Robust real-time identification for a class of linear time-varying discrete systems

The problem of robust real-time identification of linear single-input-single-output dynamic systems with stochastically time-varying parameters is considered. Two ways of constructing robust algorithms that are able to handle outliers contaminating the gaussian observation disturbance samples are discussed. The first way is based on the general formulation of dynamic stochastic approximation schemes characterized by an adequate non-linear residual transformation, as well as the step-by-step optimization with respect to the weighting matrix of the algorithm. The second way is based on the formulation of one-step optimal estimates. Monte Carlo simulation results illustrate the discussion and show the efficiency of the proposed algorithms in the presence of outliers.     

Minimal-order observer designs for linear time-varying multivariable systems

This note presents two simple approaches for the design of minimal-order observer for linear time-varying systems. Both techniques rely on canonical transformations of the given system. As a consequence, simple expressions are obtained which permit the observer constraint equation to be solved effectively. The observers possess the desirable property of having a time-invariant observer coefficient matrix with arbitrary eigenvalues. A comparison is made between the proposed methods from which conclusions are drawn. An illustrative example is also included.     

Robust controllers for uncertain linear multivariable systems

This paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems. In the first part, it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties, and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty. In the second part, in contrast with the first part, it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known. The type of plant uncertainty is the so-called “stable-factor” uncertainty, and necessary and sufficient conditions are given for robust stabilization. In the third part, the model of uncertainty is a ball in the space of rational matrices metrized by the so-called graph metric, and sufficient conditions for robust stabilization are derived.     

Design of single-functional observers for linear time-varying multivariable systems

A method is presented for the design of an observer capable of reconstructing a single linear functional of the states of a linear time-varying system. The method is based on a canonical transformation of uniformly observable time-varying multi-variable system. It is shown that this canonical form, along with certain pre-specified structures of observer coefficient matrices, leads to a systematic and straightforward procedure for designing a functional observer of minimal order.     

Robust real-time identification of linear systems with correlated noise

The problem of recursive robust identification of linear discrete-time single-input single-output dynamic systems with correlated disturbances is considered. Problems related to the construction of optimal robust stochastic approximation algorithms in the min-max sense are demonstrated. Since the optimal solution cannot be achieved in practice, several robustified stochastic approximation algorithms are derived on the basis of a suitable non-linear transformation of normalized residuals, as well as step-by-step optimization with respect to the weighting matrix of the algorithm. The convergence of the developed algorithms is established theoretically using the ordinary differential equation approach. Monte Carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms. The results indicate the most suitable algorithms for applications in engineering practice.     

Arbitrary eigenvalue assignments for linear time-varying multivariable control systems

The problem of eigenvalue assignments for a class of linear time-varying multi-variable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is ‘lexicography-fixedly controllable’ can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.     

Diagonalization algorithms for linear time-varying dynamic systems

On the basis of a mode-vector representation, we show that its time-varying amplitudes and frequencies can be obtained by diagonalizing the system matrix. Next, we reformulate an explicit diagonalizing algorithm that was earlier proposed by Wu. Then, the missing convergence proof is given. Moreover, we present a new and implicit iteration scheme that is closely related to that given by Wu. In both algorithms, the time-varying system matrix is gradually diagonalized by successive algebraic similarity transformations. It is proved that the convergence conditions are essentially the same. Although the class of systems for which the algorithms are applicable is still not fully known, the results of this paper may be of theoretical and practical interest.     

不确定时变系统的鲁棒学习控制算法

研究不确定性时变系统在有限时间区间上重复作业和在无限时间区间上周期作业的跟踪控制问题. 基于Lyapunov-like方法, 给出了形式简单的鲁棒迭代学习控制和鲁棒重复控制两种算法. 两种学习算法均可弥补单一控制算法的缺陷, 鲁棒控制部分被用来保证闭环系统中所有变量的有界性, 学习控制部分可有效消除系统跟踪误差, 改善系统的跟踪性能. 仿真结果验证了两种学习算法的有效性.     

Design of reduced-order state estimators for linear time-varying multivariable systems

The design of reduced-order state estimators for linear time-varying multivariable systems is considered. Employing the concepts of matrix operators and the method of canonical transformations, this paper shows that there exists a reduced-order state estimator for linear time-varying systems that are ‘lexicography-fixedly observable’. In addition, the eigenvalues of the estimator can be arbitrarily assigned. A simple algorithm is proposed for the design of the state estimator.     

Robust adaptive pole placement for linear time-varying systems

This paper studies the robustness properties of a basic adaptive control algorithm with respect to plant parameter variation as well as modeling errors and bounded disturbances. The algorithm consists of a projected gradient estimator and a pole assignment controller. A unified method of proof is presented for robust stability of both discrete and continuous-time adaptively controlled time-varying systems. The robust performance of an adaptive pole assignment controller is also discussed     

Robust state-derivative feedback LMI-based designs for multivariable linear systems

In some practical problems, for instance in the control systems for the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. New necessary and sufficient linear matrix inequalities (LMI) conditions for the design of state-derivative feedback for multi-input (MI) linear systems are proposed. For multi-input/multi-output (MIMO) linear time-invariant or time-varying plants, with or without uncertainties in their parameters, the proposed methods can include in the LMI-based control designs the specifications of the decay rate, bounds on the output peak, and bounds on the state-derivative feedback matrix K. These design procedures allow new specifications and also, they consider a broader class of plants than the related results available in the literature. The LMIs, when feasible, can be efficiently solved using convex programming techniques. Practical applications illustrate the efficiency of the proposed methods.     

