Decentralized stable and robust fault-tolerant PI plus fuzzy control of MIMO systems: a quadruple tank case study

This is a new laboratory process, which was designed to illustrate performance limitations due to zero location in multivariable control systems. The process is called the quadruple tank process (Johansson, 2000) and consists of four interconnected water tanks and two pumps. The system is shown in Figure 1. Its manipulated variables are voltages to the pumps and the controlled variables are the water levels in the two lower tanks. The quadruple tank process can easily be built by using two double-tank processes. The output of each pump is split into two using a three-way valve. Thus, each pump output goes to two tanks, one lower and another upper, diagonally opposite, and the ratio of the split up is controlled by the position of the valve. With change in the position of the two valves, the system can be appropriately placed either (Johansson, 2000) in the minimum phase or in the non-minimum phase. The physical parameters of the process given by Johansson (2000) are shown in Table 1. The material balance for the quadruple tank process is given by Equations (1) to (4). Note that λ₁ and λ₂ are the ratios in which the outputs of the two pumps get divided. The sampling time of the process is assumed as 1 sec.

Figure 1:

The quadruple tank level process (QTLP) is used to illustrate many concepts in MIMO systems. The quadruple tank laboratory equipment consists of four interacting tanks (1, 2, 3, and 4), two-way valves, two pumps, and a reservoir as shown in Figure 1. Tanks 1 and 2 are in the lower, while tanks 3 and 4 in the upper. The flow is delivering to tanks 1 and 3 from a reservoir by pump 1, while pump 2 sucks the flow and delivers it to the other tanks. The two-way valves after each pumping are used to divide the flow to lower and upper tanks by a factor λ_i and (1 − λ_i) which is fixed during the experiment. The input voltages U₁ and U₂ to the pumps are varied during the experiment according to the required controlled outputs (the liquid levels in the lower tanks 1 and 2). A reservoir is used to accumulate the outgoing water from tanks 1 and 2 and is present in the bottom.

The operation of QTLP can be in two phases: minimum phase and non-minimum phase. The system starts operating in non-minimum phase when the fraction of liquid entering the lower tanks is less than that of upper tanks. Otherwise, the system starts operating in the minimum phase when the fraction of liquid entering the upper tanks is less than that of lower tanks. The minimum phase and non-minimum phase can be achieved as:Minimum phase : 1 < ( λ 1 + λ 2 ) < 2 , Non - minimum phase : 1 < ( λ 1 + λ 2 ) < 1.

The nonlinear model of QTLP is based on the mass balances for each tank and the differential equations are formulated as: (1) d d t h 1 ( t ) = a 1 A 1 2 g h 1 + a 3 A 1 2 g h 3 + λ 1 k 1 A 1 υ 1 , (2) d d t h 2 ( t ) = a 2 A 2 2 g h 2 + a 4 A 2 2 g h 4 + λ 2 k 2 A 2 υ 2 , (3) d d t h 3 ( t ) = − a 3 A 3 2 g h 3 + ( 1 − λ 2 ) k 2 A 3 υ 2 , (4) d d t h 4 ( t ) = − a 4 A 4 2 g h 4 + ( 1 − λ 1 ) k 1 A 4 υ 1 , where A_i is the cross-section of tank i, i = (1, 2, 3, 4); a_i the open cross-section of the outlet line valve; h_i the water level; υ_i the voltage applied to pump; k_i v_i the flow from the pump; λ_i the position of the valve; and g the gravitational constant.

where α₁ and α₂ are the coefficients of actuator faults for CV₁ and CV₂, respectively. The value of coefficient α₁, α₂∈[0, 100%]. f₁ and f₂ are a leak flow rate from the bottom of tanks 1 and 2. The leak flow rate varies from 0 to 50.3691 cm³/min. The third fault introduced into QTLP is a sensor fault, which interprets the sensor bias fault because of sensor calibration or measurement noise. Due to sensor fault, erroneous measurement of tank height h₁ and h₂ is propagated to the controller and hence a wrong control action is taken from the controller (Himanshukumar and Vipul, 2018g). The sensor fault introduced into the QTLP via software and the value is between 0 and 40% for both sensors 1 and 2. The fault is introduced into the system QTLP via the MATLAB R2015a software package.

