President Systems Control Inc.
One issue that designers of hydraulic position-control equipment often face is the low damping inherent in such systems — which usually consist of a hydraulic actuator, servovalve, position sensor, and controller. When designers try to improve accuracy by increasing loop gain, low damping causes the system to vibrate.
Fortunately, adding acceleration or pressure feedback, or using pole-placement techniques, can remedy the situation. Here’s how.
For insight into the dynamics, let’s first look at the underlying equations that mathematically describe hydraulic systems. The following is a summary of the linearized math model for a system with a servovalve and hydraulic actuator moving a mass. The position of mass M is X, and flow Q from the servovalve is defined as:
The equation relating flow to the actuator with a cylinder-piston area A, the actuator motion, and volume under compression is:
Motion of the mass M is:
Eliminating Q and Pl from the above equations and taking the Laplace transform of the resulting equation produces an open-loop transfer function of the system:
The accompanying graphic shows a basic system block diagram.
Using block-diagram algebra gives the relationship between the desired position Xd(s) and actual mass
The above transfer function shows that a thirdorder equation governs the system’s dynamics. We can learn a lot from this equation. For example, if Kp is zero — which means servovalve flow is independent of load pressure — then the coefficient of s2 disappears, leading to severe instability.
The most critical point of operation is around the valve’s null position. For a closed-center valve, Kp is very low. This leads to extremely low damping and is a major source of instability and oscillations in hydraulic systems.
Adding acceleration feedback to the control loop can solve this by influencing the coefficient of the s2 term in the transfer function. The resulting transfer function with acceleration feedback is:
Also note that pressure feedback has the same effect as acceleration feedback, because pressure is directly proportional to acceleration.
One last important point is that introducing leakage across the actuator also adds damping to the system. This can be seen by adding leakage flow in the servovalve flow equation, which then becomes:
So leakage has the same effect of increasing the flow pressure gain and, thus, increasing system damping. The disadvantages of using leaks to damp a system are the associated energy losses and a decrease in the system’s static stiffness.
Pole placement is a powerful control technique that can improve hydraulic control-system response. Consider the control system in the previous section. Its open-loop transfer function is:
Feeding back the actuator position yields the closed-loop transfer function:
As discussed previously, Kp is usually quite low, producing an undesirable response. Designers can improve response with the following transfer function, which quite effectively uses pole placement. The desired transfer function is:
This transfer function places the closed-loop poles at optimal locations in the s-plane to obtain the desired response. This involves adding a gain block in the forward path and a compensator, as2 + bs + 1, in the feedback loop.
Using block-diagram algebra generates the closed-loop transfer function:
Equating the characteristic equation’s coefficients to the desired coefficients determines the values of K, a, and b:
Thus, control engineers can substantially improve valve-controlled actuator performance by:
- Adding acceleration feedback.
- Adding load-pressure feedback.
- Adding an intentional leakage flow across the actuator.
- Using pole-placement technique to obtain a desired response.
A = Actuator piston area, in.2
Ac = Coefficient of s3 in the desired transfer function
a = Coefficient of s2 in the compensator as2 + bs + 1
B = Acceleration feedback gain, sec2
Bc = Coefficient of s2in the desired transfer function
b = Coefficient of s in the compensator as2 + bs + 1
C = Compressibility coefficient, in.5/lb
Cc = Coefficient of s in the desired transfer function
E = Laplace transform of position error e
e = Position error, in.
G(s) = Open-loop transfer function of the system
K = Closed-loop gain
Kl = Leakage coefficient, in.5/sec-lb
Kp = Valve pressure gain, in.5/sec-lb
Kq = Valve flow coefficient, in.2/sec
M = Mass of load, lb-sec/in.2
Pl = Load pressure, psi
Q = Flow, in.3/sec
s = Laplace transform operator
t = Time, sec
x = Actuator position, in.
X(s) = Laplace transform of actuator position x
Xd = Desired actuator position
Hydraulic positioning systems with low damping have a tendency to vibrate or oscillate. Several control techniques can cure the problem.
A basic control system compares desired and actual positions and compensates for any error.
Feeding back acceleration of the mass improves stability and positioning accuracy.
Pole placement adds a gain block in the forward control path and a compensator in the feedback loop.