Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions

Nonlinear PID (N-PID) Controller for SSSP Grid Connected Inverter Control of Photovoltaic Systems

The problem of robust constrained model predictive control (MFC) of systems with polytopic uncertainty is considered in this paper. New sufficient conditions for the existence of parameter-dependent Lyapunov functions are proposed in terms of linear matrix inequalities (LMIs), which will reduce the conservativeness resulting from using a single Lyapunov function. At each sampling instant, the corresponding parameter-dependent Lyapunov function is an upper bound for a worst-case objective function, which can be minimized using the LMI convex optimization approach. Based on the solution of optimization at each sampling instant, the corresponding state feedback controller is designed, which can guarantee that the resulting closed-loop system is robustly asymptotically stable. In addition, the feedback controller will meet the specifications for systems with input or output constraints, for all admissible time-varying parameter uncertainties. Numerical examples are presented to demonstrate the effectiveness of the proposed techniques.

In this paper, techniques for analyzing transient stability of power systems with field flux decays via Lyapunov's direct method have been developed. First, Popov's stability criterion has been generalized to one for systems with multi-argument non-linearities, and a Lur'e type Lyapunov function has been constructed in a systematic method. Next, the transient stability region has been defined, and its variation has been investigated in detail. As a result, the second type of instability has been introduced. By a numerical example, stability characteristics of the power system have been illustrated, and the validity of the Lyapunov function has been proved.

This work is proposed to continue the discussion of the problems, theoretical foundations and practical features of the construction and synthesis of robust control systems with high gain, allowing us to control multidimensional nonlinear dynamic objects of high dimensional with functional uncertainties. If problems could not be solved at the level where they appeared, it is necessary to rise the level of understanding of the laws of nature, or in the words of master Lui-Shi Chun Qiu (China, 3rd century BC): "The boy of five chi growth leads the bull by the bridle and the bull obeys him in everything. This is because the person in this case follows naturalness" (the laws of nature). The judo philosophy ("soft way") is based on the principles of using the power and energy of the opponent to achieve victory. The purpose of this work is to demonstrate the theoretical aspects and practical features of the methods of synthesis of optimal control systems by the criterion of maximum reproduction accuracy using the example of robust systems, which allow to control dynamic objects with functional uncertainties, including unstable objects, no minimal-phase objects, neutral objects and objects with differentiation properties. The simplicity (at the level of the engineer) and universality, mathematical rigor and physical validity of this approach is based on the judo philosophy: suppressing the dynamics of a functionally uncertain object and external disturbances by the infinitely large gain with the finite control signal and at the same time maintaining sustainability. Theoretically exhaustive solution of the problem of robust control is given by the idea of constructing systems that are stable with an unlimited increase of the gain coefficient. The sustainability properties are valid for optimal systems that were synthesized using quadratic quality functionals that do not explicitly depend on the control signal, and using a restriction on the control signal. It is significant that in contrast to continuous systems with un-measurable disturbances and not well known control object (in which the conditions of invariance imply the use of infinitely large gain), in relay (discontinuous) systems the equivalent effect is achieved with the help of finite control signal. A nice bonus is the highest accuracy which leads to mathematically zero error of regulation, thus all error coefficients (of position, speed, acceleration, acceleration derivative, etc.) is also equal to zero in the presence of external and internal interferences. In fact, the optimal accuracy control system is equivalent to a system with astatism of the n-th order: the regulator contains n serial connected integrators.

This investigation presents a synthesis solution for stabilizing switching laws a class of parameter-varying plants represented via linear LPV systems that has all unstable subsystems. Considered class of LPV systems has state matrices as parametrically affine with parameter varying in a convex set for which all the subsystems are unstable. Stabilization design of switching laws is solved that enforce overall state trajectory that is asymptotically convergent to the equilibrium state. Via the parameter-dependent multiple Lyapunov function approach, a set of linear matrix inequalities guaranteeing the existence of parameter-dependent Lyapunov functions is derived. An illustrative example and the respective simulation results are given that demonstrate the effectiveness of this new synthesis design for this class of LPV systems.

Разработана методика построения робастных регуляторов для линейных систем с запаздыванием, грубых по отношению к параметрической неопределенности и к изменению величины запаздывания.Такие регуляторы реализуют традиционные (ПИ, ПИД, ПД) и связанные с ними законы управления.Методика базируется на критерии апериодической устойчивости

The concept of a sampled-data nonlinearity matrix is defined. It is shown that, by means of this matrix, a direct correlation can be establihed between the nonlinearity and its equivalent gain. Thus, symmetrical single-valued and/or double-valued nonlinearities, however complicated, can be transformed directly into the respective equivalent gains. This is obtained by means of linear transformations, using numerically known matrices, i.e. matrices which are independent of the nonlinearity. Similarly, in synthesis procedures, the nonlinerities are obtained from the characteristics of the desired equivalent gain. A theorem concerning these transformations is stated and proved, and some properties of the sampled-data nonlinearity matrix are emphasised. The matrices permitting direct and reciprocal transformations are given, and two non-linearities are calculated. A stability criterion is stated, and the stability of a nonlinear system is examined. Thus, it is shown that, in the design of the system, the matrix correlations and the stability criterion allow one to obtain the edesired behaviour by a very large class of nonlinear cntrol system by acting directly on the nonlinearity.

