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Engineering LibreTexts

10.2: Controllers for Discrete State Variable Models

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  • Page ID 24434

  • Kamran Iqbal
  • University of Arkansas at Little Rock

Emulating an Analog Controller

The pole placement controller designed for a continuous-time state variable model can be used with derived sampled-data system model. Successful controller emulation requires a high enough sampling rate that is at least ten times the frequency of the dominant closed-loop poles of the system.

In the following we illustrate the emulation of pole placement controller designed for the DC motor model (Example 8.3.4) for controlling the discrete-time model of the DC motor. The DC motor model is discretized at two different sampling rates for comparison, assuming ZOH at the plant input.

Example \(\PageIndex{1}\)

The state and output equations for a DC motor model are given as:

\[\frac{\rm d}{\rm dt} \left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]=\left[\begin{array}{cc} {-100} & {-5} \\ {5} & {-10} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]+\left[\begin{array}{c} {100} \\ {0} \end{array}\right]V_a , \;\;\omega =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]. \nonumber \]

The motor model is discretized at two different sampling rates in MATLAB. The results are:

\[T=0.01s: A_{\rm d} =\left[\begin{array}{cc} {0.367} & {-0.030} \\ {0.030} & {0.904} \end{array}\right],\; \; B_{\rm d} =\left[\begin{array}{c} {0.632} \\ {0.018} \end{array}\right],\; \; C_{\rm d} =\left[\begin{array}{cc} {0} & {1} \end{array}\right]. \nonumber \]

\[T=0.02s: A_{\rm d} =\left[\begin{array}{cc} {0.134} & {-0.038} \\ {0.038} & {0.816} \end{array}\right],\; \; B_{\rm d} =\left[\begin{array}{c} {0.863} \\ {0.053} \end{array}\right],\; \; C_{\rm d} =\left[\begin{array}{cc} {0} & {1} \end{array}\right]. \nonumber \]

For a desired characteristic polynomial: \(\Delta _{\rm des} (s)=s^{2} +150\,s+5000\), a state feedback controller for the continous-time state variable model was obtained as (Example 9.1.1): \(k^{T} =\left[\begin{array}{cc} {0.4} & {7.15} \end{array}\right]\).

We can use the same controller to control the corresponding sample-data system models.

The unit-step response of the closed-loop system is simulated in Figure 10.2.1, where both state variables, \(i_a\left(t\right)\) and \(\omega \left(t\right)\), are plotted.

clipboard_e29ed3918c27425662d5120784e9bbfac.png

We observe from the figure that the armature current has a higher overshoot at the lower sampling rate, though both models display similar settling time of about 100 msec.

Pole Placement Design of Digital Controller

Given a discrete state variable model \(\left\{A_{\rm d},\ B_{\rm d}\right\}\), and a desired pulse characteristic polynomial \(\Delta _{\rm des} (z)\), a state feedback controller for the system can be designed using pole placement similar to that of the continuous-time system (Sec. 9.1.1).

Let the discrete-time model of a SISO system be given as:

\[{\bf x}_{k+1} ={\bf A}_{\rm d} {\bf x}_{k} +{\bf b}_{\rm d} u_{k} , \;\; y_{k} ={\bf c}^T {\bf x}_{k} \nonumber \]

A state feedback controller for the discrete state variable model is defined as:

\[u_k=-{\bf k}^T{\bf x}_k+r_k \nonumber \]

where \({\bf k}^{T}\) represents a row vector of constant feedback gains and \(r_k\) is a reference input sequence. The controller gains can be obtained by equating the coefficients of the characteristic polynomial with those of a desired polynomial:

\[\Delta (z)=\left|z{\bf I-A}_{\rm d} \right|=\Delta _{\rm des} (z) \nonumber \]

The \(\Delta _{\rm des} (z)\) above is a Hurwitz polynomial (in \(z\)), with roots inside the unit circle that meet given performance (damping ratio and/or settling time) requirements. Assuming that desired \(s\)-plane root locations are known, the corresponding \(z\)-plane root locations can be obtained from the equivalence: \(z=e^{Ts}\).

Closed-loop System

The closed-loop system model is given as:

\[{\bf x}_{k+1} ={\bf A}_{\rm cl} {\bf x}_{k} +{\bf b}_{\rm d} r_{k} , \;\; y_{k} ={\bf c}^T {\bf x}_{k} \nonumber \]

where \({\bf A}_{\rm cl} =({\bf A}_{\rm d}-{\bf b}_{\rm d}{\bf k}^T)\).

