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This paper investigates the event-triggered control problem for switched affine systems. The presence of affine terms brings many difficulties on the exclusion of triggering Zeno behavior when ensuring asymptotic stability. We propose an event-triggered control strategy dynamically updated with triggering to solve this problem, in which the triggering condition is associated with the affine terms and given with a time-varying term that is updated at triggering instants. Based on the triggering strategy, the controllers for subsystems and a switching rule are designed to achieve asymptotic stability of the resulting closed-loop system. Moreover, due to the special feature of event-triggered switched affine systems, a new method for proving the exclusion of Zeno behavior is given. Finally, the developed results are illustrated by an electric circuit example.

This paper studies the multi-agent control problem for affine, rigid, and translational formation, with the aim of developing a unified distributed control strategy for global and finite-time convergence. Global stabilization of rigid formation in arbitrary dimensional spaces still remains open and challenging. This paper provides a general solution to it based on the sliding mode control idea. The control law design consists of two parts: the main control force regulates the trajectories of all agents to reach a sliding surface defined by an affine formation space in finite time and remain in it thereafter; the extra control force governs certain chosen leader agents towards the desired formation in the sliding surface. The paper then presents in detail two approaches for designing the extra control force, one based on distance constraints and the other based on relative position constraints. For the first time, the proposed sliding mode formation control laws solve the open problem of (almost) global and finite-time stabilization of affine, rigid and translational formations in any dimensional space.

We develop two generalizations of contraction theory, namely, semi-contraction and weak contraction theory. First, using the notion of semi-norm, we propose a geometric framework for semi-contraction theory. We introduce matrix semi-measures and characterize their properties. We show that the spectral abscissa of a matrix is the infimum over weighted semi-measures. For dynamical systems, we use the semi-measure of their Jacobian to characterize the contractivity properties of their trajectories. Second, for weakly contracting systems, we prove a dichotomy for the asymptotic behavior of their trajectories and novel sufficient conditions for convergence to an equilibrium. Third, we show that every trajectory of a doubly-contracting system, i.e., a system that is both weakly and semi-contracting, converges to an equilibrium point. Finally, we apply our results to various important network systems including affine averaging and affine flow systems, continuous-time distributed primal-dual algorithms, and networks of diffusively-coupled dynamical systems. For diffusively-coupled systems, the semi-contraction theory leads to a sufficient condition for synchronization that

This paper presents new mathematical results for exact and approximate relaxations of non-convex optimal control problems defined on disconnected control sets. The exact relaxations are based upon extreme point relaxations wherein the relaxed control set is constructed so that its extreme points belong to the original set. A system property required for exactness is normality. In the absence of normality, control set approximations and system dynamic perturbations are introduced so that normality holds. The results hold for linear dynamical systems and nonlinear systems affine in the control. It is shown that the convex relaxations enable the efficient solution of problems in aerospace engineering, sparsity-promoting optimization, over-actuated systems, and multiplexing systems. The computational guarantees associated with convex optimization make the relaxation techniques suitable for real-time control.

The present work extends recent results by second author concerning sampled-data feedback stabilization for affine in the control of nonlinear systems with nonzero drift term, under the presence of a generalized control Lyapunov function associated with appropriate Lie algebraic hypotheses concerning the dynamics of the system. The main results of present work, constitute a generalization of the well-known "Artstein-Sontag" theorem on asymptotic stabilization by means of an almost smooth feedback controller. The analysis is limited to the affine single-input nonlinear systems with nonzero drift term, however, the results can easily be extended to the multi-input case. An illustrative example of the derived results is included., Comment: 7 pages, submitted to IEEE Transactions on Automatic Control for possible publication

This article deals with the codesign of an output-dependent switching function and a full-order affine filter for discrete-time switched affine systems. More specifically, from the measured output, the switched filter has the role of providing essential information for the switching function, which must assure global practical stability of a desired equilibrium point. The design conditions are based on a general quadratic Lyapunov function and are expressed in terms of linear matrix inequalities. Moreover, whenever the system is quadratically detectable, the solution to the output feedback problem coincides with the one for the state feedback case and the associated filter admits the observer form. To the best of the authors’ knowledge, this is the first time that the dynamic output feedback control problem is treated in the context of discrete-time switched affine systems. The results can be used to cope with sampled-data control and have the property of assuring global asymptotic stability when the sampling period tends to zero. A practical application concerning the velocity control of a dc motor driven by a buck–boost converter illustrates the theoretical results.

