Continuous-time Control Research Articles

Constrained reference learning for continuous-time model predictive tracking control of autonomous systems

Abstract Often, systems need to adapt their behavior to other systems in their surroundings while obeying constraints to achieve good performance or due to safety reasons. We consider repetitive applications, where the reference for the controller stems from noisy sensor data. Including preview information of the reference, e.g. extrapolating from previous cycles or similar instances, can significantly improve the overall tracking performance and ensure constraint satisfaction. We propose a learning-supported predictive controller that uses Gaussian processes as reference generators for its control task. A Gaussian process is used to learn, filter, and predict the references. It learns references tailored to model predictive control, taking into account continuous-time system dynamics and constraints via constrained hyperparameter optimization. We illustrate the benefits concerning approximation and control performance of the informed Gaussian-process training by a cooperative, mobile robot example.

Duality and Approximation of Stochastic Optimal Control Problems under Expectation Constraints

We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is in duality with an optimization problem involving the Lagrange multiplier associated with the constraints. Then by convex analysis techniques, we provide a general existence result and some a priori estimation of the dual optimizers. We further provide a necessary and sufficient optimality condition for the initial constrained control problem. The same results are also obtained for a discrete time constrained control problem. Moreover, under additional regularity conditions, it is proved that the discrete time control problem converges to the continuous time problem, possibly with a convergence rate. This convergence result can be used to obtain numerical algorithms to approximate the continuous time control problem, which we illustrate by two simple numerical examples.

Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation

A learning approach for optimal feedback gains for nonlinear continuous time control systems is proposed and analysed. The goal is to establish a rigorous framework for computing approximating optimal feedback gains using neural networks. The approach rests on two main ingredients. First, an optimal control formulation involving an ensemble of trajectories with ‘control’ variables given by the feedback gain functions. Second, an approximation to the feedback functions via realizations of neural networks. Based on universal approximation properties we prove the existence and convergence of optimal stabilizing neural network feedback controllers.

LPV Delay-Dependent Sampled-Data Output-Feedback Control of Fueling in Spark Ignition Engines

We propose a delay-dependent sampled-data output-feedback LPV control technique to address the air-fuel ratio (AFR) regulation problem in spark ignition (SI) engines. AFR control and advanced fueling strategies are essential for maximizing fuel economy while minimizing harmful exhaust emissions. The fuel path of the SI engine, as well as the three-way catalyst (TWC) simplified dynamics, have been captured by a continuous-time linear parameter-varying (LPV) system with varying time delay, where the system dynamics rely on the engine speed, defined as the system's scheduling parameter. The interconnection of the continuous-time plant and a digital controller through analog-to-digital and digital-to-analog converter devices forms a hybrid closed-loop configuration. Therefore, in order to benefit from continuous-time control synthesis tools, the input-delay method has been employed to transform the hybrid closed-loop system into the continuous-time domain with system inherent time delay and an additional delay imposed by the mapping approach. The designed sampled-data gain scheduled output-feedback controller seeks to establish the closed-loop asymptotic stability and a prescribed level of performance for the LPV system with an arbitrarily varying time delay and varying sampling time, where the synthesis results are provided in a convex linear matrix inequality (LMI) constraint setting. Finally, several closed-loop simulation scenarios are conducted, and comparisons are provided to demonstrate the proposed methodology's performance in achieving precise reference AFR tracking and disturbance attenuation.

A Simple Approach to Increase the Maximum Allowable Transmission Interval

When designing Networked Control Systems (NCS), the maximum allowable transmission interval (MATI) is an important quantity, as it provides the admissible time between two transmission instants. An efficient procedure to compute a bound on the MATI such that stability can be guaranteed for general nonlinear NCS is the emulation of a continuous-time controller. In this paper, we present a simple but efficient modification to the well-established emulation-based approach from Carnevale et al. (2007) to derive a bound on the MATI. Whilst only minor technical changes are required, the proposed modification can lead to significant improvements for the MATI bound as compared to Carnevale et al. (2007). We revisit two numerical examples from literature and demonstrate that the improvement may amount to more than 100%.

Geometric control of algebraic systems

Abstract In this paper, we present a geometric approach for computing the controlled invariant set of a continuous-time control system. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function. We compare it with the common algebraic approach for polynomial sublevel sets and show that it is significantly more conservative than our method.

