1 Introduction

Since the beginning of the seventies, linear time-invariant singular systems have been extensively studied, and a fairly complete theory has been established, see e.g. books (Campbell, 1982; Dai, 1989) and the references therein. Linear time-varying singular systems have also received some research attention in the past decade. Campbell and Petzold(1983) have shown that analytically solvable linear time-varying singular systems can be put into standard canonical form via analytic coordinates transformation. Observability and controllability of linear time-varying singular systems have been studied in (Nichols et al., 1991; Terrell et al., 1991). Based on a given output structure associated with linear time-varying singular systems, a decomposition that decouples observable and unobservable subspaces is developed in(Terrell, 1994). Impulse elimination problem by state feedback is treated(Wang, 1996)for linear time-varying singular systems.

For linear time-invariant singular systems, disturbance decoupling problems have been addressed by several authors, such as Fletcher and Aasaraai(1989), Zhou et al.(1987). However, there is no paper dedicated to the study of the same problem for linear time-varying singular system.

In this paper, the problem of disturbance decoupling for linear time-varying singular systems are considered. Assumptions are introduced, and an algorithm is proposed for constructing a set of new coordinates in which the system assumes a simple form. A time-varying feedback control law is then constructed, which guarantees, under some conditions which are made clear later, the closed-loop system is regular, free of impulses, and its outputs are unaffected by disturbances.

2 Problem formulation and basic assumptions

Consider linear time-varying singular systems

(1)

(2)

(3)

together with an initial condition x₁(0)=x₁₀, where x_i∈R^n_i, i=1, 2, u∈R^m is the vector of inputs, y∈R^m is the vector of output, w∈R^q is the vector of disturbances, A_i, A_i+2, B_i, C_i, H_i, i=1, 2, D and N are matrices with dimensions n₁×n_i, n₂×n_i, n_i×m, m×n_i and n_i×q, m×m, and m×q, respectively.

Remark 1    A general linear singular system of the form

(4)

can be easily changed into a system of the form (1)~(2) by using the restricted equivalent transformation.

The objective of the paper is to seek sufficient conditions for the existence of a linear time-varying feedback

(5)

such that the closed-loop system

(6)

has the following properties:It is strongly regular, i.e. it has a unique solution without impulses for any piecewise inputs v and disturbance w and the initial condition x₁(0)=x₁₀; Its outputs are unaffected by disturbances.

Such a feedback is called disturbance decoupling feedback. If there exists a disturbance decoupling feedback for a system, then the disturbance decoupling problem is said to be solvable for the system.

Now we would like to introduce the basic assumptions in the paper.

(A1)    There exist integers ρ₁, …, ρ_l, such that

(7)

where M_i^k=M_i^k-1 A₁ with M_i⁰=A₃ⁱ, A₃ⁱ, A₄ⁱ, and B₂ⁱ the ith-rows of A₃, A₄, and B₂, respectively.

(A2)    The matrix[b, c]has full row rank n₂ where

with

Now for convenience, set

where

with H₂ⁱ the ith-row of H₂.

Lemma 1    if (A₁) and (A₂) are satisfied, then the vectors

are linearly independent.

Proof:    Omit this.

3 Decoupling algorithm and basic results

This section begins with the following algorithm which plays an important role in discussing the problem in question.

Algorithm 1

Step 0.    Set

where C₂ⁱ and Dⁱ are the ith-rows of C₂ and D respectively. If σ_i⁰ is equal to n₂, then by Assumption(A₁), there exists a unique vector E_i⁰ of dimension n₂ such that

Denote T_i⁰=C₁ⁱ-E_i⁰a with C₁ⁱ the ith-row of C₁. Otherwise, set r_i=0 and quit the algorithm.

Step k+1.    Assume that a sequence of T_i⁰, …, T_i^k has been defined through step 1 to k. Set

If σ_i^k+1 is equal to n₂, then there exists a unique vector E_i^k+1 of dimension n₂ such that

(8)

Denote . Otherwise, set r_i=k+1 and quit the algorithm.

Performing Algorithm 1 for i=1, …, m, produces integers r₁, …, r_m. Now set

where

with Nⁱ the ith-row of N.

Without loss of generality and for simplicity, assume that r_i>0 for i=1, …, h and r_i=0 for i=h+1, …, m.

Next assumption is made in the rest of the paper.

(A3)    The matrix

is nonsingular for any t∈[0, T].

Lemma 2    Assume that Assumptions (A₁)-(A₃) are satisfied. Then the vectors

are linearly independent for any t∈[0, T].

Proof:    Omit this.

4 Main results

This section will be devoted to constructing a disturbance decoupling feedback. To this end, we first construct a coordinates transformation in which the system assumes a simple form.

Let r=ρ₁+…+ρ_l+r₁+…+r_h. Then if r < n, then it follows from Lemma 2 that there exist(n-r) smooth functions T₁, …, T_n-r, such that the vectors

are linearly independent for any t∈[0, T]. Therefore, the matrix

constitutes a coordinates transformation.

Set

Then, it follows form (A1)~(A2) and Algorithm 1 that in the new coordinates(η, χ, ξ) system (1)~(3) is expressed as follows:

(9)

(10)

(11)

(12)

(13)

where η=[η₁, …, η_n-r]^τ, χ=[χ₁¹, …, χ₁^ρ₁, …, χ_l¹, …, χ_l^ρ_l]^τ, ξ=[ξ₁¹, …, ξ₁^r₁, …, ξ_h¹, …, ξ₁^r_h]^τ.

It is not difficult to deduce that necessary and sufficient conditions for (10) to have a impulse-free response for any piecewise continuous disturbance w are as follows:

(A4)    In [0, T]

(14)

(A5)    χ_i^k(0)=0 for k=1, …, ρ_i, i=1, …, l, that is M_i^j(0)x₁(0)=0 for k=0, …, ρ_i-1, i=1, …, l.It follows form (A4) and (A5) that χ_i^k(0)=0 for k=1, …, ρ_i, i=1, …, l, which means

(15)

As a consequence, under the assumptions (A4) and (A5), system (9)~(13) is equivalent to the following one

(16)

(17)

(18)

(19)

Next let us construct a disturbance decoupling feedback in two steps. First, for any F₂ such that b+cF₂ is invertible in [0, T] (note that such F₂ always exists by the assumption (A2)), imposing the feedback on system (16)~(19) produces the following closed-loop system

(20)

(21)

(22)

(23)

Due to the nonsingularity of b+CF, x₂ can be uniquely determined form (21) as

(24)

Substituting this into (20)~(23) yields

(25)

(26)

(27)

with for i=1, …, m

Now let

Then, since the relation

(28)

holds for any F₂ such that b+cF₂ is nonsingular for all t∈[0, T], it follows form Assumption (A3) that matrix is nonsingular for all t∈[0, T].

By imposing the feedback

That is

(29)

On system (25)~(27). It is easily seen that the closed-loop system takes the form of

(30)

(31)

(32)

Now assume that

(A6)    In [0, T]

(33)

(A7)    There exists a matrix F₂ such that b+cF₂ is nonsingular for any t∈[0, T] and .

Under (A6) and (A7), it is easily seen that system (30)~(32) becomes

(34)

(35)

(36)

It is no hard to realize that this closed-loop system has already been decoupled form disturbances. The discussion stated above can be summarized as follows:

Theorem 1    Assume that(A1)-(A7) are satisfied. Then, the disturbance decoupling problem is solvable. Moreover, the disturbance decoupling feedback is given by

with F₂ such that b+cF₂ is nonsingular for any t∈[0, T].

5 Conclusion

The disturbance decoupling problem has been considered for linear time-varying singular systems. A new algorithm has been proposed, by which a set of new coordinates, in which the system assumes a simple form, can be constructed. Sufficient conditions have been derived, which guarantees that there exists a feedback controller such that the closed-loop system has a unique solution without impulses and is decoupled form disturbances.