Polynomial Control Research Articles

Invariant sets for a class of nonlinear control systems tractable by symbolic computation

A set S is said to be controlled invariant with respect to a control system if a state feedback law exists such that the closed loop system has S as an invariant set. In the present paper we generalise results on input-affine polynomial control systems and algebraic varieties (i.e. sets described by the zeros of polynomial equations) considered in Zerz and Walcher (2012) to an extended class of vector fields. More precisely, we consider vector fields of the form f = F ° h, where F is a polynomial vector and h is a continuously differentiable function with certain (algebraic) properties, as well as sets Vh as the preimages of varieties under h. We will see that for example polynomial expressions in sine and cosine satisfy the mentioned properties. The main advantage of the considered function class is that it is accessible to symbolic computation. We give computational methods (based on the theory of Gröbner bases) to decide the controlled invariance of Vh.

Control of Unstable Systems Using a 7 DoF Robotic Manipulator

Robotic manipulators are widely used in industrial applications, and their rigidity and flexibility are very important factors during their deployment. However, their usage is not limited to repetitive point-to-point tasks and can be used for more real-time control of various processes. This paper uses a 7-degrees-of-freedom manipulator to control an unstable system (Ball and Plate) as a proof of concept. The Ball and Plate system is widely used for testing algorithms designed for unstable systems, and many recent works have dealt with robotic manipulators as a control motion system. Robots are not usually used to control unstable systems, but bipedal robots are an exception. This paper aims to design a controller capable of stabilizing an unstable system with solid robustness while keeping actuator action values as low as possible because these robots will be indented to work for a prolonged time. An algorithm for an LQ polynomial controller is described and designed, and the whole setup is tested for ball stabilization in the center. The results show that the designed controller stabilizes the ball even with large external and internal disturbances while keeping the controller effort as low as possible.

New Nonlinear Control Based on Polynomial Approach for Islanded DC Microgrid Robustness and Voltage Stability

This paper aims at developing a novel robust polynomial decentralized control approach for an islanded hybrid DC MicroGrid (MG) with saturation constraint. The investigated MG system consists of a PV unit, a battery and a SuperCapacitor (SC). Each source is controlled by a DC-DC power converter. The challenge is to design a new control approach to enhance the robustness and the stability of the DC MG subject to high nonlinear variations. The main nonlinear control objectives are to: 1) maintain the DC bus voltage to a reference value, 2) ensure the power balance in the system, and 3) warranty the system robustness against perturbations and parametric uncertainties. The Polynomial Control (PC) theory offers an adequate framework to meet the required performances. The resulting control algorithms are simple, and do not need important calculating resources. An extended version of the PC controller is also proposed to make the system Fault Tolerant (FT) w.r.t severe eventual faults, such as a short circuit case at the DC bus level. The design conditions of the developed polynomial controllers are solved using the Sum Of Squares (SOS) approach while verifying the saturation constraint. Stability conditions are ensured based on polynomial Lyapunov functions. Finally, simulations are carried out on MATLAB/Simulink to prove the pertinence and the robustness of the developed control architecture while considering variable loads and solar irradiance. Results are compared to model-based feedback linearization control and sliding mode control techniques.

Polynomial Recurrent Neural Network-Based Adaptive PID Controller With Stable Learning Algorithm

This paper introduces a novel structure of a polynomial weighted output recurrent neural network (PWORNN) for designing an adaptive proportional—integral—derivative (PID) controller. The proposed adaptive PID controller structure based on a polynomial weighted output recurrent neural network (APID-PWORNN) is introduced. In this structure, the number of tunable parameters for the PWORNN only depends on the number of hidden neurons and it is independent of the number of external inputs. The proposed structure of the PWORNN aims to reduce the number of tunable parameters, which reflects on the reduction of the computation time of the proposed algorithm. To guarantee the stability, the optimization, and speed up the convergence of the tunable parameters, i.e., output weights, the proposed network is trained using Lyapunov stability criterion based on an adaptive learning rate. Moreover, by applying the proposed scheme to a nonlinear mathematical system and the heat exchanger system, the robustness of the proposed APID-PWORNN controller has been investigated in this paper and proven its superiority to deal with the nonlinear dynamical systems considering the system parameters uncertainties, disturbances, set-point change, and sensor measurement uncertainty.

