Associate Professor Maria Seron

The concepts of ultimate bounds and invariant sets play a key role in several control theory problems, as they replace the notion of asymptotic stability in the presence of unknown disturbances. However, when the disturbances are unbounded, as in the case of Gaussian white noise, no ultimate bounds nor invariant sets can in general be found. To overcome this limitation we introduced, in previous work, the notions of probabilistic ultimate bound (PUB) and probabilistic invariant set (PIS) for discrete-time systems. This article extends the notions of PUB and PIS to continuous-time systems, studying their main properties and providing tools for their calculation. In addition, the use of these concepts in robust control design by covariance assignment is presented.

We present a new output feedback fault-tolerant control strategy for continuous-time linear systems with bounded disturbances. The strategy combines a digital nominal controller under controller-driven (varying) sampling with virtual actuator (VA)-based controller reconfiguration to compensate for abrupt actuator faults and invariant-set-based fault detection and isolation. Two independent objectives are considered: (a) closed-loop stability with setpoint tracking and (b) controller reconfiguration under faults. Our main contribution is to extend an existing fault detection and isolation and VA-based controller reconfiguration strategy to systems under controller-driven sampling in such a way that if objective (a) is possible under controller-driven sampling (without VA) and objective (b) is possible under uniform sampling (without controller-driven sampling), then closed-loop stability and setpoint tracking will be preserved under both healthy and faulty operations for any possible sampling rate evolution that may be selected by the controller.

This paper presents a fault tolerant control scheme that compensates for actuator faults in nonlinear Lure systems by adapting the controller gain based on estimates of the fault magnitude. The plant is subject to process disturbances and embedded in closed-loop with a linear state feedback tracking controller, designed for the nominal system. The control law is adapted to the fault situation diagnosed by an invariant-set-based fault detection and isolation algorithm. Conditions for correct fault detection and bounds on the fault estimation are derived. The Chua circuit is used to illustrate the proposed scheme.

In this paper, we address the problem of robust stability and set invariance of switched linear parameter varying (LPV) systems affected by bounded disturbances. A switched LPV system is a collection of single LPV subsystems with a switching rule that decides which subsystem is active at each time. This framework is useful for LPV systems with large parameter variation range for which it may be difficult to assess or guarantee stability over the whole parameter range. Our first contribution is to derive a result for a single LPV system that extends and generalises previous work for switched systems. Our second, and main, contribution is to apply this seed result in the derivation of dwell-time-type conditions on the switched-LPV switching rule, which ensure robust closed-loop stability and set invariance of the trajectories of the switched-LPV system across the whole parameter range. As an application of the results, we consider the problem of reference tracking for LPV systems using switched LPV state feedback and feedforward from an LPV reference system. Two examples are presented to illustrate the results: a two-mass-spring system with a varying spring characteristic and the nonlinear model of a coupled-tank system which is embedded into a switched LPV system description.

© 2015 IEEE. This paper presents a feedback design approach to minimise state ultimate-bounds (UBs) in linear time-invariant discrete-time systems subject to non-vanishing bounded disturbances. A target closed-loop eigenstructure is first constructed to achieve the lowest possible UBs for one or more state components of the system. Structural conditions on the system matrices are then derived to guarantee that the target eigenstructure can be assigned by state feedback. When the required structural conditions are satisfied, an eigenvector assignment procedure can be applied to simultaneously minimise multiple UBs in systems with multiple inputs. The results are illustrated by a numerical example drawn from a distributed voltage control problem in inverter-based microgrids.

In networked control, there is often an incentive to communicate only what is absolutely necessary to achieve the desired performance goals. This is true to both the downlink (between a control base station and actuators) and the uplink (between the sensors and base station). Here, we present a strategy aimed at this problem based on a singular value decomposition of the Hessian of the quadratic performance index generally considered in Model Predictive Control. The singular vectors are employed to generate an orthonormal basis function expansion of the unconstrained solution to the finite horizon optimal control problem. These are preloaded into each actuator and each sensor. On the downlink, the actuators are informed, in real-time, about which basis functions they should use. On the uplink, after a 'burn in period', the sensors need only communicate when their response departs from that pre-calculated for the given basis functions. We show that this strategy facilitates communication in both the downlink and uplink in a cost-effective fashion. We also show that the strategy can be modified so that input constraints are satisfied. We illustrate the proposed results by applying them to a simulation of the cross direction control of a paper machine. Potential extensions and other applications of the proposed strategy are also discussed.

