and Reconfigurable Control

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The two open-loop linear system models are given by Type 1, 2nd-order transfer functions.
These transfer functions are:

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Two scenarios will be considered. Scenario A will consider p_{0} to be the nominal
system and p_{1} to be a perturbation. Stability domains will be defined for this
condition, and a simultaneously stabilizing compensator will be designed for it.

Scenario B will consider p_{1} to be the nominal system and p_{0} to be a perturbation.
Stability domains will be defined for this condition also, and a second
simultaneously stabilizing compensator will be designed.

The two plant models represent the same physical system in two different operating conditions. Closed-loop stability is to be guaranteed for both systems. The stability domains are selected to provide good performance as well as stability for the nominal system and allow degraded performance but not loss of stability for the perturbed system. For this example, the stability domains are chosen rather arbitrarily.

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The starting point will be to design a compensator that will stabilize plant p_{0}
with respect to domain D_{0a}, which is the domain for p_{0} in scenario A. A simple
phase lead compensator is designed using root locus to place a dominant closed-loop pole at
s = -1.8 + j1.6. No restrictions are placed on the structure of the compensator
or on the closed-loop stabilty of plant p_{1} with this compensator. The initial
compensator for p_{0} is

The figure of the closed-loop poles for p_{0} and
p_{1}
when p_{0} is considered the nominal plant shows that this compensator does
stabilize p_{0} with respect to D_{0a}. The closed-loop poles of
p_{0} are shown as triangles, and the closed-loop poles for p_{1}
are shown as squares. Plant p_{1} is also stabilized, but not entirely
with respect to D_{1a}. If p_{1} had been stabilized with respect to
D_{1a}, then the
design process for scenario A would have been completed at this stage.

` `
Before continuing with the design process for scenario A, we will design the
initial compensator for plant p_{1} which will stabilize it with respect to
stability domain D_{1b}, which is the domain for p_{1} in scenario B. Again a root
locus design method is used to place a dominant closed-loop pole, this time at
s = -0.8 + j0.7. A compensator that accomplishes this is

The plot of the closed-loop poles for p1 and p0
when p1 is considered the nominal plant shows that this compensator does
stabilize p_{1} with respect to D_{1b}. Again, the closed-loop
poles of p_{0} are shown as triangles, and the closed-loop poles for p_{1}
are shown as squares. Plant p_{0} is also stabilized, but not entirely
with respect to D_{0b}. If p_{0} had been stabilized with respect to
D_{0b}, then the
design process for scenario B would have been completed at this stage.

` `
The design steps for scenario A will now be undertaken. The associated system p_{01}
will be formed from the coprime factorizations of the original system models and
the initial compensator c_{0}. This associated system satisfies the extended parity interlacing property
(e.p.i.p.)
with respect to domain D_{1a}, which is a necessary condition for the problem to be solved.
In order to simultaneously stabilize p_{0} and p_{1} with respect to domains
D_{0a} and D_{1a},
it is necessary and sufficient that p_{01} be stabilized with respect to
D_{1a} by a compensator
that is stable with respect to the intersection of D_{0a} and D_{1a}, which in this case is
D_{0a}.
This is termed the Double-D stabilization problem. A compensator that accomplishes this is

The closed-loop poles for (r_{0a}, p_{01a}), shown
by the triangles, are seen to lie inside D_{1a}, and the compensator poles, shown in
by the x, are in D_{0a}.
Thus, the Double-D stabilization problem has been solved. This compensator, r_{0}, will be now
used as the parameter in the Youla parametrization with the initial compensator
c_{0} in order
to get the final compensator for scenario A. This final compensator is

` `
The final closed-loop poles for p_{0} (triangles) and
p_{1} (squares) are seen to be in their respective stability regions. The
step responses of these compensated systems
indicate stability for both systems, and the performance of p_{0} will be judged
as acceptable for this example. We will assume that the overshoot for p_{1} is
too large.

