Robust MIMO water level control in interconnected twin-tanks using second order sliding mode control M. Khalid Khan and Sarah K. Spurgeon ∗ Control and Instrumentation Group, Department of Engineering University of Leicester, Leicester LE1 7RH (U.K.) Tel: +44 116 2522531; Fax: +44 116 2522619

Abstract A novel second order sliding mode control algorithm is presented for a class of MIMO nonlinear systems in input-output (I-O) form. The algorithm produces a dynamic control and does not require the derivative of the sliding variable, thus eliminating the requirement to design observers or peak detectors. The algorithm has been applied for robust control of liquid level in interconnected twin-tanks. The controller is implemented on a laboratory rig and the results presented validate the proposed theory. The implementation results show robustness to parameter variations such as tank area, the admittance coefficients of various pipes, leakage in the tanks and uncertainty in the pump dynamics. In order to contextualise the results obtained with second-order sliding mode control, the results obtained using a classical PI controller are also presented. Key words: second order sliding mode control, robust control PACS: 07.05.Dz

1

Introduction

Sliding Mode Control (SMC) is known to be a robust control method appropriate for uncertain systems. High robustness is maintained against various kinds of uncertainties such as external disturbances and measurement error [7, 23]. It is also straightforward to implement the resulting algorithms. ∗ Corresponding author Email address: [email protected] (Sarah K. Spurgeon). URL: http://www.le.ac.uk/eg/staff/spurgeon.htm (Sarah K. Spurgeon).

Preprint submitted to Elsevier Science

3 February 2005

Sliding mode control for multivariable linear systems as well as nonlinear systems is well documented in the literature [6, 7, 23]. Various SISO control techniques have been extended to MIMO systems by linearising the system and then designing linear controllers [4, 8, 22, 24]. Most of these methods rely on transformation of the nominal system to regular form. However, it is not straightforward to convert nonlinear systems to an appropriate regular form [15, 23]. Further, if an appropriate canonical form for sliding mode control of nonlinear systems is adopted, a zero dynamics will result from the sliding mode controller design and the stability of the resulting closed-loop system not only depends upon the reduced order system but also on the stability of this zero dynamics [8, 13]. Proper differential input output representations are selected as the canonical form for sliding mode controller design in this paper. It follows that dynamic sliding mode control will always result when a sliding surface having relative degree two with respect to the control is selected. Such dynamic sliding policies are desirable from the point of view of chattering removal. Dynamic sliding regimes are considered to naturally provide a dynamic control [2–4, 18, 20, 21]. However, in [21] dynamic surfaces are considered only in the traditional sliding mode sense and produce a dynamic controller only if the system contains certain control derivatives (u(β) , β ≥ 1) in the I-O representation for removal of chattering. This is not the case for all systems which are linearisable by state feedback. The work in [18–20] asymptotically linearises the plant and produces a dynamic controller provided a control dependent ∂s sliding surface is considered i.e. ∂u 6= 0, where s denotes the sliding variable. The recently developed concept of Higher Order Sliding Modes (HOSM) [9, 21] is the generalisation of so called first order sliding modes or classical sliding mode control. In HOSM control, the control acts on higher derivatives of the sliding variable. For example, the case of second order sliding modes corresponds to the control acting on the second derivative of the sliding variable, namely s¨, and the sliding set is defined as s = s˙ = 0. Several such second order sliding algorithms for SISO systems have been presented in the literature [1, 17]. A second order sliding mode control is proposed for MIMO systems in [4] where a dynamics is imposed on the sliding variable but the resulting controller requires the derivative of the sliding variable for implementation. Bartolini et al. [2] have also extended their SISO algorithm for a class of nonlinear MIMO systems. However, the algorithm still requires singular values of the sliding variable (maxima, minima or flex points) to be detected online. The method proposed in this paper uses second order sliding mode control where the sliding surface may or may not be control dependent and always produces a dynamic controller in conjunction with the canonical form representation, contrary to the work in [18] and [21]. The proposed method combines the 2

elements from [21] and [18] together with [2, 4] and is applicable to a wider class of system which are not necessarily affine in the control. Moreover, the controller does not require the derivative of the sliding surface vector to be available. Section 2 presents a brief review of the second order sliding mode control design approach for single input single output (SISO) systems. The methodology is extended and applied to the multiple input multiple output (MIMO) situation in Section 3. Section 4 describes the twin tank process of interest. Nonlinear simulation results are presented in Section 5 and the results of controller implementation on an experimental rig are presented in Section 6. The experimental results validate the proposed theoretical approach. Section 7 presents a study of the relative performance of a second-order sliding mode controller when compared to that obtained by a PI controller.

