Computing stabilising output-feedback gains for continuous-time linear time-varying systems through discrete-time periodic models

This paper focuses on the optimal control of continuous-time linear time-varying uncertain systems with completely unknown internal dynamics and proposes a novel approach which leads to an optimal controller with guaranteed stability. A model-free algorithm of adaptive dynamic programming is employed to deal with the uncertainty of system parameters, yielding an optimal feedback controller for the system subject to a predefined cost. Then the stability of the system in time-varying uncertain situation which may undergo parameter changes or jumps is analyzed from the perspective of finite-time stability. On the basis of these results, a switching control strategy is presented to ensure the stability of the time-varying uncertain system with desired optimality properties in the long run. The effectiveness of the strategy is verified by simulations on a DC torque motor servo system.

This chapter studies null controllability of continuous-time linear systems with bounded controls. Null controllability of a system refers to the possibility of steering its state to the origin in a finite time by an appropriate choice of the admissible control input. If the system is linear and is controllable, then any state can be steered to any other location, for example, the origin, in the state space in a finite time by a control input. This implies that any controllable linear system is null controllable from any point in the state space. The control inputs that are used to drive certain states to the origin, however, might have to be large in magnitude. As a result, when the control input is limited by the actuator saturation, a linear controllable system might not be globally null controllable. In this situation, it is important to identify the set of all the states that can be steered to the origin with the bounded controls delivered by the actuators. This set is referred to as the null controllable region and is denoted as C

This paper presents a novel model reference adaptive iterative learning control (ILC) for unknown continuous-time linear time-varying systems. The unknown time-varying parameters of the system are neither required to vary slowly nor to have known bounds. The system is not required to be minimum-phase, stable, controllable or observable. The input of the system is determined by a differentiator-free control law. The used reference model is time-invariant and first order and thus choosing its parameters is easily possible, even though, the system under control is high order and time variant. Almost all of the components of the system initial condition can be iteration variant. By introducing a novel kind of Lyapunov function the convergence of the proposed adaptive ILC (AILC) and achieving asymptotic tracking are proved. Also, by rigorous mathematical analysis and with the help of some mathematical key techniques such as Bellman-Gronwall lemma, it is shown that all signals and quantities in the closed-loop system are bounded in the sense of at least one norm. Finally, the effectiveness of the proposed method is verified by two simulation examples.

A procedure for identifying linear continuous-time multivariable systems is presented. The scheme is based on a discrete identification method which requires no structural identification. This is accomplished by the use of pseudo-observable forms whose structure is defined by their corresponding pseudo-observability indices. A First-Order Hold transformation of this discrete-time model to an equivalent continuous-time model is then performed. First, the state transition matrix is transformed to an equivalent continuous-time system matrix using a procedure which is insensitive to the size of the spectral radii of the system. The remaining system matrices are then transformed using a generalized First-Order Hold transformation technique which places no restrictions on the singularity of the continuous-time system.

This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a class of continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two main characteristics; namely, (a) it satisfies discrete-type Popov’s inequality at sampling instants and (b) the control law within the intersample period is generated based on its value at sampling instants being modulated by two design weighting auxiliary functions. The closed-loop continuous-time system is proved to be asymptotically hyperstable, under some explicit conditions on such weighting functions, provided that the discrete feed-forward transfer function is strictly positive real.

New results on coprime factorizations for continuous-time linear time-varying systems are presented. It is shown that a linear time-varying system admits a coprime factorization if and only if it is stabilizable and detectable, or if and only if it is internally stabilizable via dynamic output feedback. State-space formulae for coprime factorizations are given.

In this chapter, the sliding mode control (SMC) design for a class of continuous-time linear uncertain systems is considered based on the parametric approaches utilizing the Lyapunov (or Riccati-like) inequalities. SMC for the continuous-time linear systems is one of the well-known issues in control theory. However, this chapter addresses a new aspect of SMC for linear uncertain systems and establishes a systematic procedure to design a sliding hyperplane having multiple design objectives.

OBSERVABILITY OF DISCRETE TIME LINEAR SWITCHING SYSTEMS1

The stability and behavior of continuous-time nonlinear time-varying (NTV), nonlinear time-invariant (NTI) and linear time-varying (LTV) systems are generally very difficult to ascertain using current analytical tools. This paper presents sufficient conditions under which a NTV system with a bounded system matrix, A(t,X), is globally exponentially stable (GES). The continuous-time NTV system is modeled as an uncertain polytopic system even though A(t,X) may be reasonably known. It is shown that if all polytope-vertex matrices are Hurwitz, then the system is GES. Also, a method for determining the boundary estimate of a NTV system state-trajectory norm is developed. Presented is a unified stability analysis approach for a large class of NTV, NTI, and LTV systems. Even for large systems, application of the GES theorem presented is straight forward using readily available software and computational power. Examples are given that illustrate the application of the theorem offered.

The paper deals with the construction of generalized frozen-time (FT) models-for linear time-varying continuous-time systems. The generalized FT models are higher-order models generated from a given n-dimensional state model by applying a stable differential operator to the state equation. This results in FT models whose coefficients depend on the derivatives of the time-varying system coefficient matrices. The accuracy of the higher-order FT models over local time intervals is investigated in terms of truncated Taylor series. In the last part of the paper, the application of the higher-order FT models to state feedback control is considered.

This paper considers the problem of constrained stabilization of linear continuous-time systems when a quadratic cost criterion is imposed in the design of the state feedback control law. The linear quadratic regulator (LQR) is solved under positivity constraint, which means that the resulting closed-loop systems are not only optimally stable, but also positive. We focus on the class of linear continuous-time positive systems (Metzlerian systems) and use the interesting properties of Metzler matrices to provide the necessary ingredients for the main results of the paper. A two-step procedure is proposed to solve the problem. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the Metzlerian constraint, and the constrained stabilization is solved using linear programming (LP) or linear matrix inequality (LMI). Second, a sufficient condition is outlined for the existence of a solution to maintain the positivity of the first step while achieving the optimality of LQR. Finally, the robustness of the design is analyzed and possible extension of the design is proposed for an uncertain interval systems. A numerical example is included to illustrate the procedure.

In this paper, the problem of designing Robust Gain-Scheduling observers for continuous-time Linear Parameter-Varying systems via parameter-dependent Lyapunov function is addressed. The scheduling parameters are assumed to be imprecisely measured, i.e., corrupted with additive noise. Multi-simplex modeling approach is utilized to model the time-varying parameters and associated uncertainties. Sufficient conditions are given for the synthesis problem in terms of Parameterized Linear Matrix Inequalities. The designed observer not only guarantees asymptotic stability of estimation error dynamics but also minimizes the estimation errors under exogenous disturbances. Both, ℋ 2 and ℋ ∞ performances have been considered. An illustrative numerical example is presented to demonstrate the benefit of the developed approach.

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

This paper investigates the distributed consensus problem for linear continuous-time multi-agent systems subject to additive process noises. Due to the existence of noises, the asymptotic consensus cannot be achieved. Hence, a distributed consensus protocol is proposed such that all the agents can achieve consensus to some degree. Furthermore, based on Lyapunov theory, we characterize the consensus error under undirected and directed topologies. Numerical examples are presented to verify the effectiveness of two results.

Design of RLS-FIR filter using covariance information in linear continuous-time stochastic systems