Explicit solution and stability of linear time-varying differential state space systems

Linear time-varying (LTV) systems naturally arise when one linearizes nonlinear systems about a trajectory. In contrast the linear time-invariant (LTI) cases which have been thoroughly understood in the analysis and synthesis technologies, many features of the LTV systems are still limited and not clear. This paper addresses the problems of solution and stability of a general unforced LTV differential state space system. Unlike most of the work based on the Lyapunov theory, numerical simulations, or specific constraint systems, the paper proposes the spectral decompositions of the LTV systems by employing extended eigenpairs and with simple mathematical derivation. The spectral decompositions reveal the mechanisms of inherent characterization in general LTV systems, rather than a particular class. Moreover, a novel set of auxiliary equations is developed for guiding and obtaining the extended eigenpairs of its system matrix which completely characterize the LTV systems. The solutions to perform the commutative systems and the second-order systems with companion form are straightforward. The proposed innovative thinking provides a novel guided way to analyze the LTV systems. These findings are easily extended to LTI cases. Examples from the literature demonstrate the effectiveness and the superiority of the proposed approaches when compared with other methods. The proposed results may be of great interest in both for scientific research and application.     

Finite-time control with H-infinity constraints of linear time-invariant and time-varying systems

This paper presents finite-time control methods with H-infinity constraints for linear time-invariant (LTI) and time-varying (LTV) systems. The basic idea of the proposed approaches is to construct controllers for the LTI and LTV in such a way that a constant quadratic Lyapunov function and a time-varying quadratic Lyapunov function can be used to establish the finite-time stability and the H-infinity performance of the resulting closed-loop systems. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities (LMIs) and Differential Riccati Inequalities (DRIs) that are numerically feasible with commercially available software. Finally, the results are illustrated by application to the design of guidance law for a class of terminal guidance system.     

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input-output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.     

Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems

Many control applications, including feedforward and learning control, involve the inverse of a dynamical system. For nonminimum-phase systems, the response of the inverse system is unbounded. For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy. For generic linear time-varying (LTV) systems, such a dichotomy does not exist in general. The aim of this paper is to develop an inversion approach for an important class of LTV systems, namely linear periodically time-varying (LPTV) systems, which occur in, e.g. position-dependent systems with periodic tasks and non-equidistantly sampled systems. The proposed methodology exploits the periodicity to determine a bounded inverse for general LPTV systems. Conditions for existence are provided. The method is successfully demonstrated in several application cases, including position-dependent and non-equidistantly sampled systems.     

Localization based switching adaptive control for time-varyingdiscrete-time systems

In this paper a new systematic switching control approach to adaptive stabilization of linear time-varying (LTV) discrete-time systems is presented. A feature of the localization based method is its high model falsification capability, which in the case of LTI systems is manifested as the rapid convergence of the switching controller. We believe that the proposed method may help pave the way for design of practical adaptive switching controllers applicable to a wide range of linear time-invariant and time-varying systems     

State derivative feedback for adaptive cancellation of unmatched disturbances in unknown strict-feedback LTI systems

Solutions exist for the problem of canceling sinusoidal disturbances by the measurement of the state or by the measurement of an output for linear and nonlinear systems. In this paper, an adaptive backstepping controller is designed to cancel sinusoidal disturbances forcing an unknown linear time-invariant system in controllable canonical form which is augmented by a linear input subsystem with unknown system parameters. The state-derivatives of the original subsystem and the state of the input subsystem are the only measurements that are used in the design of the controller. The design is based on four steps, (1) parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters, (2) design of an adaptive disturbance observer for both disturbance and its derivative, (3) design of an adaptive controller for the virtual control input, and (4) design of the final adaptive controller by using the backstepping procedure. It is proven that the equilibrium of the closed-loop adaptive system is stable and the state of the considered original subsystem converges to zero as t→∞$t\to \infty$ with perfect disturbance estimation. The effectiveness of the controller is illustrated with a simulation example of a third order system.     

Methods for stability evaluation for linear time varying,discrete-time systems on finite time horizon

This paper develops a method for determining stability of discrete-time (DT), linear time-varying (LTV) systems defined on a finite time horizon (FTH). In our considerations we use a collection of stability definitions for linear time invariant (LTI) and LTV systems which are defined on an infinite time horizon (ITH). Based on the analysis carried out and the introduced operator-matrix notation we define four stability functions. These functions allow examination of the stability of a system described by an LTV state space model. Moreover, using these functions we introduce the time stability margin and making use of the operator-matrix notation we propose a method of determining the stability on a finite time horizon. The theoretical considerations are numerically verified on two examples of LTV systems with a variable degree of non-stationarity depending on a parameter ε. Several examples illustrate the application of the introduced concepts and definitions to examination of the system stability also depending on parameter ε.     

