Discrete-time Control Systems Research Articles

Contract composition for dynamical control systems: Definition and verification using linear programming

Designing large-scale control systems to satisfy complex specifications is hard in practice, as most formal methods are limited to systems of modest size. Contract theory has been proposed as a modular alternative, in which specifications are defined by assumptions on the input to a component and guarantees on its output. However, current contract-based methods for control systems either prescribe guarantees on the state of the system, going against the spirit of contract theory, or are not supported by efficient computational tools. In this paper, we present a contract-based modular framework for discrete-time dynamical control systems. We extend the definition of contracts by allowing the assumption on the input at a time k to depend on outputs up to time k−1, which is essential when considering feedback systems. We also define contract composition for arbitrary interconnection topologies, and prove that this notion supports modular design, analysis and verification. This is done using graph theory methods, and specifically using the notions of topological ordering and backward-reachable nodes. Lastly, we present an algorithm for verifying vertical contracts, which are claims of the form “the conjunction of given component-level contracts implies given contract on the integrated system”. These algorithms are based on linear programming, and scale linearly with the number of components in the interconnected network. A numerical example is provided to demonstrate the scalability of the presented approach, as well as the modularity achieved by using it.

On convergence of occupational measures sets of a discrete-time stochastic control system, with applications to averaging of hybrid systems

We establish that, under certain conditions, the set of random occupational measures generated by the state-control trajectories of a discrete-time stochastic system as well as the set of their mathematical expectations converge to a non-random, convex and compact set. We apply these results to the averaging a hybrid system with a slow continuous-time component and a fast discrete-time component. It is shown that the solutions of the hybrid system are approximated by the solutions of a differential inclusion. The novelty of our results is that we allow the state-control space of the fast component to be non-denumerable.

Learning-Based Model-Free Adaptive Control for Nonlinear Discrete-Time Networked Control Systems Under Hybrid Cyber Attacks.

A novel learning-based model-free adaptive control (LMFAC) approach is presented in this article for a class of unknown nonaffine nonlinear discrete-time networked control systems (NCSs) subject to hybrid cyber attacks. The aperiodic denial-of-service (DoS) attacks and persistent deception attacks are assumed to arise in feedback channels, which could result in the absence or authenticity lackness of system signals sent to the controller. With the aid of dynamic linearizaton technology, the equivalent dynamic linearized data models of considered NCSs are first established only based on I/O information instead of the knowledge of mathematical models that are commonly used under the model-based control framework. Then, an LMFAC scheme is designed on the basis of occurred maximum DoS attacks interval to adaptively tune the attenuation coefficient of the input signal for improving system performance during the next DoS attacks interval. Finally, the boundedness of tracking error is rigorously proved through the contraction mapping principle and the effectiveness of the proposed pure data-driven LMFAC method is demonstrated via simulations.

An improved repetitive controller with fractional time-delay for discrete-time linear systems: Synthesis and comparison study

This paper develops a discrete-time repetitive control (RC) system with a fractional-delay internal model. Unlike the conventional RC, the time delay for constructing the internal model is not necessarily an integer, implying that the time delay is allowed to be fractional. In this work, a fractional delay-based repetitive control system is presented. Firstly, the fractional time delay is calculated based on the information of the reference or disturbance along with the associated sampling period. Secondly, the fractional time delay is divided into two parts, i.e., the large integer and small fractional delays. Finally, the fractional part is realized by a stable and causal infinite impulse response (IIR) filter, whose coefficients are calculated using the Thiran formula. The controller structure and stability analysis are developed for three different RCs, namely a fractional-delay general RC (FD-GRC), a fractional delay-odd harmonics RC (FD-OHRC), and a fractional delay-(6l±1) RC (FD-(6l±1)RC). Experiments and comparison studies of the integer delay-based RCs are conducted to verify the effectiveness and improved tracking accuracy of the proposed method.

