National Instruments 370753C-01

Chapter 6 State-Space Design

For continuous-time systems,

the quadratic performance index takes the form:

For the discrete case where the system is defined as a multistage process:

the performance index is defined similarly except that a summation sign

replaces the integral of the preceding quadratic performance index

equation.

Rxx is a real, symmetric, positive-semidefinite matrix indicating the

weighting of the cost on the elements of the state vector x. Ruu is a real,

symmetric, positive-definite matrix indicating the weighting of the cost on

the control inputs given by the vector u. Rxu is a real matrix indicating the

cross-weighting of the cost between states and inputs; for many

applications, it will consist of all zeros if the control and states are

uncorrelated.

Bryson and Ho showed in [BH75] that the optimal control which

minimized this quadratic performance index is a linear feedback

combination of the states, u=Krx, for both the continuous and discrete

cases.

For the continuous case, Kr is defined as follows, with P solving the

continuous-time Riccati equation:

and for the discrete case, P solves the discrete Riccati equation.

·Ax Bu+=

Jx't()u't() Rxx Rxu

Rxu'Ruu

xt()

ut()dt

∞

∫

xk1+AxkBuk

KrRuu

1–B'PR

+()=

Rxx PA A′PPBR

+()Ruu

1–Rxu′B′P+()–++ 0=

A'PA A'PB Rxu

+()Ruu B'PB+()

1–B'PA R′xu

+()–Rxx

+P=

KrRuu B'PB+()

1–B'PA Rxu

+()=