Limiting performance of optimal linear discrete filters

The limiting variance of the estimation error is obtained for the optimal H₂ filter for discrete time systems when the intensity of the measurement noise tends to zero. In particular, an explicit expression is obtained for the lowest achievable mean-square error when the measurement noise is slowly varying and has intensity which tends to zero. This limitation can be characterized completely by the system order, the dimension of the process noise, the number of the system's transmission zeros, and the location of the system's unstable transmission zeros.     

Design of linear regulators optimal for time-multiplied performance indices

A procedure for the design of optimal feedback controls for linear time-invariant systems relative to time-multiplied quadratic performance indices is presented. An iterative scheme for the computation of the optimal control law is proposed. An example is given to demonstrate the control over transient behavior given by the design based on time-multiplied performance indices.     

A note on optimal linear regulators with incomplete state feedback

The design of optimal linear regulators with incomplete state feedback has been reported in a recent paper [1]. This correspondence deals with the practical utility of such controllers and the difficulties encountered in designing higher order systems.     

基于最优Hankel范数近似的线性相位ⅡR滤波器设计

提出了一类基于最优Hankel范数近似的线性相位无限脉冲响应(IIR)滤波器设计方法．首先给出了Hankel范数的相关预备知识,然后给出了离散时间单输入单数出系统Hankel范数近似的定理及证明,最后给出了线性相位IIR滤波器的设计步骤．该方法不但减小了逆矩阵求解过程中的计算量,同时给出了(?)2范数近似的误差边界,仿真结果验证了该方法的有效性．     

Asymptotically optimal controls of hybrid linear quadratic regulators in discrete time

This work develops asymptotically optimal controls for a class of discrete-time hybrid systems involving singularly perturbed Markov chains having weak and strong interactions. The state space of the underlying Markov chain is decomposed into a number of recurrent classes and a group of transient states. Using a hierarchical control approach, by aggregating the states in each recurrent class into a single state, a continuous-time quadratic limit control problem in which the resulting limit Markov chain has much smaller state space is derived. Using the optimal control for the limit problem, a control for the original problem is constructed, which is shown to be nearly optimal. Finally, a numerical example is given to demonstrate the effectiveness of the approximation scheme.     

Optimal and asymptotically optimal linear regulators resulting from a one-stage performance index

An optimal control algorithm is derived which minimizes at each time instant the one-stage k-step-ahead performance index for the state-space representation of the CARMA model. Equivalence of the input-output representation in its asymptotic form and in the Clarke-Hastings-James algorithm is proved. Formulae for computing the output and control variances at any time instant for both optimal and asymptotic algorithms are given. The results are illustrated by a numerical example.     

基于最优Hankel范数近似的线性相位IIR滤波器设计

提出了一类基于最优Hankel范数近似的线性相位无限脉冲响应(IIR)滤波器设计方法.首先给出了Hankel范数的相关预备知识,~然后给出了离散时间单输入单数出系统Hankel范数近似的定理及证明,最后给出了线性相位IIR滤波器的设计步骤.该方法不但减小了逆矩阵求解过程中的计算量,~同时给出了L_∞范数近似的误差边界.仿真结果验证了该方法的有效性.     

Comments on "The design of linear regulators optimal for time multiplied performance indices"

A procedure for the design of linear regulators optimal with respect to a time-multiplied performance index was suggested in a recent paper by Man and Smith. A counterexample, presented subsequently [2], indicated that the procedure was in error. It is shown that the optimal feedback control law must contain time-varying gains as opposed to the time-invariant gains as suggested by Man and Smith. This fact was also overlooked in [2].     

ITSE optimal controller design and achievable performance bounds

This article proposes a methodology to design optimal controllers and to compute achievable performance bounds in the control of linear, stable discrete-time SISO plants. The performance is measured using a time-weighted cost function (ITSE) of the tracking error for a step reference or a step output disturbance, combined with a measure of the energy of the incremental control. The proposed methodology relies on the use of a function space product that, for stable systems, can be defined in both the frequency and the time-domains. The solution technique requires the expansion of stable functions in orthogonal basis. An analytical expression is found for the computation of the optimal coefficients of the expansion. Numerical examples are included to illustrate the results.     

Improving model accuracy using optimal linear combinations of trained neural networks

Neural network (NN) based modeling often requires trying multiple networks with different architectures and training parameters in order to achieve an acceptable model accuracy. Typically, only one of the trained networks is selected as "best" and the rest are discarded. The authors propose using optimal linear combinations (OLC's) of the corresponding outputs on a set of NN's as an alternative to using a single network. Modeling accuracy is measured by mean squared error (MSE) with respect to the distribution of random inputs. Optimality is defined by minimizing the MSE, with the resultant combination referred to as MSE-OLC. The authors formulate the MSE-OLC problem for trained NN's and derive two closed-form expressions for the optimal combination-weights. An example that illustrates significant improvement in model accuracy as a result of using MSE-OLC's of the trained networks is included.     

Design of stochastic discrete time linear optimal regulators Part II. Extension and computational procedures

Using the relationship established in Part I, computational procedures are formulated for implementing stochastic optimal regulators based on the CARMA model. The state space configurations (including both the implicit delay and explicit delay models) allow the consideration of general quadratic cost functions, in particular the receding horizon N-stage cost function ; a simple example is used to illustrate various design steps required. The characteristics of the implicit delay and explicit delay models are exploited to give a computationally efficient scheme for system with large control delay     

Optimal linear regulators: The discrete-time case

In this paper the optimal discrete-time linear-quadratic regulator problem is carefully presented and the basic results are reviewed. Dynamic programming is used to determine the optimization equations. Special attention is given to problems unique to the discrete-time case; this includes, for example, the possibility of a singular system matrix and a singular control-effort weighting matrix. Some problems associated with sampled-data systems are also summarized, e.g., sensitivity to sampling time, and loss of controllability due to sampling. Computational methods for the solution of the optimization equations are outlined and a simple example is included to illustrate the various computational approaches.     

Maximum achievable precision of linear systems with discrete controllers

Consideration was given to the problem of maximum achievable precision of linear systems with discrete state and output controllers. The external perturbations affecting the system are the bounded step-type and harmonic (of unknown frequency) vector functions of time which the control theory regards as standard. The condition for asymptotic stability of the closed-loop system is the only requirement on the controllers aside from their physical realizability. Therefore, the conclusions of the present paper apply to the entire set of the discrete stabilizing controllers, no matter what method was used to design them.     

On zeros and responses of linear regulators and linear observers

In the pole assignment of the single input and output regulator, there is a case where the integrated square output becomes infinite and the response of the output has an infinitely large maximal amplitude when all poles of the closed loop system are assigned to negative infinity in the numericals-plane. This paper deeply investigates this problem from the linear structure point of view and applies the obtained results to the pole specification problem of observers.     

Synthesis of linear multivariable regulators

This paper generalizes the results obtained by the authors in [1] and presents new results on the construction of proper controllers for linear multivariable regulators. These results may lead to efficient computation methods that do not involve Smith forms or the order relations used in [1] to construct proper controllers. The methods are based on the recently developed theory of Λ-generalized polynomials [2]. We discuss several numerical algorithms and how they can be applied to the regulator problem. We also point out an important computational problem in need of an efficient numerical algorithm.     

Feedback properties of linear regulators

Results relating to the state feedback realization of the optimal linear regulator are reviewed. Desirable stability and parameter sensitivity properties of this regulator are discussed. Interpretations of these properties in terms of matrix return differences and comparison sensitivity functions are given. Some related open problems are discussed.     

The role of transmission zeros in linear multivariable regulators

The problem is considered of designing compensators to regulate linear multivariable systems. It is shown that the essential mechanism employed by such a compensator is the multivariable analogue of pole-zero cancellation: the compensator supplies transmission zeros to cancel the unstable poles of the disturbance and reference signals. If the compensator is additionally required to function in the presence of small variations in system parameters then it must supply transmission zeros in greater multiplicity. Finally it is shown that the transmission zeros are generated by an ‘ internal model ’, incorporated in the compensator, of the dynamics of the disturbance and reference signals.     

The optimum linear smoother as a combination of two optimum linear filters

A solution to the optimum linear smoothing problem is presented in which the smoother is interpreted as a combination of two optimum linear filters. This result is obtained from the well-known equation for the maximum likelihood combination of two independent estimates and equivalence to previous formulations is demonstrated. Forms of the solution which are convenient for practical computation are developed.