Pistikopoulos, Efstratios (London, GB)
Bozinis, Nikolaos A. (London, GB)
Dua, Vivek (London, GB)
Perking, John D. (Manchester, GB)
Sakizlis, Vassilis (London, GB)

Plaque It!

10/479100

10/07/2008

05/27/2002

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Imperial College Innovations, Ltd. (London, GB)

700/67

700/52, 700/86, 700/32, 702/189, 700/89, 700/29, 703/2

G05B13/02; G05B11/32; G05B19/42; G06F15/00; G06F7/60; G06F17/10; H03F1/26; H04B15/00

703/6, 700/52, 700/67, 700/104, 703/22, 702/189, 700/49, 700/89, 700/28-34, 700/74, 702/179, 703/2, 700/47, 700/86, 703/9, 700/44, 700/73, 700/51, 706/21, 703/7

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Bullock, Crystal Barnes

Morgan & Finnegan LLP

The invention claimed is:

1. A method of controlling operation of a system, the system having a control input variable and a parameter that can be measured or estimated, the method comprising: constructing, off-line, a system operation model based on said control input variable and measurable parameter; constructing, off-line, an optimised explicit control solution comprising a value of said control input variable as a function of said parameter using parametric optimisation, in which the control solution comprises a co-ordinate space of at least one control variable value region comprising a control variable substitution value or a control variable value as a function of a parameter value, the co-ordinate being the parameter; storing the control solution on a data storage device; and during operation of the system, measuring or estimating a parameter of the system, obtaining the control input variable based on the measured or estimated parameter from the stored control solution, and inputting the obtained control input variable to the system to control operation of the system according to the measured or estimated parameter of the system.

2. A method as claimed in claim 1 in which the system operation model is a model predictive control (MPC) model.

3. A method as claimed in claim 2 in which the explicit control solution is constructed in the continuous time domain.

4. A method as claimed in claim 3 in which the explicit control system comprises the solution of a multiparametric dynamic optimisation problem.

5. A method as claimed in claim 3 in which the control variable is obtained as a function of said parameter and time.

6. A method as claimed in claims 2 in which the MPC is formulated as a Quadratic Program.

7. A method as claimed in claim 1 in which the parameter comprises a measurable state.

8. A method as claimed in claim 1 in which the parameter comprises a system disturbance.

9. A method as claimed in claim 1 further comprising the step of analysing the control solution and constructing a revised system model based on said analysis.

10. A method as claimed in claim 1 in which the system operation model and/or the control solution includes a disturbance compensation term.

11. A method as claimed in claim 10 in which the control solution co-ordinate space further includes a disturbance as a co-ordinate.

12. A method as claimed in claim 10 in which the system operation model includes a worst-case disturbance compensation term.

13. A method as claimed in claim 12 in which the control solution co-ordinate space excludes a disturbance as a co-ordinate.

14. A method as claimed in claim 10 in which the control solution includes a disturbance estimate.

15. A method as claimed in claim 14 which the control solution is adjusted in an on-line control operation to correct the disturbance estimate.

16. A method as claimed in claim 1 in which the control solution is adjusted in an on-line control operation to correct a disturbance estimate.

17. A method as claimed in claim 1 in which the constructing are implemented by a computer program stored in a computer readable medium.

18. A computer readable medium storing a computer program for implementing a method as claimed in claim 1.

19. A method as claimed in claim 1 in which the control solution is stored in a data storage device comprising a computer readable medium.

20. A method of controlling operation of a system comprising: measuring or estimating, on-line, at a first time interval a parameter of the system; obtaining a control input variable based on the parameter from a pre-defined parametric optimised explicit control solution, the pre-defined parametric optimized explicit solution comprising a value of a control input variable of the system as a function of said parameter, in which the control solution comprises a co-ordinate space of at least one control variable value region comprising a control variable substitution value or a control variable as a function of a parameter value, the co-ordinate being the parameter value; and inputting the obtained control input variable to the system to control operation of the system according to the parameter of the system.

21. A method as claimed in claim 20 in which the measuring or estimating the on-line parameter and deriving the control variable are repeated at subsequent time intervals.

22. A method as claimed in claim 20 in which the control variable is further obtained based on a disturbance compensation term.

23. A method as claimed in claim 22 in which the control solution co-ordinate space further includes a disturbance as a co-ordinate.

24. A method as claimed in claim 22 in which the control solution includes a disturbance estimate.

25. A method of controlling operation of a system, the system having a control input variable and a parameter that can be measured or estimated, the method comprising: constructing, off-line, a system operation model based on said control input variable and measurable parameter; constructing, off-line, an optimised explicit control solution comprising a value of said control input variable as a function of said parameter using parametric optimisation; identifying a parameter of which the explicit solution is independent and eliminating measurements of said parameter; and storing the control solution in a hardware data storage device and in which the control solution comprises a coordinate space of at least one control input variable value region comprising a control input variable substitution value or a control input variable value as a function of the parameter, the co-ordinate being the parameter value, the method further comprising unifying control input variable value having the same solution, the stored control solution being employed to provide the system with the control input variable based on the parameter of the system to control operation of the system according to the parameter of the system.

FIELD OF THE INVENTION

The invention relates to improved process control, in particular improved model based process control.

BACKGROUND OF THE INVENTION

Modern automated processes such as manufacturing processes are implemented under strict and generally highly complex control regimes both to maximise efficiency and avoid error or failure. In their simplest form these regimes are typically based on a feedback system whereby a parameter of the process is monitored and fed to a controller as a control parameter for comparison with a target value, the state of the process being altered as a function of the control parameter to track the target value.

One known and widely adopted control regime is model predictive control (MPC) which is described in “Model Predictive Control; past, present and future.” Morari, M. and J. H. Lee (1999) Comput. Chem. Engng. 23, 667-682 which is incorporated herein by reference. The basic principle is illustrated in FIG. 1, adapted from Prett, D. M. and Garcia C. E. (1988) “Fundamental Process Control”, Butterworth Publishers, Boston. Using MPC, whilst the aim remains to track a target value, the controller at each control point—usually at discrete time intervals, not only monitors the existing state of the process, system or plant but estimates the state of the process at some point in the future, for example at the next time interval. References to control of a process in the following discussion can apply equally to plant or system control as appropriate.

Referring to FIG. 1, at a time t=k the current state is measured at point x _k and its future states x _k+1 to x _k+p are estimated for corresponding future times t=k to t=k+p. As can be seen the aim is for the value x to track a target value x _target . The control variable u _t is similarly varied with time, its value for the next time increment being updated as the predicted state value is updated. As a result the system takes into account the effect that the current control variable will have on future states and accommodates that accordingly. The forward projection is carried out for a finite future period which is shifted forward incrementally at each measurement interval, termed a moving horizon policy.

This operation is carried out on-line and in real time, and is illustrated in FIG. 2. MPC operates by solving an optimization problem at each time interval based on measurements 10 from the process or plant 12 at that time. An optimiser 14 includes a process or plant model used to estimate the future behavior of the plant and an objective function to determine the optimal control strategy, ie variation of the control variable or corrective action 16 , to reach the desired target.

Mathematically, the process or plant model can be represented, in state-space, by the equations:
{ x ( t+ 1)= Ax ( t )+ Bu ( t )
{ y ( t )= Cx ( t ), (1)
y _min ≦y ( t )≦ y _max
u _min ≦u ( t )≦ u _max (2)
u represents the control variables or process inputs manipulated by the controller to control the process, x the process states, ie measured values representative of the current values of one or more process variables and y the process output measurements. For multi variable processes, x, y and u are of course vectors and A and B matrices. The output and control variables are bounded by the min and max values forming constraints on the process.

The equations represent mathematically how at a time t in a process the future state of the process at a discrete, subsequent time interval t+1 is predicted as a function of the measured state x at time t and the control variables applied at time t—this calculation is then iterated forward, still at time t, for the predicted state at time t+2 based on the calculation for time t+1 and so forth. At time t, in addition, the process output is a direct function of the instantaneous process state. The process output is calculated at equation (2) to ensure it does not exceed the ouput constraints.

It has been shown that the iteration up to a time t+N _y can be posed in the form of the following problem to be optimised (for example, see Morari and Lee paper identified above):

$\begin{array}{cc}\underset{U}{\min }J\left(U,x\left(t\right)\right)=x_{1+\mathrm{Ny}}{{}_t\mathrm{Px}_{t+\mathrm{Ny}}}_t+\sum _{k=0}^{\mathrm{Ny}-1}\left(x_{t+k}^{\prime }{{}_t{\mathrm{Qx}}_{t+k}}_t+u_{t+k}^{\prime }{\mathrm{Ru}}_{t+k}\right)\mathrm{subject}\mathrm{to}{y}_{\min }\le y_{t+k}{❘}_t\le y_{\max },k=1,\dots ,N_c{u}_{\min }\le u_{t+k}\le u_{\max },k=0,1,\dots ,N_c{x}_t{❘}_t=x\left(t\right)x_{t+k+1}{{}_t={\mathrm{Ax}}_{t+k}}_t+{\mathrm{Bu}}_{t+k},k\ge 0y_{t+k}{{}_t={\mathrm{Cx}}_{t+k}}_t,k\ge 0u_{t+k}={\mathrm{Kx}}_{t+k}{❘}_t,N_u\le k\le N_y& \left(3\right)\end{array}$
where U Δ {u _t , . . . , u _t+Nu−1 }, Q=Q′ 0, R=R′ 0, P 0, N _y ≧N _u , and K is some feedback gain. The problem (3) is solved repetitively at each time t for the current measurement x(t) and the vector of predicted state variables, x _t+1|t , . . . , x _t+k|t at time t+1, . . . , t+k respectively and corresponding control actions u _t , . . . , u _t+k−1 is obtained.

Equation (3) is the iterated expansion of equation (1). The first term on the right hand comprises a quadratic function of the predicted state variables at time t+N _y , using the vector transpose x′ and constant matrix P. The second term is a summation of all the intermediate predicted state variables from time t to time t+N _y −1 using the vector transpose x′ and constant matrix Q and, over the same range, of corresponding control variables u again in quadratic form with constant matrix R. In both cases each successive state value x _t+k+1 is derived from the previous state value x _1+k using equation 1. Each value of u _t+k is derived as a function of the predicted state value for that time and gain matrix K. The constraints for y and u are controlled up to a time N _c . The function J is thus constructed.

The left hand side of equation (3) requires that an optimal solution is selected from the solution space by minimising the function J of the state x(t) and the vector U of control/input variables for all time intervals. The function J is minimised in control variable space, that is, the optimal value of U is found satisfying equation (3) including the various conditions in the “subject to” field such that J is minimum. The resulting values of u(t) and x(t) are then representable as shown in FIG. 1. The manner in which a real control situation can be represented as an MPC problem will be well known to the skilled person.

A significant problem arises in practical implementation of MPC as the processing power required for real time calculation is extremely high. Certainly if a large number of variables are required, complex control functions needed for complex processes or extremely short time intervals are desired, the burden can become cost, energy or even space inefficient. In addition, as the solution is created on-line and tailored only to instantaneous conditions, it provides no quantitative data on whether the model itself is appropriate.

Various attempts have been made to improve MPC implementation. For example in U.S. Pat. No. 6,056,781 to Wassick et al, in addition to a plant or process model, a disturbance model represents deviations from the plant model. In this version, if anything, the real time processing burden is enhanced. U.S. Pat. No. 5,347,446 to Iino et al describes a system taking into account constraints on the input or manipulated variables as well as constraints on controlled variables and change rates of the controlled variables. Again, the processing burden remains a significant concern.

U.S. Pat. No. 6,064,809 to Braatz et. al. addresses the problem of processing burden by approximating off-line the constraint set, that is; the limits or constraints on the input or manipulated variables as a mathematical construct off-line, in the form of an iteratively derived ellipsoid, effectively providing a simplified replacement for equation 3 above. However this simplified construct is then no more than an approximation which is scaled and then solved on-line in conventional MPC fashion. Accordingly Braatz et al. approached the problem by improving or simplifying the modeling process, but still requires significant on-line computational power.

SUMMARY OF THE INVENTION

According to the invention there is provided a method of deriving an explicit control solution for operation of a system, the system having a control variable and a measurable parameter, comprising the steps of constructing, off-line, a system operation model based on said control variable and measurable parameter, and constructing, off-line, an optimised explicit control solution for said control variable and measurable parameter using parametric optimisation. Because the solution is explicit and constructed off-line, on-line control of the system can be speeded up significantly using the solution. In addition the explicit solution can be reviewed off-line to provide insights into the control system/control model itself.

The explicit control solution is preferably stored on a data storage device such as a chip and the system operation model is preferably a model predictive control (MPC) model. The explicit control solution is preferably constructed in the continuous time domain allowing more accurate control. The MPC is preferably formulated as a Quadratic Program.

The control solution preferably comprises a co-ordinate space of at least one control variable value region comprising a control variable substitution value or a control variable value as a function of a measurable parameter value, the co-ordinate being the measurable parameter value. As a result a simple and manipulable parametric profile is provided. The measurable parameter may comprise a measurable state or a system disturbance. The method may further comprise the step of analysing the control solution and constructing a revised system model based on said analysis.

According to the invention there is further provided a method of controlling operation of a system comprising the steps of measuring, on-line, at a first time interval a parameter of the system, obtaining a control variable based on the measured parameter from a pre-defined parametric optimised explicit control solution and inputting the control variable to the system.

The steps of measuring the on-line state and deriving the control variable may be repeated at subsequent time intervals. The control solution preferably comprises a co-ordinate space of at least one control variable value region comprising a control variable substitution value or a control variable as a function of a measured state value, the co-ordinate being the measured state value.

According to the invention there is further provided a control system for controlling operation of a system having a control variable and a measurable state, the control system including a system state measurement element, a data storage element storing a predetermined parametric optimised explicit control solution profile, a processor arranged to obtain a control variable from said data storage element control solution based on a measured system state value and a controller arranged to input the obtained control variable to the system.

The method is preferably implemented in a computer program which can be stored in any appropriate computer readable medium. A data storage device is provided according to the invention storing a control solution derived by a method as herein described.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described by way of example with reference to the drawings, of which:

FIG. 1 shows a control strategy according to a model predictive control scheme;

FIG. 2 shows schematically an on-line optimisation strategy;

FIG. 3 shows an explicit control solution obtained by parametric optimisation;

FIG. 4 shows an explicit control solution for an alternative parametric optimisation operation;

FIG. 5 shows schematically an on-line optimisation strategy according to the present invention;

FIG. 6 shows the disturbance performance of various control strategies including those according to the present invention;

FIG. 7 shows the difference between actual and predicted disturbance in a controller according to the present invention;

FIG. 8 shows schematically a fluidised catalytic cracking unit modeled according to the present invention;

FIG. 9 shows an explicit control solution obtained for the unit of FIG. 8;

FIG. 10 a shows the time profile of the measurable states for the control strategy shown in FIG. 9;

FIG. 10 b shows the time profile of the measurable states for the control solution shown in FIG. 9;

FIG. 11 a shows the trajectory followed in state space for the control variables under the control solution of FIG. 9; and

FIG. 11 b shows the trajectory followed in state space for the control variables under the control solution of FIG. 9;

FIG. 12 shows schematically an evaporator unit modeled according to the present invention;

FIG. 13 shows a control solution according to the present invention without utilising disturbance rejection;

FIG. 14 shows the time profile of the measurable states for a control strategy according to the disturbance rejection model;

FIG. 15 shows the time profile of the measurable states for a control strategy according to the disturbance rejection model;

FIG. 16 a shows the trajectory followed in state space for the control variables according to the disturbance rejection model;

FIG. 16 b shows the trajectory followed in state space for the control variables according to the disturbance rejection model;

FIG. 17 a shows the trajectory followed for the control variables according to the disturbance rejection model;

FIG. 17 b shows the trajectory followed for the control variables according to the disturbance rejection model;

FIG. 18 shows the critical region partition for further embodiment;

FIG. 19 shows the output, control and state profile for the embodiment of FIG. 18;

FIG. 20 shows the critical region partition for further embodiment; and

FIG. 21 shows the output and control profile for the embodiment of FIG. 20.

DETAILED DESCRIPTION OF VARIOUS EXEMPLARY EMBODIMENTS

The present invention provides an alternative, improved real time solution to the problems of model-based control for example using model predictive control. In particular, by applying the technique generally known as “parametric optimisation” to the process model the invention allows control variables to be derived off-line for all operational parameters that will be encountered. The explicit analytical control solution thus constructed is embedded in a chip or other data store as a function of the operational parameters such that the process model does not have to be optimised at each time interval, hence reducing the processing burden significantly. Instead the control variables are immediately available as a function of the operational parameters and/or in look-up table form. Yet further the solution is viewable off-line and offers information on the problem to be solved.

Parametric Optimisation

The techniques of parametric or optimisation or parametric programming will be well known to the skilled person and the general principles are set out in Gal, T. (1995) “Postoptimal Analyses, Parametric Programming and Related Topics”, de Gruyter, New York and Dua, V. and E. N. Pistikopoulos (1999) “Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems”, Ind. Eng. Chem. Res. 38, 3976-3987 which are incorporated herein by reference. In essence parametric optimisation is a mathematical technique by which a problem is first formulated as an optimisation problem, an objective function and optimisation variables are obtained as a function of parameters in the optimisation problem, and the region in the parameter space where the functions are valid are also obtained. In the most general form the parametric optimisation problem can be formulated as:
z (θ)=min z =( d ^T y+f ( x )) y, x
s.t. Ey+g ( x )≦ b+Fθ
θ _min ≦θ≦θ _max
x∈X ⊂ ⁿ
y∈Y={ 0, 1} ^m
θ∈Θ ⊂ ^s , (4)
where y is vector of 0-1 binary variables, x is a vector of continuous variables, f is a scalar, continuously differentiable function of x, g is a vector of continuously differentiable functions of x, b and d are constant vectors, E and F are constant matrices, θ is a vector of parameters, θ _min and θ _max are the vectors of lower and upper bounds on θ, and X and Θ are compact and convex polyhedral sets of dimensions n and s respectively.

This most general formulation requires minimisation of z, a function of optimization or control variables, x and y, to obtain them as a function of parameters which can be for example measurable states or system disturbances represented by vector θ. The generalised formulation covers cases where the control variables are continuous value vectors x, or binary value vectors y. The parameter values θ are bounded by minimum and maximum constraints. In the example discussed below the parameter values are measurable states.

Solutions are known for various special cases. For example where there is no control variable y (ie no binary variables), f(x) is quadratic and convex in x, where f(x) is defined to be convex if for any x ₁ , x ₂ ∈X and 0≦α≦1, f[(1−α)x ₁ +αx ₂ ]≦(1−α)f(x ₁ )+αf(x ₂ ), and g(x) is linear in x, the problem is reduced to one of a multiparametric Quadratic Program (mp-QP). The solution to this case is discussed in Dua et al. “An algorithm for the solution of multi-parametric quadratic programs” (1999) tech. Rep. D00.7, Centre for Process Systems engineering, Imperial College London, which is incorporated herein by reference. An example of the form of the solution to an mp-QP problem set out in Table 1 is now discussed.

TABLE 1

mp-QP Example: Mathematical Formulation

$\begin{array}{c}z\left(\theta \right)=\underset{x_1,x_2}{\min }0.0098x_1^2+0.0063x_1{x}_2+0.00995x_2^2\\ \begin{array}{cc}s.t.& x_1\le 0.417425+3.16515{\theta }_1-3.7546{\theta }_2\\ & -x_1\le 3.582575-3.16515{\theta }_1-3.7546{\theta }_2\\ & -0.0609x_1\le 0.413225+0.17355{\theta }_1-0.2717{\theta }_2\\ & -0.0064x_1\le 0.467075+0.06585{\theta }_1+0.4714{\theta }_2\\ & x_2\le 1.0902+1.8196{\theta }_1-3.2841{\theta }_2\\ & -x_2\le 2.9098-1.8196{\theta }_1+3.2841{\theta }_2\\ & 0\le {\theta }_1\le 1\\ & 0\le {\theta }_2\le 1\end{array}\end{array}$

As can be seen the function f(x) is a quadratic function of variables x ₁ , x ₂ . Each of x ₁ and x ₂ are further constrained dependent on state parameters θ ₁ , θ ₂ , which in turn are bounded. The function is minimised in control variable x ₁ , x ₂ space, ie the values of x ₁ , x ₂ are found for which f(x) is optimised.

The solution generated according to the Dua et. al. reference above takes the form of the plot in a coordinate space θ shown in FIG. 3. The axes are the state parameters θ ₁ , θ ₂ respectively and optimisation of the mp-QP problem demonstrates that there are only four possible control variable states across the bounded range of θ, represented by feasible regions #1 to #4 respectively. Accordingly, at any point in a process represented by the model in table 1, once state parameters have been measured, the region containing the coordinates (θ ₁ , θ ₂ ) is identified and the control variable values x ₁ , x ₂ are retrieved for that region. The control variables may be stored in the form of pure numbers or as simple optimised functions of θ ₁ , θ ₂ . In the specific example, where t ₁ , t ₂ represent θ ₁ , θ ₂ respectively, region definition as functions of θ ₁ , θ ₂ and the values or functions for x ₁ , x ₂ for each region are as follows:

• • Feasible region 1
• x 1 =+0
• x 2 =+0
• −1*t 1 <=−0
• +1*t 1 <=+1
• −1*t 2 <=−0
• +1*t 1 +1.18623*t 2 <=+1.13188
• −1*t 1 +1.80485*t 2 <=+0.599143
  • Feasible region 2
• x 1 =+3.16515*t 1 +3.7546*t 2 −3.58257
• x 2 =+1.8196*t 1 −3.2841*t 2 +1.0902
• +1*t 2 <=+1
• −1.38941*t 1 −1*t 2 <=−1.19754
• +1.34655*t 1 −1*t 2 <=+0.0209892
  • Feasible region 3
• x 1 =−0.584871*t 1 +1.0556*t 2 −0.350421
• x 2 =+1.8196*t 2 −3.2841*t 2 +1.0902
• −1*t 1 <=−0
• +1*t 2 <=+1
• +1.38941*t 1 +1*t 2 <=+1.19754
• +1*t 1 −1.80485*t 2 <=−0.599143
  • Feasible region 4
• x 1 =+3.16515*t 1 +3.7546*t 2 −3.58257
• x 2 =−1.00203*t 1 −1.18864*t 2 +1.13418
• +1*t 1 <=+1
• +1*t 2 <=+1
• −1.34655*t 1 +1*t 2 <=−0.0209892
• −1*t 1 −1.18623*t 2 <=−1.13188

It can be seen that for regions 1, x ₁ and x ₂ have fixed values, whereas for regions 2 to 4 x ₁ and x ₂ are expressed as functions of the coordinates (θ ₁ , θ ₂ )—it will be noted that x ₁ and x ₂ will thus vary within a given region dependent on their position in that region.

Another special case is where f(x) and g(x) are linear in x. The problem is then presented as a multiparametric Mixed Integer Linear Program (mp-MILP) for which algorithms have been developed as discussed in Acevedo J. and Pistikopoulos E. N. “A Multiparametric Programming Approach for Linear Process Engineering Problems under Uncertainty” (1997) Ind. Eng. Chem. Res. 36, 717-728 which is incorporated herein by reference.

An mp-MILP problem is set out in Table 2

TABLE 2
mp-MILP Example: Mathematical Formulation
	z(θ) = min − 3x 1 − 2x 2 + 10y 1 + 5y 2
	y, x
	s.t. x 1 ≦ 10 + θ 1
	x 2 ≦ 10 − θ 1
	x 1 + x 2 ≦ 20
	x 1 + 2x 2 ≦ 12 + θ 1 − θ 2
	x 1 − 20y 1 ≦ 0
	x 2 − 20y 2 ≦ 0
	−x 1 + x 2 ≧ 4 − θ 2
	y 1 + y 2 ≧ 1
	x n ≧ 0, n = 1, 2
	y 1 ε {0, 1}, 1 = 1, 2
	2 ≦ θ s ≦ 5, s = 1, 2.

In this case f(x) and g(x) are linear functions of continuous variables x ₁ , x ₂ and of binary variables y ₁ , y ₂ . The solution generated according to the Acevedo and Pistikopoulos reference above takes the form of the plot of θ ₁ , θ ₂ shown in FIG. 4, in this case having five feasible regions with optimal solution as set out below:

• • Feasible region 1
• x ₁ =0
• x ₂ =0.5*t 1 −0.5*t 2 +6
• Binary vector y=[0 1]
• −1*t 1 <=−2
• −1*t 2 <=−2
• +3*t 1 −1*t 2 <=+8
• +1*t 2 <=+5
  • Feasible region 2
• x ₁ =0.333333*t 1 +0.333333*t 2 +1.333333
• x ₂ =0.333333*t 1 −0.666666*t 2 +5.333333
• Binary vector y=[1 1]
• +1*t 1 <=+5
• −1*t 2 <=−2
• +2*t 1 −1*t 2 <=+7
• −11*t 1 +1*t 2 <=−46
• +1*t 2 <=+5
  • Feasible region 3
• x ₁ =0
• x ₂ =−1*t 1 +10
• Binary vector y=[0 1]
• −1*t 2 <−2
• −3*t 1 +1*t 2 <=−8
• +11*t 1 −1*t 2 <=+46
• +1*t 2 <=+5
  • Feasible region 4
• x ₁ =−1*t 1 +1*t 2 +6
• x ₂ =−1*t 1 +10
• Binary vector y=[1 1]
• +1*t 1 <=+5
• −1*t 2 <=−2
• −2*t 1 +1*t 2 <=−7
• +1*t 1 −1*t 2 <=+2.66667
  • Feasible region 5
• x ₁ =0
• x ₂ =−1*t 1 +10
• Binary vector y=[0 1]
• +1*t 1 <=+5
• −1*t 2 <=−2
• −1*t 1 +1*t 2 <=−2.66667
  where t 1 and t 2 correspond to θ ₁ and θ ₂ respectively.

It will further be recognised that the presentation of the solution in this simple, explicit form offers insights into the control process as a whole. Most significantly, it indicates the extent to which the state measurements are required for process control. For example in more complex processes having a large number of measured states it is entirely possible that the explicit solution will turn out to be independent of one or more of the measured states as a result of which the process control hardware can be designed or modified to eliminate the corresponding state measurement device and the software to eliminate the processing of related measured state data so rendering the process control scheme for efficient and economic operation at all levels.

The skilled person will be aware of other special cases for which the parametric optimization solution is known and further discussion of those is thus not required here.

Application of Parametric Optimisation Techniques

Using these techniques a model-based problem can be optimised according to the present invention using parametric optimisation providing control actions as a function of measurements from a plant to be controlled, by defining the control variables as the optimisation variables and the state of the plant as parameters. Then, during real time operation, as soon as the measurements from the plant are obtained, this function is evaluated to obtain the control actions using the on-line optimisation routine designated generally 20 in FIG. 5.

According to this approach a model-based problem to be optimised is formulated for the plant/system/process 22 in a known manner. An explicit control solution, parametric profile 24 , is constructed off-line using parametric optimisation techniques 26 . In the simplified scheme shown, the profile is represented as a two-dimensional plot of state variables as input and control variables as output, but it will be appreciated that the profile can be multi-dimensional and include function evaluation dependent on the state variable defined region as discussed above with reference to FIGS. 3 and 4. In on-line operation of the plant 22 , therefore, the state variables are entered to the profile as inputs 28 and the control actions 30 derived or looked up instantly and fed back to the plant 22 . As a result the processing burden is shifted to the front end and on-line processing is minimised.

Returning to the MPC problem of equation (3); the function J(U, x(t)) can be formulated as a Quadratic Programming (QP) problem by expanding the formula set out in equation (3) for x _t+k+1 at time t as the summation:

$\begin{array}{cc}{x}_{t+k}{❘}_t=A^{k}x\left(t\right)+\sum _{j=0}^{k-1}{A}^j{\mathrm{Bu}}_{t+k-1-j}& \left(5\right)\end{array}$

This term is substituted into equation (3) to yield by simple algebraic manipulation the QP Problem:

$\begin{array}{cc}J\left(U,x\left(t\right)\right)=\underset{U}{\min }\frac{1}{2}{U}^{\prime }\mathrm{HU}+x^{\prime }\left(t\right)\mathrm{FU}+\frac{1}{2}{x}^{\prime }\left(t\right)\mathrm{Yx}\left(t\right)s.t.\mathrm{GU}\le W+\mathrm{Ex}\left(t\right),& \left(6\right)\end{array}$
where U Δ [u′ _t , . . . , u′ _t+Nu−1 ]′∈ ^s , s Δ mN _u , is the vector of optimisation variables, H=H ¹ 0, and H, F, Y, G, W, E are obtained from Q, R and (3) and (5).

The QP problem represented by equation (6) can be formulated as the mp-QP again using simple algebraic manipulation:

$\begin{array}{cc}\mu \left(x\right)=\underset{z}{\min }\frac{1}{2}{z}^{\prime }\mathrm{Hz}s.t.\mathrm{Gz}\le W+\mathrm{Sx}\left(t\right),& \left(7\right)\end{array}$
where z Δ U+H ⁻¹ F′x(t), z∈R ^s , represents the vector of optimisation variables, S Δ E+GH ⁻¹ F′ and x represents the vector of parameters.

It will be appreciated that the above derivation is performed based on predictions for discrete future intervals t, t+1 . . . t+k i.e. in the discrete time domain, but similar techniques can be applied to obtain the mp-QP problem in the continuous time domain, i.e. continuing to take measurements at discrete intervals t, but predicting from those measurements future continuous profiles of the state variable at a time t+k where the intermediate intervals t+1 . . . t+k−1 are infinitesimal.

In this case equation (6) is rewritten as:

$\begin{array}{cc}\hat{\varphi }\left(\theta \right)=\underset{x,v}{\min }\frac{1}{2}{x\left(t_f\right)}^T{P}_{1}x\left(t_f\right)+P_{2}x\left(t_f\right)++{\int }_{t_o}^{t_f}\left[\frac{1}{2}{x\left(t\right)}^T{Q}_{1}x\left(t\right)+Q_{2}x\left(t\right)+\frac{1}{2}{v\left(t\right)}^T{R}_{1}v\left(t\right)+R_{2}v\left(t\right)\right]dt& \left(8\right)\end{array}$

• s.t. {dot over (x)}(t)=Ax(t)+Bv(t)+Cθ(t)
• initial condition (I. C.).: x(t _o )=x _o (θ, v)
• B ₁ ·v(t _i )+A ₁ ·x(t _i )+C ₁ ·θ(t _i )+b≦0

$\begin{array}{c}y\left(t\right)=\mathrm{Dx}\left(t\right)+\mathrm{Hv}\left(t\right)+F\theta \left(t\right)\\ y_{\min }\le y\left(t\right)\le y_{\max }\text{-}\\ t_0\le t\le t_f\end{array}\rceil \begin{array}{c}\mathrm{Path}\mathrm{Constraints}\\ \end{array}$ $i=0,1,2,\dots N_f$ $\theta \in \Theta \subseteq {\Re }^s$

where A, B, C, B ₁ , A ₁ , C ₁ , H, D, F are time invariant matrices of dimensions n×n, n×m, n×s, q×m, q×n, q×s, r×m, r×n, r×s respectively and b is a vector of dimension n. Q ₁ , P ₁ , R ₁ , P ₂ , Q ₂ , R ₂ are symmetric matrices. x∈ ⁿ is the vector of the state time dependent variables, v∈ ^m is the vector of the optimisation variables, y∈ ^z is the vector of algebraic variables including measurements and θ is the vector of the uncertain parameters. Note that the terms x(t _f ) ^T P ₁ x(t _f ) and

${\int }_{t_o}^{t_f}\left[\frac{1}{2}{x\left(t\right)}^T{Q}_{1}x\left(t\right)+\frac{1}{2}{v\left(t\right)}^T{R}_{1}v\left(t\right)\right]dt$
represent the 2-norm of the state and control variables from the origin, whereas the terms ∥P ₂ x(t _f )∥ and

${\int }_{t_o}^{t_f}\left[Q_{2}x\left(t\right)++R_{2}v\left(t\right)\right]dt$
represent the 1- or ∞-norms of the state and control deviations from the set point. The time horizon t _f of this problem is considered finite. The point constraints are imposed at N _f discrete instants within the horizon. The optimisation variables, v, are called “control variables”. The initial conditions set out in the “subject to” field comprise the control problem to be solved.

Note that in (8), we have integrals and differential equations in the continuous model, whereas in (3) we had summation in the objective function and discretized model. Problem (8) is described in (i) Anderson, B. D. and J. B. Moore (1989) “Optimal Control: Linear Quadratic Methods”, Prentice Hall, Englewood Cliffs, N.J., (ii) Bryson, A. E. and Y. Ho (1975) “Applied Optimal Control”, Taylor and Francis, New York, and (iii) Kwakernak, H. and R. Sivan (1972) “Linear Optimal Control Systems”, Wiley Interscience, New York all of which are incorporated herein by reference.

The result of a parametric analysis on problem (8) is to express the optimal objective (φ) and the optimisation variables (v) as a function of the parameters (θ) for the complete space of the parameters variations. This can be achieved by transforming problem (8) to a form that can be solved with the current techniques of steady state parametric programming. Such a form in this case is the mp-QP formulation.

By adapting control vector parameterisation for solving the optimal control problem (e.g. “Off-line Computation of Optimum Controls for Plate Distillation Column” Pollard and Sargent, 1970 Automatica 6, 59-76, incorporated herein by reference), the optimisation problem can be considered only in the reduced space of v variables. This of course requires the explicit mapping of the x variables that appear in the constraints and the objective as a function of v,θ. This expression can be obtained by solving the linear ODE system in equation (8). The solution of this system is exact and in the case of time-invariant parameters and optimisation variables, is given by the analytical expression:

$\begin{array}{cc}x\left(t,v,\theta \right)=e^{A\left(t-\mathrm{to}\right)}{x}_o+e^{\mathrm{At}}{\int }_{t_o}^t{e}^{-\mathrm{At}}\left(\mathrm{Bv}\left(t\right)+C\theta \left(t\right)\right)dt& \left(9\right)\end{array}$

e ^At is the exponent of a matrix that is defined through eigenvalue or other decomposition analysis methods (“Matrix Computation”, Second Edition Colub and Van Loan 1990, The John Hopkins University Press).

The integral evaluation in equation (9) requires that the form of the profile of the uncertainty and control variables over time be specified. Control variables are usually represented as piecewise polynomial functions over time where the values of the coefficients in the polynomial are determined during the optimization (Vassiliadis, V. S., R. W. H. Sargent and C. C. Pantelides (1994) “Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints.” Ind. Eng. Chem. Res. 33, 2111-2122 incorporated herein by reference). Note that if the order of the polynomial is assumed to be zero the control profiles correspond to a piecewise constant discretization. For the case of uncertainty, θ, it is described by using periodic harmonic oscillations, i.e. sinusoidal functions over time, of uncertain amplitude or mean value (Kookos, I. K and J. D. Perkins (2000) “An Algorithm for Simultaneous Process Design and Control”, Tech. Rep. A019, Centre for Process Systems Eng., Imperial College, London incorporated herein by reference). Note that more complex, forms of periodic functions can be reduced to a linear combination of sinusoidal functions by using Fourier analysis. By substituting (9) into (8) and hence eliminating the system of differential equations the problem is transformed as follows:

$\begin{array}{cc}\varphi \left(\theta \right)\underset{v}{\min }=\frac{1}{2}{x\left(t_f,v,\theta \right)}^T{P}_{1}x\left(t_f,v,\theta \right)+P_{2}x\left(t_f,v,\theta \right)+{\int }_{t_o}^{t_f}\left[\frac{1}{2}{x\left(t,v,\theta \right)}^T{Q}_{1}x\left(t,v,\theta \right)+Q_{2}x\left(t,v,\theta \right)+\frac{1}{2}{v\left(t\right)}^T{R}_{1}v\left(t\right)+R_{2}v\left(t\right)\right]dts.t.A_1·x\left(t_j,v,\theta \right)+B_1·v\left(t_i\right)+C_1\theta \left(t_i\right)+b\le 0y\left(t\right)=\mathrm{Dx}\left(t,v,\theta \right)+\mathrm{Hv}\left(t\right)+F\theta \left(t\right)y_{\min }\le y\left(t\right)\le y_{\max }& \left(10\right)\end{array}$
By evaluating the matrix time integrals in equation (10), the objective function is expressed explicitly as a quadratic time invariant function of the parameters θ and the control variables v only. The same does not apply however, to the path constraints that are an explicit function of the control variables, the parameters and the time. The time dependence of such constraints renders problem (10) infinite dimensional. This difficulty is surpassed by applying the following methodology for converting the infinite dimensional path time-varying constraints to an equivalent finite number of point constraints:

• 1. Consider a finite number of interior point constraints that have to be satisfied only at particular points within the horizon.
• 2. Solve the optimization problem for this particular set of interior constraints.
• 3. Now test whether the same constraints are satisfied for every time instant during the complete time horizon.
• 4. If some of the constraints are violated at particular points t _k , then augment these points into Step 1 and continue until the constraints are satisfied for the complete time horizon.

This methodology transforms the path constraints that have to be satisfied throughout the complete time-horizon to an equivalent series of point constraints that hold only at particular time points and therefore, are inherently finite.

Hence the time in the path constraints of (10) is substituted by an appropriate set of fixed time instants t _k , k=1 . . . K. Consequently the constraints become a linear function of the control variables and parameters only. Thus the equivalent mathematical description of problem (10) is the following:

$\begin{array}{cc}\hat{\varphi }=\underset{v}{\min }\frac{1}{2}\left(L_1+L_2^{T}v+L_3^T\theta +v^T{L}_{4}v+{\theta }^T{L}_{5}v+{\theta }^T{L}_6\theta \right)s.t.G_{1}v\le G_2+G_3\theta & \left(11\right)\end{array}$
where L ₁ is a scalar constant and L ₂ , L ₃ , L ₄ , L ₅ and L ₆ are constant vectors and matrices of appropriate dimensions.

Problem (11) is now a multiparametric quadratic program (mp-QP) and its solution provides the control variables as a function of the state variables and uncertain parameters, similarly as in the discrete time case.

As a result, known parametric optimisation techniques for mp-QP problems can be applied to MPC problems in the discrete or continuous time domain to obtain the full solution set for z and hence u as an affine function of x of the form:
u=p ^T x+q (12)
where p is a constant vector and q is a scalar constant, for the complete feasible space of x (ie. the full available range of control variables) using appropriate algorithms and techniques such as those set out in the Dua et. al. reference identified above and as will be well-known to the skilled person. This is all performed off-line, such that on-line the only control calculations are those of evaluating or solving the functions derived for the control variables.

A limitation of the system described above is that it does not take into account model inaccuracies and constant disturbances except to the extent that the disturbances can be modeled as additional parameters. Accordingly inaccurate forecasting of the process behavior can lead to infeasabilities such as off-specification production, excessive waste disposal or even hazardous plant operation. Below there are discussed two general methodologies for compensating for disturbances.

In real processes there are usually disturbances due to sensor malfunctioning, degrading performance of the process or system due to catalyst fouling, noise, human error or mistake, cost or model inaccuracies, amongst other possible factors. These disturbances cannot always be measured and accordingly it is desirable to take such disturbances into account in deriving the optimal control law.

Rewriting equation (3) above, an appropriate formulation for deriving an explicit model-based control law for a process system is as follows:

$\begin{array}{cc}\phi \left(x_{t❘0}\right)=\underset{v_{1❘s}^N}{\min }{x}_{t❘N}^T{\mathrm{Px}}_{t❘N}+\sum _{k=0}^{N-1}\left[y_{t❘k}^T{\mathrm{Qy}}_{t❘k}+v_{t❘k}^T{\mathrm{Rv}}_{t❘k}\right]\begin{array}{c}s.t.x_{t❘k+1}=A_1{x}_{t❘k}+A_2{v}_{t❘k}\\ y_{t❘k}=B_1{x}_{t❘k}+B_2{v}_{t❘k}\\ x_{t❘0}=x^{*}\end{array}0\ge g\left(y_{k❘t},x_{k❘t},v_{k❘t}\right)=C_o{y}_{t❘k}+C_1{x}_{t❘k}+C_2{v}_{t❘k}+C_{4}k=0,1,2\dots N& \left(13\right)\end{array}$
where x∈X ⊂ ⁿ are the states, y∈ ^m , are the outputs and v∈ ^q are the controls. The outputs are the variables that we aim to control, i.e. to drive to their set-point y ^o , (which might be temperatures, concentrations) whereas the states are the variables that fully characterize the current process conditions (which might be enthalpies, internal energies, volume). v ^N denotes the sequence of the control vector over the receding horizon. Note that in (13) the outputs participate in the objective function instead of the states. Since the outputs are linear function of the states, with proper rearrangement on the weighting matrix Q this formulation (13) is equivalent to equation (3) above.

By considering the current states x _t as parameters, problem (13) is recast as a multiparametric quadratic program (mp-QP) of the form given in (7). The solution of that problem corresponds to an explicit control law for the system that expresses the optimum control action in terms of the current states. The mathematical form of this parametric controller, similar to that given in (12), is as follows:
v _ct|0 ( x *)= a _c x*+b _c ;
CR _c ¹ x*+CR _c ² ≦0
for c=1, . . . , N _c (14)
where N _c is the number of regions in the state space. Matrices a _c , CR ¹ _c and vectors b _c , CR ² _c are determined from the solution of the parametric programming problem and the index c designates that each region admits a different control law. Vector v _ct|0 is the first element of the control sequence, similar expressions are derived for the rest of the control elements.

The model-based parametric controller described in the previous paragraph does not account for the impact of persistent slowly varying unmeasured disturbances and uncertainties of the type discussed above that either enter the system as inputs or derive from modelling errors and time varying process characteristics. These disturbances result in an offset, i.e. a permanent deviation, of the output variables from their target values and additionally may cause constraint violations. As discussed below, according to the invention two possible solution techniques are introduced: Proportional Integral (PI) control, and origin re-adjustment. The suitable model readjustment according to disturbance estimates, enables the recursive on-line adaptation of the model based controller to the current conditions.

FIG. 6 illustrates the situation where a step disturbance enters and propagates in a plant controlled by the controller discussed above. In the illustration shown in FIG. 6 “conventional parco” refers to a model-based controller according to the discussion above whereas “PI-parco” refers to parametric PI. It can be seen from the resultant profile shown in FIG. 7 of the controlled output that it is driven closer to the set point but there is an off-set between the set point y _set , in this case y _set=0 and the output y:
Δ y=y−y _set (15)

The reason for the off-set is that the model has not predicted the presence of the disturbance.

Following the fist approach according to the invention, to eliminate the off-set an integral state is incorporated in the model-based optimisation framework. This provides off-line derived parametric proportional integral controller (“Parametric PI”). As discussed in more detail below, in particular an integral state is added into the dynamic system, and is also added in the objective function with a particular waiting penalty. The updated problem is solved again to obtain the model-based parametric control law discussed above.

In conventional feedback control schemes, Seborg et al., (1989) “Process Dynamics & Control”, Wiley & Sons, integral action is incorporated for attenuating any permanent deviation of the output variables from their set-points due to the presence of disturbances. This gives rise to the so-called proportional integral controller (PI-controller). Here, the incorporation of the integral action for the same purpose is achieved by introducing an integral state that is equal to the anticipated deviations of the output from its reference point, usually the origin. This state is augmented as an additional quadratic penalty in the objective function (Kwakernaak and Sivan, 1972) “Linear Optimal Control Systems”, John Wiley & Sons Ltd. This technique has also been adopted by the Generic Model Control framework (Lee and Sullivan, 1988 “Generic Model Control (GMC)” Comput. Chem. Eng. 12(6), 573-580), but in that control scheme no state constraints or other model characteristics are considered and on-line optimization is required for deriving the control actions. Equation (13) after the incorporation of the integral state is modified as follows:

$\begin{array}{cc}\phi \left(x_{t❘0}\right)=\underset{v_{k❘t}^N}{\min }{x}_{t❘N}^T{\mathrm{Px}}_{t❘N}+{\mathrm{xq}}_{t❘N}^T{P}_1{\mathrm{xq}}_{t❘N}+\sum _{k=0}^{N-1}\left[y_{t❘N}^T{\mathrm{Qy}}_{t❘k}+v_{t❘N}^T{\mathrm{Rv}}_{t❘k}+{\mathrm{xq}}_{t❘k}^T{Q}_1{\mathrm{xq}}_{t❘k}\right]\begin{array}{c}\mathrm{s1}.x_{t❘k+1}=A_1{x}_{t❘k}+A_2{v}_{t❘k}\\ y_{t❘k}=B_1{x}_{t❘k}+B_2{v}_{t❘k}\\ {\mathrm{xq}}_{t❘k+1}={\mathrm{xq}}_{t❘k}+y_{t❘k}\\ x_{t❘0}=x^{*}\\ {\mathrm{xq}}_{t❘0}={\mathrm{xq}}^{*}\\ 0\ge g\left(y_{k❘t},x_{k❘t},v_{k❘t}\right)=C_o{y}_{t❘k}+C_1{x}_{t❘k}+C_2{v}_{t❘k}+C_4\\ k=0,1,2\dots N\\ {\mathrm{xq}}_{t❘k}\in {\Re }^m\end{array}& \left(16\right)\end{array}$
where xq is the integral state; Q ₁ , P ₁ are the quadratic costs corresponding to that state. Note that the terms in bold take into account integral action and are the additional terms when comparing to (13).

Equation (16) and the associated conditions effectively introduce the additional state xq as an additional term in the objective function and as an additional condition dependent on the preceding integral state value and the respective output.

By eliminating the equalities associated with the dynamic system and considering the current pure and integral states as parameters a multiparametric quadratic program similar to equation (7) is formulated. The solution of this mp-QP corresponds to the following piecewise affine control law:
v _ct|0 ( x *)= a _c x*+b _c +d _c xq*;
CR _c ¹ x*+CR _c ² +CR _c ¹ xq* ≦0
for c=1, . . . N _c (17
which is similar to that given in equation (12).

The piecewise affine multivariable controller represented by equation (4) contains a proportional part a _c x _t (t*) with respect to the states, an integral part d _c xq _t in terms of the outputs and a bias b _c , hence it is a parametric proportional integral controller (Parametric PI). This control law is asymptotically stable, thus it guarantees no steady state offset from the target point on condition that (i) the dimension of the controls is larger or equal to the output dimension q≧m, which is the prevailing case in control problems (ii) the open-loop transfer matrix defined from the equation: H(s)=B ₁ )(sI−A ₁ ) ^−I A ₂ +B ₂ posses no zeros at the origin and (iii) the quadratic cost matrix Q ₁ that penalizes the integral error is positive-definite.

Similarly, derivative action can be accommodated in the parametric control law. An algebraic variable is introduced in the formulation that is equal to the difference between the current and the preceding output variable. This variable is penalized in the objective function as follows:

$\begin{array}{c}\phi \left(x_{t❘0}\right)=\underset{v_{k❘t}^N}{\min }{\mathrm{jx}}_{t❘N}^T{\mathrm{Px}}_{t❘N}+{\mathrm{xq}}_{t❘N}^T{P}_1{\mathrm{xq}}_{t❘N}+\sum _{k=0}^{N-1}\left[y_{t❘k}^T{\mathrm{Qy}}_{t❘k}+v_{t❘k}^T{\mathrm{Rv}}_{t❘k}+{\mathrm{xq}}_{t❘k}^T{Q}_1{\mathrm{xq}}_{t❘k}+y{d}_{t❘k}^T{Q}_{z}y{d}_{t❘k}\right]\begin{array}{c}s.t.x_{t❘k+1}=A_1{x}_{t❘k}+A_2{v}_{t❘k}k=0,1,2,\dots N\\ y_{t❘k}=B_1{x}_{t❘k}+B_2{v}_{t❘k}k=0,1,2,\dots N\\ {\mathrm{xq}}_{t❘k+1}={\mathrm{xq}}_{t❘k}+y_{t❘k}\end{array}\end{array}$