Proceedings of the 4th International Conference on Autonomous Robots and Agents, Feb 10-12, 2009, Wellington, New Zealand
Emotional Control of Inverted Pendulum System A soft switching from Imitative to emotional learning
Mehrsan Javan-Roshtkhari, Arash Arami and Caro Lucas Control and Intelligent Processing Center of Excellence School of ECE, University of Tehran Tehran, Iran
[email protected],
[email protected],
[email protected] Abstract— Model-free control of unidentified systems with unstable equilibriums results in serious problems. In order to surmount these difficulties, firstly an existing model-based controller is used as a mentor for emotional-learning controller. This learning phase prepares the controller to behave like the mentor, while prevents any instability. Next, the controller is softly switched from model based to emotional one, using a FIS1. Also the emotional stress is softly switched from the mentorimitator output difference to the combination of objectives generated by a FIS which attentionally modulated stresses. For evaluating the proposed model free controller, a laboratorial inverted pendulum2 is employed. Keywords- BELBIC, Imitative learning, Fuzzy inference system, model free control, Inverted pendulum system
I.
INTRODUCTION
Imitation is a powerful mechanism for knowledge sharing, particularly for intelligent agents. Imitative learning is an approach to artificial intelligence to transfer knowledge from an expert agent to another agent without any brain to brain transfers [1]. The imitation accelerates the learning speed and performance of the learner. In learning by imitation, an agent who wants to learn (imitator) tries to achieve the same result which mentor has been gained. It is different from mimicking, because the imitator does not act like mentor, it only performs an action which tends to the same results in environment. In many tasks in which there is an expert agent and new agent come to learn the proper action for doing the task, the imitative learning is employed. Imitative learning is widely used in robotics [2-5]. Development of new algorithms based on biological systems is an area of interest. The most important aspect of any intelligent system is its’ capability of learning. Although the learning process can be done in various ways, the main aim is adaptation of parameters to improve the performance of the system and come over the changes in the environment [6]. As the emotional behavior of humans and other animals is an important part of their intelligence, modeling this process leads to have an intelligent system with fast learning ability [7]. Although evolution mechanism codes emotional reactions in 1 Fuzzy Inference Systems 2 The Digital Pendulum Control System, crane system, manufactured by Feedback Instruments Limited, England
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animals, the mammalian can learn them very fast. In biological system, emotional reactions are utilized for fast decision making in complex environments or emergency situations. The main part of mammalians’ brain which is responsible for emotional processes is called the limbic system. Several attempts have been made to model the limbic system [8, 9]. The computational models of Amygdala and Orbitofrontal cortex which are the main parts of limbic system in the brain were first introduced in [10]. Consequently, based on works in [10], brain emotional learning based intelligent controller (BELBIC) which is an intelligent controller introduced in [11]. The fast learning ability of BELBIC makes it a powerful model free controller for many tasks. BELBIC is applied on several applications such as control of intelligent washing machines [12, 13], in [14] a modified version of BELBIC is employed for controlling heating, ventilating and air conditioning (HVAC) systems. Moreover, the BELBIC is used in time series prediction [15] and sensor-data fusion [16]. The real-time implementation of the BELBIC for interior permanent magnet synchronous motor (IPMSM) drives was first introduced in [17]. The controller was successfully implemented real-time by using a digital signal processor board for a laboratory 1-hp IPMSM and the results show fast response, simple implementation, robustness with respects to uncertainties such as manufacturing imperfections and good disturbance rejection. Another real-time implementation of BELBIC in position tracking and swing damping of laboratorial overhead crane in computer control via MATLAB external mode is described in [18]. The stability of brain emotional learning (BEL) system which is used in control as BELBIC was discussed in [19], also in order to ensure stability of the system, a general idea for choosing control parameters is described [19]. Also nonlinear combinations of objectives are used to design emotional stresses for BELBIC to control an overhead crane under uncertainties and disturbances [20]. The main drawback of model free controllers with learning ability -without any prior knowledge of the system’s dynamicssuch as reinforcement learning based controllers and BELBIC is that in early stages of learning process, they may cause the low performance, due to producing wrong control signal. This preliminary phase of learning can result in instability in some cases. After this first period of learning, if no instability occurs, the controller can learn the proper control signals to improve performance gradually. Although BELBIC shows fast learning ability, it has the same problem, but in a shorter period of time. If the system is inherently unstable, applying these controllers
Proceedings of the 4th International Conference on Autonomous Robots and Agents, Feb 10-12, 2009, Wellington, New Zealand
may cause the system become unstable, and the process must be stopped in order to prevent damages. So, BELBIC cannot be applied on such systems. To solve this problem, another approach is introduced. In this paper we employed BELBIC to control an inverted pendulum. As the pendulum angle is very sensitive to the control signal and any wrong changes in control signal makes the system oscillates and the pendulum falls down. So, if we use BELBIC as a complete model free controller that learn from scratch individually the learning phase will be impossible, and the pendulum will fall down many times. For solving the problem and accelerating the learning phase, a new approach is used. First, BELBIC learns from a classical simple controller of the system imitatively. The classical controller can be a simple one that only stabilizes the system, regardless of good performance and robustness. Then the output of BELBIC is gradually applied to the system and it replaces the initial controller. The important part in switching between controllers is changing the emotional signal of BELBIC due to change in objective. When BELBIC learns to imitate behavior of the initial controller, the objective is reducing the error between controllers output, and when BELBIC replaces the initial controller the objective is reduction of tracking and angle error. This paper organized as follows: Section 2 briefly introduces BELBIC, in Section 3 a description of the proposed controller is demonstrated, section 4 discusses the simulation results and finally section 5 concludes the paper. II.
Figure 1. Structure of BELBIC [10]
As the thalamus must provide a fast response to stimuli, in this model the maximum signal, over all sensory inputs, S, is sent directly to the Amygdala as another input (Eq. 1). Unlike other inputs to the Amygdala, the thalamic input is not projected into the Orbitofrontal cortex, so it cannot be inhibited by itself.
S th = max( S i )
(1)
In the Amygdala, each A node has a plastic connection weight V . The sensory input is multiplied by the weight and forms output of the node. Ai = SiVi
BELBIC
The BELBIC structure is a simple computational model of most important parts in limbic system of brain, Amygdala and Orbitofrontal cortex. Fig. 1 shows the schematic diagram of BELBIC structure and each part of it will be described briefly [10]. As depicted in Fig. 1, the system consists of four main parts. As it is seen, Sensory Input signals first entered in Thalamus. Thalamus is a simple model of real thalamus in the brain in which some simple pre-processing on sensory input signals is done. After pre-processing in Thalamus, the signal will be sent to Amygdala and Sensory Cortex. Sensory cortex is responsible for subdivision and discrimination of the coarse output from thalamus and then sent it to Amygdala and Orbitofrontal cortex. Amygdala is a small structure in the medial temporal lobe of brain which is thought to be responsible for the emotional evaluation of stimuli. This evaluation is in turn used as a basis of emotional states, emotional reactions and is used to signal attention and laying down long-term memories. And the last part, Orbitofrontal Cortex, is supposed to inhibit inappropriate responses from the Amygdala, based on the context given by the hippocampus [10]. In this section, functionality of these parts and the learning algorithm is based on what is stated in [10].
(2)
In the Orbitofrontal cortex, each O is similar to A nodes, and the output is calculated by applying connection weight W the input signal. Oi = SiWi
(3)
The model output can be computed as follow: E=
å A - åO i
(4)
i
Where the A nodes produce their outputs proportionally to their contribution in predicting the stress, while the O nodes inhibit the output of E if necessary. As it is observed in Fig. 1, except the thalamic signal going directly to the Amygdala, the Amygdala and the Orbitofrontal Cortex receive the same input signals. But the main difference between them is the learning rules. The connection weights Vi are adjusted proportionally to the difference between the reinforcement signal and the activation of the A nodes. The a term is a constant used to adjust the learning speed.
