A new Lyapunov design approach for nonlinear systems based on Zubov's method

The present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of Lyapunov design. In particular, it relies on a short-horizon model-based prediction and optimization of the rate of “energy dissipation” of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. The latter is computed by solving Zubov's partial differential equation based on the system's drift vector field. A nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned Lyapunov function and closed-loop performance characteristics. A set of system-theoretic properties of the proposed control law are examined as well. Finally, the proposed Lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum-phase behaviour.     

Stability assessment for cautious iterative controller tuning

This document presents new mechanisms for stability assessment in iterative controller tuning. Contrary to the usual approach adopted in robust model identification, the new mechanisms do not rely on plant models, but utilize information from the current closed-loop system to test if a newly designed controller will maintain loop stability. All controllers that satisfy the test are guaranteed to stabilize the actual plant.     

Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation

Given a continuous-time controller and a Lyapunov function that shows global asymptotic stability for the closed-loop system, we provide several results for modification of the controller for sampled-data implementation. The main idea behind this approach is to use a particular structure for the redesigned controller and the main technical result is to show that the Fliess series expansions (in the sampling period T) of the Lyapunov difference for the sampled-data system with the redesigned controller have a very special form that is useful for controller redesign. We present results on controller redesign that achieve two different goals. The first goal is making the lower-order terms (in T) in the series expansion of the Lyapunov difference with the redesigned controller more negative. These control laws are very similar to those obtained from Lyapunov-based redesign of continuous-time systems for robustification of control laws and they often lead to corrections of the well-known “-L_gV” form. The second goal is making the lower-order terms (in T) in the Fliess expansions of the Lyapunov difference for the sampled-data system with the redesigned controller behave as close as possible to the lower-order terms of the Lyapunov difference along solutions of the “ideal” sampled response of the continuous-time system with the original controller. In this case, the controller correction is very different from the first case and it contains appropriate “prediction” terms. The method is very flexible and one may try to achieve other objectives not addressed in this paper or derive similar results under different conditions. Simulation studies verify that redesigned controllers perform better (in an appropriate sense) than the unmodified ones when they are digitally implemented with sufficiently small sampling period T.     

Design of robust gain-scheduled PI controllers for nonlinear processes

Gain-scheduling has proven to be a successful design methodology in many engineering applications. However, in the absence of a sound theoretical analysis, these designs come with no guarantees of robust stability, performance or even nominal stability of the overall gain-scheduled deign.This paper presents such an analysis for one type of nonlinear gain-scheduled control system based on the process input for nonlinear chemical processes. A methodology is also proposed for the design and optimization of the robust gain-scheduled PI controller. Conditions which guarantee robust stability and performance are formulated as a finite set of linear matrix inequalities (LMIs) and hence, the resulting problem is numerically tractable. Issues of modeling error and input-saturation are explicitly incorporated into the analysis. A simulation study of a nonlinear continuous stirred tank reactor (CSTR) process indicates that this approach can produce efficient sub-optimal robust gain-scheduled controllers.     

General tuning procedure for the nonlinear balance-based adaptive controller

This paper presents the intuitive and ready-to-use, general procedure for tuning the balance-based adaptive controller (B-BAC) based on its equivalence to the controller with PI term and with additional improvements shown for the linearised approximation of the dynamics of the nonlinear controlled process. The simple formulas are suggested to calculate the B-BAC tunings based on the PI tunings determined by any PI tuning procedure chosen accordingly to the desired closed-loop performance. This methodology is verified by comparing the closed-loop performance of the equivalently tuned B-BAC and PI/PI+feedforward controllers under the same scenario, both by the simulation and practical experiments.     

Robust tuning for machine-directional predictive control of MIMO paper-making processes

This paper solves the controller tuning problem of machine-directional predictive control for multiple-input–multiple-output (MIMO) paper-making processes represented as superposition of first-order-plus-dead-time (FOPDT) components with uncertain model parameters. A user-friendly multi-variable tuning problem is formulated based on user-specified time domain specifications and then simplified based on the structure of the closed-loop system. Based on the simplified tuning problem and a proposed performance evaluation technique, a fast multi-variable tuning technique is developed by ignoring the constraints of the MPC. In addition, a technique to predict the computation time of the tuning algorithm is proposed. The efficiency of the proposed method is verified through Honeywell real time simulator platform with a MIMO paper-making process obtained from real data from an industrial site.     