Robust Constrained Drug Dosage Regulation of Acute Inflammation Response Under Disturbances

Mathematical modelling of the biological processes, diseases and organs are important for the model-based control of diseases. Due to unmodeled dynamics, unknown and external disturbances, the performance of controllers based on these models are degraded for the accurate control. Therefore, robust controllers are need especially for the applications on patients. Inflammation, the cause of many complex biological phenomena and diseases, is a nonlinear process that is difficult to control. In this paper, continuous-time sliding-mode controller has been designed for the control of acute inflammation response (AIR) and antibacterial drug infusion under external disturbances both for septic and aseptic cases. Sliding-mode controller (SMC) is mostly used to control nonlinear systems against external disturbances and parametric uncertainties. Beside the control signal generation, we propose constraints on the control signals based on the clinical experiences such that the applied control signal is suitable for the health and improves the performance of the controller. Due to the multiple equilibrium point on the behavior of the acute inflammation response, it is difficult to design such model-based controllers without input constraints. In the numerical applications, septic death case and aseptic death case with disturbances are controlled and acceptable performances are obtained for future clinical applications.

Quantized Dynamic Output Feedback Control and $L_2$-Gain Analysis for Networked Control Systems: A Hybrid Approach

In this article, the problems of stabilization and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain performance analysis for continuous-time networked control systems (NCSs) with quantization effects are addressed. The plant output and control input are quantized respectively by two different dynamic quantizers before being transmitted over networks. In order to deal with the dynamic behavior resulting from the dynamic quantizers, a hybrid system framework is adopted to remodel the NCSs. The hybrid model is flexible to the joint design of dynamic output feedback controller, nonequidistant samplers and dynamic quantizers. Furthermore, our method can effectively prevent the Zeno phenomenon on the instants of sampling and the updates of quantizer variables. By constructing a novel Lyapunov function, new sufficient conditions guaranteeing the uniformly globally asymptotic stability (UGAS) and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain performance are presented in terms of matrix inequalities, respectively. Finally, two numerical examples illustrate the effectiveness of our proposed method.

Extended dissipative sliding mode control for nonlinear networked control systems via event-triggered mechanism with random uncertain measurement

This work concentrates on addressing the sliding mode control problem of continuous-time nonlinear networked control systems. Considering the state information may not be utterly available in practice, a state observer model is designed to estimate the state information. Meanwhile, a type of discrete-time event-triggered mechanism is utilized to filter the sampled signal for reducing the occupation of network bandwidths and the transmission rate of resources. In addition, a random variable obeying the Bernoulli distribution is adopted to describe the phenomenon of uncertainties randomly occurring in the measurement. With the aid of the Lyapunov stability and sliding mode control theory, some sufficient criteria are given to both guarantee the mean-square asymptotic stability of the overall closed-loop system with an extended dissipative performance, and the reachability of predefined sliding surface. Whereafter, the event-triggered weighting matrix, and gains of sliding mode controller and with observer are obtained by solving the matrix convex optimization problem. Finally, the feasibility of the presented scheme is demonstrated through two illustrative examples.

Multivariable Unconstrained Pattern Search Method for Optimizing Digital PID Controllers Applied to Isolated Forward Converter

Most of the traditional PID tuning methods are heuristic in nature. The heuristic approach-based tuned PID controllers show only nominal performance. In addition, in the case of a digital redesign approach, mapping of the heuristically-designed continuous-time PID controllers into discrete-time PID controllers and in case of the direct digital design approach, mapping of the continuous-time plant (forward converter) into the discrete-time plant, results in frequency distortion (or warping). Besides this, nonlinear elements such as ADC and DAC, and delay in the digital control loop deteriorate the control performance. There is a need to tune conventionally-designed digital controllers to enhance performance. This paper proposes optimized discrete-time PID controllers for a forward DC–DC converter operating in continuous conduction mode (CCM). The considered conventional digital PID controllers designed on the basis of the digital redesign and direct digital approaches are tuned by one of the multivariable unconstrained pattern search methods named Hooke–Jeeves (H–J) search method to ensure excellent output voltage regulation performance against the changes in input voltage and load current. Numerical results show that the H–J-based optimized PID compensated forward converter system shows tremendous improvement in performance compared to its unoptimized counterpart and simulated annealing (SA)-based compensated system, thus justifying the applicability of the H–J method for enhancing the performance.

Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined and allows for unambiguous economic interpretation. The terms “strong equilibria” and “weak equilibria” are coined for controls under the new and standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to the Kakutani–Fan fixed-point theorem, the general existence of weak and strong equilibria is also established under an additional compactness assumption. Our theoretic results are applied to a two-state model under nonexponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon.

Discrete-time model-based output regulation of fluid flow systems

Model-based discrete-time output regulator design is proposed for fluid flow systems using a geometric approach. More specifically, a class of vortex shedding and falling thin film phenomena modelled by complex Ginzburg–Landau equation (CGLE) and Kuramoto–Sivashinsky equation (KSE) are considered. Differently from a traditional continuous-time controller design, a novel discrete-time modelling technique is proposed in a general infinite-dimensional state-space setting, which does not pertain any spatial approximation or model reduction, and preserves model intrinsic properties (such as stability, controllability and observability). Based on the time discretized plant model (CGLE and KSE systems) by the Cayley–Tustin method, discrete regulator regulation equations are established and facilitated for an output regulator design to achieve fluid flow control and manipulation. To address model instability, a spectrum analysis is utilized in stabilizing continuous-time CGLE and KSE systems, and a link between discrete- and continuous-time closed-loop system stabilizing gains is established. Finally, the proposed methodology is demonstrated through a set of simulation cases, by which the output tracking, disturbance rejection, and model stabilization are achieved for the considered CGLE and KSE systems.

Finite-time stable tracking control for an underactuated system in SE(3) in discrete time

ABSTRACT We consider tracking control of an underactuated system on the tangent bundle of the six-dimensional Lie group of rigid body motions, . We formulate a finite-time stable (FTS) tracking control scheme for this underactuated system in discrete time. This scheme is based on our recently developed theory for finite-time stability for discrete-time systems using discrete Lyapunov analysis. The proposed scheme here is developed in discrete time as it is more convenient for onboard computer implementation and ensures stability irrespective of the sampling period. This scheme guarantees a stable convergence of translational and rotational tracking errors to the desired trajectory in finite time. Furthermore, the advantages of finite-time stabilisation in discrete-time over finite-time stabilisation of a sampled continuous-time tracking control system is addressed here through a numerical comparison. This comparison is performed using numerical simulations on continuous and discrete FTS tracking control schemes applied to an unmanned aerial vehicle model.

Neural network-based fault detection for nonlinear networked systems with uncertain medium access constraint: Application to motor systems

The fault detection for a class of continuous-time nonlinear networked control systems with medium access constraint is concerned in this paper, where the occurring probability of transition from one sensor to another is allowed to be partially unknown and uncertain. First of all, a Markovian system approach is adopted to describe the access process of sensors, in which only one sensor is allowed to access the communication channel. A robust filter based residual generator is proposed to generate the residual signal such that it can be used to indicate whether the fault has occurred or not. The nonlinear term is approximated by a neural network, and the Lyapunov–Krasovskii functional is introduced to analyze the fault detection system, three sufficient conditions for the stochastic stability of fault detection error system are given, and the fault detection filter gains are calculated via solving some matrix inequalities. In the simulation, a second-order DC motor system is used to validate of the main results, which shows the effectiveness of the fault detector design.

Low-Complexity Decentralized Algorithm for Aggregate Load Control of Thermostatic Loads

Thermostatically controlled loads such as refrigerators are exceptionally suitable as a flexible demand resource. This paper derives a decentralised load control algorithm for refrigerators. It is adapted from an existing continuous time control approach, with the aim to achieve low computational complexity and an ability to handle discrete time steps of variable length -- desirable features for embedding in appliances and high-throughput simulations. Simulation results of large populations of heterogeneous appliances illustrate the accurate aggregate control of power consumption and high computational efficiency. Tracking accuracy is quantified as a function of population size and time step size, and correlations in the tracking error are investigated. The controller is shown to be robust to errors in model specification and to sudden perturbations in the form of random refrigerator door openings.

