Polytopic Set Research Articles

An algorithm to solve polytopic set optimization problem based on a partial set order relation

An algorithm to solve polytopic set optimization problem based on a partial set order relation

A Scalable Convex Min–Max Optimization Algorithm for Robust Control of Polytopic Positive Systems

ABSTRACTIn this paper, we investigate the robust control of a class of bilinear positive systems. The main objective is to minimize the norm of the controlled system while dealing with polytopic uncertainties. In contrast to traditional Lyapunov‐based robust control problems with polytopic uncertainties, we establish that minimizing the objective function over the vertices of the polytopic set, rather than considering its infinite elements, is adequate for achieving the globally optimal solution. Following this, we present a customized algorithm that relies on partial cutting plane to address the min–max problem. The introduced robust control strategy exhibits global convergence to the optimal solution and demonstrates scalability with the number of states and the number of vertices in the uncertain family. Simulation results validate the effectiveness of the proposed strategy in handling uncertainties.

Analysis and Controller Design for Parameter Varying T-S Fuzzy Systems with Markov Jump

In this paper, we investigate a novel T-S fuzzy parameter varying system with Markov jump, in which parameters depend not only on a Markov chain but also on linear parameter varying elements that take values in convex polytopic sets. Stable conditions and the gain-scheduling controller design method for this system are obtained. Applying Lyapunov function depending on the operation mode and full block S-procedure lemma, we obtain stochastic stabilization conditions. We find that this novel system has two distinct advantages. On the one hand, it inherits the advantages of traditional T-S fuzzy systems in handling nonlinear objects under the frame of T-S fuzzy systems; on the other, it obtains the advantages of dealing with time-varying characteristics from the point of linear parameter varying (LPV) systems. Finally, the theory results are illustrated via numerical simulation.

Reducing the wrapping effect of set computation via Delaunay triangulation for guaranteed state estimation of nonlinear discrete-time systems

Set computation methods have been widely used to compute reachable sets, design invariant sets and estimate system state for dynamic systems. The wrapping effect of such set computation methods plays an essential role in the accuracy of their solutions. This paper studies the wrapping effect of existing interval, zonotopic and polytopic set computation methods and proposes novel approaches to reduce the wrapping effect for these set computation methods based on the task of computing the dynamic evolution of a nonlinear uncertain discrete-time system with a set as the initial state. The proposed novel approaches include the partition of a polytopic set via Delaunay triangulation and also the representation of a polytopic set by the union of small zonotopes for the following set propagation. The proposed novel approaches with the reduced wrapping effect has been further applied to state estimation of a nonlinear uncertain discrete-time system with improved accuracy. Similar to bisection for interval and zonotopic sets, Delaunay triangulation has been introduced as a set partition tool for polytopic sets, which has opened new research directions in terms of novel set partition, set representation and set propagation for reducing the wrapping effect of set computation.

A Convex Formulation for Robust Control of Bilinear Positive Systems With Evolutionary Dynamics

In this article, optimal control of bilinear positive systems with uncertain evolutionary dynamics is studied. The first objective is to determine optimal control signals such that the effect of the disturbance signals on the outputs of the nominal dynamics is minimized in terms of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm performance index. Properties of the positive systems, Metzler matrices, and the exploitable structure of the problem are utilized to show that the underlying problem is representable as a standard convex optimization problem. Next, the optimal strategy is examined in the presence of polytopic uncertainties. In this case, the optimization problem is a semi-infinite min-max problem. It is shown that under some assumptions, the underlying problem is equivalent to a finite min-max problem, i.e., it suffices to check only the vertices of the polytopic set for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm minimization. The proposed approach for polytopic bilinear positive systems is non-iterative, does not involve parameter tuning or nonsmooth optimization algorithms, and can be solved using off-the-shelf solvers. The effectiveness of the proposed approach is investigated for the optimal combination therapy of human immunodeficiency virus (HIV).

Finite-time Boundedness of Polytopic Uncertain Systems with Guaranteed Cost Output Feedback Control

A problem of designing robust guaranteed cost reduced-order output feedback controllers for finite-time boundedness of linear time-varying uncertain systems is considered in this paper. Assuming the system parameters to lie in a polytopic set, differential linear matrix inequality (DLMI) conditions for the design of reduced-order dynamic controllers to handle worst-case exogenous disturbances are derived. The designed controller guarantees the cost on considered disturbance attenuation performance on the system to be below an upper bound, while simultaneously satisfying the conditions for finite-time boundedness and finite L2 gain. Numerical simulations are carried out to demonstrate the applicability of the proposed method in bounding the system trajectories to a predefined set.

Observer-Based Asynchronous Control of Nonlinear Systems With Dynamic Event-Based Try-Once-Discard Protocol.

This work investigates the observer-based asynchronous control of discrete-time nonlinear systems with network-induced communication constraints. To avoid the data collisions and side effects in a constrained communication channel, a novel dynamic event-based weighted try-once-discard (DEWTOD) protocol is proposed. In contrast to the existing protocols, the DEWTOD scheduling regulates whether the sampling instant to release and which node to transmit the sampling instant simultaneously. In light of a hidden Markov model, the time-varying detection probability matrix is characterized by a polytopic set. By resorting to the polytopic-structured Lyapunov functional, sufficient conditions are derived such that the closed-loop dynamic is mean-square exponentially stable, and the observer-based controller is designed. In the end, two numerical examples are provided to explicate the validity of the attained methodology.

Robust Stability Analysis of Linear Parameter-Varying Systems With Markov Jumps

This article presents conditions to assure the mean-square stability of linear parameter-varying systems with Markov jumps. The model dynamics are driven not only by a Markov chain but also by time-varying parameters that take values in a polytopic set. No assumption is imposed on how the parameters vary within the polytopic set, i.e., the variation rate can be arbitrarily fast. The proposed conditions stem from a homogeneous polynomial Lyapunov function in the state space, adapted to account for Markov jumps. The stability certificate is sought through linear matrix inequalities. Numerical examples illustrate this article’s contribution.

Invariant cover: Existence, cardinality bounds, and computation

An invariant cover quantifies the information needed by a controller to enforce an invariance specification. This paper investigates some fundamental problems concerning existence and computation of an invariant cover for uncertain discrete-time linear control systems subject to state and control constraints. We develop necessary and sufficient conditions on the existence of an invariant cover for a polytopic set of states. The conditions can be checked by solving a set of linear programs, one for each extreme point of the state set. Based on these conditions, we give upper and lower bounds on the minimal cardinality of the invariant cover, and design an iterative algorithm with finite-time convergence to compute an invariant cover. We further show in two examples how to use an invariant cover in the design of a coder–controller pair that ensures invariance of a given set for a networked control system with a finite communication data rate.

Control Minkowski–Lyapunov functions

This article considers the control Lyapunov inequality with a priori prescribed decrease over the space of Minkowski functions generated by compact convex sets containing the origin in the interior within the setting of unconstrained and constrained linear discrete time systems. Within the unconstrained setting, the article derives a novel necessary and sufficient condition, which characterizes all Minkowski functions that satisfy the considered control Lyapunov inequality. This novel necessary and sufficient condition is provided for Minkowski functions of general compact convex sets containing the origin in the interior, and it is also refined for Minkowski functions of polytopic sets containing the origin as an interior point. In addition, the article characterizes, and establishes topological properties of, the related stabilizing set-valued control map. The novel results derived for the unconstrained setting are also extended to the constrained setting.

A tracking performance analysis method for autonomous systems with neural networks

Intelligent manufacturing and automated systems several complex control tasks must be carried out. A possible way for improving the performance level of the systems is the application of machine-learning-based agents, e.g. neural networks in the control loop. A novel challenge of these complex systems is to provide analysis and synthesis methods, with which their performance levels can be assessed. The paper proposes analysis method for tracking control systems, whose control loop contains feed-forward neural networks. Through the method the asymptotic stability and the tracking performance through decay rate are assessed. The proposed method is based on the linear approximation of the closed-loop system, and thus, a polytopic set of linear systems is resulted. Using the resulted polytopic system an analysis method based on an optimization is formed, whose result approximates the decay rate of the system. The effectiveness of the method is illustrated through a benchmark example, i.e. the torque control of a one-degree-of-freedom robotic arm.

