PID CONTROL

Quantitative robust stability analysis and PID controller design

Q.-G. Wang, C.C.Hang, Y.Yang and J.B.He

Abstract: A sufficient condition for robust stability is presented. Comparing it with the small gain theorem, it incorporates both gain and phase information of the uncertain system and thus reduces the conservativeness of the theorem. The proposed quantitative robust stability criterion is 'employed to develop a robust PID controller design. Finally, simulations are presented to illustrate the proposed method.

1 Introduction

Stability criteria and stabilisation methods are important

results have been presented in the area of robust stability. The small gain theorem is a key result of robust stability

the feedback loop given in Fig. 1 will be stable provided that

1 research topics in the system and control area. Many II m) 1100 <

However, in industrial applications, the structure of the analysis [ 11. Essentially, the small gain theorem states that if a feedback loop consists of stable systems and the loop- gain product is less than unity, then the closed-loop system is stable. The criterion is widely used in the robust control area. Unfortunately, it could be very conservative in many cases because only the gain information of the open loop system is used and the phase information is ignored.

In this paper, the conservativeness of traditional robust stability results is studied. With information on both the gain and phase of the uncertain open-loop system, a new quantitative robust stability criterion is developed. The gain and phase bounds of an uncertain second-order plus time dekay (SOPTD) model are derived. Based on these new results, a new PID tuning scheme is presented using the worst gain and phase margins of the open-loop system. Simulations are given to substantiate the effectiveness of the method.

uncertainty A(s) may be known. For instance, a large class of uncertain processes can be modelled as a transfer function with interval parameters. The above criterion for robust stability may be very conservative when the struc- ture of A($) is specified.

Example 1: Consider a second-order nominal plant with transfer function

closed in a negative feedback loop with the controller K(s) = 1. Let the plant uncertainty be structured and given by

( 2 ) 1 + 6 ,

s* + (1 + 6,)s + (1 + 6,) G(s) =

where 6, ~ [ - 0 . 5 , 0.51, 6 2 ~ [ - 0 . 5 , 0.51 and S3 ~ [ - 0 . 5 , 0.51. One may easily see that the closed-loop system is robustly stable, because its characteristic polynomial 2

Consider the controlled uncertain system shown in Fig. 1. According to the small gain theorem [l], if the nominal system H(s) and the perturbation A(s) are both stable, the feedback system is robustly stable if the inequality

Review of robust control theory

p(s) = s2 + (1 + 6,)s + 2 + 6, + 6, always has positive coefficients and thus its roots lie in the open left half of the complex plane.

In order to apply the robust stability criterion (l), we express the uncertainty as an additive one:

I - (1) 1 + 6 , I r - i ( jo)Njw) I < 1, 0 E [O, 00)

A(s) = holds. Usually, one assumes that A(s) is unknown but s2+(1 + d , ) S + ( l + d , ) s 2 + s + 1 bounded in magnitude such that supw I AGO) I = M. Then, c 0 IEE, 2002 IEE Proceedings online no. 20020101 DOI 10 1049hp-cta 20020 10 I Paper received 12 November 200 1 The authors are wlth the Department of Electrical Engineering, National University of Singapore, Kent Ridge, Singapore 119260 Fig. 1 Controlled uncertain system

IEE Proc.-Control Theory AppL, &I. 149. No. I . January 2002 3

and the corresponding H in Fig. 1 is

We have

SUP I H(jw)A(jo) I >_ :F~ I H ( j o ) A W ) I w

If we choose 61 -0.5 and 63 = -0.5, or the plant is perturbed to

1.5 s2 + s + 0.5

G(s) =

this results in

sup IH(jo)AO'o) I ? 1 0

violating criterion ( I ) although the system is stable. This shows the conservativeness of criterion (1).

It should be pointed out that criterion (1) makes use of gain information only while any phase lag is permitted including infinity. It should also be pointed out that what we have said so far on the structured uncertainty is different from the existing robust stability criterion for structured perturbations, or the theory of structured singu- lar values, which have been developed for multivariable systems and its essence is the same as criterion (1). However, the focus here is on systems with single uncer- tainty and an attempt is made to develop a new robust stability criterion which employs the phase information of the perturbations to reduce conservativeness.

3 Quantitative robust stability

Let us recall the Nyquist criterion [ 11: if the controller K(s) and the plant G(s) have no poles in the right half-plane, then the closed-loop system is stable if and only if the Nyquist curve of G(jo)K(jo) does not encircle the -1 point. An uncertain process is actually a set of models. Each model gives rise to a Nyquist curve and the set thus sweeps a band of Nyquist curves. Then from the Nyquist criterion, the closed-loop system is robustly stable if and only if the band of Nyquist curves for the uncertain process and the controller does not encircle the -1 point. Conver- sely, the system is not robustly stable if and only if there exists a Nyquist curve among the band which encircles the -1 point, or there must exist a specific frequency whose corresponding uncertain region is such that both

m y ' I G(jo,)K(jwo) I} ? 1

and

hold true. Note that

and

thus we establish the following quantitative robust stability theorem.

