Design of rules for in-flight non-parametric tuning of PID controllers for unmanned aerial vehicles

Designing high performance controllers for multirotors is a rigorous task that is often solved by trial and error approach. Trial and error tuning usually results in non-optimal controller parameters. Tuning controllers based on the existing quadrotor models would result in poor performance of quadrotors due to simplifications and inaccuracies in the underlying models. In this paper optimal tuning rules for quadrotor attitude dynamics are designed, which guarantees near-optimal performance and robustness. A single in-flight run of the Modified Relay Feedback Test that takes only few seconds with guaranteed stability is enough to have near-optimal tuning of the controller. The designed tuning rule is tested experimentally in-flight on a custom-built quadrotor. The results showed significant advantages in performance and robustness due to the proposed approach.     

Quadratic stability analysis and robust distributed controllers design for uncertain spatially interconnected systems

This paper is concerned with the quadratic stability analysis and robust distributed controllers design of both continuous-time and discrete-time uncertain spatially interconnected systems (USISs), where uncertainties are modeled by linear fractional transformation (LFT). The well-posedness, quadratic stability, and contractiveness of USISs are properly defined for the first time. A sufficient condition employing the given system matrices is established to check the well-posedness, quadratic stability and contractiveness. This condition is simpler than the existing conditions based on the decomposition of system matrices. Based on the new condition derived, a sufficient condition is given for the existence of robust distributed controllers and a constructive method is then presented for the design of robust distributed controllers. The advantage of the proposed constructive approach is that it employs the given system matrices while the existing methods conduct the bilinear transformation on these matrices when design controllers, and consequently, the constructive approach in this paper is computationally more efficient than the existing methods. Several examples are included to demonstrate the simplicity, efficiency and applicability of the derived theoretical results.     

Adaptive distributed convex optimization for multi-agent and its application in flocking behavior

This paper studies adaptive optimization problem of continuous-time multi-agent systems. Multi-agents with second-order dynamics are considered. Each agent is equipped with a time-varying cost function which is known only to an individual agent. The objective is to make multi-agents velocities minimize the sum of local functions by local interaction. First, a distributed adaptive algorithm is presented, in which each agent depends only on its own velocity and neighbors velocities. It is indicated that all agents can track the optimal velocity. Then we apply the distributed adaptive algorithm to flocking of multi-agents. It is proved that all agents can track the optimal trajectory. The agents will maintain connectivity and avoid the inter-agent collision. Finally, two simulations are included to illustrate the results.     

PID controller tuning rules for integrating processes with varying time-delays

This paper discusses PID controller tuning for integrating processes with varying time-delays. Most of the existing tuning rules for the first-order lag plus integrator plus delay (FOLIPD) processes that we mainly focus on have the same general structure, and the properties of these rules are discussed in conjunction with varying time-delays. The analysis leads to novel tuning rules, where the maximum amplitude of an arbitrarily varying time-delay can be given as a parameter, which makes the use of the rules attractive in several applications. We will also extend the analysis to integrating processes with second-order lag and apply the design guidelines for a networked control application. In addition, we propose a novel tuning method that optimizes the closed-loop performance with respect to certain robustness constraints while also providing robustness to delay variance via jitter margin maximization. Further, we develop new PID controller tuning rules for a wide range of processes based on the proposed method. The new tuning rules are discussed in detail and compared with some of the recently published results. The work was originally motivated by the need for robust but simultaneously well-performing PID parameters in an agricultural machine case process. We also demonstrate the superiority of the proposed tuning rules in the case process.     

A noise-filtering event generator for PIDPlus controllers

In this paper we propose a new event generator, which has strong noise-filtering capabilities, to be used in event-based control systems with a PIDPlus controller. An approximate frequency analysis is performed in order to characterize the event generator system and tuning guidelines are provided for its design parameter. Simulation and experimental results obtained with a laboratory setup demonstrate the effectiveness of the methodology in providing a satisfactory performance related to set-point and load disturbance step responses with a total variation that is significantly reduced with respect to the standard cases.     

Optimal non-parametric tuning of PID controllers based on classification of shapes of oscillations in modified relay feedback test

In this work, tuning rules of the PID controller have been developed by categorizing a system's response into distinct classes. The classes are formed using the shapes of the test oscillations induced by the system under the Modified Relay Feedback Test (MRFT) produced by specific system models. It is proposed that a physical system can be categorized into one of the proposed classes and thus the tuning rules for a particular class can apply to any kind of system from this class. The idea of producing tuning rules that are based on the shape of the oscillations induced in the loop containing the process comes from the observations that oscillatory responses of physical systems reveal just a few different shapes depending on system dynamics. For applying the developed optimal tuning rules for an arbitrary system, first, certain system characteristics are determined using a priori knowledge of the class model. Then the system's response with the application of the MRFT is examined to classify the oscillation waveform/shape. In this work, such classification is carried out using a cross-correlation algorithm. Finally, a class tuning rules are applied.     

