Maple, in conjunction with MapleSim and the MapleSim Control Design Toolbox, provides extensive capabilities for plant modeling and advanced control system design. Maplesoft tools provide you with the ability to:
Navigate to: Plant Model and Control System Design Maplesoft's suite of tools for modeling and analyzing engineering systems provides a wide range of capabilities for advanced control system design. The tight integration that exists between Maple, MapleSim, and the MapleSim Control Design Toolbox gives engineers the ability to create detailed plant models and the analytical tools for controller development and testing. Moreover, the symbolic computation engine that lies at the core of these tools provides you with greater flexibility and accuracy in your control systems design. With these tools, engineers can dramatically reduce the time and cost of upfront analysis, virtual prototyping, and parameter optimization of their system designs. Model Linearization Using the EquilibriumPoint command, Maple gives you the ability determine the local equilibrium point of your system satisfying constraints. The command performs a local search and returns an equilibrium point closest to the initial point, either specified by the initialpoint parameter or chosen randomly. If the EquilibriumPoint command cannot find a point at which derivatives are zero, it returns a point that minimizes the derivatives. It is possible to prescribe a nonzero value to the derivatives using the optional parameter constraints. Control Design Algorithms Maple, in conjunction with MapleSim and the MapleSim Control Design Toolbox, provides you with numerous algorithms for control design. The following list provides you with an overview of Maple's capabilities in the area of standard PID tuning, advanced PID tuning, state feedback control and state estimation. Standard PID and Advanced PID Tuning:
State Feedback Control:
State Estimation:
Example: Pole Placement Method For this example, we will use the gain values for kc and ki as provided above: Using these gain values, the transfer function for the PI controller can be determined as: At this point, we can obtain the transfer function for the closedloop system. The step response of the open and closedloop system can be determined using the DynamicSysetms[ResponsePlot] command. From the plots, we can see how the including of a PI controller was able to stabilize the system. Example: Exact Pole Placement Method In this section, we will show how the exact pole placement method is used to design a PID controller for a 2nd order system. As in the previous section, the transfer function of the open loop system can be defined as follows: Let's assume that the desired closedloop pole locations are: Using these gain values, the transfer function for the PID controller can be determined: At this point, we can obtain the transfer function for the closedloop system. The step response of the open and closedloop system is shown in the following plots. The closedloop response in addition to tracking the input signal is much faster than then openloop response. Example: GainPhase Margin Method In this section, we will show how the gainphase margin method is used to design a PID controller for the following fifthorder system. The system gain margin (in decibels) and associated crossover frequency (in rad/sec) is: Similarly, the system phase margin (in degrees) and crossover frequency (in rad/sec) is: Using these gain values, the transfer function for the PID controller can be determined as: At this point, we can obtain the transfer function for the closedloop system. The gain and phase margin and crossover frequencies for the loop transfer function are shown below. These values exceed the minimum requirements specified above. The step response of the open and closedloop system is shown in the following plots. Unlike the openloop response which was more than 100% off in the steady state, the closedloop response now tracks the input signal. The bode plots of the openloop system and the loop transfer function are shown below. System Definition and Analysis Maple provides you with several options to define your system. Choose between a frequencybased representation or a timebased representation:
In addition, Maple makes it easy to define and analyze your plant and controller models. The following list outlines Maple's capabilities in this area:
Maple provides you with numerous tools to visualize the response of your plant and controller models such as:
