- Email: [email protected]
Constrained Long Range Predictive Control: Exploiting Convexity
Copyright I~') IFAC Advanced Control of Chemical Processes. Uanff. Canada. 1997
CONSTRAINED LONG RANGE PREDICTIVE CONTROL: EXPLOITING CONVEXITY Rohit S. Patwardhan
Biao Huang
Department of Chemlcal and Materials Engineering, University of Alberta, Edmonton, T6G 2G6. Email: [email protected]
Abstract. This work is concerned with the design of a long range predictive controller using a convex optimization approach. The constrained objective is first reformulated as a semidefinite program (SDP) and the resulting SDP is solved using an interior point method. A dynamic matrix controller based on the SDP formulation is experimentally evaluated on computer interfaced pilot-scale. univariate and multivariate processes. Keywords. Convex Optimization, Constrained Long Range Predictive ControL Interior Point methods, Real time Control
1. E\TRODUCTION
led to development of highly efficient and robust numerical techniques such as interior point methods. Several researchers have proposed ways of using the underlying convexity of the optimization step in LRPC algorithms through these techniques. Kothare et al. (1996) discuss robust design of model predictive controllers (MPC) , based on the min-max approach, in the linear matrix inequality (LMI) framework. Genceli and Nikolaou (1996) have considered the standard MPC optimization along with a persistent excitation constraint that current and future process inputs must satisfy over a finite moving horizon. The resulting problem is solved iteratively through a series of semidefinite programming (SDP) problems. The SDP is solved using an ellipsoid algorithm. Wright (1996) proposed an interior point algorithm for quadratic programming that exploits the special structure of the optimization step in MPC.
The term long range predictive control (LRPC) represents a family of model based predictive controllers. The LRPC algorithms are designed on the basis of a mult.istep ot.pimization objective. In general, several controller moves in the future are computed but only the first control action is implemented, hence these controllers are also referred to as receding horizon controllers. The earlier versions of LRPC are - the identificat.ion and command algorithm (IDCOM) proposed by Richalet et al. (1978) and dynamic matrix control (D~IC) algorithm due to Cutler and Ramaker (1980 ). Ot.her well known \'ariations of LRPC include: model algorithmic control (:\IAC) by Rouhani and l\lehra (1982), extended horizon adaptive control (EHAC) by Ydstie (1984) and Generalized Predictive Control (GPC) due to Clarke et al. (1987). The LRPC schemes have found widespread accpetance in the process industry due to their ability to handle interactions and constraints. The LRPC design is based on simple representations of the process using step or impulse responses.
In this study we give a novel way of casting the constrained LRPC scheme as a SDP which belongs to a particular class of convex problems. The SDP formulation of the optimization step in LRPC is outlined in a tutorial manner. Constraints on the inputs and the outputs are also incorporated. The SDP is solved using
Recent progress in the area of convex optimization has
329
This LP can be expressed as a SDP with F(r) = diag(.4..r+ b), i.e.,
an interior point method. Interior point methods offer polynomial time solutions to quadratic programs (QP) unlike quadratic programming methods which are exponential time algorithms. ~'1oreover convex optimization methods do not rely on ad-hoc termination criteria and provide guaranteed convergence to the global minima. The proposed method is experimentally evaluated on computer - interfaced pilot scale processes. The main contribution of this paper is the experimental evaluation of convex optimization techniques for real time control on univariate and multivariate processes.
Fo
Semidefinite programs can be solved very efficiently using interior point methods. Interior point methods are polynomial time algorithms for convex problems. They were first developed by Karmarkar (1984) for efficient solution of LPs. Nesterov and Nemirovsky (1994) were the first to generalize the interior point methods to general convex programming.
2.2 Long Range Predictive Control
The basic ideas involved in the design of a long range predictive control law can be summarized in the following steps: (1) Predict future plant response based on the process knowledge. (2) ~1inimize the sum of squares of the predicted control errors with some regard for the control effort, over a finite prediction horizon,
2. PRELEl,lE\ARIES 2.1 Semidefinite Programming
Convex problems are fundamentally tractable, both in theory and practice. One of their desirable properties is that the locally optimal solutions are globally optimal. SDPs are a class of convex optimization problems where the objective function is linear:
where F(r)
~
i=1
i=1
where w(k) is the setpoint at kth sampling instant. ~u(k) = u(k) - u(k - 1) ~u(k + Nu) = .. . = ~u(k + N2 - 1) = 0 (3) Compute appropriate control action that minimizes J and implement this control action.
(1) F(x)
0
In the terminology of model predictive control, N2 is the prediction horizon, Nu is the control horizon and .x is the control weighting parameter. Nu denotes the number of future incremental control moves (Nu ~ N2) .
= Fo + r1Fl + ... + xmFm
c E Rm and m + 1 symmetric matrices Fo .. · , Fm E Rm xm . The inequality sign F(x)~O implies F(x) is positive semidefinite. i.e. , 1,T F(X)l1~O for all nonzero v E Rm. In addition , the leading principal minors of F(r) must be positive. The constraint F(x)~O also defines a Ll\II in the nonstrict sense. A strict LMI implies F(x) > o (Boyd et al. , 1994) .
