regularization problems, pulse elimination problems, and so on. Taken into account the above consideration, in this article, a parametric approach via proportional plus derivative (PD) state feedback, which can provide all the degree of the design freedom, is proposed to solve the descriptor high-order PESA problem. Compared with the traditional reduced-order method to the first-order system, which may destroy the properties of the system matrix such as positive definiteness and sparsity, we choose to establish a direct, concise, and unified parametric expression directly on the framework of the descriptor high-order system.

The main contribution of this paper is proposing an effective parametric design method for the PESA problem. The specific contribution can be stated from the following three aspects. First, we classify different eigenstructures and leave the satisfied eigenstructure in the open-loop system while replacing the unsatisfied part, and the stability of the closed-loop system is guaranteed. Second, under the premise of ensuring the regularity of the closed-loop system, simple, neat, and numerical stable parametric expressions of partial eigenvector and feedback gain matrices are established. Finally, the comprehensive performance of the closed-loop system is optimized by using the freedom provided by the free parameters in the parametric method.

The full paper is divided into six sections. The section “Problem formulation and preliminaries” formulates the PESA problem for the descriptor high-order LTI system and some preliminary results are put forward in this section. The general solution to the problem with the different cases of matrix $Λ$ is presented in the section “Solution to the PESA problem.” The section also gives a design algorithm to solve the PESA problem. Two illustrative examples are investigated in the section “Two illustrative examples” to demonstrate the effectiveness of the proposed methods, and the conclusions, as well as advantages of this method, follow in the “Conclusion” section.

Notation. We present some notation that will be used throughout this paper. $R^{n}$ represents set of all real vectors of dimension $n$ . $C^{n}$ represents set of all complex vectors of dimension $n$ . $R^{n \times m}$ denotes set of all real matrices of dimension $m \times n$ . $R^{n \times m} [s]$ denotes set of all polynomial matrices of dimension $n \times m$ with real coefficients. $I_{n}$ denotes the identity matrix with $n$ dimensions; $rank A$ , $\det A$ , and $eig A$ represent the rank, determinant, and all eigenvalues of the matrix $A$ , respectively; $\deg A (s)$ denotes the degree $n$ of polynomial matrix $A (s) = A_{0} + s A_{1} + \dots + s^{n} A_{n}$ ; and $diag {s_{1}, s_{2}, \dots, s_{n_{u}}}$ indicates the diagonal matrix with diagonal elements $s_{i}$ , $i = 1, 2, \dots, n_{u}$ .

Problem formulation and preliminaries

System description

This paper considers the following dynamic descriptor high-order linear system

\sum_{i = 0}^{m} A_{i} q^{(i)} = Bu

(1)

where $q \in R^{n}$ , $u \in R^{r}$ are the state vector and the control vector, respectively, and the matrices $A_{i} \in R^{n \times n}, i = 0, 1, \dots, m$ and $B \in R^{n \times r}$ are the coefficient matrices.

Assumption 1. $rank A_{m} = n_{0}, 0 < n_{0} \leq n, rank B = r$ .

Assumption 2. The system given by equation (1) is S-controllable.

According to the definition of S-controllability, the system given by equation (1) is called R-controllable and I-controllable. Under the R-controllability, we can obtain

rank [\begin{matrix} A (s) & B \end{matrix}] = n, \forall s \in C

(2)

where

A (s) = \sum_{i = 0}^{m} s^{i} A_{i}

(3)

Denote

x = {[q^{T} {\overset{\cdot}{q}}^{T} \dots {(q^{(m - 1)})}^{T}]}^{T}

(4)

then the open-loop system given by equation (1) can be rewritten into first-order form

E_{e} \overset{\cdot}{x} = A_{e} x + B_{e} u

(5)

where

E_{e} = blockdiag {I_{n}, \dots, I_{n}, A_{m}}

A_{e} = [\begin{matrix} 0 & I_{n} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & I_{n} \\ - A_{0} & - A_{1} & \dots & - A_{m - 1} \end{matrix}], B_{e} = [\begin{matrix} 0 \\ ⋮ \\ 0 \\ B \end{matrix}]

By applying the PD feedback control law

u = \sum_{i = 0}^{m - 1} F_{i} q^{(i)}

(6)

where $F_{i} \in R^{r \times n}, i = 0, 1, \dots, m - 1$ , are the PD feedback gain matrices to be designed in this paper, and the following closed-loop system form can be obtained

A_{m} q^{(m)} + \sum_{i = 0}^{m - 1} A_{i}^{c} q^{(i)} = 0

(7)

where

A_{i}^{c} = A_{i} - B F_{i}, i = 0, 1, \dots, m - 1

(8)

Then the closed-loop system given by equation (7) can be rewritten into the following first-order space form

E_{ec} \overset{\cdot}{x} = A_{ec} x

(9)

with

E_{ec} = blockdiag {I_{n}, \dots, I_{n}, A_{m}}

A_{ec} = [\begin{matrix} 0 & I_{n} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & I_{n} \\ - A_{0}^{c} & - A_{1}^{c} & \dots & - A_{m - 1}^{c} \end{matrix}]

Obviously, the stability and performance of the closed-loop system given by equation (9) mainly depend on the matrix pair ${E_{ec}, A_{ec}}$ .

Remark 1. Based on the results of the pole assignment under R-controllability and I-controllability for the first-order descriptor linear system, we can arbitrarily assign $n_{c} = (m - 1) n + n_{0}$ finite eigenvalues and guarantee the system to be impulse-free. This is to say that we only need to consider the PESA problem of finite eigenvalues and eigenvectors (Duan, 2010).

Before we discuss the PESA problem, first, we denote the finite eigenstructure part of the Jordan matrix $A_{e}$ of the open-loop system as below

Λ_{fe} = blockdiag {Λ_{0}, Λ_{u}} \in C^{n_{c} \times n_{c}}

(10a)

with

{\begin{matrix} Λ_{u} = diag {Λ_{1}^{u}, Λ_{2}^{u}, \dots, Λ_{q_{u}}^{u}} \in C^{n_{u} \times n_{u}} \\ Λ_{i}^{u} = {[\begin{matrix} s_{i}^{u} & 1 \\ s_{i}^{u} & ⋱ \\ ⋱ & 1 \\ s_{i}^{u} \end{matrix}]}_{(p_{i}^{u} \times p_{i}^{u})} \end{matrix}

(10b)

and

{\begin{matrix} Λ_{0} = diag {Λ_{1}^{0}, Λ_{2}^{0}, \dots, Λ_{q_{s}}^{0}} \in C^{n_{s} \times n_{s}} \\ Λ_{i}^{0} = {[\begin{matrix} s_{i}^{0} & 1 \\ s_{i}^{0} & ⋱ \\ ⋱ & 1 \\ s_{i}^{0} \end{matrix}]}_{(p_{i}^{0} \times p_{i}^{0})} \end{matrix}

(10c)

with

\sum_{i = 1}^{q_{u}} p_{i}^{u} = n_{u}, \sum_{i = 1}^{q_{s}} p_{i}^{0} = n_{s}, n_{u} + n_{s} = n_{c}

In this paper, matrices $Λ_{0}$ and $Λ_{u}$ represent the satisfactory and unsatisfactory eigenstructures. Meanwhile, $p_{i}^{0}, i = 1, 2, \dots, q_{s}$ and $p_{i}^{u}, i = 1, 2, \dots, q_{u}$ represent the orders of Jordan blocks corresponding to satisfactory and unsatisfactory eigenvalues among $s_{i}^{0}, i = 1, 2, \dots, n_{s}$ and $s_{i}^{u}, i = 1, 2, \dots, n_{u}$ .

