# User:ShakespeareFan00/Sandbox1

# State Feedback

[edit | edit source]- H-infinity
- H-2
- Mixed
- Stabilization of Second-Order Systems
- LQ Regulation via H2 Control
- Controller to achieve the desired Reachable set; Polytopic uncertainty
- Controller to achieve the desired Reachable set; Norm bound uncertainty
- Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty

# D-Stability

[edit | edit source]# Optimal State Feedback

[edit | edit source]# Output Feedback

[edit | edit source]# Static Output Feedback

[edit | edit source]# Optimal Output Feedback

[edit | edit source]# Stabilizability LMI

[edit | edit source]Stabilizability LMI

A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair is shown below.

**The System**

[edit | edit source]where , , at any .

**The Data**

[edit | edit source]The matrices necessary for this LMI are and . There is no restriction on the stability of A.

**The LMI:** Stabilizability LMI

[edit | edit source]is stabilizable if and only if there exists such that

- ,

where the stabilizing controller is given by

- .

**Conclusion:**

[edit | edit source]If we are able to find such that the above LMI holds it means the matrix pair is stabilizable. In words, a system pair is stabilizable if for any initial state an appropriate input can be found so that the state asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to *approach* as whereas controllability requires that the state must *reach* the origin in a *finite* time.

**Implementation**

[edit | edit source]This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Stabilizability_LMI.m

**Related LMIs**

[edit | edit source]**External Links**

[edit | edit source]A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.

**Return to Main Page:**

[edit | edit source]# LMI for the Controllability Grammian

[edit | edit source]LMI to Find the Controllability Gramian

Being able to adjust a system in a desired manor using feedback and sensors is a very important part of control engineering. However, not all systems are able to be adjusted. This ability to be adjusted refers to the idea of a "controllable" system and motivates the necessity of determining the "controllability" of the system. Controllability refers to the ability to accurately and precisely manipulate the state of a system using inputs. Essentially if a system is controllable then it implies that there is a control law that will transfer a given initial state and transfer it to a desired final state . There are multiple ways to determine if a system is controllable, one of which is to compute the rank "controllability Gramian". If the Gramian is full rank, the system is controllable and a state transferring control law exists.

**The System**

[edit | edit source]where , , at any .

**The Data**

[edit | edit source]The matrices necessary for this LMI are and . must be stable for the problem to be feasible.

**The LMI:** LMI to Determine the Controllability Gramian

[edit | edit source]is controllable if and only if is the unique solution to

- ,

where is the Controllability Gramian.

**Conclusion:**

[edit | edit source]The LMI above finds the controllability Gramian of the system . If the problem is feasible and a unique can be found, then we also will be able to say the system is controllable. The controllability Gramian of the system can also be computed as: , with control law that will transfer the given initial state to a desired final state .

**Implementation**

[edit | edit source]This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Controllability_Gram_LMI.m

**Related LMIs**

[edit | edit source]**External Links**

[edit | edit source]A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.

**Return to Main Page:**

[edit | edit source]# LMI for Decentralized Feedback Control

[edit | edit source]LMI for Decentralized Feedback Control

In large-scale systems like a multi-agent robotic system, national economies, or chemical refineries, an actuator should act based on local information, which necessitates a decentralized or distributed control strategy. In a decentralized control framework, the controllers are distributed and each controller has only access to a subset of local measurements. We describe LMI formulations for a general decentralized control framework and then provide an illustrative example of a decentralized control design.

**The System**

[edit | edit source]In a decentralized controller design, the state feedback controller can be divided into sub-controllers .

**The Data**

[edit | edit source]A general state space representation of a linear time-invariant system is as follows:

where is a vector of state variables, is the input matrix, is the output matrix, and is called the feedforward matrix. We assume that all the four matrices, , , , and are given.

**The Optimization Problem**

[edit | edit source]We aim to solve the -optimal full-state feedback control problem using a controller .

In a decentralized fashion, the control input can be divided into sub-controllers and can be written as .

For instance, let and . Thus, there are three control inputs , , and . We also assume that u_{1} only depends on the first and the second states, while and only depend on thrid to sixth states. For this example, the controller gain matrix can be described by:

Thus, the decentralized controller gain consists of sub-matrices of gains.