2

Second order sliding mode control of SISO systems

In traditional sliding mode control, or First Order Sliding Mode (FOSM) controller design, an appropriate sliding variable, s(t), is selected such that it has relative degree one with respect to the control. This selection of sliding variable must be made to ensure that the dynamics of the system in the sliding mode, i.e. when s = 0, are desirable. The control then acts on the first derivative (with respect to time) of the sliding variable (s) ˙ to keep the system trajectories in the sliding set s = 0. Essentially, the discontinuous control signal acts on the first derivative of s to ensure the system trajectories are always directed towards s(t) = 0. Higher Order Sliding Modes (HOSM) are the generalisation of FOSM. In HOSM control, a control is sought which acts on higher derivatives of the sliding variable. For example, the case of second order sliding modes corresponds to the control acting on the second derivative of the sliding variable, namely s¨ , and the sliding set is defined as s = s˙ = 0. Higher order sliding mode control has the advantage, when compared to FOSM control, that it removes chattering effects, providing a smooth or at least piece-wise smooth control, and provides better performance with respect to switching delays in the control implementation. In this study, the second order sliding mode control algorithm first presented in Khan et al. [16] for SISO systems will be used for multivariable control of the twin tanks system. To present the backround to the algorithm, first consider an uncertain SISO nonlinear system which is affine in the control u. x˙ = φ(t, x(t)) + γ(t, x(t))u(t);

(1)

where x ∈ X ⊂ ℜn is a state vector, u ∈ U ⊂ ℜ is a bounded input and t is the independent time variable. Select a sliding surface s = S(t, x) 3

(2)

such that by zeroing it, the control objective is achieved. For the type of liquid level control problem considered in this paper, an appropriate sliding surface could be defined to be the difference between the actual and desired liquid levels in a vessel, for example. Further assume that the sliding surface, s, has relative degree two with respect to the control input i.e. ∂ S(t, x)γ(t, x) = 0. ∂x Thus the system dynamics can be written in the following form s¨ = f (t, s, s) ˙ + g(t, s, s)u ˙

(3)

The dynamics in equation (3) are assumed to satisfy the following bounding conditions 0 < Gmin ≤ g(t, s, s) ˙ ≤ Gmax ;

|f (t, s, s)| ˙ ≤ F;

and |s| ≤ s0

(4)

where Gmin , Gmax , s0 and F are some positive constants. Essentially this is a requirement that the uncertainty levels in the process are bounded and that some worse case bounds on the uncertainty can be assumed. Khan et al. [16] applied a 2-sliding control algorithm to stabilise the dynamics (3). This algorithm has the advantage that it does not require any knowledge of the derivative of the sliding variable (s). ˙ The trajectories of the resulting 2-sliding algorithm are shown on the phase portrait of the sliding variable in Fig. 1. It is seen that s and s˙ both converge to zero. The algorithm providing this evolution of the sliding variable is given as follows u(t) = −λsign(s) + u2 (t)    −ku,

u˙ 2 =  

(5)

|u| > u0

−W sign(s),

|u| ≤ u0

The simplicity of the control structure is readily apparent with only four parameters requiring selection by the designer. The corresponding sufficient conditions on these parameters for finite time convergence are u0 >

F ; λ > u0 Gmin k, W > 0

(6)

The formulation described above was developed for the case of SISO systems. The application of the algorithm to the case of MIMO systems, as required for the twin-tanks employed here, will now be explored. 4

1.5

1



0.5

0

−0.5

−1

−1.5 −0.6

−0.4

−0.2

0

0.2

s

0.4

0.6

0.8

1

1.2

Fig. 1. The phase plot

3

A MIMO problem formulation

Consider a locally observable general MIMO nonlinear system in state space form x˙ = ψ(x, u, t) y = h(x, u, t)

(7a) (7b)

where x ∈ ℜn , u ∈ ℜm , y ∈ ℜp and ψ : ℜn × ℜm × ℜ+ → ℜn and h : ℜn × ℜm × ℜ+ → ℜp are smooth vector functions. The following locally equivalent differential I-O form exists (n1 )

y1

= φ1 (ˆ y , uˆ, t) ···

(8a)

yp(np ) = φp (ˆ y , uˆ, t)

(8b)

which can be written concisely as (ni )

yi

= φi (ˆ y , uˆ, t),

i = 1, · · · , p.

(9)

(β )

where uˆ = (ˆ u1 , · · · , uˆm ), and uˆj = (uj , · · · , uj j ); j = 1, · · · , m. Similarly P (n −1) yˆ = (ˆ y1 , · · · , yˆp ) and yˆi = (yi , · · · , yi i ) with pi ni = n. This representation is the same as the Local Generalized Controller Canonical Form (LGCCF) of Fliess [11]. A differential I-O system is called proper if (a) p = m, (b) all φi , i = 1, · · · , m are C 1 functions, (c) the following regularity condition is satisfied det

"

∂(φ1 , · · · , φm ) (β )

(β )

∂(u1 1 , · · · , um m )

#

6= 0.

A large class of nonlinear systems, especially mechanical systems, are naturally in this form. Additionally, a wider class of nonlinear systems, termed 5

as ‘differentially flat systems’, can be written in this form together with dynamic compensators which may be a chain of integrators in their simplest sense [10, 12]. Definition 1 (Zero Dynamics) The corresponding zero dynamics of the system model (9) is defined as φi (0, uˆ, t) = 0;

i = 1, · · · , p

(10)

The system (9) is called minimum phase if the zero dynamics (10) is uniformly asymptotically stable. This zero dynamics is the dynamics of the control and is a generalisation of definitions in [10]. It is different from the zero dynamics defined in [13], which is the dynamics of the uncontrolled states. Only proper minimum phase systems are considered in this paper. The system (9) can be written in the following generalised canonical form ξ˙1i = ξ2i ···

(11a)

ξ˙ni i −1 = ξni 1 ξ˙ni i = φi (ξ, uˆ, t), (ni −1)

where ξ i = (ξ1i , · · · , ξni i ) = (yi , · · · , yi

(11b) i = 1, · · · , m

(11c)

), i = 1, · · · , m and ξ = (ξ 1 , · · · , ξ m ).