Robust controller design for a class of nonlinear uncertain chemical processes

This article considers the issue of designing robust controllers for single-input/single-output nonlinear chemical processes whose uncertainties satisfy the so-called generalized matching condition. The nominal system (mathematical model) is assumed to be input–output linearizable and the only assumption on uncertainties is that they are bounded. A design methodology of combining the techniques of the differential geometric feedback linearization, the sliding mode control strategy and the adaptive state feedback is presented. Based on the nominal system and the related bounds of uncertainties, a hybrid nonlinear controller, which is more practicable and easily implemented than many other existing ones in the literature, is proposed. A Lyapunov-based approach is utilized to guarantee the robust stability and behavior of the closed-loop system. For demonstrating the effectiveness and applicability of the proposed scheme, we applied it to the control of a continuously stirred tank reactor (CSTR) in the presence of uncertainties including unmodeled side reaction, measuring error, and/or extra unmeasured disturbances. The potential use of a sliding observer along with the proposed scheme is also investigated in this work. Extensive simulation results reveal that the proposed scheme appears to be a practical and promising approach to the robust control of nonlinear uncertain chemical processes.     

ISE-optimal nonminimum-phase compensation for nonlinear processes

This work concerns the optimal regulation of single-input–single-output nonminimum-phase nonlinear processes. The problem of calculation of an ISE-optimal, statically equivalent, minimum-phase output for nonminimum-phase compensation is formulated using Hamilton–Jacobi theory and the normal form representation of the nonlinear system. A Newton–Kantorovich iteration is developed for the solution of the pertinent Hamilton–Jacobi equations, which involves solving a Zubov equation at each step of the iteration. The method is applied to the problem of controlling a nonisothermal CSTR with Van de Vusse kinetics, which exhibits nonminimum-phase behaviour.     

On the region of attraction of nonlinear quadratic systems

Quadratic systems play an important role in the modelling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems, it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the region of attraction of the equilibrium. The proposed algorithm requires the solution of a suitable feasibility problem involving linear matrix inequalities constraints. An example illustrates the effectiveness of the proposed procedure by exploiting a population interaction model of three species.     

Partial asymptotic stabilization of nonlinear distributed parameter systems

The paper is devoted to stability and stabilization of a class of evolution equations arising from mathematical modeling of hybrid mechanical systems with flexible parts. A sufficient condition is obtained for partial strong asymptotic stability of nonlinear, infinite-dimensional dynamic systems in Banach spaces. This result is applied to deriving a control law that stabilizes a part of the variables describing a rotating rigid body endowed with a number of elastic beams. Results of numerical simulations are presented.     

Adaptive neural network control for strict-feedback nonlinear systems using backstepping design

This paper focuses on adaptive control of strict-feedback nonlinear systems using multilayer neural networks (MNNs). By introducing a modified Lyapunov function, a smooth and singularity-free adaptive controller is firstly designed for a first-order plant. Then, an extension is made to high-order nonlinear systems using neural network approximation and adaptive backstepping techniques. The developed control scheme guarantees the uniform ultimate boundedness of the closed-loop adaptive systems. In addition, the relationship between the transient performance and the design parameters is explicitly given to guide the tuning of the controller. One important feature of the proposed NN controller is the highly structural property which makes it particularly suitable for parallel processing in actual implementation. Simulation studies are included to illustrate the effectiveness of the proposed approach.     

Robust self-tuning PID controller for nonlinear systems

In this paper, we propose a robust self-tuning PID controller suitable for nonlinear systems. The control system employs a preload relay (P_Relay) in series with a PID controller. The P_Relay ensures a high gain to yield a robust performance. However, it also incurs a chattering phenomenon. In this paper, instead of viewing the chattering as an undesirable yet inevitable feature, we use it as a naturally occurring signal for tuning and re-tuning the PID controller as the operating regime digresses. No other explicit input signal is required. Once the PID controller is tuned for a particular operating point, the relay may be disabled and chattering ceases correspondingly. However, it is invoked when there is a change in setpoint to another operating regime. In this way, the approach is also applicable to time-varying systems as the PID tuning can be continuous, based on the latest set of chattering characteristics. Analysis is provided on the stability properties of the control scheme. Simulation results for the level control of fluid in a spherical tank using the scheme are also presented.     