The linearized state-space model about operating point is shown in Table 1 represented by the following equation: (9) d x d t = [ − 1 T 1 0 A 2 A 1 T 2 0 0 − 1 T 2 0 A 4 A 2 T 4 0 0 − 1 T 3 0 0 0 0 − 1 T 4 ] X + [ λ 1 k 1 A 1 0 0 λ 2 k 2 A 2 0 1 − λ 2 k 2 A 3 1 − λ 1 k 1 A 4 0 ] u , (10) y = [ 1 0 0 0 0 1 0 0 ] X , (11) T i = A i a i 2 h i 0 g i =1,2,3 and 4 , h}_{i}^{0}}{g}}\,i\mathrm{=1,2,3}\,{\rm{and}}\,4,\end{eqnarray}?> (12) G ( s ) = [ α 1 λ 1 T 1 s + 1 α 1 ( 1 − λ 2 ) ( T 1 s + 1 ) ( T 3 s + 1 ) α 1 ( 1 − λ 1 ) ( T 2 s + 1 ) ( T 4 s + 1 ) 2 λ 2 T 2 s + 1 ] , (13) α 1 = T i k i A i 2 h i 0 g i =1,2. h}_{i}^{0}}{g}}\,i\mathrm{=1,2.}\end{eqnarray}?>

For quadruple tank system (12), we consider the uncertainty to be a change of valve position, i.e. change of λ₁ and λ₂. The uncertainty domain is specified by three working points (Fig. 2): 1.

in the minimum phase region: WP1: λ₁ = 0.4, λ₂ = 0.8; WP2:λ₁ = 0.7, λ₂ = 0.6; WP3: λ₁ = 0.8, λ₂ = 0.8; and

Figure 2:

Open-loop response of minimum and non-minimum phases for WP2 region is found and the proposed approaches are implemented for the minimum phase and non-minimum phase region of WP2 for three faults into QTLP (Table 2).

The nominal model G_mp(s) and G_nmp(s), obtained as a model of mean parameter values, is used for control design (Figs. 3 and 4): (14) [ h 3 0 h 4 0 ] = [ − a 1 ( λ 2 − 1 ) a 3 L 1 − a 2 λ 1 a 3 L 2 − a 1 λ 2 a 4 L 1 − a 2 ( λ 1 − 1 ) a 4 L 2 ] [ h 1 0 h 2 0 ] , h}_{3}^{0}}\\ \sqrt{{h}_{4}^{0}}\end{array}]=[\begin{array}{ll}\frac{-{a}_{1}({\lambda }_{2}-1)}{{a}_{3}}{L}_{1} & \frac{-{a}_{2}{\lambda }_{1}}{{a}_{3}}{L}_{2}\\ \frac{-{a}_{1}{\lambda }_{2}}{{a}_{4}}{L}_{1} & \frac{-{a}_{2}({\lambda }_{1}-1)}{{a}_{4}}{L}_{2}\end{array}][\begin{array}{c}\sqrt{{h}_{1}^{0}}\\ \sqrt{{h}_{2}^{0}}\end{array}],\end{eqnarray}?> (15) [ υ 1 0 υ 2 0 ] = [ a 2 λ 1 2 g k 2 ( λ 2 − 1 ) L 2 a 1 a 4 2 g a 3 k 1 a 2 a 3 2 g a 4 k 2 λ 2 2 g k 1 ( λ 1 − 1 ) L 1 ] [ h 1 0 h 2 0 ] , \upsilon }_{1}^{0}}\\ \sqrt{{\upsilon }_{2}^{0}}\end{array}]=[\begin{array}{ll}\frac{{a}_{2}{\lambda }_{1}\sqrt{2g}}{{k}_{2}({\lambda }_{2}-1)}{L}_{2} & \frac{{a}_{1}{a}_{4}\sqrt{2g}}{{a}_{3}{k}_{1}}\\ \frac{{a}_{2}{a}_{3}\sqrt{2g}}{{a}_{4}{k}_{2}} & \frac{{\lambda }_{2}\sqrt{2g}}{{k}_{1}({\lambda }_{1}-1)}{L}_{1}\end{array}][\begin{array}{l}\sqrt{{h}_{1}^{0}}\\ \sqrt{{h}_{2}^{0}}\end{array}],\end{eqnarray}?> (16) G m p ( s ) = [ 2.6381 66 s + 1 1.5631 ( 25 s + 1 ) ( 65 s + 1 ) 1.4256 (33 s + 1)(92 s + 1) 2.8125 92 s + 1 ] , (17) G n m p ( s ) = [ 3.3692 ( 25 s + 1 ) ( 65 s + 1 ) 0.5352 66 s + 1 0.6812 92 s + 1 3.9856 ( 33 s + 1 ) ( 92 s + 1 ) ] .

