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We consider optimal control problems for continuous-time systems with time-dependent dynamics, in which the time-dependence arises from the presence of a known exogenous signal. The problem has been elegantly solved in the case of linear input-affine systems, for which it has been shown that the solution has a remarkable structure: It is given by the sum of two contributions; a state feedback, which coincides with the unperturbed optimal control law, and a purely feedforward term in charge of compensating the effect of the exogenous signal. The objective of this article is to extend the above result to nonlinear input-affine systems. It is shown that, while some of the relevant features of the linear case indeed rely heavily on linearity and are not preserved in the nonlinear setting, several structural claims can be proved also in the nonlinear case. [ABSTRACT FROM AUTHOR]

Differential-algebraic equations (DAEs) are a widespread dynamical model that describes continuously evolving quantities defined with differential equations, subject to constraints expressed through algebraic relationships. As such, DAEs arise in many fields ranging from physics, chemistry, and engineering. In this article, we focus on linear DAEs, and develop a theory for their minimization up to an equivalence relation. We present differential equivalence, which relates DAE variables that have equal solutions at all time points (thus requiring them to start with equal initial conditions) and extends the line of research on bisimulations developed for Markov chains and ordinary differential equations. We apply our results to the electrical engineering domain, showing that differential equivalence can explain invariances in certain networks as well as analyze DAEs, which could not be originally treated due to their size. [ABSTRACT FROM AUTHOR]

This technical note investigates the positive consensus problem of multiagent systems with directed communication topologies, where all the agents have identical continuous-time positive linear dynamics. Existing works of such a problem mainly focus on the case, where networked communication topologies are of either undirected and incomplete graphs, or strongly connected directed graphs. In contrast to them, we study this problem in which the communication topologies of the multiagent system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Based on the existing results in spectral graph theory and positive linear systems theory, several necessary, and sufficient conditions on positive consensus of the directed multiagent system are derived through using linear matrix inequality techniques. A primal-dual iterative algorithm is developed for the computation of solutions. Finally, several numerical simulations are provided to illustrate the effectiveness of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

This article presents a new optimization-based approach to determine the class of stabilizing control structures with the necessary set of feedback links for interconnected systems. The proposed approach relies on a graph-theoretic interpretation and its equivalence in terms of binary linear programs (BLPs). To carry out the primary goal, first, the stabilizability of a linear time-invariant (LTI) system under the decentralized control structure is presented in terms of a BLP. Next, two graph-based criteria are proposed to characterize stabilizing control structures with the required feedback links. Finally, all possible stabilizing control structures with the necessary feedback links are derived via solving a set of BLPs. In addition to the analysis of stabilizing control structures, the proposed graph-theoretic approach offers a versatile set of tools to find the simplest control structures capable of assigning the closed-loop spectrum in a totally arbitrary desired region. Simulation results illustrate the assessment of stabilizing control structures, the required additional feedback links, and the elegant generalization of the proposed approach to regional pole-placement. [ABSTRACT FROM AUTHOR]

This article finds the minimum cost that is needed to achieve the approximate controllability of switched infinite-dimensional linear systems (IDLSs) by a newly defined Gramian operator. Such a new cost evaluation method helps to establish a fault recoverability condition, which reveals the ability of faulty switched IDLSs to maintain the approximate controllability under given energetic constraints. An example of the swarming system is taken to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we rely on the theoretical foundations of radial basis function neural networks to form an adaptive parameter estimation problem, which we solve using the recently introduced prescribed performance control methodology. Such combination results in a compact user-configurable adaptive nonlinear system identification methodology that can be used to retrieve the open-loop nonlinear plant dynamics in any compact region of interest. [ABSTRACT FROM AUTHOR]

This article proposes an event-triggered control design for cooperative control of heterogeneous multiagent systems. The design is serial in nature and consists of the design of a distributed event-triggered observer for each follower agent that estimates the desired information of the leader system in the presence of parametric uncertainty and, based on this observer, the design of a distributed event-triggered controller. A salient feature of our observer design is that it obeys the separation principle in the sense that the event-triggered control can be designed after the observer has been designed. As a result, the observer is applicable to different follower agent dynamics and the event-triggered time sequence for the controller is asynchronous of that of the observer. We implement this design scheme on linear heterogeneous multiagent systems and uncertain Euler–Lagrange multiagent systems, and achieve results that significantly extend the existing ones. [ABSTRACT FROM AUTHOR]