Let the matrix $V_{f} \in C^{mn \times n_{c}}$ be a finite right eigenvector matrix of the matrix pair ${E_{e}, A_{e}}$ , then it can be partitioned into two parts

V_{f} = [\begin{matrix} V_{0} & V_{u} \end{matrix}]

(11)

where $V_{0} \in C^{mn \times n_{s}} and V_{u} \in C^{mn \times n_{u}}$ are both column full-rank matrices corresponding to satisfactory and unsatisfactory eigenvalues, respectively, and the following equations hold

{\begin{matrix} A_{e} V_{0} = E_{e} V_{0} Λ_{0} \\ A_{e} V_{u} = E_{e} V_{u} Λ_{u} \end{matrix}

(12)

In this paper, the goal is to retain the satisfactory eigenstructure $Λ_{0}$ as well as its corresponding matrix $V_{0}$ in the open-loop system. Conversely, the unsatisfactory part $V_{u}$ and $Λ_{u}$ will be replaced by the matrix $Λ \in C^{n_{u} \times n_{u}}$ and a full-column matrix $V_{al} \in C^{mn \times n_{u}}$ . Specifically, we let the reassigned eigenstructure part in the closed-loop matrix pair ${E_{ec}, A_{ec}}$ be similar to an arbitrary desired matrix $Λ \in C^{n_{u} \times n_{u}}$ .

To ensure the stability of the closed-loop system, $Λ$ is generally selected to be a Hurwitz matrix (Le and Wang, 2015; Wang and Zhang, 2017), that is

eig (Λ) \in C^{-} .

The right eigenvector matrix of the closed-loop system

Let the matrix $V_{al}$ be the matrix to be substituted for the matrix $V_{u}$ , and the matrix $V_{\infty}$ represent the right infinite eigenvector matrix of the matrix pair ${E_{ec}, A_{ec}}$ , then the following lemma is given.

Lemma 1. Let the matrix pair ${E_{ec}, A_{ec}}$ be given in equation (9). For the closed-loop matrix pair ${E_{ec}, A_{ec}}$ , there exists a right eigenvector matrix $V_{ec}$ that can be partitioned into the following two parts

V_{ec} = [\begin{matrix} V_{R} & V_{\infty} \end{matrix}], V_{R} \in C^{mn \times n_{c}}, V_{\infty} \in C^{mn \times (n - n_{0})}

(13)

and

\begin{matrix} V_{R} = [\begin{matrix} V_{al} & V_{0} \end{matrix}], V_{al} \in C^{mn \times n_{u}}, V_{0} \in C^{mn \times n_{s}} \\ V_{al} = {[\begin{matrix} V_{1}^{T} & V_{2}^{T} & \dots & V_{m}^{T} \end{matrix}]}^{T}, V_{i} \in C^{n \times n_{u}}, i = 1, 2, \dots, m \end{matrix}

(14)

satisfying

A_{ec} V_{al} = E_{ec} V_{al} Λ

(15)

E_{ec} V_{\infty} = 0

(16)

if and only if

A_{m} V Λ^{m} + \sum_{i = 0}^{m - 1} A_{i}^{c} V Λ^{i} = 0

(17)

and

{\begin{matrix} V_{1} = V, \\ V_{i} = V Λ^{i - 1}, i = 1, 2, \dots, m \end{matrix}

(18)

Proof. The proof is divided into two parts.

First, let

Γ = {s_{i} | s_{i} \in C, i = 1, 2, \dots, n_{u}}

which is symmetric about the real axis, be the set of unsatisfactory eigenvalues of the matrix pair ${E_{ec}, A_{ec}}$ . We denote algebraic and geometric multiplicities of $s_{i}$ by $h_{i}$ and $q_{i}$ , then in the Jordan form $Λ$ determined by the relative eigenvalues of the matrix pair ${E_{ec}, A_{ec}}$ , there are $q_{i}$ Jordan blocks associated with $s_{i}$ . Denote the orders of the $q_{i}$ number Jordan blocks associated with $s_{i}$ by $p_{ij}, j = 1, 2, \dots, q_{i}$ . Hence, the following relations hold

p_{i_{1}} + p_{i_{2}} + \dots + p_{i_{q_{i}}} = h_{i}

To eliminate the impulse response in the closed-loop system, the following requirement is necessary

h_{1} + h_{2} + \dots + h_{n_{f}} = n_{u}

Furthermore, denote the right eigenvector with respect to $s_{i}$ by $v_{ij}^{k} \in C^{n_{u}}, k = 1, 2, \dots, p_{ij}, j = 1, 2, \dots, h_{i}$ . By definition, we have

\begin{matrix} (A_{ec} - s_{i} E_{ec}) v_{ij}^{k} = E_{ec} v_{ij}^{k - 1}, v_{ij}^{0} = 0 \\ k = 1, 2, \dots, p_{ij}, j = 1, 2, \dots, q_{i} \end{matrix}

(19)

which can be written as equation (15). Then, taking $E_{ec}$ and $A_{ec}$ as equation (9) into the first part of equation (15), equations (17) and (18) can be proven easily. Hence, the first part of the proof is completed.

Second, $s_{\infty}$ represents the infinite eigenvalues of the closed-loop system with the algebraic and geometric multiplicity of $s_{\infty}$ being $n - n_{0}$ . Furthermore, the linearly independent eigenvectors corresponding to eigenvalue $s_{\infty}$ are represented as $v_{\infty_{j}}, j = 1, 2, \dots, n - n_{0}$ , which are defined as

E_{ec} v_{\infty_{j}} = 0, j = 1, 2, \dots, n - n_{0}

(20)

The above equation can be written in the form of equation (16). Thus, the second part of the proof is completed.

Finally, we complete the whole proof of this lemma.

Remark 2. Lemma 1 indicates that the replaced part Jordan matrix of the matrix pair ${E_{ec}, A_{ec}}$ is $Λ$ if and only if there exists a matrix $V \in C^{n \times n_{u}}$ satisfying equation (17). Therefore, the finite replaced part of the corresponding right eigenvector matrix of the matrix pair ${E_{ec}, A_{ec}}$ is given by

V_{al} = {[\begin{matrix} V^{T} & {(V Λ)}^{T} & \dots & {(V Λ^{m - 1})}^{T} \end{matrix}]}^{T}

(21)

Based on the above result, the entire right eigenvector matrix of the matrix pair ${E_{ec}, A_{ec}}$ can be divided into three parts as follows

V_{ec} = [\begin{matrix} V_{al} & V_{0} & V_{\infty} \end{matrix}] = {[\begin{matrix} V^{T} {(V Λ)}^{T} \dots {(V Λ^{m - 1})}^{T} \\ V_{0}^{T} \\ V_{\infty}^{T} \end{matrix}]}^{T}

(22)

Problem statement

After the above preliminary preparation, we can give the problem statement of PESA in descriptor high-order LTI systems via PD feedback.

Problem 1. Given the system shown by equation (1) satisfying Assumptions 1 and 2, the satisfactory eigenstructure ${Λ_{0}, V_{0}}$ as described previously satisfying equation (12), and a desired Jordan matrix $Λ \in C^{n_{u} \times n_{u}}$ . Find the partial right eigenvector matrix $V_{al} \in C^{mn \times n_{u}}$ and the PD feedback matrices $F_{i} \in R^{r \times n}, i = 0, 1, \dots, m - 1$ such that

A_{ec} V_{R} = E_{ec} V_{R} Λ_{R}

(23)

and

\det (s E_{ec} - A_{ec}) \neq 0

(24)

with

Λ_{R} = [\begin{matrix} Λ \\ Λ_{0} \end{matrix}]

(25)

Remark 3. The closed-loop system given by equation (9) is called regular if the relation shown by equation (24) holds, where $s \in C$ is a constant scalar. Regularity is an important property for descriptor LTI systems. Because it guarantees the existence and uniqueness of the solutions to this problem. Therefore, the relation shown by equation (24) is necessary and critical to solve problem PESA.

