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Preview Tracking Control for Continuous-Time Singular Interconnected Systems.

1. Introduction

Compared with the normal system, the singular system is a kind of dynamic system with a more extensive form and wide application background. Singular systems exist in many fields, such as economic management, bioengineering systems, aerospace technology, and robotic systems [1-3]. From the early 1970s to the end of the 1980s, fruitful achievements have been made in the research of minimum realization problem, observer design, and stability of singular systems [4-6]. Since the 1990s, research on singular systems has been developed from basis to depth, covering various topics from linear to nonlinear, from time-invariant to time-varying systems, and from linear quadratic optimal control to [H.sub.[infinity]] control [7-9]. Because the singular system model comes from the engineering practice, the research of the theory of singular systems must serve the practical application finally. In many practical engineering situations, there are usually large-scale, complex structures, many factors, and functional comprehensive of the systems, that is, large-scale systems, also known as interconnected systems, which leads to the study of singular interconnected systems. For singular interconnected systems, if the controller is designed using a centralized control scheme, it will be difficult to deal with too much information, which makes the design difficult to achieve. Therefore, a decentralized-aggregation method is adopted to design the controller [10, 11]. This is the decomposition method of large-scale systems. Firstly, the related terms are deleted artificially, and several lower-dimensional systems (also called isolated subsystems) are obtained. The controllers satisfying the requirements are designed, respectively. Then, the controller of the interconnected system is obtained through a certain method of synthesis [12, 13]. In recent years, there exist many research studies on control theory of singular interconnected systems and many significant results have been obtained [14, 15]. Wo et al. studied robust stabilization of discrete-time singular interconnected systems with parameter uncertainties [11]. Lu and Ho gave sufficient conditions for stability and decentralized stabilization of a class of singular interconnected systems utilizing linear matrix inequalities (LMIs) [16]. Employing the singular Lyapunov matrix equation, system decomposition method, and matrix theory, Chen and Ding probed the stabilization problem of singular linear interconnected systems with output feedback and provided sufficient conditions for asymptotic stability and instability of corresponding closed-loop singular interconnected systems [17].

The future values of desired tracking signals or disturbance signals of some practical control systems are partly or completely known, such as flight routes of aircraft, processing paths of numerically controlled machine tools, and driving paths of vehicles. Utilizing the known future information to improve the quality of the closed-loop system is known as the preview control problem. At present, the research on optimal preview control of linear quadratic form is in-depth. By adopting the method of augmented systems, Katayama and Hirono discussed the optimal preview control problem for continuous-time [18]. On the basis of [18], Liao et al. studied the situation where both the desired tracking signal and the disturbance signal are previewable at the same time [19]. In recent years, the theory of preview control for stochastic systems, the theory of robust preview control, and the theory of preview control for multirate systems have been developed [20-22]. Wu et al. investigate the optimal preview control problem for a class of continuous-time stochastic systems by constructing an auxiliary system [20]. Li and Liao combine parameter-dependent Lyapunov stability theory with function method and LMI techniques to solve the static output feedback preview tracking control problem for a class of polyhedral uncertain discrete-time systems [21]. At the same time, preview control has been successfully used in many practical applications, such as vehicle active suspension systems, electromechanical servo systems, robots and aircraft [23-25].

On the premise that both the theory of singular interconnected systems and preview control have made considerable progress, it is of great theoretical and practical significance to combine them. So far, there is no literature on preview control of singular interconnected systems. In view of this, this paper proposes and studies the preview tracking control problem for continuous-time singular interconnected systems.

Research contents are arranged as follows: Section 2 presents a class of preview tracking control problems for continuous-time singular interconnected systems and gives necessary assumptions. The designs of the error system controller of isolated subsystems and the controller of the singular interconnected system are discussed in Sections 3 and 4, respectively. Section 5 is numerical simulation. The method presented is not only applicable to this paper but also to general continuous-time singular systems. Finally, a brief conclusion is given in Section 6.

Throughout this paper, A [member of] [R.sup.nxm] represents A as the real matrix of n * m; Q > 0 (Q [greater than or equal to] 0) shows the matrix Q symmetric positive definite (semi-positive definite); deg(det(-)) is the degree of the determinant; and [parallel]-[parallel] represents the matrix norm derived from the Euclidean norm of a vector.

