DE19641066A1 - Direct adaptive control of non linear system - Google Patents

Abstract

The adaptive control has the process output y compared with that of a process model. The difference is used in a learning algorithm and this provides an access to a fuzzy logic controller coupled to the process. This allows the parameters to be varied to suit the process condition The learning process is subdivided into a number of part stages that allow the learning time to be reduced.

Description

1 Description and criticism of the state of the art

In industrial applications of control technology, routes are often to be controlled, which are characterized by Embossed nonlinearities are marked and where parameters and structure continue to be not exact are known. Adaptive non-linear control methods are used to control such lines. Leave this first of all divide into procedures that are tailored to a specific route structure, and widely all-round processes. The second class includes control procedures with neuronal Networks as well as adaptive fuzzy controllers. Most of these methods are about stability and convergence Given the overall system consisting of the system, controller and adaptation law, no statement is possible; therefore can these methods are only used to control lines that are not critical to stability, and it additional monitoring measures are required. In contrast, in a number of procedures a sta proof of balance possible. A distinction must be made between indirect methods, in which one is used first Identification of the route and the regulation based on the identified route model he follows; with direct methods, on the other hand, the controller is directly adapted to a given behavior of the control loop. While indirect methods generally prove stability. only for the part system route - identification model - adaptation law is possible, with direct procedures stability for the entire system consisting of the system, controller and adaptation law. It has been done so far published a number of such procedures: Tzirkel, E., and Fallside, F., A Direct Control Method for a Class of Nonlinear Systems Using Neutral Networks, Technical Report, Cambridge University Engineering Department, Cambridge, 1993; E. Tzirkel-Hancock and F. Fallside, Stable Control of Nonlinear Systems Using Neural Networks, Int. J. Robust and Nonlinear Control, Vol. 2, no. 1, 1992; Robert M. Sanner and Jean-Jacques E. Slotine, Gaussian Networks for Direct Adaptive Control, IEEE Transactions on Neural Networks, Vol. 3, no. 6, 1992; Clemens Schäffner and Dierk Schröder, An Application of General Regres sion Neural Network to Nonlinear Adaptive Control, 5th European Conference on Power Electronics and Applications (EPE), Vol. 4, Brighton, 1993; Clemens Schäffner, analysis and synthesis of neuronal regulation proceedings, dissertation, Technical University of Munich, Chair of Electrical Drive Technology, 1996; Li-Xin Wang, Slable Adaptive Fuzzy Control of Nonlinear Systems, IEEE Transactions on Fuzzy Systems, Vol. 1, 1993.

These control methods are based on the basic structure according to Fig. 1, whereby instead of the fuzzy controller, a neural network can be used as well as a non-linear function approximator. Given is a non-linear system of known order, the structure and parameters of which are completely unknown or only partially known, and a reference model that specifies the desired control behavior of the closed control loop. The reference model must be designed so that the physical conditions of the real non-linear path and the actuator are taken into account. The distance must be controlled so that its output variable y follows the reference signal y _m . u is the manipulated variable, x the state vector of the system, n the order of the system and e = yy _m the output error. The rule law is of the form

the expressions θ T | 1 ξ ( x ) and θ T | 2 ξ ( x ) represent universal nonlinear function approximators, e.g. B. neural networks with radial basic functions or adaptive fuzzy controllers (see below). The parameter vectors of these function approximators are given the form by an adaptation law

set. Here γ is the adaptation gain. With the help of the stability theory of Ljapunow it can now be shown that the overall system is stable and the error e converges to zero if the path and the input signals meet certain conditions and the coefficients p ₁ . . . p _n satisfy the so-called Lyapunov equation.

However, the processes published to date have a number of disadvantages which prevent or practically prevent their widespread practical use:

• - For both the control law and the adaptation law, n-1 derivatives of the control variable required (n = order of the route); in practice, however, this is due to the always existing Measurement disturbances in general cannot be realized.
• - The procedures do not work if there is a manipulated variable limit; the presence Such a limitation generally causes a divergence of the adapted parameters, i.e. H. Instability. In in practice, however, unlimited manipulated variables are not possible on any route; in most cases  the manipulated variable is limited so far that the limitation also occurs in real operating conditions is achieved.
• - Computing time and storage space requirements grow exponentially with the order of the route. Also the Time to learn the desired control law increases strongly with the order of the route and is generally relatively large. This time results from the dynamics of the overall system from the route, Regulator and adaptation law and depends on numerous influencing factors; already for relatively simple ones Third order routes can be several days, which is not tolerable in practice.

The following section describes an improved adaptive nonlinear direct control method wrote, which overcomes these disadvantages.

2 Presentation of the invention 2.1 Procedure

The process is again based on the system structure according to Fig. 1. The distance must be controlled so that its output variable y follows the reference signal y _m . For a certain class of lines and reference models, the approach of input-output linearization (see J. Slotine and W. Li, Applied Nonlinear Control) can be used to show that a rule exists that fulfills this task. However, this "ideal le" rule law cannot be calculated explicitly because the route is unknown; therefore, artificial intelligence methods are used to determine it. All methods that are suitable for learning static nonlinear functions, i.e. radial basis function networks (see Tzirkel, E., and Fallside, F., A Direct Control Method for a Class of Nonlinear Systems Using Neural Networks, and Robert M. Sanner and Jean-Jacques E. Slotine, Gaussian Networks for Dired Adaptive Control, General Regression Networks (see Clemens Schäffner and Dierk Schröder, An Application of General Regression Neural Network to Nonlinear Adaptive Control) or adaptive fuzzy controllers (see Li-Xin Wang, Stable Adaptive Fuzzy Control of Nonlinear Systems). The basic principle remains the same in all of the methods mentioned; The use of a fuzzy controller has the advantage that the control law is not available as a "black box", but in the form of clear IF-THEN rules. This is particularly useful when verbally formulated prior knowledge is to be used to initialize the controller. The approach using fuzzy controllers will therefore be presented in the following explanations; however, this is only intended to serve as an example and does not limit the methods presented to fuzzy control, ie the other methods can also be used.

2.1.1 Control law without differentiation of the controlled variable

The control law for calculating the manipulated variable u from the setpoint w and the state vector x can have different forms. The simplest form is

here θ ^T ξ (w, x ) is a universal nonlinear function approximator, e.g. B. a neural network with radial basic functions, a general neural regression network or a certain type of fuzzy controller (see Li-Xin Wang, Stable Adaptive Fuzzy Control of Nonlinear Systems). The vector θ contains the adjustable parameters of the function approximator, the vector ξ contains localized basic functions, each of which contributes to the approximation in a limited range of the input size space, whereas in the remaining input size space it is zero or approximately zero.

This rule of law can only be used for lines where the Lie derivative L _g L n-1 | fh ( x ) (see below) is constant. If L _g L n-1 | fh ( x ) is not constant, the control law finds

 Application. In contrast to rule laws of the form (1), these rule laws do not require differentiation the error e and thus the controlled variable y. Their applicability is based on auxiliary sentence 1. (The here The given mathematical equations are only intended to clarify the basic principle of the method. The practical implementation can be done according to these equations or by other mathematical Representations that implement the same basic principle.)  

2.1.2 Control law with a reduced number of parameters

With the two control laws just mentioned, the manipulated variable is a general non-linear function of all controller input variables. Therefore, universal function approximators are required, which have all controller input variables as input variables. For higher-order routes, this leads to problems with the computing time and storage space requirements and the learning time. One possible solution is to divide the rule into linear and non-linear sub-functions, which can then be learned separately. Such a division can be made based on previous knowledge of the route structure. In many cases it is possible to divide the differential equations of the system into known and unknown sub-functions. The ideal rule law can then be calculated for this partially known route model using the Lie derivatives. It has the general form (see auxiliary sentence 1)

where the subfunctions f * _{ki, j} ( x ) are known and the subfunctions f * _{ui, j} ( x ) are unknown. b _1m is the first counter coefficient of the reference model. The main advantage is that the unknown partial functions f * _{ui, j} ( x ) do not depend on all controller input variables, but only on some specific input variables. Which subfunctions depend on which input variables is known in advance. This prior knowledge becomes the establishment of the regular law

used. The advantage is that the unknown non-linear subfunctions θ T | i, j ξ _{i, j} ( x ) do not depend on all controller input variables. As a result, the basic functions have a lower dimension and fewer parameters are required overall (see example in section 2.2). (The mathematical equations given here are only intended to clarify the basic principle of the method. The practical implementation can be carried out according to these equations or also by other mathematical representations that implement the same basic principle.)

2.1.3 Regulatory law with the use of symmetry properties

A reduction of the number of parameters - and thus of computing time and storage space requirements as well as learning time - can be achieved by using symmetry properties. The dynamics of many lines show (approximately) symmetry properties to the point

from which it follows that

is. Consequently, all of the information required to describe the control law is contained in half of the parameter vector. To take advantage of this, the basic functions are also

of the nonlinear function approximator symmetrical to the point

arranged. The basic function vector ξ is divided into two subvectors ξ ⁺ and ξ ⁻ of the same size, so that:

(If a basic function is exactly symmetrical to the point

the associated parameter must be zero; Therefore, this basic function can be omitted.) Now, if the parameter vector also θ into two corresponding sub-vectors θ ⁺ and divided θ ⁻, a new control law as u '= θ' ^T ξ 'with θ' = θ ⁺ and ξ '= ξ ⁺ - ξ ⁻ can be written. Since θ ⁺ = - θ ⁻ (due to the symmetry of f _c ), u '= u. The new control law thus provides the same manipulated variable as (3), but manages with half the parameters. The adaptation is carried out exactly as for the original control law, whereby wobei is to be replaced by ξ ' .

The computing time efficiency can be increased by not calculating all basic functions ξ ⁺ and ξ ⁻. This is possible if the division of the basic functions takes place along a (hyper) plane orthogonal to an axis of the input quantity space (ie for example x ₁ = 0 as the separation plane). Then most of the basic functions on one side of the separation level are zero (or for functions that do not take the value zero - such as radial Gaussian basic functions - negligibly small). Only basic functions whose centers are close to or on the separation level are different from zero on both sides of the separation level. Now replace x ₁ with | x ₁ | before calculating the basic functions (it is assumed that x ₁ = 0 is the separation plane and the basic functions ξ + | k are on the side x ₁ <0), so most basic functions ξ - | k are constantly zero or very small and can be used in the calculation be omitted; ie the calculation of ξ ' is considerably simplified. To compensate for the amount of x ₁ again, the calculated manipulated variable must be multiplied by the sign of x ₁ .

For rule laws (4) and (6), symmetry can be used according to the same principle.

With control law (4), the division is made separately for the basic function vectors ξ ₁ (w, x ) and ξ ₂ ( x ). In the control law (6), the division takes place separately for the basic function vectors ξ ₁ = [ ξ T | 1,1 ( x ). f * _k1.1 ( x ), ξ T | 1.2 ( x ). f * _k1.2 ( x ) _,. . .] ^T and ξ ₂ = [ ξ T | 2.1 ( x ). f * _k2.1 ( x ), ξ T | 2.2 ( x ). f * _k2.2 ( x ) _,. . .] ^T. The different symmetry properties of the known subfunctions f * _{ki, j} ( x ) must be noted: either f * _{ki, j} ( x ) = f * _{ki, j} (- x ) or f * _{ki, j} ( x ) = -f * _{ki, j} (- x ). The following therefore applies to the combined basic function vectors ξ ₁ and gilt ₂ :

The following applies correspondingly to the parameters θ + | k = -v _k θ - | k; the modified rule law is consequently formed with θ ' = θ ⁺ and ξ' = ξ ⁺ - V ξ ⁻, V = diag (v _k ). (The mathematical equations given here are only intended to clarify the basic principle of the method. The practical implementation can be carried out according to these equations or by other mathematical representations which implement the same basic principle.)

2.1.4 Adaptation law without differentiation of the controlled variable

In order to simplify the formulas below, the following notation should be agreed: If z (t) is a time-dependent signal, then z _f (t) denotes the signal that consists of z (t) through linear filtering

 arises.

Using this notation, the adaptation law with extended error for the control law is (3)

Here are γ <0 and

the adaptation gains and e * the extended error (see J. Slotine and W. Li, Applied Nonlinear Control); the extended error is defined as

with the auxiliary error

_p and u _aux are auxiliary signals that are required for the following reason: To calculate the extended error (10), an estimate for the path gain L _g L n-1 | fh ( x ) (hereinafter referred to as k _{p for} short) is required. However, this value is not explicitly available with the rule of law used, but is implicitly contained in f _c (w, x , θ ); therefore k _p must be estimated by an internal auxiliary parameter _p . The auxiliary signal u _{aux is} required for the adaptation of _p , which is fed into the reference model and control law according to the following equations:

The following auxiliary set is required to prove stability:

Auxiliary sentence 1

Given a SISO range of the form

whose relative degree r is equal to their order n (the relative degree is defined as the order of the lowest derivative of y, which depends directly on the input variable u (see J. Slotine and W. Li, Applied Nonlinear Control). For linear systems this is synonymous with the well-known definition "number of poles minus number of zeros"), as well as a linear reference model

of order n with the degree zero. If the Lie derivatives L n | fh ( x ) and L _g L n-1 | fh ( x ) exist in a certain area of the state space and L _g L n-1 | fh ( x ) = const in the relevant area. ≠ 0, the rule of law leads

for this area of the state space to the error differential equation

The proof of this auxiliary sentence is based on the theory of input-output linearization (see J. Slotine and W. Li, Applied Nonlinear Control) and will not be discussed further here.

The following sentence ensures stability for the adaptation law (8) - (9):

Sentence 1

Given an adaptive system from a line of type (14) - (15), a reference model according to Eq. (12), an adaptive fuzzy controller according to Eq. (13) and the Adaptation Act (8) - (9). Then:

• 1. If the inherent approximation error d is sufficiently small, the parameter vector θ and the auxiliary parameter _{p are} limited.
• 2. If d ∼ is 0 and ξ and u _{aux are} constrained, then lim _{t → ∞} e = 0.

Proof of sentence 1

Using auxiliary sentence 1, the following differential equation for the error e is obtained for the system described:

ie

For the extended error e * results

here is the extended parameter error

and the extended activation vector

From the adaptation law (8) - (9) follows

With

To demonstrate the limitation of

becomes the Lyapunov function

used. Your derivative is after time

If d is sufficiently small, then ≦ 0; with that the limitation of

and thus the limitation of θ and _{p is} demonstrated.

To show that lim _{t → ∞} e = 0, we first prove that lim _{t → ∞} e * = 0. (In the following, d ∼ 0 is assumed.) For this, the second derivative of V is calculated:

From the restriction of ξ and u _aux (see prerequisite) and the restriction of

follows the limitation of

and thus the limitation of. From Barbalat's lemma (see J. Slotine and W. Li, Applied Nonlinear Control) it follows that lim _{t → ∞} = lim _{t → ∞} (-e ^{* 2} ) = 0. Hence lim _{t → ∞} e ^* = 0.

The next step is lim _{t → ∞} e _aux = 0. K. Narendra and A. Annaswamy, Siable Adaptive Systems, show that this is the case when ξ is restricted and

can be integrated into a square. The square integrability of

results from

Thus e ^{* can be} integrated into a square; there

and ξ _{f is} limited, too

square integrable. This proves that lim _{t → ∞} e _aux = 0.

Finally, the same reasoning can be used to show that lim _{t → ∞} (( _p u _aux ) _f - _p u _auxf ) = 0. With (10) follows from these partial results lim _{t → folgt} e = 0, which proves Theorem 1.

For the regulation law (4) the corresponding adaptation law is

With

The auxiliary parameter _p and the auxiliary signal u _aux can be omitted here. The adaptation of θ ₂ is to be stopped when falling below a predetermined lower limit in order to prevent division by zero in the control law.

For the regulation law (6) is the adaptation law

With

(The mathematical equations given here are only intended to illustrate the basic principle of the method clarify. The practical implementation can be done according to these equations or by others mathematical representations that implement the same basic principle.)  

2.1.5 Consideration of the manipulated variable limitation

Another problem with stable adaptive fuzzy control has so far been that it was not applicable to lines with manipulated variable limitation. The derivation of the method assumes that actuation amplitudes of any desired height are possible; Simulations show that the existence of a manipulated variable limit leads to instability (ie unlimited increase in the parameters to be adapted). The resulting control behavior is characterized by constant switching of the manipulated variable between positive and negative limitation. However, this problem could also be solved by further developing the adaptation law. The modification is essentially based on switching off the adaptation during the limitation phase and using a modified reference signal. The adaptation law (8) is modified to

the adaptation law (9) is modified accordingly. The condition (ue * ξ _f <0 ∧ | e | <ε _e ) is required in order to prevent "getting stuck" in the manipulated variable limitation if all active parameters should become <u _max (or <-u _max ) due to overshoot. In this case the error becomes <ε _e at some point and the parameters leave the limit again. Theoretically, ε _e , could be set to zero; However, simulation results have shown that a value of ε _e = (0.1 ... 1.0). max (| y _m (t) |) leads to better behavior of the parameters. If ε _e is set to zero, the adaptation for parameters that are affected by the limitation is "asymmetrical" (adaptation to zero is possible, whereas adaptation away from zero is not possible during the limitation); in connection with the inherent approximation error, this leads to oscillatory behavior of the parameters.

Furthermore, a modified reference signal y _mm must be introduced to achieve stable behavior. The modified reference model has the same dynamics as the original model, but its states y _mm , _mm , _mm,. . . y (n-1) | mm are on y,,,. . . y ^{(n-1) is} reset when u leaves the limit (In practice, the reference model is implemented in a time-discrete manner. When u leaves the limit, its states y _mm (k) ... y _mm (k - n + 1) open y (k) ... y (k - n + 1), so no differentiation of y is required). At the same time, the states of the linear filters for ξ _f and ( θ ^T ξ ) _{f are} reset to zero. (The mathematical equations given here are only intended to clarify the basic principle of the method. The practical implementation can be carried out according to these equations or by other mathematical representations which implement the same basic principle.)

2.2 Application of the method to the control of electrical drive systems with vibratory mechanics and non-linear friction characteristic

In the following, the operation of the described method will now be illustrated using an application example. A drive system with non-linear friction was used as the unknown non-linear path ( Fig. 2). The drive system is a two-mass torsional vibrator. The output variable to be controlled is the speed of the load mass x ₁ ; the other state variables are the angle of rotation of the shaft x ₂ and the motor speed x ₃ . The air gap torque u acts on the engine mass as a manipulated variable (assumption: ideal torque control); the amount of u is limited to the maximum value u _max = 5.0. A friction torque f _f (x ₁ ) acts on the load mass, which depends non-linearly on the load speed x ₁ . The friction characteristic was modeled using the arctangent function. The natural torsional frequency is approx. 50 Hz; the wave damping was assumed to be zero.

The design of the reference model was based on the double ratio method; the equivalent time constant of the reference model was specified as T _m = 20 ms.

Taking advantage of the prior knowledge that the nonlinear friction depends only on x ₁ , but not on x ₂ and x ₃ , a rule law of the form became

in which all unknown nonlinear functions only depend on x ₁ . A function approximator with 10 triangular base functions was used for each of the unknown nonlinear subfunctions.

Simulation results for this controller can be seen in Fig. 3. The controller is trained by applying a setpoint signal w (t) which assumes a new random value in the range [-0.5, 0.5] every 40 ms. A triangular test signal is applied after 300, 1800 and 7200 s to assess the learning result achieved.

In order to demonstrate that stability and convergence are independent of the initial conditions, the parameter vectors θ ₂ , θ ₃ , θ ₄ and θ ₅ from (38) are initialized to zero; θ ₁ is set to the initial value 10 ⁶ . So no previous knowledge of the parameters of the route is used. At the beginning of the adaptation there is the rule law

it is therefore only controlled with the setpoint w. As already mentioned, the adaptation process takes place relatively quickly in the initial phase. After approx. 300 s, the load speed already roughly follows the reference signal y _m . After that, however, the adaptation becomes much slower. Even after 1800 s, clear deviations between y and y _m can still be seen, especially in the range of lower speeds. After approx. 7200 s (ie two hours), the load speed and the reference signal for the triangular setpoint are practically congruent. The remaining deviations in the case of a randomly distributed setpoint curve are due to the manipulated variable limitation. The adaptation is stopped when the manipulated variable limit is reached (see 2.1.5); As you can see, this measure maintains stability and convergence, although the manipulated variable limitation (due to the quick setting of the reference model) is achieved with almost every setpoint step change.

Claims (5)

1. A method for stable direct adaptive control of non-linear paths with a structure which is not exactly known and parameters which are not exactly known, characterized in that the control law contains one or more universal non-linear function approximators with localized basic functions and that the parameters of these function approximators are set by an adaptation law, in which the time profiles of the basic functions are processed according to the principle of the extended error, which means that there is no differentiation of the controlled variable. All methods can be used as universal nonlinear function approximators with localized basic functions, in which the approximation value is calculated by scalar multiplication of a vector of adjustable parameters with a vector of basic functions arranged in the input size range, each basic function only in a specific area of the input size range is significantly different from zero. These include neural networks with radial basic functions, general neural regression networks and fuzzy controllers, the defuzzification of which is carried out using the singleton method (method according to Section 2.1.4 or a comparable method). 2. The method according to claim 1, characterized in that the controller as the input variables the setpoint and receives all state variables of the system, but no derivatives of the controlled variable. The controllers structure consists of either a universal nonlinear function approximator, which the Reg input values and which directly calculates the manipulated variable, or from two universal nonlinear function approximators to which the controller input variables are fed and their Output variables for determining the manipulated variable are divided (procedure according to section 2.1.1 or comparable procedure). 3. The method according to claim 1, characterized in that a division based on previous knowledge of the regulatory law takes place in several sub-functions, some of which are known in advance and at the rest of them know in advance which controller input variables they depend on. The rule law consists of the known sub-functions as well as universal non-linear function approximators, which learn the unknown sub-functions. These function approximators are not all controller input quantities are supplied, but only those of which the corresponding unknown really depend on the sub-functions (procedure according to Section 2.1.2 or comparable procedure). 4. The method according to claim 1-3, characterized in that by taking advantage of symmetry properties the number of adapted parameters and thus also the required learning time as well the computing time and storage space requirement is reduced. This is done by a suitable team menu of two parameters or the two associated basic functions, so that two Parameters are grouped together due to the symmetry properties against the same value or converge the same value with the opposite sign (procedure according to Section 2.1.3 or comparable procedure). 5. The method according to claim 1-4, characterized in that no adaptation of the parameters instead takes place as long as the manipulated variable is at the limit and that when the manipulated variable is exited limitation the state of the reference model is reset to the track state (procedure according to section 2.1.5 or comparable procedure).