A transfer function GK has no zeros and 3 poles in LHP (left half plane). The Ny
ID: 2081185 • Letter: A
Question
A transfer function GK has no zeros and 3 poles in LHP (left half plane). The Nyquist plot of GK is shown below.
a. Is the open loop system GK stable?
b. Draw an approximate Bode plot of GK based on the information from the Nyquist plot above.
c. Is the closed loop system GK/(1+GK) stable? Use the Nyquist criterion.
d. Looking at the Nyquist plot of GK, find the approximate values of gain and phase margins. For your convenience, a dashed unit circle is shown on the Nyquist plot.
e. Mark the gain and phase margins on the Bode plots from part above.
f. By what factor can you increase the controller gain, before the closed loop system GK/(1+GK) becomes unstable?
JE -1 Nyquist Diagram System G Real: 1.27 imag: 3.46 Frequency (rad/sec): -0.73 System G Real: 2.92 mag: -3.21 Frequency (rad/sec): 0.474 Real AxisExplanation / Answer
L(s) = 1 s L˜(s) = C0 Y N n=1 1 + s/zn 1 + s/pn , where p1 = l , zn = pn, p,n+1 = zn, and = (h/l) /N , = (h/l) (1)/N , C0 = (1/l) . If we want the approximation error |L˜(j) L(j)| to be less than some real number E > 0 in the frequency range of interest [l , h], the order of approximation N should satisfy N = & log(h/l) E 10 ( 1 1+ + 1 2 ) ' , for 1 < < 2 [15]. A good property of the this approximation method is that L˜(s) is always stable. C. Ideal Transfer Function Consider the feedback interconnection of the controller K(s) and the plant G(s) in Fig. 1. Bode, in his study on design of feedback amplifiers [5], has suggested an ideal shape of the open-loop transfer function GK(s) of the form L(s) = (c/s) (1) 0.0 5.0 10.0 15.0 20.0 0.0 0.5 1.0 1.5 2.0 Time (s) Step Response = 1.0 = 1.2 = 1.4 = 1.6 = 1.8 = 2.0 Fig. 2. Time response characteristics of the closed-loop system T(s) for c = 1 and different . for some R, where c is the gain cross-over frequency, that is, |L(jc)| = 1. The parameter determines both the slope of the magnitude curve on a log-log scale and the phase margin of the system, and may assume integer as well non-integer values. In fact, the transfer function L(s) is a fractional-order transfer function for non-integer R. The amplitude curve is a straight line of constant slope 20dB/dec, and the phase curve is a horizontal line at /2rad. The Nyquist curve consists, simply, of a straight line through the origin. Let us now consider the unit feedback system with Bode’s ideal transfer function L(s) inserted in the forward path. This choice of L(s) gives a closed-loop system with the desirable property of being insensitive to gain changes (Gain Margin = ). The variations of the gain change the cross-over frequency c but the phase margin of the system remains (1 /2)rad, independent of the gain. Next, we study the step-response of the closed-loop system consist of the fractional order transfer function L(s) given in (1) with unitary feedback T (s) = L(s)/(1 + L(s)) = 1/(1 + (s/c) ). Thus, the step-response would be y(t) = L 1 c s( c + s ) = 1 X n=0 [(ct) ] n (1 + n) , which results in y() = limt y(t) = 1, y(0+) = lim t0+ y(t) = 0. Fig. 2 shows the overshoot Mp, peak-time Tp, rise-time Tr, and settling-time Ts of the step-response of the fractionalorder transfer function T (s) for c = 1 and different . Using these results and Fig. 2, it is relatively easy to find suitable values c and based on design specifications. III. CONTROL DESIGN METHOD The control-design method consists of two separate parts: (i) loop-shaping using H-optimization, and (ii) closed-loop controller order reduction. We discuss these two parts in the subsequent subsections. A. Control Design: Loop-Shaping Using H-Optimization In the design procedure using Bode’s ideal transfer function loop shaping, first, we must find the optimal loopgain L(s) based on the design specifications and fix both and c in (1). This can be done using the results of Subsection II-C (Fig. 2). The next step is to approximate this fractional order transfer function with a rational integer order transfer function L˜(s) using the CRONE method introduced in Subsection II-B. Finally, we should make the loop-gain GK(s) as close as possible to this approximation L˜(s) in the frequency range of interest [l , h]. This part of the design procedure can be done using a weighted H-norm minimization arg min K(s) kWo(s)[GK(s) L˜(s)]Wi(s)k, (2) where the search domain should be on the set of stable controllers K(s). This optimization problem is easy to solve [17]. The weight functions Wo(s) and Wi(s) are selected based on the frequency range of interest [l , h]. This frequency range is usually dictated by the cross-over frequency c and the open-loop characteristics of the plantto-be-controlled. Furthermore, if the open-loop plant model varies under different working conditions, we may rewrite the H-optimization in (2) as arg min K(s) Wo(s)[G1K(s) L˜(s)]Wi(s) . . . Wo(s)[GnK(s) L˜(s)]Wi(s) , (3) where the transfer functions G`(s) for ` = 1, . . . , n represent the plant-to-be-controlled under different working conditions. It should be noted that, if c [l , h], both the controller and the open-loop plant are stable, and the minimization error in (2) is small enough, then the closedloop system should be stable with K(s). This is true because GK(s) is close to L˜(s), and using the properties of Bode’s ideal transfer function and the CRONE approximation, we know that the Nyquist diagram of L˜(s), and therefore, the Nyquist diagram of GK(s) does not encircle 1. B. Control Design: Controller Reduction Method The controller K(s), introduced in Subsection III-A is a high-order controller and it is not suitable for real-time implementation with low memory usage and low computational power. In this section, we use the model-order-reduction method introduced in [9] to a get a low-order controller with satisfactory closed-loop performance. Consider the closed-loop interconnection of the system and the controller in Fig. 1 with the closed-loop state-space representation x x k = A˜ x xk + B˜ r d n , y u = C˜ x xk , where A˜ = A BCk BkC Ak , B˜ = 0 B 0 Bk 0 Bk , and C˜ = C 0 0 Ck . In the above state-space realization, the matrices [A, B, C] and the matrices [Ak, Bk, Ck] are respectively the realization matrices of the plant transfer function and the controller transfer function G(s) , A B C 0 , K(s) , Ak Bk Ck 0 . Now, we can compute reachability and observability Gramians P˜ and Q˜ using the following Lyapunov equations A˜P˜ + P˜A˜T + B˜B˜T = 0, A˜TQ˜ + Q˜A˜ + C˜T C˜ = 0. (4) and extract the weighted reachability and observability Gramians for the controller K(s) using P = 0nk×n Ink×nk P˜ 0nk×n Ink×nk T , (5) Q = 0nk×n Ink×nk Q˜ 0nk×n Ink×nk T , (6) where nk is the order of the controller K(s) and n is the order of the plant. Based on these weighted Gramians, one can balance the coordinates of the controller and use the singular perturbation method to find the reduced-order controller [18]. The order of the reduced-controller r is obtained by incrementally increasing r until the design specifications are fulfilled. If the open-loop plant model varies under different working conditions, we can use this controller reduction method on the hardest plant-to-be-controlled with the design specifications. In this controller reduction method, we can choose to use a different set of inputs and outputs for the closed-loop system based on the design specifications. IV. N
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