Switched

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

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

System#

The algorithmic interconnection is

\[\begin{split}\begin{align*} w_k & \in F_k(z_k), \, \\ \mat{c}{x^N_{k+1} \hl z_k \\ y_k} &= \mat{c|cc}{A_{\theta(k)} & B_{\theta(k), \, z} & B_{\theta(k), \, u} \hl C_{\theta(k), \, z} & D_{\theta(k), \, zd} & D_{\theta(k), \, zu} \\ C_{\theta(k), \, y} & D_{\theta(k), \, yd} & D_{\theta(k), \, yu}} \mat{c}{x_k^N \hl w_k \\ u_k}, \\ \mat{c}{\xi_{k+1} \\ y_k} &= \mat{c|c}{A_{K, \theta(k)} & B_{K, \theta(k)} \hl C_{K, \theta(k)} & D_{K, \theta(k)} } \mat{c}{\xi_k \\ y_k}. \end{align*}\end{split}\]
class system.switched.opt_system_switched#

Bases: system.generic.opt_system_interface

OPT_SYSTEM_SWITCHED interconnection of network and operators polytopic setting: a cell A = sum theta_i A_i for parameters theta_i

useful for switched systems, periodic systems, and LPV systems

Constructor Summary
opt_system_switched(op, P, K, adj, bind, tracking)#

OPT_SYSTEM constructor

Property Summary
adj#

switching graph (adjacency matrix)

Method Summary
build_plant(iqc_data)#

BUILD_PLANT: form the plant to be used for analysis or synthesis

Parameters:

  • iqc_data – from manager.iqc_op_all, information about the operator iqc descriptions

  • rho – exponential weighting

Returns:

  • alg_psi – plant with filters (psi)

  • iqc_op – iqcs for the robust uncertainties

  • alg_loop – plant without filters, but after loop transformation (should be stable)

discount_schedule(ordermax)#

DISCOUNT_SCHEDULE exponential weights encountered when applying the FIR filters :param ordermax: maximum order of the IQCs

Return:

pow – Exponent sequence of discounts

Example:

[0; 1; 2] -> rho.^[0; 1; 2] for uniform exponential stability [0; 0; 1] -> rho.^[0; 0; 1] for shuffled switched stability

get_K(param)#

TODO: override this with parameters

get_P(param)#

GET_P get the plant

get_arcs()#

GET_ARCS get transitions in the adjacency matrix

get_discount()#

which subsystems are exponentially discounted?

get_tracked_opt(param)#

GET_TRACKED_OPT get the tracked position of the optimal solution

\(\eta^*_{k+1} = S_\beta \eta^*, \beta^*_{k} = R_\beta \eta_k\).

Args:

param: structure of parameters

Returns:

  • Sbeta – exosystem for optimal solution

  • Rbeta – output of optimal solution

get_type()#

get the type of the switched system

nxi()#

nxi: number of states in controller

nxn()#

nxn: number of states in network

ss_zy_wu(param)#

get state space matrices at the current parameter values

Regulator#

The subsystems for an open switched system with disturbance \(d\) and regulated error \(e\) may be described as

\[\begin{split}\begin{align} d_{k+1} &= S_j d_k, \, \\ \mat{c}{x_{k+1} \hl e_k \\ y_k} &= \mat{c|cc}{A_j & B_{j, d} & B_{j, u} \hl C_{j, e} & D_{j, ed} & D_{j, eu} \\ C_{j, y} & D_{j, yd} & D_{j, yu}} \mat{c}{x_k \hl d_k \\ u_k}. \end{align}\end{split}\]

The one-step regulator equations for this system are to find \((\Pi_j, \Gamma_j, \Phi_j)_{j=1}^{N_s}\) satisfying

\[\begin{split}\begin{align} \mat{c}{\Pi_{j} S_i \hl 0 \\ \Phi_i} &= \mat{c|cc}{A_i & B_{i, d} & B_{i, u} \hl C_{i, e} & D_{i, ed} & D_{i, eu} \\ C_{i, y} & D_{i, yd} & D_{i, yu}} \mat{c}{\Pi_i \hl I \\ \Gamma_i}. \end{align}\end{split}\]

over all arcs \((i, j)\) in the graph \(\mathcal{G}\).