This article presents a computational method to test the stabilizability of discrete-time switched linear systems. The existence of a conic cover of the space on whose elements a convex condition holds is proved to be necessary and sufficient for stabilizability. An algorithm for computing a conic partition that satisfies the new necessary and sufficient condition is given. The algorithm, which allows us to determine bounds on the exponential convergence rate, is proved to overcome the conservatism of conditions equivalent to periodic stabilizability and is applied to a 4-D system. [ABSTRACT FROM AUTHOR]

In undirected networks of identical linear oscillators coupled through dissipative and restorative connectors (e.g., pendulums undergoing small vibrations connected via dampers and springs), the relation between asymptotic synchronization and coupling structure is studied. Conditions on the interconnection under which synchronization can be achieved for some selection of coupling strengths are established. How to strengthen those conditions so that synchronization is guaranteed for all admissible parameter values is also presented. [ABSTRACT FROM AUTHOR]

This article aims to reveal an interesting phenomenon about the time delays (TDs) in control systems: stability preservation with over large intermittently or unbounded variable TDs. That is, given a system asymptotically stable for bounded TDs, it could remain to be stable when the TDs are enhanced to have peak values over large intermittently or unbounded. The investigation is carried out by theoretical analysis plus case studies with the stochastic system models. The stability of systems is the core issue; the explicit involvement of the TDs in the stability criteria is the premise for them being concerned. To these ends, a series of lemmas are established preliminarily, a new type stability theorem with its practical version is established in succession, and a stability criterion with the TDs as explicit parameters is induced. The Lyapunov function method is applied for less restriction and conservativeness. The mechanism and the approach for the emergence of the phenomenon under probe are analyzed. Implementation is considered for the linear system model. Typical variable TDs are constructed for illustrations theoretically. Finally, the obtained results are verified with numerical experiments. [ABSTRACT FROM AUTHOR]

This note presents phase conditions under which there is no suitable Zames– Falb multiplier for a given discrete-time system. Our conditions can be seen as the discrete-time counterpart of Jönsson’s duality conditions for Zames–Falb multipliers. By contrast with their continuous-time counterparts and other phase limitations in the literature, they lead to numerically efficient results that can be computed either in closed form or via a linear program. The closed-form phase limitations are tight in the sense that we can construct multipliers that meet them with equality. The numerical results allow us to conclude that the current state-of-the-art in searches for Zames–Falb multipliers is not conservative. Moreover, they allow us to show, by construction, that the set of plants for which a suitable Zames– Falb multiplier exists is nonconvex. [ABSTRACT FROM AUTHOR]

The article introduces and solves a structural controllability problem for continuum ensembles of linear time-invariant systems. All the individual linear systems of an ensemble are sparse, governed by the same sparsity pattern. Controllability of an ensemble system is, by convention, the capability of using a common control input to simultaneously steer every individual systems in it. A sparsity pattern is structurally controllable if it admits a controllable linear ensemble system. A main contribution of the article is to provide a graphical condition that is necessary and sufficient for a sparsity pattern to be structurally controllable. Like other structural problems, the property of being structural controllable is monotone. We provide a complete characterization of minimal sparsity patterns as well. [ABSTRACT FROM AUTHOR]

Complex networked systems can be modeled and represented as graphs, with nodes representing the agents and the links describing the dynamic coupling between them. The fundamental objective of network identification for dynamic systems is to identify causal influence pathways. However, dynamically related data streams that originate from different sources are prone to corruption caused by asynchronous time-stamps, packet drops, and noise. In this article, we show that identifying causal structure using corrupt measurements results in the inference of spurious links. A necessary and sufficient condition that delineates the effects of corruption on a set of nodes is obtained. Our theory applies to nonlinear systems, and systems with feedback loops. Our results are obtained by the analysis of conditional directed information (DI) in dynamic Bayesian networks. We provide consistency results for the conditional DI estimator that we use by showing almost-sure convergence. [ABSTRACT FROM AUTHOR]

This article addresses the problem of robust output feedback model predictive control for discrete-time, constrained, linear parameter-varying systems subject to (bounded) state and measurement disturbances. The vector of scheduling parameters is assumed to be an unmeasurable signal taking values in a given compact set. The proposed controller incorporates an interval observer, that uses the available measurement to update the set-membership estimation of the states, and an interval predictor, used in the prediction step of the model predictive control (MPC) algorithm. The resulting MPC scheme offers guarantees on recursive feasibility, constraint satisfaction, and input-to-state stability in the terminal set. Furthermore, this novel algorithm shows low computation complexity and ease of implementation (similar to conventional MPC schemes). [ABSTRACT FROM AUTHOR]

