Fall 2001 163119-2 So supu G lw)< y iff H has no eigenvalues on the ju-axis An equivalent test is if there exists a X >0 such that AX+XA+CC+-XBBX=0 and A+BBX is stable This is an Algebraic Riccati Equation(ARE) Typical appplication: since e =y-r, then for perfect tracking, we want e≈0 → want S≈0 sInce e=-Sr+ Sufficient to discuss the magnitude of s because the only require- ment is that it be small Direct approach is to develop an upperbound for S and then test if S is below this bound Sgo or equivalently, whether Ws(ja)S Gu)< 1, Ww Note: The state-space tests can also be used for MIMo systems but in that case, we need diffferent Frequency Domain testsFall 2001 16.31 19–2 • So supω |G(jω)| < γ iff H has no eigenvalues on the jω-axis. • An equivalent test is if there exists a X ≥ 0 such that ATX + XA + CTC + 1 γ2XBBTX = 0 and A + 1 γ2BBTX is stable. – This is an Algebraic Riccati Equation (ARE) • Typical appplication: since e = y − r, then for perfect tracking, we want e ≈ 0 ⇒ want S ≈ 0 since e = −Sr + ... – Sufficient to discuss the magnitude of S because the only require￾ment is that it be small. • Direct approach is to develop an upperbound for |S| and then test if |S| is below this bound. |S(jω)| < 1 |Ws(jω)| ∀ω? or equivalently, whether |Ws(jω)S(jω)| < 1, ∀ω • Note: The state-space tests can also be used for MIMO systems – but in that case, we need diffferent Frequency Domain tests