Traffic models of periodic event-triggered quantized control systems

Periodic Event-Triggered Quantized Control (PETQC) is a type of event-triggered control (ETC) that takes into account the quantization effect introduced by network transmission and only requires to measure the plant output periodically instead of continuously. In this work, we present an abstract model for the traffic generated by the dynamics of PETQC implementations. Construction of the abstract models is done in two steps. Our first step is to divide the state space into finite regions. Each region is then analyzed with LMIs to examine the event-triggering behavior. The state transitions among different regions are resulted from reachability analysis.     

Abstraction-based synthesis for stochastic systems with omega-regular objectives

This paper studies the synthesis of controllers for discrete-time, continuous state stochastic systems subject to omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behaviors and encompass, for example, linear temporal logic. First, we present a synthesis algorithm for minimizing or maximizing the probability that a discrete-time switched stochastic system with a finite number of modes satisfies an omega-regular property. Our approach relies on a finite-state abstraction of the underlying dynamics in the form of a Bounded-parameter Markov Decision Process arising from a finite partition of the system’s domain. Such Markovian abstractions allow for a range of probabilities of transition between states for each selected action representing a mode of the original system. Our method is built upon an analysis of the Cartesian product between the abstraction and a Deterministic Rabin Automaton encoding the specification of interest or its complement. Specifically, we show that synthesis can be decomposed into a qualitative problem, where the so-called greatest permanent winning components of the product automaton are created, and a quantitative problem, which requires maximizing the probability of reaching this component in the worst-case instantiation of the transition intervals. Additionally, we propose a quantitative metric for measuring the quality of the designed controller with respect to the continuous abstracted states and devise a specification-guided domain partition refinement heuristic with the objective of reaching a user-defined optimality target. Next, we present a method for computing control policies for stochastic systems with a continuous set of available inputs. In this case, the system is assumed to be affine in input and disturbance, and we derive a technique for solving the qualitative and quantitative problems in the resulting finite-state abstractions of such systems. For this, we introduce a new type of abstractions called Controlled Interval-valued Markov Chains. Specifically, we show that the greatest permanent winning component of such abstractions are found by appropriately partitioning the continuous input space in order to generate a bounded-parameter Markov decision process that accounts for all possible qualitative transitions between the finite set of states. Then, the problem of maximizing the probability of reaching these components is cast as a (possibly non-convex) optimization problem over the continuous set of available inputs. A metric of quality for the synthesized controller and a partition refinement scheme are described for this framework as well. Finally, we present a detailed case study.     

Existence test for solution of nonlinear systems applying affine arithmetic

A computational test is proposed for existence of solution in nonlinear systems. In this test, an interval inclusion of Newton mapping is estimated applying affine arithmetic. Numerical examples are presented to show the efficiency of this test.     

First steps toward formal controller synthesis for bipedal robots with experimental implementation

Bipedal robots are prime examples of complex cyber–physical systems (CPSs). They exhibit many of the features that make the design and verification of CPS so difficult: hybrid dynamics, large continuous dynamics in each mode (e.g., 10 or more state variables), and nontrivial specifications involving nonlinear constraints on the state variables. In this paper, we propose a two-step approach to formally synthesize controllers for bipedal robots so as to enforce specifications by design and thereby generate physically realizable stable walking. In the first step, we design outputs and classical controllers driving these outputs to zero. The resulting controlled system evolves on a lower dimensional manifold and is described by the hybrid zero dynamics governing the remaining degrees of freedom. In the second step, we construct an abstraction of the hybrid zero dynamics that is used to synthesize a controller enforcing the desired specifications to be satisfied on the full order model. Our two step approach is a systematic way to mitigate the curse of dimensionality that hampers the applicability of formal synthesis techniques to complex CPS. Our results are illustrated with simulations showing how the synthesized controller enforces all the desired specifications and offers improved performance with respect to a classical controller. The practical relevance of the results is illustrated experimentally on the bipedal robot AMBER 3.     

Multirate sampling and input-to-state stable receding horizon control for nonlinear systems

In this paper, a robust receding horizon control for multirate sampled-data nonlinear systems with bounded disturbances is presented. The proposed receding horizon control is based on the solution of Bolza-type optimal control problems for the approximate discrete-time model of the nominal system. “Low measurement rate” is assumed. It is shown that the multistep receding horizon controller that stabilizes the nominal approximate discrete-time model also practically input-to-state stabilizes the exact discrete-time system with disturbances.     

