The present work addresses the utilization of Artificial Neural Networks (NN) for the identification and control of systems, in special to control nonlinear dynamic systems or systems with some degree of uncertainty. Because NNs have an inherent ability to approximate functions and to adapt to changes in input and parameters, they can be used to control systems too complex for linear controllers, such as PID controllers. In the present work a mathematical basis for NN is presented, the mathematical representation of a process unit, or neuron, and how they can be put together in order to form nets that can learn from external data. In sequence, it is presented structures of inputs that can be used along with NN to model nonlinear systems. The most common configurations of input vectors for the training of NN are highlighted. Following, a method of control is presented that take advantage of NN, where a NN is used to build a predictive nonlinear controller using a model predictive control (MPC) structure. Two nonlinear systems were used to test the identification and control of the structures proposed. The results shows the NN used were efficient in modeling and controlling the nonlinear plants.
The modern feedback controlling systems are responsible
for the success of several operational systems and are
applied in the military, aerospace, manufacturing industry
and other fields
The operation of a complex system requires the
controller to be smart in a way, to adapt and learn from changes in
the system dynamics, noise or external output. A solution
for the control of complex system is to use control
structures inspired in biological systems. Biological systems are
adaptive and resilient to the environmental changes where
they are inserted. Bacteria constantly change their DNA
sequencing so that they remain unknown to the defense
systems of other creatures. Most animals have a neural
system that allows than to sense the environment and to
rationalize a best course of action, such as when to run
from or fight a predator, or in the case of humans, how to
solve a mathematical equation. Fuzzy logic, Evolutive
Algorithms and Artificial Neural Networks (NN) are among
the theories developed with an inspiration in biological
systems. Neural networks tries to mimic the biological
neural system, it presents an inherent capacity for learning,
adapt and parallel computing
Most NN applications are designed in an open loop,
such as designs for pattern recognitions
The present work investigates the application of NN for identification and control of systems. For the identification process, the NN is placed in parallel with the model and random step signals are generated for input. The plant’s response to the signals can be used for training the NNs. The trained NN are then used in a predictive control structure. Matlab was used to implement the NNs, plants and input signals. The goal of this work is to investigate if a system can automatically experiment with a plant, learn from the experiment and control the plant, automatically and without needing a mathematical model of the plant or a fine tune the of the controller(Nørgård, Ravn, Poulsen, & Hansen, 2000).
The processing unit of a NN is a neuron. The mathematical model of an artificial neuron tries to mimic the behavior of a biological neuron. An artificial NN is based on the approximation models of how a biological neuron processes the electric impulses it receives from other neurons or external stimuli. The model used in the present work is the perceptron of Rosenblatt (Rosenblatt, 1958). Figure 1 shows the schematic for this model of neuron and eq. 1 shows the model of a single neuron. wi0 f(.) yi where is the output of neuron , is the -th input, is the weigh given to input when it is going to neuron and 0 is the bias of the neuron. Activation function of the neuron (. ) can have several forms, such as sigmoid, linear, step or a radial basis function (S. S. Haykin, Haykin, Haykin, & Haykin, 2009).
Networks of neurons can be built by aligning neurons in single layers and by grouping the layers, forming a multilayer network. Figure 2 shows a NN with two layers of neurons.
Where are the inputs of the network, is the number of neuron in the first layer, is the weigh from -th input to -th neuron.
is the number of neuron on the secx1 x2 xn wi1 wi1 wi1 =1 u(t d ) u(t d 1) u(tdm)
y ( t ) (a) er. 0 is the bias of the -th neuron and 0 if the bias of the -th neuron.
Multilayer structures of NN, as the one shown in figure 2, are universal approximators, meaning that they can approximate or model any input pattern (Hornik, Stinchcombe, & White, 1989). This universal approximation feature makes NN feasible for modeling non-linear dynamics systems. By changing the NN input arrangements, it is possible to include temporal information about the system to be modeled. Among other input arrangement, the input structure used to model linear systems can be highlighted, such as the linear models of finite input response (FIR) and the autoregressive with exogenous input (ARX)(Nørgård et al., 2000). Figures 3 (a) and (b) shows the use of NN with input arrangement of a FIR and ARX model, respectively. plant and a series step signals, with normally distributed and output response of the plant are then arranged as an input to the NN. With the appropriate inputs, the NN are trained to best mimic the output of the plant. A validation set of input signals and plant output signals are used to test the NN on the ability to represent the plant for signals not previously used in the training. Matlab was used to implement the training inputs. Figure 4 illustrates an input signal used for identification, where the lower and upper bound of the signal is 0 and 1, and the steps last for 20 samples. performance in comparison with other methods (Moré, 1978).
Two strategies can be used for controlling dynamic
systems: feedback control and optimization. The feedback
control loop strategy includes using the controller and the
plant in the same control loop, in a way that the entire
system can described as a transfer function. Using
optimization to control systems includes defining the system as an
objective problem where the independent variables are the
inputs to the plant. For this work the optimization strategy
is used and the control structure follows that of a model
prediction control (MPC), as initially presented by Clarke
et. al.
In predictive control, the control problem is transformed into an optimization problem where the goal is to minimize the error between reference and output as well as the variability of control input at each interaction. The objective function presented in eq. 2 is minimized model, where ̂ ( ) = [ ̂( + 1), ̂( + 1 + 1), … , ̂( + (2) was also used to test the NN MPC controller. Figure 8 ( ) =
( ) √0,1 + 0,9 ( )2 (3) shows the schematics of the tank and eq. 4 illustrates the equation that models the tank ℎ( ) = 1( ) + 2( ) − 0.2√ℎ( ) ( ) = ( 1 − ( )) ℎ1(()) + ( 2 − ( )) ℎ2(( )) − (4) 1 ( ) (1+ 2 ( ))2 where ℎ( ) is the level of liquid in the tank, ( ) is the concentration of the output product, 1( ) is the flow of concentrated 1, and 2 is the flow of solvent 2. In this work, the concentration 1 and 2 is 24.9 and 0.1, respectively, following the work of Nørgård et. al. (Nørgård et al., 2000) for the same plant. The goal is to control the output concentration ( ) by varying the flow 1( ). The flow 2( ) is left at a constant rate of 0.1. Both plants are single input, single output models. In order to control the plants the first step was to model the plant using NN.
To identify the dynamics of the nonlinear valve, an input signal was generate consisting of 6000 samples, with amplitude varying from 0 to 1 at every 20 samples.
It was assumed that a NN with two layers would be sufficient to model the nonlinear dynamic of the valve. The first layer had 15 neurons and the second layer has one neuron. The second layer serves as a summation of outputs from the first layer and due that it has a linear activation function, while the first layer has a hyperbolic tangent activation function. The input structure ( ) used for the NN followed an ARX structure with delayed inputs and delayed outputs as illustrated in eq. 5 ( − 1)
⋮ ( − ) ( ) = ( − ) (5) ( − − 1)
⋮ [ ( − − + 1)] where =3, =8 and = 2.
The training performance of the NN is show in figure 9, where it can be seen that the validation performance based on the Mean Squared Error (MSE) is negligible. Figure 10 illustrates the plant output and the NN output for the same set of inputs. Notice that the NN is capable of closely represent the valves dynamic.
Using the NN model of the plant in the MPC control loop it was possible to control the output of the plant. The optimization method used is a classic levenberg-marquardt. The minimum, maximum and control prediction horizon are: 1 = 1, 2 = 7 and = 1. The penalty for signal control variation is = 10. Figure 11 shows the inputs and outputs of the system, using a MPC controller and the NN as a model for prediction. The red dotted line is the reference of the system (r), the light line is the control signal of the plant (u) and the bold line is the output of the plant.
It was assumed that a NN with two layers would be sufficient to model the nonlinear dynamic of the tank. The first layer had 12 neurons and the second layer had just one neuron. The input structure ( ) used for the NN followed an ARX structure with delayed inputs, delayed outputs and following the format in eq. 5, with = 4, =5 and = 1.
The training performance of the NN in modeling the reaction tank is shown in figure 12, where it can be seen that MSE is negligible. Figure 13 illustrates the plant’s output and NN prediction for the same set of inputs, it can be seen that both signals overlap, suggesting that the plant model the plant’s dynamic efficiently. It can be seen that despite the non-linearity of the plant, the controller was able to efficiently control the plant, with a rapid response time.
To identify the dynamics of the nonlinear reaction tank, an input signal was generate consisting of 4000 samples, with amplitude varying from 0 to 5 at every 20 samples.
The NN model was used in a MPC structure to control the reaction tank plant. The method of levenbergmarquardt was used for optimization in the MPC structure. The minimum, maximum and control prediction horizon are: 1 = 1, 2 = 7 and = 2. The penalty for signal control variation is = 0.05. Figure 14 illustrates the inputs and outputs of the system, where the red dotted line is the reference of the system (r), the light line is the control signal of the plant (u) and the bold line is the output of the plant (y).
) y & r( t22 u p t u O d n a e c n e fr e e R
Despite of the non-linearity of the plant, the controller was able to execute control of the plant. However, it can be seen that, for lower reference levels, the controller is not able to stabilize the output of the plant around that reference value. This can be due the fact that the NN did not model the dynamics of the plant for such lower reference levels. The training set used for identification of the plant should have included more data in the lower reference levels. The NN do not correctly represent the plant for such low levels of reference, therefore it predicts erroneous plant’s output. That makes the optimizer to optimize an objective function that is not representative of the plant, causing the marginal control performance in the lower reference levels.
For higher reference levels, the controller was able to control the concentration in the reaction tank. To maintain the steady levels of the plant’s output, the control signal constantly changes in the time interval from 40s to 80s. This illustrates the optimizer trying to compensate for future changes in the output and adjusting the control signal ahead of time, so that the future output would follow the reference input. In this work, the performances of the NN in a predictive control structure are in close agreement with the work of Nørgård et. al. (Nørgård et al., 2000).
In this paper, multilayer, feedforward neural networks were used to identify the dynamics of two nonlinear plants, a valve and a reaction tank. Random step signals were used as input in the identification of the plants, while their responses were recorded. The set of inputs and outputs were used to train the NNs. Despite the simplicity of the NNs used, the models proved satisfactory to represent the plants for the range of inputs used in training.
The NNs of the plants were used in a control loop with a MPC structure. Given a set of control inputs, the NN were used to provide predictions of plants outputs. The output predictions are used to calculate the error from a desired reference signal. A levenberg-marquardt optimization method was used to optimize the control inputs in order to minimize the plant’s output error.
For both plants in this work, the NN proved to be efficient in modeling the non-linearity of the plant. Additionally, the use of NN models in a MPC structure made possible the control of the nonlinear plants, where the controller would compensate for the plant’s nonlinearities. The controller had a marginal performance in controlling the reaction tank for low reference levels. What is explained by the fact that the NN did not capture the dynamics of the plant for those levels. During the training of the NN for modeling the reaction tank, more low levels of reference should be the used so that the NN could have more information about the dynamics of the plant in those levels and therefore build a more accurate model of the plant.
The use of NN in the control of systems makes it possible for the control of nonlinear systems, black box systems and system with changing dynamics. The same methods used in this work for identification and control of systems can be extended to the identification and control of other complex systems.
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