Non-classical method of calculating the integral component in regulators of multivariable discrete time control systems

Non-classical method of calculating the integral component in regulators of multivariable discrete time control systems

On Strategic Measures and Optimality Properties in Discrete-Time Stochastic Control with Universally Measurable Policies

This paper concerns discrete-time infinite-horizon stochastic control systems with Borel state and action spaces and universally measurable policies. We study optimization problems on strategic measures induced by the policies in these systems. The results are then applied to risk-neutral and risk-sensitive Markov decision processes to establish the measurability of the optimal value functions and the existence of universally measurable, randomized or nonrandomized, ϵ-optimal policies, for a variety of average cost criteria and risk criteria. We also extend our analysis to a class of minimax control problems and establish similar optimality results under the axiom of analytic determinacy. Funding: This work was supported by grants from DeepMind, the Alberta Machine Intelligence Institute (AMII), and Alberta Innovates-Technology Futures (AITF).

Design of PI Controller for a Class of Discrete Cascade Control Systems

This paper mainly studies the modeling, stability analysis and proportional-integral (PI) controller design for a class of discrete-time cascade control systems (CCSs). In the existing CCSs for solving gain of the controller, the controller mostly adopts the proportional (P) controller, but the most commonly used controller in industrial production is PI controller. Based on the actual demand in industry and in order to obtain a better control effect, the design problem of primary and secondary controllers in a class of discrete cascade control is considered, in which the primary controller adopts a PI controller and the secondary controller adopts a P controller. On this basis, the model of cascade control system (CCS) is established. Then, we can create a new Lyapunov functional, the sufficient conditions for the stability of the system is given. Then, the collaborative design method of primary PI controller and secondary P controller is given by using linear matrix inequality (LMI) technique. Finally, a simulation example of a main steam temperature with CCS structure is given to demonstrate the effectiveness of the method. The PI controller is better than the existing P controller, this method has not only the rapidity of P control, but also the ability of integral control to eliminate steady-state errors. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> — The inspiration of this paper comes from the adjustment of plant controller parameters. The PI controller is the most widely used controller in the existing industrial production, but the adjustment of proportional and integral parameters of the controller is a very troublesome thing. In order to obtain a better controller parameters, a professional engineer may spend a whole day to adjust the parameters. This paper can design the solution of the primary and secondary controller parameters to stabilize the CCS at the same time by LMI technique, which can greatly reduce the work of adjusting the controller parameters. Through simulation, we can know that this method can make the system stable, which shows this method is feasible.

A non-fragile robust [formula omitted] regulator in discrete-time control systems with nonzero reference

A non-fragile H∞ integral structure is developed in the discrete-time representation to illuminate the disturbance effects in the reference regulation of the uncertain systems. Hence, to investigate the effects of the exterior signals on the efficiency, the reference regulation problem is formulated in a discrete-time system. A controller method is addressed relying on linear matrix inequality. Then, in the worst cases, finding the controller’s parameters is transformed into a minimization issue. By such an optimization, the coefficients of the compensator would be estimated. Some examples are supplied to manifest the efficiencies of the presented H∞ integral tactic over similar approaches.

Design of Photovoltaic Power Generation Servo System Based on Discrete Adaptive Network Dynamic Surface Control Technology