Given a centralized linear time-invariant (LTI) controller, this article investigates if and when the same can be implemented in a decentralized fashion using continuous-time, high-frequency, periodic controllers, recently reported in literature, without compromising input–output performance in averaged sense. In this regard, it is shown that for a strongly connected two-channel system having each subsystem of equal sizes, there exists a relation to compute the decentralized periodic controller gains in terms of the gains of a given centralized LTI controller if the latter is minimum-phase, stable, and satisfies some relative degree conditions. A method to satisfy this relative degree constraint while designing the centralized LTI controller is developed. A suitable example is considered to show that the proposed decentralized periodic controller can mimic a centralized LTI controller in averaged sense. At the end, it is found that the proposed decentralized periodic controller, however, does not have enough number of periodic gains to obtain the above relation to compute its parameters for systems with more than two channels. [ABSTRACT FROM AUTHOR]

A finite-time observer for a class of linear time-varying systems with bounded unknown inputs is presented in this note. The design of the observer exploits structural properties of the system and, through a linear operator, allows applying the robust exact differentiator in a block form without requiring additional stabilizing terms. The proposed observer provides an exact estimate of the states after a finite transient time despite the possible instability of the system and the effects of bounded unknown inputs. [ABSTRACT FROM AUTHOR]

In this article, we present a convex optimization framework to verify the reachability of a desired set for discrete-time linear time-invariant systems. Given elliptically bounded inputs, the set of reachable states in $N$ time steps is the Minkowski sum of a finite number of ellipsoids. We formulate the inclusion verification problem as a chain of constraints in the form of linear matrix inequalities. As the time horizon grows, the number of constraints becomes unwieldy, and we present a technique to achieve a similar level of accuracy with far fewer terms, significantly reducing the computational cost of the method. Numerical examples presented in this article show that the method is highly adaptable. [ABSTRACT FROM AUTHOR]

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the Bell polynomials, the generalized Laguerre polynomials, the Lah numbers, and a suitable polynomial-based Vandermonde matrix. The results can be used to design a new controller—with time-varying gains—ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway et al. 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE–ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of time-varying kernels. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results. [ABSTRACT FROM AUTHOR]

In this article, we study how to secure distributed filters for linear time-invariant systems with bounded noise under false-data injection attacks. A malicious attacker is able to arbitrarily manipulate the observations for a time-varying and unknown subset of the sensors. We first propose a recursive distributed filter consisting of two steps at each update. The first step employs a saturation-like scheme, which gives a small gain if the innovation is large corresponding to a potential attack. The second step is a consensus operation of state estimates among neighboring sensors. We prove the estimation error is upper bounded if the filter parameters satisfy a condition. We further analyze the feasibility of the condition and connect it to sparse observability in the centralized case. When the attacked sensor set is known to be time-invariant, the secured filter is modified by adding an online local attack detector. The detector is able to identify the attacked sensors whose observation innovations are larger than the detection thresholds. Also, with more attacked sensors being detected, the thresholds will adaptively adjust to reduce the space of the stealthy attack signals. The resilience of the secured filter with detection is verified by an explicit relationship between the upper bound of the estimation error and the number of detected attacked sensors. Moreover, for the noise-free case, we prove that the state estimate of each sensor asymptotically converges to the system state under certain conditions. Numerical simulations are provided to illustrate the developed results. [ABSTRACT FROM AUTHOR]

In model-predictive control (MPC), an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. We propose an algorithm for computing which sequence of subproblems an active-set algorithm will solve, for every parameter of interest. These sequences can be used to set worst-case bounds on how many iterations, floating-point operations, and, ultimately, the maximum solution time the active-set algorithm requires to converge. The usefulness of the proposed method is illustrated on a set of QPs originating from MPC problems, by computing the exact worst-case number of iterations primal and dual active-set algorithms require to reach optimality. [ABSTRACT FROM AUTHOR]

We consider a fundamental remote state estimation problem of discrete-time linear time-invariant (LTI) systems. A smart sensor forwards its local state estimate to a remote estimator over a time-correlated multistate Markov fading channel, where the packet drop probability is time-varying and depends on the current fading channel state. We establish a necessary and sufficient condition for mean-square stability of the remote estimation error covariance in terms of the state transition matrix of the LTI system, the packet drop probabilities in different channel states, and the transition probability matrix of the Markov channel states. To derive this result, we propose a novel estimation-cycle based approach and provide new elementwise bounds of matrix powers. The stability condition is verified by numerical results and is shown more effective than existing sufficient conditions in the literature. We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or nonconvex depending on the transition probability matrix of the Markov channel states. Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator) though the smart sensor setup achieves a better estimation performance. [ABSTRACT FROM AUTHOR]

