Controller Synthesis Problem Research Articles

Advanced Model with a Trajectory Tracking Stabilisation System and Feasible Solution of the Optimal Control Problem

In this study, we consider the extended optimal control problem and search for a control function in the class of feasible functions for a real control object. Unlike the classical optimal control problem, the control function should depend on the state, not time. Therefore, the control synthesis problem for the initial-state domain should be solved, instead of the optimal control problem with one initial state. Alternatively, an optimal trajectory motion stabilisation system may be constructed. Both approaches—control and trajectory motion stabilisation system syntheses—cannot be applied to real-time control, as the task is too complex. The minimum threshold of quality criteria is searched for in the space of mathematical expression codes. Among other problems, the search space is difficult to define and the gradient is hard to determine. Therefore, the advanced control object model is used to obtain a feasible control function. The advanced model is firstly obtained before solving the optimal control problem and it already includes a trajectory motion stabilisation system; in particular, this stabilisation system is synthesised in advance at the control system design stage. When the optimal control problem appears, it is solved in real time in the classical statement, and a control function is searched for as a function of time. The advanced control object model also uses the reference model to generate the optimal trajectory. The search for the optimal control function is performed in real time and considers the synthesised stabilisation system of motion along a determined trajectory. Machine learning control via symbolic regression, namely, the network operator method, is used to directly solve the control synthesis problem. An example solution of the optimal control problem, with an advanced model moving in the environment with obstacles for a group of two mobile robots, is presented. The obtained solution is a control function for a reference model that generates a trajectory from a class of trajectories stabilised with the object’s control system.

Optimal Design of I-PD and PI-D Industrial Controllers Based on Artificial Intelligence Algorithm

This research aims to apply Artificial Intelligence (AI) methods, specifically Artificial Immune Systems (AIS), to design an optimal control strategy for a multivariable control plant. Two specific industrial control approaches are investigated: I-PD (Integral-Proportional Derivative) and PI-D (Proportional-Integral Derivative) control. The motivation for using these variations of PID controllers is that they are functionally implemented in modern industrial controllers, where they provide precise process control. The research results in a novel solution to the control synthesis problem for the industrial system. In particular, the research deals with the synthesis of I-P control for a two-loop system in the technological process of a distillation column. This synthesis is carried out using the AIS algorithm, which is the first application of this technique in this specific context. Methodological approaches are proposed to improve the performance of industrial multivariable control systems by effectively using optimization algorithms and establishing modified quality criteria. The numerical performance index ISE justifies the effectiveness of the AIS-based controllers in comparison with conventional PID controllers (ISE1 = 1.865, ISE2 = 1.56). The problem of synthesis of the multi-input multi-output (MIMO) control system is solved, considering the interconnections due to the decoupling procedure.

Hierarchical control of linear systems using extended robust approximate simulation

This paper develops a hierarchical control structure for linear systems based on extended robust approximate simulation. In numerous real-world scenarios, systems are subject to external disturbances, and there is a restriction on accessibility to all of their states. Furthermore, these systems are modeled using linear dynamics with high state dimensions. Hence, the control synthesis problem for these concrete systems is computationally challenging, and constructing an abstract model with smaller state dimensions to design the controller would be beneficial. Therefore, we introduce the notion of a extended robust approximate simulation function between the concrete system and the abstract model under a new hierarchical control structure. With this respect, a controller is designed for the abstract model and refined for the concrete system by formulating an observer-based robust interface controller based on H∞ while guaranteeing the desired performance for the concrete system. It is worth noting that an Unknown Input Observer (UIO) is employed in the controller, which circumvents the full-state accessibility of the concrete system. The applicability of the presented approach is demonstrated by controlling bus voltages on a smart grid.

A dynamic controller synthesis methodology for negative imaginary systems using the internal model control principle

This paper proposes a new controller design methodology for stable and minimum-phase Negative Imaginary (NI) systems relying on the classical Internal Model Control (IMC) principle. The closed-loop stability of the proposed scheme depends only on the DC loop gain, which is theoretically justified by the feedback stability results of the NI theory. The main objective is to design the Youla parameter of an IMC scheme, which has been cast as a Negative Imaginary (NI) controller synthesis problem. Two different methodologies have been proposed. A frequency-domain IMC design approach is first presented, which depends on solving a constrained, linear, least-square estimation problem. Then, an LMI-based methodology is developed, which can be solved by the commercially available SDP solver packages. An in-depth simulation case study on the vibration attenuation problem of a lightweight cantilever beam (a potential application of the NI theory) was carried out to demonstrate the usefulness of the NI-based IMC design methodology. Finally, the simulation results were experimentally validated on a custom-made vibration suppressor to confirm the feasibility of the proposed scheme.

