Welcome to another session of the NPTEL on nonlinear and adaptive control. I am Srikanth Sukumar from Systems and Control IIT Bombay. You folks can again see this very nice representative background image that has been ubiquitous in our course. This is sort of amalgamation of sensors, robotics, actuation and algorithms. The sort of algorithms which we choose to, I mean seek to design that will drive autonomous rovers such as these on Mars and Moon and so on and so forth. So, without further ado, we go into our lectures. So, until last time we had sort of looked at an introduction of adaptive control. We sort of saw what these, you know, building blocks mean, you know, there is a state space model block, there is a controller block. But in addition, there is also an adaptive control block, right? So, the purpose of this adaptive control block was to estimate these unknown parameters in the system, all right? So, now as we move forward, one of the things that we sort of need to remember is that the purpose of or a lot of what we do in adaptive control and non-linear control in general is this asymptotic analysis, all right? So, that is we sort of try to see what happens as time goes to infinity to different signals and functions and so on and so forth, all right? So, this is one of the key aspects of what we are going to do as we move further along in this course, right? Now, there are a few pitfalls in this sort of journey of ours. Now, one of the, you know, first few pitfalls is regarding, you know, what happens to the convergence of functions and correspondingly what happens to the convergence of its derivatives, all right? So, this is what we want to address first today in this lecture. So, suppose I consider a function, right? And let's see, if I consider a function and I have this knowledge that as t goes to infinity, the function converges to a constant, right? So, suppose I know that the function converges to a constant as t goes to infinity, right? This is what we are talking about. Then, this does not automatically imply that the derivative of the function goes to zero, right? I mean, intuitively we sort of assume that, yeah, this should be true that, you know, if function is converging to a constant, then the derivative goes to zero. If a function goes to zero, the derivative goes to zero, all of this somehow seems to make sense in our minds, right? But this is not true. This is one of the first pitfalls. So, we quote a counter example, right? I mean, of course, there are many, many counter examples possible. We quote one of them, right? And what is it? Suppose I take the function which is sin t squared divided by t, right? And suppose I take the limit of this function as t goes to infinity. Let's look carefully at what happens, right? The numerator, right, keeps oscillating between minus one and one. It does not matter what happens to time and, you know, how much time is lapsed, yeah? If I plug in something like an infinity or a very, very large number here, it doesn't matter. Sin of t squared is still going to oscillate between minus one and one, all right? So, this is, you know, not really reaching a limit. However, the denominator is going to go to infinity, right? As t becomes really, really large, the denominator is going to become very, very large, right? And therefore, it's obvious that the denominator is becoming very large while the numerator is anything between minus one to one. So, it doesn't matter what happens to the numerator. As I take a limit as t goes to infinity, we get zero, all right? So, we get that our function converges to a constant which is zero, all right? This is, you know, what it is one of the things that we need to remember, right? Now, let's look at the derivative of this function, right? So, what is the derivative? I mean, it's very standard. We just computed using the chain rule and the product rule, yeah? So, the first piece here is the derivative corresponding to the one over t term here, all right? And the second term here is the derivative corresponding to the numerator. It's very easy to compute. I mean, I really encourage you to do this computation by yourself, not to believe me, right? Then what happens? Now, we look at what happens to each of the term if I take a limit on both sides, right? What happens? Now, if I look at the first term again, it starts to look very much like this term, right?  So, it's something between plus minus one, but the denominator is again going to go to infinity. In fact, faster than the one by t, right? Which means that this term, right, is definitely going to go to 0, all right? So, this term is definitely going to go to 0, no problem, right? What happens to the second term though? What we note is that the second term now does not contain anything in the denominator. It's only a trigonometric function that is a cosine t squared in the numerator, all right? Which is not so great anymore, right? Because now I know that the cosine t squared is just going to keep oscillating and not reach a limit at all, right?  Because honestly speaking, this function is not, that is, f dot of t, in fact, will not have any limit as t goes to infinity, all right? So, the derivative of the function has no limit as t goes to infinity, yeah? So, this, whatever, seemingly counter-intuitive thing has come true that that though the function itself converges to a constant 0 in this case, the derivative does not really converge to anything at all, all right? And this is, should be baffling for us, all right? The important thing to note is that we did not really take any very badly behaved function, you know, it's not a non-smooth function, it's not a, you know, discontinuous function or anything like that, except that t equal to 0, right? The function is, in fact, c infinity, all right? What is c infinity? C infinity means infinitely times continuously differentiable, all right? So, this is infinitely times continuously differentiable except at t equal to 0. For those who have never seen this notation c infinity, I strongly encourage you to look up this notation, all right? Make sure you familiarize yourself with it because this notation is definitely going to show up regularly in our course, all right? So the important thing to notice, we did not choose any poorly behaved function to sort of prove our, you know, thesis that the function converging to a constant does not mean the derivative converges to a 0, all right? We picked a relatively nice function, okay? So the derivative, in fact, continues to oscillate between minus 2 and 2, all right? So that's what we say here, right? So the derivative doesn't go to 0, but it oscillates between minus 2 and 2, all right? So the converse also does not hold true, interestingly, all right? So what's the converse? The converse is that if the derivative goes to 0, right? Can we say that the function itself is going to go to a constant, all right? So this is