Preliminary results

Consider the following high-order generalized Sylvester equation (HGSE)

\sum_{i = 0}^{m} A_{i} V Λ^{i} = B W_{f}

(26)

where $A_{i} \in R^{n \times n}, i = 0, 1, \dots, m$ , $B \in R^{n \times r}$ and $Λ \in C^{n_{u} \times n_{u}}$ are the known coefficient matrices. The matrices $V \in C^{n \times n_{u}}$ and $W_{f} \in C^{r \times n_{u}}$ are the unknown matrices to be determined. The above Sylvester equation has a wide range of applications in the control theory (Gu et al., 2022b, 2022c; Heyouni et al., 2019).

Remark 4. The reason why the above Sylvester equation is proposed is that the PESA problem we need to solve in this paper is closely related to it, that is, equation (23) can be transformed into the Sylvester equation proposed by us through derivation. This process will be explained in detail in the next section.

Based on Assumption 2 and rank condition given by equation (2), there exists two polynomial matrices $N (s) \in R^{n \times r}$ , $D (s) \in R^{r \times r}$ , satisfying the following right co-prime factorization (RCF)

A (s) N (s) - BD (s) = 0

(27)

where $N (s)$ and $D (s)$ are called a pair of right co-prime polynomial matrices.

Denote $N (s) = [n_{ij} (s)]_{n \times r}$ , $D (s) = [d_{ij} (s)]_{r \times r}$ , and $μ = max {\deg (d_{ij} (s)), i = 1, 2, \dots, n, j = 1, 2, \dots, r}$ . Then the matrices $N (s), D (s)$ can be rewritten in the following form

{\begin{matrix} N (s) = \sum_{i = 0}^{μ} N_{i} s^{i}, N_{i} \in R^{n \times r} \\ D (s) = \sum_{i = 0}^{μ} D_{i} s^{i}, D_{i} \in R^{r \times r} \end{matrix}

(28)

Based on the above deduction, we give the following lemma for the parametric solution of HGSE given by equation (26).

Lemma 2. (Duan, 2015; Yu and Duan, 2011) Let $A_{i} \in R^{n \times n}, i = 0, 1, \dots, m$ , $B \in R^{n \times r}$ , $Λ \in C^{n_{u} \times n_{u}}$ and Assumptions 1 and 2 hold. Furthermore, let $N (s) \in R^{n \times r} [s]$ and $D (s) \in R^{r \times r} [s]$ be a pair of polynomial matrices in the form of equation (28) and satisfy RCF given by equation (27). Then all the solutions $V \in C^{n \times n_{u}}$ and $W_{f} \in C^{r \times n_{u}}$ to HGSE given by equation (26) can be obtained as

{\begin{matrix} V = \sum_{i = 0}^{μ} N_{i} Z Λ^{i} \\ W_{f} = \sum_{i = 0}^{μ} D_{i} Z Λ^{i} \end{matrix}

(29)

where $Z \in R^{r \times n_{u}}$ is an arbitrary parameter matrix.

Solution to the PESA problem

Case of $Λ$ arbitrary

With the above preliminary results, we give the following theorem for Problem 1.

Theorem 1. Let $N (s)$ and $D (s)$ be a pair of right polynomial matrices satisfying RCF given by equation (27).

(1) Problem 1 has a solution if and only if there exists an arbitrary parameter matrix $Z \in R^{r \times n_{u}}$ such that

Constraint 1. $\det V_{ec} (Z) \neq 0$ ,

where

V = \sum_{i = 0}^{μ} N_{i} Z Λ^{i}

(30)

V_{al} = \sum_{i = 0}^{μ} {[\begin{matrix} {(N_{i} Z Λ^{i})}^{T} & {(N_{i} Z Λ^{i + 1})}^{T} & \dots & {(N_{i} Z Λ^{i + m - 1})}^{T} \end{matrix}]}^{T}

(31)

and

V_{\infty} = {[\begin{matrix} 0_{(n - n_{0}) \times (m - 1) n}^{T} & {V'}_{\infty}^{T} \end{matrix}]}^{T}, A_{m} V'_{\infty} = 0

(32)

(2) When the above constraint is satisfied, all the parametric solutions of the PD feedback controller matrices $F_{i}, i = 0, 1, \dots, m - 1$ , can be obtained as

F = [\begin{matrix} F_{0} & F_{1} & \dots & F_{m - 1} \end{matrix}] = {WV}_{ec}^{- 1}

(33)

where

W = [\begin{matrix} W_{f} & 0 & W_{\infty} \end{matrix}]

(34)

$W_{f}$ is given by equation (29), $W_{\infty} \in R^{r \times (n - n_{0})}$ is an auxiliary matrix that satisfies

W_{\infty} = F V_{\infty}

(35)

where $V_{\infty}$ is given by equation (32).

Proof. The proof of the theorem consists of two steps.

Step 1: Obtain the parametric forms of the matrices $V$ , $W_{f}$ .

First, combine equation (8) with equation (17), then the following form can be easily obtained

\sum_{i = 0}^{m} A_{i} V Λ^{i} = B \sum_{i = 0}^{m - 1} F_{i} V Λ^{i}

(36)

Let

\begin{matrix} W_{f} = \sum_{i = 0}^{m - 1} F_{i} V Λ^{i} = F_{0} V + F_{1} V + \dots + F_{m - 1} V Λ^{m - 1} \\ = [\begin{matrix} F_{0} & F_{1} & \dots & F_{m - 1} \end{matrix}] [\begin{matrix} V \\ V Λ \\ ⋮ \\ V Λ^{m - 1} \end{matrix}] \\ = F V_{al} \end{matrix}

(37)

then equation (36) can be transformed into the HGSE form as follows

\sum_{i = 0}^{m} A_{i} V Λ^{i} = B W_{f}

(38)

According to Lemma 2, combining equation (21) with equation (29), we can easily get equation (31).

Second, considering equations (9) and (16), we can obtain

\begin{matrix} E_{ec} V_{\infty} = [\begin{matrix} I_{n} \\ ⋱ \\ I_{n} \\ A_{m} \end{matrix}] [\begin{matrix} V_{\infty}^{0} \\ ⋮ \\ V_{\infty}^{m - 2} \\ V_{\infty}^{m - 1} \end{matrix}] \\ = [\begin{matrix} V_{\infty}^{0} \\ ⋮ \\ V_{\infty}^{m - 2} \\ A_{m} V_{\infty}^{m - 1} \end{matrix}] = 0 \end{matrix}

(39)

which yields that $V_{\infty}^{i} = 0, i = 0, 1, \dots, m - 2$ . Denote $V_{\infty}^{m - 1} = V_{\infty}^{'}$ , then equation (32) can be proved easily. Therefore, we complete the proof of the first step.

Step 2: Derive the parametric solutions of the PD feedback controller matrix $F$ .

We introduce an auxiliary matrix as follows

W_{\infty} = F_{m - 1} V'_{\infty}

(40)

which can be written as

W_{\infty} = [\begin{matrix} F_{0} & F_{1} & \dots & F_{m - 1} \end{matrix}] {[\begin{matrix} 0^{T} & \dots & 0^{T} & {V'}_{\infty}^{T} \end{matrix}]}^{T} = F V_{\infty}

(41)

then equation (35) holds.

