Indefinite LQ Optimal Control with Terminal State Constraint for Discrete-Time Uncertain Systems

Uncertainty theory is a branch of mathematics for modeling human uncertainty based on the normality, duality, subadditivity, and product axioms. This paper studies a discrete-time LQ optimal control with terminal state constraint, whereas the weighting matrices in the cost function are indefinite and the system states are disturbed by uncertain noises.We first transform the uncertain LQproblem into an equivalent deterministic LQproblem.Then, themain result given in this paper is the necessary condition for the constrained indefinite LQ optimal control problem bymeans of the Lagrangianmultiplier method.Moreover, in order to guarantee the well-posedness of the indefinite LQ problem and the existence of an optimal control, a sufficient condition is presented in the paper. Finally, a numerical example is presented at the end of the paper.


Introduction
The linear quadratic (LQ) optimal control problem has been pioneered by Kalman [1] for deterministic systems, which is extended to stochastic systems by Wonham [2], and has rapid development in both theory and application [3].Usually, it is an assumption that the control weighting matrix in the cost is strictly definite.For stochastic LQ optimal control, it is first revealed in [4] that even if the state and control weighting matrices are indefinite the corresponding problem may be still well-posed, which evoked a series of subsequent researches in continuous time [5] and in discrete-time [6].In fact, some constraints are of considerable importance in many physical systems; the system state and control input are always subject to various constraints, so the constrained stochastic LQ issue has a concrete application background.For that reason, some researchers discussed stochastic LQ optimal problems with indefinite control weights and constraints [7,8].
As is well known, these stochastic optimal control problems have been well studied by probability theory which is based on a large number of sample sizes.Sometimes, no samples are available to estimate the probability distribution.
For such situation, we have to invite some domain experts to evaluate the belief degree that each event will occur.In order to rationally deal with belief degrees, uncertainty theory was established by Liu [9] in 2007 and refined by Liu [10] in 2010.Nowadays, uncertainty theory has become a new branch of mathematics for modeling indeterminate phenomena, which has been well developed and applied in a wide variety of real problems: option pricing problem [11], facility location problem [12], inventory problem [13], assignment problem [14], and production control problem [15].
Based on the uncertainty theory, Zhu [16] proposed an uncertain optimal control model in 2010 and gave an equation of optimality as a counterpart of Hamilton-Jacobi-Bellman equation.After that, some uncertain optimal control problems have been solved.As such, Sheng and Zhu [17] investigated an optimistic value model of uncertain optimal control problem; Yan and Zhu [18] established an uncertain optimal control model for switched systems.Inspired by the preceding work, we will tackle an indefinite LQ optimal control with terminal state constraint for discretetime uncertain systems, which is a constrained uncertain optimal control problem.The rest of the paper is organized as follows.Section 2 collects some preliminary results.In 2 Journal of Control Science and Engineering Section 3, an indefinite LQ optimal control with terminal state constraint is discussed.We present a general expression for the optimal control set in Section 4. A numerical example is applied in Section 5 to demonstrate the effectiveness of the model.We conclude the paper in Section 6.
For convenience, throughout the paper, we adopt the following notations: R  is the real -dimensional Euclidean space; R × is the set of all × matrices;   is the transpose of matrix ; and tr() is the trace of a square matrix .Moreover,  > 0 (resp.,  ≥ 0) means that  =   and  is positive (resp., positive semidefinite) definite.

Some Preliminaries
In this section, we introduce some useful definitions about uncertainty theory and Moore-Penrose pseudoinverse of a matrix.
Let Γ be a nonempty set, and let L be a -algebra over Γ.Each element Λ in L is called an event.An uncertain measure was defined by Liu [9] via the following three axioms.
An uncertain variable is defined by Liu [9] as a function  from an uncertainty space (Γ, L, M) to the set of real numbers such that { ∈ } is an event for any Borel set .In addition, an uncertainty distribution of  is defined as for any real number .
An uncertain variable  is called linear (Liu [9]) if it has a linear uncertainty distribution denoted by L(, ), where  and  are real numbers with  < .
Let  be an uncertain variable.Then, the expected value (Liu [9]) of  is defined by provided that at least one of the two integrals is finite.
Then, there exists a unique matrix  + ∈ R × such that The matrix  + is called the Moore-Penrose pseudoinverse of .

