Positive Real Control Research Articles

Comfort-Oriented Semi-Active Suspension Configuration with Inerter-Based Network Synthesis

This paper presents a comfort-oriented semi-active suspension system composed of a network-synthesized passive section and a controllable section based on a semi-active inerter. Firstly, the semi-active suspension system is divided into a passive part and a controllable part. For the passive part, first-order and second-order robust positive real controllers are designed. The problem with H2 cost is considered, and the bilinear matrix inequalities (BMI) are solved using an iterative method to obtain two admittance functions. The admittance functions are physically realized as two mechanical networks composed of mechanical passive elements such as inerter, spring, and damper (ISD). Then, the parameters of these mechanical elements in those networks are optimized by Particle Swarm Optimization (PSO). Secondly, a semi-active inerter based on Sky-hook control is introduced for the semi-active part of the suspension system. Finally, the semi-active ISD suspension structure is verified by a quarter vehicle model. The simulation results show that the first-order and second-order suspension systems optimize the RMS of the spring mass acceleration by 14.2% and 23.9%, respectively, as compared to traditional suspension systems. Furthermore, frequency-domain analysis also shows that both suspension systems effectively reduce the value of spring mass acceleration in the low-frequency band.

Passivity-based adaptive trajectory control of an underactuated 3-DOF overhead crane

This paper presents a robust passivity-based adaptive control method for payload trajectory tracking of a three degree-of-freedom overhead crane with a varying-length rigid hoist cable. A reformulation of the overhead crane’s equations of motion is used and an adaptive control law is developed to ensure the system has an output strictly passive (OSP) input–output mapping. The Passivity Theorem is used to prove that the closed-loop system is input–output stable when an OSP negative feedback controller is implemented. Convergence and boundedness of the payload trajectory tracking error is proven with two candidate OSP feedback controllers, including a constant gain and a strictly positive real (SPR) controller. Numerical simulations and experiments are performed with the proposed control method and two state-of-the-art control laws from the literature. The results demonstrate the practicality of the proposed control method and the improved performance of the proposed SPR controller in the experiments compared to the controllers from the literature.

Positive real lemmas for singular fractional‐order systems: the case

This study investigates the positive real lemmas for singular fractional-order linear time-invariant systems with the fractional order α ∈ ( 0 , 1 ) . Firstly, a novel condition for the stability and extended strictly positive realness of fractional-order systems is derived in terms of linear matrix inequalities. Secondly, a novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is obtained with two complex variables. Then, another novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is established with only one complex variable. Thirdly, the positive real controller synthesis problems are analysed and the corresponding positive real controllers are designed based on the positive real lemmas. Finally, four numerical examples are provided to show the effectiveness of the proposed results in this study.

The generalised singular perturbation approximation for bounded real and positive real control systems

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

Hybrid frequency domain control for large flexible structures

A hybrid finite frequency controller is proposed for the vibration suppression of a large flexible structure mounted with collocated sensors and actuators. The controller has passive characteristics at low frequencies and small gain characteristics at high frequencies. Compared with a strictly positive real controller based on the standard Kalman–Yakubovich–Popov lemma, the hybrid finite frequency controller has less energy consumption but can obtain approximately identical performance. Furthermore, when the plant passivity is violated at high frequencies by noncollocation of sensors and actuators, the strictly positive real controller based on the Kalman–Yakubovich–Popov lemma is no longer able to attenuate the vibration of the large flexible structures, while the hybrid finite frequency controller is effective in suppressing the vibration and avoiding spillover instability. Simulation results are presented to validate the effectiveness of the hybrid finite frequency controller.

