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**Xiaoping Liu ^{1,}**

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

^{2}Faculty 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.

**Visit for more related articles at** Journal of Electrical & Electronic Systems

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.

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

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].

Consider the problem of finding *u*(*t*) to minimize

(1)

with the constraint

(2)

Where *R* = *RT* > 0 , *Q* = *QT* ≥ 0 , *x* (*t *)∈*R ^{n}* ,

where* p* is the costate variable defined in (7) or (14),* x _{1}*(

It will prove useful to partition x_{2}, p_{2}, B_{2}, Q_{21}, Q_{22}, Qf_{21}, Qf_{22} as follows:

with

Furthermore, for* i *=1,...,*γ _{1}* , partition

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

(3)

(4)

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

(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 (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.

**Assumption 1:** 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* γ _{1}* +1,...,

The following assumptions are made so that solutions to subsystems corresponding to the Jordon blocks γ_{1} + 1, ..., γ_{2} are impulse-free.

**Assumption 2:**

**Assumption 3:**

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

**Assumption 4:**for

**Assumption 5:** for

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, ( *A _{1}*,

The necessary conditions for *J* to be minimized are

(6)

(7)

(8)

with and the boundary conditions *Ex *(0) = *Ex*_{0} and which, in the Weierstrass form, can be written as

(9)

(10)

(11)

(12)

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

(13)

(14)

(15)

which, in the Weierstrass form, can be expressed as

(16)

(17)

(18)

(19)

(20)

where i = 1,...,γ .

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 for Where is the *j*-th row of

Proof. A simple calculation shows that

for *i* =*γ _{1}* +1,...,

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

**Lemma 2:** If Assumptions 2 and 3 are satisfied, then for* i* =γ_{1} +1,...,γ_{2} .

Proof. Note that Assumption 3 implies for *i* =γ_{1} +1,...,γ_{2} . Applying this and Assumptions 2 to (5) gives

which, according to Lemma 1, means that for i =γ_{1} +1,...,γ_{2.}

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

**Lemma 3:** If Assumption 4 is satisfied, then for and for where and are the k-th rows of and respectively.

Proof. The conclusion can be drawn from the relations.

The following lemma indicates that the final values for are zeros for * i* =*γ _{1}*+1,...,

**Lemma 4:** If Assumption 5 is satisfied, then for *i* =*γ _{1}* +1,...,

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* γ _{1}* +1,...,

**Lemma 5:** If Assumptions 4-5 are satisfied, then for *i* =*γ _{1}* +1,...,

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

for *i*=*γ _{1}* +1,...,

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

A simple manipulation shows that for* i* =1,...,*γ _{1}* , (17) and (19) can be expressed as

(21)

(22)

And

(23)

(24)

where

It follows from Lemma 3 and Lemma 5 that and due to and for* i* =*γ _{1}*+1,...,

(25)

And

(26)

due to* v _{i}* = 1 for i =γ

(27)

(28)

The initial conditions (10) can be expressed as

(29)

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

(30)

(31)

for i =1,...,γ_{1} , where due to for j =γ_{1} +1,...,γ and due to *v _{j}*= 1 for

To simplify expressions, define the following matrices

with

It is straightforward to prove that

Combining (16) and (21) yields

(32)

and combining (22) and (25) produces

(33)

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

(34)

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

(35)

In addition, (28) can be rewritten as

(36)

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

(37)

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

(38)

It will prove convenient to define the following matrices

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

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

**Assumption 7:**

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

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

Therefore, and u(t) can be uniquely determined from (33), (35), and (36) as (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 as the state and costate instead of x(t) and p(t).

(32) and (34) becomes

(40)

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

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

(41)

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

(42)

for K (t ) with The general solutions is

(43)

where Y (t )∈*R ^{m×n}* is an arbitrary real matrix and U can be determined from

(44)

With P being the solution of the following Riccati 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

**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

with a constraint equation of q_{1} + 2q_{2} = 1 .

Thus, the forces of constraint are* f _{1}*=

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

This system is not impulse controllable with γ_{1} = 0, γ =γ_{2} =1 . It can be verified that Assumptions 1-7 are satisfied with initial conditions q_{10}+ 2q_{20} −1 = 0, and matrices and It is simple to verify the following: and set Then, the Riccati equation (45) for the case of f_{t} = ∞ has a unique positive definite solution of has a unique positive definite solution of It follows from (44) that and with According to (42), the feedback can be found with K_{2} being arbitrary.

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.

- Pandolfi L (1981) On the regulator problem for linear degenerate control systems. Journal of Optimization Theory and Applications 33: 241-254.
- Cobb DJ (1983) Descriptor variable systems and optimal state regulation. IEEE Transactions on Automatic Control 28: 601-611.
- Lewis FL (1985) On singular optimal control for singular systems. Proceedings of IEEE Conference, Decision and Control, pp: 266-272.
- Lovass-Nagy V, Schilling R, Yan HC (1986) A note on optimal control of generalized state-space (descriptor) systems. International Journal of Control 44: 613-624.
- Bender DJ, Laub AJ (1987) The linear-quadratic optimal regulator for descriptor systems. IEEE Transactions on Automatic Control 32: 672-688.
- Wang YY, Frank PM, Clements DJ (1993) The robustness properties of the linear quadratic regulators for singular systems. IEEE Transactions on Automatic Control 38: 96-100.
- Kunkel P, Mehrmann V (1997) The linear quadratic control problem for linear descriptor systems with variable coefficients. Mathematics of Control, Signals and Systems 10: 247-264.
- Katayama T, Minamino K (1992) Linear quadratic regulator and spectral factorization for continuous-time descriptor systems. Proceedings of IEEE Conference, Decision and Control, Tucson, Arizona pp: 967-972.
- Zhu J, Ma S, Cheng ZL (2002) Singular LQ problem for nonregular descriptor systems. IEEE Transactions on Automatic Control 47: 1128-1133.
- McClamroch NH, Wang D (1988) Feedback stabilization and tracking of constrained robots. IEEE Transactions on Automatic Control 33: 419-426.
- Verghese GC (1978) Infinite-frequency behavior in generalized dynamical systems. Ph D Thesis, Department of Electrical Engineering, Stanford University.
- Verghese GC, Levy BC, Kailath T (1981) A generalized state-space for singular systems. IEEE Transactions on Automatic Control, 26: 811-831.

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