‫دانشگاه صنعتي اميركبير‬
‫دانشكده مهندس ي پزشكي‬
‫‪MPC Stability-1‬‬
‫استاد درس‬
‫دكتر فرزاد توحيدخواه‬
‫بهمن ‪1389‬‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
Discrete-time MPC with Prescribed
Degree of Stability
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Finite Prediction Horizon: Re-visited
Example 4.1.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Condition number of the Hessian matrix increases
as the prediction horizon Np increases.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Origin of the Problem
Using Laguerre functions (for real time):
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
When there is an integrator in the system matrix A, the
norms of the matrix power ||Am|| and the convolution
sum ||φ(m)|| do not decay to zero, as m increases.
Thus, the magnitudes of the elements in Ω increase as
the prediction horizon Np increases. Hence, if the
prediction horizon Np is large, a numerical conditioning
problem occurs. This problem exists in the majority of
the classical predictive controllers formulations,
including GPC and DMC.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Traditional solution (previous chapter):
Use of an inner-loop state feedback stablization that
may compromise the closed-loop performance when
constraints become active, or the use of prediction
horizon Np and control horizon Nc as the tuning
parameters
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Idea basis:
For a large Np, a large number is divided by another
large number. This numerical problem becomes severe
when the plant model itself is unstable, or when the
dimension of the matrix A is large.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Solution:
1- Improving the numerical condition of MPC
algorithms without guaranteeing closed-loop stability.
2- Asymptotic stability
3- Create a prescribed degree of closed-loop stability
for the predictive control algorithm.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Use of Exponential Data Weighting
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Continuous-time (in the LQR design):
eλt
Discrete-time:
{αj, j = 0, 1, 2 . . .},
α = eλt with t being the sampling interval.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Cost Function:
α = 1 the cost function becomes identical to the
traditional cost function.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Exponentially Increasing Weight (α < 1):
Exponential weights α−2j , j = 1, 2, . . . ,Np, deemphasizes the state x(ki + j | ki) at the current
time and places emphasis on those at the future
time.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Exponentially Decreasing Weight (α >1):
Exponential weights α−2j , j = 1, 2, . . . ,Np, more
emphasizes the state x(ki + j | ki) at the current
time and less emphasis on those at the future
time.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Optimization of Exponentially Weighted
Cost Function
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
Weighted incremental control:
Weighted state variable:
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Theorem 4.1. The minimum solution of the exponentially
weighted cost function J can be found by minimizing:
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Example 4.2.
Consider the same double-integrator system given in
Example 4.1. Examine how the parameter α used in the
weighting affects the numerical condition and closed-loop
control performance with constraints on the amplitude of the
control signal as (only impose constraints on the first sample
of the control)
α = 1/1.2 (exponentially increasing weight), α = 1 (no
exponential weighting) and α = 1.2 (exponentially
decreasing weighting)
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
1- with exponentially increasing weighting, the Hessian
matrix is poorly conditioned even for short prediction
horizon;
2- without exponential weighting the condition number
increases rapidly as the prediction horizon increases.
3- with exponentially decreasing data weighting, the
condition number converges to a finite value and is
much smaller than the one obtained without using
exponential weighting.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Obviously, it is not feasible to use exponentially
increasing weighting in this context, as the numerical
condition rapidly deteriorates as prediction horizon
increases, when α < 1.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Interpretation of Results from Exponential Weighting
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
The key point is that by transforming the exponentially
weighted cost function to the traditional cost function,
the augmented state-space model:
maximum modulus
of all eigenvalues < 1
If
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
With this simple modification, intuitively we
understand that there is no guarantee on the closedloop stability with an arbitrary choice of α > 1.
However, when α is chosen to be slightly larger than
one for the class of stable plants with embedded
integrator, the closed-loop predictive systems are
often found to be stable with Q = CTC and a diagonal
R matrix with small positive elements.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
For the first time, the prediction horizon Np can be
selected to be sufficiently large to approximate the
infinite prediction horizon case. Thus with Q ≥ 0 and
R > 0, and sufficiently large (Np→∞), minimizing
is equivalent to the discrete-time linear quadratic
regulator (DLQR) problem.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
The traditional DLQR problem is solved using the
algebraic Riccati equation
controllable
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
observable
‫‪closed-loop system:‬‬
‫‪Because‬‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
if
closed-loop system is stable.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Second method:
For stability
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
By choosing α > 1, there is no guarantee that the
closed-loop of the original system will be stable. But, if
α is chosen to be slightly larger than unity, then
the closed-loop system A−BK would often be stable.
Indeed, a large number of simulation tests show that
this simple modification usually produces a stable
closed-loop system, if the unstable modes from the
augmented model come from the embedded
integrators. However, a proper choice of the weight
matrices Q and R is important to create the degree of
stability 1 − ε for the transformed system.
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Asymptotic Closed-loop Stability
with Exponential Weighting
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Modification of Q and R Matrices
Basic idea:
The exponentially decreasing weight α > 1 increased
the magnitudes of the actual closed-loop eigenvalues
by the α factor. If the new Q and R matrices are
selected to decrease the magnitudes of the eigenvalues of the exponentially weighted system by a
factor of α−1, then the magnitudes of the actual
closed-loop eigenvalues become unchanged
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
‫‪Theorem 4.2.‬‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
Interpretation of the Results
The essence of the results lies in the fact that the two
cost functions lead to the same optimal control.
However, the commonly used cost function is limited to
a finite prediction horizon for the class of predictive
control algorithms that have embedded integrators. In
contrast, the exponentially weighted cost function
removes the problem because the model used in the
prediction is modified to be stable using the factor α. As
a result, the prediction horizon Np can be selected to
be sufficiently large without numerical problems.
Hence, asymptotic closed-loop stability is guaranteed
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
Example 4.3.
Consider the simple double-integrator system described in
4.1
Design a MPC with an integrator for disturbance rejection,
Calculate the closed-loop eigenvalues, gain matrix via the
cost function using exponential data weighting with α = 1.6
and compare the results with the case without weighting (α
= 1)
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
With exponential data weighting
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
Without exponential data weighting (α = 1)
‫دکترتوحيدخواه‬-‫کنترل پيشبين‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫‪MIMO system‬‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫‪Example 4.4.‬‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