With the development of science and technology,the design and optimization of control systems are widely applied.This paper focuses on the application of matrix equations in linear time-invariant systems.Taking the in...With the development of science and technology,the design and optimization of control systems are widely applied.This paper focuses on the application of matrix equations in linear time-invariant systems.Taking the inverted pendulum model as an example,the algebraic Riccati equation is used to solve the optimal control problem,and the system performance and stability are achieved by selecting the closed-loop pole and designing the gain matrix.Then,the numerical methods for solving the stochastic algebraic Riccati equations are applied to practical problems,with Newton’s iteration method as the outer iteration and the solution of the mixed-type Lyapunov equations as the inner iteration.Two methods for solving the Lyapunov equations are introduced,providing references for related research.展开更多
In traditional system identification (SI), actual values of system parameters are concealed in the input and output data;hence, it is necessary to apply estimation methods to determine the parameters. In signal proces...In traditional system identification (SI), actual values of system parameters are concealed in the input and output data;hence, it is necessary to apply estimation methods to determine the parameters. In signal processing, a signal with N elements must be sampled at least N times. Thus, most SI methods use N or more sample data to identify a model with N parameters;however, this can be improved by a new sampling theory called compressive sensing (CS). Based on CS, an SI method called compressive measurement identification (CMI) is proposed for reducing the data needed for estimation, by measuring the parameters using a series of linear measurements, rather than the measurements in sequence. In addition, the accuracy of the measurement process is guaranteed by a criterion called the restrict isometric principle. Simulations demonstrate the accuracy and robustness of CMI in an underdetermined case. Further, the dynamic process of a DC motor is identified experimentally, establishing that CMI can shorten the identification process and increase the prediction accuracy.展开更多
The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and n...The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and necessary conditions of controllability and observability for such systems were derived. In terms of Lyapunov’s stability theory, using the theorems of Mittage-Leffler function in two parameters this paper directly derived the sufficient and necessary condition of stability for such systems. The results obtained are useful for the analysis and synthesis of fractional-order linear control systems.展开更多
An efficient identification algorithm is given for commensurate order linear time-invariant fractional systems. This algorithm can identify not only model coefficients of the system, but also its differential order at...An efficient identification algorithm is given for commensurate order linear time-invariant fractional systems. This algorithm can identify not only model coefficients of the system, but also its differential order at the same time. The basic idea is to change the system matrix into a diagonal one through basis transformation. This makes it possible to turn the system’s input-output relationships into the summation of several simple subsystems, and after the identification of these subsystems, the whole identification system is obtained which is algebraically equivalent to the former system. Finally an identification example verifies the effectiveness of the method previously mentioned.展开更多
In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-tim...In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-time linear time-invariant systems. It solves the perturbed linear time-invariant systems via Riccati differential equations and continuous-time algebraic Riccati equations in finite and infinite time horizons. We derive the explicit expressions of measuring the perturbation bounds of condition numbers with respect to the solution of the linear time-invariant systems. Furthermore, condition numbers and their upper bounds of Riccati differential equations and continuous-time algebraic Riccati equations are also discussed. Numerical simulations show the sharpness of the perturbation bounds computed via the proposed methods.展开更多
It is shown that the time of entropy increase, here called action time, is caused by a dynamically understood energy. It drives time by decreasing its presence per state, that is by abandoning order, information, and ...It is shown that the time of entropy increase, here called action time, is caused by a dynamically understood energy. It drives time by decreasing its presence per state, that is by abandoning order, information, and creating entropy. This mechanism can be derived from basic principles via the Lagrange-Euler formalism, just considering the properties of really experienced, oriented time and thus abandoning the paradigm of time neutrality. It describes nature driven by a dynamically understood principle of least action, which is identified as manifestation of fundamental irreversibility in nature. This readily explains the second law of thermodynamics and also yields the entropy law for non-linear irreversible thermodynamics: maximum entropy production within the restraints of the system. Dynamic energy-driven time, action time, and time asymmetry is generated via the process of erasing information and liberating its energy irreversibly as heat. It is not an illusion but information-based reality. It is the loss of information to the past and different from clock-time, which is just an artificial scale, using information for tracking real time, action time. Energy-driven fundamental irreversibility of nature can better describe experienced reality and opens the way to understand and finally imitate the self-organizing creativity in nature. It also draws far reaching consequences for understanding quantum physics, gravitation and cosmology as well as biology. From the point of view of irreversibility, nature turns out to be more elegant, simpler and rationally understandable. For the first time, it can be explained in a few words what energy and nature basically represent and why it must have been information, which has started the universe.展开更多
This paper proposes a new approach for multi-objective robust control. The approach extends the standard generalized l2 (Gl2) and generalized H2 (GH2) conditions to a set of new linear matrix inequality (LMI) constra...This paper proposes a new approach for multi-objective robust control. The approach extends the standard generalized l2 (Gl2) and generalized H2 (GH2) conditions to a set of new linear matrix inequality (LMI) constraints based on a new stability condition. A technique for variable parameterization is introduced to the multi-objective control problem to preserve the linearity of the synthesis variables. Consequently, the multi-channel multi-objective mixed Gl2/GH2 control problem can be solved less conservatively using computationally tractable algorithms developed in the paper.展开更多
A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration o...A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.展开更多
In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to d...In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to directly identify parameters of the observer-based residual generator based on a numerically reliable data equation obtained by filtering and sampling the input and output signals.展开更多
A new concept is presented to express the damping property of linear time-invariant systems, by the Lyapunov theorem in view of quadratic form-defined energy. Two definitions are introduced: damping energy function D(...A new concept is presented to express the damping property of linear time-invariant systems, by the Lyapunov theorem in view of quadratic form-defined energy. Two definitions are introduced: damping energy function D(X_0, X)=Ci∫_(x_0, x) x_idx_(i-1)and comprehensive damping coefficient η-min(Ci/a_(n-i)). It is concluded that (ⅰ) of the Hurwitz determinants, △_(x-1) is proportional to the damping effect of oscillating systems, (ⅱ) the comprehensive damping coefficients of linear time-invariant systems are derived as. piecewise rational fractions which can be easily calculated and (ⅲ) the damping torque coefficient obtained for synchronous machines is independent of ω.展开更多
We investigate the type of singularity and qualitative structure of solutions to a time-invariant linear dynamic system on time scales. The results truly unify the qualitative behaviors of the system on the continuous...We investigate the type of singularity and qualitative structure of solutions to a time-invariant linear dynamic system on time scales. The results truly unify the qualitative behaviors of the system on the continuous and discrete times with any step size.展开更多
This paper is focused on formability of multi-agent systems （MASs）. The problem is concerned with the existence of a protocol that has the ability to drive the MAS involved to the desired formation, and thus, is of ...This paper is focused on formability of multi-agent systems （MASs）. The problem is concerned with the existence of a protocol that has the ability to drive the MAS involved to the desired formation, and thus, is of essential importance in designing formation protocols. Formability of an MAS depends on several key factors： agents＇ dynamic structures, connectivity topology, properties of the desired formation and the admissible control set. Agents of the MASs considered here are described by a general continuous linear time-invariant （LTI） model. By using the matrix analysis and algebraic graph theory, some necessary and sufficient conditions on formability of LTI-MASs are obtained. These conditions characterize in some sense the relationship of formability, connectivity topology, formation properties and agent dynamics with respect to some typical and widely used admissible protocol sets.展开更多
With the rapid development of information technologies and cloud computing, sensor networks play an increasingly important role in our society. Over the past few decades, distributed observer theory has attracted unpr...With the rapid development of information technologies and cloud computing, sensor networks play an increasingly important role in our society. Over the past few decades, distributed observer theory has attracted unprecedented attention due to its wide potential applications in different areas. Meanwhile, various approaches and algorithms have been proposed and investigated. The design of distributed observers is one of the frontier topics of system and control research, which has the significant theoretical values and broad application prospects. This paper attempts to review the representative models and the corresponding approaches for distributed observer design in linear time-invariant(LTI) systems. Firstly, the research backgrounds and main advances of distributed observer designs are briefly reviewed. Then, recent results of distributed observer designs for discrete-time and continuous-time LTI multi-agent systems(MASs) are introduced in detail, respectively. Finally, the prospects and the future work directions of the design of distributed observers are put forward. The main purpose of this paper is to promote the emerging topic on the designs of distributed observers, with focuses on the interdisciplinary interest from technological sciences.展开更多
This paper focuses on the H optimal control problem in which the wholestate is available for feedback. We show that in attenuating the disturbance, the Hoptimal performance of dynamic state-feedback is no better than ...This paper focuses on the H optimal control problem in which the wholestate is available for feedback. We show that in attenuating the disturbance, the Hoptimal performance of dynamic state-feedback is no better than that of static state-feedback, which generalizes current results for linear time-invariant systems with no directtransmission from the disturbance and control input to the controlled output.展开更多