2. Problem Formulation and Basic Assumptions

Consider a continuous-time singular interconnected system as follows:

[mathematical expression not reproducible] (1)

where [mathematical expression not reproducible] are constant matrices; [A.sub.ij](i [not equal to] j) is the related matrix; and rank ([E.sub.i]) = [q.sub.i] < [n.sub.i] (i = 1j 2, ..., N).

Let [y.sub.d] (t) [member of] [R.sub.p] be the desired tracking signal or the reference signal.

Firstly, some necessary assumptions are given as follows:

A1: assume that ([E.sub.i], [A.sub.i]) is regular, that is to say, there exists [s.sub.i] [.sup..sub.C][member of], so that det ([s.sub.i][E.sub.i] - [A.sub.i]) [not equal to] [empty set] (i = 1, 2, ..., N) [26]

A2: assume that ([E.sub.i], [A.sub.i]) is impulse free, that is to say, for any s [member of] C, deg (det ([sE.sub.i] - [A.sub.i])) = rank([E.sub.i]) holds (i = 1, 2, ..., N) [26]

A3: assume that ([E.sub.i], [A.sub.i], [B.sub.i]) is stabilizable, which means rank [s[E.sub.i] - [A.sub.i] [B.sub.i]] = [n.sub.i] holds for any complex s satisfying Re (s) [greater than or equal to] 0 (i = U 2, ..., N) [26]

A4: assume that the matrix [mathematical expression not reproducible] is of full row rank (i = 1, 2, ..., N)

A5: assume that ([E.sub.i], [A.sub.i], [C.sub.i]) is detectable, which means rank [mathematical expression not reproducible] holds for any complex s satisfying Re(s) [greater than or equal to] 0 (i = 1, 2, ..., N) [26]

A6: assume that desired tracking signal [y.sub.d] (t) is a piecewise continuously differentiable function satisfying

[mathematical expression not reproducible] (2)

where [[bar.y].sub.d] is a constant vector. Besides, [y.sub.d] (t) is previewable; in other words, at the time t, [y.sub.d] ([tau]) (t [less than or equal to] t [mathematical expression not reproducible] t + [l.sub.r]) is available, where [l.sub.r] is called the preview length.

The system

[mathematical expression not reproducible] (3)

is called an isolated subsystem of (1). There are N isolated subsystems (i = 1, 2, ..., N).

Remark 1. The establishment of A1 and A2 indicates that the N isolated subsystems of System (1) are regular and impulse free. The establishment of A3 and A5 indicates that the N isolated subsystems of System (1) are stabilizable and detectable. A6 is the standard assumption of preview control.

The tracking error of System (1) is defined as follows:

e(t) = y(t)- [y.sub.d](t). (4)

The objective of this paper is to design a controller with preview action so that y(t) is able to track the desired tracking signal [y.sub.d] (t) without static error; in other words,

[mathematical expression not reproducible]. (5)

3. Controller of the Error System of Isolated Subsystems

For the sake of designing the controller, decentralized control is adopted. Firstly, the controller is designed for each isolated subsystem of System (1). Then, all the controllers are combined as the controllers of singular interconnected systems, and the Lyapunov function method is utilized to give the restriction conditions of the related terms, such that the output of closed-loop systems of singular interconnected systems is able to track [y.sub.d](t).

According to the form of System (1) output equation, the tracking error e(t) is rewritten as

[mathematical expression not reproducible], (6)

where [[alpha].sub.1] (i = 1, 2, ..., N) are constants which satisfy [[summation].sup.N.sub.i=1] [[alpha].sub.1] = 1. It is known from (6) that if for any [e.sub.i] (t) = [y.sub.i] (t) [[alpha].sub.i] [y.sub.d](t) (i = 1, 2, ..., N), there is [lim.sub.t[right arrow][infinity]] [e.sub.i](t) = 0, then [lim.sub.t[right arrow][infinity]] e(t) = 0.

