The Rinks – Irvine Inline host a number of inline hockey tournaments throughout the year. See below for more specific tournament information and registration.
The Western Inline Hockey League (WIHL) host multiple club inline hockey tournaments at The Rinks. Please visit the link to find more information and when the next WIHL tournament is coming to our rink.
The Amateur Athletic Union (AAU) host multiple club inline hockey tournaments at The Rinks. Please visit the link to find more information and when the next AAU tournament is coming to our rink.
North American Championship Roller Hockey (NARCh) host multiple club roller hockey tournaments across the United States and Canada. Please visit the link to find more information and when the next NARCh tournament is coming to our rink.
TORHS 2Hot4Ice host club roller hockey tournaments across the United States. Please visit the link to find more information and when the next TORHS tournament is coming to our rink.
State Wars Hockey host club roller hockey tournaments across the United States. Please visit the link to find more information and when the next State Wars tournament is coming to our rink.
The annual CA High School Inline Hockey State Cup has been hosted at Irvine Inline for over 10 years. Teams from Northern and Southern California compete to declare their school as the #1 school of Inline Hockey.
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text:
【EXHBITION: Lighting Art】26/8/2011
EXHBITION: Lighting Art / Exhibition of design in Ebisuminami for the world
A co-exhibition called "Exhibition : Lighting Art" takes place from 26th August until 4 th September in 3 places : Lighting Art Gallery, limArt,SOMEWHERE in Ebisuminami, Tokyo, where full of edgy independent boutiques and great galleries to go have been opening recently.
This co-exhibition shows how Yamagiwa Art Foundation has played important roles in cultural activities by realizing the collaboration with several designers such as Ettore Sottsass or Shiro Kuramata, who have crossed over the boundaries between art, architecture and design.
After an interval of twenty years, we are glad to present a series of lamp prototypes for "Light is architecture", lamp exhibition in 1990 that Yamagiwa offered to create to Sottsass. The first products from these prototypes of lamp that Sottsass has given the name of number from One to Twenty-nine, will be sold for this co-exhibition.
Also, interesting exhibits have come from "IN・SPIRATION" in 1988 that Kuramata, as a director of exhibition, asked the contemporary designers and architects from overseas to design the lighting under the support of Yamagiwa.
The co-exhibition features precious works in design history ; the ceiling lamp "Warped Plane"created by Zaha Hadid, architect wasn't well known during the period Kuramata ordered, and the Kuramata's "Acrylic Back Chair for Livina Yamagiwa". The space "meeting room for Livina Yamagiwa" designed by Kuramata in 1983 is reconstructed and arranged for this show. We are planning on creating high quality design and art scene and on developing a variety of projects.
Exhibition Dates : Friday, August 26, to Sunday, September 4, 2011
Opening Hours : 12:00-20:00 everyday
Private View : Thursday, August 25, 2011
Locations : limArt、limArt annex、SOMEWHERE
Co-organized by:Lighting Art Gallery / limArt / SOMEWHERE
Venue Production : Masato Kawai
Text : Mikio Kuranishi
This exhibition is supported by Yamagiwa Art Foundation.
limArt
2-10-3-1F Ebisuminami Shibuya-ku Tokyo / TEL: +81-3-3713-8670
limArt annex
2-15-6 Ebisuminami Shibuya-ku Tokyo / TEL: +81-3-3716-1202
2-7-1-1F Ebisuminami Shibuya-ku Tokyo / TEL: +81-3-6452-2224
Mail: info@lighting-art.jp
TEL: +81-3-6452-2224 (SOMEWHERE)
For additional information. please contact the above
The Yamagiwa Art Foundation – 50 years of efforts to improve the art of lighting
The Tokyo International Lighting Design Competition (hereafter "the Competition"), was initiated in 1968 by Yamagiwa Corporation, who foresaw the infinite possibilities of lighting and acted upon his desire to create an even more beautiful lighting environment. At that time it was an innovative international competition aimed at promoting Japanese lighting design and international exchange, as well as discovering new designers. In many ways, it was a high-level design competition, attended by eminent architects, designers, and artists both from Japan and abroad. Yusaku Kamekura, who had designed Yamagiwa's Corporate Identity before, took a central role in developing the idea of competition initially. From 1970 to 1983, Kenzo Tange, a giant of the world of architecture, was welcomed as the president of the panel of judges, and the Competition continued to be held with a different theme each year. These sterling efforts were recognized by the Agency for Cultural Affairs, and in 1973 the Yamagiwa Art Foundation (hereafter "the Foundation") was established and subsequently became the organizer of the Competition.
The degree of innovation shown by the Competition can clearly be seen from the list of judges, guests, and exhibitors, who together form quite an eminent line-up. It is interesting to note that already by 1969, when the Competition was in its second year, Italian architect and designer Ettore Sottsass was invited to sit on the panel of judges. The collaboration between Yamagiwa and Sottsass continued for a long time after that. In addition, Shiro Kuramata attended the 1972 Competition as a guest, and he also had a very close relationship with Yamagiwa in the years that followed.
The Competition was held for the 10th and final time in 1983, but its legacy lived on, and later, under Kuramata's supervision, the "In-spiration Exhibition" was held in 1988. At the same time, Yamagiwa opened a showroom / gallery in Roppongi with the same name, and in 1990, it held an exhibition of 29 Lamps designed by Ettore Sottsass, subtitled "Light is Architecture". Also, in this gallery, periodical exhibitions by active designers selected by Kuramata and Sottsass started to be held with the title "In-spiration" exhibitions under the auspices of the Foundation. Such examples make it clear that all the efforts of the Foundation and of Yamagiwa to promote lighting art started with the Competition. For example, Massimo Iosa Ghini, who held an exhibition at In-spiration, signed a contract for product development with Yamagiwa after the exhibition, which resulted in the commercialisation of a collection of floor lamp and bracket. These efforts are all part of the Competition's initial goal of promoting lighting design and international exchange, and discovering new designers.
In this exhibition "Lighting Art", we would like to present some works displayed during some of the shows held under the auspices of the Foundation, after the "In-spiration Exhibition" in 1988. They are all unique, precious, one-off works made for the exhibitions.
Lighting Art Gallery
Light is Architecture – 29 Lamps designed by Ettore Sottsass
In the 1980's, Ettore Sottsass founded the Memphis Group. After approaching Shiro Kuramata to see if he would be interested in joining forces, the two of them became even closer than before. Their efforts attracted the world's attention as their activities became ever more closely intertwined, spanning Europe and the Far East, including the launching of the San Francisco apparel brand , Esprit. Then, in 1989, following a suggestion from Kuramata, the Foundation asked Sottsass to design 15 lighting fixtures, proposing a unique exhibition that would show the way forward for new forms of lighting. Expressing his feelings in the book printed for the exhibition, Sottsass uses the expression "Light is Architecture" to name the theme of the exhibition, and explains that he could not be content with only 15 items, so eventually he designed 30 lighting fixtures, out of which 29 were exhibited (7 pendants, 9 floor lamps, 8 brackets, and 5 table stands).
The prototypes were all manufactured in Italy, and the graphics aspect of the book was also the responsibility of Sottsass Associati, and was printed in Milan. As a result, the "Light is Architecture – 29 Lamps designed by Ettore Sottsass" Exhibition was held in 1990 from April 4 to 28 at the In-spiration Gallery in Roppongi. Sottsass visited Japan from April 1 as he wanted to look over the venue before the opening, and he also gave a memorial lecture on April 5; he spent most of his time with Kuramata until he left Japan on April 7.
With the approval and the assistance of Studio Ettore Sottsass s.r.l., in charge of administrating Sottsass's works and copyright, a limited number of items from among 29 lighting fixtures exhibited have recently been selected for commercialization once again.
15 Lamps for Yamagiwa by Ettore Sottsass
Yamagiwa asked me to design, and to produce here in Italy, the prototypes of 15 lamps, for an exhibition in Tokyo.
When I started thinking about designing the lamps, I found I was never satisfied and always ended up doing another, extra lamp; so I did a lot of drawings and then chose among them and in the end decided that, to mount an exhibition, to put an idea across, 30 lamps were needed. From then on, I got into the highly complicated problem of producing prototypes.
Producing 30 prototypes of lamps actually boils down to a banal, practical matter.
The real difficulty began when I asked myself what can an object that sends light into an architectural interior, into a room, into a corridor, or onto a table, really be like today.
I personally – always – have a strong desire to check the presence of objects around me, of things, signs, memories and miscellaneous pretty, consolatory amusements. In general, I feel a strong psychic and cultural, and ultimately also a political, urge to restrain as far as possible that geological growth, that slow and normally inevitable sedimentation of products for momentary and imprecise uses and vague consumption, which life itself drags along behind it, which life deposits around us.
Therefore are people who fill rooms with faded flags that have fluttered in battles won or lost by their ancestors, or who fill corridors with swords and ancient weapons; there are people who fill rooms with photos of their children and of the children of their children, and even of the children of their sisters and brothers and who also even keep photos of their grandparents' great-grandparents when they were married and of their uncles and aunts and cousins and photo of themselves with one foot on the head of the tiger they killed and of themselves standing in front of their new car and of themselves being greeted by the Minister and so on.
There are people who deposit in their houses any thing, any residue of parties, celebrations, events, any memory, any sign of power, wealth, conquest, possession.
There are people, too, who fill their houses with lamps, chandeliers, lampshades, bracket-lamps and even candelabra for special dinners, people that is, who don't fill their homes with "light" but with sundry luminous objects, in bronze, steel, glass, paper, wood, of every kind of material and design. As I said, I personally always feel a strong desire to curb the mechanism of a culture that does nothing but deposit, that draws its energy, its actual justification, its very images, from the fact that it does nothing but deposit.
That's why it was embarrassing for me to design 15 lamps all at once, and that's why I designed 30, because as I did each one, I didn't draw it well enough to make it disappear, as I would have liked. Some additional sign always remained.
I couldn't do a clear enough design, which would disappear into architecture or become architecture itself, I mean the actual space of light. I was unable to do a design which became a wall or ceiling, or column, or the corner of a wall, or slab, or capital or architrave, instead of staying there to represent an object, just any sign in space, attached to other innumerable signs, to increase the depository of signs. Next Page
In the ancient Japanese world the problem had not been badly solved. As far as I know, ancient Japanese lamps were small or big luminous boxes that disappeared together with other wooden or lacquered boxes, together with red wooden pieces of box furniture, inside the boxes of rooms which in turn became architecture – groups of boxes with roofs. As a matter of fact, ancient Japanese lamps were small pieces of architecture, fittings, inlays of architecture, melted into architecture to sustain the total design of private, artificial space.
To give a different example, in ancient India not much care was taken over the design of private spaces. All the care was devoted to design of spaces, somehow or for some reason, sacred or, made sacred and in those spaces, in any case, there wasn't much light; there weren't any designed lamps, only pots with lights and candles in them resting here and there. But in India there was, as there still is, an immense quantity of light and the majority of problems were clearly dealt with during the day; for that matter today, too, in India, the night don't have all that many lights, they certainly don't have the light of Las Vegas, or Ginza and not even the lights of any ordinary California gas station. The nights in India are very vaguely illuminated by a few stray fluorescent tubes or by the occasional, still more ghostly, light-bulb dangling naked in the air. There's nothing else.
The fact is that in India what used to happen in ancient times, more or less, in the countryside, paddy-fields and forests, on beaches, in deserts and hills all over the Far East, still happens: the attention of design being dedicated only to objects capable of catalyzing ritual processes, objects capable of accompanying the passing of existence from the everyday, frail, provisional, illogical, anxious state, to a compact, conscious, intense, unforgettable state.
"Designed" objects in those ancient cultures, in those ancient spaces, were few and far between. They were objects situated in empty, invisible spaces, in the middle of vast silences; they were mixed into and hidden in the rituality itself.
Those objects were, however, always, very precious, on account of their rarity, of the effort, the hours and the risks undertaken to achieve that special design, but most of all because of the special vibrations which they had the capacity to transmit, because of the special role which they could assume or were intended to assume.
This idea of a different possible function, of a rarer, real and precious destiny for every object designed, or "produced" – as they say today – situated in the space of our existence, has always fascinated me deeply. It was because of this idea which I had in my head, that it seemed to me embarrassing to design 15 lamps all at once.
I finished by designing small boxes (in the Japanese manner), boxes that become columns, or beams, or stacks of boxes, with different variations.
In-spiration
After the Tokyo International Lighting Design Competition was held for the last time in 1983, the Foundation worked out a plan for new activities and a framework for their key points. The core of these efforts was "In-spiration", born out of a suggestion made by Kuramata, who developed a strong relationship of trust with Yamagiwa over the years of the Competition. It also provided an opportunity for experimenting with lighting environments in keeping with the new age. "In-spiration" was the first to try to express this concept clearly. It was an exhibition in which basic human activities such as eating, reading, and sleeping were turned into a matrix for interior design components such as the floor, the walls, and the ceiling. 21 teams of promising designers and architects from around the world were selected and each was asked to come up with a lighting design from a position they were assigned. In the foreword of the booklet published for the exhibition (IN-SPIRATION, PARCO Co.,Ltd. 1988) Kuramata wrote: "[…] the designs that should be taking shape before our eyes are those that have appeared by already betraying the demands of form and function in some way." It was a message for architects and designers who were to play a role in the future, a provocative and inflammatory message to temporarily deconstruct and then rebuild design.
