PENDULAR Control Systems – A New Strategy in Control Engineering

CĂTĂLIN NICOLAE CALISTRU “Gheorghe Asachi” Technical University of Iaşi

Department of Automatic Control & Industrial InformaticsROMANIA

Abstract: The paper presents a new class of variable structure systems so called PENDULAR Control Systems. PENDULAR (Pendulum Efficiency with Non-linear Dynamics and Unconventional Law in Achievement of Robustness) systems are Variable Structure Systems (VSS) with a simple structure. A short description of these systems, the study of stability and the essential PENDULAR system is presented in the paper. Illustrative examples are detailed. Simulation and experimental results show the efficiency of PENDULAR systems.Keywords: variable structure system, nonlinear control, robustness, PENDULAR control, control strategy

1 Introduction

The word PENDULAR comes from the similar Romanian word that means the pendulum movement. In fact PENDULAR systems also reflects the mnemonic of Pendulum Efficiency with Non-linear Dynamics and Unconventional Law in Achievement of Robustness and are a class of nonlinear control systems introduced by the author in 1996. New researches in non-linear control systems encouraged the author in obtaining new and valuable results also approved by the scientific community. However, the most important moment was when the theories were validated by experiments. The experiments made on a large class of processes shown the effectiveness of the proposed control methodology.The variable structure systems (VSS) are very interesting to be studied because often reveal surprises. Ussually the feedback control systems are closing the loop via a negative feedback. In this manner it is assumed the fact that the control system is robust (stable and performant even if different exogenous will disturb: reference variations, external disturbances, measurement noises, and plant uncertainties). The question is: "Is positive feedback always bad for a control system?”. If one considers a positive control system S with positive feedback, as in Fig. 1,

Fig.1. System with positive feedback

its input has a cumulative aspect that intuitively leads the system to instability. In the classic negative feedback control systems where y is kept close to r values the negative feedback could delay the system response. For example, let assume is analyzed "a level control system" with "automation at level 0". In other terms a human operator supervises the level in the tank closing the tap whenever the water will reach the reference level.For efficiency (and if the tank volume is large), the operator does not proceed like this: leave the tap on the drop by drop position and when the level reaches the reference value turn off the tap. He proceeds like this: turn on the tap at a large flow and when the level is closing to reference turn the tap at a small value tending to zero. Intuitively positive means feedback-large flow, negative feedback -small flow. Extending this very simple idea to the control loops the proposed system starts with positive feedback and at a certain time changes his structure becoming a negative (classic) control system. In this manner the PENDULAR control system is defined.The paper consists in the following sections: introduction, pendular control system (PCS) (here pendular control principles are detailed), stability of PCS, Essential PCS (the search for simplicity), experimental results (made on a physical plant) and conclusions.

2. PENDULAR control system

Let the control system in Fig.2,

ur y

r + y

+S

Fig. 2 PENDULAR control systemobtained by introducing a nonlinear element N on the conventional system feedback loop. The signals r, e, u, y, p, GR and Gp are respectively the reference, error, command, controlled signal, disturbance, the controller transfer function and the plant transfer function. The nonlinear element N contains a decision block and a comutator K. Initially the comutator is on “+” position. The decision block commands the comutator K, ”+” to “-“ for the very first time tc. The system changes its structure at time tc becoming a system with negative feedback. At this time the comutation condition is fulfilled:y(tc)=r. (1)The simplified control loop is depicted in Fig 3:

Fig 3 Simplified control systemwhere Gd is the open loop transfer function. The nonlinear element N is characterized by:

, (2)

and the static characteristic is depicted in Fig.4.

Fig.4. N static characteristic

Definition. The nonlinear element N with characteristic (2) is called PENDULAR nonlinear element.N leads the system to the following behaviour: till the moment tc,

, system has a positive feedback with closed loop tranfer function:

,

(3)for , system has a negative feedback (

) only if . The closed loop transfer function is:

.

(4)However, represents the differential equation Cauchy problem for the negative feedback system (the conventional one), initial condition

. If for , and for ,

the system will behave as positive feedback system (K comutes “-“ to “+”) and so on. In this manner, the controlled variable y, may be considered as output signal for the positive feedback system, then at the moment , after the very first commutation, output signal for the negative feedback system; eventually for the moment , again output signal for the positive feedback system,etc. till the controlled signal variable is stabilized at the value r.The pendulum image for the output y comes from the movement between the two classes of differential equations solutions: S+ (for positive feedback sub-system) and S_(for positive feedback sub-system) is represented in Fig 5.

Fig.5 The y oscilatory “movement”Definition . PENDULAR control system (PCS) is the system obtained by introducing a nonlinear element N on the feedback loop of a classical control systemIllustrative example Let the system depicted in Fig.6

y

0

tc1

r

tc

+ +

+r

tci

tc

_

+

N

GR(s)

K

r=y

0

r

ў

y

ў

y

Gd

N

NS+

S_

N

k

r

Fig 6.Example: PCSLet assume that r=1, a proportional controller k>0, an integrator as plant and no disturbance (p=0). One obtains:

.The system behaviour: .System starts with positive feedback

with the solution(5)

Time tc is given by:

.

From (5), , that means comutation “+”to”-“System is with negative feedback:

,

the solution:

(6

)The relation (6) shows that the system was stabilized only after a comutation and stays in this state if no disturbance is reported. The system global response is:

(7)

Let suppose that for a p. step disturbance is applied.

(8),

(9)The derivative sign is given by p sign. For

(10)

system stays on negative feedback, if:,

(11)the system changes its structure on positive feedback.The differential equation is:

(12

)

.(13)

(14)and the derivative sign depends on (2k+p). For k>0, one obtains the cases:

The system comutes for p<0, on positive feedback, then on negative feedback etc. In this case we have a infinite number of comutations the output is kept on the reference value r=1.

The system behaviour is as in the first case.

The system stays on positive feedback the output decreases continuously, system is unstable. In conclusion: 1.p>0, system behaviour is identical with the conventional one. 2. , the PCS rejects the disturbance component instantaneously, 3.p<2k, the system becomes unstableExample. PCS with k=1, at tp=10 sec disturbances 0.5, -0.5, -2.1 are applied. With Matlab-Simulink PCS response (y) and conventional system (y-)are represented. The command u is also depicted One observe the chattering fenomenon (Fig.7,8,9)

Fig 7. Disturbance applied p=0.5

Fig. 8. Disturbance applied: p=-0.5

Fig.9. Disturbance applied: p=-2.13 Stability of PCS

Definition.The complex function F(s) is called positive real if:i.F(s) is analytic in ,ii. for every s with ,iii. for every s with .The function F(s) analycity makes possible the replacement of iii:iii’. F(s) is real for every s positive real..

3.1 Popov stability criterionTheorem (Popov) The equilibrium state is globally asymptotic stable for the closed loop system (closed through h(t)) ifi. ,

ii.

(15)iii .there is , that

is a real positive

function.3.2 Circle criterionTheorem (Sandberg-Zames)Let K1 and K2 two constants . The equilibrium state is globally asymptotic stable for the closed loop system (closed through h(t)) ifi.

ii. (16)

iii. is a real positive function. Based on these two therems is very simple to analize the stability of PCS.

4 ESSENTIAL PCS

Let PCS from Fig.3, with a PI controller and a innertial plant:

. (17)

. (18)The closed loop transfer function:

(19)

The system response for r=1:

(20)The response time 5% tr- for the conventional system:

(21)

The PCS has for t[0,tc] transfer function

, (22)

,

(23)unstable for k>0, T>0.The step response:

(24)Time for the first commutation:

.

(25) “Response time 5%” :

.

(26)From (21) and (26):

. (27)

DefinitionPCS that commutes one time when no disturbance applied is called essential pendular control system EPCS.

Theorem (PENDULAR essential theorem)The system described above is EPCS. Its step response is given by:

.

(28)

Demonstration. For , the response is

given by (25).For the system response is given by the Cauchy problem:

.

(29)The general solution for (29) is:

(30)and imposing the initial condition from (29) one obtains c=0. Q.e.d.In Fig.10 the responses for the classic system (with conventional negative feedback) versus PCS system are depicted. The responses were obtained by simulation via Matlab-Simulink environment. For simplicity k=T=1.

Fig.10 Simulation results for k=T=1

If a step disturbance will be applied (at t=10 sec p=-1) an excelent behavior is reported for EPCS.From Sandberg-Zames theorem and the real positeveness theorem, the stability for PCS is very easy to prove.Theorem.PCS is globally asymptotic stable if

is a real positive function.

Application. EPCS case : Here

And this function is obviously a real positive function.

5 Experimental setup

The set-up consists in a Feedback® Discovery Product for temperature and flow control. In this paper were made tests only for the flow control.The experiments have been done using the Real-Time Workshop from Matlab® Simulink®. The process has been identified as a I order system with dead-time :

e-0.5s

depicted below

Because the process is strongly affected by disturbances, the controller used is a PI type

Creating a discrete-time model with the sample time of 0.1 seconds, using the 'zoh' discretization method, the following discrete controller was obtained: numd=[2.7643 –2.7051] and dend=[1 -1]. In order to limit the effects of the perturbations on the command, a discrete filter is used: numf=[0 0.952] and denf=[1 -0.9048].Using the PENDULAR control method, the following response is obtained.

6 Conclusions

The paper briefly presents the PENDULAR control systems. The PENDULAR control principles are sustained by a stability study. Essential PENDULAR Control systems were detailed and the simulation results are illustrative.The research is also sustained by experimental results made on different classes of systems. This new control methodology seems to have some impact over the control strategies.

References

[1]C.N Calistru., “A Robust Variable Structure System”, 4th IFAC Symposium on Advances in Control Education, ACE’97, Istanbul, 1997.

[2]C.N.Calistru., “A New Variable Structure System”, IEEE IPCAS’95 Industrial Process and Control Applications Seminar, Calcutta, 1995.

[3]C.N.Calistru, “An Inciting VSS System”, The 21st

Congress of the American-Romanian Academy of Arts and Sciences, Vancouver, Victoria, 1996.

[4]C.N Calistru., “Analysis and Control of a Variable Structure System”, International Conference on Analysis and Control of Differential Systems, Constanţa, 1997.

[5]C.N.,Calistru Lecture: PENDULAR Control: A New Strategy in Control Engineering, Università degli studi di Firenze, Florence, 2000.

[6]V.M Popov., “Absolute Stability of Nonlinear Systems of Automatic Control”, Automat. Remote Control, vol 22, no.8, pp.857-875,1961.

[7]C.N. Calistru, Increasing robustness of control systems using integral criteria and alternate feedback, Matrix Rom Publ. House, Bucharest, 2004.