Control System Design Tools
This section describes methods and tools for control system design and for analyzing stability and robustness to parameter variations. The LQR/LQG and the H-Infinity methods are commonly used for control system design. Additional methods are included for analyzing the control system’s robustness to parameter uncertainties, and also classical frequency response, singular value analysis, and root-locus tools.
The LQR/LQG Control Design Method
The Linear Quadratic Regulator (LQR) and the Linear Quadratic Gaussian (LQG) control design are easy-to-use methods for designing controls to stabilize and regulate systems. The LQR is simply statefeedback. The LQG is used when the plant states are not directly available for measurement. The Linear Quadratic Regulator (LQR) and the Linear Quadratic Gaussian (LQG) control design methods are typically used to design control laws that stabilize control systems. The LQR is simply a state-feedback. The design algorithm requires a plant model in state-space form. It produces a state-feedback control gain that stabilizes the plant and achieve good closed-loop performance of the states in response to transients. The LQG is used when the plant states are not directly available for measurement and it consists of two steps: the design of an LQR state-feedback controller, and the design of a Kalman-Filter observer in order to estimate the state vector. The estimator design also requires the plant model. The state-feedback and the estimator are then combined together to create an output feedback dynamic controller in state-space form.
The Linear Quadratic Regulator (LQR) and the Linear Quadratic Gaussian (LQG) control design methods are used to design control laws that stabilize control systems. The LQR is simply a state-feedback. The design algorithm requires a plant model in state-space form. It produces a state-feedback control gain that stabilizes the plant and achieve good closed-loop performance of the states in response to transients. The LQG is used when the plant states are not directly available for measurement and it consists of two steps: the design of an LQR state-feedback controller, and the design of a Kalman-Filter observer in order to estimate the state vector. The estimator design also requires the plant model. The state-feedback and the estimator are then combined together to create an output feedback dynamic controller in state-space form.
The H-Infinity Control Design Method
The H∞ algorithm is a powerful control synthesis method that attempts to minimize the infinity norm of the sensitivity transfer function matrix of the closed-loop system. The sensitivity function of a system is the transfer function between the disturbance inputs and some sensitive outputs that should be kept small, such as a spacecraft attitude or an airplane’s angle of attack. The H-Infinity algorithm is more powerful than the LQR because it attempts to minimize the infinity norm of the sensitivity transfer function of the closed-loop system. That is, the transfer function between the disturbance inputs and some sensitive outputs that should produce a small response to disturbances, such as a spacecraft attitude, telescope jitter or an airplane’s angle of attack in response to gusts. The infinity norm is a measure of amplitude and it is the magnitude of the largest singular value over all frequencies. The H-Infinity algorithm produces a control system than minimizes the system’s sensitivity norm and therefore, the effects of external disturbances to the plant outputs. The mathematical implementation of the H-Infinity control synthesis algorithm requires two steps. The creation of a Synthesis Model (SM) that consists of 9 matrices including plant dynamics and performance optimization requirements of the closed-loop system. The SM is then presented to the algorithm which calculates the optimal control solution that will attempt to satisfy the design requirements. The algorithm requires the solution of two Riccati equations. The H-Infinity program included in Flixan not only solves the optimization algorithm but it also includes a utility that helps the designer to create the SM interactively from the plant dynamics. The plant dynamic model is created from the vehicle modeling program. The user must separate the inputs into controls and disturbances, and the outputs into measurements and performance criteria. After separating them you end up with a SM consisting of 9 matrices that go into the H-Infinity algorithm. The SM also includes some gains that define requirements on the disturbances and the performance criteria. They trade between control bandwidth, robustness to noise and un-modeled dynamics, and sensitivity to disturbances.
