Design of an Optimal Preview Controller for Linear Discrete-Time Descriptor Noncausal Multirate Systems

The linear discrete-time descriptor noncausal multirate system is considered for the presentation of a new design approach for optimal preview control. First, according to the characteristics of causal controllability and causal observability, the descriptor noncausal system is constructed into a descriptor causal closed-loop system. Second, by using the characteristics of the causal system and elementary transformation, the descriptor causal closed-loop system is transformed into a normal system. Then, taking advantage of the discrete lifting technique, the normal multirate system is converted to a single-rate system. By making use of the standard preview control method, we construct the descriptor augmented error system. The quadratic performance index for the multirate system is given, which can be changed into one for the single-rate system. In addition, a new single-rate system is obtained, the optimal control law of which is given. Returning to the original system, the optimal preview controller for linear discrete-time descriptor noncausal multirate systems is derived. The stabilizability and detectability of the lifted single-rate system are discussed in detail. The optimal preview control design techniques are illustrated by simulation results for a simple example.


Introduction
Descriptor system theory has obtained many excellent results in the control areas; the main scholarly reports can be seen in [1,2]. In recent years, the literature [3] considered the optimal fusion problem for the state estimation of discretetime stochastic singular systems with multiple sensors and correlated measurement noise and obtained the optimal fullorder filters and smoothers for the original system. The literature [4] proposed a novel suboptimal control method for a class of nonlinear singularly perturbed systems based on adaptive dynamic programming; the literature [5] discussed finite-time robust dissipative control for a class of descriptor systems, and the control system was effectively confined within the desired state-space ellipsoid. The literature [6] provided a necessary and sufficient condition to guarantee admissibility for positive continuous-time descriptor systems. Notably, the literature [7] combined descriptor system theory with preview control theory and successfully obtained the optimal preview controller with preview action for the linear discrete-time descriptor causal system; the literature [8] derived the optimal preview controller for discrete-time descriptor causal systems in a multirate setting. The literature [9] obtained the optimal preview controller with preview feedforward compensation for linear discrete-time descriptor systems with state delay. In addition, linear quadratic optimal regulator theory for the continuous and discrete descriptor system tends to be complete as discussed in [10][11][12].
In recent years, the multirate digital control system has also obtained many new results as discussed in [13][14][15][16]. The characteristics of multirate systems are as follows. First of all, the systems are multiinput and multioutput systems. Second, the sampler and retainer of input channels and output channels have different sampling periods as discussed by Xiao [13]. For such systems, if the designed regulator satisfies appropriate multirate characteristics, it should have a better performance than that of the single-rate regulator.
The previous multirate systems have been basically studied for normal systems; however, this paper successfully constructs the optimal preview controller on the basis of the literature [8] for linear discrete-time descriptor noncausal 2 The Scientific World Journal multirate systems. The effectiveness of the proposed method is shown by simulation.

Description of the Problem and Basic Assumptions
Consider the regular linear discrete-time descriptor noncausal system described by where ( ) ∈ , ( ) ∈ , and ( ) ∈ are its state, control input, and measure output, respectively; , ∈ × , ∈ × , and ∈ × are constant matrices; here, is a singular matrix with rank( ) = < .
As [8], we need to make the following basic assumptions: (1) where = , V( ) ∈ , and ∈ × such that the closed-loop system is causal as discussed by [2]; that is, Obviously, taking advantage of the characteristic that any matrix can be transformed to a canonical form by elementary transformation, there always exist nonsingular matrices 1 , 1 , such that 11 12 21 22 ] , where 1 ( ) ∈ and 2 ( ) ∈ − . As [8], the system (4) is restricted equivalent to Because elementary transformation does not change the causality of the system, the system (7) is also a causal system. As a result, matrix 22 is nonsingular as discussed by [2]. Then the optimal preview problem for the descriptor noncausal system is transformed into the one for the descriptor causal system.
As [8], let the error signal We want to get The quadratic performance index function for the system (1) is defined as where the weight matrices satisfy > 0 and > 0. Δ is the first-order forward difference operator; that is, Δ ( ) = ( + 1) − ( ). In order to smooth the conduct of the study, we will also make the following assumptions.
is nonsingular, where the meaning of the various symbols is given in the following discussion.
is of full row rank, where the meaning of the various symbols is given in the following discussion.

The Derivation of the Single-Rate System
The system (7) is a multirate system according to the above discussion. We adopt the discrete lifting technique to convert (7) to a single-rate system.
] . (14) In order to design the optimal preview controller for linear discrete-time descriptor noncausal multirate systems (1), continue to lift the static output feedback (3).
The above equations may be represented in the matrix form: . . .

