‫دانشگاه صنعتي اميركبير‬
‫دانشكده مهندس ي پزشكي‬
‫‪Constraints in MPC‬‬
‫کنترل پيشبين‪-‬دکترتوحيدخواه‬
Feasibility in MPC
•Infeasibility implies that, for the current state, the
constraints within the MPC algorithm cannot be
satisfied.
•Without feasibility the MPC optimisation is ill posed and
there is no assurance that the answer has any useful
meaning.
•MPC looks at mechanisms for overcoming or avoiding
this.
Hard constraints
Hard constraints are constraints which must be satisfied.
Soft constraints
• Soft constraints are those which should be satisfied if
possible. Soft constraints can be violated (ignored).
• Usually soft constraints are on outputs/states although
they could also be applied to inputs. Such violations may
have no effect on nominal stability results.
Terminal constraints
Summary:
Constraints are a combination of hard, soft and
terminal constraints.
Model uncertainty
Model uncertainty may cause infeasibility because
the actual behaviour differs from the predicted
behaviour. Hence, even though the nominal
predictions could satisfy constraints over the entire
future, a small change in the model will cause the
actual behaviour to differ, and the associated
predictions at the subsequent sampling instant
could then violate constraints.
Model uncertainty (cont.)
One can form algorithms based on invariant sets
(Section 11.11) to handle model uncertainty; however,
the results are usually very conservative, as
guarantees must allow for the worst case (which will
arise with negligible probability).
A more pragmatic approach is to accept that
guarantees cannot be given where there is significant
uncertainty and make other contingencies for the rare
occasions where infeasibility arises.
•The stronger a guarantee you want, the more
conservative your control law will be.
•In practise there must be a compromise between
feasibility assurances and performance.
•Feasibility would usually be ensured by a systematic
relaxation of soft constraints. This would be
determined at a supervisory level.
•Using artificially tight constraints on future
predictions automatically builds in some slack which
can be used to retain feasibility in the presence of
moderate uncertainty. The slack should be
montonically increasing with the horizon.
Example: input limits:
Example 1. A mathematical model for an undamped
oscillator is given by:
What happens if the control amplitude is limited to
+/- 25 ?
closed-loop eigenvalues are at −0.1946, 0, 0.
Case A. Closed loop control without saturation
Case B- Closed-loop control with saturation
If we do not pay attention to the saturation of the
control, then in the presence of constraints, the
closed-loop control performance could severely
deteriorate.
Example 2:
A common practice in dealing with saturation is to let
the model know the difference in Δu(k) when
saturation becomes effective.
over-shoot in the closed-loop response is significantly reduced.
Formulation of Constrained Control Problems
There are three major types of constraints frequently
encountered in applications:
The first two types deal with constraints imposed on
the control variables u(k),
and the third type of constraint deals with output y(k)
or state variable x(k) constraints.
Constraints on the Amplitude of the
Control Variable
Constraints on the Control Variable
Incremental Variation
:‫در بيهوش ی بعنوان مثال‬
Output Constraints
Output constraints are often implemented as ‘soft’
constraints in the way that a slack variable sv > 0
is added to the constraints
Constraints in a Multi-input and
Multi-output Setting
Constraints as Part of the Optimal Solution
Numerical Solutions Using Quadratic Programming
E is assumed to be symmetric and positive definite.
Quadratic Programming for Equality Constraints
Example 4. Minimize
Solution. The global minimum, without
constraint, is at
Illustration of constrained optimal solution
Lagrange Multipliers
Example 5. Minimize
subject to:
Solution:
Without the equality constraints, the optimal solution is:
Example 6. what happens to the constrained optimal
solution when the linear constraints are dependent.
There is no feasible solution of x1 and x2
Matrix MTE−1M is not invertible
Solution:
Illustration of no feasible solution of the constrained optimization problem.
Solid-line x1 + x2 = 1; darker-solid-line 2x1 +2x2 = 6
Example 7:
How the number of equality constraints is also an
issue in the constrained optimal solution? (Ex. 5)
We add an extra constraint to the original constraints so that:
The only feasible solution:
In summary, the number of equality constraints is
required to be less than or equal to the number of
decision variables (i.e., x).
If the number of equality constraints equals the
number of decision variables, the only feasible solution
is the one that satisfies the constraints and there is no
additional variable in x that can be used to optimize
the original objective function.
Minimization with Inequality Constraints
In the minimization with inequality constraints, the number
of constraints could be larger than the number of decision
variables.
An inequality Mix ≤ γi is said to be active if Mix = γi
and inactive if Mix < γi.
Kuhn-Tucker Conditions
Example 8.
Illustration of the constrained optimization problem with inequality
constraints.Solid-line x1 + x2 = 1; darker-solid-line 3x1 +3x2 = 6
Active Set Method
Example 9
The third constraint is an inactive constraint and will
be dropped from the constrained equation set.
The first two constraints as the active constraints,
and solve the optimization problem as minimizing:
subject to:
From Example 5:
We drop the second constraint and solve the
optimization problem as:
subject to:
1.
In the case of equality constraints, the maximum number
of equality constraints equals the number of decision
variables. In this example, it is 3, and the only feasible
solution x is to satisfy the equality constraints (see (2.50)).
In contrast, in the case of inequality constraints, the
number of inequality constraints is permitted to be larger
than the number of decision variables, as long as they are
not all active. In this example, only one constraint
becomes active so it becomes an equality constraint.
Once the optimal solution is found against this active
constraint, the rest of the inequalities are automatically
satisfied.
2.
It is clear that an iterative procedure is required to solve
the optimization problem with inequality constraints,
because we did not know which constraints would become
active constraints. If the active set could be identified in
advance, then the iterative procedure would be shortened.
3.
Note that the conditions for the inequality constraints
are more relaxed than the case of imposing equality
constraints. For instance, the number of constraints is
permitted to be greater than the number of decision
variables, and the set of inequality constraints is
permitted to be linearly dependent. However, these
relaxations are only permitted to the point that the
active constraints need to be linearly independent and
the number of active constraints needs to be less than
or equal to the number of decision variables.
Primal-Dual Method
Hildreth’s Quadratic Programming Procedure
Example 10. Minimize the cost function:
Example 11