LMIs in Control/Print version

Basic Matrix Theory

Basic Matrix Notation

Consider the complex matrix $A\in \mathbb {C} ^{n\times m}$.

$A={\begin{bmatrix}a_{11}&\dots &a_{1m}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{n\times m}$

Transpose of a Matrix

The transpose of $A$, denoted as $A^{T}$ or $A'$ is:

$A^{T}={\begin{bmatrix}a_{11}&\dots &a_{n1}\\\vdots &\ddots &\vdots \\a_{1m}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $A$, denoted as $A^{*}$ is:

$A^{*}={\begin{bmatrix}a_{11}^{*}&\dots &a_{n1}^{*}\\\vdots &\ddots &\vdots \\a_{1m}^{*}&\dots &a_{nm}^{*}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$

Where $a_{nm}^{*}$ is the complex conjugate of matrix element $a_{nm}$.

Notice that for a real matrix $A\in \mathbb {R} ^{n\times m}$, $A^{*}=A^{T}$.

Important Properties of Matricies

Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $A\in \mathbb {C} ^{n\times n}$ is called Hermitian or self-adjoint if $A=A^{*}$.

If $A\in \mathbb {R} ^{n\times n}$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $A\in \mathbb {C} ^{n\times n}$ is called unitary if $A^{*}=A^{-1}$ or $A^{*}A=I$.

External Links

• 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.

Notion of Matrix Positivity

Notation of Positivity

A symmetric matrix $A\in \mathbb {R} ^{n\times n}$ is defined to be:

positive semidefinite, $(A\geq 0)$, if $x^{T}Ax\geq 0$ for all $x\in \mathbb {R} ^{n},x\neq \mathbf {0}$.

positive definite, $(A>0)$, if $x^{T}Ax>0$ for all $x\in \mathbb {R} ^{n},x\neq \mathbf {0}$.

negative semidefinite, $(-A\geq 0)$.

negative definite, $(-A>0)$.

indefinite if $A$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

• For any matrix $M$, $M^{T}M>0$.
• Positive definite matricies are invertible and the inverse is also positive definite.
• A positive definite matrix $A>0$ has a square root, $A^{1/2}>0$, such that $A^{1/2}A^{1/2}=A$.
• For a positive definite matrix $A>0$ and invertible $M$, $M^{T}AM>0$.
• If $A>0$ and $M>0$, then $A+M>0$.
• If $A>0$ then $\mu A>0$ for any scalar $\mu >0$.

External Links

• 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.

KYP Lemma (Bounded Real Lemma)

KYP Lemma (Bounded Real Lemma)

The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the $H_{\infty }$ norm of a system and is also useful for proving many LMI results.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

The Data

The matrices $A,B,C,D$ are known.

The Optimization Problem

The following optimization problem must be solved.

${\begin{aligned}&{\underset {\gamma ,\;X}{\operatorname {minimize} }}\quad \gamma \\&\operatorname {subject\;to} &X>0\\&&{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}$

The LMI: The KYP or Bounded Real Lemma

Suppose ${\hat {G}}(s)(A,B,C,D)$ is the system. Then the following are equivalent.

$1)\quad \left\|G\right\|_{H_{\infty }}\leq \gamma$

$2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$

${\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0$

Conclusion:

The KYP Lemma can be used to find the bound $\gamma$ on the $H_{\infty }$ norm of a system. Note from the (1,1) block of the LMI we know that $A$ is Hurwitz.

Implementation

Since the KYP lemma shown above is nonlinear in gamma, in order to implement it in MATLAB we must first linearize it by using the Schur Complement to arrive at the dual formulation below:

${\begin{bmatrix}A^{T}X+XA&XB&C^{T}\\B^{T}X&-\gamma I&D^{T}\\C&D&-\gamma I\end{bmatrix}}<0$.

This dual KYP LMI is now linear in both $X$ and $\gamma$.

This implementation requires the use of Yalmip and Sedumi.

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

Related LMIs

Positive Real Lemma

External Links

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.

Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

The Data

The matrices $A,B,C,D$ are known.

The LMI: The Positive Real Lemma

Suppose ${\hat {G}}(s)(A,B,C,D)$ is the system. Then the following are equivalent.

$1)\quad G\;{\text{is passive, i.e.}}\;\left\langle u,Gu\right\rangle _{L_{2}}\geq 0\;({\hat {G}}(s)+{\hat {G}}(s)^{*}\geq 0)$

$2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$

${\begin{bmatrix}A^{T}X+XA&XB-C^{T}\\B^{T}X-C&-D^{T}-D\end{bmatrix}}\leq 0$

Conclusion:

The Positive Real Lemma can be used to determine if the system $G$ is passive. Note from the (1,1) block of the LMI we know that $A$ is Hurwitz.

Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m

Related LMIs

KYP Lemma (Bounded Real Lemma)

External Links

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.

KYP Lemma for QSR Dissipative Systems

The Concept

In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state $x(t)$, its input $u(t)$ and its output $y(t)$, the input-output correlation is given a supply rate $w(u(t),y(t))$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $V(x(t))$ such that $V(0)=0$, $V(x(t))\geq 0$ and

${\dot {V}}(x(t))\leq w(u(t),y(t))$

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate $w(u(t),y(t))=u(t)^{T}y(t)$.

The physical interpretation is that $V(x)$ is the energy stored in the system, whereas $w(u(t),y(t))$ is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

The System

Consider a contiuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space realization (A, B, C, D), where ${\mathcal {A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

The Data

The matrices $A,B,C$ and $D$ which defines the state space of the system

The Optimization Problem

The system ${\mathcal {G}}$ is QSR disipative if

$\int _{0}^{T}(y^{T}(t)Qy(t)+2y^{T}Su(t)+u^{T}(t)Ru(t))\,dt\geq 0,\forall u\in {\mathcal {L}}_{2e},\forall T\geq 0$

where $u(t)$ is the input to ${\mathcal {G}},y(t)$ is the output of ${\mathcal {G}},Q\in {\mathcal {S}}^{p},S\in {\mathcal {R}}^{p\times m},$ and ${\mathcal {R}}\in {\mathcal {S}}^{m}$.

LMI : KYP Lemma for QSR Dissipative Systems

The system ${\mathcal {G}}$ is also QSR dissipative if and only if there exists $P\in {\mathcal {S}}^{n},$ where $P>0,$ such that

${\begin{bmatrix}PA+A^{T}P-C^{T}QC&PB-C^{T}S-C^{T}QD\\(PB-C^{T}S-C^{T}QD)^{T}&-D^{T}QD-(D^{T}S+S^{T}D)-R\end{bmatrix}}\leq 0.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is QSR dissipative.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2

KYP Lemma witout Feedthrough

The Concept

It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)^-1B + D is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System

Consider a contiuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, 0), where ${\mathcal {A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},$ and ${\mathcal {C}}\in {\mathcal {R}}^{m\times n},$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)\\\end{aligned}}$

The Data

The matrices The matrices $A,B$ and $C$

LMI : KYP Lemma without Feedthrough

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $p>0$ such that

${\begin{aligned}PA+A^{T}P\geq 0\\PB=C^{T}\end{aligned}}$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{aligned}AQ+QA^{T}\geq 0\\B=QC^{T}\end{aligned}}$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $p>0$ such that

${\begin{aligned}PA+A^{T}P<0\\PB=C^{T}\end{aligned}}$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{aligned}AQ+QA^{T}<0\\B=QC^{T}\end{aligned}}$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $\in {\mathcal {R}}_{>0}.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is Positive Real.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London

KYP Lemma for Descriptor Systems

The Concept

Descriptor system descriptions frequently appear when solving computational problems in the analysis and design of standard linear systems. The numerically reliable solution of many standard control problems like the solu­tion of Riccati equations, computation of system zeros, design of fault detection and isolation filters (FDI), etc. relies on using descriptor system techniques.

Many algorithm for standard systems as for example stabilization techniques, factorization methods, minimal realization, model reduction, etc. have been extended to the more general descriptor system descriptions. An important application of these algorithms is the numeri­cally reliable computation with rational and polynomial matrices via equivalent descriptor representations. Recall that each rational matrix R(s) can be seen as the transfer-function matrix of a continuous- or discrete-time descriptor system. Thus, each R(s) can be equivalently realized by a descriptor system quadruple (A-sE, B, C, D) satisfying R(S)= C(SE-A)^-1B+D

Many operations on standard matrices (e.g., finding the rank, determinant, inverse or generalized inverses), or the solution of linear matrix equa­tions can be performed for rational matrices as well using descriptor system techniques. Other important applications of descriptor techniques are the computation of inner-outer and spectral factorisations, or minimum degree and normal­ized coprime factorisations of polynomial and rational matrices. More explanation can be found in the website of Institute of System Dynamics and control

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (E, A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}E{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

The Data

The matrices The matrices $E,A,B,C$ and $D$

LMI : KYP Lemma for Descriptor Systems

The system ${\mathcal {G}}$ is extended strictly positive real (ESPR) if and only if there exists $X\in {\mathcal {R}}^{n\times n}$ and $W\in {\mathcal {R}}^{n\times m}$ such that

$E^{T}X=X^{T}E\geq 0$

$E^{T}W=0$

${\begin{bmatrix}X^{T}A+A^{T}X&A^{T}W+X^{T}B-C^{T}\\(A^{T}W+X^{T}B-C^{T})^{T}&W^{T}B+B^{T}W-(D^{T}+D)\end{bmatrix}}<0.$

The system is also ESPR if there exists ${\mathcal {X}}\in {\mathcal {R}}^{n\times n}$ such that

$E^{T}X=X^{T}E\geq 0$

${\begin{bmatrix}X^{T}A+A^{T}X&X^{T}B-C^{T}\\(X^{T}B-C^{T})^{T}&-(D^{T}+D)\end{bmatrix}}<0.$

Conclusion:

If there exist a X and W matrix satisfying above LMIs then the system ${\mathcal {G}}$ is Extended Strictly Positive Real.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
5. Numerical algorithms and software tools for analysis and modelling of descriptor systems. Prepr. of 2nd IFAC Workshop on System Structure and Control, Prague, Czechoslovakia, pp. 392-395, 1992.

Generealized KYP (GKYP) Lemma for conic Sectors

The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

Also consider $\pi _{c}(a,b)\in {\mathcal {S}}^{m}$, which is defined as

$\pi _{c}(a,b)={\begin{bmatrix}-{\tfrac {1}{b}}I&{\tfrac {1}{2}}(1+{\tfrac {a}{b}})I\\({\tfrac {1}{2}}(1+{\tfrac {a}{b}})I)^{T}&-aI\end{bmatrix}}$,

where $a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$.

The Data

The matrices The matrices $A,B,C$ and $D$. The values of a and b

LMI : Generalized KYP (GKYP) Lemma for Conic Sectors

The following generalized KYP Lemmas give conditions for ${\mathcal {G}}$ to be inside the cone $[a,b]$ within finite frequency bandwidths.

1. (Low Frequency Range) The system ${\mathcal {G}}$ is inside the cone $[a,b]$ for all $\omega \in {\omega \in {\mathcal {R}}||\omega |<\omega _{1},det(j\omega I-A)\neq 0}$, where $\omega _{1}\in {\mathcal {R}}_{>0},a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$, if there exist $P,Q\in {\mathcal {S}}^{n}$ and ${\overline {\omega }}_{1}\in {\mathcal {R}}_{>0}$, where $Q\geq 0$, such that

${\begin{bmatrix}A&B\\I&0\end{bmatrix}}^{T}{\begin{bmatrix}-Q&P\\P^{T}&(\omega _{1}-{\overline {\omega }}_{1})^{2}Q\end{bmatrix}}{\begin{bmatrix}A&B\\I&0\end{bmatrix}}-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{T}\pi _{c}(a,b){\begin{bmatrix}C&D\\0&I\end{bmatrix}}<0$.

If $\omega _{1}\rightarrow \infty ,P>0.$ and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter ${\overline {\omega }}_{1}$ is incuded in the above LMI to effectively transform $|\omega |\leq (\omega _{1}-{\overline {\omega }}_{1})$ into the strict inequality $|\omega |<\omega _{1}$

2. (Intermediate Frequency Range) The system ${\mathcal {G}}$ is inside the cone $[a,b]$ for all $\omega \in {\omega \in {\mathcal {R}}|\omega _{1}\leq |\omega |<\omega _{2},det(j\omega I-A)\neq 0}$, where $\omega _{1},\omega _{2}\in {\mathcal {R}}_{>0},a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$, if there exist $P,Q\in {\mathcal {C}}^{n}$ and ${\overline {\omega }}_{2}\in {\mathcal {R}}_{>0}$ and ${\hat {\omega }}_{2}=(\omega _{1}+{\tfrac {(\omega _{2}-{\hat {\omega }}_{2})}{2}}),$ where $P^{H}=P,Q^{H}=Q$ and $Q\geq 0$, such that

${\begin{bmatrix}A&B\\I&0\end{bmatrix}}^{T}{\begin{bmatrix}-Q&P+{\mathcal {j{\hat {\omega }}_{2}}}Q\\P-{\mathcal {j{\hat {\omega }}_{2}}}Q&\omega _{1}(\omega _{2}-{\hat {\omega }}-2)Q\end{bmatrix}}{\begin{bmatrix}A&B\\I&0\end{bmatrix}}-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{T}\pi _{c}(a,b){\begin{bmatrix}C&D\\0&I\end{bmatrix}}<0$.

The parameter ${\overline {\omega }}_{2}$ is incuded in the above LMI to effectively transform $\omega _{1}\leq |\omega |\leq (\omega _{2}-{\overline {\omega }}_{2})$ into the strict inequality $\omega _{1}\leq |\omega |<\omega _{2}$.

3. (High Frequency Range) The system ${\mathcal {G}}$ is inside the cone $[a,b]$ for all $\omega \in {\omega \in {\mathcal {R}}|\omega _{2}<|\omega |,det(j\omega I-A)\neq 0}$, where $\omega _{2}\in {\mathcal {R}}_{>0},a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$, if there exist $P,Q\in {\mathcal {S}}^{n}$, where $Q\geq 0$, such that

${\begin{bmatrix}A&B\\I&0\end{bmatrix}}^{T}{\begin{bmatrix}-Q&P\\P^{T}&\omega _{2}^{2}Q\end{bmatrix}}{\begin{bmatrix}A&B\\I&0\end{bmatrix}}-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{T}\pi _{c}(a,b){\begin{bmatrix}C&D\\0&I\end{bmatrix}}<0$.

If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict.

Conclusion:

If there exist a positive definite $q$ matrix satisfying above LMIs for the given frequency bandwidths then the system ${\mathcal {G}}$ is inside the cone [a,b].

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Exterior Conic Sector Lemma
Modified Exterior Conic Sector Lemma

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.

Discrete time Bounded Real Lemma

Discrete-Time Bounded Real Lemma

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Discrete-Time Bounded Real Lemma or the H∞ norm can be found by solving a LMI.

The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$
${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P$.

The Optimization Problem

The following feasibility problem should be optimized:

$\gamma$ is minimized while obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

H∞ norm < $\gamma$

${\begin{aligned}P\in {S^{n}};\gamma \in {R_{>0}}\;\\&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}&C_{d}^{T}\\*&B_{d}^{T}PB_{d}-\gamma I&D_{d}^{T}\\*&*&-\gamma I\end{bmatrix}}&<0,\end{aligned}}$

Conclusion:

The H∞ norm is the minimum value of $\gamma \in {R_{>0}}$that satisfies the LMI condition. If $(A_{d},B_{d},C_{d},D_{d})$ is the minimal realization then the inequalities can be non-strict.

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIs

[1] - Continuous time KYP_Lemma_(Bounded_Real_Lemma)

External Links

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.

Discrete Time KYP Lemma for QSR Dissipative System

The Concept

In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state $x(t)$, its input $u(t)$ and its output $y(t)$, the input-output correlation is given a supply rate $w(u(t),y(t))$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $V(x(t))$ such that $V(0)=0$, $V(x(t))\geq 0$ and

${\dot {V}}(x(t))\leq w(u(t),y(t))$

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate $w(u(t),y(t))=u(t)^{T}y(t)$.

The physical interpretation is that $V(x)$ is the energy stored in the system, whereas $w(u(t),y(t))$ is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$, with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$, where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$.

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$

$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$ and ${\mathcal {D}}_{d}$

The Optimization Problem

The system ${\mathcal {G}}$ is QSR disipative if

$\sum _{i\mathop {=} 0}^{K}({\mathcal {y}}_{i}^{T}Q{\mathcal {y}}_{i}+2{\mathcal {y}}_{i}^{T}S{\mathcal {u}}_{i}+{\mathcal {u}}_{i}^{T}R{\mathcal {u}}_{i})\,dt\geq 0,\forall u\in {\mathcal {l}}_{2e},\forall k\in {\mathcal {Z}}\geq 0$

where ${\mathcal {u}}_{k}$ is the input to ${\mathcal {G}},{\mathcal {y}}_{k}$ is the output of ${\mathcal {G}},Q\in {\mathcal {S}}^{p},S\in {\mathcal {R}}^{p\times m},$ and ${\mathcal {R}}\in {\mathcal {S}}^{m}$.

LMI : Discrete-Time KYP Lemma for QSR Dissipative Systems

The system ${\mathcal {G}}$ is also QSR dissipative if and only if there exists $P\in {\mathcal {S}}^{n},$ where $P>0,$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-C_{d}^{T}QC_{d}&A_{d}^{T}PB_{d}-C_{d}^{T}S-C_{d}^{T}QD_{d}\\(A_{d}^{T}PB_{d}-C_{d}^{T}S-C_{d}^{T}QD_{d})^{T}&B_{d}^{T}PB_{d}-D_{d}^{T}QD_{d}-(D_{d}^{T}S+S^{T}D_{d})-R\end{bmatrix}}\leq 0.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is QSR dissipative.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
KYP Lemma for continous Time QSR Dissipative system

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2

Discrete Time KYP Lemma with Feedthrough

The Concept

It is assumed in the Lemma that the state-space representation (A_d, B_d, C_d, D_d) is minimal. Then Positive Realness (PR) of the transfer function C_d(SI − A_d)^-1B_d + D_d is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A_d, B_d, C_d, D_d)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$, with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$, where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$.

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$

$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$ and ${\mathcal {D}}_{d}$

LMI : Discrete-Time KYP Lemma with Feedthrough

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $P>0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$

3. There exists $P\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.$

4. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $P>0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.$

3. There exists $P\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.$

4. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $\in {\mathcal {R}}_{>0}.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is Positive Real.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
KYP Lemma without Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London

Schur Complement

An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

The Schur Compliment

Consider the matricies $Q$, $M$, and $R$ where $Q$ and $M$ are self-adjoint. Then the following statements are equivalent:

1. $Q>0$ and $M-RQ^{-1}R^{*}>0$ both hold.
2. $M>0$ and $Q-R^{*}M^{-1}R>0$ both hold.
3. ${\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0$ is satisfied.

More concisely:

${\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0\iff {\begin{bmatrix}M&0\\0&Q-R^{*}M^{-1}R\end{bmatrix}}>0\iff {\begin{bmatrix}M-RQ^{-1}R^{*}&0\\0&Q\end{bmatrix}}>0$

External Links

• 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.

LMI for Eigenvalue Minimization

LMI for Minimizing Eigenvalue of a Matrix

Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.

The System

Assume that we have a matrix function of variables $x$:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$

where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ to minimize the following cost function:

${\begin{aligned}J(x)=\lambda _{\text{max}}(A(x))\end{aligned}}$

where $J(x)$ is the cost function and $\lambda _{\text{max}}(.)$ indicates the maximim eigenvalue of a matrix.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent

${\begin{aligned}\lambda _{max}(A(x))\leq t\iff A(x)-tI\leq 0\end{aligned}}$

where $t$ is defined as the maximim eigenvalue of the matrix $A$.

The LMI: LMI for eigenvalue minimization

This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)-tI\leq 0\end{aligned}}$

Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem.

Moreover, we obtain the maximum eigenvalue, $t$, of the matrix $A(x)$.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix

Related LMIs

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External Links

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

• State-space Representation of a System

• Eigenvalues and Eigenvectors of a Matrix

LMI for Matrix Norm Minimization

LMI for Matrix Norm Minimization

This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an $H_{2}$ or an $H_{\infty }$ optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.

The System

Assume that we have a matrix function of variables $x$:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$

where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ in order to minimize the following cost function:

${\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}$

where $J(x)$ is the cost function and $||.||_{2}$ indicates the norm of the matrix function $A$.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:

${\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

The LMI: LMI for matrix norm minimization

This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

${\begin{aligned}&{\text{min}}\quad t&\\&{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem and we obtain $t$ that is the norm of matrix function $A(x)$.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

Related LMIs

LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External Links

A list of references documenting and validating the LMI.

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

LMI for Generalized Eigenvalue Problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $A$ and $B$, to find the generalized eigenvector, $x$, and eigenvalues, $\lambda$, that satisfies $Ax=\lambda Bx$. If the matrix $B$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

The System

Assume that we have three matrice functions which are functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$ as follows:

$A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}$

$B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}$

$C(x)=C_{0}+C_{1}x_{1}+...+C_{n}x_{n}$

where are $A_{i}$, $B_{i}$, and $C_{i}$ ($i=1,2,...,n$) are the coefficient matrices.

The Data

The $A(x)$, $B(x)$, and $C(x)$ are matrix functions of appropriate dimensions which are all linear in the variable $x$ and $A_{i}$, $B_{i}$, $C_{i}$ are given matrix coefficients.

The Optimization Problem

The problem is to find ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that:

$A(x)<\lambda B(x)$, $B(x)>0$, and $C(x)<0$ are satisfied and $\lambda$ is a scalar variable.

The LMI: LMI for Schur stabilization

A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

${\begin{aligned}&{\text{min}}\quad \lambda \\&{\text{s.t.}}\quad A(x)<\lambda B(x)\\&\quad \quad B(x)>0\\&\quad \quad C(x)<0\end{aligned}}$

Conclusion:

The solution for this LMI problem is the values of variables $x$ such that the scalar parameter, $\lambda$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIs

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

External Links

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

LMI for Linear Programming

LMI for Linear Programming

Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.

The System

We define the objective function as:

$c_{1}x_{1}+c_{2}x_{2}+...c_{n}x_{n}$

and constraints of the problem as:

$a_{11}x_{1}+a_{12}x_{2}+...+a_{1n}x_{n}<b_{1}$

$a_{21}x_{1}+a_{22}x_{2}+...+a_{2n}x_{n}<b_{2}$

.

.

.

$a_{m1}x_{1}+a_{m2}x_{2}+...+a_{mn}x_{n}<b_{m}$

The Data

Suppose that $c_{j}\in \mathbb {R} ^{n}$, $a_{ij}\in \mathbb {R} ^{n}$, and $b_{i}\in \mathbb {R} ^{m}$ are given parameters where $i=1,2,...,m$ and $j=1,2,...,n$. Moreover, $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}$ is an $n\times 1$ vector of positive variables.

The Optimization Problem

The optimization problem is to minimize the objective function, $c^{\text{T}}x$ when the aforementioned linear constraints are satisfied.

The LMI: LMI for linear programming

The mathematical description of the optimization problem can be readily written in the following LMI formulation:

${\begin{aligned}&{\text{min}}\quad c^{\text{T}}x&\\&{\text{s.t.}}\quad a_{i}^{\text{T}}x\leq b_{i}x\\\end{aligned}}$

Conclusion:

Solving this problem results in the values of variables $x$ which minimize the objective function. It is also worthwhile to note that if $m\geq n$, the computational cost for solving this problem would be $mn^{2}$.

There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Linear-Programming

Related LMIs

LMI for Feasibility Problem

External Links

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

LMI for Feasibility Problem

LMI for Feasibility Problem in Optimization

The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.

The System

Assume that we have two matrices as follows:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\quad i=1,2,...,n\end{aligned}}$

${\begin{aligned}B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}\quad i=1,2,...,n\end{aligned}}$

which are matrix functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$.

The Data

Suppose that the matrices ${\begin{aligned}A_{0},A_{1},...,A_{n}\quad \end{aligned}}$ and ${\begin{aligned}B_{0},B_{1},...,B_{n}\quad \end{aligned}}$ are given.

The Optimization Problem

The optimization problem is to find variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that the following constraint is satisfied:

${\begin{aligned}A(x)<B(x)\end{aligned}}$

The LMI: LMI for Feasibility Problem

This optimization problem can be converted to a standard LMI problem by adding a slack variable, $t$.

The mathematical description for this problem is to minimize $t$ in the following form of the LMI formulation:

${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)<B(x)+tI\\\end{aligned}}$

Conclusion:

In this problem, $x$ and $t$ are decision variables of the LMI problem.

As a result, these variables are determined in the optimization problem such that the minimum value of $t$ is found while the inequality constraint is satisfied.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

Related LMIs

LMI for Linear Programming

External Links

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

Structured Singular Value

LMIs in Control/Print version

The LMI can be used to find a $\Theta$ that belongs to the set of scalings $P\Theta$. Minimizing $\gamma$ allows to minimize the squared norm of $\Theta M\Theta ^{-}1$.

The System

${\begin{aligned}M{\text{ with transfer function }}{\hat {M}}(s)=C(sI-A)^{-1}B+D,&&{\hat {M}}\in H_{\infty }\end{aligned}}$

The Data

The matrices $A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n},D\in R^{o\times m}$.

The Optimization Problem

${\begin{aligned}{\text{There exists }}\Theta \in \Theta {\text{ such that }}||\Theta M\Theta ^{-1}||^{2}<\gamma .\end{aligned}}$

The LMI:

${\begin{aligned}{\text{Find}}\;&X>0:\\{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\Theta \end{bmatrix}}+\gamma ^{-2}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}\Theta {\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}$

Conclusion:

${\text{The optimization problem and the LMI are equivalent. }}\gamma {\text{ must be optimized using bisection.}}$

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Eigenvalue Problem

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

Eigenvalue Problem

LMIs in Control/Print version

The maximum eigenvalue of a matrix is going to have the most impact on system performance. This LMI allows for minimization of the maximum eigenvalue by minimizing $\gamma$.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)\end{aligned}}$

The Data

The matrices $A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n}$.

The Optimization Problem

${\text{ Minimize }}\gamma {\text{ subject to the LMI below.}}$

The LMI:

${\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}-A^{T}P-PA-C^{T}C&PB\\B^{T}P&\gamma I\end{bmatrix}}>0\\\end{aligned}}$

Conclusion:

The eigenvalue problem can be utilized to minimize the maximum eigenvalue of a matrix that depends affinely on a variable.

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Structured Singular Value

External Links

• 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.

LMI for Minimizing Condition Number of Positive Definite Matrix

LMIs in Control/Print version

The System:

A related problem is minimizing the condition number of a positive-defnite matrix $M$ that depends affinely on the variable $x$, subject to the LMI constraint $F(x)$ > 0. This problem can be reformulated as the GEVP.

The Optimization Problem:

The GEVP can be formulated as follows:

minimize $\gamma$

subject to $F(x)$ > 0;

$\mu$>0;

$\mu I$< $M(x)$ < $\gamma \mu I$.

We can reformulate this GEVP as an EVP as follows. Suppose,

$M(x)$= $M_{0}$ +$\sum _{n=1}^{m}$$x_{i}M_{i}$ , $F(x)$= $F_{0}$+ $\sum _{n=1}^{m}$$x_{i}F_{i}$

The LMI:

Defining the new variables $\nu$=$1/\mu$ , ${\tilde {x}}$=$x/\mu$ we can express the previous formulation as the EVP with variables ${\tilde {x}},\nu$ and $\gamma$:

miminize$\gamma$

subject to $\nu F_{0}$+ $\sum _{n=1}^{m}$$x_{i}F_{i}$ >0; $I$ < $\nu M_{0}$+ $\sum _{n=1}^{m}$$x_{i}M_{i}$ < $\gamma I$

Conclusion:

The LMI is feasible.

References

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Continuous Quadratic Stability

LMIs in Control/Print version

To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as $t\rightarrow \infty$. A sufficient condition for this is the existence of a quadratic function $V(\xi )=\xi ^{T}P\xi$, $P>0$ that decreases along every nonzero trajectory of system . If there exists such a P, we say the system is quadratically stable and we call $V$ a quadratic Lyapunov function.

The System

${\begin{aligned}{\dot {x}}(t)&=A(\delta (t))x(t)\\\end{aligned}}$

The Data

The system coefficient matrix takes the form of

${\begin{aligned}{\dot {x}}(t)&=A_{0}+\Delta A(\delta (t))x(t)\\\end{aligned}}$

where $A_{0}\in \mathbb {R}$is a known matrix, which represents the nominal system matrix, while ${\begin{aligned}\Delta A(\delta (t))x(t)=\delta _{1}(t))A_{1}+\delta _{2}(t))A_{2}+...+\delta _{k}(t))A_{k}\end{aligned}}$ is the system matrix perturbation, where

$A_{i}\in \mathbb {R} ^{n\times n},i=1,2,..,k,$ are known matrices, which represent the perturbation matrices.

$\delta _{i}(t),i=1,2,...,k,$ which represent the uncertain parameters in the system.

$\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]^{T}$ is the uncertain parameter vector, which is often assumed to be within a certain compact and convex set : : $\Delta$ that is

$\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta ]^{T}\in \Delta$

The LMI: Continuous-Time Quadratic Stability

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+\Delta A(\delta (t))x(t))^{T}+P(A_{0}+\Delta A(\delta (t))x(t))<0\delta (t)\in \Delta \end{aligned}}$

The following statements can be made for particular sets of perturbations.

Case 1: Regular Polyhedron

Consider the case where the set of perturbation parameters is defined by a regular polyhedron as

${\begin{aligned}\Delta ={\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]\in \mathbb {R} ^{k}\mid \delta _{i}(t),{\underline {\delta _{i}}}(t),{\overline {\delta _{i}}}(t),{\underline {\delta _{i}}}\leq \delta _{i}(t)\leq {\overline {\delta _{i})}}}\end{aligned}}$

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+\Delta A(\delta (t))x(t))^{T}+P(A_{0}+\Delta A(\delta (t))x(t))<0\delta _{i}(t)\in {{\underline {\delta _{i}}},{\overline {\delta _{i}}}},i=1,2,...k.\end{aligned}}$

Case 2: Polytope

Consider the case where the set of perturbation parameters is defined by a polytope as

${\begin{aligned}\Delta ={\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]\in \mathbb {R} ^{k}\mid \delta _{i}(t)\in \mathbb {R} _{\geq 0}},\sum _{i=1}^{k}\delta _{i}(t)=1\end{aligned}}$

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+A_{i})^{T}P+P(A_{0}+A_{i})<0,i=1,2...,k.\end{aligned}}$

Conclusion:

If feasible, System is Quadratically stable for any $x\in \mathbb {R} ^{n}$

Implementation

https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities

External Links

• 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.

Exterior Conic Sector Lemma

The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.