## Robust Control Toolbox |

- Modeling of systems with uncertain parameters or neglected dynamics
- Worst-case analysis of stability margins and sensitivity to disturbances
- Automatic tuning of single-loop and multiloop control systems for nominal and uncertain plants
- Automatic tuning of gain-scheduled controllers
- Robustness analysis and controller tuning in Simulink
^{®} - H-infinity and mu-synthesis algorithms
- General-purpose LMI solvers

With Robust Control Toolbox™, you can capture not only the typical, or nominal, behavior of your plant, but also the amount of uncertainty and variability. Plant model uncertainty can result from:

- Model parameters with approximately known or varying values
- Neglected or poorly known dynamics, such as high-frequency dynamics
- Changes in operating conditions
- Linear approximations of nonlinear behaviors
- Estimation errors in a model identified from measured data

Robust Control Toolbox lets you build detailed uncertain models by combining nominal dynamics with uncertain elements, such as uncertain parameters or neglected dynamics. By quantifying the level of uncertainty in each element, you can capture the overall fidelity and variability of your plant model. You can then analyze how each uncertain element affects performance, and identify worst-case combinations of uncertain element values.

**Building and Manipulating Uncertain Models** (Example)

Build uncertain state-space models and analyze the robustness of feedback control systems that have uncertain elements.

Using Robust Control Toolbox, you can analyze the effect of plant model uncertainty on the closed-loop stability and performance of the control system. In particular, you can determine whether your control system will perform adequately over its entire operating range, and what source of uncertainty is most likely to jeopardize performance.

**Robustness of Servo Controller for DC Motor** (Example)

Model uncertainty in DC motor parameters and analyze the effect of this uncertainty on motor controller performance.

You can randomize the model uncertainty to perform Monte Carlo analysis. Alternatively, you can use more direct tools based on mu-analysis and linear matrix inequality (LMI) optimization; these tools identify worst-case scenarios without exhaustive simulation.

Robust Control Toolbox provides functions to assess worst-case values for:

- Gain and phase margins, one loop at a time
- Stability margins that take loop interactions into account
- Gain between any two points in a closed-loop system
- Sensitivity to external disturbances

These functions also provide sensitivity information to help you identify the uncertain elements that contribute most to performance degradation. With this information, you can determine whether a more accurate model, tighter manufacturing tolerances, or a more accurate sensor would most improve control system robustness.

Robust Control Toolbox lets you automatically tune centralized and decentralized MIMO control systems. The controller synthesis algorithms include H-infinity and mu-synthesis techniques, nonsmooth optimization, and LMI optimization. These algorithms are applicable to SISO and MIMO control systems. MIMO controller synthesis does not require sequential loop closure and is therefore well-suited for multiloop control systems with significant loop interaction and cross-coupling.

Most embedded control systems have a fixed architecture with simple tunable elements such as gains, PID controllers, or low-order filters. Such architectures are easier to understand, implement, schedule, and retune than complex centralized controllers. Robust Control Toolbox provides tools for modeling and tuning these decentralized control architectures. You can:

- Specify tunable elements such as gains, PID controllers, fixed-order transfer functions, and fixed-order state-space models
- Combine tunable elements with ordinary linear time-invariant (LTI) models to create a tunable model of your control architecture
- Specify and visualize tuning requirements such as tracking performance, disturbance rejection, noise amplification, closed-loop pole locations, and stability margins
- Automatically tune the controller parameters to satisfy the must-have requirements (design constraints) and to best meet the remaining requirements (objectives)
- Validate controller performance in the time and frequency domains

**Automatic Tuning of a Helicopter Flight Control System**

Automatically tune a multivariable flight control system using Control System Tuner.

In addition to tuning a fixed-structure controller for one plant model, Robust Control Toolbox lets you automatically tune a controller for a plant model with uncertain real parameters. You can use this functionality to design a controller that is robust to changes in plant dynamics due to plant parameter variations.

**Robust Tuning with the Control System Tuner App**

Automatically tune controllers to maximize performance over a range of parameter values using the Control System Tuner app from Robust Control Toolbox™.

The toolbox also lets you tune one controller against a set of plant models. This lets you design a controller that will be robust to changes in plant dynamics due to changes in operating conditions, and also to sensor or actuator failures.