The aforementioned Yamagiwa showroom gallery, designed by Kuramata, opened the same year inside the Roppongi AXIS Building, and also serves as one of the bases for propagating this concept. Exhibitions featuring predominantly up-and-coming international designers and architects, selected by Kuramata, have subsequently been held periodically.
"Eight"
Size: W62.5×D62.5×H199cm
Price: P.O.A
"Ten"
Price: 268800JPY
"Twenty"
Size: W20.5×D31.2×H36.5cm
Price: 84000JPY
"Twenty two"
"Twenty seven"
Size: W20×D19.3×H40.7cm
"Twenty nine"
Size: W26×D41×H55.3cm
"Six"
Size: Dia60×H94cm
"Seven"
"Twelve"
"Thirteen"
Size: W50×D50×H206cm
"Fourteen"
Size: W50×D50×H212.5cm
"Sixteen"
"One"
"Twenty six"
"Glass table for yamagiwa showroom"
Size: W483.3×D95×H95cm
"Acrylic back chair for livina yamagiwa"
Size: W57×D52×H89cm
"Acrylic back chair for piacere"
"Chair C"
Size: W52.5×D52.5×H75cm
"How high the moon" (early edition)
Size: W155×D83×H70cm
"Meeting room of livina yamagiwa"
"Warped plane"
"Stick&stone"
Rob Eckhardt
"Chair for livina yamagiwa"
Size: W72×D77.5×H73cm
"Chair for yamagwa president's office"
"Wagon for yamagwa president's office"
"Ettore Sottsass Advanced Studies 1986/1990"
Size: W24.7×H31cm 95pages
Price: 8000JPY

redpajama_set_name: RedPajamaCommonCrawl


Hungarian Prime Minister Viktor Orban's Fidesz party won a 2/3 supermajority in parliament.
This stunning victory proves that the Soros-led anti-government campaign was woefully ineffective and may have actually been counterproductive by strengthening Orban's appeal by reinforcing the country's developing "siege mentality". Hungarians already feel under threat from the world at large ever since the Migrant Crisis, while they've come to view Brussel's hostility to the national sovereignty policies that they've implemented in response as a sign of regional aggression. The "internal siege" by pro-Western NGOs and foundations might have pushed some on-the-fence voters to side with Orban out of fear that anything less than his party's decisive victory could endanger their national and cultural security.
The end result is that Hungary has defended its reputation as a bastion of EuroRealism and made it impossible for the EU to sanction its ideological ally Poland because of the bloc's inability to reach the necessary unanimous decision in this regard given that Orban has remained in office and is firmly opposed to this.
Therefore, the Polish-Hungarian "Three Seas" axis will only tighten in the future and continue to expand its appeal in Central and Eastern Europe after proving that it's possible to stand up to the EU's liberal-globalist elite and succeed so long as leaders have the public's sincere support like Orban does. He's been able to accomplish this through the ironically masterful employment of interwar Italian Marxist philosopher Antonio Gramsci's theory of "cultural hegemony" which essentially asserts that cultural change must always precede profound political change in any society. The New York Times recently talked about how Orban pulled this off in an article titled "How Viktor Orban Bends Hungarian Society to His Will", which despite its negative angle makes for an instructive read if objectively assessed.
The outlet describes how Orban has inserted his EuroRealist ideology into civil society, the arts, grade school textbooks, universities, and the media in order to make Hungarian culture an instrument for advancing and sustaining his revolutionary ideals. It should be remembered that left-wing and violent organizations don't have a "monopoly" on "revolutions", which can be generalized as just being the change from one organizational system to another, so it's entirely accurate to describe the Hungarian leader as a political and cultural revolutionary (or counter-revolutionary, depending on ones' perspective) in liberating his society from Brussels' liberal-globalist yoke and returning it to its conservative-nationalist roots.
The previous weekend's election can therefore be interpreted as a popular referendum on the revolution, which Orban won with flying colors in proving that the vast majority of people are firmly behind him every step of the way.


text:
Man with multiple prior convictions charged with domestic assault
Malwitz is scheduled to appear in court on Aug. 22.
Malwitz
By Emma McNamee
LUVERNE — A Luverne man is being charged with a felony count of domestic assault, for allegedly committing an act of domestic violence within ten years of two or more related convictions.
MORE CRIME NEWS
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A 50-year-old Red Wing woman has pleaded guilty to second-degree murder related to leaving her new born infant boy in the Lake Pepin as part of a plea deal.
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Former Memphis officers arrested on murder charges in death of Tyre Nichols
Officials are expected to release bodycam footage of the traffic stop on Friday.
By Tyler Clifford / Reuters
U.S. targets Google's online ad business monopoly in latest Big Tech lawsuit
The lawsuit tackles a business at Google that is responsible for 80% of its revenue. The Justice Department asked the court to compel Google to break up a key piece of its ad technology business.
By Diane Bartz and David Shepardson / Reuters
Local law enforcement was dispatched to a Luverne residence on a report of an assault. Upon arriving, they met with the victim, who had an open cut across her nose.
The victim stated that she had been arguing with her son, Kevin James Malwitz, 27, and that he'd thrown something at her. The victim repeatedly stated she did not want to get Malwitz in trouble.
When officers asked where Malwitz was, the victim told them he had left. The reporting party, who had witnessed the incident, then joined the conversation and stated that Malwitz was currently upstairs. Officers were led into the house and found Malwitz sleeping in an upstairs bedroom.
He was placed under arrest and declined to give a statement beyond that he and the victim had argued earlier. Bail, with conditions, was set at $20,000.
Malwitz has multiple past domestic violence-related convictions from 2017, 2018, and 2020. He is scheduled to make his initial appearance in court on Aug. 22.
If convicted, Malwitz faces a maximum sentence of five years imprisonment, a $10,000 fine or both.
The Southwest Crisis Center is available in our community to support and empower survivors of sexual and domestic violence. For questions about services or how you can help, contact the Southwest Crisis Center at 1-800-376-4311 or mnswcc.org .
READ MORE BY EMMA MCNAMEE
Advocates hope to re-establish Minnesota Historic Tax Credit
Minnesota's Historic Tax credit had an 11-year run before ending in 2022. Now, advocates of the program are attempting to bring it back.
LISTEN: New staffing business, new WPD officers, and new Community Ed building| Jan. 25, 2023
Follow the Globe Minute, our twice-weekly Worthington news and weather podcast, on Apple, Spotify, or Google Podcasts!
Adrian woman charged with felony drug sale and possession
Doyle is scheduled to make her initial court appearance Jan. 31.
Worthington City Council welcomes new police officers
The Center for Active Living's racquetball courts were also discussed. They will be repaired, with one of the courts to be retrofitted to better serve CAL members.
Luverne man charged with domestic assault
May is scheduled to make his initial court appearance on Jan. 23.
Windom woman charged with two counts of burglary, assault
Glynn is scheduled to make her initial court appearance on Jan. 31.
Civil rights activist advocates service, voting rights in Worthington
"We are in the throes of an important struggle in this country," McLemore said to students, faculty, and community members Tuesday. "...This battle is around the right to vote."
LISTEN: Grain fraud, Triple A winners, benefit for Heron Lake family| Jan. 13, 2023
Alleged police pursuit results in felony charges against Mountain Lake man
Khounsanthone is scheduled to make his initial court appearance Jan. 17.
Bingham Lake man charged with felony DWI after allegedly hitting two vehicles in downtown Worthington
Glidden is scheduled to make his initial court appearance Feb. 7 in Nobles County.
Related Topics: CRIMECRIME AND COURTSLUVERNEROCK COUNTY
Emma McNamee joined The Globe team in October 2021 as a reporter covering Crime & Courts, Politics, and the City beats. Born and raised in Duluth, Minn., McNamee left her hometown to attend school in Chicago at Columbia College. She graduated in 2021 with a degree in Multimedia Journalism, with a concentration in News & Feature Writing and a minor in Creative Writing.
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Wednesday's community input meeting at Worthington High School was the third of four planned by the Minnesota Attorney General's Office.
Thursday sports roundup: JCC wrestlers pin Pipestone Area


LOL!!! I always thought it was wierd too!!
Yeah.... I kinda thought it would be kinda weird at first, but my administrative login for CaparisonForum actually is the clock inlay off a Dellinger. When I first ran this place I really had no intention of using my own login so I stuck the J Custom Vine on it to differentiate things. I just ended up leaving myself logged in under my own name for so damn long that I just use my other login with the clock inlay avatar for system wide administrative purposes now.
(At first I was going to say that I wanted to promote the whole notion of being able to like other guitar brands but I'm pretty sure everyone would have seen through that BS).
I thought it was weird too but I thought it was because Budd also liked the Ibanez stuff too. Nothing wrong with that, although I do think it would be better idea that his avatar should be Caparison related. You gotta represent ya know?
I like the old floral jems. they are the only ones that come close to a Cap.
It doesn't bother me, I know a lot of people that love those Jems over any guitar, but I am not one of them.
Shall I change it back to bring back that avatar that didn't make any sense here?
Which Ebony axe is that??
An old Dellinger II I had.
I'm trying to get rid of my JEM7VWH these days but my 77BFP will stay with me no matter what. Still not even close to my Cap but I just can't let that guitar go.


text:
Blog of him
Laplace Transform
Inverse Laplace Transform
Z-Transfrom
Inverse Z-Transform
Modified Z-Transfrom
Starred Transform
Fourier Transform
State Space Representation
Continuous State Space Representation
Transfer function
Discrete State Space Representation
Controllability & Reachability
Decomposition and Realizations
Lyaponov Stability
Bounded-Input Bounded-Output Stability
Steady State Accuracy
Transient Response
Discretization and Linearization
Discretization Example
Sampling (A/D)
Reconstruction/Hold (D/A)
$s$-plane and $z$-plane
Linearization
Full State Feedback
State Estimation (Observer Design)
Notes for Control System
This note combines content from ME 564 Linear Systems & ME 561 Discrete Digital Control.
In this note, $f\in\mathbb{F}^\mathbb{G}$ stands for a function with domain in $\mathbb{G}$ and co-domain in $\mathbb{F}$, i.e. $f:\mathbb{F}\to\mathbb{G}$, $H(x)$ generally stands for Heaviside function (step function)
Please read the Algebra Basics notes first if you are not familiar with related concepts.
Definition: $F(s)=\mathcal{L}\{f(t)\}(s)=\int^\infty_0 f(t)e^{-st}\mathrm{d}t$
Note that the transform is not well defined for all functions in $\mathbb{C}^\mathbb{R}$. And the transform is only valid for $s$ in a region of convergence, which is usually separated by 0.
Laplace Transform is a linear map from $(\mathbb{C}^\mathbb{R}, \mathbb{C})$ to $(\mathbb{C}^\mathbb{C}, \mathbb{C})$ and it's one-to-one.
Properties: (see Wikipedia or this page for full list)
Derivative: $f'(t) \xleftrightarrow{\mathcal{L}} sF(s)-f(0^-)$
Integration: $\int^t_0 f(\tau)d\tau \xleftrightarrow{\mathcal{L}} \frac{1}{s}F(s)$
Delay: $f(t-a)H(t-a) \xleftrightarrow{\mathcal{L}} e^{-as}F(s)$
Convolution: $\int^t_0 f(\tau)g(t-\tau)\mathrm{d}\tau \xleftrightarrow{\mathcal{L}} F(s)G(s)$
Stationary Value: $\lim\limits_{t\to 0} f(t) = \lim\limits_{s\to \infty} sF(s), \lim\limits_{t\to \infty} f(t) = \lim\limits_{s\to 0} sF(s)$
Laplace transform is one-to-one, so we can apply inverse transform on functions in s-space
There are several ways to calculate Laplace transform, the first one is directly evaluating integration while the latter two are converting the function into certain formats that are convenient for table lookup:
(Mellin's) Inverse formula: $f(t)=\mathcal{L}^{-1}\{F(s)\}(t)=\frac{1}{2\pi j}\lim\limits_{T\to\infty} \int ^{\gamma+iT}_{\gamma-iT} e^{st}F(s)\mathrm{d}s$ where the integration is done along the vertical line $Re(s)=\gamma$ in the convex s-plane such that $\gamma$ is greater than the real part of all poles of $F(s)$.
