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本文为模型预测控制（MPC）的理论和设计提供全面而基本的处理方法，强烈推荐作为入门模型预测控制书籍。Text
Postface
Summary of change
Proposition 2. 18 Proposition 2 Removes boundedness of XN
Proposition 2.19 Remark 3
Removes boundedness of XN
Theorem 2.22 Theorem 4 Asymptotic stability with stronger KL definition
Definition B 6
Definition 9 Classical to kl definition of asymptotic stability
Theorem B11
Theorem 12 Lyapunov function and ki definition
Definition B9(e) Definition 13 Asymptotic stability with kl definition(constrained)
Theorem B 13 Theorem 14 Lyapunov function and kl definition(constrained)
Table 1: Extensions of MPC stability results in Chapter 2 and appendix b
Then the origin is asymptotically stable under Definition g with a region of attraction
XN for the system xt= f(x)
Proof, Proposition 2 extends the local upper bound in(2) to all of XN and Theo
rem 14 then gives asymptotic stability under definition 13. both Theorem 14 and
Definition 1 3 appear in Appendix b of this note
a summary of these extensions to the results of Chapter 2 and appendix b is
provided in Table 1
Positive invariance under control law KN(). Proposition 2. 1 1 correctly states
that the set XN is positive invariant for the closed-loop system x+=f(x, KN(x))
The proof follows from(2.11), and is stated in the text as
That XN is positive invariant for x+=f(x, KN(x))follows from(2. 11)
which shows that KN(. )steers every x E xN intO XN-1SXN
But notice that this same argument establishes that XN-1 is also positive invariant
for the closed-loop system, a fact that does not seem to have been noticed previ-
ously. Since XN-IE XN, this statement is a tighter characterization of the positive
invariance property. This tighter characterization is sometimes useful when estab
lishing robust stability for systems with discontinuous VN(), such as Example 2. 8
Among the feasibility sets, Mi,j=0,1,., N, the set XN is the largest positive
invariant set and XN-1 is the smallest positive invariant set for x+=f(x, KN(x))
none of the other feasibility sets, X,j=0, 1
2, are necessarily positive
invariant for xt=f(x, KN(x)) for all systems satisfying the given assumptions. A
modified Proposition 2.11 reads as follows
Proposition 2. 11'(Existence of solutions to DP recursion). Suppose Assumptions
2.2 and 23 hold Then
a)For. D20, the cost function V;( is continuous in Zj, and, for each E Xj,
the control constraint set ui (x) is compact and a solution u(x)Eu(x) to Pi(x)
exists
(b)If Xo: =X is control invariant for x*=f(x, u),uEU, then, for ec
the set x is also control invariant, X;2 Xj-1,0&j, s closeldchjED20
(c)In addition, the sets Xi and Xi-1 are positive invariant for x+= f(x,Ki(x))for
allj∈>1.
Unreachable setpoints, strong duality, and dissipativity. Unreachable setpoints
are discussed in Section 2.9.3. It is known that the optimal mPC value function in
this case is not decreasing and is therefore not a Lyapunov function for the closed
loop system. a recent paper by Diehl, Amrit, and rawlings (2011) has shown that
a modified MPC cost function, termed rotated cost, is a Lyapunov function for the
unreachable setpoint case and other more general cost functions required for op
timizing process economics. a strong duality condition is shown to be a sufficient
condition for asymptotic stability of economic MPC with nonlinear models
This result is further generalized in the recent paper Angeli, Amrit, and rawl
ings(2012). Here a dissipation inequality is shown to be sufficient for asymptotic
stability of economic MPC with nonlinear models. This paper also shows that MPC
is better than optimal periodic control for systems that are not optimally operated
at steady state.
Unbounded input constraint sets. Assumption 2. 3 includes the restriction that
the input constraint set u is compact(bounded and closed). This basic assumption
is used to ensure existence of the solution to the optimal control problem through
out Chapter 2. If one is interested in an MPC theory that handles an unbounded
input constraint set U, then one can proceed as follows. First modify assumption
2.3 by removing the boundedness assumption on U
Assumption 5(Properties of constraint sets-unbounded case). The sets X, X f
and U are closed, Xf s X; each set contains the origin
Then, to ensure existence of the solution to the optimal control problem, con
sider the cost assumption on page 154 in the section on nonpositive definite stage
costs, slightly restated here
Assumption 6(Stage cost lower bound). Consider the following two lower bounds
for the stage cost
(y,4)≥a1(|(y,4)|) for all y∈RP,∈Rn
Vr(x)≤Ⅸ2(X|)
for allx∈xr
in which i( is a o function
(b)
e(y,u)≥c1|(y,L)
for all
RA
Vf(x)≤c2|x
for all x∈X
f
in which C1, C2, a>0
e finally, assume that the system is input/output-to-state stable (IOSS). This prop
y is given in Definition 2.40(or Definition B 42). We can then state an MPC sta
bility theorem that applies to the case of unbounded constraint sets
Theorem 7(MPC stability -unbounded constraint sets
(a) Suppose that Assumptions 2. 2, 5, 2.12, 2.13, and 6(a)hold and that the system
f(x, u),y=h(x) is lOSS. Then the origin is asymptotically stable(under
Definition 9)with a region of attraction XN for the system xt=f(x, KN(x))
(b) Suppose that assumptions 2.2, 5, 2.12, 2.13, and 6(b hold and that the system
x+=f(, u),y=h(x)is 1OSS. Then the origin is exponentially stable with a region
f attraction XN for the system x*t=f(x, kN(x))
In particular, setting up the mpc theory with these assumptions subsumes the
LQR problem as a special case.
Example 1: The case of the linear quadratic regulator
Consider the linear, time invariant model x+=Ax t Bu, y= Cx with quadratic
penalties l(, u)=(1/2)(yQy+uRu) for both the finite and infinite horizon
cases. What do the assumptions of Theorem 7(b) imply in this case? Compare these
assumptions to the standard lor assumptions listed in Exercise 1.20(b)
Assumption 2.2 is satisfied for f(x, u)=Ax+ Bu for all A E Rnxn, B E Nxm
we have X=Rn, and =Rm. Assumption 6(b) implies that Q>0 andr>0
The system being IOSS implies that (A, C)is detectable(see Exercise 4.5). We can
choose xf to be the stabilizable subspace of (A, B)and Assumption 2. 13 is satisfied
The set XN contains the system controllability information. The set XN is the
stabilizable subspace of (A, B), and we can satisfy Assumption 6(a) by choosing
Vf(x)=(1/2)x'llx in which ii is the solution to the steady-state Riccati equation
for the stabilizable modes of(A, B)
In particular, if (A, B) is stabilizable, then Xf=R", XN=Rn for aln E Do
and Vf can be chosen to be vf(x)=(1/2)x'iix in which II is the solution to the
steady-state Riccati equation(1.19). The horizon n can be finite or infinite with
this choice of vf() and the control law is invariant with respect to the horiz.on
length, KN(x)=kx in which K is the steady-state linear quadratic regulator gain
given in (1.19 ) Theorem 7(b)then gives that the origin of the closed-loop system
f(x, KN(x))=(A+ BK)x is glo
tially stable
The standard assumptions for the lOR with stage cost l(, u)=(1/2)(yQy+
d'ru)are
oR>o(A, C)detectable (A, B)stabilizable
and we see that this case is subsumed by theorem 7 (b)
Chapter 6. Distributed Model Predictive Control
The recent paper (Stewart, Venkat, Rawlings, Wright, and Pannocchia, 2010) pro
vides a compact treatment of many of the issues and results discussed in Chapter
6. Also, for plants with sparsely coupled input constraints, it provides an extension
that achieves centralized optimality on convergence of the controllers' iterations
Suboptimal mPc and inherent robustness. The recent paper (pannocchia, Rawl
ings, and Wright, 2011) takes the suboptimal MPC formulation in Section 6. 1.2
also discussed in section 2. 8. and establishes its inherent robustness to bounded
process and measurement disturbances. See also the paper by lazar and heemels
(2009), which first addressed inherent robustness of suboptimal mpc to process
disturbances by (i)specifying a degree of suboptimality and (ii)using the time
varying state constraint tightening approach of Limon Marruedo, Alamo, and ca
macho(2002)to achieve recursive feasibility under disturbances
The key assumption in(Pannocchia et al, 2011) is the following
Assumption 8. For any x, x' E xN anduE UN(, there exists UN(x) such
that u-u0 for all i∈n:M. f condition⑤5)is
not satisfied, then we find the direction with the worst cost improve
ment Imax=arg maxi(up+Pu?,1-t)3, and eliminate this direction
by setting wimax to zero and repartitioning the remaining wi so that they
sum to 1. We then reform the candidate step (4)and check condition(5)
again. We repeat until(5) is satisfied. At worst, condition(5)is satisfied
with only one direction
Notice that the test of inequality(5)does not require a coordinator. Each subsystem
has a copy of the plantwide model and can evaluate the objection function inde
pendently. Therefore, the set of comparisons can be run on each controller. This
computation represents a small overhead compared to a coordinating optimization
Appendix b. stability Theory
Asymptotic stability. For several of the stability theorems appearing in the first
printings appendix b, 3 we used the classical definition of global asymptotic sta-
bility (gas), given in Definition B.6. The following stronger definition of GAS has
recently started to become popular.
Definition 9(Global asymptotic stability(Kl version). The(closed, positive invari
ant)set is globally asymptotically stable(gas)for x+= f(x) if there exists a KL
function A(·) such that, for each x∈Rn
φ(i;x)A≤B(x1n,i)VtcD0
(B.1)
inseethewebsitewww.che.wisc.edu/jbraw/mpcfortheapPendicesA-ccorrespondingtothe
st printing of the text
Notice that this inequality appears as (b.1)in appendix b
Teel and Zaccarian (2006) provide further discussion of these definitional issues
It is also interesting to note that although the kl definitions may have become
popular only recently, Hahn (1967, p. 8)used K and L comparison functions as
early as 1967 to define asymptotic stability 4 For continuous f(), we show in
Proposition B8 that these two definitions are equivalent. But we should bear in
mind that for nonlinear models, the function f()defining the closed-loop system
evolution under mPC, x+=f(x, KN(x)), may be discontinuous because the control
law Kn() may be discontinuous(see Example 2. 8 in Chapter 2 for an example)
Also, when using suboptimal MPC, the control law is a point to set map and is not a
continuous function(Rawlings and mayne, 2009, pp. 156, 417). For discontinuous
f(), the two definitions are not equivalent. Consider the following example to
make this clear
Example 2: Difference between asymptotic stability definitions(feel)
Consider the discontinuous nonlinear scalar example x+=f(x)with
f(x)
x|∈(1,2)
0
x|∈[2,∞)
The origin is attractive for all x(o)E R, which can be demonstrated as follows
For|x(0)<[0,1],|x(k)|≤(1/2)k|x(0).For|x(O)∈(1,2),|x(1)≥2 which
implies that lx(2)|=0; and for lx(O) E [2, oo), x(1)=0. The origin is lyapunov
stable, because if8≤1,then|x(0)≤δ implies|x(k)≤δ for allk. Therefore,the
origin is asymptotically stable according to the classical definition
But there is no l function b(.) such that the system meets the bound for all
x(0)∈R
lx(k)≤β(x(0),k)Vk∈0
Indeed, for initial conditions x(o) slightly less than 2, the trajectory x(k)becomes
arbitrarily large(at k=1)before converging to the origin. Therefore, the origin is
not asymptotically stable according to the Kl definition
Remark 10. Note that because of Proposition B 8, the function f()must be chosen
to be discontinuous in this example to demonstrate this difference
Proposition 11(Extending local upper bounding function). Suppose the function
V() is defined on X, a closed subset of Rn, and that V(x)0 such that Ixia s a
implies x∈Xf. For each i∈l≥l, let Si={x||x≤ia}. we define a sequence of
numbers iai as follows
sup v(x)+a(a)+i
Since si is compact for each i and X is closed their intersection is a compact subset
of X and the values ai exist for al i E 0>i because v()is bounded on every
compact subset of X. The sequence icil is strictly increasing For each i E >1, let
the interpolating function i )be defined by
φ;i(S):=(S-ta)/as∈[ia,(i+1)a]
Note that i lia)=0, il(i+1)a)=l, and that p( )is affine in [ia, (i+ 1a]. We
can now define the function阝(·) as follows
(c2/c(a))c(s)
s∈[0,a]
a+1+φ(s)(a1+2-ax+1)s∈[ia,(i+1)a] for all e∈21
It can be seen that b(0)=0, B(s)> a(s) for s E [0, al, that B(-)is continuous
strictly increasing, and unbounded, and that V(x)0, 01(0)=0, ando< oi(s)< s for s>0
But o1() may not be increasing. We modify oi to achieve this property in two
eps. First define
2(s):=maxO1(s)s∈R20
in which the maximum exists for each s E R2o because o1( is continuous. by
its definition, 02( is nondecreasing, and we next show that o2()is continuous
OnR0. Assume thatσ2(·) is discontinuous at a point c∈R≥0. Because it is a
nondecreasing function, there is a positive jump in the function 02()at c(Bartle
and Sherbert, 2000, p. 150). Define 5
lim 02(s)
2: =limo?(S)
S/C
We have that g(c)< al< ay or we violate the limit of oz from below. Since
U1(c)0and
therefore
V(φb(i+1;x)≤σ(V(φ(;x))Vx∈Ri∈0
(6)
Repeated use of(6)and then(b 3) gives
φ(;x)≤σ1(∝2(|xa) yxcRn iC20
in which o' represents the composition of o with itself i times. Using(.2)we have
that
φ(i;x)|A≤B(x1a,i)Vx∈Rnt∈0
in which
β(s,i):=1(a2(ax2(s))Vs∈R0t∈n0
For all s 2o, the sequence wi: =o(a2 (s))is nonincreasing with i, bounded below
(by zero), and therefore converges to a, say, as i-oo. Therefore both wi-a and
uUz)→aasi→∞. Sinceσ(·) is continuous we also have that o(u)→σ(a)
so o(a)=a, which implies that a=0, and we have shown that for all s 2 0
c1(σ(a2(s)→0asi→∞. Since a1(.) also is a function, we also have that
for all s>0, ai(u(a2(s))) is nonincreasing with i. We have from the properties
of functions that for all i20, a 1(ol(a2(s)))is a k function, and can therefore
conclude that b()is a L function and the proof is complete
Constrained case. Definition B 9 lists the various forms of stability for the con
strained case in which we consider X c rn to be positive invariant for xt=f(x)
In the classical definition, set a is asymptotically stable with region of attraction X
if it is locally stable in X and attractive in X. The kl version of asymptotic stability
for the constrained case is the following
PThe limits from above and below exist because o2( )is nondecreasing(Bartle and Sherbert, 2000
p. 149). If the point c=0, replace the limit from below by o2(0)
Definition 13(Asymptotic stability (constrained -KL version ). Suppose X c rn
is positive invariant for x+
f(x), that is closed and positive invariant for
x+=f(x), and that lies in the interior of X. The set A is asymptotically stable
with a region of attraction X for x+=f(x)if there exists a L function B()such
that. for each x∈X
中(i;x)|A≤β(|x1a,i)i∈0
to. Notice that we simply replace Rn with the set X in Definition 9 to obtain Defini-
tion 13. We then have the following result, analogous to Theorem B 13, connecting
a Lyapunov function to the kl version of asymptotic stability for the constrained
case
Theorem 14(Lyapunov function for asymptotic stability (constrained case -KL
version)). Suppose X c rn is positive invariant for+= f(x), that A is closed and
positive invariant for x= f(x), and that A lies in the interior of X. if there exists
a Lyapunoy function in X for the system x+=f(x)and set A with a3()aKoo
function, then A is asymptotically stable for x+=f(x)with a region of attraction
X under Definition 13
The proof of this result is similar to that of Theorem 12 with Rn replaced by x
References
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