Besides, according to equations (8), (9), (12), and (23), we can obtain $B_{e} F V_{0} = 0$ , which is equivalent to

F V_{0} = 0

(42)

Therefore, based on equations (37), (41), and (42), we have

[\begin{matrix} W_{f} & 0 & W_{\infty} \end{matrix}] = F [\begin{matrix} V_{al} & V_{0} & V_{\infty} \end{matrix}]

(43)

Since Constraint 1 holds, it can be proved that

F = [\begin{matrix} W_{f} & 0 & W_{\infty} \end{matrix}] {[\begin{matrix} V_{al} & V_{0} & V_{\infty} \end{matrix}]}^{- 1}

(44)

With the above deduction, the proof of the second step is completed.

The proof is finished.

Case of $Λ$ diagonal

In many cases, we choose the matrix $Λ$ as a diagonal form since it is often encountered in many practical applications, that is

Λ = diag {s_{1}, s_{2}, \dots, s_{n_{u}}}

where $s_{i} \in C^{-}, i = 1, 2, \dots, n_{u}$ . Under this situation, the matrices $V$ , $W_{f}$ , and $V_{al}$ can be rewritten in following forms

{\begin{matrix} V = [\begin{matrix} v_{1} & v_{2} & \dots & v_{n_{u}} \end{matrix}] \\ v_{i} = N (s_{i}) z_{i}, i = 1, 2, \dots, n_{u} \end{matrix}

(45)

{\begin{matrix} W_{f} = [\begin{matrix} w_{1} & w_{2} & \dots & w_{n_{u}} \end{matrix}] \\ w_{i} = D (s_{i}) z_{i}, i = 1, 2, \dots, n_{u} \end{matrix}

(46)

and

{\begin{matrix} V_{al} = [\begin{matrix} v_{a l_{1}} & v_{a l_{2}} & \dots & v_{a l_{n_{u}}} \end{matrix}] \\ v_{a l_{i}} = \bar{N} (s_{i}) z_{i}, i = 1, 2, \dots, n_{u} \end{matrix}

(47)

with

\bar{N} (s_{i}) = {[\begin{matrix} I & s_{i} I & \dots & s_{i}^{m - 1} I \end{matrix}]}^{T} N (s_{i}), i = 1, 2, \dots, n_{u}

(48)

and

Z = [\begin{matrix} z_{1} & z_{2} & \dots & z_{n_{u}} \end{matrix}]

(49)

Therefore, based on the above preparation, we propose the following theorem regarding Problem 1.

Theorem 2. Let $N (s)$ and $D (s)$ be a pair of right polynomial matrices satisfying RCF (27).

(1) Problem 1 has a solution if and only if there exists a group of parameter vectors $z_{i} \in C^{r}, i = 1, 2, \dots, n_{u}$ , such that

Constraint 2. $\det V_{ec} (z_{i}, s_{i}) \neq 0, i = 1, 2, \dots, n_{u}$ .

Constraint 3. $z_{i} = {\bar{z}}_{j}$ if $s_{i} = {\bar{s}}_{j}$ , $i, j = 1, 2, \dots, n_{u}$ .

(2) When the above constraints are satisfied, all the parametric solutions of the PD feedback controller matrices can be obtained as equation (33), and the matrices $V_{al}$ in equation (31) and $V$ , $W_{f}$ in equation (29) can be parameterized by columns as shown in equations (45)–(48), and $z_{i} \in C^{r}$ are a group of parameter vectors satisfying Constraint 2.

Proof. When the matrix $Λ$ is chosen as a diagonal form, $V$ and $W_{f}$ in equation (29) and $V_{al}$ in equation (31) can be written into equations (45), (46), and (47), respectively (see reference Duan, 2015). Thus, we only need to prove that equation (48) holds.

According to the diagonal form of $Λ$ , combining equations (21) and (45), the $i$ th column of the matrix $V_{al}$ can be rewritten as

\begin{matrix} v_{a l_{i}} = [\begin{matrix} N (s_{i}) z_{i} \\ s_{i} N (s_{i}) z_{i} \\ ⋮ \\ s_{i}^{m - 1} N (s_{i}) z_{i} \end{matrix}] = [\begin{matrix} I \\ s_{i} I \\ ⋮ \\ s_{i}^{m - 1} I \end{matrix}] N (s_{i}) z_{i} \\ = \bar{N} (s_{i}) z_{i}, i = 1, 2, \dots, n_{u} \end{matrix}

(50)

Clearly, equation (48) holds. Therefore, according to Theorem 1, the results of this theorem can be easily proved.

Remark 5. Some researchers have directly considered solving the gain matrix $F$ based on equations (37) and (42). However, due to the non-squareness of the matrix $[\begin{matrix} V_{al} & V_{0} \end{matrix}]$ , its solutions seem to be much more complicated. Hence, by introducing an auxiliary equation (35), we obtain an extended equation (41) with a composite matrix $[\begin{matrix} V_{al} & V_{0} & V_{\infty} \end{matrix}]$ . Such a process is called “completing the square.” Besides, the matrix $W_{\infty} \in R^{r \times (n - n_{0})}$ can be regarded as an arbitrary parameter matrix that represents an extra degree of design freedom. Therefore, different $W_{\infty}$ and arbitrary parameter matrices $Z$ can be jointly selected to meet the control requirements of relevant performance indexes such as robustness.

Remark 6. In practical systems, the robustness of the system needs to be considered due to the existence of disturbances. There are arbitrary parameters in the parametric design method proposed in this paper, so these free parameters can be used to optimize the performance index of robustness to achieve the purpose of anti-interference. Therefore, we can optimize the following index

J (Z, V'_{\infty}) = ‖ V_{ec} ‖_{2} ‖ V_{ec}^{- 1} ‖_{2}

as small as possible (Duan and Yu, 2006).

Two illustrative examples

A numerical example

Consider a third-order descriptor system in the form of equation (1) with the following coefficient matrices

\begin{matrix} \begin{matrix} A_{3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], A_{2} = [\begin{matrix} 1 & 0 & 1 \\ 0 & 2 & - 1 \\ 1 & - 1 & 3 \end{matrix}], \\ A_{1} = [\begin{matrix} 2 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{matrix}], A_{0} = [\begin{matrix} 0 & - 1 & 0 \\ - 1 & 1 & 2 \\ 0 & 1 & 1 \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}] \end{matrix} \end{matrix}

For this system, we can easily obtain the finite part of the open-loop eigenvalues as

\begin{matrix} \begin{matrix} Γ_{o} = {- 1.115957, 0.651591, \\ - 0.157491 + 1.772247 i, - 0.157491 - 1.772247 i, \\ - 0.853901 + 1.229077 i, - 0.853901 - 1.229077 i, \\ - 0.089758 + 0.237899 i, - 0.089758 - 0.237899 i} \end{matrix} \end{matrix}

It is obvious that the eigenvalue $0.651591$ lies in the right-half $s$ plane, the open-loop system is unstable (see Figure 1). Moreover, two pairs of complex-conjugate poles ${- 0.157491 \pm 1.772247 i, - 0.089758 \pm 0.237899 i}$ are located close to the imaginary axis. Then, we design the following PD control law

u = F_{0} q + F_{1} \overset{\cdot}{q} + F_{2} \overset{\cdot\cdot}{q}

such that the above unsatisfactory eigenvalues can be replaced by $s_{1} = - 1$ , $s_{2} = {\bar{s}}_{3} = - 2 + i$ , $s_{4} = {\bar{s}}_{5} = - 3 + i$ , respectively, and the rest of the eigenvalues are retained.

