text:
LMIs in Control/Click here to continue/Applications of Linear systems/Problem of Space Rendezvous and LMI Approaches

This is a Problem of Space Rendezvous and LMI Approaches

In Section 12.4 of their book LMIs in Control Systems: Analysis, Design, and Applications, Duan and Yu discuss the problem of space rendezvous and how it can be formulated into an LMI problem. Modeling and simulating space rendezvous is of importance because it is used for any cargo or passenger spacecraft traveling to and from earth-orbiting space stations and also for satellites servicing aging in-orbit satellites, and for potential missions to mine asteroids.

The System
Though Duan and Yu first mention space rendezvous in Example 7.14 of their book. In this example, they show that the relative orbital dynamic model of spacecraft rendezvous can be described by the famous Clohessy-Wiltshire equations.

\begin{align} m\ddot{r_x}-2m\omega_0\dot{r_y}-3m\omega_0^2r_x &= T_x + d_x \\ m\ddot{r_y}+2m\omega_0\dot{r_x} &= T_y + d_y \\ m\ddot{r_z}+m\omega_0^2r_z &= T_z + d_z\\ \end{align}$$

where
 * $$r_x,r_y,r_z$$ are the components of the relative position between chaser and target
 * $$\omega_0 = \pi/12$$ [rad/h] is the orbital angular velocity of the target satellite
 * $$m$$ is the mass of the chaser
 * $$T_i, (i=x,y,z)$$ is the i-th component of the control input force acting on the relative motion dynamics
 * $$d_i, (i=x,y,z)$$ is the i-th component of the external disturbance

The C-W equations give a first-order approximation of the chaser's motion in a target-centered coordinate system and is often used in planning space rendezvous problems (ISS, Salyut, and Tiangong space stations are just some examples.)

With appropriate definitions of states and variables the dynamic equations of motion for space-rendezvous can be converted into standard state-space form for LMI optimization as follows:

\begin{align} \dot{x} &= Ax + B_1u+B_2d \\ y &= Cx \\ \end{align}$$

where the vectors in the above state-space representation are defined as follows:

\begin{align} x = \begin{bmatrix} r_x & r_y & r_z & \dot{r_x} & \dot{r_y} & \dot{r_z} \\ \end{bmatrix}^T\\ \\ y = \begin{bmatrix} r_z & r_y & r_z\end{bmatrix}^T\\ \\ u = \begin{bmatrix} T_x & T_y & T_z \end{bmatrix}^T\\ \\ d = \begin{bmatrix} d_x & d_y & d_z \end{bmatrix}^T\\ \end{align}$$

and the matrices in the above state-space representation are defined as follows:

\begin{align} A = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\               0 & 0 & 0 & 0 & 0 & 1 \\                3\omega_0^2  & 0 & 0 & 0 & 2\omega_0^2 & 0 \\ 0 & 0 & 0 & -2\omega_0 & 0 & 0 \\ 0 & 0 & -\omega_0^2 & 0 & 0 & 0 \end{bmatrix}\\ \\ B_1 = B_2 = \begin{bmatrix} 0_{3x3} \\ I_3 \\ \end{bmatrix}\\ \\ C = \begin{bmatrix} I_{3} & 0_{3x3}\\ \end{bmatrix}\\ \\ \end{align}$$

The Data
The data required are the mass properties of both the target and chaser vehicles for space rendezvous. Also required is the orbital angular velocities of the target and chasers and measurements of relative kinematics between the two.

The Optimization Problem
The optimization problem is trying to attenuate the disturbance to output transfer function using either the H-inf or H2 norm.

 * Objective: Hinf or H2 norm
 * Variables: Controller Gains
 * Constraints: Relative Dynamics/Kinematics between Chaser and Target in Orbit

The LMI: Space Rendezvous LMI Optimization
The space rendezvous problem can be approached with either H-inf or H-2 optimization formulations. Both formulations can achieve closed-loop stability which ensures that rendezvous occurs because the relative distance between target and chaser eventually approaches zero. The LMIs for the H-inf and H2 optimization problem are shown below which are easily solvable because the matrices for the space rendezvous problem are available above in standard form.

Duan and Yu approach the $$H_{\infty}$$. The minimum attenuation level from disturbance to output can be found by solving the following LMI optimization problem.

\begin{align} \\ &min \gamma_{\infty} \\ \\ &\text{s.t. } X > 0 \\ \\ &\begin{bmatrix} (AX+B_1W)^T+AX+B_1W & B_2 & (C_1X+D_1W)^T \\ B_2^T & -\gamma_{H\infty} I & D_2^T \\ C_1X+D_1W & D_2 & -\gamma_{H\infty} I \end{bmatrix} < 0 \\

\end{align}$$

which is the same as Theorem 8.1 in Duan and Yu's Book, the solution to the $$ H_{\infty} $$ problem.

Conclusion:
The LMI for Space Rendezvous is a useful and interesting method to model and simulate practical problems in spacecraft engineering. Space Rendezvous usually requires very good vision-based navigation or an exceptional human operator that can close the gap for final mating of the two docking adapters.

Implementation
A link to CodeOcean or other online implementation of the LMI

Related LMIs
Links to other closely-related LMIs

 * Optimal_Output_Feedback_Hinf_LMI
 * H2OptimalOutputFeedback


text:
LMIs in Control/Click here to continue/Applications of Linear systems/LMI for Attitude Control of BTT Missiles, Roll Channel

LMI for Attitude Control of BTT Missles, Roll Channel

The dynamic model of a bank-to-burn (BTT) missile can be simplified for practical application. The dynamic model for a BTT missile is given by the same model used for nonrotating missiles. However, in this case we can assume that the missile is axis-symmetrically design, and thus Jx = Jy. We assume that the roll channel is independent of the pitch and yaw channels.

The System
The state-space representation for the pitch channel can be written as follows:

$$ \begin{align} \dot{x}(t) &= A(t)x(t) + B_{1}(t)u(t) + B_{2}(t)d(t) \\ y(t) &= C(t)x(t) + D_{1}(t)u(t) + D_{2}(t)d(t) \end{align} $$

where $$ x(t) = [\omega_{x} \quad \phi]^{\text{T}}$$ is the state variable, $$ u(t) = \delta_{x} $$ is the control input, and $$ y = \phi $$ is the output. The parameters $$ \omega_{x} $$, $$ \phi $$, and $$ \delta_{x} $$ refer to the roll angular velocity, the roll angle, and the aileron deflection, respectively.

The Data
The system can be described as:

$$ \begin{align} \begin{bmatrix} \dot{\omega}_x(t) \\ \dot{\phi}(t) \end{bmatrix} = \begin{bmatrix} -c_1(t) & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \omega_x(t) \\ \phi(t) \end{bmatrix} + \begin{bmatrix} -c_3(t) \\ 0 \end{bmatrix} \delta_x(t) \end{align} $$

$$ \begin{align} y(t) = \phi(t) \end{align} $$

which can be represented in state space form as:

$$ \begin{align} A(t) = \begin{bmatrix} -c_{1}(t) & 0 \\ 1 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} B_{1}(t) = \begin{bmatrix} -c_{3}(t) \\ 0 \end{bmatrix}, \quad B_{2}(t) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = \begin{bmatrix} 0 & 1 \end{bmatrix} \end{align} $$

$$ \begin{align} D_{1}(t) = 0, \quad D_{2}(t) = 0 \end{align} $$

where $$ c_{1}(t) $$ and $$ c_{3}(t) $$ are the system parameters.

The Optimization Problem
The optimization problem is to find a state feedback control law $$ u=Kx$$ such that:

1. The closed-loop system:

$$ \begin{align} \dot{x} &= (A+B_{1}K)x + B_{2}d \\ z &= (C + D_{1}K)x + D_{2}d \end{align} $$

is stable.

2. The $$ H_{\infty} $$ norm of the transfer function:

$$G_{zd}(s) = (C + D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2} + D_{2}$$

is less than a positive scalar value, $$ \gamma $$. Thus:

$$ ||G_{zd}(s)||_{\infty} < \gamma $$

The LMI: LMI for BTT missile attitude control
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

$$ \begin{align} &\text{min} \quad \gamma \\ &\text{s.t.} \quad X > 0 \\ &\begin{bmatrix} (AX+B_{1}W)^{T} + AX+B_{1}W & B_{2} & (CX+D_{1}W)^{T} \\ B_{2}^{T} & -\gamma I & D_{2}^{T} \\ CX + D_{1}W & D_{2} & -\gamma I \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter $$ \gamma $$ is the disturbance attenuation level. However, it should be noted that this model for the roll channel for a BTT missile is very simple and easy to handle, there is no disturbance to attenuate. This problem is presented here for completeness when used in a full BTT missile model along with the pitch/yaw channels. When the matrices $$ W $$ and $$ X $$ are determined in the optimization problem, the controller gain matrix can be computed by:

$$ K = WX^{-1} $$

Implementation
A link to MATLAB code for the problem in the GitHub repository:

https://github.com/scarris8/LMI-for-BTT-Missile-Roll-Control

Related LMIs
LMI for Attitude Control of Nonrotating Missles, Pitch Channel

LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel

Return to Main Page
LMIs in Control/Tools

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory

LMIs in system and stability Theory

 * 1) Lyapunov Inequalities
 * 2) /Continuous- time Lyapunov Stability//
 * 3) /Discrete-time Lyapunov Stability//
 * 4) /Generalized Lyapunov Stability//
 * 5) Bounded Real Lemma and Hinf Norm
 * 6) /Continuous - time Bounded Real Lemma//
 * 7) /Discrete - time Bounded Real Lemma//
 * 8) H2 Norm
 * 9) /Continuous- time H2 Norm//
 * 10) /Discrete - time H2 Norm//
 * 11) /Generalized H2 Norm//
 * 12) /Peak-to-Peak Norm//
 * 13) Kalman-Yakubovich-Popov(KYP) Lemma
 * 14) /KYP Lemma for QSR Dissipative Systems/
 * 15) /Discrete-Time KYP Lemma for QSR Dissipative Systems/
 * 16) /KYP Lemma Without Feedthrough/
 * 17) /KYP Lemma with Feedthrough/
 * 18) /Discrete-Time KYP Lemma With Feedthrough/
 * 19) /KYP Lemma for Descriptor systems/
 * 20) Conic Sectors
 * 21) /Conic Sector Lemma/
 * 22) /Exterior Conic Sector Lemma/
 * 23) /Modified Exterior Conic Sector Lemma/
 * 24) /Generalized KYP Lemma for Conic Sectors/
 * 25) Minimum Gain
 * 26) /Minimum Gain Lemma/
 * 27) /Modified Minimum Gain Lemma/
 * 28) /Discrete-Time Minimum Gain Lemma/
 * 29) /Discrete-Time Modified Minimum Gain Lemma/
 * 30) Negative Imaginary Systems
 * 31) /Negative Imaginary Lemma/
 * 32) /Discrete-time Negative Imaginary Lemma/
 * 33) /Generalized Negative Imaginary Lemma/
 * 34) /Algebraic Riccati Inequalities/
 * 35) /Discrete-time Algebraic Riccati Inequalities/
 * 36) Stabilizability
 * 37) /Continuous-time Stabilizability/
 * 38) /Discrete-time Stabilizability/
 * 39) Detectability
 * 40) /Continuous-time Detectability/
 * 41) /Discrete-time Detectability/
 * 42) Static Output Feedback Stabilizability
 * 43) /Continuous-time Static Output Feedback Stabilizability/
 * 44) /Discrete-time Static Output Feedback Stabilizability/
 * 45) Strong Stabilizability
 * 46) /Continuous-time strong stabilizability/
 * 47) /Discrete-time strong stabilizability/
 * 48) System Zeros
 * 49) /System Zeros without Feedthrough/
 * 50) /System Zeros with Feedthrough/
 * 51) D-Stability
 * 52) /Conic Sector Region Stability via the Dilation Lemma/
 * 53) /alpha-Region via the Dilation Lemma/
 * 54) /Circular Region Stability via the Dilation Lemma/
 * 55) /DC Gain of a Transfer Matrix/
 * 56) /Kharitonov-Bernstein-Haddad(KBH) Theorem/
 * 57) D-Admissibility
 * 58) /General LMI Region D-Admissibility/
 * 59) /Circular Region D-Admissibility/
 * 60) Quadratic Stability
 * 61) /Continuous-time Quadratic Stability/
 * 62) /Discrete-time Quadratic Stability/
 * 63) Stability of Time Delay Systems
 * 64) /Delay-Independent Condition/
 * 65) /Delay-Dependent Condition/
 * 66) /mu-Analysis/
 * 67) /Static Output Feedback Algebraic Loop/
 * 68) Continuous Time
 * 69) Time-Delay Systems
 * 70) /Delay Dependent Time-Delay Stabilization/
 * 71) /Delay Independent Time-Delay Stabilization/
 * 72) /Parametric, Norm-Bounded Uncertain System Quadratic Stability/
 * 73) /Stability of Structured, Norm-Bounded Uncertainty/
 * 74) /Stability under Arbitrary Switching/
 * 75) /Quadratic Stability Margins/
 * 76) /Stability of Linear Delayed Differential Equations/
 * 77) /H infinity Norm for Affine Parametric Varying Systems/
 * 78) /Entropy Bond for Affine Parametric Varying Systems/
 * 79) /Dissipativity of Affine Parametric Varying Systems/
 * 80) /Hankel Norm of Affine Parameter Varying Systems/
 * 81) /Positive Orthant Stabilizability/
 * 82) /LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control/
 * 83) /LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays/
 * 84) /Polytopic Quadratic Stability/
 * 85) /Interval Quadratic Stability/
 * 86) /Optimization Over Affine Family of Linear Systems/
 * 87) /Hurwitz Stabilizability/
 * 88) /Quadratic Hurwitz Stabilization for Polytopic Systems/
 * 89) /Output Energy Bound for Autonomous LTI Systems/
 * 90) /Transient State Bound for Non-Autonomous LTI Systems/
 * 91) /Transient Impulse Response Bound for Non-Autonomous LTI systems/

 * 1) Discrete-time
 * 2) D-stability
 * 3) /Controller D-Stability/
 * 4) /Observer D-stability/
 * 5) /Discrete-Time Lyapunov Stability/
 * 6) /LMI for Schur Stabilization/
 * 7) /L2-Gain of Systems with Multiplicative noise/
 * 8) /Discrete-Time Quadratic Stability/
 * 9) /Output Energy Bound for Discrete-Time Autonomous LTI Systems/
 * 10) /Transient Bound for Discrete-Time Non-Autonomous LTI Systems/
 * 11) /Discrete-Time Transient Impulse Response Bound/

 * 1) Lure's System
 * 2) /Stability Of Lure's System/
 * 3) /L2 Gain of Lure's Systems/
 * 4) /Output Energy Bound for Lure's Systems/
 * 5) /Stability of Quadratic Constrained Systems/


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous- time Lyapunov Stability

The System:

\begin{align} \dot x(t)&=Ax(t)\\ x(0)&=x_0 \end{align}$$

The Optimization Problem:

The LMI: The Lyapunov Inequality

\begin{align} \text{Find} \; X:&\\ \begin{bmatrix} X\end{bmatrix}&>0\\ \begin{bmatrix} A^T X+XA\end{bmatrix}&<0 \end{align}$$

Conclusion:

If feasible, System is internally exponentially Stable for any $$ x_0 \in \R^n $$.

Related LMIs:


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Lyapunov Stability

Discrete-Time Lyapunov Stability

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.

Stability of DT LTI systems can be determined by solving Lyapunov Inequality.

The System
Discrete-Time System

\begin{align} x(t)_{k+1}&=A_dx(t)_k,&A_d \in \bf{R^{n*n}} \;\\ \end{align}$$

The Data
The matrices: System $$ A_d, P $$.

The Optimization Problem
The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

\begin{align} P \in \bf{S^{n}} \;\\ \text{Find}\; &P>0,\\ \begin{bmatrix} A_d^T P A_d - P\end{bmatrix}&<0 \end{align}$$

Conclusion:
If there exists a $$P \in \bf{S^{n}}$$ satisfying the LMI then, $$|\lambda_i(A_d)|\leq 1, \forall i = 1,2,...,n;$$ and the equilibrium point $$\bar{x}=0$$  of the system is Lyapunov stable.

Implementation
A link to CodeOcean or other online implementation of the LMIMATLAB Code

Related LMIs
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Generalized Lyapunov Stability

WIP, Description in progress

The theorem can be viewed as a true essential generalization of the well-known continuous- and discrete-time Lyapunov theorems.

Kronecker Product
The Kronecker Product of a pair of matrices $$A\in \mathbb{R}^{m\times n} $$ and $$B \in \mathbb{R}^{p \times q} $$ is defined as follows:

$$A\otimes B=\begin{bmatrix} a_{11}B & a_{12}B & \cdots & a_{1n}B \\ a_{21}B & a_{22}B & \cdots & a_{2n}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}B & a_{m2}B & \cdots & a_{mn}B \end{bmatrix}\in \mathbb{R}^{mp\times nq} $$.

Lemma 1: Manipulation Rules of Kronecker Product
Let $$ A, B, C$$ be matrices with appropriate dimensions. Then, the Kronecker product has the following properties:
 * $$1\otimes A = A$$;
 * $$(A+B)\otimes C = A\otimes C + B\otimes C$$
 * $$(A\otimes B)(C\otimes D)=(AC)\otimes(BD)$$
 * $$(A\otimes B)^T = A^T \otimes B^T$$
 * $$(A\otimes B)^{-1} = A^{-1} \otimes B^{-1}$$
 * $$\lambda(A\otimes B)={\lambda_i(A)\lambda_j{B}}$$

Theorem
In terms of Kronecker products, the following theorem gives the $$\mathbb{D}$$-stability condition for the general LMI region case: Let $$\mathbb{D}=\mathbb{D}_{L,M}$$ be an LMI region, whose characteristic function is

$$F_{\mathbb{D}}=L+sM+\overline{s}M^T$$

Then, a matrix $$A\in \mathbb{R}^{n \times n}$$ is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix $$P$$ such that

$$R_{\mathbb{D}}(A, P)=L\otimes P+M\otimes (AP) +M^T\otimes(AP)^T < 0$$,

where $$\otimes$$ represents the Kronecker product.

Lemma 2
Given two LMI regions $$\mathbb{D}_1$$ and $$\mathbb{D}_2$$, a matrix $$A$$ is both $$\mathbb{D}_1$$-stable and $$\mathbb{D}_2$$-stable if there exists a positive definite matrix $$P$$, such that $$R_{\mathbb{D}_1}(A, P)<0$$ and $$R_{\mathbb{D}_2}(A, P)<0$$.

$$$$ WIP, additional references to be added


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/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 $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} &A_d \in \bf{R^{n*n}}, &B_d \in \bf{R^{n*m}}, &C_d \in \bf{R^{p*n}}, &D_d \in \bf{R^{p*m}} \;\\ \end{align}$$

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 < $$\gamma$$

$$ \begin{align} P \in {S^{n}}; \gamma \in {R_{>0}} \;\\ &P>0,\\ \begin{bmatrix}A_d^TPA_d-P & A_d^TPB_d & C_d^T\\ \end{align}$$
 * & B_d^TPB_d-\gamma I & D_d^T\\
 * & * & -\gamma I\end{bmatrix}&<0,

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 LMIMATLAB Code

Related LMIs
- Continuous time KYP_Lemma_(Bounded_Real_Lemma)


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous- time H2 Norm

$${H_2}$$-norm of System

The $${H_2}$$-norm is conceptually identical to the Frobenius (aka Euclidean) norm on a matrix. It can be used to determine whether the system representation can be reduced to its simplest form, thereby allowing its use in performing effective block-diagram algebra.

The System
Suppose we define the state-space system$$G: {L_2}\rightarrow{L_2}\text{ by }y=Gu$$ if:

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$ where $$A\in\mathbb{R}^{mxm}$$, $$B\in\mathbb{R}^{mxn}$$, $$C\in\mathbb{R}^{pxm}$$, and $$D\in\mathbb{R}^{qxn}$$ for any $$t\in\mathbb{R}$$. Then the $${H_2}$$-norm of the system can be determined as described below.

The Data
In order to determine the $${H_2}$$-norm of the system, we need the matrices $${A}$$, $${B}$$, and $${C}$$.

The Optimization Problem
Suppose we wanted to to infer properties of the system behaviour (which is represented in the form (A,B,C,D)). Then it becomes necessary to ensure that the overall system forms an algebra, as the standard use of block-diagram algebra would otherwise be invalid. The only way this is possible is by calculating $${H_2}$$ and/or $${H_\infty}$$-norms - both of which are signal norms that (in a certain sense) measure the size of the transfer function.

The LMI: The $$H_2$$ Norm
Assuming that $$\hat{P}(s)=C(sI-A)^{-1}B$$, this means that the following are equivalent:

 * $$ 1)\quad A\text{ is Hurwitz and }||\hat{P}||^2_{H_2}<\gamma$$

2) \begin{align} \begin{cases} trace(CXC^T )&<\gamma\\   AX+XA^T +BB^T &<0\\    X&>0 \end{cases} \end{align}$$

Conclusion:
The LMI can be used to minimize the $${H_2}$$-norm of the system. It is worth noting that a finite $${H_2}$$-norm does not guarantee finite $${H_\infty}$$-norm, and that in order for the block diagram algebra to be valid, $${H_\infty}$$-norm must be finite.

Implementation

 * Example Code - A GitHub link that contains code (titled "H2Norm.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * $H_2$-Filtering - LMI for $$H_2$$-Filtering
 * Discrete-Time $H_2$ Norm - LMI for $$H_2$$-norm in the Discrete-Time case.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete - time H2 Norm

Discrete-Time H2 Norm

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 LTI systems' H2 norm can be found by solving a LMI.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,0)$$ $$ \begin{align} A_d \in \bf{R^{n*n}}, \ B_d \in \bf{R^{n*m}}, \ C_d \in \bf{R^{p*n}} \end{align}$$

The Data
The matrices: System $$ (A_d,B_d,C_d,0), P, Z $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} P \in {S^{n}}; Z \in {S^{p}}; \mu \in {R_{>0}} \;\\ &P>0, &Z>0\\ \begin{bmatrix}P & A_dP & B_d\\
 * & P & 0\\
 * & * & I\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_dP\\
 * & P\end{bmatrix}&>0,\\

trZ<\mu ^2

\end{align}$$

Conclusion:
The H2 norm is the minimum value of $$ \mu \in {R_{>0}} $$that satisfies the LMI condition.

Related LMIs
- Continuous time H2 norm.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Generalized H2 Norm

Generalized $$H_{2}$$ Norm
The $$ H_2 $$ norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $$ D $$ matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $$ G $$ as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

The System
Consider a continuous-time, linear, time-invariant system $$ G : L_{2e} \to L_{2e} $$ with state space realization $$ (A,B,C,0) $$ where $$ A\in\mathbb{R}^{n\times n} $$, $$B\in\mathbb{R}^{n\times m}$$, $$C\in\mathbb{R}^{p\times n}$$, amd $$ A $$ is Hurwitz. The generalized $$ H_{2} $$ norm of $$ G $$ is:

$$ \left\| G \right\|_{2,{\infty}} = \sup_{u\in L_{2}, {u}\neq\text{zero}} $$ $$ \frac{\left\| Gu \right\|_{\infty}}{\ \left\| u \right\|_{2}}$$

The Data
The transfer function $$G$$, and system matrices $$A$$, $$B$$, $$C$$ are known and $$ A $$ is Hurwitz.

The LMI: Generalized $$H_{2}$$ Norm LMIs
The inequality $$ \left\| G \right\|_{2,{\infty}} < \mu $$ holds under the following conditions:

1. There exists $$ P\in\mathbb{S}^{n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ P > 0 $$ such that:

 * $$\begin{bmatrix} A^{T}P + PA & PB \\ * & -\mu 1 \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} P & C^{T} \\ * & \mu 1\end{bmatrix}>0 $$.

2. There exists $$ Q\in\mathbb{S}^{n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ Q > 0 $$ such that:
 * $$\begin{bmatrix} QA^{T} +AQ & B \\ * & -\mu 1 \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} Q & QC^{T} \\ * & \mu 1 \end{bmatrix}>0 $$.

3. There exists $$ P\in\mathbb{S}^{n}, V\in\mathbb{R}^{n\times n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ P > 0 $$ such that:
 * $$\begin{bmatrix} -(V+V^{T}) & V^{T}A+P & V^{T}B & V^{T} \\ * & -P & 0 & 0 \\ * & * * -\mu 1 & 0 \\ * & * & * & -P \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} P & C^{T} \\ * & \mu 1 \end{bmatrix}>0 $$.

Conclusion:
The generalized $$ H_{2} $$ norm of $$ G $$ is the minimum value of $$ \mu\in\mathbb{R}_{>0} $$ that satisfies the LMIs presented in this page.

Implementation
This implementation requires Yalmip and Sedumi.

Generalized $ H_{2} $ Norm - MATLAB code for Generalized $$ H_{2} $$ Norm.

Related LMIs
LMI for System H_{2} Norm

Return to Main Page:
LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Peak-to-Peak Norm

Peak-to-peak norm performance of a system

The System
Considering the following system:

\begin{align} \dot{x} = A x + B u\\ z     = C x + D u\\ \end{align}$$

Where $$ x(t) \in \mathbb{R}^{n} $$ is the state signal, $$ u(t) \in \mathbb{R}^{m} $$ is the input signal, and $$ z(t) \in \mathbb{R}^{p} $$ is the output. When given an initial condition $$ x(0) = 0 $$, the system can be defined to map the output and input signals for the peak-to-peak performance.

\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The Data
The matrices $$ A $$, $$ B $$, $$ C $$, and $$ D $$ are the only data sets required for this optimization problem.

The Optimization Problem
Consider a continuous-time LTI system, $$ G: L_{2e} \to L_{2e} $$, given that: $$ A $$, $$ B $$, $$ C $$, and $$ D $$ $$ A \in \mathbb{R}^{n \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$. Given that the matrix $$ A $$ is Hurwitz,The peak-to-peak norm of $$ G $$ is given as:

\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||w||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The LMI: Peak-to-Peak norm
There exists a matrix $$ P \in \mathbb{S}^{n} $$ and $$ \gamma $$, $$ \epsilon $$, $$ \mu \in \mathbb{R}_{>0} $$, where the following constraints are used: $$ \min \mu $$

\begin{align} P < 0 \\ \begin{bmatrix} A^T P + P A + \gamma P & P B \\ B' P & \epsilon I\end{bmatrix}&<0\\ \begin{bmatrix} \gamma P & 0 & C^T \\ 0 & (\mu - \epsilon) I & D^T \\ C & D & \mu I\end{bmatrix}&>0\\ \end{align}$$

Since this optimization has $$ \gamma P $$ in the constraints, this does make this optimization bi-linear. attempting to solve this LMI is not feasible unless some type of substitute is implemented to the variables $$ \gamma P $$.

Conclusion:
The results from this LMI will give the peak to peak norm of the system:

\begin{align} \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}} < \mu \end{align}$$

Implementation

 * Peak-to-Peak Norm Example


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/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 $$V(0)=0$$, $$V(x(t))\ge 0 $$ and

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

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 $$ w(u(t),y(t)) = u(t)^Ty(t) $$.

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

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

 * $$\int_0^T (y^T(t)Qy(t) + 2y^TSu(t) + u^T(t)Ru(t))\,dt \ge 0, \forall u \in \mathcal{L}_{2e}, \forall T \ge 0 $$

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}\le 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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time 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 $$V(0)=0$$, $$V(x(t))\ge 0 $$ and

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

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 $$ w(u(t),y(t)) = u(t)^Ty(t) $$.

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

 * $$\sum_{i \mathop =0}^K (\mathcal{y}_{i}^TQ\mathcal{y}_{i} + 2\mathcal{y}_{i}^TS\mathcal{u}_{i} + \mathcal{u}_{i}^TR\mathcal{u}_{i})\,dt \ge 0, \forall u \in \mathcal{l}_{2e}, \forall k \in \mathcal{Z} \ge 0 $$

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^{T}_{d}PA_{d} - P - C^{T}_{d}QC_{d} & A^{T}_{d}PB_{d} - C^{T}_{d}S - C^{T}_{d}QD_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}S - C^{T}_{d}QD_{d})^{T} &  B^{T}_{d}PB_{d} - D^{T}_{d}QD_{d} - (D^{T}_{d}S + S^{T}D_{d}) - R \end{bmatrix}\le 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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/KYP Lemma Without 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{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)\\ \end{align}$$

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{align}

PA + A^{T}P \ge 0 \\ PB = C^{T} \end{align}$$

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

 * $$\begin{align}

AQ + QA^{T} \ge 0 \\ B = QC^{T} \end{align}$$

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{align}

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

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

 * $$\begin{align}

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

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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/KYP Lemma with Feedthrough

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

LMI 1.1
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalent necessary and sufficient conditions.

 * 1) There exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that $$\begin{bmatrix} PA+A^{T}P & PB-C^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$.
 * 2) There exists $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$, such that $$\begin{bmatrix} AQ+QA^{T} & B-QC^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$

This is a special case of the KYP Lemma for QSR dissipative systems with $$Q=0$$, $$S=\frac{1}{2}\cdot1$$, and $$R=0$$.

LMIs 1.2
The system $$\mathcal{G}$$ is strictly positive real (SPR) under either of the following equivalent necessary and sufficient conditions.

 * 1) There exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that $$\begin{bmatrix} PA+A^{T}P & PB-C^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$.
 * 2) There exists $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$, such that $$\begin{bmatrix} AQ+QA^{T} & B-QC^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$

This is a special case of the KYP Lemma for QSR dissipative systems with $$Q=\epsilon1$$, $$S=\frac{1}{2}\cdot1$$, and $$R=0$$, where $$\epsilon\in\mathbb{R}_{>0}$$.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time KYP Lemma With Feedthrough

The Concept
It is assumed in the Lemma that the state-space representation (Ad, Bd, Cd, Dd) is minimal. Then Positive Realness (PR) of the transfer function Cd(SI − Ad)-1Bd + Dd  is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (Ad, Bd, Cd, Dd)=(−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 equivalent necessary and sufficient conditions.

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

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d})^{T} &  B^{T}_{d}PB_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 0.$$

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

 * $$\begin{bmatrix}

A_{d}QA^{T}_{d} - Q &  A_{d}QC^{T}_{d} - B_{d} \\ (A_{d}QC^{T}_{d} - B_{d})^{T} &  C_{d}PC^{T}_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 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^{T}_{d}\\ (PB_{d})^{T} & C_{d} & D^{T}_{d}+ D_{d} \end{bmatrix}\ge 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^{T}_{d}\\ (B_{d})^{T} & (QC^{T}_{d})^{T} & D^{T}_{d}+ D_{d} \end{bmatrix}\ge 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 equivalent necessary and sufficient conditions.

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

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d})^{T} &  B^{T}_{d}PB_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix} < 0.$$

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

 * $$\begin{bmatrix}

A_{d}QA^{T}_{d} - Q &  A_{d}QC^{T}_{d} - B_{d} \\ (A_{d}QC^{T}_{d} - B_{d})^{T} &  C_{d}PC^{T}_{d} - (D^{T}_{d}+ 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^{T}_{d}\\ (PB_{d})^{T} & C_{d} & D^{T}_{d}+ 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^{T}_{d}\\ (B_{d})^{T} & (QC^{T}_{d})^{T} & D^{T}_{d}+ 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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/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{align} E\dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$

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
 * $$ E^{T}X = X^{T}E \ge 0$$
 * $$ E^{T}W = 0 $$
 * $$\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
 * $$ E^{T}X = X^{T}E \ge 0$$
 * $$\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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Conic Sector Lemma

Conic Sector Lemma

For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.

The System
Consider a square, continuous-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 $$A \in \R^{n\times n}, B \in \R^{n\times m}, C \in \R^{m\times n},$$ and $$ D \in \mathcal{R}^{m \times m} $$. The state-space representation is:

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

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^m$$ are the system state, output, and the input vector respectively.

The Data
The system coefficient matrices $$ (A,B,C,D) $$ are required. Optionally, the parameters to define a cone, either in the form of $$ [a,b] $$ where $$ a,b \in \R, a < b $$ or a radius $$ r \in \R_+ $$ and ceter $$ c \in \R $$.

The Feasibility LMI
The system $$ \mathcal{G} $$ is inside the given cone $$[a,b]$$ if the following is feasible:

\begin{align} \text{Find: } & P \\ \text{subj. to: } & P > 0 \\ & \begin{bmatrix} PA+A^\top P+C^\top C & PB - \frac{a+b}{2} C^\top + C^\top D \\ \left(PB - \frac{a+b}{2} C^\top + C^\top D\right)^\top & D^\top D - \frac{a+b}{2}(D+D^\top) + abI \end{bmatrix} \leq 0. \end{align} $$

The above LMI can be used to also determine the cone parameters by setting $$ a $$ as a variable along with the condition $$ a < b $$, and use the bisection method to find $$ b $$.

If the given cone is represented by a center $$ c $$ and radius $$ r $$, then the following feasibility problem can be evaluated to check if $$ \mathcal{G} $$ is inside the given cone:

\begin{align} \text{Find: } & P \\ \text{subj. to: } & P > 0 \\ & \begin{bmatrix} PA+A^\top P+C^\top C & PB - c C^\top + C^\top D \\ (PB - cC^\top + C^\top D)^\top & D^\top D - c(D+D^\top) + (c^2-r^2)I \end{bmatrix} \leq 0. \end{align} $$

In order to also find the cone parameters, substituting $$ \gamma = r^2 $$ as a decision variable with additional constraint $$ \gamma \geq 0 $$ and then solving for $$ c $$ via the bisection method will give the cone in which the system $$ \mathcal{G} $$ resides if the problem is feasible.

Conclusion:
The aforementioned LMIs can be utilized to either check if $$ \mathcal{G} $$ is in the specified cone or not, or can be used to check the stability of $$ \mathcal{G} $$ by finding if a feasible cone can be obtained that encloses $$ \mathcal{G} $$. An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:

$$ Q = -1, S = frac{a+b}{2}I = cI, R = -abI = (r^2 - c^2)I $$.

Implementation
To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m

Related LMIs
Exterior Conic Sector Lemma.

KYP Lemma


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/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 $$\mathcal{D} \in \mathcal{R}^{p\times m} $$.

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

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

LMI : Exterior Conic Sector Lemma
The system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c (i.e. $$\mathcal{G} \in $$exconer(c)), where $$ r \in \mathcal{R}_{>0} $$ and $$ \in \mathcal{R}$$, under either of the following equivalent necessary and sufficient conditions.
 * 1. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P - C^{T}C &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix} \le 0.$$

 * 2. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P - C^{T}C &  PB - C^{T}(D - CI) & 0  \\ (PB - C^{T}(D - CI))^{T} &  - (D - cI)^{T}(D - cI) & rI \\ 0 & (rI)^{T} & -I \end{bmatrix} \le 0.$$

Proof, Applying the Schur complement lemma to the $$ r^{2}I $$ terms in (1) gives (2).

Conclusion:
If there exist a positive definite $$P$$ matrix satisfying above LMIs then the system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c.

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

Related LMIs
KYP Lemma State Space Stability


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Modified 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 $$\mathcal{D} \in \mathcal{R}^{p\times m} $$.

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

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

LMI : Exterior Conic Sector Lemma
The system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c (i.e. $$\mathcal{G} \in $$exconer(c)), where $$ r \in \mathcal{R}_{>0} $$ and $$ \in \mathcal{R}$$, under either of the following equivalent necessary and sufficient conditions.
 * 1. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P - C^{T}C &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix} \le 0.$$

 * 2. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P - C^{T}C &  PB - C^{T}(D - CI) & 0  \\ (PB - C^{T}(D - CI))^{T} &  - (D - cI)^{T}(D - cI) & rI \\ 0 & (rI)^{T} & -I \end{bmatrix} \le 0.$$

Proof, Applying the Schur complement lemma to the $$ r^{2}I $$ terms in (1) gives (2).

Conclusion:
If there exist a positive definite $$P$$ matrix satisfying above LMIs then the system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c.

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

Related LMIs
KYP Lemma State Space Stability


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Generalized KYP 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{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$

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 $$ a0}$$, where $$Q \ge 0$$, 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 $$|\omega| \le (\omega_{1} - \overline{\omega}_{1})$$ into the strict inequality $$|\omega| < \omega_{1}$$

 * 2. (Intermediate Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | \omega_{1} \le |\omega| < \omega_{2}, det(j\omega I - A) \neq 0} $$, where $$\omega_{1},\omega_{2} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a0}$$ and $$\hat{\omega}_{2} = (\omega_{1} + \tfrac{(\omega_{2} - \hat{\omega}_{2})}{2}),$$ where $$ P^{H} = P, Q^{H} = Q $$ and $$Q \ge 0$$, 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 $$ \omega_{1} \le |\omega| \le (\omega_{2} - \overline{\omega}_{2})$$ into the strict inequality $$\omega_{1} \le |\omega| < \omega_{2}$$.

 * 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 $$Q \ge 0$$, 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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Minimum Gain Lemma

Minimum Gain Lemma

The System
Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.

With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $$ G $$ corresponds to the following matrices $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The Data
A system of the plant $$ G $$ will be required with the following matrices: $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The LMI: Minimum Gain Lemma
The following two LMI's are equivalent, with the same variable:

Suppose there exists $$ P \in \mathbb{S}^n $$, and $$ \upsilon \in \mathbb{R}_{\geq 0} $$, Where $$ P \geq 0 $$ and:

\begin{align}

\begin{bmatrix} P A + A^T P - C^T C & P B - C^T D \\ (P B - C^T D)^T & \upsilon^2 I - D^T D \end{bmatrix}&\leq 0

\end{align}$$

Or (Obtained using Schur compliment):

\begin{align}

\begin{bmatrix} P A + A^T P - C^T C & P B - C^T D & 0 \\ (P B - C^T D)^T & -D^T D & \upsilon I \\ 0 & \upsilon I & -I \end{bmatrix}&\leq 0

\end{align}$$

Conclusion:
Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $$ \upsilon $$ obtained from the optimization.

Implementation

 * Minimum Gain Lemma Example


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Modified Minimum Gain Lemma

Modified Minimum Gain Lemma

The System
Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.

With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $$ G $$ corresponds to the following matrices $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

Essentially the same concept as the Minimum Gain Lemma; however, this modification ensure that the system being optimized is also Lyapunov stable. This Essentially means that the variable $$ Q $$ must ensure that the system is stable to this degree, otherwise the system itself is infeasible.

The Data
A system of the plant $$ G $$ will be required with the following matrices: $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The LMI: Modified Minimum Gain Lemma
The following two LMI's are equivalent, with the same variable:

Suppose there exists $$ Q \in \mathbb{S}^n $$, and $$ \upsilon \in \mathbb{R}_{\geq 0} $$, where $$ Q > 0 $$ such that:

\begin{align}

\begin{bmatrix} A Q + Q A^T & B - Q C^T D \\ (B - Q C^T D)^T & \upsilon^2 I - D^T D \end{bmatrix}&\leq 0

\end{align}$$

Or (Obtained using Schur compliment):

\begin{align}

\begin{bmatrix} A Q + Q A^T & B - Q C^T D & 0 \\ (B - Q C^T D)^T & -D^T D & \upsilon I \\ 0 & \upsilon I & -I \end{bmatrix}&\leq 0

\end{align}$$

Conclusion:
Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $$ \upsilon $$ obtained from the optimization. This system will also prove that the plant is Lyapunov stable.

Implementation

 * Minimum Gain Lemma Example


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Minimum Gain Lemma

The Concept
The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

 * $$Y(s) = P(s) U(s)$$
 * $$U(s) = C(s) E(s)$$
 * $$E(s) = R(s) - F(s)Y(s).$$

Solving for Y(s) in terms of R(s) gives

 * $$Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).$$

The expression $$H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}$$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $$|P(s)C(s)| \gg 1$$, i.e., it has a large norm with each value of s, and if $$|F(s)| \approx 1$$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

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 Minimum Gain Lemma
The system $$\mathcal{G}$$ has minimium gain γ under any of the following equivalent sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ and γ $$\in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C_{D}^{T}C_{D} & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 0.$$

 * 2. There exists $$ P \in \mathcal{S}^{n}, $$ and $$ \gamma \in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C_{D}^{T}C_{D} & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} & 0\\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d}- (D^{T}_{d}+ D_{d})  & \gamma I \\ 0 & \gamma I & I \end{bmatrix}\le 0.$$

$$proof$$ : Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.

Conclusion:
If there exist a positive definite $$P$$ for the system $$\mathcal{G}$$, then the minimum gain of the system is γ can be obtaied from above defined LMIs.

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


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Modified Minimum Gain Lemma

The Concept
The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

 * $$Y(s) = P(s) U(s)$$
 * $$U(s) = C(s) E(s)$$
 * $$E(s) = R(s) - F(s)Y(s).$$

Solving for Y(s) in terms of R(s) gives

 * $$Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).$$

The expression $$H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}$$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $$|P(s)C(s)| \gg 1$$, i.e., it has a large norm with each value of s, and if $$|F(s)| \approx 1$$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

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 Modified Minimum Gain Lemma
The system $$\mathcal{G}$$ has minimium gain γ under any of the following equivalent sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ and γ $$\in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}$$

$$Proof$$. The term $$ -C_{d}^{T}C_{d} $$ in Discrete Time Minimum Gain Lemma makes the matrix inequality more negative definite. Therefore,
 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C_{D}^{T}C_{D} & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d})\end{bmatrix} \le \begin{bmatrix} A^{T}_{d}PA_{d} - P & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}$$

 * 2. There exists $$ P \in \mathcal{S}^{n}, $$ and $$ \gamma \in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} & 0\\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d}- (D^{T}_{d}+ D_{d})  & \gamma I \\ 0 & \gamma I & I \end{bmatrix}\le 0.$$

$$proof$$ : Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.

Conclusion:
If there exist a positive definite $$P$$ for the system $$\mathcal{G}$$, then the minimum gain of the system γ can be obtaied from above defined LMIs.

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 Discrete Time Minimum Gain Lemma


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Young’s Relation-Based Properties

Young’s Relation-Based Properties
1. Consider $$X$$,$$Y\in\mathbb{R}^{n\times m}$$ and $$Z\in\mathbb{S}^m$$. The matrix inequality given by

 * $$\begin{align}

Z + XTY + YTX>0, \end{align}$$

 * is satisfied if and only if there exist $$Q\in \mathbb{S}^m$$, $$P\in\mathbb{S}^n$$,$$ G1\in\mathbb{R}^{n\times n}$$,$$

G2\in\mathbb{R}^{n\times m}$$,$$ F\in\mathbb{R}^{m\times n}$$, and $$H\in \mathbb{R}^{m\times m}$$, where $$Q > 0$$

 * and $$P > 0$$, such that

\begin{bmatrix}P&Y\\*&Q\end{bmatrix}>0 $$ and $$\begin{bmatrix}Z+Q+X^TPX & F-X^TG_2&H-X^TG_1\\*&G_1+G_1^T-P&F^T+G_2-Y\\*&*&H^T+H-Q\end{bmatrix} $$ 2. Consider $$X$$,$$Y\in\mathbb{R}^{n\times n}$$ and $$W\in\mathbb{S}^m$$, where $$X$$ is full rank and $$W>0$$. The matrix inequality given by

 * $$\begin{align}

X^T-W>0, \end{align}$$

 * is satisfied if there exist $$\lambda\in \mathbb{R}_{>0}$$ such that

\begin{bmatrix}\lambda\mathbf{1}&\lambda\mathbf{1}&\mathbf{0}\\*&\mathbf{X}+\mathbf{X}^T&\mathbf{W}^{\frac{1}{2}}\\*&*&\lambda\mathbf{1}\end{bmatrix}>0 $$


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Iterative Convex Overbounding

Iterative convex overbounding is a technique based on Young’s relation that is useful when solving an optimization problem with a BMI constraint.

The System
Consider the matrices $$ Q=Q^{T}\in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, R \in \mathbb{R}^{m\times p}, D \in \mathbb{R}^{p\times q},S \in \mathbb{R}^{q\times r} and C \in \mathbb{R}^{r\times n}$$, where S and R are design variables in the BMI given by

\begin{align} \qquad Q + BRDSC + C^{T}S^{T}D^{T}R^{T}B^{T} < 0      \qquad    (1)\\ \end{align}                              $$

The LMI:Iterative Convex Overbounding
Suppose that S0 and R0 are known to satisfy (1). The BMI of (1) is implied by the LMI
 * $$\begin{bmatrix}

Q+\phi(R,S)+\phi^{\text{T}}(R,S) & B(R-R0)U &C^{\text{T}}(S-S0)^{\text{T}}V^{\text{T}}\\ \end{bmatrix} < 0 \qquad   (2) $$ where Φ(R,S) = B(RDS0+R0DS-R0DS0)C, W>0 is an arbitrary matrix, D=UV, and the matrices U and VT have full column rank. The LMI of (2) is equivalent to the BMI of (1) when R = R0 and S = S0, and is therefore non-conservative for values of R and S and are close to the previously known solutions R0 and S0.
 * &W^{\text{-1}}&0\\
 * &*&-W\\

Alternatively, the BMI of (1) is implied by the LMI
 * $$\begin{bmatrix}

Q+\phi(R,S)+\phi^{\text{T}}(R,S) &Z^{\text{T}}U^{\text{T}}(R-R0)^{\text{T}}B^{\text{T}}+V(S-S0)C\\
 * &Z\\

\end{bmatrix} < 0 \qquad   (3) $$ where Z > 0 is an arbitrary matrix, D = UV, and the matrices U and VT have full column rank. Again, the LMI of (4) is equivalent to the BMI of (2) when R = R0 and S = S0, and is therefore non-conservative for values of R and S and are close to the previously known solutions R0 and S0.

Conclusion:
A benefit of convex overbounding compared to a linearization approach, is that in addition to ensuring conservatism or error is reduced in the neighborhood of R = R0 and S = S0, the LMIs of (2) and (3) imply (1). Iterative convex overbounding is particularly useful when used to solve an optimization problem with BMI constraints.


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Projection Lemma (Matrix Elimination Lemma)

WIP, Description in progress

A condition for eliminating a variable in an LMI using orthogonal complements is presented.

Definition 1: Orthogonal Complements
Let $$A\in \mathbb{R}^{m\times n} $$, Then, $$M_a$$ is called a left orthogonal complement of $$A$$ if it satisfies

$$ M_aA=0, \quad \text{rank}(M_a)=m-\text{rank}(A) $$;

and $$N_a$$ is called a right orthogonal complement of $$A$$ if it satisfies

$$ AN_a=0, \quad \text{rank}(N_a)=n-\text{rank}(A) $$.

Using the definition of orthogonal complements, we have the following projection lemma:

Projection Lemma
Let $$P$$, $$Q$$ and $$H=H^T$$ be given matrices of appropriate dimensions, $$N_p$$ and $$N_q$$ be the right orthogonal complement of $$P$$ and $$Q$$, respectively. Then, there exists $$X$$ such that

$$H+P^TX^TQ+Q^TXP<0$$

if and only if

$$N_p^TP^T=0, \quad N_q^TQ^T=0 $$.

WIP, additional references to be added


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Strict Projection Lemma

Introduction
Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

Strict Projection Lemma
Consider $$\Psi\in \mathbb{S}^n$$, $$ G\in \mathbb{R}^{n\times m}$$, $$\Lambda\in \mathbb{R}^{m\times p}$$, and $$ H\in \mathbb{R}^{n\times p}.$$ There exists $$\Lambda $$ such that:

$$ \begin{align} \ \Psi + G \Lambda H^T + H \Lambda^T G^T<0, \end{align} $$

if and only if,

$$ \begin{align} \ N_G^T \Psi N_G <0 \end{align} $$,

$$ \begin{align} \ N_H^T \Psi N_H <0 \end{align} $$,

where $$\mathcal{R} (N_G) =\mathcal{N} (G^T)$$ and $$\mathcal{R} (N_H) = \mathcal{N} (H^T)$$.


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Non Strict Projection Lemma

Introduction
Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

Non Strict Projection Lemma
Consider $$\Psi\in \mathbb{S}^n$$, $$ G\in \mathbb{R}^{n\times m}$$, $$\Lambda\in \mathbb{R}^{m\times p}$$, and $$ H\in \mathbb{R}^{n\times p}$$. There exists $$\Lambda $$ such that:

$$ \begin{align} \ \Psi + G \Lambda H^T + H \Lambda^T G^T \leq0, \end{align} $$

if and only if

$$ \begin{align} \ N_G^T \Psi N_G \leq0 \end{align} $$,

$$ \begin{align} \ N_H^T \Psi N_H \leq0 \end{align} $$,

and

$$ \begin{align} \ \ker G\cap\ker H\cap \{\xi\in\mathbb{R}^n\mid\xi^T \Psi \xi=0\}\subset\ker\Psi. \end{align} $$


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Reciprocal Projection Lemma

WIP, Description in progress

Another condition for eliminating a variable in an LMI called reciprocal projection lemma is introduced.

Reciprocal Projection Lemma
For a given symmetric matricx $$\Phi\in \mathbb{S}^n$$, there exists a matrix $$S\in \mathbb{R}^{n\times n}$$ satisfying

$$\Phi +S^T+ S < 0$$

if and only if, for an arbitrarily fixed symmetric matrix $$P\in \mathbb{S}^n $$, there exist a matrix $$W\in \mathbb{R}^{n\times n}$$ satisfying

$$ \begin{bmatrix} \Phi + P - (W^T + W) & S^T + W^T \\ S + W & -P \end{bmatrix} < 0 $$.

WIP, additional references to be added


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Ellipsoidal inequality

Introduction
The simplex algorithm was the first algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. The Ellipsoid algorithm is the first polynomial-time algorithm discovered for linear programming. The Ellipsoid algorithm was proposed by the Russian mathematician Shor in 1977 for general convex optimization problems and applied to linear programming by Khachyan in 1979.

The Ellipsoidal inequality constraints are a type of Lyapunov Function. They are important in process identification, parameter estimation, and statistics. Applications include the crystallization process, polymer film extrudes, and paper machines.

The Data
Formulation of the inequality depends upon the coordinates of the ellipsoid $$ (x_i) $$ and the centre of the ellipsoid $$ (x_c) $$.

The Ellipsoidal Inequality
An Ellipsoid is described by:

$$ \begin{align} \ (x-x_c)^{T} P^{-1} (x-x_c) < 1,\\ \ where, P=P^{T} > 0\\ \end{align} $$

It can be expressed as an LMI using the Schur complement lemma with $$ Q(x)=1, R(x)=P, and S(x)= (x-x_c)^{T} : $$ $$ \begin{bmatrix} 1 && (x-x_c)^{T}\\ (x-x_c) && P \end{bmatrix}$$ $$\begin{align} \geq 0\end{align}$$

Conclusion
Using this inequality, as the algorithm advances to the next step; if the value of P is positive; the volume will keep on shrinking until the LMI is fulfilled.

Implementation

 * https://github.com/Margav06/LMI-for-Ellipsoidal-Inequality

Return to Main Page
https://en.wikibooks.org/wiki/LMIs_in_Control


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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:
 * $$Q>0$$ and $$M-RQ^{-1}R^*>0$$ both hold.
 * $$M>0$$ and $$Q-R^*M^{-1}R>0$$ both hold.
 * $$\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

$$


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Strict Schur Complement

Strict Schur Complement
Consider $$A \in \mathbb{S}^n, B \in \mathbb{R}^{n \times m},$$ and $$C \in \mathbb{S}^m.$$ The following statements are equivalent.

a) $$\begin{bmatrix}A &B\\ B^T& C \end{bmatrix}< 0$$

b) $$A-BC^{-1}B^T<0,C<0$$

c) $$C-B^TA^{-1}B<0,A<0$$


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Non-Strict Schur Complement

Nonstrict Schur Complement
Consider $$A \in \mathbb{S}^n, B \in \mathbb{R}^{n \times m},$$ and $$C \in \mathbb{S}^m.$$ The following statements are equivalent.

a) $$\begin{bmatrix}A &B\\ B^T& C \end{bmatrix} \leq 0$$

b) $$A-BC^{+}B^T <0,C \leq 0, B(\mathbf{1}-CC^+)=\mathbf{0},$$ where $$C^+$$ is the Moore-Penrose inverse of $$C.$$

c) $$C-B^TA^{+}B<0,A \leq 0,B^T(\mathbf{1}-AA^+)=\mathbf{0}$$ where $$A^+$$ is the Moore-Penrose inverse of $$A.$$


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Schur Complement Lemma-Based Properties

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:
 * $$Q>0$$ and $$M-RQ^{-1}R^*>0$$ both hold.
 * $$M>0$$ and $$Q-R^*M^{-1}R>0$$ both hold.
 * $$\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

$$


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Eigen Value 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 $$x$$:

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

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

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

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

$$\begin{align} J(x) = \lambda_{\text{max}}(A(x)) \end{align}$$

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

$$\begin{align} \lambda_{max}(A(x)) \leq t \iff A(x)-tI \leq 0 \end{align}$$

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:

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Eigenvalue Problem

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

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

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


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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 $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ 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:

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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 $$x$$:

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

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

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

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

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

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:

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

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:

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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 $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ 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:

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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:

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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:

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

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

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 $$ \begin{align} A_{0}, A_{1}, ..., A_{n} \quad \end{align}$$ and $$ \begin{align} B_{0}, B_{1}, ..., B_{n} \quad \end{align}$$ are given.

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

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

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:

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Structured Singular Value

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

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

The LMI:

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

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


text:
LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Minimizing Condition Number of Positive Definite Matrix

The System:
A related problem is minimizing the condition number of a positive-definite 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.