Flexible Cable-Driven Parallel Manipulator Control: Maintaining Positive Cable Tensions

In this brief, a planar cable-driven parallel manipulator is controlled using a strictly positive real compensator, cable pretension is explicitly accounted for, and cable slack is avoided. A strictly positive real controller is implemented using actuator saturation prevention techniques and the solution to a linear programming problem to ensure that the cable tensions remain strictly positive. Pretension torque is directly accounted for when determining the input–output stability of the closed-loop system. Numerical simulations demonstrate that the controller can track high-acceleration trajectories with the same precision as a state-of-the-art controller from the literature, while maintaining positive cable tensions.

Synthesis of strictly positive real ℋ2 controllers using dilated LMIs

ABSTRACTThis paper considers strictly positive real (SPR) controller design via direct minimisation of an upper bound on the closed-loop norm. To realise a tractable design problem, an observer-based control structure is assumed and a state-feedback gain that renders the controller SPR while minimising an upper bound on the closed-loop norm is sought. A convex optimisation problem subject to linear matrix inequality (LMI) constraints is realised by using a common Lyapunov variable. By using dilated LMIs, a partial separation of design variables is possible thereby enhancing design freedom. Numerical results comparing the common Lyapunov variable solution, the dilated LMI solution and method presented in the literature are included.

Passivity-based attitude control with input quantization

A passivity-based controller with quantization for spacecraft attitude control is developed. This passive control scheme includes two parts which are a proportional controller for quaternion feedback and a strictly positive real controller for the angular velocity. To alleviate the errors caused by quantization, a special modification for the nonlinear quantized input is employed in the strictly positive real controller. Asymptotic stability can be guaranteed with the presented controller structure. A guideline for the controller parameter selection is provided with sensitivity analysis for the control scheme. Numerical simulation results demonstrate the effectiveness of the proposed controller.

An Enhanced Power Sharing Scheme for Voltage Unbalance and Harmonics Compensation in an Islanded AC Microgrid

In this paper, an enhanced hierarchical control structure with multiple current loop damping schemes for voltage unbalance and harmonics compensation (UHC) in ac islanded microgrid is proposed to address unequal power sharing problems. The distributed generation (DG) is properly controlled to autonomously compensate voltage unbalance and harmonics while sharing the compensation effort for the real power, reactive power, and unbalance and harmonic powers. The proposed control system of the microgrid mainly consists of the positive sequence real and reactive power droop controllers, voltage and current controllers, the selective virtual impedance loop, the unbalance and harmonics compensators, the secondary control for voltage amplitude and frequency restoration, and the auxiliary control to achieve a high-voltage quality at the point of common coupling. By using the proposed unbalance and harmonics compensation, the auxiliary control, and the virtual positive/negative-sequence impedance loops at fundamental frequency, and the virtual variable harmonic impedance loop at harmonic frequencies, an accurate power sharing is achieved. Moreover, the low bandwidth communication (LBC) technique is adopted to send the compensation command of the secondary control and auxiliary control from the microgrid control center to the local controllers of DG unit. Finally, the hardware-in-the-loop results using dSPACE 1006 platform are presented to demonstrate the effectiveness of the proposed approach.

Robust switched control based on strictly positive real systems and variable structure control techniques

SummaryThis paper is concerned with the design of variable structure controllers for uncertain switched linear plants. The proposed method is based on Lyapunov–Metzler inequalities and on properties of strictly positive real (SPR) systems, with the advantage that it can be applied in control of uncertain switched linear system. The first contribution shows that there exists a solution of the SPR synthesis for a class of linear uncertain switched systems using switched static output feedback, if and only if there exists a solution of this SPR synthesis with switched static state feedback. This result is a generalization of known conditions regarding this problem, for linear time‐invariant and known (without uncertainties) plants, and offers a possible procedure for output feedback SPR synthesis: first solve the state feedback SPR synthesis and then, using the obtained solution, find a solution for the output feedback SPR synthesis. Next are presented necessary and sufficient (sufficient) conditions for the SPR synthesis for a class of linear uncertain switched systems using switched static output feedback, for plants with the same number of inputs and outputs (with the number of outputs greater than the number of inputs). Two examples illustrate the effectiveness of the robust control system, including applications of the proposed methods, considering switched static state feedback and switched static output feedback, in the design of switching control strategies for active suspensions systems in road vehicles. Performance indices such as decay rate, guaranteed cost, and restriction in the norm of controller gains are considered in these designs. Copyright © 2016 John Wiley & Sons, Ltd.