Power Series: $F(s) = \sum^\infty_{n=0} \frac{n!a_n}{s^{n+1}}\xleftrightarrow{\mathcal{L}} f(t) = \sum ^\infty_{n=0} a_n t^n $
Partial Fractions: $F(s)=\frac{k_1}{s+a}+\frac{k_2}{s+b}+\ldots \xleftrightarrow{\mathcal{L}} f(t)=k_1 e^{-at} + k_2 e^{-bt} + \ldots$
To calculate partial fractions, one can use Polynomial Division or following lemma:
Suppose $F(s)=\frac{N(s)}{D(s)}=\frac{N(s)}{\prod^n_{i=1} (s-p_i)^{r_i}}$ where $\mathrm{deg}(N(s)) < \mathrm{deg(D(s))}$ and each $p_i$ is a distinct root of $D(s)$ (i.e. pole) with multiplicity $r_i$, then $F(s)=\sum^n_{i=1}\sum^{r_i}_ {j=1} \frac{k_{ij}}{(s-p_i)j}$ where $k_{ij}=\frac{1}{(r_i-j)!}\left.\frac{\mathrm{d}^{r_i-j}}{\mathrm{d}s^{r_i-j}}(s-p_i)^{r_i}F(s)\right\vert_{s=p_i}$
Definition: $F(z)=\mathcal{Z}\{f(k)_ {k\in\mathbb{N}}\}(z)=\sum^\infty_{k=0} f(k)z^{-k}$
Notice that $f$ is defined on natural numbers. In time domain, it's usually corresponding to $f(kT)$. Z-transform is also only valid for $z$ in certain region (usually separated by 1)
Laplace Transform is a linear map from $(\mathbb{C}^\mathbb{N}, \mathbb{C})$ to $(\mathbb{C}^\mathbb{C}, \mathbb{C})$ and it's one-to-one.
Accumulation: $\sum^n_{k=-\infty} f(k) \xleftrightarrow{\mathcal{Z}} \frac{1}{1-z^{-1}}F(z)$
Delay: $f(k-m) \xleftrightarrow{\mathcal{Z}} z^{-m}F(z)$
Convolution: $\sum^k_{n=0}f_1(n)f_2(k-n) \xleftrightarrow{\mathcal{Z}} F_1(z)F_2(z)$
Stationary Value: $\lim\limits_{t\to 0} f(t) = \lim\limits_{z\to \infty} F(z), \lim\limits_{t\to \infty} f(t) = \lim\limits_{z\to 1} (z-1)F(z)$
Example: Z-Transform of PID controller Assume the close-loop error input of the controller is $e(t)$, and $e(kT)$ after sampling. PID controller action in analog is $$m(t)=K\left(e(t)+\frac{1}{T_i}\int^t_0e(t)\mathrm{d}t+T_d\frac{\mathrm{d}e(t)}{\mathrm{d}t}\right)$$ We can approximate by trapezoidal rule with two point difference: $$m(kT)=K\left(e(kT)+\frac{T}{T_i}\sum^k_{h=1}\frac{e((h-1)T)+e(hT)}{2}+T_d\left(\frac{e(kT)-e((k-1)T)}{T}\right)\right)$$ Lets define $f(hT) = \frac{1}{2}\left(e((h-1)T)+e(hT)\right),\;f(0)=0$ Then $$\begin{split}\mathcal{Z}\left(\left\{\sum^k_{h=1}\frac{e((h-1)T)+e(hT)}{2}\right\}_k\right)(z)=\mathcal{Z}\left(\left\{\sum^k_{h=1}f(hT)\right\}_k\right)(z) \\ =\frac{1}{1-z^{-1}}(F(z)-F(0))=\frac{1}{1-z^{-1}}F(z)\end{split}$$ Notice that $$F(z)=\mathcal{Z}\left({f(hT)}_h\right)(z)=\frac{1+z^{-1}}{2}E(z)$$ so we can calculate the Z-transform of $m(kT)$ $$\begin{split} M(z)&=K\left(1+\frac{T}{2T_i}\left(\frac{1+z^{-1}}{1-z^{-1}}\right)+\frac{T_d}{T}(1-z^{-1})\right)E(z)\\&=K\left(1-\frac{T}{2T_i}+\frac{T}{T_i}\frac{1}{1-z^{-1}}+\frac{T_d}{T}(1-z^{-1})\right)E(z)\\&=\left(K_p+K_i\left(\frac{1}{1-z^-1}\right)+K_d(1-z^{-1})\right)E(z) \end{split}$$
Here we have
Proportional Gain $K_p=K-\frac{KT}{2T_i}$
Integral Gain $K_I=\frac{KT}{T_i}$
Derivative Gain $K_d=\frac{KT_d}{T}$
Inverse formula: $f(k)=\mathcal{Z}^{-1}\{F(z)\}(k)=\frac{1}{2\pi j}\oint _\Gamma z^{k-1}F(z)\mathrm{d}z$ where the integration is done along any closed path $\Gamma$ that encloses all finite poles of $z^{k-1}X(z)$ in the z-plane.
According to residual theorem, we can write it as $f(k)=\sum_{p_i}Res(z^{k-1}f(z), pi)$ where $p_i$ are poles of $z^{k-1}f(k)$ and residual $Res(g(z),p)=\frac{1}{(m-1)!}\left.\frac{\mathrm{d}^{m-1}}{\mathrm{d}z^{m-1}}\left((z-p)^mg(z)\right)\right\vert_{z=p}$ with $m$ being the multiplicity of the pole $p$ in $g$.
Power Series: same as inverse laplace.
Partial Fractions: same as inverse laplace.
Definition: $F(z,m)=\mathcal{Z}_m(f,m)=\mathcal{Z}(\left\{f(kT-(1-m)T)\right\} _{k\in\mathbb{N}^+})(z)$
We denote corresponding continuous form $\mathcal{L}(f(t-(1-m)T)\delta_ T(t))$ as $F^*(s,m)$
Residual Theorem: $\mathcal{Z}_m(f,m)=z^{-1}\sum _{p_i} Res(\frac{F(s)e^{mTs}}{1-z^{-1}e^{Ts}}, p_i)$
ModZ Transform is usually used when there's delay in the system, use this transform to shift the signal with proper $m$ value.
Definition: $F^* (s)=\sum^\infty_{n=0}f(n*T)e^{-nTs}$
Starred Transform is defined in continuous s-domain, but it only aggregates on discrete s values defined periodically by sampling time T, like Z-Transform. Starred Transform is usually exchangeable with Z-Transform with $z=e^{Ts}$.
Sometimes we also see * as an operator to sample a continuous signal. It converts a continuous signal to discrete delta functions. (See the "Sampler" section below)
Calculation from Laplace Transform
$F^*(s)=\sum_{p_i\in\{poles\;of\;F(\lambda)\}} Res\left(F(\lambda)\frac{1}{1-e^{-T(s-\lambda)}}, p_i\right)$
$F^*(s)=\frac{1}{T}\sum^\infty_{n=-\infty}F(s+jn\omega_s)+\frac{e(0)}{2}$ where $\omega_s=\frac{2\pi}{T}$
$F^*(s)$ is periodic in s plane with period $j\omega_s=\frac{2\pi j}{T}$
If $F(s)$ has a pole at $s=s_0$, then $F^*(s)$ must have poles at $s=s_0+jn\omega_s$ for $m\in\mathbb{Z}$
$A(s)=B(s)F^* (s) \Rightarrow A^* (s)=B^* (s)F^* (s)$, while usually $A(s)=B(s)F(s) \nRightarrow A^* (s)=B^* (s)F^* (s)$
Fourier transform is basically to substitute $s=j\omega$ into Laplace transform. Additional properties are not discussed here.
One important theorem (Shannon-Nyquist Sampling Theorem): Suppose $e:\mathbb{R}_+\to\mathbb{R}$ has a Fourier Transform with no frequency components greater than $f_0$, then $e$ is uniquely determined by the signal $e_s$ generated by ideally sampling $e$ with period $\frac{1}{2}f_0$.
A continuous-time linear state-space system can be described by following two equations: \begin{align}&\text{State equation}:\;&\dot{x}(t)&=A(t)x(t)+B(t)u(t),&\;x(t)\in\mathbb{R}^n,\;u(t)&\in\mathbb{R}^m \\&\text{Output equation}:\;&y(t)&=C(t)x(t)+D(t)u(t),&\;y(t)&\in\mathbb{R}^p\end{align}
The input $u:[0,\infty)\to\mathbb{R}^m$, state $x:[0,\infty)\to\mathbb{R}^n$, and output $y:[0,\infty)\to\mathbb{R}^p$ are all signals, i.e. functions of continuous time $t\in[0,\infty)$. The coefficients $A\in\mathbb{R}^{n\times n}$,$B\in\mathbb{R}^{n\times m}$,$C\in\mathbb{R}^{p\times n}$,$D\in\mathbb{R}^{p\times m}$
This linear time-varying (LTV) system can be written compactly as \begin{align*} \dot{x}&=A(t)x+B(t)u \\ y&=C(t)x+D(t)u\end{align*} Similarly, linear time-invariant (LTI) system can be written as \begin{align} \dot{x}&=Ax+Bu \\ y&=Cx+Du\end{align}
For non-linear system, the equation will be written as
time-varying (NLTV)
time-invariant (NTLI)
time-invariant autonomous
\begin{align*}\dot{x}&=f(x,u,t)\\y&=g(x,u,t)\end{align*}
\begin{align*}\dot{x}&=f(x,u)\\y&=g(x,u)\end{align*}
\begin{align*}\dot{x}&=f(x)\\y&=g(x)\end{align*}
Math prerequisites here:
For definition of function on matrix, see my notes for algebra basics
$e^A$ is matrix exponential, expm in MATLAB
$\frac{\mathrm{d}}{\mathrm{d}t}e^{At}=Ae^{At}=e^{At}A$
$e^{(A+B)t}\Leftrightarrow AB=BA$ (be careful when commute matrices)
$\mathcal{L}\{e^{At}\}=(sI-A)^{-1}$ (can be derived from property 1 and laplace derivative)
To calculate $e^A$
Eigenvalue decomposition
Jordan form decomposition
Directly evaluate infinite power series (converges quickly)
For more properties of the matrix function, seeMatrix Algebra
For homogeneous LTI system: $$\begin{align}x(t)=e^{A(t-t_0)}x_0\end{align}$$
"homogeneous" = zero-input, Eq.5 is also called zero input response (ZIR).
"homogeneous equation" = 齐次方程
For LTI system: $$\begin{align}x(t)=e^{A(t-t_0)}x(t_0)+\int^t_{t_0}e^{A(t-\tau)}Bu(\tau)d\tau\end{align}$$ This result requires $A$ to be time-invariant, $B,C,D$ can be time varying.
The solution consists of two parts: ZIR and ZSR (zero state response, $x(t_0)=0$), which are homogenenous solution (通解) and particular solution (特解) of the ODE.
ZIR and ZSR are both linear mapping
For homogeneous LTV system: $$\begin{align}x(t)=\Phi(t,t_0)x_0\end{align}$$
Matrix $\Phi$ is called the state transition matrix, defined as $$\begin{equation}\begin{split}\Phi(t,t_0)\equiv I+\int^t_{t_0}A(s_1)\mathrm{d}s_1+\int^t_ {t_0}A(s_1)\int^{s_1}_ {t_0}A(s_2)\mathrm{d}s_2\mathrm{d}s_1+\\ \int^t_ {t_0}A(s_1)\int^{s_1}_ {t_0}A(s_2)\int^{s_2}_ {t_0}A(s_3)\mathrm{d}s_3\mathrm{d}s_2\mathrm{d}s_1+\cdots\end{split}\end{equation}$$
Properties of $\Phi$:
$\Phi(t,t)=I$
$\frac{\mathrm{d}}{\mathrm{d}t}\Phi(t,t_0)=A(t)\Phi(t,t_0)$
(semigroup property) $\Phi(t,s)\Phi(s,\tau)=\Phi(t,\tau)$
$\forall t,\tau\geqslant 0,\;[\Phi(t,\tau)]^{-1}=\Phi(\tau,t)$
Eq.6 can be directly derived by evaluating Eq.8
For LTV system: $$\begin{align}x(t)=\Phi(t,t_0)x_0+\int^t_{t_0}\Phi(t,\tau)B(\tau)u(\tau)d\tau\end{align}$$
Some conclusions:
The solution given by Eq.9 is unique
The set of all solutions to ZIR system forms a vector space of dimension $n$
If $A(t)A(s)=A(s)A(t)$, then $\Phi(t,t_0)=e^{\int^t_{t_0}A(\tau)\mathrm{d}\tau}$
Phase Portraits: A phase portrait is a graph of several zero-input responses on the phase plane ($\dot{x}(t)$ and $x(t)$ are phase variables)
Usually in phase portraits, there are two straight lines corresponding to the eigenvector of A, other lines are growing in or opposite to the direction of the lines.
For LTI case, $\frac{Y(s)}{U(s)} = C(sI-A)^{-1}B+D$
This can be derived by take laplace transform of both sides of state equations
A discrete-time linear state-space system can be described by following two equations: $$\begin{align}&\text{State eq.}:\;&x(k+1)&=A(k)x(k)+B(k)u(k),&\;x\in\mathbb{R}^n,\;u&\in\mathbb{R}^m \\ &\text{Output eq.}:\;&y(k)&=C(k)x(k)+D(k)u(k),&\;y&\in\mathbb{R}^p\end{align}$$
The input $u:\mathbb{N}\to\mathbb{R}^m$, state $x:\mathbb{N}\to\mathbb{R}^n$, and output $y:\mathbb{N}\to\mathbb{R}^p$ are all signals, i.e. functions of continuous time $t\in\mathbb{N}$.
