Demos

Worked examples shipped with the toolbox, in demo/. Each entry below has a command you can copy into the MATLAB Command Window — openDemo followed by the demo’s path. It opens the demo in the Editor and adds the demo’s folder to your path, so you can press Run straight away.

The commands work from any current folder: openDemo finds the demo itself. Most demos are live-script-flavoured .m files, so the Editor shows their text, equations and figures inline.

Note: Requirements

Some demos need additional MathWorks products (Simulink, System Identification Toolbox, Optimization Toolbox) or third-party packages. The demo says so at the top when it does.

One tank — oneTank/

Temperature-controlled water tank, the smallest hybrid example in the toolbox.

Temperature-controlled water tank

A tank losing heat to the ambient, with a hot-water inlet regulated by an on-off (1-point) controller. The controller output is low-pass filtered to prevent Zeno behaviour and drives an inert actuator valve. The system is written as a set of difference equations plus an inequality constraint and built as a hybrid implicit multilinear model. Example from the CoDIT 2024 conference paper.

openDemo oneTank/DemoMTI_oneTank

Three tanks — threeTank/

The one-tank example scaled up to three coupled tanks competing for one inlet.

Continuous-time three-tank system

Three water tanks share a single inlet flow and exchange heat with each other and with the environment. Each tank has its own 2-point controller holding its temperature between a lower and an upper bound; when several valves open at once the inlet flow is shared uniformly between them. The model is entered as symbolic equations with inequality constraints and simulated in continuous time.

openDemo threeTank/DemoMTI_threeTanks

Three-body problem — threeBodyProblem/

Gravitational three-body dynamics as a multilinear model.

Three-body problem

The equations of motion of three gravitating bodies are written as string equations and converted into a multilinear model, using auxiliary variables and binary indices to express the inverse-cube distance terms. The demo simulates the resulting trajectories; a recorded animation of the result ships alongside it.

openDemo threeBodyProblem/DemoMTI_threeBodyProblem

Hybrid systems — IFAC_hybridSystems/

The examples accompanying the IFAC Nonlinear Analysis: Hybrid Systems 2026 contribution.

Temperature- and level-controlled water tank

Starts from an ideally-mixed water tank whose temperature is the single state, driven by inlet flow, inlet temperature, ambient temperature and an electrical heating coil, and builds it as an explicit multilinear model from its structure and parameter matrices. Further examples in the same script extend it towards hybrid, level-controlled behaviour.

openDemo IFAC_hybridSystems/DemoMTI_hybridSystemsTankExamples

Boolean-index MTI models — Demo_BooleanIndexMTI/

A subclass of MTI models whose state space grows logarithmically with the number of signals.

Boolean-index MTI models

Boolean-index MTI models enable the representation of up to $n_y$ signals by a model of order $n_x=\lceil\log_2(n_y)\rceil$ through binary encoding of the signal graph. The demo walks the full construction chain — signal graph, binary-encoded automaton, next-state OTVL, solution subset, structure and parameter matrices of an mss model — on a multi-energy system whose four measured signals become a two-state model.

openDemo Demo_BooleanIndexMTI/DemoMTI_Boolean_Index_MTI

Stochastic automata — stochasticAutomata/

Stochastic automata expressed in multilinear form.

Stochastic automata in the MTI Toolbox

Builds a stochastic automaton with 25 states, 27 inputs and 5 outputs from stored transition data, assembling the sparse structure matrix that represents its transition probabilities. Example from the IFAC World Congress 2026 contribution on multilinear modelling with the MTI Toolbox.

openDemo stochasticAutomata/DemoMTI_stochasticAutomataSimulation

Base transformation — mss2mss_literal2monomialBase/

Converting a multilinear model between its two tensor bases while keeping the input-output behavior.

Converting the MTI model base

Converts an mss model from the literal base to the monomial base and back again with mss2mss, showing how the structure matrix encodes the equations in each base.

openDemo mss2mss_literal2monomialBase/DemoMTI_literalBase2monomialBase

Multilinearization — mlinearize_demo/

Approximating an existing nonlinear Simulink model by a multilinear one.