3.1 Sliding surface design The sliding surface, s, is selected such that when it is made zero, the control objective is achieved. In the case of MIMO systems, the sliding surface is a vector which has the same dimension as that of the control vector u i.e., s ∈ ℜm . In general, the sliding surface can be selected as a nonlinear function of the system states such as s = Ψ(ξ, t). However, only linear sliding surfaces are considered here as follows si =

ξni i

+

nX i −1

cij ξji ,

j=1

i = 1, · · · , m

(12)

where the vector of constants (ci1 , ci2 , · · · , cini −1 ) are such that the polynomials λni i

+

nX i −1

cij λji = 0,

j=1

are Hurwitz. 6

i = 1, · · · , m.

(13)

It should be noted that the sliding surface in equation (12) is naturally control dependent if βi ≥ 1 for i = 1, · · · , m. However, as will be seen in this Section, for a dynamic controller to result as is required for chattering removal, no condition on β is imposed here. This is contrary to what is proposed in [21] where β ≥ 1 is required for the controller to be dynamic, and thus for chattering avoidance. Moreover, the definition (12) does not assume that the sliding surface is control dependent. Therefore, the methodology is different from what is proposed in [18], where the sliding surface is always control dependent. The first two successive time derivatives of the sliding variables (12) along the system trajectories are given by

s˙ i = φi (ξ, uˆ, t) +

cini −1 ξni i

+

nX i −2

i , cij ξj+1

i = 1, · · · , m

j=1

s¨i =

(14)

nX i −3 d i cij ξj+2 φi (ξ, uˆ, t) + cini −1 φi (ξ, uˆ, t) + cini −2 ξni i + dt j=1

βk nk m X m X X ∂φi (j+1) ∂φi k ∂φi X ξ + u + cini −1 φi (ξ, uˆ, t) + = k j+1 (j) k ∂t ∂ξ ∂u j k=1 j=0 k=1 j=1 k

+cini −2 ξni i +

nX i −3

i cij ξj+2

j=1

nk m X m βX m k −1 X ∂φi X ∂φi k ∂φi (j+1) X ∂φi (βk +1) = + ξ + u + u k k j+1 (j) (βk ) k ∂t k=1 j=1 ∂ξj k=1 j=0 ∂uk k=1 ∂uk k6=i

+cini −1 φi (ξ, uˆ, t) + cini −2 ξni i +

nX i −3

i cij ξj+2 +

j=1

(βi )

Let z i = (z1i , z2i , · · · , zβi i ) = (ui , u˙ i , · · · , ui for i = 1, · · · , m.

∂φi (β ) ∂ui i

(βi +1)

ui

(15)

(βi +1)

), z = (z 1 , · · · , z m ) and ui

= vi

The control u can be implemented using a chain of integrators with auxiliary input vi which can be written in Brunovsky canonical form as z˙ i = Gi z i + Hi vi ,

i = 1, · · · , m

Using equation (12), the system (11) can be written as ˙ ξˆi = Ai ξˆi + Bi si s¨i = fi (ξ, z, t) + gi (ξ, z, t)vi z˙ i = Gi z i + Hi vi , i = 1, · · · , m 7

(16a) (16b) (16c)

where ξˆi = (ξ1i , · · · , ξni i −1 ) and nk m m X m βX k −1 X ∂φi (βk +1) ∂φi X ∂φi k ∂φi (j+1) X uk u + + fi (ξ, z, t) = ξ + k j+1 (j) (βk ) k ∂t k=1 ∂uk k=1 j=1 ∂ξj k=1 j=0 ∂uk k6=i

+ cini −1 φi (ξ, uˆ, t) + cini −2 ξni i +

nX i −3

i cij ξj+2

j=1

gi (ξ, z, t) =

∂φi (βi )

∂ui

;

i = 1, · · · , m

and 



Ai =  

0 In−2  , −C

Gi = 

,





 0 Iβi −1 

0

 

0 Bi =    1

 

0 Hi =   1

The complete MIMO system (7) can be written as ˙ ξˆ = Aξˆ + Bs s¨ = f (ξ, z, t) + g(ξ, z, t)v z˙ = Gz + Hv

(17a) (17b) (17c)

where ξˆ = (ξˆ1 , · · · , ξˆm ) ∈ ℜn−m , s = (s1 , · · · , sm ) ∈ ℜm and v = (v1 , · · · , vm ) ∈ ℜm and A = diag(A1 , · · · , Am ) ∈ ℜ(n−m)×(n−m) , B = diag(B1 , · · · , Bm ) ∈ ℜ(n−m)×m , G = diag(G1 , · · · , Gm ) ∈ ℜβ+1×β+1 , H = diag(H1 , · · · , Hm ), f = (f1 , · · · , fm )T , g = (g1 , · · · , gm )T . Therefore, the original MIMO system produces three subsystems. The first subsystem is an (n − m) dimensional linear system driven by the sliding variable vector, s. This linear subsystem is stable as the polynomials in (12) are Hurwitz by design. The second subsystem represents the second order nonlinear uncertain dynamics of the m sliding variables. If bounding values of fi and gi (for i = 1, · · · , m) satisfy the conditions (4), the algorithm (5) can be applied for robust stabilisation of each channel independently. The third subsystem is a chain of integrators, the output of which provides the actual control. If the system (7) is linearizable by coordinate change and state feedback, the MIMO system representation in I-O form (9) will be independent of any control derivative, i.e., βj = 0,

j = 1, · · · , m 8

and therefore, the auxiliary control vector, v, which drives the sliding surface dynamics (17b) is simply the time derivative of the actual control vector u. Even though v is discontinuous, the actual control input, u, to the system will be smooth because of the integrator dynamics.

