# Simultaneous Stabilization Design Example

## Multiple Bounded Stability Domains and Reconfigurable Control

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

Two scenarios will be considered. Scenario A will consider p0 to be the nominal system and p1 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 p1 to be the nominal system and p0 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.

The starting point will be to design a compensator that will stabilize plant p0 with respect to domain D0a, which is the domain for p0 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 p1 with this compensator. The initial compensator for p0 is

The figure of the closed-loop poles for p0 and p1 when p0 is considered the nominal plant shows that this compensator does stabilize p0 with respect to D0a. The closed-loop poles of p0 are shown as triangles, and the closed-loop poles for p1 are shown as squares. Plant p1 is also stabilized, but not entirely with respect to D1a. If p1 had been stabilized with respect to D1a, 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 p1 which will stabilize it with respect to stability domain D1b, which is the domain for p1 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 p1 with respect to D1b. Again, the closed-loop poles of p0 are shown as triangles, and the closed-loop poles for p1 are shown as squares. Plant p0 is also stabilized, but not entirely with respect to D0b. If p0 had been stabilized with respect to D0b, 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 p01 will be formed from the coprime factorizations of the original system models and the initial compensator c0. This associated system satisfies the extended parity interlacing property (e.p.i.p.) with respect to domain D1a, which is a necessary condition for the problem to be solved. In order to simultaneously stabilize p0 and p1 with respect to domains D0a and D1a, it is necessary and sufficient that p01 be stabilized with respect to D1a by a compensator that is stable with respect to the intersection of D0a and D1a, which in this case is D0a. This is termed the Double-D stabilization problem. A compensator that accomplishes this is

The closed-loop poles for (r0a, p01a), shown by the triangles, are seen to lie inside D1a, and the compensator poles, shown in by the x, are in D0a. Thus, the Double-D stabilization problem has been solved. This compensator, r0, will be now used as the parameter in the Youla parametrization with the initial compensator c0 in order to get the final compensator for scenario A. This final compensator is

The final closed-loop poles for p0 (triangles) and p1 (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 p0 will be judged as acceptable for this example. We will assume that the overshoot for p1 is too large.

The same procedure is used to design the final compensator for scenario B. First the associated system p01 is formed (actually, it should be called p10), and a compensator that is stable with respect to D1b is found to stabilize the associated system with respect to D0b. 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 (r1b, p01b), shown by the triangles, are seen to lie entirely in domain D0b, and the compensator poles (shown by the x's) are seen to be in D1b. Therefore, the Double-D stabilization problem for scenario B has been satisfied.

The parameter r1 is used in the Youla parametrization with the initial c1 compensator to obtain the final compensator for scenario B. This compensator is

The final closed-loop poles for p0 (triangles) and p1 (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 p1 will be judged as acceptable for this example. We will assume that the settling time for p0 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 c0 and plant p0. At 27 seconds, the plant was switched to p1, representing a change in operating conditions or component values. At 50 seconds, the controller was switched to c1, representing the result of failure detection or system identification. At 77 seconds the plant switched back to p0, and at 99 seconds the controller was switched back to c0.

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.

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

1. 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.

2. 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.

3. Arteaga-Bravo, Francisco J., "Simultaneous Stabilization with Multiple Bounded Domains of Stability", Ph.D. Dissertation, George Mason University, May 1995.

4. 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).

5. Gibson, James D., "A Solution to the Double-D Stabilization Problem", Ph.D. Dissertation, George Mason University, May 1997.

6. 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.

7. 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.

8. 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.