Imitation learning of stabilizing policies for nonlinear systems

There has been a recent interest in imitation learning methods that are guaranteed to produce a stabilizing control law with respect to a known system. Work in this area has generally considered linear systems and controllers, for which stabilizing imitation learning takes the form of a biconvex optimization problem. In this paper it is demonstrated that the same methods developed for linear systems and controllers can be readily extended to polynomial systems and controllers using sum of squares techniques. A projected gradient descent algorithm and an alternating direction method of multipliers algorithm are proposed as heuristics for solving the stabilizing imitation learning problem, and their performance is illustrated through numerical experiments.

A Polynomial Decentralized Controller Design for a Large-Scale Nonlinear System: SOS Approach

This paper proposes a novel approach for designing a decentralized controller to stabilize the large-scale nonlinear system. Unlike the previous studies, the polynomial system framework is employed to model the nonlinear large-scale system to decrease not only the modeling error but also the complexity and computational load significantly. Especially, in case the large-scale nonlinear systems consist of non-polynomial forms (such as sine, cosine, and so on), synthesizing the controller for this system becomes much more challenging. By putting non-polynomial terms inside the matrices, a new approach for designing a decentralized polynomial controller is presented to eliminate the impacts of nonlinear terms and stabilize the system. Based on Lyapunov methodology, the conditions for synthesizing a decentralized polynomial controller expressed under the framework of Sum-of-Square (SOS) are derived in the main theorems. Finally, the effectiveness and superiority of the proposed method are demonstrated in two illustrative examples.

ELECTRIC VEHICLE INTEGRATED WITH PMSM AND REGENERATIVE BRAKING SYSTEM SPEED EVALUATION BASED ON DIVERSE CONTROL STRATEGIES

The choice of a vehicle speed controller for a permanent magnet synchronous driven battery electric vehicle (BEV), integrated with a regenerative braking system (RBS), plays a pertinent role in how far a vehicle can travel. While many studies have implemented varied control strategies, the proportional-integral (PI) based controller is predominately studied and practically used. Generally, the performance of PI controller compared to Modeled Predictive Controller; Reference Tracking Signal Controller Based on Generated Polynomial and Reference Tracking Signal Controller Based on Controlled Polynomial in a permanent magnet synchronous motor (PMSM) driven battery electric vehicle integrated with RBS speed control based on field-oriented control mechanism is lacking. In order to address that shortcoming, this paper aims to investigate the performance of a PI controller compared to generalized predictive controllers based on the difference between a reference speed and a controlled speed of the battery electric vehicle. The performance for each controller considered was evaluated in terms of the state of charge (SOC), R^2, slip ratio, minimum error metrics, and peak speed using real-time driving scenario factoring braking, throttling, wind speed, and cruise effects for 15 s. The results show that the PI controller outperforms the other controllers on most metrics. Except for the peak SOC, slip ratio, and peak speed, the Reference Tracking Signal Controller Based on Generated Polynomial seems to be better, although with a drawback of the highest error metrics. Generally, all the controllers yielded excellent regenerative braking effects in SOC and effective control of the speed trajectory.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

Polynomial Control on Weighted Stability Bounds and Inversion Norms of Localized Matrices on Simple Graphs

The (un)weighted stability for some matrices on a graph is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Banach algebras of matrices with certain off-diagonal decay is of great importance in many mathematical and engineering fields, and its inverse-closed property can be informally interpreted as localization preservation. Let $${{\mathcal {B}}}(\ell ^p_w)$$ be the Banach algebra of bounded linear operators on the weighted sequence space $$\ell ^p_w$$ on a graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a connected simple graph are inverse-closed in $${{\mathcal {B}}}(\ell ^p_w)$$ for all $$1\le p<\infty $$ and Muckenhoupt $$A_p$$ -weights w, and the Beurling norm of the inversion of a matrix A is bounded by a bivariate polynomial of the Beurling norm of the matrix A and the operator norm of its inverse $$A^{-1}$$ in $${{\mathcal {B}}}(\ell ^p_w)$$ .