3.2 Controller Design Once the sliding surface dynamics in (17b) is stabilised to s = 0, the MIMO system dynamics under sliding motion is given by ˙ ξˆ = Aξˆ

(18)

which is stable because A is a block diagonal matrix with all diagonal blocks (Ai , i = 1, · · · , m) representing the stable dynamics (13). Though the system in (17b) is coupled with the states of the system in (17a), it can be decoupled if the vector field f (ξ, z, t) and the gain matrix g(ξ, z, t), even if uncertain, satisfy the following bounding conditions in any bounded domain |fi (ξ, z, t)| < Fi Gmi ≤ gi (ξ, z, t) ≤ GMi

(19a) (19b)

and the system can be written as y˙1i = y2i y˙2i = fi (ξ, z, t) + gi (ξ, z, t)vi

(20a) (20b)

This representation contains all those uncertainties which do not violate the bounding conditions (19). The algorithm (5) can be applied because the control vi always appears linearly. The controller parameters are selected as follows u0i > sup Fi /Gmi λi > u0i ki > 0, and Wi > 0

(21a) (21b) (21c)

Sometimes, the controller (21) may be conservative. To improve the conservatism, the conditions (19) imposed on the sliding surface dynamics can be relaxed as

Gm1i

|fi (ξ, z, t)| < Fi1 + Fi2 |y1i | + Gm2i |y1i | ≤ gi (ξ, z, t) ≤ GM1i + GM2i |y1i | 9

(22a) (22b)

and the controller parameters can be selected as follows Fi1 + Fi2 |y1i | Gm1i + Gm2 i |y1i | λi > u0i ki > 0, and Wi > 0

u0i >

4

(23a) (23b) (23c)

Liquid level control in coupled-tanks

In this case study, liquid level control of two connected water tanks as shown in Figure 2 is considered. The controller has been designed based on the theory presented above and the performance achieved will be compared with that obtained from a classical PI controller.

4.1 System description and modelling

The twin-tanks system consists of two small tanks mounted above a reservoir which provides storage for the water. Water is pumped into the bottom of each tank by two independent pumps. The pump only increases the liquid level and is not responsible for pumping the water out of the tank. It is assumed that the back pressure created by the water-head does not affect the flow rate of the pump significantly. The separating wall between the two tanks has two circular holes which together form the connecting pipe with admittance coefficient k1 ; this will be referred to as pipe k1 . Each tank in the twin tanks configuration is equipped with two outlet pipes of different radius. These two outlet pipes have admittance coefficients of k2 and k3 where k3 > k2 and will be referred to as pipe k2 and pipe k3 , respectively. Pipe k3 is connected parallel to pipe k2 but not shown in the Fig 2. The admittance coefficients of various pipes are assumed constant. The pump cannot fill the tank if both pipes k2 and k3 are opened simultaneously. Therefore, only pipe k2 is considered open for designing the controller. A leak is simulated by opening pipe k3 (and pipe k2 closed) which allows more outflow than pipe k2 . The schematic diagram of the system is shown in Figure 2. 10

u1 q1

u2 q2

C1, h1

C2, h2 k1

Tank 1

Tank 2

k2

Fig. 2. The plant of a liquid-level control system

The twin-connected tanks system is a nonlinear dynamical system and the governing dynamical equations can be written as [4] 1 k1 q |h1 − h2 | sign(h1 − h2 ) + q1 h˙ 1 = − A1 A1 k2 q 1 k1 q |h1 − h2 | sign(h1 − h2 ) − h2 + q2 h˙ 2 = A2 A2 A2

(24a) (24b)

where h1 and h2 are the total water heads in Tank 1 and Tank 2 respectively, which are the two outputs of interest and q1 , q2 , are the two inflows into the tanks. It is assumed that the capacities, A1 and A2 , of Tank-1 and Tank-2 respectively remain bounded as follows Aim ≤ Ai ≤ AiM ,

i = 1, 2.

where Aim and AiM are some positive constants. For the rig in the laboratory both tanks have the same cross-sectional area i.e. A1 = A2 = A. It has been seen in [5, 14] that in chemical plants, selecting the flow rate as an input is more effective than using flow as the input. Thus, if the flow rates are considered as the inputs, i.e. q˙1 = u1 and q˙2 = u2 , the system dynamics can be written in I-O form as ˙ ˙ ¨ 1 = − k1 qh1 − h2 + 1 u1 h 2A |h1 − h2 | A

˙ ˙ k2 ˙ 1 ¨ 2 = k1 qh1 − h2 − √ h h2 + u2 2A |h1 − h2 | 2 h2 A

(25a) (25b)