Certification of robust stability and feasibility of these regulator equations are sufficient but not necessary to prove convergence. If these regulator equations fail, then there may exist a well-posed and convergent optimization algorithm for this network, but opt-syn will not find it.

class system.switched.regulator_switched#

Bases: system.generic.regulator_interface

REGULATOR_SWITCHED Regulator for switched systems

Constructor Summary
regulator_switched(sys)#

REGULATOR_SWITCHED build the regulator

Method Summary
Nss()#

NSS: number of subsystems

connect_model(plant, ind, rho)#

connect the model (nominal regulator equation)

Parameters:

  • plant – original system

  • ind – index to examine

  • rho – exponential weighting

Return:

plant_model – plant and model together

create_vars()#

CREATE_VARS: create variables that parameterize the nullspace :param param_null: should the nullspace be searched (as :type param_null: bool :param variables):

Returns:

vars_reg – structure with fields (Pi, Gam, Phi)

exosystem(param)#

get the exosystem at each mode/internal model

get_model(ind, vars_reg)#

fetch the internal model (nominal) at mode ‘ind’ with edits, allow for selection of model within feasible set :param vars_reg: variables of the problem (regulator)

Return:

model – the full-order internal model

ns()#

NS number of states of exosystem

reg_K_sys_all()#

assemble the closed-loop regulator equation system

Returns:

  • reg_mat – matrix for regulator equation

  • reg_ans – vector for regulator equation solution

reg_K_sys_indiv(param)#

control regulator equation checks (closed-loop)

reg_sys_all()#

assemble the regulator equation system

sol_K_reg_all(reg_sol)#

recover the solution to the regulator equation system

sol_reg_all(reg_sol)#

recover the solution to the regulator equation system

LMI Analysis#

class system.switched.lmi_analysis_switched#

Bases: system.generic.lmi_analysis_interface

LMI_ANALYSIS_SWITCHED analysis LMIs for algorithmic interconnections involving switched linear networks and controllers

Constructor Summary
lmi_analysis_switched(sys, config)#

LMI_DISPATCH_LTI Construct an instance of this class Detailed explanation goes here

Method Summary
Nss()#

NSS: Number of subsystems

cons_dynamic(vars, cons, diss)#

CONS form the dissipation and sign constraints

Parameters:

  • vars – variables of the problem

  • cons – accumulated constraints

  • diss (diss_data) – structure describing the dissipation constraint %

Returns:

  • cons – accumulated constraints

  • objective – term to be minimized

create_vars_spec(cons, specs)#

CREATE_VARS_SPEC declare variables for the specifications

Parameters:

  • cons – accumulated constraints

  • alg_psi – the filtered algorithmic interconnection

  • name – a name for the variable

Returns:

  • vars_spec – variables of the problem in the specifications

  • cons – accumulated constraints

create_vars_storage(cons, alg_psi, name)#

create_vars_storage create variables for the dissipation constraints. A cell of G(s) functions, one for each subsystem.

Parameters:

  • cons – accumulated constraints

  • alg_psi – the filtered algorithmic interconnection

  • name – a name for the variable

Returns:

  • vars_diss – variables of the problem in the dissipation constraints

  • cons – accumulated constraints

h2(vars, cons, diss)#

H2: certificate of stochastic performance

Parameters:

  • vars – variables of the problem

  • cons – accumulated constraints

  • diss (diss_data) – structure describing the dissipation constraint

Returns:

  • cons – accumulated constraints

  • objective – term to be minimized

  • con_M – PSD blocks for the dynamics constraint

quad(vars, cons, diss)#

QUAD: certificate of infinite-horizon quadratic performance

Parameters:

  • vars – variables of the problem

  • cons – accumulated constraints

  • diss (diss_data) – structure describing the dissipation constraint

Returns:

  • cons – accumulated constraints

  • objective – term to be minimized

  • con_M – PSD blocks for the dynamics constraint

LMI Synthesis#

class system.switched.lmi_synthesis_switched#

Bases: system.generic.lmi_synthesis_interface

LMI_SYNTHESIS_SWITCHED synthesis LMIs for algorithmic interconnections involving switched linear networks and controllers