A robust unscented transformation for uncertain moments

This paper proposes a robust version of the unscented transform (UT) for one-dimensional random variables. It is assumed that the moments are not exactly known, but are known to lie in intervals. In this scenario, the moment matching equations are reformulated as a system of polynomial equations and inequalities, and it is proposed to use the Chebychev center of the solution set as a robust UT. This method yields a parametrized polynomial optimization problem, which in spite of being NP-Hard, can be relaxed by some algorithms that are proposed in this paper.     

Design of non-parametric process-specific optimal tuning rules for PID control of flow loops

The problem of designing optimal process-specific rules for non-parametric tuning is undertaken in the paper. It is shown that producing non-parametric process-specific optimal tuning rules for PID controllers leads to the problem that can be characterized as optimization under uncertainty. This happens due to the fact that tuning rules, unlike tuning constants, are produced not for a particular process or plant model but for a set of models from a certain domain. The novelty of the proposed approach is that the problem of obtaining optimal tuning rules for a flow process is formulated and solved as a problem of optimization of an integral performance criterion parametrized through values that define the domain of available process models. The considered non-parametric tuning assumes the use of the modified relay feedback test (MRFT) recently proposed in the literature. It allows one to tune the PID controller satisfying the requirements to gain or phase margins that is achieved through coordinated selection of tuning rules and test parameters. This approach constitutes a holistic approach to tuning. In the present paper, optimal tuning rules coupled with MRFT, for flow loops, are proposed. Final results are presented in the form of tables containing coefficients of optimal tuning rules for the PI controller, obtained for a number of specified gain margins. The produced non-parametric tuning rules well agree with the practice of loop tuning.     

Coupling and disturbance characterization based robust control for manned submersibles

A novel coupling and disturbance characterization based control (CDC-BC) scheme is investigated for manned submersibles with strong couplings and disturbances. Firstly, the coupling characterization index (CCI) and disturbance characterization index (DCI) are defined to indicate whether the couplings and disturbances harm or benefit the manned submersible system. Then, the coupling and disturbance characterization based control (CDC-BC) method is developed to eliminate the detrimental couplings and disturbances, as well as to retain the beneficial ones. Moreover, it is also proved that the tracking errors of the closed-loop system will converge to zero asymptotically. Finally, some simulation results demonstrate the effectiveness of the proposed control scheme.     

A novel adaptive robust control approach for underactuated mobile robot

Underactuated mobile robot (UMR) is a typical nonlinear underactuated system with nonholonomic and holonomic constraints. Based on the model of UMR, we propose a novel adaptive robust control to control the UMR and compensate the uncertainties from the view of constraint-following. The uncertainties, which are (possibly fast) time-varying and bounded, include modeling error, initial condition deviation, friction force and other external disturbances. However, the bounds are unknown. To estimate the bounds of the uncertainties, we design an adaptive law which is of leakage type. The uniform boundedness and the uniform ultimate boundedness of the proposed control are verified by Lyapunov method. Furthermore, the effectiveness of the control is shown via numerical simulation of a case.     

Nash Bargaining Solution based rendezvous guidance of unmanned aerial vehicles

This paper addresses a finite-time rendezvous problem for a group of unmanned aerial vehicles (UAVs), in the absence of a leader or a reference trajectory. When the UAVs do not cooperate, they are assumed to use Nash equilibrium strategies (NES). However, when the UAVs can communicate among themselves, they can implement cooperative game theoretic strategies for mutual benefit. In a convex linear quadratic differential game (LQDG), a Pareto-optimal solution (POS) is obtained when the UAVs jointly minimize a team cost functional, which is constructed through a convex combination of individual cost functionals. This paper proposes an algorithm to determine the convex combination of weights corresponding to the Pareto-optimal Nash Bargaining Solution (NBS), which offers each UAV a lower cost than that incurred from the NES. Conditions on the cost functions that make the proposed algorithm converge to the NBS are presented. A UAV, programmed to choose its strategies at a given time based upon cost-to-go estimates for the rest of the game duration, may switch to NES finding it to be more beneficial than continuing with a cooperative strategy it previously agreed upon with the other UAVs. For such scenarios, a renegotiation method, that makes use of the proposed algorithm to obtain the NBS corresponding to the state of the game at an intermediate time, is proposed. This renegotiation method helps to establish cooperation between UAVs and prevents non-cooperative behaviour. In this context, the conditions of time consistency of a cooperative solution have been derived in connection to LQDG. The efficacy of the guidance law derived from the proposed algorithm is illustrated through simulations.     

Mixed order robust adaptive control for general linear time invariant systems

We provide a solution to the adaptive control problem of an unknown linear system of a given derivation order, using a reference model or desired poles defined in a possibly different derivation order and employing continuous adjustment of parameters ruled by possibly another different derivation order. To this purpose, we present an extension for the fractional settings of the Bezout’s lemma and gradient steepest descent adjustment. We analyze both the direct and indirect approaches to adaptive control. We discuss some robustness advantages/disadvantages of the fractional adjustment of parameters in comparison with the integer one and, through simulations, the possibility to define optimal derivation order controllers.     

