A dynamic allocator is proposed in order to generalize a previously introduced strategy for input redundant plants, which applies to linear plants with multiple and redundant inputs. The theory is extended here to the case of multiple linear actuators, each of them with its own dynamics, acting on a nonlinear plant with strong input redundancy. In the HEV case the two redundant inputs are the ICE and EM torques and the two actuators with different dynamics are the two propulsion systems

Cordiner, S

Galeani, S

Mecocci, F

Mulone, V

Zaccarian, L

CORDINER, STEFANO

GALEANI, SERGIO

MULONE, VINCENZO

ZACCARIAN, LUCA

English

Appendix to "Torque setpoint tracking for parallel hybrid electric vehicles using dynamic input allocation", published on IEEE Transactions on Control Systems Technology

Cineca Institutional Research Information System

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1DICII Technical report RR-13.02
Appendix to
“Torque setpoint tracking for parallel hybrid electric
vehicles using dynamic input allocation”,
published on IEEE Transactions on Control Systems
Technology
Stefano Cordiner, Sergio Galeani, Francesco Mecocci, Vincenzo Mulone and Luca Zaccarian
I. INPUT ALLOCATION WITH LINEAR DYNAMIC
REDUNDANCY
A dynamic allocator is proposed in this section to generalize
the strategy for input redundant plants introduced in [1] and
extended in [2]. The approach of [1] applies to linear plants
with multiple and redundant inputs. The theory is extended
here to the case of multiple linear actuators, each of them with
its own dynamics, acting on a nonlinear plant with strong input
redundancy. In the HEV case the two redundant inputs are
the ICE and EM torques and the two actuators with different
dynamics are the two propulsion systems.
Referring to Figure 3, the “Plant” block corresponds to the
following nonlinear system:
x˙p = fp(t, xp, Bup, d)
yp = hp(t, xp, Dup, d)
(1)
where up ∈ Rnu is the control input, d ∈ Rnd is a disturbance
input and yp ∈ Rny is the plant output. The block “Actuators”
is a diagonal and square linear time invariant (LTI) system
representing the dynamics of the nu linear actuators. These
actuators establish the following relationship (2) at the control
input of the plant, which is reported here for the easier reading:
up(s) = G(s)u(s) = diag {gi(s)}u(s) (2)
where gi(s), i = 1, ..., nu are proper asymptotically stable
transfer functions. The block “Controller” is the following a-
priori given nonlinear controller:
x˙c = fc(t, xc, yp, r)
yc = hc(t, xc, yp, r)
(3)
where r ∈ Rnr is the reference input and yc ∈ Rnu is
the controller output. The following natural assumption is
made on the closed-loop of Figure 3 in the absence of the
“Allocator” block and on the actuators transfer functions in
(2). Note that the assumption that gi(0) > 0 can always be
guaranteed as long as gi(0) 6= 0 by possibly inverting the
sign of the corresponding columns in [ BD ], namely gi(0) > 0,∀i ∈ {1, ..., nu}.
Assumption 1. The closed loop system before allocation or
unallocated closed loop, given by (1), (2), (3) with u = yc
and r = const, is well posed and has an unique globally
asymptotically stable equilibrium. Moreover, all the actuators
transfer functions in (2) are asymptotically stable and are
strictly positive at zero. ◦
Mimicking the approach of [1], it is assumed that the
matrices B and D filtering the control input up of the
nonlinear plant (1) characterize a strong input redundancy
in the sense that ker(B) ∩ ker(D) 6= ∅. Based on this
property, a nonempty full column rank matrix B⊥ can be
defined satisfying:
Im(B⊥)= ker(B) ∩ ker(D) (4)
where clearly rank(B⊥) = nu − rank([ BD ]).