Discrete LTI system is sometimes written compactly as $$\begin{align} x_{k+1}&=Ax_k+Bu_k \\ y_k&=Cx_k+Du_k \end{align}$$
For LTI case, $H(z)=C(zI-A)^{-1}B+D$ (pulse tranfer function)
Note: hereafter $\mathfrak{R}$ denotes range space, $\mathfrak{N}$ denotes null space.
Controllability: $\exists u$ that drives any initial state $x(t_0)=x_0$ to $x(t_1)=0$
Reachability: $\exists u$ that drives initial state $x(t_0)=0$ to any $x(t_1)=x_1$
Consider the continuous LTV system $\dot{x}=A(t)x+B(t)u,\;x\in\mathbb{R}^n,u\in\mathbb{R}^m$.
Reachable Subspace: Given $t_0$ & $t_1$, the reachable subspace $\mathcal{R}[t_0, t_1]$ consists of all states $x_1$ for which there exists and input $u:[t_0, t_1]\to\mathbb{R}^m$ that transfers the state from $x(t_0)=0$ to $x(t_1)=x_1$.
$\mathcal{R}[t_0, t_1]\equiv\left\{x_1\in\mathbb{R}^n\middle|\exists u(\cdot),\;x_1=\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)u(\tau)\mathrm{d}\tau\right\}$
Controllable Subspace: Given $t_0$ & $t_1$, the controllable subspace $\mathcal{C}[t_0, t_1]$ consists of all states $x_0$ for which there exists an input $u:[t_0, t_1]\to\mathbb{R}^m$ that transfers the state from $x(t_0)=x_0$ to $x(t_1)=0$
$\mathcal{C}[t_0, t_1]\equiv\left\{x_0\in\mathbb{R}^m\middle|\exists u(\cdot),\;0=\Phi(t_1,t_0)x_0+\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)u(\tau)\mathrm{d}\tau\right\}$
or $\mathcal{C}[t_0, t_1]\equiv\left\{x_0\in\mathbb{R}^m\middle|\exists u(\cdot),\;x_0=\int^{t_1}_{t_0}\Phi(t_0,\tau)B(\tau)\left[-u(\tau)\right]\mathrm{d}\tau\right\}$
Reachability Grammian: $W_\mathcal{R}(t_0, t_1)\equiv\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)B(\tau)^\top\Phi^\top(t_1,\tau)\mathrm{d}\tau$ given times $t_1>t_0\geqslant0$
The system is reachable at time $t_0$ iff $\exists t_1$ s.t. $W_\mathcal{R}(t_0,t_1)$ is non-singular.
non-singular for some $t_1$ $\Rightarrow$ non-singular for any $t_1$
$\mathcal{R}[t_0,t_1]=\mathfrak{R}(W_\mathcal{R}(t_0,t_1))$
if $x_1=W_\mathcal{R}(t_0,t_1)\eta_1\in\mathfrak{R}(W_\mathcal{R}(t_0,t_1))$, the control $u(t)=B^\top(t)\Phi^T(t_1,t)\eta_1$,$t\in[t_0,t_1]$ can be used to transfer the system from $x(t_0)=0$ to $x(t_1)=x_1$ (w/ minimum energy)
minimum energy = minimum $\int^T_0\Vert u(\tau)\Vert^2\mathrm{d}\tau$
For LTI system $W_\mathcal{R}(t_0,t_1)=\int^{t_1}_ {t_ 0}e^{A(t_1-\tau)}BB^\top e^{A^{\top} (t_1-\tau)}\mathrm{d}\tau=\int^{t_1-t_ 0}_ {0}e^{At}BB^\top e^{A^{\top}t}$
Controllability Grammian: $W_\mathcal{C}(t_0, t_1)\equiv\int^{t_1}_{t_0}\Phi(t_0,\tau)B(\tau)B(\tau)^\top\Phi^\top(t_0,\tau)\mathrm{d}\tau$ given times $t_1>t_0\geqslant0$
The system is reachable at time $t_0$ iff $\exists t_1$ s.t. $W_\mathcal{C}(t_0,t_1)$ is non-singular.
$\mathcal{C}[t_0,t_1]=\mathfrak{R}(W_\mathcal{C}(t_0,t_1))$
if $x_0=W_\mathcal{C}(t_0,t_1)\eta_0\in\mathfrak{R}(W_\mathcal{C}(t_0,t_1))$, control $u(t)=-B^\top(t)\Phi^\top(t_0,t)\eta_0$,$t\in[t_0,t_1]$ can be used to transfer the state from $x(t_0)=x_0$ to $x(t_1)=0$ (w/ minimum energy)
For LTI system $W_\mathcal{C}(t_0,t_1)=\int^{t_1}_ {t_ 0}e^{A(t_0-\tau)}BB^\top e^{A^{\top} (t_0-\tau)}\mathrm{d}\tau=\int^{t_1-t_ 0}_ {0}e^{-At}BB^\top e^{-A^{\top}t}$
Controllability Matrix: For LTI system, controllability matrix $\mathcal{C}=[B\;|\;AB\;|\;A^2B\;\cdots\;A^{n-1}B]$
The controllability matrix works for both continuous and discrete system, and it's easier to be derived from discrete LTI equations: In discrete LTI, $\mathcal{C}\mathbf{u}=-A^k x_0$ where $\mathbf{u}=\begin{bmatrix}u_{k-1} & u_{k-2} & \ldots & u_0\end{bmatrix}^\top$
For LTI, $\mathcal{R}[t_0,t_1]=\mathfrak{R}(W_\mathcal{R}[t_0,t_1])=\mathfrak{R}(\mathcal{C})=\mathfrak{R}(W_\mathcal{C}[t_0,t_1])=\mathcal{C}[t_0,t_1]$
This implies Controllability $\Leftrightarrow$ Reachability for LTI systems.
The controllable subspace $\mathfrak{\mathcal{C}}$ is the smallest A-invariant subspace that contains $\mathfrak{\mathcal{B}}$
If the controllability matrix has full rank, the LTI system (or the pair $(A,B)$) is completely controllable
PBH-Eigenvector Test: An LTI system is not controllable iff there exists a nonzero left eigenvector $v$ of $A$ such that $vB=0$
PBH-Rank Test: An LTI system will be controllable iff $[\lambda I-A \;| \;B]$ has full row rank for all eigenvalue $\lambda$
For LTI system, there exists an input $u(\cdot)$ that transfer the state from $x_0$ ito $x_1$ in finite time $T$ iff $x_1-e^{AT}x_0\in\mathfrak{R}(\mathcal{C})$
The input that transfers any state $x_0$ to any other state $x_1$ in some finite time $T$ is $u(t)=B^\top e^{A^{\top}(T-t)}W_\mathcal{R}^{-1}(0,T)[x_1 -e^{AT}x_0]$, for $t\in[0,T]$ (w/ minimum energy)
Observability: Given any input $u(t)$ and output $y(t)$ over $t\in[t_0,t_1]$, it's sufficient to determine a unique initial state $\exists !x(t_0)$.
Observability Grammian: $W_\mathcal{O}(t_0,t_1)\equiv\int^{t_1}_{t_0}\Phi^\top(t_1,\tau)C^\top(\tau)C(\tau)\Phi(t_1,\tau)\mathrm{d}\tau$
The system is observable at time $t_0$ iff $\exists t_1$ s.t. $W_{\mathcal{O}}(0,t)$ is nonsingular.
For LTI system $W_{\mathcal{O}}(t_0,t_1)=\int^{t_1}_{t_0} e^{A^{\top}(t_1-\tau)}C^\top Ce^{A(t_1-\tau)}\mathrm{d}\tau=\int^{t_1-t_0}_0 e^{A^{\top}\tau}C^\top Ce^{A\tau}\mathrm{d}\tau$
Observability Matrix: For LTI system, observability $\mathcal{O}=\begin{bmatrix}C\\CA\\CA^2\\ \vdots\\CA^{k-1}\end{bmatrix}$
The controllability matrix works for both continuous and discrete system, and it's easier to be derived from discrete LTI equations: In discrete LTI, $\Psi_{k-1}=\mathcal{O}x_0$ where $$\Psi_k\equiv\begin{bmatrix}y_0\\y_1\\y_2\\ \vdots\\ y_{k-1}\end{bmatrix}-\begin{bmatrix} D & 0 & 0 & \cdots & 0 \\ CB & D & 0 & \cdots & 0 \\ CAB & CB & D & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ CA^{k-2}B & CA^{k-3}B & CA^{k-4}B & \cdots & 0 \end{bmatrix}\begin{bmatrix}u_0\\u_1\\u_2\\ \vdots \\ u_{k-1}\end{bmatrix}$$
If the controllability matrix has full rank, the LTI system (or the pair $(A,C)$) is completely observable.
PBH-Rank Test: An LTI system will be observable iff $\begin{bmatrix}A-\lambda I \\C\end{bmatrix}$ has full column rank for all eigenvalue $\lambda$
Duality Theorem: The pair $(A,B)$ is controllable iff the pair $(A^\top, B^\top)$ is observable.
Controllability only depends on matrix $A$ and $B$ while the Observability only depends on matrix $A$ and $C$
Duality theorem is useful for proof of observability conclusions from controllability
Adjoint System:
Original System
Adjoint System
Equations $$\begin{align*} \dot{x}&=A(t)x+B(t)u \\ y&=C(t)x \end{align*}$$ $$\begin{align*} \dot{p}&=-A^*(t)p-C^*(t)v \\ z&=B^*(t)p\end{align*}$$
Initial Condition $x(t_0)=x_0$ $p(t_1)=p_1$
State Trasition Matrix $\Phi(t,t_0)$ $\Phi^*(t_1,t)=\left(\Phi^*(t,t_1)\right)^{-1}$
Zero-State Response $$\begin{split}L_u:\;&u(\cdot)\to x(t_1)\\=&\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)u(\tau)\mathrm{d}\tau\end{split}$$ $$\begin{split}P_u:\;&v(\cdot)\to p(t_0)\\=&\int^{t_1}_{t_0}\Phi^*(\tau,t_0)C^*(\tau)v(\tau)\mathrm{d}\tau\end{split}$$
Zero-Input Response $$\begin{split}L_0:\;&x_0\to y(\cdot)\\=&C(\cdot)\Phi(\cdot,t_0)x_0\end{split}$$ $$\begin{split}P_0:\;&p_1\to z(\cdot)\\=&B^*(\cdot)\Phi^*(t_1,\cdot)p_1\end{split}$$
Duality Theorem Controllable ($\rho(L_u)=n$) Observable ($\rho(P_0^*)=n$)
Observable ($\rho(L_0^*)=n$) Controllable ($\rho(P_u)=n$)
A state is reachable ($x\in\mathfrak{R}(L_u)$) A state is unobservable ($x\in\mathfrak{N}(L_0)$)
Note that ZIR and ZSR are both linear mappings and $L_u^*=P_0,\;L_0^*=P_u$
Similarity Transform of a (LTI) system: Based on Eq.3 and Eq.4, define $x=P\bar{x}$, then we have $$\begin{align}\dot{\bar{x}}&=P^{-1}AP\bar{x}+P^{-1}Bu&=\bar{A}\bar{x}+\bar{B}u \\ y&=CP\bar{x}+Du&=\bar{C}\bar{x}+Du\end{align}$$
Similarity transform doesn't affect transfer function.
Controllability Decomposition: For an uncontrollable LTI system, define matrix $V=[V_1\;V_2]$ where $V_1$ is a basis for $\mathfrak{R}(\mathcal{C})$ and $V_2$ complete a basis for $\mathbb{R}^n$, then after similarity transform with $\bar{x}=V^{-1}x$, we can partition the system like following: $$\begin{align*}\dot{\bar{x}}&=\bar{A}\bar{x}+\bar{B}u&=&\begin{bmatrix}\bar{A}_ {11}&\bar{A}_ {12} \\ \mathbf{0} & \bar{A} _{22}\end{bmatrix}\begin{bmatrix}\bar{x} _1 \\ \bar{x} _2\end{bmatrix}+\begin{bmatrix}\bar{B} _1 \\ \mathbf{0}\end{bmatrix}u \\ y&=\bar{C}\bar{x}+Du&=& \begin{bmatrix}\bar{C} _1 & \bar{C} _2\end{bmatrix}\begin{bmatrix}\bar{x} _1 \\ \bar{x} _2\end{bmatrix} + Du\end{align*}$$ Here $\bar{x}_2$ is uncontrollable, and ZSR of the system with or without $\bar{x}_2$ is the same.
