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A new control design and robustness analysis of a variable speed hydrostatic transmission used to control the velocity of a hydraulic cylinder


Controlling the velocity of a hydraulic cylinder is a common objective in fluid power industry. This objective is achieved by controlling the hydraulic fluid quantity that enters the cylinder. In this work, a control system that is used to control the velocity of a hydraulic cylinder was designed considering uncertainty in some of the design parameters. A fixed displacement pump was used to reduce the cost and complexity of the system. Furthermore, using variable speed drive eliminates the energy losses associated with valve controlled systems since no throttling or flow recirculation is needed. First, the system dynamics was modelled. Then, the stability and performance of the open loop system was studied using MATLAB/SIMULINK®. Next, controllers (PID and H-infinity) were designed and the stability and the performance of the closed loop system were studied and compared with those of the open loop system. Finally, the robustness of the system was studied considering multiplicative parametric uncertainty. Three parameters were considered as uncertain parameters which are the fluid bulk modulus, the viscous friction coefficient, and the leakage coefficient with a variation of ± 5% of their nominal values. The results showed that the open loop system is stable with a poor response in terms of input tracking and disturbance rejection. Using PID controller improved the system response. The system with the PID controller does not meet the robustness requirements. The system with H-infinity controller has better performance and satisfies the robustness requirements.

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A :

Maximum low frequency (steady state) error

A A :

Cross sectional area of side A of the actuator

A B :

Cross sectional area of side B of the actuator

b :

Viscous Drag coefficient

G F :

Transfer function of the disturbance force

E :

Error signal

F d :

Disturbance force

F o :

Spring preload

G p :

Transfer function of the plant

C :

Transfer function of the controller

K p :

Gain of the proportional part of the controller gain

K i :

Gain of the integral part of the controller

K d :

Gain of the derivative part of the controller gain

K l :

Leakage coefficient

l I :

Multiplicative parametric uncertainty

M :

Maximum high frequency error

m :

Load mass

P p :

Pump pressure

p :

Generalized plant

Q p :

Pump volumetric flow rate

S :

Sensitivity transfer function

V o :

Actuator internal volume

V r :

Reference input velocity

v c :

Velocity of the cylinder

w :

Exogenous input

w I :

Transfer function of the uncertainty rational weight

w p :

Performance weight

w u :

Effort weight of the controller

y :

Cylinder displacement

Z :

Exogenous output

β :

Bulk modulus of the hydraulic fluid

η c :

Efficiency of the hydraulic cylinder

\(\varnothing_{1,2,3,4}\) :

Nondimensional groups


Time constant

ω b :

Bandwidth frequency

\(\widehat{{\,}}\) :

Dimensionless quantity


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Correspondence to Hasan H. Ali.

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Ali, H.H., Mustafa, A.W. & Al-Bakri, F.F. A new control design and robustness analysis of a variable speed hydrostatic transmission used to control the velocity of a hydraulic cylinder. Int. J. Dynam. Control 9, 1078–1091 (2021).

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  • Hydraulic actuator
  • Velocity control
  • Variable speed drives
  • Robustness
  • Uncertainty
  • H-infinity