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APR_E1.txt
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APR_E1.txt
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TECHNISCHE UNIVERSITAT MUNCHEN
LEHRSTUHL FUR INFORMATIONSTECHNISCHE REGELUNG
ORDINARIA: UNIV.-PROF. DR.-ING. SANDRA HIRCHE
Adaptive and Predictive Control (APC)
Dr. Stefan Sosnowski / Denis Cehaji´
SS2014
14.04.2014.
EXERCISE 1
Problem 1: Parameter Identification
Consider the stationary process described by y(k) = c1 + c2 u(k) + e(k) where c1 ∈ R and c2 ∈ R
are unknown parameters, u(k) ∈ R and y(k) ∈ R are the input and output states respectively, and
e(k) ∈ R is an external disturbance. The figure below illustrates the relation between u(k) and y(k)
for k = 1, . . . , 5 and the table provides their numerical values. The unknown parameters c1 and c2
have to be found.
y(k)
u(k)
1 2 3 4 5
1 3 3 6 7
a) The least-squares estimation (LS) technique uses a loss function V (θ, k) which is minimal for a
vector of unknown parameters θ in sense of least squares. Derive Y , Φ, and θ necessary for
constructing V (θ, k).
b) Derive a general solution to minimize V (θ, k). Using the obtained expression, solve for θ using the
numerical values given in the table above.
c) A new set of data u(6) = 6 and y(6) = 7 is added. Using the recursive least-square estimation
(RLS) formulas, calculate the new estimate of the parameter vector θ(k).
d) Consider the following discrete-time dynamic system y(k) = a1 u(k − 1) + a2 u(k − 2) with unknown
parameters a1 and a2 . Represent this system in the same form as the stationary process given above
in order to solve the identification problem.
Problem 2: Discrete-Time Control Systems
Given a continuous-time linear control system, described by the ordinary differential equation (ODE)
![(1)](140605093654.png)
where yc , uc ∈ R are the system output and the control input respectively, and a, b ∈ R are constants,
its discrete-time representation, with a sampling interval T , becomes
y(k + 1) = cy(k) + du(k)
with constants c, d ∈ R. Here, y(k) is used to denote the value of y(t) at the k-th instant.
a) Determine the constants c, d so that the continuous-time system (1) is accurately represented by
the discrete model (2), or, so that yc (kT ) = y(k). How are the eigenvalues of the discretized model
related to the eigenvalues of the continuous model, and what can be concluded about stability?
b) For the dynamic system (2) with feedback u(k) = −Kp y(k), can the unknown parameters c, d be
found uniquely? Assume that y(k) ∀ k, and Kp are known.
Problem 3: Regressor Models
Write the following systems in the form of a regressor model y(t) = ϕ(t)T u(t).
a) Continuous-time transfer function G(s)
Y (s)
U (s)
s + as + b
where y(t), u(t) ∈ R are the system output and the control input respectively, and a, b, c ∈ R
are unknown constants. Assume that the derivatives of the system output up to the order 2 are
available.
\todo{What can be done, if only y(t) and u(t) are available, but not their derivatives?}
b) Non-linear system
x(k + 1) = −ax(k)2 + 2bsin(u(k))
with unknown parameters a, b ∈ R.
c) State space model
d x1
dt x2
with unknown parameters a, b, c ∈ R.