I am researching on controlling systems with time varying parametric changes. After studying few exemplar/textbook systems, I would want to apply it to a practical scenario. Thus, looking for a system matrix (A(t)) of such a robotic/real life system.
Dear Manoj,
In answer to the first part of your question, indeed there is in the field of robotics Time-varying parameters model or time-varying state constraints model. For more details and information about this subject please take a look at attached files and links on topic.
Linear Systems - MIT OpenCourseWare
https://ocw.mit.edu/...robotic-systems.../MIT2_017JF09_ch02.pdf
Control, Robotics, and Mechatronics • Electrical and Computer ...
https://www.ece.ncsu.edu/research/crm/
Robotic Networks and the Consensus Problem - Springer
https://www.springer.com/cda/.../9783319156989-c2.pdf?...0...
Best regards
To begin with start with something simple like a model for the motion of a (frictionless) point-mass in 1-D, with an acceleration input , position output, and [position; velocity] internal states (i.e. a double integrator), with system equations
A = [0 1;0 0], B = [0;1], C = [1 0], D = 0.
You could then extend this by adding a drag force, then maybe attach it to an elastic teather for some oscillatory behaviour.
Basically all systems are time varying. Wheter to model a system as time varying depends on how big are the changes of parameters or operating point of a linearized nonlinear system. A system with large parameter changes might be a light transport vehicle that is loaded with a heavy cargo.
Dear Manoj,
Please note that the general form for a time-varying linear systems is
\dot{x} = A(t) x + B(t) u
and your parameters will be in A and B.
However, wide range of robotic systems are not linear, for example a pendulum (1-DOF system). In this case your system equation is
\dot{x} = f(t,x,u)
and there is no A(t). If you want to have a your system in linear form, you should linearized it at any points (find the Jacobian matrix).
Dear Manoj,
Even if you consider the parameters of a robot/mechanical system as constant, if you work on trajectory tracking problems, the error dynamics of the system can easily become time-varying. See for example for the trajectory tracking control of a biped robot:
http://folk.ntnu.no/skoge/prost/proceedings/ifac2014/media/files/0299.pdf
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