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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 .
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.
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.
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.
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.
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.
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.
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
- Pole Placement
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- DOI: 10.1243/09596518JSCE409
- Corpus ID: 120683611
Pole assignment by state-derivative feedback for single-input linear systems
- Taha H. S. Abdelaziz
- Published 1 November 2007
- Engineering
- Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering
25 Citations
Robust pole assignment for linear time-invariant systems using state-derivative feedback, robust pole assignment by state-derivative feedback using gradient flow approach, parametric eigenstructure assignment using state-derivative feedback for linear systems, stabilization of single‐input lti systems by proportional‐derivative feedback.
- Highly Influenced
Optimal control using derivative feedback for linear systems
Robust pole placement for second-order linear systems using velocity-plus-acceleration feedback, stabilizability and disturbance rejection with state-derivative feedback, less conservative control design for linear systems with polytopic uncertainties via state-derivative feedback, state derivative feedback in second-order linear systems: a comparative analysis of perturbed eigenvalues under coefficient variation, parametric jordan form assignment by state-derivative feedback, 24 references, direct algorithm for pole placement by state-derivative feedback for multi-inputlinear systems - nonsingular case, pole-placement for siso linear systems by state-derivative feedback, robust pole assignment in linear state feedback, robust pole assignment in descriptor systems via proportional plus partial derivative state feedback, a computational algorithm for pole assignment of linear single-input systems, the parametric solutions of eigenstructure assignment for controllable and uncontrollable singular systems, efficient pole placement technique for linear time-variant siso systems, numerical algorithms for eigenvalue assignment by state feedback, an algorithm for pole assignment of time invariant linear systems, eigenstructure assignment by proportional-derivative state feedback in singular systems, related papers.
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IMAGES
COMMENTS
The state feedback pole assignment problem in control system design is essentially an inverse eigenvalue problem, which requires the determination of a matrix having given eigenvalues (cf ...
Numerical methods are described for determining robust, or well-conditioned, solutions to the problem of pole assignment by state feedback. The solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized. It is shown that for these solutions, upper bounds on the norm of ...
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).
We present a unifying computational framework to solve robust pole assignment problems for linear systems using state feedback. The new framework uses Sylvester equation based parametrizations of the pole assignment problems. The non-uniqueness of solutions is exploited by minimizing additionally sensitivity of closed-loop eigenvalues and the norm of the corresponding state feedback matrix ...
3.3 Pole Assignment for Multivariable Systems We shall now extend the pole assignment result for single-input systems developed in the previous section to the multivariable case. The generalization rests on the following result, which uses state feedback to convert the multivariable pole assignment problem back to the single-input case.
The problem of eigenvalue assignment with minimum sensitivity in multivariable descriptor linear systems via state feedback is considered. Based on the perturbation theory of generalized eigenvalues of matrix pairs, the sensitivity measures of the closed-loop finite eigenvalues are established in terms of the closed-loop normalized right and left eigenvectors.
solve robust pole assignment problems for linear sys-tems using state feedback. The new framework uses Sylvester equation based parametrizations of the pole assignment problems. The non-uniqueness of solutions is exploited by minimizing additionally sensitivity of closed-loop eigenvalues and the norm of the correspond-ing state feedback matrix.
This paper considers pole assignment and robust pole assignment problems for discrete-time linear periodic systems by using linear periodic state feedback. The monodromy matrix of the closed-loop system is represented in a special form. By combining this special form with our recent result on solutions to a class of generalized Sylvester matrix equations, a complete parametric approach for ...
By using a Sylvester equation based parametrization, the minimum norm robust pole assignment problem for linear time-invariant systems is formulated as an unconstrained minimization problem for a suitably chosen cost function. The derived explicit expression of the gradient of the cost function allows the efficient solution of the minimization problem by using powerful gradient search based ...
Abstract. In this paper, a technique for computing robust controller for multivariable time-invariant linear systems via state-derivative feedback is introduced such that the sensitivity of the closed-loop system eigenvalues to perturbations in the system and gain matrices is minimized.
This paper presents an efficient solution to the pole assignment problem using state-derivative feedback for continuous, single-input, time-invariant, linear systems. This problem is always solvable for any controllable system with some restrictions when assigning zero poles.
We propose an algorithm for the state-feedback pole assignment problem. The algorithm is the first of its kind, making direct use of the Schur form, and minimizing the departure from normality of the closed-loop poles for a given first Schur vector x 1.The robust pole assignment problem can then be solved via choosing x 1 optimally. Several numerical examples were presented to illustrate the ...
On pole assignment in linear systems with incomplete state feedback Abstract: The following system is considered: \dot{x}= Ax + Bu y = Cx where x is an n vector describing the state of the system, u is an m vector of inputs to the system, and y is an l vector ( l \leq n ) of output variables.
Abstract. It is well known that the poles of a linear system can in general not be arbitrarily placed by memoryless output feedback. On the contrary periodic memoryless output feedback for discrete time invariant linear systems is a powerful tool in the pole-assignment problem. A precise statement is made in this paper.
A controllability condensed form and a state feedback pole assignment algorithm for descriptor systems. K. Chu. Engineering, Mathematics. 1988. TLDR. A direct algorithm for the pole-assignment problem of a time-invariant, linear, multivariable, descriptor system with linear state feedback is presented, based on the controllability condensed form.
In this article, the parametric solution to the pole assignment problem for multivariable linear time-invariant systems controlled by proportional-derivative (PD) state feedback is developed. The new expressions for the PD gain controllers are derived which describe the available degrees of freedom offered by PD state feedback.
Abstract. A new algorithm for the pole-assignment problem of a controllable time-invariant linear multivariable system with linear state feedback is presented. The resulting feedback matrix is a least-squares solution and is robust in a loose sense. The method is based on the controllability canonical (staircase) form and amounts to a new proof ...
Abstract This paper presents an efficient solution to the pole assignment problem using state-derivative feedback for continuous, single-input, time-invariant, linear systems. This problem is always solvable for any controllable system with some restrictions when assigning zero poles. The proposed solution is based on the transformation to the Hessenberg matrix, which is preferable from the ...
Saad, Y., 1988, "Projection and deflection methods for partial pole assignment in linear state feedback," IEEE Transactions on Automatic Control 33, 290-297. Google Scholar. Veselic, K., 1988, "On linear vibrational systems with one dimensional damping," Applicable Analysis 29, 1-18.
Pole assignment by gain output feedback. Abstract: This short paper deals with the problem of pole assignment with incomplete state observation. It is shown that if the system is controllable and observable, and if n \leq r + m - 1 , an almost arbitrary set of distinct closed-loop poles is assignable by gain output feedback, where n, r , and m ...
On pole assignment in linear systems with incomplete state feedback. IEEE Trans. Automat. Control (1970), pp. 348-351. View in Scopus Google Scholar [5] ... H. Kimura, Pole assignment by output feedback: a longstanding open problem, Proceedings of the 33rd IEEE Conference on Decision Control, December 1994, pp. 2101-2105. ...
Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in predetermined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system.
A theorem recently proposed by Davison [1] on pole assignment with incomplete state feedback is extended to noncyclic matrices by using the results of Brasch and Pearson [2]. It is shown that the number of poles that can be arbitrarily assigned is equal to the maximum of the number of nontrivial inputs or outputs.