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Abstract

this paper proposes a new method for designing a reconfigurable controller for descriptor systems which is obtained from the second order dynamics of a robot control situation. Using mathematical tools the equations of robot control situation have been translated to descriptor systems, thus, after this change the controller has been designed for control performances. The proposed reconfigurable controller can recover the nominal closed-loop performances after fault occurrence in the system. The dynamics of descriptor systems contain infinite and finite elements, so a complete response of descriptor systems can be represented by an eigenstructure which involves finite and infinite elements. In this paper eigenstructure assignment is used to design a reconfigurable controller in general such that the reconfigured system can recover the complete response of a nominal system as much as possible. Finally, an example represents the effectiveness of the new method.

Keywords

Descriptor systems Eigenstructure Assignment Reconfigurable Controller

1. Introduction

Reconfigurable control system is a system that is capable of dealing with large variations in the system being controlled by means of adjusting or modifying the nominal control law. The stability and performance of the original closed-loop system is maintained as much as possible using the reconfigurable control systems. The reconfigurable control law must provide stability and excellent performance under conditions of failure and damage, as well as during normal operation [1]. In recent years, reconfigurable control has attracted much research attention and many new approaches have been proposed. In [2], the linear quadratic regulator method has been considered. In addition [2], pseudo inverse method [3], feedback linearization[4], the Lyapunov method [5], sliding mode control [6] are all considered, in [10] a reconfigurable controller design via output feedback in the case of post order fault is represented. [11] considers a modified approach that guarantees the stability of the closed-loop system by using an appropriate Lyapunov equation – all examples of approaches for designing reconfigurable control.

The eigenstructure assignment is one of the most powerful tools for control systems. According to the fact that the response of the system can be stated based on eigenvalues and corresponding eigenvectors, in this paper the eigenstructure assignment method is considered.

We know that the second order dynamic system is one of the most important systems in the dynamics of robot manipulators [12]. It is well known that these systems can be translated as descriptor systems by changes in variables. Designing a reconfigurable controller in these systems is important because fault occurrence in these systems may disturb the appropriate performances. Designing a reconfigurable controller in a descriptor system and high order dynamics system has been considered in some research. In [7], designing a reconfigurable controller in a second order dynamic system via p-d feedback eigenstructure assignment is considered. By changing variables in the second order system, this system is translated to a descriptor system. Based on finite eigenvalues and corresponding eigenvectors, the reconfigurable system has been designed. However, we know that the complete response of the descriptor system [8] consists of finite and infinite eigenstructures. So this method does not give any information about infinite eigenstructures.

The major contribution of this paper is to develop using eigenstructure assignment in second order dynamic systems in general case. In the proposed method there isn't any constraint on eigenvalues and eigenvectors of system, while in the previous work [7] eigenvalues are distinct. The proposed method can be used for descriptor systems which are important in network and other systems. According to the fact that the complete response of descriptor systems involves finite eigenstructure and infinite eigenstructure, the proposed method considers both finite eigenstructure and infinite eigenstructure. In this paper, a new method is suggested for the design of a reconfigurable controller in descriptor systems using finite and infinite eigenstructure assignment. Based on parametric eigenstructure assignment by state feedback in the descriptor system [9], the reconfigurable controller is designed.

This paper is organized as follows: Section 2 describes the parametric approach. In section3 the main problem is represented. Section 4 demonstrates the effectiveness of the proposed method.

