2. Classical Control System Design
2.3. Integral control
Integral control is a remarkably effective classical technique to achieve low-frequency disturbance attenuation. It moreover has a useful robustness property.
Disturbance attenuation is achieved by making the loop gain large. The loop gain may be made large at low frequencies, and indeed infinite at zero frequency, by including a factor 1/s in the loop gain L(s) = P(s)C(s). If P(s) has no “natural” factor 1/s then this is accomplished by including the factor in the compensator transfer function C by choosing
C(s)= C0(s)
s . (2.20)
The rational function C0remains to be selected. The compensator C(s) may be considered as the series connection of a system with transfer function C0(s) and another with transfer function 1/s. Because a system with transfer function 1/s is an integrator, a compensator of this type is said to have integrating action.
If the loop gain L(s) has a factor 1/s then in the terminology of §2.2(p.60) the system is of type 1. Its response to a step reference input has a zero steady-state error.
Obviously, if L0(s) contains no factor s then the loop gain
L(s)= L0(s)
2.3. Integral control
is infinite at zero frequency and very large at low frequencies. As a result, the sensitivity function S, which is given by S(s)= 1 1+ L(s) = 1 1+L0(s) s =s s + L0(s) , (2.22)
is zero at zero frequency and, by continuity, small at low frequencies. The fact that S is zero at zero frequency implies that zero frequency disturbances, that is, constant disturbances, are completely eliminated. This is called constant disturbance rejection.
e
v
z L
Figure 2.3: Feedback loop
Exercise 2.3.1 (Rejection of constant disturbances). Make the last statement clearer by prov- ing that if the closed-loop system of Fig.2.3has integrating action and is stable, then its response z (from any initial condition) to a constant disturbancev(t)= 1, t ≥ 0, has the property
lim
t→∞z(t)= 0. (2.23)
Hint: Use Laplace transforms and the final value property.
The constant disturbance rejection property is robust with respect to plant and compensator perturbations as long as the perturbations are such that
• the loop gain still contains the factor 1/s, and • the closed-loop system remains stable.
In a feedback system with integrating action, the transfer function of the series connection of the plant and compensator contains a factor 1/s. A system with transfer function 1/s is capable of generating a constant output with zero input. Hence, the series connection may be thought of as containing a model of the mechanism that generates constant disturbances, which are precisely the disturbances that the feedback system rejects. This notion has been generalized (Wonham,
1979) to what is known as the internal model principle. This principle states that if a feedback system is to reject certain disturbances then it should contain a model of the mechanism that generates the disturbances.
Exercise 2.3.2 (Typekcontrol). The loop gain of a type k system contains a factor 1/sk, with k a positive integer. Prove that if a type k closed-loop system as in Fig.2.3is stable then it rejects disturbances of the form
v(t)=t n
n!, t≥ 0, (2.24)
with n any nonnegative integer such that n≤ k − 1. “Rejects” means that for such disturbances
Compensators with integrating action are easy to build. Their effectiveness in achieving low frequency disturbance attenuation and the robustness of this property make “integral control” a popular tool in practical control system design. The following variants are used:
• Pure integral control, with compensator transfer function C(s)= 1
sTi
. (2.25)
The single design parameter Ti (called the reset time) does not always allow achieving closed-loop stability or sufficient bandwidth.
• Proportional and integral control, also known as PI control, with compensator transfer function C(s)= g 1+ 1 sTi , (2.26)
gives considerably more flexibility.
• PID (proportional-integral-derivative) control, finally, is based on a compensator transfer function of the type
C(s)= g sTd+ 1 + 1 sTi . (2.27)
Tdis the derivative time. The derivative action may help to speed up response but tends to make the closed-loop system less robust for high frequency perturbations.
Derivative action technically cannot be realized. In any case it would be undesirable be- cause it greatly amplifies noise at high frequencies. Therefore the derivative term sTd in (2.27) in practice is replaced with a “tame” differentiator
sTd
1+ sT, (2.28)
with T a small time constant.