Assuming closed-loop stability, for a constant input \(r_k=r_{\rm ss}\), the steady-state response, \({\bf x}_{\rm ss}\), of the system obeys: 

\[{\bf x}_{ss} ={\bf A}_{\rm cl} {\bf x}_{ss} +{\bf b}_{\rm d} r_{ss} ,\;\; y_{\rm ss} ={\bf c}^T {\bf x}_{ss} \nonumber \]

Hence, \(y_{\rm ss}={\bf c}^T\,({\bf A}_{\rm cl}-{\bf I})^{-1}\,{\bf b}_{\rm d}\,r_{\rm ss}\).

Example \(\PageIndex{2}\)

The discrete state variable model of a DC motor (\(T=0.02\)s) is given as: \[\left[\begin{array}{c} {i_{k+1} } \\ {\omega _{k+1} } \end{array}\right]=\left[\begin{array}{cc} {0.134} & {-0.038} \\ {0.038} & {0.816} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right]+\left[\begin{array}{c} {0.863} \\ {0.053} \end{array}\right]V_{k} , \;\; y_{k} =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right] \nonumber \]

The desired \(s\)-plane root locations for the model are given as: \(s=-50,\; -100.\)

The corresponding \(z\)-plane roots (\(T=0.02s\)) are obtained as: \(z=e^{-1} ,\; e^{-2}\).

The desired characteristic polynomial is given as: \(\Delta _{\rm des} (z)=z^{2} -0.95z+0.05.\)

The feedback gains \(k^T =[k_{1} ,\; k_{2} ]\), computed using the MATLAB ‘place’ command, are given as: \(k_{1} =0.247,\; k_{2} =4.435.\)

The closed-loop system matrix is given as: \(A_\rm d)= \left[\begin{array}{cc} {-0.080} & {-3.867} \\ {0.025} & {0.583} \end{array}\right]\).

An update rule for implementation of the controller on computer is obtained as: \(u_{k} =-0.247\, i_{k} -4.435\, \omega _{k} .\)

The closed-loop response has steady-state value of \(\omega _{\rm ss}=0.143 \;\rm rad/s\).

The step response of the closed-loop system is plotted in Figure 10.2.2, where the discrete system response was scaled to match the analog system response. The step response of the continuous-time system and that for the emulated controller gains are plotted alongside.

clipboard_e9ccc6b43c4082491c53a681502ca8cb5.png

Deadbeat Controller Design

A discrete-time system is called deadbeat if all closed-loop poles are placed at the origin \((z=0)\).

A deadbeat system has the remarkable property that its response reaches steady-state in \(n\)-steps, where \(n\) represents the model dimension.

The desired closed-loop pulse characteristic polynomial is selected as \(\Delta _{\rm des} (z)=z^{n}\).

To design a deadbeat controller, let the closed-loop pulse transfer function be defined as: \[T(z)=\frac{K(z)G(z)}{1+K(z)G(z)} \nonumber \]

The above equation is solved for \(K(z)\) to obtain: \[K(z)=\frac{1}{G(z)} \frac{T(z)}{1-T(z)} \nonumber \]

Let the desired \(T(z)=z^{-n}\); then, the deadbeat controller is given as: \[K(z)=\frac{1}{G(z)(z^{n} -1)} \nonumber \]

Example \(\PageIndex{3}\)

Let \(G(s)=\frac{1}{s+1} ;\) then \(G(z)=\frac{1-e^{-T} }{z-e^{-T} }\).

A deadbeat controller for the model is obtained as: \(K(z)=\frac{z-e^{-T} }{(1-e^{-T} )(z-1)}\).

Example \(\PageIndex{4}\)

The discrete state variable model of a DC motor for \(T=0.02\; \rm s\) is given as: \[\left[\begin{array}{c} {i_{k+1} } \\ {\omega _{k+1} } \end{array}\right]=\left[\begin{array}{cc} {0.134} & {-0.038} \\ {0.038} & {0.816} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right]+\left[\begin{array}{c} {0.863} \\ {0.053} \end{array}\right]V_{k} , \;\;y_{k} =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right] \nonumber \]

The state feedback controller is given as: \(u_{k} =-\left[k_{1} ,\, \, k_{2} \right]x_{k}\).

The closed-loop characteristic polynomial is obtained as: \[\Delta (z)=z^{2} +(0.863k_{1} +0.053k_{2} -0.95)z-0.707k_{1} +0.026k_{2} +0.111 \nonumber \]

For pole placement design, let \(\Delta _{\rm des} (z)=z^{2}\). By equating the polynomial coefficients, the deadbeat controller gains are obtained as: \(k_{1} =0.501,\; k_{2} =9.702\).