A real process is always subject to constraints on the rate and amplitude of control effort. A primary control objective is also to maintain the product quality within the desired range. The constrained predictive control objective then , is to N2
Other instances of convex problems include linearly constrained quadratic programs (LCQP) and quadratically constrained quadratic programs. A linear program (LP) is a special case of a semidefinite program as the example here illustrates ,
Min J =
L
(w(k+ i) -y(k +i))2 +.x
i=1
l'lu
L
~u(k-t- i _1)2
i=1
subject to Umin
~
u(k) ~ u max
~Umin ~ ~u(k) ~ ~umax S.t. A:r+b~O
1. .... m
where ai is the ith column of .4..
This paper is organized as follows: Section 2 gives the background material on semidefinite programming and long range predictive control. In Section 3 we discuss the formulation of the long range predictive controller design as a semidefinite program for the SISO case. The results of the the real time applications are presented and discussed in Section 4. followed by concluding remarks in Section 5.
S.t.
= diag(b), Fi = diag(ai), i =
(2)
Ymin
330
~
y(k) ~ Ymax
(3)
We shall discuss a particular formulation of LRPC namely Dynamic \latrix Control (DMC). In terms of a step response model the predicted output can be written as
FORMULATIO~
3. A SDP
OF LONG RANGE PREDICTIVE CO::\TROL
3.1 The unconstrained case .\"
y(k) =
L
sj t:::..u(k - i)
+ sssu(k
- N - 1)
(4)
In this section 've discuss the steps involved in posing the LRPC design procedure as a convex problem. Combining eqns. 6 and 7 the LRPC objective function can be rewritten as:
i=1
Similarly writing down the above relation for k+ 1. .. .. k+ N2 and expressing the resulting equations in vector form we have
J
= (w -
f - d)T(w - f - d)
-t-
~uT(GTG -t- '\I)~u-
2(w - f - d)TG~u
y=
G~u+f
(5)
We can ignore the constant term in the objective function viz. (w - f - d)T (w - f - d) . We then define a scalar rjJ such that
where. SI
S2
G=
0
. ..
SI
...
S 1\'" S/Ii,,-I
0 0
Having introduced an auxiliary variable that serves as an upper bound on the objective. we can recast our optimization problem as
SI
~u = [~u(k) ... ~u(k + Nu y = y(k + 1) ... y(k + N2)]T f = U(k + 1) ... f(k + N2) ]T
min rjJ subject to ~uT(GTG
1)f
+ '\I)~u -
In this formulation the objective function is linear with respect to rjJ; the earlier quadratic objective function appears as a quadratic constraint. This quadratic constraint can be expressed in terms of a L?vlI using the Schur Complement for the non-strict case (Vandenberghe and Boyd, 1996),
The matrix G is made up of the actual step response coefficients of the process model - it is called the dynamic matrix. The vector f is called the free response and captures the effect of the past control moves on the future outputs. The cost function of LRPC then becomes
min rjJ subject to
~uL 1 [ LT ~uT 2(w - f where w = [11.'(k
+ 1)
w(k
+ 2)
... w(k
2(w - f - dfG~u::; rjJ
]
d)T G~u + 1> ::;
0
(9)
where LT L is the Cholesky Factorization of (G T G+'\1)
+ N2) (
This is the standard form of a semidefinite program(SDP).
A more realistic prediction model includes an estimate of the disturbance
3.2 Extension to the constrained case
y = G~u + f -"- cl
cl=
(7) C sually a real process involves rate and amplitude const raints on the input, and may also require output constraints to be considered. Before proceeding we define the constraint vectors of appropriate dimensions.
[d(k + 1) d(k-t-2) ... d(I.:+.\'2)(
We assume, d(1.: ..l.. i) d(I.:). i = 1. ... , N2 where the estimate of d is obtained by finding the difference between the current measured and the predicted values of the output:
Al = [~Umin
_
A2 d(l.:) = Ym(l.:) - y(k)
(8)
... ~uminf ; BI
[~min-U(k-1)l ' :
'I1min -
331
= [~umax ... ~umaxf
_
, B2 -
u(k -
1)
[~max-'I1(k-1)l :
'I1max -
'I1 (k - 1)
A3 = [Ymin . .. Ymin]T - f -d: B3 = [Ymax ... Ymax 1T - f-d 10
R=
LMI Toolkit (Gahinet et al.. 1995). In all the cases. the LMIs were set up in Matlab 4.0 using the D.H Toolkit.
