Proceedings of the American Control Conference
Arlington, VA June 25-27, 2001
On the Coupling between the Plant and Controller Optimization Problems
Hosam K. Fathy ~, Julie A. Reyer 2, Panos Y. Papalambros 3 and A. Galip Ulsoy 3
Department of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109-2125
Abstract
This paper examines the plant and controller
optimization problems. One can solve these problems
sequentially, iteratively, using a nested (or bi-level) strategy,
or simultaneously. Unlike the nested and simultaneous
strategies, the sequential and iterative strategies fail to
guarantee system-level optimality. This is because the plant
and controller optimization problems are coupled. This
coupling is introduced using a simple experiment. To prove
it theoretically, the necessary conditions for combined plant
and controller optimality are derived. These combined
optimality conditions differ from the individual sets of
necessary conditions for plant and controller optimality by a
coupling term that reflects the plant design's influence on
the plant dynamics and control input constraints.
Table 1: Nomenclature
d
e
h, g
t
x~ u
f,q
qJ,
L
Wp~ W c
g, v,X
A, B,
C,D
Q,R
J
S
Xo
Plant design variables
Plant design objective function
Plant design equality and inequality constraints
Lagrange multipliers corresponding to h and g
Time
State variables and control inputs
State equations and control input constraints
Final state constraints and objective
Controller cost functional
Weights on plant and controller objectives
Lagrange multipliers corresponding to lrl, '~, and
f , respectively
Matrices used in the state space representation of
a linear, time-invariant (LTI) system
Weighting matrices for state/control "energy"
terms in an LQR formulation
Combined plant/controller optimization objective
Ricatti matrix
Initial conditions on state variables
to, T
Initial and final times, respectively
M,
Mass lifted by weight-lifting device and voltage
input to the device's motor
Vin
1. Introduction
When optimizing a plant (artifact) and its controller, one
could adopt a sequential strategy whereby the plant is
optimized first, followed by the controller. Alternatively, one
can adopt a systems mindset and optimize the plant and
controller simultaneously. Recent research on flexible space
structures [1-13], aeroservoelasticity [14-15], flexible robot
manipulators [16], mechatronics [17], electric motors, and
many other systems has shown that the latter, simultaneous
strategy furnishes better systems than the former,
sequential strategy. Namely, the plant and controller
optimization problems are coupled in the sense that their
sequential solution is not guaranteed to be a combined
optimum. This coupling is widely accepted in the control
community [4,8,12], but has not been extensively explored.
This paper studies such coupling formally based on a
derivation of the generalized first-order necessary conditions
for combined (system-level) plant/controller optimality.
Many authors have proposed solution strategies that
attempt to mitigate the plant/controller optimization coupling
and find true combined optima in special cases (e.g. [1-16]).
Recent work in the design community [18-20] has
addressed this problem by classifying the various
optimization strategies into sequential, iterative, bi-level
(nested), and simultaneous strategies, and by comparing
these strategies using detailed numerical examples. This
paper adopts the same classification and proceeds to show
rigorously that system-level optimality is guaranteed with the
nested and simultaneous strategies but not with the
sequential or iterative strategies.
2. Motivating Experiment
A very simple experiment was constructed to illustrate
the plant/controller optimization coupling. The setup consists
of an electric motor receiving a constant input voltage Vzn
and driving a sheave that pulls a mass M upward (Fig. 1).
Now suppose that the time taken by this weight-lifting device
to lift M from rest through a specified height is T, and
suppose that it is desired to minimize M + ] / T by varying
the mass M and the input voltage Vzn. The goal is both to
maximize the mass this device lifts and to minimize its travel
time by varying the mass and the motor's input voltage. The
problem is a very simple combined plant/controller
optimization problem where the mass M is the plant design
variable, the voltage v in is the controller design variable
(the controller in this case being a simple open-loop bangbang one), and the expression M + I / T
is an overall
system
objective.
Optimization
was
carried
out
experimentally by measuring the travel time T and
computing the system objective for different possible values
of M and v in. The results are shown in Fig. 2.
1 Graduate student.
2 Research fellow.
3 Professor.
0-7803-6495-3/01/$10.00 © 2001 AACC
1864
Pulley
Volts in
Vin ---~ M°t°r
tC°u~ng[~
~
v~
min
/ ' ~ Encoder
output
subject to:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
{~(x(T),T) + TiL(x(t),u(t),t)dtt
"u(t)'x(t)'t°'T L
to
x(t)=f(x(t),u(t),t), q(u(t),t)<o
W ( x ( r ) , r ) = 0, X(to) = Xo
H , ~5,1
Ma s s
The problem of optimizing a plant and its controller in an
integrated, system-level fashion is constructed from the
above two under the following assumptions:
M
Fig. 1: Experimental setup
1.