In recent years, under the development of the dual carbon goal, the energy crisis has become increasingly serious, and China has also experienced serious power rationing. However, the research on dynamic surface control technology in solar tracking systems in nonlinear control systems is mostly based on continuous-time systems, while adaptive dynamic surface control based on discrete-time nonlinear control systems can describe an actual control system more accurately in the production process. It can effectively suppress interference with extremely high stability and safety. To solve the problem of low efficiency in photovoltaic power generation, this research first built a photovoltaic power generation servo system model based on the parameter of uncertainty. Then, a discrete adaptive neural network dynamic surface (DANNDS) controller was designed to solve the problems in the design of the traditional backstepping method. Finally, based on the designed method of a dynamic surface controller, a discrete adaptive neural network quantization controller (DANNQC) for the photovoltaic power generation servo system was designed by introducing external disturbance. The control parameters and their studied ranges were as follows: The reference signals were or1=sin(0.1t) and or2=cos(0.1t). The parameters of the virtual control law and the final control law were m11=0.01, m22=0.01, m12=0.02, m13=0.02, and m23=0.02. The time constant of the low-pass filter was ζ12=ζ13=ζ22=ζ23=0.005. The parameters of the parameter regulation law were ρ12=ρ13=ρ22=ρ23=0.0005 and a12=a13=20, a22=a23=22. The research results show that the MTE, RMSTE, and 2NTE scores of the height angle servo motor of the DANNDS control method were 0.0026, 7.0279 × 10−4, and 0.3552, respectively. The scores for each index of the azimuth servo motor were 0.0028, 8.9237 × 10−4, and 0.4511, respectively. The height angle tracking error for the DANNQC control method was [−0.02,0.022]. The azimuth tracking error was [−0.03,0.03]. In summary, the photovoltaic power generation servo system based on the DANNQC has a better control performance. By controlling the height angle and azimuth angles, it can better track the position of the sun and adjust the position of the photovoltaic panel in real time. The sun’s rays illuminate the photovoltaic panel at an appropriate angle to achieve maximum power generation efficiency, which is of great practical significance for the development of solar technology.

The Measurable Data Rate Theorem for Discrete-Time Control Systems

The Measurable Data Rate Theorem for Discrete-Time Control Systems

Mixed H∞ and passive resilient control for discrete-time systems with Round-Robin protocol and deception attacks

This paper focuses on the mixed [Formula: see text] and passive resilient control problem for uncertain discrete-time networked control systems subject to time delays. The measurement output is considered to be quantized by the dynamic quantizer before transmission through an unreliable communication channel, where the deception attacks may occur randomly. To further reduce the communication burden, the Round-Robin (RR) protocol is adopted to schedule sensors in a predetermined transmission order. Meanwhile, a new performance index is developed to solve restriction of the same dimension of control output and the exogenous disturbance. The S-procedure combined with a two-step uncertainties elimination method is adopted to deal with the nonlinear product terms involving multiple uncertainties. Furthermore, a decoupling method is introduced for static output feedback controller design. Sufficient conditions of the resilient controller and the dynamic quantizer are provided, which can improve the robustness of resulting closed-loop system and guarantee its stochastic stability and the specified mixed [Formula: see text] and passivity performance index. Finally, an example is given to illustrate the feasibility of the theoretical results.

Learning-Ability of Discrete-Time Iterative Learning Control Systems with Feedforward

This paper considers the learning-ability for discrete-time iterative learning control (ILC) systems with feedforward. More specifically, the relation between the output realizability and the feedforward matrix is first established. Then, the learning-ability of four ILC systems is considered. It is shown that the proportional type (P-type) update law can only ensure the fully asymptotic learning-ability. By only using the feedforward matrix, a more efficient point-wise P-type update law is developed, which can ensure the fully -step learning-ability, where is the trial length. In the case that the state is measurable and controllable, it is proven that the update law with current state feedback can ensure the fully monotone learning-ability and the fully 2-step learning-ability, respectively. In addition, by only using the output data at the previous trial, a full output feedback update law is proposed, which can respectively ensure the fully 2-step learning-ability and the fully monotonic learning-ability.