It seems from equations (25a) and (25b) that there is a discontinuity at h1 = h2 . In fact, at h1 = h2 , the system model is decoupled and satisfies 11

the following dynamic equations 1 h˙ 1 = q1 A 1 k2 q h2 + q2 h˙ 2 = − A A

(26a) (26b)

which can be written as the following form as required by the following controller synthesis. ¨ 1 = 1 u1 h A 2 ¨h2 = k2 − k2 q2 + 1 u2 2A2 2A2 A

(27a) (27b)

Admittance coefficients of various pipes are estimated using the following square-root-dynamics of a gravity drained tank √ (28) Ah˙ = −k h where k is the admittance coefficient of an outflow pipe connected to the bottom of a tank with cross-sectional area A. The solution of this differential equation is given by q k 2 h0 = T A where h0 represent the initial liquid level and √T denote the time taken in draining that volume through the pipe. The 2 h0 vs. T plot is linear and the pipe admittance coefficient k can be calculated using its gradient. To calculate the admittance coefficient of a particular pipe connected into the tanks, the liquid from various levels in the tank was allowed to fall freely under gravity through the pipe separately and data collected. This was then plotted in Figures 3(a), 3(b) and 3(c).

6

√ 2 h0

√ 2 h0

8

4

5

5

4

4

√ 2 h0

10

3

2

2

2

1

1

0

0

0

0

20

40

60

T

(a) Interconnecting pipe k1

3

0

20

40

60

80

100

T

(b) Outflow pipe k2

Fig. 3. Calibration of pipes

12

0

20

40

T

(c) Outflow pipe k3

60

Error may be introduced in estimating the value of k1 and k2 because the time taken in draining out the water is not measured to a great accuracy. In addition, it is difficult to judge when the water height has become zero because the water keeps trickling out for some time. To estimate k1 , Tank 1 is drained through Tank 2. This does not directly correspond to ‘drained under gravity’ and therefore the value of k1 is likely to be in error. Moreover, both the tanks have some pipes lying inside them so the cross sectional area is not constant as assumed during the experimentation. The online measurement of water level is also inaccurate because the thread to which the float is connected does not remain vertical when the water level in the tank is rising.

4.2 System constraints

The fluid flow into the tanks (q1 and q2 ) cannot be negative because the pumps can only pump water into the tanks. Therefore constraints on the inflow are given by q1 ≥ 0 q2 ≥ 0

and

(29a) (29b)

In the steady state, for constant water level set points, the respective derivatives must be zero separately i.e., h˙ 1 = h˙ 2 = 0. Therefore, using equations (24a) and (24b) in the steady state, the following algebraic relationship holds. √ 0 = −k1 ∆h + Q1 q √ 0 = k1 ∆h − k2 h2 + Q2

(30a) (30b)

where the steady state inflows (Q1 , Q2 ) are given by √ Q1 = k1 ∆h q √ Q2 = −k1 ∆h + k2 h2

(31a) (31b)

Using equation (31a), the constraint (29a) on the input can be reformulated as a constraint on the output set point as follows √

∆h ≥ 0 ⇒ h1 ≥ h2 13

(32)

Similarly, constraint (29b) can be reformulated using (31b) as follows q √ k2 h2 ≥ k1 ∆h k22 h2 ≥ k12 (h1 − h2 ) or (k12 + k22 )h2 ≥ k12 h1 k2 ⇒ h2 ≥ 2 1 2 h1 k1 + k2

(33)

Therefore, in order to satisfy the constraints (29a) and (29b) on the inflows for given values of the plant parameters k1 , k2 , the desired liquid levels in the tanks must satisfy the constraints (32) and (33) which can be combined as follows k12 h2 ≤1 (34) ≤ 2 2 k1 + k2 h1 For the plant available in laboratory, the water level in Tank 2, h2 , must not be theoretically less than 63.89% that of h1 to satisfy positive flow constraints. Second order sliding mode controller design for the laboratory system model will now be described.

5

Simulation study

The twin-tanks model given in equations (25a) and (25b) can be simply written as ¨ 1 = f1 (h1 , h2 , h˙ 1 , h˙ 2 ) + 1 u1 h A ¨ 2 = f2 (h1 , h2 , h˙ 1 , h˙ 2 ) + 1 u2 h A

(35a) (35b)

where k1 h˙ 1 − h˙ 2 q 2A |h1 − h2 | k2 k1 h˙ 1 − h˙ 2 q − √ h˙ 2 f2 (·) = 2A |h1 − h2 | 2 h2

f1 (·) = −

(36a) (36b)

Both tanks have the same area of 155.44 cm2 and k1 = 23.45 and k2 = 17.62. The set point water level is selected as h1d = h2d = 12cm. For the given plant, the functions Af1 (·) and Af2 (·) are plotted in Figure 4 and the bounding values are calculated by maximising all the possible variations as A|f1 (·)|max = 23.4 and A|f2 (·)|max = 33.7. 14

5

10

0

8

−5

6

−10

A2 f2 (·)

A1 f1 (·)

12

4

−15

2

−20

0

−25

−2

−30

−4 250

−35 250

200

250 150

200

250 200

0

100

50

50 0

150

100

100

50 h1

200 150

150

100

h1

h2

(a) The bounding value of the function A1 f1 (·)

50 0

0

h2

(b) The bounding value of the function A2 f2 (·)

Fig. 4. Bounding values

For both the tanks, error in the water level is selected as sliding surface i.e., si = hi − hid ;

i = 1, 2.

(37)

The sliding surface variable, si , in equation (37) has relative degree two with respect to the control, vi . The complete MIMO system according to the equations (17a-17c) can be written as s¨i = fi (hi , h˙ i ) + gi (hi , h˙ i )vi u˙ i = vi i = 1, 2.

(38a) (38b)

where f1 (·) and f2 (·) are according to equation (36) and g1 (·) = g2 (·) = 1/A. The control algorithm (5) is then applied to stabilise the decoupled dynamics (38a). The controller parameters u01 and u02 should be greater than A|f1 (·)|max and A|f2 (·)|max respectively. The controller parameters selected for the simulation study shown in the following figures are u01,2 = [25, 35], λ1,2 = [100, 150], k1,2 = [1, 1] and W1,2 = [10, 10], which satisfy the selection criteria discussed in the previous section. The inflows at equilibrium, (Q1 , Q2 ), are calculated using the following equations k1 q 1 |h1 − h2 | sign(h1 − h2 ) + Q1 A A k1 q k2 q 1 |h1 − h2 | sign(h1 − h2 ) − h2 + Q2 0= A A A

0=−

which yields Q1 = 0 and Q2 = 61.07. 15

(39a) (39b)

The simulation results in Figure 5 and 6 show good stabilisation of the water levels in both tanks. The value of inflow in both tanks settles to the steady state values shown in Figure 6. 14 12

h

1

10 8 6 4

0

5

10

15 time (seconds)

20

25

30

0

5

10

15 time (seconds)

20

25

30

14 12

h

2

10 8 6 4

Fig. 5. Stabilisation of water level in both tanks 300 250 Inlet flow, q

1

200 150 100 50 0 0

5

10

15 time (seconds)

20

25

30

0

5

10

15 time (seconds)

20

25

30

300 250 Inlet flow q

2

200 150 100 50 0

Fig. 6. The flows into both tanks

6

Controller implementation

The basic block diagram of the twin tank system is shown in Figure 7. The digital value of the liquid level (y) is compared with the desired digital level (r). From the block diagram in Fig 7, it seems that the controller should 16

be digital in nature. However, if the sampling interval is very small, then a continuous controller can be implemented in this setup. In this particular case the sample rate is 2.66 × 10−4 which is quite small and hence the controller can be implemented in a continuous sense.

Fig. 7. Block diagram of the twin tank system

Two separate potentiometers are used to measure the liquid levels in both tanks. Each potentiometer is interfaced with an A/D converter. The potentiometers are calibrated such that the zero level represents the rest point of the water level i.e., when the tank is nearly empty. The D/A converter is connected to the pump via a power amplifier. The controller code is written in C++ and runs on a Windows 98 PC with CPU speed 166 MHz. The

Fig. 8. The twin tank rig in the Control Laboratory

pump interfaces between the plant and the computer. It pumps water from the storage tank to the water tank when a voltage is applied. A flowmeter is connected in series with each pump to measure the flow. The flow rate of the pump is plotted against the digital control signal in Figure 10(a) and 10(b). It has been noticed that the pump characteristics are not linear for the whole digital input range of 0 − 255. It has been calculated that the gradient of Figure 9(a) is 17.1340 digits/cm and that of Figure 9(b) is 17.2543 digits/cm. 17

300

250

250

Digital value of h2

Digital value of h1

300

200 150 100 50

200 150 100 50

0

0

−50

−50

−5

0

5 10 15 Water level h1 (cm)

20

−5

(a) Potentiometer 1

0

5 10 15 Water level h2 (cm)

20

(b) Potentiometer 2

Fig. 9. Calibration of potentiometers 200 200

q2 in cc/sec

q1 in cc/sec

150 100 50 0

150 100 50 0

0

100 Digital value of q1

200

0

(a) Pump 1

100 Digital value of q2

200

(b) Pump 2

Fig. 10. Calibration of pumps

The A/D converter is calibrated by reading the digital values for known liquid levels. Similarly, the D/A converter is calibrated by giving known digital commands to the pump and reading the corresponding flow from the flowmeter. The gradient of Figure 10(a) is 0.7913 cc/sec/digit and that of Figure 10(b) is 0.9042 cc/sec/digit. The implementation results are shown in the following figures. It has been noticed that the pumps do not respond to digital inputs less that 25 in the case of Tank 1 and 30 in case of Tank 2. To counter this dead zone, a lower saturation limit of 25 and 30 is applied to both motor inputs respectively. During practical implementation of the algorithm it is noticed that the controller used for the simulation study does not provide the good results expected due to the presence of second order actuator dynamics in the loop. Therefore the controller is further tuned to provide acceptable results. The controller parameters finally selected are λ = [150, 200], u0 = [100, 100], k = [0.1, 0.1] and W = [10, 10].

6.0.1 Stabilisation to the same level in both tanks In this case, the set points for both of the tanks are the same. The input flow (q1 ) should settle to zero as in the simulation results. However, in Figure 12, 18

it settles at 30 which means zero inflow because of the lower saturation limit of 30. In the case of unequal water levels in both tanks, the different levels are achieved robustly, the effective q1 settles to a constant value and the flow into the Tank 2, q2 , settles to zero. 10

1

Water Level h , cm

8 6 4 2 0

0

5

10

15

20 time, sec

25

30

35

0

5

10

15

20 time, sec

25

30

35

10

2

Water Level h , cm

8 6 4 2 0

Fig. 11. Implementation results for stabilisation of same water level in both tanks 300 250

Control q

1

200 150 100 50 0

0

5

10

15

20 Time, sec

25

30

35

40

0

5

10

15

20 Time, sec

25

30

35

40

300 250

Control q

2

200 150 100 50 0

Fig. 12. The control input to the pumps

6.0.2 Stabilisation to different levels in both tanks Keeping the controller parameters the same, the system has now been applied to stabilise the water levels in the tanks to two different set-point levels. The controller stabilises the water levels in both the tanks to their respective setpoints as shown in Figure 13. 19

10

1

Water Level h , cm

8 6 4 2 0

0

10

20

30

40 50 Time, sec

60

70

80

Fig. 13. Implementation results for stabilisation of different levels in both tanks

6.1 Robustness to leakage To simulate a leak in Tank 1, the outlet pipe with admittance coefficient equal to k2 is opened. In the case of Tank 2, a leak condition has been simulated by opening the outlet pipe with wider cross sectional area (admittance coefficient k3 = 26.26) than that of the pipe which has been used to model the system (k1 = 23.45). The controller robustly stabilises the water levels to the desired level in both cases. 10 8 6 4 2 0

0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

10 8 6 4 2 0

Fig. 14. Stabilisation of levels with leak in Tank 1

The transient response takes approximately 20 seconds (Fig. 11-14). There are two possible reasons for this: (1) the limited flow rate of the pumps, and (2) the small tank size, where any increase in input creates turbulence at the water surface. Even when the maximum input signal is applied (digital input of 255 in Fig. 12), the levels in Fig. ?? do not rise rapidly. Moreover, it is not possible to maintain a high flow rate near the set point as this creates turbulence which in turn makes the water level measurement inaccurate. An improved rig design would certainly provide better results. 20

7

Comparative Assessment

In this section, the performance of the second order sliding mode controller is compared with that of a PI controller. The three test conditions described in the previous section are again used for this assessment. Suitable values for the PI controller parameters are kp = 2 and ki = 0.05. These were selected to provide good reaching time with minimal overshoot across the range of operation of the rig. For all tests, the controller parameters were kept constant. In Figure 15 it is seen that both controllers exhibit similar performance when the control task requires equal heights to be maintained in both tanks. However, the second order sliding controller performs far better than the PI controller when the control task requires different liquid levels to be maintained in both tanks. The PI controller could not maintain the difference in level required between the tanks across the operating range, as demonstrated in Figure 16.

10

1

Water Level h , cm

8 6 4 2 0

0

10

20

30

40 50 Time, sec

60

70

80

70

80

(a) Water level in Tank 1

Water Level h2, cm

10 8 6 4 2 0

0

10

20

30

40 50 Time, sec

60

(b) Water level in Tank 2 Fig. 15. 2-sliding (continuous lines) and PI controller (dashed lines) performances when the same liquid level is required

21

10

1

Water Level h , cm

8 6 4 2 0

0

10

20

30

40 50 Time, sec

60

70

80

70

80

(a) Water level in Tank 1

Water Level h2, cm

10 8 6 4 2 0

0

10

20

30

40 50 Time, sec

60

(b) Water level in Tank 2 Fig. 16. 2-sliding (continuous lines) and PI controller (dashed lines) performances when different liquid levels are required

In the case of leakage in Tank 1, the second order sliding mode controller performs very well and there is no discernable drop in the water level due to the leak, which was introduced at around 35 sec on the time scale (Figure 17). However, the PI controller fails to maintain the required water levels in both tanks when the same leak in introduced at approximately 65 sec on the time scale. The comparative assessment demonstrates that a second order sliding mode control strategy, which only uses measured outputs and requires minimal tuning of a small number of parameters, has the ability to provide robust performance across a wide range of system operation. It has been demonstrated that the proposed strategy outperforms a classical PI controller, where the control gain values are fixed. Scheduling of the PI parameters would be required to achieve comparable performance to that exhibited by the second order sliding mode controller. As far as complexity of the two designs is concerned, the second order sliding mode controller has a similar structure to that of the PI controller but with appropriate switching in four operating regions: s > 0, s < 0, u > u0 and u < u0 . The second order sliding mode controller has three parameters (λ, W, k) to tune as compared to two (kp , ki ) in the PI controller. However, the three parameters of the second order sliding mode controller are easily computed using equation (4). There is thus no significant overhead in 22

terms of design for the second order sliding mode controller when compared to the PI controller. 10

1

Water Level h , cm

8 6 4 2 0

0

20

40

60 Time, sec

80

100

120

100

120

(a) Water level in Tank 1

Water Level h2, cm

10 8 6 4 2 0

0

20

40

60 Time, sec

80

(b) Water level in Tank 2 Fig. 17. 2-sliding (continuous lines) and PI controller (dashed lines) performances in the presence of a leak

