Dynamic Feedback Compensator Research Articles

Denominator Assignment, Invariants, and Canonical Forms Under Dynamic Feedback Compensation in Linear Multivariable Nonsquare Systems

In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strictly</i> proper transfer function matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)_{\alpha }^{\oplus }$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> , we define and examine the equivalent relation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {R}$</tex-math></inline-formula> relating <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> with the members of the equivalence class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[P(s)]_{R}$</tex-math></inline-formula> of the closed loop-transfer function matrices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{C}(s)$</tex-math></inline-formula> obtainable from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> by the use of a proper compensator <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C(s)$</tex-math></inline-formula> in the feedback path. For <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {R}$</tex-math></inline-formula> , we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">proper</i> internally stabilizing and denominator assigning compensators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C(s)$</tex-math></inline-formula> for the class of plants with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p=m$</tex-math></inline-formula> and having no zeros in the closed right half complex plane: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {C}^{+}$</tex-math></inline-formula> and in the case when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p&gt;m$</tex-math></inline-formula> plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> rows that give rise to a nonsingular polynomial matrix with no zeros in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {C}^{+}$</tex-math></inline-formula> .

The Multivariable Control of Carbon Dioxide Absorption System using the Proportional Integral Plus (PIP) Controller

The present article investigates the implementation of non-minimal state space (NMSS) representation with proportional-integral-plus (PIP) controller for the carbon dioxide absorption process of Shiraz petrochemical ammonia unit. The PIP controller is a logical extension of conventional PI/PID controllers with additional dynamic feedback and input compensators. PIP controller is used for multivariable control without limitation on the number of controlled variables. A Multi Input - Multi Output (MIMO) square model was extracted from step response test. In this way, input water flow rate to carbon dioxide absorption system, the heat duty of input absorbent cooler to tray (1) of absorption tower and re-boiler heat duty of stripping tower are chosen as manipulated variables (inputs), while carbon dioxide mole fraction in absorption tower vapor product, the water mole fraction in absorption tower liquid product and tray temperature No. 36 of stripping tower are determined as controlled ones (outputs). The system identification is performed with three input and three output variables using step response test. As a result, continuous and discrete time transfer function matrices and suitable NMSS model for PIP controller are reported. Finally, in order to evaluate the PIP control performance, the feed flow rate increases by 2%. The results show the proper performance of designed PIP controller for both disturbance rejection and set point tracking.

Dynamic Compensator Design of Linear Parabolic MIMO PDEs in $N$-Dimensional Spatial Domain

This article employs the observer-based output feedback control technique to deal with dynamic compensator design for a linear $N$ -D parabolic partial differential equation (PDE) with multiple local piecewise control inputs and multiple noncollocated local piecewise observation outputs. These control inputs and observation outputs are provided by only few actuators and noncollocated sensors active over partial areas (or entire) of the spatial domain. An observer-based dynamic feedback compensator is constructed for exponential stabilization of the linear PDE. Poincare–Wirtinger inequality and its variant in $N$ -D spatial domain are presented for the closed-loop stability analysis. By the Lyapunov direct method and Poincare–Wirtinger inequality and its variant, sufficient conditions on the existence of such feedback compensator of the linear PDE are developed, and presented in term of standard linear matrix inequalities. Well-posedness is also analyzed for both open-loop PDE and resulting closed-loop coupled PDEs within the $C_0$ -semigroup framework. Finally, numerical simulation results are presented to support the proposed design method.

Spatial domain decomposition approach to dynamic compensator design for linear space‐varying parabolic MIMO PDEs

This study addresses the problem of dynamic compensator design for exponential stabilisation of linear space-varying parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) subject to periodic boundary conditions. With the aid of the observer-based feedback control technique, an observer-based dynamic feedback compensator, whose implementation requires only a few actuators and sensors active over partial areas of the spatial domain, is constructed such that the resulting closed-loop coupled PDEs is exponentially stable. The spatial domain is divided into multiple subdomains according to the minimum of the actuators' number and the sensors' one. By Lyapunov direct method and two general variants of Poincare-Wirtinger inequality at each subdomain, sufficient conditions for the existence of such feedback compensator are developed and presented in terms of algebraic linear matrix inequalities (LMIs) in space. Based on the extreme value theorem, LMI-based sufficient and necessary conditions are presented for the feasibility of algebraic LMIs in space. Finally, numerical simulation results are presented to support the proposed design method.