Predictive control of discrete-time high-order fully actuated systems with application to air-bearing spacecraft simulator

This paper addresses an output tracking problem for discrete-time high-order fully actuated (DHOFA) systems and its application in the control of air-bearing spacecraft (ABS) simulator. A HOFA system model, as a novel system representation, is applied to establish the dynamics of discrete-time control systems. Accordingly, a HOFA predictive control scheme is presented to deal with this problem, which is composed of a HOFA feedback for stabilization and a HOFA predictive control for tracking. In this scheme, a Diophantine equation is exploited to construct an incremental HOFA (IHOFA) prediction model to substitute a reduced-order prediction model, and then a cost function involving tracking performance is minimized by using multi-step output predictions. A sufficient and necessary condition is proposed to discuss the stability and tracking performance of the closed-loop DHOFA systems, it is simple to utilize in system analysis and extend in practice. Two experiments of the control of ABS simulator are shown to illustrate the feasibility of the presented HOFA predictive control scheme.

High Minimum Inter-Execution Time Sigmoid Event-Triggered Control for Spacecraft Attitude Tracking With Actuator Saturation

This paper investigates the attitude tracking problem for spacecraft with limited communication, network congestion, unknown model parameters, actuator saturation, and external disturbances. To alleviate communication load, a novel sigmoid event-triggered mechanism is proposed with the special ability to guarantee a high minimum inter-execution time to avoid network congestion like package loss or time delay effectively. A neural network-based adaptive control algorithm is designed to deal with unknown model parameters. Besides, the problem of actuator saturation is tackled by introducing a dynamic loop gain function-based approach. System stability is proved by Lyapunov stability analysis and the high minimum inter-event time is substantiated by Zeno Behavior analysis with explanatory remarks. Numerical simulation results also show that a high minimum inter-event time and a high average inter-event time can be realized on the premise of high attitude tracking accuracy. Compared with all the previous studies, the ratio of the minimum inter-execution time to the average minimum inter-execution time has been improved by nearly 10 times with the proposed approach. Note to Practitioners—This paper was motivated by the problem of spacecraft attitude takeover control, but it also applies to other aperiodic discrete-time control systems. Existing event-triggered control methods have solved the problem of limited communication for attitude takeover control of spacecraft by reducing the number of times wireless communication is required. This paper proposes a new method for realizing quasi-periodic control with a large time gap between any two consistent control impulses using a hyperbolic tangent function or another sigmoid function in an event-triggered mechanism. The proposed sigmoid event-triggered mechanism (ETM) can magnify the event-triggered threshold dozens of times in a short time to avoid unnecessary frequent triggering without decreasing the precision of control. It’s feasible to employ the proposed sigmoid ETM to deal with limited communication and network congestion in other network control systems that are constructed with similar dynamic structures. Furthermore, it would be marvelous if we could transform this sigmoid ETM into a high-efficiency decision-making mechanism in some control systems without using a wireless network. Now we are studying how to realize orbital-attitude (maneuver) control with a relatively average time interval and fewer times of orbit transfer by using the proposed sigmoid ETM. Besides, to further improve the system performance, we are also seeking to further increase the ratio of the minimum inter-execution time to the average inter-execution time. For the purpose of a more even distribution of the triggering events on the time axis, we may modify the proposed sigmoid event-triggered method or associate it with other methods.

State-Space Region-Invariance of Discrete-Time Control Systems

In the paper, the so called region-invariance properties of discrete-time control systems are considered. An already known notion of positivity of control systems has been generalized (using a kind of geometric insight instead of a pure algebraic approach) to the general nonlinear discrete-time control systems, general regions in state-space, and controls from polyhedral cones in input-space. The class of such control systems has been characterized. Numerous numerical examples illustrating individual cases under consideration are presented in the paper.