Figure 3:

Figure 4:

Decentralized control is an entrenched way to compact the multi-input–multi-output MIMO plants with control. To outline a decentralized controller, matching between input and output must be dictated by RGA. The method exchanges of open-loop and closed-loop control structures are assessed for all conceivable input/output pairings. The array will be a matrix with one row for each output variable and one column for each input variable in the MIMO framework. A few principles for blending determination can be expressed: the variable pairings comparing to positive relative gains as near unity as conceivable ought to be favored. Negative relative gain much longer than unity ought to be evaded. The simplest case of control circuit of two input two output (TITO) system is shown in Figure 5.

Figure 5:

Consider a MIMO plant described by the linear model: (18) y ( s ) = G ( s ) u ( s ) , where complex vectors y(s) and u(s) are the Laplace images of output and input signal of dimensions p and m, respectively; G(s) the transfer function matrix of dimensions ×m. In the following, we assume the square system, i.e. p = m and stable plant G. Argument s is often omitted for better readability.

Our aim is to design an appropriate decentralized control, so that the overall system stability is kept (including possible uncertainties) and the required performance is achieved.

We focus on two most important steps in decentralized control design as follows: 1.

the determining of appropriate input–output pairing; and

After completing step 1, the inputs or outputs can be reordered so that the respective transfer system matrix G with reordered columns or rows has the paired elements on the main diagonal. Then, the decentralized controller can be represented by the diagonal matrix C(s) = diag(C_i). To find C(s), the so-called independent design is considered, where individual loops are designed independently (simultaneously). Local controllers C_i(s) are designed so that they: 1.

stabilize individual loops;

Note that conditions 2 and 3 are often contradictory. In the following, sensitivity is denoted as S(s) = (I + G(s)C(s))¹ and closed-loop transfer function (complementary sensitivity) is denoted as T(s) = G(s)C(s)(I + G(s)C(s))¹.

To choose appropriate pairing, several interaction measures have been proposed in the literature (RGA, dRGA, PRGA, etc.); more details can be found, e.g., in the study of Schmidt (2002). Relative gain array (RGA), frequently used in practice, is defined as follows: (19) R G A ( Λ ) = G ( 0 ) × ( G ( 0 ) − 1 ) T , where (×) denotes the element-by-element product of the two matrices (Hadamard product).

Individual subsystems are then specified by the chosen pairing, and their transfer functions are placed in the diagonal of the transfer function matrix. The structural stabilizability for the chosen pairing can be checked by the Niederlinski index: (20) I = d e t ( G ( 0 ) ) ∏ ( d i a g ( G ( 0 ) ) ) ) .

If N I < 0, the system cannot be stabilized using the chosen pairing and the pairing must be modified. It must be noted that RGA index provides limited information, e.g., for the system with one-way interconnections (when the transfer function matrix is upper or lower triangular).

The RGA concept is employed to QTLP to determine the input–output pairing for both minimum phase and non-minimum phase: (21) R G A ( Λ ) = [ 1.4 0.4 0.4 − 0.64 ] ← Minimum phase , (22) R G A ( Λ ) = [ 0.64 1.6 1.6 1.4 ] ← Non - minimum phase .

For the minimum phase system, λ₁₁ is obtained as 1.4, so the pairing is determined as y₁ − u₁ and y₂ − u₂. But for the non-minimum phase system, λ₁₁ is obtained as 0.64, so the suitable pairing is found as y₁ − u₂ and y₂ − u₁.

