# Simple Control For Open Loop Unstable Plants Accounting Essay

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The project involves developing a simple intuitive methodology for designing a controller that would stabilize an open-loop unstable plant. Two approaches, namely; Youla Parametrization and Feedback Stabilization (two-degree of freedom) have been identified through the feasibility studies by which the problem can be tackled. The Internal Model Control structure is modified in the feedback stabilization technique and simulations were implemented to confirm the information gotten from the literatures. A project plan and methodology as to how the project will be executed so as to achieve the aims and objectives within the allocated dissertation period as well as the project risk and Health & Safety Assessment involved were highlighted. The project is challenging, but from the feasibility studies report, it is feasible.

## 1.1 INTRODUCTION:

The project is titled Simple Control for Open loop unstable plants. Unstable plants can be defined as: plants that yield unbounded response when given a bounded input signal. For linear systems, they have a pole located in the open right-half plane (RHP) (Dorf and Bishop 1998, p.295). The aim of the project is to develop a controller through a simple intuitive methodology that would stabilize such plants in closed loop.

Two methods have been identified in developing a controller to stabilize such unstable systems, they are; (i) Feedback Stabilization or Two-degree of freedom (Modified Internal Model Control). (iii) Youla Parametrization

These methods will be evaluated in this study and would form a major activity during the dissertation to observe their limitations and benefits with the possibility of achieving control synthesis of the different methods to get the best control strategy.

## 1.2 BACKGROUND:

As mentioned earlier on, an unstable plant would yield an unbounded response when given a bounded input signal and for linear systems they have a pole in the open right-half plane (RHP). Examples of such systems are systems whose transfer functions are given as;

; , etc..

For the purpose of control action, the type of unstable system that would be used in this study and in the actual dissertation work is considered to be linear, continuous, time-invariant and Single-input Single-output (SISO).

Modified Internal Model Control (IMC) structure would be studied during the project to see how it was applied to stabilize unstable systems. But before then a brief introduction of the normal IMC structure for stable plants is being introduced in the next section so as to enhance a better understanding of its structure.

1.2.1 Internal Model Control: Internal model control (IMC) was developed by Garcia and Morari and they established its relationship with other control schemes, such as the Smith Predictor, LQG, etc. (Garcia and Morari, 1982).

G(s)

Q(s)

r u y d

Gm(s)

ym

Fig1.1 IMC structure by Garcia and Morari (1982, p.310)

The diagram above shows the schematic of IMC for stable plants, where; Q(s) - the IMC controller, G(s) - the plant, Gm(s) - the plant Model, r - the reference input, y - the plant output, u - the control input, ym - the model output, d - output disturbances, - feedback signal. Garcia and Morari (1982, p.310)

The feedback signal is given as;

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

If the model is the same with the plant i.e. and there are no disturbances (d = 0), then the model and the plant would have the same output which means the feedback signal will be zero. Therefore, if all the inputs of the plant are known, the control system is virtually open loop in the absence of disturbances, plant-model mismatch or uncertainty. The feedback signal is a measure of the uncertainty and disturbances in the plant. (Morari and Zafiriou, 1989, p.41)

According to Garcia and Morari (1982), IMC only allows stabilization of open loop stable plants, but our target is to see how it could be applied to open loop unstable plants.

The closed loop transfer function is given as:

â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (2) Q(s) = 1/G(s), which means that the IMC controller is given as the inverse of the plant transfer function, but to ensure that the function is proper, a filter function is added. Therefore a more general IMC controller is given as:

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3) Where G_(s) is the invertible part of the plant, - is the tuning parameter, r - positive integer to ensure properdness.

The closed loop classic feedback controller is parametrized in terms of the stable IMC controller as:

â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (4)

which is stabilizing for any stable Q(s).

Also the IMC controller can be expressed in terms of the closed loop feedback controller as:

â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (5)

which makes the two structures interchangeable.

## 1.2.2 Advantages of IMC:

According to Kou Yamada (1999), IMC has the following advantages;

The stability of the Internal model control system is only dependent on the stability of the plant G(s) and the controller Q(s). Which makes it easy to design, but for this project the plant is unstable.

Closed loop response is easily adjusted by tuning (tuning parameter) from the filter.

The IMC structure includes robustness in an explicit manner through its feedback signal given in equation (1). Garcia and Morari (1982).

It provides for online tuning which makes it attractive to the operator.

In this project, modifications of IMC controller on how it could be applied to open loop unstable systems will be studied due to its advantages listed above.

## 1.3 AIMS AND OBJECTIVES:

The aim of this project is to develop a simple, intuitive methodology of obtaining a controller that would stabilize an open loop unstable system in closed loop, ensure robustness to disturbances, easily computable and also easy to tune. Already existing methods will be studied, analyzed and evaluated to compare their strategies and come up with a control synthesis of the selected methods. The control action will be investigated through simulations and its performance will be observed. Two methods used to develop such controllers would be studied during the project period, they are:

Youla Parametrization: The Parametrization of all stabilizing controllers for unstable systems developed by Youla et al (1976), popularly called Youla Parametrization will be studied and simulations will be carried out also to see how effective the controller is.

Feedback stabilization (Two-degree of freedom): This involves designing the controller such that servo-tracking and disturbance rejection is done separately. It involves modifying the IMC structure and applying it to open loop unstable systems. Further studies will be done on this approach as well as simulations as illustrated in the literatures.

The Deliverable obtainable from this project is a control synthesis of both approaches as well as comparison, so as to develop a controller that would stabilize an open loop unstable plant in closed loop. The designed controller should be easy to design, ensure stability, good disturbance rejection and less ambiguous to the operator.

## 2.1 LITERATURE REVIEW:

According to Garcia and Morari (1982), IMC does not support the stabilization of open loop unstable plants, but could only be achieved if the if the controller cancels exactly the unstable poles of the system which they said was not possible in practice. This could be true, because in most cases the plant dynamics might not be very accurate and hence the controller might not cancel out the unstable pole.

Chiu et al. (1990) also confirmed that IMC cannot be used for unstable plants, but suggested that the conventional feedback must be used with Parametrization through the IMC structure.

But Morari and Zafiriou, (1989), Kou Yamada (1999) and Tan et al(2003) all had the same view of applying IMC to unstable plants and proposed certain conditions must be satisfied to achieve stabilization.

The Parametrization of all stabilizing controllers derived from the IMC is given as

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (6)

Where Q(s) is the IMC controller, C(s) is the closed loop controller G(s) is the plant. This controller would be stabilizing if certain conditions are fulfilled, which would be considered in the sections below.

2.1.1 Pole - Zero Cancellation with Internal Model Control: As stated earlier, to stabilize an unstable plant using IMC, the controller needs to cancel the unstable pole of the plant. In the paper by Tan et al (2003) and Morari and Zafiriou, (1989), the controller given in equation (6) is stabilizing if and only if the following expressions hold: (i) Q(s) must be stable (ii) G(s)Q(s) must be stable, this would mean that the zeros of Q(s) must cancel the right-half plane poles (RHP) of G(s). (iii) (1 - G(s)Q(s))G(s) must be stable, which means the RHP poles of P must be canceled by the zeros of (1 - G(s)Q(s)).

In the example in the paper by Tan et al (2003), implementing this for an unstable system given as;

G â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (7)

The IMC controller is designed as;

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (8)

Where Î± is a parameter that guarantees (1-G(s)Q(s)) cancels the RHP pole of G(s), Î» is the tuning parameter.

Choosing Î± = 3 and Î»=1,

The closed loop controller designed from the IMC is given as;

== â€¦. (9)

The simulation is given below:

Fig2.1 Pole zero cancelation using IMC as derived from Tan et al (2003, p. 204)

Fig2.2 Plot of Plant Output which shows the simulation of the controller designed in the paper Tan et al (2003, p. 204)

From the figure above, it can be observed that the designed controller was able to stabilize the plant and make it track set point but there was a large overshoot of about 40% and also a large settling time which can be adjusted by tuning the parameters properly. But for this study, it was not tuned because the aim was just to check stability.

This method is only considered in the feasibility studies as a an example, but will not be studied in the dissertation.

2.1.2 Feedback Stabilization (Modified Internal Model Control): This is also referred to as a two-degree of freedom controller. Huang and Chen (1997) proposed a three-element structure method that can be used to derive a two-degree of freedom controller and also the conventional PID systems. Their method is the same with that of T. Liu et al (2004) and Tan et al(2003) who did similar work and proposed the application of modified IMC for unstable processes with time delays. This approach has the advantage of solving control problems such as stabilization of the plant, servo-tracking and disturbance rejection independently. Wang et al (2004) proposed H2 optimal controller for the disturbance rejection controller, while Tan et al(2003) proposed a PD controller for the disturbance rejection controller. The Modified IMC structure according to Tan et al (2003) is shown below for unstable process with time delay.

Fig2.3 Simplified Modified IMC Structure for Unstable plant with time delay (Tan et al,(2003))

From the figure above, it can be seen that that the structure has three compensators, namely, K0, K1, and K2 and they each have influence on the overall closed loop response as follows:

K0 is used to stabilize Gm(s), the unstable plant model internally without the time delay, so as to get a stable system. K0 can be any stabilizing controller chosen by the designer, e.g Proportional controller.

K1 is an IMC controller for the stabilized plant

K2 is used to stabilize the original unstable plant with delay, and is considered as the disturbance rejection controller which is very important for the internal stability of the structure.

This method was tried out using a simple system and choosing the parameters as outlined in the paper as well as the tuning rules provided in the paper, and the result of the simulation is shown below.

The stabilized plant is given as:

, choosing K0 = 2,

K1 is the chosen as the IMC controller and is given as:

K2 is designed using tuning rules provided in the paper and is designed as a PD controller given as:

K2 = KC(TCs+1) = 2.079(0.156s+1)

The system is given below

Fig2.4 Modified IMC structure with the designed parameters (example 1: Tan et al (2003), p.209)

Fig2.5 Simulation Result replicating the simulation from example 1 in the paper: Tan et al (2003), p.209

From the figure above it can be seen that the controller was able to make the system track set point, although efforts were not made to tune the controller to achieve a better response because of the aim was just to check stability.

In other to have a better understanding of the structure, modifications were made on the structure to apply the same principle to plants without delay and also leaving out the disturbance rejection controller K2, and the structure as well as the response is shown below:

Fig2.6 Feedback stabilization with K0 and K1 alone with plant delay, Modified from example 1 in the paper Tan et al (2003)

Fig2.7 Simulation response of the Modified structure above

From the figure above it can be observed that the feedback stabilization and IMC modified controller was able to stabilize the system and make it to track set point, which shows that K2 is for disturbance rejection. But the feedback stabilizing controller K0 value affects the IMC controller value and the overall response, such that if it's changed, the IMC controller has to be recalculated to suit the new value.

## 2.1.3 Youla Parameterization:

Youla Parametrization also called the Q-Parametrization is the characterization of all stabilizing controllers that stabilize a given process.

According to Maciejowski (1989,p.315), Youla Parametrization was applied in studies of optimal control in the 1950's. Kucera and Youla et al made much improvement on the Parametrization which has found its importance in the subject of linear-feedback systems. According to him, "Kucera and Youla et al. applied the Parametrization for quadratic (H2) optimization, while Zames applied it to Hâˆž optimization problem" (Maciejowski (1989 p.315). The systems considered in the material were Multiple-input Multiple-output (MIMO) systems.

The concept of Youla Parametrization, according to (Astrom and Murray, 2008), is the characterization of all controllers that stabilize a given process. The concept was applied to Single-input Single-output (SISO) systems. The diagram showing the parametrization application to unstable plants is shown below:

Fig2.8 Block Diagram of Youla Parametrization for Unstable plant (Astrom and Murray, 2008, p. 357)

In its application to unstable process, the system is expressed in a rational polynomial transfer function form, such as;

, where a(s) and b(s) are polynomials

A stable polynomial c(s) is chosen to stabilize the process, and then the process is given as;

, where A(s) = a(s)/c(s) and B(s) = b(s)/c(s)

A controller is introduced given as;

, where G0(s) and F0(s) are stable rational functions

But the Errata supplied by S. Livingstone (15 Nov 2010) shows that there are some errors in the formula. The corrections are:

= and

Also from the diagram of the structure, there seem to be something wrong about the location of the set point input which will all be investigated during the dissertation.

For any stable rational function Q(s), the characterization of the stabilizing controller for unstable plant according to Astrom and Murray (2008) is given as;

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (10)

The Sensitivity function is given as:

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (11)

While the Complimentary Sensitivity function is given as:

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (12)

Equation 10 gives the Parametrization of all stabilizing controllers for unstable plants and the sensitivity and complementary sensitivity functions are stable if the rational function AF0 + BG0 does not have any zeros in the RHP. Therefore the controller is stabilizing for any stable Q. (Astrom and Murray, 2008, p. 357, 358)

Much work was not done on this method during this feasibility study but it constitute a major part of the dissertation work and hence would be studied in details and simulations carried out to have a better understanding of the concept and how it is used.

## 3.1 METHODOLOGY

The project aim is to develop a simple, intuitive methodology of designing a controller for open loop unstable plants. Some methods have been identified among which is the Youla Parametrization and the feedback stabilization methods.

In other to achieve this aim, the following actions will be taken with respect to the project plan, the project risk and alternative approaches may be considered where the need may arise.

Literature review of materials, journals and papers pertaining to Youla Parametrization and feedback stabilization (two-degree of freedom controller) would be studied in depth in line with the allocated period of the in the project plan, so as to have a better understanding and background of the methods.

Designing controllers based on the different methods listed above.

Hands - on simulation of the controllers and their application to unstable systems will be carried out in line with the project plan. The Simulation software to be used is Matlab of which the department has license.

Analysis of stability and robustness issues as well as the controller limitations will then be carried out.

Control synthesis of both strategies to check if there is a possibility of merging them.

Recommendations based on findings after the synthesis and possibly implementing alternative approaches to the subject.

Alternatively if the control synthesis between the Youla Parametrization and the feedback stabilization does not yield the appropriate results needed, then one of the methods may have to be used. Both methods will be compared in terms of performance criteria such as set point tracking, robustness and good disturbance rejection as well satisfying the aims and objectives of the project. The same model will be used to test the methods and the design method with a better response to the chosen model will be chosen as the control strategy. If they both do not give satisfactory responses needed, then robustness would be used as the criteria for choosing the controller. The controller that has better robustness would be chosen.

## 4.1 PROJECT PLANNING AND MILESTONE CHART

The project is scheduled to start the on the 13th of June, but I will be taking a four days holiday so I would start precisely on the 20th of June. The project through until the first week of September with the submission deadline being 5th of September. The Gantt chart provided shows the path with which the project will be implemented and how each activity is connected. A float is provided between 3rd and the 15th of August so as to accommodate any exceeded deadline for the task. The Gantt chart is shown below:

## 5.1 PROJECT RISK:

The project work will mainly involve reviews of literatures, designing of controllers, simulation of the controllers on unstable system and carrying out analysis on certain performance criteria such as robustness, set point tracking and disturbance rejection. Most of which will be done using my personal laptop. The risk involved in executing this task is outlined as follows:

There is a risk that the approach i.e. the control synthesis of both the Youla Parametrization and the feedback stabilization might not yield a controller that is simple and intuitive.

Mitigation Strategy: Then one of the methods might have to be used which produces a better performance response.

Simulations were not done on the Youla parametrization method during this study; hence there might be a risk of the method not satisfying certain conditions in the aims and objectives of the project.

Mitigation Strategy: Recommendations would be made, and possibly concentrate on the second method which some simulations had already been done during this period.

The controller designed based on the Youla Parametrization may not be very efficient on the selected model structure (SISO) and hence might affect the original objectives of the project.

Mitigation Strategy: Schedule a meeting with my supervisor to review the scope and discuss possible alternatives.

There might be a possibility of time overlap between tasks, because there are sub-tasks which might be running concurrently and some may depend on the other which may make me exceed certain deadlines and not meet up the overall dissertation deadline.

Mitigation Strategy: A float has been added to the project Gantt chart, so as to take care of such situations.

## 6.1 CONCLUSION:

The feasibility report has shown that it is feasible to stabilize an open loop unstable plant using the feedback stabilization method. Also it shows that the literatures can be sourced from the school's library.

The major software to be used for the project is Matlab of which the department has license. The Gantt chart shows the project can be executed within schedule and can therefore conclude that the project irrespective of being challenging is feasible.

## 7.1 REFERENCES

[Astrom and Murray (2008)] Astrom K. J. and Murray R. M. (2008) "Feedback Systems: An Introduction for Scientist and Engineers", Princeton University Press, p. 356-358

[Chiu et al. (1990)] Chiu, Min-Sen and Arkun, Yaman(1990) 'Parametrization of all stabilizing IMC controllers for unstable plants', International Journal of Control, 51: 2, 329 - 340

[Dorf and Bishop (1998)] Dorf R. C. and Bishop R. H. (1998) " Modern Control Systems" (Eight edition) Addison Wesley Longman Inc. p. 295

[Garcia and Morari (1982)] Garcia, C.E. and M. Morari (1982) "Internal Model Control-1 A unifying review and some new results," Ind. Eng. Chem. Process Des. & Dev., 21, pp.308-323

[Huang and Chen (1997)] Huang H. P and Chen C. C. (1997) "Control-Synthesis for Open-loop Unstable Process with Time-delay", IEE proc. Control Theory Appl, Vol. 144, No. 4, 334-346

[Kou Yamada (1999)] Kou Yamada (1999) "Modified Internal Model Control for Unstable Systems", Proceedings of the 7th Mediterranean Conference on Control and Automation Hiafa, Isreal-June 28-30"

[Maciejowski, 1989] Maciejowski, J. M (1989) "Multivariable Feedback design", Addison-Wesley Publishers Ltd, p. 315

[Morari and Zafiriou (1989)] Morari, M.; Zafiriou, E. (1989) "Robust Process Control"; Prentice-Halk Englewood Cliffs, NJ; pp 85-112.

[Tan et al. (2003)] Wen, Tan, Horacio J. Marquez. and Tongwen, Chen (2003 ) "IMC design for unstable processes with time delays" Journal of process Control,13,203-213.

[T. Liu et al. (2004)] Tao Liu, Xing He, Danying Gu, Wei Wang (2004) "A novel control scheme for typical unstable processes with time-delay" Proceeding of the 2004 American Control Conference Boston, Massachusetts