L2$$ {L}_2 $$‐gain analysis and fault‐tolerant control for nonlinear discrete‐time switched systems with time‐varying delay and actuator saturation

AbstractThe ‐gain analysis and fault‐tolerant control of a class of uncertain nonlinear discrete‐time switched systems with time‐varying delay and actuator saturation are studied by using the multiple Lyapunov functions method. The fault‐tolerant state feedback controllers and the switching law are designed such that the closed‐loop system with actuator failures satisfies the disturbance attenuation performance indicator. The problem of estimating the capacity of admissible disturbance is transformed into a constrained optimization problem to ensure that the state trajectory of the closed‐loop system is bounded under the action of external disturbances. The upper bound of restricted ‐gain is estimated by solving constrained optimization problems. Then, when the fault‐tolerant controller can be regard as the design variable, the optimization problems above are adjusted for solving control synthesis problems. Finally, the numerical example is given to verify the effectiveness of the design method.

Stability Analysis and Synthesis for Networked Systems With Noisy Sampling Intervals, Random Transmission Delays, and Packet Dropouts

In this article, we study the stability analysis and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> controller synthesis problems for networked systems with random transmission delays and packet dropouts that are subject to noisy sampling intervals. The sampling errors and transmission delays are described by an independent and identically distributed sequence of continuous random variables and the transmission delays are assumed to be smaller than the noisy sampling intervals. First, a closed-loop discrete-time stochastic framework for networked system that allows investigating noisy sampling intervals, random transmission delays and packet dropouts simultaneously is established. To render the discrete-time stochastic system suitable for stability analysis and synthesis, an equivalent yet tractable stochastic augmented model is then reformulated with the help of matrix exponential computation. Based on this augmented model, a stability condition involving the expectation of a nonlinear and random coupling matrix is derived. By recurring to the law of total expectation and Kronecker product operation, the expectation operation of the coupling matrix subject to high nonlinearity and multiple randomness is decoupled. Subsequently, a controller algorithm is designed such that the exponential mean-square stability of the original discrete-time stochastic system with a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance is guaranteed. Finally, an example is provided to illustrate the effectiveness of the controller design algorithm.

Synthesis of Optimal Control of Spacecraft Angular Momentum for Spatial Turn Taking into Account Energy Costs Using Quaternions

Background: In this paper, we propose solving the specific original problem of control synthesis of spacecraft attitude. Optimization of the control program is made with the use of a new criterion of quality that combines energy costs and duration of reorientation under restrictions on control (the presence of a time factor limits the duration of slew maneuver). Methods: The construction of optimal control for angular momentum change is based on the quaternion method and L.S. Pontryagin maximum principle. An analytical solution to the problem was obtained on the base of a differential equation relating the orientation quaternion and angular momentum of a spacecraft. Results: Key properties of the optimal solution are formulated in analytical form; the features of optimal motion are studied in detail. The control law is formulated in the form of explicit dependence between control and phase variables. In a case when the controlling torque is limited by the given restriction (at the beginning and end of a turn), analytical formulas have been written for the duration of braking and acceleration. Main relations which determine optimal values of parameters of the algorithm for control of angular momentum are given. Examples and results of mathematical modeling of spacecraft motion formed by optimal control were given. This data in addition to the theoretical descriptions illustrates the process of reorientation in evident form and demonstrates the practical feasibility of a designed method for control of angular momentum during spatial turn. Conclusion: The designed optimal algorithm of control of spacecraft motion improves the efficiency of spacecraft attitude system, and originates more economical performance of spacecraft during flight on orbit.

