Teruo Sunaga

The closed-loop expression of the minimum energy control for multi-input linear digital control systems

The minimum energy control problem for the multi-input linear time-invariant discrete-time system is discussed. Though the open-loop expression of the optimal inputs has been obtained, it is difficult to solve the two-point boundary value problem with the zero terminal state and to obtain the closed-loop expression with bounded gains. First, the general inputs which satisfy the zero terminal state condition are obtained. By using the general inputs, the system is transformed into a new time-variant system with the variable structure of the new inputs, and the zero terminal state problem is transformed into a new free terminal state problem. The performance index is the quadratic form with time-variant weighting matrices and cross terms. The optimal inputs are given by variable gain state feedback in the first phase, by linear combinations of a special deadbeat control and variable gain state feedback in the second phase, and by the special deadbeat control in the third phase. The variable gains are obtained by using the special nonstationary Riccati equations.<<ETX>>

Closed-Loop Expression of Fixed-end-point LQ Optimal Control for Digital Systems

Abstract This paper discusses the fixed-end-point linear-quadratic optimal control problem for a multi-input linear time-invariant discrete-time system. Using the general form of the inputs which satisfy the zero-end-point condition, the fixed-end-point problem is transformed into the new free-end-point problem and the special deadbeat control. The optimal inputs are made up of three phases, and they are expressed by the variable gain state feedback and the special deadbeat control. The variable gains are obtained by the special nonstationary Riccati equations, and all gains do not depend on the initial state.

A systematic design of single linear functional observers

This paper discusses the design of a single linear functional observer (SLFO) for a linear time-invariant system. The SLFO is used to estimate a scalar-valued linear function of the system state, and is generally of lower order than the Luenberger observer. Two cases are considered: (i) feedback gain included as a design parameter and designed at the same time as the SLFO (the SLFO with unfixed feedback gain), and (ii) feedback gain is previously specified, before the design of the SLFO (the SLFO with fixed feedback gain). A standard form, a canonical form and the minimal order are obtained and they are used to obtain the optimal SLFO for the single-input linear-quadratic (LQ) problem. The SLFO with unfixed feedback gain is more difficult to calculate but has the following advantages over the SLFO with fixed feedback gain: the order is generally lower, the cost for the LQ problem is smaller and the responses are improved.

New optimality condition of fixed-endpoint regulators for single-input linear digital systems and its application to fixed-endpoint LQ regulators

This note discusses the regulator problem with zero terminal state for a single-input, linear, time-invariant, discrete-time system. The fundamental structure of the regulator for a general additive performance index is obtained. The new optimality condition is expressed by the optimality condition of the induced free-endpoint regulator in the first phase and the deadbeat control in the second phase. Furthermore, a closed-loop expression for the quadratic performance index is obtained.

Singular LQ optimal control for single-input linear digital control systems

The authors discuss the singular LQ optimal control problem for single-input linear digital control systems. The singular LQ control problem can be transformed into the nonsingular one under the reachability condition. The singular LQ optimal controls are obtained for the free-end-point problem, for the fixed-end-point problem, and for the infinite control time problem. They are obtained by state feedback with bounded gains. Furthermore, the relation between the singular LQ optimal control and the nonsingular one is described. For the fixed-end-point problem, the singular LQ optimal control and the nonsingular one can be described by using the same expressions. The infinite control time problem must be regarded as the case when the terminal time approaches infinity in the fixed-end-point problem.<<ETX>>

Minimization of the input energy of regulator problems with the fixed terminal state. The case of multi-input linear time-invariant discrete-time systems.

This paper discusses the regulator problem of minimizing the input energy for the multi-input linear time-invariant discrete-time system with the zero terminal state. The performance index is the sum of the quadratic forms of the input vectors. By using the deadbeat principle, the system is transformed into the new time-variant system with the time-variant structure of the new input. The performance index is transformed into the sum of the quadratic forms of the new state vectors and the new input vectors with cross terms, and the new terminal state is free. The optimal inputs are given by the variable gain state feedback in the first phase, by linear combinations of the constant gain state feedback and the variable gain state feedback in the second phase, and by the constant gain state feedback in the third phase. The variable gains are obtained by using the special Riccati equation. Both gains are independent of the initial state.

Canonical form of state deadbeat controllers.

This paper discusses a canonical form of a state deadbeat controller for a discrete-time, linear, time-invariant system. A general form of the state deadbeat controller which shows the freedom of design has been obtained ; however, the design parameters have redundancy. To avoid the redundancy, a canonical form which is expressed by the linear combination of independent design parameters is derived for some subsets of the general form. Finally, illustrative examples are given to show the effectiveness of the proposed procedure.

Design of single linear functional filters

This paper treats the simultaneous design problem of a single linear functional filter (SLFF) and a feedback gain for the so-called single-input linear-quadratic-gaussian (LQG) problem. The SLFF is a linear unbiased filter which estimates a scalar-valued linear function of the system state, and is generally of lower order than the Kalman-Bucy filter. A canonical form, the minimal order and the optimal design procedure of an SLFF are obtained for the steady-state case. Finally, the results are compared with the Kalman-Bucy filter by an example.

Design of linear functional niters for linear stochastic systems

This paper considers the problem of designing a linear functional filter (LFF) to estimate a linear function of the state of a linear stochastic system. The LFF is a linear unbiased filter and is generally of lower order than the Kalman-Bucy filter. When the LFF is unbiased in the so-called linear-quadratic-Gaussian (LQG) problem, it is shown that the separation properties are established for the quadratic cost function and the stability. In the steady-state case, the minimal order, a canonical form and the optimal design are obtained for a single LFF which estimates a scalar-valued linear function of the state. Finally, an illustrative example is given to show the effectiveness of the LFF.