Multivariable Circle Criterion Research Articles

Radial pole path approach for fast response of affine constrained nonlinear systems with matched uncertainties

SummaryThis article proposes a novel robust feedback linearization control scheme for affine uncertain nonlinear systems subject to matched uncertainties and constraints on the control input. In this method, instead of placing the linearized system poles at exact locations, radial paths in the open left‐hand plane are selected to freely move the poles so as to enhance as much as possible the speed of response while guaranteeing satisfaction of input signal constraints. The stability of our proposed method is analyzed by means of the multivariable circle criterion and the Kalman‐Yakubovich‐Popov lemma. Simulation results demonstrate how the method significantly increases the speed of response compared to fixed pole placements.

Stability analysis of override logic system containing state feedback regulators and its application to gas turbine engines

Override logic or Min–Max selector scheme is known as an efficient manner for control of output constrained systems. Stability analysis is one of the most important issues in design of override control systems. Due to the switching behavior of this scheme, the stability of each loop does not necessarily guarantee the stability of the entire system. In this paper, a methodology is presented for stability analysis of the Min–Max structure containing state feedback regulators. For this purpose, the form of the control system is converted into the standard configuration of Lure's system and the condition for asymptotic stability is extracted using Multivariable Circle Criterion. The obtained theoretical results can also be used to analyze the stability of Min-only or Max-only arrangements containing state feedback regulators. Then, an override control system is designed for a model of a two-spool aircraft engine with high bypass ratio and using the presented procedure, the asymptotic stability of the control system is investigated and the region of attraction is determined. The operation of the designed controller in thrust control and limit management is compared to the Min–Max/SMC technique.

A Multi-loop Switching Controller for Aircraft Gas Turbine Engine with Stability Proof

In this paper, a Min-Max switching controller containing multiple state feedback regulators and fuel flow rate saturation is designed for a high bypass two-spool turbofan engine. Due to the switching nature of Min-Max algorithm and the presence of saturation function, stability analysis is an important issue in the process of controller design. Therefore, a methodology is presented to analyze the stability of the closed loop system. For this objective, the Min and Max selectors and the saturation block are replaced by their nonlinear equivalents and the structure of the control system is transformed into the canonical configuration of Lure’s system. Then, the condition for absolute stability is extracted using the Multivariable Circle Criterion. An asymptotic stability proof is achieved for the closed loop system and the performance of the designed multiregulator Min-Max controller in tracking a desired fan speed scenario and limit management is compared with the well-known Min-Max/SMC technique.

Pole Path Assignment of Constrained SISO Affine Nonlinear Systems

This paper proposes a feedback linearization control scheme for affine nonlinear systems with a constrained control signal where a fast response is required. Instead of placing the linearized system poles at exact locations, ray-like paths in the left hand plane are selected where the poles can freely vary to efficiently exploit the admissible input signal range to increase the speed of response. A stability test for the proposed method is derived via the multivariable circle criterion and the Kalman–Yakubovich–Popov lemma. Simulation results demonstrate how the method significantly increases the speed of response in spite of the input constraints compared to fixed pole placement.

A Min–Max multiregulator system with stability analysis for aeroengine propulsion control

Stability analysis is an essential issue in Min–Max multiregulator control strategy for commercial aircraft engines. In this paper, a Min–Max selector scheme along with a stability analysis method is provided for aeroengine propulsion control. It is assumed that the main regulator is a dynamic compensator and the limit regulators are constant gains. The regulators are determined in such a way that the individual control loops are stable. However, due to the switching nature of Min–Max structure, the stability of single loops does not necessarily ensure the overall system stability. In order to analyze the stability of the presented Min–Maxapproach, the architecture of the control system is transformed into the canonical form of Lure’s system and the condition for absolute stability is specified using Multivariable Circle Criterion. The theoretical results can also be applied to investigate the stability of min-only or max-only schemes. Afterwards, using the provided methodology, the global asymptotic stability is proved for the control system of a high bypass two-spool turbofan engine and the performance of the designed Min–Max controller in tracking a desired fan speed and limit protection in fault-free and fault tolerant situations is compared with the well-known Min–Max/SMC approach.