Fundamental Domains for Symmetric Optimization: Construction and Search

Symmetries of a set are linear transformations that map the set to itself. A fundamental domain of a symmetric set is a subset that contains at least one representative from each of the symmetric equivalence classes (orbits) in the set. This paper contributes a novel polynomial algorithm for constructing minimal polytopic fundamental domains of polytopic sets. Our algorithm is applicable for generic linear symmetries of the set and has linear complexity in the number of facets and dimension of the symmetric polytope, e.g., the feasible region of an optimization problem. In addition, this paper contributes a novel polynomial algorithm for mapping an element of the polytope to its representative in the fundamental domain. The algorithms are demonstrated in four examples---two illustrative and two practical. In the first practical example, we show that a minimal fundamental domain of the hypercube under the symmetric group is the set of points with sorted elements. In the second practical example, we show how the construction algorithm can be applied to the max-cut problem.

Fuzzy-Model-Based Control for Singularly Perturbed Systems With Nonhomogeneous Markov Switching: A Dropout Compensation Strategy

This article addresses the asynchronous control for singularly perturbed systems with nonhomogeneous Markov switching approximated by T–S fuzzy models. The transition probabilities of the nonhomogeneous Markov process are supposed to be time-varying and distinguished by of a polytopic set. As distinct from some reported works, to abate the effect of missing packets in unreliable communication network, a novel dropout compensation strategy is constructed, where the packet arriving rate is assumed to be uncertain. Meanwhile, to describe the asynchronizations of the dropout compensation strategy and the controller, the hidden Markov models are absorbed. By resorting to the fuzzy-rule-dependent and the parameter-dependent Lyapunov–Krasovskii functional, novel sufficient conditions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> performance are formulated and the fuzzy-based asynchronous controller gains are realized. Finally, to testify the efficiency and applicability of the proposed results, a numerical example and practical tunnel diode circuit system are provided.

Robust adaptive model predictive control: Performance and parameter estimation

SummaryFor systems with uncertain linear models, bounded additive disturbances and state and control constraints, a robust model predictive control (MPC) algorithm incorporating online model adaptation is proposed. Sets of model parameters are identified online and employed in a robust tube MPC strategy with a nominal cost. The algorithm is shown to be recursively feasible and input‐to‐state stable. Computational tractability is ensured by using polytopic sets of fixed complexity to bound parameter sets and predicted states. Convex conditions for persistence of excitation are derived and are related to probabilistic rates of convergence and asymptotic bounds on parameter set estimates. We discuss how to balance conflicting requirements on control signals for achieving good tracking performance and parameter set estimate accuracy. Conditions for convergence of the estimated parameter set are discussed for the case of fixed complexity parameter set estimates, inexact disturbance bounds, and noisy measurements.

Tube-based nonlinear model predictive control for autonomous skid-steer mobile robots with tire–terrain interactions

This work addresses the problem of robust tracking control for skid-steer mobile platforms, using tube-based Nonlinear Model Predictive Control. The strategy seeks to mitigate the impact of disturbances propagated to autonomous vehicles originated by traction losses. To this end, a dynamical model composed by two coupled sub-systems stands for lateral and longitudinal vehicle dynamics and tire behavior. The controller is aimed at tracking prescribed stable operation points of the slip and side-slip beyond the robot pose and speeds. To reach robust tracking performance on the global system, a centralized control scheme operates under a predictive control framework composed by three control actions. The first one compensates for disturbances using the reference trajectory-feedforward control. The second control action corrects the errors generated by the modeling mismatch. The third one is devoted to ensure robustness on the closed-loop system whilst compensating for deviations of the state trajectories from the nominal ones (i.e. disturbance-free). The strategy ensures robust feasibility even when tightening constraints are met. Such constraints are calculated on-line based on robust positively invariant sets characterized by polytopic sets (tubes), including a terminal region to guarantee robustness. The benefits of the controller regarding tracking performance, constraint satisfaction and computational practicability were tested through simulations with a Cat® 262C skid-steer model. Then, in field tests, the controller evidenced high tracking accuracy against terrain disturbances when benchmarking performance with respect to inherent robust predictive controllers.