0
1 1
o 1
1
4.1 S1S0 Process
1
The constrained LRPC algorithm using the LMI toolbox was evaluated on a computer interfaced temperature control process. The objective is to control the temperature in a region around the light bulb by manipulating the voltage across the bulb filament. A constant speed fan provides the cooling action and also serves as the source of a disturbance. A thermocouple measures the temperature in the region of interest. This is a simple SISO plant. The transfer function between the power(input) and temperature(output) was first identified as:
In terms of these vectors the constraints can be rewritten as (Shah. 1994): Al ::::
~u
:::: BI : Rate
A2 :::: R~u :::: B2 : Amplitude
A3 ::::
G~u
(10)
:::: B3 : Output
If we let
p
= [ -1 -R
-G 1 R G (
Q = [BI B2 B3 -AI -A2 -A 3 (
Y(z)
then the constraint set (Eqn. 10) can be represented in terms of the following inequality. P~u+Q ~ O
U(z)
(11)
The real time implementation was carried out using Realtime-Simulink workshop. The sampling time was 5 seconds. The controller tuning parameters selected were N2 = 20, Nu = 2, >. = 6. The following rate and amplitude constraints were imposed on the input: 0 :::: u(k) :::: 100, -20 :::: ~u(k) :::: 20. The nominal operating conditions were U ss = 55% ; Yss = 59 .5%.
Incorporating Eqn. 11 as an additional LMI, the earlier SDP (Eqn. 9) now becomes min Q subject to
~uL 1 [ LT ~uT 2(w - f diag(P~u + Q) ~ 0
d)TG~u + cjJ
]
0
(12)
::::
Fig. 1 shows the tracking performance and the disturbance rejection characteristics of the designed controller. Rate and amplitude constraints on the input were satisfied (see fig. 2). At the time instant 100, when the set point is changed by a large value from 60 to 80, the input saturates at its high limit and as expected the output cannot reach the desired value and a bias in the control error is observed. The same phenomenon is observed when another large set point change is effected at sampling instant 150.
This formulation also belongs to the class of SDPs. Thus we have posed a linearly constrained quadratic program as a SDP. The :ML\-IO case can be formulated as a SDP in a similar manner. Due to lack of space we do not discuss the l\n~.Jo case here but the casestudy discussed in Section 4.2 is an application of the MIl\lO approach. The SDP approach can also be extended to receding horizon t.echniques based on state space or impulse response models. etc. The convexity of the underlying optimization problem is independent of the model structure and the t~'p e of disturbance models.
4. LRPC
0.0173z- 3 + 0 .0047z- 4 + 0.0069z- 5 1 - 0.8484z- 1 + 0.08z- 2 _ 0.1484z-3 (13)
4.2 M1MO Process The SDP formulation of the long range predictive controller was tested on a temperature-level control of a mixing process. Cold water (5-10 0c) and hot water (4550°C) flow through two inlet pipes into a glass tank. The exit flO\\Tate of water from the glass tank is not controlled. The control objective was to maintain the temperature (YIl and level (Y2) of Tank 1 by manipulating the inlet hot water (u d and cold water (U2) flowrates. The input-output relations were identified using a mu ltivariable identification routine (Badmus et al. , 1996)
DESIG~ VSI~G
IKTERIOR POI;.;rT :\IETHODS
The SDP formulation described above was evaluated experimentally on two computer-interfaced pilot scale processes. A l\IPC type controller was developed and the optimization step was solved using a semidefinite programming solver. based on the projective method of ~esterO\' and l\'emirovsky (1994) available as part of the
332
SO~----------------------------------
60--------------------------------------------------~~----
,>-4(}
2i-----5~0~0~--1~0~0~0--~15~0~0--~2~00~0--~2~~~0~--=3000
1oo--------------------------~--------
100~\--------------------------------------
90-
'S5(}~
~ a; SO-'. ~ yf\. "'!
00
500
1000
1500
Q.
SO-
,""--
~2q.
Samples
I
rv-----
00
Fig. 1. Performance of constrained LMI based LRPC for the temperature control of a light bulb process.
eya-
\..___'--------
500
1000
"--
1500== 2000
=2500
3000
.
e60 "--
.:.;;=--
~ ~5o-
,,'1· .:. "';, - ;"
...
4Q-
3%
-,-
50
100
150
200
250
300
3SO
400
450
4
SOO
100------~--------------------------~
" I
~ SO- '----------
~4cJ-
400
'
600
~
, 200
400
600
I
ut
~'" ".,~\~"",; •
(P hn
800
1000
.:
1000
800
:
T
1200
:
1200
6~
20I
., v
200
-a
80:', Cl)
3000
Fig. 3. Tracking performance of the constrained LRPC for the temperature and level control of a mixing process
80------------------------------------
'0
2500
40~----------------__-------------------
5%--~5~0--~1~0~0--~1~50--~~~0--~25~0--~3~0~0--~35~0~~~
=>
2000
'S;t
" '-,
%~ . --~50--~1~00--~15~0~2~00~~25~0~3~0~0~3~50--~~0~4~50~500 Samples
Fig. 2. Temperature control of a light bulb process: Effect of constraints.
~~t 00
based on canonical variate analysis (CVA). The transfer function matrix as identified is: 1 0.02352-1 -0.1602z1 1- 0.8607z- 1 T(z-I) 1 - 0.8607z0.2043z- 1 0.2839z- 1 [ 1 - 0.9827z - 1 1 - 0.9827rl
, 200
:
200
~ 400
600
=~ 400
600
1400
j 1400
I
]
800
1000
1200
1400
800
1000
"-: 1200
1400
j
Fig. 4. Controller performance in the regulatory mode.
1 (14)
UI ss
= 29% ; U2ss = 16%; YIss
= 46.7°C.
YIss
= 16 cm
The tracking and regulatory characteristics of the designed controller are shown in figs. 3 and 4, respectively. Rate and amplitude constraints on the input were satisfied during the tracking run. :\"otice the instances in fig. 3 when the inputs saturate at the above specified limits. This illustrates that the algorithm indeed does enforce the specified limits. The hot water inlet temperature showed a time-varying trend. Fig. 4 illustrates the ability of the controller to compensate for this timevarying disturbance and maintain the outputs at their desired values.
The multi"ariable LRPC was implemented in a RealtimeSimulink environment. The sampling time was chosen as 5 seconds. The runs carried out demonstrate the applicability of the L~n based LRPC. The LRPC tuning parameters were 1\'2 = 10. Nu = 2. Al = A2 = 0.1 . The rate and amplitude constraints were imposed on both inputs were respectively 0 -s: u;(k) -s: 100 and -20 -s: 6. u i(k) -s: 20. i = l. 2. The steady state operating conditions were:
333
5. CONCLllSIO?'JS
6. REFERENCES
Badmus. O. 0. , S. 1. Shah and Fisher D. G. (1996). Case studies in system identification. Technical report. Dept. of Chem. Eng., University of Alberta. Boyd, S., 1. El. Ghaouli , E. Feron and V. Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. Vol. 15 of Studies in Applied Mathematics. SIAM, Philadelphia. Clarke, D. W., C. Mohtadi and P. S. Thffs (1987). Generalized predictive control - part 1 and 2. A utomatica 23, 137-160. Cutler. C. R. and B. 1. Ramaker (1990). Dynamic matrix control - a computer control algorithm. In: Proceedings of Joint American Control Conference. San Fransisco, USA. Gahinet, P. , A. Nemirovsky, A. J. Laub and M. Chilai (1995). LMI Control Toolbox: For use with MATLAB. MathWorks Inc. Genceli, H. and M. Nikaloau (1996). A new approach to constrained model predictive control with simultaneous identication. A IChEJ. 42, 2857-2868. Karmarkar, N. (1984). A new polynomial time algorithm for linear programming. Combinatorica 4(4). 373395. Kothare, M. V., V. Balakrishnan and M. Morari (1996) . Robust constrained model predictive control using linear matrix inequalities. Automatica 32 , 13611379. Kestervo, Y. and A. Nemirovsky (1994). Interwr Point Polynomial Methods in Convex Programming. Vol. 13 of Studies in Applied Mathematics. SIAM, Philadelphia. Richalet , J. , A. Rault , J . L Testud and J . Papan (1978) . ~lodel predictive heuristic controller applications to industrial processes. Automatica 14, 413-428. Rouhani, R. and R. K. Mehra (1982). Model algorithmic control. Automatica 18, 401-414. Shah, S. L. (1994). A tutorial introduction to constrained long range predictive control. Pulp fj Paper Canada 96(4), 148- 154. Vandenberghe, 1. and S Boyd (1996). Semidefinite programming. SIAM Review 38, 49-95. Wright, S. J. (1996). Applying new optimization algorithms to model predictive control. In: Proceedings of the CPC- V. Y dstie , B. E . (1984). Extended horizon adpative control. In: Proceedings of the 9 th IFAC World Congress. Budapest, Hungary. pp. 1771-1780.
The LRPC design problem is cast as a semidefinite program and solved using an interior point method. The proposed approach is demonstrated on SISO and MIl\10 processes- in a real time environment. To the best of our knowledge this is the first experimental evaluation of the constrained LRPC based on the convex optimization framework using interior point methods. The following additional remarks are based on our experience with this method: • The efficiency of the interior point methods can be better exploited for the multivariate case, where large dimensionality and structure play an important role. • The predictive control problem occurs naturally as a linearly constrained quadratic program. Interior point methods developed for LCQPs may prove to be superior for this case, e.g. Wright (1996) . • It should also be noted that the L:'lI control toolbox is optimized for block structured LMls with matrix variables. The tool box is quite inefficient in handling inequalities in the canonical form. This results in higher computational times as compared to standard optimization software for solving QPs. This fact is also corroborated by Kothare et al. (1996). • The performance of convex optimization methods vis-a.-vis local methods needs to be assessed. There is a need to quantify the benefits of using the above formulation . Thus, an important impediment to the widespread acceptance of the convex optimization methods is their computational efficiency. Improvement of computational efficiency of these methods is an active area of research. Inspite of these shortcomings we would like to emphasize here t hat interior point methods provide tools for dew loping very reliable design procedures. Convergence to the global minima is guaranteed and interior point methods have polynomial complexity in the worst case. As more efficient implementations are developed, the software will be able to match the efficiency promised b\' these met hods.
334
CONSTRAINED LONG RANGE PREDICTIVE CONTROL: EXPLOITING CONVEXITY Rohit S. Patwardhan
Biao Huang
Department of Chemlcal and Materials Engineering, University of Alberta, Edmonton, T6G 2G6. Email: [email protected]
Abstract. This work is concerned with the design of a long range predictive controller using a convex optimization approach. The constrained objective is first reformulated as a semidefinite program (SDP) and the resulting SDP is solved using an interior point method. A dynamic matrix controller based on the SDP formulation is experimentally evaluated on computer interfaced pilot-scale. univariate and multivariate processes. Keywords. Convex Optimization, Constrained Long Range Predictive ControL Interior Point methods, Real time Control
1. E\TRODUCTION
led to development of highly efficient and robust numerical techniques such as interior point methods. Several researchers have proposed ways of using the underlying convexity of the optimization step in LRPC algorithms through these techniques. Kothare et al. (1996) discuss robust design of model predictive controllers (MPC) , based on the min-max approach, in the linear matrix inequality (LMI) framework. Genceli and Nikolaou (1996) have considered the standard MPC optimization along with a persistent excitation constraint that current and future process inputs must satisfy over a finite moving horizon. The resulting problem is solved iteratively through a series of semidefinite programming (SDP) problems. The SDP is solved using an ellipsoid algorithm. Wright (1996) proposed an interior point algorithm for quadratic programming that exploits the special structure of the optimization step in MPC.
The term long range predictive control (LRPC) represents a family of model based predictive controllers. The LRPC algorithms are designed on the basis of a mult.istep ot.pimization objective. In general, several controller moves in the future are computed but only the first control action is implemented, hence these controllers are also referred to as receding horizon controllers. The earlier versions of LRPC are - the identificat.ion and command algorithm (IDCOM) proposed by Richalet et al. (1978) and dynamic matrix control (D~IC) algorithm due to Cutler and Ramaker (1980 ). Ot.her well known \'ariations of LRPC include: model algorithmic control (:\IAC) by Rouhani and l\lehra (1982), extended horizon adaptive control (EHAC) by Ydstie (1984) and Generalized Predictive Control (GPC) due to Clarke et al. (1987). The LRPC schemes have found widespread accpetance in the process industry due to their ability to handle interactions and constraints. The LRPC design is based on simple representations of the process using step or impulse responses.
In this study we give a novel way of casting the constrained LRPC scheme as a SDP which belongs to a particular class of convex problems. The SDP formulation of the optimization step in LRPC is outlined in a tutorial manner. Constraints on the inputs and the outputs are also incorporated. The SDP is solved using
Recent progress in the area of convex optimization has
329
This LP can be expressed as a SDP with F(r) = diag(.4..r+ b), i.e.,
an interior point method. Interior point methods offer polynomial time solutions to quadratic programs (QP) unlike quadratic programming methods which are exponential time algorithms. ~'1oreover convex optimization methods do not rely on ad-hoc termination criteria and provide guaranteed convergence to the global minima. The proposed method is experimentally evaluated on computer - interfaced pilot scale processes. The main contribution of this paper is the experimental evaluation of convex optimization techniques for real time control on univariate and multivariate processes.
Fo
Semidefinite programs can be solved very efficiently using interior point methods. Interior point methods are polynomial time algorithms for convex problems. They were first developed by Karmarkar (1984) for efficient solution of LPs. Nesterov and Nemirovsky (1994) were the first to generalize the interior point methods to general convex programming.
2.2 Long Range Predictive Control
The basic ideas involved in the design of a long range predictive control law can be summarized in the following steps: (1) Predict future plant response based on the process knowledge. (2) ~1inimize the sum of squares of the predicted control errors with some regard for the control effort, over a finite prediction horizon,
2. PRELEl,lE\ARIES 2.1 Semidefinite Programming
Convex problems are fundamentally tractable, both in theory and practice. One of their desirable properties is that the locally optimal solutions are globally optimal. SDPs are a class of convex optimization problems where the objective function is linear:
where F(r)
~
i=1
i=1
where w(k) is the setpoint at kth sampling instant. ~u(k) = u(k) - u(k - 1) ~u(k + Nu) = .. . = ~u(k + N2 - 1) = 0 (3) Compute appropriate control action that minimizes J and implement this control action.
(1) F(x)
0
In the terminology of model predictive control, N2 is the prediction horizon, Nu is the control horizon and .x is the control weighting parameter. Nu denotes the number of future incremental control moves (Nu ~ N2) .
= Fo + r1Fl + ... + xmFm
c E Rm and m + 1 symmetric matrices Fo .. · , Fm E Rm xm . The inequality sign F(x)~O implies F(x) is positive semidefinite. i.e. , 1,T F(X)l1~O for all nonzero v E Rm. In addition , the leading principal minors of F(r) must be positive. The constraint F(x)~O also defines a Ll\II in the nonstrict sense. A strict LMI implies F(x) > o (Boyd et al. , 1994) .
A real process is always subject to constraints on the rate and amplitude of control effort. A primary control objective is also to maintain the product quality within the desired range. The constrained predictive control objective then , is to N2
Other instances of convex problems include linearly constrained quadratic programs (LCQP) and quadratically constrained quadratic programs. A linear program (LP) is a special case of a semidefinite program as the example here illustrates ,
Min J =
L
(w(k+ i) -y(k +i))2 +.x
i=1
l'lu
L
~u(k-t- i _1)2
i=1
subject to Umin
~
u(k) ~ u max
~Umin ~ ~u(k) ~ ~umax S.t. A:r+b~O
1. .... m
where ai is the ith column of .4..
This paper is organized as follows: Section 2 gives the background material on semidefinite programming and long range predictive control. In Section 3 we discuss the formulation of the long range predictive controller design as a semidefinite program for the SISO case. The results of the the real time applications are presented and discussed in Section 4. followed by concluding remarks in Section 5.
S.t.
= diag(b), Fi = diag(ai), i =
(2)
Ymin
330
~
y(k) ~ Ymax
(3)
We shall discuss a particular formulation of LRPC namely Dynamic \latrix Control (DMC). In terms of a step response model the predicted output can be written as
FORMULATIO~
3. A SDP
OF LONG RANGE PREDICTIVE CO::\TROL
3.1 The unconstrained case .\"
y(k) =
L
sj t:::..u(k - i)
+ sssu(k
- N - 1)
(4)
In this section 've discuss the steps involved in posing the LRPC design procedure as a convex problem. Combining eqns. 6 and 7 the LRPC objective function can be rewritten as:
i=1
Similarly writing down the above relation for k+ 1. .. .. k+ N2 and expressing the resulting equations in vector form we have
J
= (w -
f - d)T(w - f - d)
-t-
~uT(GTG -t- '\I)~u-
2(w - f - d)TG~u
y=
G~u+f
(5)
We can ignore the constant term in the objective function viz. (w - f - d)T (w - f - d) . We then define a scalar rjJ such that
where. SI
S2
G=
0
. ..
SI
...
S 1\'" S/Ii,,-I
0 0
Having introduced an auxiliary variable that serves as an upper bound on the objective. we can recast our optimization problem as
SI
~u = [~u(k) ... ~u(k + Nu y = y(k + 1) ... y(k + N2)]T f = U(k + 1) ... f(k + N2) ]T
min rjJ subject to ~uT(GTG
1)f
+ '\I)~u -
In this formulation the objective function is linear with respect to rjJ; the earlier quadratic objective function appears as a quadratic constraint. This quadratic constraint can be expressed in terms of a L?vlI using the Schur Complement for the non-strict case (Vandenberghe and Boyd, 1996),
The matrix G is made up of the actual step response coefficients of the process model - it is called the dynamic matrix. The vector f is called the free response and captures the effect of the past control moves on the future outputs. The cost function of LRPC then becomes
min rjJ subject to
~uL 1 [ LT ~uT 2(w - f where w = [11.'(k
+ 1)
w(k
+ 2)
... w(k
2(w - f - dfG~u::; rjJ
]
d)T G~u + 1> ::;
0
(9)
where LT L is the Cholesky Factorization of (G T G+'\1)
+ N2) (
This is the standard form of a semidefinite program(SDP).
A more realistic prediction model includes an estimate of the disturbance
3.2 Extension to the constrained case
y = G~u + f -"- cl
cl=
(7) C sually a real process involves rate and amplitude const raints on the input, and may also require output constraints to be considered. Before proceeding we define the constraint vectors of appropriate dimensions.
[d(k + 1) d(k-t-2) ... d(I.:+.\'2)(
We assume, d(1.: ..l.. i) d(I.:). i = 1. ... , N2 where the estimate of d is obtained by finding the difference between the current measured and the predicted values of the output:
Al = [~Umin
_
A2 d(l.:) = Ym(l.:) - y(k)
(8)
... ~uminf ; BI
[~min-U(k-1)l ' :
'I1min -
331
= [~umax ... ~umaxf
_
, B2 -
u(k -
1)
[~max-'I1(k-1)l :
'I1max -
'I1 (k - 1)
A3 = [Ymin . .. Ymin]T - f -d: B3 = [Ymax ... Ymax 1T - f-d 10
R=
LMI Toolkit (Gahinet et al.. 1995). In all the cases. the LMIs were set up in Matlab 4.0 using the D.H Toolkit.
0
1 1
o 1
1
4.1 S1S0 Process
1
The constrained LRPC algorithm using the LMI toolbox was evaluated on a computer interfaced temperature control process. The objective is to control the temperature in a region around the light bulb by manipulating the voltage across the bulb filament. A constant speed fan provides the cooling action and also serves as the source of a disturbance. A thermocouple measures the temperature in the region of interest. This is a simple SISO plant. The transfer function between the power(input) and temperature(output) was first identified as:
In terms of these vectors the constraints can be rewritten as (Shah. 1994): Al ::::
~u
:::: BI : Rate
A2 :::: R~u :::: B2 : Amplitude
A3 ::::
G~u
(10)
:::: B3 : Output
If we let
p
= [ -1 -R
-G 1 R G (
Q = [BI B2 B3 -AI -A2 -A 3 (
Y(z)
then the constraint set (Eqn. 10) can be represented in terms of the following inequality. P~u+Q ~ O
U(z)
(11)
The real time implementation was carried out using Realtime-Simulink workshop. The sampling time was 5 seconds. The controller tuning parameters selected were N2 = 20, Nu = 2, >. = 6. The following rate and amplitude constraints were imposed on the input: 0 :::: u(k) :::: 100, -20 :::: ~u(k) :::: 20. The nominal operating conditions were U ss = 55% ; Yss = 59 .5%.
Incorporating Eqn. 11 as an additional LMI, the earlier SDP (Eqn. 9) now becomes min Q subject to
~uL 1 [ LT ~uT 2(w - f diag(P~u + Q) ~ 0
d)TG~u + cjJ
]
0
(12)
::::
Fig. 1 shows the tracking performance and the disturbance rejection characteristics of the designed controller. Rate and amplitude constraints on the input were satisfied (see fig. 2). At the time instant 100, when the set point is changed by a large value from 60 to 80, the input saturates at its high limit and as expected the output cannot reach the desired value and a bias in the control error is observed. The same phenomenon is observed when another large set point change is effected at sampling instant 150.
This formulation also belongs to the class of SDPs. Thus we have posed a linearly constrained quadratic program as a SDP. The :ML\-IO case can be formulated as a SDP in a similar manner. Due to lack of space we do not discuss the l\n~.Jo case here but the casestudy discussed in Section 4.2 is an application of the MIl\lO approach. The SDP approach can also be extended to receding horizon t.echniques based on state space or impulse response models. etc. The convexity of the underlying optimization problem is independent of the model structure and the t~'p e of disturbance models.
4. LRPC
0.0173z- 3 + 0 .0047z- 4 + 0.0069z- 5 1 - 0.8484z- 1 + 0.08z- 2 _ 0.1484z-3 (13)
4.2 M1MO Process The SDP formulation of the long range predictive controller was tested on a temperature-level control of a mixing process. Cold water (5-10 0c) and hot water (4550°C) flow through two inlet pipes into a glass tank. The exit flO\\Tate of water from the glass tank is not controlled. The control objective was to maintain the temperature (YIl and level (Y2) of Tank 1 by manipulating the inlet hot water (u d and cold water (U2) flowrates. The input-output relations were identified using a mu ltivariable identification routine (Badmus et al. , 1996)
DESIG~ VSI~G
IKTERIOR POI;.;rT :\IETHODS
The SDP formulation described above was evaluated experimentally on two computer-interfaced pilot scale processes. A l\IPC type controller was developed and the optimization step was solved using a semidefinite programming solver. based on the projective method of ~esterO\' and l\'emirovsky (1994) available as part of the
332
SO~----------------------------------
60--------------------------------------------------~~----
,>-4(}
2i-----5~0~0~--1~0~0~0--~15~0~0--~2~00~0--~2~~~0~--=3000
1oo--------------------------~--------
100~\--------------------------------------
90-
'S5(}~
~ a; SO-'. ~ yf\. "'!
00
500
1000
1500
Q.
SO-
,""--
~2q.
Samples
I
rv-----
00
Fig. 1. Performance of constrained LMI based LRPC for the temperature control of a light bulb process.
eya-
\..___'--------
500
1000
"--
1500== 2000
=2500
3000
.
e60 "--
.:.;;=--
~ ~5o-
,,'1· .:. "';, - ;"
...
4Q-
3%
-,-
50
100
150
200
250
300
3SO
400
450
4
SOO
100------~--------------------------~
" I
~ SO- '----------
~4cJ-
400
'
600
~
, 200
400
600
I
ut
~'" ".,~\~"",; •
(P hn
800
1000
.:
1000
800
:
T
1200
:
1200
6~
20I
., v
200
-a
80:', Cl)
3000
Fig. 3. Tracking performance of the constrained LRPC for the temperature and level control of a mixing process
80------------------------------------
'0
2500
40~----------------__-------------------
5%--~5~0--~1~0~0--~1~50--~~~0--~25~0--~3~0~0--~35~0~~~
=>
2000
'S;t
" '-,
%~ . --~50--~1~00--~15~0~2~00~~25~0~3~0~0~3~50--~~0~4~50~500 Samples
Fig. 2. Temperature control of a light bulb process: Effect of constraints.
~~t 00
based on canonical variate analysis (CVA). The transfer function matrix as identified is: 1 0.02352-1 -0.1602z1 1- 0.8607z- 1 T(z-I) 1 - 0.8607z0.2043z- 1 0.2839z- 1 [ 1 - 0.9827z - 1 1 - 0.9827rl
, 200
:
200
~ 400
600
=~ 400
600
1400
j 1400
I
]
800
1000
1200
1400
800
1000
"-: 1200
1400
j
Fig. 4. Controller performance in the regulatory mode.
1 (14)
UI ss
= 29% ; U2ss = 16%; YIss
= 46.7°C.
YIss
= 16 cm
The tracking and regulatory characteristics of the designed controller are shown in figs. 3 and 4, respectively. Rate and amplitude constraints on the input were satisfied during the tracking run. :\"otice the instances in fig. 3 when the inputs saturate at the above specified limits. This illustrates that the algorithm indeed does enforce the specified limits. The hot water inlet temperature showed a time-varying trend. Fig. 4 illustrates the ability of the controller to compensate for this timevarying disturbance and maintain the outputs at their desired values.
The multi"ariable LRPC was implemented in a RealtimeSimulink environment. The sampling time was chosen as 5 seconds. The runs carried out demonstrate the applicability of the L~n based LRPC. The LRPC tuning parameters were 1\'2 = 10. Nu = 2. Al = A2 = 0.1 . The rate and amplitude constraints were imposed on both inputs were respectively 0 -s: u;(k) -s: 100 and -20 -s: 6. u i(k) -s: 20. i = l. 2. The steady state operating conditions were:
333
5. CONCLllSIO?'JS
6. REFERENCES
Badmus. O. 0. , S. 1. Shah and Fisher D. G. (1996). Case studies in system identification. Technical report. Dept. of Chem. Eng., University of Alberta. Boyd, S., 1. El. Ghaouli , E. Feron and V. Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. Vol. 15 of Studies in Applied Mathematics. SIAM, Philadelphia. Clarke, D. W., C. Mohtadi and P. S. Thffs (1987). Generalized predictive control - part 1 and 2. A utomatica 23, 137-160. Cutler. C. R. and B. 1. Ramaker (1990). Dynamic matrix control - a computer control algorithm. In: Proceedings of Joint American Control Conference. San Fransisco, USA. Gahinet, P. , A. Nemirovsky, A. J. Laub and M. Chilai (1995). LMI Control Toolbox: For use with MATLAB. MathWorks Inc. Genceli, H. and M. Nikaloau (1996). A new approach to constrained model predictive control with simultaneous identication. A IChEJ. 42, 2857-2868. Karmarkar, N. (1984). A new polynomial time algorithm for linear programming. Combinatorica 4(4). 373395. Kothare, M. V., V. Balakrishnan and M. Morari (1996) . Robust constrained model predictive control using linear matrix inequalities. Automatica 32 , 13611379. Kestervo, Y. and A. Nemirovsky (1994). Interwr Point Polynomial Methods in Convex Programming. Vol. 13 of Studies in Applied Mathematics. SIAM, Philadelphia. Richalet , J. , A. Rault , J . L Testud and J . Papan (1978) . ~lodel predictive heuristic controller applications to industrial processes. Automatica 14, 413-428. Rouhani, R. and R. K. Mehra (1982). Model algorithmic control. Automatica 18, 401-414. Shah, S. L. (1994). A tutorial introduction to constrained long range predictive control. Pulp fj Paper Canada 96(4), 148- 154. Vandenberghe, 1. and S Boyd (1996). Semidefinite programming. SIAM Review 38, 49-95. Wright, S. J. (1996). Applying new optimization algorithms to model predictive control. In: Proceedings of the CPC- V. Y dstie , B. E . (1984). Extended horizon adpative control. In: Proceedings of the 9 th IFAC World Congress. Budapest, Hungary. pp. 1771-1780.
The LRPC design problem is cast as a semidefinite program and solved using an interior point method. The proposed approach is demonstrated on SISO and MIl\10 processes- in a real time environment. To the best of our knowledge this is the first experimental evaluation of the constrained LRPC based on the convex optimization framework using interior point methods. The following additional remarks are based on our experience with this method: • The efficiency of the interior point methods can be better exploited for the multivariate case, where large dimensionality and structure play an important role. • The predictive control problem occurs naturally as a linearly constrained quadratic program. Interior point methods developed for LCQPs may prove to be superior for this case, e.g. Wright (1996) . • It should also be noted that the L:'lI control toolbox is optimized for block structured LMls with matrix variables. The tool box is quite inefficient in handling inequalities in the canonical form. This results in higher computational times as compared to standard optimization software for solving QPs. This fact is also corroborated by Kothare et al. (1996). • The performance of convex optimization methods vis-a.-vis local methods needs to be assessed. There is a need to quantify the benefits of using the above formulation . Thus, an important impediment to the widespread acceptance of the convex optimization methods is their computational efficiency. Improvement of computational efficiency of these methods is an active area of research. Inspite of these shortcomings we would like to emphasize here t hat interior point methods provide tools for dew loping very reliable design procedures. Convergence to the global minima is guaranteed and interior point methods have polynomial complexity in the worst case. As more efficient implementations are developed, the software will be able to match the efficiency promised b\' these met hods.
334