Goal-driven selective attention (GDSA) is a remarkable function that allows the complex dynamical networks of the brain to support coherent perception and cognition. Part I of this two-part article proposes a new control-theoretic framework, termed hierarchical selective recruitment (HSR), to rigorously explain the emergence of GDSA from the brain's network structure and dynamics. This part completes the development of HSR by deriving conditions on the joint structure of the hierarchical subnetworks that guarantee top-down recruitment of the task-relevant part of each subnetwork by the subnetwork at the layer immediately above, while inhibiting the activity of task-irrelevant subnetworks at all the hierarchical layers. To further verify the merit and applicability of this framework, we carry out a comprehensive case study of selective listening in rodents and show that a small network with HSR-based structure can explain the data with remarkable accuracy while satisfying the theoretical stability and timescale separation requirements of HSR. Our technical approach relies on the theory of switched systems and provides a novel converse Lyapunov theorem for state-dependent switched affine systems that is of independent interest.

This article studies an optimal dynamic formation problem for heterogeneous affine nonlinear systems. The nonidenticality in agents and the requirement for dynamic spatial reconfiguration make it a challenging task to coordinate different types of agents to maintain an optimized formation shape. In an architecture of event-triggered decision and control, this article investigates how to fulfill dynamic formation by distributively optimizing a team cost function. The basic idea is to design a decision unit for each agent to generate an implicit trajectory as a servo signal, based on which a control unit is designed with a displacement-gradient-based law to achieve the desired local solution. Typical heterogeneous characteristics including different nonlinearities and nonidentical dimensions are dealt with in a unified framework. It is shown that with the proposed triggering mechanisms, the optimal dynamic formation problem can be solved by a distributed control law with only intermittent communication. In theory, the properties of convergence of trajectory tracking errors, optimality of the team solution, and Zeno-freeness of event-triggered mechanisms are proved. Two simulation examples are given to verify the proposed method.

To steer a group of agents to maneuver with the desired collective forms, this article studies the affine formation maneuver control of high-order multiagent systems using a two-layered leader-follower strategy, in which agents are divided into three types: first leader, second leader group, and followers. The first leader will decide the whole formation's maneuver parameters, and the second leader group will keep the desired relative positions with respect to the first leader. The followers aim at moving to the desired positions affinely localized by the leaders. The main feature of this two-layered leader-follower strategy is that the information interaction from the first leader to the second leader group can be realized via wireless communication, and only the measurements of local relative position and its finite-order time-derivatives are needed for the followers. To achieve the control objective under the given constraints, the distributed control algorithms are designed for the second leader group and followers, in which the backstepping, distributed estimation, $PD^m$ , and adaptive gain design techniques are employed. Finally, a simulation example with obstacle avoidance is provided to validate the effectiveness of the proposed control algorithms.

This article further considers the abstraction problem of affine control systems proposed in the paper “Consistent abstractions of affine control systems” by Pappas et al. We present a general solution for the problem. Then, constructions in the abovementioned paper are treated as a special case of our results. We also clarify the statement given in the abovementioned paper, claiming that the canonical construction on $M$ gives the desired affine distribution on $N$ , which solves the abstraction problem of smooth affine control systems. Additional assumptions are added to make the statement hold.

Accurately estimating the parameters of a gray-box linear time-invariant state-space representation is still a challenging problem especially if the number of unknowns exceeds ten. The standard nonlinear optimization-based procedures often fail because the initial guesses are not in the domain of attraction of the user-defined cost function global minimum. Herein, a new solution is developed to give access to estimated parameters which are accurate estimates of the unknown parameters directly or, at worst, good initial guesses for any standard nonlinear optimization-based method. By assuming that 1) the gray-box model to identify is minimal and structurally identifiable, 2) the parameter dependency is affine, and 3) the available datasets have been transformed into a reliable and accurate black-box state-space model, our procedure consists in determining the unique similarity transformation between the black-box and gray-box representations of the system to identify. More specifically, we introduce a technique which first extracts a system of equations linear in terms of the similarity transformation; second, it solves this system by using standard linear algebra tools; finally, completes this two-step procedure by a numerical optimization solution (dedicated now to a very small number of unknowns) if the second step does not give access to all the similarity transformation parameters effectively. In this contribution, we first describe our new procedure, prove its capabilities of giving accurate estimates under specific practical constraints, and finally demonstrates its effectiveness with simulation examples.

This note develops a distributed algorithm to solve a convex optimization problem with coupled constraints. Both coupled equality and inequality constraints are considered, where functions in the equality constraints are affine and functions in the inequality constraints are convex. Different from primal-dual subgradient methods with decreasing stepsizes for nonsmooth optimizations, our algorithm focuses on smooth problems and uses a fixed stepsize to find the exact optimal solution. Convergence analysis is derived with rigorous proofs. Our result is also illustrated by simulations.

This technical note proposes novel state feedback stability conditions based on the so called Lyapunov–Metzler inequalities for discrete-time switched affine systems, assuring practical stability of a desired equilibrium point. These conditions are obtained taking into account the volume minimization of an ellipsoidal set containing the nonconvex set of attraction to where the state trajectories are globally attracted. Compared with other recent results, the present ones are based on less conservative conditions for the existence of an attraction set. Moreover, it is not required that the equilibrium point be inside a predetermined set of attainable ones, which enlarges the range of applicability of the present technique. To the best of the authors’ knowledge, it is the first time that the Lyapunov–Metzler inequalities are adopted in the context of switched affine systems. As a second step, the associated nonconvex invariant set is provided. Academical examples illustrate the method and compare the theoretical results with recent ones available in the literature.

In this paper, we address the so-called Liouville control problem for discrete-time SISO linear systems in the class of Gaussian distributions. In particular, we propose a systematic procedure for the characterization of a dynamic output feedback policy that will transfer the output of the system, which is a known Gaussian random variable, to a goal Gaussian distribution after a finite number of stages. In the proposed approach, the Liouville control problem is reduced to two decoupled (finite-dimensional) quadratic programs, one of which is subject to a single affine constraint, which is a convex program, whereas the other one is subject to a single quadratic equality constraint. Despite the fact that the second optimization problem is not convex, one can characterize its exact solution via a systematic procedure without resorting to convex relaxation techniques, which may yield suboptimal or even infeasible solutions to the original (nonconvex) optimization problem. Finally, we present numerical simulations that illustrate the key ideas of this paper.

Adaptive control of general nonlinear systems with nonlinear parameterization is studied in this paper. Under the assumptions that the system has stable free dynamics and satisfies controllability-like conditions characterized by the Lie brackets of affine vector fields, it is proved that there exist $L_gV$ -type adaptive controllers that not only asymptotically regulate the state of the nonlinearly parameterized system but also guarantee global stability of the closed-loop system. The design of $L_gV$ -type adaptive controllers is also included. Applications of the proposed adaptive control scheme are presented, including an interesting case of a dc-microgrid with photovoltaic (PV) and battery system.

The topic of this paper is model predictive control (MPC) for max-plus linear systems with stochastic uncertainties the distribution of which is supposed to be known. We consider linear constraints on the inputs and the outputs. Due to the uncertainties, these linear constraints are formulated as probabilistic or chance constraints, i.e., the constraints are required to be satisfied with a predefined probability level. The proposed chance constraints can be equivalently rewritten into a max-affine (i.e., the maximum of affine terms) form if the linear constraints are monotonically nondecreasing as a function of the outputs. Based on the resulting max-affine form, two methods are developed for solving the chance-constrained MPC problem for stochastic max-plus linear systems. Method 1 uses Boole's inequality to convert the multivariate chance constraint into univariate chance constraints for which the probability can be computed more efficiently. Method 2 employs Chebyshev's inequality and transforms the chance constraint into linear constraints on the inputs. The simulation results for a production system example show that the two proposed methods are faster than the Monte Carlo simulation method and yield lower closed-loop costs than the nominal MPC method.

