DESIGN OF NARMA L2 NEUROCONTROLLER FOR NONLINEAR DYNAMICAL SYSTEM Kittisuk Srakaew, Viboon Sangveraphunsiri, Supavut Chantranuwathana, and Ratchatin Chancharoen Mechanical Engineering Department, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand.
[email protected]
ABSTRACT This paper proposed a technique based on NARMA L2 neurocontroller to control a trajectory of a nonlinear plant. The NARMA L2 neurocontroller was first trained to cancel both the nonlinearity and dynamic of the system. Then, it was reconfigured to become a controller. Once the NARMA L2 neurocontroller suppresses both the nonlinearlity and dynamic behaviour, the closed loop system becomes implicit algebraic relation between the input and the output. Consequently, the system is able to perfectly follow a smooth reference trajectory even it is generated in realtime. This technique was successfully implemented to control the trajectory of a nonlinear disk plant to follow a sine sweep trajectory and also to follow the input device in realtime. Thus, it is a remarkable candidate for a trajectory controller for a nonlinear system. KEY WORDS Nonlinear Trajectory Control, Neural network.
1. Introduction The determination of the trajectory of control effort that gives the output of a nonlinear system to follow a desired trajectory is a very challenging task since the output depends nonlinearly on the control effort. Since the system operates over a large working space, a controller design based on linear approximation of the system around a single equilibrium point is not well performed. The primary aim of this project is to design an accurate and practical tracking controller to control a nonlinear plant to follow a varying position reference over a large working space. In recent years, this problem draws much and increasing attention from industry since the mechanical transmission is to be replaced with “By-wire” technology in the near future to reduce cost and increase functions of the devices. For example, the acceleration pedal of some new vehicles is now using this technology, i.e., the position of the pedal, sensed by encoders, is used to control the opening of the throttle via electrical signal cables. The “steer by wire” [1] is also coming to the market. The other applications that need an accurate
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tracking controller are Master-slave devices [2] and high speed drives robotics [3] and high speed CNC machines [4]. These machines are nonlinear and operated at high speed. Thus, a tracking controller should be designed to handle both behaviors. The real mechanical systems are nonlinear. The system normally includes friction, which is hard nonlinear if dry friction included, the nonlinear gravity effect, and the varying inertia or load. The classical PID controller does not well perform when dealing with the mentioned nonlinear effects, and especially in trajectory following control. This controller cannot control the system such that the tracking error asymptotically decreases. The practical strategy to design an effective trajectory controller is a technique called “computed torque” [5]. This technique, the control effort consists of two parts, namely feedback linearization part and servo part. For the feedback linearization part, the effort is computed based on state feedback such that the closed loop system becomes linear. Once the closed loop system is linear, the superposition can be applied. Then, the servo part’s effort controls the resulting linear system such that the tracking error asymptotically decreases to zero. Both parts are superimposed to form a trajectory controller that gives a perfect tracking response. In the servo part, the effort is computed based on not only the tracking error but also its first and second derivatives. Thus, not only the position reference trajectory, but also its first and second derivatives are required as well as the actual position and its derivatives. Using this technique, the bounded input, bounded output stability is obtained and the tracking error is bounded when the bounded disturbance is presented. In case that the dynamics of the linear system is known precisely, the position reference and its first and second derivatives can be combined to generate a new single reference based on inverse dynamics of the linear system. In this way, the reference form a trajectory generator is a single signal that when used to command the linear dynamical system will give a perfect tracking. Since the closed loop system is linear, the stability is easily determined using linear system theory [6].
