Analysis of Linear Control Systems
Theorem 3.6. For the closed-loop system with an open-loop transfer function G(s) to be stable, the Nyquist plot of G(s) must encircle in a counterclockwise direction the point
3.6 Introduction to Model Reduction Techniques
3.6.4 State Space Model Reduction
Balanced realization method
Suppose that the balanced realization of the original model can be partitioned as
˙x1
and assume that the states in subvector x2are to be chopped off. Then, the reduced model is written as the following:
˙x1= [A11− A12A−122A21]x1+ [B1− A12A−122B2]u,
y = [C1− C2A−122A21]x1+ [D − C2A−122B2]u. (3.58) A functionmodred()implementing the above algorithm is provided in the Control Systems Toolbox with syntax Gr=modred(G,elim), where G is the balanced realized state space object andelimcontains the states to be dropped off. The reduced model Gr is then returned.
Example 3.43. Consider again the system model in Example 2.22. To get a second-order reduced model, the following MATLAB statements can be used to obtain the Gramian of the balanced realized system model:
>> G=tf([1,7,24,24],[1,10,35,50,24]); [Gb,g]=balreal(ss(G)) where the Gramians g = [0.5179, 0.0309, 0.0124, 0.0006]T. Clearly, the contribution to the input-output relationship from the third and fourth states is not very important.
Thus, it is safe to eliminate them to get a second-order reduced model using the following
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Magnitude (dB)
10−2 10−1 100 101 102 103
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Phase (deg)
Bode Diagram
Frequency (rad/sec)
Gr(s) G(s)
G(s) Gr(s)
(a) Bode diagrams
0 1 2 3 4 5 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Step Response
Time (sec)
Amplitude
(b) Step responses
Figure 3.36. Bode diagram and step response comparisons.
MATLAB statements:
>> Gr=modred(Gb,[3,4]); zpk(Gr), bode(G,Gr), figure, step(G,Gr)
The reduced model is then
Gr(s)=0.025974(s+ 4.307)(s + 22.36) (s+ 1.078)(s + 2.319) .
The Bode diagram and step response comparisons are shown in Figure 3.36 and it can be seen that the fitting is satisfactory. It should also be noted that the reduced-order system is not strictly proper, and hence we have the small initial jump in the step response.
Schur’s balanced realization truncation method
Schur’s balanced realization truncation functionschmr()provided in the Robust Control Toolbox can perform a model reduction task similar tomodred(). The difference between the two techniques is that an unstable system can be handled inschmr(). The syntax of schmr()is Gr=schmr(G,1,nr), where G is the original model object in state space format, nr is the expected order of the reduced model, and Gr returns the reduced-order model object also in the state space form.
Example 3.44. Consider again the plant model in Example 3.41. To apply the Schur model reduction algorithm, the state space model of the system should be obtained first. This can be done using the following MATLAB statements:
>> num=[68.6131,80.3787,67.087,29.9339,8.8818,1];
den=[0.0462,3.5338,16.5609,28.4472,21.7611,7.6194,1];
G=ss(tf(num,den)); Gr=zpk(schmr(G,1,3))
It is indicated that three states are removed, and the third-order reduced model using Schur’s method can be written as
Gr(s)=1485.3076(s2+ 0.1789s + 0.2601) (s+ 71.64)(s2+ 3.881s + 4.188) .
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Figure 3.37. Bode diagram and step response comparisons.
Optimal Hankel norm approximation
Glover presented an algorithm to find the optimal Hankel approximation to a given state space model [49]. The reduced-order model using the Hankel norm approximation algorithm can be obtained with the MATLAB functionohklmr()provided in the Robust Control Toolbox. The syntax of the function is Gr=ohklmr(G,1,k), where G is the original model object in state space format, k is the expected order of the reduced-order model, and Gr returns the reduced-order model object in state space.
Example 3.45. Consider again the plant model in Example 3.41. The third-order reduced model using the optimal Hankel norm approximation method can be obtained as follows:
>> num=[68.6131,80.3787,67.087,29.9339,8.8818,1];
den=[0.0462,3.5338,16.5609,28.4472,21.7611,7.6194,1];
G=ss(tf(num,den)); Gh=zpk(ohklmr(G,1,3))
and it is indicated that three states are removed, and a reduced-order model is then returned as
Gh(s)= 1527.8048(s2+ 0.2764s + 0.2892) (s+ 73.93)(s2+ 3.855s + 4.585) .
For the same original model, the optimal approximation can also be obtained by
>> Go=zpk(opt_app(G,2,3,0)),
bode(G,Go,Gr,Gh,{0.1,10000}), figure, step(G,Go,Gr,Gh,8) where the reduced-order model is
Go(s)= 1523.6536(s2+ 0.3492s + 0.2482) (s+ 74.85)(s2+ 3.871s + 5.052) .
The Bode diagram and step response comparisons are shown in Figure 3.37. It can be seen that they all fit satisfactorily into the original model. Among all three models, the optimal reduced-order model is significantly better than the other two reduced models.
Problems
1. Check the stability for the following systems:
(a) G(s)= 1
10.8689 1.2900 19.099 21.896 3
11 18.089799 25 2.356 9
2. Find the poles and zeros of the multivariable system and check the stability of the system. If unity negative feedback is assumed, check the stability of the closed-loop
system
3. Find the controllability index and observability index of the state space models in the previous problem. Obtain the controllable and observable staircase forms.
4. Find the controllable and observable decompositions of the systems given by
(a) ˙x =
5. Perform the Kalman decomposition of the system model given by
˙x =
and write down the transformation matrix. From the Kalman decomposition of the system, obtain the minimum realization in the state space model. Give an explanation of the minimum realization from the transfer function point of view.
6. Compute the first three time moments and Markov parameters for the models given in Problems 1 and 2.
7. Determine theH2- andH∞-norms of the following systems:
(a) G1(s)= 3s+ 5
8. Find the analytical solution to the autonomous system
Compare the results with numerical results.
9. An eighth-order model G(s) is given by
18s7+514s6+5982s5+36380s4+122664s3+222088s2+185760s+40320 s8+36s7+546s6+4536s5+22449s4+67284s3+118124s2+109584s+40320. Assume that the system has zero initial conditions. Find the analytical and numerical solutions of the system under step and impulse inputs. Also assume that the input signal is sinusoidal u(t) = sin(3t + 5). Assume again the system has zero initial conditions. Find the analytical solutions to the system response and verify the results by graphical comparison.
10. Draw the step response of the system
dif-ferent sampling intervals of T , find the equivalent discrete-time system and compare the overshoot and settling time.
11. Draw the root locus diagrams of the following systems and determine the range of K which stabilizes the open-loop system with unity negative feedback:
(a) G(s)= K(s+ 6)(s − 6)
(e)
12. Assume the plant model
G(s)=K(s− 1)e−2s (s+ 1)5 .
Find the approximate range of K which stabilizes the closed-loop system with unity negative feedback.
13. The open-loop transfer function is given by
G(s)= K
(s+2)(s+4)(s2+6s+25).
Find the range of K to make the closed-loop system with unity negative feedback stable. Also find the value of K which gives the closed-loop system a dominant damping ratio of 0.707.
14. Draw the Bode diagrams, Nyquist plots, and Nichols charts for the following systems, and check the stability of the systems under unity negative feedback control from the plots obtained. Mark the gain and phase margins on the plots obtained. Verify the results through closed-loop step responses.
(a) G(s)= 8(s+ 1)
(f) G(s)= 10s3− 60s2+ 110s + 60 s4+17s3+82s2+130s+100.
15. Draw nonlinearly transformed Nyquist plots for the systems containing integrators in the previous problems and see whether the same conclusion can be obtained.
16. Assume that a plant model is given by G(s)= 1/s2, and an optimal controller can be expressed as
Gc(s)= 5620.82s3+ 199320.76s2+ 76856.97s + 7253.94 s4+ 77.40s3+ 2887.90s2+ 28463.88s + 2817.59.
Also assume unity negative feedback. Draw the Nyquist and Nichols plots and super-impose the M and N circles on the diagrams. Plot the closed-loop frequency response of the system and confirm that the magnitude of the peak and its corresponding phase are in agreement with the deductions from the Nyquist and Nichols plots.
17. Assume that the plant model is
G(s)= 100(1+ s/2.5) s(1+ s/0.5)(1 + s/50), and a cascade controller is given by
Gc(s)= 1000(s+ 1)(s + 2.5) (s+ 0.5)(s + 50) .
Assess the closed-loop behavior of the system under unity negative feedback control.
Verify the assessment by time domain analysis.
18. For the feedback system structures with (a) G(s)= 3.5(s+ 6)
s(s+ 1)(s + 3)(s + 8), Gc(s)= (5s + 4)/s, H (s) =0.01s+ 6 2s+ 4 ,
(b) G(s)= 3.5(s+ 6)2
(s+ 1)(s + 3)(s + 8)(s2+ 3s + 6), Gc(s)=5s+ 4
6s+ 2, H (s)= 1.
By definition, the sensitivity of the feedback system can be defined as S(s)= 1/[1 + H (s)G(s)Gc(s)], and the complimentary sensitivity can be defined as T (s) = 1−S(s).
Find the sensitivity and complementary sensitivity transfer functions.
19. Find reduced-order models for the following original models using different algorithms presented in this chapter:
(a) G(s)= 10+ 3s + 13s2+ 3s2 1+ s + 2s2+ 1.5s3+ 0.5s4,
(b) G(s)=500+ 9984.3234s + 50664.9675s2+ 8169.1337s3 500+ 10105s + 52101s2+ 10520s3+ 100s4 ,
(c) G(s)= 1+ 0.4s
1+ 2.283s + 1.875s2+ 0.7803s3+ 0.125s4+ 0.0083s5, (d) G(z)= 24.1467z3− 67.7944z2+ 63.4768z − 19.8209
z4− 3.6193z3+ 4.9124z2− 2.9633z + 0.6703. 20. Consider a high-order model given by
G(s)= (1+ 2.0587s)(1 + 2.5529s + 5.4342s2)(1+ 3.2648s + 2.1476s2) (1+3.0092s+.7970s2)(1+6.8538s+0.6965s2)(1+.1394s+0.6861s2). The (2/3)-order FF-Padé reduced model given in [43] is
GF F2/3(s)= 1− 1.4257s + 4.3109s2 1+ 0.7003s + 0.8613s2+ 0.0837s3.
Meanwhile, a correspondence to the paper given by Stahl and Hippe [50] suggested a (3/4)-order model
G3/4(s)= 62.85(s+ 2.64)(s + 0.192 + 0.608j)(s + 0.192 − 0.608j) (s+ 8.09)(s + 5.75)(s + 0.1 + j1.2)(s + 0.1 − j1.2) . Using the optimum order model given in the chapter, compare the reduced-order model with the above existing reduced-reduced-order models.