interc (Robust Control Toolbox)

General multivariable interconnected system.

[acl,bcl,ccl,dcl] = interc(a,b,c,d,m,n,f)

[sscl] = interc(ss,m,n,f)

Description

interc computes state-space realization of a multivariable interconnected system closed loop, given system

and constant blocks M, N, and F representing the interconnections (see block diagram).

Figure 1-9: General MIMO Interconnection.

The resulting system closed-loop is

where X = (I - DF)^-1.

Examples

Consider a system with three subsystems (G₁, G₂, G₃) where each of the subsystems has its own state-space representation (A_x, B_x, C_x, D_x). If the overall system is interconnected as shown in the Figure 1-10, then

and the associated constant blocks M, N, F and for this problem are

Figure 1-10: Example of MIMO Interconnection.

Using the following MATLAB commands, you can easily get a state-space realization of P(s) as

[AA,BB,CC,DD] = append(A1,B1,C1,D1,A2,B2,C2,D2); 
[AA,BB,CC,DD] = append(AA,BB,CC,DD,A3,B3,C3,D3);

Then, the state-space representation (Acl, Bcl, Ccl, Dcl) of the whole system from u

y is

 [Acl,Bcl,Ccl,Dcl] = interc(AA,BB,CC,DD,M,N,F);

Note that the resulting system is not necessarily minimal; for example, pole-zero cancellations that arise as a result of the interconnection lead to such a nonminimal state-space realization. Model reduction routines such as minreal, schmr, ohklmr, or bstschmr can be helpful in removing any uncontrollable and/or unobservable modes.