# place

Pole placement design

## Syntax

``K = place(A,B,p)``
``[K,prec] = place(A,B,p)``

## 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

`$\stackrel{˙}{x}=Ax+Bu$`

`$y=Cx+Du$`

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 ABK will match the entries of `p` (up to the ordering).

example

````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.```

## Examples

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

`$A=\left[\begin{array}{cc}-1& -2\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{c}2\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0$`

Input the matrices and create the state-space system.

```A = [-1,-2;1,0]; B = [2;0]; C = [0,1]; D = 0; sys = ss(A,B,C,D);```

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

`Pol = pole(sys)`
```Pol = 2×1 complex -0.5000 + 1.3229i -0.5000 - 1.3229i ```
```figure(1) step(sys) hold on;```

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

`p = [-1,-2];`

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

```K = place(A,B,p); Acl = A-B*K; syscl = ss(Acl,B,C,D); Pcl = pole(syscl)```
```Pcl = 2×1 -2.0000 -1.0000 ```

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

```figure(1) step(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.

```p = [-2, -3]; K2 = place(A,B,p); syscl2 = ss(A-B*K2,B,C,D); figure(1); step(syscl2);```

`stepinfo(syscl)`
```ans = struct with fields: RiseTime: 2.5901 TransientTime: 4.6002 SettlingTime: 4.6002 SettlingMin: 0.9023 SettlingMax: 0.9992 Overshoot: 0 Undershoot: 0 Peak: 0.9992 PeakTime: 7.7827 ```
`stepinfo(syscl2)`
```ans = struct with fields: RiseTime: 1.4130 TransientTime: 2.4766 SettlingTime: 2.4766 SettlingMin: 0.3003 SettlingMax: 0.3331 Overshoot: 0 Undershoot: 0 Peak: 0.3331 PeakTime: 4.1216 ```

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

```A = [4,2,1;0,-1,2;0,1e-8,1]; B = [1,2;3,1;1e-6,0]; p = [-2e-13,-3e-4,3e-3]; [~,prec] = place(A,B,p)```
```prec = 2 ```

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

For this example, consider the following transfer function with complex-conjugate poles at $-2±2\mathit{i}$:

`$systf\left(s\right)=\frac{8}{{s}^{2}+4s+8}$`

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

```s = tf('s'); systf = 8/(s^2+4*s+2); sys = ss(systf);```

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

```p = [-2+2i,-2-2i]; K = place(sys.A,sys.B,p)```
```K = 1×2 0 1.5000 ```

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.

```syscl = ss(sys.A-sys.B*K,sys.B,sys.C,sys.D); step(syscl)```

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

`$A=\left[\begin{array}{cc}-1& -0.75\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{c}1\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}1& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0$`

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

```A = [-1,-0.75;1,0]; B = [1;0]; C = [1,1]; D = 0; Plant = ss(A,B,C,D);```

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

```N = 250; t = linspace(0,25,N); u = [ones(N/2,1); zeros(N/2,1)]; x0 = [1;2]; [y,t,x] = lsim(Plant,u,t,x0); figure plot(t,y); title('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`.

`L = place(A',C',[-2,-3])';`

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

```At = A-L*C; Bt = [B,L]; Ct = [C;eye(2)]; sysObserver = ss(At,Bt,Ct,0);```

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

```[observerOutput,t] = lsim(sysObserver,[u,y],t); yHat = observerOutput(:,1); xHat = observerOutput(:,[2 3]);```

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

```figure; plot(t,x); hold on; plot(t,xHat,'--'); legend('x_1','x_2','xHat_1','xHat_2') title('Comparison - Actual vs. Estimated');```

## Input Arguments

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

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.

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

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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.

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`.

## Tips

• 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`.

## References

[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