Linear Quadratic Regulator for Impulse Uncontrollable Singular Systems

Singular systems, also called descriptor systems, arise naturally in chemical processes, mechanical systems, and so on. Therefore, since the last seventies, singular systems have attracted intensive attention. Linear quadratic regulator (LQR) problems play an important role in the field of control engineering. Therefore the LQR problems for descriptor systems have been investigated in the literature. The earliest papers on this problem are those of [1] and [2]. This problem were also addressed in ref. [3,4], where a generalized Riccati equation is shown to result directly from the necessary conditions for the existence of an optimal control. The generalized Riccati equation proposed in [3] may not have solutions, which was shown in ref. [5]. Bender and Laub [5] approached this problem by performing a singular value decomposition (SVD) to isolate the dynamics from non-dynamic portion, and four equivalent Riccati equations were derived in the SVD coordinate system. [6] investigated that the robust properties of the LQR for singular systems with single input. A parametrization of the optimal feedback gains was used to show that certain robust properties can be achieved by an appropriate choice of the feedback gain. The LQR problem for linear time-varying singular systems was addressed in ref. [7]. [8] derived a Riccati equation formulated in the original state parameters under which the problem has a solution. The LQR problem for non-regular singular systems was solved by converting a non-regular singular system into a regular one with index one in ref. [9].


Introduction
Singular systems, also called descriptor systems, arise naturally in chemical processes, mechanical systems, and so on. Therefore, since the last seventies, singular systems have attracted intensive attention. Linear quadratic regulator (LQR) problems play an important role in the field of control engineering. Therefore the LQR problems for descriptor systems have been investigated in the literature. The earliest papers on this problem are those of [1] and [2]. This problem were also addressed in ref. [3,4], where a generalized Riccati equation is shown to result directly from the necessary conditions for the existence of an optimal control. The generalized Riccati equation proposed in [3] may not have solutions, which was shown in ref. [5]. Bender and Laub [5] approached this problem by performing a singular value decomposition (SVD) to isolate the dynamics from non-dynamic portion, and four equivalent Riccati equations were derived in the SVD coordinate system. [6] investigated that the robust properties of the LQR for singular systems with single input. A parametrization of the optimal feedback gains was used to show that certain robust properties can be achieved by an appropriate choice of the feedback gain. The LQR problem for linear time-varying singular systems was addressed in ref. [7]. [8] derived a Riccati equation formulated in the original state parameters under which the problem has a solution. The LQR problem for non-regular singular systems was solved by converting a non-regular singular system into a regular one with index one in ref. [9].
It is worth pointing out that all the results on the LQR problem for singular systems mentioned above were obtained with an assumption that the singular systems are impulse controllable, that is, the impulsive modes can be removed by applying a state feedback. However, many practical systems, especially constrained mechanical systems (see Example 1), can not be modelled by an impulse controllable singular system. Therefore, the LQR problem for impulse uncontrollable singular systems is not only theoretically interesting but also practically important [10].
In this paper, the LQR problem for impulse uncontrollable singular systems is addressed. In order to guarantee the existence of the cost function, solutions to the singular systems should contain no impulse, which can be achieved by a feedback if the systems are impulse controllable or by choosing initial conditions properly for impulse uncontrollable cases [11]. Moreover, the final conditions should be properly chosen as well to guarantee that the optimal solution to the LQR problem is impulse-free. In order to facilitate the analysis, the singular system is assumed to be in the Weierstrass form [12].

Problem Formulation
Consider the problem of finding u(t) to minimize and E is a square matrix of rank q n ≤ . The pencil (sE-A) is assumed to be regular, that is, Assume that there exist invertible matrices U and V so that where p is the costate variable defined in (7) or (14), ( ) = with I j being a j×j identity matrix. Note that the first λ 2 Jordon blocks are nontrivial with the nilpotency greater than 1 and the rest are trivial with the nilpotency equal 1.
The following assumptions are sufficient for solutions to the costate equation of the LQR problem to be impulse-free. In addition, in order to make the cost function finite for the case of an infinite horizon, the following assumption is necessary.

Assumption 6:
The finite modes of (2) are stabilizable, that is,

Solutions of the LQR Problem
This section is devoted to finding a solution to the two-point boundary value problem described by (16)-(20) with the boundary conditions (9)- (12). Before doing this, first let us examine the implication of Assumptions 2-3.
With the special structure of N i in mind, the following lemma can be verified using Assumption 2. It will prove useful to partition x 2 , p 2 , B 2 , Q 21 , Q 22 , Q f21 , Q f22 as follows: Furthermore, for , 22 In the Weierstrass form, (2) can be expressed as The solution to (4) is given by with I = 1, …,γ. (3)-(4) is impulse controllable (controllable at ∞) if and only if the rows of B 2 corresponding to the bottom rows of the nontrivial Jordan blocks (v i > 1) of N are linearly independent [9]. The LQR problem for impulse controllable singular systems have been solved, so we assume that (3)-(4) is not impulse controllable. Without loss of generality, the following assumption is made.
γ are linearly independent and the matrix composed of the rows of B 2 corresponding to the bottom rows of the Jordan blocks 1, …, γ 2 has rank of γ 1 .
Assumption 1 means that only impulsive modes corresponding to the nontrivial Jordon blocks 1, …, γ 1 are controllable and impulsive modes for the Jordon blocks 1 2 1,..., γ γ + are uncontrollable. With this assumption, any impulses caused by the initial conditions can be removed by a feedback for subsystems corresponding to the Jordon blocks 1, …, γ 1 .
The following assumptions are made so that solutions to subsystems corresponding to the Jordon blocks γ 1 + 1, ..., γ 2 are impulse-free.
, from which the conclusion of the lemma can be directly drawn. to be impulse-free, which will be proved in the following lemma.  (5) gives which, according to Lemma 1, means that ( ) Similar to the proof of Lemma 1, it is straightforward to prove the following lemma.
The following lemma indicates that the final values for Proof. Due to Assumption 5, (12) can be rewritten as The following lemma shows that solutions to the costate subsystems corresponding to the Jordon blocks Proof. It can be proved that solutions to (19) is given by It follows from Lemma 3 and Lemma 5 that The initial conditions (10)  Similarly, the final conditions (11) and (12) can be written as To simplify expressions, define the following matrices 22   Proof. This lemma can be easily verified by applying Assumption 1.

Assumption 7:
( ) 1  2  22  12  11  12  2  2  22  22  22  22  2= > 0 Thus, Assumption 7 implies that ( ) 2 p t , and u(t) can be uniquely determined from (33), (35), and (36) as This can then be substituted back into (32) and (34), which results in a usual two-point boundary value problem of linear-quadratic optimal control with ( ) with the boundary conditions of (37) and (38) where , By following the approach proposed by Bender and Laub [5], the optimal feedback gain can be determined by solving the equation is an arbitrary real matrix and U can be determined from 21 2 , =T It is straightforward to prove that 12 21= T Q Q .

Examples
Example 1. Consider a simple example of a planar manipulator constrained so that the end effector can only move along a straight line. The equations of motion [10] is given by Thus, the forces of constraint are 1 = f λ and 2 = 2 f λ . Set

Conclusion
In this paper, the linear quadratic regulator problem has been solved for a linear time-invariant singular systems which may not be impulse controllable. The impulse controllable and uncontrollable subsystems have been separated by transforming the singular system into the Weierstrass form. The sufficient conditions have been proposed so that the impulse uncontrollable subsystems and their corresponding costate subsystems admit a unique impulse-free solution. The necessary conditions for the minimization of the quadratic cost function have been converted to the Riccati equation.