Fault-tolerant control of underactuated MSVs based on neural finite-time disturbance observer: An Event-triggered Mechanism

This paper deals with the problem of fault-tolerant control synthesis for underactuated marine surface vessels (MSVs) with dynamic uncertainties, external time-varying disturbances, and input saturation. In control design, with the help of neural networks (NNs) to reconstruct the dynamic uncertainty of the system, a novel nonlinear finite-time disturbance observer (FTDO) is established to reconstruct the compound uncertainty online, including unknown time-varying disturbances, approximation errors, bias faults and parts of loss-of-effectiveness (LOE) faults. Combining finite-time control (FTC) theory, event-triggered control (ETC) technology and backstepping design method, a new fault-tolerant control scheme for underactuated MSVs is designed. Based on the Lyapunov stability theory, it is proved that all signals in the closed-loop tracking system are bounded, and the tracking error of the underactuated MSVs converges to a small set near the origin in finite-time. The effectiveness of the proposed novel fault-tolerant control scheme is verified by computer simulation.

Stability Analysis and Controller Synthesis Using Single-Hidden-Layer ReLU Neural Networks

This paper presents algorithms to solve analysis and controller synthesis problems for dynamical systems modeled as a recurrent single-hidden-layer rectified linear unit neural network (ReLU NN), or equivalently, a piecewise affine dynamical system. Such models are interesting since they may arise through the use of modern machine learning methods for system identification, or as closed-loop solutions in certain classes of model predictive control (MPC) problems. A key idea in the proposed approach is to use piecewise affine Lyapunov functions parametrized as ReLU neural networks, and similarly parametrized controllers. This compatible representation between the Lyapunov function and the dynamics simplifies the automation of analysis of and controller synthesis for learned models. Our method of verifying a candidate Lyapunov function is faster than methods based on mixed integer programming. We ‘learn’ controllers and Lyapunov functions using both weight updates and network architecture search, without gradients. We demonstrate the proposed algorithm on examples involving learned models, explicit MPC controllers, and constrained controller synthesis.

About one problem of optimal control synthesis

This paper tackles the problem of characterizing the natural class, or Riccati rule space, of solutions to a specific equation. Despite the significant theoretical and practical implications, there is limited research exploring the application of spectral decomposition of non-self-adjoint differential operators to explicitly solve this nonlinear Riccati equation. Therefore, investigating operator Riccati equations holds potential to validate the dynamic programming method and address the synthesis problem.

Reliable quantized sampled‐data LFC for semi‐Markov jump interconnected multi‐area power systems: Dealing with incomplete TRs

AbstractThis article aims to address the reliable quantized sampled‐data load frequency control (LFC) synthesis problem for semi‐Markov jump interconnected multi‐area power systems (IMAPSs) suffering from incomplete transition rates (TRs) and actuator failures. Primarily, the semi‐Markov process configurated with incomplete TRs is utilized to model the structural and coefficient switchings of IMAPSs, which enables a more precise representation of reality and a wider range of application. Subsequently, in order to reduce the control cost and facilitate data processing, an aperiodic quantization sampling mechanism is introduced. Furthermore, a mode‐dependent and comprehensive actuator faulty model is scheduled to portray various stochastically occurring actuator failures, which is more in line with practical application. Then, by fully utilizing the state information of sampled intervals, a two‐sided looped functional with some matrices that are not required to be strictly positive is constructed to enhance the flexibility. In what follows, on the basis of stochastic analysis technique and looped functional, sufficient conditions with two scenarios are established in the form of linear matrix inequalities (LMIs) to guarantee the stochastic stability with an performance of the resultant semi‐Markov jump IMAPSs. Finally, the effectiveness of the proposed control synthesis methodology is validated through a numerical simulation.

Online control synthesis for uncertain systems under signal temporal logic specifications

Signal temporal logic (STL) formulas have been widely used as a formal language to express complex robotic specifications, thanks to their rich expressiveness and explicit time semantics. Existing approaches for STL control synthesis suffer from limited scalability with respect to the task complexity and lack of robustness against the uncertainty, for example, external disturbances. In this paper, we study the online control synthesis problem for uncertain discrete-time systems subject to STL specifications. Different from existing techniques, we propose an approach based on STL, reachability analysis, and temporal logic trees. First, based on a real-time version of STL semantics, we develop the notion of tube-based temporal logic tree (tTLT) and its recursive (offline) construction algorithm. We show that the tTLT is an under-approximation of the STL formula, in the sense that a trajectory satisfying a tTLT also satisfies the corresponding STL formula. Then, an online control synthesis algorithm is designed using the constructed tTLT. It is shown that when the STL formula is robustly satisfiable and the initial state of the system belongs to the initial root node of the tTLT, it is guaranteed that the trajectory generated by the control synthesis algorithm satisfies the STL formula. We validate the effectiveness of the proposed approach by several simulation examples and further demonstrate its practical usability on a hardware experiment. These results show that our approach is able to handle complex STL formulas with long horizons and ensure the robustness against the disturbances, which is beyond the scope of the state-of-the-art STL control synthesis approaches.