8

Conclusions

A novel second order sliding mode control algorithm has been presented for a class of MIMO nonlinear systems. A nonlinear system in I-O form is used as the canonical form for the controller design. A large class of nonlinear systems may be modelled by square differential input-output equations. This produces a dynamic control irrespective of whether the sliding surface is control dependent or not. The auxiliary control, the highest derivative of the actual control, always appears linearly which facilitates controller design. The algorithm does not require the derivative of the sliding surfaces, thus eliminating the requirement of designing an observer or peak detector, contrary to many other sliding mode control strategies. The resulting control is dynamic and eliminates the chattering at the system input. A case study for robust control of liquid level in interconnected twin-tanks has been presented. The system has been modelled and a simulation study carried out. Actual implementation has been performed on an experimental rig. The 23

controller is implemented via a PC. The implementation results show robustness to parameter variations such as tank area and the admittance coefficients of various pipes. The pump dynamics were also ignored during controller design. The implementation results verify the proposed theory. The second order sliding mode controller is seen to perform well across the operating range of the rig. It is demonstrated that scheduling of the gain parameters in a classical PI controller would be required to obtain equivalent performance across the range of operation of the rig.

References [1] Bartolini, G., Ferrara, A., Levant, A., Usai, E., 1994. On second order ¨ sliding mode controllers. In: Young, K., Ozguner, U. (Eds.), Variable Structure Systems, Sliding Mode and Nonlinear Control. Vol. 247 of Lecture Notes in Control and Information Sciences. Springer, London, pp. 329–350. [2] Bartolini, G., Ferrara, A., Usai, E., Utkin, V., 2000. On multi-input chattering-free-second order sliding mode control. IEEE Trans. on Automatic Control 45(9), 1711–1718. [3] Bartolini, G., Levant, A., Pisano, A., Usai, E., 2002. Higher-order sliding modes for the output-feedback control of nonlinear uncertain systems. In: Yu, X., Xu, J.-X. (Eds.), Variable structure systems: towards the 21st century. Vol. 274 of Lecture Notes in Control and Optimization. Springer, London, pp. 83–108. [4] Chang, L., 1990. A MIMO sliding control with a second order sliding condition. In: ASME Winter Annual Meeting. Dallas, Texas, paper No. 90-WA/DSC-5. [5] Clark, J., Vinter, R., 2003. A differential dynamic games approach to flow control. In: 42nd IEEE Conference on Decision and Control. Hawaii, USA. [6] DeCarlo, R., Zak, S., Matthews, G., 1988. Variable structure control of nonlinear multivariable systems: A tutorial. Proc. of IEEE 76(3), 212– 232. [7] Edwards, C., Spurgeon, S., 1998. Sliding mode control: Theory and applications. Tayler & francis. [8] Elmali, H., Olgac, N., 1992. Robust output tracking control of nonlinear MIMO systems via sliding mode technique. Automatica 28(1), 145–151. [9] Emel’yanov, S., Korovin, S., Levant, A., 1996. High-order sliding modes in control systems. Computational mathematics and modelling 7 (3), 294– 318. [10] Fliess, M., 1988. Nonlinear control theory and differential algebra. In: Byrnes, C., Kurzhanski, A. (Eds.), Modelling and adaptive control. Vol. 105 of Lecture Notes in Control and Information Sciences. SpringerVerlag, New York, pp. 134–145. 24

[11] Fliess, M., 1990. Generalized controller canonical form for linear and nonlinear dynamics. IEEE Trans. on Automatic Control 35(9), 994–1001. [12] Fliess, M., 1990. What the Kalman state variable representation is good for. In: Proc. of 29th Conference on Decision and Control. Honolulu, Hawaii, pp. 1282–1287. [13] Isidori, A., 1995. Nonlinear Control Systems, 3rd Edition. Springer-Verlag London. [14] Kantor, J., 1989. Non-linear sliding mode controller and objective function for surge tanks. Int. J. Control 50, 2025–2047. [15] Khalil, H., 1996. Nonlinear Systems. Prentice Hall. [16] Khan, M., Spurgeon, S., Levant, A., 2003. Simple output feedback 2sliding controller for systems of relative degree two. In: European Control Conference (ECC). Cambridge, UK. [17] Levant, A., 1997. Higher order sliding: Collection of design tools. In: Proc. European Control Conference. Brussels. [18] Lu, X., Spurgeon, S., 1998. Asymptotic stabilization of multiple input nonlinear systems via sliding modes. Dynamics and Control 8, 231–254. [19] Lu, X., Spurgeon, S., 1998. Output feedback stabilization of SISO nonlinear systems via dynamic sliding modes. Int. J. Control 70(5), 735–759. [20] Lu, X., Spurgeon, S., 1999. Output feedback stabilization of MIMO nonlinear systems via dynamic sliding modes. Int. J. Robust Nonlinear Control 9, 275–305. [21] Sira-Ramirez, H., 1993. On the dynamic sliding mode control of nonlinear systems. Int. J. Control 57(5), 1039–1061. [22] Slotine, J., Hedrick, J., 1993. Robust input output feedback linearisation. I. J. Control 57(5), 1133–1139. [23] Utkin, V., 1992. Sliding modes in control and optimization. SpringerVerlag. [24] Weihua, X., Yuen, V., Mills, J., 1999. Application of nonlinear transformations to A/F ratio and speed control in an IC engine. In: SAE Conf. Proceedings. SAE Technical Paper 1999-01-0858.

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