Remark 2. [y.sub.i] (t) can be understood as the output of i-th subsystem. (6) means that if output [y.sub.i] (t) of i-th subsystem tracks [[alpha].sub.i] [y.sub.d](t) (i = 1, 2, ..., N), then the output y(t) = [[summation].sup.N.sub.i=1] [y.sub.i](t) of System (1) can track [y.sub.d](t). The parameter [[alpha].sub.i] (i = 1, 2, ..., N) gives us the freedom of choice. For example, let [[alpha].sub.1] = [[alpha].sub.2] =... [[alpha].sub.N] = 1/N, which means that all [y.sub.i] (t) keep track of (1/N)[y.sub.d](t). If [[alpha].sub.i] = 0, this indicates that the output of the i-th subsystem tracks the zero vector, then the task of tracking [y.sub.d] (t) is completed by the output of other subsystems cooperatively, etc.

The optimal control technique is employed to design the controller of System (3). For a given i(i = 1, 2, ..., N), the quadratic performance index function is introduced as follows:

[mathematical expression not reproducible], (7)

where [mathematical expression not reproducible] are positive definite matrices. It has been pointed out in [19] that the introduction of [mathematical expression not reproducible] (t) in the performance index function can make the controller contain integrators, which helps to eliminate the static output errors in the closed-loop system.

An error system is constructed for the isolated subsystem by utilizing the usual preview control method, so that the tracking error becomes a part of the state vector of the error system. As a result, the tracking problem of the isolated subsystem is turned into a regulation problem of the error system. Since the error system of the singular interconnected system is needed later in this paper, for the purpose of avoiding repetition, the error system of System (1) is firstly constructed, then the related term is removed to obtain the error system of isolated subsystems.

Differentiate both sides of the state equation of System (1) to obtain

[mathematical expression not reproducible] (8)

Then, differentiate [e.sub.i] (t) = [y.sub.i] (t) - [[alpha].sub.i] [y.sub.d] (t) (i = 1, 2, ..., N) to obtain

[mathematical expression not reproducible] (9)

Combining (8) and (9), we have

[mathematical expression not reproducible] (10)

where

[mathematical expression not reproducible] (11)

System (10) is the error system of singular interconnected System (1).

The error system of the isolated subsystem can be obtained by taking all the associated matrices [bar.A].sub.ij] (i [not equal to] j) in (10) as zero matrices. The error system of the i-th isolated subsystem becomes

[mathematical expression not reproducible]. (12)

Adopting the state vector of formula (12), the performance index function (7) can be written as

[mathematical expression not reproducible], (13)

where [mathematical expression not reproducible].

For the sake of making full use of the mature controller design methods and results in the optimal preview control theory of normal systems, System (12) needs to be changed into a normal system and an algebraic equation by restricted equivalent transformation [26].

Notice that if A1 and A2 are true, then ([[bar.E].sub.i], [[bar.A].sub.i]) is regular and impulse free. This is proved as follows.

In the light of [26], ([E.sup.i], [A.sub.i]) is regular and impulse free if and only if there are nonsingular matrices [S.sup.i] and [T.sub.i], such that

[mathematical expression not reproducible] (14)

Taking the nonsingular matrix [mathematical expression not reproducible] and [mathematical expression not reproducible], there are

[mathematical expression not reproducible] (15)

and [[bar.T].sub.i] are nonsingular and there are

[mathematical expression not reproducible] (16)

This means that ([[bar.E].sub.i], [[bar.A].sub.i]) is regular and impulse free [26].

In addition, the right side of the two formulas is denoted as [mathematical expression not reproducible], respectively, i.e., [mathematical expression not reproducible] and [mathematical expression not reproducible]. For System (12), introducing a nonsingular linear transformation [X.sub.i] (t) = [[bar.T].sub.i] [[bar.X].sub.i] (t) and then premultiplying a nonsingular matrix [[bar.S].sub.i] on both sides, we obtain

[mathematical expression not reproducible], (17)

where

[mathematical expression not reproducible] (18)

Denote

[mathematical expression not reproducible], (19)

where [mathematical expression not reproducible]. Blocks [mathematical expression not reproducible] are as follows:

[mathematical expression not reproducible] (20)

Then, (17) can be written as

[mathematical expression not reproducible], (21a)

[mathematical expression not reproducible]. (21b)

Using the state vector of (17) (or (21)), the performance index function (13) can be written as

[mathematical expression not reproducible], (22)

where [mathematical expression not reproducible].

