ik (*,• ) A + mjy (/,- )*, „ 0 M ' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 " Au (2.114) (2.115) 2.5 Non-minimum Phase Characteristic In this section, we investigate the non-minimum phase characteristic of a flexible link robotic manipulator. For a linear continuous-time system, if all the zeros are in the left half of the complex plane, such a system is called a minimum-phase system. A minimum-phase system with a given magnitude curve in the Bode plot will produce the smallest net change in the associated phase angle. If there are system zeros in the closed right half of the complex plane, such a system is called non-minimum phase system. In the linear discrete-time case, the 54 system is non-minimum phase if there are zeros that lie outside the unit circle. The non-minimum phase concept can be extended to a nonlinear system. By using the inverse dynamics (input-output linearization) approach, the dynamics of a nonlinear system are decomposed into an external part and an internal part, as shown in Figure 2.8. The zero-dynamics of the system are defined as the internal dynamics of the system when the control input maintains the system output at zero. A nonlinear system is minimum-phase if its zero-dynamics are asymptotically stable; otherwise, the system in non-minimum phase. Instability of the internal dynamics implies instability of the overall closed-loop system. The inverse dynamics approach cannot be applied directly to non-minimum phase systems because they cannot be inverted (inverted dynamics is unstable). This is a generalization of the linear system result that the inverse of the transfer function of a non-minimum phase linear system is unstable. Therefore, for such systems, control laws that achieve perfect (asymptotically convergent) tracking error should not be pursued in general. Instead, one should find controllers that lead to acceptably small tracking errors for the desired trajectories. Input u External Dynamics (input-output part) Internal Dynamics (unobservable part) Output y > Figure 2.8 External dynamics and internal dynamics of a nonlinear system. Next we investigate non-minimum phase characteristics of a linear system using the root locus method. The root locus gives the closed-loop pole trajectories as a function of the feedback gain K, assuming negative feedback (or some other variable parameter). Consider a single-input single-output (SISO) system with open-loop transfer function 55 G(s) = ^ - (2.116) D(s) The zeros of the system are the roots of N(s) = 0 (2.117) The closed-loop transfer function is G(s) = (2.118) D(s) + KN(s) The closed-loop poles are the roots of D(s) + KN(s) = 0 (2.119) Divided both sides of equation (2.119) by K. We have D(s) K + N(s) = 0 (2.120) From equation (2.120) and equation (2.117), we notice that as the feedback gain increases, the closed-loop poles of the system are attracted towards the open-loop zeros of the system. If the open-loop system is non-minimum phase (i.e., has RHS zeros), then the closed-loop system can become unstable under static output feedback. For multi-input multi-output (MLMO) linear systems, the transfer function G(s) becomes a transfer-function matrix G(s) . The transmission zeros (Maciejowski, 1989) of a controllable and observable m input and r output «th-order linear state-space system X = AX + Bu ( 2 m ) Y = CX + Du are defined as the values of s for which the normal rank of the system matrix drops to rank A-sln B C D < n + min(r,m) (2.122) It can be shown that the open-loop transmission zeros of the system are the eigenvalues of A - KBC as /T->oo (2.123) 56 The closed-loop characteristic roots when negative feedback of the form Kl is applied to the system are the eigenvalues of the closed-loop ' A ' matrix From equations (2.123) and (2.124) we. observe that as the static feedback gain K is increased, the closed-loop poles of the system are attracted towards the open-loop transmission zeros of the system. If the open-loop system has non-minimum phase transmission zeros (RHS zeros), then the closed-loop system can become unstable under static output feedback. Controller design for a non-minimum phase system is more difficult than that for a minimum phase system owing to the limited gain margin. The improper controller design can make the instability of the zero-dynamics to have an effect on the closed-loop system stability. The non-minimum phase characteristics of the system limit the loop bandwidth, and the achievable performance of the feedback system is reduced. For example, when using inverse dynamics algorithm, the right hand side (RHS) zeros will become unstable poles in the inverse system. The controller now has unstable poles that can cause the overall system to become unstable. One noticeable characteristic of a non-minimum phase system is the nature of its step response. For a non-minimum phase system, the step response initially starts to move in the direction opposite to final steady-state value. To illustrate the non-minimum phase characteristic of a flexible link robotic manipulator system, consider the single-link flexible manipulator shown in Figure 2.7. The nonlinear dynamic model of the system is given by equations (2.92) and (2.93). Using equation (2.113), the linearized model of the system with a single flexible mode is obtained as A c = A - KBC (2.124) Mu0,+Mn5u(t) + bg0x=Tx M i A +M225u(t) + k,5n(t) + bn8u = 0 (2.125) where 57 M 1 2 = M 2 1 = / „ , +mplx0u(l]) + Jp0'(l1) (2.126) M 2 2 =m, ,(/,)' +Jp u'(hf kx = coxx m.+m^M2 +Jp*