**The LMI:** LMI for decentralized feedback controller

[edit | edit source]The mathematical description of the LMI formulation for a decentralised optimal full-state feedback controller can be described by:

where is a positive definite matrix and such that the aforemtntioned constraints in LMIs are satisfied.

**Conclusion:**

[edit | edit source]The controller gain matrix is defined as:

where can be found after solving the LMIs and obtaining the variables matrices and . Thus,

.

**Implementation**

[edit | edit source]A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI_for_decentralized_feedback_controller/tree/master

**Related LMIs**

[edit | edit source]## External Links

[edit | edit source]A list of references documenting and validating the LMI.

- [1] - LMI in Control Systems Analysis, Design and Applications

**Return to Main Page**

[edit | edit source]# LMI for Mixed Output Feedback Controller

[edit | edit source]LMI for Mixed Output Feedback Controller

The mixed output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the controller, the channel is used to improve the robustness of the design while the channel guarantees good performance of the system.

**The System**

[edit | edit source]We consider the following state-space representation for a linear system:

where , , , and are the state matrix, input matrix, output matrix, and feedforward matrix, respectively.

These are the system (plant) matrices that can be shown as .

**The Data**

[edit | edit source]We assume that all the four matrices of the plant, , are given.

**The Optimization Problem**

[edit | edit source]In this problem, we use an LMI to formulate and solve the optimal output-feedback problem to minimize both the <> and <> norms. Giving equal weights to each of the norms, we will have the optimization problem in the following form:

**The LMI:** LMI for mixed /

[edit | edit source]Mathematical description of the LMI formulation for a mixed / optimal output-feedback problem can be written as follows:

where and are defined as the and norm of the system:

Moreover, , , , , , and are variable matrices with appropriate dimensions that are found after solving the LMIs.

**Conclusion:**

[edit | edit source]The calculated scalars and are the and norms of the system, respectively. Thus, the norm of mixed / is defined as . The results for each individual norm and norms of the system show that a bigger value of norms are found in comparison with the case they are solved separately.

**Implementation**

[edit | edit source]A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI_for_Mixed_H2_Hinf_Output_Feedback_Controller

**Related LMIs**

[edit | edit source]**External Links**

[edit | edit source]- [2] - LMI in Control Systems Analysis, Design and Applications

**Return to Main Page**

[edit | edit source]# Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

[edit | edit source]User:ShakespeareFan00/Sandbox1

If the system is quadratically stable, then there exists some and such that the LMI is feasible. The and matrices can also be used to create a quadratically stabilizing controller.

**The System**

[edit | edit source]**The Data**

[edit | edit source]The matrices .

**The LMI:**

[edit | edit source]**Conclusion:**

[edit | edit source]There exists a controller for the system with where is the quadratically stabilizing controller, if the above LMI is feasible.

**Implementation**

[edit | edit source]https://github.com/mcavorsi/LMI

**Related LMIs**

[edit | edit source]H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

**Return to Main Page:**

[edit | edit source]# H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

[edit | edit source]User:ShakespeareFan00/Sandbox1

If there exists some , and such that the LMI holds, then the system satisfies There also exists a controller with

**The System**

[edit | edit source]**The Data**

[edit | edit source]The matrices .

**The Optimization Problem**

[edit | edit source]Minimize subject to the LMI constraints below.

**The LMI:**

[edit | edit source]**Conclusion:**

[edit | edit source]The controller gains, K, are calculated by .

**Implementation**

[edit | edit source]https://github.com/mcavorsi/LMI

**Related LMIs**

[edit | edit source]Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

**Return to Main Page:**

[edit | edit source]# Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

[edit | edit source]User:ShakespeareFan00/Sandbox1

**The System**

[edit | edit source]**The Data**

[edit | edit source]The matrices .

**The LMI:**

[edit | edit source]**Conclusion:**

[edit | edit source]If the LMI is feasible, the controller, K, is calculated by .

**Implementation**

[edit | edit source]https://github.com/mcavorsi/LMI

**Related LMIs**

[edit | edit source]Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

**Return to Main Page:**

[edit | edit source]# Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

[edit | edit source]User:ShakespeareFan00/Sandbox1

**The System**

[edit | edit source]**The Data**

[edit | edit source]The matrices .

**The Optimization Problem**

[edit | edit source]subject to the LMI constraints.

**The LMI:**

[edit | edit source]**Conclusion:**

[edit | edit source]The controller is .