Intelligent Swing-Up and Robust Stabilization via Tube-based Nonlinear Model Predictive Control for A Rotational Inverted-Pendulum System

The purpose of this paper is to introduce a new robust nonlinear model-based predictive control scheme applied to a rotational inverted-pendulum system. The rotational pendulum is composed by a mechanical arm attached to a free-motion pendulum (orthogonal to the arm), namely Furuta Pendulum. In principle, a Fuzzy controller enables the robotic arm bar to lift the rotational pendulum through oscillatory swing-up motion up to automatically achieve the upper equilibrium position in a prescribed stabilizing operation range. After the pendulum reaches the operating range, an intelligent control bypass system allows the transition between the swing-up motion controller and a robust predictive controller to maintain the angular position of the pendulum around the upward critical position. To achieve robust performance, a centralized control framework combines a triplet of control actions. The first one compensates for disturbances using the regulation trajectory ?feedforward control. The second control action corrects errors produced by modelling mismatch. The third controller assures robustness on the closed-loop system whilst compensating for deviations of the state trajectories from the nominal ones (i.e, disturbance-free). The control strategy provides robust feasibility despite constraints on the arm bar and pendulum's actuators are met. Such constraints are calculated on-line based on robust positively invariant sets characterised by polytopic sets (tubes). The proposed controller is tested in a series of simulations, and experimentally validated on a high-fidelity simulation environment including a rotational inverted-pendulum built for educational purposes. The results show that robust control performance is strengthened against disturbances of the closed-loop system benchmarked to inherently-robust linear and nonlinear predictive controllers.

A Two-Stage Scheduling RPC Based on Time-Varying Coefficient Information of State-Dependent ARX Model

A two-stage scheduling robust predictive control (RPC) algorithm, which is based on the time-varying coefficient information of the state-dependent ARX (SD-ARX) model, is designed for the output tracking control of a class of nonlinear systems. First, by using the parameter variation range information of the SD-ARX, a strategy for constructing the system’s polytopic model is designed. To further reduce the conservativeness of the convex polytopic sets which are designed to wrap the system’s future dynamics, the variation range information of the SD-ARX model’s parameters is also considered and compressed. In this method, the polytopic state-space model of the system is constructed directly based on the special structure of the SD-ARX model itself, and there is no need to make such assumption that the bounds on the parameter’s variation range in the system model are known or measurable. And then, a two-stage scheduling RPC algorithm is designed for the output tracking control. A numerical example is presented to demonstrate the effectiveness of the proposed RPC strategy.

Finite-horizon suboptimal control of Markov jump linear parameter-varying systems

This paper presents a strategy to compute a suboptimal solution to the finite-horizon linear-quadratic control problem for Markov jump linear parameter-varying systems. The system parameters depend not only on a Markov chain but also on linear parameter-varying elements that take values in convex polytopic sets. The suboptimal approach represents an attempt towards solving the optimal control problem that remains unsolved in the literature. Illustrating that such a suboptimal approach can be of interest when solving practical control problems. Experimental results on the control of an automotive throttle valve subject to failures driven by a Markov chain are discussed, thus supporting the usefulness of the theoretical findings.

Set-Based State Estimation: A Polytopic Approach

This paper is concerned with guaranteed parameter estimation for discrete-time nonlinear systems subject to bounded uncertainties. The proposed approach is based on polytopic set parameterizations. Similar to other estimation and filtering approaches, the presented algorithm is based on two operations, propagation of the polytopic uncertainty through the dynamics and an update operation using the measurement. Both the propagation and the update steps are based on set-operations that use a parameterized lifted outer approximations of the polytopes. The performance of the approach is illustrated by applying it to the double integrator system, where the presented polytopic parameterization approach leads to accurate and rigorous parameter estimates.

Time optimal model predictive control for linear systems based on ellipsoidal and polytopic sets

Time optimal model predictive control (MPC) strategies are investigated in this paper. We first derive a series of controllable sets described by ellipsoids or convex polytopes offline, then obtain the control input that can steer initial state into the terminal set in the shortest time steps by solving an online optimisation problem with lower computational burden. The constraints of online optimisation problem are determined by the controllable sets. We give the methods to calculate the ellipsoidal controllable sets for linear time-invariant systems and linear time-varying systems, respectively, and propose a new method to calculate the polytopic controllable sets for linear time-varying systems. The recursive feasibility of online optimisation problem and the stability of the closed-loop system can be guaranteed. Numerical examples show that the algorithms work well.