Standard PID controllers are commercially widely available. In practice they are often tuned experimentally with the help of the rules developed by Ziegler and Nichols (see for instance
Franklin et al.(1991)). The Ziegler-Nichols rules (Ziegler and Nichols,1942) were developed under the assumption that the plant transfer function is of a well-damped low-pass type. When tuning a P-, PI- or PID-controller according to the Ziegler-Nichols rules first a P-controller is connected to the plant. The controller gain g is increased until undamped oscillations occur. The corresponding gain is denoted as g0and the period of oscillation as T0. Then the parameters of the PID-controller are given by
P-controller: g = 0.5g0, Ti = ∞, Td= 0, PI-controller: g = 0.45g0, Ti = 0.85T0, Td= 0, PID-controller: g = 0.6g0, Ti = 0.5T0, Td= 0.125T0.
The corresponding closed-loop system has a relative damping of about 0.25, and its closed-loop step response to disturbances has a peak value of about 0.4. Normally experimental fine tuning is needed to obtain the best results.
2.3. Integral control e u v z PID- controller nonlinear plant
Figure 2.4: Nonlinear plant with integral control
Integral control also works for nonlinear plants. Assume that the plant in the block diagram of Fig.2.4has the reasonable property that for every constant input u0there is a unique constant steady-state outputw0, and that the plant may maintain any constant outputw0. The “integral controller” (of type I, PI or PID) has the property that it maintains a constant output u0if and only if its input e is zero. Hence, if the disturbance is a constant signalv0then the closed-loop system is in steady-state if and only if the error signal e is zero. Therefore, if the closed-loop system is stable then it rejects constant disturbances.
Example 2.3.3 (Integral control of the cruise control system). The linearized cruise control system of Example1.2.1(p.3) has the linearized plant transfer function
P(s)= 1
T
s+1θ. (2.29)
If the system is controlled with pure integral control C(s)= 1
sTi
(2.30) then the loop gain and sensitivity functions are
L(s)= P(s)C(s) = 1 T Ti s(s+θ1), S(s)= 1 1+ L(s) = s(s+1θ) s2+1 θs+ 1 T Ti . (2.31)
The roots of the denominator polynomial s2+1
θs+ 1 T Ti
(2.32) are the closed-loop poles. Sinceθ , T and Ti are all positive these roots have negative real parts, so that the closed-loop system is stable. Figure2.5shows the loci of the roots as Ti varies from∞ to 0 (see also §2.7(p.88)). Write the closed-loop denominator polynomial (2.32) as s2+ 2ζ0ω0s+ ω20, withω0the resonance frequency andζ0the relative damping. It easily follows that ω0= 1 √ T Ti , ζ0= 1/θ 2ω0 = √ T Ti 2θ . (2.33)
The best time response is obtained forζ0=12 √
2, or
Ti = 2θ2
0
Ti ↓ 0 Im
Re −1θ −12θ
Figure 2.5: Root loci for the cruise control system. The×s mark the open- loop poles
If T = θ = 10 [s] (corresponding to a cruising speed of 50% of the top speed) then Ti = 20 [s]. It follows thatω0 = 1/
√
200≈ 0.07 [rad/s]. Figure2.6shows the Bode magnitude plot of the resulting sensitivity function. The plot indicates that constant disturbance rejection is obtained as well as low-frequency disturbance attenuation, but that the closed-loop bandwidth is not greater than the bandwidth of about 0.1 [rad/s] of the open-loop system.
Increasing Ti decreases the bandwidth. Decreasing Ti beyond 2θ2/ T does not increase the bandwidth but makes the sensitivity function S peak. This adversely affects robustness. Band- width improvement without peaking may be achieved by introducing proportional gain. (See
Exercise2.6.2, p.85.) 101 100 10−1 10−2 10−3 10−2 10−1 100 20 dB 20 dB -20 dB -40 dB |S|
angular frequency [rad/s]
Figure 2.6: Magnitude plot of the sensitivity function of the cruise control system with integrating action
Exercise 2.3.4 (PI control of the cruise control system). Show that by PI control constant disturbance rejection and low-frequency disturbance attenuation may be achieved for the cruise control system with satisfactory gain and phase margins for any closed-loop bandwidth allowed
by the plant capacity.
Exercise 2.3.5 (Integral control of a MIMO plant). Suppose that Fig.2.3(p.65) represents a stable MIMO feedback loop with rational loop gain matrix L such that also L−1is a well-defined rational matrix function. Prove that the feedback loop rejects every constant disturbance if and
only if L−1(0)= 0.
Exercise 2.3.6 (Integral control and constant input disturbances). Not infrequently distur- bances enter the plant at the input, as schematically indicated in Fig.2.7. In this case the transfer