Showing papers on "Sliding mode control published in 2020"

A New Reaching Law for Antidisturbance Sliding-Mode Control of PMSM Speed Regulation System

Abstract: In this article, in order to optimize the dynamic performance of the permanent magnet synchronous motor (PMSM) speed regulation system, a nonlinear speed-control algorithm for the PMSM control systems using sliding-mode control is developed. First, a sliding-mode control method based on a new sliding-mode reaching law (NSMRL) is proposed. This NSMRL includes the system state variable and the power term of sliding surface function. In particular, the power term is bounded by the absolute value of the switching function, so that the reaching law can be expressed in two different forms during the reaching process. This method can not only effectively suppress the inherent chattering, but also increases the velocity of the system state reaching to the sliding-mode surface. Based on this new reaching law, a sliding-mode speed controller (SMSC) of PMSM is designed. Then, considering the large chattering phenomenon caused by high switching gain, an improved antidisturbance sliding-mode speed controller method, called SMSC + ESO method, is developed. This method introduces an extended state observer to observe the lumped disturbance and adds a feedforward compensation item based on the observed disturbances to the SMSC. Finally, simulation and experimental results both show the validity of the proposed control method.

Finite-Time Sliding-Mode Control of Markovian Jump Cyber-Physical Systems Against Randomly Occurring Injection Attacks

Abstract: This paper addresses a finite-time sliding-mode control problem for a class of Markovian jump cyber-physical systems. It is assumed that the control input signals transmitted via a communication network are vulnerable to cyber-attacks, in which the adversaries may inject false data in a probabilistic way into the control signals. Meanwhile, there may exist randomly occurring uncertainties and peak-bounded external disturbances. A suitable sliding mode controller is designed such that state trajectories are driven onto the specified sliding surface during a given finite-time (possibly short ) interval. By introducing a partitioning strategy, the stochastic finite-time boundedness over the reaching phase and the sliding motion phase is analyzed, respectively. A key feature is that a set of mode-dependent sufficiently small scalars are introduced into some coupled Lyapunov inequalities such that the feasible solutions are easily obtained for the stochastic finite-time boundedness of the closed-loop systems. Finally, the practical system about a single-link robot-arm model is given to illustrate the present method.

Distributed Sliding-Mode Tracking Control of Second-Order Nonlinear Multiagent Systems: An Event-Triggered Approach

Abstract: The event-triggered tracking control problem of second-order multiagent systems in consideration of system nonlinearities is investigated by utilizing the distributed sliding-mode control (SMC) approach. An event-triggered strategy is proposed to decrease the controller sampling frequency and save the network communication resources; the triggering condition is then established for leader-following multiagent systems. In this article, by utilizing the distributed event-based sliding-mode controller, the system state of second-order multiagent systems with system nonlinearities is capable of approaching the integral sliding-mode surface in finite time. A novel integral sliding-mode surface is constructed in this article to guarantee the consensus tracking performance in the existence of system nonlinearities as the state trajectories of second-order integrator systems move on the constructed sliding manifold. By employing the Lyapunov approach, sufficient conditions are deduced to ensure that the consensus tracking performance is obtained for the closed-loop system. Furthermore, it is presented that the triggering scheme can effectively reduce state updates and eliminate the Zeno behavior. A simulation example is provided to testify the validity of our proposed methodology.

Adaptive Global Sliding-Mode Control for Dynamic Systems Using Double Hidden Layer Recurrent Neural Network Structure

Abstract: In this paper, a full-regulated neural network (NN) with a double hidden layer recurrent neural network (DHLRNN) structure is designed, and an adaptive global sliding-mode controller based on the DHLRNN is proposed for a class of dynamic systems. Theoretical guidance and adaptive adjustment mechanism are established to set up the base width and central vector of the Gaussian function in the DHLRNN structure, where six sets of parameters can be adaptively stabilized to their best values according to different inputs. The new DHLRNN can improve the accuracy and generalization ability of the network, reduce the number of network weights, and accelerate the network training speed due to the strong fitting and presentation ability of two-layer activation functions compared with a general NN with a single hidden layer. Since the neurons of input layer can receive signals which come back from the neurons of output layer in the output feedback neural structure, it can possess associative memory and rapid system convergence, achieving better approximation and superior dynamic capability. Simulation and experiment on an active power filter are carried out to indicate the excellent static and dynamic performances of the proposed DHLRNN-based adaptive global sliding-mode controller, verifying its best approximation performance and the most stable internal state compared with other schemes.

Adaptive Speed Control of PMSM Drive System Based a New Sliding-Mode Reaching Law

Abstract: In order to enhance the speed control performance of the permanent magnet synchronous motor (PMSM) with internal and external disturbances, a new adaptive terminal sliding mode reaching law (ATSMRL) is proposed with continuous fast terminal sliding mode control (CFTSMC). Firstly, the ATSMRL is presented with the aim of reducing the input control efforts, which can dynamically adopt all positive aspects in terms of the finite time convergence, high tracking precision, and reduction of the chattering in the control input of the system. Secondly, an extended sliding mode disturbance observer (ESMDO) is designed to estimate the total disturbances of the system, and then the estimated disturbance has been brought for the feed-forward compensation technique, which would enhance the disturbance rejection ability of the system. Afterwards, the close loop stability is validated by the Lyapunov function. Finally the comprehensive numerical and experimental analyses have been carried out to demonstrate the superiority of the ATSMRL method than those of conventional exponential sliding mode reaching law (ESMRL) and terminal sliding mode reaching law (TSMRL).

Sliding Mode Control for Nonlinear Stochastic Singular Semi-Markov Jump Systems

Abstract: This paper deals with the problem of sliding mode control design for nonlinear stochastic singular semi-Markov jump systems (S-MJSs). Stochastic disturbance is first considered in studying S-MJSs with a stochastic semi-Markov process related to Weibull distribution. The specific information including the bound of nonlinearity is known for the control design. Our attention is to design sliding mode control law to attenuate the influences of uncertainty and nonlinear term. First, by the use of the Lyapunov function, a set of sufficient conditions are developed such that the closed-loop sliding mode dynamics are stochastically admissible. Then, the sliding mode control law is proposed to ensure the reachability in a finite-time region. Finally, the practical system about dc motor model is given to verify the validity of the proposed method.

High-Order Disturbance-Observer-Based Sliding Mode Control for Mobile Wheeled Inverted Pendulum Systems

Abstract: In this paper, a novel high-order disturbance observer (HODO) for the mobile wheeled inverted pendulum (MWIP) system is first proposed. Based on a choice method of optimal gain matrices, the estimation accuracy of the HODO can be improved. Combining the proposed HODO and sliding mode control (SMC), a new control strategy is designed for the balance and speed control of the MWIP system. The boundness of the estimation error of HODO is proved and the stability of the closed-loop control system is achieved through the appropriate selection of sliding surface coefficients. The effectiveness of all proposed methods is verified by experiments on a real MWIP system.

Adaptive Control of Nonlinear Semi-Markovian Jump T–S Fuzzy Systems With Immeasurable Premise Variables via Sliding Mode Observer

Abstract: The issue of observer-based adaptive sliding mode control of nonlinear Takagi–Sugeno fuzzy systems with semi-Markov switching and immeasurable premise variables is investigated. More general nonlinear systems are described in the model since the selections of premise variables are the states of the system. First, a novel integral sliding surface function is proposed on the observer space, then the sliding mode dynamics and error dynamics are obtained in accordance with estimated premise variables. Second, sufficient conditions for stochastic stability with an ${H}_{\infty }$ performance disturbance attenuation level ${\gamma }$ of the sliding mode dynamics with different input matrices are obtained based on generally uncertain transition rates. Third, an observer-based adaptive controller is synthesized to ensure the finite time reachability of a predefined sliding surface. Finally, the single-link robot arm model is provided to verify the control scheme numerically.

Conventional and high order sliding mode control

Abstract: Term “Conventional” sliding mode control (SMC) was introduced in the book Sliding Mode Control and Observation [1] by the authors, working in the area of high order sliding mode (HOSM) control The term is related to all publications on n-dimensional systems with m- dimensional control and with sliding modes and state trajectories in a manifold of order n-m Most of publications on HOSM control studied a new phenomenon for systems with a scalar control (m = 1), specifically, the existence of sliding modes in manifolds of dimension lower than n-1 with a finite reaching time Along with implementation issues, it was natural to discuss to what extent the main principles of the conventional theory were to be revised (definitions, existence conditions, motion equations), what new properties of systems with HOSM can be expected Partially these questions were discussed in [2] along with several international workshops on SMC and CDC in 2018 In this paper we demonstrate that main concepts are similar for both types of SMC, discuss new developments needed for HOSM control, compare the potential to suppress chattering, complexity of the both methods

A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method

Abstract: Four-dimensional chaotic systems are a very interesting topic for researchers, given their special features. This paper presents a novel fractional-order four-dimensional chaotic system with self-excited and hidden attractors, which includes only one constant term. The proposed system presents the phenomenon of multi-stability, which means that two or more different dynamics are generated from different initial conditions. It is one of few published works in the last five years belonging to the aforementioned category. Using Lyapunov exponents, the chaotic behavior of the dynamical system is characterized, and the sensitivity of the system to initial conditions is determined. Also, systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits and bifurcation transition diagrams. Moreover, a design technique of a new fuzzy adaptive sliding mode control (FASMC) for synchronization of the fractional-order systems has been offered. This control technique combines an adaptive regulation scheme and a fuzzy logic controller with conventional sliding mode control for the synchronization of fractional-order systems. Applying Lyapunov stability theorem, the proposed control technique ensures that the master and slave chaotic systems are synchronized in the presence of dynamic uncertainties and external disturbances. The proposed control technique not only provides high performance in the presence of the dynamic uncertainties and external disturbances, but also avoids the phenomenon of chattering. Simulation results have been presented to illustrate the effectiveness of the presented control scheme.

Dynamic Terminal Sliding-Mode Control for Single-Phase Active Power Filter Using New Feedback Recurrent Neural Network

Abstract: In this article, an adaptive dynamic terminal sliding-mode controller using a double hidden layer recurrent neural network (DHL-RNN) structure for a single-phase active power filter (APF) is proposed to improve harmonic compensation performance. First, a method combining dynamic sliding mode and terminal sliding mode is proposed to solve the chattering phenomenon in traditional sliding-mode control. Then, since the nonlinear dynamics of APF is difficult to obtain accurately, the DHL-RNN is used to approximate the proposed dynamic terminal sliding-mode controller. Meanwhile, an integral robust switching term is added to eliminate the approximation error of the neural network. Simulation and experimental results proved that the proposed controller has better compensation performance and tracking effect compared with a simple terminal sliding-mode controller.

Finite-Time Control of a Linear Motor Positioner Using Adaptive Recursive Terminal Sliding Mode

Abstract: Payload variations, friction, and external disturbances deteriorate the control performance of linear motor (LM) positioners. To provide high-speed and high-precision performance for the LM, an adaptive recursive terminal sliding-mode (ARTSM) controller is proposed in this article. For the controller, a fast nonsingular terminal sliding function and a recursive integral terminal sliding function are developed in a recursive structure such that the sliding surfaces are arrived successively and ultimately the tracking error can converge to zero in a finite time. Furthermore, by setting an appropriate initial value for the integral element of the ARTSM controller, the control system is enforced to start on the sliding surface at the initial time such that the reaching time is reduced. Stability analysis is presented to prove the finite-time convergence and zero tracking error of the closed-loop system under the proposed ARTSM controller. Experimental results also demonstrate the effectiveness of the controller in terms of significantly reduced tracking errors and faster disturbance rejection in comparison with a recently reported fast nonsingular terminal sliding-mode (FNTSM) controller for the LM positioner.

RBFNN-Based Adaptive Sliding Mode Control Design for Delayed Nonlinear Multilateral Telerobotic System With Cooperative Manipulation

Abstract: Multilateral telerobotic system has potential applications in the industry environments with the advantages of cooperative manipulation for the remote and hazardous tasks, and its control design is quite challenging due to several coupling issues such as stability, position tracking, force feedback, and cooperative manipulation under time delays, various uncertainties, and external disturbance. In this paper, a novel radial basis function neural network (RBFNN) based adaptive sliding mode control design is proposed for nonlinear multilateral telerobotic system with n -master– n -slave manipulators. The environment force is modeled with a general form via the RBFNN-based environment parameters estimation in the slave side. The estimated environment parameters (nonpower signals) are transmitted to rebuild the environment dynamics in the master side and provide the good force feedback for the human operators. The RBFNN-based adaptive sliding mode controllers are designed separately for master and slave manipulators to achieve good position tracking under parameter variations and external disturbance. The coordinated force distribution algorithm is designed to achieve cooperative manipulation with the balance of force acting on the target object. The theoretical analysis is given and the comparative experiment for a nonlinear multilateral telerobotic system with 2-master–2-slave manipulators is implemented. The results show the good performance of our design.

USDE-Based Sliding Mode Control for Servo Mechanisms With Unknown System Dynamics

Abstract: This article proposes an unknown system dynamics estimator (USDE) based sliding mode control for servo mechanisms with unknown dynamics and modeling uncertainties. An invariant manifold is first constructed by introducing an auxiliary variable based on a first-order low-pass filter. This is used to design a USDE with only one tuning parameter (i.e., time constant for the filter) and a simpler structure than other estimators. The USDE is used to compensate for the effect of the lumped unknown system dynamics since it can be easily incorporated into control synthesis. Moreover, to avoid the chattering phenomenon in the conventional sliding mode control methods, a novel reaching law is designed based on hyperbolic functions to guarantee that the sliding mode variable infinitely approaches to the equilibrium point instead of crossing it. Consequently, the fast convergence and chattering-free property can be achieved simultaneously. Simulations and experiments are provided to validate the effectiveness and superior performance of the proposed method.

Active Defense-Based Resilient Sliding Mode Control Under Denial-of-Service Attacks

Abstract: This paper investigates the problem of the resilient control for cyber-physical systems (CPSs) in the presence of malicious sensor denial-of-service (DoS) attacks, which result in the loss of state information. The concepts of DoS frequency and DoS duration are introduced to describe the DoS attacks. According to the attack situation, that is, whether the attack is successfully implemented or not, the original physical system is rewritten as a switched version. A resilient sliding mode control scheme is designed to guarantee that the physical process is exponentially stable, which is a foundation of the main results. Then, a zero-sum game is employed to establish an effective mixed defense mechanism. Furthermore, a defense-based resilient sliding mode control scheme is proposed and the desired control performance is achieved. Compared with the existing results, the differences mainly lie in two aspects, that is, one where a switched model is obtained, based on which the average dwell-time like approach is utilized to derive the resilient control scheme, and the other where the zero-sum game in employed to make the attacks satisfy the concepts of DoS frequency and DoS duration. Finally, simulation results are given to demonstrate the effectiveness of the proposed resilient control approach.

Input-to-State Stabilization of Interval Type-2 Fuzzy Systems Subject to Cyberattacks: An Observer-Based Adaptive Sliding Mode Approach

Abstract: This paper focuses on the sliding mode control (SMC) problem of interval type-2 (IT2) fuzzy systems subject to the unmeasurable state and cyberattacks. A key issue is how to design a state observer under the constraint that only the bounds of membership functions are known. To this end, this paper introduces two weighting factors to construct a new membership function. Besides, the concept of input-to-state stability (ISS) is utilized to deal with the residual term resulting from the cyberattacks and external disturbances. The sufficient condition is established such that the sliding mode dynamics and the estimated error dynamics are input-to-state stable. Furthermore, by online estimating the unknown parameters in upper bounds of cyberattacks and external disturbances, an adaptive sliding mode controller is synthesized such that the reachability of the prescribed sliding surface can be guaranteed and the effect of cyberattacks on the system performance can be effectively attenuated. Finally, the validity of the proposed method is illustrated by a mass–spring–damper system.

Finite-Time Continuous Terminal Sliding Mode Control of Servo Motor Systems

Abstract: In this article, a continuous terminal sliding mode control algorithm is proposed for servo motor systems. A novel full-order terminal sliding mode surface is proposed based on the bilimit homogeneous property, such that the sliding motion is finite-time stable independent of the system's initial condition. A new continuous terminal sliding mode control algorithm is proposed to guarantee that the system states reach the sliding surface in finite-time. Not only the robustness is guaranteed by the proposed controller but also the continuity makes the control algorithm more suitable for the servo mechanical systems. Finally, a numerical example is presented to depict the advantages of the proposed control algorithm. An application in the rotary servo system is done to validate the effectiveness of the proposed control strategy.

Cross-layer congestion control of wireless sensor networks based on fuzzy sliding mode control

Abstract: Wireless sensor networks (WSNs) act as a building block of Internet of Things and have been used in various applications to sense environment and transmit data to the Internet. However, WSNs are very vulnerable to congestion problem, resulting in higher packet loss ratio, longer delay and lower throughput. To address this issue, this paper presents a fuzzy sliding mode congestion control algorithm (FSMC) for WSNs. Firstly, by applying the signal-to-noise ratio of wireless channel to TCP model, a new cross-layer congestion control model between transmission layer and MAC layer is proposed. Then, by combining fuzzy control with sliding mode control (SMC), a fuzzy sliding mode controller (FSMC) is designed, which adaptively regulates the queue length of buffer in congested nodes and significantly reduces the impact of external uncertain disturbance. Finally, numerous simulations are implemented in MATLAB/Simulink and NS-2.35 by comparing with traditional control strategies such as fuzzy, PID and SMC, which show that the proposed FSMC effectively adapts to the change of queue length and has good performance, such as rapid convergence, lower average delay, less packet loss ratio and higher throughput.

Distributed Consensus-Based Fault Tolerant Control of Islanded Microgrids

Abstract: This paper proposes a novel fault tolerant consensus-based secondary voltage and frequency restoration method considering disturbances and actuator faults by using the sliding mode control for islanded microgrids (MGs). Existing distributed methods commonly design restoration layer based on the ideal condition that the actuators of distributed generations (DGs) function healthily and there are no faults and disturbances, whereas MGs are exposed to actuator faults of biased fault and partial loss of effectiveness fault. Faults and disturbances have a great impact on control of MG, which terribly reduce the stability and quality of it. To eliminate the adverse effects of them, this paper explicitly addresses the consensus problem for frequency and voltage restoration of MGs, whereas providing stringent real power sharing, in the presence of actuation/propulsion faults and disturbances. To this end, we derive the consensus restoration proof using rigorous Lyapunov analysis. As a result, the suggested method decreases the sensitivity of the system to failures and increases its reliability. Unlike conventional distributed controllers, the proposed approach quickly reaches consensus and exhibits a more accurate robust performance. Finally, we have performed several simulation scenarios in MATLAB/SimPowerSystems Toolbox to illustrate the efficiency of the theoretical results.

Event-Triggered Sliding Mode Control for Attitude Stabilization of a Rigid Spacecraft

Abstract: Event-triggering strategy is a control implementation technique which intends to minimizing resource usage while achieving the acceptable performance of the closed-loop system. In this paper, an event-triggered sliding mode control (SMC) is designed for attitude stabilization of a rigid spacecraft subject to external disturbances and model uncertainties. A semi-global event-triggering strategy is proposed for SMC to keep the system trajectory in the vicinity of a proposed sliding surface. Since in this sliding surface, the sliding mode is asymptotic stable, the trajectories of the attitude control system remain ultimately bounded under the circumstance of disturbances and uncertainties. Moreover, the given event-triggered control implementation is only dependent on the angular velocity at sampling instant and triggering strategy with Zeno phenomenon can be avoided. Some numerical simulations are shown to illustrate the availability of the obtained results.

Self-tuning fuzzy PID-nonsingular fast terminal sliding mode control for robust fault tolerant control of robot manipulators

Abstract: In this work, a new robust controller is developed for robot manipulator based on an integrating between a novel self-tuning fuzzy proportional-integral-derivative (PID)-nonsingular fast terminal sliding mode control (STF-PID-NFTSM) and a time delay estimation (TDE). A sliding surface based on the PID-NFTSM is designed for robot manipulators to get multiple excited features such as faster transient response with finite time convergence, lower error at steady-state and chattering elimination. However, the system characteristics are hugely affected by the selection of the PID gains of the controller. In addition, the design of the controller requires an exact dynamics model of the robot manipulators. In order to obtain effective gains for the PID sliding surface, a fuzzy logic system is employed and in order to get an estimation of the unknown dynamics model, a TDE algorithm is developed. The innovative features of the proposed approach, i.e., STF-PID-NFTSM, is verified when comparing with other up-to-date advanced control techniques on a PUMA560 robot.

Lane Keeping Control of Autonomous Vehicles With Prescribed Performance Considering the Rollover Prevention and Input Saturation

Abstract: This paper investigates the lane keeping control of autonomous ground vehicles (AGVs) considering the rollover prevention and input saturation An enhanced state observer-based sliding mode control (SMC) strategy is proposed to achieve the control purpose and maintain the lane keeping errors as well as the roll angle within the prescribed performance boundaries Three contributions are made in this paper First, a prescribed performance function (PPF) is proposed in the controller design, aiming to implement the error transformation so as to constrain the controlled variables within the prescribed performance boundaries Second, a modified sliding surface is developed incorporating two nonlinear functions, whose specialities and benefits are taken advantage of: one is a barrier function to restrict the load transfer ratio (LTR) in a safe boundary to guarantee the roll stability; another is a monotonely decreasing function to adaptively change the damping ratio of the closed-loop system to improve the transient performance, including reducing the transient overshoots and steady-state errors Third, a modified multivariable adaptive SMC controller is proposed to achieve the integrated lane-keeping and roll control in the presence of the input saturation and bound-unknown disturbances The stability of the closed-loop system is rigorously proved via the Lyapunov function Finally, the effectiveness of the proposed control strategy is verified with a high-fidelity and full-car model via the CarSim platform

Double Hidden Layer Output Feedback Neural Adaptive Global Sliding Mode Control of Active Power Filter

Abstract: In this paper, a self-regulated double hidden layer output feedback neural network (DHLFNN) is presented to control an active power filter (APF) system as a current controller, which is conducive to the improvement of the response characteristic and power quality. First, a global sliding mode controller is introduced because it is effective in achieving overall robustness during the system response. A new output feedback neural structure that has two hidden layers is proposed to make the parameters adaptively adjust themselves and stabilize to their best values. A higher accuracy and stronger generalization ability can be also obtained by reducing the number of network weights and accelerating the network training speed owing to the strong fitting and presentation ability of two-layer activation functions. Furthermore, the designed feedback loops of the neural network play a significant role in possessing associative memory and rapid system convergence. This proposed double hidden layer output feedback neural based global sliding mode controller is simulated on the model of APF and the results show the excellent static and dynamic properties. Experimental results under three cases and comparisons are provided using a fully digital control system to validate the superior performance of the proposed DHLFNN controller.

Discrete-Time Fast Terminal Sliding Mode Control Design for DC–DC Buck Converters With Mismatched Disturbances

Abstract: DC–DC converter systems have drawn extensive research attentions and shown upward tendencies for industrial and military applications. In this article, a novel digital fast terminal sliding mode control (FTSMC) approach is investigated for dc–dc buck converters with mismatched disturbances. Specifically, the approximated discrete-time model of the converters with multiple disturbances is first obtained and analyzed based on the Euler's discretization method. Then, by adopting the delayed estimation technique, it is easy to obtain the accurate estimations of the lumped disturbances. Integrating disturbance compensations into the modified digital fast terminal sliding mode surface, the proposed controller is finally constructed on the basis of equivalent control method and the performance analysis is presented. Both simulation and experimental comparisons are made for the proposed digital FTSMC approach and the existing discretized linear sliding mode controllers to validate the effectiveness and feasibility of the presented controller. The proposed FTSMC approach is characterized by higher voltage tracking accuracy and better dynamic properties in different operating conditions.

An Event-Triggered Approach to Sliding Mode Control of Markovian Jump Lur’e Systems Under Hidden Mode Detections

Abstract: The asynchronous sliding mode control (SMC) problem is investigated for networked Markovian jump Lur’e systems, in which the information of system modes is unavailable to the sliding mode controller but could be estimated by a mode detector via a hidden Markov model (HMM). In order to mitigate the burden of data communication, an event-triggered protocol is proposed to determine whether the system state should be released to the controller at certain time-point according to a specific triggering condition. By constructing a novel common sliding surface, this paper designs an event-triggered asynchronous SMC law, which just depends on the hidden mode information. A combination of the stochastic Lur’e-type Lyapunov functional and the HMM approach is exploited to establish the sufficient conditions of the mean square stability with a prescribed ${H}_{\infty }$ performance and the reachability of a sliding region around the specified sliding surface. Moreover, the solving algorithm for the control gain matrices is given via a convex optimization problem. Finally, an example from the dc motor device system is provided.