In this paper, we present an invariant-set-based method for actuator and sensor fault detection and isolation in Lure systems. The Lure plant is controlled by an observer-based feedback tracking controller, designed for the nominal (fault-free) system. Suitable residual signals are constructed from measurable system outputs and estimates associated with the nominal observer. Faults are diagnosed by online contrasting the residual signal trajectories against sets of values that the residuals are shown to attain under healthy or faulty operation. These values are obtained via set-invariance analysis of the system closed-loop trajectories.

We present a novel method to compute componentwise ultimate bounds and invariant regions for a class of switching discrete-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The method has the important advantage that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a standard norm for bounding either the perturbation or state vectors, and thus may avoid conservativeness due to different perturbation or state vector components having substantially different bounds. We also establish the relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function. We illustrate the application of our method via numerical examples, including the fault tolerance analysis of the feedback control of a winding machine. © 2013 © 2013 Taylor & Francis.

We propose a fault tolerant control scheme that compensates for actuator faults by adjusting the controller gain based on an estimate of the fault magnitude. The scheme consists of a plant in closed loop with an observer-based feedback tracking controller, which is adapted to the fault situation diagnosed by a fault detection and isolation algorithm. We give conditions for correct fault detection and isolation and for robust closed-loop stability accounting for possible errors in the fault estimation.

We present a novel method to compute componentwise transient bounds, componentwise ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components. In this latter case, we also show how our componentwise method can be combined with standard techniques in order to derive bounds possibly tighter than those corresponding to either method applied individually. © 2012 Elsevier Ltd. All rights reserved.

In spite of its easy implementation, ability to handle constraints and nonlinearities, etc.; model predictive control (MPC) does have drawbacks including tuning difficulties. In this paper, we propose a refinement to the basic MPC strategy by incorporating a tuning parameter such that one can move smoothly from an existing controller to a new MPC strategy. Each change of this tuning parameter leads to a new stabilising control law, therefore, allowing one to gradually move from an existing control law to a new and better one. For the infinite horizon case without constraints and for the general case with state and input constraints, stability results are established. We also examine the practical applicability of the proposed approach by employing it in the nominal prediction model of the tube-based output feedback robust MPC method. The merits of the proposed method are illustrated by examples. © 2013 Elsevier Ltd. All rights reserved.

We propose a fault-tolerant control scheme that deals with sensor and actuator faults through the use of a virtual actuator (VA) and a bank of virtual sensors (VSs). A novel feature of the scheme is that the VSs implicitly integrate both fault detection and isolation (FDI) and - in conjunction with the VA - controller reconfiguration tasks. The VA and the bank of VSs operate in closed-loop with an observer-based tracking controller designed for a nominal (fault free) model of the plant. A switching rule that reconfigures the VA and engages the suitable VS from the bank is based on sets defined for measurable residual signals constructed directly from the VS signals. Our method handles abrupt actuator and sensor faults of arbitrary magnitude including complete outage. The overall scheme is shown to guarantee closed-loop boundedness and setpoint tracking under all considered fault situations. Enhancements of the scheme to deal with errors in the fault detection and isolation are also proposed. Applications of the scheme to a winding machine and an interconnected tank system are presented. © 2013 Taylor and Francis Group, LLC.

SUMMARY In this paper, we present a robust actuator fault-tolerant control strategy for constrained linear systems in the presence of bounded state and input disturbances. The scheme is based on a bank of state estimators that match different fault situations that can occur in the system. A fault detection and isolation unit verifies that suitable residual variables lie inside pre-computed sets and selects the estimate that matches the current plant behaviour. A bank of robustly stabilizing tube-based model predictive control laws is designed, each associated to a fault scenario, and the appropriate controller is selected among them by using the information provided by the fault detection and isolation module. By means of 'tubes' of trajectories, we ensure robust closed-loop exponential stability of the constrained system and good performance in the fault-free case and under the occurrence of abrupt actuator faults, including actuator outage and loss of effectiveness by an unknown amount. © 2012 John Wiley & Sons, Ltd.

In this paper, a novel robust sensor fault tolerant control strategy for systems with linear parameter varying (LPV) uncertainty model description is proposed. The strategy combines a robust fault detection and identification (FDI) unit based on an invariant-set approach with controller reconfiguration based on the use of a virtual sensor. The robust FDI unit employs a bank of observers that can detect faulty and healthy situations based on the separation of relevant sets, whose computation takes into account system disturbances and model uncertainty. The closed-loop system is reconfigured by means of a virtual sensor which is adapted to the fault situation detected by the FDI unit. The FDI and virtual sensor modules are designed using polytopic LPV techniques and bilinear matrix inequalities. The resulting robust fault tolerant control scheme is guaranteed to preserve boundedness of the closed-loop system trajectories under a wide range of sensor fault scenarios. The performance of the proposed scheme is illustrated by a simulation example. © 2013 Elsevier Ltd. All rights reserved.

Induction Heating Furnaces are used extensively in industry. The basic principle is that induced eddy currents are used to heat a ferromagnetic material as it passes through a series of coils. Because of the importance of such systems, there has been on-going interest in their design and operation. Past work includes model development from physical principles and optimal design of operational practices. However, previous work has invariably been based on open-loop strategies. Our work is aimed at the design of a closed-loop control strategy incorporating feedback from the available measurements. This paper reports initial work including model development and calibration together with preliminary control system design. Proposed future work includes full scale industrial implementation.© 2013 Elsevier Ltd. All rights reserved.

We present a new control design method which guarantees any prespecified componentwise ultimate bound on the state of multiple-input systems with matched perturbations. The method is based on eigenstructure assignment by state feedback and utilises a componentwise bound computation procedure recently proposed by the authors. This procedure exploits both the system and perturbation structures by performing componentwise analysis, thus avoiding the need for bounds on the norm of the perturbation. We present examples which illustrate the simplicity and generality of the method.

This paper presents an application of a diagnosis and a fault tolerant control method for an active suspension system. This method detects and identifies an actuator fault and reconfigures the controller in order to maintain the ride comfort and vehicle performance in the presence of road disturbances. The proposed scheme consists of a diagnosis module and a controller reconfiguration module. The diagnosis module is based on the Unknown Input Observer approach to detect and identify actuator faults. A bank of observers is designed to generate residual signals such that each of them matches the system in a defined actuator fault mode. The controller reconfiguration module consists of a bank of controllers, each corresponding to a fault mode of the system. The module then changes the control law after the detection and the identification of an actuator fault. Simulation results show the effectiveness of the scheme applied to the active suspension system, which keeps its properties under the occurrence of severe faults. © 2007 IEEE.

Matrix Converters (MC) have prominent advantages in AC/AC power conversion applications, for example, in aerospace and surface transport. However, due to a tight interconnection between its input and output, it is difficult to achieve satisfactory performance on both sides, simultaneously. This paper proposes a novel, optimisation based, design approach for the modulation (duty cycle) matrix associated with the MC. Under this approach, the entries of the modulation matrix are used as decision variables in a quadratic optimisation programme. The optimisation combines control requirements both at the input and output sides of the converter in a non-conflicting manner. The paper includes simulation results, which demonstrate that the proposed solution complies with the objectives of both sides. Thus high quality tracking at the MC output is achieved simultaneously with input resonance suppression and power factor control at the MC input. An estimate of the computational load is provided which suggests that a real-time implementation is easily achievable.

Progressive increase of distributed generation penetration levels into the power grid has motivated studies on network stability and control methods in order to maintain the grid in operational conditions. A key challenge to better study features of the power grid is to have a reliable network model under different circumstances. In this paper, we derive a structured nonlinear model for the network and then propose an approximate model that captures the system nonlinearities. We then design voltage control functions and derive a dynamic network model with embedded controllers. We analyse voltage stability of the controlled network for both the exact and approximate models. In particular, we give conditions for the voltage to ultimately lie within quantifiable bounds for both models. This result also allows to compute a bound on the difference between the exact and approximate model voltages.

In this paper we study frequency stabilisation in inverter-based microgrid systems with primary droop controller and three types of secondary control strategies. Particularising the microgrid to the case of two inverter buses, we propose structured models for which frequency regulation, ultimate bounds and an estimate of the region of attraction associated with the ultimate bound set are obtained in closed-form analytical expressions.

© 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.This paper analyses conditions for computing invariant sets for Lure-type nonlinear systems via two different approaches, by bounding the nonlinear terms and by embedding the nonlinear system in a linear parameter varying (LPV) model. Furthermore, we show that by an appropriate redefinition of the system's nonlinear terms, the sets obtained by both approaches can be made identical. These methods can be used to compute sets characterising faultless (nominal) and faulty system behaviour, which are applicable for fault diagnosis under specific conditions that guarantee their separation. We also present a simple mechanism for sensor fault estimation and controller compensation, thus resulting in a closed-loop fault tolerant control system. An example of a pendulum system is provided to illustrate the proposed approach.

In this contribution we propose a fault tolerant control (FTC) scheme to compensate for sensor faults in multisensor fusion-based control of linear parameter varying systems. The scheme consists of a closedloop system with an estimator-based feedback tracking controller that combines healthy sensor-estimator pairings via a fusion algorithm. Faulty sensors are detected and discarded based on a set separation criterion, which considers both model uncertainty and system disturbances. We ensure the preservation of closed-loop system boundedness for a wide range of sensor fault situations and for the whole domain of plant model uncertainty. To illustrate the performance of the proposed FTC strategy, a flexible joint robotic arm example is presented.

In this paper, we first provide a mathematical description and proof for the robust stability of systems with convex polytopic uncertainty through the construction of attractive invariant sets. Then, we present the problem of estimating the convergence time to arbitrarily tight over-approximations of an invariant set, from both a known initial condition and for initial conditions belonging to a convex polytopic set. We then propose various analytical and numerical methods for computing the aforementioned convergence time. Finally, a number of numerical examples, including a flexible link robotic manipulator, are given to illustrate the results.

Model predictive control is a well established strategy for dealing with saturating actuator problems. The essential feature of the method is a receding horizon quadratic optimal control problem which is solved subject to input constraints. On the other hand, anti-windup methods are ad-hoc procedures which achieve input saturation in an instantaneous fashion. Both methods are known to perform well in practice and each has its strong advocates. In this paper, we explore connections between the methods. In particular, we show that there are cases in which anti-windup schemes are identical to MPC schemes. In other cases, we show that anti-windup has performance which is close to that of MPC strategies.

In this paper we study frequency stabilisation in inverter-based microgrid systems with primary droop controller and decentralised secondary control strategy. For the microgrid with any configuration of inverter buses, we establish conditions for trajectory ultimate boundedness and derive an explicit description of an estimate of the region of attraction to the ultimate bound set.

Icing is regarded as a severe structural alteration affecting unmanned aerial vehicles (UAVs), since ice accretion on wings and control surfaces modifies the aircraft shape resulting in altered controllability and performance of the vehicle. We study the problem of detection of icing and estimation of its 'severity' factor in longitudinal control of UAVs. We propose to employ a bank of unknown input observers (UlOs), each designed to match a model of the aircraft under a particular level of icing taken from a quantisation of the icing's severity factor range of variation. The UIO design exploits the change in equilibrium conditions, caused by the icing effect, to identify a direction in the observer estimation error space that allows for aircraft icing detection and estimation. By selecting at each time the observer from the bank that yields the smallest value of a suitable residual signal, the icing severity factor can be estimated with an accuracy that is inversely proportional to the size of the quantisation level.

This paper is concerned with linear parameter varying (LPV) fault tolerant control (FTC) systems reconfigured using virtual actuators. Most works dealing with FTC of LPV systems in general, and in particular using virtual actuators, assume perfect knowledge of the 'varying parameter' of the LPV system description at all times. In this paper, for LPV systems with convex polytopic modelling description, we derive conditions for closed-loop stability of the overall virtual-actuator based FTC system when the varying parameter is not exactly known but measured with bounded errors.

In this paper, we present a fault tolerant control (FTC) scheme for linear parameter varying systems that utilises multiple sensor switching to compensate for sensor faults. The closed-loop scheme consists of an estimator-based feedback tracking controller and sensor-estimate switching strategy which allows for the recovery of faulty sensors. The switching mechanism tracks the transitions from faulty to healthy behaviour by means of set separation and pre-computed transition times. The selected sensor-estimate pairing is then reconfigured based on available healthy sensors. Under the proposed scheme, preservation of closed-loop system boundedness is guaranteed for a wide range of sensor fault situations. An example is presented to illustrate the performance of the FTC strategy.

A recently proposed set-theoretic approach to fault diagnosis is based on membership of residual signals to (invariant) sets that characterise healthy and faulty system behaviour. Correct diagnosis is achieved if the relevant sets are disjoint. Tighter sets that allow separation for a larger class of faults are hence desirable within this approach. In this paper, we investigate methods to compute invariant sets for a Lure system from two different perspectives: by bounding the nonlinear terms and by embedding the nonlinear system in a linear parameter varying (LPV) model. To analyse and compare the methods, we develop a scalar and a second-order system case, and provide an example of a latter system. The results show that both approaches can be used for fault diagnosis and yield comparable performance for this type of systems under specific conditions.

This paper deals with the problem of obtaining fault-tolerant guarantees of a multi-sensor switching strategy for longitudinal control. The strategy selects, at each instant of time, the sensor (belonging to a collection of sensors) that provides the best closed loop performance, as measured by a control-performance criterion. It is assumed that each sensor has an associated feedback controller that has been designed such that the sensor-controller pair stabilises the closed loop system under normal operation conditions. Recent refinements for constructing ultimate-bound invariant sets allow obtaining less conservative fault-tolerant guarantees. Stability of the switching system under fault-free operation conditions and under presence of sensor failures are established in the main results of this paper.

This paper presents the properties of a new variant of model predictive control called Reduced Parameterisation Model Predictive Control (RPMPC). The new algorithm uses the structure of the null controllable set of input constrained unstable systems to produce a closed-loop system with a region of attraction that is an arbitrarily close approximation to this set. We show that the RPMPC algorithm converges in a finite number of iterations and we establish stability of the resulting closed-loop system. In addition, we present a rigorous worst case complexity analysis together with average computational tests. Both these studies show that for long horizons RPMPC has a lower computational requirement than that of standard MPC.

This paper introduces a new variant of model predictive control called Reduced Parameterisation Model Predictive Control. The new algorithm uses the structure of the null controllable set of input constrained unstable systems to produce a closed-loop system with a region of attraction that is an arbitrarily close approximation to this set. In addition, this result is achieved with a lower complexity than that of MPC.

Model Predictive Control (MPC) is a popular advanced control strategy that is implemented by determining an optimal control sequence at each time step. One of the major contributors to MPC's computational complexity is the length of the control sequence or, equivalently, the number of optimisation parameters. This paper analyses the null controllable sets, and their related finite horizon approximation, of a class of unstable systems. The analysis reveals that the same control action can be obtained using a much smaller set of control moves via a technique which we term reduced parameterisation.

In this paper we investigate the fault tolerance properties of a multisensor switching strategy previously proposed by the authors. We show that closed-loop stability of the resulting switching scheme is guaranteed in the presence of faulty sensors if a set of conditions on the system parameters (such as bounds on the sensor noises, maximum and minimum values of the reference signal, etc.) is satisfied. This constitutes an important step in enhancing and broadening the applicability of the proposed multisensor scheme since it provides guaranteed properties such as fault tolerance and robust closed-loop stability under sensor failure. The results are applied to the problem of automotive longitudinal control.

We present a new control design method which guarantees any prespecified componentwise ultimate bound on the state of multiple-input systems with matched perturbations. The method is based on eigenstructure assignment by state feedback and utilises a componentwise bound computation procedure recently proposed by the authors. This procedure exploits both the system and perturbation structures by performing componentwise analysis, thus avoiding the need for bounds on the norm of the perturbation. We present examples which illustrate the simplicity and generality of the method.

Advanced design aims to achieve the best possible performance subject to operational constraints. Such questions can be formulated as optimization problems. The advantages of using an explicit optimization formulation include clear articulation of performance trade-offs and the provision of a clear basis for quantitative comparison of different strategies. In this paper we will briefly review aspects of modern optimization. We will also show how these tools can be deployed in scheduling, estimation and control. We will illustrate the ideas by examples drawn from the mining, mineral and metal processing field. Copyright © 2007 IFAC.

An output-feedback control scheme is proposed based on the Certainty Equivalence [CE] principle and explicit solutions to the problem are investigated. It is assumed that the full state of the system is not measured and, instead, a measured output corrupted with measurement noise is available. It is also assumed that the control inputs are subject to polyhedral constraints. The CE control implementation uses a receding-horizon control law to compute the control action based on a state estimate obtained from a moving horizon state estimator. The structure of the solution is given by a piece-wise affine function of the parameters of the problem and a partition of the parameter space into polyhedral regions. The parameterized regions and coefficients of the piece-wise affine function can be pre-computed offline, allowing a simplified approach in implementing the output-feedback controller online. © 2006 IEEE.

This paper investigates the structure present in constrained linear state estimation problems formulated as a quadratic optimization program subject to linear inequality constraints. Polyhedral constraints on the system disturbance, the measurement noise, and the initial state are considered. The result interprets the measurement data and prior estimate as parameters and the parameter space is partitioned into multiple regions. Within each region the state estimate can be calculated as a piece-wise affine function of the measurement data and prior estimate. The parameterized regions and coefficients of the piece-wise affine function can be precomputed offline allowing a simplified approach in implementing a moving horizon estimation scheme. Copyright © 2005 IFAC.

We present a receding horizon control algorithm for compensation of backlash at the input of a stable linear system under control rate constraints. The problem is first posed as a receding horizon optimal control problem by modelling backlash as a piecewise affine system with a state space partition consisting of three regions. This optimal control problem involves solving, at each step, 3N quadratic programmes, where N is the optimisation horizon. As an alternative to solving the quadratic programmes, we propose a strategy based on a recently devised suboptimal receding horizon control algorithm which utilises a singular value decomposition of the Hessian of the quadratic programme. This alternative strategy leads, at the cost of some performance degradation, to much smaller computational load since a feasible rather than optimal solution has to be obtained at each step. Copyright © 2005 IFAC.

The solution to constrained linear model predictive control (MPC) problems can be pre-computed off-line in an explicit form as a piecewise affine (PWA) state feedback law defined on polyhedral regions of the state space. Even though real-time optimization is avoided, implementation of the PWA state-feedback law may still require a significant amount of computation due to the problem of determining which polyhedral region the state lies in. In this paper, a neural network approach to this problem is investigated.

In this paper we consider an alternative approach to implementing Model Predictive Control (MPC). We show that solving a class of quadratic programming (QP) problems is equivalent to solving a class of well-posed nonlinear algebraic loops. These algebraic loops are closely related to that found in static anti-windup synthesis. It therefore follows that certain classes of MPC can be implemented in a form that is a direct generalisation of the standard anti-windup structure. Implementations for classes of MPC are then derived for both regulator and tracking problems. Several alternative iterative algorithms for solving the algebraic loop are then presented. Some simulation results are also provided, with an example computational comparison to active set and interior point based MPC.

We present two redesigns to robustify backstepping control laws against dynamic uncertainties at the input of the plant. In the first redesign we construct a passivating control law from a CLF obtained by backstepping. In the second redesign we simplify the passivation redesign which becomes cumbersome for high order systems. We also review the two main versions of backstepping: cancelation and L V-backstepping, and compare their stability margins. G