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Unformatted text preview: Example of Exact Trade-offs in Linear Controller Design Craig Barrall and Stephen Boyd ABSTRACT: The design of a linear time- invariant controller for a given linear time— invariant plant, like any engineering design, involves trade-offs among many desirable qualities, such as fast response to commands without excessive overshoot, low actuator authority, robustness low controller come plexity, and so on. Only for a few very spe- cial cases are analytic methods known for finding the exact form of these tradeeolfs. Two examples of such analytic methods are Linear Quadratic Gaussian theory, where the plant actuator and output variance can be traded off, and Nevanlinna-Pick theory, where, for instance, the achievable distur- bance rejection can be traded off in two dif- ferent bandwidths. In many cases, the limit of performance achievable with a linear time- invariant controller, and thus the exact form of the trade—oft, can be computed numeriv cally. To demonstrate this trade-off, the pa— per treats the design of a regulator for a very typical plant, a double integrator with some excess phase, The trade-offs are presented for two different measures of robustness with noise sensitivity. The exact form of these trade—offs is determined numerically by using the techniques described in the appendixes. Introduction Given a (possibly unstable) linear time— invariant plant, a basic design problem is to find a linear time-invariant controller that will stabilize the plant and meet various design objectives Beyond simple intuition, little is known about how the different design objec— tives or goals trade off. The basic character of some of the design trade—oils is obvious from simple physical reasoning, e. g. , it takes more force (actuator authority) to move a mass from one place to another more quickly (faster command re- sponse). Other trade-offs are more subtle, For example, it is well known to control engi- neers that robustness or stability margins trade off against closed—loop bandwidth for Craig Barratt and Stephen Boyd are with the In- formation Systems Laboratory in the Electrical Engineering Department at Stanford University, Stanford, CA 94305. 46 any plant with delay or other excess phase [1]. Recently [2], it was observed that the limit of performance achievable with a linear time— variant controller can be computed numeri- cally, where perfomtance reflects many practical constraints and qualities, including fast response to commands without exces— sive overshoot or undershoot, small and quick reactions to disturbances or noises, low actuator authority, and certain measures of robustness or insensitivity to unknown or un- modeled plant dynamics. Some design goals, such as low controller complexity and open— loop controller stability, cannot be included using the methods in [2]. Finding the Limit of Performance Recent theoretical results [3] show that the set of closed-loop responses achievable with stabilizing controllers is afiine. This means that if controllers C1 and C2 each stabilize a given fixed plant P, and yield closedeloop transfer functions H. and H2, then, for every real number )x, there is a controller C x~ which stabilizes P and yields the closed—loop trans fer function )xH. + (l —- MHZ. Geometri- cally, this means every closed—loop transfer function on the line through H1 and H2 is achieved by some stabilizing controller. Note that, in general, the controller Cy is not given by )xC. + (1 — )x) C3. These results allow us to express simply every closed—loop trans- fer function achievable with stabilizing con— trollers. Now consider the controller design probe lent, which is the problem of finding a con— troller that achieves particular design goals. In many cases, the design goals can be ex— pressed as convex constraints on various closed-loop transfer functions. A constraint on a closed—loop transfer function is convex if the average value of two closed-loop trans- fer functions (Hl + H2)/2 satisfies the con- straint whenever HI and H2 do (here H1 and 1-12 are the closed-loop transfer functions yielded by controllers Cl and C2). Examples of convex constraints are design goals in- volving envelope bounds on closed—loop time responses to any input (e.g., step responses and impulse responses), sensitivity to noises, 027271708/89/010070046 $01.00 © 1989 IEEE closed—loop frequency responses, and some robustness requirements. Examples of de- sign goals that cannot be expressed as convex constraints on closed-loop transfer functions are restrictions on the structure or order of the controller, e.g., the number of controller poles. In the case where every design goal can be expressed as a convex constraint on some closed-loop transfer function, the controller design problem is equivalent to finding a closed-loop transfer matrix (a collection of the appropriate closed-loop transfer func— tions) that satisfies the design goals. The simple representation