Jacquelien Scherpen | University of Groningen (original) (raw)

Papers by Jacquelien Scherpen

Network systems consist of subsystems and their interconnections, and provide a powerful framewor... more Network systems consist of subsystems and their interconnections, and provide a powerful framework for analysis, modeling and control of complex systems. However, subsystems may have high-dimensional dynamics, and the amount and nature of interconnections may also be of high complexity. Therefore, it is relevant to study reduction methods for network systems. An overview on reduction methods for both the topological (interconnection) structure of the network and the dynamics of the nodes, while preserving structural properties of the network, and taking a control systems perspective, is provided. First topological complexity reduction methods based on graph clustering and aggregation are reviewed, producing a reduced-order network model. Second, reduction of the nodal dynamics is considered by using extensions of classical methods, while preserving the stability and synchronization properties. Finally, a structure-preserving generalized balancing method for simplifying simultaneousl...

A novel technique to achieve output tracking via stable inversion of non-minimum phase linear sys... more A novel technique to achieve output tracking via stable inversion of non-minimum phase linear systems is presented wherein the desired signal is obtained from eld measurements, and hence corrupted by noise. The earlier approach to stable inversion does not take into account the noise in the system. The unknown input decoupled observer approach is appli-cable only to minimum phase systems. Moreover, the unobservable states are inadequately constructed resulting in inferior output tracking in the presence of noise. In this paper we extend this procedure to non-minimum phase systems. We present the novel Stable Dy-namic model Inversion (SDI) approach which is applicable to non-minimum phase systems, and takes into account the presence of noise in target time histories. 1

The article presents an experimental implementation of an Immersion & Invariance (I&I), [2], cont... more The article presents an experimental implementation of an Immersion & Invariance (I&I), [2], controller design to an experimental cart-pendulum system. The I&I controller was developed in [1], and in the paper under discussion the domain of attraction is proven to be the open upper half plane. The article serves merely as an experimental proof of principle for the I&I method, where in addition to the I&I controller, extra control actions are taken in order to deal with practical issues such as friction, and the fact that the velocities are not measured. It is interesting to compare these results with another controller implementation on the same set up. In [5] an IDA-PBC (Interconnection and Damping Assignment Passivity Based Control) controller was developed and implemented on the same set up. Even though it is argued in the introduction of the article under discussion that the controller of [5] is rather complicated in comparison to the I&I control, the level of complexity or simp...

crete time-varying systems In this paper we investigate the relationship between the different Ri... more crete time-varying systems In this paper we investigate the relationship between the different Riccati equations that appear in the H ∞ control problem for linear discrete time-varying systems. Once we obtain this relation we can reformulate the conditions under which the H ∞ output feedback problem is solvable. In contrary to the conditions in terms of two coupled Riccati equations found in [15], the new conditions are stated in terms of two uncoupled Riccati equations and a coupling condition. 1

This paper studies the optimal control of a micro grid of biogas prosumers equipped with local st... more This paper studies the optimal control of a micro grid of biogas prosumers equipped with local storage devices. Excess biogas can be upgraded and injected into the low-pressure gas grid or, alternatively, shipped per lorry to be used elsewhere in an effort to create revenue. The aim of the control process is to maximize the prosumers' profit from the biogas they produce, with a size restriction on the local biogas storage. The problem is solved in centralized and distributed MPC schemes in order to compare their capabilities to control a micro grid. We perform simulations with a realistic average gas usage pattern over a year to study the economic feasibility of local biogas storage.

This paper is concerned with model order reduction based on balanced realization for symmetric no... more This paper is concerned with model order reduction based on balanced realization for symmetric nonlinear systems. A new notion of symmetry for nonlinear systems was characterized recently. It plays an important role in linear systems theory and is expected to provide new insights to nonlinear systems. In this paper, we provide a novel framework of balanced realization for this class of systems and apply it for model order reduction preserving symmetry.

In this paper we develop a solution to the discrete-time H∞ output feedback control problem for L... more In this paper we develop a solution to the discrete-time H∞ output feedback control problem for Linear Time-Varying (LTV) systems. The solution is developed along the strategy set up in [Doyle et. al. 1989] and the main ingredient in its derivation is the extension of the well-known bounded real lemma to a (discrete) time-varying context, developed in [van der Veen and Verhaegen 1995]. This approach contributes to the conceptual simplicity, and hence to the accessibility, of the solution. Apart from that, we treat the infinite-horizon case for LTV system of non-uniform state dimension, and varying input and output dimension. Both situations can easily occur in practice, e.g. in multirate sampled data control systems. The algorithm that can be derived from the solution presented is then applied to the H∞ output feedback of a dynamical system changing from one operation point in its operation envelope to another.

Abstract: In the present work we are interested on the derivation of power-based passivity proper... more Abstract: In the present work we are interested on the derivation of power-based passivity properties for a certain class of non-linear mechanical systems. While for (non)-linear mechanical systems, it is of common use to adopt a storage function related to the system’s energy in order to show passivity and stabilize the system on a desired equilibrium point(e.g., IDA-PBC (Ortega et al., 1998)), we want here to obtain similar properties related to the system’s power. The motivation arises from the idea that in some engineering applications(satellite orbit motion, aircraft dynamic,etc...)seems more sensible to cope with the power flowing into the system instead of the energy that for stabilization purposes, means to consider the systems’s equilibrium the state for which the energy flow-rate(i.e.,system’s power)is minimal. In this respect, we recall first the power-based description for a certain class of (non)-linear mechanical systems given in (de Rinaldis and Scherpen, 2005) and th...

Abstract—In this paper we develop a mathematical model for the dynamics of a linear plate with pi... more Abstract—In this paper we develop a mathematical model for the dynamics of a linear plate with piezoelectric actuation. This model can then be used to design controllers with the goal of achieving a desired shape of the plate. This control scheme can be used for several applications, e.g., vibration control in structures or shape control for high precision structures like inflatable space reflectors. The starting point of the control design is modeling for control. We will do this in the framework of port-Hamiltonian (pH) modeling, since the pH modeling framework has very nice properties which can be exploited if one wants to design a controller for a specific task. One property for example is that it facilitates modeling multi physics systems or systems which consist of several systems by first modeling all parts separate and then interconnecting them. This is possible because any interconnection of pH systems yields again a pH system. Hence, the pH framework is useful for our mult...

In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. ... more In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. Given a desired trajectory, the feedforward signal is generated using a stable in-version method, and the feedback signal includes the switching term of the sliding mode control law. In this manner, we introduce robustness into the stable inversion technique. This approach is motivated by the need to replicate time signals typically in the automo-bile industry. The application of such an interpretation to a quarter car benchmark model yields encouraging results. Special attention will be given to non-minimum phase systems illustrated by a simulation example of the lunar roving vehicle. 1

In this paper the outer-J-lossless factorization for linear discrete time-varying systems is trea... more In this paper the outer-J-lossless factorization for linear discrete time-varying systems is treated. Lossless operators and its corresponding J-lossless chain-scattering operators are studied. Then the factorization is treated by first ‘taking out’ the anticausal part, and then considering the outer-J-lossless factorization of the causal part.

In this paper a method is presented to build an Euler Lagrange model for switching electrical net... more In this paper a method is presented to build an Euler Lagrange model for switching electrical networks in a structured general way, which is also applicable to ideal electrical circuits without switches. The switches make the dynamic models nonlinear. For using the Lagrangian structure for controller design, a preliminary study of the zero-dynamics of such switching network is presented. A case study, where a passivity based controller has been applied, is given for the Cuk converter.

ArXiv, 2019

In this paper we propose a new passivity-based control technique for DC power networks comprising... more In this paper we propose a new passivity-based control technique for DC power networks comprising the so-called ZIP-loads, i.e., nonlinear loads with the parallel combination of unknown constant impedance (Z), current (I) and power (P) components. More precisely, we propose a novel passifying input and a storage function based on the so-called mixed potential function introduced by Brayton and Moser, leading to a novel passivity property with output port-variable equal to the first time derivative of the voltage. Differently from the existing results in the literature, where restrictive (sufficient) conditions on Z, P and the voltage reference are assumed to be satisfied, we establish a passivity property for every positive voltage reference and every type of load. Consequently, we develop a new decentralized passivity-based control scheme that is robust with respect to the uncertainty affecting the ZIP-loads.

ArXiv, 2020

We propose a comprehensive nonlinear ODE-based thermo-hydraulic model of a district heating syste... more We propose a comprehensive nonlinear ODE-based thermo-hydraulic model of a district heating system featuring several heat producers, consumers and storage devices which are interconnected through a distribution network of meshed topology whose temperature dynamics are explicitly considered. Moreover, we present conditions under which the hydraulic and thermal subsystems of the model exhibit shifted passivity properties and discuss some of the beneficial implications for decentralized control design and stability analysis. For the former subsystem, our results draw on the monotonicity attributes manifested by the mappings involved. For the latter, we propose a storage function based on the {\em ectropy function} of a thermodynamic system, recently used in the passivity analysis of heat exchanger networks. Our formal analysis is supported with numerical simulations on a case study using realistic system parameters.

In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. ... more In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. Given a desired trajectory, the feedforward signal is generated using a stable inversion method, and the feedback signal includes the switching term of the sliding mode control law. In this manner, we introduce robustness into the stable inversion technique. This approach is motivated by the need to replicate time signals typically in the automobile industry. The application of such an interpretation to a quarter car benchmark model yields encouraging results. Special attention will be given to non-minimum phase systems illustrated by a simulation example of the lunar roving vehicle.

In this paper we derive the dynamic model of a four–cylinder double–acting wobble–yoke Stirling e... more In this paper we derive the dynamic model of a four–cylinder double–acting wobble–yoke Stirling engine introduced originally by [1, 2]. In contrast with the classical thermodynamics methods that dominate the literature of Stirling mechanisms, we present a control system perspective to obtain a useful model for the analysis and synthesis of feedback control laws. The main motivation is the application of this gas engine in a micro–combined heat and power (CHP) generation system.

The nonlinear extension of the balancing procedure requires the case of state dependent quadratic... more The nonlinear extension of the balancing procedure requires the case of state dependent quadratic forms for the energy functions, i.e., the nonlinear extensions of the linear Gramians are state dependent matrices. These extensions have some interesting ambiguities that do not occur in the linear case. Namely, the choice of the state dependent matrix in the semi-quadratic form is not unique, and therefore may result in different eigenvalues. The introduction of so-called null-matrices is useful for the analysis of this problem. Furthermore, the concept of norm-preserving transformations provides further insight on these ambiguities. This paper provides a detailed analysis of this phenomenon and outlines some future directions for research.

This paper studies model order reduction of multi-agent systems consisting of identical linear pa... more This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order networked model, which leads to a reduced-order system preserving the passivity of each subsystem. To restore the network structure, we then apply a coordinate transformation to convert the resulting reduced-order model to a state-space model of Laplacian dynamics. The proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics. Moreover, it preserves the passivity of the subsystems and allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example.

In this note we address the problem of model reduction of a particular class of linear systems, n... more In this note we address the problem of model reduction of a particular class of linear systems, namely, the linear port-Hamiltonian (PH) systems. Furthermore, we explore the preservation of the PH structure in the reduced model. Towards this end, we adopt the balanced truncation approach to reduced the order of the model, in particular, we study the use of extended Gramians to balance the linear PH systems. The latter provides degrees of freedom to impose a desired structure, in this case a PH one, to the reduced model. Moreover, for balanced truncation using extended Gramians, the error bound is well-known and is given in terms of the Hankel singular values of the truncated state.

arXiv: Optimization and Control, 2019

This paper investigates a model reduction problem for linear directed network systems, in which t... more This paper investigates a model reduction problem for linear directed network systems, in which the interconnections among the vertices are described by general weakly connected digraphs. First, the definitions of pseudo controllability and observability Gramians are proposed for semistable systems, and their solutions are characterized by Lyapunov-like equations. Then, we introduce a concept of vertex clusterability to guarantee the boundedness of the approximation error and use the newly proposed Gramians to facilitate the evaluation of the dissimilarity of each pair of vertices. An clustering algorithm is thereto provided to generate an appropriate graph clustering, whose characteristic matrix is employed as the projections in the Petrov-Galerkin reduction framework. The obtained reduced-order system preserves the weakly connected directed network structure, and the approximation error is computed by the pseudo Gramians. Finally, the efficiency of the proposed approach is illustr...

Network systems consist of subsystems and their interconnections, and provide a powerful framewor... more Network systems consist of subsystems and their interconnections, and provide a powerful framework for analysis, modeling and control of complex systems. However, subsystems may have high-dimensional dynamics, and the amount and nature of interconnections may also be of high complexity. Therefore, it is relevant to study reduction methods for network systems. An overview on reduction methods for both the topological (interconnection) structure of the network and the dynamics of the nodes, while preserving structural properties of the network, and taking a control systems perspective, is provided. First topological complexity reduction methods based on graph clustering and aggregation are reviewed, producing a reduced-order network model. Second, reduction of the nodal dynamics is considered by using extensions of classical methods, while preserving the stability and synchronization properties. Finally, a structure-preserving generalized balancing method for simplifying simultaneousl...

A novel technique to achieve output tracking via stable inversion of non-minimum phase linear sys... more A novel technique to achieve output tracking via stable inversion of non-minimum phase linear systems is presented wherein the desired signal is obtained from eld measurements, and hence corrupted by noise. The earlier approach to stable inversion does not take into account the noise in the system. The unknown input decoupled observer approach is appli-cable only to minimum phase systems. Moreover, the unobservable states are inadequately constructed resulting in inferior output tracking in the presence of noise. In this paper we extend this procedure to non-minimum phase systems. We present the novel Stable Dy-namic model Inversion (SDI) approach which is applicable to non-minimum phase systems, and takes into account the presence of noise in target time histories. 1

The article presents an experimental implementation of an Immersion & Invariance (I&I), [2], cont... more The article presents an experimental implementation of an Immersion & Invariance (I&I), [2], controller design to an experimental cart-pendulum system. The I&I controller was developed in [1], and in the paper under discussion the domain of attraction is proven to be the open upper half plane. The article serves merely as an experimental proof of principle for the I&I method, where in addition to the I&I controller, extra control actions are taken in order to deal with practical issues such as friction, and the fact that the velocities are not measured. It is interesting to compare these results with another controller implementation on the same set up. In [5] an IDA-PBC (Interconnection and Damping Assignment Passivity Based Control) controller was developed and implemented on the same set up. Even though it is argued in the introduction of the article under discussion that the controller of [5] is rather complicated in comparison to the I&I control, the level of complexity or simp...

crete time-varying systems In this paper we investigate the relationship between the different Ri... more crete time-varying systems In this paper we investigate the relationship between the different Riccati equations that appear in the H ∞ control problem for linear discrete time-varying systems. Once we obtain this relation we can reformulate the conditions under which the H ∞ output feedback problem is solvable. In contrary to the conditions in terms of two coupled Riccati equations found in [15], the new conditions are stated in terms of two uncoupled Riccati equations and a coupling condition. 1

This paper studies the optimal control of a micro grid of biogas prosumers equipped with local st... more This paper studies the optimal control of a micro grid of biogas prosumers equipped with local storage devices. Excess biogas can be upgraded and injected into the low-pressure gas grid or, alternatively, shipped per lorry to be used elsewhere in an effort to create revenue. The aim of the control process is to maximize the prosumers' profit from the biogas they produce, with a size restriction on the local biogas storage. The problem is solved in centralized and distributed MPC schemes in order to compare their capabilities to control a micro grid. We perform simulations with a realistic average gas usage pattern over a year to study the economic feasibility of local biogas storage.

This paper is concerned with model order reduction based on balanced realization for symmetric no... more This paper is concerned with model order reduction based on balanced realization for symmetric nonlinear systems. A new notion of symmetry for nonlinear systems was characterized recently. It plays an important role in linear systems theory and is expected to provide new insights to nonlinear systems. In this paper, we provide a novel framework of balanced realization for this class of systems and apply it for model order reduction preserving symmetry.

In this paper we develop a solution to the discrete-time H∞ output feedback control problem for L... more In this paper we develop a solution to the discrete-time H∞ output feedback control problem for Linear Time-Varying (LTV) systems. The solution is developed along the strategy set up in [Doyle et. al. 1989] and the main ingredient in its derivation is the extension of the well-known bounded real lemma to a (discrete) time-varying context, developed in [van der Veen and Verhaegen 1995]. This approach contributes to the conceptual simplicity, and hence to the accessibility, of the solution. Apart from that, we treat the infinite-horizon case for LTV system of non-uniform state dimension, and varying input and output dimension. Both situations can easily occur in practice, e.g. in multirate sampled data control systems. The algorithm that can be derived from the solution presented is then applied to the H∞ output feedback of a dynamical system changing from one operation point in its operation envelope to another.

Abstract: In the present work we are interested on the derivation of power-based passivity proper... more Abstract: In the present work we are interested on the derivation of power-based passivity properties for a certain class of non-linear mechanical systems. While for (non)-linear mechanical systems, it is of common use to adopt a storage function related to the system’s energy in order to show passivity and stabilize the system on a desired equilibrium point(e.g., IDA-PBC (Ortega et al., 1998)), we want here to obtain similar properties related to the system’s power. The motivation arises from the idea that in some engineering applications(satellite orbit motion, aircraft dynamic,etc...)seems more sensible to cope with the power flowing into the system instead of the energy that for stabilization purposes, means to consider the systems’s equilibrium the state for which the energy flow-rate(i.e.,system’s power)is minimal. In this respect, we recall first the power-based description for a certain class of (non)-linear mechanical systems given in (de Rinaldis and Scherpen, 2005) and th...

Abstract—In this paper we develop a mathematical model for the dynamics of a linear plate with pi... more Abstract—In this paper we develop a mathematical model for the dynamics of a linear plate with piezoelectric actuation. This model can then be used to design controllers with the goal of achieving a desired shape of the plate. This control scheme can be used for several applications, e.g., vibration control in structures or shape control for high precision structures like inflatable space reflectors. The starting point of the control design is modeling for control. We will do this in the framework of port-Hamiltonian (pH) modeling, since the pH modeling framework has very nice properties which can be exploited if one wants to design a controller for a specific task. One property for example is that it facilitates modeling multi physics systems or systems which consist of several systems by first modeling all parts separate and then interconnecting them. This is possible because any interconnection of pH systems yields again a pH system. Hence, the pH framework is useful for our mult...

In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. ... more In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. Given a desired trajectory, the feedforward signal is generated using a stable in-version method, and the feedback signal includes the switching term of the sliding mode control law. In this manner, we introduce robustness into the stable inversion technique. This approach is motivated by the need to replicate time signals typically in the automo-bile industry. The application of such an interpretation to a quarter car benchmark model yields encouraging results. Special attention will be given to non-minimum phase systems illustrated by a simulation example of the lunar roving vehicle. 1

In this paper the outer-J-lossless factorization for linear discrete time-varying systems is trea... more In this paper the outer-J-lossless factorization for linear discrete time-varying systems is treated. Lossless operators and its corresponding J-lossless chain-scattering operators are studied. Then the factorization is treated by first ‘taking out’ the anticausal part, and then considering the outer-J-lossless factorization of the causal part.

In this paper a method is presented to build an Euler Lagrange model for switching electrical net... more In this paper a method is presented to build an Euler Lagrange model for switching electrical networks in a structured general way, which is also applicable to ideal electrical circuits without switches. The switches make the dynamic models nonlinear. For using the Lagrangian structure for controller design, a preliminary study of the zero-dynamics of such switching network is presented. A case study, where a passivity based controller has been applied, is given for the Cuk converter.

ArXiv, 2019

In this paper we propose a new passivity-based control technique for DC power networks comprising... more In this paper we propose a new passivity-based control technique for DC power networks comprising the so-called ZIP-loads, i.e., nonlinear loads with the parallel combination of unknown constant impedance (Z), current (I) and power (P) components. More precisely, we propose a novel passifying input and a storage function based on the so-called mixed potential function introduced by Brayton and Moser, leading to a novel passivity property with output port-variable equal to the first time derivative of the voltage. Differently from the existing results in the literature, where restrictive (sufficient) conditions on Z, P and the voltage reference are assumed to be satisfied, we establish a passivity property for every positive voltage reference and every type of load. Consequently, we develop a new decentralized passivity-based control scheme that is robust with respect to the uncertainty affecting the ZIP-loads.

ArXiv, 2020

We propose a comprehensive nonlinear ODE-based thermo-hydraulic model of a district heating syste... more We propose a comprehensive nonlinear ODE-based thermo-hydraulic model of a district heating system featuring several heat producers, consumers and storage devices which are interconnected through a distribution network of meshed topology whose temperature dynamics are explicitly considered. Moreover, we present conditions under which the hydraulic and thermal subsystems of the model exhibit shifted passivity properties and discuss some of the beneficial implications for decentralized control design and stability analysis. For the former subsystem, our results draw on the monotonicity attributes manifested by the mappings involved. For the latter, we propose a storage function based on the {\em ectropy function} of a thermodynamic system, recently used in the passivity analysis of heat exchanger networks. Our formal analysis is supported with numerical simulations on a case study using realistic system parameters.

In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. ... more In this paper we provide a feedforward-feedback interpretation of a sliding mode control scheme. Given a desired trajectory, the feedforward signal is generated using a stable inversion method, and the feedback signal includes the switching term of the sliding mode control law. In this manner, we introduce robustness into the stable inversion technique. This approach is motivated by the need to replicate time signals typically in the automobile industry. The application of such an interpretation to a quarter car benchmark model yields encouraging results. Special attention will be given to non-minimum phase systems illustrated by a simulation example of the lunar roving vehicle.

In this paper we derive the dynamic model of a four–cylinder double–acting wobble–yoke Stirling e... more In this paper we derive the dynamic model of a four–cylinder double–acting wobble–yoke Stirling engine introduced originally by [1, 2]. In contrast with the classical thermodynamics methods that dominate the literature of Stirling mechanisms, we present a control system perspective to obtain a useful model for the analysis and synthesis of feedback control laws. The main motivation is the application of this gas engine in a micro–combined heat and power (CHP) generation system.

The nonlinear extension of the balancing procedure requires the case of state dependent quadratic... more The nonlinear extension of the balancing procedure requires the case of state dependent quadratic forms for the energy functions, i.e., the nonlinear extensions of the linear Gramians are state dependent matrices. These extensions have some interesting ambiguities that do not occur in the linear case. Namely, the choice of the state dependent matrix in the semi-quadratic form is not unique, and therefore may result in different eigenvalues. The introduction of so-called null-matrices is useful for the analysis of this problem. Furthermore, the concept of norm-preserving transformations provides further insight on these ambiguities. This paper provides a detailed analysis of this phenomenon and outlines some future directions for research.

This paper studies model order reduction of multi-agent systems consisting of identical linear pa... more This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order networked model, which leads to a reduced-order system preserving the passivity of each subsystem. To restore the network structure, we then apply a coordinate transformation to convert the resulting reduced-order model to a state-space model of Laplacian dynamics. The proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics. Moreover, it preserves the passivity of the subsystems and allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example.

In this note we address the problem of model reduction of a particular class of linear systems, n... more In this note we address the problem of model reduction of a particular class of linear systems, namely, the linear port-Hamiltonian (PH) systems. Furthermore, we explore the preservation of the PH structure in the reduced model. Towards this end, we adopt the balanced truncation approach to reduced the order of the model, in particular, we study the use of extended Gramians to balance the linear PH systems. The latter provides degrees of freedom to impose a desired structure, in this case a PH one, to the reduced model. Moreover, for balanced truncation using extended Gramians, the error bound is well-known and is given in terms of the Hankel singular values of the truncated state.

arXiv: Optimization and Control, 2019

This paper investigates a model reduction problem for linear directed network systems, in which t... more This paper investigates a model reduction problem for linear directed network systems, in which the interconnections among the vertices are described by general weakly connected digraphs. First, the definitions of pseudo controllability and observability Gramians are proposed for semistable systems, and their solutions are characterized by Lyapunov-like equations. Then, we introduce a concept of vertex clusterability to guarantee the boundedness of the approximation error and use the newly proposed Gramians to facilitate the evaluation of the dissimilarity of each pair of vertices. An clustering algorithm is thereto provided to generate an appropriate graph clustering, whose characteristic matrix is employed as the projections in the Petrov-Galerkin reduction framework. The obtained reduced-order system preserves the weakly connected directed network structure, and the approximation error is computed by the pseudo Gramians. Finally, the efficiency of the proposed approach is illustr...