For infinite time problem, the performance index is

$\displaystyle J = \sum\limits_{k = {N_0}}^{{\infty}} {\left[ {{\boldsymbol{x}^T}(k)Q\boldsymbol{x}(k) + \boldsymbol{u}^T(k)R\boldsymbol{u}(k)} \right]}$

In most cases, N₀ is considered to be 0 .

Example: Consider the dynamical system

Suppose that we want to minimize the output as well as the input with equal weightage along the convergence trajectory. Construct the associated performance index.

Since the initial condition of the system is x(0)=x₀ and we have to minimize the performance index over the whole convergence trajectory, we need to take summation from 0 to ∞.

Again, since the output and input are to be minimized with equal weightage, we can write the cost function or performance index as

Comparing with the standard cost function, we can say that here $Q=\begin{bmatrix}4 & 0 \\ 0 & 0 \end{bmatrix}$ and R=1.

In the next lecture we will discuss design of Linear Quadratic Regulator (LQR) by solving Algebraic Riccati Equation (ARE). To derive ARE, we need the following theorem.

Consider the system

$\displaystyle \boldsymbol{x}(k + 1) = A\boldsymbol{x}(k) + B\boldsymbol{u}(k)$

where $\boldsymbol{x}(k) \in {R^n}$ , $\boldsymbol{u}(k) \in {R^m}$ and $\boldsymbol{x}(0) = {\boldsymbol{x}_0}$ .

Theorem 1: If the state feedback controller $\boldsymbol{u}^*(k) = - K\boldsymbol{x}(k)$ is such that

for some Lyapunov function $V(k)=\boldsymbol{x}^T(k) P \boldsymbol{x}(k)$ , then $\boldsymbol{u}^*(k)$ is optimal. Here the cost

function is

$\displaystyle J(u) = \sum\limits_{k = 0}^\infty ({\boldsymbol{x}^T}(k)Q\boldsymbol{x}(k) + {\boldsymbol{u}^T}(k)R\boldsymbol{u}(k))$

and we assume that the closed loop system is asymptotically stable.