Abstraction-Based Verification of Approximate Preopacity for Control Systems

In this letter, we consider the problem of verifying pre-opacity for discrete-time control systems. Pre-opacity is an important information-flow security property that secures the intention of a system to execute some secret behaviors in the future. Existing works on pre-opacity only consider non-metric discrete systems, where it is assumed that intruders can distinguish different output behaviors precisely. However, for continuous-space control systems whose output sets are equipped with metrics (which is the case for most real-world applications), it is too restrictive to assume precise measurements from outside observers. In this letter, we first introduce a concept of approximate pre-opacity by capturing the security level of control systems with respect to the measurement precision of the intruder. Based on this new notion of pre-opacity, we propose a verification approach for continuous-space control systems by leveraging abstraction-based techniques. In particular, a new concept of approximate pre-opacity preserving simulation relation is introduced to characterize the distance between two systems in terms of preserving pre-opacity. This new system relation allows us to verify pre-opacity of complex continuous-space control systems using their finite abstractions. We also present a method to construct pre-opacity preserving finite abstractions for a class of discrete-time control systems under certain stability assumptions.

Review on contraction analysis and computation of contraction metrics

&lt;p style='text-indent:20px;'&gt;Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy).&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.&lt;/p&gt;

Data-Driven Synthesis of Symbolic Abstractions With Guaranteed Confidence

In this letter, we propose a data-driven approach for the construction of finite abstractions ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a.k.a.,</i> symbolic models) for discrete-time deterministic control systems with unknown dynamics. We leverage notions of so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alternating bisimulation functions</i> (ABF), as a relation between each unknown system and its symbolic model, to quantify the mismatch between state behaviors of two systems. Accordingly, one can employ our proposed results to perform formal verification and synthesis over symbolic models and then carry the results back over unknown original systems. In our data-driven setting, we first cast the required conditions for constructing ABF as a robust optimization program (ROP). Solving the provided ROP is not tractable due to the existence of unknown models in the constraints of ROP. To tackle this difficulty, we collect finite numbers of data from trajectories of unknown systems and propose a scenario optimization program (SOP) corresponding to the original ROP. By establishing a probabilistic relation between optimal values of SOP and ROP, we formally construct ABF between unknown systems and their symbolic models based on the number of data and a required confidence level. We verify the effectiveness of our data-driven results over two physical case studies with unknown models including (i) a DC motor and (ii) a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> jet engine compressor. We construct symbolic models from data as appropriate substitutes of original systems and synthesize policies maintaining states of unknown systems in a safe set within infinite time horizons with some guaranteed confidence levels.

Constructing MDP Abstractions Using Data With Formal Guarantees

This letter is concerned with a data-driven technique for constructing finite Markov decision processes (MDPs) as finite abstractions of discrete-time stochastic control systems with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unknown</i> dynamics while providing formal closeness guarantees. The proposed scheme is based on notions of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic bisimulation functions</i> (SBF) to capture the probabilistic distance between state trajectories of an unknown stochastic system and those of finite MDP. In our proposed setting, we first reformulate corresponding conditions of SBF as a robust convex program (RCP). We then propose a scenario convex program (SCP) associated to the original RCP by collecting a finite number of data from trajectories of the system. We ultimately construct an SBF between the data-driven finite MDP and the unknown stochastic system with a given confidence level by establishing a probabilistic relation between optimal values of the SCP and the RCP. We also propose two different approaches for the construction of finite MDPs from data. We illustrate the efficacy of our results over a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> jet engine compressor with unknown dynamics. We construct a data-driven finite MDP as a suitable substitute of the original system to synthesize controllers maintaining the system in a safe set with some probability of satisfaction and a desirable confidence level.

Homogeneity for control systems in discrete time

Homogeneity, as a generalization of linearity to nonlinear systems, has proven to be a very powerful in systems and control. Nevertheless, only recently a notion of homogeneity was proposed for discrete-time control systems. However, this so-called D-Homogeneity directly couples the stability behaviour with the degree of homogeneity—in contrast to the continuous-time case. As an alternative, we propose the notion of S-Homogeneity, which avoids this coupling. S-Homogeneity uses a state-dependent time step that is compatible with sampling and discretization in time. We show that this concept preserves a contraction property and null-controllability for state-dependent sampling. For fixed sampling time, it yields (practical/semi-global) null controllability for sufficiently fast sampling, depending on the degree of homogeneity.