Energy balance equations

We begin with notation for analyzing the DISSIPATIVITY of the systems obtained by connecting f, g to F, G in several different ways. The energy function on the statespace is denoted by . HWUY below is the preHamiltonian of the two decoupled systems where inputs are W, U and Y.

H W U Y = out - W² + P f(z,Y) + p F(x,W,U)

SetNonCommutative[F,G1,G2,f,g,p,P];
HWUY = tp[out1]**out1-tp[W]**W+
                       (p**F[x,W,U]+tp[F[x,W,U]]**tp[p])/2+
                                     (P**f[z,Y]+tp[f[z,Y]]**tp[P])/2;

Connecting the two systems in feedback, that is tieing off U and Y, mathematically means

Y G(x,W,U)         U g(z,Y).

and the preHamiltonian with this substitution becomes H

H=HWUY/. {Y -> G2(x,W,U),   U -> g(z,Y)}

By definition the closed loop system being DISSIPATIVE corresponds to the energy balance function H above being negative. This is the same as

Ham := max H  0

We refer to Ham as the Hamiltonian for the problem. Since H is quadratic in W we can optimize W by hand and write down a formula for Ham. The main problem becomes:

( CNTRL) Is there a system a, b, c, d, so that Ham 0 has a solution 0 ?

Theory (see [PAJ]) tells us that (CNTRL) has a positive solution only if

( CNTRL=) There is a system a, b, c, d, so that Ham = 0 has a solution 0.

Note that the function Ham contains both state and dual variables. Dual variables are defined by the gradient of the energy function, as follows:

In a moment we shall write down Ham explicitly, but first we shall make assumptions to simplify the exposition. They are not essential for the demo. The outline here should work for nonlinear as well as linear systems and at a high level of generality:

d[z]=0

The system is linear.

The storage function is quadratic. Thus it corresponds to the

We now write out energy balance formulas for the plant (1.4) and compensator (1.5) (under these assumptions) purely in terms of state space variables x and z (designated by the prefix s for state space). This can be done in SYStems using a command which specializes systems to the linear case. Ham is a quadratic form on the statespace (x,z) of the closed loop system and so it can be expressed by its components , , , with respect to the x statespace and z statespace. We run NCAlgebra to find that these are the polynomials defined in §. }