Construction of the Descriptor Augmented Error System
As [8], we take advantage of the first-order forward difference operator Δ: Construct the vector . . .
where is a × matrix; notice the identity ( + 1) = ( ). Using the identity and (31), we obtain This is the constructed descriptor augmented error system. The dimension of the system (34) is + + , and

Design of an Optimal Regulator for Descriptor Augmented Error Systems
As [8], we convert the performance index (10) as follows By (A6),̃2̃2 > 0. Then, adopting the first-order forward difference operator on both sides of (21), the performance index (36) can be written as If we denotẽ=̃1̃1 −̃1̃2[̃2̃2] −1̃2̃1 , it is easy to seẽ≥ 0. Let Substituting (43) into (34), we get Then, the problem becomes an optimal control problem for a normal system (44) under the performance index (42). According to the results in Duan [17], we immediately get the following.
is detectable, the optimal regulator of the system (44) minimizing the performance index (42) is given by where is the unique symmetric semipositive definite solution of the algebraic Riccati equation: (46)

The Existence Conditions of the Optimal Regulator
We will verify the existence conditions of the optimal regulator for (44).
Proof. Notice that the system (44) is derived from the system (34) under the state feedback (43). We know that the state feedback does not change the stabilizability of the system as discussed by [17], so the system (44) is stabilizable if and only if the system (34) is stabilizable; that is, This completes the proof.
is of full row rank.
Proof. First, we have Noticing the structure of Φ and , Theorem 5 can be proved by Lemma 1(a) in Liao et al. [14]. Here we omit the proof.
Note that the matrix in (47) is Ψ in (A7).
By using formula (6) and the nonsingularity of (49) So, the system (7) is stabilizable if and only if the system (4) is stabilizable. (4) is stabilizable if and only if the system (1) is stabilizable; that is, (A1) holds.
In summary, this completes the proof.

Remark 7.
This theorem also proves that the systems (7) and (1) have the same stabilizability.
Combining Theorems 4, 5, and 6, if the original system (1) is stabilizable and Ψ in (A7) is of full row rank, the final formal system (44) is also stabilizable. Furthermore, the condition is both necessary and sufficient. These conditions ensure that the state feedback gain in Theorem 3 exists.

Theorem 8. If (A2) holds, the system (4) is detectable.
Proof. Since the output feedback does not change the detectability of the system as discussed by [2], this completes the proof. ] .

(51)
This shows that the systems (4) and (7) have the same detectability.
Again from [8], the system (7) is detectable if and only if ( 1 1 ) is detectable.
In summary, this completes the proof.
This completes the proof.

The Optimal Preview Controller for the Original System
Returning to the optimal control input (45) of the descriptor augmented error system and the related formula (43), we get (21) and (54), we continue to get . . .
( ), , and are decomposed into The Scientific World Journal 9 The above equation can be further written as That is, If is substituted by − 1, we obtain the control input of the most important theorem. . . .

Numerical Example
Consider the following regular linear discrete-time descriptor noncausal system in the form of (1): respectively. Through calculating, the above system satisfies all conditions required in the paper. By MATLAB simulation, the gain matrix in output feedback is taken as = 2, and the coefficient matrices in (7) By MATLAB simulation, the output response of the linear discrete-time descriptor noncausal multirate system (with preview action and no preview action) is shown in Figure 1. The error signals are shown in Figure 2. Note that the preview action significantly reduces the error. In particular, the error signal is asymptotically zero.  The output responses are shown in Figure 3. The error signals are shown in Figure 4.
From Figures 1-4, we can easily see the effectiveness of the present controller of this paper. On the one hand, when using preview control, the output curve can track the desired signal faster; on the other hand, the overshoot is smaller.

Conclusion
This paper studied the optimal preview controller for linear discrete-time descriptor noncausal multirate systems. By making use of the characteristics of causal controllability and causal observability, the original system was converted into a descriptor causal closed-loop system. Then, using the characteristics of a causal system and a discrete lifting technique, the descriptor causal closed-loop multirate system was changed into a single-rate normal system. Taking advantage of the conventional method of the error system in preview control theory, a descriptor augmented error system is constructed, and the problem is transformed into a regulator problem. Finally, the optimal preview controller is designed according to the related theory of preview control. From preview control theory, the obtained closed-loop system contains an integrator so that the response of the system does not have static error. The numerical simulation showed the effectiveness of the proposed preview control system.