vibration control of civil engineering structures via
TRANSCRIPT
Vibration control of Civil Engineering structures
via Linear Programming
P. Rentzos, G.D. Halikias and K.S. Virdi
Control Engineering Research Centre
School of Engineering and Mathematical Sciences
City University, London EC1V 0HB, UK
Email: [email protected], [email protected], [email protected]
Abstract
The paper presents a novel active-control design approach which minimizes the peak
response of regulated signals rather than, e.g., r.m.s or energy levels optimized by
traditional control techniques. This objective is more relevant for active control of civil-
engineering structures, as failure occurs after a maximum displacement is exceeded in a
structural member, while control constraints typically arise from hard saturation limits
on the actuator signal and its rate. The design method is formulated in discrete-time
and involves the parametrization of all finite settling-time stabilizing controllers. This
leads to a linear programming optimization framework, in which the peak response of
the structure is directly minimized, subject to linear constraints on the actuator’s peak
level signal and its rate. The design method is illustrated via a simulation study based
on a simple model corresponding to a benchmark design problem. The simulation
results compare favourably to those obtained via LQG active control. Finally, some
practical implementation issues related to the method are discussed.
1
1. Introduction
Structural control aims at protecting structures from severe natural hazards such as
earthquakes and large wind loads. The simplest structural control scheme involves
a Tuned-Mass-Damper (TMD), which consists of a pendulum vibrating at the same
frequency as the natural frequency of the structure, opposing its movement to mitigate
the response. Control schemes such as passive, active, semi-active or hybrid have been
proposed and implemented with various degrees of success [15].
Over the last few years a wide range of design methodologies have been proposed in the
area of structural control, including non-linear/sliding-mode control, pole-placement
and observer-based methods, adaptive control, fuzzy/neural-based methods, reliability-
based control and optimal control [9]. Optimal control appears to be the design method
increasingly favoured by most researches, mainly due to important recent theoretical
advances in this field and to the design flexibility that this method offers. The two
most important optimal control paradigms, around which most other optimal-control
methods cluster, are LQR/LQG optimal control and H∞ optimization methods. In
addition, new design optimal algorithms have been proposed to account for different
objectives, assessed via analytical, simulation and experimental results.
Optimal control design methods are typically formulated as optimization problems
involving the minimization of a norm, such as the H2 or H∞ norm, of the closed-loop
transfer function between an input disturbance signal and the regulated outputs. For
example, the H2 norm measures the expected power of the regulated signal (mean-
square value). Normally, the input disturbance signal in this case is assumed to be a
random white-noise process. Weighting factors or filters can be employed to emphasize
specific frequency ranges of the input or output spectrum.
Consider the diagram shown in Figure 1, where n(t) represents a zero-mean white-noise
vector signal, i.e.
E[n(t)n′(t)] = I (1)
Suppose that T (s) is a stable transfer-function with T (∞) = 0 and let e(t) represent
2
T(s)n(t) e(t)
Figure 1: Transfer function with white noise input
the response of T (s) when n(t) is applied to its input. Then the H2-norm of T (s) is
defined as
‖T (s)‖2 = E
(limt→∞
1
2t
∫ t
−t
‖e(t)‖2dt
)1/2
(2)
where ‖e(t)‖ denotes the Euclidean-norm of e(t), i.e. ‖e‖2 = e′e. Typically, T (s) is
an implicit function of the structure and the active controller. The H2-optimal control
problem is to choose the controller which stabilizes T (s) (internally) and minimizes
(2). The regulated (vector) signal e(t) typically includes the control effort as one of its
components.
The H2-problem is intimately related to the deterministic Linear Quadratic Regulator
(LQR) problem, which involves the minimization of:
J [u] =
∫ ∞
0
(xT Qx(t) + uT Ru + 2xT Nu)dt (3)
where x(t) denotes the system’s state-vector, u(t) is the control signal, Q = Q′ ≥ 0,
R = R′ > 0 and N are appropriate weighting matrices penalizing the state-vector
and control signal. It is well known [12] that the solution of the H2 problem can
be decomposed in two separate sub-problems. The first subproblem is to optimally
estimate the state-vector (in the mean-square sense), whose solution is provided by
Kalman-filtering theory. The second sub-problem is to find the control signal which
minimizes the deterministic cost of (3), subject to constraints in terms of the system’s
dynamics x = Ax + Bu. The solution is to let the control signal u(t) be a linear
function of the state:
u(t) = −Kcx(t) (4)
where Kc is the optimal state-feedback matrix, defined via the solution of an algebraic
3
Riccati equation. Then, the so-called separation principle (or certainty equivalence
principle) guarantees that the overall optimal solution of the H2 problem is still
obtained when the optimal state-feedback Kc is applied to the state estimates (obtained
from the Kalman filter), rather than the states themselves.
The H∞ optimal control problem assumes bounded-energy disturbance signals and
minimizes the maximum input-output energy transfer, given by the infinity-norm of
their transfer function, i.e.
min ‖T (s)‖∞ = minK∈S
maxω∈R
σ(T (jω)) (5)
where ‖ · ‖∞ is the infinity-norm, S denotes the set of all stabilizing controllers and
σ(·) is the largest singular value of a matrix. H∞-optimal control is essentially a
frequency-domain design methodology and is especially powerful in dealing with model
uncertainties. Thus, it can result in more robust designs than H2-optimal control,
although it can be conservative if the disturbances are naturally modelled as (filtered)
white-noise signals.
In this paper a novel approach is presented for minimizing the peak value of the
regulated signal, subject to peak magnitude and rate constraints on the control
signal. The method is developed in discrete time, using a finite-settling time (dead-
beat) parametrization, leading to a linear-programming optimization framework.
The method is particularly relevant to active vibration control of civil engineering
structures: Structural members fail after a maximum displacement is exceeded, and
thus direct optimization of peak output levels is more significant than, say, rms or
output energy levels. In addition, control constraints for systems of this type normally
arise in the form of hard saturation limits on actuator signals and their rates. Again,
using the proposed method such constraints can be directly addressed. In contrast,
in the LQG or H∞ design framework the designer can only penalize control signal
power or energy (possibly frequency weighted). Minimization of peak responses in
the context of active vibration control has been investigated by various researchers,
e.g. [10], using an adaptive bang-bang control methodology. The proposed method is
4
more straightforward as it relies on a fixed parameter-controller which does not require
on-line tuning. Since the problem is solved via Linear Programming, the resulting
controller will be denoted as LPOC (Linear Programming-Optimal Controller).
2. Structural model
The design algorithm is described in a step-by-step procedure. A simple benchmark
design problem from the area of active vibration control is presented alongside the
algorithm for illustration purposes.
The structure model chosen for the example employs active tendon control, since this is
reported in the literature to achieve the best results (disregarding cost considerations)
[14]. A model structure described in [13] was proposed as a benchmark problem and
has been investigated by a number of researchers. The model represents a simple and
regular 3-storey structure. A schematic of the structure is shown in Figure 2 below and
its parameters summarized in Table 1. The parameters of the linear actuator (force
constant kf , back-emf constant ke and armature resistance R) are defined in Table 2.
For simplicity only the ground and first floors are initially considered. The tendons
are connected between the ground and first floor and produce a pair of equal and
opposite forces. The structure is a scaled-down version of a real building with small
masses and dimensions, suitable for experimental work. A high value is assumed for
the base stiffness to account for the interaction between the base of the building and
the surrounding ground. The main objective of the controller is to minimize first-floor
acceleration when subjected to a force at the base.
The structure is idealized as a mass-spring-damper system shown in figure 2. In this
diagram, us is the actuator force and ν is the external-disturbance acceleration signal
(representing an earthquake) assumed to act at its base. The main design objective is
to minimize the peak value of the first regulated signal, chosen to represent first-floor
acceleration. This is equivalent to minimizing the l∞ norm of the impulse response of
5
Table 1: Structural parameters
Floor mi (Kg) ci (Ns/m) ki (N/m)
Base (i = 0) 5 100 16000
First (i = 1) 1.72 0.078 2600
Table 2: Actuator parameters
kf (N/A) ke (Vs/m) R (Ω)
2.0 2.0 1.5
the system, corresponding to the transfer function between the external disturbance
and first-floor acceleration. Constraints on the amplitude and rate on the actuation
signal will be subsequently imposed.
3. The design algorithm
The overall design procedure consists of the following steps:
(i) Definition of generalized plant: The two regulated signals are chosen as first-
floor acceleration and the control signal u representing actuator’s input voltage. It is
required that the controller stabilizes the system and
minK∈S
maxt≥0
|x1| (6)
subject to:
|u(t)| ≤ umax for all t ≥ 0 (7)
In addition, to avoid highly discontinuous or high-rate signals we may impose
constraints on the derivative of the control, i.e.,
|u(t)| ≤ umax for all t ≥ 0 (8)
6
c
k
c
k1
1
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0
0
xxxx
xxxx
x
x
0
1
us
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxx
xxxxx
xxxxx
xxxxxxxx
xxxxxxxxxxx
xxxxxxxxxxxx
xxxxxx
m0
m1
ν
Figure 2: One-storey structure
P
K
z
z
uy
1
2
ν
Figure 3: Generalized plant
Choosing as state-variables the displacements and velocities x0, x1, x0 and x1, a state-
space description of the model is given as x = Ax + B1ν + B2u where ν denotes
the disturbance input and u is the input actuator voltage. The state-space matrices
defining the model are given as:
A =
0 0 1 0
0 0 0 1
−k0+k1
m0
k1
m0
c1m0
+kf ke
Rm0
c1m0
+kf ke
Rm0
k1
m1− k1
m1
c1m1
+kf ke
m1R− c1
m1+
kf ke
m1R
(9)
7
and
B1 =
0
0
1
0
, B2 =
0
0
− kf
m0R
kf
m1R
(10)
Choosing as the only measurement the first-floor acceleration signal, defines the output
equation of the system as y = Cx + Du, where
C =[−k0+k1
m0
k1
m0− c1+c0
m0+
kf ke
Rm0
c1m0
+kf ke
m0R
](11)
and
D =kf
m1R(12)
Choosing the vector of regulated signals as z = (x1 u)′, the generalized plant (see figure
3) has a state-space description:
x = Ax + B1ν + B2u (13)
z1
z2
y
=
C
0
C
x +
0
0
0
ν +
D
1
D
u (14)
Note that there is no direct feed-through term from the disturbance ν to z or y.
Signals z1 and z2 define the two regulated outputs, in this case first-floor acceleration
and control input effort u, respectively (i.e. z1 = x1 and z2 = u). Variable y represents
the measured output, in this case also first-floor acceleration (i.e. y = x1).
(ii) System discretization: The solution to the optimization problem will be
obtained in discrete-time. Thus we first need to discretize the system using an
appropriate sampling interval. The zero-order hold discretization can be employed
using the standard procedure for transforming between continuous and discrete-time
state-space models [5]. The sampling period was chosen as Ts = 0.01s. The
corresponding Nyquist frequency fN = fs/2 = 50 Hz is significantly higher than the
frequencies of all system modes. We will still use the same notation for the discrete-
time state-space realization of the generalized plant, by appropriately redefining the
state-space matrices.
8
(iii) Youla parametrization of all stabilizing controllers: All stabilizing
controllers and the corresponding closed-loop transfer functions between disturbance
and regulated signals can be defined in terms of two matrices F and H (stabilizing state-
feedback and output injection matrices, respectively). Matrices F and H can be any
two matrices such that A+B2F and A+HC are asymptotically stable (all eigenvalues
inside unit circle). The parametrisation proceeds by first expressing the discrete plant
G(z) as the ratio of two stable, relatively prime transfer functions. Note that the
procedure is identical in the continuous and discrete domains, with the exception that
“stability” needs to be defined appropriately in each domain. In addition, note that in
defining the parametrization we have complete freedom in the choice of state-feedback
and output injection matrices, as long as A + B2F and A + HC are asymptotically
stable; here F and H will be chosen so that all eigenvalues of A + B2F and A + HC
are placed at the origin; this is always possible under appropriate controllability and
observability assumptions, which are satisfied in this case. The algorithm is described
next:
Pole placement at the origin: The problem of pole placement is stated as follows:
Find F such that all eigenvalues of A + B2F are placed at the origin. This can
be achieved with a standard method via Ackermann’s formula [2], given that the
pair (A,B2) is controllable, by transforming it via a state-space transformation T to
controllable canonical form (Ac, Bc) and selecting Fc such that det(sI−Ac−BcFc) = sn.
The state feedback matrix F is easily obtained from the inverse transformation T−1.
This can be achieved by the following steps:
• Define characteristic polynomial of matrix A,
p(s) = det(sI − A) = sn + αn−1sn−1 + αn−2s
n−2 + ... + a0 (15)
9
• Define Ac, Bc in canonical controllable form:
Ac =
−αn−1 −αn−2 · · · −α1 −α0
1 0 · · · 0 0
0. . .
......
. . . 0
0 0 1 0
, Bc =
1
0...
0
(16)
Note that the first row of Ac contains the coefficients of the characteristic
polynomial in descending order with negative signs.
• Define matrices Γ, Γc, T via
Γ = [B AB A2B . . . An−1B] (17)
Γc = [Bc AcBc A2cBc . . . An−1B] (18)
and
T = Γ−1c Γ (19)
• Obtain the state-feedback matrix F as
F = [αn−1 αn−2 . . . α0]T (20)
The problem of selecting H so that A + HC has all eigenvalues at the origin is dual to
the state feedback problem described above.
The set of all stabilizing controllers can now be parametrized in bilinear (linear-
fractional) form, while the set of corresponding closed-loop systems is given in linear
(more precisely affine) form, i.e.
T (z−1) = T1(z−1)− T2(z
−1)Q(z−1)T3(z−1) (21)
where Q(z−1) is a free stable parameter. Concrete state-space realizations of Ti(z−1)
can be found in [4].
10
(iv) Formulation of optimisation problem in terms of linear constraints: First
partition the closed-loop equations as:
y(z−1)
u(z−1)
=
t11(z
−1)
t21(z−1)
−
t12(z
−1)
t22(z−1)
q(z−1)t3(z
−1)
ν(z−1) (22)
where y(z−1) and u(z−1) are the regulated output responses to a discrete pulse
ν(z−1) = 1 and tji (z−1) ∈ H∞. Note that we have also set q(z−1) = Q(z−1) to emphasize
the fact that in this case the free parameter is scalar. Equation (22) can be alternatively
written as: y(z−1)
u(z−1)
=
t11(z
−1)
t21(z−1)
−
t12t3(z
−1)
t22t3(z−1)
q(z−1) (23)
Hence the transfer function between ν(z−1) → y(z−1) can be written as:
T (z−1) =y(z−1)
ν(z−1)=
b(z−1) + c(z−1)q(z−1)
a(z−1)(24)
where we have defined b(z−1) = t11(z−1) and c(z−1) = −t12(z
−1)t3(z−1). Note that under
the assumptions made earlier (all eigenvalues of A + B2F and A + HC2 placed at the
origin), we have that a(z−1) = 1. The degree of both b(z−1) and c(z−1) is r, where r
denotes the number of state variables (in this example r = 4). Parametrize q(z−1) as
a finite-impulse-response filter of degree p, i.e.
q(z−1) = q0 + q1z−1 + q2z
−2 + . . . + qpz−p (25)
Also write:
b(z−1) = b0 + b1z−1 + b2z
−2 + . . . + brz−r (26)
c(z−1) = c0 + c1z−1 + c2z
−2 + . . . + crz−r (27)
y(z−1) = y0 + y1z−1 + y2z
−2 + . . . + yNz−N (28)
Then:
y(z−1) = b0 + . . . + brz−r + (c0 + . . . + crz
−r)(q0 + . . . + qpz−p) (29)
11
so that deg[y(z−1)] = N = r + p. The equations can be written in matrix form as:
y0
y1
...
yr
yr+1
...
yN
=
b0
b1
...
br
0...
0
+
c0 0 · · · 0
c1 c0...
.... . .
...
cr cr−1 · · · c0
0 cr · · · c1
.... . .
...
0 · · · 0 cr
q0
q1
...
qp
(30)
Note that the response is forced to be dead-beat, i.e. yr+p is the last non-zero sample
of the regulated output. This is due to the restriction on q(z−1) which is taken to be
an FIR filter and may lead to a conservative solution unless r is taken to be large.
Ideally r should be selected to make NTs, a reasonable transient before the structure
is fully stabilized. It is expected (and can be established formally) that in the limit
N → ∞ the deviation from optimality can be made arbitrarily small. The equations
can be written compactly in matrix form as y = b + Cq where vector q contains the
coefficients of the polynomial q(z−1) and where C is a Toeplitz matrix.
(v) Formulation into a linear programming problem: Since all constraints are
linear, the minimization of the peak response of the regulated signal can be formulated
as a linear programming problem of the form:
min c′x subject to Ax ≤ b (31)
Let δ be the maximum absolute value of the regulated signal (first-floor acceleration)
that we wish to minimize. Then:
−δ ≤ yk ≤ δ for all 0 ≤ k ≤ N (32)
Now yk = ck′x + bk, where c′k denotes the k-th row of the C-matrix, and bk = bk for
0 ≤ k ≤ r and bk = 0 for k > r. Thus, separating the two equations we can write:
−ck′q− δ ≤ bk and ck
′q− δ ≤ −bk (33)
12
for all 0 ≤ k ≤ N , which can be written in matrix form as:
−1 C
−1 −C
δ
q
≤
b
−b
(34)
where 1 represents a column vector of ones. Setting x = (δ q)′, the problem is now in
the standard linear programming form:
min δ =[
1 0 · · · 0]x (35)
subject to (34). The solution to the problem will result in the optimal peak-value of
the regulated signal and the coefficients of the optimal q(z−1) , from which the optimal
controller can be recovered via the Youla parametrization in bilinear form.
(vi) Introducing constraints to the problem: In the above formulation, the
peak value of the regulated output is minimized for an impulsive loading without any
constraints on the size or rate of the control input. This is unrealistic and may result
in highly discontinuous control signals that would be difficult to implement or could
cause stability problems, especially in the presence of model uncertainty, due to the
potentially excessive bandwidth of the closed-loop system. The first constraint limits
the magnitude of the control signal and corresponds to the actuator’s saturation limits.
Hence we require that:
|uk| ≤ umax for all k ≥ 0 (36)
Now using Youla parametrization and the fact that the control effort has been chosen
as the second regulated output, u(z−1) may be written in the form:
u(z−1) = β(z−1) + γ(z−1)q(z−1) (37)
where we have defined β(z−1) := t21(z−1) and γ(z−1) := −t22(z
−1)t3(z−1). Note again
that β(z−1), γ(z−1) and q(z−1) are all polynomials in z−1 (the first two due to special
type of parametrization, and the third due to truncation). Similarly to the last section
13
the polynomial equation can be written in matrix form:
u0
u1
...
ur
ur+1
...
uN
=
β0
β1
...
βr
0...
0
+
γ0 0 · · · 0
γ1 γ0...
.... . .
...
γr γr−1 · · · γ0
0 γr · · · γ1
.... . .
...
0 · · · 0 γr
q0
q1
...
qp
(38)
where βi and γi are the coefficients of β(z−1) and γ(z−1), respectively. Writing the
equation in compact form u = β + Γq as before, and its k-th row as uk = βk + γ′kq,
the constraints |uk| ≤ umax for all k, may be expressed by a pair of linear inequalities:
−γ′kq ≤ umax + βk and − γ′
kq ≤ umax − βk (39)
for all k. Equivalently, this can be written in matrix form as: 0 −Γ
0 Γ
δ
q
≤
umax1 + β
umax1− β
(40)
In order to make the response “smoother” an additional constraint needs to be included
limiting the rate of actuator signal, u (slew-rate constraint). Now,
∆uk = uk+1 − uk
= βk+1 + γ′k+1q− βk − γ′
kq
= (βk+1 − βk) + (γ′k+1 − γ′
k)q
and we require
|∆uk| ≤ (∆u)max for all k ≥ 0 (41)
This may be written as a pair of linear inequalities:
(γ′k+1 − γ′
k)q ≤ (∆u)max − (βk+1 − βk) (42)
and
−(γ′k+1 − γ′
k)q ≤ (∆u)max + (βk+1 − βk) (43)
14
for all k, or, in matrix form as: 0 −(Γ− Γ)
0 Γ− Γ
δ
q
≤
(∆u)max1− (β − β)
(∆u)max1− (β + β)
(44)
where Γ and β denote the matrix Γ and vector β with the first row eliminated, while Γ
and β denote the matrix Γ and vector β with the last row eliminated. The inequalities
can now be augmented to the previous set of linear inequalities (40), and solved in a
linear programme to impose additional rate constraints on the control signal.
Using the constraints of the previous section, we can write:
−1 C
−1 −C
0 −Γ
0 Γ
0 −(Γ− Γ)
0 Γ− Γ
δ
q
≤
b
−b
umax1 + β
umax1− β
(∆u)max1− (β − β)
(∆u)max1− (β − β)
(45)
which is the overall set of inequalities of the LP optimization problem.
4. Application results and Discussion
The LP design method was first applied to the structure without any control constraints
with a filter length of r = 200 samples, corresponding to a deadbeat response of
approximately 2 seconds. The two regulated signals (1st floor acceleration and actuator
voltage) are shown in Figures 4 and 5 respectively.
In order to assess the effectiveness of the LPOC controller, its responses are compared
with those obtained via Linear Quadratic Regulator (LQR) design. The design involves
a quadratic cost-function consisting of two penalty terms, acceleration and control
effort. Both weighting factors were set to 1, penalizing equally the two terms. The
design was carried out both in continuous and discrete-time (with a sampling rate of
100 Hz), producing almost identical results.
15
0 0.5 1 1.5 2 2.5−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.251st floor acceleration
time sec
x 1 m/s
2
Figure 4: First-floor acceleration (unconstrained LP design)
0 0.5 1 1.5 2 2.5−30
−20
−10
0
10
20
30u
time sec
volts
Figure 5: Actuator voltage (unconstrained LP design)
16
0 0.1 0.2 0.3 0.4 0.5 0.6
−10
−5
0
5
10
1st floor acceleration
time sec
x 1 m/s
2
LPLQR
Figure 6: First floor acceleration (Constrained LP and LQR design)
The objective of LQR is to minimise the performance index defined in equation (3)
subject to plant dynamic constraints x = Ax + Bu. The terms included in the cost
function are:
• First floor acceleration x1(t) and
• Control input u(t)
resulting in an optimization index of the form:
J [u] =
∫ ∞
0
(x21 + ρu2)dt (46)
Here x1 represents acceleration of mass m1, u is the control input effort (actuator
voltage), and ρ is a penalty coefficient initially taken as 1. The performance index can
be formulated in the standard form of equation (3) by defining Q = CC ′, R = D2 + ρ
and N = C ′D.
Subsequently, the LP design was again carried out, this time with control constraints
on the peak control signal and its rate. The peak-magnitude control constraint was set
at 15 Volts, slightly less than the peak control signal obtained from the LQR simulation
(around 16 Volts) and the maximum rate constraint was set at 40 Volts/s. The two
17
0 0.1 0.2 0.3 0.4 0.5 0.6
−15
−10
−5
0
5
10
15
u
time sec
volts
LPLQR
Figure 7: Actuator voltage (Constrained LP and LQR design)
8 10 12 14 16 18 20 22 24 26 28 30
−200
−100
0
100
200
Unc
ontr
olle
d
8 10 12 14 16 18 20 22 24 26 28 30−100
−50
0
50
100
Acc
eler
atio
nLQ
R
8 10 12 14 16 18 20 22 24 26 28 30−100
−50
0
50
100
time (s)
LPO
C
LPOC
LQR
Uncontrolled
Figure 8: First floor acceleration (Constrained LP and LQR design for Earthquake
input)
18
8 10 12 14 16 18 20 22 24 26 28 30−150
−100
−50
0
50
100
150LPOC and LQR voltage (volts)
Vol
tage
(V
olts
)
8 10 12 14 16 18 20 22 24 26 28 30−150
−100
−50
0
50
100
150
time (s)
LQR
LPOC
Figure 9: Actuator voltage (Constrained LP and LQR design for Earthquake input)
Table 3: Comparison between LP and LQR methods
LQR LP Constrained LP
|x|max(m/s2) 13 0.21 5.5
|u|max(volt) 16 24 15
regulated signals resulting from the two designs (LQR and constrained LP) are shown
in figures 6 and 7. The main results of all simulations are also summarized in Table 3.
The unconstrained LP method yields excellent results in terms of optimizing the peak
signal level. The maximum acceleration is about 60 times smaller than the peak
acceleration resulting from the original LQR design, the peak voltage control level
increasing by a factor of 1.5. However, the resulting acceleration profile (Figure 12)
clearly indicates that the response is unrealistic for practical implementation. The
acceleration reaches its peak positive value of 0.21 m/s2 extremely fast and swings to
to its minimum negative value 0.21 m/s2 almost 10 ms later, requiring a huge slew-rate
from the actuator. Subsequently, the acceleration fluctuates between the two extreme
values for a few cycles of progressively increasing frequency before decaying to zero
19
after about 1.5 seconds (0.5 seconds earlier than the set deadbeat horizon) exhibiting
highly-damped oscillations. This behavior can be explained as follows: The maximum
acceleration is reached very fast (first peak) because the disturbance is an impulse. To
counter the acceleration increasing excessively, the controller produces a large negative
force, followed by a large positive force a few milliseconds later, to limit acceleration
increase in the opposite direction. After the maximum acceleration is reached the
controller’s primary goal is to keep it at the same level and thus gradually reduces
the applied forces. Finally, the acceleration reaches zero as the system settles to its
equilibrium. Thus the method works theoretically in the sense that it succeeds to
minimize peak acceleration, as indicated by the flat regions of the acceleration signal
at positive and negative peaks of the same magnitude. However, the response is clearly
unrealistic and thus the controller cannot be implemented in practice. The high rate
of the control signal (especially in the early part of the response) means that even if
this control profile could be generated by the actuator, the resulting closed-loop system
would have an unrealistically large bandwidth, and hence the system would have poor
stability margins and would be highly susceptible to model uncertainties.
By setting an acceptable limit in the rate of change of the control signal (|u|max = 40
Volts/s which is about ten times less than the fast rates of the early response observed
in Figure 5) the response of the system to the impulsive loading becomes acceptable.
The maximum acceleration for the constrained LP design is almost 2.5 times less than
the peak value obtained by LQR, while the controller peak signal (15 volts) is slightly
less than the peak value obtained from the LQR design (16 Volts). This improvement is
made despite the fact that the LQR controller is based on state-feedback (all four states
assumed measurable), whereas the LP controller uses output feedback only (first-floor
acceleration being the only measurement).
Next the impulsive load was replaced by a real earthquake signal consisting of the east-
west acceleration component of the Loma Prieta earthquake. The simulation results
with the LPOC and LQR controller are shown in figures 8 and 9. Note that both
rms and peak responses of the acceleration signal are reduced using LPOC control for
20
comparable levels of the control signal. The parameters chosen here are: Sampling
time Ts = 0.015 s, umax = 0.2 Volts, ∆umax = 0.3 Volts/s and N = 200 samples.
The LPOC technique was applied to the design of the three-storey building using the
same actuator arrangement described previously, the objective being to minimize the
1st floor accelerations which is assumed to be the only measurement. The state space
model used is given as: x = Ax + B1ν + B2u, the states being the displacement and
velocity variables x0, x1, x2, x3, x0, x1, x2 and x3. The state-space matrices A, B1 and
B2 in this case are:
A =
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
−k0+k1
m0
k1
m00 0 c1
m0+
kf ke
Rm0
c1m0
+kf ke
Rm00 0
k1
m1−k1+k2
m1
k2
m10 c1
m1+
kf ke
m1R− c1+c2
m1− kf ke
m1Rc2m1
0
0 k2
m2−k2+k3
m2
k3
m20 c2
m2
c2+c3m2
c3m2
0
0 0 k3
m3− k3
m30 0 c3
m3− c3
m3
(47)
and
B1 =
0
0
0
0
1
0
0
0
, B2 =
0
0
0
0
− kf
m0R
kf
m1R
0
0
(48)
Choosing as the only measurement the first-floor acceleration signal, defines the output
equation of the system as y = Cx + Du, where
C =[
k1
m1−k1+k2
m1
k2
m10 c1
m1+
kf ke
m1R− c1+c2
m1− kf ke
m1Rc2m1
0]
(49)
21
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−15
−10
−5
0
5
10
15
1st floor acceleration
time sec
x 1 m/s
2
LPOCLQR
Figure 10: LQR and LPOC Acceleration of 3-storey building
and
D =kf
m1R(50)
The following values were used for the simulation: Ts = 0.015 s, umax = 0.2 Volts,
∆umax = 0.2 Volts/s and N = 200 samples. The responses compare favourably with
those obtained by LQR and are shown in Figure 10 and 11. Note that the LPOC is
capable of achieving significant reduction in peak acceleration levels using a significantly
reduced (peak) level voltage.
5. Further design considerations
The optimization problem described in previous sections minimizes the peak response
of one regulated signal (first floor acceleration) subject to peak and rate constraints of
another regulated signal (control input). Although the external disturbance is assumed
to be a unit impulse, the method can be easily modified to take into account any
disturbance of finite-duration. Assuming, for example that the external disturbance
signal has a z-transform,
ν(z−1) = ν0 + ν1z−1 + . . . + νlz
−l (51)
22
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20
−15
−10
−5
0
5
10
15u
time (s)
volts
LPOCLQR
Figure 11: LQR and LPOC voltage of 3-storey building
equation (24) can be still written as:
y(z−1) = b(z−1) + c(z−1)q(z−1) (52)
by redefining b(z−1) ← b(z−1)ν(z−1) and c(z−1) ← c(z−1)ν(z−1) (and similarly for
u(z−1)).
Once the optimal “free parameter” q(z−1) is obtained in the form of an FIR filter via
the LP programme solution, the corresponding optimal controller can be recovered in
bilinear (lower linear fractional) form as:
Kopt = Fl(J, qopt) = J11 + J12qopt(I − J22qopt)−1J21 (53)
where
J =
J11 J12
J21 J22
=
A + B2F + HC + HDF −H B2 + HD
F 0 I
−(C + DF ) I −D
(54)
(see [4]). The corresponding closed loop transfer functions between the external
disturbance (ν(z−1)) and the regulated outputs (y(z−1) and u(z−1)) are obtained in
affine form via equation (23). The frequency response of the closed-loop system between
23
0 50 100 150 200 250 300−60
−40
−20
0
20
40
60
Angular frequency (rads/s)
Mag
nitu
de (
dB’s
)
Uncontrolled, LQR and LPOC frequency Bode plot
UncontrolledLQRLPOC
Figure 12: Bode plots of LPOC
ν(z−1) and y(z−1) in figure 12 shows that the gain is significantly reduced over all
frequencies. This suggests that the controller will be effective for arbitrary disturbance
inputs, not just an impulse.
A disadvantage of the method is that it results in high order controllers (of degree N =
r + p). This can result in a heavy computational load and implementation difficulties
and hence order-reduction techniques should be applied for practical purposes. Suitable
approximation techniques include balanced truncation or Hankel-norm approximation
methods [18], [7]. In this case, since the bulk of the controller complexity is due to the
high degree of the FIR filter q(z−1), model reduction techniques developed specifically
for systems of this type may be more appropriate. A Hankel-norm approximation
method of FIR systems by low-order infinite-impulse-response (IIR) systems was
developed in [8] based on a model-reduction technique applicable to general discrete-
time descriptor systems [1]. A nice aspect of this approach is that the reduced order
controller is guaranteed to be stabilizing (due to Youla parametrization), irrespective
of the approximation order. However, the performance properties of the design may
deteriorate, especially for approximations of a low degree.
24
6. Conclusions
The paper presents a LP-based algorithm aiming to minimize the peak value of a
regulated signal, an objective which is especially relevant for the design of active
vibration control of civil engineering structures. Linear constraints are introduced
to limit the magnitude of the control signal and its rate, resulting in “smooth”
responses and low-bandwidth control schemes which can be implemented in practice.
The design algorithm was developed in parallel to a simple example involving a
scaled-down scalar benchmark model of a one-storey building, although extensions
to the multivariable case and multiple regulated signals are straightforward. It was
demonstrated via simulations that the design method is capable to reduce significantly
the peak acceleration response of the model compared to LQR designs, even after the
introduction of constraints on the control-signal. Other advantages of the method
include the ability to formulate realistic constraints involving the magnitude and
rate of regulated signals (rather than rms or energy content) and to provide indirect
control of the overall damping by specifying the settling-time horizon. Although the
disturbance signal was assumed to be an impulse, more general disturbance models
can be accommodated.
A number of issues related to the design require further investigation. These include
a full robustness analysis and the possibility of incorporating the method within a
larger multi-objective optimization framework (e.g. using multiple regulated signals
and a mixture of linear and quadratic constraints, which can be tackled via quadratic
programming). Another important issue is related to controller complexity. The design
method tends to produce high-order controllers, in the form of a bilinear transformation
of a high-order FIR filter. This can be approximated by a low-order IIR filter resulting
in an overall low-order controller (which is still stabilizing) using a recently derived
Hankel-norm model-reduction algorithm for discrete-time descriptor systems [1], [8].
The approximation order should be chosen so that small magnitude and phase errors
are introduced in the controller’s frequency response, especially in the cross-over region
which determines the gain and phase margins of the feedback loop. Alternatively,
25
direct closed-loop approximation techniques with quantifiable measures of performance
deterioration can be considered [6].
Alternative control design methods aiming to minimize the peak response of the
regulated signal have recently been reported both in the area of active vibration
control [10] and also in the general control literature [16], [17], [3]. Reference [10]
is based on an adaptive bang-bang methodology, which clearly offers advantages in
the case of uncertainty about the disturbance-signal, but is also difficult to apply
in practice. A systematic general approach is l1 optimal control, which attempts to
minimize the peak amplification gain between disturbance input and regulated output
[3], [11]. Interestingly, the method also results in a Linear Programming optimization
framework. However, as l1 is an induced norm, all bounded disturbance signals are
taken into account in the formulation of the optimization problem. As a result, the
design may be conservative, unless the method can be restricted to specific models of
disturbance signals that are likely to arise in practice, i.e. signal classes whose spectral
content is similar to typical seismic acceleration signals. Assessing the potential merits
of this method relative to the proposed approach needs further investigation.
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28