Local exponential stabilization via boundary feedback controllers for a class of unstable semi-linear parabolic distributed parameter processes

This paper addresses the problem of local exponential stabilization via boundary feedback controllers for a class of nonlinear distributed parameter processes described by a scalar semi-linear parabolic partial differential equation (PDE). Both the domain-averaged measurement form and the boundary measurement form are considered. For the boundary measurement form, the collocated boundary measurement case and the non-collocated boundary measurement case are studied, respectively. For both domain-averaged measurement case and collocated boundary measurement case, a static output feedback controller is constructed. An observer-based output feedback controller is constructed for the non-collocated boundary measurement case. It is shown by the contraction semigroup theory and the Lyapunov’s direct method that the resulting closed-loop system has a unique classical solution and is locally exponentially stable under sufficient conditions given in term of linear matrix inequalities (LMIs). The estimation of domain of attraction is also discussed for the resulting closed-loop system in this paper. Finally, the effectiveness of the proposed control methods is illustrated by a numerical example.     

Data-driven subspace predictive control: Stability and horizon tuning

Data-driven Subspace Predictive Control (SPC) is an advanced model-free process control strategy in the presence of system constraints. Efficient implementation of SPC requires appropriate tuning of the controller horizons, which are called Prediction Horizon and Control Horizon. This tuning is a critical step to guarantee the SPC closed-loop stability and to enhance the closed-loop performance and robustness. In this paper we propose an optimal tuning method for unconstrained SPC, which can guarantee stability, computational efficiency and optimality of the unconstrained SPC closed-loop system and is applicable to non-minimum phase open-loop stable or marginally stable systems. Derivation of general form of closed-loop transfer function for unconstrained SPC, and providing a necessary and sufficient condition of the closed-loop stability is the primary contribution of this work. In addition, the stability analysis enabled us to propose an algorithm to determine the shortest-feasible-prediction-horizon and the feasible range of prediction horizon. Consequently, these results are used in proposing a new algorithm to determine the SPC horizons in optimal manner. Simulation results illustrate effectiveness and importance of our proposed stability analysis and horizons tuning algorithm for unconstrained SPC.     

A nonlinear PID controller based novel maximum power point tracker for PV systems

This paper proposes a novel application of Nonlinear Proportional-Integral-Derivative (NPID) controller to effectively attain Maximum Power Point Tracking (MPPT) in Photovoltaic (PV) systems. The proposed controller is based on the basic structure of the PID controller wherein, its integral term gain is varied at run time according to instantaneous error. The performance of the NPID controller is assessed in terms of undershoot, settling time and ripple which have been evaluated under varying realistic irradiation and temperature profiles. The Teaching Learning Based Optimization (TLBO) tuned NPID controller is found to be superior to TLBO tuned PID, Perturb and Observe and Incremental Conductance classical MPPT methods for all the considered environmental profiles. Therefore, based on the presented comprehensive investigations, it is concluded that the proposed NPID controller is a promising MPPT technique.     

TS fuzzy robust L1 control for nonlinear systems with persistent bounded disturbances

In this paper, a novel technique for Takagi–Sugeno (TS) model-based robust L₁ controller design of nonlinear systems is proposed. Two synthesis methods based on quadratic and non-quadratic Lyapunov functions are considered. To design the robust stabilizing controller, a new approach for deriving sufficient conditions associated with the L₁ performance criterion in terms of strict linear matrix inequality is proposed. This novel technique results in less pre-chosen scalar design variables and calculation burden. Furthermore, deriving the controller synthesis conditions via a non-quadratic Lyapunov function (NQLF) relaxes the obtained conditions. Therefore, the proposed approaches not only efficiently minimize the effect of persistent bounded disturbance, but also are applicable for wider classes of TS systems. Furthermore, some new lemmas are proposed to facilitate strict LMI formulation and to provide more degrees of freedom. Finally, several numerical and practical examples are presented to show the merits of this paper.     

A robust global approach for LPV FIR model identification with time-varying time delays

Robust identification of the linear parameter varying (LPV) finite impulse response (FIR) model with time-varying time delays is considered in this paper. A robust observation model based on Laplace distribution is established to deal with the output data contaminated with the outliers, which are commonly existed in modern industries. A Markov chain model is utilized to model the correlation between the time delays as they do not simply change randomly in reality. A transition probability matrix and an initial probability distribution vector are used to govern the switching mechanism of the time delays. Since it is difficult to optimize the complex log likelihood function directly, the derivations of the proposed algorithm are performed under the framework of Expectation-Maximization (EM) algorithm. A numerical example and a chemical process are utilized to verify the effectiveness of the proposed approach.