On the use of bifurcation analysis for robust controller tuning for nonlinear systems

This paper presents a new approach to robust controller tuning for nonlinear systems. The main idea is to apply numerical bifurcation analysis to the closed-loop process, using the controller tuning parameters, the set points, and parameters describing model uncertainty (parametric as well as unmodeled dynamics) as bifurcation parameters. By analyzing the Hopf bifurcation and saddle-node bifurcation loci, bounds on the controller parameters are identified. These bounds depend upon the type as well as the degree of mismatch that exists between the plant and the model used for controller design.The method is illustrated by tuning a state feedback linearizing controller for an unstable reactor as well as by comparing a proportional-integral (PI) controller and a state feedback linearizing controller applied to a continuously operated fermenter. The feedback linearizing controller can result in better performance than the PI controller if a very accurate model of the process is known and if the operating conditions vary over a significant range. However, stability properties of systems controlled by feedback linearizing controllers can degrade significantly as the mismatch between the plant and the model increases. This is illustrated in the fermenter example by showing that bounds on the tuning parameter of the feedback linearizing controller are significantly tighter than the ones for the PI controller.     

A nonlinear hybrid controller for swinging-up and stabilizing the Furuta pendulum

The conventional switching strategy for solving the inverted pendulum control problem is based on two steps: swinging-up and stabilization. In this note, first, a new strategy for swinging the Furuta pendulum up towards the desired upright position is designed using the Speed-Gradient method, which uses only directly measured coordinates. Then, a nonlinear controller, based on the Forwarding approach, stabilizes the upright position. As a new contribution the latter leads to a nonlinear stabilizer around the upright position, whose Lyapunov function yields a larger size estimation of the domain of attraction than the one obtained with the traditional linear approach. This estimation allows us to use it in a global switching strategy in the practical implementation and guarantees almost-global asymptotic stability of the equilibrium. Successful experimental results are reported with the available laboratory Furuta pendulum.     

Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.     

Stability of nonlinear asynchronous systems

In this work, we focus on a class of nonlinear asynchronous systems defined by two different modes of operation, one stable and the other one unstable. The switching between the two modes of operation is driven by external asynchronous events. It is assumed that on any time interval of a given length, the maximum time in which the system evolves in the unstable mode is bounded. This property is given in the form of a rate constraint. Under this assumption, we study the behavior of this class of systems and provide existential results of conditions on this rate constraint under which various types of stability of the origin of the nonlinear asynchronous system can be assured.     

A power system nonlinear adaptive decentralized controller design

In this paper, a novel excitation control is designed for improvement of transient stability of power systems. The control algorithm is based on the adaptive backstepping method in a recursive way without linearizing the system model. Lyapunov function method is applied in designing the controller to ensure the convergence of the power angle, relative speed of the generator and the active electrical power delivered by the generator when a large fault occurs. Compared with the existing nonlinear decentralized control approaches, the proposed controller has no requirement for the bounds of interconnections in the power system. And the new approach does not need the existence of solution of a designed algebraic Riccati equation. Furthermore, the transient stability performance of power systems can also be improved by the designed control approach. The efficacy of the designed controller has been demonstrated in a multimachine power system. Simulation results show transient stability enhancement of a power system in the face of a large sudden fault.     

约束非线性系统构造性模型预测控制

研究了连续时间约束非线性系统模型预测控制设计.利用控制Lyapunov函数离线构造单变量可调预测控制器,再根据性能指标在线优化可调参数,其中该参数近似于闭环系统的"衰减率".同时,控制Lyapunov函数保证了算法的可行性和闭环系统的稳定性.最后通过数值仿真验证了该算法的有效性.     

A method for construction of stability regions by Lyapunov functions

This article proposes a new method of constructing a sequence of Lyapunov functions which provides more adequate approximation for the exact stability region via the regions which are bounded by a level surfaces of the functions.