Robust Control of Linear Systems via Switching

This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.

We investigate the design of a static output feedback control for switching discrete time systems. Looking at the problem of stability analysis, we give two conditions allowing to check the existence of a particular class of Lyapunov functions. Although one can recover the first condition as a particular case from the second one, it is proved that the two conditions are equivalent for stability analysis. Then we show that when one is faced with a static output feedback control problem, the second condition leads to a sufficient solution to build a stabilizing static output feedback controller which is less conservative than the solution based on the first condition.

This paper addresses the stability, as well as an upper bound on the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain of linear switched state-multiplicative stochastic systems. The switching law is assumed to obey a dwell time constraint. The results are achieved by assigning each subsystem its own Lyapunov function. The Lyapunov function is time varying during the dwell time and becomes time invariant afterwards. The Lyapunov function is required to be non-increasing at the switching instances. If the dwell time approaches zero-the standard quadratic conditions are recovered. Stability and performance are guaranteed in the mean-square sense by requiring the expected value of the Lyapunov function to decrease, both for nominal systems and systems that entail polytopic-type parameter uncertainties. The theory is then extended to state-feedback controller design. The conditions are all given in terms of linear matrix inequalities.

This article considers the problems of robust stability and stabilization for a class of fractional-order linear time-invariant systems via 0&lt;α&lt;1 subject to input saturation with convex polytopic uncertainties, by a kind of [Formula: see text]–proportional–integral–derivative control synthesis. A new saturation-based stability condition is suggested to estimate the stable region ([Formula: see text]) and region of attraction (ε) using the ellipsoid approach. The main concept of the presented strategy is based on transforming the proportional–integral–derivative controller design problem into the static output feedback control problem. Gronwall–Bellman lemma and sector bounded condition of the saturation function is used for stability investigation and stabilization of such systems. The static output feedback control laws are obtained with the aid of a non-iterative linear matrix inequality implication. From the static output feedback controller, the proportional–integral–derivative controller is then restored. Stability problem is addressed as an optimization problem to obtain the controller gains and the stable region, as well as the largest region of attraction. Some theorems and numerical examples are provided to illustrate the results of proposed mechanism.

Sufficient conditions for the stability of linear-switched systems with dwell-time and polytopic-type parameter uncertainties are presented. A Lyapunov function, in quadratic form, which is nonincreasing at the switching instants is assigned to each subsystem. During the dwell time, this function varies piecewise linearly in time after switching occurs and it becomes time invariant afterward. This function leads to asymptotic stability conditions for the nominal set of the subsystems that can be readily extended to the case where these subsystems suffer from polytopic-type parameter uncertainties. The method proposed is then applied to stabilization via state feedback, both for the nominal and the uncertain cases.

This paper investigates the static output feedback control problem for discrete-time positive linear systems subject to state delay, actuator constraints, and resource limitations in communication. To address these challenges simultaneously, an event-triggered control framework is developed. A novel event-triggering condition is proposed, based on a component-wise comparison where the triggering error is checked against state-dependent bounds. Sufficient conditions for ensuring positivity, exponential stability, and satisfaction of input constraints are derived. Crucially, these design conditions are formulated as a set of solvable linear programming problems, making the approach computationally tractable. Furthermore, conditions are established to demonstrate a provably reduced control update frequency compared to traditional time-triggered control, quantifying the resource-saving benefits of the proposed approach. Two numerical examples, including one based on a population dynamics model, illustrate the effectiveness of the proposed approach.

ABSTRACTThis paper investigates the static output feedback control problem for linear systems. The sensor and actuator may simultaneously suffer the total/partial loss of effectiveness. An adaptive fault-tolerant control scheme is developed by combination of linear matrix inequality method and adaptive mechanism. By estimating the inverses of failure factors online, the adverse effects of sensor and actuator failure can be compensated for. Finally, simulation results on the ‘Raptor-90’ helicopter verify the effectiveness of the proposed approaches.

This paper studies the composite anti-disturbance static output feedback control problem of networked nonlinear Markov jump systems with general transition probabilities subject to multiple disturbances and deception attacks. The transition probabilities cover the known, uncertain with known bounds, and unknown cases. The unmatched disturbance and deception attacks are attenuated by the static output controller, while the matched disturbance is observed and compensated by the disturbance observer. Then, a composite anti-disturbance static output controller, including a linear part and a nonlinear part, is constructed to satisfy the stochastic H∞ stability. By using the Finsler lemma, sufficient conditions formed as symmetric linear matrix inequalities are derived to design the gains of disturbance observer and the output feedback controller. Finally, some simulations are given to illustrate the feasibility of the developed strategy.

Output feedback [formula omitted] finite-time control of switched positive delayed systems with MDADT

The static output feedback control problem for discrete linear, singularly perturbed, stochastic systems is studied. A recursive algorithm is presented to solve the corresponding coupled non-linear algebraic equations. The algorithm removes the ill-conditioning by decomposing the higher order equations into lower order equations and develops a technique which enables us to obtain an arbitrary accuracy, that is, 0(ej) (j = 1, 2, 3, …)a approximation for this problem. A numerical example is presented to support the theoretical results. Communicated by Professor A. A. Ioannou.

Improved stability criterion and output feedback control for discrete time-delay systems

This article investigates the robust exponential stabilization of a class of switched positive systems with uncertainties by co‐designing controllers for subsystems and dwell time switching strategy. The uncertainties refer to interval uncertainties. One of the distinguishing features is that none of the forced individual subsystems is assumed to be stabilized. The other feature is that the stabilization for the unforced switched system can not be solvable by designing dwell time switching signal. First, for switched positive systems without uncertainties, a type of multiple time‐varying co‐positive Lyapunov functions is used, and the computable sufficient conditions on the state feedback controller for each subsystem are derived in the framework of dwell time strategy. Moreover, the exponential stabilization problem is solved by confining the lower and upper bounds of the dwell time, restricting the upper bound of derivative of considered Lyapunov function of the active subsystem, and decreasing the Lyapunov function values at successive switching instants of the overall switched system. Then, the results are extended to the robust exponential stabilization case for switched positive systems with uncertainties, and sufficient conditions are given in the same framework of the dwell time for that of the forced switched systems by further designing state feedback controllers. Finally, illustration examples are given to illustrate the effectiveness of the proposed methods.

A linear parameter varying approach for designing a constant output feedback controller for a linear time-invariant retarded system with stochastic multiplicative Wiener-type noise, that achieves a minimum bound on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance level is introduced. The stochastic uncertainties appear in the dynamic matrices, which correspond to the delayed and non-delayed states of the system, and in the measurement matrix of the system. The solution of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> static output-feedback control problem is solved, for the stationary case, via the input-output approach where the system is replaced by a non-retarded system that contain, instead, deterministic norm-bounded uncertainties. In this problem, a cost function is defined which is the expected value of the standard H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance cost with respect to the stochastic parameters. We extend the results achieved for the nominal case, to the uncertain case where the system matrices lie within a given polytop.

Stability analysis for a class of switched systems with state constraints via time-varying Lyapunov functions

Robust stability of switched positive linear systems with interval uncertainties via multiple time-varying linear copositive Lyapunov functions