Robustness analysis and performances

has all its roots in the open left half plane (LHP), wherep is the degree ofN(s).

For the SOPTD process, it follows from (5.39) that the closed-loop stability requiresH(s) =s³θ_n1³ T+ (9θ_n1² T+θ³_n1+θ³_n1T)s²+ (18T θ_n1+ 6θ_n1² + 6θ_n1² T+θ_n1³ )s+ 6T+ 6θ_n1+ 6T θ_n1+ 3θ²_n1 be stable where T = τ₂/τ₁. Then, the constant term 6T+ 6θ_n1+ 6T θ_n1+ 3θ²_n1

> 0 is necessary, which leads to θn1 >√

1 +T²−T−1. Thus, the controllerG_cd1could stabilizeGp1if and only ifθn1 >√

1 +T²−T−1.

Similarly, for UFOPTD process, it follows from (5.39) that the the closed-loop stability requires H(s) =s²θ³_n1 + (6θ_n1² −θ_n1³ )s+ 6θ_n1−3θ²_n1 be stable. Then, the zero order term 6θ_n1−3θ_n1²

> 0 is necessary, which requires θ_n1 < 2. Therefore, the controller G_cd1 can stabilize G_p1 if and only if θ_n1 <2.

Using the same technique, the stabilization conditions for the remaining primary loop processes are derived and summarized in the Table 5.4.

Table 5.4: Stabilizability results

Sl. No. Process Condition on the normalized time delay

1. FOPTD ∀θn1>0

2. SOPTD θn1>

q

1 + (τ2/τ1)²−(τ2/τ1)−1

3. UFOPTD θn1<2

4. IPTD ∀θn1>0

5. DIPTD ∀θn1>0

5.4.2 Stabilization of secondary loop controller, G_cd2

As the secondary process is considered as FOPTD, ∀θ_n2>0 the secondary process is stabilizable by the PID controllerG_cd2.

5.5 Robustness analysis and performances

The robust stability of the closed loop can be analyzed taking into account that G_p1(s) belongs to a family of linear models described as G_p1(s) =G_pn1(s) (1 +δG_p1(s)) [129]. The nominal model is represented by G_pn1(s) and δG_p1(s) is a multiplicative description of the modeling errors. Consider the closed-loop characteristic equation

1 +G_cd1(s)G_p1(s) = 1 +G_cd1(s) (G_pn1(s) (1 +δG_p1(s))) = 0

For all the plants, it is necessary that

|δG_p1(jω)|< dG_p1(ω) = |1 +G_cd1(jω)G_pn1(jω)|

|G_pn1(jω)| ∀ω≥0 (5.40)

for the bounds of the multiplicative uncertainties (δG_p1(ω)) and where

δGp1(ω)< dGp1(ω) ∀ω ≥0 (5.41) The functiondG_p1(ω) is the upper bound of the multiplicative modelling error of the plant to guarantee stability and it can be used as a measure of the controller robustness. Here, dG_p1(ω) is a rational function because the Smith predictor has eliminated the dead time. This, in general, allows a better trade-off to be achieved between robustness and performance. The types of uncertainties considered here are the parametric uncertainties such as uncertainty in process gain, time constant and time delay. According to the Small gain theorem [93], the closed loop system for the load disturbance rejection is robustly stable if and only if

kδm(jω)T_d(jω)k<1 ∀ω∈(−∞,∞) (5.42) where T_d(s = jω) is the closed loop complementary sensitivity function and δ_m(jω) is the process multiplicative uncertainty bound i.e. δ_m(jω) = |(G_p(jω)−G_m(jω))/G_m(jω)|. The complementary sensitivity function of the closed loop system consists of G_cd1 whose parameters Kc1,Ti1 and T_d1 are functions of the tuning parametersωgcand Ψ1. For the process gain uncertainty, the tuning parameter should be selected in such a way that

kT_d(jω)k∞< 1

|∆k|/k ∀ω >0 (5.43)