Observability Decomposition: For an unobservable LTI system, define matrix $U=\begin{bmatrix}U_1\\U_2\end{bmatrix}$ where $U_1$ is a basis for $\mathfrak{R}(\mathcal{O}^\top)$ and $U_2$ complete a basis for $\mathbb{R}^n$, then after similarity transform with $\hat{x}=Ux$, we can partition the system like following: $$\begin{align*}\dot{\hat{x}}&=\hat{A}\hat{x}+\hat{B}u&=&\begin{bmatrix}\hat{A}_ {11}&\mathbf{0} \\ \hat{A}_ {21} & \hat{A} _{22}\end{bmatrix}\begin{bmatrix}\hat{x} _1 \\ \hat{x} _2\end{bmatrix}+\begin{bmatrix}\hat{B} _1 \\ \hat{B} _2\end{bmatrix}u \\ y&=\hat{C}\hat{x}+Du&=& \begin{bmatrix}\hat{C} _1 & \mathbf{0}\end{bmatrix}\begin{bmatrix}\hat{x} _1 \\ \hat{x} _2\end{bmatrix} + Du\end{align*}$$
Realization: $\Sigma$ (system with Eq.3 and Eq.4) is a realization of $H(s)$ iff $H(s)=C(sI-A)^{-1}B+D$.
Equivalent: Two realizations are said to be equivalent if they have the same transfer function
Algebraically Equivalent: Two realizations have same transfer function and $n$ (dimension of states). (this implies a similarity transform between them)
Minimal Realization: $\Sigma$ is a minimal realization of $H(s)$ iff there doesn't exists an equivalent realization $\bar{\Sigma}$ with $\bar{n}< n$
$\Sigma$ is a minial realization iff $\Sigma$ is completely controllable and observable.
Kalman Cannonical Structure Theorem (aka. Kalman Decomposition): suppose $\rho(\mathcal{C})< n$ and $\rho(\mathcal{O})< n$, $\mathfrak{R}(\mathcal{C})$ are the controllable states, $\mathfrak{N}(\mathcal{O})$ are the unobservable states, define subspaces:
Subspaces
Controllable
$V_2\equiv\mathfrak{R}(\mathcal{C})\cup\mathfrak{N}(\mathcal{O})$ Yes No
$V_1$ s.t. $V_1\oplus V_2=\mathfrak{R}(\mathcal{C})$ Yes Yes
$V_4$ s.t. $V_4\oplus V_2=\mathfrak{N}(\mathcal{O})$ No No
$V_3$ s.t. $V_1\oplus V_2\oplus V_3\oplus V_4=\mathbb{C}^n$ No Yes
Then let $$\begin{align*}\tilde{x}=\begin{bmatrix}\tilde{x}_ 1\\ \tilde{x}_ 2\\ \tilde{x}_ 3\\ \tilde{x}_ 4\end{bmatrix},\;\tilde{A}=&\begin{bmatrix}A_ {\mathrm{co}} &&A_{13}&\\A_{21}&A_{\mathrm{c\bar{o}}}&A_{23}&A_{24}\\&&A_{\mathrm{\bar{c}o}}&\\&&A_{43}&A_{\mathrm{\bar{c}\bar{o}}} \end{bmatrix},\;\tilde{B}=\begin{bmatrix}B_{\mathrm{co}}\\ B_{\mathrm{c\bar{o}}} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \\ \tilde{C}=&\begin{bmatrix}C_{\mathrm{co}}&\mathbf{0}\quad&C_{\mathrm{\bar{c}o}}&\mathbf{0}\quad\end{bmatrix}\end{align*}$$
$$\tilde{\Sigma}:\begin{cases} \dot{\tilde{x}}=A_{\mathrm{co}}\tilde{x}_ 1+B_{\mathrm{co}}u_1\\ y=C_{\mathrm{co}}\tilde{x}_1\end{cases}$$
$\tilde{\Sigma}$ is completely controllable and completely observable.
Consider SISO systems $$H(s)=\frac{b(s)}{a(s)}=\frac{b_{n-1}s^{n-1}+\ldots+b_1s+b_0}{s^n+a_{n-1}s^{n-1}+\ldots+a_1s+a_0}=\frac{\sum^{n-1}_{j=0} b_js^j}{s^n+\sum^{n-1}_{i=0} a_is^i}=\frac{Y(s)}{U(s)}$$
Controllable Cannonical Form: $$\begin{align}\dot{x}&=\begin{bmatrix} 0&1&&\\ \vdots & & \ddots & \\ 0&&&1 \\-a_0&-a_1&\cdots&-a_{n-1}\end{bmatrix}x+\begin{bmatrix}0\\ \vdots \\ 0 \\ 1\end{bmatrix}u&=&A_cx+B_cu\\ y&=\begin{bmatrix}\quad b_0 &\quad b_1 &\cdots & \quad b_{n-1}\end{bmatrix}x&=&C_cx \end{align}$$
$(A_c, B_c)$ is controllable
$(A_c, C_c)$ is observable if $a(s)$ and $b(s)$ have no common factors
Observable Cannonical Form: $$\begin{align}\dot{x}&=\begin{bmatrix} 0&&&&-a_0\\ 1 & \ddots &&&-a_1 & \\ &\ddots&\ddots&&\vdots \\&&\ddots&0&-a_{n-2} \\ &&&1&-a_{n-1}\end{bmatrix}x+\begin{bmatrix}b_0\\ b_1 \\ \vdots \\ b_{n-2} \\ b_{n-1}\end{bmatrix}u&=&A_ox+B_ou\\ y&=\begin{bmatrix}0 & \;\cdots &\;\cdots & 0 & \quad 1\qquad \end{bmatrix}x&=&C_ox \end{align}$$
$(A_o, C_o)$ is observable
$(A_o, B_o)$ is controllable if $a(s)$ and $b(s)$ have no common factors
Model Cannonical Forms: Do Spectral Decomposition (eigen-decomposition) or Jordan Decomposition, and then use the modal matrix (matrix of eigenvectors) to do similarity transform.
The Gilbert Realization: Let $G(s)$ be a $p\times m$ rational transfer function with simple poles (nonrepeated) at $\lambda_i,\;i=1,2,\ldots,k$. Calculate partial fraction expansion $$G(s)=\sum^k_{i=1}\frac{R_i}{s-\lambda_i},\qquad \text{Residue}\;R_i=\lim_{s\to\lambda_i}(s-\lambda_i)G(s)$$ Let $r_i=\rho(R_i)$, now write $R_i=C_iB_i$ where $C_ i\in\mathbb{R}^ {p\times r_ i},\;B_ i\in\mathbb{R}^ {r_ i\times p}$, then write $$A=\mathrm{blkdiag}\{\lambda_i I_{r_i}\},\;B^\top=[B_1^\top \;\cdots\; B^\top_k],\;C=[C_1\; \cdots\;C_k]$$, then $(A,B,C)$ is a realization of $G(s)$ with order $n=\sum^k_1 r_i$
For MIMO system the cannonical forms with be quite complex:
Controllable Cannonical Form (for MIMO): Here we provide a way to convert from controllable LTI system to controllable. The collection of independent columns of $\mathcal{C}$ may be expressed as $$M=[b_1\;Ab_1\; \cdots\;A^{\mu_1-1}b_1\;|\;b_2\;Ab_2\;\cdots\;A^{\mu_2-1}b_2\;|\;\cdots\;|\;b_p\;Ab_p\;\cdots\;A^{\mu_p-1}b_p]$$ Construct $M^{-1}$ and then $T$:$$M^{-1}=\left[m_{11}^\top\;m_{12}^\top\;\cdots\;m_{1\mu_1}^\top\;\middle|\;\cdots\;\middle|\;m_{p1}^\top\;m_{p2}^\top\;\cdots\;m_{p\mu_p}^\top \right]^\top$$ $$T=\left[m_{1\mu_1}^\top\;(m_{1\mu_1}A)^\top\;\cdots\;\left(m_{1\mu_1}A^{\mu_1-1}\right)^\top\;\middle|\;\cdots\;\middle|\; m_{p\mu_p}^\top\;(m_{p\mu_p}A)^\top\;\cdots\;\left(m_{p\mu_p}A^{\mu_p-1}\right)^\top\right]^\top$$ Perform similarity transform with $\bar{x}=Tx$ and the canonical form will be obtained like following: $$\bar{A}=\begin{bmatrix}\bar{A}_ {\mu_1\times\mu_1}&\mathbf{0}_ {\cdot\cdot}&\cdots&\mathbf{0}_ {\cdot\cdot} \\ \mathbf{0}_ {\cdot\cdot}&\bar{A}_ {\mu_2\times\mu_2}&\cdots&\mathbf{0}_ {\cdot\cdot}\\ \vdots&\vdots&\ddots&\vdots \\ \mathbf{0}_ {\cdot\cdot}&\mathbf{0}_ {\cdot\cdot}&\cdots&\bar{A}_ {\mu_ p\times\mu_ p} \end{bmatrix},\quad\bar{B}=\begin{bmatrix}\mathbf{0}_ {\cdot n}\\ \mathbf{0}_ {\cdot (n-1)}\\ \vdots\\ \mathbf{0}_ {\cdot 1}\end{bmatrix}$$ Here $\bar{A}_ {\mu_ i\times\mu_ i}$ is the same structure as in SISO, $\mathbf{0}_ {\cdot\cdot}$ is a zero matrix except the last row, $\mathbf{0}_ {\cdot i}$ is a zero matrix except for the last row, and in the last row there are $i$ non-zeros on the right with the first element being 1.
System Characteristic Equation: The polynomical with the roots equal to the poles of the output that are independent of the input.
System Type: A plant $G$ can always be written as $G(s)=\frac{K\prod^m_{i=1}(s-s_i)}{s^N\prod^p_{j=1}(s-s_j)},\;z_i,z_j\neq 0$ or $G(z)=\frac{K\prod^m_{i=1}(z-z_i)}{(z-1)^N\prod^p_{j=1}(z-z_j)},\;z_i,z_j\neq 1$. Here $N$ is called the system type of $G(z)$.
Properties that matters for a controller:
Exogenous disturbance rejection
Bounded control effort
Stability means when the time goes to infinity, the system response is bounded.
A system is stable if all its poles lies in the left half of $s$-plane or all inside the unit circle of $z$-plane.
A system is marginally stable if one of the pole is on the imaginary axis of $s$-plane or on the unit circle of $z$-plane.
Stability of linear systems is independent of input
The stability of a linear system can be evaluated by its characteristic equation $1-G_{op}(z)=0$, where $G_{op}$ is the open-loop transfer function (transfer function when input is eliminated, or feedback route is cut off).
Methods to evaluate stability
Routh-Hurwitz Criterion: $s$-plane (omited here, see Wikipedia)
For discrete system, a strategy is use bilinear transformation: $z=e^{\omega T}\approx \frac{1+\omega T/2}{1-\omega T/2}$
Jury Criterion: $z$-plane (see Wikipedia)
Root Locus Method: both $s$- and $z$-plane (see Wikipedia, rlocus in MATLAB)
Nyquist Criterion: both $s$- and $z$-plane (see Wikipedia, nyquist in MATLAB)
It works for both continuous and discrete systems, the difference is that in $s$-plane the detour point is at $s=0$ while in $z$-plane the detour point is at $z=1$.
Bode Diagrams: draw frequency response for (pulse) transfer function, works for both $s$- and $z$-plane (see Wikipedia, bode in MATLAB)
A review of the stability judgement method here at 知乎
Lyaponove Stability is only concerned with the effect of initial conditions on the response of the system (ZIR)
Equilibrium Point $x_e$: Consider NLTV $\dot{x}(t)=f(x(t),u(t),t)$, equilibrium point satisfies $x(t_0)=x_e,\;u(t)\equiv 0\Rightarrow x(t)=x_e,\;\text{i.e. }f(x_e,0,t)=0,\;\forall t>t_0$
For discrete system, it's $x(k+1)=x(k)=x_e$
For LTI system, $x_e$ can be calculated from $Ax_e=0$, so the origin $x=0$ is always an equilibrium point.
Set of equilibrium points in LTI systems are connected.
Lyapunov stability: An equilibrium point $x_e$ of the system $\dot{x}=A(t)x$ is stable (in the sense of Lyapunov) iff $\forall \epsilon>0,\;\exists \delta(t_0,\epsilon)>0$ s.t. $\Vert x(t_0)-x_e\Vert<\delta\Rightarrow\Vert x(t)-x_e\Vert <\epsilon,\;\forall t>t_0$
$x_e$ is uniformly stable if $\delta=\delta(\epsilon)$ (regardless of $t_0$)
$x_e$ in LTI is stable $\Rightarrow x_e$ is uniformly stable
$x_e$ is asymptotically stable if $\Vert x(t)-x_e\Vert\to 0$ as $t\to 0$
$x_e$ is exponentially stable if $\Vert x(t)-x_e\Vert \leqslant \gamma e^{-\lambda(t-t_0)}\Vert x(t_0)-x(e)\Vert$
$x_e$ in LTI is asymptotically stable $\Rightarrow x_e$ is exponentially stable
$x_e$ is globally stable if $\delta$ can be chosen arbitrarily large
For LTV system, the system is stable (the zero solution is stable) iff $\Phi(t,t_0)$ is bounded by $K(t_0)$.