2. Parametric Eigenstructure Assignment

Consider the second order dynamic system

E_{1} \ddot{q} - A_{1} \dot{q} - C_{1} q = B_{1} u_{1}

(1)

Where q ∈ R^n₁ and u ∈ R^r₁ are state and input vectors respectively. Suppose that rank(E₁) = n₁, rank (B₁) = r₁. Consider the following transformation

E = [\begin{matrix} I & 0 \\ 0 & E_{1} \end{matrix}], A = [\begin{matrix} 0 & I \\ C_{1} & E_{1} \end{matrix}], B = [\begin{array}{l} 0 \\ B_{1} \end{array}], x = [\begin{array}{l} q \\ \dot{q} \end{array}]

(2)

By applying the above transformation the following descriptor system is obtained

\begin{array}{l} E \dot{x} = A x + B u \\ y = C x \end{array}

(3)

Where x ∈ Rⁿ, u ∈ R^r, n = 2n₁, r = 2r₁, are respectively state vector and control vector E, A,B and C are the real matrices with appropriate dimensions, such that the closed-loop finite eigenvalues λ_i and (i = 1,…,m'), infinite eigenvalues and corresponding eigenvectors are designed to satisfy the special control specifications. Note that the closed system (E, A + BK) is regular, i.e., DET(sE- A – BK) is not identically zero [9] where rank (E) = m ≥ n, rank (B) = r, also suppose that, (1) is controllable, i.e., rank = [sE-A B] = n for all s ∈ C.

Assume the following state feedback which is design to satisfy special performances

u = k x

(4)

Suppose that due to the fault occurrence in the system, the dynamics of the nominal system (3) is changed. The new model of the faulty system is described as follows

\begin{array}{c} E_{f} {\dot{x}}_{f} = A_{f} x_{f} + B_{f} u_{f} \\ y_{f} (t) = C_{f} x_{f} \end{array}

(5)

Where A_f, E_f, B_f are real matrices with appropriate dimension and x_f ∈ Rⁿ, u_f ∈ R^r, y_f ∈ R^p, also rank(Ef) = m ≤ n, rank(B_f) = r and the faulty system is controllable. Our objective is to design a new state feedback

u_{f} = k_{f} x_{f} (t)

(6)

such that the performances of the nominal system are recovered. The closed-loop system is given by

E_{f} {\dot{x}}_{f} = (A_{f} + B_{f} K_{f}) x_{f}

(7)

In order to solve this problem, first we describe the parametric eigenstructure assignment by state feedback which is presented in [9].

Let θ = {λ₁^f, λ₂^f,…λ_m^f}, where θ is the set of finite eigenvalues of the closed-loop system (E_f, A_f + B_fK_f). The eigenvalue λ_i^f has the algebraic multiplicity m_i and a geometric multiplicity q_i. The Jordan canonical form J_f of (E_f, Af + B_fK_f) has $\sum_{i = 1}^{m^{'}} q_{i}$ Jordan blocks. The order of Jordan block corresponds to λ_i^f is p_ij (j = 1, 2,…, q_i) According to the above definition the following equation is satisfied [9]

P_{i 1} + P_{i 2} + … + P_{i q_{i}} = m_{i}

(8)

Also m₁ + m₂ +… m_m' = m according to definition of eigenvalues and eigenvector in descriptor system, the following equations are satisfied

\begin{array}{c} (A_{f} + B_{f} K_{f} - λ_{i}^{f} E_{f}) g_{i j}^{k} = E g_{i j}^{k - 1} g_{i j}^{0} = 0 \\ k = 1, 2, … p_{i j}, j = 1, 2, …, q_{i} \\ i = 1, 2, …, m^{'} \end{array}

(9)

Where g_ij^k and λ_i^f are eigenvalue and corresponding eigenvector of faulty system. Based on [9], (9) can be written as

\begin{array}{c} A_{f} G_{f} + B_{f} M_{f} = E_{f} V_{f} \\ M_{f} = K G_{f} \end{array}

(10)

Where

\begin{array}{c} G_{f} = [G_{1}, G_{2}, …, G_{m^{'}}] \\ G_{i} = [G_{i 1}, G_{i 2}, …, G_{i q}], m_{i j}^{k} = k_{f} g_{i j}^{k} \\ G_{i j} = [g_{i j}^{1}, g_{i j}^{2}, …, g_{i j}^{p_{i j}}] \end{array}

(11)

By applying singular value decomposition to the matrix [A_f – λ_i^f E_f B_f], the following equation is obtained