Multilinear approximation of a CSTR

Approximates a nonlinear Simulink model of a continuous stirred-tank reactor — four states, two inputs — by an explicit multilinear (mss) model, using the sparse-grid interpolation performed by mlinearize. The multilinear model is then simulated with msim and compared against the original nonlinear model over the same input trajectory. Needs Simulink and the Sparse Grids MATLAB Kit.

openDemo mlinearize_demo/DemoMTI_mlinearize

System identification — mlgreyest_example/

Estimating multilinear models from data with mlgreyest.

Parameter estimation with a known structure

Identifies the parameter matrix of a multilinear model when its structure matrix is already known, comparing the estimate against the parameter matrix the data was generated from.

openDemo mlgreyest_example/DemoMTI_system_identificationPhi

Identification from building data — ALS method

Estimates a discrete-time multilinear model of a test room from measured building data, using the alternating least-squares method with a rank-4 parameter tensor. The identified model is validated by simulation against a held-out part of the measurement record.

openDemo mlgreyest_example/DemoMTI_system_identification_Als

Identification from building data — nonlinear optimization

The same test-room identification problem solved by nonlinear optimization (fmincon) with a rank-2 parameter tensor and four inputs, again validated against held-out data.

openDemo mlgreyest_example/DemoMTI_system_identification_nonlin

State estimation — EKF_demo/

Extended Kalman filtering on a multilinear model.

EKF for a multilinear Lorenz system

Runs an Extended Kalman Filter for state estimation on a multilinear representation of the chaotic Lorenz attractor, scoring the filter against the trajectory that generated the measurements and keeping the covariance update symmetric positive semidefinite throughout. Two prediction steps are available — a local fixed-step Dormand-Prince step and msim with adaptive step control — and the demo measures what each one costs. Shows how the toolbox’s Jacobian and output evaluation functions are used inside an estimator.

openDemo EKF_demo/DemoMTI_EKF

Moving horizon estimation — MHE_demo/

Large-scale state and disturbance estimation by successive affine linearization.

MHE for a chain of 100 Van der Pol oscillators

Estimates 200 states and 100 unmeasured disturbance inputs of a closed-loop chain of 100 Van der Pol oscillators from the measured outputs. Each oscillator is an implicit MTI (mdss) model taking the previous oscillator’s output plus an unknown disturbance, with the last oscillator fed back to the first. The estimator starts as an Extended Kalman Filter and switches to the moving horizon estimator once enough measurements are available. Companion to the IFAC 2026 contribution.

openDemo MHE_demo/DemoMTI_MHE_VanDePol

HVAC — HVAC/

Component-based modelling of a full air-conditioning plant.

Air-conditioning system with dehumidification

A constant-air-volume partial air-conditioning plant supplying a production space: mixing chamber with heat recovery, preheater, cooler with dehumidification modelled as a four-cell heat exchanger, reheater, supply and exhaust fans, and a PLC unit with saturated PI controllers and anti-windup feedback. Each component is built as its own submodel in a separate script and the plant is assembled from them, demonstrating component-based hybrid implicit-MTI modelling.

openDemo HVAC/DemoMTI_AirConditioningSystem

District heating networks — DistrictHeatingNetworks/

Thermo-hydraulic pipe networks, and exact model reduction on them.

Discretized pipe: algebraic elimination on a segment chain

Builds a single district-heating pipe as a chain of finite-volume segments, each carrying a mass flow, a pressure and a temperature, driven by a constant inlet flow with a +30 K step on the inlet temperature. The resulting MTI model is reduced with algebraicElimination and the reduced model is checked to reproduce the states and outputs of the original.

openDemo DistrictHeatingNetworks/DemoMTI_Discretized_Single_pipe

Branched network: build, reduce and validate

Assembles a small branched district-heating network from individual pipes and junctions, simulates it against measured input data from the AIT test rig, then applies algebraicElimination and compares the outlet temperatures of the original and reduced models to confirm the reduction is exact.

openDemo DistrictHeatingNetworks/DemoMTI_DHN_Pongau

Multi-stack electrolyzer control — mELY_mpc/

Supervisory model predictive control of a multi-stack PEM electrolyzer.

Two-stage MPC for a multi-stack electrolyzer

A multi-stack PEM electrolyzer powered by a variable power profile is controlled by a two-stage nonlinear MPC built on an implicit MTI model of all stacks. Stage 1 decides which stacks run, relaxing the on/off signals to continuous variables and quantizing them afterwards; stage 2 allocates power among the running stacks. The result is compared against the classic Daisy Chain and Equal rule-based allocation strategies. This main file drives the whole chain. Companion to the paper: Aline Luxa, Richard Hanke-Rauschenbach, Gerwald Lichtenberg: Model predictive supervisory control for multi-stack electrolyzers using multilinear modeling, International Journal of Hydrogen Energy, Volume 185, 2025, 151847, ISSN 0360-3199, https://doi.org/10.1016/j.ijhydene.2025.151847.

openDemo mELY_mpc/DemoMTI_mELY_sup_control

Power systems — PowerSystems/

Converter-dominated grids: modelling, simulation and small-signal stability analysis.

Phase-locked loop

Models a nonlinear synchronous-reference-frame phase-locked loop as a second-order multilinear model derived with trigonometric identities, then simulates it and compares the response against the nonlinear model.

openDemo PowerSystems/PhaseLockedLoop/DemoMTI_powerSystemsDemoPLL

Small-signal stability analysis of an essential system

An essential system — a grid-following converter, a grid-forming converter and a load on a Thévenin-equivalent grid. Operating points are initialized from the nonlinear Simulink model, an implicit MTI model is built and its Jacobian formed to give a linear descriptor state-space model, and stability is assessed from the generalized eigenvalues. The nonlinear, multilinear and linear descriptor responses are then compared, along with the computational cost of each.

openDemo PowerSystems/SmallSignalAnalysis/DemoMTI_powerSystemThreeBusSSA

Multilinearizing grid-following converter control

Takes the essential system of the small-signal analysis demo and multilinearizes the nonlinear part of the grid-following converter with mlinearize, producing an mss object that is brought back into the Simulink model through an S-function block. The bounds of the operating region are taken from the nonlinear simulation.

openDemo PowerSystems/MultilinearizeGFLcontrol/DemoMTI_mlinearizeGFL

Nine-bus converter-based network (ECC 2024)

A nine-bus converter-based power network with grid-forming droop-controlled voltage source converters, written symbolically and converted into an implicit multilinear (mdss) model, with its response compared against the corresponding nonlinear model. The demo can either build the models symbolically or load previously built ones from disk.

openDemo PowerSystems/InverterBasedNetwork/DemoMTI_inverter_nine_bus

WSCC nine-bus system — small-signal analysis and simulation

The WSCC nine-bus system — one synchronous generator, one grid-forming and one grid-following converter — built as an MTI model directly from its Excel case files. The demo walks the complete workflow: model construction in both qd and abc coordinates, small-signal analysis (linearization at the operating point, eigenvalues, mode table, participation factors), and time-domain simulation of a load step. Every result is compared against the STAMP tool, whose linear state-space model is the small-signal reference and whose nonlinear Simulink model is the time-domain reference.

openDemo PowerSystems/NineBusNetwork/DemoMTI_powerSystemNineBus

WSCC nine-bus system — computation profiling

Expert companion to the demo above: it runs the same workflow and profiles the execution time of every step in both tool chains, reporting per-step wall times and model sizes (states, inputs, algebraic variables, sparsity, Simulink block count, file sizes). It also compares the cost of the two eigenvalue problems — STAMP’s standard eig(A) against the descriptor model’s generalized eig(A, E), which additionally carries the algebraic variables.

openDemo PowerSystems/NineBusNetwork/DemoMTI_powerSystemNineBusProfiler

Grey-box identification of a grid-forming inverter

Identifies a multilinear model of the nonlinear subsystem of a grid-forming converter, isolated from a one-bus grid made up of a nonlinear diesel synchronous generator and a ZIP load. The subsystem is first multilinearized with mlinearize over bounds taken from the nonlinear simulation, which gives the structure matrix; mlgreyest then estimates the parameter matrix from normalized input and state data, using the first 70 % of the record for training. The identified model is simulated over the full record and its states compared against the nonlinear ones, before and after rescaling. Needs Simulink, the System Identification Toolbox and the Sparse Grids MATLAB Kit. The inverter grid model was provided by Prof. Byungkwon Park and colleagues at Soongsil University, Seoul, South Korea, and is described in Wong et al., 2026.

openDemo PowerSystems/SysIDGFMinverter/DemoMTI_greyestGFM

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