Note that the restricted equivalence transformation does not change the dynamic characteristics of singular systems, such as regularity, impulse free, stability, stabilization, and detectability [26]. Therefore, System (12) can be studied through System (17) (or (21)). That is, only the controller of System (21a) needs to be designed.

By utilizing the state vector of formula (21a), the performance index function (22) can be written as

[mathematical expression not reproducible], (23)

where [mathematical expression not reproducible]

From the known conclusion [19], Theorem 1 can be proved directly.

Theorem 1. If ([mathematical expression not reproducible]) is stabilizable, ([mathematical expression not reproducible]) is detectable and A6 holds; then, the controller of System (21a), which minimizes the performance index function (14), has the form of

[mathematical expression not reproducible], (24)

where

[mathematical expression not reproducible]. (25)

[mathematical expression not reproducible] is a stable matrix and its expression is

[mathematical expression not reproducible]. (26)

[P.sub.i] is the positive definite solution of the following Riccati equation:

[mathematical expression not reproducible]. (27)

Next, the optimal preview controller for System (12) and the performance index (13) is given. Firstly, the following two lemmas are established.

Lemma 1 (see [26]). For a given i(i = 1, 2, ..., N), ([mathematical expression not reproducible]) is stabilizable if and only if ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.B].sub.i]) is stabilizable.

The proof of Lemma 1 is shown in Theorem 8.4.1 in literature [26].

Remark 3. Lemma 1 shows that the stabilization of singular System (12) is determined only by normal System (21a) obtained through its restricted equivalent transformation.

Lemma 2. For a given [mathematical expression not reproducible] is detectable if and only if ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.B].sub.i.sup.1/2]) is detectable.

Proof. This lemma can be obtained according to Theorem 8.3.2 in literature [26] and by the method of proving Theorem 8.4.1 in literature [26]. It is omitted here.

Under the performance index function (13), the optimal preview controller of System (12) is given by Theorem 2.

Theorem 2. If ([E.sub.i], [A.sub.i]) is regular and impulse free, ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.B].sub.i]) is stabilizable, ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.Q].sup.1/2.sub.i]) is detectable, and A6 holds; then under the performance index function (13), the optimal preview controller of System (12) is

[mathematical expression not reproducible], (28)

where [mathematical expression not reproducible] is admissible.

The so-called admissibility refers to regularity, impulse free, and stabilization [26]. Therefore, [mathematical expression not reproducible] is admissible which means that [mathematical expression not reproducible] is regular, impulse free, and stable.

Proof. When ([E.sub.i], [A.sub.i]) is regular and impulse free, the restricted equivalent transformation of the above system (13) holds. In addition, according to Lemma 1 and Lemma 2, the condition of Theorem 2 can guarantee that Theorem 1 is true. Therefore, the controller (24) is obtained according to Theorem 1. Because [[bar.T].sub.i] is an invertible matrix in the transformation [X.sub.i] (t) = [[bar.T].sub.i] [[bar.X].sub.i] (t), the optimal preview controller (24) for System (21a) and the performance index (23) is the optimal preview controller for System (12) under the performance index function (13).

Thanks to [mathematical expression not reproducible], there is [mathematical expression not reproducible]. Therefore,

[mathematical expression not reproducible] (29)

Substitute (29) into (24) to get (28).

The following proves that [mathematical expression not reproducible] is admissible.

Taking [mathematical expression not reproducible] are not singular. Because there is

[mathematical expression not reproducible] (30)

[mathematical expression not reproducible] (31)

This indicates that ([mathematical expression not reproducible]) is regular and impulse free [26]. Notice that [mathematical expression not reproducible] is stable, which means that ([mathematical expression not reproducible]) is also stable [26]. Thus, Theorem 2 is proved.

4. Design of a Preview Controller for Singular Interconnected Systems

[mathematical expression not reproducible] given by equation (28) are employed to construct a vector

[mathematical expression not reproducible], (32)

as the controller of error System (10). Next, determine the conditions in which the related term should satisfy so that the state vector [X.sub.i](t) (i = 1,2,...,N) of the closed-loop system of System (10) is asymptotically stable to zero vector. Firstly, the following lemma is given.

Lemma 3 (see [27]). For singular systems, the following propositions are equivalent:

(i) [mathematical expression not reproducible] is admissible

(ii) For any given positive definite matrix W, there exists a matrix [bar.P] so that

[mathematical expression not reproducible], (33)

holds

(iii) The singular Lyapunov function V(Ex) = [x.sub.T][E.sub.T] [bar.P]x satisfies

dV(Ex(t))/(t))/dt < 0, (34)

where x(t) [not equal to] [empty set], [E.sub.T] [bar.P] = [[bar.P].sup.T] E [greater than or equal to] 0, and rank ([E.sub.T] [bar.P]) = rank (E)

Substitute (32) into System (10) to get the closed-loop system:

[mathematical expression not reproducible] (35)

where

[mathematical expression not reproducible]. (36)

Theorem 3 can be obtained, which gives the condition that the state vector of System (35) is asymptotically stable to the zero vector.

Theorem 3. If

(I) ([E.sub.i], [A.sub.i]) is regular and impulse free, (([[bar.E].sub.i], [[bar.A].sub.i], [[bar.B].sub.i])) is stabilizable, and (([[bar.E].sub.i], [[bar.A].sub.i], [[bar.Q].sup.1/2.sub.i])) is detectable (i = 1,2,...,N); (II) A6 holds;

(III) [parallel][[bar.P].sup.T.sub.i] [[bar.A].sub.ij][parallel] [less than or equal to] [[beta].sub.ij] (i,j = 1,2,...,N,i [not equal to] j);

(IV) matrix

[mathematical expression not reproducible] (37)

is a nonsingular M matrix, then the state vector [X.sub.i](t) (i = 1, 2,..., N) of System (35) (that is, the closed-loop system of (10)) is asymptotically stable to zero vector, where Y[[gamma].sub.i] = [[lambda].sub.min] ([W.sub.i]), in which Wi is any given positive definite matrix (i = 1,2,...,N). [[bar.P].sub.i] is the solution of the following equation:

[mathematical expression not reproducible] (38)

Proof. Firstly, it is proved that the homogeneous system

[mathematical expression not reproducible] (39)

corresponding to (35) is admissible.

In the light of Theorem 2, [mathematical expression not reproducible] is admissible if (I) and (II) of this theorem hold. According to Lemma 3, given matrix [W.sub.i] > 0, there exists matrix [[bar.P].sub.i] satisfying (38). Employing [[bar.E].sub.i] and [[bar.P].sub.i] to construct [V.sub.i] ([[bar.E].sub.i] [[bar.X].sub.i]) = [X.sup.T.sub.i] [[bar.P].sup.T.sub.i] [[bar.P].sub.i] [X.sub.i], it is clear that there is [V.sub.i] ([[bar.E].sub.i] [X.sub.i]) [greater than or equal to] 0. Differentiate [V.sub.i] ([[bar.E].sub.i] [X.sub.i]) along System (39) trajectory to get

[mathematical expression not reproducible] (40)

Note that [mathematical expression not reproducible] to obtain an estimate of [V.sub.i][|.sub.(39)] as follows:

[mathematical expression not reproducible] (41)

Continuously, utilizing the properties of norms and (III), we obtain

[mathematical expression not reproducible] (42)

Let i = 1,2,..., N to get

[mathematical expression not reproducible] (43)

Since (IV), that is, H is a nonsingular M matrix, there exists K = diag ([k.sub.1], [k.sub.2],..., [k.sub.N]) > 0 so that KH + [H.sub.T]K > 0 [28].

Using the diagonal elements of matrix K and taking

[mathematical expression not reproducible], (44)

as the Lyapunov function of System (39), the total derivative of V along System (39) trajectory is

[mathematical expression not reproducible]. (45)

Substitute (43) into (45) to obtain

[mathematical expression not reproducible] (46)

When [lambda] is utilized to represent the minimum eigenvalue of (KH + [H.sub.T] K)/2, there must be [lambda] > 0. Therefore, we obtain

[mathematical expression not reproducible]. (47)

X(t)= [[[X.sup.T.sub.1] (t) [X.sup.T.sub.2](t)... [X.sup.T.sub.N](t)].sup.T] is the state vector of System (39). Therefore, the total derivative of V along the trajectory of System (39) is negative definite. According to Lemma 3, System (39) is admissible.

