WO2004012021A9 - System and method for nonlinear dynamic control based on soft computing with discrete constraints - Google Patents
System and method for nonlinear dynamic control based on soft computing with discrete constraintsInfo
- Publication number
- WO2004012021A9 WO2004012021A9 PCT/US2003/023671 US0323671W WO2004012021A9 WO 2004012021 A9 WO2004012021 A9 WO 2004012021A9 US 0323671 W US0323671 W US 0323671W WO 2004012021 A9 WO2004012021 A9 WO 2004012021A9
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- WIPO (PCT)
- Prior art keywords
- control
- control system
- entropy
- plant
- controller
- Prior art date
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
- G05B13/0285—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion using neural networks and fuzzy logic
Definitions
- the invention relates in generally to nonlinear electronic control system optimization.
- Description of the Related Art Feedback control systems are widely used to maintain the output of a dynamic system at a desired value in spite of external disturbances that would displace it from the desired value.
- a household space-heating furnace controlled by a thermostat
- the thermostat continuously measures the air temperature inside the house, and when the temperature falls below a desired minimum temperature the thermostat turns the furnace on. When the interior temperature reaches the desired minimum temperature, the thermostat turns the furnace off.
- the thermostat-furnace system maintains the household temperature at a substantially constant value in spite of external disturbances such as a drop in the outside temperature. Similar types of feedback controls are used in many applications.
- a central component in a feedback control system is a controlled object, a machine or a process that can be defined as a "plant", whose output variable is to be controlled.
- the "plant” is the house
- the output variable is the interior air temperature in the house
- the disturbance is the flow of heat (dispersion) through the walls of the house.
- the plant is controlled by a control system.
- the control system is the thermostat in combination with the furnace.
- the thermostat-furnace system uses simple on-off feedback control system to maintain the temperature of the house.
- a feedback control based on a sum of proportional feedback, plus integral feedback, plus derivative feedback is often referred as a P1D control.
- a PID control system is a linear control system that is based on a dynamic model of the plant. In classical control systems, a linear dynamic model is obtained in the form of dynamic equations, usually ordinary differential equations. The plant is assumed to be relatively linear, time invariant, and stable. However, many real-world plants are time varying, highly non-linear, and unstable.
- the dynamic model may contain parameters (e.g., masses, inductance, aerodynamics coefficients, etc.), which are either only approximately known or depend on a changing environment. If the parameter variation is small and the dynamic model is stable, then the PID controller may be satisfactory. However, if the parameter variation is large or if the dynamic model is unstable, then it is common to add adaptive or intelligent (Al) control functions to the PID control system. Al control systems use an optimizer, typically a non-linear optimizer, to program the operation of the
- a stepping motor moves by stepping in controlled increments and cannot be arbitrarily moved from a first shaft position to a second shaft position without stepping through all shaft positions in between the first shaft position and the second shaft position.
- Prior art control systems based on soft computing with genetic analyzer are not necessarily well suited for plants that must be changed or controlled in a stepwise fashion, in part because of the operation of the genetic analyzer.
- the chromosomes of a genetic analyzer are typically coded with the values of one or more control parameters used to control the plant.
- the genetic optimizer finds new control parameters without regard to the value of the previous control parameters or the constraints imposed by the plant in moving from the previous control parameters to the new control parameters.
- a Genetic Algorithm with step-coded chromosomes is used to develop a teaching signal that provides good control qualities for a controller with discrete constraints, such as, for example, a step-constrained controller.
- the step-coded chromosomes are chromosomes where at least a portion of the chromosome is constrained to a stepwise alphabet.
- the step-coded chromosome can also have portion which are position coded (i.e., coded in a relatively more continuous manner that is not stepwise constrained).
- the control system uses a fitness (performance) function that is based on the physical laws of minimum entropy.
- the genetic analyzer is used in an off-line mode to develop a teaching signal for a fuzzy logic classifier system that develops a knowledge base.
- the teaching signal can be approximated online by a fuzzy controller that operates using knowledge from the knowledge base.
- the control system can be used to control complex plants described by nonlinear, unstable, dissipative models.
- the step-constrained control system is configured to control stepping motors, stepwise actuators, or other step-constrained systems.
- the control system comprises a learning system, such as a neural network that is trained by a genetic analyzer.
- the genetic analyzer uses a fitness function that maximizes sensor information while minimizing entropy production.
- a suspension control system uses a difference between the time differential (derivative) of entropy from the learning control unit and the time differential of the entropy inside the controlled process (or a model of the controlled process) as a measure of control performance.
- the entropy calculation is based on a thermodynamic model of an equation of motion for a controlled process plant that is treated as an open dynamic system.
- the control system is trained by a genetic analyzer that generates a teaching signal for each solution space.
- the optimized control system provides an optimum control signal based on data obtained from one or more sensors. For example, in a suspension system, a plurality of angle and position sensors can be used.
- fuzzy rules are evolved using a kinetic model (or simulation) of the vehicle and its suspension system.
- Data from the kinetic model is provided to an entropy calculator that calculates input and output entropy production of the model.
- the input and output entropy productions are provided to a fitness function calculator that calculates a fitness function as a difference in entropy production rates for the genetic analyzer constrained by one or more constraints.
- the genetic analyzer uses the fitness function to develop training signals for the off-line control system.
- Control parameters (in the form of a knowledge base) from the off-line control system are then provided to an online control system in the plant that, using information from the knowledge base, develops a control strategy.
- One embodiment includes a method for controlling a nonlinear object (a plant) by obtaining an entropy production difference between a time differentiation ⁇ dSJdt) of the entropy of the plant and a time differentiation ⁇ dScldtj of the entropy provided to the plant from a controller.
- a step-coded genetic algorithm that uses the entropy production difference as a fitness (performance) function evolves a control rule in an off-line controller.