Theorem 7. If the indefinite LQ problem (10) is solvable by a feedback control
where   are constant crisp matrices, then it is equivalent to the following deterministic optimal control problem: for  = 0, 1, . . .,  − 1.
(i) If   = 0, we obtain Therefore, we have Substituting ( 21) into ( 16) produces the following state matrix: The associated cost function reduces to min and the constraint [x   x  ] =  becomes tr[  ] = .

A Necessary Condition for State Feedback Control.
In this subsection, a necessary condition for the optimal linear state feedback control with deterministic gains to the indefinite LQ problem (10) is obtained by applying the deterministic matrix maximum principle [22].Theorem 9.If the indefinite LQ problem (10) is solvable by a feedback control where   are constant crisp matrices, then there exist symmetric matrices   and a nonnegative  ∈ R 1 solving the following constrained difference equation: for  = 0, 1, . . .,  − 1.Moreover, with   ∈ R × ,  = 0, 1, . . .,  − 1, being any given crisp matrices.

Special Cases.
We have obtained that   ≥ 0 in the constrained difference equation (25) of Theorem 9.The following corollaries are special cases of the above result if we have   > 0 and   = 0.
Corollary 10.The indefinite LQ problem (10) is uniquely solvable if and only if   > 0 for  = 0, 1, . . .,  − 1.Moreover, the unique optimal control is given by Proof.By using Theorem 9, we immediately obtain the corollary.
3.5.Well-Posedness of the Indefinite LQ Problem.In the following, it is shown that the solvability of the constrained difference equation ( 25) is sufficient for the well-posedness of the indefinite LQ problem and the existence of an optimal control.Moreover, any optimal control can be represented explicitly as a linear state feedback by the solution of (25).
Theorem 12.The indefinite LQ problem (10) is well-posed if there exist symmetric matrices   and  ∈ R 1 satisfying the constrained difference equation ( 25).Moreover, the optimal control is given by Furthermore, the optimal cost of the indefinite LQ problem (10) is Proof.Let   and  ∈ R 1 satisfy (25).Then, Since   ≥ 0, from (51), we can easily deduce that the cost function of problem ( 10) is bounded from below by Hence, the indefinite LQ problem (10) is well-posed.It is clear that it is solvable by the feedback control Furthermore, by using tr[  ] =  and   =   +  which we have obtained in Theorems 7 and 9, (52) indicates that the optimal value of problem (10) equals

General Expression for the Optimal Control Set
In this part, we will present a general expression for the optimal control set based on the solution to (25).
A sufficient and necessary condition that u  is in the set of all optimal feedback controls for indefinite LQ problem (10) is that where   ∈ R × and   ∈ R  are arbitrary variables with appropriate size. Proof.

Numerical Example
In this section, application of Theorem 9 to solve constraint optimal control problem is illustrated.We present a twodimensional indefinite LQ problem with terminal state constraint for discrete-time uncertain systems.A set of specific parameters of the coefficients are given as follows: (67) Note that, in this example, the state weight  0 is negative semidefinite,  1 is negative definite, and  2 is positive semidefinite and the control weights  0 and  1 are negative definite.
In order to find the optimal controls and optimal cost value of this example, we have to solve the following equations: Secondly, by applying Theorem 9, we obtain the optimal feedback control and optimal cost value as follows.

3. Indefinite LQ Optimal Control with Constraints
Let matrices L, M, and N be given with appropriate sizes.Then, the matrix equation  =  has a solution X if and only if  +  + = .Moreover, any solution to  =  is represented by  =  +  + +  −  +  + , where  is a matrix with an appropriate size.