One difficulty of using the conventional computed torque technique is that the trajectory generator must provide the position reference and also its first and second derivatives and/or the dynamic model of the closed loop system must be known accurately. In case that the input device is used as the position reference generator, the first and second derivatives of the position reference must be computed real time and thus requiring computation capability of the hardware and increase cost. The intelligence is also required to compute and generate a single reference based on inverse dynamics of the system to command the dynamical system to perfectly follow a smooth trajectory. The other problem is that the nonlinearity must be modeled with sufficient accuracy which normally required an extensive effort. To overcome these problems, the control effort is designed such that it does not only eliminate the nonlinearity, but also the dynamics of the system. The closed loop system becomes implicit algebraic relation between the control effort and the output. The position reference can be used as reference directly. In this way, only the position reference is required and the system still perfectly tracks a smooth trajectory. This technique promotes higher complexity to the state feedback controller but make everything else simple. The controller is also design based on discrete-time neural network model in order to lessen an effort to model the nonlinearity and to construct the controller. The discretetime neural network is trained offline using the data pairs from the operation of the system and then reconfigured to be a controller. The NARMA L2 controller is designed based on this concept and is a potential candidate for a tracking controller. In this project, the performance of the NARMA L2 neurocontroller is investigated. The NARMA L2 is designed to control the trajectory of the nonlinear pendulum plant. The controller is designed such that it eliminates nonlinear, including hard nonlinear dry friction, and gravity and also eliminates the dynamic behavior. Since the NARMA L2 based on neural network is trained using the input-output data pairs, the nonlinearity and the dynamics of the system is easily but efficiently modeled in a single activity. The NARMA L2 neurocontroller is first trained, and then reconfigured to form a controller that efficiently suppresses the nonlinearity and dynamics of the system. The NARMA L2 controller is capable of control the pendulum plant such that it perfectly tracks the varying position reference as demonstrated by real time experimentation. This technique is practical and not too difficult to construct and implement on real time controller. Thus, it is a remarkable candidate for a universal trajectory controller for nonlinear system.
2. Narma L2 Neurocontroller NARMA stands for nonlinear autoregressive moving average. It is a discrete-time representation of the nonlinear dynamical system in neighborhood of the equilibrium state. When the system is nth order nonlinear SISO and has a relative degree d.The companion form of NARMA can be written as [5] y(k + d ) = F [y(k ), y(k −1),..., y(k − n + 1), u(k), u(k −1),..., u(k − n + 1)] where
u(k) ∈ R is the sequence of control effort y(k) ∈ R is the sequence of output F: R2n→R and F ∈ C∞
This model implies that an input at time k affects the output only at time d later. In other word, the relative degree, d, represents the delay of the system from control effort, u, to the output, y. This model is not convenient for determination of the control effort, u(k), such that the plant output tracks a desired reference since it is a solution of the inverse dynamics problem of nonlinear system. To overcome this problem, two linear approximations [5] of NARMA are proposed, namely NARMA L1 and NARMA L2. The relation between the input, u(k), and the output, y(k), appears linearly in both approximations, and thus, the control effort, u(k), is easily determined from algebraic linear equation. The difference between these two approximations is that the Taylor expansion for NARMA L1 is around (y(k), y(k-1), …, y(kn+1), u(k)=0, u(k-1)=0, …, u(k-n+1)=0) while the Taylor expansion for NARMA L2 is around the scalar u(k)=0. These two approximation models are given as NARMA L1 y ( k + d ) = f [ y ( k ), y ( k − 1),..., y ( k − n + 1)]
+
∑
n −1 i =1
gi [ y ( k ), y ( k − 1),..., y (k − n + 1)]u ( k − i )
Where f = F [( y ( k ), ..., y ( k − n + 1), 0, 0,..., 0] gi =
∂F ∂u(k − i) ( y(k ),..., y(k −n+1); u(k )=0,..., u(k −n+1)=0)
NARMA L2 y ( k + d ) = f [ y ( k ), y ( k − 1),..., y ( k − n + 1),
u ( k − 1),..., u ( k − n + 1)] + g[ y ( k ), y ( k − 1),..., y ( k − n + 1), u ( k − 1),..., u ( k − n + 1)] ⋅ u ( k ) Where
f = F [( y ( k ), ..., y ( k − n + 1), 0, u ( k − 1),..., u ( k − n + 1)]
g=
∂F ∂u(k ) ( y(k ),..., y(k −n+1); u(k )=0, u(k −1),..., u(k −n+1))
2.2 Identification of the NARMA-L2
The identification of the system to be controlled is the first step to construct the NARMA L2 neurocontroller. The neurocontroller can be trained offline in batch from using set of input-output data pairs to approximate the nonlinear function of f and g.
In NARMA L1 model, the f and g functions are only functions of the past values of the output, y, and the control effort, u, and its past values appear linearly in the equation. In contrast, in NARMA L2 model, the f and g functions are the functions of the past values of both the output, y, and control effort, u, and thus, the relation between the next value of output, y(k+d), and control effort at the current step, u(k), appears linearly. NARMA L2 is preferred to be a universal tracking controller since its realization is simpler compared to NARMA L1. NARMA L2 consists of only two nonlinear functions, f and g, which can be modeled using two neural subnetworks. Both functions have (2n-1) inputs which are the past values of output, y, and control effort, u. Once the nonlinear system is approximated and modeled as NARMA L2, the control effort, u(k), is designed such that it eliminates both f and g and also the output, y(k+d), is equal to the reference, yr(k+d). Since the relation between the control effort, u(k), and the output, y(k+d), is in the form of algebraic linear equation (not finite difference equation), the sequence of control effort, u(k), that give the system output, y(k+d), equal to the reference, yr(k+d), is easily determined. 2.1 Structure of NARMA L2 neurocontroller
Fig. 2 Identification of the NARMA-L2 2.3 NARMA-L2 Controller
Since the relation between the control effort and the output in the NARMA L2 model is linear, the control effort that gives the output equal to the position reference is easily determined. The control law for this controller is
u(k +1) =
yr (k + d) − f [ y(k),..., y(k − n +1), u(k),..., u(k − n + 1)] g[ y(k),..., y(k − n +1), u(k),..., u(k − n +1)]
Once the nonlinear function of f and g are modeled, the neurocontroller is then simply the rearrangement of these two subnetworks of the plant model such that the control law is obtained. The closed loop system can be viewed as shown in FIG. 3
Z-1
yt+1
Z-1 Z-1 Z-1
W b W W W b
+
+
W b
+
W b
+
Fig. 1 The structure of NARMA L2 neurocontroller
yrt+2
The NARMA L2 neurocontroller [FIG. 1] consists of two subnetworks, f and g. These two subnetworks are arranged such that the controller cancels not only the nonlinearity but also dynamics of the system. The resulting closed loop system becomes implicit algebraic model, i.e., y(k+d) = yr(k+d). This means that the system perfectly tracks a smooth trajectory.
Reference Trajectory
To construct the neurocontroller, the number of delayed plant inputs and outputs are chosen based on structural model. The size of the hidden layer is chosen such that the network can accurately approximate the nonlinearity of the system. There are two steps to construct the NARMA L2 controller: identification and control.
g()
ut+1
+ f()
Nonlinear Plant
Fig. 3 Closed Loop system 2.4 Stability
The closed loop system is stable if the plant is a minimum phase system and the relative degree is well defined. However, the proof of the controllability, observability, and stability is not within this paper.
If the sequence of control effort that gives the nonlinear system to follow a desired trajectory is known to exist and unique, this technique is capable of solving to find those sequence.
The Matlab® NARMA L2 blockset implemented on the Matlab® xPC realtime target PC is used to control the trajectory of the disk plant. The target PC is Pentium processor with 100 MB RAM with the AD 8133 and AD 726 cards installed. The drivers for all cards are developed such that the highest speed is obtained. The sampling is set to 0.005 second.
3. The Nonlinear Disk Plant 3.1 The Implementation of NARMA L2 Neurocontroller
The plant model 220 from Educational Control Products (ECP) [FIG. 4] is modified and then used to demonstrate the performance of NARMA L2 controller. The plant has several mechanisms for adjusting the load inertia, and gear ratio and can simulate several types of friction, backlash, and drive flexibility. The modified plant consists of two disks. The smaller disk is designed to be an input device to generate the position reference in real time while the bigger disk is the nonlinear plant. The driven motor is a brushless DC servo motor coupled with 4000 PPR encoder. The DSP controller that comes with the plant is replaced by The Matlab’s xPC real time controller, which is used to implement the neurocontroller and to control the system in real time.
y (t ) + f ( y (t )) − mgl sin(y (t ))=u (t ) This is a second order nonlinear plant which can be modeled as n-dimensional discrete time as
[
] [
]
y(k + 2)=f y(k + 1), y(k ) + g y(k + 1), y(k ) .u(k )
The control input to the plant is in the form of:
u(k +1) =
yr (k + d) − f [ y(k), y(k −1)] g[ y(k), y(k −1)]
Thus, the parameter of the NARMA L2 neurocontroller is chosen as The number of delayed plant inputs:0 The number of delayed plant outputs:2 The size of hidden layer: 5 The sampling interval: 0.005
Input (volts)
The bigger disk, which is used as the nonlinear plant, is installed with rubber to generate real dry friction and the brass mass is installed such that the gravity effect is nonlinear.
The model of the nonlinear plant can be written as
Pendulum Position (rad)
a) Mechanical Plant
b) Matlab® xPC Real time Controller Fig. 4 The ECP model 220 Disk Plant
Fig. 5 Training data pairs The NARMA L2 neurocontroller is trained offline using the 12000 data pairs [FIG. 5] collected from the operation
of the plant. The data pairs consist of random input signal and y is sampled at 100 Hz sampling frequency. Data pairs are divided into two parts; training data intended for training the NARMA-L2 model and validate the model. The training epoch is set to 1000 and the LevenbergMarquardt backpropagation technique [8] is used to train the network. Once the NARMA L2 neurocontroller is configured and trained, it is implemented on the Matlab’s xPC real time controller as described earlier. The sampling is set to 0.005 second.
4. Experimental Results In the first experiment, the trajectory is sine sweep internally generated by real time controller. The initial position of the system is not on the trajectory at the beginning. The NARMA L2 neurocontroller quickly bring the system to the trajectory but the chattering is observed. This is because the computed control effort to rapidly bring the system to the trajectory is quite high and exceeds the hardware limit. The control effort is then saturate at the maximum possible effort. However, the control effort decreases when the system is near the reference trajectory. The chattering is also decreasing. When the computed control effort is within the hardware limit, the neurocontroller successfully control the system to stay on the reference trajectory. The tracking response to the sine sweep is both in phase and magnitude ratio is one. The NARMA L2 is a linear approximation about u(k)=0 and gives a perfect response when the system is on the trajectory. At this time, the control effort, u, is around the zero value.
controller when controlling a nonlinear system in the trajectory following mode [FIG. 6]. In order to suppress the chattering, Pukrittayakame et al. [9] suggested an adjunction of a linear feedback to the NARMA-L2 controller to smooth the control action. The addition of a linear feedback to the NARMA-L2 controller is equivalent to the feedback linearization control methodology. This technique installs the dynamic behavior back to the closed loop system. The drawback is that the position reference should be corrected in order to give a perfect tracking. In the second experiment, the reference trajectory is generated in real time using input disk. The input disk is manually rotated and thus the reference trajectory is arbitrary. This experiment demonstrates that NARMA L2 neurocontroller is capable of control the system to track the input reference [FIG. 7]. Again, the tracking response is both in phase and true magnitude. The NARMA L2 neurocontroller is an outstanding candidate for trajectory controller and obviously outperform the classical PID controller. Tracking Response Reference Trajectory Pendulum Trajectory using NARMA L2
3 2 1 0 -1 -2
Pendulum Trajectory using PID Tracking error, NARMA L2
-3 0
1
2
3
4
5
6
7
8
9
10
Time (s)
Fig. 7 Tracking to an arbitary input reference
5. Conclusion
Fig. 6 Tracking to sine sweep reference The tracking response of the classical PID controller demonstrates that the PID is not an efficient trajectory controller. The phase of the tracking response lags the reference trajectory and there is also an error in the magnitude that depends on the frequency of the reference. The NARMA L2 obviously outperforms the PID
The NARMA L2 neurocontroller is a discrete-time controller that has internal feedback of past control effort and output, designed to eliminate not only the nonlinearity but also the dynamic behavior of a nonlinear system. It is able to transform a nonlinear dynamical system into an implicit algebraic model and thus easily but efficiently control the trajectory of the system. The control effort is computed based on the position reference directly and its derivatives are not required. In this project, the NARMA L2 neurocontroller successfully implemented to control the real nonlinear
mechanical system, the pendulum system. The results demonstrate that the NARMA L2 neurocontroller perfectly control the nonlinear system to follow the smooth desired trajectory, even though the trajectory is generated in real time. This technique effectively integrates the discrete-time model and the discrete-time controller techniques such that signal conversion is minimized and thus also error. Derivatives of signals are also effectively managed. This results in an effective but simple trajectory controller.
Acknowledgement This research is sponsor by the Chulalongkorn university century academic development project.
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