Passivity-Based Control of Sampled-Data Systems on Lie Groups with Linear Outputs

Abstract: We present a method of stabilizing a sampled-data system that evolves on a matrix Lie group using passivity The continuous-time plant is assumed passive with known storage function, and its passivity is preserved under sampling by redefining the output of the discretized plant and keeping the storage function. We show that driftlessness is a necessary condition for a sampled-data system on a matrix Lie group to be zero-state observable. The closed-loop sampled-data system is stabilized by any strictly passive controller, and we present a synthesis procedure for a strictly positive real LTI controller. The closed-loop system is shown to be asymptotically stable. This stabilization method is applied to asymptotic tracking of piecewise constant references.

107 正実なコントローラによるセミアクティブ振動制御系設計(GS2-2 ロボティクス・メカトロニクス)

107 正実なコントローラによるセミアクティブ振動制御系設計(GS2-2 ロボティクス・メカトロニクス)

Passive Mechanical Control With a Special Class of Positive Real Controllers: Application to Passive Vehicle Suspensions

This paper presents an efficient H2 optimization method for passive mechanical control problems with a special class of positive real controllers. In particular, the problem of designing passive vehicle suspensions based on a full-car model is taken as an example, where both the positive real constraint and the constraint imposed on the static stiffness are considered. An unconstrained nonlinear programming problem is formulated by using the structured H2 optimization framework, and the Lagrange matrix multiplier method is employed to derive a set of necessary conditions for the optimization so that the time-efficient gradient-based algorithms can easily be implemented. The proposed method can also effectively deal with the fixed static stiffness optimization problem and it is shown in the numerical examples that the proposed method cannot only recover the existing fixed-structure configuration, but also introduce new (optimal) configurations with respect to the specific weighting factors, which demonstrates the effectiveness of the proposed method.

Generalised Kalman–Yakubovich–Popov lemma with its application in finite frequency positive realness control for two‐dimensional continuous‐discrete systems in the Roesser model form

This study is concerned with the problem of generalised Kalman–Yakubovich–Popov (GKYP) lemma and its application to two-dimensional (2D) continuous-discrete systems described by Roesser model. On the basis of the feature of states in the system, a rectangular finite frequency range is characterised by a linear matrix inequality and then combined with 𝒮-procedure, the GKYP lemma is developed for 2D continuous-discrete systems in Roesser model. As special cases of this lemma, 2D continuous-discrete case finite frequency bounded realness and positive realness are investigated as well. Furthermore, the finite frequency 2D positive realness control problem via state-feedback controllers are considered based on the developed lemma. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Positive real control for discrete-time singular systems with affine parameter dependence

This paper presents a less conservative and numerically tractable solution to the static state feedback positive real control problem for affinely parameter dependent discrete-time singular systems. Relied on the use of auxiliary matrices and a positive scalar decision variable, a novel necessary and sufficient condition of positive realness is first derived in terms of a strict matrix inequality for linear time-invariant discrete-time singular systems. This characterization leads to a numerically efficient and reliable way for the controller design synthesis. Then, the results are further expanded to parameter dependent singular systems whose coefficient matrices are affine functions of a time-invariant uncertain parameter vector. Both robust analysis and robust controller synthesis are addressed. Numerical examples are included to illustrate the effectiveness of the present results.

