Generic Controllability Research Articles

Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

AbstractThe present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

Generic observability for port-Hamiltonian descriptor systems

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

Relative genericity of controllablity and stabilizability for differential-algebraic systems

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices (E,A,B)in mathbb {R}^{ell times }times mathbb {R}^{ell times n}times mathbb {R}^{ell times m}, while for relative genericity we allow the restricted set Sigma _{ell ,n,m}^{le r} := {(E,A,B)in mathbb {R}^{ell times n}times mathbb {R}^{ell times n}times mathbb {R}^{ell times m} ,big vert ,textrm{rk},_{mathbb {R}} E le r}, where rin {mathbb {N}}. Our main results are characterizations of generic controllability and generic stabilizability in Sigma _{ell ,n,m}^{le r} in terms of the numbers ell , n, m, r.

Comments on “On structural accessibility of network nonlinear systems”

In a recent paper (Meng et al., 2021), it was claimed that for meromorphic nonlinear systems linear in control input, generic controllability of the linearized system is necessary and sufficient for generic accessibility of the original system, and this result was used to prove equivalence between structural accessibility and structural controllability of the linearized system. We point out a mistake in the proof of the results of Meng et al. (2021) and show that controllability of the linearized system is only sufficient for accessibility of the original nonlinear system. The main result of the paper (Meng et al., 2021) about the relation between structural accessibility of the system and structural controllability of its linearized system can partially be remedied.

Minimal Perturbations for Zero Controllability of Discrete-Time Linear Systems: Complexity Analysis

This article deals with computational complexity of various problems related to the zero controllability of a discrete-time linear time-invariant system, assuming that purely structural conditions at the level of the connections between the system states (i.e., state-connections) and the connections from the inputs to the states (i.e., input-connections) are known. Given a generically zero controllable system, we consider the following problems: i) find a minimal set of input-connections whose removal makes the resulting system not generic zero controllability; ii) identify a minimal cost set of input-connections that must be retained from the given set of input-connections while preserving generic zero controllability property; and iii) given a not generically zero controllable system, find a smallest set of state-connections whose removal makes the resulting system generically zero controllable. Problem i) is polynomially solvable. Problems ii) and iii) are NP-hard and approximation results are provided for them. The results of i) and iii) provide clues to analyze the fragility and hardness involved in modifying a system structure. Problem ii) is useful to ensure an accurate discrete-time linear approximation of a large-scale system by maintaining generic zero controllability of the linear system.

Generic Controllability of Networks With Identical SISO Dynamical Nodes

Recently, controllability of linear systems has been studied for various applications in large networks in biology, physics, and engineering. To address scale and lack of precise knowledge of the network parameters issues, tools from structured system theory are well indicated for such study. Precisely, generic properties, i.e., properties that are true for almost all values of the parameters, can be stated. While most existing results concern networks of single integrators, in this article, we show that generic controllability of a network of identical single-input-single-output controllable and observable systems is ensured if and only if the structural controllability conditions are satisfied for the graph representing the network. This result constitutes an important generalization of the famous Lin's theorem stated in the 1970s. It is even the broadest generalization of this theorem.

Actuator Placement for Symmetric Structural Controllability With Heterogeneous Costs

In this letter, we consider the problem of optimal actuator placement for networked linear time-invariant systems with underlying undirected topologies. We aim to leverage the undirected topology for the design of an actuator deployment strategy that guarantees symmetric structural controllability , i.e., generic controllability of the system subject to symmetric weights. Naturally, the objective is to make this design as cost effective and computationally efficient as possible. Therefore, we focus on two related (yet different) problems: 1) to determine the subset of state nodes that need to be actuated to ensure symmetric structural controllability at the lowest cost incurred by the actuators and 2) given a symmetrically structurally controllable system (whose input nodes may actuate on more than one state node), to determine a subset of its input nodes that still ensure symmetric structural controllability, at the lowest possible cost. We show that both problems can be solved in polynomial time by implicitly providing an efficient algorithm that determines their solutions. Several numerical experiments are provided to illustrate our main results.

Exact Controllability in Projections of the Bilinear Schrödinger Equation

We consider the bilinear Schrodinger equation with discrete-spectrum drift. We show, for n ∈ N arbitrary, exact controllability in projections on the first n given eigenstates. The controllability result relies on a generic controllability hypothesis on some associated finite-dimensional approximations. The method is based on Lie-algebraic control techniques applied to the finite-dimensional approximations coupled with classical topological arguments issuing from degree theory.

The study of structural properties of linear time-invariant systems and their dependability: a Monte Carlo simulation approach

ABSTRACTStructural properties such as the controllability and observability are of major importance in automatic control. Indeed, the controllability property of a system, for example, guarantees that this system can be controlled to carry out the specified mission it is designed for. These properties rely on the system's structure and components more than the numerical value of its internal parameters. However, these may be prone to failures that can make the system lose its initial properties, thus affecting its function. This paper proposes to use the Monte Carlo simulation and Kalman algebraic criteria to study the generic controllability and observability of linear time-invariant systems represented by structured models. The proposed approach allows, on the one hand, to compute the set of components which are necessary for these structural properties and, on the other hand, to assess the dependability of these properties.

The single-input Minimal Controllability Problem for structured systems

This paper considers the Minimal Controllability Problem (MCP), i.e. the problem of controlling a linear system with an input vector having as few non-zero entries as possible. We focus on structured systems which represent an interesting class of parameter dependent linear systems and look for structural controllability properties based on the sparsity pattern of the input vector. We show first that the MCP is solvable when a rank condition is satisfied and show that generically one non-zero entry in the input vector is sufficient to achieve controllability when there is no specific system structure. According to the fixed zero/non-zero pattern of the state matrix entries, we give an explicit characterization of the minimum number and the possible location of non-zero entries in the input vector to ensure generic controllability. The analysis based on graph tools provides with a simple polynomial MCP solution and highlights the structural mechanisms that make it useful to act on some variables to ensure controllability.

Generic Controllability of 3D Swimmers in a Perfect Fluid

We address the problem of controlling a dynamical system governing the motion of a 3D weighted shape changing body swimming in a perfect fluid. The rigid displacement of the swimmer results from the exchange of momentum between prescribed shape changes and the flow, with the total impulse of the fluid-swimmer system being null at every time. We prove the following tracking results: (i) Synchronized swimming: Up to an arbitrarily small change of its density, any swimmer can approximately follow any given trajectory while, in addition, undergoing approximately any given shape changes. In this statement, the control consists in arbitrarily small superimposed deformations. (ii) Freestyle swimming: Up to an arbitrarily small change of its density, any swimmer can approximately track any given trajectory by suitably combining at most five basic movements that can be generically chosen (no macro shape changes are prescribed in this statement).

Well-posedness and controllability of a wind turbine tower model

We derive a model for a wind turbine tower in the plane of the turbine blades, which comprises an Euler–Bernoulli beam coupled with a nacelle (rigid body) and a two-mass drive-train model (with gearbox). This model has two possible control inputs: the torque created by the electrical generator and the force created by an electrically driven mass located in the nacelle. First, we consider the case of only torque control and a possibly non-uniform tower. Using the theory of coupled linear systems (one infinite dimensional and one finite dimensional) developed by us recently, we show that this wind turbine tower model is well-posed and regular on either the energy state space Xc or the domain of its generator on Xc, denoted by Xc 1. We also show that generically, this model is exactly controllable on X c 1 in arbitrarily short time. More precisely, for every T > 0, we show that if we vary a certain parameter in the model, then exact controllability in time T holds for all except three values of the parameter. In the case of using both force and torque control, we derive similar well-posedness, regularity and generic exact controllability results on a state space that is larger than Xc 1 but smaller than X c. In this second case, we assume that the tower is uniform.

Generic Controllability Properties for the Bilinear Schrödinger Equation

In [16] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schrödinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schrödinger operator. The aim of this paper is to show that these conditions are generic with respect to the uncontrolled and the controlled potential, denoted respectively by V and W. More precisely, we prove that the Schrödinger equation is approximately controllable generically with respect to W when V is fixed and also generically with respect to V when W is fixed and non-constant. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.

Local controllability bifurcations in families of bidynamical systems on the plane

We classify generic local controllability bifurcations in two-parameter families of bidynamical systems on the plane at points with nonzero velocity indicatrix.

Controllability of Systems on Solvable Lie Groups: The Generic Case

A Lie group G with Lie algebra \frak g is called SID-controllable if there exist A,B \in {\frak g} such that the (Single Input with Drift) control system \dot{g} = g(A + uB), u \in {\Bbb R}, is controllable. This is equivalent to saying that the semigroup S(A, \pm B) generated by \exp({\Bbb R}^{+} A) \cup \exp({\Bbb R} B) is all of G. This definition is due to Sachkov who also classified SID-controllable solvable Lie algebras, cf. [7]-[9], [11]. It turns out that SID-controllability is actually a property of the Lie algebra (rather than of a control system): if a solvable \frak g is SID-controllable, then a generic SID-system will be controllable. In this paper we generalize this result to systems with multiple inputs and drifts: G is I_{n}D_{m}-controllable if there exist “inputs” B_{1} \dots B_{n} and “drifts” A_{1] \dots A_{m} such that S(A_{1}, \dots, A_{m}; \pm B_{1}, \dots, \pm B_{n}) = G. A Lie algebra is called I_{n}D_{m}-controllable if the corresponding simply connected group has this property. We will show that every solvable Lie algebra has a generic controllability rank r \in {\Bbb N} and a generic controllability type (i,d) \in {\Bbb N} \times {\Bbb N}_{0} such that: (GCR) \frak g is not I_{r-1}-controllable, \frak g is I_{r}-controllable, and the latter is generic. In particular, \frak g is I_{n}D_{m}-controllable if n \geq r or n + m \gt r; (GCT) \frak g is I_{i}D_{d}-controllable, a generic I_{i}D_{d}-system is controllable, i + d = r, and i is minimal. Determination of these invariants is one of our goals. Our major tools will be reduction arguments which are of independent interest: we show that every solvable Lie algebra \frak g has a maximal ideal \frak i which is completely irrelevant for all controllability questions. Passing to the factoralgebra {\frak g} / {\frak i} and analyzing its structure is the key step in solving our problem.

Structured time-delay systems: A graph approach

In this paper, linear structured systems with delays are defined and a simple graph representation for these systems is given. Then, the generic controllability of time-delay systems is introduced and a graph criterion for controllability is provided. The generic row-by-row decoupling problem without prediction of such systems is then studied and graphical necessary and sufficient solvability conditions for this problem is derived.

On the controllability of distributed systems.

To "control" a system is to make it behave (hopefully) according to our "wishes," in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed-i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our "wish" is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any "neighborhood" of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is "unstable" (chaotic, turbulent) the "simplest," or the "cheapest," it is to achieve exact or approximate controllability. When the PDEs are the Navier-Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.

Dynamic path controllability in economic models: From linearity to nonlinearity

In this paper we study the dynamic path controllability of discrete-time nonlinear economic models in state-space form. First we present an algorithmic procedure for testing this property for time-varying linear systems. Secondly we extend this method to general nonlinear models and we show that under generic conditions path controllability around a specific trajectory of the nonlinear system is equivalent to path controllability of the corresponding linearized model. The results are illustrated with two economic examples.

On the generic controllability of continuous generalized state-space systems

In this paper using a duality property between regular and generalized state-space systems, the necessary and sufficient conditions for the generic controllability of continuous generalized state-space systems are established. the necessary and sufficient conditions for the generic observability of such systems are also derived.

A further simplification in the proof of the structural controllability theorem

The structural controllability theorem characterizes generic controllability of parameter-dependent systems. A further simplification of its proof is presented.