After the appropriate pairing has been determined, the decentralized control law is to be designed. There are various approaches to find the respective diagonal controller matrix C(s). We adopt independent design as a simple possibility to design single loops so that the overall stability and performance requirements are kept, i.e. that interactions do not introduce instability and do not significantly deteriorate performance. Let us turn to stability condition for a system with decentralized control. Matrix G(s) can be split into its diagonal and off-diagonal parts: G(s) = G_D(s) + G_M (s).

The uncertainties can be included into G_M (s). For stable open-loop system G(s)C(s), the closed-loop system stability condition based on small gain theorem is given in the next Lemma (Vesel and Harsnyi, 2008): Lemma 1.

(Vesel and Harsnyi, 2008) Consider stable system G(s) with a decentralized controller C(s). The respective closed-loop system T (s) is stable if:(23)(24)

Inequality (24) can be reformulated into: (25) ‖ G D − 1 T D ‖ < M 0 = 1 ‖ G M ‖ , G}_{D}^{-1}{T}_{D}\Vert \lt {M}_{0}=\frac{1}{\Vert {G}_{M}\Vert },\end{eqnarray}?> where T_D = G_D C(I + G_D C)⁻¹.

Condition (25) can be used for stable system without or with RHP zeros (both for minimum and non-minimum phases case). However, the above condition can be rather limiting in low frequencies, where ||T_D ||≈1; for a stable system with no RHP zeros, this may be too restrictive. The alternative condition for this case is in Lemma 2 (Skogestad and Postlethwaite, 2009): Lemma 2.

(Skogestad and Postlethwaite, 2009) Consider a stable system G(s) with a decentralized controller C(s). Assuming that neither G nor G_D has RHP zeros, the overall closed-loop system is stable if and only if (I − ES_D)⁻¹ is stable, where:

The above condition can be reformulated as follows: (I + ES_D)⁻¹ stable means det(I + ES_D)⁻¹ ≠ 0. The sufficient stability condition is then ||ES_D|| < 1 or (26) ‖ G − 1 S D ‖ < M 0 = 1 ‖ G M ‖ .

Either of alternatives (25) or (26) must be satisfied for all frequencies.

The performance relative gain array (PRGA) has been introduced (Hovd and Skogestad, 1992) and shown to provide information for appropriate pairing, but also performance limits for system with decentralized control. PRGA is defined as: (27) P R G A ( G ) = Γ = G D ( s ) G ( s ) − 1 .

There is a close relationship between PRGA and closed-loop system performance specified by bounds on control error (offset) and disturbance presented as follows: (28) | e i ( j w ) / r j ( j w ) | = | S i ( j w ) | < 1 / | w r i ( j w ) | ∀ w , i , j , (29) | e i ( j w ) / z j ( j w ) | = | S G z | i k ( j w ) < 1 / | w z i ( j w ) | ∀ w , i , k , where r_j denotes the jth set-point change; S_ij is the respective element of sensitivity function S; z_k is the expected disturbance; G_z is the transfer function; w_ri, w_zi are the performance weights for control error and disturbance, respectively.

For frequencies where a feedback is effective (w ¡ w_B, w_B denotes bandwidth), it is assumed that S = (I + GC)¹ ≈ (GC)¹ yield the following bounds for individual loop: (30) | g i i ( j w ) C i ( j w ) | > | η i j w r i ( j w ) | ∀ w < w B , ∀ i , j , where η_ij are elements of the PRGA index Γ. (31) | g i i ( j w ) C i ( j w ) | > | δ i k w z i ( j w ) | ∀ w < w B , ∀ i , k , where δ_ik are the elements of Γ G_z.

Inequalities (30) and (31) determine performance limits lower bounds on single-loop modules to achieve the required control error and disturbance attenuation, and the former is discussed in the control design stage.