again another important question, yeah? What is the question? If the derivative goes to 0, is the function itself going to go to a constant? We again present our hypothesis and say that, no, this is not true. And again, we present some counter example, yeah? Please ignore this. I have deliberately cancelled this because this is erroneous, this is wrong, right? But look at this example right here. If I take a function f of t, which is log of t, logarithm of t, then the derivative is 1 over t, yeah? We all know what is the derivative of the logarithmic function, right? Simply 1 over t, all right? And we know very well that if I take the limit as t goes to infinity for f dot of t, then it's 0, right? Because the t goes to, the denominator goes to infinity and therefore, the limit is 0, right? On the other hand, if I take the limit of f of t, right, bad things happen, yeah? Because t goes to infinity and logarithm of course, goes to infinity, may be slower than t, but still it does go to infinity, right? So therefore, limit of f of t is in fact, infinity, right? Wow, that's bad, all right? In fact, it's not a bounded function at all. However, the derivative is actually going to go to 0, right? Again, if you note, this function is nicely behaved everywhere except at the origin, yeah? This function is nicely behaved, right? So remember that we are only sort of interested in the behavior, at least as far as asymptotic analysis is concerned, at infinity and that is at very, very large time, yeah? So we are not really interested in small time behavior, right? So therefore, the function not being well defined at the origin and so on should not matter so much. Always tweak the function so that it's well behaved at the origin and everything goes through. I mean, how would I do that? Very simple. I will, you know, say make this instead of log t, I will make log t plus 1, right? And now the function's derivative is 1 over t plus 1, right? And I still have the exact same result that is 1 over t plus 1 goes to 0 as t goes to infinity. Log t plus 1 blows up as t goes to infinity, right? In fact, this is now very well behaved, I mean, very well behaved at the origin and since the time is always considered to be non-negative that starts at 0. So I have no problem. This is a very valid, nice function in the domain of my interest, right? So I have a very nice function f of t, nicely behaved at the origin and everywhere else beyond t equal to 0 and still it does not satisfy this sort of intuitive idea maybe that if the derivative converges to 0 then the function itself converges to a constant, okay? So I really want all of you to drive this notion out of your minds that if you have a derivative converging to 0 then the function goes to a constant and vice versa that if the function is converging to a constant then the derivative goes to 0, right? Where this lacunae happens is because we are not looking at the function itself, right? We are not really looking at the function itself, we are looking at the limit of a function, okay? We are not saying that the function is a constant, it is obvious that if the function is a constant the derivative is 0 and vice versa, that is obvious but we are not saying that we are saying that if the function converges to a constant, right? So the limit in these, I mean in all our discussion this limit is what somehow messes up things for you, okay? So this is very critical, please keep this in mind, alright? So I really strongly, strongly encourage all of you to come up with more counter examples, I know you can, yeah? So give it a shot, just take hints from what I have done. The more you sort of come up with counter examples, the more you will convince yourself and also actually develop the habit of coming up with counter examples, yeah? In applied mathematics and mathematics, coming up with counter examples is one of the most challenging things, you know? Most mathematicians, a lot of mathematicians actually write papers out of coming up with counter examples of some other result that somebody seems to have proven, just to claim that, yeah, that result is probably not correct, yeah? So, yeah, you can actually make a business out of coming up with counter examples, yeah? So it is an interesting exercise, just I really would recommend that all of you try to come up with these counter examples, alright? Once we have sort of understood, not understood, but we have sort of got rid of some myths, some temptations in a symptotic analysis, we come to the next important set of notions that are really, really critical to what we want to study, okay? Yeah? And what is this? These are the notions of norms, alright? There are two things we are really interested in. The fact that we do asymptotic analysis means that limits and convergence in these notions become really important for us. And second is, because we are always dealing with vectors, signals, vectors and signals, because all the states, all the control, outputs, everything is a vector. So we are always working in some kind of a vector space, at least for the purposes of this course, these are vector spaces, yeah? There may be other more complicated structures in other courses. But right in this course, we are dealing with vector spaces and hence all the objects that we are dealing with are also vectors, alright? Since we have vectors and then we have matrices which are operating on these vectors, it is important for us to sort of have a measure of the size of these vectors, yeah? Because we eventually want a lot of these vectors to go to zero, right? Or we want one vector to dominate another vector. We want to actually have a sense of how these notions can be created. Like in real numbers, it is very easy to say that three is greater than two, right? But if I give you a vector, for example, if I just said, you know, I have a vector three, one and minus one, five, right? Can you tell me which one is greater than which one is less than the other, right? There is no obvious way of doing these things. So norms actually gives you one such obvious way because it reduces any vector to a scalar, yeah? I am not saying that this is the way, yeah, but this is in particular way. And this, in fact, gives us a lot of ability to do a lot of mathematics with vector spaces, yeah? So like I said, well, I did not say it yet, but the norm is essentially a measure of length, yeah? In a vector space. A norm is a measure of length in a vector space, yeah? So keep this in mind. But of course, we have a more formal definition of what a norm is. So norm is a function in our case from Rn to R. So we are keeping it a little bit simple. So we are only working with real vector space, right? So norm is a function from Rn to R, yeah? And it's a valid norm if it's non-negative, right? So this is the notation, the two vertical brackets. This is the notation, right? Get used to this if you have not seen this before. I would