The update rule for controller implementation is given as: \[u_{k} =0.501\, \, i_{k} +9.702\, \, \omega _{k} \nonumber \]

The step response of the deadbeat controller (Figure 10.2.3) settles in two time periods. The response was scaled to match that of the continuous-time system.

An approximate deadbeat design can be performed by choosing distinct closed-loop eigenvalues close to the origin, e.g., \(z=\pm {10}^{-5}\), and using the 'place' command from the MATLAB Control Systems Toolbox.

The feedback gains for the approximate design are obtained as: \(k_{1} =0.509,\; k_{2} =9.702\). The resulting closed-loop system response is still deadbeat.

clipboard_ebdfc98111c6648a5cf1713d4bf80c26c.png

Feedforward Tracking System Design

A tracking system was previously designed by using feedforward cancelation of the error signal (Section 9.2.1). A similar design can be performed in the case of discrete systems.

Towards this end, let the discrete state variable model be given as: \[{\bf x}_{k+1} ={\bf A}_{\rm d} {\bf x}_{k} +{\bf b}_{\rm d} u_{k} , \;\;y_{k} ={\bf c}^T {\bf x}_{k} \nonumber \]

A tracking controller for the model is defined as: \[u_k=-{\bf k}^T{\bf x}_k+k_rr_k \nonumber \] where \({\bf k}^{T}\) represents a row vector of feedback gains, \(k_r\) is a feedforward gain, and \(r_k\) is a reference input sequence.

Assuming that a pole placement controller for the discrete system has been designed, the closed-loop system is given as: \[{\bf x}_{k+1}=\left({\bf A}_{\rm d}-{\bf b}_{\rm d}{\bf k}^T\right){\bf x}_k+{\bf b}_{\rm d}k_rr_k \nonumber \]

The closed-loop pulse transfer function is obtained as: \[T\left(z\right)={\bf c}^T_{\rm d}{\left(z{\bf I-A}_{\rm d}+{\bf b}_{\rm d}{\bf k}^T\right)}^{-1}{\bf b}_{\rm d}k_r \nonumber \] where \({\bf I}\) denotes an identity matrix. The condition for asymptotic tracking is given as: \[T\left(1\right)={\bf c}^T_{\rm d}{\left({\bf I-A}_{\rm d}+{\bf b}_{\rm d}{\bf k}^T\right)}^{-1}{\bf b}_{\rm d}k_r=1 \nonumber \]

The feedforward gain for error cancelation is obtained as: \(k_r=\frac{1}{T\left(1\right)}\).

Example \(\PageIndex{5}\)

The discrete state variable model of a DC motor (\(T=0.02\)s) is given as: \[\left[\begin{array}{c} {i_{k+1} } \\ {\omega _{k+1} } \end{array}\right]=\left[\begin{array}{cc} {0.134} & {-0.038} \\ {0.038} & {0.816} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right]+\left[\begin{array}{c} {0.863} \\ {0.053} \end{array}\right]V_{k} , \;\;y_{k} =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right] \nonumber \]

A state feedback controller for the motor model was previously designed as: \(k^T =[k_{1} ,\; k_{2} ]\), where \(k_{1} =0.247,\; k_{2} =4.435.\)

The closed-loop system is defined as: \[T\left(z\right)=\frac{0.367z+0.179}{z^2-0.503z+0.05}k_r \nonumber \]

From the asymptotic condition, the feedforward gain is solved as: \(k_r=6.98\).

The step response of the closed-loop system is shown in Figure 10.2.4.

clipboard_edca113b5d7dfbf45077bd40bafc3153a.png

Example \(\PageIndex{6}\)

The discrete state variable model of a DC motor (\(T=0.02\)s) is given as:

\[\left[\begin{array}{c} {i_{k+1} } \\ {\omega _{k+1} } \end{array}\right]=\left[\begin{array}{cc} {0.134} & {-0.038} \\ {0.038} & {0.816} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right]+\left[\begin{array}{c} {0.863} \\ {0.053} \end{array}\right]V_{k} , \;\;y_{k} =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_{k} } \\ {\omega _{k} } \end{array}\right] \nonumber \]

A dead-beat controller for the motor model was designed as: \(k^T =[k_{1} ,\; k_{2} ]\), where \(k_{1} =0.501,\; k_{2} =9.702\).

