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Extended state observer-based pressure control for pneumatic actuator servo systems

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Abstract

A pneumatic actuator is a fast and economical tool that converts compressed air into mechanical motion. In this paper, an extended state observer (ESO)-based sliding mode controller (SMC) is developed to adjust the air pressure of the actuator for accurate position control. Specifically, an impedance control module is established to produce desired air pressure based on the relationship between forces and desired positions. Then, the ESO-based SMC is implemented to adjust the air pressure to the required level despite the presence of system uncertainties and disturbances. As a result, the position of the actuator is controlled to a setpoint through the regulation of pressure. The performance of ESO-based SMC is compared with that of a classic active disturbance rejection controller (ADRC) and a SMC. Simulation results demonstrate that the ESO-based SMC shows comparable performance to ADRC in terms of precise pressure control. In addition, it requires the least control effort necessary to excite valves among the three controllers. The stability of ESO-based SMC is theoretically justified through Lyapunov approach.

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Abbreviations

P 1(t):

Absolute air pressures for chamber 1

P 2(t):

Absolute air pressures for chamber 2

F L(t):

Payload force

F f(t):

Coulomb friction

\(\beta\) :

Viscous friction coefficient

\(M_{{\text{P}}}\) :

Mass of the piston

\(M_{{\text{L}}}\) :

Mass of the load

\(P_{{\text{a}}} (t)\) :

Atmospheric pressure

\(A_{{\text{p}}}\) :

Area of smaller rod portion of the piston

\(A_{1}\) :

Area of larger disc portion of the piston

\(A_{2}\) :

Difference between \(A_{1}\) and \(A_{{\text{p}}}\)

x(t):

Piston position

\(\dot{Q}(t)\) :

Heat rate for the air volume

\(\dot{m}_{i} (t)\) :

Mass flow rate for each chamber (\(i = 1 \, {\text{or}} \,2\))

\(\dot{m}_{{\text{l}}}\) :

Mass flow rate of air leakage

\(\dot{E}(t)\) :

Rate of change of total energy

\(\dot{W}(t)\) :

Rate of work delivered to piston

\(h_{n}\) :

Enthalpy of gas that enters the chamber

\(h_{x}\) :

Enthalpy of gas that exits the chamber

\(v_{n}\) :

Velocity of gas that enters the chamber

\(v_{x}\) :

Velocity of gas that exists the chamber

V i :

Volume of each chamber (\(i = 1 \,{\text{or}} \, 2\))

C v :

Specific heat at constant volume

C p :

Specific heat at constant pressure

R :

The difference between Cp and Cv, i.e., R = Cp − Cv

T 1 :

Temperature of chamber 1

T 2 :

Temperature of chamber 2

K :

The ratio between Cp and Cv, i.e., \(k = C_{{\text{p}}} /C_{{\text{v}}}\)

T :

Constant temperature

P i(t):

Air pressure for each chamber (\(i = 1 \;{\text{or}} \;2\))

d(t):

External disturbance caused by air leak

K i :

On/off state of each valve

P s(t):

Supply pressure

q m :

Function of Ps(t) and Pa(t)

x d :

Desired position

\(P_{{\text{D}}}\) :

Desired pressure

u(t):

Control signal

\(F_{{\text{D}}} (t)\) :

Total force produced by air pressure

\(M\) :

Total mass of piston and load

\(\tilde{M}\) :

Reference mass

\(\tilde{B}\) :

Reference damping coefficient

\(\tilde{K}\) :

Reference stiffness coefficient

\(P_{{{\text{1D}}}} (t)\) :

Desired pressures for chamber 1

\(P_{{{\text{2D}}}} (t)\) :

Desired pressure for chamber 2

\(P_{{{\text{sum}}}} (t)\) :

Total supplied pressure

\(f(t)\) :

Generalized disturbance

b(t):

Coefficient of controller

\(b_{{{\text{min}}}}\) :

Minimum value of b(t)

x(t):

Stable variable (pressure)

y(t):

Pressure output

A :

State matrix

B :

Input matrix

C :

Output matrix

z 1(t):

Observed pressure

z 2(t):

Observed generalized disturbance

\(\hat{y}(t)\) :

Output of extended state observer

\(\omega_{o}\) :

Observer bandwidth

h(t):

Derivative of generalized disturbance

L :

Extended state observer gain vector

\(\tilde{e}_{1} (t)\) :

Estimation error between actual and observed pressure

\(\tilde{e}_{2} (t)\) :

Estimation error between actual and observed \(f( \cdot )\)

\(s(t)\) :

Sliding surface

V(t):

Lyapunov function

\(\overline{f}(t)\) :

An approximate of \(f(t)\)

\(P_{{{\text{avg}}}} (t)\) :

Average pressure

\(k_{{{\text{SMC}}}} \;{\text{and}}\;\eta_{{{\text{SMC}}}}\) :

Positive controller gains for SMC

\({\text{sgn}}(s(t))\) :

Sign function of s(t)

\(F\) :

Bound of \(f(t) - \overline{f}(t)\)

\(\varPhi (t)\) :

Bound of sliding surface

\(u_{{{\text{ESMC}}}}\) :

Control input of ESO-based SMC

\(P_{{{\text{ESMC}}}} {\text{and}}\;\eta_{{{\text{ESMC}}}}\) :

Controller gains of ESO-based SMC

\(\hat{s}(t)\) :

Difference between observed and desired pressures

\(u_{{{\text{ADRC}}}} (t)\) :

Control signal of ADRC

\(\omega_{{\text{c}}}\) :

Controller bandwidth for ADRC

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Authors and Affiliations

  1. Delta Systems Inc., Streetsboro, OH, 44241, USA

    John Mohorcic

  2. Department of Electrical Engineering and Computer Science, Cleveland State University, Cleveland, OH, 44115, USA

    Lili Dong

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  1. John Mohorcic
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Correspondence to Lili Dong.

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Mohorcic, J., Dong, L. Extended state observer-based pressure control for pneumatic actuator servo systems . Control Theory Technol. 19, 64–79 (2021). https://doi.org/10.1007/s11768-021-00038-y

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