Based on the theory of Markov performance potentials and neuro-dynamic programming(NDP) methodology, we study simulation optimization algorithm for a class of continuous timeMarkov decision processes (CTMDPs) under ra...Based on the theory of Markov performance potentials and neuro-dynamic programming(NDP) methodology, we study simulation optimization algorithm for a class of continuous timeMarkov decision processes (CTMDPs) under randomized stationary policies. The proposed algo-rithm will estimate the gradient of average cost performance measure with respect to policy param-eters by transforming a continuous time Markov process into a uniform Markov chain and simula-ting a single sample path of the chain. The goal is to look for a suboptimal randomized stationarypolicy. The algorithm derived here can meet the needs of performance optimization of many diffi-cult systems with large-scale state space. Finally, a numerical example for a controlled Markovprocess is provided.展开更多
A new discretization scheme is proposed for the design of a fractional order PID controller. In the design of a fractional order controller the interest is mainly focused on the s-domain, but there exists a difficult ...A new discretization scheme is proposed for the design of a fractional order PID controller. In the design of a fractional order controller the interest is mainly focused on the s-domain, but there exists a difficult problem in the s-domain that needs to be solved, i.e. how to calculate fractional derivatives and integrals efficiently and quickly. Our scheme adopts the time domain that is well suited for Z-transform analysis and digital implementation. The main idea of the scheme is based on the definition of Grünwald-Letnicov fractional calculus. In this case some limited terms of the definition are taken so that it is much easier and faster to calculate fractional derivatives and integrals in the time domain or z-domain without loss much of the precision. Its effectiveness is illustrated by discretization of half-order fractional differential and integral operators compared with that of the analytical scheme. An example of designing fractional order digital controllers is included for illustration, in which different fractional order PID controllers are designed for the control of a nonlinear dynamic system containing one of the four different kinds of nonlinear blocks: saturation, deadzone, hysteresis, and relay.展开更多
Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of qua...Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrdinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrdinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrdinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.展开更多
The variation of plant dead-time deeply a?ects the stability of the predictive PI controlsystem. It is important for designing and applying the PPI controller to calculate the delay margin.A criterion of stability for...The variation of plant dead-time deeply a?ects the stability of the predictive PI controlsystem. It is important for designing and applying the PPI controller to calculate the delay margin.A criterion of stability for the PPI system and the quantitive relationship among the delay margin,the time constant of the closed-loop system, and the dead-time of the model are given. A graphicalgorithm to compute the delay margin is also developed. The phenomenon that there exist morethan one stability delay zones is discussed. The algorithm is shown to be precise by some simulations.展开更多
The H∞ hybrid estimation problem for linear continuous time-varying systems is in-vestigated in this paper, where estimated signals are linear combination of state and input. Designobjective requires the worst-case e...The H∞ hybrid estimation problem for linear continuous time-varying systems is in-vestigated in this paper, where estimated signals are linear combination of state and input. Designobjective requires the worst-case energy gain from disturbance to estimation error be less than a pre-scribed level. Optimal solution of the hybrid estimation problem is the saddle point of a two-playerzero sum di?erential game. Based on the di?erential game approach, necessary and su?cient solvableconditions for the hybrid estimation problem are provided in terms of solutions to a Riccati di?e-rential equation. Moreover, one possible estimator is proposed if the solvable conditions are satisfied.The estimator is characterized by a gain matrix and an output mapping matrix that re?ects theinternal relations between the unknown input and output estimation error. Both state and unknowninputs estimation are realized by the proposed estimator. Thus, the results in this paper are alsocapable of dealing with fault diagnosis problems of linear time-varying systems. At last, a numericalexample is provided to illustrate the proposed approach.展开更多
The problem of robust H ∞ fuzzy state feedback control for uncertain fuzzy descriptor systems with state delay is solved. In the case that time-varying uncertainties are in all parameter matrices, sufficient conditi...The problem of robust H ∞ fuzzy state feedback control for uncertain fuzzy descriptor systems with state delay is solved. In the case that time-varying uncertainties are in all parameter matrices, sufficient conditions for the existence of fuzzy state feedback controller are presented in terms of linear matrix inequality (LMI). The proposed robust H ∞ control laws guarantee that the resulting closed-loop system is regular, impulse free, and stable with prescribed H ∞ norm bounded constraint for all admissible uncertainties. Finally, a numerical example is provided to demonstrate the validity of the proposed method.展开更多
The design objective of a mixed H2/H∞ control is to find the H2 optimal control law trader aprescribed disturbance attenuation level. This paper addresses a optimal PID control law on the basis of the newsolution to ...The design objective of a mixed H2/H∞ control is to find the H2 optimal control law trader aprescribed disturbance attenuation level. This paper addresses a optimal PID control law on the basis of the newsolution to mixed H2/H∞ optimal control problem that provide much more flexible design compared to the existing works. Then a closed-form PID controller to mixed H2/H∞ robotic tracking problem is simply constructed and hence the design procedure is presented. Finally, numerical simulations illustrate the effectiveness of the optimal PID feedback design proposed in this paper via a two-link robotic manipulator.展开更多
A new simple linear tracking-differentiator (T-D) is presented. Stability and convergence of the T-D are thoroughly proved, which avoids inconvenience of stability proof for nonlinear T-D with switch function. Through...A new simple linear tracking-differentiator (T-D) is presented. Stability and convergence of the T-D are thoroughly proved, which avoids inconvenience of stability proof for nonlinear T-D with switch function. Through error analysis, the presented T-D can extract differential of any piecewise smooth nonlinear signals to reach a high accuracy. Compared with the nonlinear T-D with switch function, shaking phenomenon is avoided at the switching point of the function. By further modifying the linear T-D, it can effectively restrain differential impulses produced by initial errors of signals or by signals suddenly changing. Finally, analysis of several simulation examples shows the effectiveness of the proposed theory.展开更多
In this paper, we address the problem of robust H∞ filter design for a class of nonlinear discrete-time systems with norm-bounded parameter uncertainty, unknown state delay, and unknown nonlinear disturbance. Firstly...In this paper, we address the problem of robust H∞ filter design for a class of nonlinear discrete-time systems with norm-bounded parameter uncertainty, unknown state delay, and unknown nonlinear disturbance. Firstly, the suffident condition of the existence of the robust filter is analyzed by the modified Riceati inequalities, such that the filtering process remains robustly stable and a prescribed H∞ performance lever is achieved. Then the filter is characterized in the terms of positive solutions of two algebraic Riccati-like equations, irrespective of the uncertainties, time delays and nonlinearities. The simulation example further shows the validity and the applicability of the proposed approach.展开更多
The sufficient conditions of stability for uncertain discrete-time systems with state delay have been proposed by some researchers in the past few years, yet these results may be conservative in application. The stabi...The sufficient conditions of stability for uncertain discrete-time systems with state delay have been proposed by some researchers in the past few years, yet these results may be conservative in application. The stability analysis of these systems is discussed, and the necessary and sufficient condition of stability is derived by method other than constructing Lyapunov function and solving Riccati inequality. The root locations of system characteristic polynomial, which is obtained by augmentation approach and Laplace expansion, determine the stability of uncertain discrete-time systems with state delay, the system is stable if and only if all roots lie within the unit circle. In order to analyze robust stability of system characteristic polynomial effectively, Kharitonov theorem and edge theorem are applied. Example shows the practicability of these methods.展开更多
This paper presents brief science history and applications developments of life system of life science about fluid system simulation with some typical examples to show their meanings and results.Finally to discuss the...This paper presents brief science history and applications developments of life system of life science about fluid system simulation with some typical examples to show their meanings and results.Finally to discuss the coming era of nano-bio-info-cogno trends of the 21 st century development.展开更多
Based on steady-state Kalman filter and white noise estimators, and according to thepole-assignment principle of the control theory, the pole-assignment fixed-interval steady-stateKalman smoother and Wiener smoother a...Based on steady-state Kalman filter and white noise estimators, and according to thepole-assignment principle of the control theory, the pole-assignment fixed-interval steady-stateKalman smoother and Wiener smoother are presented. They avoid computation of the initial opti-mal smoothing estimates and can rapidly eliminate the effects of the initial smoothing estimates byassigning the poles of the smoothers, so that they have a practical stability in the finite fixed inter-val. A simulation example shows their effectiveness.展开更多
A novel variable structure control (VSC) with new rapid-smooth reaching law (RSRL) and new rapid-convergent sliding mode (FCSM) is proposed, which is based on analysis of normal VSC system. When it is used for an MIMO...A novel variable structure control (VSC) with new rapid-smooth reaching law (RSRL) and new rapid-convergent sliding mode (FCSM) is proposed, which is based on analysis of normal VSC system. When it is used for an MIMO nonlinear system, we combine the method of Input/Output linearizing (I/O L) with VSC. After analyzing the robustness of the MIMO nonlinear system, we use this novel controller for Precision One Robot position control system. Simulation provides a quite satisfactory performance with uncertainties and external disturbances.展开更多
In order to solve the constraint satisfied problem in the genetic algorithm, the partheno-genetic algorithm is designed. And then the schema theorem of the partheno-genetic algorithm is proposed to show that the high ...In order to solve the constraint satisfied problem in the genetic algorithm, the partheno-genetic algorithm is designed. And then the schema theorem of the partheno-genetic algorithm is proposed to show that the high rank schemas at the subsequent generation decrease exponentially even though its fitness is more optimal than the average one in the population and the low rank schemas at the subsequent generation increase exponentially when its fitness is more optimal than the average one in the population. In order to overcome the shortcoming that the optimal high rank schema can be deserted arbitrarily, the HGA (hybrid partheno-genetic algorithm) is proposed, that is, the hill-climbing algorithm is integrated to search for a better individual. Finally, the results of the simulation for facility layout problem and no-wait schedule problem are given. It is shown that the hybrid partheno- genetic algorithm is of high efficiency.展开更多
It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the u...It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the unknown parameters and controlling the chaos, which is not closely related to the particular chaotic system to be controlled. The global uniform boundedness of estimated parameters and the asymptotical stability of the tracking errors are proved by Lyapunov stability theory and LaSalleYoshizawa theorem. The suggested method enables stabilization of chaotic motion to a steady state ad well as tracking of any desired trajectory to be achieved in a systematic way. Computer simulation on a complex chaotic system illustrtes the effectiveness of the proposed control method.展开更多
The paper consists of three topics on control theory and engineering applications, namely bifurcation control, manufacturing planning, and formation control. For each topic, we summarize the control problem to be addr...The paper consists of three topics on control theory and engineering applications, namely bifurcation control, manufacturing planning, and formation control. For each topic, we summarize the control problem to be addressed and some key ideas used in our recent research. Interested readers are referred to related publications for more details. Each of the three topics in this paper is technically independent from the other ones. However, all three parts together reflect the recent research activities of the first author, jointly with other researchers in different fields.展开更多
This paper proposes a novel drifting modeling (DM) method. Briefly, we first employ an improved SVMs algorithm named weighted support vector machines (W.SVMs), which is suitable for locally learning, and then the DM m...This paper proposes a novel drifting modeling (DM) method. Briefly, we first employ an improved SVMs algorithm named weighted support vector machines (W.SVMs), which is suitable for locally learning, and then the DM method using the algorithm is proposed. By applying the proposed modeling method to Fluidized Catalytic Cracking Unit (FCCU), the simulation results show that the property of this proposed approach is superior to global modeling method based on standard SVMs.展开更多
The dynamics of 2DOF spherical inverted pendulum system is analyzed. The motion of the pendulum may be projected onto the orthogonal planes in the Cartesian Space. In this way the system can be decoupled into two clas...The dynamics of 2DOF spherical inverted pendulum system is analyzed. The motion of the pendulum may be projected onto the orthogonal planes in the Cartesian Space. In this way the system can be decoupled into two classical cart-pendulum systems and the design of controllers aimed at each subsystem separately are proposed. The linear quadratic optimal control strategy is applied in order to balance the pendulum system at the 'inverted' status. The method proposed is verified by the simulation and actual system experiments and the performance of the controller is discussed.展开更多
This paper presents an approach that is useful for the identification of a fuzzy model in SISO system. The initial values of cluster centers are identified by the Hough transformation, which considers the linearity an...This paper presents an approach that is useful for the identification of a fuzzy model in SISO system. The initial values of cluster centers are identified by the Hough transformation, which considers the linearity and continuity of given input-output data, respectively. For the premise parts parameters identification, we use fuzzy-C-means clustering method. The consequent parameters are identified based on recursive least square. This method not only makes approximation more accurate, but also let computation be simpler and the procedure is realized more easily. Finally, it is shown that this method is useful for the identification of a fuzzy model by simulation.展开更多
The controllability for a class of fractional-order linear control systems is mainly investigated. The generalizations of the usual complete solution formulae of the fractional-order linear control systems are derived...The controllability for a class of fractional-order linear control systems is mainly investigated. The generalizations of the usual complete solution formulae of the fractional-order linear control systems are derived not only for time-invariant case but also for time-varying case. Several sufficient and necessary conditions for state controllability of such systems are established and the corresponding criteria for fractional-order time-invariant continuous-time systems are also obtained. The results obtained will be help for future study of fractional-order control systems.展开更多
文摘Based on the theory of Markov performance potentials and neuro-dynamic programming(NDP) methodology, we study simulation optimization algorithm for a class of continuous timeMarkov decision processes (CTMDPs) under randomized stationary policies. The proposed algo-rithm will estimate the gradient of average cost performance measure with respect to policy param-eters by transforming a continuous time Markov process into a uniform Markov chain and simula-ting a single sample path of the chain. The goal is to look for a suboptimal randomized stationarypolicy. The algorithm derived here can meet the needs of performance optimization of many diffi-cult systems with large-scale state space. Finally, a numerical example for a controlled Markovprocess is provided.
文摘A new discretization scheme is proposed for the design of a fractional order PID controller. In the design of a fractional order controller the interest is mainly focused on the s-domain, but there exists a difficult problem in the s-domain that needs to be solved, i.e. how to calculate fractional derivatives and integrals efficiently and quickly. Our scheme adopts the time domain that is well suited for Z-transform analysis and digital implementation. The main idea of the scheme is based on the definition of Grünwald-Letnicov fractional calculus. In this case some limited terms of the definition are taken so that it is much easier and faster to calculate fractional derivatives and integrals in the time domain or z-domain without loss much of the precision. Its effectiveness is illustrated by discretization of half-order fractional differential and integral operators compared with that of the analytical scheme. An example of designing fractional order digital controllers is included for illustration, in which different fractional order PID controllers are designed for the control of a nonlinear dynamic system containing one of the four different kinds of nonlinear blocks: saturation, deadzone, hysteresis, and relay.
文摘Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrdinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrdinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrdinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.
文摘The variation of plant dead-time deeply a?ects the stability of the predictive PI controlsystem. It is important for designing and applying the PPI controller to calculate the delay margin.A criterion of stability for the PPI system and the quantitive relationship among the delay margin,the time constant of the closed-loop system, and the dead-time of the model are given. A graphicalgorithm to compute the delay margin is also developed. The phenomenon that there exist morethan one stability delay zones is discussed. The algorithm is shown to be precise by some simulations.
文摘The H∞ hybrid estimation problem for linear continuous time-varying systems is in-vestigated in this paper, where estimated signals are linear combination of state and input. Designobjective requires the worst-case energy gain from disturbance to estimation error be less than a pre-scribed level. Optimal solution of the hybrid estimation problem is the saddle point of a two-playerzero sum di?erential game. Based on the di?erential game approach, necessary and su?cient solvableconditions for the hybrid estimation problem are provided in terms of solutions to a Riccati di?e-rential equation. Moreover, one possible estimator is proposed if the solvable conditions are satisfied.The estimator is characterized by a gain matrix and an output mapping matrix that re?ects theinternal relations between the unknown input and output estimation error. Both state and unknowninputs estimation are realized by the proposed estimator. Thus, the results in this paper are alsocapable of dealing with fault diagnosis problems of linear time-varying systems. At last, a numericalexample is provided to illustrate the proposed approach.
文摘The problem of robust H ∞ fuzzy state feedback control for uncertain fuzzy descriptor systems with state delay is solved. In the case that time-varying uncertainties are in all parameter matrices, sufficient conditions for the existence of fuzzy state feedback controller are presented in terms of linear matrix inequality (LMI). The proposed robust H ∞ control laws guarantee that the resulting closed-loop system is regular, impulse free, and stable with prescribed H ∞ norm bounded constraint for all admissible uncertainties. Finally, a numerical example is provided to demonstrate the validity of the proposed method.
文摘The design objective of a mixed H2/H∞ control is to find the H2 optimal control law trader aprescribed disturbance attenuation level. This paper addresses a optimal PID control law on the basis of the newsolution to mixed H2/H∞ optimal control problem that provide much more flexible design compared to the existing works. Then a closed-form PID controller to mixed H2/H∞ robotic tracking problem is simply constructed and hence the design procedure is presented. Finally, numerical simulations illustrate the effectiveness of the optimal PID feedback design proposed in this paper via a two-link robotic manipulator.
文摘A new simple linear tracking-differentiator (T-D) is presented. Stability and convergence of the T-D are thoroughly proved, which avoids inconvenience of stability proof for nonlinear T-D with switch function. Through error analysis, the presented T-D can extract differential of any piecewise smooth nonlinear signals to reach a high accuracy. Compared with the nonlinear T-D with switch function, shaking phenomenon is avoided at the switching point of the function. By further modifying the linear T-D, it can effectively restrain differential impulses produced by initial errors of signals or by signals suddenly changing. Finally, analysis of several simulation examples shows the effectiveness of the proposed theory.
文摘In this paper, we address the problem of robust H∞ filter design for a class of nonlinear discrete-time systems with norm-bounded parameter uncertainty, unknown state delay, and unknown nonlinear disturbance. Firstly, the suffident condition of the existence of the robust filter is analyzed by the modified Riceati inequalities, such that the filtering process remains robustly stable and a prescribed H∞ performance lever is achieved. Then the filter is characterized in the terms of positive solutions of two algebraic Riccati-like equations, irrespective of the uncertainties, time delays and nonlinearities. The simulation example further shows the validity and the applicability of the proposed approach.
基金This project was supported by National "863" High Technology Research and Development Program of China (2001-AA413130) and the National Key Research Project (2001-BA201A04).
文摘The sufficient conditions of stability for uncertain discrete-time systems with state delay have been proposed by some researchers in the past few years, yet these results may be conservative in application. The stability analysis of these systems is discussed, and the necessary and sufficient condition of stability is derived by method other than constructing Lyapunov function and solving Riccati inequality. The root locations of system characteristic polynomial, which is obtained by augmentation approach and Laplace expansion, determine the stability of uncertain discrete-time systems with state delay, the system is stable if and only if all roots lie within the unit circle. In order to analyze robust stability of system characteristic polynomial effectively, Kharitonov theorem and edge theorem are applied. Example shows the practicability of these methods.
文摘This paper presents brief science history and applications developments of life system of life science about fluid system simulation with some typical examples to show their meanings and results.Finally to discuss the coming era of nano-bio-info-cogno trends of the 21 st century development.
文摘Based on steady-state Kalman filter and white noise estimators, and according to thepole-assignment principle of the control theory, the pole-assignment fixed-interval steady-stateKalman smoother and Wiener smoother are presented. They avoid computation of the initial opti-mal smoothing estimates and can rapidly eliminate the effects of the initial smoothing estimates byassigning the poles of the smoothers, so that they have a practical stability in the finite fixed inter-val. A simulation example shows their effectiveness.
文摘A novel variable structure control (VSC) with new rapid-smooth reaching law (RSRL) and new rapid-convergent sliding mode (FCSM) is proposed, which is based on analysis of normal VSC system. When it is used for an MIMO nonlinear system, we combine the method of Input/Output linearizing (I/O L) with VSC. After analyzing the robustness of the MIMO nonlinear system, we use this novel controller for Precision One Robot position control system. Simulation provides a quite satisfactory performance with uncertainties and external disturbances.
文摘In order to solve the constraint satisfied problem in the genetic algorithm, the partheno-genetic algorithm is designed. And then the schema theorem of the partheno-genetic algorithm is proposed to show that the high rank schemas at the subsequent generation decrease exponentially even though its fitness is more optimal than the average one in the population and the low rank schemas at the subsequent generation increase exponentially when its fitness is more optimal than the average one in the population. In order to overcome the shortcoming that the optimal high rank schema can be deserted arbitrarily, the HGA (hybrid partheno-genetic algorithm) is proposed, that is, the hill-climbing algorithm is integrated to search for a better individual. Finally, the results of the simulation for facility layout problem and no-wait schedule problem are given. It is shown that the hybrid partheno- genetic algorithm is of high efficiency.
文摘It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the unknown parameters and controlling the chaos, which is not closely related to the particular chaotic system to be controlled. The global uniform boundedness of estimated parameters and the asymptotical stability of the tracking errors are proved by Lyapunov stability theory and LaSalleYoshizawa theorem. The suggested method enables stabilization of chaotic motion to a steady state ad well as tracking of any desired trajectory to be achieved in a systematic way. Computer simulation on a complex chaotic system illustrtes the effectiveness of the proposed control method.
基金Supported in part by Ford Motor Company, U.S. Air Force Research Laboratory, and National Science Foundation
文摘The paper consists of three topics on control theory and engineering applications, namely bifurcation control, manufacturing planning, and formation control. For each topic, we summarize the control problem to be addressed and some key ideas used in our recent research. Interested readers are referred to related publications for more details. Each of the three topics in this paper is technically independent from the other ones. However, all three parts together reflect the recent research activities of the first author, jointly with other researchers in different fields.
文摘This paper proposes a novel drifting modeling (DM) method. Briefly, we first employ an improved SVMs algorithm named weighted support vector machines (W.SVMs), which is suitable for locally learning, and then the DM method using the algorithm is proposed. By applying the proposed modeling method to Fluidized Catalytic Cracking Unit (FCCU), the simulation results show that the property of this proposed approach is superior to global modeling method based on standard SVMs.
文摘The dynamics of 2DOF spherical inverted pendulum system is analyzed. The motion of the pendulum may be projected onto the orthogonal planes in the Cartesian Space. In this way the system can be decoupled into two classical cart-pendulum systems and the design of controllers aimed at each subsystem separately are proposed. The linear quadratic optimal control strategy is applied in order to balance the pendulum system at the 'inverted' status. The method proposed is verified by the simulation and actual system experiments and the performance of the controller is discussed.
基金This project was supported by the Natural Science Foundation of Heilongjiang province and Doctor Foundation of Yanshan U-niversity.
文摘This paper presents an approach that is useful for the identification of a fuzzy model in SISO system. The initial values of cluster centers are identified by the Hough transformation, which considers the linearity and continuity of given input-output data, respectively. For the premise parts parameters identification, we use fuzzy-C-means clustering method. The consequent parameters are identified based on recursive least square. This method not only makes approximation more accurate, but also let computation be simpler and the procedure is realized more easily. Finally, it is shown that this method is useful for the identification of a fuzzy model by simulation.
文摘The controllability for a class of fractional-order linear control systems is mainly investigated. The generalizations of the usual complete solution formulae of the fractional-order linear control systems are derived not only for time-invariant case but also for time-varying case. Several sufficient and necessary conditions for state controllability of such systems are established and the corresponding criteria for fractional-order time-invariant continuous-time systems are also obtained. The results obtained will be help for future study of fractional-order control systems.
