Abstract— This paper introduces a novel control law that is generic in nature and stabilizes linear and a class of non-linear systems. The parameters of the proposed controller give the same flexibility of tuning the transient response of a system as the standard PID does. The superiority of the Output-Prediction based Proportional Switching (OPPS) controller lies in the fact that it addresses the fundamental implementation problems of the PID such as integrator windup, high frequency gain of the derivative term and especially sacrifice of rise-time as system damping is further enhanced using the derivative term. Extensive simulation results are presented implementing OPPS on different linear and non-linear systems that show its generic nature and strength. Finally OPPS is implemented to stabilize a 2-DOF platform stabilization scheme and results are compared with those of a PID.

I. INTRODUCTION

ODAY more than 90% of the closed-loop industrial processes incorporate conventional proportional-

integral-derivative (PID) controllers with or without some slight modifications [1], [2]. Since the invention of PID in 1910, there has been an unbridled increase in acceptance and application of the PID in industry. Easy to use tuning methods of Ziegler-Nichols (Z-N), published in 1942 have contributed a great deal to the success story of PID [3]. With such a long history of academic research encompassing the PID and as the knowledge base matures and enters into an area of diminishing returns, PID still holds an uncanny facet to many plant operators and engineers around the world. A standard PID controller is also known as a “three-term” controller whose transfer function is generally written either in a parallel form or a series form [4]. Well known issues regarding use and tuning of a PID relate mostly to its integral and derivative terms. Adding integral term to a pure proportional term decreases the gain margin (GM) and phase margin (PM) of the system i.e. the system will become potentially unstable [5]. Moreover in applications where the controlled actuator has a range limit; techniques of “anti-windup” must be introduced to offset the saturation [6]-[7]. The derivative term is perceived to increase stability and though it increases the PM of the system but at the same

Manuscript received December 31, 2008. This work was supported in

part by the National University of Sciences & Technology, Pakistan. Mansoor Shaukat is with the Centre for Advanced Studies in

Engineering, Islamabad, 44000, Pakistan (phone: 00923215194279; e-mail: mansoor@carepvtltd.com).

Khalid Munawar is with the Electrical Engineering Department, National University of Sciences & Technology, Pakistan (e-mail: munawar@ceme.edu.pk).

time it decreases GM [5]. This property generates a very tricky choice for a range of derivative gains which ensure system stability. This is also a reason why 80% of the PID controllers in use have their derivative part switched off [8]. The failure of the derivative term to restrict high-frequency gains makes the introduction of low pass filter mandatory [9]-[10]. Furthermore, in an attempt to completely kill overshoots, system damping must be increased. The derivative gain contributes to the damping factor of a system and increasing which causes the rise time to be sacrificed [11]-[13]. This renders task of tuning a PID very tedious especially when the plant model is unknown. Problems associated with the PID that have been discussed relate to the fundamental structure of a PID. Various tuning methods have been developed that address these intricacies of implementation as well as help in improving stability. Moreover these methods also improve robustness, tracking and regulation performance and noise attenuation using analytic, heuristic, frequency response, optimization and adaptive tuning methods [4], [6], [14]-[17]. However, these tuning methods are generally complex in nature and most of them need to work offline. Such systems generally work as an expert operator by first considering that a PID controller is difficult to tune and try to deliver both a short rise-time and a low overshoot. Most PID patents filed so far focus on automatic tuning techniques with intelligent system identification and a knowledge base of the process to work with. Although complex and computationally taxing structures have been built over the years; still there is a lack of generic methods that can be viably applied on an embedded system. In this paper, we propose a discrete nonlinear controller which addresses the fundamental problems of a standard PID structure without incorporating a complex structure. Output-Prediction based Proportional Switching (OPPS) consists of two control modes about which the control law is switched. The name "switching controller" comes through this approach of switching between these modes. The Output Update (OU) mode and the Output Prediction (OP) mode constitute OPPS. OU is a very simple control mode which resembles with a conventional integral controller. OU governs the control law most of the time while OP intervenes only when there is an indication of an overshoot based on rate of change of system output. OP is based on a very simple overshoot suppression rule base that is computationally just a fraction of the complex fuzzy and adaptive structures [18]-[19].

Discrete Output-Prediction based Proportional Switching Controller

Mansoor Shaukat, Member, IEEE and Khalid Munawar

T

17th Mediterranean Conference on Control & AutomationMakedonia Palace, Thessaloniki, GreeceJune 24 - 26, 2009

978-1-4244-4685-8/09/$25.00 ©2009 IEEE 682

Conventional switching controllers especially employing Bang-Bang control, time-optimal control [20]-[22] or switched output feedback [23]-[24] have generally a solitary control law whose output switches between predefined constants. OPPS on the other hand has an integrator at heart which switches to a pre-defined constant in case of an overshoot prediction. The action resembles that of re-setting the control output in case of integrator windup.

In Section II, OPPS is presented with the two control modes discussed in detail and the control law is formulated. Extensive simulations on different linear and nonlinear systems incorporating OPPS are given in Section III; where ability of OPPS to stabilize any linear and a class of nonlinear systems is established. 2-DOF platform stabilization, achieved with OPPS is given in Section IV and the results are compared with those of a PID on the same system. Finally, Section V concludes the implementation findings.

II. CONTROLLER STRUCTURE

The purpose of this research was to come up with a compensator which has the ability of stabilizing a generic linear/nonlinear system without incorporating a complex control structure like the ones discussed in Section I. The block diagram of the novel OPPS controller is shown in feedback configuration in Fig.1, where e is the error between the current output y and the desired output yd, d/dt is the discrete differential operator and u is the OPPS control output which is switched in between OU module’s output, u1 and OP module’s output, u2 depending on the operating conditions.

Fig.1. Block diagram of controller in closed loop feedback configuration.

Both OU and OP control modes can be switched between depending on the set of rules based principally on the idea of suppression of overshoots and are defined on the basis of output rate and error.

A. Output Update (OU) module

The OU module is active almost all the time. The OPPS calculates the suggested differential u in the output u.

)()( keu K ku (1)

where, Ku is the positive proportional gain constant and k is the time index.

The suggested differential is then subtracted or added to the previous u depending on the negative or positive sign of the error e respectively, where:

)()()( kykd y ke (2)

The partial control law consisting of OU module will be:

)()1(

)()1()(1 kuku

kuku ku (3)

OU resembles with an analog integrator in s-domain where:

)()( s U sEs

iK (4)

or expressed in time domain as:

)()( tu teiK (5)

So, OU is nothing more than a discrete integrator which computes the proposed differential in u based on the magnitude and sign of the error and moreover has the capability of zeroing down the error on its own.

B. Output Prediction (OP) module

The OP module in contrast to the OU module only gets activated when there is a prediction of an overshoot in the output y with respect to the desired output yd. The idea is to prevent overshoot before it actually happens. OPPS continuously predicts the approximated future output

)1(ˆ ky as yp(k).

)}()({)()1(ˆ kyky kpyky (6)

where, β is a constant prediction gain and y = y(k) – y(k-1). In case an overshoot is predicted then the OP module intervenes and modifies the control law dictated previously by the OU module. The condition whether yp is greater or lesser than yd is actually the indication for an overshoot. If e in (2) is positive, yp > yd indicates an overshoot; while when e is negative, yp < yd indicates an overshoot. Both the conditions are catered for in the rule base in (8). Unlike complex prediction algorithms, a very simple prediction mechanism based on rate of change of output is used. After extensive simulations on different linear and nonlinear systems it was concluded that β=1, generally suffices with the exception of higher prediction gains for systems with process dead time or transport delay as in (7).

LseTs

K sG

1)( (7)

where, K is the process gain, T is the process time constant and L is the process dead-time or transportation delay. The modified control law is given in (8), where uc is a constant, to which control output is switched to, provided the need for overshoot suppression. The property of OP mode to switch to some other desirable output, the magnitude of which is dependent on the type of system and will be discussed in section III, is in itself an effective integrator anti-windup technique. is a positive constant that defines the error boundary inside which OP module cannot have an impact. This is to make

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sure smooth transitioning of output y to desired output yd. Switching OP module, if active within certain error boundary will give rise to undesired chattering while e tends to zero. This is to let integrator role of OU to zero down the error smoothly inside the error boundary. The sign of the output differential y in (8) indicates whether the output is diverging from the target position or converging towards it. It has to be noted that the given conditions are themselves sub-conditions of the error sign condition.

( ) ( ) 0&[{ ( ) 0}

&{ ( ) ( )}

&{| ( ) | }]( )2 ( ) ( ) 0&[{ ( ) 0}

&{ ( ) ( )}

&{| ( ) | }]

u k if e kcif y k

y k y kp de ku k u k if e kc

if y ky k y kp de k

(8)

C. Control law

The OPPS’s overall control law then can be listed as follows:

( 1) | ( ) | ( ) 0

( ) ( ) 0&[{ ( ) 0}

&{ ( ) ( )}&{| ( ) | }]

( )( 1) | ( ) | ( ) 0

( ) ( ) 0&[{ ( ) 0}

&{ ( ) ( )}&{| ( ) | }]

c

p d

c

p d

u k u k if e k

u k if e kif y k

y k y ke k

u ku k u k if e k

u k if e kif y k

y k y ke k

(9)

III. SIMULATIONS

In this section, OPPS will be applied to different system categories in the closed loop configuration as given in Fig.1 and the ability of stabilization will be presented and transient performance based on selection of varying control parameters will be discussed. Linear systems on which OPPS will be applied will be over-damped, under-damped and unstable respectively. OPPS will finally be implemented on a monotonic nonlinear system.

A. Linear Systems

1) Over-damped system: OPPS is first applied to an over-damped system given in (10) with only OU control mode active.

)1)(3(

2)(

ss

s sG (10)

Fig.2 shows the step response of (10) for varying proportional gain Ku. Fig.2 also shows faster settling times for increased Ku and higher frequency of oscillations.

Further if the OP module is also made active, it can be shown that better transient performance can be achieved as shown in Fig.3. Step responses are shown by keeping OU control parameters constant and varying OP control parameters. For OP mode β is kept unity as discussed earlier while error boundary for over-damped system (delta) = 0.01 in which OP mode cannot have an impact and uc is varied from 2 in Fig.3 to 2.5 in Fig.4. As soon as the OP mode gets triggered control output switches and the new control output due to u2 pulls back the system. In Fig.4 a little softer switching approach (selection of uc) ensures a much smoother transitioning of the system output to the desired output.

Fig.2. Step response of system in (10) with OPPS parameters: Ku = 0.2, 0.4, and 1. (OP module: inactive) Fig.5 shows the control output of the closed loop system implemented in Fig.4. Control output switches to uc at the instant OP takes over control from OU. OP control mode resets the saturated u to uc as is done by the anti-windup technique for PID. Control output then gradually smoothes off due to integration action of OU control mode. Another characteristic shown in Fig.6 is that though both the closed loop systems of Fig.3 and Fig.4 have a lot of difference in their respective overshoots but their rise-times are identical. Therefore, OPPS gives the user the flexibility to modify overshoots without sacrificing rise-times. Fig.7 shows the sinusoid for the the over-damped system. Fig.8 shows the improved response of the system once the proportional constant Ku of OU module is increased significantly while other parameters are unchanged.

2) Under-damped system: The controller was further applied on an under-damped system in a closed loop configuration as in Fig.1 with G(s) given in (11).

)5.0)(5.0(

1)(

jsjs sG

(11)

Fig.9 shows the step response for the under-damped system. Further Fig.10 shows the sinusoid following for the system.

3) Unstable system: OPPS is then applied to an unstable system with G(s) given in (12) in the same configuration of Fig.1.

)1)(3(

2)(

ss

s sG (12)

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Fig.11 shows the step response for the system (12) and the sinusoid following is given in Fig.12.

Fig.3. OPPS applied to system in (10) with OPPS parameters: Ku = 1.00; β = 1; (delta) = 0.01; uc = 2.0

Fig.4. OPPS applied to system in (10) with controller parameters: Ku = 1.00; β = 1; (delta) = 0.01; uc = 2.5

Fig.5. Control output u for closed loop system as in Fig.4.

Fig.6. Comparison of rise-time for closed loop systems in Fig.3 and Fig.4 with different control switching output.

B. Nonlinear system

Finally, OPPS is applied as in Fig.1 to a monotonic nonlinear system as given in (13).

3)( u uf (13)

Fig.13 shows the step response for (13) with OPPS applied and control parameters for both modes stated. Fig.14 shows the sinusoid following for a non-linear system.

Fig.7. Sinusoid following for system in (10) with controller parameters: Ku = 10.00; β = 1; = 0.01; uc = 4.0.

Fig.8. Sinusoid following system in (10) with controller parameters: Ku = 100.0; β = 1; = 0.01; uc = 4.0.

Fig.9. Step-response for system in (11) with controller parameters: Ku = 100.0; β = 1; = 0.01; uc = 4.0.

IV. 2-DOF PLATFORM STABILIZATION

After extensive simulations in section VI, OPPS is applied on a 2 Degree-of-Freedom (DOF) platform to test its performance in the real world. Control was implemented

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only on 1DOF (azimuth). A Micro Electro-Mechanical Systems (MEMS) gyro [25] was used to extract information for change in orientation of the platform base and two servo-motors in each DOF were implemented in a “drive – anti-drive” fashion to compensate for gear backlash [26]. The system can be approximated by (7). Fig.15 shows the PID compensated step response of the platform in azimuth DOF with settling time, Ts=1.32s. Fig.16 shows PID's performance with varying desired angle and its vulnerability to high frequency inputs. Fig.17 shows the OPPS compensated step response of the platform in azimuth DOF with varying values of β with Ts=1.035s. Higher values of β are used because of the data transportation lag due to the PIC-SERVO boards (used to control motors) incorporating a serial protocol [27]. Fig.18 shows OPPS compensated platform stabilization. Even at a much higher frequency inputs than in Fig.16, system output never blows as does for a PID compensated system and OPPS tracks the fast changing input with system time constant being the limitation.

Fig.10. Sinusoid following for system in (11) with controller parameters: Ku = 100.0; β = 1; = 0.001; uc = 0.05.

Fig.11. Step response for system in (12) with controller parameters: Ku = 100.0; β = 1; = 0.001; uc = 0.0.

V. CONCLUSIONS

OPPS’s simple computational structure and ability to suppress overshoots while maintaining best rise-times, display no high frequency gains and an inherent integrator anti-windup technique gives it a much agreeable choice over the fundamental PID structure. OPPS ability to address all these fundamental problems of a PID without incorporating any fuzzy-like expert rule base, learning algorithm or an

adaptive tuning mechanism, makes OPPS easier to be implemented on an embedded system. Different types of systems (plants) compensated by OPPS show strength of its generic control law as well as satisfying transient and steady-state performance. The control parameters strictly lay in a typical range for a particular type of a system which results in easy tuning procedures. Fundamentally, having the structure of an integrator and an overriding prediction control mode based on fundamental concepts of overshoot occurrence; OPPS is a simple solution to most of the industrial processes. This paper reports very initial results and future work encompasses development of stability analysis, sensitivity to the choice of parameters and noise rejection capability of the controller. The controller will also be applied to more real world systems.

Fig.12. Step response for system in (12) with controller parameters: Ku = 20; β = 1; = 0.001; uc = 1.0.

Fig.13. Step-response for system in (13) with controller parameters: Ku = 0.05, 0.06; β = 1; = 0.1; uc = 0.0.

Fig.14. Sinusoid following system in (13) with controller parameters: Ku = 0.06; β = 1; = 0.1; uc = 0.0

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Fig.15. Finally tuned step response of the platform with PID implemented

Fig.16. Platform response with PID, for varying reference angle of rotations. High frequency instability is evident.

Fig.17. Step response with OPPS applied with varying beta, β: 0, 8, and 16. Other parameters: Ku = 12; = 0.5; uc = 0.

Fig.18. Platform response with OPPS, to varying reference angle of rotation.

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