Fuzzy logic control is derived from the fuzzy set theory introduced by Zadeh (1965). In the fuzzy set theory, the transition between membership and non-membership can be gradual. Therefore, boundaries of fuzzy sets can be vague and ambiguous, making it useful for an approximate system. Combining multi-valued logic, probability theory, and knowledge base, FLC is a digital methodology that simulates human thinking by incorporating the imprecision inherent in all physical systems. Fuzzy logic controller is an attractive choice when precise mathematical formulations are not possible (Buckley and Ying, 1989; Driankov et al., 1993). The decentralized fault-tolerant fuzzy plus PI control structure includes two fuzzy MISO controllers and two PI controllers. In the proposed control method for the quadruple tank process, two fuzzy logic controllers are used separately for controlling the level outputs. The structure of the proposed decentralized fault-tolerant fuzzy plus PI controller is shown in Figure 6.

Figure 6:

The characteristics of the designed decentralized fault-tolerant fuzzy plus PI control system are shown in Figures 7 to 12 for minimum phase configuration of QTLP. Figure 7 shows that stability condition (27) is not satisfied; in this case, the black line is for low frequencies above the red one; however, the overall stability is guaranteed by (for this case) the less restrictive condition (28 and 29), which is satisfied since the blue line is below the red one for all frequencies.

Figure 7:

Figure 8:

Figure 9:

Figure 10:

Figure 11:

Figure 12:

Figure 13:

Performance bounds: upper bounds on weights obtained from (30) are depicted in Figure 8.

The simulation results of proposed controller compared with those of controllers in four different cases. The PI controller parameters designed for individual loops are: P₁ = 1.30, I₁ = 0.053; P₂ = 1.38, I₂ = 0.049. The characteristics of the designed decentralized control system are shown in Figures 7 to 11. Figure 7A shows that stability condition (25) is not satisfied; in this case, the blue line is for low frequencies above the red one, however, the overall stability is guaranteed by (for this case) the less restrictive condition (26), which is satisfied since green line is below the red one for all frequencies:

Case i: QTLP with process disturbances.

The proposed controller is tested on a QTLP nonlinear system subject to three different faults (i.e. actuator, sensor, and system component fault) with minimum phase configuration. All three faults are given to QTLP one after the other at the same time, t = 600 sec. The simulation results of the proposed decentralized controller compared with decentralized fuzzy control are proposed in the study of Suja and Thyagarajan (2008), which is depicted in Figures 9 to 12. The simulation results clearly show that the proposed controller gives guaranteed stability with superior steady-state and transient response as compared to decentralize fuzzy control. The proposed controller is capable enough to accommodate all three possible faults into QTLP efficiently. A comparison of the integral error indices for non-minimum phase configuration is depicted in Figure 13.

This configuration is characterized by the existence of transient RHP zeros (while individual transfer functions have no RHP zeros), which complicates the decentralized controller design. We illustrate the impact of interactions on two different designs of control loops.

In the first case, taking the same decentralized controller as in minimum phase case, the overall stability condition is not satisfied, though the individual loops indicate stable performance. Step responses in Figure 13 show significant differences between individual loops (both are stable and damped) and the overall system, which is unstable.

The next (detuned) case: P₁ = 0.208, I₁ = 0.0039; P₂ = 0.238, I₂ = 0.0030 show that as soon as condition (25) is satisfied (Fig. 14A: blue line below the red one), the overall system responses are similar to single-loop ones Figures 16 to 19; performance indicators are still satisfactory for low frequencies (Fig. 14B).

Figure 14:

Figure 15:

Figure 16:

Figure 17:

Figure 18:

Figure 19:

Now the proposed decentralized fault-tolerant control is applied on QTLP with three faults and process disturbances (Fig. 15):

Case i: QTLP with process disturbances.

The proposed controller is tested on a QTLP nonlinear system subject to three different faults (i.e. actuator, sensor, and system component fault) with a non-minimum phase configuration. All three faults are given to the QTLP one after the other at the same time t=650 sec. The simulation results of the proposed decentralized controller are compared with the decentralized fuzzy control proposed in the study of Suja and Thyagarajan (2008), which is depicted in Figures 16 to 19. The simulation results clearly show that the proposed controller gives guaranteed stability with superior steady-state and transient response as compared to decentralize fuzzy control. The proposed controller is capable enough to accommodate all three possible faults into QTLP efficiently. The comparison of integral error indices for non-minimum phase configuration is depicted in Figure 19.