Right-Half-Plane zeros

Note the initial negative deviation in response to a positive forcing disturbance, caused by the right half plane zero in the transfer function. 

The first chapter of Part II illustrates all these aspects by considering a coaxially coupled inverted pendula system that is unstabilisable when torque actuation is applied but is controllable in case of velocity actuation although there is a right-half plane zero. 

Abstract A bond graph method is used to examine qualitative aspects of a class of unstable under-actuated mechanical systems. It is shown that torque actuation leads to an unstabilisable system, whereas velocity actuation gives a controllable system which has, however, a right-half plane zero. The fundamental limitations theory of feedback control when a system has a right-half plane zero and a right-half plane pole is used to evaluate the desirable physical properties of coaxially coupled inverted pendula. An experimental system which approximates such a system is used to illustrate and validate the approach. 

Figure 5.5 shows the Quanser [29] self-balancing seesaw apparatus. A dc-motor-driven cart moves along a track forming the top of a seesaw pivoted at a point below the track. As the pivot is below the track, the seesaw is unstable as the acceleration of the cart is associated with a reaction force opposite to the acceleration direction, the system has a right-half plane zero. The cart had a fast inner loop to control velocity (relative to the seesaw) together with a low-pass filter to implement (5.24). 

Morari [2] identified the relationship between the invertibility of transfer function matrix of a system and its ability to move fast and smoothly from one operating condition to another and to deal effectively with disturbances. Therefore, plants that are easier to invert should be easier to control and possess greater resilience. Four fundamental factors were identified preventing the inversion of the process right-half-plane zeros, time delays, constraints on the variables, and model uncertainty. The limitations imposed by these factors on the operability of multivarible systems were explored by Morari, coworkers, and others in a series of papers [24-28]. 

Inverse response or overshoot can be expected whenever two physical effects act on the process output variable in different ways and with different time scales. For the case of reboiler level mentioned above, the fast effect of a steam pressure increase is to spill liquid off the trays above the reboiler immediately as the vapor flow increases. The slow effect is to remove significant amounts of the liquid mixture from the reboiler through increased boiling. Hence, the relationship between reboiler level and reboiler steam pressure can be represented approximately as an overdamped second-order transfer function with a right-half plane zero. 

Note that Eq. 6-22 indicates that K and K2 have opposite signs, because ti > 0 and T2 > 0. It is left to the reader to show that. S > 0 when K > and that X < 0 when K < 0. In other words, the sign of the overall transfer function gain is the same as that of the slower process. Exercise 6.5 considers the analysis of a right-half, plane zero in the transfer function. 

Figure 6.7a illustrates the response of the 1/1 and 2/2 Fade approximations to a unit step input. The first-order approximation exhibits the same type of discontinuous response discussed in Section 6.1 in connection with a first-order system with a right-half plane zero. (Why ) The second-order approximation is somewhat... 

The specification of the desired closed-loop transfer function, (Y/Y p), should be based on the assumed process model, as well as the desired set-point response. The FOPTD model in Eq. 12-6 is a reasonable choice for many processes but not all. For example, suppose that the process model contains a right-half plane zero term denoted by (1 - t ) where > 0. Then if Eq. 12-6 is selected, the DS controller will have the (1 - T ) term in its denominator and thus be unstable, a very undesirable feature. This problem can be avoided by replacing (12-6) with (12-15) ... 

Note that the IMC controller in Eq. 12-20 based on the invertible part of the process model, G-, rather than the entire process model, G. If G had been used, thQ controller could contain a prediction term c if G+ contains a time delay 0), or an unstable pole (if G+ contains a right-half plane zero). Thus, by employing the factorization of (12-19) and using a filter of the form of (12-21), the resulting IMC controller G is guaranteed to be physically realizable and stable In general, the noninvertible part of the model, G+,... 

This example has demonstrated that even an exact cancellation of an unstable pole leads to instability. Consequently, an unstable pole should never be canceled with a right-half plane zero. In contrast, open-loop unstable systems can be stabilized with feedback control, as was demonstrated in Example 11.8. 

However, having determined qualitative properties, it is necessary to turn to actual system parameters for the design of controllers for specific systems. As discussed in the textbooks (for example, [25, 26]) system (qualitative) structure together with parameters implies fundamental limitations on the performance of feedback control systems. In particular, right-half plane poles and zeros impose constraints on the achievable sensitivity function. 

As shown in Section 5.2, the system has two real poles at s = p and two real zeros at i = z. Thus the system has both a right-half plane (RHP) pole and an RHP zero as discussed in the textbooks [25, 26], this imposes fundamental restrictions on feedback control performance. In particular, as discussed by Skogestad and Postlethwaite [26, Section 5.9, p. 185], system bandwidth (as measured by the critical frequency, coc, [26, Section 2.4.5, p. 36]) should be approximately bounded by... 

The resultant linearised system displays a single right-half plane pole/zero pair which imply fundamental limitations of achievable performance. The quantitative properties of the CCIP model are analysed from this point of view. 

Unfortunately, a plant always has fundamental limitations which restrict the highest bandwidth (Ob that a feedback control system can achieve, even with the best possible control law. Fundamental limitations stem from the process itself, e.g., in the form of time delays 6 and right half plane (RHP) zeros z > 0. In addition, the phase lag of a plant imposes a limitation when low order controllers, such as PID-controllers, are employed. Assume the plant model can be written on the form... 

Note that this time-delay approximation is a right-half plane (RHP) zero at s = +0. An alternative first-order approximation consists of the transfer function,... 

Suppose that the numerator of a transfer function contains the term 1 - ts, with t > 0. As shown in Section 6.1, a right-half plane (RHP) zero is associated with an inverse step response. The frequency response characteristics of G(s) = 1 - t5 are... 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

The time-independent term on the right-hand side of Eq. (73) gives zero because the contour C may be closed in the upper half-plane where Y00(2) is everywhere analytic (see Section II-D) we are then left with the time-dependent term, which in the limit t — oo only contributes at its pole 2 = 0. From Eqs. (72) and (73), we then get ... 

The brackets contain two terms the first (between parentheses) gives zero. Indeed, since the variables zv z2> and z in it are independent we have the right to close the contours corresponding to zx or zz in the upper half plane and we do not then include any poles. The other term must be treated in detail. 

B, APPLICATION OF THEOREM TO CLOSEDLOOP STABILITY. To check the stability of a system, we are interested in the roots or zeros of the characteristic equation. If any of them lie in the right half of the s plane, the system is unstable. For a closedloop system, the characteristic equation is... 

We can use this theorem to find out if the closedloop characteristic eqga-tion has any roots or zeros in the right half of the s plane. The s variable follows a closed contour that completely surrounds the entire right half of the s plane. Since the closedloop characteristic equation is given in Eq. (16.1), the function that we are interested in is... 

The contour of is plotted in the F plane. The number of encirclements of the origin made by this plot is equal to the difference between the number of zeros and the number of poles of F(j) in the right half of the s plane. 

If the process is openloop stable, none of the transfer functions in Qu(,> tll have any poles in the right half of the s plane. And the feedback controllers in are always chosen to be openloop stable (P, PI, or PID action), so has no poles in the right half of the s plane. Clearly, the poles of are the poles of Thus if the process is openloop stable, the F(,) function has no poles in the right half of the s plane. So the number of encirclements of the origin made by the F,jj function is equal to the number of zeros in the right half of the s plane. 

Although by definition the system is stable, the phase margin is so close to zero and the gain margin so close to unity that any slight variation in any of the control system parameters or, indeed, in the process conditions, could make the system unstable, i.e. could cause a pole or poles of the system closed-loop transfer function to move into the right half of the complex plane (Section 7.10.1). 

If we pick a contour that goes completely around the right half of the s plane and plot 1 + Gm(s)Gc s) Eq. (11.1) tells us that the number of encirclements of the origin in this (1 + G Gq) plane will be equal to the difference between the zeros and poles of 1 + GmGq that lie in the RHP. Figure 11.2 shows a case where there are two zeros in the RHP and no poles. There are two encirclements of the origin in the (1 + GmGc) plane. 

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