WO2010065866A2 - Adaptive control based on retrospective cost optimization - Google Patents

Abstract

A discrete-time adaptive control law for stabilization, command following, and disturbance rejection that is effective for systems that are unstable, MIMO, and/or nonminimum phase. The adaptive control algorithm includes guidelines concerning the modeling information needed for implementation. This information includes the relative degree, the first nonzero Markov parameter, and the nonminimum-phase zeros. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant's spectral radius, the required zero information can be approximated by a sufficient number of Markov parameters. No additional information about the poles or zeros need be known. Numerical examples are presented to illustrate the algorithm's effectiveness in handling systems with errors in the required modeling data, unknown latency, sensor noise, and saturation.

Description

ADAPTIVE CONTROL BASED ON RETROSPECTIVE COST OPTIMIZATION

GOVERNMENT SUPPORT

[0001] This invention was made with government support under Grant No. NNX08AB92A awarded by NASA. The government has certain rights in the invention.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. Utility Application No. 12/630,004, filed on December 3, 2009 and claims the benefit of U.S. Provisional Application No. 61/201 ,035 filed on December 5, 2008. The entire disclosure of the above applications are incorporated herein by reference.

[0003] This application is also related to U.S. Patent No. 6,208,739, issued on March 27, 2001 , which is incorporated herein by reference.

FIELD

[0004] The present disclosure relates to methods and control systems for using a digital adaptive control algorithm and, more particularly, relates to methods and control systems for using a digital adaptive control algorithm that are based on a retrospective correction feedback filter.

BACKGROUND AND SUMMARY

[0005] This section provides background information related to the present disclosure which is not necessarily prior art. This section also provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.

[0006] Unlike robust control, which chooses control gains based on a prior, fixed level of modeling uncertainty, adaptive control algorithms tune the feedback gains in response to the true dynamical system (or "plant"), and commands and disturbances (collectively "exogenous signals"). Generally speaking, adaptive controllers require less prior modeling information than robust controllers, and thus can be viewed as highly parameter-robust control laws. The price paid for the ability of adaptive control laws to operate with limited prior modeling information is the complexity of analyzing and quantifying the stability and performance of the closed-loop system, especially in light of the fact that adaptive control laws, even for linear plants, are nonlinear. [0007] Stability and performance analysis of adaptive control laws often entails assumptions on the dynamics of the plant. For example, a widely invoked assumption in adaptive control is passivity, which is restrictive and difficult to verify in practice. A related assumption is that the plant is minimum phase, which may entail the same difficulties. In fact, sampling may give rise to nonminimum-phase zeros whether or not the continuous-time system is minimum phase, which must ultimately be accounted for by any adaptive control algorithm implemented digitally in a sampled-data control system. Beyond these assumptions, adaptive control laws are known to be sensitive to unmodeled dynamics and sensor noise, which necessitates robust adaptive control laws.

[0008] In addition to these basic issues, adaptive control laws may entail unacceptable transients during adaptation, which may be exacerbated by actuator limitations. In fact, adaptive control under extremely limited modeling information, such as uncertainty in the sign of the high-frequency gain, may yield a transient response that exceeds the practical limits of the plant. Therefore, the type and quality of the available modeling information as well as the speed of adaptation must be considered in the analysis and implementation of adaptive control laws.

[0009] Adaptive control laws have been developed in both continuous-time and discrete-time settings. In the present application we consider discrete-time adaptive control laws since these control laws can be implemented directly in embedded code for sampled-data control systems without requiring an intermediate discretization step that may entail loss of stability margins.

[0010] According to some prior art, references on discrete-time adaptive control include a discrete-time adaptive control law with guaranteed stability developed under a minimum-phase assumption. Extensions based on internal model control and Lyapunov analysis also invoke this assumption. To circumvent the minimum-phase assumption, the zero annihilation periodic control law uses lifting to move all of the plant zeros to the origin. The drawback of lifting, however, is the need for open-loop operation during alternating data windows. An alternative approach, is to exploit knowledge of the nonminimum-phase zeros. Knowledge of the nonminimum-phase zeros is used to allow matching of a desired closed-loop transfer function, recognizing that minimum-phase zeros can be canceled but not moved, whereas nonminimum-phase zeros can neither be canceled nor moved. Knowledge of a diagonal matrix that contains the nonminimum- phase zeros is used within a MIMO direct adaptive control algorithm. Finally, knowledge of the unstable zeros of a rapidly sampled continuous-time SISO system with a real nonminimum-phase zero is used in some instances.

[0011] Motivated by the adaptive control laws given in some instances, the goal of the present application is to develop a discrete-time adaptive control law that is effective for nonminimum-phase systems. In particular, we present an adaptive control algorithm that extends the retrospective cost optimization approach. This extension is based on a retrospective cost that includes control weighting as well as a learning rate, which can be used to adjust the rate of controller convergence and thus the transient behavior of the closed-loop system. Unlike some instances, which use a gradient update, the present application uses a Newton-like update for the controller gains as the closed-form solution to a quadratic optimization problem. No off-line calculations are needed to implement the algorithm or control system. A key aspect of this extension is the fact that the required modeling information is the relative degree, the first nonzero Markov parameter, and nonminimum-phase zeros, if any. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant's spectral radius, we show that the required zero information can be approximated by a sufficient number of Markov parameters from the control inputs to the performance variables. No matching conditions are required on either the plant uncertainty or disturbances.

[0012] In some embodiments, a goal of the present application is to develop the RCF adaptive control algorithm and demonstrate its effectiveness for handling nonminimum-phase zeros. To this end we consider a sequence of examples of increasing complexity, ranging from SISO, minimum-phase plants to MIMO, nonminimum-phase plants, including stable and unstable cases. We then revisit these plants under off-nominal conditions, that is, with uncertainty in the required plant modeling data, unknown latency, sensor noise, and saturation. These numerical examples provide guidance into choosing the design parameters of the adaptive control law in terms of the learning rate, data window size, controller order, modeling data, and control weightings.

[0013] According to the principles of the present teachings, a discrete-time adaptive control law or algorithm for stabilization, command following, and disturbance rejection that is effective for systems that are unstable, MIMO, and/or nonminimum phase. The adaptive control algorithm includes guidelines concerning the modeling information needed for implementation. This information includes the relative degree, the first nonzero Markov parameter, and the nonminimum-phase zeros. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant's spectral radius, the required zero information can be approximated by a sufficient number of Markov parameters. No additional information about the poles or zeros need be known. We present numerical examples to illustrate the algorithm's effectiveness in handling systems with errors in the required modeling data, unknown latency, sensor noise, and saturation.

[0014] Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.

DRAWINGS

[0015] The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure. [0016] FIG. 1 depicts a closed-loop system including adaptive control algorithm with the retrospective correction filter (dashed box) for ^ ⁼ .

[0017] FIG. 2 depicts roots of ^²⁰ W for the stable, nonminimum-phase plant in Example 5.1. The dashed line denotes P^ ' ^~ . Note that the roots outside

P^ ' are close to the outer nonminimum-phase zeros ^~l-5 and 1-25. The remaining roots are either located at the origin or form an approximate ring with radius close to P(A)

[0018] FIG. 3 depicts roots of ^ W f_or f_ne unstable, nonminimum-phase plant in Example 5.2. The dashed line denotes P(^A) = ^{1 Λ} . Note that the root of ^²⁵ fø) outside P^ ' is close to the outer nonminimum-phase zero ^~l-5 . However, the nonminimum-phase zero 1-25 is not approximated by a root of P^w) -|-_πe remaining roots are either located at the origin or form an approximate ring with radius close to P(A)

[0019] FIG. 4 depicts roots of ^w) f_or ^_πe unstable, nonminimum-phase plant in Example 5.3. The dashed line denotes P^ ' ⁼ , where A j_s the dynamics matrix of a minimal realization of ^zu . Note that the roots outside P^ ' are close to the inner and outer nonminimum-phase zeros of ^m . The remaining roots are either located at the origin or form an approximate ring with radius close to P^ ' .

[0020] FIG. 5 depicts closed-loop response of the unstable, minimum-phase, SISO plant in Example 7.1 using the nonminimum-phase-zero-based construction of z^u . The control is turned on at k = 0. The controller order is ⁿ° ^~ with parameters p = 1 and α(k) ≡ 10.

[0021] FIG. 6 depicts closed-loop response of the unstable, minimum-phase, SISO plant in Example 7.2 with a step command. The control is turned on at k = 200.

The controller order is ⁿ° ^~ with parameters p = 5, α(k) ≡ 5, and r = 10 with ^zu given by (56).

[0022] FIG. 7 depicts closed-loop response of the stable, minimum-phase, SISO plant in Example 7.1 with a step command and sinusoidal disturbance. The control is turned on at ^ = 200 jh_e controller order is ⁿ° ^~ with parameters P ^~ , a(k) ≡ 50 _{^ and r} = 3 _with ^B _zu g_{jven by (56}). [0023] FIG. 8 depicts closed-loop disturbance-rejection response of the stable, minimum-phase, SISO plant in Example 7.4. The control is turned on at ^ ^{= 2}OO .

The controller order is "^{c = 15} with parameters ^{p = 1} , ^W ^{≡ 25} , and ^{r = 3} with ^B™ given by (56).

[0024] FIG. 9 depicts time history of the components of ^ > for the stable, minimum-phase, SISO plant in Example 7.4. The control is turned on at ^ ^{= 2}OO .

[0025] FIG. 10 depicts bode magnitude plot of the adaptive controller in

Example 7.4 at £ = 1000 samples. The adaptive controller places poles at the disturbance frequencies ^{l ~ 7I>} rad/sample and "2 - ¹^⁵O _rad/sample. The controller magnitude ^c^^e ' is plotted for Ω _up to the Nyquist frequency ^{Nyq ~} rad/sample.

[0026] FIG. 1 1 depicts closed-loop disturbance-rejection response of the stable, nonminimum-phase, SISO plant in Example 7.5. The control is turned on at k = 200 jhe controller order is "^{c = 15} with parameters ^{p = 1} , ^a^ ^{≡ 25} , and ^r = ⁷ with ^zu given by (56). [0027] FIG. 12 depicts closed-loop disturbance-rejection response of the stable, nonminimum-phase, SISO plant in Example 7.5. The control is turned on at k = 200. The controller order is ⁿ< ^{= 15} with parameters ^{p = 1} , ^) ^{≡ 250}O , _and

— π B ^~ r - ' with ^zu given by (56). Compared to Figure 1 1 , the initial transient is reduced at the expense of convergence speed.

[0028] FIG. 13 depicts closed-loop disturbance-rejection response of the unstable, minimum-phase, SISO plant in Example 7.6. The control is turned on at k = 200. _The controller order is "^{c = 15} with parameters ^{p = 1} , ^a^ ^{≡ 25} , and ^r = ¹⁰ with ^zu given by (56). [0029] FIG. 14 depicts closed-loop disturbance-rejection response of the stable, minimum-phase, two-input, two-output plant in Example 7.7. The control is turned on at £ = 200. The controller order is "^{c = 15} with parameters P = ¹ , ot(k) ≡ \ _^ and ^r = 10 with ^zu given by (56).

[0030] FIG. 15 depicts closed-loop disturbance-rejection response of the stable, nonminimum-phase, two-input, two-output plant in Example 7.8. The control is turned on at k = ²OO . The controller order is ⁿ° ^~ with parameters P ^~ , ^a^ ' , and ^r = ^ with ^zu given by (56).

[0031] FIG. 16 depicts closed-loop disturbance-rejection response of the unstable, nonminimum-phase, two-input, two-output plant in Example 7.9. The control is turned on at ^k = ²⁰° . The controller order is "^{c = 10} with parameters ^{p = 1} , ^a^ ^{≡ 1} , and ^r = 10 with ^zu given by (56).

[0032] FIG. 17 depicts closed-loop performance comparison of the stable, nonminimum-phase, SISO plant in Example 7.5 with multiplicative error in B . We take

"^{c = 10} , P ^{= 1} , and ^a^ ^{≡ 100}° . The multiplicative error *7 , which is used to obtain the Markov parameters for ^zu given by (56) with ^{r =} 10 , is varied between 0.3 _anc| 5 The best performance is obtained for ^ , which corresponds to the true value of B .

[0033] FIG. 18 depicts closed-loop performance comparison of the stable, nonminimum-phase, SISO plant in Example 8.3 with a multiplicative error in the nonminimum-phase zero ² . We take ^{Hc ~} , P ⁼ , and ^a^ ' ⁼ . The nonminimum- phase-zero multiplicative error ¹I , which is used to construct ^zu given by (52), is varied between 0-75 and 2-5 jh_e best performance is obtained for ¹I ^~ , which is close to the true value of the nonminimum-phase zero.

[0034] FIG. 19 depicts closed-loop response of the unstable, minimum- phase, SISO plant in Example 7.6 with random white noise added to the measurement.

The control is turned on at k = 0 jh_e controller order is ^{Hc ~} with parameters P ^~ , a(k) = 25 _{^ anc}| _{r =} 3 _wjth ^.„ gj_{ven bv} ^y _The p_erf_ormance variable is degraded to the level of the additive sensor noise ^v' -* .

[0035] FIG. 20 depicts closed-loop disturbance-rejection response of the stable, minimum-phase, SISO plant in Example 7.4, where both the actuator and sensor are saturated at ± 2 . The control is turned on at ^ ⁼ 200 _ jhe controller order is ⁿ° ^~ with parameters ^ ^{= 1} , «(^) = 25 _{j anc}| _r = 3 _wjtn ^.„ g_{jven bv} ^g^ _Tne saturations degrade steady-state performance.

[0036] FIG. 21 depicts closed-loop step-command-following responses of the stable, minimum-phase, SISO plant in Example 7.4 with and without actuator saturation at ±0.1 jh_e control is turned on at ^ ⁼ 200 jh_e controller order is ^{Hc ~} with parameters ^{p = 1} , ^a^ ^{≡ 25} , and ^r = ³ with ^B™ given by (56).

[0037] FIG. 22 depicts model reference adaptive control problem with performance variable z . [0038] FIG. 23 depicts closed-loop model reference adaptive control of

Boeing 747 longitudinal dynamics. The controller order is ^{Hc ~} with parameters ^{p = 1} , "W ≡ ⁴⁰ , and ^{r = 1}° with ^B™ given by (56). The controller is turned on at t = Osec _; and the performance variable converges within about 20 sec.

[0039] FIG. 24 depicts closed-loop model reference adaptive control of missile longitudinal dynamics. The control effectiveness ^ = I , and thus the plant and reference model are identical. Therefore, the adaptive control input ^{Mac ~} .

[0040] FIG. 25 depicts missile longitudinal dynamics with control effectiveness ^ = °-50 _and adaptive controller turned off, that is, autopilot-only control. [0041] FIG. 26 depicts closed-loop model reference adaptive control of missile longitudinal dynamics with control effectiveness ^ = 0.50 jh_e augmented controllers provide better performance than the autopilot-only simulation.

[0042] FIG. 27 depicts closed-loop model reference adaptive control of missile longitudinal dynamics with control effectiveness ^ ⁼ 0-25 After a transient, the augmented controllers stabilize the system, whereas the autopilot-only simulation fails. Note that the system is stabilized despite the total control input ^u reaching the actuator saturation level of ^{± 3Ode}§ .

[0043] FIG. 28 depicts closed-loop model reference adaptive control of missile longitudinal dynamics with control effectiveness ^ = 0.25 jh_e adaptive controller is initialized with the converged gains from the 50% control effectiveness case. The initial transient is reduced as compared with initializing the control gains to zero. In this case, the actuator saturation level is never reached.

[0044] Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.

DETAILED DESCRIPTION

[0045] Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments are provided so that this disclosure will be thorough, and will fully convey the scope to those who are skilled in the art. Numerous specific details are set forth such as examples of specific components, devices, and methods, to provide a thorough understanding of embodiments of the present disclosure. It will be apparent to those skilled in the art that specific details need not be employed, that example embodiments may be embodied in many different forms and that neither should be construed to limit the scope of the disclosure.

[0046] The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. As used herein, the singular forms "a", "an" and "the" may be intended to include the plural forms as well, unless the context clearly indicates otherwise. The terms "comprises," "comprising," "including," and "having," are inclusive and therefore specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. The method steps, processes, and operations described herein are not to be construed as necessarily requiring their performance in the particular order discussed or illustrated, unless specifically identified as an order of performance. It is also to be understood that additional or alternative steps may be employed.

PROBLEM FORMULATION

[0047] We begin by first considering the MIMO discrete-time system jc(Jfc + l) = Ax(Ic) + Bu(Ic) + D₁W(Ic), (1 ) y(k) = Cx(k) + D₂w(k), (2) z(k) = E₁X(Ic) + E_ow(k), (3)

where *<*> ^{e R}" , **> ^{e R} " , ^z^ ^{e R}'^z , ^M ^ ^{e R}'^u , ^w^ ^{e R}'^w , and * ≥ 0 . _Our goal is to develop an adaptive output feedback controller under which the performance variable z is minimized in the presence of the exogenous signal ^w . In (1 )-(3), ^w can represent either a command signal to be followed, an external disturbance to be rejected, or both. For example, if A = ⁰ _and ° ^≠ , then the objective is to have the output ^Ei^x follow the command signal ^{~ E}°^w . On the other hand, if A ≠ ° and ^E° ⁼ ° , then the objective is to reject the disturbance ^w from the performance variable ^lX . The combined command-following and disturbance-rejection problem is addressed when i and ° are suitably partitioned matrices. More precisely, if ^{1 "} L n J ₁

follow the command ⁰^² while rejecting the disturbance ^llWl . Lastly, if ^l and ° are zero matrices, then the objective is output stabilization, that is, convergence of z to zero. We assume that ^ ' ^ is stabilizable, ^ ' -* and ^ ' ^Ei ' are detectable, and that measurements of ^y and z are available for feedback. If the command signal is included as a component of ^y , then the adaptive controller has a feedforward architecture. For disturbance-rejection problems, the controller does not require measurements of the external disturbance ^w .

ARMAX MODELING

[0048] Consider the ARMAX representation of (1 ), (3), given by

where _{We define the re|ative}

degree ^ ^≥ 1 as the smallest positive integer * such that the ' th Markov parameter

_{js nonzero Note that>} then whereas, if

[0049] Letting the data window size

be a positive integer, we define the

extended performance vector

and

by

where T The data window size P has a small but noticeable effect on

transient behavior. Now, (4) can be written in the form

and

Note that ^z includes modeling information about the plant poles and exogenous input path, whereas

includes modeling information about the plant zeros. Both

and

B, have block-Toeplitz structure.

CONTROLLER CONSTRUCTION

[0050] To formulate an adaptive control algorithm for (1 )-(3), we use a strictly proper time-series controller of order

such that the control

is given by

where, for all '

and

The controller order

is determined by standard control guidelines in terms of stabilization and disturbance rejection. The control (10) can be expressed as

where

is the controller gain matrix, and the regressor vector

is given by

[0051] We define the extended control vector

by

where

. Note that, if then From (1 1 ), it follows that the

extended control vector U can be written as

where

[0052] Next, we define the retrospective performance vector

by

where

is the surrogate controller gain matrix, is the

surrogate input matrix, and

is the recomputed extended control vector. Substituting (6) into (17) yields

Note that the expression for given by (19) does not depend on either the

exogenous signal ^w or the matrix

which includes information about the open-loop poles as well as the transfer function from ^w to z . Hence, we do not need to know this model data, and, when ^w represents a disturbance, we do not need to assume that ^w is known. However, when ^w represents a command, then ^w can be viewed as an additional measurement ^y , and thus the controller has feedforward action. The matrix z^u is discussed further below.

[0053] Note that (19) can be rewritten as

where

vec is the column-stacking operator, and ® represents the Kronecker product. [0054] Now, consider the retrospective cost function

w

definite, and (24)

[0055] Substituting (20) into (23) yields (25) where

(26) (27)

(28)

Since ^ j_s positive definite, has the strict global minimizer gjven

by

Equation (29) is the adaptive control update law. Note that ^zu (which appears in ^■> (*' and

must be specified in order to implement (29). Furthermore, (29) requires the on-line inversion of a positive-definite matrix of size n

[0056] In the special case

where

[0057] Using the matrix inversion lemma it follows that

Consequently, in this case, the update law (29) requires the on-line inversion of a positive-definite matrix of size

We use the weightings (30), (31 ) for all of the examples in the present application. The weighting parameter

introduced in (31 ) is called the learning rate since it affects the convergence speed of the adaptive control algorithm. As ^a^ > is increased, a higher weight is placed on the difference between the previous controller coefficients and the updated controller coefficients, and, as a result, convergence speed is lowered. Likewise, as

is decreased, convergence speed is raised. By varying ^a^ ' , we can effect tradeoffs between transient performance and convergence speed.

[0058] We define the retrospective performance variable by

In the particular case using in place of

in the regressor vector (13) yields faster convergence. Therefore, for we redefine (13) as

[0059] The novel feature of the adaptive control algorithm given by (1 1 ) and (29) is the use of the retrospective correction filter (RCF) (19), as shown in Figure 1 for

RCF provides an inner loop to the adaptive control law by modifying the extended performance vector

in terms of the difference between the actual past control inputs

and the recomputed control inputs

MARKOV-PARAMETER POLYNOMIAL

[0060] By recursively substituting (1 ) into (3), it follows that

can be represented by

where and, for all . In terms of the backward-

shift operator (38) can be rewritten as

Shifting (39) forward by ^r steps gives

where

and

We call PrW the Markov-parameter polynomial. Note that PrW j_{s a} matrix polynomial in the MIMO case and a polynomial in the SISO case. Furthermore, since

_when d > 2 , it follows for all

_can be written as

[0061] The Markov-parameter polynomial PrW contains information about the relative degree d and, in the SISO case, the sign of the high-frequency gain, that is, the sign of ^d . We show below that PrW _a|_so contains information about the transmission zeros of

₃ which is given by

[0062] In order to relate the transmission zeros of ^m to PrW _; the Laurent series expansion o

about

is given by

[0063] This expansion converges uniformly on all compact subsets of

where p(

A) is the spectral raduius radius of A . By truncating the summation in

(45), we obtain the truncated Laurent expansion

of , given by

[0064] Consequently, the Markov-parameter polynomial

is closely related to the truncated Laurent expansion of .

Approximation of Outer Nonminimum-Phase Zeros

[0065] In the case of MIMO systems,

j_{s a} matrix polynomial and thus does not have roots in the sense of a polynomial. We therefore require the notion of a Smith zero. Specifically,

is a Smith zero of PrW if the rank of

is less than the normal rank of

that is, the maximum rank of Z

taken over all . [0066] Definition 5.1 - Let be a transmission zero of Then, is

an outer zero of

if . Otherwise, ' is an inner zero of ^zu .

[0067] The following result shows that the Smith zeros of the Markov- parameter polynomial

asymptotically approximate each outer transmission zero of

[0068] Fact 5.1 - Let be an outer transmission zero of

. For each denote the set of Smith zeros of P

. Then, there exists a

sequence

that converges to

[0069] The following specialization to SISO transfer functions shows that the roots of

asymptotically approximate each outer zero of

[0070] Fact 5.2 Consider

and let

^ be an outer zero of

For each } be the set of roots of Then, there exists a

{ sequence that converges to

as

[0071] The following examples illustrate Fact 5.2 by showing that, as ^r increases, roots of the Markov-parameter polynomial P

M\ and hence, roots of the numerator of the truncated transfer function

, asymptotically approximate each outer nonminimum-phase zero of

. The remaining roots of

are either located at the origin or form an approximate ring with radius close to

These roots are spurious and have no effect on the adaptive control algorithm. [0072] Example 5.1 (SISO, nonminimum-phase, stable plant). Consider the plant

poles _minimum._phase

zeros _{and o}ut_er nonminimum-phase zeros Table 1

lists the approximated nonminimum-phase zeros obtained as roots of

as a function of Note that as ^r increases, the outer nonminimum-phase zeros are more

closely approximated by the roots of . See Figure 2.

Table 1 : Approximated nonminimum-phase zeros obtained as roots of PrW as a function of ^r for the stable, nonminimum-phase plant in Example 5.1. As ^r increases, the outer zeros are more accurately modeled.

[0073] Example 5.2 (SISO, nonminimum-phase, unstable plant). Consider the plant ^G- with d = 2 , H₂ = I _po|es 0.5 ± 0.5J,-0.5 ± 0.5J,±0.7J,-0.95,1.4 _! minimum-phase zeros U.3 - U.7J,-U.7 _ UJJ ^ _out_er nonminimum-phase zero ^~l-5 , and inner nonminimum-phase zero 1-25 . Figure 3 shows the roots of P²ⁱW . Note that the root of P²⁵W outside ^ ' is close to the outer nonminimum-phase zero ^~l-5 . However, the inner nonminimum-phase zero 1-25 js not approximated by a root of

P²⁵W jh_e remaining roots are either located at the origin or form an approximate ring with radius close to P(A)

Approximation of Inner Nonminimum-Phase Zeros [0074] Example 5.2 illustrates that the roots of P^rW approximate each

C C outer nonminimum-phase zero of ^zu . However, inner nonminimum-phase zeros of ^zu are not approximated by roots of PrW . To overcome this deficiency, we can use information about the plant's unstable poles to create a modified Markov-parameter polynomial PrW whose roots approximate each nonminimum-phase zero of ^zu . For illustration, assume that the SISO plant ^zu has a unique unstable pole

whose

absolute value is greater than all other poles of . Then, we define

where, for are the modified Markov parameters, and

. By repeating this operation for each unstable pole of the roots of the

modified Markov-parameter polynomial

can approximate each nonminimum-phase zero of The following example

illustrates this process.

[0075] Example 5.3 (Ex. 5.2 with pole information) Reconsider Example 5.2, where the inner nonminimum-phase zero 1-25 js not approximated by a root of

Using knowledge of the unstable pole 1-4 to construct

given by (48), Figure 4 shows the roots of

Note that the roots outside P

where ^ is the dynamics matrix of a minimal realization of

are close to the nonminimum-phase zeros of ^zu . The remaining roots are either located at the origin or form an approximate ring with radius close to

CONSTRUCTION OF

[0076] We present four constructions for

based on the available modeling information. Based Construction

[0077]

given by (8) is known, then, with

can be chosen to be equal to ^zu . In this case,

becomes

[0078] This construction of captures information about the relative degree d , the first nonzero Markov parameter (since

and exact values of all transmission zeros of , that is, both minimum-phase and nonminimum-phase

transmission zeros.

Nonminimum-Phase-Zero-Based Construction [0079] Consider

and assume that

and the nonminimum- phase zeros of

are known. Then we define the nonminimum-phase-zero polynomial

to be the polynomial whose roots are equal to the nonminimum-phase zeros of

that is,

where is the number of nonminimum-phase zeros in

, and . If

, that is,

is minimum phase, then

with

, the nonminimum- phase-zero-based construction of

is thus given by

wwhheerree

This construction of

captures information about the relative degree d _; the first nonzero Markov parameter, and exact values of all nonminimum-phase zeros of . In the minimum-phase case, the only required modeling information is

. This construction of can be extended to the MIMO

case by replacing each minimum-phase zero in the Smith-McMillan form of by a

zero at z = 0

r -MARKOV-Based Construction

[0080] Replacing

with j_n (4) _and substituting the resulting relation

back into (4) yields a 2-MARKOV model. Repeating this procedure ^{r ~}1 times yields the r -MARKOV model of (1 )-(3)

where, for the coefficients are given

by

Note that

. We represent (52) with

as the r -MARKOV transfer function

[0081] The system representation (54) is nonminimal since its order is and thus (54) includes poles that are not present in the original model.

Furthermore, note that the coefficients of the terms

through zⁿ in the denominator are zero. These facts are irrelevant for the following development. Using the numerator coefficients of (54), the ^r -MARKOV-based construction of

with is given by

H₁ H_r β_r,2 - £_,„ O_1. x'.. ⁰KxL

O_{1 xl} ' ^{■ ■} β =

(55)

[0082] This construction of β ^z.^u captures information about the relative degree d , the first nonzero Markov parameter, and exact values of all transmission zeros of ^zu , that is, both minimum-phase and nonminimum-phase transmission zeros.

Markov-Parameter-Based Construction

[0083] Using the numerator coefficients of (46), the Markov-parameter-

D based construction of ^zu with ^μ° ^{~ Hc} ^ ' ^± is given by

[0084] The Markov parameters are the numerator coefficients of a truncated

Laurent series expansion of ^zu about z = °° . The Markov parameters contain information about the relative degree d and, as shown by Fact 5.2 for the SISO case, approximate values of all outer nonminimum-phase zeros of ^zu . The advantage in using ^zu given by (56) rather than (55) is that "^{r 2}'^{- "}' "^{r »} need not be known. If,

C C however, ^zu has inner nonminimum-phase zeros and the unstable poles of ^zu whose absolute values are greater than at least one inner nonminimum-phase zero are known,

TT TT then we can replace the Markov parameters ⁿ^--^^rir j_n (56) by the modified Markov parameters ⁿ ffi^>- - -^{> n} ffr given in (47). If these poles are not known, then B ^zu can be chosen to be either ^zu , the nonminimum-phase-zero form (51 ), or the >- -MARKOV form (55). [0085] Note that, if the order ⁿ of the system is known and 2w + l Markov parameters are available, then a state-space model of the system can be reconstructed by using the eigensystem realization algorithm. However, the examples considered in sections below use substantially fewer Markov parameters.

NUMERICAL EXAMPLES: NOMINAL CASES

[0086] We now present numerical examples to illustrate the response of the

RCF adaptive control algorithm under nominal conditions. We consider a sequence of examples of increasing complexity, ranging from SISO, minimum-phase plants to MIMO, nonminimum-phase plants, including stable and unstable cases. Each SISO example is constructed such that ^{d ~} . All examples assume ^{y = z} with Ψ^) _gj_ven ^y (37^ and, in all simulations, the adaptive controller gain matrix ' -* is initialized to zero.

Unless otherwise noted, all examples assume *' -* ^~ .

[0087] Example 7.1 (SISO, minimum-phase, unstable plant, stabilization) Consider the plant ^m with ^ = I , poles ^¹-⁵ , and inner nonminimum-phase zero

1 o< D E B

^{" 1}^³ . For stabilization, we take ^l and ° to be zero matrices. Let ^zu be given by

TT 1

(51 ), which is constructed using the first nonzero Markov parameter ⁿι ^{~ ι} and the location of the nonminimum-phase zero -1-25 _; that is, ^ (^<?) = ^<? + 1-²⁵ . VVe take ^{Hc ~} , P ^~ , and ^W = IU jhg closed-loop response is shown in Figure 5 for 40) = [0.1 0.4]^τ .

[0088] Example 7.2 (SISO, minimum-phase, unstable plant, command following). Consider the double integrator plant G_zu with d = 3, poles 0.5 ± 0.5J, -0.5 ± 0.5J,1,1, and a minimum-phase zeros 0.3 ± 0.7 j , 0.5. We consider a command-following problem with step command w(k) = 1. With the plant realized in controllable canonical form, we take D₁ = Q and E₀ = -1. We take n_c= 10, p = 5, a(k) ≡ 5, and r= 10 with B_zu given by (56). The closed-loop response is shown in Figure 6.

[0089] Example 7.3 (SISO, minimum-phase, stable plant, command following and disturbance rejection). Consider the plant G_zu with d = 3, poles 0.5 ±0.5/, -

0.5±0.5y, ±0.9, ±0.7/, and minimum-phase zeros 0.3 ± 0.7/, 0.7 ± 0.3/, 0.5. We consider a combined step-command-following and disturbance-rejection problem with command w _λ and disturbance w₂ given by

where rad/sample. With the plant realized in controllable canonical form, we

A take

and

The disturbance, which is not matched, is assumed to be unknown, and the command signal is not used directly. We take a_{nd with} ^B g_iven by (56). The closed-loop response

is shown in Figure 7.

[0090] The following examples are disturbance-rejection simulations, that is,

, with the unknown two-tone sinusoidal disturbance

where

rad/sample and

rad/sample. With each plant realized in

D controllable canonical form, we take and, therefore, the disturbance is not matched.

[0091] Example 7.4 (SISO, minimum-phase, stable plant, disturbance rejection) Consider the plant ^G- with ^ = 3 , poles

and minimum-phase zeros

_{Taking ^}

and

with

given by (56), the closed-loop response is shown in Figure 8. The control algorithm converges (see Figure 9) to an internal model controller with high gain at the disturbance frequencies, as seen in Figure 10.

[0092] Example 7.5 (SISO, nonminimum-phase, stable plant, disturbance rejection) Consider the plant with poles

minimum-phase zeros

_and outer nonminimum-phase zero 2 . We take and The Markov-parameter polynomial used to

construct K as in (56) is given by _{wjth r0Qts}

N_{ote that the root} 1.94 _{approxjmates the zero 2}. The closed-

loop response is shown in Figure 1 1.

[0093] To illustrate the effect of the learning rate

the closed-loop response is shown in Figure 12 for

and all other parameters unchanged. Note that, with

the initial transient is reduced at the expense of convergence speed.

[0094] Example 7.6 (SISO, minimum-phase, unstable plant, disturbance rejection). Consider the plant

with

poles a_nd minimum-phase zeros

_{We take and with}

given by (56). The closed-loop response is shown in Figure 13.

[0095] Example 7.7 (MIMO, minimum-phase, stable plant, disturbance rejection). Consider the two-input, two-output plant

where and

Consequently,

has poles -

_and minimum-phase transmission zeros _{We take}

and with given by (56). The closed-loop response is shown in

Figure 14.

[0096] Example 7.8 (MIMO, nonminimum-phase, stable plant, disturbance rejection) Consider the two-input, two-output plant G

where A is given in Example 7.7, _and

Consequently,

has poles

_i minimum-phase transmission zero 0.5 _; and outer nonminimum-phase transmission zero 2 . We take

, , _{^ and}

_{with given by} (₅₆). _The

closed-loop response is shown in Figure 15.

[0097] Example 7.9 (MIMO, nonminimum-phase, unstable plant, disturbance rejection) Consider the two-input, two-output plant

where

₄(z) z z z , , Consequently,

has poles m_inimum._{phase tranS}mission zero 0.5 , _and

outer nonminimum-phase transmission zero 2 . We take ⁿ and

with

given by (56). The closed-loop response is shown in Figure 16.

NUMERICAL EXAMPLES: OFF-NOMINAL CASES

[0098] We now revisit the numerical examples of the preceding section to illustrate the response of the RCF adaptive control algorithm under conditions of uncertainty in the relative degree and Markov parameters as well as measurement noise and actuator and sensor saturation. In each example, the adaptive controller gain matrix

' -* is initialized to zero. Unless otherwise noted, all examples assume *' -* .

[0099] Example 8.1 (Ex. 7.5 with Markov-parameter multiplicative error) Reconsider Example 7.5 with Markov-parameter multiplicative error. For controller implementation, we use the estimate B = ηB , where ηε R is varied between 0-3 and

⁵ . For ^{z = 1}'- "' ^r , the estimated Markov parameters ' ^~ are used to construct

^B™ given by (56). Taking ⁿc ^{= 15} , P = ¹ , r = 10 _{> an}d αW ≡ lOOO _{^ the dosed}.|_oop performance is compared in Figure 17. In each case, the control is turned on at k = 0 _^ and the performance metric is given by Jt₀ = min{Jt > 9 : — ∑ I z(Jt -i) l< 0.01 }, (59)

that is, ° is the minimum time step ^ such that the average of * ^z^ ^{~ 1}' '^l=0 is less than 0.01. Figure 17 shows that the best performance is obtained for ^ , which corresponds to the true value of B . As ⁷I is decreased, convergence slows significantly. [0100] In the case where the sign of the first nonzero Markov parameter (the

TT TT sign of the high-frequency gain) is wrong, that is, ^{3 ~ 3} , the simulation fails. These simulations suggest that performance degradation due to an unknown scaling of the Markov parameters provides a useful measure of adaptive gain margin. These findings are consistent with the adaptive gain-margin results.

[0101] Example 8.2 (Ex. 7.5 with unknown latency). A known latency of ^ steps can be accounted for by replacing d by d + l in the construction of ^zu . However, we now assess the effect of unknown latency in Example 7.5, which is equivalent to uncertainty in the relative degree ^ . The system has relative degree d = 3 p_or controller implementation, we use the erroneous estimate d ₀\ d _anc| t_ak_e "<= ^~ P = I _t a(k) ≡ 1000 _{; and} r = 10 _with ^B _z« _g\_{ven by (56}). _Letting d be either 2, 3, 4, 5, or 6, Table 2 compares both the performance metric (59) and the maximum value of z' -* for each estimate d of ^ . In each case, the control is turned on at k = 0 jh_e best performance is obtained for d = d = 3

Table 2: Closed-loop performance comparison of the stable, nonminimum-phase, SISO plant in Example 7.5 with unknown latency. For controller implementation, we use the erroneous estimate d of ^d and take ⁿ' ^{= 15} , P = ¹ , a(k) ≡ 1000 _{^ and r} = 10 _with ^B _Zu given by (56). The best performance is obtained for d = d = 3

[0102] These simulations show the sensitivity of the adaptive controller to unknown errors in the relative degree ^ , which provides a useful measure of adaptive phase margin.

[0103] Example 8.3 (Sensitivity to nonminimum-phase-zero uncertainty). Consider the plant ^zu with ^ = I , i , poles ^υ'^υ-⁵ , and outer nonminimum-phase zero 2 . The plant is subject to disturbance ^w^ ' given by (58), and thus, with the plant realized in controllable canonical form, we take ^{l = 2} and ° ^~ . Furthermore, we assume ^{y ~ z} and let ^' -* be given by (37). To illustrate the sensitivity of the adaptive control algorithm to knowledge of the nonminimum-phase zero, we let ^zu be given by

TT 1 (51 ), which is constructed using the first nonzero Markov parameter ^{λ ~} , the nonminimum-phase zero 2 , and a multiplicative error ^e R , that is, '^ ^{~ Q ~}^V _m We

vary *7 between °-⁷⁵ and ²-⁵ with ^ = ¹⁰ , P = ¹ , and ^a^ ^{≡ 25}. A closed-loop performance comparison is shown in Figure 18. In each case, the control is turned on at k = 0 _j and the performance metric is given by (59). The best performance is obtained for ⁷I ⁼ , which is close to the true value of the nonminimum-phase zero. Note that the adaptive control algorithm is more robust to larger values of ⁷I than smaller values. [0104] Example 8.4 (Ex. 7.6 with stabilization and noisy measurements). Reconsider Example 7.6 with no commands or disturbances. For stabilization, we take

D F i and ° to be zero matrices. To assess the performance of the adaptive algorithm with added sensor noise, we modify (2) and (3) by y(k) = z(k) = E₁x(k) + E_Qw(k) + v(k), ₍₆₀₎

where v(k) G R'^z is Gaussian white noise with mean v = 2 _anc| standard deviation

σ = 0.1. we take "^{c = 15} , ^{p = 1} , ^a^ ^{≡ 25} , and ^r = ³ with ^B™ given by (56). For the initial condition -1.67 0.13 0.29 -1.15 1.19 1.19

the closed-loop response is shown in Figure 19.

[0105] Example 8.5 (Ex. 7.4 with actuator and sensor saturation). Reconsider Example 7.4 with the additional assumption that both the control input and sensor measurement are subject to saturation at ± 2 . vVe take ⁿ° ^~ , P ^~ , a(k) = 25 _{^ anc}| _r = 3 vvith ^zu given by (56). The closed-loop response shown in Figure 20 indicates that the saturations degrade steady-state performance.

[0106] Example 8.6 (Ex. 7.4 with command following and actuator saturation). Reconsider Example 7.4 with step command given by ¹⁴W - I wrtn the plant realized in controllable canonical form, we take ^l and ° . Taking ⁿ* ^{= 15} , P = ¹ , ^αW ^{≡ 25} , and ^r = ³ with ^B™ given by (56), the closed-loop responses are shown in Figure 21 with and without actuator saturation at ±0.1 with actuator saturation, the performance variable reflects the capability of the saturated control.

MODEL REFERENCE ADAPTIVE CONTROL [0107] Model reference adaptive control (MRAC), as illustrated in Figure 22, is a special case of (1 )-(3), where z = y₁ - y_m is the difference between the measured output ^¹ of the plant ^ and the output ^^m of a reference model ^m . For MRAC, the exogenous command ^w is assumed to be available to the controller as an additional measurement variable ^². Unlike standard MRAC methods, retrospective cost adaptive control does not depend on knowledge of the reference model ^m .

[0108] We now present numerical examples to illustrate the response of the RCF adaptive control algorithm for model reference adaptive control (see Figure 22).

Unless otherwise noted, the adaptive controller gain matrix ^θ^) j_s initialized to zero.

Boeing 747 longitudinal dynamics

[0109] Consider the longitudinal dynamics of a Boeing 747 aircraft, linearized about steady flight at 40,000 ft and 774 ft/sec. The inputs to the dynamical system are taken to be elevator deflection and thrust, while the output is the pitch angle. The continuous-time equations of motion are thus given by

-0.003 0.039 0 0.010 1

-0.065 -0.319 7.74 -0.180 -0.040

0.020 -0.101 -0.429 -1.160 0.598

0 0 1 0 0

(61 )

z = y₁ - y_m, (63) where ^w is the exogenous command and ^y>n is the output of the reference model

Y_m(s) 0.0131

GM = W(s) s² +0.16s + 0.013l - (64)

[0110] We discretize (61 )-(64) using a zero-order hold and sampling time s ^{~ - sec} . The reference command is taken to be a 1- ^e§ step command in pitch angle. The controller order is ⁿ^ ^{= 1}° with parameters ^{p = 1} , ^α(^) ^{≡ 40} _; and ^{r = 1}° with ^zu given by (56). The closed-loop response is shown in Figure 23 for zero initial conditions. Missile Longitudinal Dynamics

[0111] We now present numerical examples for MRAC of missile longitudinal dynamics under off-nominal or damage situations. The missile longitudinal plant is derived from the short period approximation of the longitudinal equations of motion, given by

where

and ^ ' -I represents the control effectiveness. Nominally, ^ - 1.

[0112] The open-loop system (65), (66) is statically unstable. To overcome this instability, a classical three-loop autopilot is wrapped around the basic missile longitudinal plant. The adaptive controller then augments the closed-loop system to provide control in off-nominal cases, that is, when λ < \ jh_e autopilot and adaptive controller inputs are denoted ^Wap and ^Wac , respectively. Thus, the total control input

- ^a _P ^ac -fh_e reference model ^m consists of the basic missile longitudinal plant with ^ = I and the classical three-loop autopilot. An actuator amplitude saturation of _30deg = _0.524 _ra(_j _js j_nc|_uc|_ec| j_{n t}h_{e mo}del, but no actuator or sensor dynamics are included.

[0113] The goal is to have the missile follow a pitch acceleration command ^w consisting of a 1-g amplitude 1-Hz square wave. The performance variable z is the difference between the measured pitch acceleration z and the reference model pitch

A * acceleration z , that is, z = A -A^* The closed-loop response is shown in Figure 24 for ^ = I . Since the plant and reference model are identical in the nominal case, the adaptive control input ^{Wac ~} .

[0114] All of the following examples use zero initial conditions and the same adaptive controller parameters. The adaptive controller is implemented at a sampling rate of ³⁰° Hz. We take "^{c = 3} , ^{p = 1} , and ^{r = 2}° with ^B™ given by (56). A time- varying learning rate ^a(k) = 75k + l _{js usec}| _{S(jch thatj} j_nj_tj_a||_Vj controller adaptation is fast, and, as performance improves, the adaptation slows. The learning rate is identical for each simulation. System identification using the observer/Kalman filter identification

(OKID) algorithm is used to obtain the ^O Markov parameters required for controller implementation. The offline identification procedure is performed with a nominal simulation (^ = 1) by injecting band-limited white noise at the adaptive controller input

^Wac and recording the performance variable z while the autopilot is in the loop. No external disturbances are assumed to be present during the identification procedure.

[0115] Example 9.1 (50% control effectiveness). Consider ^ = 0.50 Figure 25 shows simulation results with the adaptive controller turned off, that is, autopilot-only control. Now, with the autopilot augmented by the adaptive controller, simulation results are shown in Figure 26. After a transient, the augmented controllers provide better performance than the autopilot-only simulation.

[0116] Example 9.2 (25% control effectiveness). Consider ^ = 0.25 with the adaptive controller turned off, that is, autopilot-only control, the simulation fails. With the autopilot augmented by the adaptive controller, simulation results are shown in

Figure 27. After a transient, the augmented controllers stabilize the system, whereas the autopilot-only simulation fails.

[0117] Figure 27 shows that the total control input ^u reaches the actuator saturation level of --^Oαeg j_{Q rec}|_uce ^_e j_njtj_a| transient, we initialize the adaptive controller with the converged control gains ^ from the 50% control effectiveness case. As shown in Figure 28, the initial transient is reduced as compared with initializing the control gains to zero. In this case, the actuator saturation level is not reached.

CONCLUSION

[0118] We presented the RCF adaptive control algorithm, system, and method and demonstrated its effectiveness in handling nonminimum-phase zeros through numerical examples illustrating the response of the algorithm under conditions of uncertainty in the relative degree and Markov parameters, measurement noise, and actuator and sensor saturations. Bursting was not observed in any of the simulations. We also suggested metrics that can serve as gain and phase margins for discrete-time adaptive systems. Development of Lyapunov-based stability and robustness analysis of the RCF adaptive control algorithm as well as development of a theoretical foundation for analyzing broadband disturbance-rejection properties of the controller is anticipated.

[0119] The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the invention, and all such modifications are intended to be included within the scope of the invention.

Claims

CLAIMS What is claimed is:

1 . A method for adaptive control of nonminimum-phase systems, said method comprising: receiving input data; distinguishing between said input data and modified data; determining control gains from said input data and estimates of nonminimum phase zeros using a modified quadratic cost criterion that uses a retrospective performance vector; calculating a control signal based on said control gains, said input data, and said retrospective performance vector; and outputting said control signal.

2. The method according to Claim 1 wherein said determining control gains comprises determining control gains using estimates of a relative degree, a first nonzero Markov parameter, and nonminimum-phase zeros.

3. The method according to Claim 1 wherein said determining control gains comprises determining control gains using an explicit minimizer of said modified quadratic cost criterion to achieve a minimizer in a single operational step.

4. The method according to Claim 1 wherein said determining control gains comprises determining control gains using a one-step learning penalty.

5. The method according to Claim 1 wherein said determining control gains from said input data and estimates of nonminimum phase zeros using a modified quadratic cost criterion comprises a disturbance rejection response.

6. The method according to Claim 1 wherein said determining control gains from said input data and estimates of nonminimum phase zeros using a modified quadratic cost criterion comprises a command following response.

7. The method according to Claim 1 wherein said determining control gains from said input data and estimates of nonminimum phase zeros using a modified quadratic cost criterion comprises a stabilization response.

8. The method according to Claim 1 , wherein said determining control gains further comprises adjusting a rate of convergence.

9. A system for adaptive control of nonminimum-phase systems, said system comprising: a device receiving input data; a device distinguishing between said input data and modified data; a controller determining control gains from said input data and estimates of nonminimum phase zeros using a modified quadratic cost criterion, said controller calculating a control signal based on said control gains and said input data and outputting said control signal.

10. The system according to Claim 9 wherein said controller determines control gains using a relative degree, a first nonzero Markov parameter, and nonminimum-phase zeros.

1 1 . The system according to Claim 9 wherein said controller determines control gains using estimated Markov parameters to obtain estimates of a relative degree, a first nonzero Markov parameter, and nonminimum-phase zeros.

12. The system according to Claim 9 wherein said controller determines control gains using an explicit minimizer of said modified quadratic cost criterion to achieve a minimum in a single operational step.

13. The system according to Claim 9 wherein said controller determines control gains using a one-step learning penalty.

14. The system according to Claim 9 wherein said controller employs a modified quadratic cost criterion to define a disturbance rejection response.

15 The system according to Claim 9 wherein said controller employs a modified quadratic cost criterion to define a command following response.

16. The system according to Claim 9 wherein said controller employs a modified quadratic cost criterion to define a stabilization response.

17. The system according to Claim 9, wherein said controller comprises a device for adjusting a rate of convergence.