expect most of you have seen some version of this, yeah? But all right. So the first property is that it is non-negative. The second is that it's 0 only if the vector itself is the 0 vector, OK? The third is the scalar multiplication property that is the norm of alpha x is actually equal to mod of alpha times norm of x for any scalar alpha. And the final and probably the most critical property is the triangle inequality, which is that norm of x plus y is less than equal to norm x plus norm y, OK? So these are the standard properties for norms. Any function that you can come up with that maps from Rn, that is your vector space, to real numbers, to scalars, yeah? With these properties will be a norm, a valid norm, OK? Will be a valid norm. Now what are examples of a valid norm? One is the, I mean, the commonly used norms are the infinity norm, right? Which is just the maximum of all the elements, OK? So if you just take the max of all the elements of a vector, then you have the infinity norm for the vector. And then you have the p norm, OK? What's the p norm? The p norm is simply taking the absolute value to the power p of each component, summing them up and then taking the pth root, OK? So that is what is the p norm, OK? Now I'm going to go forward and then back again, yeah? So these are, you know, rather important notions and these are rather important norms, OK? So if for example, if I take a vector, say, something like this, right? What is that? This is essentially a vector x in R4, all right? Which is 3, 2, 7, 5. These four elements are 4, right? Now suppose I want to compute these norms, right? So what will be the infinity norm? Infinity norm is just the largest component. So it should be obvious to you that this is the largest component. Therefore the infinity norm is 7, right? What about some p norms, right? There are some useful p norms, right? The two norm is actually the Euclidean distance, like I said, yeah? What all of you are used to in measuring distance is the Euclidean distance, is actually the two norm. How do you compute it? You take the absolute value of every element. Well, I mean, the example I've chosen, everything is positive, so absolute value does not change anything. I take the pth power, that is the squared in this case and then I take a sum and then I take a square root, all right? So I've not actually shown what the answer is, but it should be pretty straightforward for you to compute, okay? Finally, what is the one norm? The one norm is again very similar to compute. You take the absolute value, add them up, take the first root, which is basically do nothing, and then you get the answer 3 plus 2 plus 7 plus 5, which is just 17, all right? So this is some key vector norms, all right? And these are sort of some of the ideas, yeah? These are some of the examples of norms that you have that will be very useful as we go forward, okay? We also have the notion of a matrix-induced norm. So like I said, we have vectors, right? And then we have matrices, right? So since we have both vectors and then matrices which operate on these vectors, we will see how, don't worry about it now, but the point is we also have to define a norm. So the way we do this is that we use the vector norm itself to somehow generate a matrix norm, okay? So this is again something very normal in mathematics is that whenever you want to solve a new problem, you try to get motivated by a previous problem that has already been solved or a simpler problem that has already been solved, yeah? So here we already have the notion of a vector norm, right? So a typical mathematician would think, yeah, why not use the vector norm itself to generate a matrix norm, yeah? Smart, yeah? So what is it? So how do I do that? So the matrix-induced norm, that's why it's called the induced norm because it's induced by a vector norm, is simply measuring a maximum magnification of a vector, of any vector by a matrix, okay? So that's why you see this definition. It says it's supremum, which is a generalization of the maximum, okay? So the supremum is a generalization of the maximum, right? And so it's taking the supremum over all x in Rn, right? So it's the supremum over all possible vectors, right? And the numerator is the norm of Ax and the denominator is norm of x, right? So it's somehow measuring the magnification due to the matrix A of all possible vectors in the vector space, okay? So it's also important to notice that this matrix A is not a square matrix, right? Not necessarily a square matrix. You can define a norm for any matrix. It doesn't have to be a square matrix, right? Now the interesting thing to note is that the numerator is therefore a different sized vector and the denominator is a different sized vector. But only because of the norm, right? Again, you should sort of become conversant with the power of the norm, right? Only because of the norm, you are able to sort of compare vectors in two different vector spaces, right? Because R2 and R4, for example, are two different vector spaces. But just because you have this norm operation, you're able to compare the two, okay? So if you know, if you have some idea of, you know, eigenvalues and eigenvectors, you can make some guesses as to, you know, what will be the maximum magnification and things like that, all right? So I would leave you to try to figure out what would be this maximum magnification of a matrix, right? I mean, so the important thing to note is that this, depending on what norm, vector norm I choose here, like, say if I put a P here, the P here, I get a corresponding P matrix norm, okay? So depending on what vector norm I use, of course I have to use the same vector norm in the numerator and denominator, then I get a corresponding P induced norm. All right? It's as simple as that, okay? So now, before sort of discussing what is the supremum and so on and so forth, we sort of want to understand what is the structure that is given to a vector space by a norm, all right? So this is something that is critical, right? There's something that is very critical. So I'm, of course, assuming that all of you have seen vector spaces or some variant of it, all right? If you've not, again, this is something that you really should see in a linear systems course, like with a state space linear systems course, right? Because whenever we talk linear systems, we are saying that the system itself is evolving on a vector space, right? The notion is the idea of spaces, subspaces, planes and so on, planes and subplanes, right? Hyperplanes, yeah? So that's vector space is essentially generalization of these hyperplanes, right? So these are spaces where superposition principle is satisfied, okay? So again, I am being very vague about it, yeah? But I would expect that all of you do know what is a vector space, yeah? Because otherwise, you do not follow what is a normed linear space, right? Whenever I say linear space, linear space and vector space are identical, yeah? Linear space, linear vector space, vector space, these are used almost analogously, yeah? So what is a normed linear space, yeah? So a normed linear space is essentially a linear vector space with an associated norm, okay? That's it. If you have a vector space or a linear space or a linear vector space, like I said, and you have a norm on it, then the two together denoted as x, this norm form a normed linear space, yeah? Okay? Now, the good thing is most of the spaces we are working with, which is RKs, RPs and so on and so forth, they are all normed linear spaces, right? So basically, any Rn with any of the aforementioned vector norms, right? That is the x infinity, x1, x2. That is Rn with infinity norm, Rn with one norm, Rn with two norm. These are all normed linear spaces, okay? So just the fact that you're able to define a norm in this space makes it a normed linear space, all right? So remember this, yeah? So this is a rather nice set of notions, okay? The notion of a normed linear space, we will continue to use this on a regular basis, yeah? So again, so what is it that we are looking at is some notions on, you know, on sort of pitfalls in asymptotic analysis that we want to avoid, yeah? That's the first set of things that we did look at today. The next set of ideas is on the fact that we want to deal with vectors, because states, controls, outputs, everything are vectors, right? And we want to look at sizes of these, yeah? We want to look at what happens to the sizes of these vectors as time evolves, right? So therefore, in order to sort of make sense of all these notions and compare two vectors if you want to, right? So we define vector norms, matrix norms also are something which are very critical because these matrices will eventually operate on these vectors, right? And we want to sort of assess how this operation affects the vector. So therefore, we have taken the liberty of defining a matrix norm using a vector norm and this is called the induced norm, right? And this matrix induced norm also has some nice properties which we will look at next time. And then we also saw that these norms on a vector space, in fact, give us rather nice structure. I mean, as of now we are simply making definitions out of it, yeah? But the fact that you have a norm linear space is a rather serious object that you have, all right? I mean, it's a serious amount of structure that you have in this space, yeah? First of all, you already have a vector space which is linear in the sense that you are in hyper planes and there is a superposition principle that is followed. On top of it, you have a notion of a distance on this vector space, yeah? Which becomes really, really useful in all sorts of analysis, all sorts of mathematical analysis that we are very, very keen on exploring, right? Even the matrix induced norm is the norm that we have developed for the matrix also makes the space of matrices a normed linear space, okay? We don't discuss it at this stage. But yeah, I mean, this is an additional tidbit for you, if you may, that these norms that we have defined for the matrix, that is the matrix induced norm also makes the space of matrices, for example, Rn by n matrix along with the induced norm also gives you a non-linear space, all right? All right. With that, we will stop.


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I was born in Zimbabwe, that's where I attended school and grew up. I have three kids, Crystal Bell, Clarence and Clarice. They've recently joined me in Australia and it has been a very exciting time for me because I've been separated from them for more than four and a half years and the process of getting to know each other again has been amazing. I work for the Australian Red Cross as a case worker in the migration support program. I've met with abundant and tremendous profound kindness from the Australian community. Even in the process of trying to bring my kids here, I didn't have enough money to pay for them. People just came in a big way and facilitated the process and it culminated in my being united with my kids. So I decided that getting the skills and the knowledge in social work would capacitate me to give back to humanity by way of helping others in need. Having made that decision to become Australian, when I stood up to seeing the Australian anthem, I sang it with pride. I didn't feel like a sense of guilt at all as if I was sailing off my heritage and forgetting my past. No, I felt it was the best decision that I've ever made in my life. Australia is a lovely and wonderful country which is welcoming to everyone. I think it's something to admire and be part and parcel of. So I would encourage anyone who wants to become Australian to do so without even looking back.


Ogovee. On that point, I think that care navigation is still very much needed and I think that health boards or GP practices in different areas have greater or lesser extent of that. What we are noticing certainly from the Allied Health Profession Federation perspective is that we are seeing more Allied Health professionals working as part of first contact. Practitioners within GP practices, from short-term evaluation projects that are being undertaken at the moment, we see quite positive response and positive impact of those first contact practitioners that are making at the moment. I think that it is just really around care navigation of patients and signposting them to who else can meet their needs other than the GPs. A supplementary question on this from Sandesh Gohani. Thank you, convener. This is really for Dr Thompson and Dr Beust. With 85 times percent of all patient contacts happening in primary care, it is quite clear with the demand happening in primary care, there is going to be patients who are desperate in going to A&E, but with patients being redirected from A&E back to GP, are we seeing a set of patients who are just being passed between primary care and A&E and what can we do to stop that from happening? I think that, Sandesh, that is clearly a risk. The redirection policy is that I support it, but it needs to be introduced sensitively with a degree of flexibility of the senior clinical decision maker at the front door. In an area, once it has been established for some time, the public get to know that turning up at A&E with a sore back you have had for two months is not going to get you seen. In areas such as Tayside where they have used this for a number of years, patients stop doing that and they will come to their GD instead as appropriately for a problem like that. I worry about a merry-go-round of people getting passed around and why I am very strongly against the policy that is being pursued just now on the redesign of urgent care, where patients who would previously have attended A&E are supposed to phone 111 NHS 24. Sometimes they have to wait up to 25-30 minutes to get that first call answered and