before tuned
Chapter 3 H ∞ Design for the Network Delay SystemsDelay Systems
3.3 H ∞ NCS Design Procedures
3.3.1 Performance Target
A design flow for H∞ NCS controller is applied to a servo motor system [17] with an identified plant P that
P (s) = V (s) × 1
s = 180.1s + 10000 s2+ 190s + 1000 ×1
s (3.19)
where V (s) is the build-in velocity-loop plant. The actuator generates output voltage calculated from the controller output to the motor and the rotor position is fed back from sensor. When the measurement of the rotor position is transmitted through network to a remote controller, an equivalent delay is determined by network transmission. Before H∞ controller is designed, since delay decreases the phase margin and causes instability, a lead controller is used to restore phase margin. Let a lead controller be designed as
Klead = 6.442 s + 7.7495
s + 18.36244 (3.20)
that it satisfies system performance specifications listed in Table 3.1. An H∞ controller is designed to further improve performance of the lead controller Klead by specifying
performance specifications.
3.3.2 Design Weighting Functions
Specification of this example is provided in Table 3.1. From Eq. (3.8), since the larger peak value of kSk easily causes ripples, the maximum value is set as Ms = 1. According to the requirement that the bandwidth must be larger than 2.4(Hz), ωs is set to be 2.32 with εs = 10−6 so that the sensitivity function approaches zero at a low frequency. The calculated Ws becomes
Ws = s + 2.32
s + 2.32 × 10−6. (3.21)
According to the requirement of disturbance rejection and Eq. (3.9), let ωr = 72 and assign εs an arbitrary small number with Mr = 1. Besides, a low-pass filter is cascaded to template Wr because there is an integrator already in P (s). Thus, the weighting function Wr becomes
Gather the designed weighting functions and apply to MATLAB function ’mixsyn’ with plant transfer function P , a H∞ controller is generated that
Khinf = 7.727e − 4s5+ 5564s4 + 3.332e6s3+ 6.39e5s2+ 6063s + 0.06057
s6+ 781.9s5+ 1.238e5s4+ 8.632e6s3+ 3.994e6s2 + 3.916e4s + 0.09083 (3.24) with γ = 1.956. System performances are listed in Table 3.1. It can be found that the system with the controller Khinf fits the bandwidth specification, but delay tolerance and disturbance rejection requirements are unsatisfied. This means weighting functions need to be further adjusted.
3.3.3 Adjust H ∞ Algorithm Property
Generally when the stability of a NCS is considered, a control strategy is to maintain a proper phase margin for the closed-loop system since the delay time may vary in a NCS.
Table 3.1: Specification for system performances.
Klead desired Khinf freq. crossover(rad/sec) 2.87 12.5
phase margin 101.3◦ 80.2◦
max Td(sec) 0.616 0.6 0.11 bandwidth(rad/sec) 2.39 >2.4 15.6
|KS(j100)|(dB) 16.28 <0 19.1
p l a n t controller
r y
n e t w o r k + u
-f o r w a r d c o m p e n s a t o r
Figure 3.6: New NCS control structure with a forward compensator.
However, due to the fact that the phase lagged caused by network delay as
∠e−jωθ = −jωθ, (3.25)
the delay-induced phase lag increases as frequency increases. To maintain the same phase margin, it needs a lead compensation in the higher frequency range. Controllers designed by H∞ algorithm only considers magnitude response and it actually decreases open-loop gain to maintain system stability. When open-loop gain within some frequency band is lower than one, the input sinusoidal signals within this frequency band can tolerate arbitrarily large network delay without causing system instability in closed-loop system responses.
However, reducing open-loop gain slows system response and there is a trade-off between delay tolerance and bandwidth, even with the H∞ algorithm. To conquer this problem, a forward compensator is applied before system input node, as shown in Fig.
3.6. Therefore, H∞ algorithm is used to satisfy the delay tolerance requirement, and a forward compensator is designed to improve system performance.
Firstly, adjust weighting functions to meet stability requirement. Let ωs = 0.1 and
−50
Figure 3.7: Sensitivity functions and weighting functions. (Solid: weighting functions;
dotted: sensitivity functions.)
θmax= 36.4, and a new controller is generated that Khinf_mod = P aisi
P bisi (3.26)
with γ = 1.4527 and
ai = h
0.0015 1.082e4 2.056e6 1.083e8 7.026e6 5.949e4 0.5942 i
, bi = h
1 299.6 3.409e6 1.747e6 3.37e7 6.505e7 8.373e5 0.08372 i
. (3.27)
System performance indices are listed in Table 3.2. It can be found that delay tolerance is enlarged with the trade-off as a low bandwidth. Frequency responses for sensitivity and weighting functions are shown in Fig. 3.7.
3.3.4 Order Reduction for the Controller
The order of controller designed by applying the H∞ algorithm equals to the summation of orders of plant P and all weighting functions Ws, Wr and Wt, and is generally much
−80
−60
−40
−20 0 20
Magnitude (dB)
10−8 10−6 10−4 10−2 100 102 104
−180
−90 0 90
Phase (deg)
Bode Diagram
Frequency (rad/sec) original
reduced
Figure 3.8: Frequency responses before and after controller order reduction. (Solid: before reduction; dotted: after reduction.)
higher than it requires. By observing magnitude in the frequency response, it can be found that not all poles/zeros play major on control performance or system stability.
Therefore, other reduction methods can be also applied, such as pole-zero cancellation.
From the Bode plot of the controller, there are two pairs of nearby pole and zero in high frequency(> 102(rad/sec)) and low frequency(< 10−4(rad/sec)) that they have no significant effects on system performance and stability. Performing pole zero cancellation to these two pairs of pole/zero can reduce the controller order without degrading system responses. After pole-zero cancellation, a fourth-order controller is acquired
Khinf_loop = 104.9764(s + 0.05495)(s2 + 190s + 10000)
(s + 40.84)(s + 2.147)(s2+ 153.3s + 7128) (3.28) The frequency responses for controller and after order reduction is shown in Fig. 3.8.
Their similarity within specified frequency bands can be found.
Figure 3.9: Illustration for H∞ NCS controller design.
Table 3.2: Performance comparison.
desired without pre-filter with pre-filter
freq. crossover(rad/sec) 2.74 2.74
phase margin 121◦ 121◦
max Td(sec) 0.6 0.772 0.772
bandwidth(rad/sec) >2.4 0.07 6.42
|KS(j100)|(dB) <0 -1.6 -1.6
3.3.5 Forward Compensator Design
To enhance delay tolerance, the H∞ algorithm tents to lower open-loop gain and causes a magnitude lack in the high frequency region, as shown in Fig. 3.9. It can be compensated by a forward compensator. After H∞ controller order reduction, a forward compensator is then designed that
Khinf_pre = s/0.009561 + 1 s/0.01 + 1
s/0.03578 + 1 s/0.05495 + 1
(3.29) which is basically a high-pass filter with two pole-zero pairs as shown in Fig. 3.10, and aims on compensating magnitude response lack within some frequency range. The maximum gain value is 4.12(dB), and system performance with a forward compensator is listed in Table 3.2.