Next, it is proven that [lim.sub.t[right arrow][infinity]] [[eta].sub.i](t) = 0 and [[eta].sub.i] (t) is bounded on [0, +[infinity](i = 1,2,..., N). Note the Hypothesis A6, the desired tracking signal [y.sub.d] (t) is piecewise continuously differentiable, so [mathematical expression not reproducible] (t) has only the discontinuity of first kind at most. Naturally, [mathematical expression not reproducible] (t) is a bounded function. Besides, considering the hypothesis of [mathematical expression not reproducible], we only need to prove that [lim.sub.t[right arrow][infinity]] [g.sub.i](t) = 0 and [g.sub.i] (t) is bounded.

Because [mathematical expression not reproducible] is a stable matrix, there exist constants [[omega].sub.i] > 0 and [[alpha].sub.i] > 0, so that [mathematical expression not reproducible] holds for all [mathematical expression not reproducible] there is

[mathematical expression not reproducible] (48)

Substitute [mathematical expression not reproducible] to get

[mathematical expression not reproducible](49)

For any given [epsilon] > 0, due to [mathematical expression not reproducible], there exists [T.sub.0] > 0 such that when t [greater than or equal to] [T.sub.0], there is [mathematical expression not reproducible], so when t [greater than or equal to] [T.sub.0], there is

[mathematical expression not reproducible] (50)

This proves [lim.sub.t[right arrow][infinity]] [g.sub.i](t) = 0.

Secondly, exp [less than or equal to] has only the discontinuity of the first kind at most on the basis of the property of [mathematical expression not reproducible] is a continuous function [29]. As a result, [g.sub.i] (t) is bounded on [0 + [infinity]), since [g.sup.i][.sub. ](t) is continuous and [lim.sub.t[right arrow][infinity]] [g.sub.i](t) = 0 [30]. This proves that [lim.sub.t[right arrow][infinity]] [[eta].sub.i](t) = 0 and [[eta].sub.i] (t) is bounded on [0,+ [infinity]]

Because System (39) is admissible, [lim.sub.t[right arrow][infinity]] [[eta].sub.i](t) = 0 and [[eta].sub.i] (t) is bounded on [0, + [infinity], according to [31], there is [lim.sub.t[right arrow][infinity]] [X.sub.i](t) = 0 (i = 1,2,...,N) in System (35). Hence,

Theorem 3 is proved.

Below, we adopt the relevant parameters of singular interconnected System (1) to give the conditions, which ensure that ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.B].sub.i]) is stabilizable and ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.Q].sup.1/2.sub.i])) is detectable.

Lemma 4 (see [32]). The sufficient and necessary condition for ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.B].sub.i])) to be stabilizable is that rank [mathematical expression not reproducible] p and (Ei,Ai,Bi) is stabilizable (i = 1,2,...,N).

Lemma 5 (see [32]). The sufficient and necessary condition for ([[bar.E].sub.i], [[bar.A].sub.i], [[bar.Q]sup.1/2.sub.i])) to be detectable is that (Ei,Ai,Ci) is detectable (i = 1,2,...,N).

Lemmas 4 and 5 can be proved by a method similar to that in reference [32], so the proof is omitted here.

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

Theorem 4. Suppose

(I) A1-A6 hold

(II) [parallel][[bar.P].sup.T.sub.i] [[bar.A].sub.ij][parallel] [less than or equal to] [[beta].sub.ij] (i,j = 1,2,...,N,i [not equal to] j);

(III) H is a nonsingular M matrix

(IV)[mathematical expression not reproducible]

(V) Let [y.sub.d] (k) = 0, [x.sub.i] (t) = 0, [u.sup.i] (t) = 0 (i = 1,2,...,N) for t < 0

Then, the controller with the preview effect, which enables the output signal of (1) to track the desired tracking signal asymptotically, is given as

u(t) = [[[u.sup.T.sub.1](t) [u.sup.T.sub.2](t)... [u.sup.T.sub.N](t)].sup.T], (51)

where

[mathematical expression not reproducible] 52)

in which [mathematical expression not reproducible] the expression [mathematical expression not reproducible].

