Fuzzy Automaton with Kalman State-Smoothing

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1 Fuzzy Automaton with Kalman State-Smoothing Scott Imhoff and Palak Thakkar Raytheon Intelligence and Information Systems E. Centretech Parkway Aurora Colorado Abstract - This paper presents a fuzzy automaton/kalman filter hybrid algorithm (Kalmaton). Automata are of pivotal importance in artificial intelligence (AI), being used in knowledge engines such as Wolfram Alpha. Also they are applicable to emerging needs for predictive situational awareness involving data mining and data fusion. Kalman filters have many applications which need predictive validity, including GPS. Fuzzy transitions from different learning states associated with fuzzy automata along with estimation of the true state of a system using a Kalman filter creates an algorithm that achieves good stability and reaches a decision boundary early. Keywords: Kalman filtering, automaton, fuzzy logic, change detection, state smoothing, machine learning. 1 Introduction Formal Languages provide an advantageous framework in applications where the essential behavior of a system is language-like. A very approachable treatment of Formal Languages is given by Revesz [1]. An automaton is a kind of language receiver. Automata are not limited to spoken languages but can be used with nearly any sequence of symbols. As an automaton interacts with its environment receiving symbols, it undergoes transitions of its internal state. The internal state of the automaton determines, in part, what the output (response) of the automaton is. Automata learn from the environment what their internal state should be. If you think of an automaton as a machine, you may look inside the machine to see what it is made of. If you run the automaton, and let it interact with the environment, when you open the box again and look you will typically see a different machine. It is often advantageous to use a fuzzy automaton. When I say yester, you go partly into a state where you are ready to hear day. But this is not the whole story. You also go partly (slightly) into a state where you are ready to hear year because yesteryear is also a word you might expect to hear occasionally. Fuzzy automata allow you to work with such situations. They allow your state membership to straddle two or more states. In addition to fuzzy states, fuzzy automata also have fuzzy transitions and fuzzy output mappings (responses). Klir and Yuan provide an excellent introduction to fuzzy automata [2]. Fuzzy automata are able to learn from actions produced in the environment by making fuzzy transitions from learning state to learning state until an admissible response is achieved. This response corresponds to reaching a decision boundary for labelling in active learning. These decisions about what has been detected or acquired (the label) inform the user of what is present in the environment. A pernicious problem with fuzzy automata is the tendency in applications for the internal state to be unstable. The presently discussed new algorithm addresses this problem by integrating components of a Kalman filter to smooth the internal state of the fuzzy automaton. Sometimes the state of a system cannot be known perfectly. For example, in GPS it is not possible to know the position and velocity perfectly owing to noise and environmental effects. A Kalman filter is used to smooth the state vector (position, velocity, etc.) to provide an optimal approximation to the true state. A Kalman filter is an algorithm for estimating the true state of a system at iteration k by using previous estimates and measured data. State estimates are formed using the previous state estimate and measured data as well other variables including Kalman Gain and covariance. An engineering approach to Kalman Filtering, with a MATLAB disk, is provided by Grewal and Andrews [3] while a mathematical discussion is presented in a Dover book by Stengel [4]. The present paper presents a new algorithm which addresses the stability problem of the fuzzy automaton by smoothing its internal state using a Kalman filter. An example application is change detection. An example of the application of Kalman state-smoothing of a fuzzy automaton is presented in Figure 1. The inputs and outputs of the fuzzy automaton are shown. In Figure 1, in the top of the plot, an input signal over two channels (for example, I and Q) has a hidden, low entropy, signal in one of the channels (the solid line channel). The output (automaton response) is three signals that appear in the lower half of the diagram. (Think of them as three voters.). Here the input actions (the -. and black curves) produce output ( -,., and + ). A smooth, early arrived at, change detection is demonstrated.

2 Example decision boundaries could be the horizontal lines at y =.6. The sequence of activities of the algorithm at each iteration are as follows: 1) An input from the environment is aggregated (minmax) with the matrix of fuzzy transitions to form fuzzy intermediate matrix. (1) 2) is aggregated with and the result of this aggregation is called measurement. Where the operator is defined as: (2) Figure 1: Smooth, early change detection using fuzzy automaton with Kalman state-smoothing. The output signals rise in response to the hidden signal in the solid component of the input. 2 Description of the Algorithm This paper presents a fuzzy automaton/kalman filter hybrid algorithm ( Kalmaton ). The algorithm is an iterative predictor device which steps through a sequence of steps from to, taking input at each step and producing output at each step. The elements that occur together in this algorithm are: 1) An input matrix of actions from the environment external to the device. 2) A matrix of outputs returned from the device and presented to the user. 3) An estimate internal state of the system which is updated every iteration,. 4) A three-dimensional matrix of fuzzy state transitions. 5) An aggregate matrix which is derived from the input and the matrix of fuzzy state transitions at each iteration k. 6) A Kalman Gain matrix which is updated each. 7) A symmetric covariance matrix which is updated each using a congruence transform. 8) A transition matrix. 9) A fuzzy response matrix. 10) An observation matrix. 11) An internal measurement. 12) A white noise covariance matrix. 3) The Kalman gain,, is updated using the covariance,, observation matrix,, and white noise covariance,. 4) The covariance is updated to become by using: 5) The state estimate update,, is then formed by using the Kalman weighted average with serving as the measurement. 6) Finally, is aggregated with response matrix to form the output. A point of novelty is that the fuzzy intermediate matrix,, is aggregated with the state estimate from the Kalman filter,. This aggregation amalgamates the state of the automaton with the Kalman filter state so that the state can be smoothed in order to increase the stability. A second point of novelty is that the aggregate,, is used as the input to the Kalman state update operation. This allows an innovation process that smoothes the output, making it easier to place decision boundaries. A third point of novelty is that the output,, to the user is formed by aggregating the state estimate with the fuzzy response matrix. This allows more flexibility to make the output in a format that is conducive to setting decision boundaries. (3) (4) (5) (6) (7)