` `
The same procedure is used to design the final compensator for scenario B. First
the associated system p_{01} is formed (actually, it should be called p_{10}), and a compensator that
is stable with respect to D_{1b} is found to stabilize the associated system with
respect to D_{0b}. The e.p.i.p. for this system
is satisfied, and the following compensator provides the stabilization (the {pole, zero}
pair at {-3.6, -3.7} is only used to provide a compensator as the same order as the one
in Scenario A for simulation convenience; it is not need to stabilize the associated system):

The closed-loop poles for (r_{1b}, p_{01b}),
shown by the triangles, are seen
to lie entirely in domain D_{0b}, and the compensator poles (shown by
the x's) are seen to be in D_{1b}.
Therefore, the Double-D stabilization problem for scenario B has been satisfied.

` `
The parameter r_{1} is used in the Youla parametrization with the initial
c_{1} compensator
to obtain the final compensator for scenario B. This compensator is

` `
The final closed-loop poles for p_{0} (triangles) and
p_{1} (squares) are seen to be in their respective stability regions. The
step responses of these compensated systems
indicate stability for both systems, and the performance of p_{1} will be judged
as acceptable for this example. We will assume that the settling time for p_{0}
is too long.

` `
Each of these two final compensators provides closed-loop stability for both system
models and acceptable performance for the corresponding nominal model. System identification
or failure detection procedures may be used to determine which of the two models is
correct, and the appropriate compensator may then be selected. Even if the wrong
compensator is being used, closed-loop stability is not lost.

` `
To show the stability robustness of the controllers and the performance improvement
that reconfiguration provides, a MATLAB simulation was performed. The controller and
system model were in cascade with unity feedback. The reference input was a square wave
with a period of 12 seconds. The initial configuration was controller c_{0} and plant
p_{0}.
At 27 seconds, the plant was switched to p_{1}, representing a change in operating conditions
or component values. At 50 seconds, the controller was switched to c_{1}, representing the
result of failure detection or system identification. At 77 seconds the plant switched
back to p_{0}, and at 99 seconds the controller was switched back to c_{0}.

These switching times are not meant to model an actual situation. They do provide an illustration that closed-loop stability is maintained even when the incorrect controller is being used. The simulation also shows the performance improvement that results when the correct controller is switched in. In the following time response plot the vertical dotted lines indicate the switching times. The current {controller, plant} pair is shown between the lines at the top of the figure. State space models were used to represent the plant and controller. The models were in observer canonical form so that the output signal would be continuous at the instant a switch in the model occurred.

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Some references that apply to the techniques used in this example are:

- Ammeen, Edward S., "A New Simultaneous Stabilization Approach Using an Iterative
Combining of System Models for the Design of Robust Control Algorithms", Ph.D.
Dissertation, George Mason University, January 1995.
- Ammeen, E.S., Gibson, J.D., and Beale, G.O., "Robust Depth-Keeping by Means of Simultaneous
Stabilization,"
*Proc. of 4th IFAC Conference on Maneuvering and Control of Marine Craft*, September 10-12, 1997, pp. 42-47, Brijuni, Croatia. - Arteaga-Bravo, Francisco J., "Simultaneous Stabilization with Multiple Bounded
Domains of Stability", Ph.D. Dissertation, George Mason University, May 1995.
- Beale, G.O. and Arteaga, F.J., "Simultaneous Stabilization with Multiple Bounded Stability
Domains,"
*(KoREMA)Automatika*, 37(1996) 3-4, pp. 91-98, May 1997 (selected from KoREMA '96 as an original scientific paper). - Gibson, James D., "A Solution to the Double-D Stabilization Problem", Ph.D.
Dissertation, George Mason University, May 1997.
- Gibson, J.D. and Beale, G.O., "Stabilizing a Nonlinear System Over a Range of Operating
Points,"
*Proc. of 4th IFAC Conference on Maneuvering and Control of Marine Craft*, September 10-12, 1997, pp. 65-68 Brijuni, Croatia.

- Gibson, J.D. and Beale, G.O., “A Geometric Solution to the Simultaneous
Bounded Domain Stabilization Problem,”
*International Journal of Control*, Vol. 73, No. 17, pp. 1536-1547, 2000.

- Arteaga-Bravo, F., Beale, G.O., and Montilla, O.,
“Multiple
Bounded Domains for Simultaneous Stabilization: Interpolation Conditions for Two
Plants,”
*SIAM Conference on Linear Algebra in Signals, Systems and Control*, August 11-15, 2001, Boston, MA.

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