Robust optimization of multivariable feedback systems with time-varying nonlinear uncertainties

A design criterion is developed to achieve the input-output decoupling of multivariable feedback systems and the robust stabilization of systems with time-varying nonlinear uncertainties. Moreover, an effective design algorithm is derived to achieve the robust optimization of multivariable feedback systems subjected to time-varying nonlnear uncertainties. The theory of minimum H^∞ -norm and the optimal interpolation technique are employed to solve this robust optimization problem. Since the requirements of internal stability are satisfied, this design algorithm performs appropriately, even if the plant is unstable and/or non-minimum phase. From the result of the robust optimization, we can predict the maximum sector bounds of nonlinear uncertainties that can be tolerated in the multivariable feedback system.     

Persistency of excitation criteria for linear,multivariable, time-varying systems

For continuous-time, multiple-input, multiple-output, linear systems, we present conditions under which the persistency of excitation of one regression vector implies the persistency of another regression vector derived from the first via a linear, dynamical transformation. We then introduce a definition of sufficient richness for vector input signals in the form of a persistency of excitation condition on a basis regression vector. Finally we establish input conditions which guarantee the persistency of excitation of a large class of regression vectors obtained from both time-invariant and time-varying systems. The work reported was performed while both authors were at the Department of Systems Engineering, Research School of Physical Sciences, Australian National University, G.P.O. Box 4, A.C.T. 2601, Australia.     

Robust adaptive regulation of linear time-varying systems

Given reference signals generated by a stable, linear time-invariant exosystem, to be tracked by the output, the problem of robust adaptive regulation by output feedback is addressed for single-input, single-output linear systems with unknown time-varying parameters and unmodeled bounded additive disturbances. Without requiring parameters or disturbances to be slowly varying or to have known bounds, a robust adaptive regulator is designed that ensures boundedness of closed-loop signals and arbitrarily improved transient performance for the regulation error. Asymptotic regulation is achieved when parameters belong to a compact set determined by the controller parameters' time derivatives are L₁∩L_∞, and disturbances and parameter variations from unknown constant nominal values are L₂∩L_∞     

Robust exponential stability of time-varying linear systems

This paper investigates the robustness of time-varying linear systems under a large class of complex time-varying perturbations. Previous results⁸ which were restricted to bounded linear perturbations of output feedback type are generalized to unbounded and nonlinear perturbations of multi-output feedback type. We establish a lower bound for the stability radius of these systems and show how it may be possible to improve the bound using time-varying scalar transformations of the state, input and output variables. The results are applied to derive Gershgorin type stability criteria for time-varying linear systems.     

Subspace identification of multivariable linear parameter-varying systems

A subspace identification method is discussed that deals with multivariable linear parameter-varying state-space systems with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of system is the enormous dimension of the data matrices involved. To overcome the curse of dimensionality, we suggest using only the most dominant rows of the data matrices in estimating the model. An efficient selection algorithm is discussed that does not require the formation of the complete data matrices, but processes them row by row.     

Canonical forms for the identification of multivariable linear systems

The advantage of using a unique parameterization in a numerical procedure for the identification of a system from operating records has been well established. In this paper several sets of canonical forms are described for state space models of deterministic multivariable linear systems; the members of these sets having therefore the required uniqueness property within the equivalence classes of minimal realizations of the system. In the identification of a stochastic system, it is shown how the problem depends also upon determining a unique factorization of the spectral density matrix of the system, and the sets of canonical forms obtained for the deterministic system are extended to this case.     

Optimal instrumental-variable methods for identification of multivariable linear systems

The identification of multivariable linear systems using instrumental-variable (IV) methods is discussed. The emphasis is on accuracy properties. The IV estimates of multi-input-multi-output system parameters are proved to be asymptotically Gaussian distributed under weak conditions. An explicit expression for the covariance matrix of the parameter estimates is given. It is then shown how this matrix can be optimized by appropriately choosing the IV variant. The optimal accuracy so obtained is for some model structures equal to that corresponding to a prediction error method.     

Design of multivariable linear time-varying tracking systems with stochastic disturbances

A new designing method of multivariable linear time-varying tracking systems with stochastic disturbances for polynomial command inputs is proposed. It has been shown that it is possible to choose the matrices of gain elements placed in feedbacks and parallel paths of the closed-loop system so that the output y(t) tracks the polynomial command input v(t) with (E expectation operator).     

Robust boundary control for linear time-varying infinite dimensional systems

The problem of robust boundary control for a class of infinite dimensional systems under mixed uncertainties is addressed. A strong solution of the Dirichlet boundary problem corresponding to the perturbed evolution operator is introduced. The Lyapunov function approach is used for proving that there is a controller which stabilizes this class of systems under the presence of smooth enough internal and external perturbations and guarantees some tolerance level for the joint cost functional. The derived control consists of two parts: a compensating one and a linear feedback controller with a gain operator which is a positive inverse solution of a corresponding operator Riccati equation. A heating boundary control process is given as an illustration of the suggested approach