Discrete-time Linear-quadratic Optimal Control Research Articles

Linear-quadratic optimal control for discrete-time stochastic descriptor systems

<p style='text-indent:20px;'>In this paper, an optimal control model ruled by a class of linear discrete-time stochastic descriptor systems is considered under quadratic index performance. Employing dynamic programming method, a recurrence equation to simplify the optimal control problem is presented provided that the descriptor systems are both regular and impulse-free. When the objective function is quadratic, according to the recurrence equation, a discrete-time linear-quadratic optimal control problem is completely settled, that is, optimal controls and optimal values of the problem are both obtained through analytical expressions. At last, a numerical example about linear-quadratic optimal control for a discrete-time stochastic descriptor system is provided to illustrate the validness of the results derived.</p>

The Novel Current Control for Virtual Synchronous Generator

In recent years, power generation using renewable energy sources has been developed as a solution to the global warming problem. Among these power generation methods, wind power generation is increasing. However, as the penetration level of wind power generation increases, the low inertia and lack of synchronous power characteristics of the penetrated power system can have a significant impact on the transient stability of the grid. The virtual synchronous generator provides the ability of virtual inertia and synchronous power to interconnected inverters. The interconnected inverter with the virtual synchronous generator ability uses, in general, PI control based current controller. This paper proposes a new current-control method and compares it with conventional methods. The proposed current control is a method that follows virtual synchronous generator model that changes every moment by solving the discrete-time linear quadratic optimal control problem for each sampling time interval. The new method follows the conventional method, and therefore the reactive power fluctuation can be suppressed and the interconnected inverter will be downsized.

Indefinite Linear Quadratic Optimal Control: Strict Dissipativity and Turnpike Properties

We investigate strict dissipativity and turnpike properties for indefinite discrete-time linear quadratic optimal control problems in the presence of constraints on state and input. Previous results provide a geometric characterization of these properties in case that the stage cost is convex. We generalize these results to indefinite cost functions using two approaches: First, we show that the existing framework can be extended to indefinite state weighting if the stage cost accumulated over multiple consecutive time steps is convex. As a second contribution, we study the strict dissipation inequality by taking the particular shape of the constraints into account. This allows us to state sufficient conditions for strict dissipativity and turnpike properties where the occurrence of turnpikes on the boundary of the constraints is directly related to negative eigenvalues of the cost.

Discrete-Time Linear Quadratic Optimal Control with Fixed and Free Terminal State via Double Generating Functions

A double generating functions based method for the discrete-time linear quadratic optimal control problem with fixed and free terminal state, i.e., hard constraint problem and soft constraint problem, is developed in this paper. By this method, the optimal state and input can be expressed by functions only in terms of the boundary conditions and pre-computed coefficients. Accordingly, the whole optimal computation can be divided into two parts in which the off-line part calculates the coefficients and the on-line part generates the optimal solutions by these proposed functions. In view of this, the developed method is useful in the on-line iterative computation for optimal control problem with a number of different boundary conditions. Examples demonstrate the effectiveness of the double generating functions method for the both hard and soft constraint problems with different boundary conditions.

An optimal linear-quadratic digital tracker for analog neutral time-delay systems

In this paper, an optimal hybrid tracking control problem for continuous neutral time-delay systems is formulated and studied. An optimal linear integral quadratic cost function that has a high-gain property is used for tracking control specification. Two interpolation methods are applied to directly convert the original analog neutral time-delay system into an equivalent digital retarded time-delay system, and meanwhile convert the continuous-time quadratic cost function into a discretized form. Then, an extended state vector is constructed for an associate extended discrete-time optimal control problem without time delay. Using the standard discrete-time linear-quadratic optimal control theory and an indirect digital redesign technique with a predictive feature, an effective digital tracker is designed for the original analog neutral time-delay system. An example is finally given for illustrating the effectiveness of the new tracker design method.

Optimal feedback solution of a constrained stochastic one-storage model

We describe a method for obtaining the optimal feedback solution to a constrained discrete time linear-quadratic optimal control problem. The dimensions of the state and control vectors are one and the stochasticity is represented by a finite number of possible outcomes at each stage. The method is based on dynamic programming and exploits that the optimal value is piecewise quadratic and that the optimal feedback solution is piecewise linear. Also the special case with linear criterion function is considered. Approximation methods which allow a trade-off between computation times and solution accuracy are discussed. The method is applied to two cases, viz., hydropower scheduling and temperature control of a greenhouse. Comparative studies on these two cases were made with a mathematical programming formulation solved by a standard code. On the cases studied the new method was found to be around 100 times faster and to display a better solution stability. © 1997 John Wiley & Sons, Ltd.

Optimal feedback solution of a constrained stochastic one‐storage model

We describe a method for obtaining the optimal feedback solution to a constrained discrete time linear-quadratic optimal control problem. The dimensions of the state and control vectors are one and the stochasticity is represented by a finite number of possible outcomes at each stage. The method is based on dynamic programming and exploits that the optimal value is piecewise quadratic and that the optimal feedback solution is piecewise linear. Also the special case with linear criterion function is considered. Approximation methods which allow a trade-off between computation times and solution accuracy are discussed. The method is applied to two cases, viz., hydropower scheduling and temperature control of a greenhouse. Comparative studies on these two cases were made with a mathematical programming formulation solved by a standard code. On the cases studied the new method was found to be around 100 times faster and to display a better solution stability. © 1997 John Wiley & Sons, Ltd.

Spectral theory of the linear-quadratic optimal control problem: A new algorithm for spectral computations

In a new formulation of the discrete-time linear-quadratic optimal control problem, the spectrum of a bounded self-adjoint Hilbert space operator, which can be decomposed as the sum of a Toeplitz operator and a compact perturbation, has been shown to elucidate the underlying structure of the problem. A new and efficient algorithm for computing this spectrum is presented.

Fixed bed reactor kalman filtering and optimal control—II Experimental investigation of discrete time case with stochastic disturbances

Discrete time linear quadraticoptimal control is investigated experimentally on a catalytic fixed-bed reactor with stochastic upstream disturbances. It is demonstrated that the discrete time variance evaluated with time intervals equal to the control intervals oscillates along the reactor axis due to the time discretization of the control action. However, if the variance is evaluated with much shorter time intervals than used for control, the oscillations are not present. The control system behaviour is demonstrated with both Gaussian and step disturbances.

Spectral theory of the linear-quadratic optimal control problem: Discrete-time single-input case

The single-input discrete-time linear-quadratic optimal control problem, together with the classical time-domain and frequency domain conditions, is reviewed and treated from a novel point of view. We show that when the quadratic cost is not positive semidefinite, the problem of the boundedness of the infimum in both the zero and the free terminal state cases is related to the positivity of a self-adjoint Hilbert space operator. The structure of the spectrum of the operator in question, in both cases, is investigated, leading to a clarification of the linkage between the time-domain and the frequency-domain conditions for boundedness. More over, it is shown that the spectra of the operators in question reflect the relevant properties of the control problem.