A negative imaginary lemma and state-feedback negative imaginary synthesis for commensurate fractional-order systems

This article focuses on negative imaginariness of commensurate fractional-order linear time-invariant systems. Enlightened by the generalized Kalman–Yakubovic–Popov lemma, minimal state-space realization based necessary and sufficient conditions are developed to characterize negative imaginariness of commensurate fractional-order systems for both 0<α<1 and 1<α<2 cases. Meanwhile, the problem of negative-imaginary state-feedback controller synthesis for fractional-order systems is addressed. A fractional-order viscoelastic system, a fractional-order RLC circuit network and a numerical example are used as illustrative examples to demonstrate the main developed theory.

T-S fuzzy model based event-triggered change control for product and supply chain systems

Product and supply chain systems are a class of complex nonlinear systems due to the uncertain demand fluctuation and unexpected events, and Takagi–Sugeno (T-S) fuzzy systems are suitable for modelling such product and supply chain systems with different production rates. According to the uncertainty customer demands, different subsystems which describe the corresponding production-inventory processes are orchestrated by the T-S fuzzy rule. Change control is designed to handle the supply chain blocking and event breaking caused by uncertain factors. Then, sufficient conditions are given such that the exponential stability with a prescribed bullwhip effect attenuation level can be guaranteed, and the control synthesis problem is transformed into the solvability of linear matrix inequalities for the product and supply chain systems. An example is introduced to verify the validness and efficiency of the proposed approach.

HMM-based asynchronous synthesis for Interval type-2 fuzzy semi-Markovian jump systems with its applications: A bilateral looped functional methodology

This article mainly addresses the hidden-Markovian model-based asynchronous sampled-data control (SDC) synthesis problem for interval type-2 (IT2) fuzzy semi-Markovian jump systems (S-MJSs) with stochastic actuator failures (SAFs). Herein, the asynchronous factor resulted from twofold: (1) hidden Markovian model (HMM) and (2) imperfectly matched membership functions (IMMFs) between the original plant and the controller. Aiming at approximating the nonlinearity along with randomness with high accuracy and broadening practical application background, the IT2 fuzzy S-MJSs associated with IMMFs is established. At the beginning, a HMM is borrowed to portray the discrepancy between the corresponding plant and the desired controller. Additionally, in comparison with majority of some previous sampled-data control mechanisms, a intrinsic lag signal and gain fluctuations are purposefully introduced in the double asynchronous control strategy, which exploited initially to settle the control synthesis issue. By considering the information of sampling intervals x(tk)→x(t)→x(tk+1) along with x(tk−τ)→x(t−τ)→x(tk+1−τ) adequately, a fresh looped Lyapunov functional called bilateral looped Lyapunov functional is innovatively constructed. Whereafter, by suitably incorporating with the skills of stochastic analysis and relaxed integral inequality (including generalized free matrix-based integral inequality and canonical Bessel–Legendre integral inequality) together with symmetric slack matrices, numerically testable sufficient conditions are established in the form of a group of feasible multi-parameterized linear matrix inequalities (M-PLMIs). The theoretical developments are implies the hierarchy character, and the conservatism decreases with the increase of the degree of hierarchy. Conclusively, the design methodology brought forward herein is demonstrated by two practical examples, which show the potency and practicability.

