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Pole placement design

Description

Pole placement is a method of calculating the optimum gain matrix used to assign closed-loop poles to specified locations, thereby ensuring system stability. Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. For more information, see Pole Placement .

System with state matrix A, input matrix B, output matrix C, and feedthrough matrix D, with full state feedback matrix K

From the figure, consider a linear dynamic system in state-space form

x ˙ = A x + B u

y = C x + D u

For a given vector p of desired self-conjugate closed-loop pole locations, place computes a gain matrix K such that the state feedback u = – Kx places the poles at the locations p . In other words, the eigenvalues of A – BK will match the entries of p (up to the ordering).

K = place( A , B , p ) places the desired closed-loop poles p by computing a state-feedback gain matrix K . All the inputs of the plant are assumed to be control inputs. place also works for multi-input systems and is based on the algorithm from [1] . This algorithm uses the extra degrees of freedom to find a solution that minimizes the sensitivity of the closed-loop poles to perturbations in A or B .

[ K , prec ] = place( A , B , p ) also returns prec , an accuracy estimate of how closely the eigenvalues of A – BK match the specified locations p ( prec measures the number of accurate decimal digits in the actual closed-loop poles). A warning is issued if some nonzero closed-loop pole is more than 10% off from the desired location.

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Pole Placement Design for Second-Order System

For this example, consider a simple second-order system with the following state-space matrices:

A = [ - 1 - 2 1 0 ] B = [ 2 0 ] C = [ 0 1 ] D = 0 Spate-space matrices

Input the matrices and create the state-space system.

Compute the open-loop poles and check the step response of the open-loop system.

Figure contains an axes object. The axes object contains an object of type line. This object represents sys.

Notice that the resultant system is underdamped. Hence, choose real poles in the left half of the complex-plane to remove oscillations.

Find the gain matrix K using pole placement and check the closed-loop poles of syscl .

Now, compare the step response of the closed-loop system.

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent sys, syscl.

Hence, the closed-loop system obtained using pole placement is stable with good steady-state response.

Note that choosing poles that are further away from the imaginary axis achieves faster response time but lowers the steady-state gain of the system. For instance, consider using the poles [-2,-3] for the above system.

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent sys, syscl, syscl2.

Pole Placement Precision

For this example, consider the pole locations [-2e-13,-3e-4,-3e-3] . Compute the precision of the actual poles.

A precision value of 2 is obtained indicating that the actual pole locations are precise up to 2 decimal places.

Pole Placement using Complex Poles

For this example, consider the following transfer function with complex-conjugate poles at - 2 ± 2 i :

s y s t f ( s ) = 8 s 2 + 4 s + 8 Transfer function of the system

Input the transfer function model. Then, convert it to state-space form since place uses the A and B matrices as input arguments.

Next, compute the gain matrix K using the complex-conjugate poles.

The values of the gain matrix are real since the poles are self-conjugate. The values of K would be complex if p did not contain self-conjugate poles.

Now, verify the step response of the closed-loop system.

Figure contains an axes object. The axes object contains an object of type line. This object represents syscl.

Pole Placement Observer Design

For this example, consider the following SISO state-space model:

A = [ - 1 - 0 . 7 5 1 0 ] B = [ 1 0 ] C = [ 1 1 ] D = 0 SISO State-Space Model

Create the SISO state-space model defined by the following state-space matrices:

Now, provide a pulse to the plant and simulate it using lsim . Plot the output.

Figure contains an axes object. The axes object with title Output contains an object of type line.

For this example, assume that all the state variables cannot be measured and only the output is measured. Hence, design an observer with this measurement. Use place to compute the estimator gain by transposing the A matrix and substituting C' for matrix B . For this instance, select the desired pole locations at -2 and -3 .

Use the estimator gain to substitute the state matrices using the principle of duality/separation and create the estimated state-space model.

Simulate the time response of the system using the same pulse input.

Compare the response of the actual system and the estimated system.

Figure contains an axes object. The axes object with title Comparison - Actual vs. Estimated contains 4 objects of type line. These objects represent x_1, x_2, xHat_1, xHat_2.

Input Arguments

A — state matrix nx -by- nx matrix.

State matrix, specified as an Nx -by- Nx matrix where, Nx is the number of states.

B — Input-to-state matrix Nx -by- Nu matrix

Input-to-state matrix, specified as an Nx -by- Nu matrix where, Nx is the number of states and Nu is the number of inputs.

p — Closed-loop pole locations vector

Closed-loop pole locations, specified as a vector of length Nx where, Nx is the number of states. In other words, the length of p must match the row size of A . Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. For an example on selecting poles, see Pole Placement Design for Second-Order System .

place returns an error if some poles in p have multiplicity greater than rank(B) .

In high-order problems, some choices of pole locations result in very large gains. The sensitivity problems attached with large gains suggest caution in the use of pole placement techniques. See [2] for results from numerical testing.

Output Arguments

K — optimum gain ny -by- nx matrix.

Optimum gain or full-state feedback gain, returned as an Ny -by- Nx matrix where, Nx is the number of states and Ny is the number of outputs. place computes a gain matrix K such that the state feedback u = – Kx places the closed-loop poles at the locations p .

When the matrices A and B are real, K is

real when p is self-conjugate.

complex when the pole locations are not complex-conjugates.

prec — Accuracy estimate of the assigned poles scalar

Accuracy estimate of the assigned poles, returned as a scalar. prec measures the number of accurate decimal digits in the actual closed-loop poles in contrast to the pole locations specified in p .

You can use place for estimator gain selection by transposing the A matrix and substituting C' for matrix B as follows, as shown in Pole Placement Observer Design . You can use the resultant estimator gain for state estimator workflows using estim .

[1] Kautsky, J., N.K. Nichols, and P. Van Dooren, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, 41 (1985), pp. 1129-1155.

[2] Laub, A.J. and M. Wette, Algorithms and Software for Pole Assignment and Observers , UCRL-15646 Rev. 1, EE Dept., Univ. of Calif., Santa Barbara, CA, Sept. 1984.

Version History

Introduced before R2006a

lqr | rlocus | estim

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Robust pole assignment for linear control systems in a circular region using novel global harmony search algorithm

  • Published: 01 June 2016
  • Volume 14 , pages 713–722, ( 2016 )

Cite this article

  • Junchang Zhai 1 , 2 ,
  • Liqun Gao 3 &
  • Steven Li 4  

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3 Citations

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This paper presents a new approach to the problem of robust pole assignment in a circular region using novel global harmony search (NGHS) algorithm. Based on geometric principles, the position information of poles in the circular region is depicted and the rules of dynamic selection of poles from the circular region are determined. This ensures the algorithm select poles dynamically from the circular region. In order to get a set of poles and the state feedback controller which allow the system to have a maximum allowable perturbation or uncertainty, the upper bound of perturbation or uncertainty is optimized by the NGHS algorithm for the poles in the circular region. In contrast to most existing methods, an optimization method using NGHS algorithm for the dynamic selection of poles, makes the closed-loop system show better robustness. Finally, the simulation results demonstrate the effectiveness of the proposed approach.

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Authors and affiliations.

College of Information Science and Technology, Bohai University, Jinzhou, Liaoning, 121013, P. R. China

Junchang Zhai

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110819, P. R. China

Graduate School of Business and Law, RMIT University, Melbourne, VIC, 3000, Australia

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Corresponding author

Correspondence to Junchang Zhai .

Additional information

Recommended by Associate Editor Shengyuan Xu under the direction of Editor Myotaeg Lim. This work is partially supported by the Funds of National Science of China (Grant nos. 61104106 and 61104029), the Natural Science Foundation of Liaoning Province (Grant nos. 201202156 and 2013020144).

Junchang Zhai received the B.S. degree in mathematics from Bohai University in 2002, and he obtained his M.S. degree in computer software and theory from Bohai University, Jinzhou, China in 2009. He is currently working towards Ph.D. degree in the field of control theory & control Engineering at Northeastern University. His current research interests are robust control and evolutionary algorithms.

Liqun Gao received the M.S. and Ph.D. degrees in automatic control from Northeastern University, Shenyang, China, in 1985 and 1991, respectively. Currently, he is a professor in the control theory and navigation technology department, Northeastern University. His current research interests are artificial intelligence, control theory and control methods, and pattern recognition.

Steven Li is a Professor of Finance in RMIT University, Melbourne, Australia. He holds a PhD in applied mathematics from Delft University of Technology, The Netherlands, an MBA from Melbourne Business School and a Bachelor of Science degree from Tsinghua University, China. He has previously taught at University of South Australia, Queensland University of Technology, Edith Cowan University and Tsinghua University. His current research interests are mainly in Quantitative Finance and Financial Management. He has published extensively in both finance and applied mathematics.

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Zhai, J., Gao, L. & Li, S. Robust pole assignment for linear control systems in a circular region using novel global harmony search algorithm. Int. J. Control Autom. Syst. 14 , 713–722 (2016). https://doi.org/10.1007/s12555-014-0302-2

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Received : 28 July 2014

Revised : 29 March 2015

Accepted : 08 July 2015

Published : 01 June 2016

Issue Date : June 2016

DOI : https://doi.org/10.1007/s12555-014-0302-2

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Chen, W. (September 1, 1997). "Robust Control Design for Pole Assignment of Uncertain Systems." ASME. J. Dyn. Sys., Meas., Control . September 1997; 119(3): 588–590. https://doi.org/10.1115/1.2801301

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This Technical Brief addresses the problem of robust pole assignment for uncertain dynamic systems. A control design methodology is proposed to determine a fixed controller assigning all poles of uncertain systems in a pre-specified disk. The robust pole assignment problem is reduced to a convex optimization problem. The proposed method is applied in design of a flight control system for a small aircraft successfully.

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Pole assignment for linear and quadratic systems with time-delay in control

Research output : Contribution to journal › Article › Research › peer-review

We consider the pole assignment problems for time-invariant linear and quadratic control systems, with time-delay in the control. Closed-loop eigenvectors in X=[x 1 ,x 2 ,...] are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms of X. Condition of the problems is also investigated. Our approach extends the well-known Kautsky, Nichols, and VanDooren algorithm. Consequently, the results are similar to those for systems without time-delay, except for the presence of the 'secondary' eigenvalues and the condition of the problems. Simple illustrative numerical examples are given.

  • Linear system
  • Pole assignment
  • Quadratic system

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  • 10.1002/nla.817

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  • Link to publication in Scopus

T1 - Pole assignment for linear and quadratic systems with time-delay in control

AU - Chu, E. K W

PY - 2013/3

Y1 - 2013/3

N2 - We consider the pole assignment problems for time-invariant linear and quadratic control systems, with time-delay in the control. Closed-loop eigenvectors in X=[x1,x2,...] are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms of X. Condition of the problems is also investigated. Our approach extends the well-known Kautsky, Nichols, and VanDooren algorithm. Consequently, the results are similar to those for systems without time-delay, except for the presence of the 'secondary' eigenvalues and the condition of the problems. Simple illustrative numerical examples are given.

AB - We consider the pole assignment problems for time-invariant linear and quadratic control systems, with time-delay in the control. Closed-loop eigenvectors in X=[x1,x2,...] are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms of X. Condition of the problems is also investigated. Our approach extends the well-known Kautsky, Nichols, and VanDooren algorithm. Consequently, the results are similar to those for systems without time-delay, except for the presence of the 'secondary' eigenvalues and the condition of the problems. Simple illustrative numerical examples are given.

KW - Linear system

KW - Pole assignment

KW - Quadratic system

KW - Robustness

KW - Time-delay

UR - http://www.scopus.com/inward/record.url?scp=84873694519&partnerID=8YFLogxK

U2 - 10.1002/nla.817

DO - 10.1002/nla.817

M3 - Article

AN - SCOPUS:84873694519

SN - 1099-1506

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

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Home Theater & Automation Installers in Elektrostal'

Location (1).

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  • Elektrostal', Moscow Oblast, Russia

Featured Reviews for Home Theater & Automation Installers in Elektrostal'

  • Reach out to the pro(s) you want, then share your vision to get the ball rolling.
  • Request and compare quotes, then hire the Home Automation & Home Media professional that perfectly fits your project and budget limits.
  • Home Automation
  • Home Security Companies & Installation
  • Home Theater Installation
  • Security Camera Installation
  • Surround Sound Installation
  • TV Installation
  • Home Theater Design

To set up a home theater system in your home, it’s best to get help from a professional home theater installation company.

  • Find a Reliable Company, for example on this Houzz page.
  • Get in touch with the company and schedule a meeting. Talk to them about what you want and how much you can spend.
  • The experts from the company will create a unique plan for your home theater based on your space and preferences.
  • Work together to select the right equipment, like speakers, a TV or projector, a receiver, and more.
  • The company will take care of the installation process. They will set up everything properly and make sure it all works correctly.
  • Once everything is installed, they will do calibration and testing to get the best sound and picture quality.

By getting help from a professional in Elektrostal', you can have a great home theater system without the hassle of setting it up yourself.

Home automation can bring many benefits and add value to your home in Elektrostal'.

  • Easy Control: you can easily control things like lights, temperature, and security using your smartphone or voice commands.
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  • Personalization: customize the system to fit your needs and preferences, controlling what matters most to you.
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In summary, home automation offers easy control, energy savings, enhanced security, personalization, increased home value, and an improved lifestyle. It’s a smart investment that makes daily living easier and more efficient.

What is a home media designer?

Questions to ask home media designers or home automation companies:, find home theater & automation companies near me on houzz, business services, connect with us.

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IMAGES

  1. Schematic diagram of pole assignment strategy with the identical

    pole assignment in control systems

  2. PPT

    pole assignment in control systems

  3. Schematic diagram of the pole assignment strategy with the identical

    pole assignment in control systems

  4. PPT

    pole assignment in control systems

  5. (PDF) Robust Pole Assignment in Singular Control Systems

    pole assignment in control systems

  6. l7v2 Introduction to pole assignment and general closed loop canonical

    pole assignment in control systems

VIDEO

  1. Urgent Requirements for Reputed gulf company 14/09/23 विदेश जाने का सुनहरा मौका DUBAI KUWAIT SAUDI

  2. The Concept of Pole Placement in Classical and Modern Control, 30/3/2016

  3. Loss of Controllability Due to a Pole-Zero Cancellation in the LH s-plane (Example 9-13), 23/3/2016

  4. ASSIGNMENT CONTROL 2 GROUP 4 QUESTION B

  5. L5E21

COMMENTS

  1. PDF POLE ASSIGNMENT FOR LINEAR SYSTEMS

    Pole Assignment Problem Given an nth degree monic polynomial with real coefficients r(s), find a matrix K such that pK(s) = det(sI − A + BK) = r(s). 45 If for every given r(s), there exists a K such that pK(s) = r(s), we say that the closed-loop poles can be arbitrarily assigned.

  2. Pole placement design

    Pole placement is a method of calculating the optimum gain matrix used to assign closed-loop poles to specified locations, thereby ensuring system stability. Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. For more information, see Pole Placement.

  3. Robust partial pole assignment problem for high order control systems

    In this paper, we consider the partial pole assignment problem (PPAP) for high order control systems. It is shown that solving the PPAP is essentially solving a pole assignment for a linear system of a much lower order, and the robust PPAP is then concerning the robust pole assignment problem for this linear system.

  4. PDF Controller Design by Pole placement

    PowerPoint Presentation Controller Design by Pole placement Introduction to control Design of two position controller Control design by pole placement Control design by PID control Introduction to Control So far we have modeled systems ( mechanical, electromechanical and electric) and analyzed their time-response behavior.

  5. Partial pole assignment using rank-one control and ...

    Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey 1 Introduction The control of linear dynamic systems represented by second-order models has been an exciting research topic in the last four decades.

  6. Robust pole assignment in singular control systems

    Robust pole assignment in singular control systems. Necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems (descriptor systems) to have a solution which is regular and nondefective. For a robust solution, such that the assigned closed-loop poles are insensitive to perturbations in the ...

  7. Consistent systems and pole assignment: I

    Abstract. We study the consistency property of a linear time-invariant control system with incomplete feedback and of a bilinear time-invariant control system in connection with the pole assignment problem. We show that consistency is a sufficient and, in some cases, necessary condition for the arbitrary pole assignability of systems with ...

  8. Pole assignment by state-derivative feedback for single-input linear

    AbstractThis paper presents an efficient solution to the pole assignment problem using state-derivative feedback for continuous, single-input, time-invariant, linear systems. ... Part I: Journal of Systems and Control Engineering. VIEW ALL JOURNAL METRICS. Article usage * Total views and downloads: 24 * Article usage tracking started in ...

  9. Robust pole assignment for linear control systems in a ...

    This paper presents a new approach to the problem of robust pole assignment in a circular region using novel global harmony search (NGHS) algorithm. Based on geometric principles, the position information of poles in the circular region is depicted and the rules of dynamic selection of poles from the circular region are determined. This ensures the algorithm select poles dynamically from the ...

  10. Robust Control Design for Pole Assignment of Uncertain Systems

    A control design methodology is proposed to determine a fixed controller assigning all poles of uncertain systems in a pre-specified disk. The robust pole assignment problem is reduced to a convex optimization problem. The proposed method is applied in design of a flight control system for a small aircraft successfully. Issue Section:

  11. Pole-zero assignment by the receptance method: multi-input active

    Pole-zero assignment by the receptance method: multi-input active vibration control Dario Show more Add to Mendeley Vibration suppression is a goal intensively pursued over the decades to improve the performances of lightweight flexible systems.

  12. Robust pole assignment for synthesizing fractional-order control

    In this paper, a neurodynamic optimization approach is proposed for robust pole assignment of fractional-order control systems. Compared with integral-order systems, the pole assignment of fractional-order systems is more challenging due to variability of stability region. The robust pole assignment is formulated as a constrained optimization problem, and a robustness measure is derived as a ...

  13. On pole assignment in multi-input controllable linear systems

    On pole assignment in multi-input controllable linear systems. W. Wonham. Published in IEEE Transactions on… 1 December 1967. Engineering. It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable ...

  14. Robust pole assignment in singular control systems

    21 Robust Pole Assignment via State Feedback for Uncertain High-order System Ya‐Jun Gao G. Duan Engineering 2022 4th International Conference on Control and… 2022 The problem of robust pole assignment for the uncertain high-order system is formulated.

  15. Pole assignment for linear and quadratic systems with time-delay in control

    We consider the pole assignment problems for time-invariant linear and quadratic control systems, with time-delay in the control. Closed-loop eigenvectors in X=[x 1,x 2,...] are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms of X. Condition of the problems is also investigated.

  16. Robust pole assignment for linear control systems in a circular region

    This paper presents a new approach to the problem of robust pole assignment in a circular region using novel global harmony search (NGHS) algorithm based on geometric principles, and the rules of dynamic selection of poles from the circular region are determined. This paper presents a new approach to the problem of robust pole assignment in a circular region using novel global harmony search ...

  17. On Pole-Zero Assignment of Vibratory Systems by Multi-Input Feedback

    Vibrating system, pole and zero assignment, receptance method, linear matrix Inequalities ... [43] that the poles of a rst-order control system can be assigned via state feedback if and only if the system is controllable. In [21], some numerical algorithms were developed for the robust state-feedback pole assignment problem in the rst-order control

  18. Pole Assignment in Vibratory Systems by Multi-input Control

    POLE ASSIGNMENT IN VIBRATORY SYSTEMS BY MULTI-INPUT CONTROL Y.M. RAM a , S. ELHAY b Add to Mendeley https://doi.org/10.1006/jsvi.1999.2622 Get rights and content The problem of reassigning some poles of a vibratory system, while keeping the other poles unchanged, is considered.

  19. Creating a Model of Voltage Control in Vacuum Arc Furnace. Report 1

    The control objective was determined, which is to maintain the integral of the difference between the desired and the actual voltage in the range, not exceeding the value predetermined by technologists. ... The schedule and system of equations describing the gradual formation of droplets was proposed. В целях изучения ...

  20. ST-01 Speed gate with a built-in card capture reader

    protection rating. motor drive. scheme. ST-01 Speed gate with a built-in card capture reader is designed for sites with increased requirements for design and convenience. The built-in card capture reader allows automating the collection of guest passes. Description. Delivery set: two sections with built-in electronics.

  21. Home Theater & Automation Installers in Elektrostal'

    Before you hire a home theater or home automation service in Elektrostal', Moscow Oblast, browse through our network of over 224 local home theater & automation companies. Read through customer reviews, check out their past projects and then request a quote from the best home theater & automation companies near you.

  22. Design of pole-assignment controller for systems with algebraic

    Obviously, our problem is a pole-assignment problem with the control structure constraint. We call it as the constrained pole-assignment problem (CPAP). The control law (3) is a solution of the CPAP if and only if the corresponding feedback gain matrix K satisfies Eq. (5) and the resulting closed-loop system (4) has the desired poles.

  23. ST-01 Speed gate with a built-in barcode scanner

    ST-01 Speed gate with a built-in front panel for the barcode scanner is designed for sites with high aesthetic and convenience requirements. The barcode is used as an identifier at facilities with high pedestrian flow, including paid access systems. The barcode can also be used as a guest identifier in business centers, offices, and enterprises.