then the idea is that they are maybe given an appointment time to go to A&E. That could be with a cut to their leg or maybe a broken wrist. I just think that it is a policy that is full of flaws and unintended consequences, one of which is the merry-go-round that you mentioned. It was good to hear from Dr Buest the mention of Tayside, where you have got the NHS working together with local authorities to roll out the enhanced community model there, especially around older people at home. My question, though, is really directed towards A&E and perhaps John Mooney as well. In the past, we have done quite a bit of work as far as integrated joint boards were concerned of looking at, you know, actually preventing admissions to hospital and making sure that we get discharged as quickly as possible, which kind of fits in with the Tayside model there. I am just wondering what your views are on how helpful that can be going forward, how much of that we need to incorporate going forward. Thanks for that question. It is an interesting one for our constituency of interest because most third sector providers are supporting people who have sometimes very complex social care needs but they are not in and out of hospital. That goes back to what I was saying at the top of the meeting about seeing social care through the prism of the extent to which it acts as a pressure valve for the NHS. That is where your question is going. Most of our members are supporting people who have learning disability, perhaps mental health issues, who are not in the position of being admitted and discharged from hospital. They use the NHS just like you and I would use. What you are referring to there probably is delayed discharges for older people. Certainly, there is an awful lot more that could be made, in my view, of the third sector there. I would recommend that you would speak to organisations like, for example, British Red Cross, who have got some fantastic home from hospital services. They are not registered care services in the same way that we conceptualise social care, but they do a huge amount also with volunteers. The whole staffing and recruitment issue is entirely different for them. The delayed discharge issue tends to focus around old people who are admitted, sometimes in emergencies, unplanned admissions and then are not able to be discharged because social care packages are not available for them. There are a number of ways to tackle that. For the purposes of our membership, that is not really where their main activity is located. On the A&E issue, we have found the question from Gillian Mackay. I am concerned particularly about regional variations in waiting times between health boards. For example, NHS Forth Valley in my region for the weekend in the 24th of October, only 51.8 per cent of people attending A&E were seen within four hours. It is a considerable improvement from 41 per cent on the weekend in the 10th of October, but it is still considerably lower than the national average. I know that Forth Valley will be subject to the same acute pressures elsewhere. Dr Thomson, in particular, had an insight as to why particular health boards seem to be struggling with that strain more than some of the others. I think that you have raised a very valid point. The demands on emergency departments are unrelenting and the four-hour standard performance for the month ending September is the lowest ever since records began. We have more patients waiting greater than eight hours and more patients waiting greater than 12 hours than we have ever had before. That equates to crowded and unsafe emergency departments. For the weekend ending the 31st of October, there was no major emergency department in Scotland that did not have patients waiting beyond eight hours. All bar one had a number of patients waiting greater than 12 hours. We know in some departments that some patients are waiting many days for a bed. What that reflects, quite simply, is inadequate capacity within the system for patients who need admitted to hospital. The emergency departments on average admit about 30 per cent of the patients that attend emergency departments. The vast majority are seen and discharged, but that 30 per cent are waiting far, far longer than is required for a bed within the system. We know quite simply that patients waiting that length of time come to harm. Do we have a sense of why certain health boards are maybe struggling with this more than others? Is it the number of consultants that there are in particular health boards? Is it particular health board geography in particular? The fourth valley was the example. I use it as a relatively urban health board comparatively with NHS Lanarkshire next door.  Do you have any particular insights as to why some health boards are struggling more than others? You mentioned variation across Scotland. Some emergency departments will deal only with emergency department patients and other emergency departments. That is the conduit for all admissions to the hospital to come through. Those patients who are referred from primary care in some hospitals will go directly to the ward and other boards will go to the emergency department and wait there for a bed. There is significant variation across the country. The main issue that is causing the very poor performance equates to a very poor patient experience in terms of the length of time patients are waiting within the emergency departments is a lack of bed capacity within the system. We estimate that we are approximately 1,000 acute beds short nationally. As we head into winter, the expectation is that, unfortunately, patients will be spending longer and longer within emergency departments and coming to more harm. That is a very useful insight. We have zeroing in on some of the specific areas, colleagues. If you can direct your questions to individuals, I will come to some questions following up on social care from Evelyn Tweed. My question is about the social care sector and capacity. I know that, in my constituency in Stirling, we have an acute shortage at present, but can you give us a flavour of the picture across Scotland in general? Can you also tell us something about how we can help the issues that are being experienced in remote areas in particular? I think that capacity is a big issue. I mentioned earlier in the meeting that we surveyed providers on their recruitment issues. There were some findings that addressed capacity. I shall just flip them up and let you know what they said. We had 30 providers responding to our survey, and they are all pretty large organisations. They are the ones that are operating in multiple authority areas. 63 per cent of them said that they had already had to reduce capacity for service delivery as a result of recruitment shortages. That is just shy of two thirds across the country, which surprised us. I have to say that it was as high as that. We also had 53 per cent of those providers saying that they either have or would have to refuse any new care packages, even if commissioners came to them and asked them to do it. Again, that is over half of providers saying that. Those are pretty significant numbers. We have not seen anything like that before this particular period, which is 18 months into a pandemic approaching the winter. Yes to serious, what we can do about it is very much hinged on how we get more people into the workforce. The social care workforce is very different to the NHS workforce. Typically, the NHS model is that you train, you qualify and then you start work.  We do it the other way round. A lot of recruitment is more about values based. What kind of person are you? Are you the right person to do this kind of work? If so, let's have you in. Then we start training and qualifying you. The kinds of issues that colleagues were talking about in terms of how long it might take for nursing staff to come through the system or medical staff, that is not really the case with social care. If it was an attractive enough option for people, they could start tomorrow. We need to make it more attractive for people to start tomorrow. In rural areas, that means being able to compete with retail and hospitality purely on pay. Overall, it is back to fair work to make it something that people want to do, to understand what it is. I talked at the beginning of the meeting about the value that is placed on social care and what people think it is. It is not just a pressure belt for the NHS. It is not just time and task personal care. It is standing alongside people and supporting them to live their best lives. That is what social care is about. We do not hear a lot of that. Social care is mentioned in the list of public services that people like to speak about when they are on platforms, doctors, nurses, teachers, etc. Social care is nowhere. There is a whole awareness raising that needs to be done. In the immediate term, we could get people in the door tomorrow if it was attractive enough and particularly if it was attractive enough in pay terms. It is really as simple as that. Thank you, Annie. That was really helpful. Thank you, convener. Maybe just continuing with Annie for a moment. I have been hearing a lot about unpaid carers being in crisis because they feel that they cannot access the packages that they need or they have been told by their local authority that their package will have to be scaled back or cut back. Particularly around care at home and not only for older people but particularly for people who have learning disabilities. Again, from the work that you do with providers and your survey work, is that a sense that there is a crisis for unpaid carers as well? Yes, I think so. Some of that is also a kind of hangover from the pandemic. I said that we are not a sector that needs to be remobilising because we have been here all the time. There are exceptions to that and the exceptions to that are congregate care settings. We know about care homes but there has been very little focus on things such as daytime opportunities, which are buildings-based, and especially on short breaks for carers, what we used to call respite. A lot of those had to close at the beginning of the pandemic simply because of the restrictions on the numbers of people that were allowed to meet indoors and the social distancing. Those were buildings-based congregate settings that just couldn't admit people anymore. What that has meant for a lot of family carers is that they just haven't had a break at all for 18 months, none, no support whatsoever. One of the things that I was going to ask the committee to consider is that you have a lot of providers including me. That is who I am speaking on behalf of, but there are a number of user-led organisations that have a lot of information on that. Inclusion Scotland would be one of them, and they have been tracking carers coalition. They have been tracking what has been happening to people's support. It is not particularly happy picture. I would thoroughly recommend to the committee that at some point you have a session, not for me to tell you your business, but you would really get some very rich information from those organisations, which would come from the people who have lost out on receiving social care. I can talk about the staff, providers and the impact on the sector and all the rest, but the really important thing is what is happening to people who rely on social care. The capacity issues that we have in the sector have had quite an impact on them for sure. I would like to move on to questions around the general improving outcomes. It has been alluded to by many of you throughout everything that we have talked about so far. That is really the crux of the matter, how we improve health outcomes for people over winter. You have pointed to quite a few areas, but Emma Harper has some specific questions on that. As we are looking at planning for winter, which this session is about, how do we look at improving outcomes? I am the co-convener of a few health care cross-party groups, and health inequality is one. We know that we need to improve the outcomes for many people. We heard about the women's health plan earlier from Minister Marie Todd. I wonder if any members of the panel have specific proposals about improving outcomes, not just for winter but in the future as well. I think that our recipe for improving outcomes was contained in our submission to Derek Feeley's review of social care. There is a whole range of approaches for that. For us, fundamentally, it means that we just drop the whole idea of competitive tendering for social care. That takes us nowhere in relation to improving outcomes. It is about a skilled workforce that is discouraged by competitive tendering and current commissioning arrangements. It is about standing alongside people and letting them make their own decisions. Interestingly, the national care service consultation did not have enough to say about that and the importance of self-directed support. In social care, that has been legislated for eight years now. The whole point of self-directed support is that people identify their own outcomes and the role of social care is to support them to achieve it. That, for me, is the absolutely essential part of all that. We need to be really serious about self-directed support, we need to implement it properly and we need to put more resources behind it and get it moving. That is certainly what Derek Feeley said in his report. At the moment, there is a little bit of a mismatch between what Derek Feeley said and the proposals for the national care service as they stand. I think that the job that we have all got to do over the next few months is to make sure that they can realign on that because the best people to tell you what outcomes they want to achieve are the people that we support. Unison's long-time stance is really straightforward. A highly trained and valued workforce is the best thing that it can do to improve standards within social care. It is something that we have been pushing throughout the Feeley review and also into the national care service review. I think that that work has really got to be centred around the fair work principles. That is certainly the direction of travel in which we should be going in. When you are talking about an overnight fit for the staff shortage stuff and lots of the fair work principles, I am not going to deliver an overnight success. In the medium to long-term areas that we should be going down, I am probably not going to surprise you by saying that the conflict between private profit in-care and providing a top-notch service is clear for unison to see. I was disappointed in the national care service proposals that private profit is something that appears to be accepted going forward. We have got lots of members responding to our survey speaking about even though they are struggling to provide the services that they currently provide. There are still organisations out there that are tendering for new services. Speaking about the fact that, essentially, if they had the level of training that they need, things would be a lot better for service users on the ground. I think that we need to shift some of the resources or think more about how we can develop resources around prevention and early intervention. We have seen that prior to the pandemic. We have begun to see the shift, but because of the pandemic and the needs of the population have changed and the requirement for more therapeutic intervention and longer rehab, a lot of the allied health professions workforce has been diverted to address those acute issues, rather than focus on prevention and early intervention. For example, how might we support people with low or medium-level frailty in the community so that they do not develop or access any acute services? I think that we need to shift some resources to that preventative agenda. I know that I read in the submission from the Royal College of Emergency Medicine that the data showed that, for every 67 patients waiting 60 to 12 hours, one of them will come to an avoidable harm. That is an issue that we need to look at. How can that be avoided? For every harm that occurs, there is a data entry system that requires that to be tracked and dealt with. I think that it is a data system and I know it because I am a former nurse who used to enter adverse events into the data system. I am interested to know how we will make sure that we have a wider ability for our GP's or doctors to utilise a system or use that to learn so that harm can be avoided in the future. Dr John Thomson, first, please. On the issue that you have raised, there is clear evidence, adjusting for all other confounding factors such as age and deprivation, that a weight in an emergency department of 8 to 12 hours increases mortality on that admission for patients. For every 67 patients waiting between 8 to 12 hours for admission to hospital, there will be one avoidable death related purely to that excess weight in emergency departments. That is absolutely unconscionable and completely avoidable with the correct capacity in the system. What we do not know is what harm that does not result in death is happening to those patients who are waiting a significant length of time. Emergency department staff are not trained to look after patients for many hours after their initial assessment and immediate management. Therefore, those patients, despite best efforts of all our staff, are receiving poorer quality care than they would if they were in the correct bed under the correct in-patient specialty. However, there is no doubt that, as we head into winter, if we see a continuation in the long waits of patients waiting for beds within emergency departments, those patients are not waiting to be seen. Patients are being seen within an appropriate timescale. Those patients are waiting many hours to move to an appropriate in-patient bed. We have a couple more questions that we want to ask of you all before we go away. Sorry, staff welfare has been mentioned many times in this session. I am just looking at my colleagues. I have a number of colleagues that want to ask specific questions on staff welfare if you can make your questions direct and quick. I had a very long bit, but I will make this as succinct as possible. When it comes to nurses working in the acute sector, how feasible is it to really provide them with the opportunity to do for flexible working? I am just with the challenges and the restrictions in their wellbeing and retaining those members of staff. I guess we go to the RCN for that, Colin. I think that it is absolutely possible. I do not think that there should be restrictions. I think that we should look at all flexible working options for any individual who wants to look at that. If it is going to retain them in the service, that is a way to do it. If it is going to help them to maintain their health and wellbeing, that is a way to do it. I do not think that there are any barriers to not look at flexible working within the acute sector, as well as across the whole of the health and social care sector. We need to get much more in working with our workforce around what suits their work-life balance. That means looking at flexibility of opportunity and flexibility of employment. It is hugely important. Far rather, we had individuals who could work some of the time and even extend their career. I am going to use the word flog them until they feel they have to leave. I do not think that there are barriers. It is more around about choice and about making those choices available. That is great to hear. One of the things that the Government is looking to do is to attract people back who have maybe retired early. Do you think that flexible approach would attract people back who have maybe felt that they wanted to take early retirement? That is a huge point, convener, because we have not only got the people who have left who we want to attract back. There was some pension implications, but that was changed with the pandemic with emergency measures. Currently, we know that we have an aging workforce. We have some changes to the pension provisions going forward. We also know that there are a lot of people who are looking at retirement now, and we need to make sure that we keep that expertise in the workforce for mentoring and for support, as well as providing patient care. It is a part of both, but I do think that if we could get that right, we have seen a number of people come back during the pandemic. Not as many as we would all have wished or hoped, but we have seen a number. We have also got to ask those individuals what have we done that has made it work for you so that we can improve what the offer is and maintain people in the workforce. Those individuals who choose to retire take such an experience and we need them to support our new workforce that comes in and new qualified nurses and to help them to develop the best practitioners that they can be. It is something that we have talked about for a number of years, as many of you will know, but we are not on that right yet. Just to support that, obviously in social care but also in the NHS from colleagues, staff are looking for a degree of flexibility in their work. It is our understanding that many people are considering or have already moved on to things like the bank, and the only real reason for doing that is to give people more control over the shifts at the worker, how they work and what they do. Clearly, there needs to be a more modern approach, some way to meet in the middle where a service division can be maintained but also people can get the work-life balance that they want. I think that we really need to focus on going down that route in the future. The general question is about wellbeing and welfare of staff. As I have said before, fair work is top of the list and access to wellbeing resources. For our sector, it would help enormously with morale if staff and commission services were not treated as second-class citizens compared to the rest of the system. I put in our written submission that the uplift to £10.02 as a minimum wage for care workers is absolutely brilliant, fantastic announcement. It is now being entangled in a whole load of implementation problems that we think are going to, for some organisations at least, make it worse rather than better. We are simply not viewed as an equal part of the system compared with public sector employees. That is a hugely demoralising thing for staff in our sector. The same goes for organisations. Our submission talks briefly about some of the financial support that has been made available to our sector. Again, hugely important. We are very grateful for it but the way it is being administered is an absolute catastrophe. We all feel like we are slightly outside the loop as third sector organisations. Parity of esteem would help tremendously with staff wellbeing and for all of us working here in the third sector. We are trusted enough to provide very intensive personal support to arrange hundreds of thousands of people but somehow we are not trusted enough to deal with the money. The support in the same way that other organisations are. That really needs to stop now. We are coming to the end of our session but Paul Cain has some lessons learned from dealing with Covid last year. Over to you Paul to end the session. Thank you very much, convener. I appreciate that we are tight for time here at the end. My question would really be whether we are still living through the pandemic. We had a winter last year that was unprecedented. I suppose that I am keen to understand what are the feelings on lessons learned from last winter. Has the Government learned the lessons of what didn't work so well and perhaps what did work? Has that been factored into the winter preparedness plan? The winter demands that we see year on year are entirely predictable. In my experience we put short term measures in to mitigate over this period and we do not put longer term solutions in that allow us to deal with that fluctuation demand. For example, it is entirely normal over this period to reduce the amount of elective surgery to accommodate the increased bed base that is required for unscheduled care admissions and we do that every winter. As number of admissions decreased, we revert back to normal. I think that it will be the same again. It will be short term mitigation that is required and I do not see anything that has changed that has made things any different for this coming winter from any previous winter in terms of this element of cross your fingers and hope that we cope. I think that there is a sense there around every winter that we see these pressures and I take the point about what you said about elective surgeries but do we feel that because of the unusual circumstances of the pandemic having already cancelled more and more elective surgery that creates a problem for us at the other end of winter? I think that any remobilisation plan that has been discussed or published in regards to Covid recovery, particularly around waiting lists and surgery, has not taken into consideration unscheduled care and without doing that is likely to fail. There needs to be a single overview of the capacity that is required for unscheduled care. Although it is unscheduled, it is relatively predictable. We know at what points of the year that we are particularly busy and at what points of the year that we are not. I guess that this is the concern this year is that during autumn, which is a normally relatively quiet time heading into winter, we are at our worst ever performance, far worse than any previous winter and we are not yet in winter. I think that what will happen again is that medical patients will be admitted to surgical beds because that is where the capacity is within hospitals to allow these patients to get to bed. I have a long come to terms with the reality that, on behalf of our sector, I tend to say the same things that I said last year and indeed the year before that. That is also true this year. However, staying positive, the FELE recommendations and the national care service proposals would go a long way to supporting social care in the way that it needs supporting in the winter or indeed at any other time, but they are a very long way off. In terms of the Scottish Government learning lessons and understanding what needs to be done, I think that there is a very broad understanding of what needs to be done. We are not doing it yet. It comes back to what I was saying about the long-term prospects being quite positive, but in the short term we need some very rapid action around some of those things. Unfortunately, a lot of that will take money. There has already been quite a lot of money allocated. The £10.2 is brilliant. We are all very pleased about that. As I have said, some of the implementation methodology around that and some of the other financial support that we have is just not doing it. We need a bit of an injection of urgency and speed into some of the solutions that we already know will work. Unfortunately, we have really gone over our time. I want to thank everyone who has given us evidence this morning. Is there anything specific that you want to follow up on? The committee is always here to take any of your emails and letters to us about some of the specific issues that you might feel that you did not get time to express, but thank you very much. Everything that you have said has been extremely helpful to us. That next meeting in 16 November, the committee will take evidence from the Minister for Mental Well-being and Social Care on session 6 priorities. That concludes the public part of our meeting today.