Finite-Time Control of Continuous-Time Networked Dynamical Systems

Finite-time stability (FTS) analysis and control problems are investigated in this paper for continuous-time networked dynamical systems constituted by many subsystems. Interactions among these subsystems are arbitrary and time-varying, and every subsystem has different and time-varying dynamics. Several differential linear matrix inequality (LMI) formed methods are presented for the FTS analysis and control, which avoid the inverse computation of matrices with large dimensions and efficiently utilize the block diagonal characteristic of system parameters and the sparseness of the subsystem connection matrix. Sufficient conditions only depending on the parameters of each individual subsystem are then derived. Furthermore, LMI-based conditions are also provided for the FTS analysis and control of linear time-invariant networked systems. Numerical simulations show that they are computationally effective to analyze and synthesize a large-scale networked dynamical systems.

Data-Driven Model-Free Sliding Mode and Fuzzy Control with Experimental Validation

The paper presents the combination of the model-free control technique with two popular nonlinear control techniques, sliding mode control and fuzzy control. Two data-driven model-free sliding mode control structures and one data-driven model-free fuzzy control structure are given. The data-driven model-free sliding mode control structures are built upon a model-free intelligent Proportional-Integral (iPI) control system structure, where an augmented control signal is inserted in the iPI control law to deal with the error dynamics in terms of sliding mode control. The data-driven model-free fuzzy control structure is developed by fuzzifying the PI component of the continuous-time iPI control law. The design approaches of the data-driven model-free control algorithms are offered. The data-driven model-free control algorithms are validated as controllers by real-time experiments conducted on 3D crane system laboratory equipment.

Optimization and discrete approximation of sweeping processes with controlled moving sets and perturbations

This paper addresses a new class of optimal control problems for perturbed sweeping processes with measurable controls in additive perturbations of the dynamics and smooth controls in polyhedral moving sets. We develop a constructive discrete approximation procedure that allows us to strongly approximate any feasible trajectory of the controlled sweeping process by feasible discrete trajectories and also establish a W1,2-strong convergence of optimal trajectories for discretized control problems to a given local minimizer of the original continuous-time sweeping control problem of the Bolza type. Employing advanced tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete optimal solutions under fairly general assumptions formulated entirely in terms of the given data. The obtained results give us efficient suboptimality (“almost optimality”) conditions for the original sweeping control problem that are illustrated by a nontrivial numerical example.

Sampled-data output feedback control for a class of high-order switched nonlinear systems

In this paper, we design new sampled-data output feedback controllers to globally stabilise a class of high-order switched nonlinear systems in finite time under arbitrary switching. The switching signals exist in both the system powers and the nonlinear terms. First, we adopt the common Lyapunov function method to design a continuous-time output feedback controller. Then, the emulation method is used to construct the sampled-data output feedback controllers with a sampling period. Based on the homogeneity theory, Hölder inequality and Gronwall–Bellman inequality, the relationships between the sampling states (controllers) and the current states (controllers) are revealed. In the meantime, we obtain the sampling period for the sampled-data output feedback controllers to guarantee the global stabilisation of the closed-loop systems in finite time. Finally, a numerical example and a practical example are given to prove the efficiency of the proposed control method.

Wait Time Based Pricing for Queues with Customer-Chosen Service Times

This paper studies a pricing problem for a single-server queue where customers arrive according to a Poisson process. For each arriving customer, the service provider announces a price rate and a system wait time, and the customer decides whether to join the queue, and, if so, the duration of the service time. The objective is to maximize either the long-run average revenue or social welfare. We formulate this problem as a continuous-time control model whose optimality conditions involve solving a set of delay differential equations. We develop an innovative method to obtain the optimal control policy, whose structure reveals interesting insights. The optimal dynamic price rate policy is not monotonic in wait time. In particular, in addition to the congestion effect often reported in the literature, i.e., the optimal price rate increases in the queue length (measured by the wait time in our setting), we find a compensation effect, meaning that the service provider should lower the price rate when the wait time is longer than a threshold. Compared with the prevalent flat pricing policy, our optimal dynamic pricing policy improves the objective value through admission control, which, in turn, increases the utilization of the server. We use a real data set obtained from a public charging station to calibrate our model with an objective of maximizing the average revenue. We find that our optimal pricing policy outperforms the best flat pricing policy, especially when the arrival rate is low and drivers are impatient. Interestingly, our revenue-maximizing pricing policy also improves social welfare over the flat pricing policy in most of the tested cases.