Proof. According to Lemmas 4 and 5, when (I)-(IV) is true here, all conditions of Theorems 1 to 3 are satisfied, so the conclusion is valid. Thus, (32) gives the controller of the error system, in which the component [mathematical expression not reproducible] (i = 1,2,...,N) is determined by (28) of Theorem 2. We now derive the controller of the original system from (32) and (28).

Note that (28) is

[mathematical expression not reproducible] (53)

Integrate on [0, t] to get

[mathematical expression not reproducible] (54)

By exchanging the integration order of the last item on the right side of the upper formula and then integrating it, we can get (52). Then, combining [u.sub.i] (t) (i = 1,2,..., N), the controller of System (1) is attained, that is, (51).

Remark 4. In (52), [mathematical expression not reproducible] is the state feedback, [mathematical expression not reproducible] is the integrator, and [[bar.g].sub.i](t) is the preview feed forward of reference information; moreover, [u.sub.i] (O) is the initial value of input and [mathematical expression not reproducible] is the compensation of initial value.

5. Numerical Simulation

5.1. Numerical Simulation Method. In this section, the numerical simulation of a singular system

[mathematical expression not reproducible] (55)

is discussed. Here, matrices A, B, and C have appropriate dimensions, A is a square matrix, and E is a singular matrix and satisfies rank (E) = q < n. Let the state feedback be u(t) = Fx(t).

Because (55) contains the singular matrix E, it is impossible to calculate the state value of the next moment from System (55) and u(t) = Fx(t) by the usual direct discretization method, as normal systems do. Therefore, an algorithm is proposed to solve this problem.

Taking the sampling interval as h, it is noted that

[mathematical expression not reproducible]. (56)

At t = (k + 1)h, equation (55) is discretized to obtain

E x(kh) - x((k + 1)h)/-h = Ax(k + 1)h) + Bu((k + 1)h), (57)

that is,

(E - Ah) * ((k + 1)h) = Ex(kh) + Bhu((k + 1)h). (58)

For equation (58), an appropriate sampling interval h can be chosen so that matrix (E - Ah) is nonsingular. Then, (58) can be written as

x ((k + 1)h) = [(E - Ah).sup.-1] [Ex (kh) + Bhu ((k + 1)h)]. (59)

Unfortunately, since the right side of (59) contains the term u((k + 1)h), besides, u(t) = Fx(t), it is known that u ((k + 1)h) is related to x ((k + 1)h); therefore, equation (59) cannot be calculated. To overcome this difficulty, we take u(kh) as the approximate value of u((k + 1)h) and then substitute it into (59) to obtain the following iteration scheme:

x((k + 1)h) = [(E - Ah).sup.-1] [Ex(kh) + Bhu(kh)]. (60)

The rationality of replacing u((k + 1)h) with u(kh) is explained below. Notice that if the output of the closed-loop system can track the reference signal, there are x([infinity]) and u([infinity]) such that

[mathematical expression not reproducible] (61)

By adopting (56), (6[degrees]) can be written as [mathematical expression not reproducible]. Letting k [right arrow] [infinity], we get the same relation. Therefore, this method is reasonable. In other words, when k is large, there is u((k + 1)h) [approximately equal to] u(kh).

Notice that the output equation is y(t) = Cx(t), and then the iterative scheme of (55) is

[mathematical expression not reproducible] (62)

The convergence condition of the iterative scheme (62) is given below. The state feedback u(kh) = Fx(kh) is substituted into the equation of state to obtain the following closed-loop system:

x ((k + 1)h) = [(E - Ah).sup.-1] (E - hBF) * (kh). (63)

Obviously, a sufficient condition for the convergence of the iteration scheme (62) is that the spectral radius of [(E - Ah).sup.-1] (E - hBF) (i.e., the maximum value of the absolute value of the eigenvalue) is less than 1 [33].

5.2. Simulation Example. In this section, the effectiveness of the designed controller is verified by numerical simulation.

Considering the singular interconnected system with two subsystems (i.e., N = 2) [n.sub.1] = 3 and [n.sub.2] = 2, the coefficient matrices are

[mathematical expression not reproducible] (64)

Let the weight matrix of the performance index function (7) be

[mathematical expression not reproducible] (65)

[y.sub.d] (t) can be chosen as

[mathematical expression not reproducible], (66)

Take the initial state be [x.sub.1] (0) = [[0.02 0 0].sub.T], [u.sub.1] (0) = 0, [x.sub.1] (0) = [0 0.01 ].sup.T], and [u.sub.2] (0) = 0. Select [[alpha].sub.1] = 0.3 and [[alpha].sub.1] = 0.7. It is easy to verify that the given system and the reference signal meet A1-A6.