Learning from neural control of nonlinear systems in normal form

A deterministic learning theory was recently proposed which states that an appropriately designed adaptive neural controller can learn the system internal dynamics while attempting to control a class of simple nonlinear systems. In this paper, we investigate deterministic learning from adaptive neural control (ANC) of a class of nonlinear systems in normal form with unknown affine terms. The existence of the unknown affine terms makes it difficult to achieve learning by using previous methods. To overcome the difficulties, firstly, an extension of a recent result is presented on stability analysis of linear time-varying (LTV) systems. Then, with a state transformation, the closed-loop control system is transformed into a LTV form for which exponential stability can be guaranteed when a partial persistent excitation (PE) condition is satisfied. Accurate approximation of the closed-loop control system dynamics is achieved in a local region along a recurrent orbit of closed-loop signals. Consequently, learning of control system dynamics (i.e. closed-loop identification) from adaptive neural control of nonlinear systems with unknown affine terms is implemented.     

IQC‐based robustness analysis of discrete‐time linear time‐varying systems

The paper investigates the robust stability and performance of uncertain linear time‐varying (LTV) systems using an integral quadratic constraint (IQC) based analysis approach. Specifically, previous theoretical work on IQC‐based robustness analysis of linear time‐invariant (LTI) systems is extended to discrete‐time LTV systems. In the case of a general LTV nominal system, the analysis solution is provided in terms of an infinite‐dimensional convex optimization problem. This optimization problem reduces into a finite‐dimensional semidefinite program when the nominal system in question is finite horizon, periodic, or, more generally, eventually periodic. Finally, the results are applied to an unmanned aircraft control system executing an aggressive maneuver, where the developed techniques are used to find the region in which the aircraft is guaranteed to reside at the end of its planned trajectory. Copyright © 2017 John Wiley & Sons, Ltd.     

On using least-squares updates without regressor filtering in identification and adaptive control of nonlinear systems

In continuous-time system identification and adaptive control the least-squares parameter estimation algorithm has always been used with regressor filtering, which adds to the dynamic order of the identifier and affects its performance. We present an approach for designing a least-squares estimator that uses an unfiltered regressor. We also consider a problem of adaptive nonlinear control and present the first least-squares-based adaptive nonlinear control design that yields a complete Lyapunov function. The design is presented for linearly parametrized nonlinear control systems in ‘normal form’. A scalar linear example is included which adds insight into the key ideas of our approach and allows showing that, for linear systems, our Lyapunov-LS design with unfiltered regressor, presented in the note for unnormalized least-squares, can also be extended to normalized least-squares.     

ROBUST LTV FEEDBACK SYNTHESIS FOR SISO NON-LINEAR PLANTS

Given a feedback system with uncertain nonlinear plant, it is required that the plant's output, to a set of command inputs, will satisfy certain specifications, i.e., will be bounded between a maximum response β(t) and a minimum response α(t). A rigorous synthesis technique to solve this problem using an LTV controller is developed. A design example is included, and it is shown that the LTV design has much lower bandwidth as compared to LTI designs. All design steps utilize the well-established QFT technique for LTI SISO uncertain systems. The methodology also suits rejection of disturbances at the plant's input or output, and for output specifications for nonzero initial conditions. © 1997 by John Wiley & Sons, Ltd.     

On the destabilizing effect of cross-coupling in decentralized adaptive control

The stability of square multivariable linear time-invariant (LTI) systems coupled with an adaptive controller for which the process cross-coupling terms have been neglected is studied in this paper. Our aim is to find conditions over the reference signals richness and the process transfer matrix frequency response such that, for sufficiently small adaptation rates, the stability of the adaptive system is insured. It is shown that if the adaptive system without interconnections is stable then stability of the overall system is preserved if either: (1) The frequency bands of the reference signals are non-overlapping. (2) When there is a spectrum overlap each output of the LTI system to input signals in this spectrum has an average energy where the contribution from its corresponding input is larger than the one due to the other inputs. Global results are obtained for a simple single-parameter adaptation scheme. The extension to general adaptive controllers is straightforward in a local context [5].     