P_{f} (λ_{i}^{f}) [A_{f} - λ_{i}^{f} E_{f} B] = [ψ_{f} (λ_{i}^{f}) 0]

(12)

Where P_f (λ_i^f) ∈ C^n*n and Q_f (λ_i^f) ∈ C^(n+r)*(n+r) are orthogonal matrices, and ψ_f (λ_i^f) is a non singular diagonal matrix, such that the diagonal elements are singular values of [A_f – λ_i^fE_f B_f]. The parametric form of eigenstructure assignment is

[\begin{array}{l} g_{i j}^{k} \\ m_{i j}^{k} \end{array}] = [\begin{array}{l} N_{1}^{f} (λ_{i}^{f}) \\ D_{1}^{f} (λ_{i}^{f}) \end{array}] h_{i j}^{k} + [\begin{array}{l} N_{2}^{f} (λ_{i}^{f}) \\ D_{2}^{f} (λ_{i}^{f}) \end{array}] h_{i j}^{k - 1} + …. [\begin{array}{l} N_{k}^{f} (λ_{i}^{f}) \\ D_{k}^{f} (λ_{i}^{f}) \end{array}] h_{i j}^{1}

(13)

k = 1,2,… p_ij, j = 1,2,…q_i, i = 1,2,‥m'

Where h_ij^k ∈ C^r are parameter vectors and N_k^f (λ_i^f), D_k^f (λ_i^f) are obtained as follows

\begin{matrix} N_{k}^{f} (λ_{i}^{f}) = {(Q_{11}^{f} (λ_{i}^{f}) \sum^{- 1} (λ_{i}^{f}) P_{f} (λ_{i}^{f}) E_{f})}^{k - 1} Q_{12}^{f} (λ_{i}^{f}) \\ D_{k}^{f} (λ_{i}^{f}) = {\begin{array}{l} Q_{22}^{f} (λ_{i}^{f}) & k = 1 \\ (Q_{21}^{f} (λ_{i}^{f}) \sum^{- 1} P_{f} (λ_{i}^{f}) E_{f}) {(Q_{11}^{f} (λ_{i}^{f}) \sum^{- 1} P_{f} (λ_{i}^{f}) E_{f})}^{k - 2} Q_{12}^{f} (λ_{i}^{f}) & k \neq 1 \end{array} \\ Q_{f} (λ_{i}^{f}) = [\begin{array}{l} Q_{11}^{f} (λ_{i}^{f}) & Q_{12}^{f} (λ_{i}^{f}) \\ Q_{21}^{f} (λ_{i}^{f}) & Q_{22}^{f} (λ_{i}^{f}) \end{array}], Q_{22}^{f} (λ_{i}^{f}) \in C^{r \times r} \end{matrix}

(14)

Assume that the infinite eigenvalue of (E_f, A_f + B_fK_f) is shown by λ_∞^f, so $s_{\infty}^{f} = \frac{1}{λ_{\infty}^{f}} = 0$ is the infinite eigenvalue of (A_f + B_fK_f, E_f).

According to definition of eigenstructure assignment, we have

(E^{f} - s_{\infty}^{f} (A^{f} + B^{f} K^{f} j) g_{\infty j} = 0 j = 1, 2, …. (n - m)

(15)

Note that the algebraic and geometric multiplicities are n – m. Corresponding eigenvector of S_∞ is g_∞j. Consider

m_{\infty j} = k_{f} g_{\infty j} j = 1, 2, …, n - m

(16)

Based on the results of [9], the parametric infinite eigenvector is

[\begin{array}{l} g_{\infty j} \\ m_{\infty j} \end{array}] = [\begin{array}{l} N_{\infty}^{f} \\ D_{\infty}^{f} \end{array}] h_{\infty j} j = 1, 2, …, n - m

(17)

Where N_∞^f, D_∞^f are obtained from the following equations

\begin{array}{c} E_{f} S_{\infty}^{f} = 0 r a n k (s_{\infty}^{f}) = n - m \\ [\begin{array}{l} N_{\infty}^{f} \\ D_{\infty}^{f} \end{array}] = [\begin{matrix} S_{\infty}^{f} & 0 \\ 0 & I_{r} \end{matrix}] \end{array}

(18)

Define the matrix G_∞^f = [g_∞1, g_{∞2, …},g_∞n-m] and M_∞ = [m_∞1, m_∞2,…,m_∞n-m]

m_{\infty j} = k g_{\infty j} j = 1, 2, …, n - m

(19)

Similar to (10), we have

M_{\infty} = K_{f} G_{\infty}

(20)

By combining (10) and (20) the following equation is obtained

[M M_{\infty}] = K [G G_{\infty}]

(21)

If det[V_fV_∞] ≠ O then K_f is obtained as follows

k_{f} = [M M_{\infty}] {[G G_{\infty}]}^{- 1}

(22)

In order to calculate the matrix K_f, the following constraints must be satisfied

g_ijⁱ ∈ R^r for real eigenvalue λ_i^f, whereas g_ij^k = ḡ_ij^k ∈ C^r for complex conjugate of eigenvalues λ_i^f, λ_i^f = λ̄_i^f

det[G G_∞] ≠ 0

det [F_∞h_∞1F_∞h_∞2…F_∞h_∞2] ≠ 0, F_∞ = (T_∞^f)^T A_fN_∞ + (T_∞^f)^T B_fD_∞^f

Where T_∞^f is the left eigenvector matrix of the closed-loop system associated with the finite closed-loop eigenvalue s_∞.

T_{\infty}^{f} E_{f} = 0 r a n k (T_{\infty}^{f}) = n - m .

(23)

3. Main problem

Based on [8], the closed-loop response of descriptor systems (1) using state feedback and in terms of eigenstructure assignment can be stated as follows

\begin{array}{l} x_{f} (t) = x_{f}^{i n i t i a l} + x_{f}^{e x t e r n a l} (t) \\ x_{f}^{i n i t i a l} (t) = G e^{J_{f} t} T_{f}^{T} E_{f} x (0) \\ x_{e x t e r n a l} = - G_{\infty} {({(T_{\infty}^{f})}^{T} A_{c l}^{f} G_{\infty})}^{- 1} {(T_{\infty}^{f})}^{T} B_{f} u_{e}^{f} + \\ + \int_{0}^{t} G e^{J_{f} (t - τ)} B_{f} u_{e}^{f} d τ \end{array}

(24)

Where T_f and T_∞^f are left eigenvector matrices of the closed-loop system associated with the finite closed-loop eigenvalues λ_i^f and left eigenvector matrices of the closed-loop system associated with the finite closed-loop eigenvalue s_∞^f respectively, such that

T_{f}^{T} E_{f} G = I

(25)

The complete response of closed-loop system can be rewritten as follows

\begin{array}{l} x_{c}^{f} (t) = x_{f i n}^{f} + x_{\inf}^{f} \\ x_{f i n}^{f} = G e^{J_{f} t} T_{f}^{T} E_{f} x (0) + \int_{0}^{t} G e^{J_{f} (t - τ)} B_{f} u_{e}^{f} d τ \\ x_{\inf}^{f} = - G_{\infty} {({(T_{\infty}^{f})}^{T} A_{c l}^{f} G_{\infty})}^{- 1} {(T_{\infty}^{f})}^{T} B_{f} u_{e}^{f} \end{array}

(26)

Clearly, this response consists of two parts, the finite part X_fm^f that related to finite eigenstructure and the infinite part X_inf^f related to infinite eigenstructures.