A multiagent formation control task usually consists of two subtasks. The first is to steer the agents to form a desired geometric pattern, and the second is to achieve desired collective maneuvers so that the centroid, orientation, scale, and other geometric parameters of the formation can be changed continuously. This paper proposes a novel affine formation maneuver control approach to achieve the two subtasks simultaneously. The proposed approach relies on stress matrices, which can be viewed as generalized graph Laplacian matrices with both positive and negative edge weights. The proposed control laws can track any target formation that is a time-varying affine transformation of a nominal configuration. The centroid, orientation, scales in different directions, and even geometric pattern of the formation can all be changed continuously. The desired formation maneuvers are only known by a small number of agents called leaders, and the rest of the agents called followers only need to follow the leaders. The proposed control laws are globally stable and do not require global reference frames if the required measurements can be measured in each agent's local reference frame.

In part I we considered the problem of convergence to a saddle point of a concave–convex function in $C^2$ via gradient dynamics and an exact characterization was given to their asymptotic behavior. In part II we consider a general class of subgradient dynamics that provide a restriction in a convex domain. We show that despite the nonlinear and nonsmooth character of these dynamics their $\omega$ -limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.

This paper deals with the certification problem for robust quadratic stability, robust state convergence, and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters. We consider both continuous-time (CT) and discrete-time (DT) parameter-varying systems. In this paper, we provide a uniform method for this certification problem in both cases and we show that, contrary to what was claimed previously, the DT case requires a significantly different treatment compared to the existing CT results. In the established uniform approach, quadratic Lyapunov functions, that are affine in the parameter, are used to certify robust stability, robust convergence rates, and robust performance in terms of linear matrix inequality feasibility tests. To exemplify the procedure, we solve the certification problem for $\mathscr{L}_2$-gain performance both in the CT and the DT cases. A numerical example is given to show that the proposed approach is less conservative than a method with slack variables., 8 pages, 3 figures

A fundamental question in systems biology is what combinations of mean and variance of the species present in a stochastic biochemical reaction network are attainable by perturbing the system with an external signal. To address this question, we show that the moments evolution in any generic network can be either approximated or, under suitable assumptions, computed exactly as the solution of a switched affine system. We then propose a new method to approximate the reachable set of such switched affine system. A remarkable feature of our approach is that it allows one to easily compute projections of the reachable set for pairs of moments of interest, without requiring the computation of the full reachable set, which can be prohibitive for large networks. As a second contribution, we also show how to select the external signal in order to maximize the probability of reaching a target set. To illustrate the method we study a renown model of controlled gene expression and we derive estimates of the reachable set, for the protein mean and variance, that are more accurate than those available in the literature and consistent with experimental data., IEEE Transactions on Automatic Control, 63 (11), ISSN:0018-9286, ISSN:1558-2523

This paper considers linear parameter-varying (LPV) control within a passive systems framework. A very strictly passive (VSP) controller is presented in state-space form where the state-space matrices have an affine dependence on plant parameters. The resulting controller is a VSP-LPV controller guaranteed to stabilize nonlinear passive systems via the passivity theorem. An ${\boldsymbol{\mathcal {H}_{\infty} }}$ -inspired control synthesis method is presented along with an experimental comparison between the proposed VSP-LPV controller and a self-scheduled LPV controller. Experimental results demonstrate effective joint angle and rate tracking of a two-link flexible-joint manipulator when the proposed VSP-LPV controller is used.