( [ (
D V i = a max 0, S i stress -
å A )]) j
(5)
As mentioned before, the task of the Amygdala is learning the associations between the sensory and the emotional input to generate an output. But the Eq 5 is mainly different from similar associative learning systems, because this weight adaptation rule is monotonic, i.e., the weights V cannot be decreased. At first, it may seem as a drawback of learning rule,
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Proceedings of the 4th International Conference on Autonomous Robots and Agents, Feb 10-12, 2009, Wellington, New Zealand
but this adaptation rule has biological reasons. According to what occurs in Amygdala, once an emotional reaction is learned, this should become permanent. The Orbitofrontal cortex inhibits inappropriate reactions of Amygdala.
controller, the controller is replaced with BELBIC. Due to the capability of learning, BELBIC will learn to enhance the system’s performance.
The Orbitofrontal cortex learning rule is very similar to the Amygdala rule:
A. Proposed Controller According to idea of hierarchical controller structure, to design a controller to satisfy various objectives, at first it is assumed that the objectives can be decoupled and then a separate controller is designed to satisfy each objective. After that, outputs of these controllers must be fused together. Fig. 2 shows the proposed BELBIC structure. As there are two major objectives, position tracking and pendulum angle regulating, two BELBICs are employed. The cart position error and its first derivation are defined as sensory signals for one of BELBICs and the pendulum angel and its first derivation are for the other. In most of the previously reported structures of BELBIC, they have only one neuron, because the sensory input signal was one dimensional. In our structure, as each BELBIC has two sensory inputs, they must have more than one neuron and for this task two neurons seems to be adequate.
D W i = b ( S i ( E - stress )
(6)
The reinforcement signal for the O nodes is defined as difference between model output E and the stress signal. In other words, the O nodes compare expected and received reinforcement signal, and inhibit output of the model if there is a mismatch. The main difference between adaptation rule of Orbitofrontal cortex and Amygdala, is that the Orbitofrontal connection weight can be increased and decreased as needed to track the required inhibiting of Amygdala. Parameter b is another learning rate constant. As discussed, BELBIC learns from its emotional signal and produce its output based on sensory inputs and connection weights. In [19] the stability of BELBIC is demonstrated by using cell to cell mapping method. III.
CONTROLLER DESIGN
As mentioned before, the test bed for evaluating the controller performance is an inverted pendulum which is a well known SIMO system. Controlling the inverted pendulum is a hard and interesting control task. The control task is tracking reference signal and stabilizing the pendulum. The system which is used to evaluate the controller performance is a nonlinear model of a laboratorial inverted pendulum system provided by Feedback Ltd. Due to nonlinearity of system’s state equations and nonlinear properties of driving motor and friction, designing a model based controller is a hard task. We used BELBIC as a model free controller to control the inverted pendulum. The main challenge in using BELBIC as a model free controller in unstable systems or stable systems with unstable equilibrium point such as our test bed is the learning phase at the beginning. As BELBIC has no information of system’s dynamics, performance of the controlled system may seems to be awful at the beginning of learning process and the pendulum falls down. BELBIC has fast learning ability; and theoretically in the short time it should learn the proper control action according to its sensory inputs and emotional stress. But in this task the pendulum angle is very sensitive to the control signal, and any wrong changes in control signal make the system unstable. First we used BELBIC as the only controller of the system. It is possible for BELBIC to learn the proper control strategy, but in our simulation we find that this process will take too long or probably impossible in real applications. To accelerate the learning process and avoiding making the system unstable, we proposed a new approach. This approach consists of two parts, in the first part, a simple stabilizing controller used as the main control system and BELBIC learns to imitate the behavior of this controller. In the second, after BELBIC imitatively learned to stabilize the system from initial
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Figure 2. Structure of controller
The emotional stress signal which will be described, couples the two separate controllers. Also by employing this kind of stress signal there will be no need to use complex fusion block to combine the output of controllers, and just a summation operator is adequate [16]. And the computational cost of output fusion is reduced to the cost of fusing some main and auxiliary objectives in stress generator block. Also to change the control objectives, and switching from imitative learning to normal learning, there is no need to change the controller structure and only changing the emotional stress signal is enough. B. Stress generation As stated before, BELBIC can show various behaviors by applying different stress signal on it. So, to satisfy different control objective, proper stress signal must be defined based on each objective. The ability of achieving more different objectives can be obtained by defining different stress signals. 1) Imitative learning In imitative learning, the objective is that BELBIC produces a similar control signal to the initial controller. So reducing the difference between these two control signals is
Proceedings of the 4th International Conference on Autonomous Robots and Agents, Feb 10-12, 2009, Wellington, New Zealand
the main goal in this part and there is no more control objectives. So the emotional stress signal is consists of two signals, error of control signal and first derivation of this error. (7)
The reason to add e&u in above sum is that in sometimes
Stress
Stress = w1eu + w2 e&u + w3 eu e&u
0.1 0.08
a) Fuzzy stress generation In order to enhance the behavior of system the major objectives are defined. Moreover some extra objectives should be attended. Tracking error of the cart position and error of pendulum angle are the main concerns which needed to be decreased as much as possible. One of the extra objectives is to avoid reaching edges of the rail which leads to breaking the operation. To impose this behavior to the cart, closeness to the edges of rail must be punished via stress signal. Another important index which must be considered in every control tasks is energy of control force and its variations. This objective is imposed to the stress generation unit by such a behavior that when the stresses of previous parts are small, BELBIC tries to decrease the control forces. Also when these stresses are significant, the limiting of control force is relaxed to increase the possibility of fast responses. Figure 3 shows the stress generating function. By use of a set of linguistic rules the most salient objectives can be reinforced to generate emotional stress with respect to contemporary situation. To generate the stress signal, we used linguistic rules and then import them to Sugeno fuzzy inference system [21]. Using this method to generate stress signal makes BELBIC capable to attend important parts of stress at any time. As it is mentioned before, four effective variables, errors of the cart position and pendulum angle, control force and first derivation of it and 16 rules are employed for generating the emotional stress. Fig. 3 shows some of the resulted fuzzy surface. The inputs of the stress generation function are angle and position error, control force and its first derivation.
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To satisfy more than two objectives, more complicated combination of stress signals which are associated with each objective is necessary. In order to make BELBIC capable to satisfy more objectives, the emotional stress signal by combining of stresses is applied, as describe as follow. To generate the proper stress signal for all objectives, at any time, the more important objective must be attended more than the others. To generate this attention mechanism we use linguistic rules.
0.04 0.02
eu may become zero, while these control signals may have completely different behaviors. 2) Improving the performance After BELBIC imitatively learns the control action from initial controller, it becomes the main controller of the system and the initial controller is replaced with it. At this time the control objective changes and reducing position tracking error and angle error become new control objectives. So the stress signal must be modified.
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0.06 0.04 0.02 1 0
Derivation-of-Control-Force
-1
-1
-0.5
0
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Figure 3. Resulted fuzzy surfaces
C. Switching between controllers and stress signals As we observed in experimental results, hard switching between controllers and changing stress signals makes the BELBICs become unstable. So instead of hard switching, a soft switching must be employed, and the BELBIC control system must gradually replace the initial controller and at the same time its emotional stress must be gradually changed. To do this, we employed a fuzzy inference system to make soft switching, as it common solution for soft switching. The human linguistic rules can be imported to Sugeno fuzzy inference system [21] easily. Fig. 4 shows the fuzzy surface for this task. The inputs of this fuzzy system are the two mentioned stresses (for imitative learning and improving performance), the error in imitative learning phase (difference between initial controller output and BELBIC output). The mentioned fuzzy switch is used for switching between both controllers and stresses. IV.
RESULTS
To validate the result of proposed controller, the results are compared with the original supplied controller, which consists of two PID controllers [22]. The initial controller for imitative learning phase is the mentioned original controller. In addition, without employing imitative learning, BELBIC did not learn
Proceedings of the 4th International Conference on Autonomous Robots and Agents, Feb 10-12, 2009, Wellington, New Zealand
the proper control signal in more than 150 seconds of training. Also the pendulum falls down many times in this duration. Cart Position
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Figure 5. Results Results of originally supplied controller (Double PID) in presence of disturbance
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Figure 6. Results of BELBIC with fuzzy stress in presence of disturbance
40 30
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Figure 4. Resulted fuzzy surfaces
In order to evaluate the ability of controller to reject disturbances, a random voltage produced by a Gaussian distribution with zero mean and 0.1 of variance is applied to the motor from in some instances. The time of applying this voltage is random variable which obtained from a uniform distribution. The mentioned disturbance is applied 8 times from the 55th seconds until the end of simulation. The results of original PID controller and proposed controller are depicted in Figs 5 and 6 respectively. As it is seen BELBIC can imitate the behavior of original controller in about 10 second from starting time completely. After that, based on fuzzy switch structure, from 30 second to 50 the both controller are controlling the system and after it BELBIC controls the system individually. It is clear that after imitative learning, BELBIC performance in reducing tracking and angle error is far better. As it can be seen, BELBIC clearly shows better performance in tracking and disturbance rejection which is the results of its learning capability.
To have a meaningful comparison these controllers, four performance measures are defined as follow and calculated to both control systems, originally supplied controller and BELBIC. As the disturbance applied in randomly selected times, the experiments carried out 20 times and the statistical moments of the following parameters (Mean and standard deviation) are calculated. IAE: Integral Absolute Error (for cart position and pendulum angle) IACF: Integral of Absolute values of Control Force IADCF: Integral of Absolute values of derivation of Control Force (shows the fluctuations of the control force) These performance measures are calculated for the two mentioned controllers, in normal operation and without applying disturbance and the results are demonstrated in Table I. TABLE I.
PERFORMANCE MEASURES OF VARIOUS CONTROLLERS WITHOUT DISTURBANCE
No-Disturbance BELBIC- fuzzy stress Double PID
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IAE (position) 3.301 3.925
IAE (angle) 0.367 0.457
IACF 9.432 12.219
IADCF 9.262 14.353
Proceedings of the 4th International Conference on Autonomous Robots and Agents, Feb 10-12, 2009, Wellington, New Zealand
From the Table I it is seen that BELBIC shows the fast learning ability for tracking. Also the control force signal which is penalized by stress signal is lower than control force in other controllers and has less oscillation. In the presence of disturbance, the above mentioned measures are calculated for the controllers and the results are presented in Table II. As it is seen, in presence of disturbance BELBIC again shows better performance although its performance decreases slightly in comparison with normal operation. But it shows better disturbance rejection and robustness. TABLE II.
[4]
[5]
[6]
[7] [8]
PERFORMANCE MEASURES OF VARIOUS CONTROLLERS WITH
Employing Disturbance
DISTURBANCE
BELBICfuzzy stress Double PID
IAE (position)
IAE (angle)
IACF
[9]
MADCF
E
STD
E
STD
E
STD
E
STD
5.699
0.915
0.476
0.174
73.247
3.152
151.26
6.245
11.725
2.141
1.692
0.371
82.705
3.247
160.58
7.831
V.
[10]
[11]
[12]
CONCLUSIONS
In this paper a new approach in employing model free controller with learning ability is introduced based on a soft switching between two phases of learning. Although BELBIC has rapid and powerful learning capability, it cannot simply used to control systems with unstable equilibriums. The experimental results show that by employing imitative learning at first phase, BELBIC can rapidly learn to produce proper control signal for controlling a system with unstable equilibrium point. After BELBIC learns imitatively from a simple classical designed controller, by gradually alter the objectives and stress signals, we can reduce the tracking and angle error. Moreover it shows more robustness with disturbances. Another advantage of BELBIC is that it produces smoother control force with lower energy. The fuzzy stress generation leads to superior performance in terms of tracking and angle error than alternative method for stress generation. Another interesting result is that BELBIC has better performance in presence of disturbance than the originally supplied controller, which is a model based controller that is well tuned especially for this task. This is the effect of learning capability of BELBIC, which can produce more appropriate control force at various working conditions.
[13]
[14]
[15]
[16]
[17]
[18]
[19]
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