Adaptive Synthesized Control for Solving the Optimal Control Problem

The development of artificial intelligence systems assumes that a machine can independently generate an algorithm of actions or a control system to solve the tasks. To do this, the machine must have a formal description of the problem and possess computational methods for solving it. This article deals with the problem of optimal control, which is the main task in the development of control systems, insofar as all systems being developed must be optimal from the point of view of a certain criterion. However, there are certain difficulties in implementing the resulting optimal control modes. This paper considers an extended formulation of the optimal control problem, which implies the creation of such systems that would have the necessary properties for its practical implementation. To solve it, an adaptive synthesized optimal control approach based on the use of numerical methods of machine learning is proposed. Such control moves the control object, optimally changing the position of the stable equilibrium point in the presence of some initial position uncertainty. As a result, from all possible synthesized controls, one is chosen that is less sensitive to changes in the initial state. As an example, the optimal control problem of a quadcopter with complex phase constraints is considered. To solve this problem, according to the proposed approach, the control synthesis problem is firstly solved to obtain a stable equilibrium point in the state space using a machine learning method of symbolic regression. After that, optimal positions of the stable equilibrium point are searched using a particle swarm optimization algorithm using the source functional from the initial optimal control problem statement. It is shown that such an approach allows for generating the control system automatically by computer, basing this on the formal statement of the problem and then directly implementing it onboard as far as the stabilization system has already been introduced.

Abstraction of Continuous-Time Systems Based on Feedback Controllers and Mixed Monotonicity

In this paper, we consider the problem of the computation of efficient symbolic abstractions for continuous-time control systems. The new abstraction algorithm builds symbolic models with the same number of states but fewer transitions in comparison to the one produced by the standard algorithm. At the same time, the new abstract system is at least as controllable as the standard one. The proposed algorithm is based on the solution of a region-to-region control synthesis problem. This solution is formally obtained using the theory of viscosity solutions of the dynamic programming equation and the theory of differential equations with discontinuous righthand side. In the new abstraction algorithm, the symbolic controls are essentially the feedback controllers that solve this control synthesis problem. The improvement in the number of transitions is achieved by reducing the number of successors for each symbolic control. For a certain class of control systems, with a suitable set of discretization parameters, the new algorithm may even produce deterministic abstract systems or systems with a singleton input alphabet. The approach is illustrated by examples that compare the two abstraction algorithms.

Comparative Analysis of H2 and H∞ Robust Control Design Approaches for Dynamic Control Systems

This paper discusses using H2 and H∞ robust control approaches for designing control systems. These approaches are applied to elementary control system designs, and their respective implementation and pros and cons are introduced. The H∞ control synthesis mainly enforces closed-loop stability, covering some physical constraints and limitations. While noise rejection and disturbance attenuation are more naturally expressed in performance optimization, which can represent the H2 control synthesis problem. The paper also applies these two methodologies to multi-plant systems to study the stability and performance of the designed controllers. Simulation results show that the H2 controller tracks a desirable closed-loop performance, while the H∞ controller guarantees robust stability for the closed-loop system. The validation of the techniques is demonstrated through the robust and performance gamma index, where the H∞ controller achieved a robust gamma index of 0.8591, indicating good robustness and the H2 controller achieved a performance gamma index of 2.1972, indicating a desirable performance. The robust control toolbox of MATLAB is used for simulation purposes. Overall, the paper shows that selecting a suitable, robust control strategy is crucial for designing effective control systems, and the H2 and H∞ robust control approaches are viable options for achieving this goal.

Logarithmically completely monotonic rational functions

This paper studies the class of logarithmically completely monotonic (LCM) functions. These functions play an important role in characterizing externally positive linear systems which find applications in important control problems such as non-overshooting reference tracking. Conditions are proposed to ensure a rational function is LCM, a result that enables the known space of linear continuous-time externally positive systems to be enlarged and an efficient and optimal pole-placement procedure for the monotonic tracking controller synthesis problem to be developed. The presented conditions are shown to be less conservative than existing approaches whilst being computationally tractable.

Finite-time contractive boundedness analysis and controller synthesis for interval positive linear systems under [formula omitted]-gain performance

This paper investigates the finite-time contractive boundedness (FTCB) analysis and controller synthesis problems for interval positive linear systems under L1-gain performance. Firstly, both FTCB and L1-gain FTCB definitions are extended to positive systems with disturbances. Subsequently, FTCB and L1-gain FTCB criteria are established for the considered positive systems by using the linear copositive Lyapunov function approach. Furthermore, the state feedback controller is designed to guarantee the positivity and L1-gain FTCB of the resulting closed-loop systems, and a convex optimization-based algorithm is given to derive the optimal L1-gain and controller parameter. Finally, a numerical example and an application example about pest control are used to verify the effectiveness of the proposed conditions.