Examples

Complex shape metal beam is being deflected using piezoceramic actuators attached to subdomains Ω1 – Ω3. Distributed transient characteristics (transient responses) are numerically calculated in ANSYS software. The dynamics of control inputs comprises of actuating member transfer functions SA(s), input generator transfer functions SG(s) and shaping units T. The control task is to bring the beam into a new steady state shape by means of PID control synthesis.

Legend

Ω – definition domain of the metal beam
Ωi – i-th subdomain of the piezo element (i-th boundary condition region)
SAi – i-th transfer function of the actuating member (voltage source time-dynamics)
SGi – i-th transfer function of the distributed input generator (piezo element time-dynamics)
Ui – i-th lumped input quantity (i-th control input, voltage)
Ti – i-th shaping unit (load distribution over subdomain region)
HLDS – lumped input / distributed output system (voltage / displacement)
K – sampling (Simulink inherited)

Model data

Material – aluminium alloy
Number of lumped inputs – 3, individually controlled
Ti (x, y, z), i = 1,2 and 3 – conical shape
SAi , i = 1,2,3 – const = 1 (neglected time-dynamics, immediate reactions)
SGi , i = 1,2,3 – const = 1 (neglected time-dynamics, immediate reactions)

Distributed transient characteristics (system dynamics identification)

U₁ – control voltage step change / displacement transient response

U₂ – control voltage step change / displacement transient response

U₃ – control voltage step change / displacement transient response

Control process (reaching a new steady-state displacement field)

Glass melting furnace is being heated by gas burners represented by triangular submains Ω1 – Ω4. Distributed transient characteristics (transient responses) are numerically calculated in Matlab PDE Toolbox. The control task is to reject disturbances caused by changing operating conditions in the gas furnace by means of adaptive control synthesis.

Furnace diagram

Legend

Y(x, y, t) – Temperature field of the glass furnace arch
Ui (t) – i-th lumped input quantity (i-th control input, servo unit control signal)
T1-11 – temperature sensors (thermocouples)

Model data

a = 0.00491 (m2 s-1), b = 0.00026 (m s-1), c = 1418 (J kg-1 K-1), rho = 2741 (kg m-3)
Number of lumped inputs – 4, individually controlled
Ti (x, y), i = 1,2,3 and 4 – pyramidal shape
SAi , i = 1,2,3 and 4 – const = 1 (neglected time-dynamics, immediate reactions)
SGi , i = 1,2,3 and 4 – const = 1 (neglected time-dynamics, immediate reactions)

Distributed transient characteristics (system dynamics identification)

U₁ – gas flow step change / temperature transient response

U₂ – gas flow step change / temperature transient response

U₃ – gas flow step change / temperature transient response

U₄ – gas flow step change / temperature transient response

Control process (reaching a new steady-state temperature field)

Groundwater remediation process is to be designed for a given area using 10 extraction wells (see dots on the map), arranged in two lines (HS, HO). Distributed transient characteristics (transient responses) are numerically calculated in Modflow software. The control task is to follow the desired shape of groundwater hydraulic-head over remediation area by means of extraction wells.

Map of groundwater remediation area

Legend

solid lines – groundwater hydraulic-head iso-lines (m)
hydraulic head – height of the groundwater surface relative to the sea level (m)
HS – first line of extraction wells (dots)
HO – second line of extraction wells (dots)

Distributed transient characteristics (system dynamics

U1 – extraction rate step change / hydraulic-head transient response