**Fault-Tolerant Control of a Passenger Jet** (Example)

Tune a fixed-structure controller for multiple operating modes of the plant.

Robust Control Toolbox provides several algorithms for synthesizing robust MIMO controllers directly from frequency-domain specifications of the closed-loop responses. For example, you can limit the peak gain of a sensitivity function to improve stability and reduce overshoot, or limit the gain from input disturbance to measured output to improve disturbance rejection. Using mu-synthesis algorithms, you can optimize controller performance in the presence of model uncertainty, ensuring effective performance under all realistic scenarios. H-infinity and mu-synthesis techniques provide unique insight into the performance limits of your control architecture, and let you quickly develop first-cut compensator designs.

**Robust Control of an Active Suspension** (Example)

Design a robust controller for an active suspension system using H-infinity and mu-synthesis methods.

Gain scheduling is a linear technique for controlling nonlinear or time-varying plants. It involves computing linear approximations of the plant at various operating conditions, tuning controller gains at the operating condition, and scheduling controller gains as the plant changes operating conditions. Robust Control Toolbox provides tools for automatically computing gain schedules for fixed-structure control systems. You can:

- Automatically trim and linearize Simulink models at multiple operating conditions (using Simulink Control Design
^{™}) - Parameterize controller gain surfaces as functions of scheduling variables
- Construct a linear parameter-varying (LPV) model representing the system throughout its operating range
- Specify tuning requirements such as tracking and disturbance rejection
- Automatically tune gain surface coefficients to satisfy tuning requirements at all operating conditions
- Update parameters of the Simulink Look Up Table or Interpolation blocks implementing the controller with tuned gain values

Robust Control Toolbox eliminates the need to tune one controller at a time for each operating condition and instead provides globally tuned gain surfaces that generate smooth transitions from one operating condition to another.

**Tuning of Gain-Scheduled Three-Loop Autopilot** (Example)

Generate smooth gain schedules for a three-loop autopilot.

Robust Control Toolbox provides tools for performing robustness analysis and for tuning controllers modeled in Simulink.

The toolbox lets you model and analyze uncertainty in Simulink models. You can:

- Introduce uncertainty into a Simulink model by using an Uncertain State Space block or by specifying block linearization for any Simulink block
- Linearize a Simulink model to create an uncertain system that represents the whole Simulink model
- Analyze the resulting uncertain system for stability and performance

**Linearization of Simulink Models with Uncertainty** (Example)

Compute uncertain linearizations.

Robust Control Toolbox lets you automatically tune decentralized controllers modeled in Simulink. Because Robust Control Toolbox operates on linear models, you can use Simulink Control Design to automatically compute and store a linearization of your Simulink model. Simulink Control Design automatically creates a tunable model of the control architecture specified in a Simulink model. You can:

- Specify Simulink model blocks that should be tuned
- Specify tuning requirements
- Automatically tune specified blocks to satisfy the must-have requirements (design constraints) and to best meet the remaining requirements (objectives)
- Validate your design by running nonlinear simulations

Using this approach, you can automatically tune complex multivariable controllers that are modeled using Simulink blocks. For example, you can automatically tune inner-loop and outer-loop PID controllers in a multiloop control system without changing the control system architecture.

**Automatic Tuning of a Multivariable Fixed-Structure Simulink Controller**

Design a decoupling controller for a distillation column.

Detailed first-principles or finite-element plant models often have a large number of states. Similarly, H-infinity and mu-synthesis algorithms tend to produce high-order controllers with superfluous states. Robust Control Toolbox provides algorithms that let you reduce the order (number of states) of a plant or controller model while preserving its essential dynamics. As you extract lower-order models, which are more cost-effective to implement, you can control the approximation error.

The model reduction algorithms are based on Hankel singular values of the system, which measure the energy of the states. By retaining high-energy states and ignoring low-energy states, the reduced model preserves the essential features of the original model. You can use the absolute or relative approximation error to select the order, and use frequency-dependent weights to focus the model reduction algorithms on specific frequency ranges.

**Simplifying Higher-Order Plant Models** (Example)

Approximate higher-order plant models with simpler, lower-order models.