An Adaptive Super Twisting Nonlinear Fractional Order PID Sliding Mode Control of Permanent Magnet Synchronous Motor Speed Regulation System Based on Extended State Observer

In this article, a novel adaptive super-twisting nonlinear Fractional-order PID sliding mode control (ASTNLFOPIDSMC) strategy using extended state observer (ESO) for the speed operation of permanent magnet synchronous motor (PMSM) is proposed. Firstly, this paper proposes a novel nonlinear Fractional-order PID (NLFOPID) sliding surface with nonlinear proportion term, nonlinear integral term and nonlinear differential term. Secondly, the novel NLFOPID switching manifold and an adaptive super-twisting reaching law (ASTRL) are applied to obtain excellent control performance in the sliding mode phase and the reaching phase, respectively. The novel ASTNLFOPIDSMC strategy is constructed by the ASTRL and the NLFOPID sliding surface. Due to the utilization of NLFOPID switching manifold, the characteristics of fast convergence, good robustness and small steady state error can be ensured in the sliding mode phase. Due to the utilization of ASTRL, the chattering phenomenon can be weakened, and the characteristics of high accuracy and strong robustness can be obtained in the reaching phase. Further, an ESO is designed to achieve dynamic feedback compensation for external disturbance. Furthermore, Lyapunov stability theorem and Fractional calculus are used to prove the stability of the system. Finally, comparison results under different controllers demonstrate that the proposed control strategy not only achieves good stability and dynamic properties, but also is robust to external disturbance.

Fuzzy Approximation Based Asymptotic Tracking Control for a Class of Uncertain Switched Nonlinear Systems

The problem of asymptotic tracking control for a class of uncertain switched nonlinear systems under fuzzy approximation framework is solved in this paper. Superior to most existing results based on fuzzy adaptive control strategy that can only achieve bounded error tracking performance, our proposed control scheme can guarantee the local asymptotic tracking performance for the uncertain switched nonlinear systems under consideration. This is accomplished by constructing a nonsmooth Lyapunov function and introducing a novel discontinuous controller with dynamic feedback compensator in the design procedure. Meanwhile, some concepts, such as differential inclusion and set-valued map, are introduced to theoretically verify the local asymptotic tracking performance of the systems with our proposed controller. With the help of set-valued Lie derivative, the common virtual control functions, the desired controller, and the adaptive laws can be precisely constructed. Finally, simulation results are given to show the effectiveness of the proposed method.

A Lyapunov-based design of dynamic feedback compensator for linear parabolic MIMO PDEs

Abstract This paper discusses dynamic feedback compensator design for a linear parabolic partial differential equation (PDE) with multiple inputs and multiple outputs. Actuating control inputs are provided by actuators distributed over partial areas (or active at specified positions) of the spatial domain, and observation outputs are taken from the non-collocated sensors distributed over partial areas of the spatial domain. An observer-based dynamic feedback compensator is constructed via the observer-based feedback control technique to exponentially stabilize the multi-input–multi-output PDE in the spatial $\mathscr{L}^2$ norm. By constructing an appropriate Lyapunov function candidate and using two variants of Poincaré–Wirtinger inequality, sufficient conditions on the existence of such observer-based dynamic feedback compensator are developed and presented in terms of linear matrix inequalities. The well posedness of the closed-loop coupled PDEs is also analyzed within the framework of $C_0$ semigroup theory. Finally, numerical simulation results are given to show the effectiveness of the proposed method.

