# Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder
## 2 Background and Method  
Adaptive systems have long been used for system identification and parameter estimation. One of the first methods is described in [4]. Another common method is the least mean squares algorithm, which was developed in [5]. An example of this is shown in Equations (1)-(3). Given the linear system:  
\[Y =\theta^{T}\cdot X \tag{1}\] \[E =Y-\dot{\theta}^{T}\cdot X\] (2) \[\dot{\dot{\theta}} =\gamma\cdot X\cdot E^{T} \tag{3}\]  
where  
\(Y\) = system output;  
\(\theta\) = system parameters;  
\(X\) = system input;  
\(E\) = estimation error;  
\(\dot{\theta}\) = estimated parameters;  
\(\gamma\) = adaptation gain, constant. The estimated parameters will converge towards the system parameters. The idea of using the sign function in the adaptive law comes from the sign-sign least mean squares algorithm, and was first introduced by [3]. Equation (3) then becomes:  
Figure 1: Electro-hydraulic motion control system with feedforward. \[\hat{\theta}=\gamma\cdot\text{sign}(X)\cdot\text{sign}(E^{T}) \tag{4}\]  
By taking the sign of the estimation error and system input, the adaptation becomes insensitive to the magnitudes of \(E\) and \(X\), and as such only the adaptation gain \(\gamma\) sets the adaptation speed. The MIT rule is also used for adaptive control, and is described in [2]. A typical application is model reference adaptive control, shown in Figure 2. Based on the model output \(y_{m}\), an additional control output \(\hat{u}\) is multiplied with the command signal \(u_{c}\) to shape the plant output \(y\). The equations for the model reference adaptive control is shown in Equations (5) and (6). \[\dot{u} =-\gamma\cdot y_{m}\cdot(y-y_{m}) \tag{5}\] \[u =u_{c}\cdot\hat{u} \tag{6}\]  
where  
\(u\) = control output;  
\(\hat{u}\) = adaptive control output;  
\(u_{c}\) = command signal;  
\(y_{m}\) = model output;  
\(y\) = plant output. Early work in adaptive control can be found in [6; 7; 8; 9; 10]. Other work on adaptive control include [11] which investigates adaptive feedback and feedforward control of robot manipulators, Reference [12] which models and implements adaptive control of a flexible arm, and [13] which uses model reference adaptive control on linear time-varying plants. Adaptive fuzzy sliding mode control is investigated and implemented on an inverted pendulum in [14]. Newer applications of adaptive control systems include adaptive friction compensation with an adaptive velocity estimator to compensate for the estimated non-linear friction force [15]. In [16], a fuzzy model reference adaptive control of an active magnetic bearing for a milling process is investigated to reduce the milling dynamics. Adaptive integral robust control of an electro-hydraulic servo system is investigated in [17], using parameter estimation and integral control to compensate for disturbances and plant uncertainties. Adaptive control of quadrotors is investigated in [18], which uses an cerebellar model arithmetic computer to adapt to model uncertainties and disturbances. In [19], adaptive control based on least-mean-fourth is implemented for a three-phase grid connected solar system, which is able to provide load balancing and power factor correction. As for motion control of hydraulic systems, different approaches have previously been investigated, including vector control [20], pressure control [21; 22], force control [23; 24], and feedforward control [25]. Figure 2: Model reference adaptive control based on MIT-rule. To the knowledge of the authors, adaptive feedforward motion control of hydraulic cylinders has not previously been investigated, and this paper will focus on this novel concept. In this paper, two adaptive controllers have been tested on a hydraulic differential cylinder and compared to a fixed gain feedforward controller. Based on a typical fixed gain feedforward controller, an adaptive controller can be made by extending it with the MIT rule.