Cart-pendulum System Research Articles

Stability and Stabilization of a Class of Discrete-Time Fuzzy Systems With Semi-Markov Stochastic Uncertainties

This paper addresses the problems of stability and stabilization for a class of discrete-time Takagi–Sugeno fuzzy systems with semi-Markov stochastic uncertainties. By means of the semi-Markov kernel, the probability density function of the sojourn time for different modes in describing the underlying semi-Markov stochastic uncertainties can be mode-dependent in contrast with the previous studies. Both the sojourn-time-independent and sojourn-time-dependent Lyapunov functions are proposed, by which the stability criteria are obtained in terms of a new $\boldsymbol {\sigma }$ -error mean square stability concept, and the latter is demonstrated to be less conservative and more practical than some existing results. Then, control synthesis problem is investigated and the sufficient conditions on the existence of admissible mode-dependent state-feedback stabilizing controller are developed. A numerical example and a cart-pendulum system are given to show the effectiveness and potential of the new design techniques.

Stabilization of Mechanical Systems with Backlash by PI Loop Shaping

Backlash is one of several discontinuities found in different kinds of systems, it can be found in actuators of different types, such as mechanical and hydraulic, giving way to unwanted effects in the system behavior. PI loop shaping control design implementing a describing function to find the limit cycle oscillations and the appropriate control gain is developed. Therefore a frequency domain approach is implemented for the control of nonlinear system of any kind such as robotics, mechatronics, other kind of mechanisms, electrical motors etc. Finally, in order to corroborate the theoretical background explained in this article, the stabilization of a cart-pendulum system with the proposed control strategy is shown.

Robust constrained Model Predictive Control of fast electromechanical systems

A major drawback hinders the application of Model Predictive Control (MPC) to the regulation of electromechanical systems or, more generally, systems with fast dynamics: the time needed for the online computation of the control is often too long with respect to the sampling time. This paper shows how this problem can be overcome by suitably implementing the MPC technique. The main idea is to compute the control law using the discrete-time Euler Auxiliary System (EAS) associated with the continuous-time plant, and apply the control obtained for the discrete-time system to the continuous-time system. In this way the implementation sampling time can be much smaller than the EAS time parameter, which leads to significant savings in computation time. Theoretical results guarantee stabilisation, constraint satisfaction and robustness of such a control strategy, which is applied to the control of an electric drive and a cart-pendulum system.

Modeling and Stabilization Fuzzy Control Strategies of a Nonlinear Cart-Pendulum System: a Comparative Study

In this paper, we propose and validate an unsimplified nonlinear state-space model of the cart-pendulum system. Then, we present several fuzzy control strategies in order to stabilize and control both the pendulum angle in upright position and the cart position into a desired position. These strategies are built upon two separate fuzzy logic controllers (FLC), having the same characteristics. To achieve this goal, we used various structures of two-input fuzzy controllers such as fuzzy proportional-derivative (FPD), fuzzy proportional-integral (FPI), FPI plus FPD and FPI derivative (FPID), designed by two basic fuzzy inference systems, i.e., Mamdani or zero-order Takagi-Sugeno. Then, the effectiveness of different strategies was tested and compared using several performance integration indices. Many tests were carried out and the obtained simulation results proved that the strategy, based on the zero-order Takagi-Sugeno FPID structures, could be applied to control the coupled variables of a nonlinear system. Furthermore, the proposed unsimplified nonlinear state-space model of the cart-pendulum system, might be used for validating new nonlinear control strategies.

State and output feedback control for Takagi–Sugeno systems with saturated actuators

SummaryIn the present paper, a Takagi–Sugeno (TS) model is used to simultaneously represent the behaviour of a nonlinear system and its saturated actuators. With the TS formalism and the Lyapunov approach, stabilization conditions are expressed as linear matrix inequalities for different controller designs. Static parallel distributed compensation (PDC) state feedback and static and dynamic PDC output feedback controllers for nonlinear saturated systems are proposed. The descriptor approach is used to obtain relaxed conditions to compute the controller gains. The nonlinear cart–pendulum system is used to illustrate the proposed control laws. Copyright © 2015 John Wiley & Sons, Ltd.

Dynamic quantization of uncertain linear networked control systems

This paper is concerned with the stability analysis of networked control systems with dynamic quantization, variable sampling intervals and communication delays. A time-triggered zooming algorithm for the dynamic quantization at the sensor side is proposed that leads to an exponentially stable closed-loop system. The algorithm includes proper initialization of the zoom parameter. More precisely, given a bound on the state initial conditions and the values of the quantizer range and error, we derive conditions for finding the initial value of the zoom parameter, starting from which the exponential stability is guaranteed by using “zooming-in” only. Polytopic type uncertainties in the system model can be easily included in the analysis. The efficiency of the method is illustrated on an example of an uncertain cart–pendulum system.

State feedback with fractional integral control design based on the Bode’s ideal transfer function

State feedback technique through a gain matrix has been a well-known method for pole assignment of a linear system. The technique could encounter a difficulty in eliminating the steady-state errors in some states. Introducing an integral element can effectively eliminate these errors. State feedback with fractional integral control is proposed, in this work, for pole placement of a linear time invariant system. The proposed method yields simple gain formulae. The paper presents the derivation of the design formulae. The method is applied to stabilise an inherently unstable inverted pendulum-cart system. Simulation and experimental results show the effectiveness of the proposed method for set-point tracking, disturbance rejection and stabilising the inverted pendulum. Comparison with the results obtained from applying Achermann’s formula is also presented.

Robust Takagi-Sugeno Fuzzy Dynamic Regulator for Trajectory Tracking of a Pendulum-Cart System

Starting from a nonlinear model for a pendulum-cart system, on which viscous friction is considered, a Takagi-Sugeno (T-S) fuzzy augmented model (TSFAM) as well as a TSFAM with uncertainty (TSFAMwU) is proposed. Since the design of a T-S fuzzy controller is based on the T-S fuzzy model of the nonlinear system, then, to address the trajectory tracking problem of the pendulum-cart system, three T-S fuzzy controllers are proposed via parallel distributed compensation: (1) a T-S fuzzy servo controller (TSFSC) designed from the TSFAM; (2) a robust TSFSC (RTSFSC) designed from the TSFAMwU; and (3) a robust T-S fuzzy dynamic regulator (RTSFDR) designed from the RTSFSC with the addition of a T-S fuzzy observer, which estimates cart and pendulum velocities. Both TSFAM and TSFAMwU are comprised of two fuzzy rules and designed via local approximation in fuzzy partition spaces technique. Feedback gains for the three fuzzy controllers are obtained via linear matrix inequalities approach. A swing-up controller is developed to swing the pendulum up from its pendant position to its upright position. Real-time experiments validate the effectiveness of the proposed schemes, keeping the pendulum in its upright position while the cart follows a reference signal, standing out the RTSFDR.

Risk-Sensitive Model Predictive Control with Gaussian Process Models

This paper proposes the use of risk-sensitive costs in a model predictive controller (MPC) with Gaussian process (GP) models, for more effective online learning and control. Being a probabilistic model, a GP incorporates the uncertainty information due to imperfect knowledge of the system. The MPC then utilises this uncertainty information in a risk-sensitive, especially risk-seeking fashion, to balance the exploration of the unknown characteristics and the exploitative control actions simultaneously. Comparison of MPCs with the risk-seeking cost and the risk-neutral cost, i.e. the standard quadratic cost, on the swing-up control of a cart-pendulum system demonstrates that the risk-seeking cost exhibits an effective exploratory behaviour which leads to a better learning of the unknown system and in turn gives improved control performance.

