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Lentis/Eyjafjallajökull 2010

Eyjafjallajökull is an Icelandic ice cap that covers the caldera of an active stratovolcano located near the villages of Skógar and Mýrdalsjökull. In the spring of 2010, most of Europe faced great consequences arising from several significant eruptions and their subsequent ash cloud formations. These impacts were diverse and broad in scope, affecting not only the physical environment damaged by the eruptions, but also local economies and government policies as they reacted to protect communities and their interests.

Eruption Timeline
The buildup to the eruptions started late in 2009 in the form of increased seismic activity near the volcano. This activity continued to increase in intensity until a small eruption began on 20 March 2010. Rated as a 1 on the volcanic explosivity index, the eruption was essentially a stable stream of lava flow until a second phase of eruptions began on 14 April 2010. This phase saw a much greater increase in volcanic activity that was responsible for the formation of an ash cloud that covered virtually all of Europe for about a week. Twenty European nations closed their airspaces from 15 to 20 April 2010, leading to catastrophic financial losses in the airline and travel industries and millions of tourists being left stranded.

Lava flow and ash production gradually stopped by 21 May 2010, but a new crater with elevated seismic activity appeared near the original eruption site on 6 June 2010, and so the region was continually monitored until October 2010, when the eruption was finally declared to be over.

Environmental Impacts
In terms of volcanic activity, geologists rate the peak of the 2010 eruptions of Eyjafjallajökull as a 4, or cataclysmic eruption, on the volcanic explosivity index. In terms of destructive ability, the eruptions were actually relatively insignificant, with the total ejecta bulk reaching less than a tenth of the total bulk expelled by the 1980 eruption of Mount St. Helens. The impacts of the ash produced by the Eyjafjallajökull eruptions were much greater, as pieces of the 250 million cubic meters of total expelled ash were found over 200 kilometers away from the volcano itself. The eruptions combined would also release around 150,000 tons of CO2, but the continental halting of air travel during the eruption period would result in a net savings of up to 2.8 million tons of CO2 from reduced fuel emissions. The melting of nearby ice caps during the eruption also led to significant flooding in the regions right beside the volcano, with rivers being raised by ten feet and highways being temporarily impassable. Around 800 people were evacuated from these areas.

Preliminary Measures
Following detection of consistently elevated seismic activity in the volcanic region, the regional Chief of Police and Icelandic Civil Protection Office organized emergency meetings shortly prior to the occurrence of the March eruption to inform local communities of a potential eruption in the area. As a result of this preventive measure and quick evacuations following subsequent eruptions, no human casualties were reported across the entire eruption period. A quick response to ban air travel in Europe following the April eruptions also contributed to passenger safety during this period, but at the expense of the airline industry losing over one billion euros over passenger compensation costs.

Farmers
When the initial eruption occurred, many farmers were hesitant to evacuate and abandon their farms. Previous evacuation plans built from the Katla eruptions were focused on human evacuation only and did not include the agricultural departments were not included in the planning or response. Although the Icelandic farming minister released a task force to evaluate the effect of the eruption on livestock and crops, there was no immediate decision made. Instead, the Farmers Association of Iceland, a farming union, had to make frequent posts to their website to provide advice tailored for farmers.

Livestock
The region around the eruption site contained 15% of Iceland's livestock and around 6% of its sheep. As farmers weren't adequately prepared for the eruption, many decided to release their livestock during the evacuation as that was recommended during evacuation exercises. However, following the heavy ash fall from the eruption, the Farmers Association released a memo recommending farmers to shelter their livestock to prevent them from inhaling the ash. This contradicted the advice from the Icelandic government and led to confusion among the farmers.

After the initial emergency response, the Farmers Association of Iceland noted that the ash-fall greatly hurt the sheep, pig, poultry, and milk industries. Despite promises from the government to provide full financial support for the recultivation of farmland, farmers only received a small portion of the cost.

Wheat
The ash fall also impacted wheat crops. Farmers were told to cut their pastures higher than usual to avoid capturing ash in the hay bales, losing 30% of their regular harvest. To avoid shortages of food and fodder in the winter, farmers banded together to establish a 'hay bank' to store excess wheat from farms around the country. However, this loss was also offset after farmers discovered that the ash had a fertilizer effect, leading to a 50% increase in wheat yield and restoring damaged fields.

Farmers were upset about the lack of a proper response plan and pushed the government to incorporate agriculture into future evacuation plans. Officials realized that the plans created from the Katla eruption focused on saving lives would be insufficient for all future eruptions and adopted the advice from the Farmers Association in emergency response plans. The data collected from the aftermath of the eruption was also used to provide farmers with comprehensive information for future eruptions.

Icelandic Government
For updates on the situation, the Icelandic Meteorological Office (IMO) and the Institute of Earth Sciences (IES) published 3-hourly and daily reports detailing plume activity. Launched on the second day of the summit eruption, additional pages containing relevant background and overview information were added to the website, alongside real-time monitoring data from IMO. IES also considered it essential to make this primary scientific data open to the public as soon as possible due to the international extent of Eyjafjallajökull’s impacts. FUTUREVOLC, a conjoined project by the IMO and IES, serves Iceland's goal of providing seismic data to the public. Their website displays real-time data on volcanic activity in Iceland and alerts citizens of any local dangers through SMS texts.

Recognizing the impact the eruption would have on the local tourism industry, the Icelandic government compensated businesses through the Icelandic CAT-fund. The government invested 4.3 million euros in a tourism campaign called Inspired by Iceland, which resulted in a 17% increase in the number of tourists from the previous year.

British Government
Like many other European nations, the United Kingdom closed its airspace on April 15th. However, they also launched Operation Cunningham to assist stranded passengers by using navy vessels. Despite this program, the British response could have been more cohesive as the eruption was not included in its risk assessment report despite being already identified as a level 2 emergency.

Residents
Residents were generally satisfied with the Icelandic government's evacuation response. In a survey after the evacuation, residents liked the real-time updates provided by IMO and IES. However, they mentioned that communication during the evacuation could have been improved.

Health Concerns
During the eruption, residents were disappointed in the amount of information available regarding health concerns from ash exposure. Survey respondents reported that the published impacts of the eruption were "misleading" as they downplayed the effect the ash had as many still had respiratory problems long after the initial eruption. In addition, the stress caused by living with ash fall lead to a greater risk of psychological issues like post-traumatic stress disorder that were not addressed. The Icelandic Department of Health has since committed to providing more information about the dangers of ash.

Media
Local residents were angry that the media was able to access the evacuation zone after everyone was evacuated. Many were frustrated that scientists and media were able to freely travel while they were not. Interviewees were also upset that the media dramatized the eruption by only publishing images of residents in shock or distress and exaggerating the amount of ash. The international media also drew criticism for focusing on the impacts on air travel over the local residents. Another issue was that the media and local government kept comparing the eruption with Katla, stating that Eyjafjallajökull eruptions were “a show and nothing in comparison to what Katla can do”, even though the residents were suffering.

After the evacuation, community interest in emergency management meetings rose and residents began contributing to the creation of emergency response plans. The Icelandic government also aimed to provide better communication regarding ash and established two media centers to better serve the local residents.

Air Travelers
With the airspace in Europe closed, people sought alternatives forms of transportation. Increased demand for trains and ferries by the stranded passengers resulted in increased services to accommodate. The introduction of social media lead to a trending hashtag campaign (#ashtag) with those looking for rides and support. Business conferences were also replaced with video conferencing – it became so popular that some companies doubled the use of online meetings. The switch to online conferencing during the eruption is similar to the situation during the 2020 COVID-19 pandemic and the emergence of Zoom.

Airlines
In 1982, a plane lost engine control in an ash cloud, leading to the creation of a zero-tolerance policy for ash by the airline industry. Airplane engine manufacturers followed the zero tolerance policy until pressure from the monetary losses from the Eyjafjallajökull eruption warranted both the CAA and the manufacturers to create new tolerances for ash. The tolerances were increased and Time Limited Zones, areas where ash is detected but doesn't exceed airplane engine manufacturer's tolerance levels, were created, enabling commercial airlines to fly in low ash density airspace.

Similar Events
From March 24-30 of 2021, the volcano Pacaya erupted spewing ash 2.9-4.5 km into the sky and traveling 15-20 km. This resulted in the International Guatemala Airport closing due to ash covering the runway and the fear of the large quantities of ash damaging the engines.

Starting September 19th 2021, the volcano Cumbre Vieja erupted throwing ash 2.4-4.6 km into the sky and drifting 55 km S. The ash and lava had forced the airport on La Palma to cancel flights, however the Spanish government has vowed to spend over $348 million USD to reconstruct the island for tourism.

Conclusion
Following the events of the 2010 Eyjafjallajökull eruptions, the Icelandic government, local residents, and international committees are working together to better face future eruptions. Practices such as access to real-time data, health information regarding ash, and changes to air space regulation have been adopted by Iceland and the international community. By learning from the ash-fall and lightning storms of Eyjafjallajökull, communities are now better prepared to face future volcanic eruptions.

title: Lentis/Eyjafjallajökull 2010


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User:Margav06/sandbox/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

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


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User:Margav06/sandbox/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


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User:Margav06/sandbox/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


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Negative Imaginary Lemma

Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

The System
Consider a square continuous time Linear Time invariant system, with the state space realization $$(A,B,C,D)$$

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

The Data
$$A\in\mathbb R^{n\times n}, B\in\mathbb R^{n\times m}, C\in\mathbb R^{m\times n}, D\in\mathbb S^m$$

The LMI: LMI for Negative Imaginary Lemma
According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions

 * There exists a $$P$$ ∈ $$\begin{align}\mathbb{S}\end{align}$$n ,where $$P \geq 0$$, such that,

\begin{bmatrix} A^TP + PA & PB-A^TC^T \\ B^TP-CA & -(CB+B^TC^T) \end{bmatrix}$$$$\begin{align} \leq 0\end{align}$$

 * There exists a $$Q$$ ∈ $$\begin{align}\mathbb{S}\end{align}$$n ,where $$Q \geq 0$$, such that,

\begin{bmatrix} QA^T + AQ & B-QA^TC^T \\ B^T- CAQ & -(CB+B^TC^T) \end{bmatrix}$$$$\begin{align} \leq 0\end{align}$$

Conclusion
The system is strictly negative if det($$A$$) $$\neq$$ 0 and either of the above LMIs are feasible with resulting $$P>0$$ or $$Q>0$$

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs
Positive Real Lemma


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Negative Imaginary Lemma

The System
Given a square, discrete-time LTI system G: L2e --> L2e with state-space realization (Ad, Bd, Cd, Dd) where

$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{m x n}$$, and $$D_d \in \mathbb{R}^{m x m}$$.

In this system, $$C_d(z1 - A_d)^{-1} B_d + D_d = B_d^T (z1 - A_d^T)^{-1} C_d^T + D_d^T$$ and $$det(1+A) \neq 0$$  and $$det(1-A) \neq 0$$.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{m x n}$$, and $$D_d \in \mathbb{R}^{m x m}$$

The LMI:
The system G posed above is considered to be negative imaginary under either of the sufficient and necessary conditions:

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

$$A_d^T P A_d - P \leq 0,$$

$$C_d + B_d^T (A_d^T - 1)^{-1} P (A_d +1) = 0$$

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

$$A_d Q A_d^T - Q \leq 0,$$

$$B_d + (A_d -1)^{-1} Q (A_d^T + 1)C_d^T = 0$$

Conclusion
By using the LMI described above, a discrete LTI system can be evaluated for the negative imaginary condition.

Implementation
This LMI can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like MOSEK.

Related LMIs

 * Negative Imaginary Lemma


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Algebraic Riccati Inequalities

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System

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

The Data
Following matrices are needed as Inputs:.

 * $$ A,B,N$$.

The Optimization Problem
In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

The LMI: Algebraic Riccati Inequality
Title and mathematical description of the LMI formulation.

\begin{align} P>0, \\Q>0, \\ R>0.\\ \text{ The Algebraic Riccati inequality is given by}\;\\ A^T P+PA - (PB+N^T)R^-1(B^TP+N)+Q & \geq 0\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^TP+PA+Q && PB+N^T\\ \star && R\\ \end{bmatrix}& \geq 0 \end{align}$

Conclusion:
If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:
Matlab code for this LMI in the Github repository:

 * 1) REDIRECT []- CODE


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Algebraic Riccati Inequalities

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Ricatti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System
The system is defined as:

$$y_t = Ay_{t-1} + Bu_{t-1},$$

The Data
A typical specification of the discrete-time linear quadratic control problem is to minimize

 * $$\sum_{t=1}^T (y_t^T Qy_t + u_t^T Ru_t)$$

subject to the state equation

 * $$y_t = Ay_{t-1} + Bu_{t-1},$$

where y is an n × 1 vector of state variables, u is a k × 1 vector of control variables, A is the n × n state transition matrix, B is the n × k matrix of control multipliers, Q (n × n) is a symmetric positive semi-definite state cost matrix, and R (k × k) is a symmetric positive definite control cost matrix.

The LMI: Discrete-Time Algebraic Riccati Inequality
Title and mathematical description of the LMI formulation.

\begin{align} \text{ Consider}\; A_d \in \mathbb{R}^{n \times m}, B_d \in \mathbb{R}^{n \times m}, P, Q \in \mathbb{S}^{n}, \text {where}\;\\ P>0, \\Q \geq 0, \\ R>0.\\ \text{ The Discrete-Time Algebraic Riccati inequality is given by}\;\\ A^T_d PA_d - A^T_d PB_d(R + B^T_d P B_d)^{-1} B^T_d P A_d + Q - P & \geq 0,\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^T_d - P + Q && A^T_d P B_d\\ \star && R+B^T_d P B_d\\ \end{bmatrix}& \geq 0 \\ \end{align}$$

Equivalently, this discrete-time algebraic Riccati inequality is satisfied under any of the following necessary and sufficient conditions.

1. There exist $$P, Q \in \mathbb{S}^{n}, \text {and} \; R \in \mathbb{S}^m,$$ where $$ P>0, Q \geq 0,and R>0, $$ such that

$$ \begin{bmatrix} Q && 0 && A^T_d P && P\\ \star && R && B^T_d P && 0\\ \star && \star && P && 0 \\ \star && \star && \star && P \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

2. There exist $$X, Q \in \mathbb{S}^{n}, \text {and} \; R \in \mathbb{S}^m,$$ where $$ X>0, Q \geq 0,and R>0, $$ such that

$$ \begin{bmatrix} Q && 0 && A^T_d && 1\\ \star && R && B^T_d P && 0\\ \star && \star && X && 0 \\ \star && \star && \star && X \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

Conclusion:
If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:
Matlab code for this LMI in the Github repository:


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time Stabilizability

Stabilizability LMI

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

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$.

The Data
The matrices necessary for this LMI are $$A$$ and $$B$$. There is no restriction on the stability of A.

The LMI: Stabilizability LMI
$$(A,B)$$ is stabilizable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T+BB^T<<0$$,

where the stabilizing controller is given by
 * $$u(t)=-\frac{1}{2} B^TX^{-1}x(t)$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,B)$$ is stabilizable. In words, a system pair $$(A,B)$$ is stabilizable if for any initial state $$x(0)=x_0$$ an appropriate input $$u(t)$$ can be found so that the state $$x(t)$$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach $$x(t)=0$$ as $$t\rightarrow \infty$$ whereas controllability requires that the state must reach the origin in a finite time.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Stabilizability

Discrete-Time Stabilizability

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 can be made stable using controller gain K, which can be found using LMI optimization, such that the close loop system is stable.

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 $$ (A_d,B_d,C_d,D_d), P, W $$.

The Optimization Problem
The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints. Then K is found.

The LMI:
Discrete-Time Stabilizability

The LMI formulation

$$ \begin{align} P \in {S^{n}}; W \in {R^{m*n}}\;\\ &P>0\\ \begin{bmatrix}P & A_dP+B_dW\\
 * & P\end{bmatrix}&>0,\\

K_d = WP^{-1}

\end{align}$$

Conclusion:
The system is stabilizable iff there exits a $$P$$, such that $$P>0$$. The matrix $$A_d+B_dK_d$$ is Schur with $$K_d = WP^{-1}$$

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

Related LMIs
- Continuous Time Stabilizability


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time Detectability

Detectability LMI
Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair $$(A,C)$$ is shown below.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$.

The Data
The matrices necessary for this LMI are $$A$$ and $$C$$. There is no restriction on the stability of $$A$$.

The LMI: Detectability LMI
$$(A,B)$$ is detectable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T-B^TB<0$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,C)$$ is detectable. In words, a system pair $$(A,C)$$ is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input $$u(t)$$ and output $$y(t)$$.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Stabilizability LMI

Hurwitz Stability LMI

Controllability Grammian LMI

Observability Grammian LMI


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Detectability

Discrete-Time Detectability

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 can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.

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 $$ (A_d,B_d,C_d,D_d), P, W $$.

The Optimization Problem
The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints. Then L is found.

The LMI:
Discrete-Time Detectability

The LMI formulation

$$ \begin{align} P \in {S^{n}}; W \in {R^{m*n}}\;\\ &P>0\\ \begin{bmatrix}P & A_d^TP+C_d^TW\\
 * & P\end{bmatrix}&>0,\\

L = P^{-1}W

\end{align}$$

Conclusion:
The system is detectabe iff there exits a $$P$$, such that $$P>0$$. The matrix $$A_d+LC_d$$ is Schur with $$L = P^{-1}W$$

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

Related LMIs
- Continuous time Detectability


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time Static Output Feedback Stabilizability

In view of applications, static feedback of the full state is not feasible in general: only a few of the state variables (or a linear combination of them, $$y= Cx(t)$$, called the output) can be actually measured and re-injected into the system. '''So, we are led to the notion of static output feedback '''

The System
Consider the continuous-time LTI system, with generalized state-space realization  $$(A,B,C,0)$$

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

\end{align}$$

The Data

 * $$A\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{n\times m}, C\in\mathbb{R}^{p\times n}$$

 * $$x\in\mathbb{R}^n, y\in\mathbb{R}^p, u\in\mathbb{R}^m$$

The Optimization Problem
This system is static output feedback stabilizable (SOFS) if there exists a matrix F such that the closed-loop system $$ \dot x = (A-BKC)x $$ (obtained by replacing $$u = -Ky$$ which means applying static output feedback) is asymptotically stable at the origin

The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability
The system is static output feedback stabilizable if and only if it satisfies any of the following conditions:

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$P\in\mathbb{S}^n$$, where $$P>0$$, such that

\begin{bmatrix} A^TP + PA - PBB^TP & PB+C^TK^T \\ KC+B^TP & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$Q\in\mathbb{S}^n$$, where $$Q>0$$, such that

\begin{bmatrix} QA^T + AQ - QC^TCQ & BK+QC^T \\ CQ^T+K^TB^T & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$Q\in\mathbb{S}^n$$, where $$Q>0$$, such that

\begin{bmatrix} QA^T + AQ - BB^T & B+QC^TK^T \\ B^T+KCQ^T & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$P\in\mathbb{S}^n$$, where $$P>0$$, such that

\begin{bmatrix} A^TP + PA - C^TC & PBK+C^T \\ K^TB^TP & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

Conclusion
On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix $$P$$ (or $$Q$$) and $$K$$

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Discrete time Static Output Feedback Stabilizability Static Feedback Stabilizability


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Static Output Feedback Stabilizability

The static output feedback (SOF) problem has been investigated and analyzed by many people and the literature concerning this topic is vast. In practicality, it is not always possible to have full access to the state vector and only a partial information through a measured output is available. This explains why this problem has challenged many researchers in control theory. Here is a systematic approach for the SOF control design for discrete time linear systems.

The System
Consider a discrete-time LTI system, with state-space realization $$(Ad, Bd,Cd, 0)$$,

 * $$\begin{align}

x_{k+1} = A_dx_k+B_du_k\\ y_k=C_dx_k \end{align}$$

$$x_k\in\mathbb{R}^n$$ is the state, $$y_k\in\mathbb{R}^p$$ is the measured output, $$u_k\in\mathbb{R}^m$$ is the control input.

The Data
$$A_d\in\mathbb{R}^{n\times n}, B_d\in\mathbb{R}^{n\times m}, C_d\in\mathbb{R}^{p\times n}$$ are the constant matrices of appropriate dimensions.

The Optimization Problem
The full state is not measurable and only partial information is available through $$y_k$$ which can used for control purposes.

We have to find a static output feedback gain with respect to
 * $$u_k = -K_dy_k,$$

where $$K_d\in\mathbb{R}^{m\times p}$$ is the output feedback gain such that the final closed loop system is asymptotically stable.

The LMI: LMI for Discrete-Time Static Output Feedback Stabilizability
The discrete time system considered is static output feedback stabilizable under any of the following equivalent necessary or sufficient conditions.

 * There exists a $$K_d\in\mathbb{R}^{m\times p}$$ and $$P \in\mathbb{S}^n$$ where $$P>0$$ such that

\begin{bmatrix} -P & (A_d+B_dK_dC_d)P \\ ((A_d+B_dK_dC_d)P)' & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}.$$

 * There exists a $$K_d\in\mathbb{R}^{m\times p}$$ and $$P \in\mathbb{S}^n$$ where $$P>0$$ such that

\begin{bmatrix} -A_dPPA_d^T & A_dP+B_dK_dC_d & A_dP \\ (A_dP+B_dK_dC_d)' & -1 & 0 \\ PA_d^T & 0 & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}.$$

Conclusion
If it is feasible we obtain a output feedback gain matrix $$K_d$$ such that the closed loop system is asymptotically stable. While implementing the optimization problem the following conditions are assumed to be satisfied
 * The Transfer matrix and its inverse are both analytical a s=0
 * The matrix $$C_dA_d^{-1}B_d$$ is non-singular
 * The triple $$(A_d,B_d,C_d)$$ are reachable and observable.

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Continuous-time Static Output Feedback Stabilizability


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time strong stabilizability

The System
Consider the continous-time LTI system, $$G:L_{2e}\rightarrow L_{2e}$$ with state-space realization (A,B,C,0)

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

where $$A\in \R^{n\times n}$$, $$B\in \R^{n\times m}$$, $$C\in \R^{p\times n}$$, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable, and the transfer matrix $$G(s)=C(s1-A^{-1})B$$ has no poles on the imaginary axis.

The Data
The matrices $$ A,B,C $$.

The Optimization Problem
The system G is strongly stabilizable if there exist $$P \in \mathbb S^n$$, $$Z\in \R^{n\times p}$$, and $$\gamma\in \R_{>0}$$, where $$P > 0$$, such that

\begin{align} PA+A^T+ZC+C^TZ^T<0\\ {\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^TP+ZC+C^TZ^T&&-Z&&-XB\\ \end{align}$$
 * &&-\gamma I && 0\\
 * && * && -\gamma I \end{bmatrix}}<0\end{aligned}}}\\

Conclusion:
where $$F=-B^TX$$ and $$X \in S_n$$, $$X \geq 0$$ is the solution to the Lyapunov equation given by

\begin{align} XA+A^TX-XBB^TX=0 \end{align}$$ Moreover, a controller that strongly stabilizes G is given by the state-space realization

\begin{align} \dot x_c = (A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\ y_C=-B^TXx(t) \end{align}$$

Implementation

 * Example Code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time strong stabilizability

The System
Consider the continous-time LTI system, $$G:L_{2e}\rightarrow L_{2e}$$ with state-space realization ($$A_d,B_d,C_d,0$$)

\begin{align} \dot x(k)&=A_dx(k)+B_du(k),\\ y(k)&=C_dx(k), \end{align}$$

where $$A_d\in \R^{n\times n}$$, $$B_d\in \R^{n\times m}$$, $$C_d\in \R^{p\times n}$$, and it and it is assumed that ($$A_d,B_d $$),is stabilizable, ($$A_d, C_d$$) is detectable, and the transfer matrix $$G(s)=C_d(s1-A_d)^{-1})B_d$$ has no poles on the imaginary axis.

The Data
The matrices $$ A_d,B_d,C_d $$.

The Optimization Problem
The system G is strongly stabilizable if there exist $$P \in \mathbb S^n$$, $$Z\in \R^{n\times p}$$, and $$\gamma\in \R_{>0}$$, where $$P > 0$$, such that

\begin{align}

{\displaystyle {\begin{aligned}{\begin{bmatrix}A^T_dPA_d-P-A_d^TZC_d-C^T_dZ^TA_d && C_d^TZ^T\\ {\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11} &&(A_d +B_dF)^TZ&& XB_d &&C_d^TZ^T\\ \end{align}$$
 * &&-P\\
 * && * && -\gamma I \end{bmatrix}}<0\end{aligned}}}\\
 * && -\gamma I && 0 && Z^T\\
 * && * && -\gamma I && 0\\
 * &&*&&*&&-P\end{bmatrix}}<0\end{aligned}}}\\

Conclusion:
where $$N_{11}=(A_d+B_dF)^TP(A_d+B_dF)-P+(A_d+B_dF)^TZC_d^TZ^T(A_d+B_dF), F=-B_d^TX,X=Y$$ and $$X \in S_n$$, $$X \geq 0$$ is the solution to the discrete-time Lyapunov equation given by

\begin{align} A_dXA^T_d+X-B_dB_d^TX=0 \end{align}$$ Moreover, a controller that strongly stabilizes G is given by the state-space realization

\begin{align} \dot x_{c,k+1} = (A_d+B_dF+P^{-1}ZC_d)x_k-P^{-1}Zu_k\\ y_{c,k}=-B_d^TXx_k \end{align}$$

Implementation

 * -example code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIsinControl/StabilityAnalysis/ContinuousTime/StrongStabilizability - Continuous Time Strong Stabilizability


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/System Zeros without Feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros without feedthrough, we take the assumption that $$ D=0 $$.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,0) $$

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

The Data
The matrices:

\begin{align} A \in \mathbb{R}^{n \times n} \\ M \in \mathbb{R}^{n \times q} \\ N \in \mathbb{R}^{q \times n} \\ \end{align}$$

The LMI: System Zeros without feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B $$ are the eigenvalues of $$ NAM $$, where $$ N \in \mathbb{R}^{q \times n}, M \in \mathbb{R}^{n \times q}, CM = 0, NB = 0, NM = 1. $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} PNAM + M^T A^T N^T P <0 \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.

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


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/System Zeros with Feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take $$ D $$ as full rank.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,D) $$

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

The Data
The matrices needed as inputs are:

\begin{align} A \in \mathbb{R}^{n \times n} \\ B \in \mathbb{R}^{n \times m} \\ N \in \mathbb{R}^{p \times n} \\ \end{align}$$ In this case, $$ m \leq p $$

The LMI: System Zeros with feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B + D $$ are the eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} P(A-B(D^T D)^{-1} D^T C) + (A-B(D^T D)^{-1} D^T C)^T P < 0 \\ \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$ then gives the zeros of the system.

Related LMIs
LMIs_in_Controls/pages/systemzeroswithoutfeedthrough

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


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Conic Sector Region Stability via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

Conic Sector Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$k\in\mathbb{R}_{>0}$$.

The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{P}(k)$$, where $$\mathcal{P}(k):=\{\lambda\in\mathbb{C}:\left\vert Im( \lambda) \right\vert0}$$, where $$X>0$$, such that

$$\begin{bmatrix} k(AX+XA^{T}) & AX-XA^{T} \\ * & k(AX+XA^{T}) \end{bmatrix}<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{P}(k)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$ and $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -kX & X & 0 \\ * & 0 & 0 & -X \\ * & * & 0 & -kX \\ +He\begin{Bmatrix} \begin{bmatrix} A &0 \\ 1 & 0 \\ 0 & 1 \\ 0 & A \end{bmatrix} \begin{bmatrix} F &0 \\ 0 & F \end{bmatrix} \begin{bmatrix} k1 & -\epsilon k1 & \epsilon1 & 1 \\ -1 & -\epsilon1 & \epsilon k1 & k 1 \end{bmatrix}\end{Bmatrix}<0$$.
 * & * & * & 0\end{bmatrix}

Moreover, for every $$X$$ that satisfies

$$\begin{bmatrix} k(AX+XA^{T}) & AX-XA^{T} \\ * & k(AX+XA^{T}) \end{bmatrix}<0$$,

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -kX & X & 0 \\ * & 0 & 0 & -X \\ * & * & 0 & -kX \\ +He\begin{Bmatrix} \begin{bmatrix} A &0 \\ 1 & 0 \\ 0 & 1 \\ 0 & A \end{bmatrix} \begin{bmatrix} F &0 \\ 0 & F \end{bmatrix} \begin{bmatrix} k1 & -\epsilon k1 & \epsilon1 & 1 \\ -1 & -\epsilon1 & \epsilon k1 & k 1 \end{bmatrix}\end{Bmatrix}<0$$
 * & * & * & 0\end{bmatrix}


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/alpha-Region via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

α-Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$\alpha\in\mathbb{R}_{>0}$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$, where $$\mathcal{H}(\alpha):=\{\lambda\in\mathbb{C}:Re(\lambda)<-\alpha\}$$ if and only if there exist $$X\in\mathbb{S}^{n}$$ and $$\epsilon\in\mathbb{R}_{>0}$$, where $$X>0$$, such that

$$AX+XA^{T}+2\alpha X<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$, $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -X & X \\ * & 0 & 0 \\ * & * & -\frac{1}{2}\alpha^{ -1}X \end{bmatrix} +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 \end{bmatrix} \end{Bmatrix}<0$$.

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}+2\alpha X<0$$

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -X & X \\ * & 0 & 0 \\ * & * & -\frac{1}{2}\alpha^{ -1}X \end{bmatrix} +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 \end{bmatrix} \end{Bmatrix}<0$$.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Circular Region Stability via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

Circular Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$, $$r\in\mathbb{R}_{>0}$$, and $$c\in\mathbb{R}_{<0}$$, where $$c<-r$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{G}(c,r)$$, where $$\mathcal{G}(c,r):=\{\lambda\in\mathbb{C}:\left\vert \lambda-c \right\vert0}$$, where $$X>0$$, such that

$$AX+XA^{T}-\frac{c^{2}-r^{2}}{c}X-\frac{1}{c}AXA^{T}<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{G}(c,r)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$, $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -X & X & 0 \\ * & 0 & 0 & -X \\ * & * & \frac{c}{c^{2}-r^{2}}X & 0 \\ +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 & 1 \end{bmatrix} \end{Bmatrix}<0$$
 * & * & * & cX \end{bmatrix}

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}-\frac{c^{2}-r^{2}}{c}X-\frac{1}{c}AXA^{T}<0$$

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -X & X & 0 \\ * & 0 & 0 & -X \\ * & * & \frac{c}{c^{2}-r^{2}}X & 0 \\ +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 & 1 \end{bmatrix} \end{Bmatrix}<0$$
 * & * & * & cX \end{bmatrix}


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/DC Gain of a Transfer Matrix

The continuous-time DC gain is the transfer function value at the frequency s = 0.

The System
Consider a square continuous time Linear Time invariant system, with the state space realization $$(A,B,C,D)$$ and $$\gamma \in\mathbb{R}_{>0}$$

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

The Data
$$A\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{n\times m}, C\in\mathbb{R}^{p\times n}, D\in \mathbb{R}^{p\times m}$$

The LMI: LMI for DC Gain of a Transfer Matrix
The transfer matrix is given by$$G(s)=C(sI-A)^{-1}B+D$$ The DC Gain of the system is strictly less than $$\gamma$$ if the following LMIs are satisfied.

\begin{bmatrix} \gamma I & -CA^{-1}B+D \\ (-CA^{-1}B+D)' & \gamma I \end{bmatrix}$$$$\begin{align} > 0\end{align}$$  OR

\begin{bmatrix} \gamma I & -B^TA^{-T}C^T+DT \\ (-B^TA^{-T}C^T+DT)' & \gamma I \end{bmatrix}$$$$\begin{align} > 0\end{align}$$

Conclusion
The DC Gain of the continuous-time LTI system, whose state space realization is give by ($$A,B,C,D$$), is $$K = D - CA^{-1}B$$
 * Upon implementation we can see that the value of $$\gamma$$ obtained from the LMI approach and the value of $$K$$ obtained from the above formula are the same

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Kharitonov-Bernstein-Haddad(KBH) Theorem

WIP, Description in progress

Kharitonov-Bernstein-Haddad (KBH) Theorem
Consider the set of matrices

$$\mathcal{A}=\{\textbf{A}=\begin{bmatrix}\textbf{0}_{(n-1)\times 1} & & \textbf{1}_{(n-1)\times(n-1)} \\ -a_0 & \cdots & -a_{n-1} \end{bmatrix}| \underline{a_j}\leq a_j \leq \overline{a_j}, j=0,1,2,\cdots,n-1\}$$

Every matrix in the set $$\mathcal{A}$$ is Hurwitz if and only if there exist $$\textbf{P}_i \in mathbb{S}^n, i = 1, 2, 3, 4$$ where $$\textbf{P}_i>0, i = 1, 2, 3 ,4,$$ such that

$$\textbf{P}_iA_i + \textbf{A}_i^T\textbf{P}_i<0, \quad i = 1, 2, 3, 4, $$

where

$$\textbf{A}_i =\begin{bmatrix} \begin{bmatrix}\textbf{0}_{(n-1)\times 1} & \textbf{1}_{(n-1)\times(n-1)} \end{bmatrix} \\ \textbf{a}_i \end{bmatrix}, \quad i = 1, 2, 3, 4,$$

$$ \textbf{a}_1 = -[\underline{a}_0 \quad \underline{a}_1 \quad \overline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \underline{a}_{n-4} \quad \underline{a}_{n-3} \quad \overline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_2 = -[\underline{a}_0 \quad \overline{a}_1 \quad \overline{a}_2 \quad \underline{a}_3 \quad \cdots \quad \underline{a}_{n-4} \quad \overline{a}_{n-3} \quad \overline{a}_{n-2} \quad \underline{a}_{n-1} ], $$

$$ \textbf{a}_3 = -[\overline{a}_0 \quad \underline{a}_1 \quad \underline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \underline{a}_{n-3} \quad \underline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_3 = -[\overline{a}_0 \quad \underline{a}_1 \quad \underline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \underline{a}_{n-3} \quad \underline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_4 = -[\overline{a}_0 \quad \overline{a}_1 \quad \underline{a}_2 \quad \underline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \overline{a}_{n-3} \quad \underline{a}_{n-2} \quad \underline{a}_{n-1} ]. $$

WIP, additional references to be added


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Delay-Independent Condition

Stabilization of Time-Delay Systems - Delay Independent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system independent of the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:

\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_d(t-d)+Bu(t),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} &A\, A_d\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{nxr}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Independent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A, A_d,$$ and $$B$$.

The Optimization Problem
Suppose - for the time-delayed system given above - we were asked to design a memoryless state-feedback control law where $$u=Kx$$ such that the closed-loop system:

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Independent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix$$W\in\mathbb{R}^{r\times n}$$ and 2 symmetric matrices $$X>0$$ and $$Y>0$$ that satisfy the following:

\begin{align} \begin{bmatrix}X{A^T}+AX+BW+{W^T}{B^T}+Y&&{A_d}X\\X{A_d}^T&&-Y\end{bmatrix}&<0\\ \end{align}$$

Conclusion:
Given the resulting feedback gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable while simultaneously ensuring that the solution is delay-independent from the time-delay system where this gain matrix was derived.

Implementation

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

Related LMIs

 * ../Delay Dependent Time-Delay Stabilization/ - Equivalent LMI for delay-dependent time-delay stabilization.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Delay-Dependent Condition

Stabilization of Time-Delay Systems - Delay Dependent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system while being dependent on the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:

\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_d(t-d)+Bu(t),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} &A\text{, }{A_d}\in\mathbb{R}^{n\times n}\text{, }{B}\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Dependent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A\text{, }{A_d}\text{, and }B$$.

The Optimization Problem
Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where $$u(t)=Kx(t)$$ such that the closed-loop system:

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Dependent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a scalar $$0<\beta<1$$, a symmetric matrix $$X>0$$ and a matrix $$W$$ that satisfy the following:

\begin{align} \begin{bmatrix}\Phi(X,W)&&\bar{d}(X{A^T}+{W^T}{B^T})&&\bar{d}X{A^T_d}\\ \bar{d}(AX+BW)&&-\bar{d}{\beta}I&&0\\ \bar{d}{A_d}X&&0&&-\bar{d}(1-\beta)I\end{bmatrix}&<0\\ \end{align}$$

where$$\Phi(X,W)=X{(A+{A_d})^T}+(A+{A_d})X+BW+{W^T}{B^T}+\bar{d}{A_d}{A^T_d}$$

Conclusion:
Given the resulting stabilizing control gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable from the time-delay system where this gain matrix was derived.

Implementation

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

Related LMIs

 * ../Delay Independent Time-Delay Stabilization/ - Equivalent LMI for delay-independent time-delay stabilization.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/mu-Analysis

Mu Synthesis. The technique of $$ \mu $$ synthesis extends the methods of $$ H\infin $$ synthesis to design a robust controller for an uncertain plant. You can perform $$ \mu $$ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. $$ \mu $$ analysis is an extremely powerful multivariable technique which has been applied to many problems in the almost every industry including Aerospace, process industry etc.

The System:
Consider the continuous-time generalized LTI plant with minimal states-space realization

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

where it is assumed that $$ D $$ is Invertible.

The Data
The matrices needed as inputs are only, $$ A $$ and $$ D $$.

The LMI: $$ \mu $$- Analysis
The inequality $$ \overline{\sigma}(DAD^{-1})< \gamma $$ holds if and only if there exist $$ X \in \mathbb{S}^n $$ and $$ \gamma \in \mathbb{R}_{>0}$$, where $$ X>0 $$, satisfying:

\begin{align}

A^TXA - \gamma^2X <0 \end{align}$$

Conclusion:
The inequality $$ \overline{\sigma}(DAD^{-1}) < \gamma $$ holds for $$ D= X^{1/2}, $$ where X satisfies the above Inequality.

Implementation

 * https://github.com/Ricky-10/coding107/blob/master/Mu%20Analysis - $$ \mu $$-Analysis


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User:Margav06/sandbox/Click here to continue/Optimal control systems/H2-optimal Dynamic output Feedback control

Discrete-Time H2-Optimal Dynamic Output Feedback Control

A Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w$$ and performance output $$z$$.

The System
Continuous-Time LTI System with state space realization $$(A,B,C,D)$$

\begin{align} \dot{x}&=A_x + B_{1}w + B_{2}u\\ z&=C_{1}x + D_{11}w_k+ D_{12}u\\ y&=C_{2}x + D_{21}w+ D_{22}u\\ \end{align}$$

The Data
The matrices: System $$ (A,B_{1},B_{2},C_{1},C_{1},D_{11},D_{12},D_{21},D_{22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{c},B_{c},C_{c},D_{c}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Solve for $$A_{n} \in {R^{n_x \times n_x}}, B_{n} \in {R^{n_x \times n_y}}, C_{n} \in {R^{n_u \times n_x}}, D_{n} \in {R^{n_u \times n_y}},X_{1}, Y_{1}\in {S^{n_x}}, Z \in {S^{n_z}},$$ and $$\nu \in {R_{>0}}$$ that minimize $$\mathcal{J}(\nu)<\nu$$ subject to $$X_{1}>0, Y_{1}>0,Z>0,$$

\begin{align}

\begin{bmatrix} AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T &A+A_n^T+B_2D_nC_2 &B_1+B_2D_nD_{21}\\
 * &X_1A+A^TX_1+B_nC_2+C_2^TB_n^T &X_1B_1+B_nD_{21}\\
 * & * &-\mathbf{1}\end{bmatrix}&>0,\\

\begin{bmatrix} X_1 &\mathbf{1} &Y_1C_1^T+C_n^TD_{12}^T\\
 * &Y_1 &C_1^T+C_2^TD_n^TD_{12}^T \\
 * & * &Z \end{bmatrix}&>0,\\

D_{11}+D_{12}D_{n}D_{21}=0\\

\begin{bmatrix}X_{1} & \mathbf{1}\\
 * & Y_{1}\end{bmatrix}&>0,\\

trZ<\nu \end{align}$$

The controller is recovered by

$$ \begin{align} &A_{c} = A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\ &B_{c} = B_{k}(1-D_{c}D_{22})\\ &C_{c} = (1-D_{c}D_{22})C_{k}\\ &D_{c} = (1+D_{k}D_{22})^{-1}D_{k}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{k} & B_{k}\\ C_{k} & D_{k}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{2}\\ \mathbf{0} & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{n} & B_{n}\\ C_{n} & D_{n}\end{bmatrix} -\begin{bmatrix}X_{1}AY_{1} & \mathbf{0}\\ \mathbf{0} & \mathbf{0}\end{bmatrix}) \begin{bmatrix}Y_{2}^T & \mathbf{0}\\ CY_{1} & \mathbf{1}\end{bmatrix}^{-1}

\end{align}$$ and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T = 1 - X_{1}Y_{1}$$. If $$D_{22}=0,$$ then $$A_c=A_k,B_c=B_k,C_c=C_k$$ and $$D_c=D_k.$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{11} = 0$$, $$D_{12} \neq 0,$$ and $$D_{21} \neq 0$$, then it is often simplest to choose $$D_{n} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Continuous-Time H2-Optimal Dynamic Output feedback controller is the system $$ (A_{c},B_{c},C_{c},D_{c}) $$

Related LMIs
Discrete Time H2 Optimal Dynamic Output Feedback Control

Continuous Time H∞ Optimal Dynamic Output Feedback Control


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User:Margav06/sandbox/Click here to continue/Optimal control systems/Hinf-optimal Dynamic output Feedback control

Discrete-Time H∞-Optimal Dynamic Output Feedback Control

In this section, a Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Continuous-Time LTI System with state space realization $$(A,B,C,D)$$

\begin{align} \dot{x}&=Ax + B_1w + B_2u\\ z&=C_1x + D_{11}w+ D_{12}u\\ y &=C_2x + D_{21}w+ D_{22}u\\ \end{align}$$

The Data
The matrices: System $$ (A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12},D_{21},D_{22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{c},B_{c},C_{c},D_{c}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Solve for $$A_{n} \in {R^{n_x*n_x}}, B_{n} \in {R^{n_x*n_y}}, C_{n} \in {R^{n_u*n_x}}, D_{n} \in {R^{n_u*n_y}}, X_{1}, Y_{1}\in {S^{n_x}}, and \gamma \in {R_{>0}};$$ that minimize $$\mathcal{J}(\gamma) = \gamma$$ subject to $$X_1>0,Y_1>0,$$ $$\begin{align}

\\ \begin{bmatrix}N_{11} &A+A_n^T+B_2D_nC_2 &B_1+B_2D_nD_{21} &Y_1^TC_1^T+C_n^TD_{12}^T \\
 * &X_1A+A^TX_1+B_nC_2+C_2^T+B_n^T &X_1B_1+B_nD_{21} &C_1^T+C_2^TD_n^TD_{12}^T \\
 * & * &-\gamma \mathbf{1} &D_{11}^T+D_{21}^TD_n^TD{12}^T \\
 * & * &* &-\gamma \mathbf{1}\end{bmatrix}&>0,\\

\begin{bmatrix}X_{1} & \mathbf{1}\\
 * & Y_{1}\end{bmatrix}&>0,\\

\end{align}$$

where $$N_{11}=AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T.$$ The controller is recovered by

$$ \begin{align} &A_{c} = A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\ &B_{c} = B_{k}(1-D_{c}D_{22})\\ &C_{c} = (1-D_{c}D_{22})C_{k}\\ &D_{c} = 1+D_{k}D_{22})^{-1}D_{k}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{k} & B_{k}\\ C_{k} & D_{k}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{n} & B_{n}\\ C_{n} & D_{n}\end{bmatrix} -\begin{bmatrix}X_{1}AY_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T = 1 - X_{1}Y_{1}$$. If $$D_{22}=0$$, then $$A_c=A_k,B_c=B_k,C_c=C_k,$$ and $$D_c=D_k$$.

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

Conclusion:
The Continuous-Time H∞-Optimal Dynamic Output Feedback Controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

Related LMIs
Discrete Time H∞ Optimal Dynamic Output Feedback Control

Continuous Time H2 Optimal Dynamic Feedback Control


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User:Margav06/sandbox/Click here to continue/Optimal control systems/Hinf-optimal Full-state Feedback Control

Full State Feedback Optimal $$H_{\infty}$$ Control
Full State Feedback is a control technique that attempts to place the system's closed loop system poles in specified locations based off of performance specifications given. $$H_{\infty}$$ methods formulate this task as an optimization problem and attempt to minimize the $$H_{\infty}$$ norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the $$H_{\infty}$$ we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The lower linear fractional transformation (LFT) is used to implement a controller $$K$$ into the system. The lower LFT is denoted as $$\underline{S}(P,K)$$ and is formed by $$\underline{S}(P,K)=P_{11}+P_{12}(I-KP_{22})^{-1}KP_{21}$$ with $$\begin{bmatrix} z\\y \end{bmatrix}=\begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w\\u \end{bmatrix} $$. For full-state feedback we consider a controller of the form $$u(t)=Fx(t)$$. This is a special case where $$y(t)=x(t)$$ and results in a controller of the form $$ K= \begin{bmatrix} 0 & 0 \\ 0 & F \end{bmatrix} $$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI:Full State Feedback Optimal $$H_{\infty}$$ Control LMI
The following are equivalent.

1) There exists a $$F$$ such that $$||\underline{S}(P,K(0,0,0,F)||_{H_{\infty}}\leq \gamma$$

2) There exists $$Y>0$$ and $$Z$$ such that

 * $$\begin{bmatrix} YA^T+AY+Z^TB_2^{T}+B_2Z & B_1 & YC^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -\gamma I & D^{T}_{11} \\ C_1 Y + D_{12} Z & D_{11} & -\gamma I \end{bmatrix}<0

$$.

Then $$F=ZY^{-1}$$.

Conclusion:
The above LMI, if feasible, will determine the bound $$\gamma$$ on the $$H_{\infty}$$ norm of the system. In addition to this $$F$$ is also determined allowing the closed loop system to be determined using the controller $$\hat{K}(0,0,0,F)$$ found during the optimization.

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

Related LMIs
Full State Feedback Optimal H2 LMI


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User:Margav06/sandbox/Click here to continue/Optimal control systems/H2-optimal Full-state Feedback control

Full State Feedback Optimal $$H_2$$ Control
Full State Feedback in general has the goal of positioning a system's closed loop poles in a desired location. This allows us to specify the performance of the system such as requiring stability or bounding the overshoot of the output. By minimizing the $$H_2$$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications, particularly when there is information about the distribution of the noise.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI: Optimal Output Feedback $$H_{\infty}$$ Control LMI
The following are equivalent.

1) There exists a $$K$$ such that $$||S(K,P)||_{H_{2}}<\gamma$$

2) There exists $$X>0$$, $$Z$$ and $$W$$ such that

 * $$\begin{bmatrix} A & B_2 \end{bmatrix} \begin{bmatrix}

X \\ Z \end{bmatrix}+ \begin{bmatrix} X & Z^T \end{bmatrix} \begin{bmatrix} A^T \\ B_2^T \end{bmatrix}+ B_1B_1^T <0 $$
 * $$\begin{bmatrix}

X & *^T \\ C_1X+D_{12}Z & W \end{bmatrix} > 0 $$
 * $$trace(W)<\gamma^2$$

where $$K=ZX^{-1}$$

Conclusion:
This LMI solves the $$H_2$$ optimal full state feedback problem and finds the upper bound of the $$H_2$$ norm of the system, $$\gamma$$. In addition to this the controller $$K$$ is also found in the process.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_H2.m

Related LMIs
Full State Feedback Optimal $H_{\infty}$ LMI


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Discrete-time H2-Optimal Full-state Feedback control

Discrete-Time H2-Optimal Full-State Feedback Control

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.

A full state feedback controller design for a Discrete Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d12}u_k\\ y_k &= x_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},D_{d12}), P, Z, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H2-Optimal Full-State Feedback Control

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} P \in {S^{n_x}}; Z \in {S^{n_z}}; &F_d \in {R^{n_u*n_x}}; \mu \in {R_{>0}} \;\\ &P>0\\ &Z>0\\ \begin{bmatrix}P & A_dP+B_{d2}F_d & B_{d1}\\
 * & P & 0\\
 * & * & I\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_{d1}P+D_{d12}F_d\\
 * & P\end{bmatrix}&>0,\\

&trZ<\mu ^2

\end{align}$$

Conclusion:
The H2-optimal full-state feedback gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time H2 Optimal Full State Feedback Control


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Discrete-time H2-optimal Dynamic output Feedback control

Discrete-Time H2-Optimal Dynamic Output Feedback Control

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.

A Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d11}w_k+ D_{d12}u_k\\ y_k &=C_{d2}x_k + D_{d21}w_k+ D_{d22}u_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H2-Optimal Full-State Feedback Control

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} X_{1}, Y_{1}\in {S^{n_x}}; Z \in {S^{n_z}}; \mu \in {R_{>0}} \;\\ A_{dn} \in {R^{n_x*n_x}}; B_{dn} \in {R^{n_x*n_y}}; C_{dn} \in {R^{n_u*n_x}}; D_{dn} \in {R^{n_u*n_y}};\\ &X_{1}>0\\ &Y_{1}>0\\ &Z>0\\ \begin{bmatrix}X_{1} & 1 & X_{1}A_d+B_{dn}C_{d2} & A_{dn} & X_{1}B_{d1}+B_{dn}D_{d21}\\
 * & Y_{1} & A_d+B_{d2}D_{dn}C_{d2} & A_dY_{1}+B_{d2}C_{dn} & B_{d1}+B_{d2}D_{dn}D_{d21}\\
 * & * & X_{1} & 1 & 0\\
 * & * & * & Y_{1} & 0\\
 * & * & * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_{d1}+D_{d12}D_{dn}C_{d2} & C_{d1}Y_{1}^T+D_{d12}C_{dn}\\
 * & X_{1} & 1\\
 * & * & Y_{1}\end{bmatrix}&>0,\\

D_{d11}+D_{d12}D_{dn}D_{d21}=0\\

\begin{bmatrix}X_{1} & 1\\
 * & Y_{1}\end{bmatrix}&>0,\\

trZ<\mu ^2 \end{align}$$

The controller is recovered by

$$ \begin{align} &A_{dc} = A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\ &B_{dc} = B_{dk}(1-D_{dc}D_{d22})\\ &C_{dc} = (1-D_{dc}D_{d22})C_{dk}\\ &D_{dc} = 1+D_{dk}D_{d22})^{-1}D_{dk}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{dk} & B_{dk}\\ C_{dk} & D_{dk}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{dn} & B_{dn}\\ C_{dn} & D_{dn}\end{bmatrix} -\begin{bmatrix}X_{1}A_{d}Y_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{d2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrixes $$X_2$$ and $$Y_2$$ satisfies $$X_2Y_2^T = 1 - X_{1}Y_{1}$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{d11} = 0$$, $$D_{d12}$$ ≠ 0, and $$D_{d21}$$ ≠ 0, then it is often simplest to choose $$D_{dn} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Discrete-Time H2-Optimal Dynamic Output feedback controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

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

Related LMIs
- Continuous Time H2 Optimal Dynamic Output Feedback Control


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Discrete-time Hinf-Optimal Full State Feedback Control

Discrete-Time H∞-Optimal Full-State Feedback Control

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.

A full state feedback controller design for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d12}u_k\\ y_k &= x_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},D_{d12}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H∞-Optimal Full-State Feedback Control

The LMI formulation

H∞ norm < $$\gamma$$

$$ \begin{align} P \in {S^{n_x}}; &F_d \in {R^{n_u*n_x}}; \gamma \in {R_{>0}} \;\\ &P>0\\ \begin{bmatrix}P & A_dP-B_{d2}F_d & B_{d1} & 0\\
 * & P & 0 & PC_{d1}^T-F_{d}^TD_{d12}^T\\
 * & * & -\gamma I & D_{d11}^T\\
 * & * & * & -\gamma I\end{bmatrix}&>0,\\

\end{align}$$

Conclusion:
The H∞-optimal full-state feedback gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time H∞ Optimal Full State Feedback Control


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Discrete-time Hinf-optimal Dynamic output Feedback control

Discrete-Time H∞-Optimal Dynamic Output Feedback Control

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.

A Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d11}w_k+ D_{d12}u_k\\ y_k &=C_{d2}x_k + D_{d21}w_k+ D_{d22}u_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H∞-Optimal Dynamic Output Feedback Control

The LMI formulation

H∞ norm < $$\gamma$$

$$ \begin{align} X_{1}, Y_{1}\in {S^{n_x}}; Z \in {S^{n_z}}; \gamma \in {R_{>0}} \;\\ A_{dn} \in {R^{n_x*n_x}}; B_{dn} \in {R^{n_x*n_y}}; C_{dn} \in {R^{n_u*n_x}}; D_{dn} \in {R^{n_u*n_y}};\\ &X_{1}>0\\ &Y_{1}>0\\ &Z>0\\ \begin{bmatrix}X_{1} & 1 & X_{1}A_d+B_{dn}C_{d2} & A_{dn} & X_{1}B_{d1}+B_{dn}D_{d21} & 0\\
 * & Y_{1} & A_d+B_{d2}D_{dn}C_{d2} & A_dY_{1}+B_{d2}C_{dn} & B_{d1}+B_{d2}D_{dn}D_{d21} & 0\\
 * & * & X_{1} & 1 & 0 & C_{d1}^T+C_{d2}^TD_{dn}^TD_{d12}^T\\
 * & * & * & Y_{1} & 0 & Y_{1}C_{d1}^T+C_{dn}^TD_{d12}^T\\
 * & * & * & * & -\gamma I & D_{d11}^T+D_{d21}^TD_{dn}^TD_{d12}^T\\
 * & * & * & * & * & -\gamma I\end{bmatrix}&>0,\\

\begin{bmatrix}X_{1} & 1\\
 * & Y_{1}\end{bmatrix}&>0,\\

\end{align}$$

The controller is recovered by

$$ \begin{align} &A_{dc} = A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\ &B_{dc} = B_{dk}(1-D_{dc}D_{d22})\\ &C_{dc} = (1-D_{dc}D_{d22})C_{dk}\\ &D_{dc} = 1+D_{dk}D_{d22})^{-1}D_{dk}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{dk} & B_{dk}\\ C_{dk} & D_{dk}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{dn} & B_{dn}\\ C_{dn} & D_{dn}\end{bmatrix} -\begin{bmatrix}X_{1}A_{d}Y_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{d2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrixes $$X_2$$ and $$Y_2$$ satisfies $$X_2Y_2^T = 1 - X_{1}Y_{1}$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{d11} = 0$$, $$D_{d12}$$ ≠ 0, and $$D_{d21}$$ ≠ 0, then it is often simplest to choose $$D_{dn} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Discrete-Time H∞-Optimal Dynamic Output Feedback Controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

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

Related LMIs
- Continuous Time H∞ Optimal Dynamic Output Feedback Control


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Mixed H2-Hinf-optimal Full-state Feedback Control

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Control

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.

A full-state feedback controller $$K = K_d \in {R^{n_u*n_x}}$$ (i.e., $$uk = K_dx_k$$) is to be designed to minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$ from the exogenous input $$w_{1,k}$$ to the performance output $$z_{1,k}$$ while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma_{d}$$.

The System
Discrete-Time LTI System with state space realization

$$ \begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k}+B_{d2}u_{k},\\

\begin{bmatrix} Z_{1,k}\\ Z_{2,k}\end{bmatrix} &= \begin{bmatrix} C_{d1,1}\\ C_{d1,2}\end{bmatrix}x_{k}+ \begin{bmatrix} 0 & D_{d11,12}\\ D_{d11,21} & D_{d11,22}\end{bmatrix} \begin{bmatrix} w_{1,k}\\ w_{2,k}\end{bmatrix}+ \begin{bmatrix} D_{d12,1}\\ D_{d12,2}\end{bmatrix}u_{k}\\

y_k &= x_k\\ \end{align} $$

The Data
The matrices: System $$ (A_d,B_{d1,1},B_{d1,2},B_{d2},C_{d1,1},C_{d1,2},D_{d11,12},D_{d11,21},D_{d11,22},,D_{d12,1},D_{d12,2}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

Minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$, while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ is less than $$\gamma_{d}$$, while obeying the LMI constraints.

The LMI:
Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Controller is synthesized by solving for $$P \in S^{n_{x}}, Z \in S^{n_{w}}, F_{d} \in R^{n_{u} * n_{x}}$$, and $$\mu \in R_{>0}$$ that minimize $$\mu$$ subject to $$P>0, Z>0$$

The LMI formulation

H∞ norm < $$\gamma_{d}$$

H2 norm < $$\mu$$

$$ \begin{align}

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0\\

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,2} & 0\\
 * & P & 0 & PC_{d1,2}^T-F_{d}^TD_{d12,2}^T\\
 * & * & \gamma_{d} I & D_{d11,22}^T\\
 * & * & * & \gamma_{d} I\end{bmatrix}&>0\\

\begin{bmatrix} Z & C_{d1,1}P-D_{d12,1}\\
 * & P\end{bmatrix}&>0\\

trace(Z) &< \mu^2

\end{align}$$

Conclusion:
The H2-optimal full-state feedback controller gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time Mixed H2-H∞ Optimal Full State Feedback Control


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Discrete-Time Mixed H2-Hinf Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

The System

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k},\\

y_{k}&=C_{c2}x_{k} + D_{d21,1}w_{1,k}+ D_{d21,2}w_{2,k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n * n}$$  and is the state matrix,  $$ B \in R^{n * r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m * n}$$  and is the output matrix,  $$ D \in R^{m * r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

$$ A \in R^{n * n}$$

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} * n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2}) e_{k} + (B_{d1,1} - L_{d}D_{d21,1})w_{1,k} + (B_{d1,2} - L_{d}D_{d21,2})w_{2,k}$$,

and the performance output is defined as

$$ \begin{bmatrix} Z_{1,k} \\ Z_{2,k}\end{bmatrix} = \begin{bmatrix} C_{d1,1} \\ C_{d1,2} \end{bmatrix}e_{k} + \begin{bmatrix} 0 & D_{d11,12} \\ D_{d11,21} & D_{d11,22}\end{bmatrix}\begin{bmatrix} w_{1,k} \\ w_{2,k} \end{bmatrix} $$.

The observer gain $$L_{d}$$ is to be designed to minimize the $$H_{2}$$ norm of the closed loop transfer matrix $$T_{11}(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma _ {d}$$, where

$$ \begin{align} T_{11}(z) &= C_{d1,1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,1} - L_{d}D_{d21,1}),\\ T_{22}(z) &= C_{d1,2}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,2} - L_{d}D_{d21,2}) + D_{d11,22} \end{align} $$

The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer
The discrete-time mixed-$$H_{2}H_{inf}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x} * n_{y}}$$, and $$v \in R_{>0}$$ that minimize J$$(v) = v$$ subject to $$P>0, Z>0$$,

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,1}-G_{d}D_{d21,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,2}-G_{d}D_{d21,2} & 0\\
 * & P & 0 & C^{T}_{d1,2}\\
 * & * & \gamma_{d} 1 & D^{T}_{d11,22}\\
 * & * & * & \gamma_{d} 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1,1}\\
 * & P\end{bmatrix}&>0,\\

trZ<v

\end{align}$$

where $$tr$$ refers to the trace of a matrix.

Conclusion:
The mixed-$$H_{2}H_{inf}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$, the $$H_{2}$$ norm of $$T_{11}(z)$$ is less than $$\mu = \sqrt{v}$$, and the $$H_{inf}$$ norm of $$T_{22}(z)$$ is less than $$\gamma_{d}$$. This result gives us a matrix of observer gains $$L_{d}$$ that allow us to optimally observe the states of the system indirectly as:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

Related LMIs
Discrete-Time_Hinfinity-Optimal_Observer

Discrete-Time_H2-Optimal_Observer


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems/Discrete-time H2-Hinf-optimal Dynamic output Feedback control

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

The System

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k},\\

y_{k}&=C_{c2}x_{k} + D_{d21,1}w_{1,k}+ D_{d21,2}w_{2,k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n * n}$$  and is the state matrix,  $$ B \in R^{n * r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m * n}$$  and is the output matrix,  $$ D \in R^{m * r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

$$ A \in R^{n * n}$$

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} * n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2}) e_{k} + (B_{d1,1} - L_{d}D_{d21,1})w_{1,k} + (B_{d1,2} - L_{d}D_{d21,2})w_{2,k}$$,

and the performance output is defined as

$$ \begin{bmatrix} Z_{1,k} \\ Z_{2,k}\end{bmatrix} = \begin{bmatrix} C_{d1,1} \\ C_{d1,2} \end{bmatrix}e_{k} + \begin{bmatrix} 0 & D_{d11,12} \\ D_{d11,21} & D_{d11,22}\end{bmatrix}\begin{bmatrix} w_{1,k} \\ w_{2,k} \end{bmatrix} $$.

The observer gain $$L_{d}$$ is to be designed to minimize the $$H_{2}$$ norm of the closed loop transfer matrix $$T_{11}(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma _ {d}$$, where

$$ \begin{align} T_{11}(z) &= C_{d1,1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,1} - L_{d}D_{d21,1}),\\ T_{22}(z) &= C_{d1,2}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,2} - L_{d}D_{d21,2}) + D_{d11,22} \end{align} $$

The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer
The discrete-time mixed-$$H_{2}H_{inf}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x} * n_{y}}$$, and $$v \in R_{>0}$$ that minimize J$$(v) = v$$ subject to $$P>0, Z>0$$,

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,1}-G_{d}D_{d21,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,2}-G_{d}D_{d21,2} & 0\\
 * & P & 0 & C^{T}_{d1,2}\\
 * & * & \gamma_{d} 1 & D^{T}_{d11,22}\\
 * & * & * & \gamma_{d} 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1,1}\\
 * & P\end{bmatrix}&>0,\\

trZ<v

\end{align}$$

where $$tr$$ refers to the trace of a matrix.

Conclusion:
The mixed-$$H_{2}H_{inf}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$, the $$H_{2}$$ norm of $$T_{11}(z)$$ is less than $$\mu = \sqrt{v}$$, and the $$H_{inf}$$ norm of $$T_{22}(z)$$ is less than $$\gamma_{d}$$. This result gives us a matrix of observer gains $$L_{d}$$ that allow us to optimally observe the states of the system indirectly as:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

Related LMIs
Discrete-Time_Hinfinity-Optimal_Observer

Discrete-Time_H2-Optimal_Observer