The closed-loop system is defined as: \[T\left(z\right)=\frac{0.672z+0.328}{z^2}k_r \nonumber \]

From the asymptotic condition, the feedforward gain is solved as: \(k_r=12.77\).

The step response of the closed-loop system is shown in Figure 10.2.5.

clipboard_eaf6c75fb62a0c3b9c3294150ad9f04e5.png

Tracking PI Controller Design

A tracking PI controller for the discrete state variable model is designed similar to the design of continuous-time system (Figure 9.3.1). The tracking PI controller places an integrator in the feedback loop, thus ensuring that the tracking error goes to zero in the steady-state.

In the case of continuous-time system, the tracking PI controller was defined as: \(u=-{\bf k}^{T} {\bf x}+k_{i} \int (r-y)\rm dt\).

Using the forward difference approximation to the integrator, given as: \(v_k=v_{k-1}+Te_k\), an augmented discrete-time system model including the integrator state variable is formed as:

\[\left[\begin{array}{c} {{\bf x}(k+1)} \\ {v(k+1)} \end{array}\right]=\left[\begin{array}{cc} {{\bf A}_{\rm d} } & {\bf 0} \\ {-{\bf c}^T T} & {1} \end{array}\right] \left[\begin{array}{c} {{\bf x}(k)} \\ {v(k)} \end{array}\right]+\left[\begin{array}{c} {{\bf b}_{\rm d} } \\ {0} \end{array}\right]u+\left[\begin{array}{c} {\bf 0} \\ {T} \end{array}\right]r \nonumber \]

The state feedback controller for the augmented system is defined as:

\[u(k)=\left[\begin{array}{cc} {-{\bf k}^T } & {k_ i } \end{array}\right]\, \left[\begin{array}{c} {{\bf x}(k)} \\ {v(k)} \end{array}\right] \nonumber \]

where \(k_ i\) represents the integral gain. With the addition of the above controller, the closed-loop system is described as:

\[\left[\begin{array}{c} {{\bf x}(k+1)} \\ {v(k+1)} \end{array}\right]=\left[\begin{array}{cc} {{\bf A}_{\rm d} -{\bf b}_{\rm d} k^{T} } & {{\bf b}_{\rm d} k_{i} } \\ {-{\bf c}^T T} & {1} \end{array}\right] \left[\begin{array}{c} {{\bf x}(k)} \\ {v(k)} \end{array}\right]+\left[\begin{array}{c} {\bf 0} \\ {T} \end{array}\right]r(k) \nonumber \]

The closed-loop characteristic polynomial of the augmented system is formed as:

\[{\mathit{\Delta}}_a\left(z\right)=\left| \begin{array}{cc} z{\bf I-A}_{\rm d}+{\bf b}_{\rm d}k^T & -{\bf b}_{\rm d}k_i \\ -{\bf c}^T_{\rm d}T & z-1 \end{array} \right| \nonumber \]

where \({\bf I}\) denotes an identity matrix of order \(n\).

Next, we choose a desired characteristic polynomial of \((n+1)\) order, and perform pole placement design for the augmented system. The location of the integrator pole in the \(z\)-plane may be selected keeping in view the desired peformance criteria for the closed-loop system.

\[\left[ \begin{array}{c} i_{k+1} \\ {\omega }_{k+1} \end{array} \right]=\left[ \begin{array}{cc} 0.134 & -0.038 \\ 0.038 & 0.816 \end{array} \right]\left[ \begin{array}{c} i_k \\ {\omega }_k \end{array} \right]+\left[ \begin{array}{c} 0.863 \\ 0.053 \end{array} \right]V_k,\ \ {\omega }_k=\left[ \begin{array}{cc} 0 & 1 \end{array} \right]\left[ \begin{array}{c} i_k \\ {\omega }_k \end{array} \right] \nonumber \]

The control law for the tracking PI controller is defined as:

\[u_k=-k_1i_k-k_2{\omega }_k+k_iv_k \nonumber \]

where \(v_{k} =v_{k-1} +T(r_{k} -\omega _{k} )\) describes the output of the integrator. The augmented system model for the pole placement design using integral control is given as:

\[\left[ \begin{array}{c} i_{k+1} \\ {\omega }_{k+1} \\ v_{k+1} \end{array} \right]=\left[ \begin{array}{ccc} 0.134 & -0.038 & 0 \\ 0.038 & 0.816 & 0 \\ 0 & -0.02 & 1 \end{array} \right]\left[ \begin{array}{c} i_k \\ {\omega }_k \\ v_k \end{array} \right]+\left[ \begin{array}{c} 0.863 \\ 0.053 \\ 0 \end{array} \right]V_k+\left[ \begin{array}{c} 0 \\ 0 \\ 0.02 \end{array} \right]r_k \nonumber \]

The desired \(z\)-plane pole locations for a desired \(\zeta=0.7\) are selected as: \(z=e^{-1} ,\; e^{-1\pm j1}\).