**Implementation**

[edit | edit source]https://github.com/mcavorsi/LMI

**Related LMIs**

[edit | edit source]Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

**Return to Main Page:**

[edit | edit source]# Optimal Output Controllability for Systems With Transients

[edit | edit source]Optimal Output Controllability for Systems With Transients

This LMI provides an optimal output controllability problem to check if such controllers for systems with unknown exogenous disturbances and initial conditions can exist or not.

**The System**

[edit | edit source]where is the state, is the exogenous input, is the control input, is the measured output and is the regulated output.

**The Data**

[edit | edit source]System matrices need to be known. It is assumed that . are matrices with their columns forming the bais of kernels of and respectively.

**The Optimization Problem**

[edit | edit source]For a given , the following condition needs to be fulfilled:

**The LMI:** Output Feedback Controller for Systems With Transients

[edit | edit source]**Conclusion:**

[edit | edit source]Solution of the above LMI gives a check to see if an optimal output controller for systems with transients can exist or not.

**Implementation**

[edit | edit source]A link to CodeOcean or other online implementation of the LMI

**Related LMIs**

[edit | edit source]Links to other closely-related LMIs

## External Links

[edit | edit source]- LMI-based -optimal control with transients Link to the original article.

**Return to Main Page:**

[edit | edit source]# Quadratic Polytopic Stabilization

[edit | edit source]A Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about , , and matrices.

**The System**

[edit | edit source]where , , at any .

The system consist of uncertainties of the following form

where ,, and

**The Data**

[edit | edit source]The matrices necessary for this LMI are , , and

**The Optimization and LMI:**LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability

[edit | edit source]There exists a K such that

is quadratically stable for if and only if there exists some P>0 and Z such that

**Conclusion:**

[edit | edit source]The Controller gain matrix is extracted as

Note that here the controller doesn't depend on

- If you want K to depend on , the problem is harder.
- But this would require sensing in real-time.

**Implementation**

[edit | edit source]This implementation requires Yalmip and Sedumi. https://github.com/JalpeshBhadra/LMI/blob/master/quadraticpolytopicstabilization.m

**Related LMIs**

[edit | edit source]Quadratic Polytopic Controller

Quadratic Polytopic Controller

**External Links **

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

# Quadratic D-Stabilization

[edit | edit source]Continuous-Time D-Stability Controller

This LMI will let you place poles at a specific location based on system performance like rising time, settling time and percent overshoot, while also ensuring the stability of the system.

**The System**

[edit | edit source]Suppose we were given the continuous-time system

whose stability was not known, and where , , , and for any .

Adding uncertainty to the system

**The Data**

[edit | edit source]In order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:

- matrices , , ,
- rise time ()
- settling time ()
- percent overshoot ()

Having these pieces of information will now help us in formulating the optimization problem.

**The Optimization Problem**

[edit | edit source]Using the data given above, we can now define our optimization problem. In order to do that, we have to first define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:

**Rise Time**:

**Settling Time**:

**Percent Overshoot**:

Assume that is the complex pole location, then:

This then allows us to modify our inequality constraints as:

**Rise Time**:

**Settling Time**:

**Percent Overshoot**:

which not only allows us to map the relationship between complex pole locations and inequality constraints but it also now allows us to easily formulate our LMIs for this problem.

**The LMI:** An LMI for Quadratic D-Stabilization

[edit | edit source]Suppose there exists and such that

for

**Conclusion:**

[edit | edit source]Given the resulting controller , we can now determine that the pole locations of satisfies the inequality constraints , and for all

**Implementation**

[edit | edit source]The implementation of this LMI requires Yalmip and Sedumi https://github.com/JalpeshBhadra/LMI/blob/master/quadraticDstabilization.m

**Related LMIs**

[edit | edit source]- Continuous Time D-Stability Observer - Equivalent D-stability LMI for a continuous-time observer.

## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

**Return to Main Page:**

[edit | edit source]# Quadratic Polytopic Full State Feedback Optimal Control

[edit | edit source]**Quadratic Polytopic Full State Feedback Optimal Control**

[edit | edit source]For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based off of performance specifications given. methods formulate this task as an optimization problem and attempt to minimize the norm of the system.

**The System**

[edit | edit source]Consider System with following state-space representation.

where , , , , , , , , , , , , , for any .

