Design of Output Feedback Compensator

Design of Output Feedback Compensator

Vanita Jain 1, B.K.Lande 2

1Professor, Bharati Vidyapeeths College of Engineering, Paschim Vihar, New Delhi-110063

2Principal, Shah and Anchor Kutchhi Engineering College, Chembur, Mumbai 400 088

Abstract

This paper discusses the problem of stabilization of physical systems of a very high order and the reduced order model of the higher order systems. The methodology uses the concept of parameterization of all compensators that stabilize a given plant. The output feedback compensator is parameterized by one parameter. It stabilizes the higher order system as well as its reduced order model. The compensator design method does not involve higher order system for its computation.

Keywords: Output feedback; Compensator; Reduced order model.

1. Introduction

   The mathematical models of many engineering systems are frequently of high dimension. These models are often too large for the purpose of, analysis and controller design. The cost and complexity of the controller increases as the system order increases [1-3]. This problem can be overcome if a good reduced order model is available for the original higher order system and if it is possible to design a controller using the reduced model, which will stabilize the original higher order system when placed in the closed-loop. Hence for the cost and time saving in design, and for simplifying implementation, reduced order models are highly desirable for engineers in analysis, synthesis, and simulation of complicated high order system.

   Model reduction concept originated towards the end of the 19th century when Pade approximations were introduced in [4]; interest of researchers in this very important field was spurred after the work of Rosenbrock [5] on distillation columns. Today, there exist a variety of concepts and techniques that have a common goal of reducing the dimension of the mathematical model of a large-scale system in order to simplify the design of control and estimation schemes. Among the important methods are Pade approximation [6], Routh approximations [7], moment matching techniques [8] and balanced realization [9,10]. A good review on the available techniques and comparison can be found in

   Mahmoud [11] and Fortuna [12]. Linear matrix inequality based model reduction techniques have been developed in [13] and improved Routh-Pade approximants has been given in [14]. Its limitation is that it works for stable systems only. Reduced order controller has been designed for nonlinear systems also [ Astolfi]. A very less effort has been put to use the reduced models for designing controller for the higher order system. This is primarily due to the fact that if a stabilizing controller is designed from the reduced model and applied to the higher order system, it does not guarantee the stability of the closed- loop system [15,16]. In [3], Lamba shows that, if a state feedback control is designed from the Davison [17] and applied to the higher order system, it results in a stable closed-loop system. But the method needs the system states to be fully available for feedback. Such a controller cannot be implemented for a system where some of the state variables are not accessible. Some other methods are reported in [18-20].

   This paper describes a technique for designing a stabilizing controller for a higher order system using its reduced order model. The method uses the parameterization of all compensators that stabilize a given plant [21]. It is shown that the output feedback compensator, which is obtained from the reduced model, not only stabilizes the reduced model but also the higher order system.

2. Problem Statement

   Let a stable single-input single-output (SISO) linear time-invariant system of order 'n' is represented as

   G(s) np (s) / dp (s)

   (1.1)

   G s

   represents a higher order system whose reduced order model of order 'm' is

   represented by

   M (s) nm (s) / dm (s)

   (1.2)

   We assume that there are no polezero cancellations. The problem is to find a controller

   using the reduced model that will stabilize both the higher order system G s and its

   reduced order model M s and the order of compensator should be less than the order of the higher order system G s .

   The technique of parameterization of all stabilizing controllers for a system [21] is used to design the controller for higher order system via its reduced model. Assume that a given system P(s) (s) and is expressed as a ratio of rational function as:

   P(s) n(s) / d (s); n(s), d (s) S (1.3)

   Let C' (s) (s) be a stabilizing controller of system P s expressed as a ratio of

   rational functions

   c c c c

   c c c c

   C' (s) n' (s) / d ' (s); n' (s), d ' (s) S

   (1.4)

   where s is the set of real rational functions in the variable s and S is the subset of

   s consisting of all rational functions that are bounded at infinity and whose all poles

   have negative real parts .

   The system (1.3) and the stabilizing controller (1.4) are expressed as a ratio of two stable rational functions, which is called factorization approach [21]. Fractional factorization of

   rational functions in S is non unique [21,22]. The stable rational functions in (1.3) and (1.4) are selected such that they satisfy Bezout identity.

   c c

   c c

   n' (s)n(s) d(s)d(s) 1

   Assume that P(s) (s) n!

   and let C(s) (s) be a stabilizing controller of

   (1.5)

   r !n r !

   P(s) . Let

   n(s) / d (s) and

   n' (s) / d ' (s)

   be fractional factorizations of

   P(s) and C(s) in

   c c

   c c

   S. The set of all stabilizing controllers of P(s) is given by

   Stab(P(s)) nc (s) rd (s) : r S and d (s) rn(s) 0

   (1.6)

   d (s) rn(s) c

   c

   To each r S corresponds a stabilizing controller and to each stabilizing controller corresponds a stable rational function r S .

3. Compensator Design

   In this section, the above concept of controller parameterization is used to derive a controller for higher order system using its reduced order model.

   As the higher order system is stable we can choose rational functions as:

   n1 (s) G(s), d1 (s) 1, x1 (s) 0

   and

   y1 (s) 1,

   then they satisfy Bezout identity

   x1 (s)n1 (s) d1 (s) y1 (s) 1.

   Similarly for reduced order plant, if we choose

   n2 (s) M (s), d2 (s) 1, x2 (s) 0 and y2 (s) 1

   then they satisfy Bezout identity

   n2 (s)x2 (s) d2 (s) y2 (s) 1 .

   We can take

   G /1and

   M /1as the fractional factorization of the system and its reduced

   model respectively. As G and M are stable, there exists constants

   k1 and

   k2 which

   k1

   k1

   k2

   k2

   stabilize the system and the model respectively as shown in Fig. 1.1 and Fig. 1.2.

   Input

   G Output Input

   M Output

   Fig 1.1: Higher order system with Fig 1.2: Reduced order model with

   constant gain k1 constant gain k2

   Let the fractional factorization of

   k1 and

   k2 be

   k1 /1 and

   k2 /1 are trivial factorization

   respectively. From (1.6), the set of stabilizing controller for the system G(s) is

   Stab(G(s)) k1 r1 : r S

   (1.7)

   1

   1

   1 r1G

   Similarly the set of stabilizing controller for the reduced model is

   Stab(M (s)) k2 r2 : r S

   (1.8)

   2

   2

   1 r2 M

   Given the plant and reduced model G(s) and M(s), which are to be stabilized,the

   parameterization of two compensators, which individually stabilize those plants, can be

   equated to find the common stabilizing compensator. We get,

   k1 r1

   1 r1G

   k2 r2

   1 r2 M

   Solving for r1 ,

   r k2 r2 k1r2 M k1

   (1.9)

   1 1 r M Gk r G

   Theorem

   2 2 2

   If dmdp nmdp npdmk2 npdm is Hurwitz then there exists an mth order compensator

   C(s) (k2 1)dm

   dm nm

   , which stabilizes the higher order system and its reduced order model.

   Proof: From (1.9) we have

   r k2 r2 k1r2 M k1

   1 1 r M Gk r G

   2 2 2

   for simplicity we take r2 1 S then

   r k2 1 k1M k1

   as M nm / dm

   and

   1

   G np / dp , we get

   1 M Gk2 G

   r (k2 dm dm k1nm k1dm )dp

   (1.10)

   1 d d n d n d k n d

   m p m p p m 2 p m

   Hence if dmdp nmdp npdmk2 npdm is Hurnitz then r1 S . From (1.8).

   Stab(M (s)) k2 r2

   : r S as r2 1 and M nm

   1 r M 2 d

   2 m

   so mth order compensator

   C(s) (k2 1)dm

   1 nm

   (1.11)

   stabilizes both the system and the reduced order model of the system.

   Remark 1: From (1.11) we see that, for designing a controller for higher order system, only a reduced order model and checking Hurwitzness of a certain polynomial is involved in computation. Hence, the complexity in obtaining a controller for higher order system is greatly reduced and further we get a controller of lower order.

   Remark 2: The compensator in (1.11) is parameterized by one parameter k2 . The range of k2 for which the feedback system shown in Fig.1.2 is stable can be obtained. Similarly the range of k2 for which the polynomial dmdp nmdp npdmk2 npdm is stable can be

   obtained. From these the range of k2

   can be obtained for which C(s) (k2 1)dm

   1 nm

   stabilizes the system and the reduced order model. In this approach k1 is not required to be computed.

4. Example

   Consider the continuous time system having three poles of relatively different order of magnitude described by the following transfer function [23]

   G(s) 1

   (s 5)(s 1)(s 0.1)

   The transfer function of reduced order model is

   M (s) 0.1835s 0.1938

   s 0.0969

   The range of k2 for which reduced model is stable is k2 0.5 .

   From the polynomial dmdp nmdp k2npdm npdm

   are: 0.1662k2 5.8622 0,

   ,the constraints on k2

   for the stability

   0.1662k 2 4.9977k

   5.8046

   and

   0.0485 0.0969k 0

   2 2 0 2

   0.1662k2 5.8622

   To satisfy all the constraints k2 0.5

   Thus a compensator C(s) (k2 1)(s 0.0969) , k

   (1.1835s 0.0969) 2

   0.5

   is a stabilizing controller for the given higher order system and its reduced order model.

   If k2 1 k

   then C(s)

   k(s 0.0969)

   (1.1835s 0.0969)

   , k 0.5

   The closed-loop responses of the higher order system and its reduced order model with this compensator for k 1.6 is shown in Fig. 1.3.

5. Conclusions

The effectiveness of a reduced order model is established, as a controller designed from the reduced model works satisfactory when applied to the higher order system. The controller design method does not involve higher order system for its computation and it ensures the stability of the closed-loop system. Another feature of this method is that the order of the compensator is equal to the order of the reduced model. The efforts required to design the compensator through the reduced model are very less as compared to the one, which will involve higher order system for its computation.

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