On the LQ Optimization Subject to Descriptor System under Disturbance

MATERIALS AND METHODS

In order to find the desired result, let us define a new vector z such that = ∈. From the relation ∈ = (y-y_d), the system (1) can be written as follows:

(5)

Since the disturbance ω and the desire output y_d are constant vectors, the differentiation of Eq. 5 for t result:

(6)

By defining:

(7)

the problem is Ω equivalent to the following optimization problem:

(8)

(9)

It is obvious that rank (L) = p+r<n+r. Since the system (1) is regular, the system (9) is also regular, i.e., there exist such that det . It is also obvious that Eq. 8 and 9 constitute a LQ optimization problem subject to descriptor system without disturbance with state ς and control ϑ. By using the theory in Cobb⁸ and Duan⁹, the optimal control for optimization problem Eq. 8 and 9 exist if and only if system (9) is impulse controllable and stabilizable⁷.

Under the assumption controllable impulse of the system (9), there exist a matrix K∈ℝ^m×(n+r) such that:

(10)

By choosing a feedback control ϑ = Kς+v, for some new control v∈ℝ^m and apply it to system (9), we have:

(11)

Thus we have the normal decomposition of the form:

for some nonsingular matrices⁸ Q₁, P₁∈ℝ^(n+r)×(n+r). Denoting:

(12)

the system (11) is equivalent to:

(13)

(14)

Using Eq. 13, the objective function (8) becomes:

(15)

where:

(16)

(17)

By using the substitution:

(18)

where:

(19)

the LQ optimization problem (8) and (9) can be converted into the following normal LQ optimization problem:

(20)

(21)

where:

(22)

and:

(23)

RESULTS AND DISCUSSION

It is obvious that Eq. 20-23 constitute the normal LQ optimization problem with state ς₁ and control w. Based on the theory of normal LQ optimization problem, the optimal control for Eq. 20 and 21 exist and unique if the system (21) is stabilizable³. Note that the stabilizability of the system (9) implies the system (13) is stabilizable. Since Eq. 21 is closed-loop system resulted in by applying a state feedback with gain matrix to the system (13) and the fact that the state feedback does not change stabilizability, we get the system (21) is stabilizable.

It follows that the solution of the problem (20) and (21) is given by:

(24)

where, ς₁ is the solution of the following initial value problem:

(25)

and the matrix P is the unique symmetric positive definite solution of the following algebraic Riccati equation:

(26)

with the minimum value is Moreover, the solution of initial value problem (25) is stable, in the sense that ς₁(t)→0 if t→∞.

Using Eq. 18 we have:

(27)

and using Eq. 19, 12 and 14, we have:

(28)

Moreover, from Eq. 7 and 12 we also have:

(29)

Therefore, the optimal solution of the optimization problem Ω are u^opt and x^opt, where, u^opt satisfies (28) and x^opt satisfies (29). Moreover, since ς₁(t)→0 when t→∞ we also obtain ∈ = (y-y_d)→0 when t→∞.

Thus we have proved the following Theorem that constitutes the main result of this study.

Theorem 3.1: If the system (9) is impulse controllable and stabilizable, then the optimal control u^opt and the optimal state x^opt for the optimization problem Ω exist and unique, where, u^opt satisfies the Eq. 28 and x^opt satisfies (29). Moreover, y→y_d when t→∞.

This result constitute a new contribution in field of the LQ optimization subject to descriptor system area that require condition ∈ = (y-y_d)→0 when t→0. It is obvious that this result different to the result of Chen⁶ and Fang et al.⁷. This difference is due to the additional condition ∈ = (y-y_d)→0 when t→∞ in our problem. The findings show that this sufficient condition constitutes a method to find the pair the optimal control and optimal state of the LQ optimization problem subject to descriptor system under disturbance.

CONCLUSION

A sufficient condition that guaranteeing the existence of pair the optimal control and optimal state of the LQ optimization problem subject to descriptor system under disturbance has been established, namely as given in Theorem 3.1. The findings show that this sufficient condition constitutes a method to find the pair the optimal control and optimal state of the LQ optimization problem subject to descriptor system under disturbance. This result constitute a new contribution in the dynamic optimization area.

SIGNIFICANCE STATEMENT

The study discovers a sufficient condition that guaranteeing the existence of pair the optimal control and optimal state of LQ optimization problem subject to descriptor system under disturbance. The sufficient condition can be beneficial for both practitioners and researches. This study will help the researcher to solve the model optimization of the form linear quadratic governed by the descriptor system.