Add uncertainty to system matrices

New state-space representation

**The Optimization Problem:**

[edit | edit source]Recall the closed-loop in state feedback is:

This problem can be formulated as optimal state-feedback, where K is a controller gain matrix.

**The LMI:**

[edit | edit source]An LMI for Quadratic Polytopic Optimal
State-Feedback Control

**Conclusion:**

[edit | edit source]The Optimal State-Feedback Controller is recovered by

Controller will determine the bound on the norm of the system.

**Implementation:**

[edit | edit source]https://github.com/JalpeshBhadra/LMI/tree/master

**Related LMIs**

[edit | edit source]Full State Feedback Optimal Controller

**External Links **

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

# Quadratic Polytopic Full State Feedback Optimal Control

[edit | edit source]User:ShakespeareFan00/Sandbox1

**Quadratic Polytopic Full State Feedback Optimal Control**

[edit | edit source]For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the norm of this system we are minimizing the effect noise has on the system as part of the performance specifications.

**The System**

[edit | edit source]Consider System with following state-space representation.

where , , , , , , , , , , , , , for any .

Add uncertainty to system matrices

New state-space representation

**The Data**

[edit | edit source]The matrices necessary for this LMI are

**The Optimization Problem:**

[edit | edit source]Recall the closed-loop in state feedback is:

This problem can be formulated as optimal state-feedback, where K is a controller gain matrix.

**The LMI:** An LMI for Quadratic Polytopic Optimal

[edit | edit source]State-Feedback Control

**Conclusion:**

[edit | edit source]The Optimal State-Feedback Controller is recovered by

**Implementation:**

[edit | edit source]https://github.com/JalpeshBhadra/LMI/blob/master/H2_optimal_statefeedback_controller.m

**Related LMIs**

[edit | edit source]Optimal State-Feedback Controller

**External Links **

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

# Continuous-Time Static Output Feedback Stabilizability

[edit | edit source]User:ShakespeareFan00/Sandbox1

In view of applications, static feedback of the full state is not
feasible in general: only a few of the state variables (or a linear
combination of them,
, called the output) can be
actually measured and re-injected into the system.

**So, we are led to the notion of static output feedback**

**The System**

[edit | edit source]Consider the continuous-time LTI system, with generalized state-space realization

**The Data**

[edit | edit source]**The Optimization Problem**

[edit | edit source]This system is static output feedback
stabilizable (SOFS) if there exists a matrix
F such that the closed-loop system

**(obtained by replacing which means applying static output feedback)**

is asymptotically stable at the origin

** The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability**

[edit | edit source]The system is static output feedback stabilizable if and only if it satisfies any of the following conditions:

- There exists a and , where , such that

- There exists a and , where , such that

- There exists a and , where , such that

- There exists a and , where , such that

**Conclusion**

[edit | edit source]On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix (or ) and

**Implementation**

[edit | edit source]A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

** Related LMIs**

[edit | edit source]Discrete time Static Output Feedback Stabilizability

Static Feedback Stabilizability

** External Links**

[edit | edit source]- [3] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.

**Return to Main Page:**

[edit | edit source]# Multi-Criterion LQG

[edit | edit source]User:ShakespeareFan00/Sandbox1

The Multi-Criterion Linear Quadratic Gaussian (LQG) linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a state space system with gaussian noise based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.

**The System**

[edit | edit source]The system is a linear time-invariant system, that can be represented in state space as shown below:

where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input, is and are the output matrices, and and are and are the output and the output of interest, respectively.

and , and the system is controllable and observable.

**The Data**

[edit | edit source]The matrices and the noise signals .

**The Optimization Problem**

[edit | edit source]In the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.

There are multiple outputs of interest for this problem. They are defined by

For each of these outputs of interest, we associate a cost function:

Additionally, the matrices and must be found as the solutions to the following Riccati equations:

The optimization problem is to minimize over u subject to the measurability condition and the constraints . This optimization problem can be formulated as:

over , with:

**The LMI:** Multi-Criterion LQG

[edit | edit source]over , subject to the following constraints:

**Conclusion:**

[edit | edit source]The result of this LMI is the solution to the aforementioned Ricatti equations:

**Implementation**

[edit | edit source]This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/multicriterionquadraticproblems.m

**Related LMIs**

[edit | edit source]## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

**Return to Main Page:**

[edit | edit source]# Inverse Problem of Optimal Control

[edit | edit source]User:ShakespeareFan00/Sandbox1

In some cases, it is needed to solve the inverse problem of optimal control within an LQR framework. In this inverse problem, a given controller matrix needs to be verified for the system by assuring that it is the optimal solution to some LQR optimization problem that is controllable and detectable. In other words: in this inverse problem, the controller is known and the LQR gain matrices are to be calculated such that the controller is the optimal solution.