of the achievable closed-loop transfer functions means that this new problem can be solved numerically to arbitrary accuracy [2]. A point on a trade—off curve can be found by constraining all but one of the design goals and minimizing the remaining one (it is as— sumed that all design goals are expressed so that “smaller” is “better"). The corre- sponding optimization problem is convex. This means there are no local minima. Such convex optimization problems can be solved numerically to arbitrary accuracy. The Plant We consider the design of a regulator for a single—input/single-output plant that con— sists of a double integrator with some excess phase, Pm = (1/52) (4 — s)/(4 + s) The all—pass term (4 — s)/(4 + s) approxi— mates a 0.5-sec delay (exp (is/2)) at low frequencies. We may think of the all-pass term as accounting for any and all of a va— riety of sources of excess phase in a real control loop, e.g., small delays. antialias fil- ters, equivalent excess phase contributed by a sample-and-zero—order-hold plant input, and so on. The idea of using a double inte— grator plant with some excess phase as a simple but realistic typical plant with which to explore control design trade-offs is taken from a study presented by Gunter Stein in [41‘ The plant is discretized using a zero—order lEEE Control Systems Magazine hold at 10 Hz, giving the following transfer function: P (z) —0.00379(z2 — 0.7241z — 1.1457) _ Z3 — 2.67032,2 + 2.3406z — 0.6703 _ —0.00379(z ~ 1.492) (2 + 0.7679) A (z , 1f (2 — 0.6703) The objective is to investigate some trade- ofis in the design of a regulator C for this plant, where the closed—loop system is shown in Fig. 1. The discrete-time inputs w and v represent actuator (input—referred process noise) and sensor noise, which are taken to be independent white noise. The discrete— time outputs u and y are the actuator and plant outputs. LQG: An Analytically Computable Trade-off Consider the steady—state noise at u and y in Fig. 1 due to the injection of noises at v and w. As usual, assume that v and w are zero mean, white, independent, with root— mean-square (nns) values of 1 for U and 10 for w. The trade—oilP between the steady—state out- put variance limb a, Eyf and the steady—state actuator variance lim,Hm Eufi is analytically computable from Linear Quadratic Gaussian (LQG) theory [5]. Given a fixed positive p, the LQG optimal regulator minimizes (over all stabilizing regulators) the weighted cost function J, which is a linear combination of the actuator and output variance, J = lim E{yi + pug} k—> on Appendix B gives an example where the dis- crete—time LQG optimal regulator is found for the plant with p equal to 0.0001. Note that for each p, the LQG optimal regulator gives the best noise sensitivity tie. the minimum J) that can be achieved by any linear time-invariant regulator (of any com- plexity or structure) that stabilizes the plant P. Thus, the rms values of u and )1 achieved by the LQG regulator give a point on the trade—off curve between these quantities. Let us justify this assertion. If some other linear time-invariant regulator stabilized P and achieved better rtns values of both it and y, then it would achieve a cost J smaller than the LQG regulator (recall that p is greater than 0). This is not possible, since the LQG optimal regulator gives the smallest J achievable by any linear time-invariant reg— ulator. As the number p varies. the nns values of u and y achieved by the corresponding LQG January 7989 Fig. l. optimal regulator sweep out the trade-off curve. For our plant. the trade-off between the rrns values of u and y is shOWn in Fig. 2. The interpretation of the curve in Fig. 2 is as follows. No linear time-invariant reg— ulator C that stabilizes P can achieve rms values ofu and y, which lie below the curve. This is true for regulators of any order, de— signed by any method. We can restate this as: every regulator C that stabilizes P achieves rrns values of u and y, which lie in the shaded region, on or above the trade-off curve. For example, the following simple lead-lag regulator stabilizes P and achieves n'ns values of u and y of 27.55 and 5.98, respectively. C(z) = 10(z * 0.98)/(z - 0.56) This property is shown in Fig. 3. This reg- ulator achieves closed-loop performance, which lies in the shaded region of Fig. 2, as it must. Any regulator C that gives closed—loop performance in the shaded region in Fig. 3 will perform the same or better (in terms of rrns values of u and y) than the simple lead— lag regulator C. This means that C will achieve the same or better output regulation (rms y no worse than 5.98) and use the same or less actuator effort (rms u no worse than 27.55). One family of such regulators is the LQG optimal regulators with p between 000135 and 2.31. These regulators lie on the boundary of the shaded region. There are infinitely many other regulators that give closed—loop performance in the shaded re— gion. This idea gives us a second interpretation of the trade-off curve in Fig. 2 in terms of achievable design goals. Any design goals (i.e., upper bounds on the rrns values of u and y) that lie in the shaded region in Fig. 2 can be met or exceeded by some linear time- invariant regulator C, which stabilizes P. Design goals outside of the shaded region cannot be achieved by any linear time-in— variant regulator C, which stabilizes the plant Regulator system. 