Choose the diagonal matrix $Λ$ with the desired values

Λ = diag {- 1, - 2 + i, - 2 - i, - 3 + i, - 3 - i}

The retained eigenvector part $V_{0}$ in equation (11) can be given as

V_{0} = [\begin{matrix} v_{3} & v_{4} & v_{5} \end{matrix}]

(51)

where

\begin{matrix} v_{3} = {\bar{v}}_{4} = [\begin{matrix} - 0.169512 + 0.310000 i \\ - 0.189776 + 0.344090 i \\ - 0.178106 + 0.080378 i \\ - 0.236268 - 0.473053 i \\ - 0.260863 - 0.527067 i \\ 0.053294 - 0.287540 i \\ 0.783168 + 0.113550 i \\ 0.870558 + 0.129442 i \\ 0.307901 + 0.311033 i \end{matrix}], v_{5} = [\begin{matrix} 0.042312 \\ - 0.802980 \\ - 0.564041 \\ - 0.047218 \\ 0.896092 \\ 0.629445 \\ 0.052693 \\ - 1.000000 \\ - 0.702434 \end{matrix}] \end{matrix}

With the coefficient matrices $A_{i}, i = 0, 1, \dots, 3$ , we have

A (s) = [\begin{matrix} s^{3} + s^{2} + 2 s & - 1 & s^{2} \\ - 1 & s^{3} + 2 s^{2} + s + 1 & - s^{2} + 2 s + 2 \\ s^{2} & - s^{2} + 2 s + 1 & 3 s^{2} + s + 1 \end{matrix}]

and

rank A_{3} = 2, rank [\begin{matrix} A (s) & B \end{matrix}] = 3, \forall s \in C

Therefore, Assumptions 1 and 2 hold. It can be verified that the above system is R-controllable and I-controllable.

Furthermore, a pair of $N (s)$ and $D (s)$ satisfying RCF given by equation (27) can be obtained as

\begin{matrix} {\begin{matrix} N (s) = [\begin{matrix} s^{3} + 2 s^{2} + s + 1 & - s^{2} + 2 s + 2 \\ 1 & 0 \\ 0 & 1 \end{matrix}] \\ D (s) = [\begin{matrix} d_{11} (s) & d_{12} (s) \\ d_{21} (s) & d_{22} (s) \end{matrix}] \end{matrix} \end{matrix}

with

\begin{matrix} {\begin{matrix} d_{11} (s) = s^{6} + 3 s^{5} + 5 s^{4} + 6 s^{3} + 3 s^{2} + 2 s - 1 \\ d_{12} (s) = - s^{5} + s^{4} + 2 s^{3} + 7 s^{2} + 4 s \\ d_{21} (s) = s^{5} + 2 s^{4} + s^{3} + 2 s + 1 \\ d_{22} (s) = - s^{4} + 2 s^{3} + 5 s^{2} + s + 1 \end{matrix} \end{matrix}

The design parameters $z_{i} \in C^{2}, i = 1, 2, \dots, 5$ , $V'_{\infty}$ and $W_{\infty}$ , are required to satisfy Constraints 2 and 3.

Non-optimized solution

Simply choose the free parameters as follows

\begin{matrix} z_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}], z_{2} = {\bar{z}}_{3} = [\begin{matrix} 1 + i \\ 1 - i \end{matrix}], z_{4} = {\bar{z}}_{5} = [\begin{matrix} 2 + i \\ 2 - i \end{matrix}], \\ {V^{'}}_{\infty} = {[\begin{matrix} 0 & 0 & 3 \end{matrix}]}^{T}, W_{\infty} = {[\begin{matrix} 1 & 2 \end{matrix}]}^{T}, \end{matrix}

(52)

It yields the following particular solutions

\begin{matrix} \begin{matrix} V = [\begin{matrix} 1 & 18 i & - 18 i & - 39 - 54 i & - 39 + 54 i \\ 1 & 1 + i & 1 - i & 2 + i & 2 - i \\ 0 & 1 - i & 1 + i & 2 - i & 2 + i \end{matrix}] \\ W_{f} = [\begin{matrix} - 3 & - 164 - 62 i & - 164 + 62 i & - 556 - 1743 i & - 556 + 1743 i \\ - 1 & 57 + 35 i & 57 - 35 i & 5 + 613 i & 5 - 613 i \end{matrix}] \\ V_{al} = [\begin{matrix} 1 & 18 i & - 18 i & - 39 + 54 i & - 39 - 54 i \\ 1 & 1 + i & 1 - i & 2 + i & 2 - i \\ 0 & 1 - i & 1 + i & 2 - i & 2 + i \\ - 1 & - 18 - 36 i & - 18 + 36 i & 63 - 201 i & 63 + 201 i \\ - 1 & - 3 - i & - 3 + i & - 7 - i & - 7 + i \\ 0 & - 1 + 3 i & - 1 - 3 i & - 5 + 5 i & - 5 - 5 i \\ 1 & 72 + 54 i & 72 - 54 i & 12 + 666 i & 12 - 666 i \\ 1 & 7 - i & 7 + i & 22 - 4 i & 22 + 4 i \\ 0 & - 1 - 7 i & - 1 + 7 i & 10 - 20 i & 10 + 20 i \end{matrix}] \end{matrix} \end{matrix}

and

V_{\infty} = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \end{matrix}]}^{T}

Then, the PD feedback gain matrices can be obtained as

\begin{matrix} \begin{matrix} F_{0} = [\begin{matrix} - 14.585176 & 37.322256 & - 47.118978 \\ 2.709735 & 2.898170 & - 6.944050 \end{matrix}] \\ F_{1} = [\begin{matrix} - 19.342302 & 53.536512 & - 45.998244 \\ 0.921505 & 8.910472 & - 11.288694 \end{matrix}] \\ F_{2} = [\begin{matrix} - 6.871077 & 15.328207 & 0.333333 \\ 1.094982 & 2.129089 & 0.666666 \end{matrix}] \end{matrix} \end{matrix}

With the above controller, the closed-loop system in equation (7) can be obtained as

\begin{array}{l} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] \overset{⃛}{q} + [\begin{matrix} 7.871077 & - 15.328207 & 0.666667 \\ 0 & 2 & - 1 \\ - 0.094982 & - 3.129090 & 2.333333 \end{matrix}] \overset{\cdot\cdot}{q} \\ + [\begin{matrix} 21.342302 & - 53.536512 & 45.9982440 \\ 0 & 1 & 2 \\ - 0.921505 & - 6.910472 & 12.288694 \end{matrix}] \dot{q} \\ + [\begin{matrix} 14.585176 & - 38.322256 & 47.118978 \\ - 1 & 1 & 2 \\ - 2.709734 & 1.898170 & 7.944050 \end{matrix}] q = 0 . \end{array}

Denoting the non-optimized index $J$ as $J_{u}$ , we can obtain $J_{u} = 1.060721 \times 10^{4}$ . The closed-loop eigenvalues are assigned to

\begin{matrix} \begin{matrix} Γ_{c_{1}} = {- 1.115957, - 0.999999, \\ - 1.999999 + 1.000000 i, - 2.000000 - 0.999999 i, \\ - 0.853901 + 1.229077 i, - 0.853901 - 1.229077 i, \\ - 3.000000 + 1.000000 i, - 2.999999 - 1.000000 i} \end{matrix} \end{matrix}