Discrete‐time network‐based control under scheduling and actuator constraints

SummaryThis paper is concerned with the solution bounds for discrete‐time networked control systems via delay‐dependent Lyapunov–Krasovskii methods. Solution bounds are widely used for systems with input saturation caused by actuator saturation or by the quantizers with saturation. The time‐delay approach has been developed recently for the stabilization of continuous‐time networked control systems under the round‐robin protocol and under a weighted try‐once‐discard protocol, respectively. Actuator saturation has not been taken into account. In the present paper, for the first time, the time‐delay approach is extended to the stability analysis of the discrete‐time networked control systems under both scheduling protocols and actuators saturation. The communication delays are allowed to be larger than the sampling intervals. A novel Lyapunov‐based method is presented for finding the domain of attraction. Polytopic uncertainties in the system model can be easily included in our analysis. The efficiency of the time‐delay approach is illustrated on the example of a cart–pendulum system. Copyright © 2014 John Wiley & Sons, Ltd.

Nonlinear stabilization by dynamic parameter adaptation: Algebraic-Ricatti-Equation-based approach

A dynamic-gain parameterized controller is proposed to stabilize a class of nonlinear systems subject to norm-bounded uncertainties. The stabilizing controller is designed as a standard linear feedback with polynomial dynamic parameters. The expression of the dynamic parameters is defined from the solution of an Algebraic Ricatti Equation. The pendulum cart system with other examples are given as illustrative case studies to show the simplicity, the straightforwardness and the efficiency of the proposed design.

Interactive Multiobjective Optimization for a Grab-Shift Unloader Crane

In this paper, multiobjective optimization (MOO) is applied to an optimal control problem for a grab-shift unloader crane. The crane is modeled as a cart-pendulum system with varying rope length and the trajectory of the grab is limited by the ship, the quay, and the crane structure. The objectives to minimize are chosen as time, energy and maximal instantaneous power. The optimal control problem is solved using a direct simultaneous optimal control method. The study shows that MOO can be an efficient tool when choosing a good compromise between conflicting objectives such as time and energy. Furthermore, navigation among the Pareto optimal solutions has proven to be very useful when a user wants to learn how the control variables interact with the process.

Fusion Control of Flexible Logic Control and Neural Network

Based on the basic physical meaning of errorEand error varietyEC, this paper analyzes the logical relationship between them and usesUniversal Combinatorial Operation ModelinUniversal Logicto describe it. Accordingly, a flexible logic control method is put forward to realize effective control on multivariable nonlinear system. In order to implement fusion control with artificial neural network, this paper proposes a new neuron model ofZero-level Universal Combinatorial OperationinUniversal Logic. And the artificial neural network of flexible logic control model is implemented based on the proposed neuron model. Finally, stability control, anti-interference control of double inverted-pendulum system, and free walking of cart pendulum system on a level track are realized, showing experimentally the feasibility and validity of this method.

Stabilization of an inverted pendulum-cart system by fractional PI-state feedback

This paper deals with pole placement PI-state feedback controller design to control an integer order system. The fractional aspect of the control law is introduced by a dynamic state feedback as u(t)=Kpx(t)+KIIα(x(t)). The closed loop characteristic polynomial is thus fractional for which the roots are complex to calculate. The proposed method allows us to decompose this polynomial into a first order fractional polynomial and an integer order polynomial of order n−1 (n being the order of the integer system). This new stabilization control algorithm is applied for an inverted pendulum-cart test-bed, and the effectiveness and robustness of the proposed control are examined by experiments.

A remark on Controlled Lagrangian approach

A new approach to trajectory planning for underactuated mechanical systems is presented and discussed based on analysis of feasible behaviors of a standard 2-DOF benchmark example — the cart–pendulum system. Following the Controlled Lagrangians approach of Bloch et al. (2000) [7], we present and re-establish known conditions and forms of feedback control laws for this example, which are leading to an equivalent completely integrable closed-loop Euler–Lagrange system; and then extend them. As shown, full integrability and, in particular, the presence of a linear in velocities first integral of dynamics plays the key role in an elegant new procedure for trajectory planning.

Stabilization of Inverted Cart-Pendulum System Using Controller: A Frequency-Domain Approach

This paper focuses on the angular stabilization of inverted cart-pendulum system using controller. The tuning of controller is formulated as a nonlinear optimization problem, in which the objective function is composed of five design conditions in frequency domain. Particle swarm optimization technique has been used for optimizing parameters. Also a PID controller has been designed based on same specifications, and a comparative study has been carried out. All the responses have been calculated using FOMCON toolbox of Matlab/Simulink.

Integral backstepping sliding mode control for underactuated systems: Swing-up and stabilization of the Cart–Pendulum System

In this paper an integral backstepping sliding mode controller is proposed for controlling underactuated systems. A feedback control law is designed based on backstepping algorithm and a sliding surface is introduced in the final stage of the algorithm. The backstepping algorithm makes the controller immune to matched and mismatched uncertainties and the sliding mode control provides robustness. The proposed controller ensures asymptotic stability. The effectiveness of the proposed controller is compared against a coupled sliding mode controller for swing-up and stabilization of the Cart–Pendulum System. Simulation results show that the proposed integral backstepping sliding mode controller is able to reject both matched and mismatched uncertainties with a chattering free control law, while utilizing less control effort than the sliding mode controller.

Experimental evaluation of regulated non‐linear under‐actuated mechanical systems via saturation‐functions‐based bounded control: the cart–pendulum system case

The authors are concerned in this study by bounded control of single input non-linear under-actuated mechanical systems. The authors focus the exposition on a feedback-based stabilisation-bounded control action shaped by saturation functions, and the proposed approach was illustrated via the design and the experimental evaluation of a simple stabilising controller for the cart–pendulum system, a well-known control benchmark. The proposed simple control strategy is built around a lumped linear continuous time-invariant description of the concerned under-actuated non-linear system. Namely, a model consisted of a cascade non-linear dynamical system constituted by a chain of four integrators affected by a high-order smooth non-linear perturbation. Assuming initialisation of the under-actuated system to the upper-half plane, the proposed feedback-based regulation design procedure involves the simultaneous combination of two control actions: one bounded linear and one bounded quasi-linear. Control boundedness is provided in both involved control actions by specifically designed saturation functions. The first bounded control action brings the non-actuated coordinate near to the upright position and keep it inside of a well characterised small vicinity, whereas the second bounded control action asymptotically brings the whole state of the dynamical system to the origin. The necessary closed-loop stability analysis uses standard linear stability arguments as well as the traditional well-known Lyapunov method and the LaSalle's invariance principle. The proposed control law ensures global stability of the closed-loop system in the upper-half plane.

A new discrete mechanics approach to swing-up control of the cart-pendulum system

This paper develops a new swing-up control method for the cart-pendulum system via discrete mechanics. The swing-up control law consists of two parts: the swing-up stage and the stabilization one. In the swing-up stage, we use a controller based on a discrete Lyapunov function and it can swing up the pendulum. Then, in the stabilization stage, we utilize a stabilizing controller based on the linearized system and discrete-time optimal regulator theory. In addition, transformation methods from discrete control inputs into continuous zero-order hold inputs are introduced. From some simulation results, we can confirm that the cart-pendulum system is swung up and stabilized by our new method.

Reduced-order stable controllers for two-link underactuated planar robots

This paper concerns reduced-order stable controllers in a unified way for two typical examples of underactuated robots called the Acrobot and the Pendubot, which are two-link planar robots moving in a vertical plane with a single actuator at joints 1 and 2, respectively. The main contributions of this paper are threefold. First, with the aid of properties of the mechanical parameters of two-link planar robots, this paper proves constructively that stable controllers exist for stabilizing the UEP (Upright Equilibrium Point with two links being still in the upright position) of each robot by designing a single output with an adjustable parameter. Second, the paper presents a design method to obtain explicitly a formula of second-order (reduced-order) stable controllers by choosing the adjustable parameter. Third, in terms of strong stabilization, some new features of the Acrobot and the Pendubot are revealed and are compared with those of the cart–pendulum system. Simulation results for the two robots are presented to validate the theoretical results and to demonstrate that the presented stable stabilizing controllers achieve a better control performance than several other controllers.