10 rms value of y 1 ’ 10 100 rms value of u Fig. 2. Trade-off between achievable in» put and output nns noise sensitivities, on a log—log scale. p 20.00135 rms value of y .t + _. 10 100 rms value of 11 Fig. 3. Simple regulator C, which achieves performance above the trade-oilr curve. P. For example, consider the design goal that the rms value of u should be less than 10 and the rms value of y should be less than 4. This design goal can be achieved by a linear timevinvariant regulator that stabilizes the plant P, whereas the design goal that the rrns value of u should be less than 3 and the mis value of y should be less than 5 cannot be achieved by any linear time-invariant reg— ulator that stabilizes the plant P. For p equal to 10—4, the details of the LQG design are given in Appendix B. The LQG optimal (current estimator) regulator is shown to be 45.9748 — 72.79z2 + 28.138z C(z) — z3 — 0.8061:2 + 0.7107z — 0.1071 45.974z(z — 0.91305) (z — 0.6703) (z — 0.17897) (z — 0.3136 + 0.7072j) (z - 0.3136 — 07072)) 47 For this regulator. the rms values of u and y are 63.73 and 2.0184, respectively. The value of the cost function J is 4.48. Two Measures of Robustness Tolerance of Additive Loop Perturbations One measure of robustness of a control system, which combines the gain and phase margins, is the M-circle radius, defined as the minimum distance from the Nyquist plot of the loop gain PC (exp 10) to the critical point — 1. Mathematically, this can be writ- ten as M l l dist (PC(exp jfl), — 1) min |1 + PC(expj9)| Osnsr The M—circle radius gives a measure of the sensitivity to additive loop perturbations. If the distance M is small, then slight variations in the loop gain PC could change the number of net clockwise encirclements of _ 1 by the loop gain, resulting in an unstable closed- loop system. The M—circle radius is related to the max- imum sensitivity of the control system. The Mecircle radius is the inverse of the maxi— mum sensitivity M=1/lSll°. where S = 1/(1 + PC) is the sensitivity transfer function, and the maximum of the sensitivity transfer function over all frequenv cies is (In this notation, HHIIQ, denotes the maxi— mum magnitude over real frequencies of a transfer function H.) Thus. a small M—circle radius corresponds to peaking of the sensi- tivity function at some frequency. It also fol- lows that constraining the maximum mag- nitude of the closed»loop transfer function S to be at most 1 over Mm,“ is equivalent to constraining the M—circle radius to be at least Mm. For example, the p = 10’4 LQG regulator gives ilSllm = 3.34, so that M Z 0.30 is the closest the Nyquist plot comes to the crit- ical point #1. This can be seen from the Nyquist plot in Fig. 4. The M-circle of ra- dius 0.30 is also shown in Fig. 4. The mag- nitude of the sensitivity transfer function S for the p = 10 '4 LQG regulator is shown in Fig. 5. Note that its maximum is 10.5 dB (which is equal to —20 log M). Tolerance of Additive Plant Perturbations The second measure of robustness we will consider concerns the ability of the regulator 48 lm(PC) -15 '-2 —1.5 .1 -0.5 0 0.5 1 Re(PC) Fig. 4. Nyquist plot of the LQG regulator. Note: p = 10‘4 and the M 2 0.30 circle is shown. 10 i {:11 f\//, s / 9 0.1 § 0.01 ' ‘ ,7 0 0.5 1 1.5 2 2.5 3 Q: (OT Fig. 5. Magnimde of sensitivity transfer function of LQG regulator. Note: p = 10*. to maintain closed—loop stability in the face of stable additive plant perturbations AP. as shown in Fig. 6. The plant perturbation could represent errors in modeling the plant, com- ponent variations. or deliberately ignored plant dynamics. Intuitively, if AP is small at all frequen— cies, then we would expect that if the reg- ulator C stabilizes the plant P, then C should stabilize P + AP as well. This intuition is indeed correct, a consequence of the small gain theorem [6]. [7]. We may ask, what is the smallest (in the sense of maximum magnitude of frequency response) stable additive plant perturbation AP that will destabilize the system in Fig. 6? Let us define the quantity D as the small- est plant perturbation AP that will destabi- lize the closed»loop system, D = min IIAPH,” APdcstabilizcssystem It is not difficult to derive that the inverse of D is equal to the maxinrum of the closed— loop transfer function from the reference in Fig. 6. Additive plant perturbation. put r to the actuator output u, l/D = IlC/(l + PC) II... This observation is due to Doyle and Stein [6]. The positive number D can be interpreted as the largest size of stable additive plant perturbation the control system can be guar— anteed to withstand. A regilator that yields small values of D corresponds to a system that can be destabilized by a small stable additive plant perturbation. It also follows that constraining the maximum magnitude of the closed—loop transfer function C/(l + PC) to he at most l/Dmm is equivalent to con- straining the additive plant perturbation tol- erance D to be at least Dmi“. For the p = 10'4 LQG optimal regulator, the peak of the closed-loop transfer function C/(l + PC) is 83.02; so D = 0.0121. Hence, there are stable plant perturbations AP that destabilize the system in Fig. 6, with HAPIIm as small as 0.0121. One destabiv lizing perturbation is shown for which IIAPIIW = 1/80, which is greater than D. APO _ 0.2152 x 1040 W z) “ v 22 v 0.985z + 0.9801 This AP destabilizes the system in Fig. 6. The frequency—response magnitude of AP is plotted in Fig. 7. The transfer function AP might represent a mechanical resonance, which was ignored when the plant P was modeled for the LQG design. In general, the robustness requirements that M and D be large are independent. A system may have good margins (i.e., large M) but be quite sensitive to additive plant perturbations (i.e., small D) and vice versa. .0 _. tude of A P (expj Q) o 9 0.001 o Magn '00010 0.5 1 1.5 2 2.5 3 Frequency .0 Fig. 7. Frequency response of a destabilizing AP. IEEE Control Systems Magozme Trade-off Curves Involving Noise Sensitivity and Robustness We wish to minimize the noise sensitivity of the system and determine how this trades otf with the two different robustness require— ments described in the previous section. Spe- cifically, how does the noise sensitivity .I trade off against M when we require D 2 D,,,,,,, and how does it trade off against D when we require M 2 Mmm? The design problem can be expressed mathematically as Jmin 2 min J thuimp l/M= H l/(l + PC) It , s l/Mm," l/D:i|C/(1+PC)H, 5 no... Unlike LQG, no exact analytical solution of this minimization problem is known, al- though some work has been done [8]. Never- theless, this minimization problem can be solved numerically since it can be cast as an (infinite-dimensional) convex optimization problem [2]. In general, the optimizing con— trollers are of high order. In general, no method is known for finding low»order con~ trollers that achieve close to the optimal per- formance. For three fixed values of Dmm, the trade— off between 1",,“ and l/Mmin is shown in Fig. 8. The p = 10—4 LQG regulator is also shown. Since this regulator achieves l/D = 8302, it lies below the I/Dmin = [0 curve and on the l/Dmin = 83.02 curve. All trade- off curves with l/Dm,n 2 83.02 will pass through the LQG performance point and be horizontal to the right of it. Note the interesting fact that by allowing the M-circle radius to be less than 0.5, only modest improvement in the noise response is gained, with the same tolerance D to ad— ditive plant perturbations. For four fixed values of Mmm, the trade- ol’f between Jmi“ and l/Dm,n is shown in Fig. 9. The p = 10—4 LQG regulator is also 6 \\ 5: \\\\ \\\fl55 4i \ \\ A” i \ \\1/0510 .3» ,2, 2 \\\1/a:_3-02 LQG 1 0 _. 1.5 2 2.5 3 3.5 1/M Fig. 8. Trade-off between rms noise and M—circle radius. January V989 6 #-__,V 2v 2 \ 5 \\ \ \\\ \\\ t/M <15 4 \ \\ \\ ’ 22ml \ \\. 1/M <17 «IN \ \ x ,2 2*, 777 7 7 33 \\\ \\ 1/M_<_2 \\» 2‘WLAE 33"” 2 LQG ( 1 . 0O 170 20 30 40 50 60 70 80 90100 1/D Fig. 9. Trade—off between rms noise and additive plant sensitivities. shown. Since this regulator achieves l/M = 3.34. it lies below the l/Mm,n : 2.0 curve and on the l/Mmm = 3.34 curve. All trade- off curves with l/Mmin 2 3.34 will pass through the LQG performance point and be horizontal to the right of it. From Fig. 9 we can draw some interesting conclusions. Consider the UM S 3.34 curve. which corresponds to regulators that yield the same or larger M-circle radius as the p = 10—4 LQG regulator. The curve is relatively flat for [ID 2 20, meaning that D can be increased to about 0.05 with a rela- tively small increase in ms noise response and the same or larger M-circle radius (0.3). For the p = 10‘4 LQG regulator, this rep— resents an increase in additive plant perturv bation tolerance D of a factor of 4. Of course, all regulators that stabilize P yield a noise sensitivity J 2 JLQG, so that all curves lie on or above the horizontal asymptote J 1130 = 2.117. Imposing further constraints on the two measures of robust ness naturally will increase the minimum noise sensitivity J achievable with linear time»invariant stabilizing regulators. What is neither intuitively obvious nor analytically computable is how much the minimum noise sensitivity J must increase when we impose various constraints on the two measures of robustness. Figures 8 and 9 show this trade— ofl" precisely. Let us give two examples of specific con clusions we may draw from Figs. 8 and 9. First, the following design goals can be achieved with a linear time—invariant regu- lator that stabilizes P (this point is marked “x" in Fig. 9). 11/2 s 3, M 2 0...
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