WO1990015369A1

WO1990015369A1 - A controller for a system

Info

Publication number: WO1990015369A1
Application number: PCT/AU1990/000244
Authority: WO
Inventors: Bruce Penfold; Rob Evans
Original assignee: The University Of Newcastle Research Associates Limited
Priority date: 1989-06-05
Filing date: 1990-06-04
Publication date: 1990-12-13

Abstract

A method of controlling a system (12) using a controller (11). A trajectory reference model T (13) establishes an absolute goal to which the behaviour reference system R (14) will tend by virtue of a constant KR (15). The controller comprises a time local prediction means and a behaviour algorithm resetting means which are both arranged to be utilized to produce an output which ensures that the subject system (12) behaves as though it was the behaviour reference system R (14). The time local prediction means is arranged to predict from a past input from the system to the controller, a control system condition at the next control time based on the assumption that the output applied at a first control time will again be applied at the next control time. The behaviour algorithm resetting means is arranged at the first control time from the input received at the first control time to set the initial conditions of a model equal to the condition of the system indicated by the input from the system.

Description

A CONTROLLER FOR A SYSTEM

FIELD OF THE INVENTION

The present inventions relates to a method of controlling a system with applications varying from radar systems and chemical processes to control of robots and medical apparatuses.

BACKGROUND OF THE INVENTION

Typically system controllers are designed based on firstly developing a mathematical model of the system to be controlled and then using a control design technique to derive a controller which achieves the desired system performance. This approach. to control system design

produces very good controllers when the mathematical model for the system is well known and reasonably simple.

However, if the system cannot be mathematically modelled or if the system parameters are constantly changing with time these traditional control design techniques break down.

To overcome the second of these problems adaptive control techniques have been attempted but they still rely on being able to represent the system with linear

parameterized models.

Traditional approaches to system control are also deficient in that they are unable to exploit current

technology to its fullest. In this respect many control strategies are based on simple recursive equations which can be coded into half a dozen lines and executed on a

conventional Von Neumann achitecture computer. Such

approaches are only able to exploit the speed advantage of modern computers and are unable to exploit the complexity that can now be put onto an integrated circuit chip.

Technology's ability to replicate tens of

thousands of simple cells on a chip to accomplish the function of a controller enables the utilization of a new approach to control system design. DISCLOSURE OF THE INVENTION

According to the present invention there is provided a controller for a system, the controller

comprising a time local prediction means and a behaviour algorithm resetting means which are both arrangedto be utilized to produce an output, at a first control time, to a system, based on a reference input to the controller, the controller being arranged to receive an input from the system which input is indicative of the condition of the system after it receives the output; the time local

prediction means being arranged to predict from a past input from the system to the controller, a controlled system condition at the next control time, based on the assumption that the output applied at a first controlled time will again be applied at the next control time; and the

behavioural algorithm resetting means being arranged, at the first control time, from the input received at the first control time, to set the initial conditions of a model which behaves, up until the next control time, in a manner desired of the system, equal to the condition of the system

indicated by the input.

According to another aspect of the present

invention there is provided a method of controlling a system comprising the steps of producing an output, at a first control time, from a controller to the system, based on a reference of the controller; receiving at the controller an input from the system which input is indicative of the condition of the system after it has received the output; utilizing a time local prediction and a behaviour algorithm resetting to produce a further output to the system at the next control time; the time local prediction being a

prediction made from a past input from the system, of a controlled system condition at the next control time, based on the assumption that the output applied at the first control time will again be applied at the next control time; and the behaviour algorithm resetting involving, at the first control time, from the input received at the first control time, setting the initial conditions, of a model which behaves, up until the next control time, in a manner desired of the system, equal to the condition of the system indicated by the input. Preferably the model is an

algorithm.

According to another aspect of the present

invention there is provided a discrete time or sampled data controller which includes a means by which a prediction of a controlled system condition at the next control time, under the assumption that the present control output persists, is formed and utilized; and a means by which an algorithm or process which behaves in the manner desired of the process being controlled is reinitialized to the conditions of the system being controlled at the current control time, which means are employed to compute the control output for the system. Preferably the conditions are measured or imputed.

Preferably each of the above controller is a continuous time controller.

It is preferred that the reference is an internal reference.

Preferably the controller is a data processor which is arranged to be connected with a system.

According to one embodiment of the present invention there is provided a controller for controlling a system, S to behave as though it were a reference system R, wherein, the controller is arranged to produce a control output to the system such that at a particular control time t(k), the system state time vector V_S at a first point

[y_S(k), Sk), t(k)] is used to set

the reference system state time vector V_R at the first point [y_R(k), (k), t(k)],

such that from a past measurement of the output of the system S at time t(k-1), the control output U_S which is arranged to be applied to the system S throughout the kth time step which is of duration δ, is U_S(k) = U_S(k-1) Δ _R-Δ

where Δ - (k+1) - _R(k)

Δ

=

(k+1) - _S(k)

Preferably the system is a 2nd order system.

Preferably the system state time vector V_S at the first point is set equal to the reference system state time vector V_R at the first point.

The above controller may be used in the control of temperature control systems, chemical processors, robots or their components, materials handling devices or systems, electric motors, electric generators, medical apparatus, position control systems, radar systems, navigation systems and manufacturing machinery.

Preferred embodiments of the present invention will now be described by way of example only with reference to the accompanying drawings in which:

A DESCRITPION OF THE DRAWINGS

Figure 1 shows a block diagram of a controller according to a first embodiment of the present invention.

Figure 2 shows a diagrammatical representation of a plant S and behaviour reference R in a state time vector field.

Figure 3 is a block diagram of a 2nd order

continuous time controller and

Figure 4 shows a hardware implementation of the controller of figure 3 and

Appendix A is a computer program for a discrete time controller.

DETAILED DESCRIPTION

To assist the reader in understanding the present inventive concept two examples will be described, The first example having regard to a discrete time controller and the second example with regard to a continuous time controller. The theory behind the controllers described hereinafter is outlined as follows.

Control is a 'local' problem. By this it is meant that the control problem is concerned only with the present state of the system, its probable next state (without intervention), and our preference as to its next state. The notion of global behaviour is deliberately avoided since it introduces a degree of complexity irrelevant to the problem of control in the very next time instant. Given this view there is a need to introduce a systematic element into the 'local' decision. This is achieved by the following

principle.

It is held that the basic objective of control is to persuade a system (subject system), by means of a

'clever' series of inputs, to behave as though it were some other system (the reference system), which has behaviour which is in some sense preferable. The usual way this is achieved is to augment the subject system with external dynamics in order that the 'closed-loop' system has the required behaviour. The rationale of this invention

achieves the same effect, but only through control inputs. It is an important feature of this rationale that no

explicit extra dynamics are added to the subject system.

Simply put, we require that the subject system behave at all times, and from all initial conditions as though it were the reference system. This is, of course, different from requiring that the subject system follow some particular trajectory of the reference system without error. The usual requirement that the control process achieves regulation of the subject system is separately accounted for in a familiar manner. The important

distinction in this regard, however, is that the regulation requirement is implemented on the reference system (that is, one open to choice), and not directly on the subject system.

The mathematical framework in which the foregoing principles are simultaneously dealt with comprises a

state-time vector field which represents all possible evolutions of the system's state over time. The formulation of the problem in this way allows a particularly simple definition of the desired control applicable during any time instant. It is the control which minimizes the magnitude of the vector (outer) product of the incremental state-time vector for the subject system, and that for the reference system. Both vectors have the same origin in the state-time domain, and the increment in the time-direction is the sample step size.

Referring to figure 1, a structure 10 is shown including a controller 11 for controlling system 12.

A trajectory reference model T, 13 establishes an absolute goal to which the behaviour reference system R, 14 will tend by virtue of a constant K_R, 15.

The controller 11 ensures that the subject system

12, or plant, always behaves as though it were R. Using an analogy, it is as though T is a manager giving policy, R is a supervisor with a proclivity to follow that policy, and S is an employee closely supervised by R but not unduly penalized if he strays from his appointed task.

In order to calculate the control which ensures that S behaves as though it were R for ease of visualization the case of a second order system will be considered.

The augmented state time vectors (y,y,t) lie in a vector field in R ³ which appears as an exponential

ellipsoid, as shown in figure 2. Control inputs to the system are manifested as a translation in the y-direction.

Point 1: (y_S(k), (k), t(k)) =

(y_R(k), (k),t(k)); Point 2: (y_S(k + 1), _S(k + 1),

t(k + 1)); Point 3: (y_R(k + 1), (k + 1), t(k + 1).

If the state time vector fields of systems S and R are denoted as V_S and V respectively, then the ideal control sequence for a behaviour modification purpose is that sequence which causes V_S to map into V_R for all t.

For a particular time increment δ and from common initial conditions in V_R and V_S. It is necessary to choose a control which minimizes the magnitude of the vector product of the incremental vectors [Δy_S,Δ

_S,δ] and

[Δy_R,Δ

_R,δ]. If, as is the case in adaptive

control, states of S are yet not known for the time step of interest, they are replaced in the above by their

predictions. The control to be applied to the subject

system S, throughout the kth time step, which is of duration δ, is u_S(k)=u_S(k-1)+

where

Δ = (k+1)-y_R(k)

Δ = (k+1)-y_S(k)

A first example of a controller will now be described in a discrete time case. For simplicity an

algorithm will be described for a controller of a second order. The same principles, however, may be extended to higher order controllers. Having regard to figure 1, the algorithm may be outlined as follows.

Step 1. Denote the required tracking model, the forward

simulation of which gives the required trajectory, as the 'trajectory reference system', T.

Step 2. Denote an arbitrary, but well behaved,

second-order system which, with a simple feedback control law, will follow the above system as the 'behaviour reference system' R.

Step 3. Denote the unknown plant as the 'subject system' S.

In the following, the association of variables with one or other of the above systems will be indicated by the appropriate subscript. Step 4. From the required initial conditions, and with the appropriate external input u_in, simulate one time step of each of system R and T. For the first step the initial conditions of R and T are the same, but as described later, this will not necessarily be true for subsequent steps.

Step 5. From the outputs of R and T, y_R and y_T, form a

control adjustment to R. uR^=uin^+KR^(yT-^yR⁾

Step 6. With the control u_R applied to R, and from the

same initial conditions for R as used in Step 4 above, repeat the simulation of R for the same time step.

Step 7. From past measurements of the output of systems S, y_S, form a prediction of the value of

; a necessary assumption at this point is that

the control that was applied to S during the last time step will again be applied in the next time step. Step 8. From the incremental trajectories in y for the

behaviour reference and the subject system compute

Δ _R= (k+l)- (k)

Δ _S= (k+l)- (k)

Note that

(k+1) is known exactly from Step 6, that _g(k+1) was calculated in Step 7. As

described later, the initial conditions on R and S for the kth step are in fact the same, and R(k)= (k) Step 9. Compute the control to be applied to the subject system S, during the kth time step, which is of duration δ, as u_S(k)=u_S(k-1)₊

Apply this input to the subject system during the kth time step. Because of the resetting procedure in Step 10 the control is given by u_S(k)=u_S(k-1) Λ

Step 10. At the end of the kth time step, from measurements of y_S, estimate , and set the initial

conditions of the behaviour reference system R, for the (k+l)th step equal to the final conditions of the subject system S, at the kth step, i.e. y_R(k)=y_S(k)

(k)= _S(k)

Step 11. The initial conditions of the trajectory reference system T, are simply the final conditions of the previous step. The subject system S of course, does this without our intervention.

Step 12. Repeat the algorithm from Step 4.

Having regard to the algorithm described above there are weak restrictions on the choice of the behaviour reference system R.

In the implementation reported here K_R=1,

however, there is scope for choice in this regard. Because R and T are perfectly known-indeed specified-K_R could be more complex; perhaps a linear optimal control law. It is clear that increasing K_R (within the limits of stability of R) will compel the trajectories of S to follow more closely those of T. Also K_R may incorporate a derivative function in ramp tracking applications.

The prediction of (k) used for the results in this work may be based on a simple linear predictor s(k)=y_S(k-1)+δ (k-1)

Whilst the algorithm is tolerant of prediction error, a more sophisticated predictor would enhance the

robustness of the algorithms, especially for non-linear

systems with large rates of change of trajectory curvature.

There are, of course, many sophisticated methods to obtain short-term local predictions of system behaviour from noisy past samples including and the use of recursive Kalman-type predictors.

A second example of a controller will now be described based on a continuous time case utilizing the

block diagram shown in figure 1.

Assume systems T and R are specified to be of the form + α₁ + α₂y_T - α₂u_in

+ a_{1 R} + α₂y_R = α₂u_R

New u _R = u_in + K_R(y_T-y_R)

So

_R =

-α_{1 R}-[α₂(1+K_R)]y_R+α₂u_in+α₂K_Ry_T-(1) Similarly if the inputs from the plant are measured with delay, a modification to the illustrated embodiment is made as

indicated by the theory disclosed previously. Behaviour algorithm resetting required that yR ^{= y}S where y_S and

_S

y_R =

_S are measured without delay.

Thus

_R =

-α_{1 S}-α₂(1+K_R)y_S+α₂[u_in+K_Ry _T] where

s is a laplace variable. where y_T

Time local prediction requires that, for some small time step, δ

U_(K+1)δ=u_Kδ+1[(

_R,_(k+1)δ-

_,kδ)-

_,(k+1)δ-

_,kδ)] where _(k+1)δ is the anticipated plant state at the

next time step.

Subtracting u_Kδ from both sides and dividing by δ, then allowing δ 0 for continuous time gives

Where K - lim 1 (ε) where ε is dependent on errors at the previous step then u = K )dt -(2)

In the case where y_S and y_S are measured with delay, D, i.e. the actual states of δ are D seconds in advance of measured values, (1) becomes subject to the behaviour algorithm resetting

y_R = y_S + D

So (1) becomes - -α_l + D )-α₂(l+K_R) (y_S + D _S)

+ α₂ [u_in + K_R y_T]

- = α₂(u_in + )-α₂(1+K_R)y_S

T1 T2

- α_l s y_S - D s α₂(1+K_R)y_S - (1+α₁D)s²y_S

T3 T4 T5 and u = K )dt

This may be realised by the block diagram shown in figure 3.

A specific hardware implementation of the block diagram shown in figure 3, is shown in figure 4 for

exemplification only, and is not discussed further as a person skilled in the art could derive this from figure 3 without further inventive input.

In the above description time local prediction may alternatively be achieved by use of a one-step estimation of the predicted value of the system vector field at the current control time and the use of the valve together with the measured state at this time.

Thus, + f_S(y,

,t) = u_in

is replaced by + f_R(y, ,t) = u_in

Claims

CLAIMS :

1. A controller for a system, the controller

comprising a time local prediction means and a behaviour algorithm resetting means which are both arranged to be uti lized to produce an output , at a f irst control time, to a system, based on a reference input to the controller, the controller being arranged to receive an input from the system which input is indicative of the condition of the system after it receives the output; the time local

indicated by the input.

2. A controller according to claim 1, wherein the model is an algorithm.

3. A method of controlling a system comprising the steps of producing an output, at a first control time, from a controller to the system, based on a reference of the controller; receiving at the controller an input from the system which input is indicative of the condition of the system after it has received the output; utilizing a time local prediction and a behaviour algorithm resetting to produce a further output to the system at the next control time; the time local prediction being a prediction made f rom a past input f rom the system, of a controlled system

condition at the next control time, based on the assumption that the output applied at the first control time will again be applied at the next control time; and the behaviour algorithm resetting involving, at the first control time, from the input received at the first control time, setting the initial conditions, of a model which behaves, up until the next control time, in a manner desired of the system, equal to the condition of the system indicated by the input. Preferably the model is an algorithm.

4. A method according to claim 3, wherein the model is an algorithm.

5. A method according to claim 3 or claim 4, wherein the steps in the method are repeated for a predetermined number of control times.

6. A method according to anyone of claims 3 to 5, wherein the time local prediction is preformed by a data processing means.

7. A discrete time or sampled data controller which includes a means by which a prediction of a controlled system condition at the next control time, under the

assumption that the present control output persists, is formed and utilized; and a means by which an algorithm or process which behaves in the manner desired of the process being controlled is reinitialized to the conditions of the system being controlled at the current control time, which means are employed to compute the control output for the system.

8. A discrete time or sample data controller

according to claim 6, wherein the controller is a continuous time controller.

9. A controller for controlling a system, S to behave as though it were a reference system R, wherein, the

controller is arranged to produce a control output to the system such that at a particular control time t(k), the system state time vector V_S at a first point [y_S(k),

S(k), t(k)] is used to set

the reference system state time vector V_R at the first point [y_R(k),

(k), t(k)],

such that from a past measurement of the output of the system S at time t(k-1), the control output U _S which is arranged to be applied to the system S throughout the kth time step which is of duration δ, is U_S(k) = U_S(k-1) + Δ

_R-Δ

where Δ _R - _R(k+1) - _R(k)

Δy

_S -

_S(k+1) -

_S(k)

10. A controller according to claim 9, wherein the system state time vector V_g and the reference system state time vector V_R, are both nth order system state time vectors.