If bounded by constant $K$, then the system is uniformly stable.
If bounded by constant $K$ and $\Vert\Phi(t,0)\Vert\to 0$ as $t\to 0$, then the system is asymptotically stable.
For LTI system $\dot{x}=Ax$, it is Lyapunov stable iff $\mathrm{Re}(\lambda_i)\leqslant 0$ or $\mathrm{Re}(\lambda_i)=0,\;\eta_i=1$. ($\eta_i$ is the multiplicity of $\lambda_i$)
If $\mathrm{Re}(\lambda_i)<0$, then the system is asymptotically stable
Internal stability: concerns the state variables
External stability: concerns the output variables
Notes for contents below:
Positive definite (pd.) function: function $V$ is pd. wrt. $p$ if $V(x)>0,\;x\neq p$ and $V(x)=0,\;x=p$
$C^n$ denotes the set of continuous and at least n-th differentiable functions
Lyapunov's Direct Method: Let $\mathcal{U}$ be an open neighborhood of $p$ and let $V:\mathcal{U}->\mathbb{R}$ be a countinuous positive definite $C^1$ function wrt. $p$, we have following two conclusions:
If $\dot{V}\leqslant 0$ on $\mathcal{U}\backslash\{p\}$ then $p$ is a stable fixed point of $\dot{x}=f(x)$
If $\dot{V}< 0$ on $\mathcal{U}\backslash\{p\}$ then $p$ is an asymptotically stable fixed point of $\dot{x}=f(x)$
Lyapunov Function:
A function satisfying conclusion 1 is called a Lyapunov function
A function satisfying conclusion 2 is called a strict Lyapunov function
A function that is $C^1$ and pd. is called a Lyapunov function candidate
The energy function usually can be used as Lyapunov function. If it's only semi-positive definite, one can use LaSalle's Theorem
For LTI system, the zero solution of $\dot{x}=Ax$ is asymptotically stable iff $\forall$ pd. hermitian matrices $Q$, equation $A^*P+PA=-Q$ has a unique hermitian solution $P$ that is positive definite.
$A^*P+PA=-Q$ is called Lyapunov's Matrix Equation
here $V(x)=x^* Px=\int^\infty_0 x^*(t)Qx(t)dt$, which can be also called cost-to-go, or generalized energy
Lyapunov's Indirect Method (Lyapunov's First Method / Lyapunov's Linearization Theorem): The nonlinear system $\dot{x}=f(x)$ is (locally) asymptotically stable near the equilibrium point $x_e$ if the linearized system $\dot{x}_L=\frac{\partial f}{\partial x}(x_e)x_L$ is asymptotically stable.
BIBO stability is only concerned with the response of the system to the input (ZSR).
Bounded-Input Bounded-Output (BIBO) stability: The LTV system is said to be (uniformly) BIBO stable if there exists a finite constant $g$ s.t. $\forall u(\cdot)$, its forced response $y_f(\cdot)$ satisfies $$ \sup_{t\in[0,\infty)}\Vert y_f(t)\Vert \leqslant g \sup_{t\in[0,\infty)} \Vert u(t)\Vert $$
The impulse response can be analyzed to assess BIBO stability The LTV system is uniformly BIBO stable iff every entry of $D(t)$ is bounded and $\sup_{t\geqslant 0}\int^t_0|g_{ij}(t,\tau)|d\tau <\infty$ for every entry $g_{ij}$ of the matrix $C(t)\Phi(t,\tau)B(\tau)$.
BIBO stability is related with the stability descibed in classical control theory.
Exponential Lyapunov Stability $\Rightarrow$ BIBO stability
Steady state accurary can be derived from the property of Laplace/Z-transform as mentioned above (assuming stability) $$\lim\limits_{t\to \infty} f(t) = \lim\limits_{z\to 1} (z-1)F(z) = \lim\limits_{s\to 0}sF(s)$$
Some measurements of transient response (with step input):
Rise time $t_r$: time from 10% to 90% of steady state value
Peak overshoot: $M_p$ for overshoot magnitude and $t_p$ for time
Settling time $t_s$: time after which the magnitude fall in $1-d$ to $1-d$ final value. $d$ is usually %2~5.
Given a transfer function $H(z)$ with parameter $\Theta\in\mathbb{R}$, then sensitivity is defined as $S_H=\frac{\partial H}{\partial \Theta}\cdot\frac{\Theta}{H} = \frac{\partial H/H}{\partial \Theta/\Theta}$
The following image shows a minial example of sampling and hold.
Ideal sampler (a.k.a impulse modulator) converts a continuous signal $e: \mathbb{R}_+ \to \mathbb{R}$ to a discrete one $\hat{e}: \mathbb{N}\to \mathbb{R}$, such that $$ \hat{e}=e(t)\delta(t-kT)=e(t)\delta_T(t); \forall k\in \mathbb{N} $$
Ideal sampler is actually applying starred transform.
How to sample? A rule of thumb used to select sampling rates is chosing a rate of at least 5 samples per time constant.
The $\tau$ appearing in the transient response term $ke^{-t/\tau}$ of a first order analog system is called the time constant.
If the sampling time is too large, it can make the system unstable.
Zero order hold (ZOH): $ZOH(\{e(k)\}_{k\in\mathbb{N}})(t) = e(k)\;for\;kT\leq t\leq (k+1)T$
Alternative form: $ZOH(\{e(k)\})=\sum^\infty_{k=0}e(k)(H(t-kT)-H(t-(k+1)T))$
Its Laplace Transform: $G_{ZOH}(s)=\frac{1-e^{-Ts}}{s}$
First order hold (FOH): (delayed version) $$FOH(\{e(k)\}_ {k\in\mathbb{N}})(t)=\sum_ {k\in\mathbb{N}}\left[e(kT)+\frac{t-kT}{T}(e(kT)-e((k-1)T)) \right]\left[H(t-kT)-H(t-(k+1)T) \right]$$
Suppose we are given the LTI continuous system $$\begin{align*} \dot{x}(t) &= Ax(t)+Bu(t) \\ y(t)&=Cx(t)+Du(t) \end{align*}$$ If the input is sampled and ZOH and the output is sampled, then $$\begin{align*} x(k+1)&=\bar{A}x(k)+\bar{B}u(k) \\ y(k)&=\bar{C}x(k)+\bar{D}u(k)\end{align*}$$ where $\bar{A}=\Phi((k+1)T,kT)=e^{AkT}$
$\bar{B}=\int^{(k+1)T}_{kT}\Phi((k+1)T,\tau)B\mathrm{d}\tau=A^{-1}(e^{AT}-I)$
$\bar{C}=C$ and $\bar{D}=D$
Steps to apply conversion:
Dervice SS model for analog system
Calculate discrete representation (c2d in MATLAB)
Calculate pulse transfer function (ss2tf in MATLAB) If there is a complex system with multiple sampling and holding, a general rule is
Each ZOH output is assumed to be an input
Each sampler input is assumed to be an output and then create continuous state space from analog part of the system, then discretize them to generate discrete equations
When converting $s$ to $z$, the complex variables are related by $z=e^{Ts}$. Suppose $s=\sigma+j\omega$, then $z=e^{T\sigma}\angle \omega T$
Note: if frequencies differ in integer multiples of the sampling frequency $\frac{2\pi}{T}=\omega_s$, then they are sampled into the same location in the $z$-plane.
For transient response relationship, suppose $s$-plane poles occur at $s=\sigma\pm j\omega$, then the transient response if $Ae^{\sigma t}\cos(\omega t+\varphi)$. When sampling occurs at $z$-plane poles, then the transient response if $Ae^{\sigma kT}\cos(\omega kT+\varphi)$.
Example: 2nd order transfer function $$G(s)=\frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}$$ The $z$-plane poles occur at $z=r\angle\pm\theta$ where $r=e^{-\xi\omega_n T}$ and $\theta=\omega_n T\sqrt{1-\xi^2}$.
Then we can get the inverse relationship
$\xi=-\ln( r)/\sqrt{\ln^2( r)+\theta^2}$
$\omega_n=(1/T)\sqrt{\ln^2( r)+\theta^2}$
(time constant) $\tau=-T/\ln( r)$
Jacobian Linearization: linearize $\dot{x}=f(x,u)$ at an equilibrium $(x_e, u_e)$ is $$\frac{\mathrm{d}z}{\mathrm{d}t}=Az+Bv,\quad\text{where}\;A=\left.\frac{\mathrm{d}f}{\mathrm{d}x}\right|_ {\begin{split}x=x_e\\u=u_e\end{split}},\;B=\left.\frac{\mathrm{d}f}{\mathrm{d}u}\right|_ {\begin{split}x=x_e\\u=u_e\end{split}},\;z=(x-x_e),\;v=(u-u_e)$$
change $(x_e, u_e)$ to a trajectory $(x_e(t), u_e(t))$ we can linearize the system about a trajectory.
This method can be used for both continuous and discrete systems, just make sure to use corresponding method for choosing correct closed-loop transfer function.
For state space systems, with access to all of the state variables, we can change the $A$ matrix and thereby change the system dynamics by feedback.
Consider SISO LTI system ($u\in\mathbb{R},y\in\mathbb{R}$), we define the input as $u\equiv Kx+Ev$ where $K\in\mathbb{R}^{1\times n},\;E\in\mathbb{R}$ is an input matrix and $v(t)\in\mathbb{R}^\mathbb{R}$ is the exogeneous (externally applied) input. The new system will be $$\begin{align}\dot{x}&=(A+BK)x+BEv\\y&=(C+DK)x+DEv\end{align}$$ The mission is to find a state update matrix $A_{\mathrm{CL}}\equiv A+BK$ with desired set of eigenvalues, therefore we can construct $A_{\mathrm{CL}}$ with specific eigenvalues and then calculate $K$. This process will be quite easy if the system is already in controllable cannonical form. (which can be constructed directly from transfer function or using similarity transform)
Another way (SISO only) to calculate $K$ without controllable cannonical form is using the following formulae given the desired characteristic polynomial $\phi^{\star}(s)=s^n+\sum^{n-1}_ {i=0} a^\star_i s^i$ and original characteristic polynomial $\phi(s)=s^n+\sum^{n-1}_ {i=0} a_i s^i$
Ackermann's Formula: $K=-e^\top_n\mathcal{C}^{-1}\phi^\star(A)$ (here $e_i$ is unit vector with 1 at i-th position)
Bass-Gura's Formula: $$K=-[(a^\star_{n-1}-a_{n-1}) \;\cdots\;(a^\star_0-a_0)]\begin{bmatrix}1&a_{n-1}&a_{n-2}&\cdots&a_1\\&1&a_{n-1}&\cdots&a_2\\ &&\ddots&\ddots&\vdots \\ &&&1&a_{n-1}\\ &&&&1\end{bmatrix}^{-1}\mathcal{C}^{-1}$$
Note that the zeros of transfer function will not be affected by state feedback.
Some times we don't have the direct access to the state, we need construct an observer. For stochastic version, please check my notes for stochastic system.
Assume a plant $\Sigma$ and an (Luenberger) observer $\hat{\Sigma}$: $$\Sigma:\begin{cases}\dot{x}=Ax+Bu\\ y=Cx\end{cases},\quad \hat{\Sigma}:\begin{cases} \dot{\hat{x}}=A\hat{x}+Bu+L(y-\hat{y})\\ y=C\hat{x}\end{cases}$$
Subtract observer dynamics from plant dynamics and define $e\equiv x-\hat{x}$, the dynamics for $e$ is $\dot{e}=(A-LC)e$ and $y-\hat{y}=Ce$. This error dynamic $A_e=A-LC$ can be easily changed with observable cannonical form. (which similarly can be constructed directly from transfer function or using similarity transform)
Reduced-order Observer: If the state length of the system $n$ is large while $n-p$ is small, split the system and let $x_1$ holds the states that can be measured directly while $x_2$ holds states that are to be estimated, (i.e. $y=x_1+Du$). Define $z=\hat{x}_2-Lx_1$ then the system runs like $$\begin{align*}\begin{bmatrix}x_1 \\ \hat{x}_2\end{bmatrix}&=\begin{bmatrix} y-Du\\ z+Lx_1 \end{bmatrix}\qquad\begin{split}&\text{measurement} \\ &\text{observer}\end{split} \\ u&=K\begin{bmatrix}x_1 \\ \hat{x}_2 \end{bmatrix} + v \qquad\text{control law}\end{align*}$$ And then the error we care about is only $e=x_2-\hat{x}_2$.
Ackermann's Formula: $L=\phi^\star(A)\mathcal{O}^{-1}e_n$ ($\phi^\star$ is the desired characteristic function for $A_e$)
Separation Principle: If a stable observer and stable state feedback are designed for an LTI system, then the combined observer and feedback will be stable.