The controller gains, obtained using the MATLAB ‘place’ command, are given as: \(k_{1} =0.43,k_{2} =15.44,\; k_{i} =-297.79.\)

An update rule for controller implementation on computer is given as:

\[u_k=-0.43i_k-15.44{\omega }_k+297.8v_k \nonumber \]

\[v_k=v_{k-1}+0.02\left(r_k-{\omega }_k\right) \nonumber \]

The step response of the closed-loop system is plotted in Figure 10.2.6. The step response of the continuous-time system (Example 9.1.1) is plotted alongside. The output in both cases attains steady-state value of unity in about 0.12sec.

clipboard_ea8f695d9639c458f7619cc7b76b4d269.png

Fusion of Engineering, Control, Coding, Machine Learning, and Science

Aleksandar Haber

Pole Placement With Integral Control Action to Eliminate Steady-State Error (State-Space Control Design)

admin

In this post, we explain how to integrate an integral control action into a pole placement control design in order to eliminate a steady-state error. The YouTube video accompanying this post is given below

First, we explain the main motivation for creating this tutorial. When designing control algorithms, we are primarily concerned with two design challenges. First of all, we have to make sure that our control algorithm behaves well during the transient response. That is, as designers we specify the acceptable behavior of the closed-loop system during the transient response. For example, we want to make sure that the rise time (settling time) is within a certain time interval specified by the user. Also, we want to ensure that the system’s overshoot is below a certain value. For example, below 10 or 15 percent of the steady-state value. Secondly, we want to make sure that the control algorithm is able to eliminate the steady-state control error.

In your introductory course on control systems, you have probably heard of a pole placement problem and the solutions. The classical pole placement method is used to stabilize the system or for improving the transient response. This method finds a state feedback control matrix that assigns the poles of the closed-loop system to desired locations specified by the user. However, it is not obvious how to use the pole placement method for set-point tracking and for eliminating steady-state error in set-point tracking design problems. This tutorial explains how to combine the pole placement method with an integral control action in order to eliminate steady-state error and achieve a desired transient response. The technique presented in this lecture is very important for designing control algorithms.

Consider the following state-space model:

\begin{align*}\dot{\mathbf{x}} & =A\mathbf{x}+Bu  \\y&=C\mathbf{x}\end{align*}

is stable and that the poles are placed at the desired locations (that are specified by the designer). The issue with this approach is that although we can place the poles at the desired locations, we do not have full control of steady-state error.

The basic idea for tackling this problem is to augment the original system ( 2 ) with an integrator of the error:

\begin{align*}x_{i}=\int \big(r-C\mathbf{x} \big) \text{dt}\end{align*}

We can write ( 5 ) compactly

\begin{align*}\begin{bmatrix}\dot{\mathbf{x}} \\ \dot{x}_{i}  \end{bmatrix} &=\begin{bmatrix}A & 0 \\ -C & 0  \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}+\begin{bmatrix} B \\ 0  \end{bmatrix}u+\begin{bmatrix} 0 \\ 1   \end{bmatrix}r  \\y&=\begin{bmatrix}C & 0  \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}\end{align*}

From the last system of equations, we can observe that we have formed a new state-space model, with the state variable:

\begin{align*}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}\end{align*}

The state-feedback controller now has the following form

\begin{align*}u=- \underbrace{\begin{bmatrix} K_{x} & K_{i} \end{bmatrix}}_{K}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix} =-K_{x} \mathbf{x} - K_{i}x_{i}\end{align*}

By substituting the feedback control algorithm ( 8 ) in the state-space model ( 6 ), we obtain the following system

\begin{align*}\begin{bmatrix}\dot{\mathbf{x}} \\ \dot{x}_{i}  \end{bmatrix} &=\begin{bmatrix} A & 0 \\ -C & 0  \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}-\begin{bmatrix}B \\ 0 \end{bmatrix}\begin{bmatrix} K_{x} & K_{i} \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}+\begin{bmatrix} 0 \\ 1  \end{bmatrix}r \\ \begin{bmatrix}\dot{\mathbf{x}} \\ \dot{x}_{i}  \end{bmatrix} &=\begin{bmatrix} A & 0 \\ -C & 0  \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}-\begin{bmatrix} BK_{x} & BK_{i} \\0 & 0 \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}+\begin{bmatrix} 0 \\ 1  \end{bmatrix}r \\\begin{bmatrix}\dot{\mathbf{x}} \\ \dot{x}_{i}  \end{bmatrix} &=\begin{bmatrix} A-BK_{x} & -BK_{i} \\ -C & 0  \end{bmatrix}\begin{bmatrix} \mathbf{x} \\ x_{i}  \end{bmatrix}+\begin{bmatrix} 0 \\ 1  \end{bmatrix}r \end{align*}

The new system matrix

\begin{align*}A_{\text{new}}=\begin{bmatrix} A-BK_{x} & -BK_{i} \\ -C & 0  \end{bmatrix}\end{align*}

Next, we present MATLAB codes for implementing this control approach. The following code lines define the system, compute eigenvalues of the open-loop system, perform basic diagnostics, and compute the open-loop response.

The open-loop step response is given below.

pole placement control design

The open-loop system is asymptotically stable. However, we can observe that we have a significant steady-state error. To eliminate the steady-state error, we use the developed approach.

Next, we define the augmented system for pole placement, define desired closed-loop poles, compute the feedback control gain, define the close-loop system, compute the transfer function of the system, and compute the step response. The following code lines are used to perform these tasks.

The resulting step response of the system is shown in the figure below.

pole placement control design

A few comments about the presented code are in order. In code line 13, we define the closed-loop poles. We shift the open-loop poles to left compared to their original location in order to ensure a faster response. Also, we set an additional pole corresponding to the introduced integral action to be a shifted version of the open-loop pole with the minimal real part (maximal absolute distance from the imaginary axis). From the step response of the closed-loop system, we can observe that the steady-state error has been eliminated. Consequently, our control system with the additional integral action is able to successfully track reference set points. The computer transfer function of the closed-loop system is

\begin{align*}W(s)=\frac{614.6}{s^3 + 46.14 s^2 + 427.7 s + 614.6}\end{align*}

That would be all. A related post to this post and tutorial is a tutorial on how to compute a Linear Quadratic Regulator (LQR) optimal controller. That tutorial can be found here .

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pole placement control design

Introduction: State-Space Methods for Controller Design

In this section, we will show how to design controllers and observers using state-space (or time-domain) methods.

Key MATLAB commands used in this tutorial are: eig , ss , lsim , place , acker

Related Tutorial Links

  • LQR Animation 1
  • LQR Animation 2

Related External Links

  • MATLAB State FB Video
  • State Space Intro Video

Controllability and Observability

Control design using pole placement, introducing the reference input, observer design.

There are several different ways to describe a system of linear differential equations. The state-space representation was introduced in the Introduction: System Modeling section. For a SISO LTI system, the state-space form is given below:

$$
\frac{d\mathbf{x}}{dt} = A\mathbf{x} + Bu
$$

To introduce the state-space control design method, we will use the magnetically suspended ball as an example. The current through the coils induces a magnetic force which can balance the force of gravity and cause the ball (which is made of a magnetic material) to be suspended in mid-air. The modeling of this system has been established in many control text books (including Automatic Control Systems by B. C. Kuo, the seventh edition).

pole placement control design

The equations for the system are given by:

$$
m\frac{d^2h}{dt^2} = mg - \frac{Ki^2}{h}
$$

From inspection, it can be seen that one of the poles is in the right-half plane (i.e. has positive real part), which means that the open-loop system is unstable.

To observe what happens to this unstable system when there is a non-zero initial condition, add the following lines to your m-file and run it again:

pole placement control design

It looks like the distance between the ball and the electromagnet will go to infinity, but probably the ball hits the table or the floor first (and also probably goes out of the range where our linearization is valid).

$u(t)$

Let's build a controller for this system using a pole placement approach. The schematic of a full-state feedback system is shown below. By full-state, we mean that all state variables are known to the controller at all times. For this system, we would need a sensor measuring the ball's position, another measuring the ball's velocity, and a third measuring the current in the electromagnet.

pole placement control design

The state-space equations for the closed-loop feedback system are, therefore,

$$
\dot{\mathbf{x}} = A\mathbf{x} + B(-K\mathbf{x}) = (A-BK)\mathbf{x}
$$

From inspection, we can see the overshoot is too large (there are also zeros in the transfer function which can increase the overshoot; you do not explicitly see the zeros in the state-space formulation). Try placing the poles further to the left to see if the transient response improves (this should also make the response faster).

pole placement control design

This time the overshoot is smaller. Consult your textbook for further suggestions on choosing the desired closed-loop poles.