We consider a linear chain of masses, each coupled to its two nearest neighbors by elastic springs. The maximal characteristic frequency of this dynamical system is a strictly convex function of certain parameters that depend on the masses and spring elasticities. Minimizing the maximal characteristic frequency under an affine constraint on these parameters is thus a convex optimization problem. For a homogeneous affine constraint, we prove that the mass and elasticity values that minimize the maximal characteristic frequency have a special structure: They are symmetric with respect to the middle of the chain and the optimal masses [spring elasticities] increase [decrease] toward the center of the chain. Intuitively speaking, this means that in order to minimize the maximal characteristic frequency we need to “fix” the center of the chain, by increasing [decreasing] the masses [spring elasticities] there. We further show that minimizing the maximal characteristic frequency of the linear chain is equivalent to maximizing the steady-state protein production rate in an important model from systems biology called the ribosome flow model.

This technical note focuses on stability analysis and control design of switched affine systems in discrete-time domain. The stability conditions are obtained by taking into account that the system trajectories, governed by a certain switching function, converge to a set of attraction ${\boldsymbol{\mathcal V}}$ containing a desired equilibrium point. These conditions follow from the adoption of a general quadratic Lyapunov function whose time variation is bounded above by a concave-convex function with center determined by minimax theory. Our main contribution is to provide a stabilizing state feedback switching function and an invariant set of attraction ${\boldsymbol{\mathcal V}}_*$ with minimum volume as far as this class of quadratic Lyapunov functions is adopted. The results are applied to sampled-data control of continuous-time switched affine systems with chattering avoidance. The speed control of a DC motor is presented.

The network routing literature contains many results showing that tolls can be used to improve the efficiency of network traffic routing. These results typically require toll-designers to have an exact characterization of the network and user population. We relax this strict informational dependence and present a simple setting in which scaled marginal-cost tolls can be guaranteed to provide significant efficiency improvements over the un-tolled case, even if the toll-sensitivities of the users are unknown.

In this technical note, we introduce the study of the in-block controllability (IBC) of affine systems on polytopes. In particular, we study whether all the states in the interior of a given polytope are mutually accessible through its interior by applying uniformly bounded inputs. The proposed study formalizes controllability under state constraints, and it has strong connections to the study of controllability of piecewise affine hybrid systems. We provide easily checkable necessary conditions for IBC. Then, for an important class of polytopes, simplicial polytopes, we show that these conditions are also sufficient.

This paper introduces a new multi-agent control problem, called an affine formation control problem, with the objective of asymptotically reaching a configuration that preserves collinearity and ratios of distances with respect to a target configuration. Suppose each agent updates its own state using a weighted sum of its neighbor's relative states with possibly negative weights. Then the affine control problem can be solved for either undirected or directed interaction graphs. It is shown in this paper that an affine formation is stabilizable over an undirected graph if and only if the undirected graph is universally rigid, while an affine formation is stabilizable over a directed graph in the $d$ -dimensional space if and only if the directed graph is $(d + 1)$ -rooted. Rigorous analysis is provided, mainly relying on Laplacian associated with the interaction graph, which contain both positive and negative weights.

Prediction-based transformation is applied to control-affine systems with distributed input delays. Transformed system state is calculated as a prediction of the system's future response to the past input with future input set to zero. Stabilization of the new system leads to Lyapunov-Krasovskii proven stabilization of the original one. Conditions on the original system are: smooth linearly bounded open-loop vector field and smooth uniformly bounded input vectors. About the transformed system which turns out to be affine in the undelayed input but with input vectors dependent on the input history and system state, we assume existence of a linearly bounded stabilizing feedback and quadratically bounded control-Lyapunov function. If all assumptions hold globally, then achieved exponential stability is global, otherwise local. Analytical and numerical control design examples are provided.

This paper addresses the problem of determining switching rules for affine switched systems such that the system state is driven to a desired point and a guaranteed cost is minimized. The switching rule is determined by solving an LMI problem and global asymptotic stability of the tracking error dynamics is guaranteed even if sliding motions occur on any switching surface of the system. The potential of the results is illustrated on a true refrigeration system, namely a domestic refrigerator, where the purpose is to control the temperature in the fresh food and the freezer compartments.