Errors from state estimation
Inaccurate knowledge of $A$ and $B$
Initial condition uncertainty
Disturbance or sensor error It's advised to choose observer poles to be 2-4x faster than closed loop poles
Motivation: handle control constraints and time varying dynamics with performance metric (ideas of optimal control) Note: $x^\top Ax$ is called a quadratic form, $x^\top Ay$ is called a bilinear form
A quadratic function $f(x)=x^\top Dx+C^\top x+c_0$ has one minimizer iff $D\succ 0$, or multiple minimizers iff $D\succeq 0$.
(discrete finite time) Linear Quadratic Regulator (LQR): the control problem is defined as $$\begin{align*}\min_{u\in\left(\mathbb{R}^m\right)^{\{0,\ldots,N\}}} J_{N}(u,x_0)&=\frac{1}{2}\sum^{N}_{k=0}(x^\top(k)Q(k)x(k)+u^\top(k)R(k)u(k)) \\ \mathrm{s.t.}\qquad x(k+1) &= A(k)x(k) + B(k)u(k)\quad \forall k\in\{0,\ldots,N-1\}\\ y(k)&=C(k)x(k)\\ x(0)&=x_0\end{align*}$$ where $Q(k)\succ 0$ and $R(k)\succ 0$
Bellman's Principle of Optimality: If a closed loop control $u^\star$ is optimal over the interval $0\leqslant k\leqslant N$, it's also optimal over any subinterval $m\leqslant k\leqslant N$ where $m\in\{0,\ldots,N\}$
The Minimum Principle: The optimal input to the LQR problem satisfies the following backward equations: $$\begin{align*}u^\star(k)&=-K(x)x(k) \\ K(k)&=\left[B^\top(k) P(k+1)B(k)+\frac{1}{2}R(k)\right]^{-1}B^\top(k)P(k+1)A(k) \\ P(k)&=A^\top(k)P(k+1)[A(k)- B(k)K(k)]+\frac{1}{2}Q(k)\end{align*}$$ and $P(N)=Q(N),\;K(N)=0$. The optimal cost is $J^\star_N=x^\top(0)P(0)x(0)$
For infinite horizon, $K(k)$ start becoming constants. The optimal input for LQR problem (assuming the system became LTI when $N\to\infty$) is $u^*(k)=-Kx(k)$ where $$K=(B^\top PB+R/2)^{-1}B^\top PA$$ and $P\succ 0$ is the unique solution to the discrete-time algebraic Riccati Equation: $$P=A^\top PA-A^\top PB\left(B^\top PB+R/2\right)^{-1}B^\top PA+Q/2$$
#Math#Control
Notes for Algebra Basics
Notes for Stochastic System
Notes for Probability Theory (Basics)
Theme Stacked designed by Jimmy, modified by Jacob


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CDF: Complementary Norms of the Apostolic Constitution "Anglicanorum coetibus"
Holy Father Approves Updated Norms for Former Anglicans and Episcopalians
April 09, 2019 17:25ZENIT StaffCatholic Church
The Congregation for the Doctrine of the Faith on April 9, 2019, issued Complementary Norms of the Apostolic Constitution "Anglicanorum coetibus".
In an article published in today's "L'Osservatore Romano" regarding the Complementary Norms of the Apostolic Constitution "Anglicanorum coetibus", the author Nicola Gori explains that in 2009 Benedict XVI signed the Apostolic Constitution Anglicanorum Coetibus on the institution of personal Ordinariates for Anglicans who enter into full communion with the Catholic Church, in response to requests from some Anglican groups who wished to be received, also corporately, in full Catholic communion. Currently, there exist three Ordinariates: Our Lady of Walsingham in England, the Chair of Saint Peter in the United States of America, and Our Lady of the Southern Cross in Australia.
The 2009 constitution provides general norms, integrated with complementary norms. During the ten years that have passed since its publication, some suggestions and theological indications have been received, both ecumenical and in relation to canon law, to make the application of the provisions more consistent with the spirit of the Apostolic Constitution. This led to a new version of these norms, decided by the Congregation for the Doctrine of the Faith and approved by Pope Francis on 8 March this year.
"In more detail", Gori explains, "in Article 4, where reference is made to the Ordinariate which has the faculty of incardinating into the Ordinariate the Anglican ministers who have entered into full communion with the Catholic Church, in the revised complementary provisions particular reference is made to those who are already incardinated in a diocese by virtue of the Pastoral Provision and the candidates belonging to the Ordinariate promoted by him to the Sacred Orders". In addition, "clerics who are about to be incardinated in the Ordinariate must be uncardinated from their original dioceses".
As Gori reiterates, the Pastoral Provision, authorized by Saint John Paul II in 1980, was created in 1980 in the United States of America to receive married former Anglican priests in the Catholic ordained ministry.
"With regard to the faithful of the Ordinariate, dealt with by Article 5, the new Complementary Norms specify the case of a person who has been validly baptized in another ecclesial community outside the Catholic Church", the author continues. "When such a person expresses the wish to enter into full communion with the Catholic Church, thanks to the efforts and the evangelizing mission of the Ordinariate, he may be admitted and belong to the Ordinariate from the moment in which he enters into full communion and receives the sacraments of Confirmation and the Eucharist. Furthermore, it should be underlined that this is applied also to those who have not been validly baptized, but who again thanks to the evangelizing action of the Ordinariate have discovered the faith so as to receive all the sacraments of initiation".
In Article 10, regarding the clergy, a further novelty is introduced. While in previous paragraph 2, it was specified that the "candidates may receive other aspects of priestly formation at a seminary program or house of formation established, with the consent of the Governing Council, expressly for the purpose of transmitting Anglican patrimony". In the new Complementary Norms, the same paragraph 2 states that "Ordinariate Seminarians will receive their theological formation with other seminarians at a seminary or theological faculty in agreement with the Diocesan Bishop or Bishops concerned". By adding that such candidates "may receive other aspects of priestly formation", the norm is simplified and adapted to current needs. Again in Article 10, regarding the clergy, in paragraph 5 it is added that the Ordinariate, for the purpose of the continuing formation of its clergy, will promote participation both "in local programs for formation provided by the Episcopal Conference and the Diocesan Bishop" and "in their own programs of ongoing formation".
In the new Complementary Norms, an entire article has been added, number 15, dedicated to the celebration of Divine Worship. It is acknowledged that the Missal proper to the personal Ordinariates, entitled "Divine Worship", namely the form approved by the Holy See for use by the Ordinariate, expresses and preserves for Catholic worship "the worthy Anglican liturgical patrimony, understood as that which has nourished the Catholic faith throughout the history of the Anglican tradition and prompted aspirations towards ecclesial unity".
This is the reason for the emphasis that public liturgical worship following Divine Worship is limited to the personal Ordinariates, as established by the Apostolic Constitution Anglicanorum coetibus, explains Gori. "Any priest incardinated in the Ordinariate is authorized to celebrate using Divine Worship. This applies outside the parishes of the Ordinariate when the priest celebrates Mass without the participation of the faithful, and also publicly with the permission of the rector or parish priest of the church or of the parish concerned. Furthermore, when pastoral needs demand it, or in the absence of a priest incardinated in an Ordinariate, if requested, any priest incardinated in the diocese or in an institute of consecrated life or of a society of apostolic life can celebrate in accordance with Divine Worship for the members of the Ordinariate. Finally, it is granted to any priest incardinated in the diocese or in an institute of consecrated life or in a society of apostolic life to concelebrate following Divine Worship".
Complementary Norms
for the Apostolic Constitution Anglicanorum coetibus
Jurisdiction of the Holy See
Each Ordinariate is subject to the Congregation for the Doctrine of the Faith. It maintains close relations with the other Roman Dicasteries in accordance with their competence.
Relations with Episcopal Conferences and Diocesan Bishops
§1. The Ordinary follows the directives of the national Episcopal Conference insofar as this is consistent with the norms contained in the Apostolic Constitution Anglicanorum coetibus §2. The Ordinary is a member of the respective Episcopal Conference.
The Ordinary, in the exercise of this office, must maintain close ties of communion with the Bishop of the Diocese in which the Ordinariate is present in order to coordinate its pastoral activity with the pastoral program of the Diocese.
§1. The Ordinary may be a bishop or a presbyter appointed by the Roman Pontiff ad nutum Sanctae Sedis, based on a terna presented by the Governing Council. Canon Law 383-388, 392-394, and 396-398, of the Code of Canon Law, apply to him.
§2. The Ordinary has the faculty to incardinate in the Ordinariate former Anglican ministers who have entered into full communion with the Catholic Church; particularly those already incardinated in a diocese by virtue of the Pastoral Provision; as well as candidates belonging to the Ordinariate and promoted to Holy Orders by him. Clerics incardinating into the Ordinariate must excardinate from their former Diocese.
§3. Having first consulted with the Episcopal Conference and obtained the consent of the Governing Council and the approval of the Holy See, the Ordinary can erect as needed territorial deaneries supervised by a delegate of the Ordinary covering the faithful of multiple personal parishes.
The Faithful of the Ordinariate
§1. The lay faithful originally of the Anglican tradition who wish to belong to the Ordinariate, after having made their Profession of Faith and received the Sacraments of Initiation, with due regard for Canon 845, are to be entered in the apposite register of the Ordinariate. Those who have received all of the Sacraments of Initiation outside the Ordinariate are not ordinarily eligible for membership, unless they are members of a family belonging to the Ordinariate.
§2. A person who has been baptized in the Catholic Church but who has not completed the Sacraments of Initiation, and subsequently returns to the faith and practice of the Church as a result of the evangelizing mission of the Ordinariate, may be admitted to membership in the Ordinariate and receive the Sacrament of Confirmation or the Sacrament of the Eucharist or both.1
§3. A person, who has been validly baptized in another Ecclesial Community outside of the Catholic Church, and subsequently desires to enter into full communion with the Catholic Church through the evangelizing mission of the Ordinariate, may be admitted to membership in the Ordinariate upon reception into full communion and conferral of the Sacraments of Confirmation and Eucharist. Also, this applies to the case of those not being validly baptized that have come to the faith through the evangelizing mission of the Ordinariate and therefore receive in it all of the sacraments of initiation.
§4. Lay faithful and members of Institutes of Consecrated Life and Societies of Apostolic Life, when they collaborate in pastoral or charitable activities, whether diocesan or parochial, are subject to the Diocesan Bishop or to the pastor of the place; in which case the power of the Diocesan Bishop or pastor is exercised jointly with that of the Ordinary and the pastor of the Ordinariate.
§1. In order to admit candidates to Holy Orders the Ordinary must obtain the consent of the Governing Council. In consideration of Anglican ecclesial tradition and practice, the Ordinary may present to the Holy Father a request for the admission of married men to the presbyterate in the Ordinariate, after a process of discernment based on objective criteria and the needs of the Ordinariate. These objective criteria are determined by the Ordinary in consultation with the local Episcopal Conference and must be approved by the Holy See.
§2. Those who have been previously ordained in the Catholic Church and subsequently have become Anglicans, may not exercise sacred ministry in the Ordinariate. Anglican clergy who are in irregular marriage situations may not be accepted for Holy Orders in the Ordinariate.
§3. Presbyters incardinated in the Ordinariate receive the necessary faculties from the Ordinary.
§1 The Ordinary must ensure that adequate remuneration be provided to the clergy incardinated in the Ordinariate, and must provide for their needs in the event of sickness, disability, and old age.
§2 The Ordinary will enter into discussion with the Episcopal Conference about resources and funds which might be made available for the care of the clergy of the Ordinariate.
§3. When necessary, priests, with the permission of the Ordinary, may engage in a secular profession compatible with the exercise of priestly ministry (cf. CIC, can. 286).
§1. The presbyters, while constituting the presbyterate of the Ordinariate, are eligible for membership in the Presbyteral Council of the Diocese in which they exercise pastoral care of the faithful of the Ordinariate (cf. CIC, can. 498, §2).
§2. Priests and Deacons incardinated in the Ordinariate may be members of the Pastoral Council of the Diocese in which they exercise their ministry, in accordance with the manner determined by the Diocesan Bishop (cf. CIC, can. 512, §1).
§1. The clerics incardinated in the Ordinariate should be available to assist the Diocese in which they have a domicile or quasi-domicile, where it is deemed suitable for the pastoral care of the faithful. In such cases they are subject to the Diocesan Bishop in respect to that which pertains to the pastoral charge or office they receive.
§2. Where and when it is deemed suitable, clergy incardinated in a Diocese or in an Institute of Consecrated Life or a Society of Apostolic Life, with the written consent of their respective Diocesan Bishop or their Superior, can collaborate in the pastoral care of the Ordinariate. In such case they are subject to the Ordinary in respect to that which pertains to the pastoral charge or office they receive.
§3. In the cases treated in the preceding paragraphs there should be a written agreement between the Ordinary and the Diocesan Bishop or the Superior of the Institute of Consecrated Life or the Moderator of the Society of Apostolic Life, in which the terms of collaboration and all that pertains to the means of support are clearly established.
§1. Formation of the clergy of the Ordinariate should accomplish two objectives: 1) joint formation with diocesan seminarians in accordance with local circumstances; 2) formation, in full harmony with Catholic tradition, in those aspects of the Anglican patrimony that are of particular value.
§2. Ordinariate Seminarians will receive their theological formation with other seminarians at a seminary or theological faculty in agreement with the Diocesan Bishop or Bishops concerned. Candidates may receive other aspects of priestly formation at a seminary program or house of formation established, with the consent of the Governing Council, expressly for the purpose of transmitting Anglican patrimony.
§3. The Ordinariate must have its own Program of Priestly Formation, approved by the Holy See; each house of formation should draw up its own rule, approved by the Ordinary (cf. CIC, can. 242, §1).
§4. The Ordinary may accept as seminarians only those faithful who belong to a personal parish or community of the Ordinariate or who were previously Anglican and have established full communion with the Catholic Church.
§5. The Ordinariate sees to the continuing formation of its clergy, through their participation in local programs for formation provided by the Episcopal Conference and the Diocesan Bishop, as well as in their own programs of ongoing formation.
Former Anglican Bishops
§1. A married former Anglican Bishop is eligible to be appointed Ordinary. In such a case he is to be ordained a priest in the Catholic Church and then exercises pastoral and sacramental ministry within the Ordinariate with full jurisdictional authority.
§2. A former Anglican Bishop who belongs to the Ordinariate may be called upon to assist the Ordinary in the administration of the Ordinariate.
§3. A former Anglican Bishop who belongs to the Ordinariate and who has not been ordained as a bishop in the Catholic Church, may request permission from the Holy See to use the insignia of the episcopal office.
§1. The Governing Council, in accord with Statutes which the Ordinary must approve, will have the rights and responsibilities accorded by the Code of Canon Law to the College of Consultors and the Presbyteral Council.
§2. In addition to these responsibilities, the Ordinary needs the consent of the Governing Council to:
admit a candidate to Holy Orders;
erect or suppress a personal parish;
erect or suppress a house of formation;
approve a program of formation.
§3. The Ordinary also consults the Governing Council concerning the pastoral activities of the Ordinariate and the principles governing the formation of clergy.
§4. The Governing Council has a deliberative vote:
when choosing a terna of names to submit to the Holy See for the appointment of the Ordinary;
when proposing changes to the Complementary Norms of the Ordinariate to present to the Holy See;
when formulating the Statutes of the Governing Council, the Statutes of the Pastoral Council, and the Rule for houses of formation.
§5. The Governing Council is composed according to the Statutes of the Council. Half of the membership is elected by the priests of the Ordinariate.
The Pastoral Council
§1. The Pastoral Council, constituted by the Ordinary, offers advice regarding the pastoral activity of the Ordinariate.
§2. The Pastoral Council, whose president is the Ordinary, is governed by Statutes approved by the Ordinary.
The Personal Parishes
§1. The pastor may be assisted in the pastoral care of the parish by the parochial vicar, appointed by the Ordinary; a pastoral council and a finance council must be established in the Parish.
§2. If there is no vicar, in the event of absence, incapacity, or death of a pastor, the pastor of the territorial parish in which the church of the personal parish is located can exercise his faculties as pastor so as to supply what is needed.
§3. For the pastoral care of the faithful who live within the boundaries of a Diocese in which no personal parish has been erected, the Ordinary, having heard the opinion of the local Diocesan Bishop, can make provisions for quasi-parishes (cf. CIC, can. 516, §1).
The Celebration of Divine Worship
§1. Divine Worship, the liturgical form approved by the Holy See for use in the Ordinariate, gives expression to and preserves for Catholic worship the worthy Anglican liturgical patrimony, understood as that which has nourished the Catholic faith throughout the history of the Anglican tradition and prompted aspirations towards ecclesial unity.
§2. Public liturgical celebration according to Divine Worship is restricted to the Personal Ordinariates established under the Apostolic Constitution Anglicanorum coetibus. Any priest incardinated in an Ordinariate may celebrate according to Divine Worship outside the parishes of the Ordinariate when celebrating Mass without a congregation, or publicly with the permission of the rector/pastor of the corresponding church or parish.
§3. In cases of pastoral necessity or in the absence of a priest incardinated in an Ordinariate, any priest incardinated in a Diocese or in an Institute of Consecrated Life or Society of Apostolic Life may celebrate according to Divine Worship for members of the Ordinariate who request it. Any priest incardinated in a Diocese or in an Institute of Consecrated Life or Society of Apostolic Life may concelebrate Mass according to Divine Worship.
The Supreme Pope Francis, at the Audience granted to the undersigned Cardinal Prefect on March 8, 2019, approved this updated version of the Complementary Norms for the Apostolic Constitution Anglicanorum coetibus, adopted in the Ordinary Session of the Congregation, and ordered its publication.
Rome, from the Offices of the Congregation for the Doctrine of the Faith, March 19, 2019, the Solemnity of St. Joseph, Spouse of the Blessed Virgin Mary and Patron of the Universal Church.
Luis F. Card. LADARIA, S.I.
X Giacomo MORANDI
Archbishop tit. of Cerveteri
1 This paragraph was added to the text of the Complementary Norms according to a decision of the Ordinary Session held on 29 May 2013, approved by Pope Francis on 31 May 2013.
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At the present time the world is poised on the brink of serious economic collapse and the best brains in our universities, who have been complicit in the development of this catastrophic state, are remaining silent whilst still collecting their government pay checks. Chaos will inevitably follow the shattering of belief in materialistic science and unlimited economic progress. I am convinced that the only possible way out of the mess is to revive the traditional structure of the British Magical Monarchy, tolerant Anglican Christianity, and responsible Imperialism all of which were still providing motivation to millions a mere 60-70 years ago. This ideological and mythological complex of beliefs and practices does not depend on ever-increasing material wealth and consumerism but on Tory respect for the land and traditional Christian communal values. After all it was this mytho-dynamic system deriving from the Arthurian myths which gave the people of the British Empire the amazing courage to stand alone against the combined forces of Hitler and Stalin who were both "secular-scientific", anti-religious, national socialists.
different peoples under their control. They were both were destroyed in the Twentieth Century by the rise of international capitalism and internal rebellion fermented by rival nationalistic powers. Their revival, if it could be achieved, is the only hope we have of saving the world from an inevitable collapse into chaos. I refer to the British and Ottoman Empires. Ultimately I think that a revival of the Chinese Empire is equally important. We all have to watch out for the totalitarian Russian Imperial revival taking place under our noses.
Those who think that this proposal is absurd may wish to propose a more realisable solution. Irresponsible international capitalism or globalisation, and the underlying ideology that drives it, has to be carefully brought under control, and we have had recent experience that intolerant racial or linguistic anti-imperialistic nationalism devoid of traditional values only plunges us into totalitarianism and violent conflict. Without a vision the people perish.


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Chicago Theater Critics in the Time of COVID – Part 2
Melissa Schmitz2021-08-05T13:36:46-05:00July 1st, 2020|Chicago Theater, Interviews, Latest News|
(Pictured L to R: Colin Douglas, Regina Victor, Kerry Reid)
(Editor's Note: These interviews were conducted in May 2020, prior to the murder of George Floyd and the protests that followed, so it is not included in the discussion of current events.)
Since the closing of theaters due to COVID-19, it's become more evident that it takes a whole community to bring theater to life—from the artists, actors, and designers, to the administrators, to the stagehands, to the audience, and even the press. We asked some Chicago critics for their thoughts on the current state of theater, theater criticism, and where they think the artform is headed from here. This is part two of a two-part series. Read part 1 here.
How do you feel about the rise of virtual theater? Is this something that will continue into the future for theater companies?
Kerry Reid, Chicago Reader: I think particularly in the realm of teaching, it will be more common for companies that offer classes, such as Annoyance, to continue offering classes in some kind of online format. That is a silver lining, as I think it makes work more accessible for those who live someplace where, under normal times, theater and theater classes just aren't readily available. I suspect that every company will rethink how they create archival videos so that they have material on the shelf that is technically more solid than the "camera with mike at the back of the house" set-up that mostly seems to exist to capture a show for in-house use, grant applications, etc.
The big question will be how Actors Equity, the Dramatists Guild, etc. approach virtual theater in terms of protecting the rights and royalties of artists. I think this is something they are wrestling with right now.
Regina Victor, Rescripted: I think it will have to, we don't know what will be coming in the days ahead and I think finding a way to connect digitally is necessary. However I hope we don't think of this as a way to replace theater, but rather as an accessible part of our work we can maintain after we are back in our buildings.
Colin Douglas, Chicago Theatre and Concert Reviews: I think it's going to have to for a while. Until everyone (artists and audience, alike) can be tested and be assured that they're virus free, and vaccinated for the disease, I don't see theater returning to what we used to think of as normal.
Have you been reading any plays or taking in the local streaming opportunities?
Kerry Reid: I did see the last live performance of Teenage Dick at Theater Wit and then the streaming recorded version. I also saw Otherworld Theatre's archived online version of Super Richard World III and TimeLine Theatre's To Master the Art, from their remount at the Broadway Playhouse. And several others, including a Houston production of Chicago playwright and Theater Oobleck founder Mickle Maher's There Is a Happiness That Morning Is, which is one of my favorite plays from the last decade or so in Chicago.
Regina Victor: Yes! Kill Move Paradise by James Ijames at TimeLine Theatre directed by Wardell Julius Clark was excellent. I think because of the way it was filmed. The intimacy of their space came through on camera which is difficult to do with archival footage.
Colin Douglas: I'm not very tech savvy so I don't understand how to "stream" anything. I've read on Playbill.com about musicals and plays being available somewhere, somehow, but it's all Greek to me.
How do you see theater criticism evolving in the wake of virtual theater or theater in a post-COVID world?
KR: I honestly have no idea. It was on life support as a viable career before this. There will always be people who want to write about and discuss theater and I don't see that changing. I think, pandemic or not, we need to be more aware than we have been of inequities in artistic representation and opportunities and not be afraid of addressing that as part of the context in which we all create theater, see it, and evaluate it.
The disproportionate death tolls that we're seeing in communities of color didn't arise in a vacuum. Those numbers are directly linked to our history of segregation and discrimination along race and class lines in this city and country.
What we value in life and what we value in art speak about our communal values, or lack thereof. And critics should be bolder about speaking to that.
Will we see a funding environment that prizes supporting working artists over supporting brick-and-mortar real estate and naming opportunities? Will we see critics who can speak to the lack of support for arts in underserved and marginalized communities? I cannot possibly say at this point."
RV: I hope that we can recognize the urgency of underrepresented voices. Theater criticism is a record. If we were to make no more theater today, would you feel that all of Chicago theater has been properly represented by critics that reflect the people making it? I don't, and that bothers me.
CD: If productions are going to just be virtual, I'd better figure out how to stream them. Then it'll be just like reviewing film or TV, I imagine.
Has this pause provoked you to contemplate the relevance of theater and live performance to a society? To Chicago? Or to yourself personally?
KR: Of course. I have no doubt it will return someday. Whether I will be there to cover it in the same way is a big question for me personally, but hardly important in the bigger picture. But ultimately, theater will continue to exist in some form.
RV: It definitely has, it's been alarming to realize we are not considered essential. The theater I make as an artist I hope is all essential to the civic conversation and I think we should all ask ourselves about that. If I had to guess I would say that we will be more relevant the more we open ourselves to reflecting our city's experiences as a whole, rather than a privileged few.
CD: It's simply reinforced the time honored idea that man needs stories and storytelling. That is what theater is all about. We're entertained, lifted out of our daily problems, sometimes offered a moral to take home but, in the end, we are treated to a good story well-acted and produced.
When we finally open the houses again, what are you most looking forward to?
KR: Just chatting with seatmates and patrons at intermission about what else they've seen.
RV: Seeing all my friends at an opening night! I miss that feeling of celebration and community so much and that will be the most fulfilling moment.
CD: Simply being back in excitement that is Chicago. Also, I had to cancel my Spring trip to NYC, so I'm dreaming of rescheduling a trip to the Big Apple.
Final thoughts from Kerry Reid: Maybe we should start adding healthcare workers to the category of people who get reduced-rate tickets, as we do veterans and educators. And maybe we should think about what accessibility really means in a post-COVID world. There are always people with immune disorders or other disabilities, visible or otherwise, who have a tough time attending shows. As Brian Balcom, director of Teenage Dick pointed out in the talkback the first night the show streamed, for people with disabilities, virtual theater may be their only option.
And the safety of artists ALSO needs to be paramount, so while it's fine to talk about what would make PATRONS feel safe in returning to the theater, I care far less about that than I do whether or not the actors and crew are able to protect themselves in the close quarters most theaters have backstage.
If you're like these critics and also missing theater, check out the streaming options happening now in the BTD!
Follow Kerry Reid on Twitter and Facebook.
Follow Regina Victor on Twitter and Instagram.
Friend Colin Douglas on Facebook.
Special thanks to James Juliano and SHOUT Marketing & Media Relations for their contribution to this post.
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