Iterative stability analysis for general polynomial control systems

The main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

Controllability of Hypergraphs

In this paper, we develop a notion of controllability for hypergraphs via tensor algebra and polynomial control theory. Inspired by uniform hypergraphs, we propose a new tensor-based multilinear dynamical system representation, and derive a Kalman-rank-like condition to determine the minimum number of control nodes (MCN) needed to achieve controllability of even uniform hypergraphs. We present an efficient heuristic to obtain the MCN. MCN can be used as a measure of robustness, and we show that it is related to the hypergraph degree distribution in simulated examples. Finally, we use MCN to examine robustness in real biological networks.

Isogeometric MITC shell

This paper proposes an isogeometric formulation of the Reissner–Mindlin shell, using the Mixed Interpolation of Tensorial Components (MITC) technique to alleviate shear locking and membrane locking. Instead of over each Non-Uniform Rational B-Spline (NURBS) element, kinematics of the shell is directly formulated on the entire NURBS patch, with its displacements decoupled to the motions of control points and of director vectors tied to the control points. Since control points are in general not located on the surface, those control-point director vectors are interpolated from normal vectors at a set of pre-defined integration points on the patch. The assumed in-plane membrane strain field and the assumed transverse shear strain field are built as linear combinations of NURBS basis functions with degrees lower than those employed by the displacement interpolation, and tied to the original covariant strain fields at a set of well-selected tying points. Integration points for the definition of control-point director vectors and tying points of the assumed covariant strain fields are provided by the optimal and reduced macro-element quadrature rules, respectively. In this way, the classical MITC technique is fully incorporated into the isogeometric framework with arbitrary polynomial orders or patch configurations. Thanks to the high smoothness of NURBS basis functions, the geometry-preserving nature of isogeometric transformation, and the locking-resistive capability of MITC technique, the isogeometric MITC shell formulation shows superior convergence behavior, minimal geometrical error, and good robustness against patch distortion. In particular, its convergence property is insensitive to NURBS polynomial order or control point density. These advantages are demonstrated through a number of well-established benchmark problems, validating the proposed isogeometric MITC formulation being an accurate and efficient solution for linear analysis of shell structures.

Digital control of structurally unstable power facilities using a polynomial approach

The problem of improving the quality of control of structurally unstable power facilities is viewed based on a polynomial controller with a dynamic change in settings using the inverse pendulum on a carriage as an example. The mathematical model of the control object was built; a polynomial controller was synthesized using the method of symbolic computation of systems of differential equations with an additional input for specifying the geometric mean root from the outside. Reverse pendulum model on a carriage and a model of regulators were created using the MATLAB Simulink visual modeling environment to analyze the quality and parameters of the created control system. As a result of modeling, direct dependence of the position setting time of pendulum on the value of the geometric mean root was obtained which confirmed the possibility of external control of the speed of the control system. The possibility to control the control actions level to prevent saturation of executive devices was investigated and confirmed. In fact, a system of external parametric control of the processes dynamics was created and the described regulator does not require adaptation and training.

Verified Polynomial Controller Synthesis for Disturbed Nonlinear Systems

Abstract We introduce a new control synthesis approach to solve reach-avoid problems occurring in many fields, such as autonomous driving or robot motion planning. Our control approach steers a set of initial states as close as possible to a given target state while provably satisfying constraints on both inputs and states. We compute a control law consisting of a state-dependent, piecewise constant feedforward controller and a continuous feedback controller. The feedforward controller can be polynomial in the state and steers the initial set of states as close as possible to the target state for an undisturbed system, while the feedback controller minimizes the effect of disturbances and abstraction errors. Compared to other formal synthesis approaches, our approach can be verified in polynomial time and generates nonlinear feedforward control laws. The achievable control performance is demonstrated by two use cases.