Note: If you want to place two or more poles at the same position, place will not work. You can use a function called acker which achieves the same goal (but can be less numerically well-conditioned):

K = acker(A,B,[p1 p2 p3])

Now, we will take the control system as defined above and apply a step input (we choose a small value for the step, so we remain in the region where our linearization is valid). Replace t , u , and lsim in your m-file with the following:

pole placement control design

The system does not track the step well at all; not only is the magnitude not one, but it is negative instead of positive!

$K\mathbf{x}$

and now a step can be tracked reasonably well. Note, our calculation of the scaling factor requires good knowledge of the system. If our model is in error, then we will scale the input an incorrect amount. An alternative, similar to what was introduced with PID control, is to add a state variable for the integral of the output error. This has the effect of adding an integral term to our controller which is known to reduce steady-state error.

$y = C\mathbf{x}$

From the above, we can see that the observer estimates converge to the actual state variables quickly and track the state variables well in steady-state.

Published with MATLAB® 9.2

pole placement control design

Book cover

International Joint conference on Industrial Engineering and Operations Management

IJCIEOM 2023: Industrial Engineering and Operations Management pp 263–274 Cite as

Design of a Controller by Pole Placement Applied to a Production and Inventory System

  • Verónica Olvera 4 &
  • Esther Segura 4  
  • Conference paper
  • First Online: 21 December 2023

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 431)

In recent decades there have been several studies and proposals based on the theory of automatic control as a mechanism for the analysis and representation of the dynamic conditions of production models. The model of Automatic Pipelines Inventory and Order-Based Production Control System (APIOBPCS) model considers inputs, demand, and the desired inventory level. In the present work, the theory of control is applied to guarantee the profitability of the inventory starting the analysis in an open loop and later in a closed loop. This production and inventory system is represented by a block diagram and equations of state, and from this, control is designed by the feedback of states and applying the technique of design by pole placement. With the simulation, it is verified that the found value of these leads the system to maintain the level of inventory desired by the company. The behavior of the system is analyzed by assigning theoretical values which can be replaced by real data from a production system.

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Olvera, V., Segura, E. (2023). Design of a Controller by Pole Placement Applied to a Production and Inventory System. In: Gonçalves dos Reis, J.C., Mendonça Freires, F.G., Vieira Junior, M. (eds) Industrial Engineering and Operations Management. IJCIEOM 2023. Springer Proceedings in Mathematics & Statistics, vol 431. Springer, Cham. https://doi.org/10.1007/978-3-031-47058-5_21

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IMAGES

  1. Pole placement design

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  2. Pole placement design

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  3. Pole Placement Controller Structure

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  4. Class 22 Pole Placement: Control Canonical Form and Pole Placement

    pole placement control design

  5. The general flowchart of the pole placement control strategy

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  6. Self-tuning pole placement control structure.

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VIDEO

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

  2. Introducing the Rapid Pole

  3. Pole Placement in Matlab using the "place" Command, 11/4/2016

  4. pole placement compensator design 1 least degree solution بالعربي

  5. Pole placement part 3

  6. Example problem on Control system design by pole placement

COMMENTS

  1. 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.

  2. Pole Placement

    This design technique is known as pole placement, which differs from root locus in the following ways: Using pole placement techniques, you can design dynamic compensators. Pole placement techniques are applicable to MIMO systems. Pole placement requires a state-space model of the system (use ss to convert other model formats to state space).

  3. PDF Controller Design by Pole placement

    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.

  4. 9.1: Controller Design in Sate-Space

    Pole Placement Design. Pole placement design refers to the selection of the feedback gain vector that places the poles of the characteristic polynomial at the desired locations. The control law for the pole placement design is expressed as: \[u(t)=-{\bf k}^{T} {\bf x}(t)+r(t), \nonumber \]

  5. PDF Pole Placement Design

    Pole Placement Design Introduction Simple Examples Polynomial Design State Space Design Robustness and Design Rules Model Reduction Oscillatory Systems Summary Theme: Be aware where you place them! Introduction A simple idea Strong impact on development of control theory The only constraint is reachability and observability The robustness debate