Constructor Summary
lmi_synthesis_switched(sys, config)#

LMI_SYNTHESIS_SWITCHED Constructor

Method Summary
Nss()#

NSS: Number of subsystems

common()#

is a common storage function used?

con_spread(cons, vars)#

CON_SPREAD increase numerical conditioning by separating the primal and dual blocks. Invoke this over multiple subsystems

Parameters:

  • cons – accumulated constraints

  • GX – primal storage matrix

  • GY – dualstorage matrix

Returns:

cons – accumulated constraints

connect_model(diss, rho)#

connect the plant to the internal model :param diss: information about dissipation relation :type diss: diss_data :param rho: discount rate

Returns:

P_model – generalized plant with internal model attached

cons_dynamic(vars, cons, diss)#

CONS form the dissipation and sign constraints

Parameters:

  • vars – variables of the problem

  • cons – accumulated constraints

  • diss (diss_data) – structure describing the dissipation constraint

Returns:

  • cons – accumulated constraints

  • objective – term to be minimized

create_vars_controller(cons, alg_psi, name)#

CREATE_VARS_CONTROLLER create the nonlinearly-transformed controller matrices :param cons: accumulated constraints :param alg_psi: the filtered algorithmic interconnection :param name: a name for the variable :param D_mask: sparsity pattern for D of the controller

Returns:

  • vars_K – controller variables [Ak, Bk, Ck, Dk], or some subset if elimination is used.

  • cons – accumulated constraints

create_vars_storage(cons, alg_psi, name)#

create_vars_storage create variables for the dissipation constraints. A cell of G(s) functions, one for each subsystem.

Parameters:

  • cons – accumulated constraints

  • alg_psi – the filtered algorithmic interconnection

  • name – a name for the variable

Returns:

  • vars_diss – variables of the problem in the dissipation constraints

  • cons – accumulated constraints

get_D_mask()#

GET_D_MASK get the direct feedthrough terms

Returns:

D_mask – sparsity pattern for D of the controller

get_storage_slack(vars_diss, vars_reg)#

GET_STORAGE_SLACK get the slack storage function matrix Gslack. Used in the extended LMI characterization of stability

Parameters:

  • vars_diss – variables of the problem in the dissipation constraints

  • vars_reg – variables for regulator equation

Returns:

G – the closed-loop storage matrix (warped)

get_vars_involved(vars, ind)#

GET_VARS_INVOLVED get variables involved in the current mode :param vars: variables of the problem :param ind: index of subsystem/mode

Returns:

vars_inv – variables (diss, reg) at subsystem ind

h2(vars, cons, diss)#

H2: certificate of stochastic performance

Parameters:

  • vars – variables of the problem

  • cons – accumulated constraints

  • diss (diss_data) – structure describing the dissipation constraint

Returns:

  • cons – accumulated constraints

  • objective – term to be minimized

  • con_M – PSD blocks for the dynamics constraint

quad(vars, cons, diss)#

QUAD: certificate of infinite-horizon quadratic performance

Parameters:

  • cons – accumulated constraints

  • specs – performance specifications

Returns:

vars_spec – variables for performance specification cons: accumulated constraints

recover_subcontroller(alg_psi, P_trans, sol)#

RECOVER_SUBCONTROLLER recover the subcontroller of the current mode/control

Parameters:

  • alg_psi – the filtered algorithmic interconnection

  • P_trans – the transformed generalized plant before IQC

  • sol – solution structure

Returns:

sol – solution structure

recover_subcontroller_warp(P_trans, vars_rec)#

RECOVER_SUBCONTROLLER_WARP recover the nonlinearly warped controller :param alg_psi: the filtered algorithmic interconnection :param P_trans: the transformed generalized plant before IQC :param sol: solution structure

Output:

K_nofeed: subcontroller without direct feedthrough Gcl: closed-loop storage matrix (original) Ycl: similarity transformation/nonlinear warping

validate_recovery_gain(alg_trans, iqc_op_all)#

VALIDATE_RECOVERY validate that the system obeys the stability constraint (not yet supported)

Parameters:

  • alg_trans – the plant with confirmed performance by LMIs

  • iqc_op_all – all IQCs

Return:

gain – [Passivity index, H-infinity index].