3 The Kalman filter is integrated with the fuzzy automaton as a hybrid algorithm, rather than being a Kalman filter and automaton in series or parallel. The block diagram of the Fuzzy Automaton with Kalman State-Smoothing is presented in Figure 2. The variables are identified in Table 1. Lower-case denotes a vector while upper-case denotes a matrix. Table 1: List of Variables Variable 3 Application of Algorithm Description Iteration Number Estimate State Input from Environment Output to User Fuzzy Intermediate Matrix Kalman Gain Matrix of Fuzzy Transitions State Covariance Response Matrix Measurement of State The algorithm described by the diagram in Figure 2 was demonstrated by coding it as a MATLAB script. That script was executed on an HP EliteBook notebook computer, generating the plot appearing in Figure 1. The plot shows a 20x2 matrix input actions,, represented by the.- and solid curves, and a 20x3 output matrix,, which is represented by the dot, dash, and plus curves. The meaning of the plot is that there is a smooth response to very noisy input actions while an early change detection capability is demonstrated as the output signals rise at about time k = 14. For the reduction to practice the authors used the following state transition matrix : The white noise covariance that was used is given by the following: The observation matrix,, that was used is: The fuzzy response,, that was used is: (9) (10) (11) (8) (12) The input actions,, were given by: The fuzzy state transition used is a 4x4x2 matrix given by the following:

4 Figure 2: Fuzzy Automaton with Kalman State-Smoothing block diagram.

5 (13) The initial state of the system was: (14) As a basis of comparison, the Kalman filter was extricated from the algorithm and the same experiment was run. As can be seen in the example in Figure 3, the automaton broke down and failed to respond to the hidden signal. Figure 4. Average of 20 sets of automata show that utomaton breakdown is typical for the input signal of Figure 3. Figure 5. Average Kalmaton output shows response to hidden signal is typical. Figure 3. Automaton breaks down and fails to respond to the hidden signal. The average of 20 Automaton responses shows that the automaton breakdown is typical. See Figure 4. Here, the same input generator was used. The average of output responses of the Kalmaton for the same input signal generator is shown in Figure 5. The response to the hidden signal is evident. 4 Summary A Kalman filter can be incorporated into a fuzzy automaton and used to smooth the internal state. The resulting Kalmaton can produce a stable automaton response while retaining the ability to perform an effective change detection. 5 References [1] Gyorgy E. Revesz, Introduction to Formal Languages, Dover, [2] George J. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall, 1995.

6 [3] Mohinder S. Grewal and Angus P. Andrews, Kalman Filtering: Theory and Practice Using MATLAB, 3 rd Edition, Wiley, [4] Robert F. Stengel, Optimal Control and Estimation, Dover, Appendix: MATLAB Code % % Scott Imhoff, PhD 15 June 2011 % Palak Thakkar % M = 20; Actions = rand(m,2) for k = M/2+1:M Actions(k,1) = sin(2*k/m); Actions(k,2) = Actions(k-M/2,2)^2; end figure plot(1:m,actions(:,1)+1,'k') hold on plot(1:m,actions(:,2)+1,'k-.') I = [ ; ; ; ]; phi = rand(4,4); phi(2,1) = phi(1,2); phi(3,1) = phi(1,3); phi(4,1) = phi(1,4); phi(3,2) = phi(2,3); phi(4,2) = phi(2,4); phi(4,3) = phi(3,4); P = rand(4,4) B_buffer = zeros(m,3); for k = 1:M end A = Actions(k,:); [SA] = maxmin(a,s); z = aggregate(x,sa)'; x = phi*x; P = phi*p*phi'; % Update K = P*H'*inv(H*P*H' + Rho); % Kalman gain P = (I - K*H)*P; % Update covariance x = x + K*(z - H*x); % Update state [B] = aggregate(x,r); B_buffer(k,:) = B(:); plot(1:m,b_buffer(:,1),'k+') plot(1:m,b_buffer(:,2),'k.') plot(1:m,b_buffer(:,3),'k-.') Rho = rand(4,4) H = rand(4,4) R = rand(4,3) S = zeros([4 4 2]); S(:,:,1) = rand(4,4); S(:,:,2) = rand(4,4); C = rand(1,4) x = rand(4,1)