A Sliding Mode Control with Nonlinear Fractional Order PID Sliding Surface for the Speed Operation of Surface‐Mounted PMSM Drives Based on an Extended State Observer

A novel sliding mode controller (SMC) with nonlinear fractional order PID sliding surface based on a novel extended state observer for the speed operation of a surface‐mounted permanent magnet synchronous motor (SPMSM) is proposed in this paper. First, a new smooth and derivable nonlinear function with improved continuity and derivative is designed to replace the traditional nonderivable nonlinear function of the nonlinear state error feedback control law. Then, a nonlinear fractional order PID sliding mode controller is proposed on the basis of the fractional order PID sliding surface with the combination of the novel nonlinear state error feedback control law to improve dynamic performance, static performance, and robustness of the system. Furthermore, a novel extended state observer is designed based on the new nonlinear function to achieve dynamic feedback compensation for external disturbances. Stability of the system is proved based on the Lyapunov stability theorem. The corresponding comparative simulation results demonstrate that the proposed composite control algorithm displays good stability, dynamic properties, and strong robustness against external disturbances.

Robustness evaluation and robust design for proportional-integral-plus control

ABSTRACTProportional-integral-plus (PIP) control provides a logical extension to conventional two- or three-term (proportional-integral-derivative) industrial control, with additional dynamic feedback and input compensators introduced when the process has second order or higher dynamics, or time delays. Although PIP control has been applied in a range of engineering applications, evaluation of closed-loop robustness has generally relied on empirical methods. In the present article, expressions for the norm of two commonly used PIP control implementations, the feedback and forward path forms, are used, for the first time, to quantify closed-loop robustness. It is shown that the forward path form is not robust for unstable plants. Additional expressions for the norm that encompass frequency weightings of generalised disturbance inputs are also determined. Novel analytical expressions to minimise the norm are derived for the simplest plant, while simulation results based on numerical optimisation are provided for higher order examples. We show that, for certain plants, there are (non-unique) sets of PIP control gains that minimise the norm. The norm is introduced in these cases to determine the controller that balances performance with robustness. Finally, the norm is used as a design parameter for a practical example, namely control of airflow in a 2 m by 1 m by 1 m forced ventilation chamber. The performance of the new PIP controller is compared to previously developed PIP controllers based on pole placement and linear quadratic design.

Asymptotic Regulation of Time-Delay Nonlinear Systems With Unknown Control Directions

This paper studies the problem of global state regulation with stability for time-delay nonlinear systems with unknown control directions. Using a dynamic gain-based method for counteracting time-delay nonlinearity and the Nussbaum-gain function for dealing with unknown control directions, we develop a dynamic state feedback control strategy that solves the problem. A novel construction of Lyapunov–Krasovskii functionals is presented and plays a key role in handling nonlinearity with delayed states and unknown control directions simultaneously. The proposed dynamic state feedback compensators are shown to guarantee 1) global asymptotic convergence of the system state to the origin and 2) global boundedness of the resulting closed-loop systems.

ADRC Attitude Controller Design for Hypersonic Vehicle based on MIMO-ESO

For the hypersonic vehicle nonlinear attitude mode in reentry process with a strong coupling, aerodynamic parameter perturbations and non-deterministic, combine extended state observer and nonlinear law state error feedback, design the hypersonic vehicle MIMO-ESO ADRC attitude controller. Put interference such as uncertainty, coupling and parameter perturbations as “the sum of interference” ,use the extended state observer to estimate and dynamic feedback compensation, use nonlinear law state error feedback to inhibit residual of compensation. ADRC controller is charged without a precise model of vehicle , and without precise perturbation boundaries of aerodynamic parameters.Simulation results show that the MIMO-ESO ADRC attitude controller can overcome the impact of large-scale perturbations of interference and aerodynamic parameters, have good dynamic qualities and tracking capabilities, also have strong robustness.