- the nonlinear stability characteristics of the plant are evaluated using a Lyapunov function.
- the genetic analyzer minimizes entropy and maximizes sensor information content.
- the control method also includes evolving a control rule relative to a variable of the controller by means of a genetic algorithm.
- the genetic algorithm uses a fitness function based on a difference between a time differentiation of the entropy of the plant (dSJd ⁇ and a time differentiation ⁇ dSJdt) of the entropy provided to the plant.
- the variable can be corrected by using the evolved control rule.
- the invention comprises a self-organizing control system adapted to control a nonlinear plant.
- the control system includes a simulator configured to use a thermodynamic model of a nonlinear equation of motion for the plant.
- the thermodynamic model is based on a Lyapunov function (V), and the simulator uses the function V to analyze control for a state stability of the plant.
- the control system calculates an entropy production difference between a time differentiation of the entropy of the plant ⁇ dSJdt) and a time differentiation [dSJdt) of the entropy provided to the plant by a low-level controller that controls the plant.
- the entropy production difference is used by a genetic algorithm to obtain an adaptation function wherein the entropy production difference is minimized in a constrained fashion.
- the teaching signal is provided to a fuzzy logic classifier that determines one or more fuzzy rules by using a learning process.
- the fuzzy logic controller is also configured to form one or more control rules that set a control variable of the controller in the vehicle.
- the invention includes a new physical measure of control quality based on minimum production entropy and using this measure for a fitness function of genetic algorithm in optimal control system design.
- This method provides a local entropy feedback loop in the step-constrained control system.
- the entropy feedback loop provides for optimal control structure design by relating stability of the plant (using a Lyapunov function) and controllability of the plant (based on production entropy of the step- constrained control system).
- the stepc-constrained control system is applicable to a wide variety of control systems, including, for example, control systems for mechanical systems, bio-mechanical systems, robotics, electro-mechanical systems, etc. Brief Description of the Figures Figure 1 shows the general Structure of a Self-Organization Intelligent Control System Based on Soft
- Figure 2 shows the Simulation System of Control Quality (SSCQ).
- Figure 3 shows the general Structure of a Self-Organization Intelligent Control System Based on Soft Computing with the Constraint Control System.
- Figure 4 shows the Simulation System of Control Quality for Constraint Control System.
- Figure 5A shows the block scheme of simulation of a dynamic system model with the SSCQ.
- Figure 5B shows the temporal representation of the SSCQ modes.
- Figure 6 is a flowchart of the SSCQ.
- Figure 7A shows normally coded chromosomes.
- Figure 7B shows step-based coding of the chromosomes.
- Figure 7C is a graph showing output of the step-constraint control system.
- Figure 8A shows a schemata where gene 5 is a position-coded wildcard.
- Figure 8B shows outputs of a position-coded control system corresponding to the schemata of Figure 8A.
- Figure 9A shows a schemata where gene 5 is a step-coded wildcard.
- Figure 9B shows outputs of a step-coded control system corresponding to the schemata of Figure 9A.
- Figure 10 is a flowchart of coding and evaluation operations of the genetic algorithm.
- Figure 11 is a flow chart of the roulette wheel (Monte-Carlo) selection operation
- Figure 12 is a flow chart of crossover operation.
- Figure 13 is a flowchart of mutation operation.
- Figures 14A-14N are graphs that show the nature of the attractor of the Holmes-Rand (Duffing-Van der Pol) dynamic system.
- Figures 15A-15C are graphs that show the nature of the 3D attractor of the Holmes-Rand (Duffing-Van der Pol) dynamic system.
- Figures 16A-16N are graphs that show the nature of the attractor of Holmes-Rand (Duffing-Van der Pol) dynamic system.
- Figure 17A-17C are graphs that show the nature of the 3D attractor of the Holmes-Rand (Duffing-Van der Pol) dynamic system under stochastic excitation with normal probability distribution.
- Figure 18 shows GA optimization dynamics of SSCQ control of the Holmes-Rand dynamic system with a position-encoded GA (with coding according to Table 1).
- Figure 19 shows GA optimization dynamics of SSCQ control of the Holmes - Rand dynamic system with a step-based GA (with coding according to Table 2).
- Figures 20A-20N are graphs that show results of control signal optimization using the GA coding method according to Table 1.
- Figures 21A-21C are graphs that show 3D results of control signal optimization using the GA coding method according to Table 1.
- Figures 22A-22N are graphs that show results of control signal optimization using the GA coding method according to Table 2.
- Figures 23A-23C are graphs that shows results of control signal optimization using the GA coding method according to Table 2.
- Figures 24A-24C are graphs that show comparisons, on an interval between 0 and 60 seconds, of control using the coding method according to Table 1 and the coding method according to Table 2.
- Figures 25A-25C are graphs that show comparisons of control, on an interval between 20 and 15 seconds, using the coding method according to Table 1 and the coding method according to Table 2
- Figures 26A-26F are graphs that show control errors and control signals obtained using the coding method according to Table 1.
- Figures 27A-27F are graphs that show control errors and control signals obtained using the coding method according to Table 2.
- Figures 28A-28C are graphs that show comparison of control error accumulation.
- Figures 29A-29B are graphs that show stochastic excitation (band limited white noise with a mean of 0.0 and a variance of 1.0).
- Figures 30A-30F are graphs that show results of control, on an interval between 0 and 15 seconds, of an automotive suspension system model.
- Figures 31A-31 D are graphs that show optimal control signals, on the interval between 0 and 15 seconds, obtained to control the automotive suspension system model.
- Figures 32A-32F are graphs that show control, on an interval between 5 and 7 seconds, from Figures
- Figure 33A-33D are graphs that show control, on the interval between 5 and 7 seconds, from Figures 31A-31 D.
- Figures 34A-34D are graphs that show fitness function component accumulation.
- Figure 35A shows a control damper layout for a suspension-controlled vehicle having adjustable dampers.
- Figure 35B shows an adjustable damper for the suspension-controlled vehicle.
- Figure 35C shows fluid flow for soft and hard damping in the adjustable damper from Figure 8B.
- Figure 36 shows damper force characteristics for the adjustable dampers illustrated in Figures 35A-C.
- Figure 1 is a block diagram of a control system 100 for controlling a plant based on soft computing.
- a reference signal y is provided to a first input of an adder 105.
- An output of the adder 105 is an error signal ⁇ , which is provided to an input of a Fuzzy Controller (FC) 143 and to an input of a Proportional-lntegral-Differential (PID) controller 150.
- An output of the PID controller 150 is a control signal if, which is provided to a control input of a plant 120 and to a first input of an entropy-calculation module 132.
- a disturbance m(t) 110 is also provided to an input of the plant 120.
- An output of the plant 120 is a response x, which is provided to a second input of the entropy-calculation module 132 and to a second input of the adder 105.
- the second input of the adder 105 is negated such that the output of the adder 105 (the error signal ⁇ ) is the value of the first input minus the value of the second input.
- An output of the entropy-calculation module 132 is provided as a fitness function to a Genetic
- Analyzer (GA) 131 An output solution from the GA 131 is provided to an input of a FNN 142. An output of the FNN 132 is provided as a knowledge base to the FC 143. An output of the FC 143 is provided as a gain schedule to the PID controller 150.
- the GA 131 and the entropy calculation module 132 are part of a Simulation System of Control Quality (SSCQ) 130.
- the FNN 142 and the FC 143 are part of a Fuzzy Logic Classifier System (FLCS) 140.
- FLCS Fuzzy Logic Classifier System
- the genetic algorithm 131 generates sets of "chromosomes" (that is, possible solutions) and then sorts the chromosomes by evaluating each solution using the fitness function 132.
- the fitness function 132 determines where each solution ranks on a fitness scale. Chromosomes (solutions) that are more fit, are those which correspond to solutions that rate high on the fitness scale. Chromosomes that are less fit are those which correspond to solutions that rate low on the fitness scale. Chromosomes that are more fit are kept (survive) and chromosomes that are less fit are discarded (die). New chromosomes are created to replace the discarded chromosomes. The new chromosomes are created by crossing pieces of existing chromosomes and by introducing mutations.
- the PID controller 150 has a linear transfer function and thus is based upon a linearized equation of motion for the controlled "plant" 120.
- Prior art genetic algorithms used to program PID controllers typically use simple fitness functions and thus do not solve the problem of poor controllability typically seen in linearization models, As is the case with most optimizers, the success or failure of the optimization often ultimately depends on the selection of the performance (fitness) function. Evaluating the motion characteristics of a nonlinear plant is often difficult, in part due to the lack of a general analysis method. Conventionally, when controlling a plant with nonlinear motion characteristics, it is common to find certain equilibrium points of the plant and the motion characteristics of the plant are linearized in a vicinity near an equilibrium point.
- Control is then based on evaluating the pseudo (linearized) motion characteristics near the equilibrium point. This technique is scarcely, if at all, effective for plants described by models that are unstable or dissipative.
- Computation of optimal control based on soft computing includes the GA 131 as the first step of global search for optimal solution on a fixed space of positive solutions.
- the GA 131 searches for a set of control weights K for the plant.
- the weight vector K ⁇ k x , ... , k n ⁇ is used by a conventional proportional- integral-differential (PID) controller 150 in the generation of the signal ⁇ ( ⁇ ) which is applied to the plant.
- PID proportional- integral-differential
- the entropy S( ⁇ ( ⁇ )) associated to the behavior of the plant on this signal is assumed as a fitness function to minimize,
- the GA 131 is repeated several times at regular time intervals in order to produce a set of weight vectors.
- the vectors generated by the GA 131 are then provided to the FNN 142 and the output of the FNN 142 to the fuzzy controller 143.
- the output of the fuzzy controller 143 is a collection of gain schedules for the PID controller 150.
- One embodiment of the SSCQ 130, shown in Figure 2 is an off-line block that produces the teaching signal Kj for the FLCS 140.
- Figure 10 shows the structure of an SSCQ 230.
- the SSCQ 230 is one embodiment of the SSCQ 130.
- Figure 2 also shows a stochastic excitation signal generator 210, a simulation model 220, a PID controller 250, and a timer 20.
- the SSCQ 230 includes a mode selector 229, a buffer 2301, a GA 231, a buffer 8 2302, a PID controller 234, a fitness function calculator 232, and an evaluation model 236.
- the Timer 200 controls the activation moments of the SSCQ 230.
- An output of the timer 200 is provided to an input of the mode selector 2304.
- the mode selector 2304 controls operational modes of the SSCQ 230.
- a reference signal y is provided to a first input of the fitness function calculator 232.
- An output of the fitness function calculator 232 is provided to an input of the GA 231.
- a CGS ⁇ output of the GA 231 is provided to a training input of the PID controller 234 through the buffer 2302.
- An output U e of the controller 234 is provided to an input of the evaluation model 236.
- An X e output of the evaluation model 236 is provided to a second input of the fitness function calculator 232.
- a CGS' output of the GA 231 is provided (through the buffer 2301) to a training input of the PID controller 250.
- a control output from the PID controller 250 is provided to a control input of the suspension system simulation model 220.
- the stochastic excitation generator 210 provides a stochastic excitation signal to a disturbance input of the simulation model 220 and to a disturbance input of the evaluation model 236.
- a response output X' from the system simulation model 220 is provided to a training input of the evaluation model 236.
- the output vector K 1 from the SSCQ 230 is obtained by combining the CGS 1 output from the GA 1031 (through the buffer 2301) and the response signal X' from the system simulation model 220.
- the stochastic excitation signal generator 210 generates the excitation signal.
- the excitation signal can be excitation obtained using stochastic simulations, or it can be real measured environment.
- the block 210 produces the excitation signal for each time moment generated by the timer 200.
- the simulation model 220 is a kinetic model of the control object.
- Simulation system of control quality (SSCQ) 230 is a main optimization block on the structure. It is a discrete time block, and its sampling time is typically equal to the sampling time of the control system.
- Entropy production rate is calculated inside the evaluation function 232, and its values are included in the output (X e ) of the evaluation model.
- the SSCQ 130 can be used to perform optimal control of different kinds of nonlinear dynamic systems, when the control system unit is used to generate discrete impulses to the control actuator, which then increases or decreases the control coefficients depending on the specification of the I control actuator ( Figure 3).
- FIG. 4 The structure of an SSCQ 430 for a discrete or step-constrained control actuator is shown in Figure 4 and described in more detail in the text in connection with Figure 4.
- the SSCQ 430 is similar to the SSCQ 230 and thus the flowchart of Figure 5A can be used to describe both.
- the simulation model 220 is integrated using the stochastic excitation signal from the stochastic excitation generator 210 and the control signal CGS ⁇ T) 2303 from the first time interval t° and it generates the output X 1 204 3.
- the output of X 1 204 with the output of the CGS ⁇ T) 2301 is saved into the data file as a teaching signal K'. 4.
- Sequence 1-4 is repeated until the end of the stochastic signal is reached,
- the SSCQ has two operating modes: 1. Updating the buffer 1 2301 using genetic algorithm 231 ; and 2. Extraction CGS ⁇ T) from buffer 2301.
- the current mode of the SSCQ 2304 controlled by the mode selector 2304 using the information of the current time moment T, as it is presented in Figure 5B.
- Figure 6 is a flowchart of SSCQ calls, and it combines the following steps: 1.
- buffer 1 2301 and the buffer 6 2302 The structure of the buffer 1 2301 and the buffer 6 2302 is presented in Table 1 below as a set of row vectors, where first element in each row is time value, and the other elements are the control parameters associated with the time value.
- buffer 1 2301 stores optimal control values for the evaluation time interval t e to control the simulation model
- the buffer 3 2302 stores temporal control values for evaluation on the interval t e for calculation of the fitness function.
- the models used for the simulation 220 and for the evaluation 236 are typically similar models. There are many different methods for numerical integration of systems of differential equations. Practically, these methods can be classified into two main classes: a) Variable-step integration methods with control of integration error; and b) Fixed-step integration methods without integration error control. Numerical integration using methods of type a) are very precise, but time-consuming. Methods of type b) are faster, but with smaller precision. During each SSCQ call in GA mode, the genetic algorithm 231 calls the fitness function hundreds of times. Each calculation generally includes integration of the model of the dynamic system, which results in exponential computational complexity growths.
- variable step solvers By choosing a small enough integration step size, it is possible to adjust the fixed step solver so that the integration error on a relatively small time interval (like the evaluation interval t e ) will be small, thus allowing the use of fixed step integration in the evaluation loop for integration of the evaluation model 236.
- the results of high-order variable step integration in the simulation model 220 As initial conditions for evaluation model integration.
- variable step solvers To obtain better precision, it is typically advantageous to use variable step solvers to integrate the evaluation model. Unfortunately, such variable step solvers can be prohibitively slow, especially if the dynamic model is relatively complicated, as in the case of a suspension system model.
- the fitness function calculation block 232 computes the fitness function using the response (X e ) 237 of the evaluation model 236 on the control signal CGS e (t e ) and the reference signal Y 238.
- the fitness function is considered as a vector of selected components of a matrix (x e ) of the response of the evaluation model 236. Its squared absolute value in general form is the following:
- such coefficients can be used to represent the importance of the corresponding elements from a human comfort viewpoint.
- the weighting factors can have some empirical values and they can then be adjusted using experimental results. Extraction of frequency components can be done using standard digital filtering design techniques for obtaining the parameters of filter. The standard difference equation can be applied to the x column elements of the matrix X e :
- n b n ⁇ .
- Every electromechanical control system has a certain time delay, which is usually caused by the analog to digital conversion of the sensor signals, computation of the control gains in the computation unit, by mechanical characteristics of the control actuator, and so on. Additionally, many control units do not have continuous characteristics. For example, when the control actuators are step motors, such step motors can change only one step up or one step down during a control cycle.
- such a stepwise constraint can constrain the search space of the genetic algorithm 131 in the SSCQ 130.
- N the number of possible N positions each time the stepper motor position is updated. It is enough to check only the cases when the stepper motor position is going change one step up, one step down, or hold position. This gives only 3 possibilities, and thus reduces the search space from the size of N points to three points. Such reduction of the search space will lead to better performance of the genetic algorithm 131, and thus will lead to better overall performance of the intelligent control system.
- the SSCQ 130 can be used to perform optimal control of different kinds of nonlinear dynamic systems, when the control system unit is used to generate discrete impulses to the control actuator, which then increases or decreases the control coefficients depending on the specification of the control actuator (see e.g., Figure 3).
- the conventional PID controller 150 in the control system 100 shown in Figure 1
- a PID controller 350 with discrete constraints as shown in Figure 3 to create a new control system 300.
- This type of control is called step-constraint control.
- the structure of the SSCQ 130 for step-constraint control is shown in Figure 4, which is a block diagram of a step-constrained SSCP 430.
- the SSCQ 430 is similar to the SSCQ 230 in many respects.
- the difference between the SSCQ structure 230 described in Figure 2 and the SSCQ structure 430 in Figure 4 lies in the structures of the buffer 2301 and the buffer 6 2302 and the addition of constraints to the PID controllers 234 and 350,
- the PID controllers in the SSCQ 430 are constrained by discrete constraints and at least a portion of the chromosomes of the GA 231 in the SSCQ 430 are step-coded rather than position-coded.
- the SSCQ buffers 2301 and 2301 have the structure presented in the Table 2, and can be realized by a new coding method for discrete constraints in the GA 131.
- Time column corresponds to time assigned after decoding of a chromosome
- STEP denotes the changing direction values from the stepwise alphabet ⁇ -1,0,1 ⁇ corresponding to (STEP UP, HOLD, STEP DOWN) respectively.
- Figure 7A the chromosome generated by the genetic algorithm 231 is presented, for the control of the proportional step-constraint controller, with a step value 1 , a minimum value 0, and a maximum value 8 (ordinate axis), and with a controller sampling time T c of one second.
- the evaluation time T e in Figures 7A-7C is 20 seconds.
- Figure 7B shows values of a chromosome generated by the GA 231 using step-based coding.
- the chromosome presented in Figure 7A and Figure 7B is then passed through the controller, and the actual control signal accepted by the controller is presented on the Figure 7C.
- the signal accepted by the controller is the same for both the chromosomes shown in Figures 7A and 7B.
- the chromosome described in Figure 7A is shown when position 5 is a wildcard. Concentric circles at position five are used in Figures 8A and 8B to indicate each combination which can be obtained, changing the value in the position number five. In totality, there are 9 different combinations as shown in Figure 8A, but mechanical constraints of the control system prohibit all of the combinations except the three shown in Figure 8B. Note that the GA can check all 9 combinations independently, even if they give the same controller output. Also note that the step constraints of the controller will cause a relatively rapid vanishing of the effect of the wildcard. Thus, the genetic operations such as mutation, described later, typically only produce local effects. In Figure 9, the chromosome described in Figure 7B is presented, when the position 5 is a wildcard.
- Each concentric circle indicates a combination, which can be obtained by changing the value in the position number five.
- there are 3 different combinations (Figure 9A), and all of them will affect on the output of the control system ( Figure 9B).
- the GA 430 can check only three combinations and they will have different fitness values. The vanishing of the effect of the wildcard in this case can be caused only by reaching the limits of the control signal range.
- the genetic operations, such as mutation will have global effect, and the building blocks of such a chromosome correspond better to the Goldberg principles for effective coding.
- Step-based coding reduces the search space of the GA 430. In this case, the range of the control signal for the GA search space is relatively unimportant.
- the relatively more important parameters are the evaluation time T e , the controller's sampling time T c , and the number of control parameters for optimization.
- GA Theory are stochastic search algorithms, based on the mechanism of natural evolution. A population of individuals evolves from one generation to another such that the best (most fit) individuals will increase their number and quality, and the worst (less fit) will vanish.
- C is a coding system
- F is a fitness function (quality criteria)
- P° is an initial population
- ⁇ is a size of the population
- ⁇ is a selection operation
- P CR is a probability of the crossover operation
- ⁇ is a mutation operation
- P MU is a probability of the mutation operation
- the coding system C is a unitary transformation, which defines the map between physical solution space and the mathematical space in which evolutionary (genetic) operations are defined.
- the initial population P° is a set of the elements defined in the coding space. Usually, it is taken from uniform distribution among all elements of the search space, in one embodiment of the strings from B 1 .
- the size of the population ⁇ is the number of individuals (chromosomes) in the population.
- the fitness function F is a quality criterion, which defines how good a certain individual of the population is relatively to other elements of the population.
- the fitness function is usually calculated as a response of the certain function or of the certain dynamic system to the information coded in the corresponding individual (chromosome). For complex optimization problems evolving a number of calculations, the fitness function calculation is the most time-consuming part of any optimization algorithm, including evolutionary algorithms. In most cases, after calculating the absolute values of the fitness functions, the relative (normalized) values are used to distribute the importance inside population.
- Figure 10 is a flowchart of the coding and evaluation operations of a genetic algorithm. Selection
- ⁇ is a probabilistic operation, defined to reproduce more chromosomes with high fitness values into the next generation.
- selection selects a set of individuals from initial population into a set known as a mating pool.
- the mating pool is usually an intermediate population of the size ⁇ from which the individuals will be taken for the genetic operations.
- the selection is based on Monte-Carlo method also known as the roulette wheel method shown in Figure 11.
- Each sector of the roulette wheel represents an individual of the population, and the size of each sector is proportional to the relative fitness of the corresponding individual. If the wheel is rotated, the probability of the individuals with higher relative fitness to be selected is proportional to its relative fitness (size of the sector of the roulette wheel).
- FIG 12 is a flowchart of the crossover operation of a typical genetic algorithm.
- the crossover operation r is a probabilistic operation, aimed to share information between best individuals of the population.
- the probability of the crossover operation P CR is usually predefined.
- the input to the crossover operation r is a mating pool, obtained from the selection operation ⁇ .
- the mechanism of the crossover operation is the following: 1. Select two chromosomes from the mating pool; 2. Generate a uniformly distributed random number from [0 1]. If the generated number is less than P CR , then proceed with the following steps; 3. Generate a crossover point (uniformly distributed random integer number from [1 / ]); 4. Replace the right parts of the selected chromosomes, starting from the crossover point; 5.
- FIG. 13 is a flowchart of the mutation operation.
- the mutation operation ⁇ is a probabilistic operation, aimed to introduce new information into a population.
- the probability of mutation operation P MU is predefined, depending on the parameters of the search space. Usually, the probability of mutation is less than the probability of crossover.
- the mutation operation ⁇ is applied after crossover.
- the input to mutation is the new generation, obtained after the crossover operation.
- the mechanism of the mutation operation is the following: 1. Select the first chromosome from the new generation; 2. Generate a uniformly distributed random number form [0 1] and if this number is less than
- P MU proceed with step 3, otherwise go to the step 5; 3.
- ⁇ of the genetic algorithm can be predefined in different ways depending on the problem. In one embodiment, it can be a certain number of generations on which population should be evolved.
- the convergence criterion can be based on a difference between the best fitness value of theo current and of the previous generation. If this difference is less than a predefined number, then algorithm ) stops.
- the algorithm is assumed to be converged when an entire generation includes the same or similar individuals. Every genetic algorithm has a certain set of parameters, including: ⁇ the size of the population; GN the maximum number generations; P CR the crossover operation probability; and P m the mutation5 operation probability. These parameters depend on the parameters of the search space. The parameters are obtained using principles from the Holland theorem of schemata. The description of the Holland theorem requires some additional definitions, described as follows.
- a schemata is a representation of the group of the individuals of the population, with similar genotypes, (e.g., selecting a common part of the chromosomes), and replacing the0 different parts with asterisks (to denote wildcards).
- An individual who matches the schemata in its defined positions is called a representative of the schema.5
- N s the total number of schemata, N s is 2 *- ⁇ N5 ⁇ n 2 . Since each string can represent many schemata, it means that GA operations defined on a population of strings posses a much larger number of schemata in parallel. This property is called implicit parallelism of GA.
- the probability p(x t ) that an individual x, will be copied into the next generation depends upon the ratio of its fitness value f(x ⁇ ) to the total fitness F of all individuals in the population:
- each representative S t of schema S is copied to a mating pool Nf(S l )/ F times.
- the average fitness of schema S which is as follows:
- f(P) F/N which is the average of the fitness over all string in the population.
- Equation (15) corresponds to theorem 2 of Holland, which states that the growth of a schema depends upon the value of the fitness functions of its representatives, as well as its defining length. Thus, a schema with a high fitness function and short defining length is more prolific. Mutation Each bit in a string has a chance of mutation to its opposite value. Let P MU be the probability of mutation in any bit position.
- the probability that a bit does not change is ( 1- P MU ). If a string represents a schema that has o(S) bits that are either 0 or 1, then the probability that all corresponding o(S) bits in the string do not mutate is (i - p MU ⁇ - s) . in other words, the probability that a string remains a representation of a schema after mutation is: Since P MU is usually small, then formula (16) can be simplified as: (l - o(S)P MU ) (17) The combined effects of reproduction, crossover and mutation can be written as follows. Combining Equations (15) and (17) gives the following schema theorem of Holland.
- n(S, t + 1) [n(S, t)f(S) I f(P)][l - P CR ⁇ (S) 1(1 - l)
- Equation (18) the following designations are used: n(S,t) is the number of representatives of schema S at time f; f(S) is the average fitness function for schema S; f(P) is the average fitness function over the population; P CR is the crossover probability; / is a length of a string; S(S) is the defining length of schema S; o(S) is the order of schema S; and P MU is the probability of mutation at any bit position. Coding Problems The above disclosure dealt with a simple coding scheme using a binary alphabet without examining whether the binary coding is optimal or effective. Experimental observations show that a GA is robust and typically produces good results despite differences in coding.
- the coding method can have a significant effect on the accuracy and efficiency of the GA.
- Goldberg in 1989 proposes two principles for effective coding: 1. The principle of minimal alphabets; and 2. The principle of meaningful building blocks: a user should select a coding so that short defining length, low-order schemata are relevant to the given problem. A small alphabet tends to produce a long string. In this case, the efficiency of the implicit parallelism property is increased. A binary alphabet yields the maximum number of schemata for a string of a given length, making the binary alphabet the optimal choice for coding. But, binary coding is not a requirement. Other coding methods can be used in the GA. For example, real-coded chromosomes having real or integer numbers.
- the shortest alphabet for coding of such a control signals is a step-like alphabet such as ⁇ -1 ,0,1 ⁇ , which can be interpreted as ⁇ "one step down", “hold”, “one step up” ⁇ .
- control signal u(t) can be generated as a PID control signal with the following control function: u(t) - k p (i)e + k D (t)e + k j (t) ⁇ e(t)dt (20) where e - y(t) - x(t) is the control error, and y(i) is the reference signal, defined by the user.
- control gains k p ,lc D ,k j have stepwise constraints.
- the entropy production rate of the plant is:
- dt D Kinetic energy is: rA (23) 2 .
- Figures 14A-14N show the simulation results of free dynamic and thermodynamic motion of the nonlinear dynamic system in Equation (19).
- Figure 14A shows the dynamic evolution of coordinate x of the system (19).
- Figure 14B shows the dynamic evolution of the velocity x of the system (19), where x-axis is a system coordinate x and the y-axis is a system velocity x .
- Figure 14D shows the kinetic energy evolution of the Holmes-Rand oscillator.
- Figure 14E shows the derivative of the potential energy of the Homes-Rand oscillator phase portrait.
- Figure 14F shows the velocity phase portrait of kinetic energy.
- Figure 14G shows the potential energy evolution of the Holmes-Rand oscillator.
- Figure 14H shows coordinate phase portraits of potential energy of the Holmes-Rand oscillator.
- Figure 141 shows the total energy evolution of the Holmes-Rand oscillator, and corresponding coordinate and velocity phase portraits of total energy are shown in Figures 14J and 14K, respectively.
- Figure 14L shows the plant entropy production evolution.
- Figure 14M shows the plant entropy evolution.
- Figure 14N shows the phase portrait of the plant entropy of the Holmes-Rand oscillator.
- Figures 15A-15C show simulation results of free dynamic and thermodynamic motion of the nonlinear dynamic system (19).
- the x-axis and y-axis on the graphs of the Figure 15A-15C are the coordinate x and velocity .
- Figure 15A shows the total energy of the attractor.
- Figure 15B shows the entropy production of the attractor.
- Figure 15C shows the entropy of the attractor.
- Figures 16A-16N show the simulation of dynamic and thermodynamic motion of the nonlinear dynamic system (19) under stochastic excitation.
- the excitation ⁇ (t) is a band-limited white noise with zero mean.
- Figure 16A shows the dynamic evolution of coordinate x of the system (19).
- Figure 16B shows the dynamic evolution of the velocity x of the system of Equation (19), where x-axis is a system coordinate x and the y-axis is a system velocity .
- Figure 16D shows the kinetic energy evolution of the Holmes-Rand oscillator.
- Figure 16E shows the derivative of the potential energy of the Homes-Rand oscillator phase portrait.
- Figure 16F shows the velocity phase portrait of kinetic energy.
- Figure 16G shows the potential energy evolution of the Holmes-Rand oscillator.
- Figure 16H shows coordinate phase portraits of potential energy of the Holmes-Rand oscillator.
- Figure 161 shows the total energy evolution of the Holmes-Rand oscillator, and corresponding coordinate and velocity phase portraits of total energy are shown in Figures 16J and 16K, respectively.
- Figure 16L shows the plant entropy production evolution.
- Figure 16M shows the plant entropy evolution.
- Figure 16N shows the phase portrait of the plant entropy of the Holmes-Rand oscillator.
- Figures 17A-17C show simulation results corresponding to Figures 16A-16N.
- the x-axis and y-axis on the graphs of the Figures 17A-17C are the coordinate x and velocity x .
- Figure 17A shows the total energy of the attractor.
- FIG 17B shows the entropy production of the attractor.
- Figure 17C shows the entropy of the attractor.
- Figure 18 shows the performance of the GA 231 used in the SSCQ 230.
- the fitness function is presented in the following form: (25)
- the evaluation time T e of the SSCQ 230 is taken as 4 seconds.
- the GA 231 population size is set to 50 chromosomes, and the number of generations the GA 231 should evolve is set to 10.
- the performance of the GA 231 is shown on the time interval t e ⁇ ,7/ £ ] (first SSCQ call) is presented in Figure 18, where the search space of the GA is specified according to Table 1.
- Figure 16 shows the corresponding case where the GA search space is specified according to Table 2,
- the x-axis represents the generation number.
- the y-axis represents the values of the fitness function (25). Each point represents the fitness value of the single chromosome.
- Figure 20B shows the dynamic evolution of the velocity x of the system of Equation (19), where the x-axis is a system coordinate x and the y-axis is a system velocity x .
- Figure 20D shows the kinetic energy evolution of the Holmes-Rand oscillator.
- Figure 20E shows the derivative of the potential energy of the Holmes-Rand oscillator phase portrait.
- Figure 20F shows the velocity phase portrait of kinetic energy.
- Figure 20G shows the potential energy evolution of the Holmes-Rand oscillator.
- Figure 20H shows coordinate phase portraits of potential energy of the Holmes-Rand oscillator.
- Figure 201 shows the total energy evolution of the Holmes-Rand oscillator, and corresponding coordinate and velocity phase portraits of total energy are shown in Figures 20J and 20K, respectively.
- Figure 20L shows the plant entropy production evolution.
- Figure 20M shows the plant entropy evolution.
- Figure 20N shows the phase portrait of the plant entropy of the Holmes-Rand oscillator.
- Figures 21A-21C show simulation results corresponding to Figures 20A-20N.
- the x-axis and y-axis on the graphs of the Figures 21 A-21C are the coordinate x and velocity x .
- Figure 21 A shows the total energy of the attractor.
- Figure 21 B shows the entropy production of the attractor.
- Figure 21 C shows the entropy of the attractor.
- Figure 22A shows the dynamic evolution of coordinate x of the system (19).
- Figure 22B shows the dynamic evolution of the velocity x of the system of Equation (19), where the x-axis is a system coordinate x and the y-axis is a system velocity x .
- Figure 22D shows the kinetic energy evolution of the Holmes-Rand oscillator.
- Figure 22E shows the derivative of the potential energy of the Holmes-Rand oscillator phase portrait.
- Figure 22F shows the velocity phase portrait of kinetic energy.
- Figure 22G shows the potential energy evolution of the Holmes-Rand oscillator.
- Figure 22H shows coordinate phase portraits of potential energy of the Holmes-Rand oscillator.
- Figure 22I shows the total energy evolution of the Holmes-Rand oscillator, and corresponding coordinate and velocity phase portraits of total energy are shown in Figures 22J and 22K, respectively.
- Figure 22L shows the plant entropy production evolution.
- Figure 22M shows the plant entropy evolution.
- Figure 22N shows the phase portrait of the plant entropy of the Holmes-Rand oscillator.
- Figures 23A-23C show simulation results corresponding to Figures 22A-22N.
- the x-axis and y-axis on the graphs of the Figure 23A-23C are the coordinate x and velocity x .
- Figure 23A shows the total energy of the attractor.
- Figure 23B shows the entropy production of the attractor.
- Figure 23C shows the entropy of the attractor.
- Figures 20A-N, 21A-C, 22A-N and 23 -C show the good performance of SSCQ control of the system (19) in the presence of noise using both approaches. More analysis of these results is presented in Figures 24-28.
- Figures 24A-24C compare the dynamic results using position encoding and step coding over the interval 0 to 60 seconds.
- Figure 24A shows the evolution of the x coordinate for the position-encoded and the step-encoded processes.
- Figure 24B shows the evolution of the velocity x for the position-encoded and the step-encoded processes.
- Figure 24C shows the evolution of the squared control error signal for the position- encoded and step-encoded processes.
- Figures 25A-25C compare the dynamic results using position encoding and step coding from Figures 24A-24C over the interval 20 to 25 seconds.
- Figure 25A shows the evolution of the x coordinate for the position-encoded and the step-encoded processes.
- Figure 25B shows the evolution of the velocity x for the position-encoded and the step-encoded processes.
- Figure 25C shows the evolution of the squared control error signal for the position-encoded and step-encoded processes.
- Figures 25C and 25C show using step-coding instead of position coding results in a smaller control error and thus better control performance. The smaller control error results from improved performance of the GA 430, caused by the search space reduction using the coding presented in Table 2.
- Figures 26A-26F show the control error evolution and the gain coefficient evolution of the PID controller for position-encoded control using the approach presented in Table 1.
- Figures 27A-27F show the control error evolution and the gain coefficient evolution of the PID controller for step-encoded control using the approach presented in Table 2.
- Figures 26A and 27A show the error signal.
- FIGS. 27B show the time derivative of the error signal.
- Figures 26C and 27C show the time integral of the error signal.
- Figures 26D and 27D show the proportional gain Ki of the PID controller.
- Figures 26E and 27E show the derivative gain K 2 of the PID controller.
- Figures 26F and 27F show the integral gain K 3 of the PID controller.
- Figures 28A-28C show the squared control error accumulation using both position coding and step coding. The y-axis values in Figures 28A-28C are calculated as a cumulative sum of the corresponding squared control errors along the simulation time.
- Figure 28A shows the squared error accumulation.
- FIG. 28B shows the squared differential error accumulation.
- Figure 28C shows the squared integral error accumulation.
- the step coding approach described in the Table 2 gives approximately a 50% performance improvement, as compared to the position coding approach described in Table 1.
- Figure 29A shows the stochastic excitation signal used in the Holmes-Rand oscillator simulations is presented above.
- Figures 29A shows the probability distribution function of the stochastic excitation signal of Figure 29A.
- Figure 35A shows a vehicle body 3510 and left-side wheels 3532 and 3534, and right-side wheels 3531 and 3533.
- Figure 8A also shows dampers 3501-3504 configured to provide adjustable damping for the wheels 3531-3534 respectively.
- the dampers 3501-3504 are electronically-controlled dampers controlled by an electronic controller 3536.
- a stepping motor actuator on each damper controls an oil valve. Oil flow in each rotary valve position determines the damping factor provided by the damper.
- Figure 35B shows an adjustable damper 3517 having an actuator 3518 that controls a rotary valve
- a hard-damping valve 3511 allows fluid to in the adjustable damper 3517 to produce hard damping.
- a soft-damping valve 3513 allows fluid to flow in the adjustable damper 3517 to produce soft damping.
- the rotary valve 3512 controls the amount of fluid that flows through the soft-damping valve 3513.
- the actuator 3518 controls the rotary valve 3512 to allow more or less fluid to flow through the soft-damping valve 3513, thereby producing a desired damping.
- the actuator 3518 is a stepping motor.
- the actuator 3518 receives control signals from the controller 3510.
- Figure 35C shows fluid flow through the soft-damping valve 3513 when the rotary valve 3512 is opened.
- Figure 35C also shows fluid flow through the hard-damping valve 3510 when then rotary valve 3512 is closed.
- Figure 36 shows damper force characteristics as damper force versus piston speed characteristics when the rotary valve is placed in a hard damping position and in a soft damping position.
- the valve is controlled by the stepping motor to be placed between the soft and the hard damping positions to generate intermediate damping force.
- the following example demonstrates the results of the step-based coding approach to control of the vehicle dampers 3501-3504 (i.e., the vehicle shock absorbers). This approach is illustrative for suspension system application since each controlled valve can change one step position in each control cycle. In the illustrated example, the maximum controlled valve has a range of 0 to 8 and the control system has a 7.5 ms sampling period. The following results are indicating the improved performance of the step-based coding approach on the control of the semi-active vehicle suspension.
- the fitness function optimized in both cases composed of the components shown in Table 3: Table 3 Omission of the value of the weight numerator means the exclusion of the parameter form the optimization. According to Table 3, and the generalized fitness function equation (1), the fitness function of the automotive suspension system control can be described as:
- x i e t can represent the frequency components of the corresponding state l o variables. Such frequency components can be selected according to comfort requirements of the passengers.
- the simulation results of the optimization of the Fitness function (26) with parameters taken from the Table SS1 are presented on the Figures 30A-30F.
- Figure 30A shows Heave acceleration output z 0 .
- Figure 30D shows heave velocity output z 0 .
- Figure 30B shows pitch acceleration output ⁇ 0 .
- Figure 30E shows pitch velocity output ⁇ 0 .
- Figure 30C shows roll acceleration output ⁇ 0 ,
- Figure 30F shows roll velocity output
- Figures 31 A-31 D show the respective control signals obtained according to the code method presented in Table 1 and according the coding method presented in Table 2 for the front-left, front-right, rear- left, and rear-right shock absorbers.
- Figures 32A-32F are the zoomed versions of the Figures 30A-30F, respectively for the time interval 20 5-7 seconds.
- Figures 33A-33D are the zoomed versions of the Figures 31 A-31 D, respectively for the time interval 5-7 seconds.
- Figures 30A-30F and 32A-32F show the improvements of the optimization results when the control signal is coded using the step-based method. The amplitude of the optimized fitness function components is smaller in the step- based coding case.
- Figures 34A-34D Comparisons of the optimization performance are shown in Figures 34A-34D based on cumulative error in the control signal.
- Figures 34A and 34B show the squared components of the fitness function, with the weight coefficients taken from the Table 3. The result indicates that in case of complicated fitness functions depending on many system parameters and state variables, the step-based coding approach described in Table 2 give improvement of the optimization performance comparing with the normal, direct coding method.
Abstract
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AU2003256969A8 (en) | 2004-02-16 |
AU2003256969A1 (en) | 2004-02-16 |
US6950712B2 (en) | 2005-09-27 |
CN1672103A (en) | 2005-09-21 |
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