Performance=
M+I/T
10-
2.
Vin =
(~
3.
2.5V
C)
///
~i~ ~ 5V vi~
,
10
0.1
0.2
0.3
0.4
Mass M (kg)
0.5
= o30V
4.
0.6
Suppose now that one chooses to optimize the weightlifting device sequentially. One can assume some initial
controller design, say vi,, = 2 . 5 V , then optimize the
combined objective with respect to the plant design ( M )
keeping the controller fixed, and finally optimize the
combined objective with respect to the controller design
keeping the plant fixed. The resulting sequential design is
indicated as point P1 in Fig. 2. The true system optimum is
significantly better, indicated as point P2. Sequential
optimization has failed to furnish this system optimum
because it assumed that the plant and controller
optimization problems are separable, effectively neglecting
their coupling.
3. Formulating the Combined Plant/Controller
Optimization Problem
The traditional sequential optimization strategy involves
solving two optimization problems in tandem: a plant
optimization problem and a controller optimization problem.
The former problem is typically expressed as a static
optimization problem of the following form [21]:
subject to:
h(d)=0, g(d)_< 0
The combined plant/controller design must satisfy all
the constraints on the individual plant and controller
designs. Hence, the set of constraints for the combined
problem is the union of the two sets of constraints for its
subproblems.
Similarly, the set of variables for the combined problem
is the union of the two sets of variables for its two
subproblems.
The influence of the plant's design (e.g. mass, stiffness,
actuator selection, etc.) on the control must be
accounted for by replacing
Fig. 2 Experimental Results
mine(d)
(2)
(1)
d
The latter problem is commonly formulated as a dynamic or
variational optimization problem as shown below [22,23]:
x(t)=f(x(t),u(t),t)
and
q(u(t), t) _<0
with
x(t) = f(x(t),u(t),t,d)
and
q(u(t), t, d) < 0, respectively.
Due to the multiobjective nature of the combined
problem, its optimal solution will be a Pareto set (a set
of plant/controller designs for which the plant objective
cannot be improved without compromising the controller
objective, and vice versa). To generate this set, a
scalarization must be used to combine the (vector of)
two individual objectives into a so-called scalar
substitute objective function. The exact form of the
scalarization is not critical as long as the Pareto set is
identified. In this paper, the scalar substitute objective
function will be defined as the weighted sum of the plant
and controller objectives, and points on the Pareto set
will be generated by varying the weights.
Given these conditions, the combined plant/controller
optimization problem becomes:
min
d,u(t ),x( t ),t o ,T
{
{
Wpe(d)+ wc
subject to: h(d)=O, g(d)_< O,
}}
L(x(t),u(t),t)dt
(I)(x(r),r) +
to
x(t)=ffx(t),u(t),t,d),
q(u(t),t,d)_ 0, W(x(T),T)
(3)
0, x(t o) = x o
This formulation assumes that all the state variables can
be directly measured and that the plant and controller
optimization objectives are separable. In addition, sections
4.3 and 7 will further assume the existence of a feasible
combined plant/controller design (i.e. a feasible solution to
all the plant/controller constraints) for every feasible plant
(i.e. every solution to h(d) 0 and g(d)_<0), effectively
ignoring the backward coupling from the control problem to
the plant design problem. Reyer and Papalambros [18-20]
studied the scenario where this backward coupling does
exist for a non-variational formulation of the optimal control
problem. Extending those results and the results found
herein to the more general case where the two objective
functions are not separable and not all the state variables
1865
can be directly measured is an interesting topic for future
research.
x(t) = f ( x ( t ) , u ( t ) , t , d ) ,
W(x(T),T) O,
4. Strategies for Combined Plant/Controller
Optimization
Following
Reyer
[20],
we
classify
combined
plant/controller optimization strategies into sequential,
iterative, bi-level, and simultaneous strategies (Fig. 3).
q(u(t),t,d) _<O,
x(t o) = x o ,
IT~(x(T)'T) +
< -~(x°
+ T (T°)'T°)
o
] ~L(x(t),u(t), t ) d t - fL(x o (t),u o (t), t)dt
Lto
too
2.
3.
Given the plant design obtained in step 1, find the
optimal controller by solving Eq. (2) above.
Return to step 1.
4.3. Bi-level (nested): In a bi-level plant/controller
optimization strategy, two nested optimization loops are
used. The outer loop optimizes the scalar substitute
objective function by changing only the plant design. The
role of the inner loop is to generate the optimal controller for
each plant selected by the outer loop. For example, the
outer loop could optimize some weighted sum of the mass
of a structure and the energy needed to actively control its
vibrations by varying the structural design only, and the
inner loop's role would be to generate the minimum-energy
controller for each structure chosen by the outer loop. Bilevel strategies are commonly used in the integrated
plant/controller
optimization
literature
[1,2,10,11 ].
Mathematically, they involve solving the following problem:
1. Outer loop:
Sequential
Iterative
Simultaneous
Bi-level
Fig. 3: Strategies for
Plant/Controller Optimization
subject to: h(d) = 0, g(d) < 0, u(t) = output of inner loop
2. Inner loop: Solve problem from Eq. (2) above.
4.1. Sequential: These strategies solve the plant and
controller optimization problems serially, as follows:
1. Optimize the plant by solving Eq. (1) above.
2. For the optimized plant, optimize the controller by
solving the problem in Eq. (2) above.
As already asserted, sequential optimization often leads
to non-optimal system designs due to plant/controller
optimization coupling. To overcome this, some authors have
developed modified plant optimization methodologies that
incorporate open-loop measures of the ease of controlling a
given plant (e.g. open-loop eigenvalue locations [16]). This
design for ease of control approach is still essentially
sequential, since the plant and controller optimization stages
still proceed in tandem.
4.2. Iterative: Given some initial plant/controller design,
iterative plant/controller optimization strategies, such as
LMI-based strategies [3,12], attempt to improve on that
initial design by first improving the plant design without
compromising control performance, then optimizing the
controller design without compromising plant performance:
1. Given a plant d o and a controller (Uo,Xo(t),To,too),
solve the problem:
mine(d)
d
4.4.
Simultaneous:
Simultaneous
plant/controller
optimization involves finding the optimal system design by
solving Eq. (3) above. The simultaneous strategy can be
mathematically and computationally challenging for several
reasons. The simultaneous plant/controller optimization
problem in Eq. (3) is a hybrid static/variational problem.
Also, even when the plant and controller optimization
subproblems are convex, the combined problem is not
guaranteed to be also convex. To see this, consider one of
the coupling constraints, say x(t)=f(x(t),u(t),t,d). If the
plant is linear and time-invariant, this constraint becomes
x(t) = A(d)x(t)+ B(d)u(t).
Furthermore,
assume
the
coefficient matrices A and B are linear in the plant design
variables d. Despite these benign assumptions this
coupling constraint remains bilinear and non-convex.
5. Necessary Conditions for Plant, Controller,
and Combined Optimality
Consider the problem of optimizing a plant's design,
Eq.(1). Every solution to this problem must satisfy the
following first order necessary conditions for optimality,
commonly known as the Karush-Kuhn-Tucker (KKT)
conditions [21], where e~ and 13are Lagrange multipliers:
h(d) = O, g(d) _<O, 13T g(d) = O,
subject to:
h(d) = O, g(d) < O,
3 ( u ( t ) , x ( t ) , T , t o)
:
Oe
(4)
(Oh] T
+(Og] T
~-~ + ~,-~-,) '~ t,-~-)
1866
I=o
13->O,
(6)
Now consider the controller optimization problem in
Eq.(2). Its first-order necessary conditions for optimality,
commonly known as the Pontryagin conditions, are
commonly expressed in terms of the Hamiltonian
H = L - XT f as follows [22,23]:
x(t)
=
f(x(t),u(t),t,d)
,
q(u(t),t,d) _ 0,
W(x(T),T) = 0, x(t o) = x o ,
t
T
T
'
Hu-quTla=0, ;[(t)=Hx,l aTlrl=0, la---0
(7)
The necessary conditions for combined plant/controller
optimality can now stated as follows:
x(t) = f(x(t),u(t),t,d)
W(x(T),T) = O,
,
q(u(t),t,d)_ O,
x(t o) = x o ,
(((I) t + WTv + L-zT fiT dT + (~x + ~Tv + x)T T dx(T)/=0 ,
H u - q uT ~ = 0 , ~.(t)=H x, l.tTq=0, ~_<0,
(8)
h(d)=0, g(d)_<0, ~r g(d)= 0, 132_0,
T
--+
~d
o~+
(-~)
f f X+q~ ~t
]3(-'~)
Wpt
=0
°
Using these conditions, one can gain insight into the
plant/controller optimization coupling.
6. The Coupling Between the Plant and
Controller Optimization Problems
As already mentioned, the plant and controller
optimization problems are coupled in the sense that their
sequential solution is not guaranteed to be a combined
optimum. This is now stated as Theorem 1.
for plant and controller optimality. The difference is the
coupling term
Wc j ' ; [ f f X+q~" t~]dt
Wp
This coupling term depends on the derivatives of the two
coupling constraints with respect to the plant design
variables. Therefore, in the trivial case where the coupling
constraints do not depend on the plant design, the plant and
controller optimization problems become separable, and
their sequential solution will furnish combined optima.
An obvious but interesting observation is given in
Theorem 2, namely that when a plant and its controller are
optimized simultaneously, the resulting controller is the
optimal controller (i.e. the solution to the problem defined by
Eq.(2)) for the resulting plant.
Theorem 2: Let d and
( u ( t ) , x ( t ) , T , t o) be plant/controller
designs which together satisfy the conditions in Eq.8. Then
d and (u(t),x(t),T,to)also satisfy the conditions in Eq.7,
i.e., substituting Eq.8 into Eq.7 produces identities•
Another interesting observation comes from the Pareto
set concept. As already mentioned, for different values of
the plant and controller optimization weights Wp and w c,
simultaneous optimization will generate different members
of this set. Theorem 3 states that the sequential
plant/controller optimization procedure also generates a
member of this set, specifically, the member corresponding
to the case where the plant design objective is infinitely
more important than the controller design objective.
Theorem 3: Let d I and (Ul(t),Xl(t),Tl,tOl)
be a pair of
sequentially optimized plant/controller designs (i.e., a pair of
solutions to equations (1) and (2)). Furthermore, let d 2 and
combined
plant/controller
design
given
by
plant/controller
design
some
optimal
for given weights
combined
wpand
w c.
Assume that the plant and controller objective functions are
bounded. Then
lira
e(dl)+
w p __>oo[
wc
d z and
(Ul(t),Xl(t),Tz,toz) will not in general satisfy the combined
optimality conditions in Eq.(8).
Proof: Substituting Eq.(6) and Eq.(7)in Eq.8 reduces the
latter set of conditions to identities plus the following
equation:
be
(u2(t),x2(t),T2,t02)
Theorem 1: Let d z be a plant design satisfying the plant
optimality conditions in Eq.(6), and let (u z(t),x z(t),Tz,tol) be
a controller design that satisfies the controller optimality
conditions in Eq.(7) for the plant design d]. Then the
(10)
o
".
lim
e(d2)+
w p _~oo[
wc
W c
Wp
~
Wc
Wp
~(Xl(T1),T1)+
L(Xl(t),Ul(t),t)dt
=
tol
(I)(x2(T2),T2)+
L(xz(t),u2(t),t)dt
to2
Proof: The assumption that the plant and controller
objectives are bounded implies that the minimum value of
the scalar substitute objective function is Lipschitz
continuous in Wp/W c . Hence, one can exchange the limit
T
and minimization operators, which completes the proof~l
Wp
In general, d I and
to
(Ul(t),Xl(t),Tl,tol)
will not satisfy Eq.94
According to Theorem 1, the set of conditions for
combined optimality is not the union of the sets of conditions
The three theorems in this section were simple
illustrations of how the plant and controller coupling can be
explored via the optimality conditions, particularly for
sequential strategies. The following sections will examine
how bi-level and iterative strategies addrress this coupling.
1867
7. B i - l e v e l
Plant/Controller
Optimization
Unlike the sequential strategy, if the bi-level
plant/controller optimization strategy is convergent, then it is
guaranteed to converge to an optimal combined
plant/controller design, according to Theorem 4 below.
Theorem 4: Let
dc
:=
~T (t)
XT
regulator (LQR) [23]. This solution has the special property
that the optimal control objective can be expressed as an
implicit function of the plant design through the Algebraic
Ricatti Equation [23]. Using these results the bi-level
problem, shown to be equivalent to the simultaneous
problem, reduces to the following:
min {w pe(d)+
d,S
(t) t o T~
a T (d)S + SA - SB(d)R -1B T (d)S
rank[B AB ... A n-l B J= dim x
T
E(d c ) := q:)(x(T),T) +
~L(x(t),u(t),t)dt,
to
I
and
q(u(t), t,,t) _<o, W(x(r), r) = o, X(to) = Xo
f
N : = d,d c :h(d)=0,g(d)_<0,d* = a r g
min
d~:(d,d~)eO
E(d )
c
t
/
\
every solution to the design
constraints h(d)=0 and g(d)___0, the following two sets,
(argmin(wpe(d)+ wcE(dc)} and
{argn~n(wpe(d) + wcE(dc)) }
(12)
Iterative plant/controller re-optimization approaches,
such as LMI-based ones, are intended as a simpler,
computationally cheaper alternative to simultaneous
plant/controller optimization. They have many advantages,
including guaranteed convergence in some cases because
they divide the non-convex simultaneous optimization
problem into a series of c o n v e x optimization problems
[3,12]. Yet these algorithms are not guaranteed to converge
to optimal combined plant/controller designs, as Theorem 5
states.
Theorem 5: Let (d z,u z(t),x l(t),T 1,tol) be a solution to the
are identical.
The proof follows directly from the definition of a minimum
and is omitted for brevity~l
Theorem 4 states that the nested optimization strategy is
guaranteed to furnish system optima only if there exists a
feasible system design for every feasible plant design. This
means that within the constraints h(d)=O and g(d)___O
there must be some constraints that guarantee the
existence of a feasible controller. For nonlinear systems,
these special constraints might be difficult to determine. For
linear systems, however, these constraints are the familiar
controllability constraints, which, fortunately, can be
expressed directly in the form h(d) = 0.
The importance of nested optimization is that it helps
reduce the complexity of some important combined
plant/controller optimization problems without compromising
system-level (combined) optimality. To see this, consider the
special case where the plant is linear and time-invariant
(LTi), there are no control input constraints, and the control
objective can be expressed as an infinite horizon quadratic
cost functional of the form:
dO
j'(xT"Qx + uT"Ru)dt, Q _ O , R > O
= 0,
,
8. Iterative Plant/Controller Optimization
be the vector of controller design variables, the controller
design objective, the set of all feasible plant/controller
designs, and the set of plant/controller designs searched by
the outer loop of the nested optimization strategy,
respectively. Then, assuming that the set N contains at
least one pair / d , d * ) f o r
h(d) = 0 , g(d)<_ 0
Note the addition of an explicit controllability constraint to
satisfy the assumption in Theorem 4. The above problem is
a static optimization problem as opposed to a
static/variational, two-point boundary value problem.
Nevertheless, the solution to the above problem, because it
is derived using the bi-level approach, is guaranteed to be a
combined optimum. This explains why the optimization of a
system by varying its plant assuming an LQR controller is a
popular and successful method in the literature [1,2,10,11].
.,..c :
• := h(d) = O,g(d) < O,x(t) = f(x(t),u(t),t,d),
[
wcX ov Sx ° } subject to:
iterative plant/controller optimization problem (defined in
Eq.(4) above). Then, in general, (d 1,u l(t),x z(t),T1,toz) does
not satisfy the conditions in Eq.(8).
Proof: Assuming the iterative strategy converges, one can
show from its definition that it will converge to a Paretooptimal design that solely depends on the initial guess.
Since the set of Pareto-optimal designs is not guaranteed to
be a singleton, it follows that the iterative strategy will
generally not converge to that Pareto-optimal design
corresponding to the chosen weights w p and w c
An interesting example of a scenario where iterative
plant/controller optimization fails to generate an optimal
system design is the case where the initial guess is the
output of a sequential plant/controller optimization. Given
this initial guess, the iterative process will not change the
plant since, for the given controller and for all controllers, it
is the optimal plant and hence no better plant can be found.
Furthermore, the iterative process will not change the
controller since it is already the optimal controller for the
given plant. Hence, the output of the iterative process will be
the s e q u e n t i a l
optimum,
which, according to Theorem 1,
does not necessarily equal the simultaneous optimum.
(11)
9. Discussion and Conclusions
0
Under t h e s e special assumptions, the solution to the
controller optimization problem
is the linear quadratic
The plant and controller optimization problems are
in the sense that when solved sequentially or
coupled
1868
iteratively, as opposed to simultaneously or using the nested
approach, the results are not guaranteed to constitute a
combined system optimum. This has been shown both using
a general theoretical derivation and a simple experiment.
The derivation revealed that the reason for the coupling is
that the combined plant/controller optimality conditions are
different from the combined set of plant and controller
optimality conditions. The difference is the plant/controller
coupling term, which depends directly on the gradients of
the coupling constraints with respect to the plant design
variables.
These results were derived using a variational
formulation of the optimal control problem, where the
controller was not assumed to have a particular structure.
Showing that these results still apply if the controller is
assumed to have a particular structure (e.g. LTI full-state
feedback) is an interesting topic for future research.
Furthermore, it was assumed in the derivations that all the
state measurements were available for feedback control.
Eliminating this assumption leads to an interesting problem
worth
investigation,
namely
the
integrated
plant/observer/controller optimization problem.
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