Robust suboptimal control of nonlinear systems

In this paper, we consider a nonlinear dynamic system with uncertain parameters. Our goal is to choose a control function for this system that balances two competing objectives: (i) the system should operate efficiently; and (ii) the system’s performance should be robust with respect to changes in the uncertain parameters. With this in mind, we introduce an optimal control problem with a cost function penalizing both the system cost (a function of the final state reached by the system) and the system sensitivity (the derivative of the system cost with respect to the uncertain parameters). We then show that the system sensitivity can be computed by solving an auxiliary initial value problem. This result allows one to convert the optimal control problem into a standard Mayer problem, which can be solved directly using conventional techniques. We illustrate this approach by solving two example problems using the software MISER3.     

Discrete model reference adaptive control of continuous-time linear multi-input multi-output systems via multirate sampled-data controllers

The use of multirate sampled-data controllers for linear multivariable time-invariant systems with unknown parameters is investigated. Such controllers contain periodically time-varying elements and a multirate sampling mechanism with different sampling periods at each system input. Their application to unknown continuous-time linear multi-input, multi-output systems results in a sampled closedloop system for which an arbitrary discrete-time transfer function matrix can be assigned, as is shown in the present paper. The contribution of the present paper is twofold: the use of multirate sampled-data controllers in the area of model reference adaptive control; and the application, for the first time, of periodically varying controllers for model reference adaptive control of multi-input, multi-output systems.The work described in this paper has been partialy funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.     

Variance-constrained dissipative observer-based control for a class of nonlinear stochastic systems with degraded measurements

This paper is concerned with the variance-constrained dissipative control problem for a class of stochastic nonlinear systems with multiple degraded measurements, where the degraded probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over a given interval. The purpose of the problem is to design an observer-based controller such that, for all possible degraded measurements, the closed-loop system is exponentially mean-square stable and strictly dissipative, while the individual steady-state variance is not more than the pre-specified upper bound constraints. A general framework is established so that the required exponential mean-square stability, dissipativity as well as the variance constraints can be easily enforced. A sufficient condition is given for the solvability of the addressed multiobjective control problem, and the desired observer and controller gains are characterized in terms of the solution to a convex optimization problem that can be easily solved by using the semi-definite programming method. Finally, a numerical example is presented to show the effectiveness and applicability of the proposed algorithm.     

A derivative-free algorithm for systems of nonlinear inequalities

Recently a new derivative-free algorithm has been proposed for the solution of linearly constrained finite minimax problems. This derivative-free algorithm is based on a smoothing technique that allows one to take into account the non-smoothness of the max function. In this paper, we investigate, both from a theoretical and computational point of view, the behavior of the minmax algorithm when used to solve systems of nonlinear inequalities when derivatives are unavailable. In particular, we show an interesting property of the algorithm, namely, under some mild conditions regarding the regularity of the functions defining the system, it is possible to prove that the algorithm locates a solution of the problem after a finite number of iterations. Furthermore, under a weaker regularity condition, it is possible to show that an accumulation point of the sequence generated by the algorithm exists which is a solution of the system. Moreover, we carried out numerical experimentation and comparison of the method against a standard pattern search minimization method. The obtained results confirm that the good theoretical properties of the method correspond to interesting numerical performance. Moreover, the algorithm compares favorably with a standard derivative-free method, and this seems to indicate that extending the smoothing technique to pattern search algorithms can be beneficial.     

A new control method and its robustness for a class of nonlinear systems

For a class of time-varying nonlinear systems described by the equation , the precalculating control is not available if the input matrixg(x,t) is not invertible. With Lyapunov's second method, a stabilizing controller which makes the system practically stable is constructed in this paper. It is shown that the implementation of this scheme depends on some so-called posi-invertibility conditions forg(x,t). In case the system is partly stable, the method, named part-calculating control, can simplify the on-line computations. Without the assumption that the nominal system is asymptotically stable, the method is applied to the problems of control for the corresponding uncertain system that satisfies the matching condition. When the matching condition is not satisfied, the mismatching control problem is also studied with Lyapunov's second method.This work was supported by the Science Fund of the Chinese Academy of Science.     

Quasi-Newton ABS methods for solving nonlinear algebraic systems of equations

This paper is concerned with the solution of nonlinear algebraic systems of equations. For this problem, we suggest new methods, which are combinations of the nonlinear ABS methods and quasi-Newton methods. Extensive numerical experiments compare particular algorithms and show the efficiency of the proposed methods.The authors are grateful to Professors C. G. Broyden and E. Spedicato for many helpful discussions.     

Successive approximation procedure of optimal tracking control for nonlinear similar composite systems

The optimal tracking control (OTC) problem for a class of affine nonlinear composite systems with similar structure is considered. By using a modeling technique, the nonlinear similar composite system is first transformed into some quasi-decoupled subsystems. Then the high-order, strongly coupled, nonlinear two-point boundary value (TPBV) problem is transformed into a sequence of linear decoupled TPBV problems through a successive approximation procedure. The obtained OTC law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence. A suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term.     

Resource limited event-triggered model predictive control for continuous-time nonlinear systems based on first-order hold

This paper investigates a resources-limited situation in the event-triggered model predictive control (ETMPC) for continuous-time nonlinear system with first-order hold fashion. In consideration of limited bandwidth in data transmission through wireless network under actual operation, our strategy divides the prediction horizon, and applies linear interpolation instead of zero-order hold fashion to obtain a better system performance, so that the reduction of resources and the optimization of strategy can be guaranteed. Furthermore, in actual industry processes, quadratic cost function cannot be implemented in all operations, then general cost function is adopted in this paper. Based on the first-order hold method and general cost function, the feasibility of the ETMPC algorithm and the stability of dynamical systems are analyzed. At last, a practical example is given to show the advantages of our method.     

A new method for control of nonlinear state-dependent switched systems

In this paper, a new method for the control of input-affine nonlinear switched systems is introduced. The system switching conditions are assumed to be state-dependent, rather than the simpler input-dependent case. The main contribution of this research is that the effects of switched dynamics are interpreted as a model uncertainty bounded within a polynomial of states norms, with unknown coefficients. In order to prevent extra conservativeness, coefficients are tuned adaptively, so that a minimal state-varying bound could be achieved. This is unlike the conventional sliding mode control (SMC) scheme, where the existence of a constant and usually large upper bound must be presumed. To address the challenge of coping with such a new concept of uncertainty, an extended form of the original adaptive fuzzy sliding mode control scheme is proposed. Adaptation laws are used to tune a fuzzy controller and also real-time estimation of the instantaneous bound of uncertainties. Closed-loop stability is guaranteed by proposing a group of multiple Lyapunov functions (MLF) with tunable parameters. Except for the mild condition that the largest difference between the magnitudes of the sub-manifolds of the switched system is bounded by a polynomial of states with uncertain coefficients, the proposed method has the distinct advantage that no information about the dynamic equations or switching conditions is required in the control design stage. The proposed method is applied to the two challenging case studies, depicting the outstanding effectiveness of the method.     

Canonical forms for positive discrete-time linear control systems

In this paper, the properties of reachability, controllability and essential reachability of positive discrete-time linear control systems are studied. These properties are characterized in terms of the directed graph of the state matrix. From these characterizations canonical forms of those properties are deduced.     

A NARMAX model-based state-space self-tuning control for nonlinear stochastic hybrid systems

A novel state-space self-tuning control methodology for a nonlinear stochastic hybrid system with stochastic noise/disturbances is proposed in this paper. via the optimal linearization approach, an adjustable NARMAX-based noise model with estimated states can be constructed for the state-space self-tuning control in nonlinear continuous-time stochastic systems. Then, a corresponding adaptive digital control scheme is proposed for continuous-time multivariable nonlinear stochastic systems, which have unknown system parameters, measurement noise/external disturbances, and inaccessible system states. The proposed method enables the development of a digitally implementable advanced control algorithm for nonlinear stochastic hybrid systems.     

Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems

This paper concerns the nonfragile guaranteed cost control problem for a class of nonlinear dynamic systems with multiple time delays and controller gain perturbations. Guaranteed cost control law is designed under two classes of perturbations, namely, additive form and multiplicative form. The problem is to design a memoryless state feedback control law such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Based on the linear matrix inequality (LMI) approach, some delay-dependent conditions for the existence of such controller are derived. A numerical example is given to illustrate the proposed method.     

Adaptive fuzzy backstepping control design for a class of pure-feedback switched nonlinear systems

In this paper, an adaptive fuzzy output tracking control approach is proposed for a class of single input and single output (SISO) uncertain pure-feedback switched nonlinear systems under arbitrary switchings. Fuzzy logic systems are used to identify the unknown nonlinear system. Under the framework of the backstepping control design and fuzzy adaptive control, a new adaptive fuzzy output tracking control method is developed. It is proved that the proposed control approach can guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) and the tracking error remains an adjustable neighborhood of the origin. A numerical example is provided to illustrate the effectiveness of the proposed approach.