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Pole assignment problem
Let $R$ be a commutative ring and let $(A,B)$ be a pair of matrices of sizes $(n \times n)$ and $(n \times m)$, respectively, with coefficients in $R$. The pole assignment problem asks the following. Given $r_1,\ldots,r_n$, does there exist an $(m \times n)$-matrix $F$, called a feedback matrix, such that the characteristic polynomial of $A+BF$ is precisely $(X-r_1)\cdots(X - r_n)$? The pair $(A,B)$ is then called a pole assignable pair of matrices. The terminology derives from the "interpretation" of $(A,B)$ as (the essential data of) a discrete-time time-invariant linear control system: \begin{equation}\label{eq:a1} x(t+1) = Ax(t) + Bu(t) \end{equation} where $x(t) \in R^n$, $u(t) \in R^m$, or also, when $R = \mathbf{R}$ or $\mathbf{C}$, a continuous-time time-invariant linear control system: \begin{equation}\label{eq:a2} \dot x(t) = Ax(t) + Bu(t) \end{equation} where $x(t) \in R^n$, $u(t) \in R^m$.
In both cases, state feedback (see Automatic control theory ), $u \mapsto u + Fx$, changes the pair $(A,B)$ to $(A+BF,B)$.
The transfer function of a system \eqref{eq:a1} or \eqref{eq:a2} with output $y(t) = C x(t)$ is equal to \begin{equation}\label{eq:a3} T(s) = C(sI-A)^{-1}B \end{equation} and therefore the terminology "pole assignment" is used.
The pair $(A,B)$ is a coefficient assignable pair of matrices if for all $a_1,\ldots,a_n \in R$ there is an $(m\times n)$-matrix $F$ such that $A+BF$ has characteristic polynomial $X^n + a_1X^{n-1} + \cdots + a_n$.
The pair $(A,B)$ is completely reachable , reachable , completely controllable , or controllable if the columns of the $(n\times nm)$-reachability matrix \begin{equation}\label{eq:a4} (B,AB,\ldots,A^{n-1}B) \end{equation} span all of $R^n$. All four mentioned choices of terminology are used in the literature. The reachability matrix \eqref{eq:a4} is also called the controllability matrix. This terminology also derives from the "interpretations" \eqref{eq:a1} and \eqref{eq:a2} of a pair $(A,B)$, see again Automatic control theory .
A cyclic vector for an $(n\times n)$-matrix $M$ is a vector $v\in R^n$ such that $(v,MV,\ldots,M^{n-1}v)$ is a basis for $R^n$, i.e., such that $(M,v)$ is completely reachable. Now consider the following properties for a pair of matrices $(A,B)$:
a) there exist a matrix $F$ and a vector $w \in R^m$ such that $Bw$ is cyclic for $A+BF$;
b) $(A,B)$ is coefficient assignable;
c) $(A,B)$ is pole assignable;
d) $(A,B)$ is completely reachable.
Over a field these conditions are equivalent and, in general, a)$\Rightarrow$b)$\Rightarrow$c)$\Rightarrow$d). In control theory, the implication d)$\Rightarrow$a) for a field $R$ is called the Heyman lemma, and the implication d)$\Rightarrow$c) for a field $R$ is termed the pole shifting theorem.
A ring $R$ is said to have the FC-property (respectively, the CA-property or the PA-property) if for that ring d) implies a) (respectively, d) implies b), or d) implies c)). Such a ring is also called, respectively, an FC-ring, a CA-ring or a PA-ring. As noted, each field is an FC-ring (and hence a CA-ring and a PA-ring). Each Dedekind domain (cf. also Dedekind ring ) is a PA-ring. The ring of polynomials in one indeterminate over an algebraically closed field is a CA-ring, but the ring of polynomials in two or more indeterminates over any field is not a PA-ring (and hence not a CA-ring) [a4] .
For a variety of related results, see [a1] , [a2] , [a3] , [a5] .
