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Linear Quadratic Regulator for Impulse Uncontrollable Singular Systems | OMICS International
ISSN: 2332-0796
Journal of Electrical & Electronic Systems
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Linear Quadratic Regulator for Impulse Uncontrollable Singular Systems

Xiaoping Liu1,2* and Yufu Jia2

1Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, P7B 5E1, Canada

2Faculty of Electronics and Information Engineering, Liaoning University of Science and Technology, Anshan, Liaoning Province, China

*Corresponding Author:
Xiaoping Liu
Department of Electrical Engineering
Lakehead University
Thunder Bay, ON, P7B 5E1 Canada
Tel: 807-343-8613
E-mail: [email protected]

Received Date: November 08, 2016 Accepted Date: November 28, 2016 Published Date: December 05, 2016

Citation: Liu X, Jia Y (2016) Linear Quadratic Regulator for Impulse Uncontrollable Singular Systems. J Electr Electron Syst 5: 207. doi:10.4172/2332-0796.1000207

Copyright: © 2016 Liu X, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This paper investigates the linear quadratic regulator problem for singular systems which may contain some impulse uncontrollable modes. By transforming to the Weierstrass form, impulse controllable and uncontrollable subsystems can be separated. Sufficient conditions are proposed, under which solutions to both state and costate equations for the linear quadratic regulator problem are impulse-free. The necessary conditions for the minimization of the quadratic cost function are converted to a two-point boundary value problem, which can be solved by either computing its back-time state transition matrix or solving a Riccati equation. Two examples are provided to illustrate the main results.

Keywords

Singular systems; Descriptor systems; Linear quadratic regulator; Optimal feedback

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 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

Equation (1)

with the constraint

Equation (2)

Where R = RT > 0 , Q = QT ≥ 0 , x (t )∈Rn , x (t )∈Rn , u (t )∈Rm , and E is a square matrix of rank qn . The pencil (sE-A) is assumed to be regular, that is, det (λ EA) ≠ 0 for some λ. Let ρ = deg(det (λ EA)). Assume that there exist invertible matrices U and V so that

Equation

where p is the costate variable defined in (7) or (14), x1(t) ∈ , EquationEquationEquationis a nilpotent matrix of the nilpotency v. Without loss of generality, assume that N takes a Jordan normal form of N=diag{N ,..., } with the Jordon block Ni being a shift matrix of size Equationwhere Equationwith Ij being a j×j identity matrix. Note that the first λ2 Jordon blocks are nontrivial with the nilpotency greater than 1 and the rest are trivial with the nilpotency equal 1.

It will prove useful to partition x2, p2, B2, Q21, Q22, Qf21, Qf22 as follows:

Equation

with EquationEquation

Furthermore, for i =1,...,γ1 , partition

Equation

Equation

Equation

Equation

Equation

Equation

Equation

In the Weierstrass form, (2) can be expressed as

Equation (3)

Equation (4)

with i = 1,...,γ . The solution to (4) is given by

Equation (5)

with I = 1, …,γ. (3)-(4) is impulse controllable (controllable at ∞) if and only if the rows of B2 corresponding to the bottom rows of the nontrivial Jordan blocks (vi > 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.

Assumption 1:Equation are linearly independent and the matrix composed of the rows of B2 corresponding to the bottom rows of the Jordan blocks 1, …, γ2 has rank of γ1.

Assumption 1 means that only impulsive modes corresponding to the nontrivial Jordon blocks 1, …, γ1 are controllable and impulsive modes for the Jordon blocks γ1 +1,...,γ2 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, …, γ1

The following assumptions are made so that solutions to subsystems corresponding to the Jordon blocks γ1 + 1, ..., γ2 are impulse-free.

Assumption 2:Equation

Assumption 3:Equation

The following assumptions are sufficient for solutions to the costate equation of the LQR problem to be impulse-free.

Assumption 4:Equationfor Equation

Assumption 5:Equation for Equation

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, ( A1,B1) is stabilizable.

Necessary Conditions

The necessary conditions for J to be minimized are

Equation (6)

Equation (7)

Equation (8)

with Equation and the boundary conditions Ex (0) = Ex0 andEquation which, in the Weierstrass form, can be written as

Equation (9)

Equation (10)

Equation (11)

Equation (12)

where i =1,...,γ . It follows from (6)-(8) that

Equation (13)

Equation (14)

Equation (15)

which, in the Weierstrass form, can be expressed as

Equation(16)

Equation (17)

Equation (18)

Equation (19)

Equation (20)

where i = 1,...,γ .

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 Ni in mind, the following lemma can be verified using Assumption 2.

Lemma 1: If Assumption 2 is satisfied, then Equation for EquationEquationWhereEquation is the j-th row of Equation

Proof. A simple calculation shows that

Equation

for i =γ1 +1,...,γ2 , from which the conclusion of the lemma can be directly drawn.

Assumptions 2-3 are sufficient for solutions of the subsystems corresponding to the Jordon blocks γ1+1,...,γ2to be impulse-free, which will be proved in the following lemma.

Lemma 2: If Assumptions 2 and 3 are satisfied, then Equation for i1 +1,...,γ2 .

Proof. Note that Assumption 3 implies Equation for i1 +1,...,γ2 . Applying this and Assumptions 2 to (5) gives

Equation

which, according to Lemma 1, means that Equation for i =γ1 +1,...,γ2.

Similar to the proof of Lemma 1, it is straightforward to prove the following lemma.

Lemma 3: If Assumption 4 is satisfied, then Equation forEquation and Equation for EquationEquationwhere Equation and Equation are the k-th rows ofEquation and Equationrespectively.

Proof. The conclusion can be drawn from the relations.

Equation

Equation

The following lemma indicates that the final values for Equation are zeros for i =γ1+1,...,γ2if Assumption 5 is satisfied.

Lemma 4: If Assumption 5 is satisfied, then Equation for i =γ1 +1,...,γ2.

Proof. Due to Assumption 5, (12) can be rewritten as

Equation

The following lemma shows that solutions to the costate subsystems corresponding to the Jordon blocks γ1 +1,...,γ2 are impulse-free if Assumptions 4-5 are satisfied.

Lemma 5: If Assumptions 4-5 are satisfied, then Equation for i =γ1 +1,...,γ2 .

Proof. It can be proved that solutions to (19) is given by

Equation

Equation

for i=γ1 +1,...,γ2. It follows from Lemma 4 that Equation for i =γ1 +1,...,γ2 . As a result, with Assumption 4, Equation can be simplified as

 

Equation

which, together with Lemma 3, implies that the lemma holds.

A simple manipulation shows that for i =1,...,γ1 , (17) and (19) can be expressed as

Equation (21)

Equation (22)

And

Equation (23)

Equation (24)

where EquationEquation

It follows from Lemma 3 and Lemma 5 that Equation andEquation due to Equation andEquation for i =γ1+1,...,γ2. As a result, (17) and (19), for i =γ1 +1,...,γ , are equivalent to

Equation (25)

And

Equation(26)

due to vi = 1 for i =γ2 +1,...,γ.(18) and (20) can be described as

Equation (27)

Equation (28)

The initial conditions (10) can be expressed as

Equation (29)

Similarly, the final conditions (11) and (12) can be written as

Equation (30)

Equation (31)

for i =1,...,γ1 , where Equation due to Equation for j =γ1 +1,...,γ and Equation due to vj= 1 for j =γ2 +1,...,γ are used.

To simplify expressions, define the following matrices Equation

Equation

 

Equation

 

Equation

 

Equation

with

Equation

It is straightforward to prove that Equation

Combining (16) and (21) yields

Equation (32)

and combining (22) and (25) produces

Equation(33)

Similarly, (27), and (24) can be combined to get

Equation (34)

and (23) and (26) can be put together to obtain

Equation (35)

In addition, (28) can be rewritten as

Equation (36)

The initial conditions (9) and (29) can be written as

Equation (37)

The final conditions (30) and (31) can be described as

Equation (38)

It will prove convenient to define the following matrices

Equation

Equation

Equation

Lemma 6: If Assumption 1 is satisfied, then the matrix Equation has full row rank.

Proof. This lemma can be easily verified by applying Assumption 1.

Assumption 7:Equation

Lemma 7: If Assumptions 1 and 7 are satisfied, then Equation is invertible. Proof. Note that

Equation

Equation

 

is invertible. Therefore, according to Lemma 6, Equation is invertible.

Therefore, Equation and u(t) can be uniquely determined from (33), (35), and (36) as Equation (39)

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 Equation as the state and costate instead of x(t) and p(t).

(32) and (34) becomes

Equation (40)

with the boundary conditions of (37) and (38) where EquationEquation

The solutions to the two-point boundary value problem (40) can be solved by computing its back-time state transition matrix

Equation (41)

By following the approach proposed by Bender and Laub [5], the optimal feedback gain can be determined by solving the equation

Equation (42)

for K (t ) with Equation The general solutions is

 

Equation (43)

where Y (t )∈Rm×n is an arbitrary real matrix and U can be determined from

Equation (44)

With P being the solution of the following Riccati equation

Equation (45)

The main results can be summarized by the following theorem.

Theorem 1. Assume that the singular system is in the Weierstrass form. If Assumptions 1-7 are satisfied, then

(1) there exists a unique, impulse-free solution of the necessary conditions (16)-(20), together with the boundary conditions (9)-(12), which can be found from (39) and (40);

(2) the optimal feedback gain can be determined by (43);

(3) the optimal cost J from (1) is given by Equation

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

Equation

with a constraint equation of q1 + 2q2 = 1 .

Thus, the forces of constraint are f1= λ and f2 = . Set

Equation

Then, the manipulator can be expressed as the form of (3)-(4) with

Equation This system is not impulse controllable with γ1 = 0, γ =γ2 =1 . It can be verified that Assumptions 1-7 are satisfied with initial conditions q10+ 2q20 −1 = 0, Equation and matricesEquationEquation and Equation It is simple to verify the following: EquationEquationand EquationEquation set Equation Then, the Riccati equation (45) for the case of ft = ∞ has a unique positive definite solution of has a unique positive definite solution ofEquation It follows from (44) thatEquation andEquation withEquation According to (42), the feedback Equation can be found with K2 being arbitrary.

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.

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