Definition 1. Two constant vectors yc, u0 ∈ Rnu are statically
compatible with respect to the strongly input redundant plant
(1) and the actuators (2) if they satisfy relation (8), which is
reported here for the easier reading:[
B
D
]
G(0)(yc − u0) = 0 (5)
◦
The structure of the proposed dynamic allocator is shown
in Figure 4 and corresponds to the linear dynamics (6),
represented in the Laplace domain, which is reported here for
the easier reading:
w =
1
s
satM (KB
′
⊥G¯(0)W (s)(u− u0)) (6a)
δu = −G¯(s)B⊥w (6b)
u = yc + δu (6c)
where w ∈ Rna is the state of the dynamic allocator and δu
is the input variation imposed by the allocator. The function
satM : R
na → Rna is a symmetric decentralized vector
saturation having saturation limits M =
[
m1 . . . mna
]′
,
2which specify the maximum allocation rates. The diagonal
matrix K > 0 on the other hand specifies the local exponential
convergence rate of the allocator. Moreover, G¯(s) and W (s)
are diagonal and proper matrix transfer functions selected in
such a way that there exists a scalar asymptotically stable
transfer function κ(s) satisfying (7) (here reported for the
easier reading):
G(s)G¯(s) = G¯(s)G(s) = Iκ(s) (7a)
κ(0) 6= 0 (7b)
W (0) = W0 > 0, (7c)
where equality (7a) easily follows from the diagonal structure
of G(s) and G¯(s).
Remark 1. From the first equation in (7), and the definition of
B⊥ in (4), if two vectors yc and u0 are statically compatible,
then there exists a third vector w∗ satisfying:
yc − u0 = G¯(0)B⊥w∗, (8)
indeed from (5) there exists w¯ such that B⊥w¯ = G(0)(yc−u0)
and (8) holds also using (7b), with w∗ = w¯/κ(0). ◦
Remark 2. If G¯(s) = I and W (s) = W0 > 0 diagonal, then
the dynamic allocator (6) reduces to the structure proposed in
[1]. ◦
The following theorem formalizes two fundamental proper-
ties of the allocated closed loop (1), (2), (3), (6) and repre-
sented in Figure 3. These properties hold under the assumption
that the closed loop in Figure 4 satisfies the following property.
Later in Procedure 1, we will provide constructive technique
for selecting G¯(s) and W (s) to guarantee this property when-
ever the actuators dynamics is minimum phase.
Property 1. The parameters of allocator (6) satisfy (4) and
(7) and are such that G¯(s) and W (s) are proper1. Moreover
for any constant selection of yc and u0, it has a globally
asymptotically stable (GAS) and locally exponentially stable
(LES) equilibrium point. ◦
The first item of Theorem 1 ensures that the allocator
does not enforce any modification at the plant state xp and
output yp but only affects the actuator input u (and the plant
input up). The second property ensures that the allocator
is capable of asymptotically tracking constant actuator input
references u0 as long as they are statically compatible with the
controller output yc (in the sense of Definition 1) generated
by the controller before allocation. The proof of the theorem
is reported in Section II.
Theorem 1. Consider the allocated closed-loop (1), (2), (3),
(6) satisfying Property 1 and the unallocated closed-loop (1),
(2), (3), u = yc satisfying Assumption 1. Initializing the
allocator dynamics (6) at zero, the following holds:
1) using the same plant-controller initial conditions and the
same external signals r, d, the plant state and output
responses (xp, yp) produced by the unallocated closed-
loop and by the allocated closed-loop coincide;
1The properness assumption on G¯(s) and W (s) ensures that (6) is causal,
hence physically realizable.
2) given any constant u0 and a steady-state controller
output yc generated by the unallocated closed-loop if
(yc, u0) is a statically compatible pair (in the sense of
Definition 1), then the plant input response u of the
allocated closed-loop from the same initial conditions
satisfies lim
t→∞u(t) = u0.
◦
In the following procedure, under the assumption that all
the diagonal elements in (2) are minimum phase, a procedure
is proposed for the selection of the diagonal matrix transfer
functions G¯(s) and W (s) in (6), guaranteeing Property 1.
Procedure 1.
Inputs: G(s) = diag {gi(s)} diagonal and proper matrix
transfer function where gi(s), i = 1, ..., nu, have
all poles and zeros with strictly negative real part;
B⊥ satisfying (4); K > 0 and W0 > 0 diagonal
matrices.
Step 1: (Least common multiple). Define g¯i(s) =
∏
j 6=i
gj(s),
i = 1, ..., nu and select G¯(s) := diag {g¯i(s)}. If
the allocator (6) satisfies Property 1 2, then select
W (s) = W0 and terminate, otherwise repeat the
following steps for i = 1, ..., nu.
Step 2: (Relative degree). If the relative degree of g¯i(s) 1s
is equal to one3 then select w1,i(s) as the i−th
diagonal component of W0. Else define w1,i(s)
as the i-th diagonal component of W0 times a
number of stable zeros such that w1,i(s)g¯i(s) 1s has
relative degree one and, without loss of generality
w1,i(0) = 1. Go to Step 3.
Step 3: (Positive realness). If w1,i(s)g¯i(s) 1s is positive real
then w2,i(s) := 1. Else define w2,i(s) as a cascade
of biproper, stable and minimum phase lead and/or
lag compensators such that w2,i(s)w1,i(s)g¯i(s) 1s
is positive real and without loss of generality
w2,i(0) = 1. Go to Step 4.
Step 4: (Proper W (s)). If w1,i(s)w2,i(s) is proper then
w3,i(s) := 1. Else define w3,i(s) with a sufficient
number of high - frequency stable poles such that
w3,i(s)w2,i(s)w1,i(s) is proper.
Output: G¯(s), W (s) := diag {w3,i(s)w2,i(s)w1,i(s)} and
κ(s) =
nu∏
i=1
gi(s)
?
The following theorem establishes that Procedure 1 always
leads to an allocator satisfying Property 1. The proof is
passivity-based and hinges upon the property that the con-
struction in Procedure 1 ensures that the lower system in
Figure 1 (which is an equivalent representation of the closed
loop in Figure 4) is positive real. Then, from the passivity
theorem, there exists a single GAS and LES equilibrium for its
interconnection with the upper system (a saturation function).
The proof of the theorem is reported in the Section II.
2As shown in the proof, g¯i(s) 1s positive real ∀i = 1, ..., nu is a sufficient
condition to have an allocator (6) respecting the Property 1.
3Note that, due to the selection of g¯i(s) in Step 1, g¯i(s)/s is always strictly
proper.
3Fig. 1. Equivalent block diagram of the internal feedback of the proposed
dynamic allocator for the case yc − u0 = 0.
Theorem 2. If each diagonal element of G(s) in (2) is
minimum phase and asymptotically stable, then Procedure 1
always leads to a set of parameters G¯(s), W (s) and κ(s)
satisfying Property 1. ◦
Remark 3. Since the proof of Theorem 2 is based on the
passivity of the interconnection in Figure 1, then, as long as
the procedure did not terminate at Step 1 , the allocator gain
K = K ′ > 0 can be selected arbitrarily large while preserving
closed-loop stability. This property allows for a useful tuning
of the dynamic allocator speed without needing to worry about
stability. ◦
Remark 4. The main extension given here from the dynamic
allocator proposed in [1] is the presence of the new dynamics
G¯(s) in Figure 4. This transfer matrix is based on a model
of the actuators dynamics and has the goal of making w
completely invisible from [ BD ]up. This goal is obtained by the
property in equation (7) which ensures that w is filtered by
some common multiple κ(s) =
nu∏
i=1
gi(s) of all the actuators
dynamics. ◦
II. PROOFS OF THE MAIN RESULTS
Proof 1. Item 1. By linearity of the actuators and of the
allocator (6), it is sufficient to show that the transfer function
from w to [ BD ]up is zero. As a matter of fact, the allocator
contribution at the plant input will consequently be zero,
thus fully preserving the unallocated closed-loop response
from the point of view of the plant state and output. This
property follows from the following calculations arising from
the second equation in (6), from (7), (4) and (2):[
B
D
]
G(s)G¯(s)B⊥w = κ(s)
[
B
D
]
B⊥w = 0 ∀w ∈ Rna
(9)
Item 2. By Property 1, for each constant selection of yc, u0,
all signals in (6) converge. Then also w converges to a steady-
state value w¯. In particular, using the fact that w˙ = 0, at the
steady-state (namely replacing s by 0 in the transfer functions):
0 = satM (KB
′
⊥G¯(0)W (0)(yc − u0 − G¯(0)B⊥w¯)) (10a)
= satM (KB
′
⊥G¯(0)W (0)G¯(0)B⊥(w
∗ − w¯)) (10b)
where, according to Remark 1, w∗ satisfies (8). Since K > 0,
G¯(0)W (0)G¯(0) > 0 from (7), and B⊥ is full column rank by
assumption, then the matrix KB
′
⊥G¯(0)W (0)G¯(0)B⊥ is non
singular, which together with (10) implies w¯ = w∗. Finally,
using (8), the steady-state value of u can be computed as u¯ =
yc − δ¯u = yc − G¯(0)B⊥w∗ = u0 which completes the proof.
◦
Proof 2. To prove the theorem, it will be first shown in
Step A that the procedure always terminates with parameters
satisfying (7), with proper, therefore causal, G¯(s) and W (s).
Then it is shown in Step B that for any constant exogenous
signal, the dynamics (6) with W (s) = diag {w2,i(s)w1,i(s)}
has a single GAS and LES equilibrium. Finally in Step C it
will be shown that the same property holds with W (s) coming
from Procedure 1, thus completing the proof of Property 1.
Step A. If the procedure terminates at Step 1, then (7a) holds
with κ(s) =
nu∏
i=1
gi(s) by the definition of G¯(s), (7b) holds by
Assumption 1, (7c) holds by definition of W0 and the proof is
completed because Property 1 holds. Otherwise, Step 2 of the
procedure can be completed because all the functions gi(s)
(therefore also g¯i(s) 1s ) are proper. Step 3 can be completed
because, by Assumption 1, all the functions w1,i(s)g¯i(s)) are
asymptotically stable and minimum phase and the residual of
w1,i(s)g¯i(s)
1
s relative to s = 0 is w1,i(0)g¯i(0) > 0. Then
it is always possible to correct the phase of w1,i(s)g¯i(s) 1s
by suitable biproper stable minimum phase lead or lag filters
and the relative degree property ensured at Step 2 together
with the phase property ensured at Step 3, guarantees that
w1(s)w2(s)w3(s)g¯i(s)
1
s is positive real. Finally Step 4 can
always be performed, and it is ensured that W (s) and G¯(s)
are proper in addition to stable. Note that (7) still holds since
(7a) and (7b) only depend on G¯(s) and (7c) holds because
wj,i(0) > 0, j = 1, 2, 3.
Step B. Since the proof has already been completed in the
case when the procedure stops at Step 1, in the following it is
assumed that Steps 2-4 are performed. First note that, referring
to Figure 4, given the response of (6) with yc − u0 constant
and some fixed initial conditions, it is always possible to find
another set of initial conditions for w, the block −G¯(s)B⊥
and W (s) such that the state response of (6) with yc−u0 = 0
is the same. Then from this equivalence, in the following it
will be assumed that yc − u0 = 0. Now consider the function
Hˆ(s) := KB
′
⊥G¯(0)diag {w2,i(s)w1,i(s)} G¯(s)B⊥ 1s , define
Smin as its minimal realization with state x and consider
its negative interconnection with satM (·) (which is shown in
Figure 1 when y3 = 0 and S2, S3 are not considered). Since
Hˆ(s) is positive real by the construction at Step 3, then (see
[3, Chapter 6]) there exists a matrix P = P ′ > 0 such that
the function V (x) := x′Px satisfies V˙ (x) ≤ 0. This function
is also a Lyapunov function proving stability and passivity of
the negative feedback interconnection of Hˆ(s) and satM (·).
V˙ is only negative semi-definite in x but the largest invariant
set in
{
x : V˙ (x) = 0
}
is the origin because Hˆ(s) consists in
a pool of integrators times a nonsingular matrix transfer func-
tion. Then, from LaSalle’s invariance principle [3, Corollary
4.2], the origin is GAS and LES for the negative feedback
interconnection between satM (·) and the minimal realization
of Hˆ(s). Consider now the actual (non-necessarily minimal)
realization of Hˆ(s) composed by Smin and S2, S3 which
represent, respectively, the uncontrollable and unreachable
exponentially stable (ES) parts. Since S2, S3 are ES so that
their outputs converge to zero, GAS and LES of the cascade
4system can be proven by establishing global boundedness and
applying [4]. To establish global boundedness, first note that
the saturation output y1 is bounded by |y1| ≤ √na|M |∞
therefore the trajectories of the exponentially stable system
S2 remain bounded. Regarding the remaining states, note that
from passivity of Smin, there exists U(x) positive definite and
radially unbounded such that U˙(x) ≤ u2y2. Moreover, since
by the properties of satM (·) it follows that u1y1 ≥ 0, then:
U˙ ≤ u2y2 + u1y1 = y1y3 ≤ |y1||y3|. (11)
Denote now by x3,0 the initial conditions of S3, it follows
from ES of S3 that, for some positive scalars k3, λ3, |y3| ≤
k3exp(−λ3t)|x3,0|. Therefore, integrating (11) it follows that
U(x(t)) ≤ U(x0) + (√na|M |∞k3|x3,0|)/λ3, which implies
global boundedness of x.
Step C. Similar to Step B, define H(s) :=
KB
′
⊥G¯(0)W (s)G¯(s)B⊥
1
s and note that its negative
interconnection with satM (·) is equivalent to the scheme
of Figure 4 with yc − u0 = 0. H(s) differs from Hˆ(s)
only by a number of high frequency asymptotically stable
poles corresponding to diag {w3,i(s)} constructed at Step
4. Then the closed-loop between H(s) and satM (·) is a
perturbation of the asymptotically stable closed loop between
Hˆ(s) and satM (·) by way of a high frequency strictly proper
filter. Then, since H(s) is strictly proper (due to Step 2
of Procedure 1), so that there is no algebraic loop in the
feedback interconnection, as long as the high-frequency poles
of diag {w3,i(s)} are sufficiently fast, asymptotic stability of
the feedback interconnection is preserved by the results in [5,
Section 4.7] (see also a similar discussion in [6]). ◦
REFERENCES
[1] L. Zaccarian, “Dynamic allocation for input-redundant control systems,”
Automatica, vol. 45, pp. 1431–1438, April 2009.
[2] G. D. Tommasi, S. Galeani, A. Pironti, G. Varano, and L. Zaccarian,
“Nonlinear dynamic allocator for optimal input/output performance trade-
off: application to the JET tokamak shape controller,” Automatica, vol. 47,
no. 5, pp. 981–987, 2011.
[3] H. K. Khalil, Nonlinear systems, 3rd edition. Prentice Hall, 2001.
[4] E. Sontag, “Remarks on stabilization and input-to-state stability,” in Proc.
CDC, (Tampa, Florida), pp. 1376–1378, Dec. 1989.
[5] A. Isidori, Nonlinear control systems. Springer, third ed., 1995.
[6] F. Forni, S. Galeani, and L. Zaccarian, “An almost anti-windup scheme
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Appendix to "Torque setpoint tracking for parallel hybrid electric vehicles using dynamic input allocation", published on IEEE Transactions on Control Systems Technology

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