According to the above discussions, similarly the closed-loop response of the nominal system contains two parts X_inf, X_fin. Based on (24) and (26), it is clear that the response of the system is related to its eigenstructure, so in order to recover nominal performance, the state feedback controller K_f must be designed such that the behaviour of the faulty system with this controller is closed to nominal system much as possible. The reconfiguration objective can be translated to the following equations

\begin{array}{l} λ_{i} = λ_{i}^{f} \\ \min [{(g_{i j}^{k} - v_{i j}^{k})}^{T} (g_{i j}^{k} - v_{i j}^{k})] \\ \min [{(g_{\infty j} - v_{\infty j})}^{T} (g_{\infty j} - v_{\infty j})] \end{array}

(27)

Where λ_i is eigenvalue of the nominal system and v_ij^k is corresponding eigenvector. By substituting the parametric form of g_ij^k and g_j_∞ from (12) and (17), the above equations change to

\begin{array}{l} J_{f i n} = \min [(N_{1}^{f} (λ_{i}^{f}) h_{i j}^{k} + N_{2}^{f} (λ_{i}^{f}) h_{i j}^{k - 1} + … \\ + N_{k}^{f} (λ_{i}^{f}) h_{i j}^{1} - v_{i j}^{k} - v_{i j}^{k - 1} - … - v_{i j}^{1})]^{T} … \\ [(N_{1}^{f} (λ_{i}^{f}) h_{i j}^{k} + N_{2}^{f} (λ_{i}^{f}) h_{i j}^{k - 1} + … \\ + N_{k}^{f} (λ_{i}^{f}) h_{i j}^{1} - v_{i j}^{k} - v_{i j}^{k - 1} - … - v_{i j}^{1})] . \end{array}

(28)

J_{\inf} = \min [{(N_{\infty}^{f} h_{j \infty} - v_{\infty j})}^{T} (N_{\infty}^{f} h_{j \infty} - v_{\infty j})]

(29)

Where v_ij^k are eigenvectors of closed-loop of nominal system. Note that the parametric form of v_ij^k obtains similar to g_ij^k. In order to solve the above problem, taking gradient with respect to h_ij^k, h_j_∞ and then set to zero we have

[\begin{array}{l} \frac{\partial J_{f i n}}{\partial h_{i j}^{k}} \\ \frac{\partial J_{f i n}}{\partial h_{i j}^{k - 1}} \\ ⋮ \\ \frac{\partial J_{f i n}}{\partial h_{i j}^{1}} \end{array}] = \bar{0}, \frac{\partial J_{\inf}}{\partial h_{j \infty}} = 0

(30)

In order to describe the method, consider k = 2 so the J _fjn is stated as follows

\begin{array}{l} J_{f i n} = \min {[(N_{1}^{f} (λ_{i}^{f}) h_{i j}^{2} + N_{2}^{f} (λ_{i}^{f}) h_{i j}^{1} - v_{i j}^{1} - v_{i j}^{2})]}^{T} \\ [(N_{1}^{f} (λ_{i}^{f}) h_{i j}^{2} + N_{2}^{f} (λ_{i}^{f}) h_{i j}^{1} - v_{i j}^{1} - v_{i j}^{2})] \end{array}

(31)

According to (30)

[\begin{array}{l} \frac{\partial J_{f i n}}{\partial h_{i j}^{2}} \\ \frac{\partial J_{f i n}}{\partial h_{i j}^{1}} \end{array}] = \bar{0}

(32)

Taking the above gradient and then by simplifying the results, we have

{\begin{cases} {(N_{2}^{f} (λ_{i}^{f}))}^{T} (N_{1}^{f} (λ_{i}^{f})) h_{i j}^{2} + {(N_{2}^{f} (λ_{i}^{f}))}^{T} (N_{2}^{f} (λ_{i}^{f})) h_{i j}^{1} = {(N_{2}^{f} (λ_{i}^{f}))}^{T} v_{i j}^{1} + {(N_{2}^{f} (λ_{i}^{f}))}^{T} v_{i j}^{2} \\ {(N_{1}^{f} (λ_{i}^{f}))}^{T} (N_{1}^{f} (λ_{i}^{f})) h_{i j}^{2} + {(N_{1}^{f} (λ_{i}^{f}))}^{T} (N_{2}^{f} (λ_{i}^{f})) h_{i j}^{1} = {(N_{1}^{f} (λ_{i}^{f}))}^{T} v_{i j}^{1} + {(N_{1}^{f} (λ_{i}^{f}))}^{T} v_{i j}^{2} \end{cases}

(33)

h_ij¹, h_ij² can be calculated from the above equations. In the special case of distinct finite eigenvalues (m_i = q_i = 1, i = 1,2,m), K_f can be calculated as follows

\begin{array}{l} K_{f} = [D_{1}^{f} (λ_{1}^{f}) h_{11}^{1} … D_{1}^{f} (λ_{m}^{f}) h_{m 1}^{1} D_{\infty}^{f} h_{\infty 1} … \\ D_{\infty}^{f} h_{\infty, m - n}] \times [N_{1}^{f} (λ_{1}^{f}) h_{11}^{1} … \\ N_{1}^{f} (λ_{m}^{f}) h_{m 1}^{1} N_{\infty}^{f} h_{\infty 1} … N_{\infty}^{f} h_{\infty, m - n}]^{- 1} \end{array}

(34)

The parametric eigenstructure of distinct eigenvalues is computed as

\begin{array}{l} g_{i 1}^{1} = N_{1}^{f} (λ_{i}^{f}) h_{i 1}^{1} \\ g_{\infty i} = N_{\infty} h_{\infty i} \end{array}

(35)

By substituting (35) in (28) and then take gradient the solution is obtained as

h_{i 1}^{1} = {[{(N_{1}^{f} (λ_{i}^{f}))}^{T} N_{1}^{f} (λ_{i}^{f}) h_{i 1}^{1}]}^{- 1} {(N_{1}^{f} (λ_{i}^{f}))}^{T} v_{i 1}^{1}

(36)

Where V_i1¹ is eigenvector of faulty system in distinct case.

One of the most important specifications that should be recovered by the reconfigured system is steady state response of the nominal systems. The following theorem states how a feedforward matrix should be designed to minimize the difference between the after fault and prefault steady state response

Theorem1: For the faulty system (5), consider the control law

u_{f} (t) = K_{f} x_{f} + L r_{f} (t)

(37)

Where L ∈ R^m×m is feedforward matrix. The steady state response of the nominal system to step input can be recovered if R is selected as

L = {(R^{T} R)}^{- 1} R^{T} H

(38)

Where

\begin{array}{l} H = C {(A + B K)}^{- 1} B \\ R = C_{f} {(A_{f} + B_{f} K_{f})}^{- 1} B_{f} \end{array}

(39)

Proof: The closed-loop systems of nominal and faulty systems are given by

{\begin{cases} E \dot{x} (t) = (A + B K) x (t) + B r (t) \\ y = C x (t) \end{cases}, u (t) = K x (t) + r (t)

(40)

{\begin{cases} E_{f} {\dot{x}}_{f} (t) = (A_{f} + B_{f} K_{f}) x_{f} (t) + B_{f} L r (t) \\ y_{f} = C_{f} x_{f} (t) \end{cases}

(41)

By applying Laplace transformation to (40) and (41), we have

{\begin{cases} s E X (s) = (A + B K) X (s) \\ X (s) = {(E s - (A + B K))}^{- 1} B R (s) \end{cases}

(42)

{\begin{cases} s E_{f} X_{f} (s) = (A_{f} + B_{f} K_{f}) X_{f} (s) + B_{f} L R (s) \\ X_{f} (s) = {(E_{f} s - (A_{f} + B_{f} K_{f}))}^{- 1} B_{f} L R (s) \end{cases}

(43)

In the above equation suppose that (E, A + BK) and (E_f, A_f + B_fK_f) are regular. The steady state output of the nominal closed-loop system to a unit step input is given by

y_{s s} = \lim_{s \to 0} s [C (s E - {(A + B K)}^{- 1} B \frac{1}{s}] = - C {(A + B K)}^{- 1} B

(44)

In addition, for the faulty system

\begin{array}{l} y_{s s}^{f} = \lim_{s \to 0} s [C_{f} (s E_{f} - {(A_{f} + B_{f} K_{f})}^{- 1} B_{f} L \frac{1}{s}] = \\ = - C_{f} {(A_{f} + B_{f} K_{f})}^{- 1} B_{f} L \end{array}

(45)

So for recovering of the steady state response of the nominal system we have to design L such that

\min ‖ y_{s s} - y_{s s}^{f} ‖ \to \min ‖ H - R L ‖ = {(H - R L)}^{T} (H - R L)

(46)

The solution is described in (38).

4. Example

Consider the following fifth order descriptor [9]

\begin{array}{l} E = [\begin{matrix} 0 & 0 & 0 & 1.72 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ - 0.82 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}], & A = [\begin{matrix} 0 & 1.1 & 0 & 0 & 0 \\ 0 & 0 & 1.56 & 0 & 0 \\ 1.23 & 0 & 0 & 1.98 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1.01 & 0 & 0 \end{matrix}] \\ B = [\begin{matrix} 0 & 0 & 0 \\ 1.55 & 0 & 0 \\ 0 & 1.07 & 0 \\ 0 & 0 & - 1.11 \\ 0 & - 2.5 & 0 \end{matrix}] \end{array}

According to [9], considering the assignment of the following closed-loop eigenstructure

θ = {- 0.5, - 1, - 2}, m = 3, m_{i} = q_{i} = 1, i = 1, 2, 3

So the corresponding parameters are as follows

\begin{array}{l} T_{\infty} = {[\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{matrix}]}^{T}, S_{\infty} = {[\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}]}^{T} \\ N_{\infty} = [\begin{matrix} S_{\infty} & 0_{5 \times 3} \end{matrix}], D_{\infty} = [\begin{matrix} 0_{3 \times 2} & I_{3} \end{matrix}] \\ v_{\infty 1} = N_{\infty} f_{\infty 1}, v_{\infty 2} = N_{\infty} f_{\infty 2} \\ w_{\infty 1} = D_{\infty} f_{\infty 1}, W_{\infty 2} = D_{\infty} f_{\infty 2} \\ N_{1} (- 0.5) = [\begin{matrix} 0.2347 & - 0.1417 & - 0.8465 \\ - 0.0282 & 0.0371 & - 0.3022 \\ - 0.6584 & 0.0070 & - 0.1263 \\ 0.0361 & - 0.0475 & 0.3866 \\ 0.0969 & 0.9683 & - 0.0780 \end{matrix}] \\ N_{1} (- 1) = [\begin{matrix} 0.1798 & - 0.1679 & - 0.9125 \\ - 0.1540 & 0.2164 & - 0.3328 \\ - 0.6589 & - 0.0860 & - 0.0566 \\ 0.0985 & - 0.1384 & - 0.2128 \\ 0.0461 & 0.8879 & - 0.0533 \end{matrix}] \\ N_{1} (- 2) = [\begin{matrix} 0.2232 & - 0.0159 & - 0.8853 \\ - 0.4416 & 0.6051 & 0.1408 \\ - 0.5890 & - 0.2660 & - 0.2009 \\ 0.1412 & - 0.1935 & - 0.0450 \\ 0.0777 & 0.5743 & - 0.2185 \end{matrix}] \\ D_{1} (- 0.5) = [\begin{matrix} 0.6627 & - 0.0071 & 0.1271 \\ - 0.2466 & 0.1965 & - 0.0666 \\ 0 & 0 & 0 \end{matrix}] \\ D_{1} (- 1) = [\begin{matrix} 0.6631 & 0.0866 & 0.1271 \\ - 0.2477 & 0.3204 & - 0.0442 \\ 0 & 0 & 0 \end{matrix}] \\ D_{1} (- 2) = [\begin{matrix} 0.5928 & 0.2677 & 0.2022 \\ - 0.1758 & 0.3520 & - 0.2559 \\ 0 & 0 & 0 \end{matrix}] \end{array}

