MINIMUM ENERGY CONTROL OF LINEAR TIME-INVARIANT SYSTEMS WITH INPUT DERIVATIVES

Linear time-invariant systems of the form dz/dt = Az + B0u + B1u(1) ... + B(k)u(k), u(i) = d(i)u/dt(i) are considered, and the problem is as follows. Find a control vector u(t) that will drive the state vector z(t) from a fixed z(0) to a prescribed z(t(f)) in some fixed final time tf while minimizing a cost functional of the form J = 1/2 integral-tf/0 (u(T) R0u + u(1)T R1u(1) + ... + u(j)T R(j)u(j)) dt First, the state-variable system is re-written in the descriptor-variable form diag {I(rho), 0} dx/dt = Fx + [0, I(mu)]u where the order of diag {I(rho), 0} is rho + mu. Then, using a variational method developed previously, a formula is obtained that yields a set of input vectors making the first variation of J zero. Substitution of this input formula into J leads to a scalar-valued functional of several scalar variables that is minimized by the standard Lagrange-multiplier method. The results of the paper may also be used to find inputs that satisfy some prescribed boundary conditions while minimizing J.