The H-Infinity method is more powerful than the LQR because it attempts to minimize the infinity norm of the sensitivity transfer function of the closed-loop system. That is, the transfer function between the disturbance inputs and the sensitivity outputs that should have small response to disturbances, such as a spacecraft attitude, telescope jitter or an airplane’s angle of attack response to gusts. The infinity norm is a measure of amplitude and it is the magnitude of the largest singular value over all frequencies. The H-Infinity algorithm produces a control system than minimizes the system’s sensitivity norm, and therefore, the effects of external disturbances on the plant outputs. The H-Infinity algorithm requires a 9-matrices Synthesis Model which consists of plant model and performance optimization parameters.
Robustness Method for Analyzing Parameter Uncertainties
Robustness is the ability of the control system to tolerate external disturbances and also variations or uncertainties in the vehicle parameters. In this section we shall develop dynamic models that can be used to analyze robustness to parameter uncertainties and also to synthesize robust control laws using µ-tools.
Robustness is the ability of the control system to tolerate external disturbances and also variations or uncertainties in vehicle parameters. In this section we will develop dynamic models that can be used to analyze robustness to parameter uncertainties and also models that can be used to synthesize robust control laws. Sensitivity is analyzed by the ability of the control system to tolerate the effects of external disturbances on some sensitive outputs, for example, optical sensors or structural load sensors. Parameter uncertainties can be seen as imprecise knowledge of the plant model parameters, such as: the mass, moments of inertia, aerodynamic coefficients, vehicle altitude, dynamic pressure, center of gravity, etc. Variations of internal vehicle parameters relative to nominal values are called “Structured Uncertainties”. The question is, how much of parameter variations is a stable closed-loop system able to tolerate before it becomes unstable, or stops performing properly? In this section we present a method that is used to create state-space models for analyzing the robustness of systems that have uncertain parameters, but the variations relative to the nominal values are bounded.
Each parameter variation is “pulled out” of the uncertain plant model and it is placed inside a diagonal block D that contains only uncertainties and it is connected to the known plant via some additional inputs and outputs. The remaining plant is assumed to be known and is created from the nominal vehicle parameters. The Flixan program creates a vehicle system where the internal parameter variations are converted to external inputs and outputs. An input/ output pair for each parameter variation. The inputs are treated as disturbances and the outputs as performance criteria. In addition, the inputs and outputs of the system connecting to the uncertainty block D are scaled to assume that each variation in the uncertainty block D varies between -1 to +1.
Robustness is the ability of the control system to tolerate external disturbances and also parameter variations. In this video we will develop dynamic models that can be used to analyze robustness to parameter uncertainties and also models that can be used to synthesize robust control laws. Parameter uncertainties can be seen as imprecise knowledge of the plant model parameters, such as: the mass, moments of inertia, aerodynamic coefficients, vehicle altitude, dynamic pressure, center of gravity, etc. The question is, how much of parameter variations is a system able to tolerate before it becomes unstable, or stops performing properly? In this section we present a method that is used to create state-space models for analyzing the robustness of systems that have uncertain parameters, but the variation is bounded. Each parameter variation is “pulled out” of the uncertain plant model and it is placed inside a diagonal block D that contains only uncertainties. The remaining plant is assumed to be known. The Flixan program creates a vehicle system where the internal parameter variations are converted to external inputs and outputs. The inputs are treated as disturbances and the outputs as performance criteria.
Frequency Domain Control Analysis Tools
The frequency response analysis program calculates the frequency response of a system that is described in state-space form by a set of four matrices (A,B,C,D), either continuous or discrete at fixed sampling rate. This section describes the frequency response analysis tools. There is a frequency response analysis program, a Singular Values (Sigma) program that calculates the SV frequency responses of systems, and a Pole/ Zero program that also calculates the root-locus by varying the control gain. The programs read the state-space data from system files and they plot the results on the screen. The systems are either continuous or discrete.
This video describes the frequency response analysis tools. There is a frequency response program, a Singular Values (Sigma) program that calculates the SV frequency responses of systems, and a Pole/ Zero program that also calculates the root-locus by varying the control gain. The programs read the state-space data from system files and they plot the results on the screen. The systems are either continuous or discrete.