Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances

Abstract

This paper presents the optimal quadratic-Gaussian controller for uncertain stochastic polynomial systems with unknown coefficients and matched deterministic disturbances over linear observations and a quadratic criterion. As intermediate results, the paper gives closed-form solutions of the optimal regulator, controller, and identifier problems for stochastic polynomial systems with linear control input and a quadratic criterion. The original problem for uncertain stochastic polynomial systems with matched deterministic disturbances is solved using the integral sliding mode algorithm.

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Optimal Controller for Uncertain Stochastic Polynomial Systems with
Deterministic Disturbances
Michael Basin Dario Calderon-Alvarez
Abstract—This
Gaussian controller for uncertain stochastic polynomial systems
with unknown coefficients and matched deterministic distur-
bances over linear observations and a quadratic criterion. As
intermediate results, the paper gives closed-form solutions of
the optimal regulator, controller, and identifier problems for
stochastic polynomial systems with linear control input and a
quadratic criterion. The original problem for uncertain stochas-
tic polynomial systems with matched deterministic disturbances
is solved using the integral sliding mode algorithm.
paperpresentstheoptimal quadratic-
I. INTRODUCTION
Although the optimal quadratic-Gaussian controller prob-
lem for linear systems was solved in 1960s, based on the
solutions to the optimal filtering [1] and optimal regulator
[2], [3] problems, the optimal controller for nonlinear sys-
tems has to be determined using the nonlinear filtering theory
(see [4], [5], [6]) and the general principles of maximum
[2] or dynamic programming [7], which do not provide an
explicit form for the optimal control in most cases. There is
a long tradition of the optimal control design for nonlinear
systems (see, for example, [8]–[13]) and the optimal closed-
form filter design for nonlinear [14], [15], [16], and in
particular, polynomial ([17]–[20]) systems, as well as the
robust filter design for stochastic nonlinear systems (see,
for example, [21]–[23]). However, the optimal quadratic-
Gaussian controller problem for nonlinear, in particular,
polynomial, systems with unknown parameters has not even
been consistently treated. Indeed, the optimal solution is not
defined, if some parameters are undetermined. The problem
becomes even more complicated, if the plant is affected by
deterministic disturbances. Nonetheless, the problem state-
ment starts making sense, if unknown parameters are mod-
eled and deterministic disturbances are matched, in other
words, belong to a controllable subspace. Taking into account
the stochastic Gaussian specifics of the optimal quadratic-
Gaussian problem, the unknown parameters are represented
as stochastic Wiener processes. The extended state vector
consists of the real unmeasured states and unknown param-
eters, and the obtained extended state equations are polyno-
mial with respect to the extended state vector. The integral
sliding mode algorithm for unmeasured states (see [25] for
the original version and [26] for a modification) is used
for compensating the matched deterministic disturbances
The authors thank the Mexican National Science and Technology Council
(CONACyT) for financial support under Grants 55584 and 52953.
M. Basin and D. Calderon-Alvarez are with Department of Physical
and Mathematical Sciences, Autonomous University of Nuevo Leon, San
Nicolas de los Garza, Nuevo Leon, Mexico mbasin@fcfm.uanl.mx
dcalal@hotmail.com
optimally with respect to the observations. Other recent
developments in the sliding mode theory and applications
can be found in [27]–[32].
This paper presents solution to the optimal quadratic-
Gaussian controller problem for uncertain stochastic polyno-
mial systems with unknown coefficients and matched deter-
ministic disturbances over linear observations and a quadratic
criterion. First, the paper recalls the optimal solution to
the quadratic-Gaussian controller problem for incompletely
measured stochastic polynomial states with linear control
input and a quadratic criterion [33]. Next, the paper provides
the optimal solution to the quadratic-Gaussian controller
problem with unknown parameters, which is based on the
preceding result. Finally, the integral sliding mode algorithm
yields a solution to the original quadratic-Gaussian controller
problem for uncertain stochastic polynomial systems with
deterministic disturbances, that is conditionally optimal with
respect to the observations.
II. OPTIMAL CONTROLLER PROBLEM
A. Problem statement
Let (Ω,F,P) be a complete probability space with an
increasing right-continuous family of σ-algebras Ft,t ≥ t0,
and let (W1(t),Ft,t ≥ t0) and (W2(t),Ft,t ≥ t0) be indepen-
dent Wiener processes. The Ft-measurable random process
(x(t),y(t)) is described by a nonlinear differential equation
with a polynomial drift term including an unknown vector
parameter θ(t) for the system state
dx(t) = f(x,θ,t)dt +B(t)u(t)dt +h(t)dt +b(t)dW1(t),
x(t0) = x0,(1)
and a linear differential equation for the observation process
dy(t) = (A0(t)+A(t)x(t))dt +G(t)dW2(t).(2)
Here, x(t) ∈ Rnis the state vector, u(t) ∈ Rlis the control
input, and y(t) ∈ Rmis the linear observation vector, m ≤ n,
and θ(t) ∈ Rpis the vector of unknown parameters. The
initial condition x0∈ Rnis a Gaussian vector such that x0,
W1(t)∈Rr, and W2(t)∈Rqare independent. The observation
matrix A(t) ∈ Rm×nis not supposed to be invertible or
even square. It is assumed that G(t)GT(t) is a positive
definite matrix, therefore, m ≤ q. All coefficients in (1)–
(2) are deterministic functions of appropriate dimensions.
The system (1),(2) is assumed to be uniformly controllable
and observable; the definitions of uniform controllability
and observability for nonlinear systems can be found in
[34]. The plant operates under deterministic disturbances
2009 American Control Conference
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h(t) and stochastic noises dW1(t) and dW2(t) represented
as weak mean square derivatives (see [24]) of the Wiener
processes, that is, white Gaussian noises. The function h(t)∈
Rnrepresents matched disturbances such that h(t)=B(t)γ(t),
γ ∈ Rl, and the norm ?γ(t)? is bounded by
?γ(t)? ≤ qa(t),
qa(t) > 0,(3)
where qa(t) is a finite time-dependent function; ?x? =
?(xTx) denotes the Euclidean 2-norm of a vector x ∈ Rl.
The nonlinear function f(x,θ,t) is considered polynomial
of n variables, components of the state vector x(t)∈Rn, with
time-dependent coefficients. Since x(t) ∈ Rnis a vector, this
requires a special definition of the polynomial for n > 1.
In accordance with [19], a p-degree polynomial of a vector
x(t) ∈ Rnis regarded as a p-linear form of n components of
x(t)
f(x,t) = a0(θ,t)+a1(θ,t)x+a2(θ,t)xxT+
...+as(θ,t)x...s times...x,
where a0 is a vector of dimension n, a1 is a matrix of
dimension n×n, a2is a 3D tensor of dimension n×n×n,
asis an (s+1)D tensor of dimension n×...(s+1) times...×n,
and x ×...s times...×x is a pD tensor of dimension n×
...s times...×n obtained by p times spatial multiplication
of the vector x(t) by itself. Such a polynomial can also be
expressed in the summation form
fk(x,t) = a0 k(θ,t)+∑
i
a1 ki(θ,t)xi(t)
+∑
ij
a2 kij(θ,t)xi(t)xj(t)+...
+∑
i1...is
as ki1...is(θ,t)xi1(t)...xis(t),
k,i, j,i1... is= 1,...,n.
The dependence of a0(θ,t), a1(θ,t),...,as(θ,t) on θ
means that those structures contain unknown compo-
nents a0i= θk(t), k = 1,...,p1≤ n, a1 ki= θk(t), k =
p1+ 1,...,p2≤ n × n + n,...,as ki1...is= θk(t), k = ps+
1,...,p ≤ n+n2+,...,+ns, as well as known components
a0i(t),a1 ki(t),...,as ki1...is(t), whose values are known func-
tions of time.
It is considered that there is no useful information on
values of the unknown parameters θk(t), k = 1,...,p. In
other words, the unknown parameters can be modeled as
Ft-measurable Wiener processes
dθ(t) = β(t)dW3(t),(4)
with unknown initial conditions θ(t0) = θ0∈ Rp, where
(W3(t),Ft,t ≥ t0) is a Wiener process independent of x0,
W1(t), and W2(t), and β(t) ∈ Rp×pis an intensity function.
The quadratic cost function J to be minimized is defined
as follows
J =1
2E[xT(T)Φx(T)+
?T
t0
uT(s)R(s)u(s)ds+
?T
t0
xT(s)L(s)x(s)ds],(5)
where R is positive definite and Φ, L are nonnegative definite
symmetric matrices, T > t0 is a certain time moment, the
symbol E[f(x)] means the expectation (mean) of a function
f of a random variable x, and aTdenotes transpose to a
vector (matrix) a.
The optimal controller problem is to find the control
u∗(t), t ∈ [t0,T], that minimizes the criterion J along with
the unobserved trajectory x∗(t), t ∈ [t0,T], generated upon
substituting u∗(t) into the state equation (1).
B. Problem Reduction
To deal with the stated controller problem, the equations
(1) and (4) should be rearranged. For this purpose, a vector
α0(t) ∈ R(n+p), matrix α1(t) ∈ R(n+p)×(n+p), cubic tensor
α2(t) ∈ R(n+p)×(n+p)×(n+p), and other k+1-dimensional ten-
sors αk(t) ∈ R(n+p)×...(k+1) times...×(n+p), k = 0,...,s+1, are
introduced as follows.
The equation for the i-th component of the state vector is
given by
dxi(t) = (a0 i(θ,t)+∑
k
a1 ik(θ,t)xk(t)
+∑
jk
a2 ijk(θ,t)xj(t)xk(t)+...
+∑
k1...ks
as ik1...ks(θ,t)xk1(t)...xks(t))dt +∑
j
Bij(t)uj(t)dt
+∑
j
Bij(t)γj(t)dt +∑
j
bij(t)dW1j(t),
i, j,k1... ks= 1,...,n, xi(t0) = x0i. Then:
1. If the variable a0i(t) is a known function, then the i-
th component of the vector α0(t) is set to this function,
α0i(t)=a0i(t); otherwise, if the variable a0i(t) is an unknown
function, then the (i,n+i)-th entry of the matrix α1(t) is set
to 1.
2. If the variable a1ij(t) is a known function, then the (i, j)-
th component of the matrix α1(t) is set to this function,
α1ij(t) = a1ij(t); otherwise, if the variable a1ij(t) is an
unknown function, then the (i,n+ p1+k, j)-th entry of the
cubic tensor α2(t) is set to 1, where k is the number of
this current unknown entry in the matrix a1(t), counting the
unknown entries consequently by rows from the first to n-th
entry in each row.
...
3. If the variable as ik1...ks(t) is a known function, then
the (i,k1,...,ks)-th component of the sD tensor αs(t) is set
to this function, αsi,k1,...,ks(t) = as ik1...ks(t); otherwise, if the
variable as ik1...ks(t) is an unknown function, then the (i,n+
ps+k,...,ks)-th entry of the (s+1)D tensor αs+1(t) is set
to 1, where k is the number of this current unknown entry in
the tensor as(t), counting the unknown entries consequently
by rows from the first to s-th dimension and from the first
to n-th entry in each row.
4. All other unassigned entries of the tensors αk(t) ∈
R(n+p)×...(k+1) times...×(n+p), k = 0,...,s+1, are set to 0.
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Using the introduced notation, the state equations (1),(4)
for the vector z(t) = [x(t),θ(t)] ∈ Rn+pcan be rewritten as
dz(t) = g(z,t)dt +[B(t) | 0p×l]u(t)dt +[B(t)γ(t) | 0p×l]dt+
diag[b(t),β(t)]d[WT
1(t),WT
3(t)]T, z(t0) = [x0,θ0],(6)
where the polynomial function g(z,t) is defined as
g(z,t) = α0(t)+α1(t)z+α2(t)zzT+
...+αs+1(t)z...(s+1) times...z.
Thus, the equation (6) is polynomial with respect to the
extended state vector z(t) = [x(t),θ(t)].
C. Optimal Estimate Design
Let us replace the unmeasured bilinear state z(t) =
[x(t),θ(t)], satisfying (6), with its optimal estimate m(t) =
[ˆ x(t),ˆθ(t)] over linear observations y(t) (2), which is ob-
tained using the following optimal filter for polynomial states
over linear observations (see [19] for the corresponding
filtering problem statement and solution)
dm(t) = E(g(z,t) | FY
t)dt +[B(t) | 0p×l]u(t)dt+
[B(t)γ(t) | 0p×l]dt +P(t)[A(t),0m×p]T×(7)
(G(t)GT(t))−1(dy(t)−(A0(t)+A(t)ˆ x(t))dt).
m(t0) = [E(x(t0) | FY
t),E(θ(t0) | FY
t)],
dP(t) = (E((z(t)−m(t))(g(z,t))T| FY
t)+
E(g(z,t)(z(t)−m(t))T) | FY
t)+(8)
diag[b(t),β(t)]diag[b(t),β(t)]T−P(t)[A(t),0m×p]T×
(G(t)GT(t))−1[A(t),0m×p]P(t))dt,
P(t0) = E((z(t0)−m(t0))(z(t0)−m(t0))T| FY
t),
where 0m×pis the m× p - dimensional zero matrix; P(t) is
the conditional variance of the estimation error z(t)−m(t)
with respect to the observations Y(t).
Recall that ˆ z(t) = m(t) = [ˆ x(t),ˆθ(t)] is the optimal esti-
mate for the state vector z(t) = [x(t),θ(t)], based on the
observation process Y(t) = {y(s),t0≤ s ≤t}, that minimizes
the Euclidean 2-norm H = E[(z(t)− ˆ z(t))T(z(t)− ˆ z(t)) | FY
at every time moment t. Here, E[ξ(t) | FY
conditional expectation of a stochastic process ξ(t)=(z(t)−
ˆ z(t))T(z(t)− ˆ z(t)) with respect to the σ - algebra FY
ated by the observation process Y(t) in the interval [t0,t]. As
known [24], this optimal estimate is given by the conditional
expectation ˆ z(t) = m(t) = E(z(t) | FY
z(t) with respect to the σ - algebra FY
observation process Y(t) in the interval [t0,t]. As usual, the
matrix function P(t) = E[(z(t)−m(t))(z(t)−m(t))T| FY
the estimation error variance.
Remark 1. The equations (7) and (8) do not form a
closed system of equations due to the presence of polynomial
terms depending on x, such as E(g(z,t) | FY
m(t))gT(z,t))|FY
t), which are not expressed yet as functions
of the system variables, m(t) and P(t). However, as shown in
t]
t] means the
t gener-
t) of the system state
generated by the
t
t] is
t), and E((z(t)−
[17]-[20], the closed system of the filtering equations can be
obtained for any polynomial state (6) over linear observations
(2), using the technique of representing of superior moments
of the conditionally Gaussian random variable z(t)−m(t) as
functions of only two its lower conditional moments, m(t)
and P(t) (see [17]-[20] for more details of this technique).
Apparently, the polynomial dependence of g(z,t) and (z(t)−
m(t))gT(x,t) on z is the key point making this representation
possible.
D. Optimal control problem solution: Measured state
To handle the optimal control problem for the designed
optimal estimate (8), let us first give the solution to the
general optimal control problem for a polynomial system
with linear control input and a quadratic cost function.
Consider a polynomial system with linear control input
dx(t) = f(x,t)dt +B(t)u(t)dt +b(t)dW1(t), x(t0) = x0,
(11)
where x(t) ∈ Rnis the state vector, u(t) ∈ Rlis the control
input, the polynomial drift function f(x,t) is defined by
f(x,t) = a0(t)+a1(t)x+a2(t)xxT+...+as(t)x...s times...x,
and the assumptions made for the system (1) hold. The
quadratic cost function J to be minimized is defined by (5).
The optimal control problem is to find the control u∗(t), t ∈
[t0,T], that minimizes the criterion J along with the trajectory
x∗(t), t ∈ [t0,T], generated upon substituting u∗(t) into the
state equation (11). The solution to the stated optimal control
problem is given by the following theorem [33].
Theorem 1. The optimal regulator for the polynomial
system (11) with linear control input with respect to the
quadratic criterion (5) is given by the control law
u∗(t) = R−1(t)BT(t)[Q(t)x(t)+ p(t)],(12)
where the matrix function Q(t) is the solution of the Riccati
equation
˙Q(t) = L(t)−[a1(t)+2a2(t)x(t)+3a3(t)x(t)xT(t)+...
(13)
+sas(t)x(t)...s−1 times...x(t)]TQ(t)−
Q(t)[a1(t)+a2(t)x(t)+a3(t)x(t)xT(t)+...
+as(t)x(t)...s−1 times...x(t)]−Q(t)B(t)R−1(t)BT(t)Q(t),
with the terminal condition Q(T) = −ψ, and the vector
function p(t) is the solution of the linear equation
˙ p(t)=−Q(t)a0(t)−[a1(t)+2a2(t)x(t)+3a3(t)x(t)xT(t)+...
+sas(t)x(t)...s−1 times...x(t)]Tp(t)−(14)
Q(t)B(t)R−1(t)BT(t)p(t),
with the terminal condition p(T) = 0. The optimally con-
trolled state of the polynomial system (11) is governed by
the equation
dx(t) = f(x,t)dt +B(t)R−1(t)BT(t)[Q(t)x(t)+ p(t)]dt+
+b(t)dW1(t),
x(t0) = x0.(15)
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E. Optimal controller problem solution: Unmeasured state
Based on the result of Theorem 1, the solution of the
optimal controller problem for the polynomial state (11) over
linear observations (2) with a quadratic criterion (5) is given
as follows [33]. The corresponding optimal control law takes
the form
u∗(t) = R−1(t)BT(t)[Q(t)ˆ x(t)+ p(t)],(16)
where ˆ x(t) = E(x(t) | FY
solution of the Riccati equation
t), the matrix function Q(t) is the
˙Q(t)=L(t)−[c1(t)+2c2(t)ˆ x(t)+3c3(t)ˆ x(t)ˆ xT(t)+... (17)
+scs(t)ˆ x(t)...s−1 times... ˆ x(t)]TQ(t)−
Q(t)[c1(t)+c2(t)ˆ x(t)+c3(t)ˆ x(t)ˆ xT(t)+...
+cs(t)ˆ x(t)...s−1 times... ˆ x(t)]−Q(t)B(t)R−1(t)BT(t)Q(t),
with the terminal condition Q(T) = −ψ, and the vector
function p(t) is the solution of the linear equation
˙ p(t)=−Q(t)c0(t)−[c1(t)+2c2(t)ˆ x(t)+3c3(t)ˆ x(t)ˆ xT(t)+...
+scs(t)ˆ x(t)...s−1 times... ˆ x(t)]Tp(t)−(18)
Q(t)B(t)R−1(t)BT(t)p(t),
with
c0(t),c1(t),...,cs(t) are the coefficients in the representation
of the term E(f(x,t) | FY
t) as a polynomial of ˆ x, that is,
theterminalcondition
p(T) = 0,where
E(f(x,t) | FY
t) = c0(t)+c1(t)ˆ x+c2(t)ˆ xˆ xT+
...+cs(t)ˆ x...s times... ˆ x.
Upon writing down the optimal estimate equation for the
polynomial state (11) over the linear observations (2) (see
[20] and also the equations (7),(8)) and substituting the
optimal control (16) into its right-hand side, the following
optimally controlled state estimate equation is obtained
dˆ x(t)=E(f(x,t)|FY
t)dt+B(t)R−1(t)BT(t)[Q(t)ˆ x(t)+p(t)]dt
(19)
+P(t)AT(t)(G(t)GT(t))−1(dy(t)−(A0(t)+A(t)ˆ x(t))dt).
with the initial condition ˆ x(t0) = E(x(t0) | FY
Thus, the optimally controlled state estimate equation (19),
the gain matrix constituent equations (17) and (18), the
optimal control law (16), and the corresponding variance
equation give the complete solution to the optimal controller
problem for polynomial systems with linear control input and
a quadratic cost function. This solution is not yet written in
a closed form due to non-closeness of the filtering equations
in the general situation; however, as noted in Remark 1, the
closed-form solution can be obtained for any specific form
of the polynomial drift f(x,t) in the equation (11).
t).
F. Solution of optimal controller problem with deterministic
disturbances
The next step is to give the solution to the optimal con-
troller problem for a polynomial stochastic system with linear
control input and a quadratic cost function, that operates
under influence of matched deterministic disturbances (3).
Consider a polynomial system (11) with deterministic
disturbances
dx(t) = f(x,t)dt +B(t)u(t)dt +B(t)γ(t)dt +b(t)dW1(t),
x(t0) = x0,(20)
where the assumptions made for the system (11) hold, and
the quadratic cost function J to be minimized is defined by
(5). The deterministic disturbance γ(t) satisfies the condition
(3). The optimal control problem is to find the control u∗(t),
t ∈ [t0,T], that minimizes the criterion J along with the
trajectory x∗(t), t ∈ [t0,T], generated upon substituting u∗(t)
into the state equation (11).
If the realization of the Wiener process W1(t), affecting
the state (20), and the initial condition x0are exactly known,
the optimal control u∗(t) is represented as [25]
u∗(t) = u∗
0(t)+u∗
1(t).
Here, u∗
(11), which is given by (12), and u∗
mode control u∗
1(t) = −Ksign[s(t)], where the sliding mode
manifold is defined as
0(t) is the optimal control for the nominal system
1(t) is the integral sliding
s(t) = (BT(t)B(t))−1BT(t)(x(t)−
[x0+
?t
t0
(f(x,s)+B(s)u∗
0(s))ds+
?t
t0
b(s)dW1(s)]).
Note that s(t)∈Rl; therefore, sign[s(t)] is defined as a vector:
{sign[s1(t)],...,sign[sl(t)]}.
The key idea is as follows: the sliding mode control u∗
leads the state trajectory to the sliding manifold s(t), where
the deterministic disturbances h(t) = B(t)γ(t) are absent and
the state is therefore regulated optimally by the control
u∗
0(t). If the current realization of the Wiener process W1(t)
and the initial condition x0 are exactly known, then there
is no reaching phase and the sliding mode motion on the
manifold s(t) starts from the initial moment t0(see [25] for
substantiation of the integral sliding mode technique).
If the current realization of the Wiener process W1(t) and
the initial condition x0are unknown and the state x(t) is not
measurable directly but only through the observations (2), the
best estimate of the sliding mode manifold s(t) is given (see
[24]) by its conditional estimate with respect to observations
(2), i.e.,
1(t)
ˆ s(t) = E[s(t) | FY
t] = (BT(t)B(t))−1BT(t)(ˆ x(t)−(21)
[ˆ x(t0)+
?t
t0
(E(f(x,s) | FY
s)+B(s)u∗
00(s))ds]),
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where ˆ x(t) = E[x(t) | FY
corresponding to the case of a directly unmeasurable state
(20). Therefore, the optimal control is given by
t], and u∗
00(s) is the optimal control
u∗(t) = u∗
00(t)+u∗
11(t),(22)
where u∗
surable system (20), which is given by (16), and u∗
−Ksign[ˆ s(t)]. More discussion of the integral sliding mode
technique for unmeasurable systems can be found in [26].
Taking into account the preceding considerations, the
following optimally controlled state estimate equation is
obtained
00(t) is the optimal control for the nominal unmea-
11(t) =
dˆ x(t) = E(f(x,t) | FY
t)dt +B(t)(R(t))−1BT(t)×
[Q(t)ˆ x(t)+ p(t)]dt −Bsign[ˆ s(t)]dt−(23)
P(t)AT(t)(G(t)GT(t))−1(dy(t)−(A0(t)+A(t)ˆ x(t))dt).
with the initial condition ˆ x(t0) = E(x(t0) | FY
given by (21).
t), where ˆ s(t) is
G. Solution of optimal controller problem with deterministic
disturbances and uncertain parameters
Based on the result of the preceding subsections, the
solution of the original optimal controller problem for the
polynomial state (1) with deterministic disturbances (3) and
uncertain parameters θ over linear observations (2) with a
quadratic criterion (5) is given as follows. The corresponding
optimal control law takes the form (22), where the matrix
function Q(t) is the upper left corner of the matrix ¯Q(t) =
diag[Q(t),0p×p], which is the solution of the Riccati equation
˙¯Q(t) = diag[L(t),0p×p]−[γ1(t)+2γ2(t)m(t)+3γ3(t)×
m(t)mT(t)+...+(s+1)γs+1(t)m(t)...s times...m(t)]T¯Q(t)−
(24)
¯Q(t)[γ1(t)+γ2(t)m(t)+γ3(t)m(t)mT(t)+...
+γs+1(t)m(t)...s times...m(t)]−
¯Q(t)[B(t) | 0p×l]R−1(t)[B(t) | 0p×l]T¯Q(t),
with the terminal condition ¯Q(T) = diag[−Φ | 0p×p], and
the vector function p(t) is the upper subvector of the vector
¯ p(t)=[p(t)|0p], which is the solution of the linear equation
˙¯ p(t) = −¯Q(t)γ0(t)−[γ1(t)+2γ2(t)m(t)+3γ3(t)×
m(t)mT(t)+...+(s+1)γs+1(t)m(t)...s times...m(t)]T¯ p(t)−
(25)
¯Q(t)[B(t) | 0p×l]R−1(t)[B(t) | 0p×l]T(t) ¯ p(t),
with
γ0(t),γ1(t),...,γs+1(t)
representation of the term E(g(z,t) | FY
hand side of (6) as a polynomial of m, that is,
the terminalcondition
are
¯ p(T) = 0n+p,
coefficients
t) in the right-
where
the thein
E(g(z,t) | FY
t) = γ0(t)+γ1(t)m+γ2(t)mmT+
...+γs+1(t)m...s+1 times...m.
Upon substituting the optimal control (22) into the equa-
tion (7), the following optimally controlled state estimate
equation is obtained
dm(t) = E(g(z,t) | FY
t)dt +[B(t) | 0p×l]R−1(t)BT(t)× (26)
[Q(t)ˆ x(t)+ p(t)]dt −[B(t) | 0p×l]sign[ˆ s(t)]dt +P(t)×
[A(t),0m×p]T(G(t)GT(t))−1(dy(t)−(A0(t)+A(t)ˆ x(t))dt).
with the initial condition m(t0) = E(x(t0) | FY
is given by (21).
Thus, the optimally controlled state estimate equation
(26), the gain matrix constituent equations (24) and (25),
the optimal control law (22), and the variance equation (8)
give the complete solution to the optimal controller problem
for an uncertain polynomial system (1) with deterministic
disturbances (3) over linear observations (2) and a quadratic
cost function (5). This solution is not yet written in a closed
form due to non-closeness of the filtering equations (7),(8)
in the general situation. In the next subsection, the closed-
form optimal solution is obtained for the particular case of
a third degree polynomial drift g(z,t), which corresponds to
a second degree polynomial function f(x,t).
1) Optimal controller problem solution for third degree
polynomial state: Let the function
t), where ˆ s(t)
g(z,t) = α0(t)+α1(t)z+α2(t)zzT+α3(t)zzzT
(27)
be a third degree polynomial, where z is an (n + p)-
dimensional vector, a0(t) is an (n+ p)-dimensional vector,
a1(t) is a (n+ p)×(n+ p)-dimensional matrix, a2(t) is a
3D tensor of dimension (n+ p)×(n+ p)×(n+ p), a3(t)
is a 4D tensor of dimension (n+ p)×(n+ p)×(n+ p)×
(n+ p). In this case, taking into account the representations
for E(g(z,t) | FY
functions of m(t) and P(t) (see the results obtained in [17]-
[19] for third degree polynomial functions), the following
filtering equations for the optimal estimate m(t) and the error
variance P(t) are obtained
t) and E((z(t) − m(t))(g(z,t))T| FY
t) as
dm(t) = (α0(t)+α1(t)m(t)+α2(t)m(t)mT(t)+α2(t)P(t)+
(28)
3α3(t)P(t)m(t)+α3(t)m(t)m(t)mT(t)+[B(t)|0p×l]u(t))dt+
P(t)[A(t),0m×p]T(G(t)GT(t))−1(dy(t)−(A0(t)+A(t)ˆ x(t))dt),
dP(t) = (α1(t)P(t)+P(t)αT
1(t)+2α2(t)m(t)P(t)+
2(α2(t)m(t)P(t))T+3(α3[P(t)P(t)+P(t)m(t)mT(t)])+
3(α3[P(t)P(t)+P(t)m(t)mT(t)])T+(29)
diag[b(t),β(t)]diag[b(t),β(t)]T−
P(t)[A(t),0m×p]T(G(t)GT(t))−1[A(t),0m×p]P(t))dt,
with the same initial conditions as in (7),(8).
Taking into account the representation (24): γ0(t) =
α0(t)+α2(t)P(t), γ1(t) = α1+3α3(t)P(t)(t), γ2(t) = α2(t),
γ3(t) = α3(t), the equations (24) and (25) take the following
particular forms in the case of a third degree polynomial (27)
˙¯Q(t) = diag[L(t),0p×p]−[α1(t)+3α3(t)P(t)+
782