Soft Variable Structure Control of Linear Systems via Desired Pole Paths

In this article, a novel method of soft variable structure control of linear systems, desired pole paths, is proposed. The proposed method is helpful to reach a fast response when a continuous control signal and satisfaction of some constraints are desired. The method selects a desired path for closed loop poles instead of the exact location of poles, then the pole placement in this path is determined by solving an optimization problem subject to a control signal constraint leading to a suboptimal control structure. The stability of the proposed method is provided based on the multivariable circle criterion and the Kalman-Yakubovich-Popov lemma. A design parameter is also introduced in this paper which can adjust a tradeoff between speed of response and smoothness of the control signal. Simulation of a satellite model shows an improvement in shortening the settling time and softening the control signal compared to published soft variable structure control schemes. DOI: http://dx.doi.org/10.5755/j01.itc.47.3.18805

Radial Pole Paths SVSC for Linear Time Invariant Multi Input Systems with Constrained Inputs

AbstractIn this article a novel soft variable structure control method is proposed to reduce the settling time of liner time‐invariant multi‐input systems with constrained inputs. The method is a generalized pole placement technique which extends the desired pole locations to radial pole paths to reduce settling time while satisfying input constraints. The control signal obtained by this method is soft and is therefore suited to industrial applications. The paper provides a test for the stability of the control system based on the multivariable circle criterion and the Kalman‐Yakubovich‐Popov lemma. Simulation results where the method is applied to an aviation example, and comparison with pole placement, demonstrate how the method significantly increases the speed of response in spite of the input constraints.

机械手臂之LQG/LTR最佳控制综合设计

本研究综合计算扭矩法与LQG/LTR对非线性机械手臂系统进行多变量强健控制设计，解决存在于系统内部的不确定性与受到随机干扰情况下的非线性机械手臂系统之控制设计问题，使得非线性机械手臂控制系统具有良好的强健性与满足性能要求。文中首先使用计算扭矩法对非机械手臂系统中各项的估计值进行控制律的设计与回授线性化，并使用变异渐进法对回授系统进行适当的加权扩增；接着使用LQG/LTR设计，使得输出回授控制器(output feedback controller)能够趋近于预先设计的目标回授回路(target feedback loop)。至于非线性机械手臂闭回路系统在形成Lu’re-type问题后，可讨论非线性项之稳定性容许在一定的上界与下界，根据多变量圆稳定准则理论(multivariable circle criterion)探索此控制器之强健性能。文末则以非线性机械手臂系统为范例，进行计算机仿真，验证控制器的有效性与可行性。 In this thesis, the multivariable robust control of nonlinear manipulator systems is based on the compute torque method and the LQG/LTR design procedure was proposed. This controller is able to handle the system that have modeling errors and external disturbances while it keeps the close-loop system robustness and satisfied the prescribed performance. In this research, the computed toque method is applied to design the proposed control law to form the main control structure by using the benefit of its feedback linearization strategy. The error dynamics of the plant is then formulated to the standard H2/H∞ control problem, which is easy to be applied by the LQG/LTR design procedure to find the optimal control gain and observer gains in the process of matching the target loop. With regard to the non-canceling nonlinear terms, the closed-loop system is formulated to the Lu’re-type problem form with sector-bounded uncertainties, which is then analyzed by the Multivariable Circle Criterion to discuss the stability and robustness. To verify the feasibility of proposed controller, one example with various external disturbances and parameter uncertainties is made and its computer simulation result shows the efficiency and feasibility of the proposed design methodology.

A SYNCHRONIZATION CONDITION FOR COUPLED NONLINEAR SYSTEMS WITH TIME-DELAY: A FREQUENCY DOMAIN APPROACH

This paper considers the synchronization problem for nonlinear systems with time-delay couplings. We assume that the error dynamics can be rewritten as a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for time-delay coupled systems by applying the multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain stability condition and avoids the use of the Lyapunov–Krasovskii approach. As a result, the condition based on the circle criterion does not contain the conservativeness caused by the Lyapunov–Krasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling.

Curved Trajectory Tracking for Surface Vessels under Constant External Disturbances

Controlling a vessel in the presence of constant, unknown disturbances requires control methods capable of adjusting to this situation. In this paper we use the multivariable circle criterion, extended to linear time-varying systems, and propose a method for curved trajectory tracking based on conditional integrators. This control method combines the robustness of sliding mode control and integral action, while avoiding their inherent drawbacks like chattering and integrator wind-up. The problem of curved trajectory tracking in the horizontal plane is discussed, showing a separation between the design of the tracking control law and the disturbance rejection control law. Simulation results are provided to illustrate the effectiveness of the proposed method.

A NonLinear Observer of an Electrical Transformer: A Bond Graph Approach

A bond graph model of an electrical transformer including the nonlinear saturation is presented. A nonlinear observer for the transformer based on multivariable circle criterion in the physical domain is proposed. In order to show the saturation and hysteresis effects on the electrical transformer, simulation results are obtained. Finally, the paper describes that convergence of the estimates to the true states is achieved.

An Inner-Loop Controller Guaranteeing Robust Transient Performance for Uncertain MIMO Nonlinear Systems

An output-feedback controller has been recently proposed that has the following features: 1) it is an inner-loop controller so that it can be added on the existing closed-loop system working in harmony with a pre-designed (possibly non-robust) outer-loop controller; 2) it robustifies the closed-loop system in a way that the uncertain plant under external disturbance behaves like a disturbance-free nominal plant; and 3) it recovers the trajectory of the nominal closed-loop system in time domain. However, it is restricted to the single-input-single-output nonlinear systems. In this technical note, we extend this result for a class of multi-input-multi-output (MIMO) nonlinear systems having the same number of inputs and outputs. The tools used in this synthesis are the singular perturbation theory and the multi-variable circle criterion. An example shows the effectiveness of the proposed method.

Control for Actuator Arrays

Actuator arrays are planar arrangements of simple actuators that cooperate to translate and orient objects. This paper derives the equations of motion for manipulating an object in stick/slip contact with the actuators. A controller derived from kinematics considerations is presented and its stability analyzed with the multivariable circle criterion. Stability results are verified through simulation and experimental results on a 20 unit macroscopic actuator array.

A Synchronization Condition for Coupled Nonlinear Systems with Time-delay -A Circle Criterion Approach-

This paper considers the synchronization problem for coupled chaotic systems with time-delay. We assume that the error dynamics can be rewritten by a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for coupled systems with delay by applying a multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain condition and can avoid the use of the Lyapunov-Krasovskii approach. As a result, the condition does not contain the conservativeness caused by the Lyapunov-Krasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling.

An Inner-loop Controller Guaranteeing Robust Transient Performance for Uncertain MIMO Linear Systems

An output-feedback controller has been recently proposed that has the following features: (1) it is an inner-loop controller so that it can be added on the existing closed-loop system working in harmony with a pre-designed (possibly non-robust) outer-loop controller, (2) it robustifies the closed-loop system in a way that the uncertain plant under external disturbance becomes a nominal plant without any disturbance, (3) it recovers the trajectory of the nominal closed-loop system in time domain. However, it is restricted to the single-input-single-output systems. In this paper, we extend this result for a class of multi-input-multi-output (MIMO) linear systems having the same number of inputs and outputs. The used tools in this synthesis are the singular perturbation theory and the multi-variable circle criterion.

A sufficient condition for the stability of optimizing controllers with saturating actuators

The quadratic programme that must be solved with certain output–feedback model predictive controllers can be expressed as a continuous sector-bounded nonlinearity together with two linear transformations. Thus, the multivariable circle criterion gives a simple test for stability, with or without model mismatch. In particular, it may be applied if the open-loop plant is stable and the actuators are subject to simple saturation constraints. In the case of single horizon model predictive control, it suffices to check for positive realness a transfer function matrix whose dimension corresponds to the number of inputs. For an arbitrary length receding horizon it suffices to check the poles of a low dimension transfer function matrix and the eigenvalues (over an appropriate range of operator values) of a matrix whose dimension is independent of the horizon length. Copyright © 2005 John Wiley & Sons, Ltd.

Synthesis of MIMO controllers for extending the stability range of periodic solutions in forced nonlinear systems

This paper deals with a central issue in bifurcations and chaos control applications, i.e., the stabilization of periodic motions in sinusoidally forced nonlinear systems. Specifically, the problem of designing multi-input–multi-output (MIMO) finite-dimensional linear time-invariant controllers maximizing the amplitude of the sinusoidal input for which the corresponding periodic solutions are guaranteed to be stable, is considered. By exploiting linearization techniques and the multi-variable circle criterion, a synthesis algorithm is developed to determine the controller which maximizes a suitable lower bound of the amplitude of the input. The algorithm requires the solution of a sequence of linear matrix inequalities (LMIs) of increasing size. The Brusselator oscillator is employed as a case study to show that the synthesized controllers, though optimizing a lower bound, provide quite satisfactory control performance.

Graphical and numerical approach to robust stability analysis of fuzzy modeled systems with parametric uncertainty and disturbance

In this paper, robust stability analysis methods for the fuzzy feedback control systems are presented, the graphical method via multivariable circle criterion and the numerical method via linear matrix inequalities (LMI). The well-known Takagi–Sugeno fuzzy model is used as the nonlinear plant model. Uncertainties are assumed to be included in the model structure with known bounds. For these structured parametric uncertainties, L 2 robust stability analysis is performed by taking external disturbance as input and system state as output. The effectiveness of the proposed methods is illustrated by examples.

Feedback of passive systems: synthesis and analysis of linear robust control systems

The feedback interconnection of passive systems always leads to a stable control loop. This fact can be used for synthesis and analysis of linear robust control systems in a uniform framework. In the case of slowly varying uncertainties the plant can be represented by a linear time-invariant multimodel. Rendering this multimodel passive and using a strong strictly passive controller ensures robust stability of the closed loop. The paper refines this idea to the design of robust observer-based controllers using parametric controller design. The new method allows the inclusion of a very extensive multimodel. Based on the well-accepted concept of pole assignment, the procedure yields robust controllers whose orders do not exceed the order of the plant. For rapidly varying uncertainties the plant must be modelled as a linear time-variant system. To obtain an uncertain control loop that can also be analysed with the feedback theorem the LTV-components must be separated from the LTI system. It is shown that this can be done by establishing an affine relation between time varying parameters and the state-space matrices. The resulting closed-loop structure allows the application of the multivariable circle criterion which is used here in LMI form. Hence one can profit from the benefit of convex optimisation.

Stability analysis of nonlinear multivariable Takagi-Sugeno fuzzy control systems

Points out how the nonlinearities involved in multivariable Takagi-Sugeno (T-S) fuzzy control systems could originate complex behavior phenomena, such as multiple equilibrium points or limit cycles, that cannot be detected using conventional stability analysis techniques. In the paper, the application of MIMO frequency-domain methods to predict the existence of multiple equilibria and of limit cycles are presented. The proposed method is based on the formulation of a Lur'e problem from the original structure of a T-S fuzzy system with a fuzzy controller. Furthermore, this technique makes straightforward the application of input-output stability techniques such as the multivariable circle criterion, also called the conicity criterion, and the harmonic balance method. Moreover, in the paper, the application of the harmonic balance method has been generalized to the case of a MIMO fuzzy system with asymmetric nonlinearities and improved by the decreasing conservatism. A new and more general stability index which could be used to perform a bifurcation analysis of fuzzy control systems is presented. The paper includes a collection of examples where the advantages of the proposed approach are made explicit comparing it to the input-output conicity criterion and the Lyapunov direct method.