The above parameter is computed based on (36). Suppose that in a special case the free parameters are computed as follows

\begin{array}{l} f_{11}^{1} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], f_{21}^{1} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], f_{31}^{1} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] \\ f_{\infty 1} = [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}], f_{\infty 2} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}] s \end{array}

Then the controller is

K = [\begin{matrix} 0.6374 & 0 & 0 & 3.0598 & 0.2361 \\ - 0.1938 & 0 & 0 & - 1.0072 & 0.1252 \\ 0.2489 & 1 & 0 & 2.6630 & 0.1287 \end{matrix}]

Consider due to the fault the dynamics of the system are changed as follows

\begin{array}{l} E_{f} = [\begin{matrix} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}], A_{f} = [\begin{matrix} 0 & 1.1 & 0 & 0 & 0 \\ 0 & 0 & 0.5 & 1 & 0 \\ 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.5 & 0 & 0 \end{matrix}] \\ B_{f} = [\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1.5 & 0 \\ 0 & 0 & - 1 \\ 0 & - 2 & 0 \end{matrix}] \end{array}

According to the proposed method in this work the reconfigured controller is obtained as follows. The following controller is computed based on (…14)

\begin{array}{l} N_{1}^{f} (- 0.5) = [\begin{matrix} - 0.1497 & - 0.1497 & - 0.9175 \\ - 0.0139 & 0.0916 & - 0.1167 \\ - 0.6486 & 0.6420 & - 0.0196 \\ 0.0306 & - 0.2016 & 0.2568 \\ 0.6851 & 0.6329 & - 0.1265 \end{matrix}] \\ N_{1}^{f} (- 1) = [\begin{matrix} - 0.3210 & - 0.1240 & - 0.9386 \\ 0.0988 & 0.2695 & - 0.0696 \\ - 0.5905 & 0.6805 & 0.1121 \\ - 0.1087 & - 0.2965 & 0.0763 \\ 0.5852 & 0.4504 & - 0.2595 \end{matrix}] \\ N_{1}^{f} (- 2) = [\begin{matrix} - 0.4026 & 0.1872 & - 0.6749 \\ 0.5156 & 0.4361 & 0.1662 \\ - 0.4294 & 0.6581 & 0.4391 \\ - 0.2836 & - 0.2399 & - 0.0914 \\ 0.2170 & 0.2801 & - 0.4378 \end{matrix}] \\ K_{f} = [\begin{matrix} - 0.1083 & 0 & - 0.7715 & 0 & 0.5683 \\ - 0.0243 & 0 & 0.0125 & 0 & 0.4852 \\ 0.2061 & 1.0000 & 0.6340 & 0 & - 0.7391 \end{matrix}] \end{array}

Because the eigenvalues are distinct, K_f is computed based on 34). So we can simulate the response of this system to the initial conditions x(0) = [1 1 0.9 1 0.9]^T as follows

Figure 1.

Response of first state

Figure 2.

Response of second state

Figure 3.

Response of third state

Figure 4.

Response of fourth state

Figure 5.

Response of fifth state

In these pictures dashed lines relate to the reconfigured system, which is simulated by using the proposed controller K_f, and the solid lines relate to the nominal system. From the above figures it can be seen that the proposed reconfigurable control system can recover the nominal system response.

5. Conclusion

An eigenstructure assignment-based method is suggested to design a reconfigurable controller for descriptor systems via state feedback. The controller can be reconfigured to compensate for the effect of change in the system dynamics. This work deals with the control reconfiguration in the general case where the complete response of systems includes the infinite and finite eigenstructure. Finally, an example demonstrates the effectiveness of the proposed method via simulation.

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Reconfigurable Controller Design in Descriptor Systems Obtained from Second Order Dynamic Systems via State Feedback Eigenstructure Assignment