Interval set‐membership estimation for continuous linear systems

This article proposes a mixed interval set‐membership estimation (ISME) method for continuous linear time‐invariant (LTI) systems by combining the positive system theory and the set theory. The proposed ISME method gives a new mixed interval‐set estimation framework for continuous LTI systems, whose benefit consists in that it has potential to achieve a balance of computational complexity and robust state estimation conservatism with respect to the interval observer (IO) and the set‐valued observer (SVO) for continuous LTI systems. Particularly, the proposed ISME method first uses a coordinate transformation such that the original system is transformed into an equivalent system. Second, the equivalent system is partitioned into two subsystems, where the first subsystem has a Meztler and Hurwitz subsystem matrix and then an IO is designed for the first subsystem based on the positive system theory. Because it is not guaranteed that the second subsystem also has a Meztler and Hurwitz subsystem matrix, a zonotopic SVO is further designed for the second subsystem based on the set theory. Consequently, an integration of the two steps above provides the whole SE results for the original system. At the end of this article, an example is used to illustrate the effectiveness of the proposed ISME method.     

Feedforward controller synthesis for linear time variant systems

This paper develops a framework for the synthesis of a feedforward controller for both discrete and continuous linear time variant (LTV) systems. The proposed method determines the structure of a feedforward controller, which is cascaded with the known time varying process, so that the ensemble behaves as a linear time invariant (LTI) system, satisfying certain design requirements. The obtained controller is simple, consisting in a parallel bank of first or second order LTI filters, followed by amplifiers with time variable gain. Simulation results are included to illustrate theoretical considerations. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust Performance Analysis for Structured Linear Time-Varying Perturbations With Bounded Rates-of-Variation

The robust performance analysis problem is considered for linear time-invariant (LTI) systems subject to block-diagonal structured and bounded linear time-varying (LTV) perturbations with specified maximal rates-of-variation. Analysis methods are developed in terms of semidefinite programming for the computation of upper and lower bounds for the optimum robust performance level. The upper bound computation is based on an integral quadratic constraint (IQC) test developed using a generalized version of the so-called swapping lemma. The lower bound computation method employs an extended version of the power distribution theorem together with a generalized version of the Kalman-Yakubovich-Popov (KYP) lemma and serves as a means to assess the conservatism of the computed upper bounds in the case of dynamic LTV perturbations. As corollaries of the underlying result for lower bound computation, it is shown for general block-diagonal uncertainty structures that thefrequency-dependent/constant D-scaling tests are exact for robust performance analysis against arbitrarily slow/fast dynamic LTV perturbations, respectively     

Stabilising compensators for linear time-varying differential systems

In this paper, we describe a constructive test to decide whether a given linear time-varying (LTV) differential system admits a stabilising compensator for the control tasks of tracking, disturbance rejection or model matching and construct and parametrise all of them if at least one exists. In analogy to the linear time-invariant (LTI) case, the ring of stable rational functions, noncommutative in the LTV situation, and the Ku?era–Youla parametrisation play prominent parts in the theory. We transfer Blumthaler's thesis from the LTI to the LTV case and sharpen, complete and simplify the corresponding results in the book ‘Linear Time-Varying Systems’ by Bourlès and Marinescu.     

Pinning control and controllability of complex dynamical networks

In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output (SISO) or multi-input multi-output (MIMO) dynamical systems, and could be non-identical and nonlinear in general but will be specified to be identical linear time-invariant (LTI) systems here in the study of network controllability. Both state and structural controllability problems will be discussed, illustrating how the network topology, node-system dynamics, external control inputs and inner dynamical interactions altogether affect the controllability of a general complex network of LTI systems, with necessary and sufficient conditions presented for both SISO and MIMO settings. To that end, the controllability of a special temporally switching directed network of linear time-varying (LTV) node systems will be addressed, leaving some more general networks and challenging issues to the end for research outlook.     

Reduced order LQG controllers for linear time varying plants

The problem of reduced order LQG optimization is addressed in a finite horizon, linear time-varying (LTV) system setting. First-order necessary conditions for local optimality in the parameter space is provided in terms of four coupled matrix differential equations. This result provides a transparent generalization of the optimal projection equations of Hyland and Bernstein for the optimal, steady-state compensation of linear time-invariant (LTI) plants.     

Switching control of linear systems subject to asymmetric actuator saturation

In this paper, we study the saturation control problem for linear time-invariant (LTI) systems subject to asymmetric actuator saturation under a switching control framework. The LTI plant with asymmetric saturation is first transformed to an equivalent switched linear model with each subsystem subject to symmetric actuator saturation, based on which a dwell-time switching controller augmented with a controller state reset is then developed by using multiple Lyapunov functions. The controller synthesis conditions are formulated as linear matrix inequalities (LMIs), which can be solved efficiently. Simulation results are also included to illustrate the effectiveness and advantages of the proposed approach.