This technical note presents stability analysis and stabilization for a general class of switched systems characterized by nonlinear functions. The proposed approach is composed of approximating the switched nonlinear system with a switched affine system that has a mixture of controlled and autonomous switching behavior. Utilizing a joint polyhedral partitioning approach, a stabilizing switching law based on quadratic Lyapunov functions and with considering the autonomous switching between polyhedral regions is proposed. To ensure the decrease of the overall Lyapunov function, two approaches are proposed: 1) guarantee continuity of the Lyapunov function over boundaries of polyhedral regions and 2) relax the continuity requirement by using additional matrix inequalities. The second approach is less conservative but with more variables and matrix inequalities than in the first method. With fixing one scalar variable, the stabilization conditions will have the form of linear matrix inequalities (LMIs). Further, the sufficient conditions for stabilizing the original switched nonlinear system using the proposed switching schemes are presented. Finally, through two examples, the performance of the proposed stabilization methods is demonstrated.

Continuous piecewise affine (PWA) functions arise in many aspects of control. For this kind of function, we propose the minimal conjunctive normal expression (CNE). The CNE can be expressed as the minimum of a collection of terms, each of which is the maximum of a set of affine functions. The minimal CNE is defined to contain the smallest number of parameters. Analogous to Boolean algebra, we propose implicants and prime implicants for continuous PWA functions. After obtaining all prime implicants, the problem of finding minimal CNEs can then be cast as a binary programming problem. A sharp bound on the number of boolean variables in the binary programming problem is given. In two worked examples, minimal CNEs are derived for given continuous PWA functions.

For general nonlinear autonomous systems, a Lyapunov characterization for the possibility of semi-global asymptotic stabilizability by means of a time-varying sampled-data feedback is established. We exploit this result in order to derive a Lie algebraic sufficient condition for sampled-data feedback semi-global stabilizability of affine in the control nonlinear systems with non-zero drift terms. The corresponding proposition constitutes an extension of the “Artstein-Sontag” theorem on feedback stabilization.

We study a class of $N$ -player finite-horizon linear-quadratic difference games with linear constraints. We introduce constrained open-loop information structure and derive necessary conditions for the existence of constrained open-loop Nash equilibria. We show that these conditions lead to a weakly coupled system of parametric two-point boundary value problem and a set of linear complementarity problems. By restricting the costate variables to be affine in the state variable, we show that these necessary conditions can be reformulated as a single large-scale linear complementarity problem. Then we provide sufficient conditions under which a solution of the linear complementarity problem constitutes a constrained open-loop Nash equilibrium.

This technical note studies the control design problem for a class of affine nonlinear descriptor systems with actuator saturation, and presents several new results. A stabilization controller design method is first proposed by state feedback and compensation, and a robust $H_{\infty}$ controller and a robust adaptive $H_{\infty}$ controller are respectively designed for the discussed system when external disturbances and/or parametric perturbations are taken into account. Simulation results on a nonlinear descriptor circuit system illustrate the efficiency of the controllers presented in this technical note.

International audience; This paper addresses the problem of modes detection for switching discrete-time linear and affine systems. A unified presentation of the model-based methods with the corresponding modes detectors is carried out and a comparative study is conducted. Next, the different notions of discernibility proposed in the literature are presented. Such notions are central since they guarantee the uniqueness of the solution delivered by the modes detectors. A complete step-by-step procedure of modes detection is proposed. The paper includes, whenever necessary, some extensions of the existing results.

This paper focuses on the task of safety controller synthesis, that is, designing a controller that will take a system from any point within a compact set of initial states to a point inside a set of acceptable goal states, while never entering any state that is deemed unsafe. To do this we use a human generated trajectory-based approach. We introduce the control autobisimulation function, which is the analog of the control Lyapunov function for approximate bisimulation. We consider a class of hybrid systems and use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness for underlying dynamics that are linear affine, feedback linearizable, and differentially flat. This property ensures that any trajectory of the closed-loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input. This feedback control and input can be used as the controller for some subset of the initial states. We demonstrate how to combine multiple trajectories into a synthesized controller that satisfies the safety problem.

This note deals with solving scalar coupled algebraic Riccati equations. These equations arise in finding linear feedback Nash equilibria of the scalar $N$ -player affine quadratic differential game. A numerical procedure is provided to compute all the stabilizing solutions. The main idea is to reformulate the Riccati equations into an extended eigenvalue-eigenvector problem for a specific parametrized matrix $U\in I\!\!R^{2^{N}\times 2^{N}}$ . Since the size of $U$ increases exponentially on $N$ , the algorithm only applies for games where the number of players is not too large.

Checking validity of a model is a crucial step in the process of system identification. This is especially true when dealing with switched affine systems since, in this case, the problem of system identification from noisy data is known to be generically NP-Hard and can only be solved in practice by using heuristics and relaxations. Therefore, before the identified models can be used for instance for controller design, they should be systematically validated against additional experimental data. In this paper we address the problem of model (in)validation for multi-input multi-output switched affine systems in output error form with unknown switches. As a first step, we prove that necessary and sufficient invalidation certificates can be obtained by solving a sequence of convex optimization problems. In principle, these problems involve increasingly large matrices. However, as we show in the paper by exploiting recent results from semialgebraic geometry, the proposed algorithm is guaranteed to stop after a finite number of steps that can be be explicitly computed from the a priori information. In addition, this algorithm exploits the sparse structure of the underlying optimization problem to substantially reduce the computational burden. The effectiveness of the proposed method is illustrated using both academic examples and a non-trivial problem arising in computer vision: activity monitoring.

Classical affine root locus applies to controllers that are linear in a parameter $k$ and yield affinely parameterized closed-loop denominator polynomials. This paper presents root locus rules for controllers that are rational in $k$ and yield quadratically parameterized closed-loop denominator polynomials. We show that the quadratic root locus has several advantages relative to the classical affine root locus. Specifically, the quadratic root locus is high-parameter stabilizing for minimum-phase systems that are relative degree 1, 2, or 3. Moreover, the quadratic root locus admits a wider variety of asymptote angles, which provides more controller design flexibility. The quadratic root locus is illustrated on several examples that are difficult to handle using affine root locus, such as high-parameter stabilization of the triple integrator.

We study the dynamic scheduling of a multi-item machine with setup times and static priorities. We consider a class of base-stock policies for this system and seek sharp stability conditions that allow us to respect the item priorities over a large region of the decision space. We show that requiring the existence of a linear Lyapunov function that decreases between production runs reduces to an intuitive stability condition, which can nevertheless be too conservative for this policy class. We then sharpen this result by showing that, if the original condition is satisfied for the highest-priority $N\!\!-\!\!1$ part types, the system with $N$ items is stable. Finally, we develop stability conditions based on affine Lyapunov functions.

For a class of nonlinear systems, we investigate the problem of under what conditions there exists a coordinate transformation that yields a state affine linear system up to output injection with implicit outputs. In particular, we provide necessary and sufficient conditions for time-varying linearization up to multi-output injection. We highlight that if the conditions hold, as a consequence, it is possible in the new coordinates to construct an observer with linear error dynamics. We propose a methodology to find the coordinate transformation. Several examples illustrate the proposed procedure.

This paper analyzes the passivity and feedback passivity of discrete-time-switched nonlinear systems with passive and nonpassive modes that are affine in the control input. When a nonpassive mode is active, the increase in storage function is not necessarily bounded by the energy supplied to the switched system at every time step. Therefore, a switched system with at least one nonpassive mode is defined to be nonpassive in the classical passivity theory. In this paper, we propose a framework to analyze the passivity of such switched systems in a more general sense. We consider switched nonlinear systems which are affine in the control input and may consist of passive, feedback passive modes, and modes which cannot be rendered passive using feedback. In the proposed framework, we prove that a switched nonlinear system is locally feedback passive if and only if its zero dynamics are locally passive. A lower bound on the ratio of total activation time between (feedback) passive and nonfeedback passive modes is obtained to guarantee passive zero dynamics. Finally, we prove that two important properties of classical passivity still hold for the proposed passivity definition, that is: 1) output feedback control can be used to stabilize the switched system, and 2) parallel and negative feedback interconnections of two such passive systems are also passive.

