Time-Varying Dynamical Systems

A trajectory \((x, w, z)\) of a Linear Time-Varying (LTV) dynamical system with matrix representation \((\Acl_k, \Bcl_k, \Ccl_k, \Dcl_k)_{k \in \N}\) obeys the relation

\[\begin{split}\mat{c}{x_{k+1} \\ z_k} = \mat{c|c}{\Acl_k & \Bcl_k \hl \Ccl_k & \Dcl_k } \mat{c}{x_k \\ w_k}, & & \forall k \in \N.\end{split}\]

Linear Time Invariant (LTI) systems are a special case of LTV systems where \((\Acl, \Bcl, \Ccl, \Dcl)\) are constant in time. The opt_system class in the System page involves networks and controllers that are only Linear Time Invariant (LTI).

This page documents time-varying dynamical systems that are supported by opt-syn.

In all types of dynamical systems listed on this page, one of the network or controller can be LTI. If both the network and controller are LTI, then opt_system should be used instead of the broader time-varying system class.

Periodic

A linear time-varying system is periodic if there exists an integer \(h\) such that

\[\begin{align*} \mat{c|c}{\Acl_k & \Bcl_k \hl \Ccl_k & \Dcl_k } = \mat{c|c}{\Acl_{k+h} & \Bcl_{k+h} \hl \Ccl_{k+h} & \Dcl_{k+h} } & & \forall k \in \N. \end{align*}\]

Systems with periodic networks and controllers can be specified using the command

sys_per = opt_system_periodic(Operator_Class, Network, Controller);

An \(h\)-periodic network is stored as an \(h\)-length cell of genplant objects, or as a genplant_poly object.

If the network is \(h\)-periodic, then the controller can either be LTI (a single ss) or \(h\)-periodic (an \(h\)-length cell of ss objects). If the network is LTI, then the controller can be an \(h\)-length cell of ss objects.

A periodic system can be lifted into an LTI system by

sys_lti = sys_per.periodic_lift();

Periodic-Orbit

A periodic-orbit linear system is an \(h\)-periodic linear system in which there exists matrices \((M_x, M_w, M_z)\) such that

\[\begin{split}M_x^h &= M_x, \qquad \ M_w^h = M_w, \qquad M_z^h = M_z, \\ \mat{c|c}{\Acl_k & \Bcl_k \hl \Ccl_k & \Dcl_k } &= \mat{c c}{M_x & 0 \\ 0 & M_y}^{-k} \mat{c|c}{\Acl_{0} & \Bcl_{0} \hl \Ccl_{0} & \Dcl_{0} } \mat{c c}{M_x & 0 \\ 0 & M_u}^k, & & \qquad \forall k \in \N.\end{split}\]

Cyclic Coordinate-Descent algorithms are instances of periodic-orbit algorithms satisfying this simplified structure.

The opt-syn implementation of periodic-orbit systems is simplified. The matrix \(M \in \R^{c \times c}\) must satisfy \(M = M^\top, M^h = M\), where \(c\) is the coordinate dimension of the Kronecker Structure. All other matrices follow from \(M\) by Kronecker structure (e.g. \(M_x = I_{n_x/c} \otimes M\)).

Systems with periodic-orbit networks and controllers can be specified using the command

sys_orbit = opt_system_periodic_orbit(Operator_Class, Network, Controller, M);

Periodic-orbit systems can be enumerated into periodic systems, and can then be lifted into LTI systems:

sys_per = sys_orbit.export_periodic();
sys_lti = sys_orbit.periodic_lift();

Switched

A switched linear system with \(N_s\) modes is described by a collection of \(N_s\) LTI systems (modes), and a directed, unweighted adjacency graph \(\mathcal{G}\) with \(N_s\) vertices. Each subsystem \(j\) of the switched system can be represented as

\[\begin{split}\mat{c}{x_{k+1} \\ z_k} = \mat{c|c}{\Acl_j & \Bcl_j \hl \Ccl_j & \Dcl_j } \mat{c}{x_k \\ w_k}.\end{split}\]

The function \(\theta: \N \rightarrow \{1, \ldots, N_s\}\) chooses the active subsystem at time \(k\). A trajectory \((x, w, z, \theta)_{k \in \N}\) of a switched system satisfies the relations

\[\begin{split}\begin{align*} & \mat{c}{x_{k+1} \\ z_k} = \mat{c|c}{\Acl_{\theta(k)} & \Bcl_{\theta(k)} \hl \Ccl_{\theta(k)} & \Dcl_{\theta(k)} } \mat{c}{x_k \\ w_k} & & \forall k \in \N, \\ \\ & (\theta(k), \theta(k+1)) \in \text{Edges}(\mathcal{G}). \end{align*}\end{split}\]

Switched systems can model network phenomena such as time-varying delays and communication drops. Periodic and periodic-orbit systems are specific instances of switched systems.

A system with switched system networks and controllers can be specified using the command

sys_per = opt_system_switched(Operator_Class, Network, Controller, G);

where G is the \(\{0, 1\}\) adjacency matrix for the graph \(\mathcal{G}\).