This article discusses the reachability analysis (RA) of interval max-plus linear (IMPL) systems, a subclass of continuous-space, discrete-event systems defined over the max-plus algebra. Unlike standard max-plus linear systems, where the transition matrix is fixed at each discrete step, IMPL systems allow for uncertainty on state matrices. Given an initial and a target set, we develop algorithms to verify the existence of IMPL system trajectories that, starting from the initial set, eventually reach the target set. We show that RA can be solved by encoding the IMPL system, as well as initial and target sets, into linear real arithmetic expressions, and then checking the satisfaction of a resulting logical formula via a satisfiability modulo theory (SMT) solver. The performance and scalability of the developed SMT-based algorithms are shown to drastically outperform state-of-the-art RA algorithms applied to IMPL systems, which promises to usher their use in practical, industrial-sized IMPL models. [ABSTRACT FROM AUTHOR]

This article addresses the problem of estimating the domain of attraction of positive systems under a saturated linear feedback. Compared with the analysis of general saturated linear systems, one remarkable difference is that only the domain of attraction inside the first orthant is of interest. Hence, we tackle this problem by exploiting different shapes of invariant sets confined in the first orthant, including hyperpyramids and hyperrectangles, which are, respectively, the level sets of sum-separable and max-separable Lyapunov functions. For an open-loop positive system, a sufficient condition is given to ensure that a pyramid (or a hyperrectangle) is contractively invariant for the closed loop. This condition is nonconservative for the single-input case. Moreover, for a system without open-loop positivity, we establish conditions such that the closed loop has a hyperrectangular invariant set. Examples show the superiority of the results, in comparison with those existing ones using invariant ellipsoids. [ABSTRACT FROM AUTHOR]

This article provides the formulas connecting the Laguerre spectrum of the output signal of a linear discrete-time time-invariant delay system to the Laguerre spectrum of its input signal. The Laguerre-domain system representation is meaningful when the input signal is square summable, and the system is stable. A certain type of polynomials arising in the evaluation of the output spectrum due to the presence of time delay is defined, and key properties of these are investigated. The polynomials are characterized by a three-term recurrence relation that also facilitates their numerically reliable calculation. [ABSTRACT FROM AUTHOR]

This article is concerned with the state estimation of linear dynamic systems when some sensors are corrupted by attackers. This problem is known as resilient state estimation (RSE), and aims to achieve, under some conditions, the estimation of the true state despite the malicious attacks on sensors. The state-of-art RSE methods provides a bound on estimation errors when external disturbance exists. However, it is shown in this article that the effect of the disturbance on estimation error may be larger than that for conventional observers, or even worse, resiliency may be lost for the disturbance that exceeds the bound. To resolve this issue, unknown input observer (UIO) mechanism is adopted in RSE for the purpose of estimating true plant state under both sensor attacks and external disturbance. Also achieved in this work is the method of partial state UIO synthesis, which relaxes the design requirement for full state UIO. In relation to resiliency, it is shown that 2 $q$ redundant detectability is a necessary condition for robust and resilient state estimator in order to tolerate up to $q$ sensor attacks. Numerical examples are given to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, a new time-varying adaptivedynamic programming (ADP) algorithm is developed to solve finite-horizon optimal control problems for a class of discrete-time affine nonlinear systems. Inspired by the pseudolinear method, the nonlinear system can be approximated by a series of time-varying linear systems. In each iteration of the time-varying ADP algorithm, the optimal control law for the time-varying linear system is obtained. For an arbitrary initial state, it is proven that states of the time-varying linear systems converge to the states of discrete-time affine nonlinear systems. It is also shown that the iterative value functions and the iterative control laws converge to the optimal value function and the optimal control law, respectively. Finally, numerical results are presented to verify the effectiveness of the present method. [ABSTRACT FROM AUTHOR]

Time-distributed optimization (TDO) is an approach for reducing the computational burden of model predictive control (MPC) and a generalization of the real-time iteration scheme. When using TDO, optimization iterations are distributed over time by maintaining a running solution estimate and updating it at each sampling instant. In this article, TDO applied to input-constrained linear-quadratic MPC is studied in detail, and an analytic bound for the number of optimization iterations per sampling instant required to guarantee closed-loop stability is derived. Further, it is shown that the closed-loop stability of TDO-based MPC can be guaranteed using multiple mechanisms, including increasing the number of solver iterations, preconditioning the optimal control problem, adjusting the MPC cost matrices, and reducing the length of the receding horizon. These results in a linear system setting also provide insights and guidelines that could be more broadly applicable, for example, to nonlinear MPC. [ABSTRACT FROM AUTHOR]

Set-membership filter (SMF) has been extensively studied for state estimation in the presence of bounded noises with unknown statistics. Since it was first introduced in the 1960s, the studies on SMF have used the set-based description as its mathematical framework. One important issue that has been overlooked is the optimality of SMF. In this article, we put forward a new mathematical framework for SMF using concepts of uncertain variables. We first establish two basic properties of uncertain variables, namely, the law of total range (a nonstochastic version of the law of total probability) and the equivalent Bayes’ rule. This enables us to put forward a general SMFing framework with established optimality. Furthermore, we obtain the optimal SMF under a nonstochastic Markov condition, which is shown to be fundamentally equivalent to the Bayes filter. Note that the classical SMF in the literature is only equivalent to the optimal SMF we obtained under the nonstochastic Markov condition. When this condition is violated, we show that the classical SMF is not optimal and it only gives an outer bound on the optimal estimate. [ABSTRACT FROM AUTHOR]

In this article, we propose a dual set membership filter for nonlinear dynamic systems with additive unknown but bounded noises, and it has three distinct advantages. First, the nonlinear system is translated into the linear system by leveraging a semi-infinite programming, rather than linearizing the nonlinear function. The semi-infinite programming is to find an ellipsoid bounding the nonlinear transformation of an ellipsoid, which aims to compute a tight ellipsoid to cover the state. Second, the duality result of the semi-infinite programming is derived by rigorous analysis; then, a first-order Frank–Wolfe method is developed to efficiently solve it with a lower computation complexity. Third, the proposed filter enjoys stability for some special nonlinear dynamic systems and succeeds the advantages of the classic linear set membership filter. Finally, two illustrative examples in the simulations reveal the effectiveness of the dual set membership filter. [ABSTRACT FROM AUTHOR]

Recent studies have shown how regularization may play an important role in linear system identification. An effective approach consists of searching for the impulse response in a high-dimensional space, e.g., a reproducing kernel Hilbert space (RKHS). Complexity is then controlled using a regularizer, e.g., the RKHS norm, able to encode smoothness and stability information. Examples are RKHSs induced by the so-called stable spline or tuned-correlated kernels, which contain a parameter that regulates impulse response exponential decay. In this article, we derive nonasymptotic upper bounds on the $\ell _2$ error of these regularized schemes and study their optimality in order (in the minimax sense). Under white noise inputs and Gaussian measurement noises, we obtain conditions which ensure the optimal convergence rate for all the class of stable spline estimators and several generalizations. Theoretical findings are then illustrated via a numerical experiment. [ABSTRACT FROM AUTHOR]

We consider minimal controllability problems (MCPs) on linear structural descriptor systems. We address two problems of determining the minimum number of input nodes such that a descriptor system is structurally controllable. We show that MCP0 for structural descriptor systems can be solved in polynomial time. This is the same as the existing results on typical structural linear time-invariant (LTI) systems. However, the derivation of the result is considerably different because the derivation technique of the existing result cannot be used for descriptor systems. Instead, we use the Dulmage–Mendelsohn decomposition. Moreover, we prove that the results for MCP1 are different from those for usual LTI systems. In fact, MCP1 for descriptor systems is an NP-hard problem, while MCP1 for LTI systems can be solved in polynomial time. [ABSTRACT FROM AUTHOR]

Using the notion of exponential QSR-dissipativity (i.e., dissipativity with respect to a quadratic supply rate given in terms of real matrices Q, S,R), this article presents necessary and sufficient conditions for exponential stabilizability of nonlinear systems by linear static output feedback (SOF). It is shown that, under mild assumptions, the exponential stabilization of the closed-loop system around the origin is equivalent to the exponential QSR-dissipativity of the plant. Furthermore, whereas strict QSR-dissipativity is only sufficient for SOF asymptotic stabilization, it is proved to be necessary and sufficient for full state feedback control. New necessary and sufficient conditions for SOF stabilizability of linear systems are presented as well, along with a linear and noniterative semidefinite strategy for controller design. Necessary linear matrix inequality conditions for stabilization are also introduced. Some examples illustrate the usefulness of the proposed approach. [ABSTRACT FROM AUTHOR]

This article considers the problem of output feedback stabilization for a class of nonlinear planar systems with unknown structures and measurements, which prevent the construction of conventional state observers. By taking advantage of the stability-increasing capability of a lead compensator, we propose a dynamic output feedback controller to globally stabilize the uncertain planar systems. For the special case of linear planar systems with unknown coefficients, a finite-time output feedback stabilizer based on a nonlinear compensator is constructed for a faster convergence rate. [ABSTRACT FROM AUTHOR]

In this article, we investigate the robust stabilization of linear time-invariant (LTI) systems with external disturbances via event-triggered impulsive control (ETIC). Especially, in order to suppress the disturbances effectively, the impulsive instant sequence is generated through the sliding-variable-based event-triggering mechanism (ETM). Based on the proposed ETM, some sufficient conditions are derived to ensure the robust stabilization of the LTI systems and to exclude Zeno phenomenon. It is shown that the system states are ultimately bounded under the ETIC, and the ultimate bound can be arbitrarily small by selecting the appropriate parameter of ETM. More interestingly, it is proved that our obtained results can well deal with the case of unknown bounded external disturbances, which is often addressed by the adaptive control in the previous results. Then, the theoretical results are applied to the motor servo systems in the presence of external disturbances. Finally, two illustrative examples are presented to show the validity of the results. [ABSTRACT FROM AUTHOR]

The purpose of this article is to synthesize a linear quantum system, which is strictly stable and has a steady thermal state. Specifically, we give a parameterization of a class of stable linear quantum systems that have $V=\tau I/2$ , $\tau > 1$ , as their steady covariance matrsices. This is physically important since the covariance matrix $\tau I/2$ , $\tau > 1$ , corresponds to a quantum thermal state. Hence, we can say that these systems will asymptotically evolve into a quantum thermal state. An extension to the case where $V=S\operatorname{diag}(\Lambda,\Lambda) S^{\top }/2$ with $\Lambda > I$ being a diagonal matrix and $S$ being a symplectic matrix will also be considered. Physically, a covariance matrix of the form $V=S\operatorname{diag}(\Lambda,\Lambda) S^{\top }/2$ , $\Lambda > I$ , corresponds to a mixed Gaussian quantum state. So, we can alternatively say that the corresponding linear quantum systems will asymptotically evolve into a mixed Gaussian quantum state. [ABSTRACT FROM AUTHOR]

We consider a version of the pole placement problem for one-sided linear discrete time-varying linear systems. Our purpose is to prove that uniform complete controllability is equivalent to possibility of arbitrary assignment of the dichotomy spectrum. [ABSTRACT FROM AUTHOR]

This article addresses the problem of sliding-mode control (SMC) of linear uncertain systems with impulse effects. The difficulty in solving such problem lies in that the continuity property of the well-used linear sliding function is lost under the intermittent impulsive action. In order to overcome this difficulty, a piecewise linear sliding function considering the dynamics properties of impulses is introduced, which turns out to be continuous along the trajectories of the impulsive system. Then, a suitable integral SMC law with switching feedback gains is constructed to guarantee the reachability of the designed sliding surface in a finite time. The resulting sliding-mode dynamics is modeled by an impulsive switched system whose stability is analyzed by applying a piecewise discontinuous Lyapunov function. Next, a sufficient condition for the existence of integral SMC law is derived in terms of linear matrix inequalities. Finally, a numerical example with several different types of impulses is provided to validate the theoretical results, which shows that the switching gain-based design contributes to the robustness of the sliding-mode controller. [ABSTRACT FROM AUTHOR]

We consider the following communication scenario. An encoder causally observes the Wiener process and decides when and what to transmit about it. A decoder estimates the process using causally received codewords in real time. We determine the causal encoding and decoding policies that jointly minimize the mean-square estimation error, under the long-term communication rate constraint of $R$ bits per second. We show that an optimal encoding policy can be implemented as a causal sampling policy followed by a causal compressing policy. We prove that the optimal encoding policy samples the Wiener process once the innovation passes either $\sqrt{\frac{1}{R}}$ or $-\sqrt{\frac{1}{R}}$ and compresses the sign of innovation (SOI) using a 1-bit codeword. The SOI coding scheme achieves the operational distortion-rate function, which is equal to $D^{\mathrm{op}}(R)=\frac{1}{6R}$. Surprisingly, this is significantly better than the distortion-rate tradeoff achieved in the limit of infinite delay by the best noncausal code. This is because the SOI coding scheme leverages the free timing information supplied by the zero-delay channel between the encoder and the decoder. The key to unlocking that gain is the event-triggered nature of the SOI sampling policy. In contrast, the distortion-rate tradeoffs achieved with deterministic sampling policies are much worse: we prove that the causal informational distortion-rate function in that scenario is as high as $D_{\mathrm{DET}}(R) = \frac{5}{6R}$. It is achieved by the uniform sampling policy with the sampling interval $\frac{1}{R}$. In either case, the optimal strategy is to sample the process as fast as possible and to transmit 1-bit codewords to the decoder without delay. We show that the SOI coding scheme also minimizes the mean-square cost of a continuous-time control system driven by the Wiener process and controlled via rate-constrained impulses. [ABSTRACT FROM AUTHOR]

This article deals with the stability of discrete-time switched linear systems whose all subsystems are unstable and the set of admissible switching signals obeys prespecified restrictions on switches between the subsystems and dwell times on the subsystems. We derive sufficient conditions on the subsystems matrices such that a switched system is globally exponentially stable under a set of purely time-dependent switching signals that obeys the given restrictions. The main apparatuses for our analysis are (matrix) commutation relations between certain products of the subsystems matrices and graph-theoretic arguments. [ABSTRACT FROM AUTHOR]

We establish formulas relating the state of a continuous-time system with that of the transformed one in a Takenaka–Malmquist–Kautz orthogonal basis. We use such relation to give a simple proof that if the original system is dissipative, then the transformed one is also dissipative with the same storage function. We apply our results to continuous-time subspace system identification. [ABSTRACT FROM AUTHOR]

In this article, a robust nonlinear model reference adaptive control (MRAC) is proposed for disturbed linear systems, i.e., linear systems with parameter uncertainties, and external time-dependent perturbations or nonlinear unmodeled dynamics matched with the control input. The proposed nonlinear control law is composed of two nonlinear adaptive gains. Such adaptive gains allow the control to counteract the effects of some perturbations and nonlinear unmodeled dynamics ensuring asymptotic convergence of the tracking error to zero, and the boundedness of the adaptive gains. The nonlinear controller synthesis is given by a constructive method based on the solution of linear matrix inequalities. Besides, the simulation results show that, due to the nonlinearities, the rate of convergence of the proposed algorithm is faster than that provided by a classic MRAC. [ABSTRACT FROM AUTHOR]

We address the problem of synthesizing provably correct controllers for linear systems with reach-avoid specifications. Discrete abstraction-based controller synthesis techniques have been developed for linear and nonlinear systems with various types of specifications. However, these methods typically suffer from the state space explosion problem. Our solution decomposes the overall synthesis problem into two smaller, and more tractable problems: one synthesis problem for an open-loop controller, which can produce a reference trajectory, and a second for synthesizing a tracking controller, which can enforce the other trajectories to follow the reference trajectory. As a key building-block result, we show that, once a tracking controller is fixed, the reachable states from an initial neighborhood, subject to any disturbance, can be overapproximated by a sequence of ellipsoids, with shapes that are independent of the open-loop controller. Hence, the open-loop controller can be synthesized independently to meet the reach-avoid specification for an initial neighborhood. Moreover, we are able to reduce the problem of synthesizing open-loop controllers to satisfiability problems over quantifier-free linear real arithmetic. The number of linear constraints in the satisfiability problem is linear to the number of hyperplanes as the surfaces of the polytopic obstacles and goal sets. The overall synthesis algorithm, computes a tracking controller, and then iteratively covers the entire initial set to find open-loop controllers for initial neighborhoods. The algorithm is sound and, for a class of robust systems, is also complete. We implement this synthesis algorithm in a tool RealSyn ver 2.0 and use it on several benchmarks with up to 20 dimensions. Experiment results are very promising: RealSyn ver 2.0 can find controllers for most of the benchmarks in seconds. [ABSTRACT FROM AUTHOR]

The event-triggered control problem for stochastic nonlinear systems is addressed in the presence of stochastic transmission protocols. A stochastic event-triggered transmission strategy is first proposed, which implies that the triggering condition is detected at stochastically distributed detecting instants. The entire system is modeled as a stochastic hybrid system and a new Lyapunov function is constructed to analyze stability of the resulting system. Stability conditions are derived and explicit design of the parameters is also presented. Examples of linear systems and a nonlinear system are provided to illustrate the effectiveness. [ABSTRACT FROM AUTHOR]

A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal innerapproximation to the region of attraction (ROA). The first theorem addresses linear time-invariant systems, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the system to include perturbations such as unmodeled dynamics, slope-restricted nonlinearities, and time delay, using integral quadratic constraint (IQCs) to capture their input/output behavior. This in turn allows for off-by-one IQCs to refine the description of activation functions by capturing their slope restrictions. Both results rely on semidefinite programming to approximate the ROA. The method is illustrated on systems with neural networks trained to stabilize a nonlinear inverted pendulum as well as vehicle lateral dynamics with actuator uncertainty. [ABSTRACT FROM AUTHOR]

This article investigates the cluster consensus control problem for general linear multiagent systems. The focus of this article is especially to solve the following problem: in what kinds of connection scenarios can agents achieve cluster consensus regardless of the coupling strength (CCRCS). Different from existing literature, this article extends the case to general partition graphs, including both acyclic and cyclic partition cases. New analysis methods are put forward. It is proved that, under some basic assumptions, the multiagent systems can achieve CCRCS if and only if the interaction topology satisfies the nonvicious circle condition, which is proposed for the first time. An algorithm is presented which can check whether a given interaction topology satisfies the nonvicious circle condition. The result can also be extended to heterogeneous cases that agents from different clusters possess nonidentical system dynamics. [ABSTRACT FROM AUTHOR]

In this article, we consider controllability of discrete-time bilinear systems. A classical sufficient condition was obtained in this journal in 1973, which, unfortunately, is not algebraically verifiable, and the required control inputs to achieve state transition are unknown. However, based on a recently established notion, near-controllability, we propose a seeking nonzero entries approach to derive fully algebraically verifiable controllability criteria for discrete-time bilinear systems. More specifically, by our approach, the verification of controllability of the systems is transformed into the problem of seeking nonzero entries of a sequence of matrices generated by the two system matrices. In addition, the control inputs steering the controllable systems between any given pair of states can be easily computed by proposing two algorithms based on near-controllability. As applications, the seeking nonzero entries approach is generalized to construct arbitrary-finite-dimensional controllable discrete-time bilinear systems and also to prove positive-controllability of discrete-time bilinear systems, where algebraic criteria are obtained by the root locus theory and positive-near-controllability that is a new notion introduced in this article. Examples are provided to illustrate the obtained algebraic controllability criteria and algorithms. [ABSTRACT FROM AUTHOR]

We address the output regulation problem for a general class of linear stochastic systems. Specifically, we formulate and solve the ideal full-information and output-feedback problems, obtaining perfect, but noncausal, asymptotic regulation. A characterization of the problem solvability is deduced. We point out that the ideal problems cannot be solved in practice because they unrealistically require that the Brownian motion affecting the system is available for feedback. Drawing from the ideal solution, we formulate and solve approximate versions of the full-information and output-feedback problems, which do not yield perfect asymptotic tracking but can be solved in a realistic scenario. These solutions rely on two key ideas: First, we introduce a discrete-time a posteriori estimator of the variations of the Brownian motion obtained causally by sampling the system state or output; second, we introduce a hybrid state observer and a hybrid regulator scheme which employ the estimated Brownian variations. The approximate solution tends to the ideal as the sampling period tends to zero. The proposed theory is validated by the regulation of a circuit subject to electromagnetic noise. [ABSTRACT FROM AUTHOR]

This article is concerned with robust stability analysis of feedback interconnections of nonlinear systems using integral quadratic constraints (IQCs). Its main purpose involves reconciling hard (a.k.a. unconditional) IQC and soft (a.k.a. conditional) IQC-based analysis. In particular, it is shown that the hard IQC stability result can be recovered from the soft IQC theory. The hard IQC approach is closely related to the dissipativity theory, whereas the soft IQC approach makes use of homotopies that are continuous in the gap distance measure. [ABSTRACT FROM AUTHOR]

Distributed event- and self-triggered controllers are developed for approximate leader-follower consensus with robustness to adversarial Byzantine agents for a class of homogeneous multi-agent systems (MASs). A strategy is developed for each agent to detect Byzantine agent behaviors within their neighbor set and then selectively disregard their transmission. Selectively removing Byzantine agents results in time-varying discontinuous changes to the network topology. Nonsmooth dynamics also result from the use of event/self-triggered strategies and triggering condition estimators that enable intermittent communication. Nonsmooth Lyapunov methods are used to prove approximate consensus of the MAS consisting of the remaining cooperative agents. Simulations are included to validate the result and to outline the tradeoff between communication and performance. [ABSTRACT FROM AUTHOR]