For the process time dealy uncertainty, the tuning parameter should be selected in such a way that kT_d(jω)k_∞< 1

|e^−j∆θω−1| ∀ω >0 (5.44) If uncertainty exists in both process gain and time delay, the tuning parameter should be selected in such a way that

kT_d(jω)k_∞< 1

(1 + ^∆k_k )e^−j∆θω−1

∀ω >0 (5.45)

5.5 Robustness analysis and performances

According to robust control theory [91], the closed loop performance for load disturbance rejection is robust, if

kδ_m(jω)T_d(jω) +w₁(jω)(1−T_d(jω))k_∞<1 (5.46) where w₁(jω) is the weight function of the sensitivity function, S_d(jω) = 1−T_d(jω). Therefore, the tuning parameters should be selected such that the resulting controller satisfy the robust nominal performance and robust stability constraints.

5.5.1 Maximum sensitivity (M_s) to modeling error

The maximum sensitivity has a nice geometrical interpretation in the Nyquist diagram for ro- bustness and stability. M_s is the inverse of the shortest distance from the Nyquist curve of the loop transfer function to the critical point (-1, 0) on the complex plane. The maximum sensitivity

M_s1 = max

ω |S₁(jω)|= max

ω

1

|1 +G_cd1G_p1| (5.47)

for primary loop and

M_s2 = max

ω |S₂(jω)|= max

ω

1

|1 +G_cd2G_p2| (5.48)

for secondary loop. For a single loop system, the recommended values forM_s are typically within the range 1-2 [130]. The use of the maximum sensitivity as a robustness measure, has the advantage that lower bounds to the gain and phase margins can be assured according to

A_m> M_s

M_s−1 (5.49)

φ_m >2 arcsin 1

2Ms

(5.50) A small value of M_s indicates that the stability margin of the control system is large. Responses obtained with M_s≈1 show little or no overshoot, as is desirable in process control. Faster responses are obtained with M_s ≈ 2. The settling time at load disturbances is significantly shorter with the larger value ofMs. On the other hand, these responses are oscillatory with larger overshoots.

5.5.2 Model mismatch consideration

As the modified Smith predictor has been used in the outer loop of parallel cascade control struc- ture, it is necessary to analyze the model mismatch condition. The over sensitivity problem during

design. If the controller tuning is too tight, the closed-loop system may become unstable with a small model mismatch in the model parameters. Now, we will examine the closed-loop stability property of the proposed controller design with some model mismatch of G_m1 and G_m2.

Assuming the true transfer function of the secondary process is G_p2 = ^k_τ^2e−θ2s

2s+1 and that of primary IPTD process is G_p1 = ^k1e−θ_s ^1s. Due to lack of precise knowledge of the actual system parame- ters, the model transfer functions are guessed wrong with gains, time constants and time delays:

G_m2 = ^(k2+∆k_(τ ^{2)e−(θ2+∆θ2)s}

2+∆τ2)s+1 and G_m1 = ^(k1+∆k1)e_s^{−(θ1+∆θ1)s}. Kharitonov’s theorem is the well-known and simplest tool for robust stability analysis [110, 112]. This theorem has been used for the robust- ness analysis considering parametric uncertainties in the model parameters. Now, consider a cascade system with primary and secondary process transfer functions (see Example-5.3): Gp1 = ^2e−2s_s and Gp2 = ^4e_s+1^−s. The model transfer functions are assumed with +10% change in gains, time constants and time delays simultaneously: G_m1 = ^2.2e−2.2s_s and G_m2 = ^4.4e_1.1s+1^−1.1s. From (5.1), the closed-loop characteristic equation of the primary loop is given by

(1 +Gm) (1 +Gp1G_cd1+Gp2G_cd2+Gp1G_cd1Gm2G_cd2) = 0 The four Kharoitonov polynomials are obtained as

K₁(s) = 0.0001 + 0.007s+ 0.2196s²+ 2.6045s³+ 16.2225s⁴+ 66.8261s⁵+ 197.5344s⁶+ 405.2495s⁷ +582.2001s⁸ + 620.8209s⁹+ 477.2083s¹⁰+ 234.4039s¹¹+ 68.5753s¹²+ 11.0409s¹³+ 0.7056s¹⁴ K₂(s) = 0.0001 + 0.0073s+ 0.2109s²+ 2.5023s³+ 16.8847s⁴+ 69.5537s⁵+ 189.7879s⁶+ 389.3573s⁷

+605.9633s⁸ + 646.1606s⁹+ 458.4943s¹⁰+ 225.2116s¹¹+ 71.3743s¹²+ 11.4915s¹³+ 0.6779s¹⁴ K₃(s) = 0.0001 + 0.007s+ 0.2109s²+ 2.6045s³+ 16.8847s⁴+ 66.8261s⁵+ 189.7879s⁶+ 405.2495s⁷

+605.9633s⁸ + 620.8209s⁹+ 458.4943s¹⁰+ 234.4039s¹¹+ 71.3743s¹²+ 11.0409s¹³+ 0.6779s¹⁴ K₄(s) = 0.0001 + 0.0073s+ 0.2196s²+ 2.5023s³+ 16.2225s⁴+ 69.5537s⁵+ 197.5344s⁶+ 389.3573s⁷

+582.2001s⁸ + 646.1606s⁹+ 477.2083s¹⁰+ 225.2116s¹¹+ 68.5753s¹²+ 11.4915s¹³+ 0.7056s¹⁴ The coefficients of Kharitonov polynomials are checked for Hurwitz condition. It is observed that all the roots of the Kharitonov polynomials (Table 5.5) have negative real part i.e. all roots are in the left half of the complex plane. From the Figure 5.5, it is clear that Kharitonov rectangles move

5.5 Robustness analysis and performances

point (0,0). Since the origin is excluded from the Kharitonov rectangles (Figure 5.5) it is concluded that the closed-loop control system is robustly stable. By following a similar procedure as above the robustness analysis considering parametric uncertainties in the model parameters for FOPTD, UFOPTD, DIPTD and SOPTD can be checked.

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

x 10²⁶ -0.5

0 0.5 1 1.5 2 2.5

3x 10²⁶

Real Axis

Imaginary Axis

(0, 0)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 5.5: Kharitonov’s rectangle for +10% estimation error in the parameters ofGm1andGm2

Table 5.5: The roots of Kharitonov polynomials

K1(s) K2(s) K3(s) K4(s)

−6.60 −7.41 −5.19 +j1.65 −8.01

−2.35 +j2.03 −4.76 −5.19−j1.65 −3.28

−2.35−j2.03 −0.53 +j1.58 −2.62 −1.10 +j1.98

−2.14 −0.53−j1.58 −1.33 −1.10−j1.98

−0.12 +j0.84 −1.72 −0.27 +j1.17 −1.21

−0.12−j0.84 −0.92 −0.27−j1.17 −0.03 +j0.63

−0.84 −0.01 +j0.50 −0.62 −0.03−j0.63

−0.51 −0.01−j0.50 −0.03 +j0.39 −0.67

−0.07 +j0.30 −0.49 −0.03−j0.39 −0.39

−0.07−j0.30 −0.27 −0.38 −0.10 +j0.20

−0.27 −0.12 +j0.12 −0.16 +j0.06 −0.10−j0.20

−0.16 −0.12−j0.12 −0.16−j0.06 −0.19

−0.02 +j0.02 −0.03 +j0.02 −0.02 +j0.02 −0.02 +j0.02

5.5.3 Performance

To evaluate the closed-loop performance, we consider three popular performance specifications based on integral error (e(t) = r(t)−y(t)) such as the integral absolute error (IAE=

R∞

0 |e(t)|dt), the integral time-weighted absolute error (ITAE=

R∞ 0

t|e(t)|dt) and the integral square error (ISE=

∞

R

0

e(t)²dt) criteria.

To evaluate the manipulated input, we compute the total variation (TV) of the input u(t) i.e.TV=

∞

P

i=1|u_i+1−u_i|, which should be as small as possible. The TV is a good measure of smoothness of a signal [94].