This paper proposes a robust adaptive filter based on the exponent sin cost to improve the capability against Gaussian or multiple types of non-Gaussian noises of the adaptive filtering algorithm when dealing with tim...This paper proposes a robust adaptive filter based on the exponent sin cost to improve the capability against Gaussian or multiple types of non-Gaussian noises of the adaptive filtering algorithm when dealing with time-varying/time-invariant linear systems function exponent sin(ExpSin).Then a variable step-size(VSS)-ExpSin algorithm is extended further.Besides,the stepsize,the convergence,and the steady-state performance of the proposed algorithm are validated experimentally.The Monte Carlo simulation results of linear system identification illustrate the principle and efficiency of this proposed adaptive filtering algorithm.Results suggest that the proposed adaptive filtering algorithm has superior performance when estimating the unknown linear systems under multiple-types measurement noises.展开更多
This paper is concerned with the parameter estimation of deterministic autoregressive moving average(DARMA)systems with quantization data.The estimation algorithms adopted here are the least squares(LS)and the forgett...This paper is concerned with the parameter estimation of deterministic autoregressive moving average(DARMA)systems with quantization data.The estimation algorithms adopted here are the least squares(LS)and the forgetting factor LS,and the signal quantizer is of uniform,that is,with uniform quantization error.The authors analyse the properties of the LS and the forgetting factor LS,and establish the boundedness of the estimation errors and a relationship of the estimation errors with the size of quantization error,which implies that the smaller the quantization error is,the smaller the estimation error is.A numerical example is given to demonstrate theorems.展开更多
In this paper,several equivalent forms of the well-known Brockett's second example system are firstly presented.The stabilization of the system is then treated in the fully actuated system approach.A simple contin...In this paper,several equivalent forms of the well-known Brockett's second example system are firstly presented.The stabilization of the system is then treated in the fully actuated system approach.A simple continuous time-invariant sub-stabilizing controller is designed,and the corresponding region of attraction is characterized.As a result,all trajectories of the system starting from the characterized region of attraction are driven exponentially to the origin.Since the region of attraction is very large,the designed sub-stabilizing controller can be directly useful in many practical situations.In cases where the initial values are indeed needed to be chosen out of the region of attraction,extremely simple pre-controllers can be designed,which drive the system trajectories into the designed region of attraction.A simulation of the designed control system is carried out to show the effect of the proposed approach.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.12571388)the Visiting Scholar Program of National Natural Science Foundation of China(Grant No.12426616)Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications(Grant No.NY223127).
文摘With the development of science and technology,the design and optimization of control systems are widely applied.This paper focuses on the application of matrix equations in linear time-invariant systems.Taking the inverted pendulum model as an example,the algebraic Riccati equation is used to solve the optimal control problem,and the system performance and stability are achieved by selecting the closed-loop pole and designing the gain matrix.Then,the numerical methods for solving the stochastic algebraic Riccati equations are applied to practical problems,with Newton’s iteration method as the outer iteration and the solution of the mixed-type Lyapunov equations as the inner iteration.Two methods for solving the Lyapunov equations are introduced,providing references for related research.
基金Supported by the National Natural Science Foundation of China(61605218)National Defense Science and Technology Innovation Foundation of Chinese Academy of Sciences(CXJJ-17S023)
文摘In traditional system identification (SI), actual values of system parameters are concealed in the input and output data;hence, it is necessary to apply estimation methods to determine the parameters. In signal processing, a signal with N elements must be sampled at least N times. Thus, most SI methods use N or more sample data to identify a model with N parameters;however, this can be improved by a new sampling theory called compressive sensing (CS). Based on CS, an SI method called compressive measurement identification (CMI) is proposed for reducing the data needed for estimation, by measuring the parameters using a series of linear measurements, rather than the measurements in sequence. In addition, the accuracy of the measurement process is guaranteed by a criterion called the restrict isometric principle. Simulations demonstrate the accuracy and robustness of CMI in an underdetermined case. Further, the dynamic process of a DC motor is identified experimentally, establishing that CMI can shorten the identification process and increase the prediction accuracy.
基金Shanghai Science and Technology Devel-opm ent Funds ( No.0 1160 70 3 3)
文摘The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and necessary conditions of controllability and observability for such systems were derived. In terms of Lyapunov’s stability theory, using the theorems of Mittage-Leffler function in two parameters this paper directly derived the sufficient and necessary condition of stability for such systems. The results obtained are useful for the analysis and synthesis of fractional-order linear control systems.
基金Sponsored by 863 Project (Grant No.2002AA517020) Developing Fund of Shanghai Science Committee (Grant No.011607033).
文摘An efficient identification algorithm is given for commensurate order linear time-invariant fractional systems. This algorithm can identify not only model coefficients of the system, but also its differential order at the same time. The basic idea is to change the system matrix into a diagonal one through basis transformation. This makes it possible to turn the system’s input-output relationships into the summation of several simple subsystems, and after the identification of these subsystems, the whole identification system is obtained which is algebraically equivalent to the former system. Finally an identification example verifies the effectiveness of the method previously mentioned.
文摘In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-time linear time-invariant systems. It solves the perturbed linear time-invariant systems via Riccati differential equations and continuous-time algebraic Riccati equations in finite and infinite time horizons. We derive the explicit expressions of measuring the perturbation bounds of condition numbers with respect to the solution of the linear time-invariant systems. Furthermore, condition numbers and their upper bounds of Riccati differential equations and continuous-time algebraic Riccati equations are also discussed. Numerical simulations show the sharpness of the perturbation bounds computed via the proposed methods.
文摘It is shown that the time of entropy increase, here called action time, is caused by a dynamically understood energy. It drives time by decreasing its presence per state, that is by abandoning order, information, and creating entropy. This mechanism can be derived from basic principles via the Lagrange-Euler formalism, just considering the properties of really experienced, oriented time and thus abandoning the paradigm of time neutrality. It describes nature driven by a dynamically understood principle of least action, which is identified as manifestation of fundamental irreversibility in nature. This readily explains the second law of thermodynamics and also yields the entropy law for non-linear irreversible thermodynamics: maximum entropy production within the restraints of the system. Dynamic energy-driven time, action time, and time asymmetry is generated via the process of erasing information and liberating its energy irreversibly as heat. It is not an illusion but information-based reality. It is the loss of information to the past and different from clock-time, which is just an artificial scale, using information for tracking real time, action time. Energy-driven fundamental irreversibility of nature can better describe experienced reality and opens the way to understand and finally imitate the self-organizing creativity in nature. It also draws far reaching consequences for understanding quantum physics, gravitation and cosmology as well as biology. From the point of view of irreversibility, nature turns out to be more elegant, simpler and rationally understandable. For the first time, it can be explained in a few words what energy and nature basically represent and why it must have been information, which has started the universe.
基金Project supported by the National Natural Science Foundation ofChina (No. 60374028) and the Scientific Research Foundation forReturned Overseas Chinese Scholars Ministry of Education (No.[2004]176)
文摘This paper proposes a new approach for multi-objective robust control. The approach extends the standard generalized l2 (Gl2) and generalized H2 (GH2) conditions to a set of new linear matrix inequality (LMI) constraints based on a new stability condition. A technique for variable parameterization is introduced to the multi-objective control problem to preserve the linearity of the synthesis variables. Consequently, the multi-channel multi-objective mixed Gl2/GH2 control problem can be solved less conservatively using computationally tractable algorithms developed in the paper.
基金This work is supported by the National Natural Science Foundation of China （No.21373064）, the Program for Innovative Research Team of Guizhou Province （No.QKTD[2014]4021）, and the Natural Sci- entific Foundation from Guizhou Provincial Department of Education （No.ZDXK[2014]IS）. All the calculations were performed at Guizhou Provincial High- Performance Computing Center of Condensed Mate- rials and Molecular Simulation in Guizhou Education University.
文摘A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.
基金This work was supported was supported in part by the European Union under grant NeCST.
文摘In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to directly identify parameters of the observer-based residual generator based on a numerically reliable data equation obtained by filtering and sampling the input and output signals.
文摘A new concept is presented to express the damping property of linear time-invariant systems, by the Lyapunov theorem in view of quadratic form-defined energy. Two definitions are introduced: damping energy function D(X_0, X)=Ci∫_(x_0, x) x_idx_(i-1)and comprehensive damping coefficient η-min(Ci/a_(n-i)). It is concluded that (ⅰ) of the Hurwitz determinants, △_(x-1) is proportional to the damping effect of oscillating systems, (ⅱ) the comprehensive damping coefficients of linear time-invariant systems are derived as. piecewise rational fractions which can be easily calculated and (ⅲ) the damping torque coefficient obtained for synchronous machines is independent of ω.
文摘We investigate the type of singularity and qualitative structure of solutions to a time-invariant linear dynamic system on time scales. The results truly unify the qualitative behaviors of the system on the continuous and discrete times with any step size.
基金supported by the National Nature Science Foundation of China under Grants Nos.60934006 and 61104136the Shandong Provincial Natural Science Foundation under Grant No.ZR2010FQ002+1 种基金the School Foundation of Qufu Normal University under Grant No.XJ200913the Scientific Research Foundation of Qufu Normal University
文摘This paper is focused on formability of multi-agent systems （MASs）. The problem is concerned with the existence of a protocol that has the ability to drive the MAS involved to the desired formation, and thus, is of essential importance in designing formation protocols. Formability of an MAS depends on several key factors： agents＇ dynamic structures, connectivity topology, properties of the desired formation and the admissible control set. Agents of the MASs considered here are described by a general continuous linear time-invariant （LTI） model. By using the matrix analysis and algebraic graph theory, some necessary and sufficient conditions on formability of LTI-MASs are obtained. These conditions characterize in some sense the relationship of formability, connectivity topology, formation properties and agent dynamics with respect to some typical and widely used admissible protocol sets.
基金supported in part by the National Key Research and Development Program of China(Grant No.2018AAA0101100)in part by the National Natural Science Foundation of China(Grant Nos.61621003,92067204 and 61903017)in part by the China Postdoctoral Science Foundation(Grant Nos.2020TQ0027 and 2020M680285)。
文摘With the rapid development of information technologies and cloud computing, sensor networks play an increasingly important role in our society. Over the past few decades, distributed observer theory has attracted unprecedented attention due to its wide potential applications in different areas. Meanwhile, various approaches and algorithms have been proposed and investigated. The design of distributed observers is one of the frontier topics of system and control research, which has the significant theoretical values and broad application prospects. This paper attempts to review the representative models and the corresponding approaches for distributed observer design in linear time-invariant(LTI) systems. Firstly, the research backgrounds and main advances of distributed observer designs are briefly reviewed. Then, recent results of distributed observer designs for discrete-time and continuous-time LTI multi-agent systems(MASs) are introduced in detail, respectively. Finally, the prospects and the future work directions of the design of distributed observers are put forward. The main purpose of this paper is to promote the emerging topic on the designs of distributed observers, with focuses on the interdisciplinary interest from technological sciences.
文摘This paper focuses on the H optimal control problem in which the wholestate is available for feedback. We show that in attenuating the disturbance, the Hoptimal performance of dynamic state-feedback is no better than that of static state-feedback, which generalizes current results for linear time-invariant systems with no directtransmission from the disturbance and control input to the controlled output.
文摘This paper proposes a robust adaptive filter based on the exponent sin cost to improve the capability against Gaussian or multiple types of non-Gaussian noises of the adaptive filtering algorithm when dealing with time-varying/time-invariant linear systems function exponent sin(ExpSin).Then a variable step-size(VSS)-ExpSin algorithm is extended further.Besides,the stepsize,the convergence,and the steady-state performance of the proposed algorithm are validated experimentally.The Monte Carlo simulation results of linear system identification illustrate the principle and efficiency of this proposed adaptive filtering algorithm.Results suggest that the proposed adaptive filtering algorithm has superior performance when estimating the unknown linear systems under multiple-types measurement noises.
基金supported by National Key R&D Program of China under Grant No.2018YFA0703800the National Natural Science Foundation of China under Grant No.61877057。
文摘This paper is concerned with the parameter estimation of deterministic autoregressive moving average(DARMA)systems with quantization data.The estimation algorithms adopted here are the least squares(LS)and the forgetting factor LS,and the signal quantizer is of uniform,that is,with uniform quantization error.The authors analyse the properties of the LS and the forgetting factor LS,and establish the boundedness of the estimation errors and a relationship of the estimation errors with the size of quantization error,which implies that the smaller the quantization error is,the smaller the estimation error is.A numerical example is given to demonstrate theorems.
基金partially supported by Shenzhen Key Laboratory of Control Theory and Intelligent Systems under Grant No.ZDSYS20220330161800001the Science Center Program of the National Natural Science Foundation of China under Grant No.62188101the Major Program of National Natural Science Foundation of China under Grant Nos.61690210 and 61690212。
文摘In this paper,several equivalent forms of the well-known Brockett's second example system are firstly presented.The stabilization of the system is then treated in the fully actuated system approach.A simple continuous time-invariant sub-stabilizing controller is designed,and the corresponding region of attraction is characterized.As a result,all trajectories of the system starting from the characterized region of attraction are driven exponentially to the origin.Since the region of attraction is very large,the designed sub-stabilizing controller can be directly useful in many practical situations.In cases where the initial values are indeed needed to be chosen out of the region of attraction,extremely simple pre-controllers can be designed,which drive the system trajectories into the designed region of attraction.A simulation of the designed control system is carried out to show the effect of the proposed approach.
