Suboptimality analysis of receding horizon predictive control with terminal constraints

SUBOPTIMALITY ANALYSIS OF RECEDING HORIZON PREDICT...

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C:opyright © t 999 IF AC J4th Triennial \Vorld Congress, Beijing, P.R. China

F-2d-07-5

SUBOPTIMALITY ANALYSIS OF RECEDING HORIZON PREDICTIVE CONTROL WITH TERMINAL CONSTRAINTS

Xiaojun Geng-

and

Yugeng Xi

*'"

Institute ofAuto/nation, Shanghai Jiao Tong University,

Shanghai 200030, P. R. China Email: *weng@hotmail com' **vgxjtiiJmail sjtu edu en

Abstract: In this paper, the suboptimality of predictive control ""Tith terminal constraints is analyzed. and compared with traditional optimal control. First, by studying the property of receding finite-horizon optimization, the upper bound of suboptimality of predictive control for nonlinear system is deduced. The result is then applied to linear system, and a quantitative measure of 5uboptimality is obtained. Copyright© 1999 IFAC. Key\vords.: predictive control; optimal control; suboptimal systems; nonlinear systems.

1. INTRODUCTION The optimal control theory has put forward the modem control theory. However, its application to the industry was limited because the industrial process is always time.. varying and has uncertainty. Predictive control, developed in 70's, improved the optimal control by adopting on-line receding horizon optimization and feedback information from rea)

process,

and

has

complicated industrial Stemming from

proved

successful

to

the

(Mayne, et al.~ 1990b). Alamir and Bomard (1994) gave some stability results of CRHPC for nonlinear discrete-time systems. Meadows et. al (1995) further extended it to feedback Iinearizable systems. Later, in order to overcome feasibility difficulties resulting from the terminal constraints, some effective approaches to relax the terminal condition are proposed (see Rossiter, 1994; Rossiter and et. aI, 1996). Those works are mainly based on General Predictive Control (GPC).

processes~

receding horizon concept \vith

terminal constraint, Kwon and Pearson (1977, 1978) designed a stable feedback controller for linear discrete time-varying systems, and analyzed the relation between it and the standard optimal control strategy in terms of the properties of Riccati equations~ Later, Mayne et. al. (1990a) showed that Receding Horizon Predictive Control with Tenninal Constraints (CRHPC) can yield a stable closed-loop system when applied to continuous nonlinear system, and gave an implementable control algorithm

Freeman and Kokotovic (1996) pointed out that optimal control can guarantee the stability of system if an infinite-horizon optimization index is adopted, whereas with finite horizon optimization index the conclusion does not hold. However} based on receding finite horizon optimization, predictive control can stabilize system under certain conditions. CRHPC, constraining the state to a designed value in the terminal of predictive horizon.. is a stable control strategy and has attracted many researchers. The study on CRHPC can be classified into two categories, one is based on system 1/0 model and

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SUBOPTIMALITY ANALYSIS OF RECEDING HORIZON PREDICT...

required to track a expected output, the other is based on state space model and required to drive all the states to origin. Suppose that the system is observable, then the substance of above two problems is congruous. In order to highlight the approach in this paper, only the state-regulating problem is considered.

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The optimal control problem at k form:

then takes the

minJ1(k) 1rN

(k)

s.t.

Zj+t(k)=f(zj(k),vj(k)

(3)

zo(k) = X(k),ZN(k) == 0 It's obvious that optimal control achieves global optimal because the whole control horizon is considered. However, in receding horizon control, at each stop only a finite horizon optimization is considered, so receding horizon control can't guarantee global optimality. Thus, the suboptimality of CRHPC deserves our attention. In this paper, based on above stable strategies, the suboptimality of CRHPC for discrete nonlinear systems with respect to the traditional optimal control is analyzed. Section 2 gives the description of receding horizon predictive control with terminal constraints. In section 3, the formulation of suboptimality is presented, and the main result of this paper, that is, the upper bound of suboptimality is given. The result is applied to linear systems in section 4 and a quantitative measure for suboptimality is obtained. FinallY;t some concluding remarks are given in section 5.

where Zj(k),vi(k) denote the predictive state and predictive control concerned in the course of current finite horizon optimization respectively. In addition, L is continuous in (0,0) and has the following properties (Meadows et at. 1995):

a) b)

L(O,O)=O There exists a non-decreasing r : [O~C()) ~ [0,00), such that y(O)

o < Y(llx,u\\) for all (x,u)

Consider the described by

general discrete

x(k + 1)

nonlinear system

= f(x(k), u(k))

where x(O) is known, and x(k)

f

E

d)

1C N

(k)

represent a N step control sequence:

1!N(k) == {vo(k),vt(k), .. ,vN_t(k)}

vi(k)

E

R

m

*" (0,0). <1(·,,,11

represents a norm on

L(x,u) L(x,u)

0

-)0

:=

0

=> (x,u)

~

0

:::::> (x,u) = (0,0)

1r~(k) = {v; (k), v;" (k),

... , V~_l (k)}

by solving the problem (3), and implement the current control move

is continuous in origin satisfying f(O,O) = 0 .

At time k, given current state x(k) , let

~ L(x,u)

Predictive control as a receding horizon optimization strategy is described as: at time k, obtain the optimal control sequence

(1)

Rn ,u(k) E R m .

and

Rn X R m ). These lead to the following additional properties of L: c) L(x,u) > 0; \;f(x,u)*-(O~O) e)

2. RECEDING HORIZON PREDICTIVE CONTROL WITH TERMINAL ONSTRAINTS

=0

u(k) = v~ (k)

(4)

to the system. In the next step k + 1, repeat the above procedure based on the new initial state x(k + 1), and so on. The control strategy yields a closed-loop stable system~ The stability results concerned can be found in Alamir et. al. (1994) and Meadows et. al (1995).

The state resulted from ( 1) with input sequence 1i N

(k) is given as:

{zo(k), Zl (k),·· ·,ZN (k)},z; (k)

E

3.. THE SUBOPTIMALITY OF CRHPC FOR NONLINEAR SYSTEMS

Rn

And the corresponding fmite horizon optimization index is defined by Lt

J l(k) == J(x(k), re N (k)

N-l

= LL(z

j

(k), V j (k))

)=0

(2)

To analyze the suboptimality of CRHPC, the optimal control performance with inrmite horizon is taken as a standard. Consider system (I), if we can obtain a control sequence {u( k), k = 0,1,· ..}, such that the following performance index

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SUBOPTIMALITY ANALYSIS OF RECEDING HORIZON PREDICT...

co

J

L L(x(k), u(k))

0==

.7!N(k+I)

(5)

= {vo (k + 1),··" V N-2 (k + 1), V N-J (k + I)} = {v;(k), ... ,V~_l(k),O}

k=O

exists and is minimized. Then the control sequence is optimal, where L is congruous with that in section 2. Note that the performance index of infinite horizon optimal contro I satisfies co

J; = L L(x· (k),u· (k)) <

{zi(k+l)'···Z}l(k+l)}

Let

resulted by

00 .

1!/ol

the

state

(k + 1) . Clearly,

(k + 1) =

ZI

denote

z; (k) ,

k=O

Z .~r -I

lim L(x'" (k), u + (k)) == 0 in terms of

There exists

k4d:J

L. Then (x· (k),u· (k)

(0,0)

==;

z~. (k)

=0 ~

ZN

(k + 1) = /(Zh'-l (k + 1),

ZlV

(k + 1) = !(ZjV-I (k + 1),

the property c) of

~

(k + 1)

V N - l (k

+ 1)) == /(0,0) = 0

so the control system is stable·. satisfies the terminal constraints~ so (k + 1) is a feasible control sequence with respect

which In order to analyze the suboptimality of predictive control strategy above, it is necessary to study the

J;

relation between

and the global performance

index of predictive control J 10

,

to the optimization problem (3) at time k + 1. The corresponding performance index is dermed as J] (k + 1). Comparing J 1 (k + 1) with the optimal performance index in k + 1, there exists

cl)

== LL(x(k)~ u(k)

JJO

1t N

(6)

J. (k

k=O

where x(k), u(k) is the real state and control value during the whole receding horizon predictive control. Starting from this point, the main results will be presented be lo~r.

+ 1) ~ Jt (k + 1)

Note that:

J 1 (k + 1) N-l

== LL(zj(k+l),v j (k+l)) )=0

}/-2

Solving the finite-horizon optimization problem at time k, we can obtain N step optimal control

1l",~ (k) "" {v; (k),

corresponding

v; (k),···, V~_1 (k)}.

state

sequence

==

j=D

and the

described

N-l

= L L(z~ (k), v; (k)) -

as

{z;(k),z;(k), ... ,z;(k)}. where z;(k) == O. as Jt(k).

= J;~ (k) -

is

implemented,

that

is

u(k) == v~ (k) ~ After that the state value at time k + 1 is obtained as .Y"(k + 1) = f(x(k)~ v; (k)), VtJ1hich is

z; (k) . By repeating the procedure above ~

we get 7l"~r ek + 1) and

and implement v; (k

v; (k)

L(x(k), u(k))

We have,

J 1• (k) -

Based on receding optimization, only v;(k) as the control

L(z; (k),

J=O

The optimal performance index in this step is defined

current

LL(z.~+l (k), V;+l (k))

+ 1)

J; (k + 1),

in addition,

(7)

where the items on the left-hand represent the finitehorizon optimal performance index (2) in step k and k + 1 respectively, and that on the right-hand corresponds to the real value implemented~ Summing (7) from k=O to k= k *, then

+ 1) , and so on. time k + 1 defined

J; (0) -

as u(k

Consider a control sequence at as follows:

J; (k + 1) 2 L(x(k),u(k))

k·-]

J 1 (k*) 2: LL(x(k), u(k») $

(8)

i==O

When

k* ~

cIj

~

v-'e

get

lim JI~ (k"') == 0

k"'~oo

because the RHC controller is a stable control

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SUBOPTIMALITY ANALYSIS OF RECEDING HORIZON PREDICT...

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strategy (Alamir and Bomard , 1994; Meado\vs, et al. ] 995). Thus

11l-1

JtN == J lA1 = min

l: L(Zl , Vi) 2 Jl~U /=0

~

J; (0) ~ ~L(x(k),u(k))

(9)

M-l

== min LL(Zl' Vi)

k==(J

i:=:O

where the term on the right-hand is just the global performance index of receding horizon predictive controL Then we have the following:

It shows the greater the optimization horizon, the smaller the frrst step finite horizon optimal index, so the better the sllboptimality of the CRHPC.

Theorem I Considering CRHPC for discrete nonIinear system (1), if control law (4) is adopted, there exists a upper bound for global performance index (6), which is the finite-horizon optimal index of the first step optimization of (3).

Corollary 2 The CRHPC can approach the global optimal control performance arbitrarily. 00

Proof

=L

Let J 10

L(x(k),u(k)). It follows

k=O

Remarks: In practice, the controller is designed not only to guarantee the stability of closed... loop system~ but also to meet some requirements of performance index. Considering linear systems with uncertainty, Petersen and Mcfarlane (1994) approached so...called optimal guaranteed cost control to guarantee the closed-loop index J ~

xJ PX o '

from theorem 1 that

J;CO) ~ J IO ~ where

there

defined as

JZ"~ (0) == {v~ (0),···, V~_l (D)}, then the

corresponding

state

resulted

{Z;(O),Z;(O), ..• ,Z~(O)},

performance index is JtN (0)

and

from the

it

is

N

If

-4

J; ; that nleans, for any small

exists

N*

a

!J

10 -

so

>0,

N > N· ,

if

that

f;

00

J~I ~ !J; (0) - J~I < E:

SO, J 1D can approach

J;

with arbitrary precision.

They are equivalent when N ~

4.

00 .

THE SUBOPTIMALITY OF CRHPC FOR LINEAR SYSTEM

Apply theorem 1 to the following linear discrete system

x(k + 1) = Ax(k) + Bu(k) and (A) B) is controllable. The optimization problem defined as

fmite

(1 J)

horizon

l'i-l

optimal

minJ] = min

L {ZT (i)Qz(i) + v

T

(i) Rv(i)}

t=O

0

SJ4

Tl1en take the optimization horizon as Al(M > N) , and define the first step optimal perfonnance index

as JtM (0). Similar to the previous, the M-step control sequence at time 0

7rM(O)={V;(O), ... ,V~_I(O) ,O'''''O}

control.

IJ]* (0) - J;/ < li. Hence

Corollary 1 The suboptimality of CRHPC relates to the optimization horizon N. The greater N, the better the 8uboptimality. Proof First, take the optimization horizon as N, we obtain the optimal control sequence through solving the first step finite horizon optimization,

optimal

JJ. (0) ~

(10)

denotes the global performance index of

traditional

where Xv is the

initial state value and P is a positive-definite matrix satisfying some certain conditions. This inequality is similar to our suboptimality relation. From this point of view, CRHPC also realizes a guaranteed cost control of state feedback actually, for its closed-loop cost, which is the global performance index, is smaller than a certain value. Moreover, thjs upper bOWld is dependent on optimization horizon, which will be explained in the following:

J;

J;

is

a

feasible control with respect to the terminal constraints and its performance index is defined as J]A1 . Hence,

z(O)

= x(k)

zeN) = 0 z(k + 1) = Az(k) + Bv(k) On one hand, in terms of the knowledge of linear systems we know that

J;~ (x o ) == x~ P(O)X D where X o

= x(O)

and

P(O) is the solution of the

fo)lowing discrete-time Riccati equation

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SUBOPTIMALITY ANALYSIS OF RECEDING HORIZON PREDICT...

5.

P(k):::; ATp(k+l)A +Q

CONCLUSION

T

- AT P(k + l)B(B P(k + l)B + R)-l ET P(k + l)A P(N)

==:. 00

(12) Since peN) = 00 will lead to computation, we can derive a

P(t) = K(t)-'

to calculate

confusion in recursion for

pea)

(Kwon and

Pearson, 1978). On the other hand, for infinite horizon optimal control, we obtain

J~(XO)=XOTpXO =llxoll~

,

The control performance obtained by optimal control is optimal globally. However, predictive control strategy is suboptimal for it uses rolling optimization strategy with finite horizon index. In this paper, for general discrete nonlinear systems, we analyzed the property of _RHC with terminal constraints and deduced the upper bound of suboptimality. In particular, we got a quantitative measure of suboptimality of CRHPC for linear systems. The result is congruous \vith that of K won' s. Clearly, these results are valuable for predictive control theoretically.

where P is the solution of the algebra Riccati equation. It follows from the monotonicity of the solution of Riccati equation

Ilxoll~ < I!xo!I~(OJ From theorem 1, we obtain 2

J 1D ~ J]* (X o) == If X o JfJp(D) To sum up, we have the following

results~

Theorem 2 Considering RHC with tenninal constraints for linear discrete systems, there exists a upper bound

Ilx(}II~(OJ

for global perfonnance index

corresponding to optimal control, \vhich X o denotes the initial state value and P( 0) denotes the so lution of discrete Riccati equation (12) . Remarks: According to the above analysis, the 5uboptimality of CRHPC for linear discrete systems can be measured by (13)

P

where

Pk

::::

~+l

denotes the solution of (12) with == P . The result is identical with that of

Kwon~s

obtained through analyzing the properties of Ricatti equation (Kwon and Pearson, 1978); The latter method is only suitable for linear system, whereas the results of this paper can be applied to a larger scope.

ACKNOWLEDGEMENT This paper was supported by the Chinese National Natural Science Foundation.

REFERENCES:

Alamir M. and BOTIlard G. (1994), On the Stability of Receding Horizon Control of Nonlinear Discrete-Time Systems, Systems & Control Letters, Vol.23, pp.291-296. Edward S. Meadows, Michael A. H., and et al. (1995), Receding Horizon Control and Discontinuous State Feedback Stab i lization, Int, J. Control, Vol. 62, pp. 1217-1299. Freeman and P. V. Kokotovic ( 1996). Inverse optimality in robust stabilization, SIMA J Control and Optimization, Vol.. 34, pp. 13651391. Kwon, W. H., and A. E. Pearson (1977), A Modified Quadratic Cost Problem and Feedback StabilizatioIl of a Linear System, IEEE Trans. Automat. Contr.. VoJ. 22~ pp. 838...842. Kwon, W. H.~ and A. E. Pearson (1978), On Feedback Stabilization of Time-varying Discrete Linear Systems, IEE_E Trans. Automat. Contr.~ Vat 23, pp. 479-481. Mayne, D. Q.. , and H. Michalska (1990a), Receding Horizon Control of Nonlinear Systems, IEEE Trans. on Automa. Confr., Vol. 35, pp. 814-824. Mayne, D.. Q.~ and H. Micha)ska (1990b), An Implementable Receding Horizon Control for Stabilization of Nonlinear Systems, the 29th IEEE Conference on Decision and Control, pp.3396-3397. Petersen, 1. R., D . C. Mcfar)ane (1994), Optimal Guaranteed Cost Control and Fjltering for Uncertain Linear Systems, IEEE Trans. on ~4utoma. Contr.) VoJ.39,pp. 1971-1977~ Rossiter ( 1994), GPC controllers with guaranteed stability and mean-level control of unstable plant, Proc. 33Td Conf. Decision Contr.~ pp. 3579-3580. Rossiter, et al (1996), Infinite Horizon Stable Predictive Control, IEEE Trans. on Automa. Contr., Vol. 41, pp. 1522-1527.

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