Multi-Sensor Obstacle Detection System Via Model-Based State-Feedback Control in Smart Cane Design for the Visually Challenged

Smart canes are usually developed to alert visually challenged users of any obstacles beyond the canes’ physical lengths. The accuracy of the sensors and their actuators’ positions are equally crucial to estimate the locations of the obstacles with respect to the users so as to ensure only correct signals are sent through the associated audio or tactile feedbacks. For implementations with low-cost sensors, however, the users are very likely to get false alerts due to the effects from noise and their erratic readings, and the performance degradation will be more noticeable when the positional fluctuations of the actuators get amplified. In this paper, a multi-sensor obstacle detection system for a smart cane is proposed via a model-based state-feedback control strategy to regulate the detection angle of the sensors and minimize the false alerts to the user. In this approach, the overall system is first restructured into a suitable state-space model, and a linear quadratic regulator (LQR)-based controller is then synthesized to further optimize the actuator’s control actions while ensuring its position tracking. We also integrate dynamic feedback compensators into the design to increase the accuracy of the user alerts. The performance of the resulting feedback system was evaluated via a series of real-time experiments, and we showed that the proposed method provides significant improvements over conventional methods in terms of error reductions.

Performance Resilience Analysis of Dynamic Feedback Controllers for both Multiplicative and Additive Gain Perturbations

In this work, a procedure is presented for performance resilience analysis of continuous-time systems controlled by full-order dynamic feedback compensators. The resilience property is defined in terms of defining maximum perturbations on both the controller and observer gains that will maintain controller and observer eigenvalues in disjoint regions in the complex plane so that the closed loop system will sustain certain performance characteristics. The desired performance is characterized by the response speed and magnitude bounds on the response. Therefore, the intersection of two regions are chosen, vertical strips and sector regions, to guarantee upper and lower bounds on the settling (or response) time and an upper bound on the percent overshoot simultaneously. Maximum allowable gain perturbations are obtained based on the designer’s choices of closed loop eigenvalues and the regions in which eigenvalues remain when the gains are perturbed. The linear matrix inequality technique is used throughout the analysis process. Illustrative examples are included to demonstrate the effectiveness of the proposed methodology.

Dynamic partial state feedback control of cascade systems with time-delay

This paper investigates the problem of global stabilization by partial state feedback for a class of cascade nonlinear systems with time-delay. Under suitable ISS conditions imposed on zero-dynamics, a delay-free, dynamic partial state feedback compensator is presented for achieving global state regulation. The controller is constructed by employing a dynamic gain based design method, together with the ideas of changing supply rates and adding an integrator. With appropriate choices of Lyapunov–Krasovskii functionals, it is shown that all the states of the time-delay cascade system can be regulated to the origin while maintaining boundedness of the closed-loop system. Two examples are given to illustrate the effectiveness of the proposed dynamic partial state feedback control scheme.

Global Regulation of Time-Delay Cascade Systems via Partial State Feedback

Abstract: The problem of global stabilization is studied by partial state feedback for a class of cascade systems with time-delay. Under suitable ISS conditions on inverse dynamics, a delay-free, dynamic partial state feedback compensator is designed to achieve global state regulation. This is made possible by developing a dynamic gain based design method, together with the ideas of changing supply rates and backstepping. By constructing appropriate Lyapunov-Krasovskii functionals, we prove that all the states of the time-delay cascade system can be regulated to the origin while maintaining boundedness of the closed-loop system.

Nonlinear fractional order proportion-integral-derivative active disturbance rejection control method design for hypersonic vehicle attitude control

Near space hypersonic vehicle model is nonlinear, multivariable and couples in the reentry process, which are challenging for the controller design. In this paper, a nonlinear fractional order proportion integral derivative (NFOPIλDμ) active disturbance rejection control (ADRC) strategy based on a natural selection particle swarm (NSPSO) algorithm is proposed for the hypersonic vehicle flight control. The NFOPIλDμ ADRC method consists of a tracking-differentiator (TD), an NFOPIλDμ controller and an extended state observer (ESO). The NFOPIλDμ controller designed by combining an FOPIλDμ method and a nonlinear states error feedback control law (NLSEF) is to overcome concussion caused by the NLSEF and conversely compensate the insufficiency for relatively simple and rough signal processing caused by the FOPIλDμ method. The TD is applied to coordinate the contradiction between rapidity and overshoot. By attributing all uncertain factors to unknown disturbances, the ESO can achieve dynamic feedback compensation for these disturbances and thus reduce their effects. Simulation results show that the NFOPIλDμ ADRC method can make the hypersonic vehicle six-degree-of-freedom nonlinear model track desired nominal signals accurately and fast, has good stability, dynamic properties and strong robustness against external environmental disturbances.

Airship horizontal trajectory tracking control based on Active Disturbance Rejection Control (ADRC)

Aiming at flight property of airship, a trajectory tracking controller of airship horizontal model is designed based on active disturbance rejection control (ADRC). The six Degree of Freedom (DOF) dynamic model of airship is simplified at a horizontal plane. ADRC is used to realize the decoupling control for the multivariable system. The uncertain items of the model and external disturbances are estimated by the extended state observer (ESO) and dynamic feedback compensation is carried on at real time. The disturbance of wind is added to the simulation environment. The simulation results show that the designed tracking controller can overcome the influences of uncertain items of the model and external disturbances, and track the desired trajectory rapidly and steadily, and possess good robustness and control performances.

Visual Trajectory-Tracking Model-Based Control for Mobile Robots

In this paper we present a visual-control algorithm for driving a mobile robot along the reference trajectory. The configuration of the system consists of a two-wheeled differentially driven mobile robot that is observed by an overhead camera, which can be placed at arbitrary, but reasonable, inclination with respect to the ground plane. The controller must be capable of generating appropriate tangential and angular control velocities for the trajectory-tracking problem, based on the information received about the robot position obtained in the image. To be able to track the position of the robot through a sequence of images in real-time, the robot is marked with an artificial marker that can be distinguishably recognized by the image recognition subsystem. Using the property of differential flatness, a dynamic feedback compensator can be designed for the system, thereby extending the system into a linear form. The presented control algorithm for reference tracking combines a feedforward and a feedback loop, the structure also known as a two DOF control scheme. The feedforward part should drive the system to the vicinity of the reference trajectory and the feedback part should eliminate any errors that occur due to noise and other disturbances etc. The feedforward control can never achieve accurate reference following, but this deficiency can be eliminated with the introduction of the feedback loop. The design of the model predictive control is based on the linear error model. The model predictive control is given in analytical form, so the computational burden is kept at a reasonable level for real-time implementation. The control algorithm requires that a reference trajectory is at least twice differentiable function. A suitable approach to design such a trajectory is by exploiting some useful properties of the Bernstein-Bézier parametric curves. The simulation experiments as well as real system experiments on a robot normally used in the robot soccer small league prove the applicability of the presented control approach.

Bond graph modelling of hydraulic six-degree-of-freedom motion simulator

Stewart-type six-degree-of-freedom motion simulator is a multidisciplinary system containing multiple mechanical, hydraulic and electric components. This article proposed a unified bond graph representation for the simulator. First, dynamics equations of the upper platform are developed using Newton–Euler method and its bond graph model is established. Each hydraulic actuator is then modelled according to the three basic equations of valve-controlled cylinder. An equivalent approach is further employed to treat the inertial effects of each cylinder. This approach projects the forces caused by the cylinder inertia equivalently onto the joint point of the upper platform, which relatively simplifies the modelling of the piston and the tube. The whole simulator takes independent close-loop position feedback control on each valve-controlled cylinder actuator. A proportional controller with dynamic pressure feedback and feed-forward compensation is proposed. The bond graph model of the overall simulator is finally completed with 20-Sim software, and further simulations and experiments are carried out to verify the model. The model provides another reference for motion simulator modelling besides the multibond graph approach.

Denominator Assignment, Invariants and Canonical Forms Under Dynamic Feedback Compensation in Linear Multivariable Systems

A result originally reported by Hammer for linear time invariant (LTI) single input-single output systems and concerning an invariant and a canonical form of the transfer function matrix of the closed loop system under dynamic feedback compensation is generalized for LTI multivariable systems. Based on this result, we characterize the class of transfer function matrices that are obtainable from an open loop transfer function matrix via the use of proper dynamic feedback compensators and show that if the closed loop transfer function matrix Pc(s) has a desired denominator polynomial matrix which satisfies a certain sufficient condition, then there exists a proper compensator giving rise to an internally stable closed loop system, whose transfer function matrix is Pc(s).