Elsevier

Systems & Control Letters

Volume 101, March 2017, Pages 28-36
Systems & Control Letters

Fast modes in the set of minimal dissipation trajectories

https://doi.org/10.1016/j.sysconle.2016.02.009Get rights and content

Abstract

In this paper, we study the set of trajectories satisfying both a given LTI system’s laws and also laws of the corresponding ‘adjoint’ system: in other words, trajectories in the intersection of the system’s behavior and that of the adjoint system. This intersection has important system theoretic significance: for example, it is known that the trajectories in this intersection are the ones with minimal ‘dissipation’. Underlying the notion of adjoint is that of a power supply: it is with respect to this supply rate that the trajectories in the intersection are known to be ‘stationary’. In this paper, we deal with half-line solutions to the differential equations governing both the system and its adjoint. Analysis of half-line solutions plays a central role for example in initial value problems and in well-posedness studies of an interconnection. We interpret the set of half-line trajectories allowed by a system and its adjoint as an interconnection of these two systems, and thus address issues about well-posedness/ill-posedness of the interconnection. We formulate necessary and sufficient conditions for this intersection to be autonomous. For the case of an ill-posed interconnection and resulting autonomous system, we derive conditions for existence of initial conditions that lead to impulsive solutions in the states of the system. We link our conditions with the strongly reachable and weakly unobservable subspaces of a state space system. We show that absence of impulsive initial conditions is equivalent to the well-known subspace iteration algorithms for these subspaces converging in one step.

Introduction

For an LTI system, the intersection of the sets of trajectories allowed by the system and its ‘adjoint’ (dual) system has significance in various areas: as ‘stationary’ trajectories in the context of LQ control (see  [1], [2]), as Hamiltonian systems (see  [3]), and as trajectories of minimal dissipation (see  [4]). Under suitable regularity assumptions, this intersection exhibits desirable properties—like, autonomy, having McMillan degree equal to twice the McMillan degree of the original system. One or both of these properties are lost when the regularity assumptions are relaxed. Consequently, under non-satisfaction of the regularity assumptions, the usage of the interconnection of the system and its adjoint, in control problems, becomes subject to major modifications. For example, in singular LQ control, the intersection of the system and its adjoint may or may not contain impulsive optimal solutions: see  [5], [1], [6] for a related exposition. In this paper, we go beyond the intersection and view the same as an ‘interconnection’. While the interconnection point of view does not provide, for the regular case, any significant leverage over that of the intersection, the former point of view can handle the singular case better than the latter; this is because the singular case is nothing but an ‘ill-posed’ interconnection of the system and its adjoint.

Following the tradition of the study of ill-posedness in the interconnection paradigm, in this paper, we study half-line solutions of the interconnection of the system and its adjoint. Further, we investigate the issue of whether this interconnection, when ill-posed, contains impulsive modes. For the purpose of this paper: ‘impulsive’ modes are those trajectories that contain one or more derivatives of the Dirac delta δ. ‘Fast’ modes include impulsive modes and jumps.

Without dwelling on the essential preliminaries (which are elaborated below in Section  2), we first list the main questions we address in this paper. Let B be the behavior of the system, that is, the collection of all the allowable trajectories under the system’s dynamical equations. Further, let Σ, a constant real symmetric matrix, induce the quadratic supply rate wTΣw on trajectories wB. Let BΣ denote the adjoint of B with respect to the supply rate wTΣw. We address the following issues:

  • 1.

    Given Σ and controllable/observable state space representations of a system B and its adjoint system BΣ, when is the interconnection BBΣ an autonomous system?

  • 2.

    Find conditions on B under which the interconnection is an ill-posed interconnection.

  • 3.

    If the interconnection is autonomous and ill-posed: find conditions under which there are no initial state-space conditions causing impulsive solutions.

  • 4.

    Find conditions on the system B under which the external system variables exhibit impulsive solutions: relate these conditions to those in Item 3 above.

  • 5.

    Can there be situations under which one or more of the states of the interconnected system are impulsive, but the external system variables are not impulsive? Does ‘impulse unobservability’ or ‘unobservability at infinity’ resolve this?

In this paper we formulate necessary and sufficient conditions for resolving some of the above questions and we provide counter-examples for the unresolved ones. When studying the interconnection of B and BΣ, there are three important representations for the interconnected system: BBΣ:
  • 1.

    the (possibly singular descriptor) state space system obtained from the minimal state space representations of B and BΣ,

  • 2.

    the kernel representation of BBΣ obtained by using the kernel representations of B and BΣ, and

  • 3.

    the latent variable representation w=M(ddt) and M(ddt)TΣM(ddt)=0.

Note that, while the various representations listed above all lead to the same set of solutions for the case of well-posed interconnection between B and BΣ, it is ill-posed interconnection that results in difference in the fast solution sets of the various representations: this paper focuses only on the fast modes. In this context, it turns out that even when the state space of the interconnected system has impulsive initial conditions, the external system variables do not necessarily have impulsive modes. See also [7]. In the later part of the paper, we describe numerical examples with these features, and further, investigate if impulse unobservability can explain why the system is ‘impulse unobservable’.

A brief overview of the main results in this paper and the paper organization are as follows. The following section contains definitions pertaining to the behavioral approach, quadratic differential forms (QDFs), and preliminary results on well-posedness of interconnection and the notion of zeros at infinity of a polynomial matrix and its relation to inadmissible initial conditions, i.e. those initial conditions that cause impulsive solutions. In Section  3, we summarize the assumptions used in this paper and also their system-theoretic justifications. Section  4 contains new results on ill-posedness of interconnection of a system B and its dual BΣ, and conditions for the interconnection to be autonomous. Section  5 contains another main result of this paper: necessary and sufficient conditions for the ill-posed interconnection case under which the interconnected system has no impulsive initial conditions. Section  6 raises questions about how the presence/absence of impulsive solutions need not be the same for the case of state variables, manifest system variables and the latent variable used in an image representation. Section  7 contains some concluding remarks.

We use standard notation in this paper: R and C stand for the fields of real and complex numbers respectively. The ring of polynomials in the indeterminate ξ with coefficients from R is denoted as R[ξ], while matrices with entries from R[ξ] and having p rows and m columns are denoted by R[ξ]p×m, which for polynomials is also Rp×m[ξ]. The spaces C(R,Rw) and L1loc(R,Rw) stand for the spaces of infinitely often differentiable functions and locally integrable functions each from R to Rw. In this paper, we also need C(R+,Rw), where R+ stands for (0,). The set of those elements in C(R,Rw) which have compact support is denoted by D(R,Rw). When the co-domain is clear from the context, then we drop the co-domain and write C(R+), for example. Further, when both domain and co-domain are clear, we write just C or L1loc.

Section snippets

Preliminaries

This section deals with the preliminaries that are required for this paper. The following subsection reviews required results from the behavioral approach to dynamical systems.

Assumptions and justifications

In this section we list the assumptions we make throughout this paper. We also give system theoretic justification for the assumptions.

Ill-posed interconnection BBΣ

In this section, we obtain a state space representation of BBΣ for the case that the interconnection is not well-posed, i.e.  (IpDDT) is singular.

The main result needs the notions of the weakly unobservable subspace V and the strongly reachable subspace W as proposed in  [27]. The weakly unobservable subspace V is defined as the set of all initial conditions x0 for which there exists an input uC(R+) such that the corresponding output y(t) is identically zero on [0,). The strongly

Impulsive initial conditions

In this section we formulate necessary and sufficient conditions for the interconnected system BBΣ to have inadmissible initial conditions, i.e. initial conditions that cause impulsive solutions.

The following result is one of the main results of this paper: necessary and sufficient conditions on B for the interconnection BBΣ to have no inadmissible initial conditions. The relation of the conditions with all-pass characteristics of a MIMO system is elaborated in Remark 5.2. Also compare the

Impulsive solutions in the manifest/system variables

The last section (in particular, Theorem 5.1) formulated conditions under which the state-space of the interconnected system (with the states being that of B and BΣ) has impulsive initial conditions. In this section we investigate further into the case when the manifest variables w have impulsive solutions in BBΣ.

Recall again from  [14, Theorem 4.32] that a necessary and sufficient condition for absence of inadmissible initial conditions for an autonomous system P(ddt)w=0 with P square and

Concluding remarks

We studied the interconnection of B and BΣ and studied half-line solutions in the interconnected system. While the full-line solutions are the same for three different representations of this set (namely, the latent variable representation, the kernel representation and the state space representation), the fast-modes in the half-line solutions set need not be the same. We formulated necessary and sufficient conditions for the interconnected system to be well-posed and for it to be autonomous.

References (31)

  • J.C. Willems et al.

    Singular optimal control: a geometric approach

    SIAM J. Control Optim.

    (1986)
  • R.K. Kalaimani et al.

    Singular LQ control, optimal PD controller and inadmissible initial conditions

    IEEE Trans. Automat. Control

    (2013)
  • J.W. Polderman et al.

    Introduction to Mathematical Systems Theory: A Behavioral Approach

    (1998)
  • J.C. Willems et al.

    On quadratic differential forms

    SIAM J. Control Optim.

    (1998)
  • G. Verghese et al.

    A generalized state-space for singular systems

    IEEE Trans. Automat. Control

    (1981)
  • This work was supported in part by IRCC, IIT Bombay, SERB, DST, and BRNS, India.

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