I have modeled a multi-story building structure equipped with active tendons systems. I am using sliding mode control method to decrease the dynamical responses of building in MATLAB.
Using the sliding mode control law in the state space form of the systems as dZ/dt=A*Z+Bu*U+Br*ag, I can calculate the sliding mode control "force" as U=-inv(P*Bu)*(P*A*Z+Br*ag)-eta*sign(P*Z).
I wonder if I need to consider the actuator properties of active tendon system in order to obtain the voltage/current signal. If so, considering my work as a numerical study, I would appreciate any comments or references on this issue.
Regards,
Keyvan
Hi Keyvan Aghabalaei Baghaei, if you need to obtain the control signal in voltage or current, then you have to include the active tendon system dynamics.
The control law U in dZ/dt = A*Z + Br*ag + Bu*U, as you claimed, only gives you the desired 'force' that the active tendon actuator needs to produce to compensate for the seismic force. To produce that desired 'force', you need to apply certain voltage or current to the ATS.
Hi Keyvan,
As Yew-Chung Chak said, once you have the desired control force, you need to consider the actuator's dynamics in order to obtain the input voltage/current.
The simplest way to do it, it would be considering the dynamics of the main system as not affected by the actuator's dynamics. In this case you can look at it as an additional transfer function (TF) whose input is the voltage/current and whose output is the force. Then, the actuator's TF would be placed in series with the main system.
If the main system's dynamics is affected by the actuator's, the implementation is a bit trickier because the two systems are coupled. In this case the actuator should be taken into account within the main system model.
Best,
G.
Dear Giuseppe Pennisi ,
Thank you for your comment. I would appreciate if you could introduce references in this regard, so I could study it in detail.
Dear Yew-Chung Chak Giuseppe Pennisi ,
Thank you for you comment. Considering my work with no experimental set, I could not find any proper paper detailing on the active tendon systems dynamics. I would appreciate your suggestions on this issue.
Hi Keyvan Aghabalaei Baghaei, if you are conducting the research with a theoretical framework on the active tendon control, then I think you should focus on studying the effective applications of different types of active tendon forces on the multi-story building structure.
Hence, you can leave the physical actuation of the active tendon system to the hardware guys or Electrical engineers. It's their job to ensure that the ATS meets Earthquake engineer's technical specifications. In most cases, the actuation control can be designed separately, if it doesn't cause an internal disturbance to the system itself. But if you use a conventional SMC, the unwanted chattering (infinite switching frequency) may excite the unmodeled vibration mode in the building structure.
By the way, when the proposed control law, Usmc = − inv(P*Bu)*(P*A*Z + Br*ag) − η*sign(P*Z), is applied, what will become of the closed-loop dynamics?
Hi Keyvan Aghabalaei Baghaei, based on the info from Smart Structures: Innovative Systems for Seismic Response Control, if you assume that the K-bracing is rigid without elastic deformation, then the actuator-piston displacement Δa(t) = – z(t). I have expanded the dynamics for Eq. (5.12) added the electrical system for the actuator.
There are five state variables, namely
z(t) = structural displacement,
f(t) = actuator force,
c(t) = servo valve-piston displacement,
θ(t) = servomotor position,
i(t) = servomotor armature current.
The 6th-order linear model of the building structure with servo valve-controlled hydraulic actuator as Active Tendon System is given by:
M*z''(t) + C*z'(t) + K*z(t) – ag = f(t)
f'(t) = (2*β*A2/V)*Δa'(t) + (β*A*KV*√(2Ps)/V)*c(t)
c(t) = N*θ(t)
J*θ''(t) + b*θ'(t) = Km*i(t) = Tm(t)
L*i'(t) + R*i(t) + Km*θ'(t) = Vm(t)
where
Tm = servomotor torque
N = gear ratio (assume that c(t) is proportional to θ(t))
J = moment of inertia of the rotor
b = motor viscous friction constant
Km = motor torque constant
L = electrical inductance
R = electrical resistance
Vm = voltage source applied to the motor's armature.
Since the model is in strict-feedback form, you can apply the backstepping procedure to develop the control law for the Voltage, Vm(t). You can also obtain the entire plant transfer function Gp(s) = z(s)/Vm(s).
For more info on the modeling of servomotor, please refer to
- Modern Control Engineering, 5e by Katsuhiko Ogata
- Modern Control Systems, 13e by Dorf and Bishop
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