Finite-time anti-synchronization for delayed inertial neural networks via the fractional and polynomial controllers of time variable

<abstract> This paper focuses on the finite-time anti-synchronization for a class of delayed master-slave inertial neural networks. By means of using the property of quadratic inequality of one variable and designing the fractional and polynomial controllers of time variable, two sufficient conditions to assure the finite-time anti-synchronization for the master-slave delayed inertial neural networks are established. Our controllers designed related to time variable t and the study method on the finite-time anti-synchronization are different from these in the existing papers. </abstract>

Offline Multiobjective Optimization for Fast Charging and Reduced Degradation in Lithium-Ion Battery Cells Using Electrochemical Dynamics

The rapid charging of lithium ion battery cells while minimizing degradation is a key challenge in battery management. Offline optimal control frameworks employing physics-based models provide a first-principles approach to this problem that provides critical insights and benchmarks for real-time oriented online optimization algorithms. In this letter, we approach the optimal control problem of minimizing both charging time and degradation of a NMC battery using the single shooting optimization framework, employing a coupled electrochemical-thermal-aging model. The aging mechanism used captures growth of the SEI layer, solvent diffusion. The effectiveness of the single shooting method using a piecewise polynomial control and the low sensitivity of the method to chosen numerical parameters in the multi-objective problem is illustrated. The tradeoff between charging time and degradation is investigated numerically.

Neural network predictive control of vibrations in tall structure: An experimental controlled vision

This article presents the use of neural network predictive controller as a novel technique for vibration control of tall structures employing single degree of freedom active tuned mass damper (ATMD). Additionally, the proposed technique is compared with two modern control techniques: pole-placement controller and adaptive neuro-fuzzy inference controller. A scaled-down laboratory model is used to validate the control techniques. A linear and a nonlinear auto-regressive exogenous (ARX) models are identified for the constructed structure. A neural network predictive controller is designed using the nonlinear ARX model. Polynomial and state-space pole-placement controllers are designed using the linear ARX model. A fuzzy logic controller is designed for the structure and trained using adaptive neuro fuzzy inference system (ANFIS). Hardware-in-the-loop implementation of these controllers demonstrates that the neural network predictive controller combines the advantages of both pole-placement and the ANFIS controllers and reduces the settling time of the mass damper six times with same amplitude mitigation.

Invariance of a class of semi-algebraic sets for polynomial systems with dynamic compensators

We derive a constructive sufficient condition for the existence of a compensator such that a prescribed semi-algebraic set is invariant for the resulting closed-loop system. The construction procedure consists of symbolic computations. Although the target system is restricted to polynomial control systems, non-polynomial smooth compensators can be designed. We also derive the procedure for extracting static controllers from the set of dynamic compensators. Moreover, we explain the similarities and differences between our method and the conventional slack-variable method of Jacobson and Lele (1969).

Development and practical implementation of automatic tuning methods for mechatronic system digital controllers

Currently, the great potential abilities of digital state controllers widely applied in mechatronic systems may be fully realized only with the presence of automatic tuning instruments. The importance of automatic tuning in this case is related to high number of variable parameters for state controllers as well as to parametric indeterminacy of the majority mechatronic controlled objects. The application of optimization methods for automatic tuning of such controllers in real time mode is complicated by a large number of experiments, a significant influence of initial conditions on the result, and the local extremes of the fitness function due to the nonlinearity of real objects.Methods of state space, modal control, digital system design, numerical optimization algorithms and simulation in MatLab environment were used.The development, implementation and experimental research of automatic tuning methods for digital polynomial controllers are performed during the studying. The proposed approach based on application of reference model, obtained by optimizing the system according to a complex quality criterion, as well as using a priori information about a mechatronic object. The tuning method includes variation of the object unknown parameters in accordance with the accepted optimization algorithm and calculation of the controller parameters values by modal control method at each iteration.As a result of computer simulation and field experiments, the efficiency of the proposed methods for automatic tuning of digital polynomial controllers of mechatronic systems was confirmed. Using the reference model of the system and a priori information about the control object allows us to solve tuning problems with a minimum number of experiments in real time.

On polynomial real-time control policies in stochastic AC optimal power flow

In traditional power system operations real-time control policies are based on simple control policies such as automatic generation control and/or local voltage control. Although these simple policies have worked well in the past, the increased variability associated with higher penetrations of renewable energy strengthens the case for more general approaches. In this paper we analyze the design of real-time control policies through the lens of stochastic AC-OPF. Using a chance-constrained approach in combination with a reformulation based on polynomial chaos expansion, we show that the solution corresponds to a general polynomial control policy. By restricting this general policy, we are able to recover the traditional (affine) control policies as a special case. This allows to quantify the value of implementing non-affine policies, and to analyze their impact on the system behavior, as demonstrated in the numerical example.