[a1] | J.W. Brewer, J.W. Bunce, F.S. van Vleck, "Linear systems over commutative rings" , M. Dekker (1986) |
[a2] | J. Brewer, D. Katz, W. Ullery, "Pole assignability in polynomial rings, power series rings, and Prüfer domains" , (1987) pp. 265–286 |
[a3] | R. Bumby, E.D. Soutrey, H.J. Sussmann, W. Vasconcelos, "Remarks on the pole-shifting theorem over rings" , (1981) pp. 113–127 |
[a4] | A. Tannenbaum, "Polynomial rings over arbitrary fields in two or more variables are not pole assignable" , (1982) pp. 222–224 |
[a5] | J. Brewer, T. Ford, L. Kingler, W. Schmale, "When does the ring $K[g]$ have the coefficient assignment property?" , (1996) pp. 239–246 |
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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.
collapse all
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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Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey
- TO THE MEMORY OF G.A. LEONOV
- Published: 23 December 2019
- Volume 52 , pages 349–367, ( 2019 )
Cite this article
- M. M. Shumafov 1
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This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for solving Brockett’s stabilization problem are provided. The effective necessary and sufficient conditions for stabilization of two- and three-dimensional controllable linear systems are given in terms of the system parameters. The pole assignment problem and the related issues for linear systems are considered.
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Shumafov, M.M. Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey. Vestnik St.Petersb. Univ.Math. 52 , 349–367 (2019). https://doi.org/10.1134/S1063454119040095
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Accepted : 13 June 2019
Published : 23 December 2019
Issue Date : October 2019
DOI : https://doi.org/10.1134/S1063454119040095
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Pole-assignment problem for discrete-time linear periodic systems
- V. Hernández , A. Urbano
- Published 1 August 1987
- Engineering, Mathematics
- International Journal of Control
46 Citations
Output stabilization via pole placement of discrete-time linear periodic systems, eigenvalue assignment for periodic continuous-time systems by sampled output periodic hold control, pole-placement problem for discrete-time linear periodic systems, pole assignment in descriptor periodic systems, a note on robust pole assignment for periodic systems, on feedbacks for positive discrete-time singular systems, characteristic multiplier assignment in continuous-time linear periodic systems, a stability problem for discrete-time linear periodic systems, structural properties of linear periodic discrete-time systems: the polynomial approach, optimal control for linear periodic systems by using multirate piecewise constant sampled state feedback, 8 references, eigenvalue assignment in linear periodic discrete-time systems†, robust control of linear time-invariant plants using periodic compensation, controllability and pole assignment for discrete time linear systems defined over arbitrary fields, stability conditions derived from spectral theory: discrete systems with periodic feedback, discrete-time linear periodic systems: gramian and modal criteria for reachability and controllability†, a unified analysis of multirate and periodically time-varying digital filters, frequency domain and state space methods for linear systems, on the matrices ab and ba, related papers.
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A corollary of Pole assignment theorem
The well-known Pole Assignment Theorem is
Pole Assignment Theorem. Given $A\in \mathbb{R}^{n\times n},B\in \mathbb{R}^{n\times m}$ , for any target pole set $P=\{\lambda_1,...,\lambda_n\}$ which is closed under complex conjugate, there exists a matrix $F\in \mathbb{R}^{m\times n}$ such that $\lambda(A+BF)=P$ if and only if $(A,B)$ is controllable.
If $\mu$ is an eigenvalue of $A$ and $\mu$ is still an eigenvalue of $A+BF$ for any $F\in \mathbb{R}^{m\times n}$ , $\mu$ is called an uncontrollable pole of $A$ , my question is
Question. The pole assignment problem has a solution i.e. given a target pole set $P$ , there exists a matrix $F\in \mathbb{R}^{m\times n}$ such that $$\lambda(A+BF)=P$$ if and only if all uncontrollable poles of $A$ are in the target pole set $P$ .
by real Schur decomposition, we can choose an orthogonal matrix $Q$ such that $$Q^TAQ=\begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}$$ where $\lambda(A_{11})$ consists of all the uncontrollable poles of $A$ . Let $$Q^TB=\begin{bmatrix} B_1 \\B_2 \end{bmatrix}$$ but I do not have idea that how to continue... any help will be appreciated.
- linear-algebra
- matrix-calculus
- control-theory
This is closely related to the Kalman decomposition , from which it can be noted that $\lambda(A_{22})$ instead of $\lambda(A_{11})$ should contain all uncontrollable poles of $A$ . You want to show that the eigenvalues of $A + B\,F$ and thus also $Q^\top(A + B\,F)\,Q$ always contain the uncontrollable poles of $A$ . Using this what can you say about $B_1$ and $B_2$ ? Hint: use $F\,Q = \begin{bmatrix}F_1 & F_2\end{bmatrix}$ and look at the eigenvalues of $Q^\top(A + B\,F)\,Q$ .
- $\begingroup$ Thank you very much! I got it! The key is Kalman decomposition... $\endgroup$ – Xin Fu Commented Dec 11, 2019 at 3:32
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of the pole assignment problem, and give a constructive procedure for finding K. 3.2 Pole Assignment for Single-Input Systems We first solve the pole assignment problem for single-input systems of the form x˙ = Ax +bu The control law is of the form u = −kTx (3.4) for some column vector k, with closed-loop system given by x˙ = (A −bkT)x ...
R. Bumby, E.D. Soutrey, H.J. Sussmann, W. Vasconcelos, "Remarks on the pole-shifting theorem over rings" J. Pure Appl. Algebra, 20 (1981) pp. 113-127 [a4] A. Tannenbaum, "Polynomial rings over arbitrary fields in two or more variables are not pole assignable" Syst. Control Lett. , 2 (1982) pp. 222-224
Abstract This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for ...
Abstract. 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.
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. ... , Algorithms and Software for Pole Assignment and Observers, UCRL-15646 Rev. 1, EE Dept., Univ. of Calif., Santa Barbara, CA, Sept. 1984 . Version ...
A Schur Approach to Pole Assignment Problem. A new approach to the pole assignment in linear systems is proposed which is based on unitary or orthogonal transformation of the closed loop system matrix to its Schur canonical form. The method has a number of advantages over the other known methods. In particular it does not require the ...
The problem of pole assignment by gain output feedback or by low-order dynamical compensator is considered from a geometrical point of view. This allows unification of a general framework for most of the existing pole assignment methods formulated in a state-space context, such as the minimal-order observers, the F.M. Brasch and J.B. Pearson (1970) compensator, the methods proposed by H ...
Pole assignment problems are special algebraic inverse eigenvalue problems. In this paper, we research numerical methods of the robust pole assignment problem for second-order systems. The problem is formulated as an optimization problem. Depending upon whether the prescribed eigenvalues are real or complex, we separate the discussion into two cases and propose two algorithms for solving this ...
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 ...
The problem of maintaining the stability of second-order system by proportional-plus-derivative feedback subjected to parameter perturbations has been an active area of research; see [8 - 11]. Furthermore, the robust pole assignment problem for first-order systems have been well-studied in literature; see [37, 40 - 43].
The problem of modifying the invariant polynomials of a linear system by dynamical output feedback is considered. A new necessary condition which the invariant polynomials must satisfy is derived. The sufficiency condition of Rosenbrock and Hayton is proved in an alternative way. The proof is based on polynomial matrix equations and provides a simple construction of the feedback which affects ...
Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey. October 2019. Vestnik St Petersburg University Mathematics 52 (4):349-367. DOI: 10.1134/S1063454119040095. Authors:
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 ...
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 are the numbers of state variables, inputs and outputs, respectively.
The pole-zero assignment problem for high-order systems with time delay is addressed. The approach described here uses measured receptances absolutely without requirements for the system matrices. Our solution is easy to achieve and need not solve the Sylvester equation or turn high-order systems into the first-order form. The method is supplemented by a series of illustrative numerical ...
Pole assignment by gain output feedback. 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 are the ...
Abstract—This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered.
Gianluca Bianchin. Abstract—The exact pole placement problem concerns com-puting a static feedback law for a linear dynamical system that will assign its poles at a set of pre-specified locations. This is a classic problem in feedback control and numerous methodologies have been proposed in the literature for cases where a model of the system ...
Published 1 August 1987. Engineering, Mathematics. International Journal of Control. Abstract This paper considers the pole-assignment problem for discrete-time linear periodic systems through the use of linear periodic state-variable feedback control. It is shown that if the N-periodic system with m inputs and n states is completely reachable ...
The pole assignment problem has a solution i.e. given a target pole set P P, there exists a matrix F ∈Rm×n F ∈ R m × n such that. λ(A + BF) = P λ ( A + B F) = P. if and only if all uncontrollable poles of A A are in the target pole set P P. by real Schur decomposition, we can choose an orthogonal matrix Q Q such that.
Abstract: In this paper a new result in the problem of pole assignment by gain output feedback is given. Roughly speaking, this result says that arbitrary pole assignment is possible for almost all systems if n < r + m + \nu - 1, r > \mu, m \geq \nu.Here n, r and m are the number of states, of inputs and of outputs, respectively, and ν and μ are the so-called controllability index and the ...
The periodic state feedback pole assignment problem of high-order periodic discrete systems is investigated, and the pole assignment problem for such systems is transformed into a class of problems for resolving periodic Sylvester matrix equations with constraints. Using the technique of cyclic lifting, such equations can be transformed into ...
Waymo issued a recall for its 672 driverless cars to make them less likely to drive into telephone poles. The recall follows a May 21 accident in Phoenix, Arizona, in which a Waymo driverless car ...
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 ...