Dead-beat servo problem for 2-dimensional linear systems

Necessary and sufficient conditions for the existence of a solution to the deadbeat servo problem for single-input single-output 2D linear systems are given. An algorithm is presented for finding a control law such that the tracking error vanishes in the shortest time possible. The algorithm is illustrated by a simple example.     

A pole-assignment algorithm for linear state feedback

A new algorithm for the pole-assignment problem of a controllable time-invariant linear multivariable system with linear state feedback is presented. The resulting feedback matrix is a least-squares solution and is robust in a loose sense. The method is based on the controllability canonical (staircase) form and amounts to a new proof for the existence of a solution of the pole-assignment problem. Illustrative examples are given.     

Dynamic programming for 2-D discrete linear systems

The authors calculate the optimal control of 2-D discrete linear systems using a dynamic programming method. It is assumed that the system is described with Roesser's state-space equations for which a 2-D sequence of inputs minimizing the given performance criterion is calculated. The method is particularly suitable for problems with bounded states and controls, although it can also be applied for unbounded cases. One numerical example is given     

An approximation to bounded phase coordinate control problem for linear discrete systems

This paper considers the penalty function method to obtain an approximate solution to the bounded phase coordinate optimal control problem for linear discrete systems with essentially quadratic cost functionals. The penalty function assumes positive values outside the phase constraint set, and zero inside the phase constraint set. The problem is to find an optimal control from a convex compact control restraint set such that the cost functional is minimum, and the sum of the penalty function along the response is smaller than a prescribed constant. It is shown that the maximum principle is a necessary and sufficient condition for an optimal control in a number of cases, and an analytic method of finding an optimal control is given. Also, the existence of an optimal control is proved.     

Solutions of minimum time problem and minimum fuel problem for discrete linear admissible control systems

In this paper the minimum time problem and the minimum fuel problem for discrete linear admissible control systems arc formulated (is one linear programming problem with two different objective functions. It is found that the obtained problem is very similar to the dual form of the linear programming problem for the discrete linear L₁ approximation problem. An efficient dual simplex algorithm for the latter problem is used with the necessary modification* to obtain the solution of either of the former problems. Numerical results show that the present, method compares favourably with other known methods.     

A note on aperiodicity condition for linear discrete systems

In this note conditions for the roots of a real polynomial to lie in the segment [0, 1), i.e., aperiodicity condition is obtained. The conditions obtained are based on a nonlinear transformation which transforms the segment [0, 1) onto the periphery of the unit circle. The results of this note correct the earlier proposed corollary obtained by Szaraniec using this transformation and present a test for verifying the aperiodicity condition.     

A new on-line identification method for discrete multivariable linear systems

An improved method for on-line identification of discrete data linear multivariable systems based on stability theory is presented. This method makes use of data collected at the present and past instants and exhibits fast convergence.     

Lyapunov inequalities for discrete linear Hamiltonian systems

In this paper, we present some Lyapunov type inequalities for discrete linear scalar Hamiltonian systems when the coefficient c(t) is not necessarily nonnegative valued and when the end-points are not necessarily usual zeros, but rather, generalized zeros. Applying these inequalities, we obtain some disconjugacy and stability criteria for discrete Hamiltonian systems     

离散线性周期系统的模型匹配

考虑离散线性周期系统的模型匹配问题, 提出一种基于参数化极点配置的模型匹配方法. 该方法从时域的角度出发, 采用周期状态反馈, 使得闭环系统充分接近目标系统. 由于所采用的参数化极点配置算法提供了充分的自由度, 所提出的方法能够实现零误差匹配. 数值算例验证了所提出算法的有效性.

     

On stabilization for discrete linear time-varying systems

In this paper, we consider the stabilization problem for discrete linear time-varying systems in an operator-theoretic framework. By using the complete finiteness of a certain discrete nest algebra, we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation. This result extends the Youla parametrization theorem by only requiring a strong left or a strong right representation but not both. The strong stabilization problem is also discussed.     

Insensitive observers for discrete time linear systems

This paper considers the problem of designing an observer insensitive to system parameter variations in discrete time linear multivariable systems. A K-insensitive observer is defined as an observer that can reconstruct a linear function of the state vector in spite of the variations of system parameters, provided the initial state of the observer is suitably chosen. A deadbeat observer is defined to be an observer that reconstructs the linear function for an arbitrary initial condition of the observer. Then, the existence of the K-insensitive observer is examined, and the class of K-insensitive observers is characterized. A necessary and sufficient condition is derived under which the K-insensitive deadbeat observer can be designed, and a simple algorithm is proposed to design the observer. The resulting observer is shown to be stable. The order of the observer is evaluated. The condition for generic solvability of the problem is also given.     

Asymptotical stability of 2-D linear discrete stochastic systems

The main goal of the present paper is to find computable stability criteria for two-dimensional stochastic systems based on Kronecker product and nonnegative matrices theory. First, 2-D discrete stochastic system model is established by extending system matrices of the well-known Fornasini–Marchesini?s second model into stochastic matrices. The elements of these stochastic matrices are second-order, weakly stationary white-noise sequences. Second, a necessary and sufficient condition for 2-D stochastic systems is presented, this is the first time that has been proposed. Third, computable mean-square asymptotic stability criteria are derived via Kronecker product and the nonnegative matrix theory. The criteria are only sufficient conditions. Finally, illustrative examples are provided.     

Extended Nehari problem for time-variant discrete systems

In this paper a suboptimal solution to the time-variant discrete version of the extended Nehari problem is given. The main tool used to obtain such a solution, with prescribed tolerance, is the so-called anticausal stabilizing solution to the reverse discrete-time Riccati equation.     

A simple method for determining the reachable set for linear discrete systems

A simple method is described for determining the reachable set RNof annth-order linear discrete systemN > nfor a single bounded input. The method proposed reduces the problem from solving a set of (n - 1) simultaneous equations to a simpler one, requiring only the evaluation of two inner products.     

Suboptimal control for discrete linear constant stochastic systems

The regulation of discrete-time, constant, linear stochastic systems with quadratic performance criterion is considered. A fixed-dimensional, linear time-invariant controller is used. Algebraic matrix equations are derived whose solutions are the gains of the fixed-dimensional optimal controller.     

Optimal fault detection for linear discrete time-varying systems

This paper deals with the problem of observer-based fault detection for linear discrete time-varying (LDTV) systems. A problem formulation is first proposed to address the optimization of the fault detection filter (FDF) design, which is expressed in terms of maximizing a finite horizon H_∞/H_∞ or H₋/H_∞ performance index. This formulation can be applied to FDF design of LDTV systems subject to l₂-norm bounded unknown inputs or stochastic noise sequences. It is shown that a unified optimal solution to the FDF can be obtained by solving the discrete time Riccati equation and the optimal FDF is not unique. A numerical example is given to illustrate the proposed method.