Saturated control of flexible-joint manipulators using a Hammerstein strictly positive real compensator

SUMMARYIn this paper the control of flexible-joint manipulators while explicitly avoiding actuator saturation is considered. The controllers investigated are composed of a bounded proportional control term and a Hammerstein strictly positive real angular rate control term. This control structure ensures that the total torque demanded of each actuator is bounded by a value that is less than the maximum torque that each actuator is able to provide, thereby disallowing actuator saturation. The proposed controllers are shown to render the closed-loop system asymptotically stable, even in the presence of modeling uncertainties. The performance of the controllers is demonstrated experimentally and in simulation.

Conic-sector-based control to circumvent passivity violations

In order to ensure input–output stability for plants experiencing passivity violations, this paper explores the use of the conic sector theorem for both stability analysis and controller design. Given a previously designed controller and a plant that has experienced a (partially unknown) passivity violation, a novel conic bound selection procedure is presented. This procedure can be used to assess input–output stability of the violated plant and original controller via the conic sector theorem. Should input–output stability not be ensured, two original controller synthesis methods are suggested: one is designed to mimic the controller, and the other is inspired by strictly positive real controller synthesis. Both methods guarantee input–output stability by selecting controllers within appropriate conic sectors, and involve only the evaluation of readily solvable linear matrix inequalities and algebraic Riccati equations. Along with details on computational considerations, numerical simulations are provided, demonstrating the effectiveness of the proposed methods.

Robust and resilient state-dependent control of discrete-time nonlinear systems with general performance criteria

A novel state-dependent control approach for discrete-time nonlinear systems with general performance criteria is presented. This controller is robust for unstructured model uncertainties, resilient against bounded feedback control gain perturbations in achieving optimality for general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. For the system model, unstructured uncertainty description is assumed, which incorporates commonly used types of uncertainties, such as norm-bounded and positive real uncertainties as special cases. By solving a state-dependent linear matrix inequality (LMI) at each time step, sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this paper unify existing results on nonlinear quadratic regulator, H∞ and positive real control to provide a novel robust control design. The effectiveness of the proposed technique is demonstrated by simulation of the control of inverted pendulum.

Robust and resilient state-dependent control of continuous-time nonlinear systems with general performance criteria

A novel state-dependent control approach for continuous-time nonlinear systems with general performance criteria is presented in this paper. This controller is optimally robust for model uncertainties and resilient against control feedback gain perturbations in achieving general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. For the system model, unstructured uncertainty description is assumed, which incorporates commonly used types of uncertainties, such as norm-bounded and positive real uncertainties as special cases. By solving a state-dependent linear matrix inequality at each time, sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this paper unify existing results on nonlinear quadratic regulator, H∞ and positive real control. The efficacy of the proposed technique is demonstrated by numerical simulations of the nonlinear control of the inverted pendulum on a cart system.

Passivity-Based Attitude Control on the Special Orthogonal Group of Rigid-Body Rotations

Set-point regulation and tracking control of a rigid body is considered. The rigid body, which could be a spacecraft, underwater vehicle, or unmanned aerial vehicle, is equipped with various sensors, including those that provide unit-length vector measurements. At no point is the rotation matrix associated with the rigid-body’s orientation parameterized. The control algorithms developed in this paper are posed directly on the special orthogonal group of rigid-body rotations . The set-point controller presented is composed of a proportional control term and an angular velocity control term. The proportional control term is a function of the vector measurements and a set of desired vector measurements that are used to compute the orientation error of the rigid body. Passive systems theory is used to motivate the use of a strictly positive real angular velocity controller. The set-point regulator is robust to modeling errors associated with the mass distribution of the body. Tracking control is also considered. To ensure robust tracking control, an adaptive scheme is developed in which the controller is composed of feedback and feedforward terms. The feedback term is composed of a proportional term and a strictly-positive-real-based angular velocity term, similar to the set-point regulator. The feedforward term is a function of an estimate of the spacecraft mass distribution. The synthesis of strictly positive real controllers to be used within the set-point and tracking controllers is discussed. Numerical simulation results that demonstrate successful set-point regulation and tracking control are presented.