**System description**

This article is about altitude control of the ballistic missile. From the control theory point of view, the system is unstable with one input and one output. The input manipulated variable is thrust chamber deflection and the output controlled variable is the ballistic missile altitude.

**Physical quantities:**

- δ [-] thrust chamber deflection

- y [m] altitude

**Mathematical model**

The transfer function model relating missile altitude (*y*) to thrust chamber deflection (*δ*) has two RHP (unstable) poles and a negative zero:

**Simulation files**

{phocadownload view=file|id=12|text=Download model - MATLAB/Simulink, ver. 6.5 (R13)|target=s}

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**Information sources**

BLAKELOCK, John H. *Automatic Control of Aircraft and Missiles*. 2 ed. New York: John Wiley, 1991, 676 p. ISBN 04-715-0651-6.

PADMA SREE, R., CHIDAMBARAM, M. *Control of Unstable Systems*. Oxford: Alpha Science, 2006, 297 p. ISBN 1-84265-287-7.

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**System description**

A fluidized bed reactor is a type of reactor device that can be used to carry out a variety of multiphase chemical reactions.

The presented reactor is carrying out oxidation of benzene to maleic anhydride. The reaction is highly exothermic and the temperature of the bed (*T*) is carefully controlled by regulating the coolant flow rate (*Wc*).

**Physical quantities:**

- T [K] temperature of the bed

- Wc [l/s] coolant flow rate

**Mathematical model**

The transfer function relating the reactor bed temperature *T* to coolant flow rate *Wc* is given by:

**Simulation files**

{phocadownload view=file|id=13|text=Download model - MATLAB/Simulink, ver. 6.5 (R13)|target=s}

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**Information sources**

KENDI, Thomas A.; DOYLE III, Francis J. *Nonlinear Control of a Fluidized Bed Reactor Using Approximate Feedback Linearization*. Industrial & Engineering Chemistry Research. 1996, 35 (3), pp 746-757. DOI 10.1021/ie950334a. Available from: http://pubs.acs.org/doi/abs/10.1021/ie950334a

PADMA SREE, R., CHIDAMBARAM, M. *Control of Unstable Systems*. Oxford: Alpha Science, 2006, 297 p. ISBN 1-84265-287-7.

**System description**

The system consists of a cart which can be moved along a metal guiding bar. An aluminium rod with a cylindrical weight is fixed to the cart by an axis. This system is unstable and non-linear with one input and two outputs. Input signal is control voltage of a DC motor which can change position of the cart. The outputs are cart position and angle of the pendulum rod. Both outputs are measured by incremental encoders.

1 - Servo amplifier, 2 - Motor, 3 - Drive wheel, 4 - Transmission belt, 5 - Metal guiding bar

6 - Cart, 7 - Pendulum weight, 8 - Guide roll, 9 - Pendulum rod

**Parameters of the real system AMIRA PS600:**

All the used constants were either taken from the producer (Amira, 2000) or identified by experiments (Chalupa & Bobál, 2008; Marholt, Gazdoš & Dostál, 2011):

Parameter |
Symbol |
Value & Unit |
Parameter |
Symbol |
Value & Unit |

Cart weight | mc | 4 kg | Inertia moment | Θ | 0.08433 kg.m2 |

Pendulum weight | mp | 0.36 kg | Cart friction | Fr | 6.5 kg/s |

Total weight | m | 4.36 kg | Pendulum friction | C | 0.00652 kg.m2/s |

Pendulum length | l | 0.42 m | Rate constant | ka | 7.5 N/V |

**Mathematical model**

The system can be described by the following nonlinear differential equations (Amira, 2000):

where *F* represents input signal, which is the force produced by the DC motor. Output signals are *r* - cart position (*r*' - denotes cart speed) and *φ* - pendulum angle (*φ*' - denotes pendulum angular speed). Symbol *g* is the gravity acceleration constant. All other constants and symbols are clearly defined in the presented table above.

The nonlinear differential equations were linearized in the operating point *φ* = 0 (top unstable position of the pendulum rod). A transfer function of the pendulum angle for this operating point then takes the following form (Marholt, Gazdoš & Dostál, 2011):

**Simulation files**

{phocadownload view=file|id=14|text=Download model - MATLAB/Simulink, ver. 6.5 (R13)|target=s}

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**Information sources**

AMIRA. *PS600 Laboratory Experiment Inverted Pendulum*. Duisburg : Amira GmbH, 2000, 351p.

**System description**

The magnetic levitation system CE 152 depicted in the figure below represents a laboratory-scale model designed by TQ Education and Training Ltd for studying system dynamics and experimenting with control algorithms. It demonstrates control problems associated with nonlinear unstable systems.

The system consists of a coil levitating a steel ball in the magnetic field with the position sensed by an inductive linear sensor connected to an A/D converter. The coil is driven by a power amplifier connected to a D/A converter. A basic control task is to control the position of the ball freely levitating in the magnetic field of the coil. From the control theory point of view, the magnetic levitation system is a nonlinear unstable system with one input and one output.

**Parameters of the real system CE 152:**

The values of all important parameters are listed in the following table:

Parameter |
Symbol |
Value & unit |
Parameter |
Symbol |
Value & unit |

A/D converter gain | k_{AD} |
0.2 MU/V |
Coil constant | kc |
1.769 x 10-6 Nm2/A2 |

D/A converter gain | k_{DA} |
20 V/MU | Ball mass | mk |
8.27 x 10-3 kg |

Damping constant | kfv |
0.02 Ns/m | Coil offset | x_{0} |
7.6 x 10-3 m |

Position sensor gain | kx |
821 V/m | Gravity constant | g |
9.81 m/s2 |

Power amplifier gain | ki |
0.3 A/V | Position sensor offset | y_{0} |
0.0183 V |

MU ... voltage converted by the data acquisition card and scaled to ±1 machine unit (MU).

**Mathematical model**

A mathematical model of the system including both D/A and A/D converters can be derived in the following form of a second-order nonlinear differential equation (Humusoft, 2002; Gazdoš, Dostál & Pelikán, 2009):

where *y* denotes the controlled variable - ball position [MU] and *u* is the control input [MU], proportional to the voltage from the D/A converter. All constants and symbols are clearly defined in the previous table.

**Simulation files**

{phocadownload view=file|id=16|text=Download model - MATLAB/Simulink, ver. 6.5 (R13)|target=s}

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**Information sources**

Humusoft. *CE 152 Magnetic levitation model – educational manual. *2002. Prague: Humusoft s.r.o.

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**System description**

This process is represented by a continuous stirred-tank reactor (CSTR) with nonideal mixing described by Cholette’s model. The process can be illustrated according to the following figure.

**Parameters:**

Presented parameters are described in detail in the mathematical model given below.

n = m = 0.75, V = 1 l, k1 = 10 s-1, k2 = 10 l/mol, q = 0.033 l/s

A nominal operating point (for the linearization):

cf = 6.484 mol/l, ce = 1.8 mol/l, c = 0.2387 mol/l

**Mathematical model**

A simplified mathematical model of the process dynamics can be described by the following nonlinear differential formulas (Liou & Chien, 1991; Padma Sree & Chidambaram, 2006):

where *C*(*t*) is the concentration of the reactant in the well-mixed zone, *C _{e}*(

Linearization of the nonlinear model around the above given nominal operating point gives the transfer function model as:

**Simulation files**

{phocadownload view=file|id=18|text=Download model - MATLAB/Simulink, ver. 6.5 (R13)|target=s}

Downloading files is possible only for registered users.

**Information sources**

PADMA SREE, R., CHIDAMBARAM, M. *Control of Unstable Systems*. Oxford: Alpha Science, 2006, 297 p. ISBN 1-84265-287-7.

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**System description**

The X-29 was an experimental aircraft that tested a forward-swept wing, canard control surfaces, and other novel aircraft technologies. It was deliberately designed with static instability to increase its manoeuverability and speeds of command response. Considerable effort has been devoted to the design of the flight control system for this airplane (Rogers & Collins, 1992; Clarke et al., 1994; Stein, 2003). One of the design criterion is phase margin should be greater than 45° for all flight conditions.

**Mathematical model**

At one flight condition the transfer function model si given by:

**Simulation files**

{phocadownload view=file|id=15|text=Download model - MATLAB/Simulink, ver. 6.5 (R13)|target=s}

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**Information sources**

ROGERS, W.L.; COLLINS, D.J. X-29 H_{∞} controller synthesis. *Journal of Guidance, Control, and Dynamics*, vol. 15, no. 4, pp. 962-967, 1992.

CLARKE, R.; BURKEN, J.J.; BOSWORTH, J.T.; BAUER, J.E. X-29 flight control system—lessons learned. *International Journal of Control*, vol. 59, no. 1, pp. 199-219, 1994.

STEIN, G. Respect the unstable. *IEEE Control systems magazine*, vol. 23, no. 4, pp. 12-25, 2003.

PADMA SREE, R.; CHIDAMBARAM, M. Control of unstable systems. Oxford: Alpha Science, 2006. ISBN 1-84265-287-7.

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