Optimized solution

Consider the optimization index in Remark 6, under the condition of keeping $W_{\infty}$ constant and choosing the initial values in equation (52), the optimized parameters can be obtained as by the fminsearch function in MATLAB Optimization Toolbox^® as follows

\begin{matrix} \begin{matrix} z_{1} = [\begin{matrix} 1.300570 \\ 0.001374 \end{matrix}], z_{2} = {\bar{z}}_{3} = [\begin{matrix} 0.239038 + 0.239038 i \\ - 0.161589 - 0.161589 i \end{matrix}], \\ z_{3} = {\bar{z}}_{4} = [\begin{matrix} 0.036716 + 0.018358 i \\ 0.051209 - 0.025605 i \end{matrix}], V'_{\infty} = {[\begin{matrix} 0 & 0 & 5.812714 \end{matrix}]}^{T} \end{matrix} \end{matrix}

and it yields the following optimized solutions

V = [\begin{matrix} 1.2991 & - 0.4006 - 0.1042 i & - 0.4006 + 0.1042 i & - 0.8319 + 1.1942 i & - 0.8319 - 1.1942 i \\ 1.3005 & 0.2390 + 0.2390 i & 0.2390 - 0.2390 i & 0.0367 + 0.0183 i & 0.0367 - 0.0183 i \\ 0.0013 & - 0.1615 + 0.1615 i & - 0.1615 - 0.1615 i & 0.0512 - 0.0256 i & 0.0512 + 0.0256 i \end{matrix}],

W_{f} = [\begin{matrix} - 3.8975 & 2.0622 - 2.4009 i & 2.0622 + 2.4009 i & - 12.7434 - 37.9404 i & - 12.7434 + 37.9404 i \\ - 1.2978 & - 4.0024 + 4.3600 i & - 4.0024 - 4.3600 i & 0.5772 + 13.1666 i & 0.5772 - 13.1666 i \end{matrix}],

V_{al} = [\begin{matrix} 1.2991 & - 0.4006 - 0.1042 i & - 0.4006 + 0.1042 i & - 0.8319 + 1.1942 i & - 0.8319 - 1.1942 i \\ 1.3005 & 0.2390 + 0.2390 i & 0.2390 - 0.2390 i & 0.0367 + 0.0183 i & 0.0367 - 0.0183 i \\ 0.0013 & - 0.1615 - 0.1615 i & - 0.1615 + 0.1615 i & 0.0512 - 0.0256 i & 0.0512 + 0.0256 i \\ - 1.2991 & 0.9054 - 0.1922 i & 0.9054 + 0.1922 i & 1.3014 - 4.4146 i & 1.3014 + 4.4146 i \\ - 1.3005 & - 0.7171 - 0.2390 i & - 0.7171 + 0.2390 i & - 0.1285 - 0.0183 i & - 0.1285 + 0.0183 i \\ - 0.0013 & 0.1615 - 0.4847 i & 0.1615 + 0.4847 i & - 0.1280 + 0.1280 i & - 0.1280 - 0.1280 i \\ 1.2991 & - 1.6187 + 1.2898 i & - 1.6187 - 1.2898 i & 0.5101 + 14.5454 i & 0.5101 - 14.5454 i \\ 1.30056 & 1.6732 - 0.2390 i & 1.6732 - 0.2390 i & 0.4038 - 0.0734 i & 0.4038 + 0.0734 i \\ 0.0013 & 0.1615 + 1.1311 i & 0.1615 - 1.1311 i & 0.2560 - 0.5120 i & 0.2560 + 0.5120 i \end{matrix}],

and

V_{\infty} = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5.812713 \end{matrix}]}^{T}

Then, the optimized PD feedback gain matrices can be obtained as

\begin{matrix} \begin{matrix} F_{0} = [\begin{matrix} - 42.391933 & 34.964804 & - 16.000946 \\ - 8.353998 & 10.906570 & - 9.590059 \end{matrix}] \\ F_{1} = [\begin{matrix} - 31.618484 & 19.144955 & 5.643118 \\ - 4.777658 & 8.706741 & - 6.203333 \end{matrix}] \\ F_{2} = [\begin{matrix} - 8.812141 & 0.770902 & 0.172037 \\ 0.189064 & 0.189195 & 0.344073 \end{matrix}] \end{matrix} \end{matrix}

With the above controller, the closed-loop system in equation (7) can be obtained as

\begin{matrix} \begin{matrix} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] \overset{\cdot\cdot\cdot}{q} + [\begin{matrix} 9.812141 & - 0.770902 & 0.827963 \\ 0 & 2 & - 1 \\ 0.810936 & - 1.189195 & 2.655927 \end{matrix}] \overset{\cdot\cdot}{q} \\ + [\begin{matrix} 33.618484 & - 19.144955 & - 5.643118 \\ 0 & 1 & 2 \\ 4.777658 & - 6.706741 & 7.203333 \end{matrix}] \overset{\cdot}{q} \\ + [\begin{matrix} 42.391933 & - 35.964804 & 16.0009460 \\ - 1 & 1 & 2 \\ 8.353998 & - 9.906570 & 10.590059 \end{matrix}] q = 0 \end{matrix} \end{matrix}

Denote the optimized index $J$ as $J_{o}$ . Under this condition, the index is $J_{o} = 4.256105 \times 10^{2}$ . The closed-loop eigenvalues are assigned to

\begin{matrix} \begin{matrix} Γ_{c_{1}} = {- 1.115957, - 0.999999, \\ - 2.000000 + 1.000000 i, - 2.000000 - 1.000000 i, \\ - 0.853901 + 1.229077 i, - 0.853901 - 1.229077 i, \\ - 2.999999 + 1.000000 i, - 3.000000 - 0.999999 i} \end{matrix} \end{matrix}

Simulation and comparison

To more intuitively show the effectiveness of the parametric method proposed in this article, the comparative simulation between the non-optimized solution and optimized solution is carried out in the MATLAB environment.

Choose the initial value as

{\begin{matrix} q_{0} = {[\begin{matrix} 0.1 & 0.2 & 0.3 \end{matrix}]}^{T} \\ {\overset{\cdot}{q}}_{0} = {[\begin{matrix} - 0.1 & - 0.2 & 0.3 \end{matrix}]}^{T} \\ {\overset{\cdot\cdot}{q}}_{0} = {[\begin{matrix} 0.1 & - 0.2 & 0.3 \end{matrix}]}^{T} \end{matrix}

then, the simulation results are as shown in Figures 1 to 5 and Table 2.

Table 1.

A design algorithm for problem 1.

Design algorithm PESA for descriptor high-order linear systems

Step 1: Divide the right finite eigenvector matrix

V_{f}

of the matrix

A_{e}

into two parts satisfying equation (12).

Step 2: Choose a Hurwitz matrix

Λ

in equation (25) with desired eigenstructure.

Step 3: Find a pair of polynomial matrices

N (s)

and

D (s)

satisfying RCF given by equation (27).

Step 4: Select a group of parameters

z_{i}, i = 1, 2, \dots, n_{u}

, satisfying Constraints 2 and 3.

Step 5: Based on Remark 6, form an optimization problem to find the proper parameters

Z

and

V'_{\infty}

Step 6: Compute the matrices

V

W_{f}

, and

V_{al}

according to equations (29) and (31) or equations (45)–(48).

Step 7: Obtain the feedback gain matrix

F

according to equation (33) based on the solutions of Step 6.

Table 2.

Comparison of rapidity between two solutions.

Average error	$q < 10^{- 3}$	$\overset{\cdot}{q} < 10^{- 3}$	$\overset{\cdot\cdot}{q} < 10^{- 3}$
Optimized solution (in seconds)	8.627	9.214	9.186
Non-optimized solution (in seconds)	40.863	37.957	30.206

It can be observed from the three diagrams as shown in Figures 3 –5 that the state of each variable can tend to zero in a short time, which indicates that the closed-loop system eventually tends to be stable compared to the open-loop system in Figure 1. Meanwhile, the optimized solutions reduce the amplitude of oscillation and have a faster convergence time than non-optimized solutions. Besides, from Figure 2, we see that all the control input curves can keep up with the change of the response curve below and the optimized solution are less than that of the non-optimized solution, which illustrates that the optimized solution leads to a better control performance at the cost of less energy. Therefore, after the above analysis, the parametric approach we proposed in this paper is considered to be effective.

A flexible-joint robot system

Consider a flexible-joint mechanism shown in Figure 6. In some cases, when the rotating rate is high enough, the dynamical equation for this flexible-joint mechanism needs to be modeled by the following descriptor third-order system (Duan, 2015; Spong and Vidyasagar, 2008)

A_{3} \overset{\cdot\cdot\cdot}{q} + A_{2} \overset{\cdot\cdot}{q} + A_{1} \overset{\cdot}{q} + A_{0} = Bv

(53)

with

\begin{matrix} A_{3} = [\begin{matrix} 0 & 0 \\ 0 & J \end{matrix}], A_{2} = [\begin{matrix} I & 0 \\ 0 & D_{2} - a_{0} J \end{matrix}], A_{1} = [\begin{matrix} D_{1} & 0 \\ - k & k - a_{0} D_{2} \end{matrix}], \\ A_{0} = [\begin{matrix} k & - k \\ a_{0} k & - a_{0} k \end{matrix}], B = [\begin{matrix} 0 \\ 1 \end{matrix}] \end{matrix}

When the system parameters are chosen as

{\begin{matrix} I = J = 0.0004 Nm s^{2} / rad \\ D_{1} = 0.0 Nms / rad \\ D_{2} = 0.015 Nms / rad \\ k = 0.8 Nm / rad \\ a_{0} = - 0.95 \end{matrix}

we have

\begin{matrix} A_{3} = [\begin{matrix} 0 & 0 \\ 0 & 0.0004 \end{matrix}], A_{2} {= 10}^{- 3} \times [\begin{matrix} 0.4 & 0 \\ 0 & 15.38 \end{matrix}], \\ A_{1} = [\begin{matrix} 0 & 0 \\ - 0.8 & 0.81425 \end{matrix}], A_{0} = [\begin{matrix} 0.8 & - 0.8 \\ - 0.76 & 0.76 \end{matrix}], B = [\begin{matrix} 0 \\ 1 \end{matrix}] \end{matrix}

The finite eigenvalues of the open-loop system can be easily given by

\begin{matrix} Γ_{o} = {0, - 0.949999, - 20.538733, \\ - 8.480633 + 59.830727 i, - 8.480633 - 59.830727 i} \end{matrix}

It can be seen that the eigenvalue $0$ is on the imaginary axis and $- 0.949999$ lies close to the imaginary axis, which may cause the system to produce constant amplitude oscillation and ultimately affect the stability of the system. Therefore, based on the above deduction, we assign the eigenvalues ${0, - 0.949999}$ to $s_{1} = - 1, s_{2} = - 2$ , respectively, while retaining other eigenvalues in the open-loop system. Then, we design the following PD control law

u = F_{0} q + F_{1} \overset{\cdot}{q} + F_{2} \overset{\cdot\cdot}{q}

Choose the diagonal matrix $Λ$ with the desired values

Λ = diag {- 1, - 2}

The retained eigenvector part $V_{0}$ in equation (11) can be given as

V_{0} = [\begin{matrix} v_{1} & v_{2} & v_{3} \end{matrix}]

where

\begin{matrix} v_{1} = {\bar{v}}_{2} = 10^{- 4} \times [\begin{matrix} 1.935702 - 1.14410 i \\ - 2.039841 - 0.119645 i \\ 52.036463 + 125.517169 i \\ 24.457567 - 121.030496 i \\ - 7951.085686 + 2048.914314 i \\ 7033.926924 + 2489.729276 i \end{matrix}], \\ v_{3} = [\begin{matrix} 0.001958 \\ 0.002371 \\ - 0.040208 \\ - 0.048688 \\ 0.825819 \\ 1.000000 \end{matrix}] \end{matrix}

With the coefficient matrices $A_{i}, i = 0, 1, \dots, 3$ , we have

A (s) = 10^{- 3} \times [\begin{matrix} 0.4 s^{2} + 800 & - 800 \\ - 800 s - 760 & - 0.4 s^{3} + 15.38 s^{2} + 814.25 s + 760 \end{matrix}]

(54)

and

rank A_{3} = 1, rank [\begin{matrix} A (s) & B \end{matrix}] = 2, \forall s \in C

Therefore, Assumptions 1 and 2 hold. It can be verified that the above system is R-controllable and I-controllable.

Furthermore, a pair of $N (s)$ and $D (s)$ satisfying RCF given by equation (27) can be obtained as

{\begin{matrix} N (s) = [\begin{matrix} 2000 \\ s^{2} + 2000 \end{matrix}] \\ D (s) = s (0.0004 s^{4} + 0.01538 s^{3} + 1.6143 s^{2} + 31.52 s + 28.5) \end{matrix}

Thus, according to Theorem 2, the general solution to the HGSE is given by

v_{i} = [\begin{matrix} 2000 \\ s_{i}^{2} + 2000 \end{matrix}] z_{i}, i = 1, 2

\begin{matrix} w_{i} = s_{i} (0.0004 s_{i}^{4} + 0.01538 s_{i}^{3} + 1.6143 s_{i}^{2} + 31.52 s_{i} + 28.5) z_{i}, \\ i = 1, 2 \end{matrix}

and

\begin{matrix} v_{al}^{i} = [\begin{matrix} 2000 \\ s_{i}^{2} + 2000 \\ 2000 s_{i} \\ s_{i} (s_{i}^{2} + 2000) \\ 2000 s_{i}^{2} \\ s_{i}^{2} (s_{i}^{2} + 2000) \end{matrix}] z_{i}, i = 1, 2 \end{matrix}

Non-optimized solution

Particularly choosing the free parameters as follows

z_{1} = z_{2} = 1, V'_{\infty} = {[\begin{matrix} 4 & 0 \end{matrix}]}^{T}, W_{\infty} = 2

(55)

yields the following particular solutions

\begin{matrix} \begin{matrix} V = [\begin{matrix} 2000 & 2000 \\ 2001 & 2004 \end{matrix}], W_{f} = [\begin{matrix} 1.420680 & 56.408480 \end{matrix}], \\ V_{al} = [\begin{matrix} 2000 & 2000 \\ 2001 & 2004 \\ - 2000 & - 4000 \\ - 2001 & - 4008 \\ 2000 & 8000 \\ 2001 & 8016 \end{matrix}], V_{\infty} = {[\begin{matrix} 0 & 0 & 0 & 0 & 4 & 0 \end{matrix}]}^{T} \end{matrix} \end{matrix}

Then, the PD feedback gain matrices can be obtained as

\begin{matrix} F_{0} {= 10}^{3} \times [\begin{matrix} 1.001640 & - 1.001670 \end{matrix}] \\ F_{1} {= 10}^{- 4} \times [\begin{matrix} - 8.001384 & - 315.553032 \end{matrix}] \\ F_{2} {= 10}^{- 4} \times [\begin{matrix} 5000.000000 & - 8.201377 \end{matrix}] \end{matrix}

With the above controller, the closed-loop system in equation (7) can be obtained as

\begin{matrix} \begin{matrix} [\begin{matrix} 0 & 0 \\ 0 & 0.0004 \end{matrix}] \overset{\cdot\cdot\cdot}{q} + [\begin{matrix} 0.0004 & 0 \\ - 0.5 & 0.0162 \end{matrix}] \overset{\cdot\cdot}{q} + [\begin{matrix} 0 & 0 \\ - 0.799199 & 0.845805 \end{matrix}] \overset{\cdot}{q} \\ + 10^{3} \times [\begin{matrix} 0.0008 & - 0.0008 \\ - 1.0024 & 1.00243 \end{matrix}] q = 0 \end{matrix} \end{matrix}

Denote the non-optimized index $J$ as $J_{u}$ . In this case, it can be easily obtained that $J_{u} = 3.3759 \times 10^{7}$ , and the closed-loop system eigenvalues are assigned to

\begin{matrix} Γ_{c_{1}} = {- 1.00000176, - 2.00003425, - 20.538733, \\ - 8.480633 + 59.830727 i, - 8.480633 - 59.830727 i} \end{matrix}

Optimized solution

Similarly, consider the optimization index in Remark 6. Choosing the initial parameter values in equation (55) and keeping $W_{\infty}$ constant, using the fminsearch function in MATLAB Optimization Toolbox^® to optimize the index $J$ , we can get a group of parameters as follows

z_{1} = 10^{- 4} \times 1.051470, z_{2} = 10^{- 4} \times 0.540298, V'_{\infty} = {[\begin{matrix} 1.306054 & 0 \end{matrix}]}^{T}

and the corresponding optimized solutions

\begin{matrix} \begin{matrix} V = [\begin{matrix} 0.203094 & 0.108059 \\ 0.203195 & 0.108275 \end{matrix}], W_{f} = [\begin{matrix} 0.000144 & 0.003047 \end{matrix}], \\ V_{al} = [\begin{matrix} 0.203094 & 0.108059 \\ 0.203195 & 0.108275 \\ - 0.203094 & - 0.216119 \\ - 0.203195 & - 0.216551 \\ 0.203094 & 0.432238 \\ 0.203195 & 0.433102 \end{matrix}], V_{\infty} = {[\begin{matrix} 0 & 0 & 0 & 0 & 1.306053 & 0 \end{matrix}]}^{T} \end{matrix} \end{matrix}

Then, the optimized PD feedback gain matrices can be obtained as

\begin{matrix} F_{0} {= 10}^{3} \times [\begin{matrix} 3.064302 & - 3.064332 \end{matrix}] \\ F_{1} {= 10}^{- 4} \times [\begin{matrix} - 8.001384 & - 315.553032 \end{matrix}] \\ F_{2} {= 10}^{- 4} \times [\begin{matrix} 15313.307083 & - 8.201377 \end{matrix}] \end{matrix}

With the above controller, the closed-loop system in equation (7) can be obtained as

\begin{matrix} \begin{matrix} [\begin{matrix} 0 & 0 \\ 0 & 0.0004 \end{matrix}] \overset{\cdot\cdot\cdot}{q} + [\begin{matrix} 0.0004 & 0 \\ - 1.531331 & 0.0162 \end{matrix}] \overset{\cdot\cdot}{q} & + [\begin{matrix} 0 & 0 \\ - 0.799199 & 0.845805 \end{matrix}] \overset{\cdot}{q} \end{matrix} \\ + 10^{3} \times [\begin{matrix} 0.0008 & - 0.0008 \\ - 3.065062 & 3.065092 \end{matrix}] q = 0 \end{matrix}

Denote the optimized index $J$ as $J_{o}$ . Under this circumstance, the optimized index $J_{o} = 6.6365 \times 10^{3}$ . The closed-loop system eigenvalues are assigned to

\begin{matrix} \begin{matrix} Γ_{c_{2}} = {- 1.00000176, - 2.00003425, - 20.538733, \\ - 8.480633 + 59.830727 i, - 8.480633 - 59.830727 i} \end{matrix} \end{matrix}

Based on the above optimization process, we have

J_{o} = 6.6365 \times 10^{3}, J_{u} = 3.3759 \times 10^{7}

$J_{o} < J_{u}$ , which means that the robustness of the system is improved effectively through the optimization based on the data results. To more intuitively reflect this point of view, the simulation results and comparison will be given in the next chapter.

Simulation and comparison

Choosing the initial value as

{\begin{matrix} q_{0} = {[\begin{matrix} - 1 & 0.5 \end{matrix}]}^{T} m \\ {\overset{\cdot}{q}}_{0} = {[\begin{matrix} 1 & - 1.5 \end{matrix}]}^{T} m / s \\ {\overset{\cdot\cdot}{q}}_{0} = {[\begin{matrix} - 1.5 & - 1 \end{matrix}]}^{T} m / s^{2} \end{matrix}

we have the following as shown in Figures 7 –10.

Based on the above results, it is not difficult to draw the following conclusions. First, compared with the open-loop system, the state variables $q_{1} (t)$ and $q_{2} (t)$ in the closed-loop system are eventually stable after PESA. Meanwhile, the remaining state variables in the closed-loop system are still stable whether the optimization is carried out or not. Second, compared with the non-optimized solution and the open-loop system, the optimized solution reduces the amplitude of oscillation and has a faster convergence speed to a certain extent (Table 3). Finally, the control input in the optimized case leads to a better control performance at the cost of less energy than others. It can be seen that the parametric method we proposed is very feasible and effective.

Table 3.

Comparison of rapidity between three solutions.

Average error	$q < 10^{- 2}$	$\overset{\cdot}{q} < 10^{- 2}$	$\overset{\cdot\cdot}{q} < 10^{- 2}$
Optimized solution (in seconds)	6.33	6.531	5.682
Non-optimized solution (in seconds)	9.031	9.044	9.02
Open-loop system (in seconds)	$\infty$	9.495	9.446

Conclusion

In this paper, the problem of PESA for a type of descriptor high-order LTI systems is investigated. Based on the S-controllability condition, a complete parametric approach via PD state feedback is presented. Specifically, we establish a direct and closed general parametric expression of the partial right closed-loop eigenvector associated with the new assigned finite closed-loop eigenvalues and give a simple parametric solution of the state feedback gain matrices. The proposed method possesses several advantages:

The unsatisfactory finite closed-loop eigenvalues with any given geometric and algebraic multiplicities can be arbitrarily assigned, so all possible initial time impulse responses can be eliminated.

It is not required that the open-loop system is regular, but the regularity of the closed-loop system can be well-guaranteed.

There are no restrictions on the closed-loop finite eigenvalues, such as the dissimilarity between the closed-loop finite eigenvalues. More importantly, it can well-keep the eigenvalues that do not need to be assigned in the open-loop system.

All design degrees of freedom can be provided. These degrees of freedom are represented by a group of parameter vectors and a parameter matrix $Z$ . It is very simple, convenient, and has good numerical stability, which is the core advantage of parameterized methods.

The future research can be carried out in the following two aspects. First, the normalization of closed-loop systems needs to be further considered. Second, when the state vectors cannot be measured directly, the PESA problem can be dealt with through static output feedback or dynamic compensator.

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 Scientific Research Foundation for the Doctor of Northeast Electric Power University in China under Grant No. BSJXM-2022210 and also by the Science Center Program of the National Natural Science Foundation of China under Grant No. 62188101.

ORCID iDs

Da-Ke Gu

Yin-Dong Liu

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Partial eigenstructure assignment for descriptor high-order linear systems via proportional plus derivative state feedback: A parametric approach

Abstract

Keywords

Introduction

Problem formulation and preliminaries

System description

The right eigenvector matrix of the closed-loop system

Problem statement

Preliminary results

Solution to the PESA problem

Case of Λ arbitrary

Case of Λ diagonal

Two illustrative examples

A numerical example

Non-optimized solution

Optimized solution

Simulation and comparison

A flexible-joint robot system

Non-optimized solution

Optimized solution

Simulation and comparison

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Case of $Λ$ arbitrary

Case of $Λ$ diagonal