**The System**

[edit | edit source]The system is a linear time-invariant system, that can be represented in state space as shown below:

where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input, is and are the output matrices, and and are and are the output and the output of interest, respectively.

**The Data**

[edit | edit source]The matrices that define the system, and a given controller for which the inverse problem is to be solved.

**The Optimization Problem**

[edit | edit source]In this LMI, the following cost function is to be minimized for a given controller K by finding an optimal input:

the solution of the problem can be formulated as a state feedback controller given as:

**The LMI:** Inverse Problem of Optimal Control

[edit | edit source]the inverse problem of optimal control is the following: Given a matrix , determine if there exist and , such that is detectable and is the optimal control for the corresponding LQR problem. Equivalently, we seek and such that there exist nonnegative and positive-definite satisfying

**Conclusion:**

[edit | edit source]If the solution exists, then is the optimal controller for the LQR optimization on the matrices and

**Implementation**

[edit | edit source]This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/inverseprob.m

**Related LMIs**

[edit | edit source]## External Links

[edit | edit source]- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

**Return to Main Page:**

[edit | edit source]# Nonconvex Multi-Criterion Quadratic Problems

[edit | edit source]User:ShakespeareFan00/Sandbox1

The Non-Concex Multi-Criterion Quadratic linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a non-convex state space system based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.

**The System**

[edit | edit source]The system for this LMI is a linear time invariant system that can be represented in state space as shown below:

where the system is assumed to be controllable.

where represents the state vector, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input.

for any input, we define a set cost indices by

Here the symmetric matrices,

- ,

are not necessarily positive-definite.

**The Data**

[edit | edit source]The matrices .

**The Optimization Problem**

[edit | edit source]The constrained optimal control problem is:

subject to

**The LMI:** Nonconvex Multi-Criterion Quadratic Problems

[edit | edit source]The solution to this problem proceeds as follows: We first define

where and for every , we define

then, the solution can be found by:

subject to

**Conclusion:**

[edit | edit source]If the solution exists, then is the optimal controller and can be solved for via an EVP in P.

**Implementation**

[edit | edit source]This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/multicriterionquadraticproblems.m

**Related LMIs**

[edit | edit source]## External Links

[edit | edit source]A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

**Return to Main Page:**

[edit | edit source]# Static-State Feedback Problem

[edit | edit source]User:ShakespeareFan00/Sandbox1

We are attempting to stabilizing The Static State-Feedback Problem

**The System**

[edit | edit source]Consider a continuous time Linear Time invariant system

**The Data**

[edit | edit source]are known matrices

** The Optimization Problem**

[edit | edit source]The Problem's main aim is to find a feedback matrix such that the system

and

is stable
Initially we find the matrix such that is Hurwitz.

** The LMI: Static State Feedback Problem**

[edit | edit source]This problem can now be formulated into an LMI as Problem 1:

From the above equation and we have to find K

The problem as we can see is bilinear in

- The bilinear in X and K is a common paradigm
- Bilinear optimization is not Convex. To Convexify the problem, we use a change of variables.

Problem 2:

where and we find

The Problem 1 is equivalent to Problem 2

**Conclusion**

[edit | edit source]If the (A,B) are controllable, We can obtain a controller matrix that stabilizes the system.

**Implementation**

[edit | edit source]A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

** Related LMIs **

[edit | edit source]Hurwitz Stability

** External Links**

[edit | edit source]- [4] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- [5] -Mathworks reference to DC Gain

**Return to Main Page:**

[edit | edit source]# Mixed H2 Hinf with desired pole location control

[edit | edit source]**LMI for Mixed with desired pole location Controller**

The mixed output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the controller, the channel is used to improve the robustness of the design while the channel guarantees good performance of the system and additional constraint is used to place poles at desired location.

**The System**

[edit | edit source]We consider the following state-space representation for a linear system: