The gateway network selects one of the experts if appropriate, or defaults back to case-based reasoning. The utility of the hybrid algorithm is demonstrated on the problem of nonlinear control, regulation of the set value in the ball and beam system. The contribution of this thesis is to formalize the synthesis of the above decision processes using hybrid case-based reasoning and a neural network algorithm and apply it to a nonlinear control problem.
In the implementation of the case-based module, experiences are stored as vectors of variables known as cases in a set called a caseba8e. A second network (known as a gate) is simultaneously trained to select one of the experts when appropriate or otherwise revert to case-based reasoning. Chapter four describes the ball and beam control problem and defines the actual implementation of the hybrid system.
Chapter 2 Background
- Case-based Reasoning
- Mathematical Analysis of 1D Function Approximation
- Neural Networks
- Basic Computational Elements
- Feedforward Networks
- Competitive Networks (Winner-Take-All)
- Mixture of Experts Networks
- Learning Models
- Control Theory
- System Identification
- Linear System Analysis
- Nonlinear System Analysis
- Neural Network Control
- Related Work
The last equation states that the maximum error is proportional to the size of the input interval. Signals are encoded by the sequential timing of impulses (frequency) along with loss based on the location of the synapse relative to the soma (amplitude) of the receiving neuron. The output of this element is a measure of the distance (norm) between the input and the center (mean) of the base function.
The index of the neuron with the minimum distance, i.e., the "winner", is the output value of the network. This controller model uses a neural network to approximate the inverse input-output map of the system [37] and has been applied to many control problems including robot control [38]. The inputs to the neural network of the inverse model are the actual outputs of the system.
The desired output of the neural network is the actual applied input to the system. The desired output of the model network is the corresponding output of the installation.
Chapter 3 Hybrid Reasoning System
Case-based Reasoning Model
- Case Structure
- Input Acquisition
- Case Selection
- Case Modification
- Case Application
- Result Evaluation
The resulting output is analyzed by the output evaluation module to determine if some type of unusual or anomalous event occurred during the execution of the input. The starting point of any reasoning system involves obtaining data that describe the current state of the system. For any algorithm to work, the input data must sufficiently describe the current state of the system.
By compiling these two pieces of information into the desired outcome vector, Y d , a decision can be made about what action is necessary to achieve the desired outcome given the current state. Once the desired outcome vector is constructed, the second step in the reasoning process involves selecting the base case, denoted C*, that represents the past experience that is most similar to the current situation. Since recent instances represent examples of the input-output map at the present time, they may be more suitable for determining an appropriate action than distant past experiences, even if the most recent output vector is not the "best " in weighted l2. meaning norm.
Sometimes due to natural symmetry within the system, additional cases can be inferred from the existing ones. Therefore, the system can directly implement the base case (u = u*) thereby recognizing that significant error may exist in the output, or modify the input vector from the base case in an attempt to produce a better output. A common method of modifying the base case is to use first-order gradient information within a local neighborhood.
By numerically approximating the gradient using cases close to the base case, a first-order linear correction factor can be calculated. Depending on the dimensionality of the system and the density of instances in the local neighborhood, higher-order corrections can also be calculated. After changing the inputs from the base case, the final input, u, is then applied to the system.
However, abnormal cases are often stored in a separate case collection so that modified cases can be compared against them to prevent the system from making a similar error.
3.1. 7 Casebase Augmentation
- Neural Network Generalization
- Hybrid Algorithm
- ID Case-based Reasoning Example
- Input Acquisition
- Case Selection
- Case Modification
- Case Application
- Result Evaluation
- Casebase Augmentation
- Chapter 4 Experimental Setup
- Descript io n of Experimental Systems
- Case-based Reasoning System
The purpose of this work is to combine a discrete variation of the mixture of experts technique, known as a gated expert network, with case-based reasoning. The center of each region, i.e. the mean value of the corresponding gate network neuron, is denoted by p,. The algorithm used in this work also delimits the areas of expertise of the expert networks to a local area where there is data by setting a threshold, denoted by .
This delimitation allows the gate to set the decision to the case-based reasoning module if the desired result is not within the expert areas of any of the expert networks. The purpose of the control is to rotate the beam in such a way that the position of the ball follows a desired trajectory. Given the moment of inertia of the beam Jb; the mass, radius and moment of inertia of the ball M, R and J respectively; and the acceleration due to gravity G;.
Thus, the actual dynamics of the servo beam when transitioning from one beam angle to another is hidden from the controller. A computer simulation calculates the dynamics of the above equations using the fourth-order Runge-Kutta method. Constraints are placed on the position of the ball, which takes into account that the ball reaches the end of the beam.
This setup allows for accurate prediction of the performance of the hybrid control algorithm when applied to the physical system. The first mode is a full-state mode in which the reasoning algorithm is provided with the exact position and velocity of the ball at each update time. One of the beam's rods is made of threaded nylon wrapped with resistive Ni-chromium wire giving an end-to-end resistance of about 100 ohms.
The details of the implemented case-based control algorithm will now be presented using the structure presented in the third chapter.
- Input Acquisition Procedure
- Case Selection Procedure
- Case Modification Procedure
- Case Application Procedure
- Result Evaluation Procedure
- Case base Augmentation Procedure
- Neural Network Generalization Procedure
- Chapter 5 Results
- Initial Cases
- Analytic Control- Hand Tuned PD Controller
- Full State Simulation
- Position Only Simulation
- Noisy Si m ulation
- Physical Hardware System
- Neural Network Generalization
The initial state variables of the system and the resulting output variables are stored in a vector, y, defined as . Additionally, the transition velocity, Vrn, is modified based on the terminal velocity of the base case, the desired terminal velocity, and the ratio of the second beam angles with. A neuron (and expert) is assigned for each distinct combination of position change and starting position. The center of each neuron, denoted by J1-{i), represents the mean value of the class corresponding to that particular combination.
A threshold, denoted by t, is set for the output of the gate network neurons to ensure that the areas of expertise are bounded and mutually exclusive, i.e. produces distinct classes for all inputs. As desired outcome vectors, Yd, are subsequently presented to the gate network, the outputs of the gate neurons are compared to the threshold. If the output of a neuron is less than the threshold (meaning that the desired result vector is within the corresponding expert's area of expertise), then the final output of the hybrid algorithm, u, is the output of that expert given the desired result vector.
Finally, unless otherwise stated, the desired final velocity is assumed to be zero (Vd = 0), i.e. the ball is stopped at the set point. From the final state of the first case, (yj1), v?»), the second case tries to return the ball to Yo through beam angles of about one quarter maximum, again with final velocity Vd. The final transition velocity is calculated based on the final velocity after the completion of the previous case (refer to Section 4.2.4) as.
It should also be noted that the speed at the completion of almost all transitions was negligible, i.e. the algorithm was able to stop the ball in the final position. To demonstrate the flexibility of the algorithm, a second simulation was performed with a different terminal velocity (Vd = 0.05). Therefore, velocity is calculated using the slope of a linear regression fit to a moving window of the last fifty position readings (250 ms).
The algorithm attempts to filter out some of the noise from the position data using a moving average window of the last five data values (25 ms). Figure 5.7 shows plots of the effective values for these errors as a function of noise level. Based on the classification of the trained gate network, the case base was segmented into six individual subcase bases.
5. 8 Hybrid System
Chapter 6 Discussion and Conclusions
- Discussion of Results
- Advantages and Disadvantages of the Hybrid Algorithm
- Further Extensions
The applicability of the hybrid algorithm is demonstrated on a nonlinear control problem, set point regulation in a ball and beam system. The algorithm performs input gathering to gather relevant information about the current state of the system to form the desired result vector. Result evaluation then determines whether something abnormal happened during the application of the action; and if not, casebase augmentation updates the casebase with the new experience.
The trained gating network selects an expert if the desired result lies within one of the expert's regions of expertise, otherwise it selects case-based reasoning to generate the hybrid algorithm output. The algorithm was still able to achieve the desired results even with noise levels greater than those present in a raw physical system. The most obvious advantage of the hybrid algorithm is the elimination of the requirement for system identification.
As long as a functional form of the inputs is provided, the reasoning algorithm requires no additional knowledge of the system dynamics. The disadvantages of hybrid reasoning are of the same nature as those of most model-free adaptive systems. Likewise, neural network experts only provide a local approximation to the inverse input-output map, which does not provide any information about the dynamics of the underlying system.
This limitation makes it difficult to ensure robustness and performance, limiting the algorithm's usefulness for critical systems. An extension would be to have the algorithm select the entire input sequence, but then possibly adjust the second beam angle based on the state of the system at the transition point. This extension would improve learning time because the algorithm would be able to dynamically adjust both parts of the input sequence independently rather than having to wait for another similar condition to arise.
This procedure was performed for simplicity in both the simulation code and for real-time performance evaluation of the algorithm.
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