We study the reach control problem (RCP) for a single input affine system with a simplicial state space. We extend previous results by exploring arbitrary triangulations of the state space; particularly allowing the set of possible equilibria to intersect the interior of simplices. In the studied setting, it is shown that closed-loop equilibria, nevertheless, only arise on the boundary of simplices. This allows to define a notion of reach controllability which quantifies the effect of the control input on boundary equilibria. Using reach controllability we obtain necessary and sufficient conditions for solvability of RCP by affine feedback.

We present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances. We take as a performance metric either an expected-value or minimax discounted cost over an infinite horizon, and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case. Our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances, and we show that the two bounds are closely related. In particular, the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set. We show that our performance bounds can be computed via solution of a finite-dimensional convex optimization problem, and provide numerical examples to illustrate the efficacy of our method.

The paper studies the problem of making an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope. The focus is on solvability by continuous piecewise affine feedback, and we formulate a variant of the problem in which trajectories exit in a monotonic sense. This allows to obtain necessary and sufficient conditions for solvability in certain geometric situations. Next, we show that, generically, solvability via arbitrary triangulations is equivalent to monotonic solvability. In contrast with existing simplex-based methods, this provides an avenue for reach control on polytopes that does not depend on the choice of triangulation of the polytope.

The article presents a state-space based Fault Diagnosis (FD) method for discrete-time, affine Linear Parameter Varying (LPV) systems. The goal of the technical note is to develop a robust and dynamic inversion based technique for systems with parameter varying representations when an additive, exogenous disturbance signal perturbs the system. After applying geometric concepts for explicit fault inversion, a robust strategy is proposed to attenuate the effect of the unknown disturbance input signal on the fault estimation error. The proposed robust observer is derived as a solution of off-line Linear Matrix Inequality (LMI) conditions. The technical note demonstrates the viability of the novel methodology through a numerical example.

We propose a fast algorithm for the linear-quadratic control problem with probabilistic constraints that is repeatedly solved in stochastic model predictive control. Under the assumption of affine state feedback and Gaussian noise, the finite horizon control problem is converted to an equivalent deterministic problem using the mean and covariance matrix as the state. A line search interior point method is proposed to solve this optimization problem, where the step direction can be quickly computed via a Riccati difference equation. Numerical examples show that this algorithm has linear complexity in the horizon length.

This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and some minimal a priori information about the set of admissible plants, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information, while minimizing either the number of switches or subsystems. For the case where it is desired to minimize the number of switches, the key idea of the paper is to reduce this problem to a sparsification form, where the goal is to maximize sparsity of a suitably constructed vector sequence. Our main result shows that in the case of l∞ bounded noise, this sparsification problem can be exactly solved via convex optimization. In the general case where the noise is only known to belong to a convex set N, the problem is generically NP-hard. However, as we show in the paper, efficient convex relaxations can be obtained by exploiting recent results on sparse signal recovery. Similarly, we present both a sparsification formulation and a convex relaxation for the (known to be NP hard) case where it is desired to minimize the number of subsystems. These results are illustrated using two non-trivial problems arising in computer vision applications: video-shot and dynamic texture segmentation.

A continuous robust adaptive control method is designed for a class of uncertain nonlinear systems with unknown constant time-delays in the states. Specifically, a robust adaptive control method and a delay-free gradient-based desired compensation adaptation law (DCAL) are utilized to compensate for unknown time-delays, linearly parameterizable uncertainties, and additive bounded disturbances for a general nonlinear system. Despite these disturbances, a Lyapunov Krasovskii-based analysis is used to conclude that the system output asymptotically tracks a desired time varying bounded trajectory.

Motivated by long-time dynamic analysis of hybrid systems and safety verification problems, this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications. Necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long-time oscillatory dynamic analysis. A linear program based algorithm is proposed to verify these conditions, and its computational complexity is analyzed. The positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems. Further, the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets.