pole assignment problem

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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] .

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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 )

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• 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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M. M. Shumafov

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Translated by E. Oborin

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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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Received : 09 May 2019

Revised : 13 June 2019

Accepted : 13 June 2019

Published : 23 December 2019

Issue Date : October 2019

DOI : https://doi.org/10.1134/S1063454119040095

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• linear control system
• unstable system
• stabilization
• asymptotic stability
• pole assignment
• static output feedback
• Brockett’s problem.
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