Open-loop characteristic equation

We want to look at the stability of the closedloop system with a proportional controller Gc(j) = K- First, however, let us check the openloop stability of this system. The open-loop characteristic equation is the denominator of the openloop transfer function set equal to zero. 

We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + KCG0 = 0, where KCG0 is referred to as the "open-loop" transfer function, G0l- The proportional gain is Kc, and G0 is "everything" else. If we only have a proportional controller, then G0 = GmGaGp. If we have other controllers, then G0 would contain... 

Let say we have a simple open-loop transfer function G0 of the closed-loop characteristic equation... 

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

Example 7.2 If we have only a proportional controller (i.e., one design parameter) and real negative open-loop poles, the Routh-Hurwitz criterion can be applied to a fairly high order system with ease. For example, for the following closed-loop system characteristic equation ... 

Equation 7.119 is the characteristic equation of the system shown in Fig. 7.34 and is dependent only upon the open-loop transfer function G (j)H(j) and is therefore the same for both set point and load changes (equations 7.109 and 7.110). 

The roots of the characteristic equation may be real and/or complex, depending on the form of the open-loop transfer function. Suppose at to be complex, such that ... 

Define the following terms open-loop transfer function, characteristic equation, closed-loop poles. 

Suppose that the two feedback controllers Gci and Gc2 are tuned separately (i.e., keeping the loop under tuning closed, and the other open). Then we cannot guarantee stability for the overall control system, where both loops are closed. The reason is simple Tuning each loop separately, we force the roots of the characteristic equations (24.12) for the individual loops to acquire negative real parts. But the roots of these equations are different from the roots of the characteristic equation (24.13), which determines the stability of the overall system with both loops closed. 

Timing each loop separately. The characteristic equation of loop 1 (when loop 2 is open) is given by... 

Representing the roots of the characteristic equation in the complex domain offers a simple way to perform a stability analysis. The system is stable if and only if all the poles are located in the open left-half-plane (LHP). If there is at least one pole in the right-half-plane (RHP), the system is unstable. The representation is similar wiA the well-known root-locus plot used to evaluate the stability of a closed-loop system. 

Mechanism calibration consists of identifying the geometric parameters to improve the model accuracy by reducing the system error [1]. To do this, the calibration procedure tries to minimize the error between the measured pose of the end-effector and the calculated pose. This characteristic requires two kinds of equations in parallel mechanism calibration a) those transformations that relate the platform location with the reference system of the base by means of open-loop kinematic chains, and b) those closed-loop transformations that contain the constraints imposed by the closed-loop chains [2]. 

From Eqs. (15)-(16) it is easily seen that the feedback will induce instability if the characteristic equation 1 —g2i s)gR s) = 0 has any roots in the RHP. Provided the open loop system is stable, the Bode criterion applies, i.e., the feedback will induce instability provided there exist some frequency (0 for which the loop-gain... 

To find the gain and phase characteristics of a loop away from its natural period, the vector equation for the inner loop must be solved for various values of input period t i. This entails first finding the gain and phase of the open loop, g gj. This vector must then be added to the vector 1.0, LO to form the denominator of the equation. Then the closed-loop gain is the quotient of the magnitude of the two vectors, and its phase is the difference between their phase angles. 

Equation (7.20) can be solved by faetoring into two parts, gsXu and 1/(1 + gcigiXii). The latter is a characteristic of the closed loop and can be evaluated by summing the open-loop vector gdgiXu with 1.0, followed by inversion. The solution for the general case of a dead-time plus integrating eombination adjusted to g jgi = 0.5 at the natural period is plotted in Fig. 7.7. 

Digital controllers of the Direct Synthesis type share yet one more characteristic namely, they contain time-delay compensation in the form of a Smith predictor (see Chapter 16). In Eq. 17-61, for Gc to be physically realizable, (Y/Y ) must also contain a term equivalent to which is z, where N = 0/Ar. In other words, if there is a term z in the open-loop discrete transfer function, the closed-loop process cannot respond before NAt or 0 units of time have passed. Using YIYsp)d of this form in Eq. 17-63 yields a Gc containing the mathematical equivalence of time-delay compensation, because the time delay has been eliminated from the characteristic equation. 

Nyquist Stability Criterion. Consider an open-loop transfer function Gol( ) that is proper and has no unstable pole-zero cancellations. Let N he the number of times that the Nyquist plot for Gol( ) encircles the (-1, 0) point in the clockwise direction. Also let P denote the number of poles of Gql( ) that lie to the right of the imaginary axis. Then, Z = N - - P where Z is the number of roots (or zeros) of the characteristic equation that lie to the right of the imaginary axis. The closed-loop system is stable if and only if Z = 0. 

The block diagrams presented in Figs 2.1 - 2.3 are characteristic for the open systems and differ from one another only in the number of inertial objects. The dynamic objects are not always arranged in series. In many cases, as a result of self-arrangement of the objects and their configurations, we must consider the set of differential equations presented by the block diagrams of closed-loop systems. For example, for the calorimeter described by the differential equation... 

---