We conducted numerical simulation for [l.sub.r] = 0 (without reference signal preview), [l.sub.r] = 0.5, and [l.sub.r] = 1.2, respectively. The solution of the Riccati equation for two isolated subsystems and the feedback gain matrix of the controller is obtained by using MATLAB as follows:

[mathematical expression not reproducible] (67)

Let us take

[mathematical expression not reproducible] (68)

Obviously, there are [gamma].sub.1] = [[lambda].sub.min]([W.sub.1]) = 10 and [gamma].sub.2] = [[lambda].sub.min]([W.sub.2]) = 10. The solution of the singular Lyapunov equation is obtained by using MATLAB:

[mathematical expression not reproducible] (69)

Let [mathematical expression not reproducible]. So,

[mathematical expression not reproducible] (70)

The eigenvalues of H are 18.570590721965054 and 1.429409278034949, which means that H is a nonsingular M matrix. In conclusion, the conditions of Theorem 4 are all satisfied.

Selecting the sampling interval h = 0.01, the calculation shows that E - Ah is nonsingular and the spectral radius of [(E - Ah).sup.-1] (E - hB[F.sub.x]) is less than 1. The tracking effect of System (1) is shown in Figure 1.

As can be seen from Figure 1, with the gradual increase in time t, the output signals of the singular interconnected system (1) can asymptotically track the desired tracking signal under different preview lengths. In addition, with the increase in the preview length, overshoot and adjustment time are decreasing. The tracking error of singular interconnected systems is given in Figure 2. It can be found from Figure 2 that, compared with the controller with no preview effect (i.e., [l.sup.r] = 0), the controller with the preview effect can reduce the overall tracking error.

Figures 3 and 4 depict the control input component diagram of the singular interconnected System (1). It can be seen from the graph that the increase in the preview length will not make the control input change dramatically but makes the control input change smoothly.

6. Conclusion

In this paper, the basic theory of preview control is extended to continuous-time singular interconnected systems and the problem of preview tracking control for such systems is studied. With the help of decomposition theory of largescale systems, several isolated subsystems are obtained by deleting related terms. Furthermore, the common methods of preview control theory are adopted to construct error systems for isolated subsystems, and the tracking problem is transformed into a regulation problem. For each isolated error system, the preview controller is designed and the obtained controllers are combined as the controllers of the singular interconnected system error system. By constructing Lyapunov functions and using the properties of nonsingular M-matrices, the stability of closed-loop error interconnected systems is discussed, and the criterion theorem to ensure its stability is given. Finally, the sufficient conditions for the existence of preview controllers and controllers in the original singular interconnected system are derived.

Furthermore, a numerical simulation algorithm for continuous-time singular systems is proposed, which does not depend on the restricted equivalent transformation and is suitable for the numerical simulation of all continuoustime singular systems. The theoretical results and numerical simulation show that the designed controller is able to make the output of the system to track the desired tracking signal without a static error, and the tracking performance is improved with the increase in the preview length.

https://doi.org/10.1155/2019/6175837

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

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Hao Xie, (1) Fucheng Liao [ID], (1) and Jiamei Deng (2)

(1) School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

(2) School of Built Environment, Engineering and Computing, Leeds Beckett University, Leeds LS6 3QS, UK

Correspondence should be addressed to Fucheng Liao; fcliao@ustb.edu.cn

Received 28 September 2019; Accepted 26 November 2019; Published 19 December 2019

Academic Editor: Hung-Yuan Chung

Caption: Figure 1: The output response of singular interconnected systems.

Caption: Figure 2: The tracking errors of singular interconnected systems.

Caption: Figure 3: Control input components [u.sub.1] (t) of singular interconnected systems.

Caption: Figure 4: Control input components [u.sup.2] (t) of singular interconnected systems.
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Title Annotation:Research Article
Author:Xie, Hao; Liao, Fucheng; Deng, Jiamei
Publication:Mathematical Problems in Engineering
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Date:Dec 31, 2019
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