[Update] State-PID Feedback for Pole Placement of LTI Systems | ratchamangkala university – Sathyasaith

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Abstract

Pole placement problems are especially important for disturbance rejection and stabilization of dynamical systems and regarded as algebraic inverse eigenvalue problems. In this paper, we propose gain formulae of state feedback through PID-elements to achieve desired pole placement for a delay-free LTI system with single input. Real and complex stable poles can be assigned with the proposed compact gain formulae. Numerical examples show that our proposed gain formulae can be used effectively resulting in very satisfactory responses.

1. Introduction

Pole placement has been an important design method of a linear control system [18]. One approach is to use state feedback in which the gain matrix is calculated via Ackermann’s formula [4]. Regarding this, the original state model can be transformed into the bidiagonal-Frobenius canonical form to achieve the desired pole placement [9]. Based on the Frobenius form, the gain matrices can be readily computed for SISO and MIMO systems [1012]. The concept of using state-derivative feedback was introduced in 2003-2004 [13, 14]. One advantage over the conventional state feedback is that it results in smaller gains. In practical control, for example, vibration control, the derivative signal can be derived from an accelerometer output as the concept has been successfully implemented [15, 16]. A linear quadratic regulator to achieve the state-derivative feedback was also developed [17]. Recently, stabilizability and disturbance rejection issues have been investigated for an LTI system with state-derivative vector as its output [18]. The state-derivative feedback is useful for stabilization and rejection of dynamic disturbances not for set-point regulation or tracking control. The approach of state-derivative feedback leads to a possibility of using state-PID feedback.

Consider a delay-free LTI system having single input of the form

where , , and are and real coefficient matrices, respectively.

The linear system under consideration must possess a complete controllability property. Therefore, the controllability matrix must have rank and can be formed from

The Frobenius canonical form of the system is
where , , , and . The vector in which . For the state-PID feedback, the control is of the form
where are row gain vectors for the P, I, and D feedback elements, respectively. Guo et al. (2006) [21] proposed a pole placement method consisting of 3 separated steps. The pole placement by a state-P feedback is conducted first leading to an intermediate system. Secondly, a state-I feedback is performed; another intermediate system resulted. Finally, the closed-loop system with the desired characteristic polynomial is realized by a state-D feedback incorporated as the final design stage. These steps may be utilized purposefully to design state-P, -PD, and -PI feedback elements. According to this, the control of the P-feedback is

where to achieve a desired characteristic polynomial
The closed-loop system at this interim stage has its characteristic polynomial
Equating (1.6) and (1.7) results in the gain matrix . For the I-feedback, the control is
where to achieve a desired characteristic polynomial
The second interim system has its closed-loop characteristic polynomial
Equating (1.9) and (1.10) leads to the gain matrix . For the D-feedback, the control is
where to achieve a desired characteristic polynomial
Equating (1.12) and the closed-loop characteristic polynomial in (1.13) results in the last set of

gain matrix, . This previous method requires three sets of poles to be assigned. Two sets are fictitious, and only the last set is the prescribed characteristic polynomial. In [21], there are no recommendations for selection of these intermediate pole sets. One may attempt arbitrarily chosen poles during the separated design phases. In due course, the calculation procedures are quite awkward. Besides, the paper [21] contains no proof of the proposed theorem.

This paper begins by presenting derivation of the gain matrices for the state-PID feedback in rigorous manner. It also presents the gain matrices for the state-PI and -PD feedback cases. Section 3 presents the analysis of disturbance rejection property of the proposed method. Such complete treatment has not appeared elsewhere before. Three numerical examples are shown in Section 4 to illustrate the effectiveness of our proposed gain formulae in comparison with the use of Ackermann’s formula [4, 19] and the methods by Guo et al. [21] and Kuo [20], respectively. Moreover, we show by simulations in the example shown in Section 4.3 that using the method [21] can result in very large controller gains although the final pole sets remain unchanged. Conclusion follows in Section 5.

2. The PID Gain Matrices

Without loss of generality, the single-input LTI system (1.1) is assumed to be completely controllable, and is of full column rank. The next proposition is the main result presenting the state-PID feedback gain matrices. Note that due to the integral element, one additional closed-loop pole is needed. This imposes a condition for derivation of the gain matrices and results in an increase in the order of the system by one.

Proposition 2.1.
(i)(ii)

The characteristic polynomial of the closed-loop system can be expressed as

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where I is an identity matrix,

that is,

or

It can be observed that the order of the closed-loop system is increased by 1 due to the integral element. By equating (2.6) with the desired characteristic polynomial, the following relations can be obtained for an order system:

Therefore, a desired pole placement can be achieved via the state-PID feedback using the gain matrices in (2.1) and (2.2).
This completes the proof.

The following are 2 immediate consequences of Proposition 2.1.

Corollary 2.2.

Corollary 2.3.

The design procedures are as follows:(1)calculate the transformation matrix for an order LTI plant using where , , and ,(2)calculate the matrices and using and for the Frobenius form of (1.3),(3)assign the closed-loop pole locations of an order:(i)for state-PID feedback, add one negative real pole having a fast time-constant (i.e., a negative real pole with a large magnitude),(ii)for state-PI feedback, add one negative real pole having a fast time constant,(iii) for state-PD feedback, no additional pole is needed,(4)determine the prescribed characteristic polynomial having the order of or corresponding to step 3,(5)calculate the gain matrices: (i)for state-PID feedback, use (2.1) or (2.2),(ii)for state-PI feedback, use (2.8),(iii)for state-PD feedback, use (2.9).

3. Disturbance Rejection

Disturbance rejection is an important property of the proposed state-PID feedback. This section provides the analysis of such property. There are three propositions, one of which has been proposed by [18] and denoted as proposition 3.1 for state-D feedback. The other propositions denoted as Propositions 3.2 and 3.3 are newly developed to confirm the disturbance rejection property accomplished by the state-P and -I feedback components, respectively.

Proposition 3.1.

(see in [18], Lemma  3.2) An immediate conclusion from this proposition according to (3.3) is that the state-D feedback cannot attenuate the influence of in in controlled systems because (3.3) is independent of the state-D matrix .

Proposition 3.2.

From (3.1) and (3.4), note that

Applying the Laplace transform to (3.6), observe that

Thus,

This completes the proof.

From Proposition 3.2, it can be concluded according to (3.8) that the state-P feedback can attenuate the influence of in in controlled systems because (3.8) is dependent on the state-P matrix .

Proposition 3.3.

From (3.1) and (3.9), note that

Applying the Laplace transform to (3.11), observe that

Thus,

This completes the proof.

An immediate conclusion from Proposition 3.3 is that with the state-I feedback a complete rejection of disturbance in controlled systems can be achieved due to (3.13).

4. Illustrative Examples

Three illustrative examples are presented with focusing on stabilization and disturbance rejection issues. Results are compared with those designed by the previous methods including Ackermann’s formula [4, 19], Guo et al. [21], and Kuo [20], respectively.

4.1. Magnetic Ball Suspension

The magnetic ball suspension system [20] represented by the diagram in Figure 1 is adopted as the first example. This 3rd-order system is described by
where , and . The system is originally unstable with its poles at and −100. It is desirable to have the closed-loop poles at , and such that the characteristic polynomial is . The Frobenius canonical form is
The obtained gain matrices are , , and .

The gain matrix due to Ackermann’s formula is . Figure 2 shows the responses and the control input according to the proposed method in which the initial conditions are , and the states are disturbed by 1 unit at the time . It can be observed that using the proposed method the states possess very good transient responses, the disturbances are completely dampened out, and the control input is reasonable. With the conventional pole placement method, some states contain a large amount of steady-state errors due to disturbance as depicted in Figure 3.

(a) Responses of system states
(b) Control signal

(a) Responses of system states
(b) Control signal

By applying the method [21] to achieve the same closed-loop poles, the design requires the following fictitious sets of poles: and . As a result, the gain matrices are , , and with which a combined state-PID feedback controller is derived. The two intermediate systems denoted as and are given in Frobenius canonical forms as follows:
where . Under the same simulation situations previously described, similar state responses to those in Figure 2 are achieved due to the closed-loop poles located at the same locations ,. Notice that some of the gains designed by the proposed method are somewhat larger but in reasonable ranges for implementation using either analog or digital technology. The proportional gains of the proposed method are smaller than those obtained using the method [21]. This means that the proposed controller draws less energy to achieve its control action.

Based on the method [20], the gains can be obtained to place the closed-loop poles at , −20 with the desired characteristic polynomial of . Hence, (4.4) describes the closed-loop system
Figure 4 shows the state responses having the initial conditions , and the states are disturbed by 0.05 and 1 unit at the time and 3 s. Noticeably, a large amount of steady-state errors in some states due to the disturbances still remain.

(a) Disturbance of 0.05 units at t = 2 s
(b) Disturbance of 1 unit at t = 3 s

4.2. Inverted Pendulum

The inverted pendulum system in [19] is adopted as the second example and represented by the diagram in Figure 5. Its state model is expressed by
where , and . With its poles at and , the system is inherently unstable. It is desirable to place the closed-loop poles at , and −100 such that the characteristic polynomial is

The Frobenius canonical form is

With the proposed state-PID feedback, the gain matrices are , , and.

The gain matrix due to the Ackermann’s formula is. Figures 6 and 7 show the responses and the control inputs in which the initial conditions are , and the states are disturbed by 1 unit at the time . These comparative results show a similarity to those of the first example. Very good transient responses with zero steady-state errors are achieved by the proposed method. With the method [21] to achieve the same closed-loop poles, it requires two fictitious pole sets designated as and . The following feedback gains are obtained, respectively, as , , and with which a combined state-PID feedback controller is derived. During the design process, the two intermediate systems and are calculated and expressed in Frobenius canonical forms as follows:
Under the same simulation situations, similar responses to those in Figure 6 are achieved because the system possesses the same closed-loop pole locations. As a result, the proposed method gives small proportional gains meaning that the proposed controller draws less energy in comparison with that of the method [21]. The magnitude of the integral and the derivative gains are in reasonable ranges for implementation.

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(a) Responses of system states
(b) Control signal

(a) Responses of system states
(b) Control signal

4.3. Mechanical Vibration

The mechanical vibration system in [14] is adopted as the third example. The diagram in Figure 8 represents the system, which is described by
where , and . The open-loop poles of the system are and . It is desirable to have the closed-loop poles at , , and −1000 such that the characteristic polynomial is. The Frobenius canonical form is
With the proposed pole placement method, the obtained gain matrices are , , and. The gain matrix due to the Ackermann’s formula is . Figures 9 and 10 show the responses and the control inputs in which the initial conditions are , and the states are disturbed by 1 unit at the time . Similar to the other examples, using the proposed method, the transient responses of the system states are reasonably good with moderate control input, and all the states converge to origin without steady-state errors. As shown in Figure 10, using the method [4, 19], some states cannot converge to origin properly although the control input is not high. As a result of applying the method [21], the same closed-loop pole locations can be placed through the use of two fictitious pole sets, namely, and . This leads to two intermediate systems, respectively, denoted as
The obtained gain matrices are , , andwith which a combined state-PID feedback controller is derived. Again, similar simulation results to those in Figure 9 are obtained. In order to show that arbitrarily chosen intermediate poles affect the gains, more results of applying the method [21] are included. All cases aim to achieve the same closed-loop pole locations at . Two fictitious pole sets being considered are and . This leads to two intermediate systems denoted as
The following three feedback gains are obtained: , , and . Next, assume the following fictitious pole sets: and resulting in the following intermediate systems:
The obtained gain matrices are,, and . As the final case, the fictitious pole sets are and resulting in two intermediate systems as follows:
The obtained gain matrices are ,, and.


(a) Responses of system states
(b) Responses of system states (continue)
(c) Control signal

(a) Responses of system states
(b) Responses of system states (continue)
(c) Control signal

The above case studies serve to show the effects of fictitious pole locations required by the method [21] on the designed gains, the magnitudes of which can be very large. Selection of fictitious poles is a critical problem of this previous method, which has been neither solved nor considered.

5. Conclusion

A new design method for pole placement via state-PID, -PI, and -PD feedback has been proposed. The method has two distinctive features: (i) compact design formulae and (ii) disturbance rejection property. The analyses of these features have been elaborated through relevant propositions. The paper also describes the design procedures and presents some illustrative examples including a magnetic ball suspension, an inverted pendulum, and mechanical vibration systems, respectively. The simulation results reflect that the proposed method is promising for a real-world application. A future work will be design optimization to achieve minimum gains subject to nonlinear restriction in control input and specified performance constraints.

Acknowledgments

The authors gratefully acknowledge the financial supports by Ratchamangkala University of Technology Isarn and Suranaree University of Technology, Thailand. The authors’ appreciations are also due to the reviewers for their constructive comments.

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Implementation of the NBSAP

Thailand has developed three NBSAPs to date, with the third formulated in the light of achieving the 2010 Biodiversity Target. National policies, measures and plans are grouped into 5 strategies and 17 action plans, which serve as a framework for implementing the NBSAP over a five-year period (2008-2012). The five strategies aim to: 1) protect the components of biodiversity 2) encourage the sustainable use of biodiversity 3) minimize threat to biodiversity 4) promote research, training, education and public awareness and networking on biodiversity and 5) strengthen national capacity for implementing biodiversity-related international agreements.

A process to further update the NBSAP, which will include a suite of national targets, was initiated in May 2011. Notably, the Aichi Biodiversity Targets have been integrated into Thailand’s 11th National Economic and Social Development Plan (2012-2016).

Actions taken to achieve the 2020 Aichi Biodiversity Targets

Among the major achievements toward the 2020 Aichi Biodiversity Targets is the increase of protected areas. At least 20% of the marine and coastal areas in Thai waters have been designated as protected areas and at least 5 sites of wetlands of international importance have been designated as Ramsar sites. Regarding threatened species, at least 10 endangered endemic species have been protected and restored in situ and increased in population. In addition, 98 areas were identified as important plant areas. Awareness-raising campaigns through various activities and media have promoted knowledge on biodiversity issues. The Biodiversity Clearing-House Mechanism (CHM) and the Biosafety Clearing-House (BCH) are fully operational and linked to one another.

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The National Center for Genetic Engineering and Biotechnology has cooperated with the Ratchamangkala University of Technology to implement a project on rice seed production technology transfer, with the full participation of local farmers, using fragrant rice with high resistance to blast disease. The Center has also cooperated with several universities and research institutions to implement a project on the management and improvement of hot chili varieties, resulting in a collection of 752 species of hot chili from around the world. The Office of Natural Resources and Environmental Policy and Planning (ONEP) has compiled, documented and published a series of 17 books (checklists) entitled “Biodiversity Series”, containing the names of and some information on important taxa in Thailand. The development of a national inventory on traditional knowledge and/or local wisdom related to the conservation and sustainable use of biodiversity is ongoing. The royalty-initiated projects with institutions of higher education have synthesized existing knowledge into practical models to be used as methodologies in plant conservation, using biotechnology for the reintroduction of endangered plants.

Support mechanisms for national implementation (legislation, funding, capacity-building, coordination, mainstreaming, etc.)

In 2009, Thailand adopted the National Biodiversity Policy which focuses on the protection and restoration of conservation areas. Further mechanisms include the Country Management Plan (2008-2011), promoting the sustainable use of biodiversity, and the NBSAP (2008-2012). Thailand has also established practical guidelines to facilitate activities related to biodiversity impact assessment, sustainable use and access and benefit-sharing. The country has also established criteria and regulations to control and mitigate threats from invasive alien species, namely through quarantine measures. The Biodiversity Bureau serves as a national focal point for access to and transfer of biological resources. Some departments have put mechanisms in place for access and benefit-sharing, through implementing provisions in relevant laws and policies, such as the Plant Varieties Act, Fisheries Act and the Protection and Promotion of the Thai Traditional Medical Intelligence Act.

Many departments, institutions and organizations in Thailand are emphasizing community engagement and empowerment. The Community Organization Development Institution (CODI) is the leading agency to promote and mobilize capacity-building and empower indigenous and local communities. CODI has developed a tool for strengthening local capacities called “community mapping”. As a result, local communities are able to identify the status and trends of traditional knowledge, as well as the vision and priorities required for protecting traditional practices. Thailand’s Research Fund (TRF) provides funds for research activities and facilitates the exchange of experiences for biodiversity conservation at local levels. In addition, the Population and Community Development Association (PDA) is promoting community participation in forestation by establishing local community networks, facilitating the exchange of experiences, promoting organic farming and soil conservation and providing training to local communities.

Mechanisms for monitoring and reviewing implementation

Although endowed with rich and diverse biodiversity in various tropical ecosystems, Thailand is unfortunately faced with threats due to the loss of natural habitats, overexploitation, as well as from the lack of public concern and inconsistent and non-systematic biodiversity research and study. To solve these problems and facilitate the achievement of the 2010 Biodiversity Target, the ONEP has cooperated with the Kasetsart University in the conduct of a survey and inventory of Biodiversity Important Areas (BIAs) and Biodiversity Hotspots in Thailand, based on seven thematic ecosystems, with particular emphasis put on species contained in Thailand’s Red List, and endemic species. The National Center for Genetic Engineering and Biotechnology (BIOTEC) has collaborated with the Department of National Parks, Wildlife and Plant Conservation to implement a project on the Inventory and Collection of Flowering and Ornamental Plants in the Forests of the Southern Region, and established the “Cooperative Natural Research Center” at Hala Bala Peatswamp Forest, to facilitate research and networking among all related organizations. The main purpose of the project is to facilitate sustainable development in areas based on scientifically- and technologically-sound research and study. The BIOTEC group has also implemented and supported projects related to insect fungus.


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Walking in Suansunandha Ratjabhat University : King RamaV Old Palace

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Nelliglla temple
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