Delay-dependent robust stability analysis and stabilization of linear systems using a simple delay-discretization approach

This paper focuses on the robust stability and the memory feedback stabilization problems for a class of uncertain switched nonlinear systems with multiple time-varying delays. Especially, the considered time delays depend on the subsystem number. Based on a novel common Lyapunov functional, the aggregation techniques, and the Borne and Gentina criterion, new sufficient robust stability and stabilization conditions under arbitrary switching are established. Compared with existing results, the proposed criteria are explicit, simple to use, and obtained without finding a common Lyapunov function for all subsystems through linear matrix inequalities, considered very difficult in this situation. Moreover, compared with the memoryless one, the developed controller guarantees the robust stability of the corresponding closed-loop system with more performance by minimizing the effect of the delays in the system dynamics. Finally, two numerical simulation examples are shown to prove the practical utility and the effectiveness of the proposed theories.

This paper is concerned with the stability of networked control systems in the discrete-time domain. A new bounding technique is proposed to estimate some finite-sum terms appearing in the forward difference of the chosen Lyapunov functional. This new bounding technique can provide tighter upper bounds for some finite-sum terms and avoid overly bounding for the finite-sum terms. Then a novel delay-dependent stability criterion is derived by using this new bounding technique. Compared with some existing stability criteria in the published literature, the novel stability criterion is proven theoretically to be less conservative and is shown to be of smaller numerical complexity.

An input-output approach to the stability and stabilization of uncertain Takagi-Sugeno (T-S) fuzzy systems with time-varying delay is proposed in this paper. The time-varying parameter uncertainties are assumed to be norm-bounded, and the delay is intervally time varying. A novel method is employed to approximate the time-varying delay, based on which the considered system is transformed into a feedback interconnection form. The new formulation of the system is comprised of a forward subsystem with constant time delay and a feedback subsystem embedding the uncertainties. By applying the scaled small-gain theorem to the converted system, less conservative stability and stabilization criteria are obtained. Moreover, the applicability of the proposed approach to the robust case is simpler since both delay and parameter uncertainties are processed in a unified framework. Numerical experiments are performed to illustrate the advantage of the proposed techniques.

Synthesis of the Flexible Flight Control Algorithms Via Reflection Onto Integral Auxiliary Surfaces

Abstract. To meet the needs of drilling rescue, a variable structure detection robot for underground rescue was designed for the motion requirements of a small crawler robot in underground unstructured environments. The robot can improve the stability and anti-overturning ability of the system through its own configuration changes. The existing stability criteria are computationally complex and cannot meet the requirements of real-time control. Therefore, a stability criterion based on the contact force between the robot and the ground is proposed, and a robot adaptive stability control system was established based on back propagation (BP) neural network. The system can change the robot configuration in real time to ensure the overturning stability. The criterion is analyzed and compared with the existing stability methods, and the influence of the control system on the dynamic stability of the robot in three typical terrains is discussed. The robot simulation test model is established to simulate the robot's driving process on unstructured road and the process of adjusting the arm configuration on the slope. Finally, the stability control processes of the robot on a longitudinal slope and a transverse slope are experimentally analyzed. The analysis results show that the stability criterion is accurate and efficient, and the adaptive control system can improve the stability of the robot.

The paper presents stabilization of a Networked Control System (NCS) with uncertain, time-varying delays using two stabilization approaches namely, Parameter Dependant Lyapunov Function (PDLF) and Quadratic Lyapunov Function (QLF). The PDLF approach depends upon the time varying delay parameters. Thus, for larger delays the system's stability is ensured. In NCS, the uncertainties arising out from the variations in time-delay is modeled using polytopic approach and the stability region of the system is presented in terms of controller gain and maximum tolerable delay. The overall NCS is represented in discrete domain. The proposed PDLF stabilization results have been compared with that of the QLF approach. Lyapunov-based stability criterion is presented in Linear Matrix inequalities (LMIs). The stabilized state feedback controller is designed using LMIs and considering the bounded time-varying delays. The proposed approach is illustrated by two numerical examples. From the results obtained, it is observed that the PDLF approach improves the stability region of the system.

By now it is known that several problems in control and matrix theory are NP-hard. These include matrix problems that arise in control theory, as well as other problems in the robustness analysis and synthesis of control systems. These negative results force us to modify our notion of “solving” a given problem. If we cannot solve a problem exactly because it is NP-hard, then we must settle for solving it approximately. If we cannot solve all instances of a problem, we must settle for solving “almost all” instances of a problem. An approach that is recently gaining popularity is that of using randomized algorithms. The notion of a randomized algorithm as defined here is somewhat different from that in the computer science literature, and enlarges the class of problems that can be efficiently solved. We begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters. It is argued that, in order to assess the performance of a controller as the plant varies over a prespecified family, it is better to use the average performance of the controller as the objective function to be minimized, rather than its worstcase performance, as the worst-case objective function usually leads to rather conservative designs. Then it is shown that a property from statistical learning theory known as uniform convergence of empirical means (UCEM) plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems. In particular, whenever the UCEM property holds, there exists an efficient (i.e., polynomial-time) randomized algorithm. Using very recent results in VC-dimension theory, it is shown that the UCEM property holds in several problems such as robust stabilization and weighted H∞-norm minimization. Hence it is possible to solve such problems efficiently using randomized algorithms. The paper is concluded by showing that the statistical learning methodology is also applicable to some NP-hard matrix problems.