  6. PDF Pole Placement Design Technique 8.2 State Feedback and Pole Placement

    Pole Placement Design Technique 8.2 State Feedback and Pole Placement Consider a linear dynamic system in the state space form In some cases one is able to achieve the goal (e.g. stabilizing the system or improving its transient response) by using the full state feedback, which represents a linear combination of the state variables, that is

  7. PDF Module 2-1: Pole Placement

    Module 2-1: Pole Placement Linear Control Systems (2020) Ding Zhao Assistant Professor College of Engineering School of Computer Science Carnegie Mellon University Table of Contents PID Feedback Control Design (State) Feedback Control Design (Luenberger) observer Separation Principle Reduced Order Observer MIMO Systems Table of Contents

  8. PDF Controller design by pole placement

    In this chapter, we examine state-space methods of pole placement design for linear constant coefficient systems of equations (and hence for the physical systems which those systems purportedly model).

  9. Controller design by pole placement

    In classical methods of control law design, the designer introduces a structure for the controller and then several parameters within that structure are chosen to yield a response that meets specifications. ... Controller design by pole placement. In: Handbook of Control Systems Engineering. The Springer International Series in Engineering and ...

  10. 10.2: Controllers for Discrete State Variable Models

    The pole placement controller designed for a continuous-time state variable model can be used with derived sampled-data system model. Successful controller emulation requires a high enough sampling rate that is at least ten times the frequency of the dominant closed-loop poles of the system.

  11. Pole Placement Method

    An assisted pole placement method has been proposed to help the designers in their choice of the design parameters. Once the desired rise time, settling time and percentage of overshoot of the step response of the closed-loop system with respect to a reference change have been specified, the computation of the controller is performed automatically.

  12. Pole Placement Optimization for SISO Control System

    5.4. Control Design by the Pole Placement Optimization. In the proposed method, the order of the controller ( 4) is taken to be \ (m=n-1=3 \). The desired characteristic polynomial of the closed-loop system ( 20) of order \ (2n-1=7 \) can contain up to three pairs of complex conjugate roots.

  13. Pole Placement With Integral Control Action to Eliminate Steady-State

    The classical pole placement design finds the feedback control matrix and the control law such that the closed-loop system (2) is stable and that the poles are placed at the desired locations (that are specified by the designer).

  14. Introduction: State-Space Methods for Controller Design

    Control Design Using Pole Placement Introducing the Reference Input Observer Design Modeling There are several different ways to describe a system of linear differential equations. The state-space representation was introduced in the Introduction: System Modeling section. For a SISO LTI system, the state-space form is given below: (1) (2)

  15. Simultaneous Linear Quadratic Pole Placement (Lqpp) Control Design

    The pole placement (PP) technique for design of a linear state feedback control system requires specification of all the closed-loop pole locations even though only a few poles dominate the system's transient response characteristics.

  16. Full state feedback

    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 pre-determined locations in the s-plane. [1]

  17. Chapter 9.2.2

    design goals for the PI controller: (1) the closed-loop system is stable; (2) steady- state error is minimized; (3) settling time does not exceed; and (4) maximum overshoot does not exceed . The first design goal is achieved by ensuring that all poles lie within the unit circle. The second goal is achieved by using a PI

  18. Pole Placement Design for Load Frequency Control (Lfc) of An Isolated

    Modern control design is especially useful in multivariable systems. One approach in modern control system accomplished by the use of state feedback is known as Pole-Placement Design. The pole-placement design allows all roots of the system characteristic equation to be placed in desired locations.

  19. Design of a Controller by Pole Placement Applied to a ...

    The control theory applied to a production and inventory system lows the system to remain stable in the presence of uncertainties [ 1 ], approaches based on control theory are generally proposed for a single product in a system [ 2 ]. Control theory has generally been applied to reduce inventory variation and thereby minimize inventory costs ...

  20. Control Design

    Design a full-state feedback controller using pole placement using Control System Toolbox™. You can use pole placement technique when the system is controllable and when all system states can be measured. Using the pole placement technique, you can design a controller so that closed-loop system poles are placed in desired locations to meet ...

  21. Watch Intuitive Machines' private Odysseus lander attempt historic moon

    Odysseus (nicknamed "Odie"), a 14-foot-tall (4.3 meters) lander built and operated by Houston company Intuitive Machines, will attempt to land close to the moon's south pole today at 6:24 p.m. EST ...

  22. State-Space Control Design

    Design an LQG servo controller using a Kalman state estimator. Design LQR Servo Controller in Simulink. Design an LQR controller for a system modeled in Simulink ®. Pole Placement. Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations.