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In cases where we do not have the exact parameter values of a mathematical model, we often have at least some structural information, e.g., that some parameters are nonzero. Such information can be captured by so-called pattern matrices, whose symbolic entries are used to represent the available information about the corresponding parameters. In this letter, we focus on pattern matrices with three types of symbolic entries: those that represent zero, nonzero, and arbitrary parameters. We formally define and study addition and multiplication of such pattern matrices. The results are then used in the algebraic characterization of three strong structural properties. In particular, we provide sufficient conditions for controllability of linear descriptor systems, necessary and sufficient conditions for input-state observability, and sufficient conditions for output controllability of linear systems.

This letter studies reachability and null-controllability for difference inclusions involving convex processes. Such difference inclusions arise, for instance, in the study of linear discrete-time systems whose inputs and/or states are constrained to lie within a convex cone. After developing a geometric framework for convex processes relying on invariance properties, we provide necessary and sufficient conditions for both reachability and null-controllability in terms of the spectrum of dual processes.

We consider output processes which are realizable by stochastic linear time-invariant (LTI) systems. Such processes can always be realized by LTI systems in forward innovation form, and we study the transfer matrices of such LTI realizations. We show that such a transfer matrix is consistent with an acyclic directed graph if and only if the edges of this graph represent Granger-causality relations among the components of the output process. By consistency we mean that if there is no edge between two vertices of the graph, then the corresponding block of the transfer matrix is zero. Under this assumption, conditional Granger non-causality between the components of the process is equivalent with a zero block in the transfer matrix.

Projected dynamical systems (PDS) are discontinuous dynamical systems obtained by projecting a vector field on the tangent cone of a given constraint set. As such, PDS provide a convenient formalism to model constrained dynamical systems. When dealing with vector fields, which satisfy certain monotonicity properties, but not necessarily with respect to usual Euclidean norm, the resulting PDS does not necessarily inherit this monotonicity, as we will show. However, we demonstrate that if the projection is carried out with respect to a well-chosen norm, then the resulting 'oblique PDS' preserves the monotonicity of the unconstrained dynamics. This feature is especially desirable as monotonicity allows to guarantee important (incremental) stability properties and stability of periodic solutions (under periodic excitation). These properties can now be guaranteed based on the unconstrained dynamics using 'smart' projection instead of having to carry out a difficult a posteriori analysis on a constrained discontinuous dynamical system. To illustrate this, an application in the context of observer re-design is presented, which guarantees that the state estimate lies in the same state set as the observed state trajectory.

This paper studies consensus problems for multi-agent systems defined on directed graphs where the consensus dynamics involves general nonlinear and discontinuous functions. Sufficient conditions, only involving basic properties of the nonlinear functions and the topology of the underlying graph, are derived for the agents to converge to consensus.

This paper studies nature of solutions of scalar linear complementarity systems. The main result asserts that such systems do not exhibit Zeno behavior, that is there can only be finitely many switchings within any finite time interval.

In this paper, we study a class of set-valued dynamical systems that satisfy maximal monotonicity properties. This class includes linear relay systems, linear complementarity systems, and linear mechanical systems with dry friction under some conditions. We discuss two numerical schemes based on time-stepping methods for the computation of the periodic solutions when these systems are periodically excited. We provide formal mathematical justifications for the numerical schemes in the sense of consistency, which means that the continuous-time interpolations of the numerical solutions converge to the continuous-time periodic solution when the discretization step vanishes. The two time-stepping methods are applied for the computation of the periodic solution exhibited by a power electronic converter and the corresponding methods are compared in terms of approximation accuracy and computation time.

In this paper we study the question under which conditions a linear multi-modal system is disturbance decoupled. We establish necessary and sufficient geometric conditions from which the existing results on switched linear systems and conewise linear systems can be recovered as special cases. Also, we apply these conditions to a class of linear complementarity systems in order to obtain a more crisp characterization.

This paper deals with robust synchronization of directed and undirected multi-agent networks with uncertain agent dynamics. Given a network with identical nominal dynamics, we allow uncertainty in the form of coprime factor perturbations of the transfer matrix of the agent dynamics. These perturbations are assumed to be stable and have H-infinity-norm that is bounded by an a priori given desired tolerance. We derive state space equations for dynamic observer based protocols that achieve robust synchronization in the presence of such uncertainty. We obtain an achievable interval, i.e. an interval such that for each value of the tolerance contained in this interval there exists a robustly synchronizing protocol. (C) 2016 Elsevier B.V. All rights reserved.

Çamlıbel, Mehmet Kanat (Dogus Author) This paper deals with a class of dynamical systems obtained from interconnecting linear systems with static set-valued relations. We first show that such an interconnection can be described by a differential inclusions with a maximal monotone set-valued mappings when the underlying linear system is passive and the static relation is maximal monotone. Based on the classical results on such differential inclusions, we conclude that such interconnections are well-posed in the sense of existence and uniqueness of solutions. Finally, we investigate conditions which guarantee well-posedness but are weaker than passivity.

In this paper, we deal with the well-posedness (in the sense of existence and uniqueness of solutions) and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting. First, we show that the conditions guaranteeing uniqueness of Filippov solutions in the context of general differential inclusions are quite restrictive when applied to bimodal piecewise affine systems. Later, we present a set of necessary and sufficient conditions for uniqueness of Filippov solutions for bimodal piecewise affine systems. We also study the so-called Zeno behavior (possibility of infinitely many switchings within a finite time interval) for Filippov solutions. (C) 2014 Elsevier B.V. All rights reserved.

This paper deals with model reduction by balanced truncation of switched linear systems (SLS). We consider switched linear systems whose dynamics, depending on the switching signal, switches between finitely many linear systems with a common state space. These linear systems are called the modes of the SLS. The idea is to seek for conditions under which there exists a single state space transformation that brings all modes of the SLS in balanced coordinates. As a measure of reachability and observability of the state components of the SLS, we take the average of the diagonal gramians. We then perform balanced truncation by discarding the state components corresponding to the smallest diagonal elements of this average balanced gramian. In order to carry out this program, we derive necessary and sufficient conditions under which a finite collection of linear systems with common state space can be balanced by a single state space transformation. Among other things, we derive sufficient conditions under which global uniform exponential stability of the SLS is preserved under simultaneous balanced truncation. Likewise, we derive conditions for preservation of positive realness or bounded realness of the SLS. Finally, in case that the conditions for simultaneous balancing do not hold, or we simply do not want to check these conditions, we propose to compute a suitable state space transformation on the basis of minimization of an overall cost function associated with the modes of the SLS. We show that in case our conditions do hold, this transformation is in fact simultaneously balancing, bringing us back to the original method described in this paper. IEEE Control Systems Society

In this paper we consider the problem of approximating a consensus network by a less complex network, by removing cycles from the original network graph. The consensus network consists of agents that exchange relative state information with their neighbors in the network. We assume the agents have single-integrator dynamics and the network graph is undirected. The network used to approximate the original system has the same nodes as the original graph, but its edge set is a strict subset of the original edge set. We obtain a priori upper bounds on the absolute approximation error, depending on the length of the removed cycles, the algebraic connectivity of a chosen spanning tree of the network graph, and the largest eigenvalue of the Laplacian matrix of that spanning tree.

This paper presents a complete analogue of the well-known Kalman-Yakubovich-Popov lemma for descriptor systems, i.e. necessary and sufficient linear matrix inequality conditions for passivity and positive realness of descriptor systems. Also a full characterization of extended strictly positive realness is given for this class of systems. Some of the earlier related results are recovered from the presented results. (C) 2009 Elsevier B.V. All rights reserved.

Lately, with the increasing use of automated manual transmissions (AMT) the engagement control of the dry clutch becomes more important. The engagement control plays a crucial role, since different and conflicting objectives have to be satisfied: preservation of driver comfort, fast engagement and small friction losses. In this paper two optimal control strategies for clutch engagement, based on hybrid control principles, are compared. For developing a useful clutch control scheme, the driveline is modelled as a piecewise linear system. The first control strategy is widely known as explicit MPC. However, it seems that it is not suitable (yet) for this type of problem. The second strategy is a piecewise LQ controller, based on piecewise quadratic Lyapunov functions. Simulation results obtained with both strategies are presented and discussed.

Çamlıbel, Mehmet Kanat (Dogus Author) This paper studies the disturbance decoupling problem for multi-agent systems with single integrator dynamics and a directed communication graph. We are interested in topological conditions that imply the disturbance decoupling of the network, and more generally guarantee the existence of a state feedback rendering the system disturbance decoupled. In particular, we will develop a class of graph partitions, which can be described as a "topological translation" of controlled invariant subspaces in the context of dynamical networks. Then, we will derive sufficient conditions in terms of graph partitions such that the network is disturbance decoupled, as well as conditions guaranteeing solvability of the disturbance decoupling problem. The proposed results are illustrated by a numerical example.

Çamlıbel, Mehmet Kanat (Dogus Author) In this paper, we generalize all previously known results on the controllability of discrete-time linear systems with conic input and/or state constraints. In addition, we single out two cases of the problem which did not appear in the literature before. We provide a characterization for the first one of these new cases. For the second newly introduced case, we show that it is rare and pathological. Moreover, we show that the classical results cannot be extended to this last pathological case. These results altogether lead to an almost complete spectral characterization of controllability for discrete-time linear systems with conic input and/or state constraints.

We consider the problem of fault detection and isolation for a class of linear dynamical systems defined by a graph containing faulty vertices and observer vertices. Using a geometric approach, we provide a characterization of the smallest conditioned invariant subspaces generated by faults in terms of the underlying graph structure. Based on this characterization, we give graph-theoretic conditions guaranteeing fault detectability. In addition, we provide a condition under which fault detectability fails.

In this paper we study the disturbance decoupling problem for continuous piecewise affine systems. We establish a set of necessary conditions and a set of sufficient conditions, both geometric in nature, for such systems to be disturbance decoupled. Furthermore, we investigate mode-independent state feedback controllers for piecewise affine systems and provide sufficient conditions for the solvability of the disturbance decoupling problem by state feedback.

This paper studies the reachability problem for discrete-time conewise linear systems. Controllability of continuous-time conewise linear systems has already been studied in the literature and admits elegant spectral characterizations that resemble the well-known Popov-Belevitch-Hautus test for linear systems. However, the techniques that are employed for continuous-time systems are not applicable in the context of discrete-time due to the lack of convexity of a certain set of reachable states. To overcome this obstacle, we employ a combination of methods from geometric control theory and convex analysis and provide spectral conditions for the reachability of discrete-time conewise linear systems.

Piecewise-linear (PL) modeling is often used to approximate the behavior of nonlinear circuits. One of the possible PL modeling methodologies is based on the linear complementarity problem, and this approach has already been used extensively in the circuits and systems community for static networks. In this paper, the object of study will be dynamic electrical circuits that can be recast as linear complementarity systems, i.e., as interconnections of linear time-invariant differential equations and complementarity conditions (ideal diode characteristics). A mathematically precise framework is developed that formalizes the mixed discrete and continuous behavior of these switched networks. Within this framework, the fundamental question of well-posedness (existence and uniqueness of solution trajectories given an initial condition) is studied and additional properties of the behavior are derived. For instance, a full characterization is presented of the inconsistent states.

The recent interest in hybrid systems has given rise to a large variety of model classes and to many different notions of solution trajectories. In this paper we enumerate several solution concepts and compare them on the basis of some examples displaying Zeno behaviour. The relation to well-posedness is also discussed.

We consider the class of dynamical systems that arises when inputs and outputs of a linear system are connected pairwise by means of piecewise linear algebraic relations. It is not assumed that these relations define inputs in terms of outputs or vice versa; in particular, the relations need not be Lipschitzian. We obtain conditions for existence and uniqueness of solutions of such dynamical systems in the class of piecewise Bohl functions. (C) 2002 Elsevier Science Inc. All rights reserved.

In this brief, we will study the computation of transient solutions of a class of piecewise- linear (PL) circuits. The network models will be so-called linear complementarity systems, which can be seen as dynamical extensions of the PL modeling structure as proposed by [1]. In particular, the numerical simulation will be based on a time-stepping method using the well-known backward Euler scheme. It will be demonstrated, by means of an example, that this widely applied time-stepping method does not necessarily produce useful output for arbitrary linear dynamical systems with ideal diode characteristics. Next the consistency of the method will be proven for PL networks that can be realized by linear passive circuit elements and ideal diodes by showing that the approximations generated by the method converge to the true solution of the system in a suitable sense. To give such a consistency proof, the fundamental framework developed in [2] is indispensable as it proposes a precise definition of a "solution" of a linear complementarity system and provides conditions under which solutions exist and are unique.

This paper studies the interaction between the notions of passivity of systems theory and complementarity of mathematical programming in the context of complementarity systems. These systems consist of a dynamical system (given in the form of state space representation) and complementarity relations. We study existence, uniqueness, and nature of solutions for this system class under a passivity assumption on the dynamical part. A complete characterization of the initial states and the inputs for which a solution exists is given. These initial states are called consistent states. For the inconsistent states, we introduce a solution concept in the framework of distributions.

In this paper we study a class of set-valued dynamical systems that satisfy maximal monotonicity properties. This class includes linear relay systems, linear complementarity systems, and linear mechanical systems with dry friction under certain conditions. We discuss two numerical time-stepping schemes for the computation of periodic solutions of these systems when being periodically excited. For these two schemes we will provide formal mathematical justifications and compare them in terms of approximation accuracy and computation time using a numerical example.

We introduce bimodal higher-order piecewise linear systems, i.e. the sets of solutions of two n-th order linear differential equations with n ≥ 1, coupled with an inequality constraint defined by a polynomial differential operator acting on the system trajectories. Under suitable assumptions on the characteristic polynomials of the differential equations and the polynomial associated with the inequality constraint, we prove that a solution always exists and is unique given the initial conditions, that no forward Zeno-behavior occurs, and that the trajectories are continuous together with their first n − 1 derivatives. Moreover, we prove that such systems are quadratically stable, and we provide an algorithm based on polynomial algebra to compute a Lyapunov function.

This paper introduces the new class of incrementally port-Hamiltonian systems. This class can be obtained from standard port-Hamiltonian systems by replacing the composition of the Dirac structure and energy-dissipating relation by a maximal monotone relation. After introducing this new class of systems, we study their compositions and show that incrementally port-Hamiltonian systems are closed under composition. Also, we study existence and uniqueness os state trajectories for such systems as well as an energy-based state re-initialization principle.

We construct systematically a class of uncontrollable diffusively coupled multiagent systems with a single leader and multichain topologies. For studying the controllability of diffusively coupled multiagent systems, such identified uncontrollable systems serve as counterexamples that prove the need to modify the existing sufficient condition using graph partitioning characterization. The uncontrollability of the constructed multichain structures can be preserved when the structures are further augmented to get better connected. The paper also provides an algorithm to obtain the minimal leader-invariant relaxed equitable partition for the graph associated with any diffusively coupled multiagent system guided by a single leader.

This paper deals with robust synchronization of undirected multi-agent networks with uncertain agent dynamics. Given an undirected network with identical nominal dynamics for each agent, we allow uncertainty in the form of coprime factor perturbations of the transfer matrix of the agent dynamics. We assume that these perturbations are stable and have H∞-norm that is bounded by some a priori given desired tolerance. In this paper, we derive state space equations for dynamic observer based protocols that achieve synchronization in the presence of such perturbations. We show that this robust synchronization of the network by the dynamic protocol is equivalent to robust stabilization of a single linear system by all controllers from a related finite set of feedback controllers. Our protocols are expressed in terms of real symmetric solutions to certain algebraic Riccati equations, and contain weighting factors depending on the eigenvalues of the graph Laplacian. We show that in this class of dynamic protocols, one can achieve a guaranteed tolerance that is proportional to the square root of the quotient of the smallest and the largest eigenvalue of the graph Laplacian.

In the context of continuous piecewise affine dynamical systems, we study the Zeno behavior, i.e., infinite number of mode transitions in finite time interval, in this note. The main result reveals that piecewise affine dynamical systems do not exhibit Zeno behavior. A direct benefit of the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis of piecewise affine systems. IEEE Control Systems Society

Çamlıbel, Mehmet Kanat (Dogus Author) In this paper we present observer and output-based controller design methods for linear complementarity systems (LCS) employing a passivity approach. Due to various inherent properties of LCS, such as the presence of state jumps, mode dynamics described by differential and algebraic equations (DAEs), and regions for certain modes being lower dimensional, various observer and control design schemes that have been proposed for other classes of (hybrid) dynamical systems do not apply to LCS. In particular, we present an observer design method for LCS which is effective even in the presence of state jumps. We show the well-posedness of the observer, in the sense of existence and uniqueness of solution trajectories for the estimated state, and prove the global exponential stability of the observation error. These two properties guarantee that the estimated state exponentially recovers the state of the system. For the problem of stabilization based on output measurements only, we adopt an observer-based control approach in which we apply a state feedback law to the estimated state obtained from the observer. We prove that the resulting closed-loop system is well-posed and globally exponentially stable. In order to show the well-posedness of the closed loop, novel well-posedness results for LCS based on low-index properties are presented.

In this paper, first, balanced truncation of linear systems is revisited. Then, simultaneous balancing of multiple linear systems is investigated. Necessary and sufficient conditions are introduced to identify the case where simultaneous balancing is possible. The validity of these conditions is not limited to a certain type of balancing, and they are applicable for different types of balancing corresponding to different equations, like Lyapunov or Riccati equations. The results obtained are used for model order reduction of switched linear systems (SLS) by simultaneous balanced truncation. Finally, we give conditions under which global uniform exponential stability is preserved after simultaneous balanced truncation of the original switched linear system.

Çamlıbel, Mehmet Kanat (Dogus Author) -- Conference full title: 2010 49th IEEE Conference on Decision and Control, CDC 2010; Atlanta, GA; United States; 15 December 2010 through 17 December 2010 Continuous-time linear constrained optimal control problems are in practice often solved using discretization techniques, e.g. in model predictive control (MPC). This requires the discretization of the (linear time-invariant) dynamics and the cost functional leading to discrete-time optimization problems. Although the question of convergence of the sequence of optimal controls, obtained by solving the discretized problems, to the true optimal continuous-time control signal when the discretization parameter (the sampling interval) approaches zero has been addressed in the literature, we provide some new results under less restrictive assumptions for a class of constrained continuous-time linear quadratic (LQ) problems with mixed state-control constraints by exploiting results from mathematical programming extensively. As a byproduct of our analysis, a regularity result regarding the costate trajectory is also presented.

The main object of study in this paper is the class of switched linear systems, i.e. a collection of linear input/state/output systems and a set of switching signals that determine the active subsystem from the collection at any given time instant. We address the problem of left-invertibility for these systems. Left-invertible switched linear systems are those for which one can recover the switching signal (up to an equivalence class) and the input uniquely from a given output of the system. After introducing an equivalence relation for switching signals, we give precise definition of left-invertibility based on this equivalence relation. By taking a geometric approach, the paper presents a set of necessary and sufficient conditions for left-invertibility. These conditions are linear in nature and can be checked by using the techniques from geometric approach to linear systems.

Çamlıbel, Mehmet Kanat (Dogus Author) -- Conference full title: 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009; Shanghai; China; 15 December 2009 through 18 December 2009 Instantaneous charge/flux transfers may occur in switched electrical circuits when the switch configuration changes. Characterization of such state discontinuities is a classical issue in circuit theory which, typically, is based on the so-called charge and flux conservation principle. This paper proposes a general state transfer principle for arbitrary switching port-Hamiltonian systems. This new principle coincides with the charge and flux conservation principle in the special case of linear RLC circuits, but also covers circuits with nonlinear capacitors and inductors, and of arbitrary topology. Moreover, the new principle is applied to switching mechanical systems.

This paper studies open-loop stabilization problem for bimodal systems with continuous vector field. It is based on the earlier work of the authors on the controllability problem for the same class of systems. A full characterization of stabilizability is established by presenting algebraic necessary and sufficient conditions. It is also shown that controllability implies stabilizability for these systems in a very similar fashion to the linear case. (c) 2008 Published by Elsevier Ltd.

The problem of checking certain controllability prop erties of even very simple piecewise linear systems is known to be undecidable. This paper focuses on conewise linear systems, i.e. systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions. For this class of systems, we present algebraic necessary and sufficient conditions for controllability. We also show that the classical results of controllability of linear systems and input-constrained linear systems can be recovered from our main result. Our treatment employs tools both from geometric control theory and mathematical programming. IEEE Control Systems Society

This paper presents necessary and sufficient conditions for null controllability of discrete-time linear systems subject to both input and state constraints. The classical results for linear systems without constraints by Kalman and Hautus and for linear systems with only input constraints by Evans, Nguyen and Sontag can be obtained from our main result as particular cases.

This paper presents necessary and sufficient conditions for controllability of linear systems subject to input/state constraints.

The notion of convergent systems is a powerful tool both in the analysis and synthesis of nonlinear systems. Sufficient conditions for convergence have been under investigation for smooth systems and for classes of non-smooth switching systems in the literature. In this paper, we look at a very particular class of non- smooth systems, namely complementarity systems. These systems have the capability of capturing the non-smooth dynamics of various interesting applications from different fields of engineering. The main contribution of this paper is to show that a linear complementarity system is convergent if the underlying linear dynamics possesses a certain positive realness property.

This paper studies controllability of bimodal systems that consist of two linear dynamics on each side of a given hyperplane. We show that the controllability properties of these systems can be inferred from those of linear systems for which the inputs are constrained in a certain way. Inspired by the earlier work on constrained controllability of linear systems, we derive necessary and sufficient conditions for a bimodal piecewise linear system to be controllable.

The object of study of this paper is the class of hybrid systems consisting of so-called linear complementarity (LC) systems, that received a lot of attention recently and has strong connections to piecewise affine (PWA) systems. In addition to PWA systems, some of the linear or affine submodels of the LC systems can 'live' at lower-dimensional subspaces and re-initializations of the state variable at mode changes is possible. For LC systems we study the stability and controllability problem. Although these problems received for various classes of hybrid systems ample attention, necessary and sufficient conditions, which are explicit and easily verifiable, are hardly found in the literature. For LC systems with two modes and a state dimension of two such conditions are presented. IEEE; SIAM; Control Syst Soc; SICE; Informs; Honeywell Lab; Natl Instruments; XEROX; MathWorks

A modeling framework is proposed for circuits that are subject both to externally induced switches (time events) and to state events. The framework applies to switched networks with linear and piecewise-linear elements, including diodes. We show that the linear complementarity formulation, which already has proved effective for piecewise-linear networks, can be extended in a natural way to also cover switching circuits. To achieve this, we use a generalization of the linear complementarity problem known as the cone-complementarity problem. We show that the proposed framework is sound in the sense that existence and uniqueness of solutions is guaranteed under a passivity assumption. We prove that only first-order impulses occur and characterize all situations that give rise to a state jump; moreover, we provide rules that determine the jump. Finally, we show that within our framework, energy cannot increase as a result of a jump, and we derive a stability result from this. IEEE Circuits and Systems Society

We study the notion of passivity in the context of complementarity systems, which form a class of nonsmooth dynamical systems that is obtained from the coupling of a standard input/output system to complementarity conditions as used in mathematical programming. In terms of electrical circuits, the systems that we study may be viewed as passive networks with ideal diodes. Extending results from earlier work, we consider here complementarity systems with external inputs. It is shown that the assumption of passivity of the underlying input/output dynamical system plays an important role in establishing existence and uniqueness of solutions. We prove that solutions may contain delta functions but no higher-order impulses. Several characterizations are provided for the state jumps that may occur due to inconsistent initialization or to input discontinuities. Many of the results still hold when the assumption of passivity is replaced by the assumption of "passifiability by pole shifting". The paper ends with some remarks on stability.

In this paper, the so-called Zeno phenomenon is addressed for linear complementarity systems which are interconnections of linear systems and complementarity conditions. We present some sufficient conditions for absence of Zeno behavior. It is also shown that the zero state, which is the most obvious candidate for being a Zeno state, cannot be a Zeno state in certain cases.

Linear complementarity systems are used to model discontinuous dynamical systems such as networks with ideal diodes and mechanical systems with unilateral constraints. In these systems mode changes are modeled by a relation between nonnegative, complementarity variables. We consider approximating systems obtained by replacing this non-Lipschitzian relation with a Lipschitzian function and investigate the convergence of the solutions of the approximating system to those of the ideal system as the Lipschitzian characteristic approaches to the (non-Lipschitzian) complementarity relation. It is shown that this kind of convergence holds for linear passive complementarity systems for which solutions are known to exist and to be unique. Moreover, this result is extended to systems that can be made passive by pole shifting.