Theorem 1: Assume stability of the nominal closed-loop system and the controller; the uncertain closed-loop system is robustly stable if either of the following inequalities holds:

(3)

or

arg{K(jo)) > -71 - min{arg( G(jw)}} (4) G

Alternatively, we may transform into the diagram of Fig. I , where H(s) is the equivalent nominal controlled system, A(s) is the uncertainty of the system. If we modify the inequalities (3) and (4) as follows:

( 5 )

and

arg{H(jw)] > --.n - min(arg(A(jo)}} (6) A

then we have the corollary below.

Corollary 1: If the nominal controlled system H(s) and the uncertainty A(s) are all stable, then the uncertain closed- loop system remains stable if either (5) or (6) hold for every o.

Typically, one may use an additive uncertainty model G = Go + Aa or a multiplicative uncertainty model G = Go( 1 + Am) to describe the uncertain process. Then corresponding nominal controlled system H is given, respectively, by

or

For a designed control system, the robust stability criterion (3) and (4) can be tested graphically as follows:

(i) Draw Bode plots for magnitude l/max{ I G( jw) I } and phase --71 - min{arg{ G ( j w ) } } respectively, according to the given structure of the uncertain plant. (ii) Draw Bode plots for K ( j o ) in the same diagram. (iii) Check if condition (3) or (4) is satisfied.

4 Second-order uncertain model

In most practical situations, it is reasonable to approximate high-order processes by low-order plus delay models [2, 31. Reduced-order models are often required to simplify the design and implementation of control systems [4]. For this reason, model reduction has attracted much attention in engineering sciences, especially in model-based control. However, real processes are usually not of first order or second order, and sometimes may be perturbed by different operating conditions or other disturbances. Thus model errors are inevitable in practical applications if only the nominal model is used. One way to avoid problems arising from modelling errors is to use a model set to express the desired closed-loop specifications. The first-order plus

IEE Proc.-Control Theory Appl., Vol. 149, No I , January 2002 4

dead time (FOPDT) model is relatively simple but may not achieve the accuracy required in many cases as it has only real poles, hence is unable to generate peaks in the frequency response of oscillatory processes. Therefore, it is desirable to use a second-order plus dead time model to describe uncertain systems.

Consider an uncertain second-order plus dead time model

b e-Ls G(s) = s2 + als + a2

where the uncertain parameters are 'b E [b- b+], al E [UT, a t ] , a2 E [a:, a;] and L E [L-, L+], b- > 0, a c 7 0, a; > 0 and L- > 0. For a fixed w, the gain of the process is

I G(jw) I = b / d ( a , w ) Z f (a2 - 02)2 (7)

For this process, max{ I G(jo) I}, min{ I G(jw) I } can be derived as

maxi I W w ) I I =

( E a ; 0 7

and

min(IG(jw)l) =

In a similar way, the phase of the second-order system can be obtained as

arg(G(jo)) = -arg(a2 - w2 + ju, w) - Lw

Consider the uncertainty of the parameters, we obtain

min[ arg(G( jo))) =

and

max{arg(G(jw))) =

Example 2: In example 1 , small gain theory fails to verify the stability of the uncertain system. We now use the proposed method to examine if the uncertain system is stable.

Reconsidering (2), we obtain

m a 4 I G(jo) I I = 1.5

, 0 E 10, a] (0 .50)~ + (0.5 - w2)2 I J I = 0.50 '

w E [m, a 1

1.5 I o E [ m , w ] li (o.sw)2 + (1 .s - w2)2

and

min(arg(G(jw))] =

-arg(0.5 - w2 + j l SO), w E [o, ,4731 I -arg(O.5 - w2 +j0.5w), w E [a, 001

As in example 1, the unit feedback controller is used, that is K(jw) = 1. Then we can use theorem 1 to check the stability of the system. Though the gain condition (3) is violated when w < 1.66 in the gain plot of Fig. 2, one sees that the phase condition (4) will be satisfied for all frequencies in the phase plot of Fig. 2 . Thus the uncertain system is robustly stable.

5 Robust PID controller design

In the context of controller synthesis, one usually has a controller selected according to the nominal plant model. If the uncertain range of the plant model is known, then the above idea of robust stability may be used to design a controller for robust performance.

Since proportional-plus-integral-plus-derivative (PID) controllers are widely used in the process control industry and are easy to implement [5-71, the robust quantitative stability criterion proposed in Section 3 is used to tune a PID controller such that the desired closed-loop specifica- tions are satisfied. Gain and phase margins are typical control loop specifications associated with the frequency response technique [8]. Here, a PID tuning algorithm is developed which guarantees the required gain and phase margins of the closed-loop system for the whole family of the uncertain plant. The transfer function of the PID controller is given by

lo-' 100 frequency

Fig. 2 Plot ofmax{IGGm)I} and min{arg{GGw)}} in example 2

10'

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According to the preceding section, we can build a model G, representing the worst case of the uncertain process using (8) and (IO), that is

G,(jo) = max{ I G(jo)l)e' mln(arg(G(Jcu))l (1 3)

Let the desired closed-loop gain margin be A, and the phase margin be 4,, then the following two equations follow from the definition of gain and phase margins:

(14) 1

and

where wp and wg are the phase and gain crossover frequency of the cascaded process-plus-controller system respectively. The tuning objective is to find parameters Kp, Ki and Kd so that the given gain and phase margins are achieved, i.e. (14) and (15) are satisfied. It is noted that there are altogether five unknowns, namely, K p , K,, Kd, og and up, in (14) and (15). Since the equations are complex, they can be broken down into four real equations. Because the number of unknown exceeds the number of equations, there is no unique solution to (14) and (15) unless one more constraint is added.

It is well accepted in engineering practice that the closed-loop bandwidth should be close to its open-loop value. Thus the value of wp can be assigned near the plant phase-crossover frequency Wp, that is,

0 = a w P

The default value for a is 1. The solution to (1 4) and (1 5 ) is then [8]

I-

and

where os satisfies

To test the performance robustness of the closed-loop system for the tuned PID controller, we simply check both gain and phase margins of the worse case. The following corollary is derived from theorem 1.

Corollary 2: The phase margin of the uncertain process is larger than 4, if

arg(K(jo)l + m$4arg(G(jw)l) 2 - E + 4n, when maxG { IG(jo) I } 1 KGw) I > 1 and the gain margin of the uncertain process is larger than A, if

In summary, given an uncertain plant G(s), the PID controller K(s) in the form of (1 2) can be tuned for the worse case to meet the given gain and phase margins of A , and 4, as follows:

(i) Obtain the phase-crossover frequency of the worst- case model of the uncertain process from the frequency response. (ii) Set the op near that frequency. (iii) Compute K,, from (16). (iv) Find os from (1 8) by searching through the frequency range from cop downwards. (VI Compute K, and Kd from (17).

Example 3: Consider a second-order plus dead time uncertain process

G(s) = b e-Ls

s2 + a1s + a2

with the uncertain parameters as b E [0.8, 1.21, a l E [1.2, 1.81, a2 E [0.8, 1.21 andL E [4,6]. We obtain from (8)-(1 I),

1.2 1 0 E [O, 2/08]

(1.20)~ + (0.8 - ~ I . I ~ ) ~ I J {E 1.20 ' 0 E [a, m]

1.2 , w E [ m , W l I,; (1.242 + (1.2 - w2)2

min(IG(jco)l) =

0.8

(1 .2w)2 + (1.2 - w q 2

0.8 I,; (1 .2w)2 + (w2 - 0.q2

min{ arg(G(jw)) } =

-arg(0.8 - co2 +j1.80) - 6 0 ,

-arg(jl.20 + 0.8 - 02) - 6 0 ,

0 E [O, m] 0 E [m, co]

and

max{arg(G(jo))} =

-arg( 1.2 - w2 + j ~ 20) - 40,

-arg(jl.gw + 1.2 - w2) - 40,

0 E [o, w E [m, 001

The worst-case model is then set according to (13). The phase-crossover frequency of G,(s) is 0.3890 rad s-'. If we set the gain and phase margins for the nominal plant to be 3 and 60" respectively, and choose cop the same as the plant phase-crossover frequency, cop = 0.3890. Then the propor- tional gain K, can be readily calculated using (15) to be

-l ] =0.222 3 x G,(j0.3890)

Kp = Re when arg{K('jw)} +mint {arg{G('jw)}} < --n.

6 IEE Proc.-Control Theory Appl., Vol. 149, No. 1. January 2002

The designed PID controller is hence

0.071 K(s) = 0.222 + - + 0.4784s

S

-50 r

” 6 -200 - 2 -250 - E-300 - -350 - -400

10-2 lo-’ 1 00 frequency

Fig. 3 example 3

Plot of max{lG(jm)K(j‘w)l} and min{arg{G(p)K(jw)I}} in

1 0 -

08-

0 6 -

0 4 -

0 2 -

I

0 10 20 30 40 50 60 70 80 90 100

Fig. 4 Step response of the uncertain system in example 3

The frequency range from below o = op is then searched and it is found that wg = 0.1047, which satisfies (1 8). The values of Ki and Kd are then calculated using (17) to be

r 1 7 - 1 0.3890

0.1047 0.1047

0.0710

= [ 0.47841

Then the robustness of the designed controller is checked. In the gain plot of Fig. 3, it can be seen that (G,(jw)K(jw)l > 1 when o < 0.1047. In the phase plot of Fig. 3, one sees that at the same frequency interval 0<0.1047, arg{G,(jo)K(jo)} 2 -71++qh,=-120°. According to corollary 2, the uncertain system satisfies the phase margin specification. We can check that the gain margin specification is also satisfied using a similar method. Thus the tuned controller meets the condition for robust performance of the closed-loop system. With different values for b, a , , a2 and L, the step responses of the uncertain system are shown in Fig. 4.

6 Conclusion

In this paper, a new criterion for robust stability has been developed which makes use of both the gain and phase information of the open-loop system in order to reduce the conservativeness of the small gain theory. Based on it, a robust PID design is presented for a SOPDT uncertain model. Simulation shows the effectiveness of the method.

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