Fast modes in the set of minimal dissipation trajectories☆
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 . ‘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 be the behavior of the system, that is, the collection of all the allowable trajectories under the system’s dynamical equations. Further, let , a constant real symmetric matrix, induce the quadratic supply rate on trajectories . Let denote the adjoint of with respect to the supply rate . We address the following issues:
- 1.
Given and controllable/observable state space representations of a system and its adjoint system , when is the interconnection an autonomous system?
- 2.
Find conditions on 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 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?
- 1.
the (possibly singular descriptor) state space system obtained from the minimal state space representations of and ,
- 2.
the kernel representation of obtained by using the kernel representations of and , and
- 3.
the latent variable representation and .
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 and its dual , 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: and stand for the fields of real and complex numbers respectively. The ring of polynomials in the indeterminate with coefficients from is denoted as , while matrices with entries from and having rows and columns are denoted by , which for polynomials is also . The spaces and stand for the spaces of infinitely often differentiable functions and locally integrable functions each from to . In this paper, we also need , where stands for . The set of those elements in which have compact support is denoted by . When the co-domain is clear from the context, then we drop the co-domain and write , for example. Further, when both domain and co-domain are clear, we write just or .
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
In this section, we obtain a state space representation of for the case that the interconnection is not well-posed, i.e. is singular.
The main result needs the notions of the weakly unobservable subspace and the strongly reachable subspace as proposed in [27]. The weakly unobservable subspace is defined as the set of all initial conditions for which there exists an input such that the corresponding output is identically zero on . The strongly
Impulsive initial conditions
In this section we formulate necessary and sufficient conditions for the interconnected system 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 for the interconnection 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 and ) has impulsive initial conditions. In this section we investigate further into the case when the manifest variables have impulsive solutions in .
Recall again from [14, Theorem 4.32] that a necessary and sufficient condition for absence of inadmissible initial conditions for an autonomous system with square and
Concluding remarks
We studied the interconnection of and 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.
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This work was supported in part by IRCC, IIT Bombay, SERB, DST, and BRNS, India.